POSITION AND TEMPERATURE MEASUREMENTS OF A SINGLE ATOM
VIA RESONANT FLUORESCENCE
by
RICHARD WAGNER JR.
A DISSERTATION
Presented to the Department of Physics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2019
DISSERTATION APPROVAL PAGE
Student: Richard Wagner Jr.
Title: Position and Temperature Measurements of a Single Atom via Resonant
Fluorescence
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Physics by:
Michael Raymer Chair
Daniel A Steck Advisor
Benjamı́n J. Alemán
George Nazin Institutional Representative
and
Janet Woodruff-Borden Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate
School.
Degree awarded June 2019
ii
©c 2019 Richard Wagner Jr.
This work is licensed under a Creative Commons
Attribution-NonCommercial-NoDerivs (United States) License.
This work may be shared freely with attribution to the author and
institution. It may not be edited or used for any commercial purpose.
iii
DISSERTATION ABSTRACT
Richard Wagner Jr.
Doctor of Philosophy
Department of Physics
June 2019
Title: Position and Temperature Measurements of a Single Atom via Resonant
Fluorescence
The magneto-optical trap (MOT) has been an important tool in quantum optics
research for three decades. MOTs allow for hundreds of thousands to millions of atoms
to be cooled to micro-Kelvin temperatures for use in a wide variety of experiments.
For nearly as long, MOTs with just a single atom have been of some interest to the
research community. We have developed an algorithm, based on Bayesian statistics,
to carefully measure small numbers of atoms in a MOT.
Many techniques have been developed to measure the temperature of atoms
in a MOT, including some that can translate to single atoms. We propose a
new technique to measure the temperature of a single atom without releasing the
atom from the MOT. Temporal modulations in a spatially dependent magnetic field
encode information about the position of an atom through associated variation in its
fluorescence rate. Measuring this variation reveals the atom’s position distribution
and therefore its temperature. The technique is examined for a variety of MOT
parameters. Measurements with the technique are an order of magnitude larger than
predicted by theory and potential routes for future study are offered.
iv
CURRICULUM VITAE
NAME OF AUTHOR: Richard Wagner Jr.
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR
Miami University, Oxford, OH
DEGREES AWARDED:
Doctor of Philosophy in Physics, 2019, University of Oregon
Master of Science in Physics, 2016, University of Oregon
Bachelor of Science in Physics, 2009, Miami University
PROFESSIONAL EXPERIENCE:
Graduate Research Fellow, Department of Physics, University of Oregon, Eugene
OR, Fall 2012, AY 2012-2013, Winter & Spring 2014, Fall 2015, Fall 2016,
Spring 2016, Fall 2017, Spring 2016, AY 2017-2018
Science Literacy Program Fellow, University of Oregon, Eugene OR, Winter &
Spring 2012, Fall 2014
Graduate Teaching Fellow, Department of Physics, University of Oregon, Eugene
OR, AY 2009-2010, AY 2019-2011, Winter & Spring 2015, Winter 2016, Winter
2017
Adjunct Instructor, Lane Community College, Eugene OR, AY 2016-2017, AY
2017-2018
GRANTS, AWARDS AND HONORS:
Science Literacy Program Fellow, University of Oregon, Winter & Spring 2012,
Fall 2014
Astronaut Scholar, Astronaut Scholarship Foundation, 9/2008
Provost’s Academic Achievement Award, Miami University Office of the Provost,
9/2008
v
PUBLICATIONS:
Richard Wagner and James P. Clemens. Fidelity of quantum teleportation based
on spatially and temporally resolved spontaneous emission. Journal of the
Optical Society of America B, 27(6):A73-A80, 2010.
N. Souther, R, Wagner, P. Harnish, M. Briel, and S, Bali. Measurements of light
shifts in cold atoms using Raman pump-probe spectroscopy. Laser Physics
Letters, 7(4):321-327, 2010.
Richard Wagner and James P. Clemens. Performance of a quantum teleportation
protocol based on collective spontaneous emission. Journal of the Optical
Society of America B, 26(3):541-548, 2009.
Richard Wagner Jr. and James P. Clemens. Performance of a quantum
teleportation protocol based on temporally resolve photodetection of collective
spontaneous emission. Physical Review A, 79:042322, 2009.
vi
For Mom and Dad.
Thank you.
vii
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Single Atom MOTs . . . . . . . . . . . . . . . . . . . . . . . 2
1.2. MOT Temperature Measurements . . . . . . . . . . . . . . . 4
1.3. Optical Trap Oscillations . . . . . . . . . . . . . . . . . . . . 7
1.4. Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . 8
II. ATOM OPTICS AND MAGNETO-OPTICAL TRAPS . . . . . . . . . 9
2.1. A Single Atom . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2. Atoms and Light . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3. Atoms and Magnetic Fields . . . . . . . . . . . . . . . . . . 21
2.4. The Fg = 0→ Fe = 1 Atom . . . . . . . . . . . . . . . . . . 25
2.5. Magneto-Optical Traps . . . . . . . . . . . . . . . . . . . . . 35
III. EXPERIMENTAL SETUP . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1. Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2. Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3. Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4. Photon Collection . . . . . . . . . . . . . . . . . . . . . . . . 80
IV. OUR SINGLE ATOM MOT . . . . . . . . . . . . . . . . . . . . . . . . 90
4.1. Detecting A Single Atom MOT . . . . . . . . . . . . . . . . 91
4.2. Bayesian Algorithm . . . . . . . . . . . . . . . . . . . . . . . 100
V. ATOMIC FORCES IN A MOT . . . . . . . . . . . . . . . . . . . . . . 114
5.1. 3D and 87Rb Hamiltonians . . . . . . . . . . . . . . . . . . . 114
5.2. Matching Simulation to Experiments . . . . . . . . . . . . . 123
viii
Chapter Page
5.3. 3D and 87Rb Calculations . . . . . . . . . . . . . . . . . . . 128
5.4. Escape Channels . . . . . . . . . . . . . . . . . . . . . . . . 134
5.5. Recovering Potential . . . . . . . . . . . . . . . . . . . . . . 143
VI. POSITION AND TEMPERATURE MEASUREMENTS . . . . . . . . 146
6.1. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.2. Analysis of Photon Arrivals . . . . . . . . . . . . . . . . . . 156
6.3. Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.4. Multiple Atom Fluorescence Amplitudes . . . . . . . . . . . 184
6.5. Parametric Resonances . . . . . . . . . . . . . . . . . . . . . 185
6.6. Non-Sinusoidal Waveforms . . . . . . . . . . . . . . . . . . . 198
VII. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
APPENDICES
A. HOW A FG = 0→ FE = 1 ATOM BECOMES A V-ATOM . . . . . 208
B. DUAL POLARIZER NOISE REDUCTION . . . . . . . . . . . . . . 212
C. MOT COIL WATER COOLING RATE . . . . . . . . . . . . . . . . 215
D. BAYESIAN EVOLUTION DERIVATION . . . . . . . . . . . . . . . 222
E. GAUSSIAN SAMPLED OSCILLATION AMPLITUDE . . . . . . . 225
F. PARAMETRIC RESONANCE DERIVATION . . . . . . . . . . . . 229
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
ix
LIST OF FIGURES
Figure Page
1.1. The magneto-optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1. 87Rb D2 Transition Level Diagram . . . . . . . . . . . . . . . . . . . . . 22
2.2. Fg = 0→ Fe = 1 Atom Level Diagrams . . . . . . . . . . . . . . . . . . 26
2.3. Optical molasses beam arrangement . . . . . . . . . . . . . . . . . . . . 32
2.4. V-atom forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5. Magneto-Optical Trap Level Diagram . . . . . . . . . . . . . . . . . . . 36
2.6. Atom with Mutlple Ground States . . . . . . . . . . . . . . . . . . . . . 41
2.7. Sisyphus Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1. Experimental Diagram of Vacuum Chamber . . . . . . . . . . . . . . . . 46
3.2. Installed LIAD system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3. LED circuit for LIAD setup . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4. (Re)baking the vacuum chamber . . . . . . . . . . . . . . . . . . . . . . 52
3.5. Beam Path of MOT Trapping Lasers . . . . . . . . . . . . . . . . . . . . 54
3.6. Beam Path of MOT Repumping Lasers . . . . . . . . . . . . . . . . . . 54
3.7. Pyramid MOT Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.8. MOT Frequency Controlling Beam Path . . . . . . . . . . . . . . . . . . 61
3.9. 87Rb D-2 line spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.10. Fiber Polarization Mount . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.11. Permanent Magnet Magnetic Field Calculation Methods . . . . . . . . . 68
3.12. Permanent Magnet Quadrupole Field . . . . . . . . . . . . . . . . . . . . 69
3.13. Single Atom MOT Magnetic Field Designs . . . . . . . . . . . . . . . . . 70
x
Figure Page
3.14. Plumbing Diagram for water cooling the MOT coils . . . . . . . . . . . . 74
3.15. Water Cooled Coils Protection Circuit . . . . . . . . . . . . . . . . . . . 75
3.16. MOT Coils Current Monitoring Circuit . . . . . . . . . . . . . . . . . . 77
3.17. Helmholtz Coil Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.18. Single Photon Collect Lens System . . . . . . . . . . . . . . . . . . . . . 81
3.19. BNC-Header Adaptor for DE2 FPGA . . . . . . . . . . . . . . . . . . . 84
3.20. FPGA implementation for counting photons . . . . . . . . . . . . . . . . 85
3.21. FPGA implementation for timing photons . . . . . . . . . . . . . . . . . 86
3.22. APD Protection Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 88
4.1. Sample Photon Collection Data . . . . . . . . . . . . . . . . . . . . . . . 92
4.2. Histograms of sample photon collection data . . . . . . . . . . . . . . . . 95
4.3. Variances of sample photon collection data . . . . . . . . . . . . . . . . . 98
4.4. Atom counting with CCD camera . . . . . . . . . . . . . . . . . . . . . . 99
4.5. Sample data for bayesian fluorescence number estimation . . . . . . . . . 104
4.6. Bayesian algorithm flow chart . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1. Excited state populations for 87Rb and V-atom . . . . . . . . . . . . . . 130
5.2. 1D forces on 87Rb and V-atom . . . . . . . . . . . . . . . . . . . . . . . 133
5.3. 3D MOT forces for an assortment of beam phases . . . . . . . . . . . . . 135
5.4. 3D MOT phase dependence . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.1. Sampled oscillation spectrum . . . . . . . . . . . . . . . . . . . . . . . . 159
6.2. Assorted measured spectra . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.3. Single-atom temperatures with release-recapture method . . . . . . . . . 168
6.4. Fluorescence amplitude scaling with current amplitude . . . . . . . . . . 168
6.5. Simulated 1D modulation slopes . . . . . . . . . . . . . . . . . . . . . . 170
xi
Figure Page
6.6. Fluorescence amplitude spectrum . . . . . . . . . . . . . . . . . . . . . . 174
6.7. Lorentzian fluorescence amplitude spectrum . . . . . . . . . . . . . . . . 176
6.8. Fluorescence amplitude scaling with field gradient . . . . . . . . . . . . . 177
6.9. Simulated fluorescence amplitude scaling with field gradient . . . . . . . 178
6.10. Fluorescence amplitude as a function of laser detuning . . . . . . . . . . 180
6.11. Fluorescence amplitude with background magnetic fields . . . . . . . . . 182
6.12. Spectra of multiple atoms in a MOT. . . . . . . . . . . . . . . . . . . . . 185
6.13. Images of Single Atom Parametric Resonance . . . . . . . . . . . . . . . 189
6.14. Light Distribution for Parametric Oscillating Atom . . . . . . . . . . . . 191
6.15. Fitted Light Distributions for Parametric Oscillating Atoms . . . . . . . 193
6.16. Light intensity loss due to parametric oscillations . . . . . . . . . . . . . 195
6.17. Power spectra for different waveforms . . . . . . . . . . . . . . . . . . . 199
B.1. Laser power control with polarizers . . . . . . . . . . . . . . . . . . . . . 213
C.1. Water cooling channel dimensions . . . . . . . . . . . . . . . . . . . . . . 219
xii
LIST OF TABLES
Table Page
3.1. Permanent Ring Magnet Parameters . . . . . . . . . . . . . . . . . . . . 67
3.2. Helmholtz Coil Field Gradients . . . . . . . . . . . . . . . . . . . . . . . 79
3.3. Photon Collection Lens System Collection Efficiency . . . . . . . . . . . 83
5.1. Repumping field and populations of 87Rb D2 energy levels . . . . . . . . 115
5.2. MOT Beam Circular Polarizations . . . . . . . . . . . . . . . . . . . . . 120
6.1. High-gradient MOT parametric resonance conditions . . . . . . . . . . . 187
6.2. Variance compared to amplitude for common waveforms . . . . . . . . . 201
C.1. Water-cooled MOT coil parameters . . . . . . . . . . . . . . . . . . . . . 218
xiii
CHAPTER I
INTRODUCTION
The magento-optical trap (MOT) has become one of the bedrocks for research
on the quantum behavior of atoms. The MOT can produce millions of atoms with
temperatures on the order of microkelvins. Such small temperatures are necessary
to limit atomic motion for studying their classical and quantum dynamics. The
magneto-optical trap uses multiple laser fields whose frequencies are often a few MHz
smaller (red-detuned) than an atomic resonance of the atomic species being trapped.
The light is lower in frequency so that atoms moving towards the laser source sees
a Doppler shift moving the light from that laser closer to resonance. This Doppler
shift makes the atom more likely to absorb photons from the laser (due the reduced
detuning), resulting in radiation pressure that pushes the atom in the propagation
direction of the light. With D + 1 lasers for a MOT in D-dimensions, this can result
in a cooling force as the lasers damp the motion of the atoms. This process creates
what is often referred to as optical molasses and was originally conceived in the 1970s
[1, 2] and experimental verified in the following decade [3].
Doppler cooling is only responsible for cooling atoms; it does not trap them.
In addition to the laser fields, a MOT requires a (quadrupole) magnetic field which
produces a spatially dependent Zeeman shift of atomic energy levels. The Zeeman
shifts provide in an additional preferential excitation of the atoms by the laser,
creating a restoring force that traps atoms near the location where the magnetic
field vanishes. Together with optical molasses, the quadrupole fields impart a force
on the atom which causes it behave as a damped harmonic oscillator. There are a
number of configurations for a magneto-optical trap, but this work focuses on the
trap shown in Figure 1.1. Here, three pairs of counter-propagating lasers push the
1
a) ẑ b)
current
+ ŷ

x̂
+ 
+

current
FIGURE 1.1. The magneto-optical trap. a) Schematic drawing. b) Photograph of
our MOT setup around the experiment vacuum cell. For both images, blue arrows
show the six-counter propagating MOT beams (with their appropriate polarizations
in the schematic drawing). Orange loops are anti-Helmholtz coils that generate the
linear magnetic field near the origin. The MOT loads at the origin (red dot).
atom towards their intersection point and the magnetic field is generated by a pair of
anti-Helmholtz coils—coaxial coils with currents traveling in opposite directions. At
the midpoint between the two coils on their central axis, the magnetic field vanishes,
establishing the equilibrium position for the atom.
1.1 Single Atom MOTs
Not long after the first MOTs were developed [4], they were extended to allow
capture of small numbers of atoms, primarily by greatly increasing the strength of
the confining magnetic field. This produced traps on order of tens of atoms [5] and
quickly down to individual atoms [6]. Since then, single- or few-atom MOTs have
largely been used as efficient sources of single atoms for loading into other optical
systems [7]. These systems include cavity QED experiments [8, 9], which allow for
strong coupling between the atom and cavity optical field modes; optical dipole traps
2
[10, 11], which generally have tighter confinement of atomic motion than MOTs and
are far-detuned from atomic excitation; and one-dimensional optical lattices [12, 13],
which allow for targeted experimental interaction with multiple atoms. There have
been a broad field of research looking at atomic counting and MOT characterization
with single (and low-numer) MOTs [14, 15]. Other uses of single atom MOTs have
included studies of cross-atomic-species cold-atom interactions [16, 17], rare-isotope
separation [18, 19], and detailed studies of the high-gradient MOT loading and loss
mechanisms [20, 21].
Besides these studies of loading and loss mechanisms, there is little experimental
research on the dynamics of a few atoms in a magneto-optical trap. Additionally,
these have investigated loss rates statistically as opposed to the dynamics which
causes atomic loss in the traps. For few-atom MOTs, these loses are due to collisions
between atoms in the MOT resulting in atoms exiting the trap [15]. Some atomic
collisions coincide with atomic energy transitions that provide enough kinetic energy
for the atoms to escape the MOT [22]. The only experiments that have looked at the
dynamics of a single atom in a magneto-optical trap examined correlations between
photons emitted by the atom [23, 24]. These works reveal temporal-correlations
that reflect both internal atomic dynamics (Rabi oscillations) and external dynamics
(position-dependent electric field intensity and polarization) over several orders of
magnitude.
The work discussed in this dissertation adds to this little-explored topic by
looking at position-dependent oscillations of the atom in a MOT magnetic field. The
fluorescence rate of an atom depends on the detuning from resonance of the exciting
laser field. Because of the linear magnetic field of the quadrupole, the Zeeman shifted
energy levels have detunings that vary spatially. Modulating this magnetic field
3
introduces oscillations in the detuning and thus the rate of fluorescence from the
atom. Measuring the fluorescence oscillations can produce a time-averaged position
distribution for the atom in the MOT. The position distribution of the atom is very
closely related to its potential energy, which can be used, via the equipartition theorem
[25–27], to measure the atomic temperature. Therefore, this work functions as an in-
situ temperature measurement of the atom in addition to examining the averaged
motion of the atom in the MOT.
1.2 MOT Temperature Measurements
There are already a few established methods for measuring atomic temperatures
in a MOT. Some of the methods, like the method proposed here, compare
measurements of the atom(s) in the MOT to models of the MOT in order to
extract potential energy information about the atom. Others methods more directly
measure the temperature through the atomic kinetic energy, but these methods are
lossy, requiring releasing the atom(s) from the MOT and reloading new atoms for
experiments.
One such lossy method is the release-recapture method which was used to
estimate the MOT temperature in the first successful MOT publication [4]. This
method turns off the trapping fields in order to allow the trapped atoms to expand
from the trap ballistically. Turning the trap on again, after a given amount of time,
recaptures just a fraction of the atoms. Comparing trap-off times with recapture
fraction gives an estimate of average atomic velocity, and hence temperature in the
MOT. In addition to measuring the temperature of traps with large numbers of atoms,
this technique has be used to measure the temperature of a single atom both in a
MOT and in a dipole trap [16, 28, 29].
4
The time of flight technique releases atoms from a MOT and allows them to fall
under gravity through a near-resonant laser field [30]. As the atoms pass through the
field, the light they scatter is measured. Observing how this fluorescence changes as a
function of time from release (atoms moving directly downward initially pass through
the laser before atoms that were initially moving directly upward) gives an estimate of
the average velocity of the atoms, and hence their temperature. This technique can be
used for other systems including atoms in an optical lattice [31]. Other configurations
of the time of flight technique use an additional laser beam to push the atoms in some
direction where the probe beam has been located [30]. Pushing the atoms vertically
upward takes advantage of converting kinetic energy to gravitation potential energy in
order to measure a maximum height for the MOT atoms to reach, giving a measure of
their initial kinetic energy. Pushing the atoms horizontally lets gravity drag the atoms
downward, under the probe beam, to measure a travel distance for the atoms—and
therefore a maximum horizontal velocity distribution.
Modified time-of-flight techniques have been used to measure the temperature of
single atom [29], although due to the difficulty of imaging scatter from a single atom,
they are less commonly used than release-recapture methods. For the single-atom
measurement, instead of detecting the light of an atom as it passes through a nearby
beam, the position of the atom is detected on a CCD after a resonant imaging pulse.
Repeating the test provides information of the spatial distribution of the atom after
release, allowing velocity and temperature to be estimated for an atom initially inside
the MOT.
Another lossy method adiabatically reduces the optical potential in which an
atom resides [16, 28, 32]. Measuring the probability that the atom remains in
the trap at a given potential energy gives an estimate of its kinetic energy. This
5
method is similar in concept to evaporative cooling techniques used, as an example,
in Bose-Einstein condensates [33]. While evaporative cooling is used to decrease
the temperature and increase the density of an atomic cloud, the adiabatic lowering
technique just probes the temperatures of a small number of atoms. For temperature
measurements, this technique is mostly used in dipole traps where the potential can
be easily reduced by lowering the intensity of the trapping laser [4, 16, 34].
One method that preserves the number of atoms in the MOT looks at the
frequency spectrum of photons emitted by the atoms. The motion of atoms in the trap
will broaden the wavelength of the emitted light via the Doppler effect. Measuring
this broadening allows for an estimate of the velocity of the atoms [35]. This is also
used in measuring the temperature of trapped ions [36]. Ion traps can also make
use of quantized motion to measure spectra and temperature [36]. Spectra from ions
(including single ions [37]) reveal sidebands of resonance peaks. The number and
relative intensities of the sidebands are related to the average vibrational mode of
the ion in the trap, and hence to temperature. This technique can also be used to
measure temperatures of neutral atoms in optical lattices [38].
Another number-preserving method takes advantage of the harmonic-oscillator
model of the MOT and the equipartition theorem. This method uses an external
force (created either an oscillating, external uniform magnetic field [26] or additional
laser beam [27]) to drive oscillations in the center of mass of the MOT. Measuring the
amplitude responses at different frequencies gives the natural frequency, and thus the
spring constant, of the restoring force in the MOT. With a measurement of the RMS
radius of the MOT via pictures of it, the average potential energy of the atoms is
revealed. As a temperature estimate, this method is similar to our proposed technique
6
as it measures temperature via external oscillations, but this method is not applicable
to single-atom MOTs where there is not a well defined size of the atomic cloud.
1.3 Optical Trap Oscillations
In addition to measuring MOT spring constants with oscillations, oscillations
of laser fields have been used to measure properties of particles in other forms of
optical traps. For beads in an optical dipole trap (see Section 2.2.4), oscillating
the power of the laser which confines the atom can excite resonances in the bead
[39]. As with the MOT spring measurements above, Imaging the bead’s motion with
these oscillations reveals the trapping strength on the bead. Similar experiments have
probed the trapping potential for atoms in an optical lattice, in which the interference
of multiple laser fields creates a periodic lattice of positions where atoms are trapped
[40, 41]. Here, modulations of the lattice’s trapping beam power also modulates the
trapping potential for the atoms. Measuring the populations of atoms still present in
the lattice after being driven at various frequencies can reveal the vibrational states
of the lattice [40]. Additional, the modulations can be seen in changes in power
measured from beams diffracted by the atoms arranged in the lattice [41].
For these two purely optical traps, a magnetic field is not necessary, requiring
that modulations be driven by oscillations in laser power. Attempting to detect
oscillations in fluorescence from the trapped particles, then, would be difficult as the
signal would be swamped by oscillations in background fluorescence levels. Instead,
these experiments (save for the diffraction experiment in [41]) directly imaged the
particles in the trap with a camera to observe oscillations. This can be challenging
for a single atom, although our experiment does reveal similar oscillations for the
single atom when imaged with a CCD camera (see section 6.5). Without modulating
7
the beam power, as in our experiment, we instead detect oscillations directly through
measurements of photon arrivals from an atom.
1.4 Dissertation Outline
The layout of this dissertation is as follows. Chapter II has a theoretical
description of the interaction between atoms, light and magnetic fields, building to a
description of the functioning of a MOT. Chapter III describes the experimental
apparatus, focusing on relevant changes made for the single-atom experiments
described in later Chapters. Chapter IV discusses our single-atom MOT and examines
a new technique for monitoring and controlling experiments based on a single atom.
Chapter V expands on the theory in Chapter II to look more closely at the behavior
of atoms with the complete electronic structure of the D2 transition for rubidium.
Chapter VI looks at position and temperature measurement experiments performed
on our single atoms. Finally, Chapter VII draws conclusions for our experiment and
briefly lays out the path forward for future investigation.
8
CHAPTER II
ATOM OPTICS AND MAGNETO-OPTICAL TRAPS
In this Chapter, the interaction of atoms with light and magnetic fields is
sketched using a standard semiclassical picture in which the atom’s internally energy
is quantized but its external motion and external fields are treated classically.
Calculations are done in one-dimension with the atom is treated as having a single
ground state and a small number of excited states when appropriate. Details of the
calculation are given with an eye toward the three-dimensional picture in Chapter V
with a full D2 transition of
87Rb. After examining atomic interactions with electric
and magnetic fields individually, an atom inside a magneto-optical trap is discussed.
2.1 A Single Atom
Until the broader discussion of magneto-optical traps in Section 2.5, the atom
will be treated as a qubit with energy separation ~ω0. Under this assumption, the
atomic Hamiltonian should have the form
ĤA = ~ω0 |e〉〈e| (2.1)
with the quantum state of the atom in the form
|ψ〉 = ce|e〉+ cg|g〉 (2.2)
where |e〉 and |g〉 are the atomic excited and ground states, respectively. In this
definition, the ground state energy is defined to be zero. Rather than working with
the atomic wavefunction, later calculations are simplified by using the atomic density
9
operator [42]. defined by
   
|c |2 c c∗ ρ ρ
ρ = |ψ〉〈ψ|  e e g  e,e e,g=   =   . (2.3)
c ∗ 2gce |cg| ρg,e ρg,g
s In the Schrödinger picture, the density operator evolves under the equation
( ) ( )
d d
ρ = |ψ〉 〈ψ|+ | dψ〉 〈ψ|
dt dt dt
i i
= − Ĥ |ψ〉〈ψ|+ |ψ〉〈ψ| Ĥ
~ [ ] ~d i
ρ = − Ĥ, ρ . (2.4)
dt ~
Note that this equation is identical to the time evolution of an operator in the
Heisenberg picture. The use of density matrices must be implemented to look at mixed
states—quantum states which cannot be simple written as a linear superposition
of eigenstates of a Hamiltonian [43]. Such states appear, for example, in analysis
of entanglement [44], teleportation [45], and when looking at quantum trajectories
[46, 47]. The density operator also is beneficial as operator expectation values are
calculated simply by tracing the atomic states over the product of the operator applied
to the atomic density operator as
[ ] ∑
〈Â〉 = Tr Âρ = 〈n|Âρ|n〉 (2.5)
n
where |n〉 form a complete basis to describe eigenstates of the system.
For analysis of an atom in the MOT, the use of the density operator is important
in modeling spontaneous emission of photons from the atom through the Lindblad
10
superoperator [48] defined as
1 ( )L [σ̂] ρ = σ̂ρσ̂† − σ̂†σ̂ρ+ ρσ̂†σ̂ (2.6)
2
where σ̂ = |g〉〈e| and σ̂† = |e〉〈g| are the atomic raising and lowering operators
respectively. For the two level atom, the superoperator simplifies to
L [σ̂] ρ = ρe,e|g〉〈g| −
1
(|e〉〈e|ρ+ ρ|e〉〈e|) . (2.7)
2
Including this operator, the evolution of an atom which undergoes spontaneous
emission follows
d − i
[ ]
ρ = ĤA, ρ + ΓL [σ̂] ρ, (2.8)
dt ~
where Γ is the decay rate of the atom.
2.2 Atoms and Light
A few simplifying assumptions are made to analyze the interaction of an atom
with light, following the methodology of [48]. The light, for now, is treated as a
linearly polarized electric field of a single mode. The atom is treated as small enough
that the spatial variation of the electric field can be ignored. Thus the light field has
the form
E~ (t) = ̂E0 cos(ωt+ φ), (2.9)
where ω is the frequency of the light, assumed to be close to the transition frequency
of the atom, ω0, and ̂ is the polarization direction of the light. These assumptions
are sufficient for finding a form for the interaction Hamiltonian between an atom and
11
light, but removing some of these assumptions leads to interesting results, which are
discussed when appropriate.
2.2.1 Interaction Hamiltonian
The interaction between the atom and electric field is treated as a dipole
interaction with an atomic dipole operator
( )
d̂ = 〈e|d~|g〉 σ̂† + σ̂ , (2.10)
where d~ = e~r is the dipole moment of the atom (~r is the position operator for the
atom’s electron). This definition of d̂ derives from treating the electron position
operator as that of a harmonic oscillator, where r̂ is proportional to the sum of the
oscillator raising and lowering operators [49], which here correspond to atomic raising
and lowering operators, σ̂† and σ̂. The expectation value of the dipole operator is
( )
〈d̂〉 = 〈e|d~|g〉 〈σ̂†〉+ 〈σ̂〉 . (2.11)
With the atom treated as a dipole, we can find an interaction Hamiltonian by
comparison tot the energy of an electric dipole interacting with a field. This provides
an interaction Hamiltonian
ĤAF = −E~ · d̂. (2.12)
In the absence of the electric field, the excited state population evolves as e−iω0t and
〈σ̂〉 evolves in this same way. With this, the dipole operator can be written in the
form
d̂ = d̂+ − d̂+ (2.13)
12
with d̂− ≡ 〈e|d~|g〉σ̂† ∼ eiω0t and d̂+ ≡ 〈e|d~|g〉σ̂ ∼ e−iω0t. Writing the electric field in
terms of complex exponentials gives
~ E0
[ ]
E(t) = ̂ e+iωt+φ + e−iωt−φ ≡ E~− + E~+ (2.14)
2
and is used to write the interaction Hamiltonian as
( )
HAF = − d̂− · E~− + d̂− · E~+ + d̂+ · E~− + d̂+ · E~+ . (2.15)
In terms of the exponentials, these four terms are proportional to ei(ω0+ω)t, ei(ω0−ω)t,
e−i(ω0−ω)t and e−i(ω0+ω)t respectively. In the limit where the frequency of light is
very close to that of the atomic energy, the terms with the frequency differences
oscillate much more slowly than the terms with their sum (the first and last term).
The rotating wave approximation ignores these quickly oscillating terms, so that the
interaction Hamiltonian becomes
H = −d̂− · E~+AF − d̂+ · E~−. (2.16)
Written in terms of atomic raising and lowering operators, the Hamiltonian is
~ ( ∗ − )HAF = Ω σ̂e iωt + Ωσ̂†e+iωt , (2.17)
2
where
〈g|̂ · d~|e〉E e−iφ0
Ω = (2.18)
~
13
is named the Rabi frequency. The excited state population for any two level system at
rest and interacting with a non-resonant electric field will oscillate with this frequency
[42, 50, 51].
In an alternate view of the rotating wave approximation, the electric field
is quantized as a harmonic oscillator. Under this view, the (single frequency,
polarization and mode) electric field follows
E~ ∝ f~(~r)â+ f~∗(~r)â†, (2.19)
where f~(~r) are the spatial mode functions of the field, and ↠is the photon creation
operator for this mode (expanding to a general electric field requires summing over this
term for each frequency, mode, and polarization). With this picture, the interaction
Hamiltonian is
Ĥ ∝ âσ̂ + âσ̂† + â†σ̂ + â†AF σ̂†. (2.20)
The first term of this equation removes a photon from the field and lowers the atom
from excited to ground state. The last term adds a photon to the field and raises
the atom into the excited state. Both of these are non-energy conserving and can be
dropped1. These two terms are the same as the quickly rotating terms which were
dropped previously.
1These terms, while ignorable here, can be viewed as atomic interaction with the quantum vacuum
[52]. This interaction leads to phenomenon such as the Casimir-Polder affect [53] and the Lamb shift
[42, 54].
14
2.2.2 Optical Bloch Equations
The evolution of the atomic wavefunction in the Schrödinger picture is
d | iψ〉 = − (H +H
~ A AF
) |ψ〉 (2.21)
dt
which produces equations
iωt
ċg = − iΩe c2 e
(2.22)
− iΩ∗ iωtċe = iω e0ce + c .2 g
These can be simplified in a rotating frame where ce is transformed to c̃ee
−iωt to
produce
ċ = − iΩg c2 e
(2.23)
ċ = −i(ω − ω)c + iΩ∗e 0 e c2 g
(note, here I’ve dropped the c̃e notation). Eventually moving to this frame is what
motivated writing the dipole operator in terms of complex exponentials in Equation
2.13. These same equations could have been derived from an atomic Hamiltonian of
the form
ĤA = −~∆ |e〉〈e| (2.24)
and interaction Hamiltonian
~ ( ∗ †)ĤAF = Ω σ̂ + Ωσ̂ , (2.25)
2
where ∆ = ω − ω0 is the detuning of the electric field from the atomic resonance
frequency. These are the forms of the atomic and interaction Hamiltonians that will
be used throughout this text.
15
From these forms, the time evolution of the atomic density operator, calculated
from Equation 2.8, produces the optical bloch equations [42]
ρ̇e,e = −Γρ ie,e − (Ωρ ∗g,e − Ω ρ2 e,g)
( )
ρ̇ = − Γ iΩe,g + i∆ ρe,g − (ρ2 2 g,g − ρe,e)
( ) (2.26)∗
ρ̇ Γ iΩg,e = − − i∆ ρg,e + (ρ2 2 g,g − ρe,e)
ρ̇ i ∗g,g = Γρe,e + (Ωρg,e − Ω ρe,g) .2
In the steady state, these have analytic solutions
ρSS = |Ω|
2/Γ2
e,e 1+(2∆/Γ)2+2|Ω|2/Γ2
ρSS = − iΩ 1+2i∆/Γe,g Γ 1+(2∆/Γ)2+2|Ω|2/Γ2
(2.27)
SS iΩ∗ρ = 1−2i∆/Γg,e Γ 1+(2∆/Γ)2+2|Ω|2/Γ2
1+(2∆/Γ)2ρSSg,g = .1+(2∆/Γ)2+2|Ω|2/Γ2
Recall that ρe,e is the population of the excited state ρ
2
e,e = |ce| from Equation 2.2.
With this definition, the rate that an atom scatters photons is
R = Γρe,e. (2.28)
16
2.2.3 Radiation Pressure
The force acting on an atom is the time evolution of the atom’s (classical)
momentum p~. Following the Heisenberg picture, this is
~ i
[ ]
F = Ĥ, p~ = −∇~ Ĥ. (2.29)
~
To look at the force of light on the atoms, we use the Hamiltonian 2.25. This gives
~ −~
( )
F = ∇~ Ω∗σ̂ +∇~ Ωσ̂† . (2.30)
2
From the definition of the Rabi frequency in Equation 2.18, Ω ∝ E iφ0e , where E0 is
the field magnitude and φ is its phase. Both of these can depend on space, but for
now consider the electric field to be a plane wave propagating in the +ẑ direction so
that the electric field strength has the form
E0(~r) = E
+ikz
0e . (2.31)
Using this in Equation 2.30 gives a straightforward equation for the force
( )
~ −~ 2dE0F = ∇~ e−ikz 2dE0σ̂ + ∇~ e+ikzσ̂†
2 ( ~ ~− )~ 2dE e ikz 2dE e+ikz
= − − 0 0ikẑ σ̂ + ikẑ σ̂†
2 ~ ~
(
F~
ik~
= − −Ω∗
)
σ̂ + Ωσ̂† ẑ. (2.32)
2
17
Taking the expectation value of the force gives
〈~ 〉 −ik~
[( ) ]
F = Tr −Ω∗σ̂ + Ωσ̂† ρ ẑ
2
−ik~= (−Ω∗ρe,g + Ωρg,e) ẑ (2.33)
2
after using Equation 2.5. The evolution for the excited state population from the
optical bloch equations in 2.26 can be solved in steady state to get
i ( )
ΓρSS = − −Ω∗ρSSe,e + ΩρSS .2 e,g g,e
Comparing this to the expectation value for the force, the steady-state force on the
atoms is
〈F~ 〉SS = ~kΓρSSe,e ẑ. (2.34)
This is an insightful equation as Γρe,e is just the scatter-rate of photons by the atom
as in Equation 2.28, and ~kẑ is the momentum carried by each photon. Thus the force
felt by atom is just the average rate it absorbs momentum from scattered photons.
The momentum change from emitting photons goes to zero in the limit of many
absorption-emission events as the emissions have random directions. This force due
to absorbed photon momentum is commonly referred to as radiation pressure.
In addition to being the basis for optical molasses (see Section 2.4.1 below),
radiation pressure has been used experimentally to launch atoms in atomic clocks
[55, 56], cool micromechanical resonators [57, 58], and even macroscopic objects such
as the first solar sail successfully flown by the Japan Aerospace Exploration Agency
[59, 60].
18
2.2.4 Dipole Traps
Returning to Equation 2.30, if the assumption is made that the electric field
magnitude, rather than the phase, depends on space, the equation for force can be
written as
( − )iφ iφ
F~ = −~ 2de ∇~ 2deE0σ̂ + ∇~ E σ̂†0
2 ( ~ ~ )
−~ 2dE e
−iφ
0 ∇~ E0 2dE0eiφ∇~ E0
= σ̂ + σ̂†
2 ~ E0 ~ E0
~ ( )
= − Ω∗∇~ log[E0]σ̂ + Ω∇~ log[E ]σ̂†0 . (2.35)
2
As done previously, the expectation value for the force is
〈F~ 〉 −~= ∇~ log[E0] (Ω∗ρe,g + Ωρg,e) . (2.36)
2
This equation is real, as the sum of the complex terms produces just 2Re [Ωρg,e].
From the steady state Bloch equations, this is
( ) 4∆ |Ω|2∗ ss ss /Γ2Ω ρe,g + Ωρg,e = (2.37)
1 + (2∆/Γ)2 + 2 |Ω|2 /Γ2
where, again, ∆ is the detuning of the electric field from the atomic energy, Ω is the
Rabi frequency and Γ is the atomic decay rate. Here, it is convenient to introduce a
saturation parameter defined as
2 |Ω|2 /Γ2
s = (2.38)
1 + (2∆/Γ)2
19
so that
( ∗ )ss ss 2∆sΩ ρe,g + Ωρg,e = . (2.39)1 + s
Then, the force becomes
〈~ 〉 − ~∆sF = ∇~ log[E0]. (2.40)
1 + s
The definition of s is proportional to the field intensity, which goes as E20 . As field
intensity is a more common parameter for experiments than field magnitude (the
(integrated) electric field intensity of a laser is proportional to beam power), writing
∇~ log[E0] in terms of the saturation parameter comes from
∇~ s = 2s∇~ log[E0]. (2.41)
This produces a force equation just in terms of the saturation parameter
~
〈~ 〉 − ~∆∇s −~∆F = = ∇~ log[1 + s]. (2.42)
2(1 + s) 2
For large detunings where there is little excitation of the atom by the field, s  1.
This simplifies the force to
~
〈F~ 〉 ≈ − ~∆∇s = −~∆∇~ s, (2.43)
2(1 + s) 2
so that the force on the atom is directed towards regions of high intensity (large s).
A common field where the beam intensity depends on position is a pair of tightly
focused gaussian beams. This field has electric field intensity [61]
[ ]
[ 2P ] −(2 (x
2 + y2)
I(~r) = exp ) (2.44)
πw20 1 + (z/z )
2
0 w20 1 + (z/z0)
2
20
where P is the total beam power, w0 is the beam waist, the beam radius at its focus,
and z0 is the Rayleigh range of the beam, which gives a measure of the length of the
focus size along the beam axis and is defined as the distance where the beam radius
√
grows from its minimum by a factor of 2. For this field, the force is given by
[ ( ) ]
〈~ 〉 2~∆s w
2 2(x2 + y2)
F = xx̂+ yŷ + 0 − zẑ . (2.45)
w20 [1 + (z/z
2
0) ] z2 z20 0 [1 + (z/z
2
0) ]
With the laser detuned below resonance, ∆ < 0, this is clearly a restoring force. This
specific arrangement of focused gaussian beams is often called a dipole trap.
2.3 Atoms and Magnetic Fields
The interaction of an atom with a magnetic field is a magnetic dipole interaction
based on the total angular momentum of the atom, including the orbital angular
momentum L~ , spin S~ and nuclear angular momentum I~. The combination of these
give the hyperfine structure of the atom, quantized with
F~ = L~ + S~ + I~. (2.46)
The D2 transition of
87Rb, on which our MOT is based, has transitions between the
52S1/2 ground state and the 5
2P3/2 excited states. Both of these states have S = 1/2.
The ground state has L = 1/2 and the excited state has L = 3/2. Together with
87Rb’s nuclear spin of I = 1/2, the excited state can have total angular momentum
F with values between 0 and 3, and the ground state can have values of either 0
or 1 [62]. This is shown in the level diagram of Figure 2.1. The frequency splitting
between the hyperfine levels is on the order of 102 MHz while energy shifts due to
the magnetic field (Zeeman shifts) of the MOT is on the order of MHz. With such
21
|E; F = 3i
52P3/2
|E; F = 2i
|E; F = 1i
|E; F = 0i
|G; F = 2i
52S1/2
|G; F = 1i
FIGURE 2.1. 87Rb D2 Transition Level Diagram. Labeled at the transitions address
for our MOT.
a small shift of the levels due to the magnetic field, the total angular momentum F~
is a fair quantum number to use to study the interaction between the atom and the
magnetic field [62]. Here, we’ll treat the |G;F = 2〉 and |E;F = 3〉 levels as the only
two levels of our atom. A fuller picture of the hyperfine atom is discussed in Chapter
V.
The atom is treated as a magnetic dipole that is aligned with its angular
momentum
µ~ = µ ~BgFF , (2.47)
where µB is the Bohr magneton.
Much like the electric dipole interacting with the electric field of light, the
interaction between the magnetic dipole and magnetic field is then [63]
Ĥz = −µ~ ·B~ . (2.48)
22
Trapping
Repump
Working in one dimension, assuming that B~ lies along the z-axis, the interaction
simplifies to
Ĥz = −µBgF F̂zBz
when the z-axis is also the angular momentum quantization axis of the atom. F̂z
is the projection operator of the atomic angular momentum along the quantization
axis. Writing this explicitly gives
∑∑∑
Ĥz = −µBBz gFmF |s;F,mF 〉〈s;F,mF | (2.49)
s=e,g F mF
where s is the excited or ground state of the atom. From this, it is clear that the
magnetic field is responsible for shifts of the energy levels of atoms, the Zeeman shifts
[64].
The quantities gF are Landé g-factors [62]. They result from perturbative
approximations made when the shift in atomic energy levels due to the magnetic
field is much smaller than the hyperfine splitting. In this case, the total atomic
angular momentum F~ , as defined in 2.46, serves as a good quantum number. In
87Rb, the hyperfine splitting is on the order of gigahertz for the ground states and
hundreds of MHz for the excited states. The Zeeman splitting (per Gauss) is on the
order of 1MHz/G. With magnetic fields in the MOT on the order of tens of Gauss,
this condition is met,which is good since all For 87Rb’s D2 transition, all excited states
have gFe = 2/3 and the ground states have gFg=2 = 1/2 and gFg = −1/2 [62].
When a 3D model of the atom is discussed in Chapter V, this assumption about
the magnetic interaction will remain true—the atom’s quantization axis will align
with the magnetic field direction at every location in space. This will require rotation
of the lab frame to align the z-axis of the lab frame with the magnetic field. This is
23
the largest complication of the 3D model as the polarization directions for each of the
six MOT beams must be correctly written in a circular and linear polarization basis.
This rotation is discussed in Section 5.1.3.
2.3.1 Magnetic Trapping
A particle with a magnetic dipole momentum µ~ in a magnetic field will have a
potential energy U = −µ~ · B~ . If this energy is spatially varying, the particle will
experience a magnetic force
F~ = ∇~ (µ~ ·B~ ) (2.50)
As done above (and as will be assumed while discussing the MOT), when the dipole
∣∣ ∣∣
moment is aligned with the field the interaction energy is U = µ ∣B~ ∣. Under these
circumstances, the force is then
~ − ∇~ ∣
∣ ∣
F = µ ∣B~ ∣∣ , (2.51)
so that the force is zero where the magnetic field strength vanishes. This force
must also apply to an atom with magnetic dipole moment [65, 66] and is critical
to evaporative cooling for Bose-Einstein condensates [67].
As done for radiation pressure, the force is examined quantum mechanically by
Equation 2.29 with the Hamiltonian 2.49. An atom in the steady state experiences a
force ( )
〈~ 〉 ∂B
∑∑∑
z ∣∣ ∣∣2F = µB ẑ g ssFmF cs,m , (2.52)∂z F
s=e,g F mF
where the energy level populations are written as their wavefunction coefficients
rather than density operator elements (strictly for simplification of subscripts). The
24
relationship to the classical magnetic force of 2.51 is a bit unclear as it appears
uniform. However, the steady state populations have spatial dependence (see Figures
5.1 for a graph of the populations as a function of position), which allows for magnetic
trapping.
For a linear magnetic field B~ = −B′zzẑ, as will be discussed for the MOT, the
force is
∑∑∑
′ ∣∣ ∣F~ = −µ B ẑ g m css ∣2B z F F s,m . (2.53)F
s=e,g F mF
For states where gF > 0, when z < 0 (so that B > 0), the energy levels with mF < 0
will be preferentially populated as their energies will be reduced by Zeeman shifting.
This results in the double sum producing a negative value, given an overall force in
the positive z-direction. When z > 0, the mF > 0 energy levels are preferentially
populated and produce a force in the negative z-direction. Thus, the overall force is
to locate the atom near z = 0, just as in the classical case. This results holds true
when gF < 0, with the preferential population switching positive and negative values
for mF .
It is important to note here that this magnetic trapping is distinct from the
magnetic confinement discussed in Section 2.5.1. The trapping here results from the
minimization of the magnetic dipole energy. The confinement trapping results from
spatially preferential photon absorption due energy level Zeeman splitting.
2.4 The Fg = 0→ Fe = 1 Atom
For the remainder of this Chapter, the focus will be on a Fg = 0→ Fe = 1 atom,
that is one having no ground state angular momentum and an excited state with a
total angular momentum of 1. With the angular momentum formalism in Equation
25
|E; mF = 1 i |E; mF = 0 i |E; mF = +1 i |i |0 i |+ i |i |+ i
⌦0
⌦ ⌦ + 
+
|G; mF = 0 i |g i |g i
(a) (b) (c)
FIGURE 2.2. Fg = 0 → Fe = 1 Atom Level Diagrams. (a) Full labeling for states
of the atom. (b) Simplified naming conventions used in the text. The electric fields
coupling excited and ground states are labeled with their Rabi frequencies as give in
Equation 2.56. (c) V-atom reduction when there is no electric field to excite the |0〉
state.
2.46, this could correspond to a spin-1/2 atom (S = 1/2), orbital angular momentum
L = 1/2 and no nuclear spin (I = 0). With no nuclear spin, the angular momentum
vector J~ = L~ + S~ is often used rather than F~ , but F~ used here for more directed
comparison to the full atomic energy levels discussed in Chapter V. The level diagram
for such an atom is shown in Figure 2.2a.
For this atom, the levels will be labeled as shown in Figure 2.2b. The atom has
density operator  
 ρ−− ρ−0 ρ−+ ρ−g 


ρ0− ρ  00
ρ0+ ρ0g
ρ =  

 . (2.54) ρ+− ρ+0 ρ++ ρ+g 
ρg− ρg0 ρg+ ρgg
Each excited states has its own raising operator defined, the three being defined
as
σ†− = |−〉〈g| ,
σ†0 = |0〉〈g| , and (2.55)
σ†+ = |+〉〈g| .
26
The excited and ground states are coupled via electric fields E~ , E~+ 0 and E~−,
named relative to the excited states they couple (note this names are opposite the
naming conventions in Equation 5.9). These fields are circularly polarized or linearly
polarized. Following the derivation of Equation 2.25, this leads to an atom-field
coupling Hamiltonian
~ ( ) ~ ( ) ~ ( )
ĤAF = − Ω∗ †−σ̂− + Ω−σ̂− − Ω∗ † ∗ †2 2 0σ̂0 + Ω0σ̂0 − Ω σ̂+ + Ω+σ̂ . (2.56)2 + +
It is also assumed that the electric fields are all detuned from resonance ω0 by some
amount. Following the derivation in 2.24, the atomic Hamiltonian can be written as
ĤA = −~∆+ |+〉〈+| − ~∆0 |0〉〈0| − ~∆− |−〉〈−| . (2.57)
The magnetic field Hamiltonian is given directly from Equation 2.49 and for the
Fg = 0→ Fe = 1 atom is
Ĥz = µBgFBz (|+〉〈+| − |−〉〈−|) , (2.58)
as evident from the mF values shown in Figure 2.2a. Defining ∆B = µBgFBz/~, this
can be written as
Ĥz = ~∆B |+〉〈+| − ~∆B |−〉〈−| . (2.59)
Each of the excited states can spontaneously decay and it assumed this occurs at the
same rate, Γ, for each state. The evolution of the atomic density operator follows the
equation
d i [ ]
ρ = − ĤA + ĤAF + Ĥz, ρ + ΓL [σ̂−] ρ+ ΓL [σ̂0] ρ+ ΓL [σ̂+] ρ (2.60)
dt ~
27
based on the density operator evolution definition of 2.8. The form of the Lindblad
superoperator of Equation 2.7 reveals that spontaneous emission from one excited
state of the atom depends only on the populations of that excited state. With these
separate decay paths, the impact on the evolution of the state of the atom from each
state is independent as written above. The time evolution of each density operator
element ρi,j is listed in Appendix A.2.
Our magneto-optical trap consists of pairs of counter-propagating, circularly
polarized lasers, as shown in Figure 2.5. The counter-propagating beam have the
same circular polarization in their reference frame, but in the reference frame of an
atom (with the beams moving towards it from different directions) the two beams have
opposite polarizations, corresponing to only the E~− and E~+ electric fields. Matching
the model to our experiment sets E~0 → 0. As discussed in Appendix A, doing this
effectively simplifies the atom to the V-atom, shown in Figure 2.2c. Without the
linear field component to excite the |0〉 state, the x‘x‘population of this state will
decay quickly to zero and can be ignored. The remainder of this Chapter will assume
this simplification of the atomic structure.
The equations of motion for the internal structure of the V-atom are given in
equations A.4. In these equations, for simplicity, the excited state energies Zeeman
energy shifts are left out. They can be returned to the solutions by allowing ∆± →
∆±±∆B. These solutions are not particularly enlightening other than it is nice that
there is a analytic solution and we can use the two-level atom steady-state solution
to check their validity. To do this, remove one electric field by letting Ω∓ → 0 and
∆∓ → 0, then the population in the |±〉 state is
ss |Ω̃±|2ρ±,± = (2.61)
1 + 4δ2± + 2|Ω̃±|2
28
where, as defined in Appendix A, Ω̃± = Ω±/Γ and δ± = ∆±/Γ. These agree with the
population of the excited state from the two-level atom optical Bloch equations 2.27,
as they should: as was the case for the |0〉 state, when there is no field coupling an
excited state to the ground state, that excited state can be ignored as its population
will decay to zero. Thus, with only one laser, the atom should behave as a two-level
atom excited by just one field and the solutions should agree exactly with the optical
bloch equations.
The quantity Ω̃ = Ω/Γ is often written in terms of the ratio of the electric field
intensity relative to a saturation intensity as
I ∣∣ ∣∣ ∣
2 2
= 2 Ω̃∣ 2 |Ω|= . (2.62)
I 2sat Γ
This is the saturation parameter, Equation 2.38, with ∆ = 0. For the electric field
with amplitude E0, the intensity of the field is I = 0cE
2
0/2. Together with the
definition of Ω in Equation 2.18, the saturation intensity is
 c~2Γ20
Isat = ∣∣ ∣2 . (2.63)
4 ∣〈g|̂ · ∣d~|e〉∣
This is referred to as the saturation intensity, as when I  Isat (i.e. |Ω| >> Γ), the
two-level atom excited state population in Equation 2.27 saturations to 1/2. This
quantity clearly has different values for different electric field polarizations ̂ (the
dominator will be changed). When the saturation intensity is referred to throughout
this text, it is in reference to circular polarized light exciting MOT trapping transition
which has a value 1.669mW/cm2 for the atomic species of rubidium used in our
experiments [62].
29
2.4.1 Optical Molasses
The idea of radiation pressure can be extended further to look at the effects of
identical counter-propagating plane waves interacting with an atom. These two fields
are defined as
E+(~r) = E
−ikz
0e and
(2.64)
E−(~r) = E e
+ikz
0 .
These are defined so the field propagating in the negative z-direction excites only
the σ+ transition and the field propagating in the positive z-direction excites only
the σ− transition, as shown in 2.3. For each of these beams, atoms moving against
their propagation direction should see the frequency of the light shifted to a higher
frequency due to the Doppler effect. An atom with velocity v then sees each beam
having velocity
ω+ = ω + kv
(2.65)
ω− = ω − kv,
where ω is the rest frequency of the light. The atom then sees light that is detuned
from the resonant frequency ω0 by
∆+ = ω+ − ω0 = (ω − ω0) + kv = ∆L + kv
(2.66)
∆− = ω− − ω0 = (ω − ω0)− kv = ∆L − kv,
where ∆L is the (red) detuning of the laser from the atomic resonance. Following the
atomic force derivation in Section 2.2.3, the force operator is
( )
~ −ik~ ∗ − † − ik~
( )
F = Ω σ̂ ∗ †+ + Ω+σ̂+ ẑ −Ω2 2 −σ̂− + Ω−σ̂− ẑ, (2.67)
30
which calculated in steady-state gives
F ss(v) = [−~kΓρ(v)+.+ + ~kΓρ(v)−,−] ẑ, (2.68)
where the density operator elements are written to note that they explicitly depend on
the velocity of the atom through the Doppler shifted detuning of the lasers. Allowing
√
Ω+ = Ω− = Γ/ 2 and ∆L = −Γ, the force (in units of ~kΓ) is plotted as a function
of velocity (in units of |∆L| /k) in Figure 2.4a in red.
This result is easy to explain. An atom moving towards one of the fields, sees
a Doppler-shifted beam (at higher frequency) that is closer to resonance than the
beam it is travel along with. A smaller detuning (recall, ∆L < 0) allows the opposing
beam to more easily excite the atom. This is shown with the different length arrows
coupling the ground and excited states in the level diagram of Figure 2.3. The photons
absorbed from the in-tune beam apply a larger force than the opposing beam, giving
a net force pushing the away in the opposite direction of its motion. With counter-
propagating beams, each beam slows atoms moving towards it. This gives an overall
decrease in speed, and thus temperature of the atom. This arrangement of lasers is
known as optical molasses and are a well studied method to cool atoms [1–3].
For the D2 transition of
87Rb, the force scale is 3.2×10−20 N and the velocity scale
is 4.7 m/s in Figure 2.4a. A speed of 4.7 m/s corresponds to an atomic temperature
of 78mK [62]. Typically MOT temperatures are order of 10 µK, corresponding to
velocities of order 10 cm/s (This difference in expected and measured temperatures
is discussed in Section 2.5.3). The graph of Figure 2.4b rescales the axes to 10−22 N
and cm/s. Clearly from this graph, in the range of velocities for atoms in the MOT,
the force is very close to linear. Expanding Equation 2.68 to first order in v gives a
31
 +
|i |+ i
|g i
FIGURE 2.3. Optical molasses beam arrangement. An atom moving towards a beam
see its frequency shifted closer to resonance.
F [~k]
(a)
0.2
v [|L| /k]
0.1
" or #
~ |L|
z
µ g @BB F @z
-3 -2 -1 1 2 3
-0.1
-0.2
(b) ⇥ (c) ⇤ ⇥  ⇤F 10 22N 30 F 10 22N
5 20
10
v [cm/s] z [µm]
-40 -20 20 40 -40 -20 20 40
-10
-5 -20
-30
FIGURE 2.4. V-atom forces. (a) The red curve shows forces of a 1-dimensional
V-atom and the blue curve shows the force on the extended two-level atom. Graph
shown is for both atom in optical molasses as a function of velocity or atom in a MOT
as a function of position. Peaks occur where Doppler shift (molasses) or Zeeman shift
(MOT) match the laser detuning, ∆L. (b) Damping force for velocities in range of
atoms in a MOT. (c) Restoring force for positions on scale of atomic motion in a
MOT.
32
damping force F~ (v) = F~0ẑ − βvẑ where
( )
~kΓ |Ω̃ 2p| − |Ω̃m|2
F0 = (2.69)
1 + 4δ2L + 2|Ω̃ |2m + 2|Ω̃ 2p|
and
{
2 2 2
~ 2 | | × (|Ω̃m| +|Ω̃
2
p| )(|Ω̃ 2 2m| −[|Ω̃p| ) +4(|Ω̃ 2m| +|Ω̃ 2p| )β = 8 k δL ]+
[1+4δ2 2
2
2 2 2 2
L+2|Ω̃m| +2|Ω̃p| ] 16δL+(2+|Ω̃m| +|Ω̃p| )} (2.70)
4(|Ω̃ |2+|Ω̃ |2)(1+[ 4δ2 )+16|Ω̃ |2m p L m |Ω̃p|2 ]
2 .
[1+4δ2L+2|Ω̃m|2+2|Ω̃p|2] 16δ2L+(2+|Ω̃ |2m +|Ω̃p|2)
The sign of this constant force F0 depends just on the two field’s intensities
(recall, the Rabi frequencies are proportional to the field strength). So, this constant
force is just an overall force caused by one beam having more power, and thus pushing
harder, on the atom. For most cases, this is dropped by balancing the beams, as
plotted in Figure 2.4a. In this case, the damping force reduces to F~ = −βV-atomvẑ
with ∣∣ ∣∣2
16k2~ |δL| ∣Ω̃∣
βV-atom = −[ ∣ ∣ ] [ ] . (2.71)∣∣ ∣∣
2
2 2 ∣
∣ ∣2
∣ ∣∣ ∣
∣ ∣
∣ ∣
4
1 + 4 Ω̃ + 4δL 1 + 4δL + 2 Ω̃ + Ω̃∣
A common alternative derivation of this damping is done by using the two-level
atom result for ρsse,e in Equation 2.27 for both excited state populations in Equation
2.68. This is taking the two equations of 2.61 and treating them both as accruate,
while they are in only true in the presence of just one excitation field. With these,
and making sure to use the the appropriate detunings from Equation 2.66, as done
in [25, 48], the atomic damping force is
[ ]
F = −~kΓρSS (∆ + kv) + ~kΓρSSe,e L e,e (∆L − kv) ẑ.
33
The results of this extended two-level atom is shown in Figure 2.4 as the blue curves.
This calculation does not rely on oppositely polarized fields. Two fields with linear
polarization will work for the extended two-level model.
The extended two-level method overestimates the force from one beam. It
assumes too large of an excited state population, as it ignores effects on the atomic
population due the opposite beam. With the V-atom, an individual excited state
population is reduced from its two-level atom population, as some of that population is
shifted into the other excited state. At large speeds, the agreement between methods
is better. In this case, the fast-moving atoms are much closer to resonance with one
of the beams, allowing it to dominate the atomic state. The V-atom then behaves
much like the two-level atom at large speeds. In the small speed range, the force
calculated by [25, p. 88] for the extended two-level atom is
∣ ∣2
16~k2 | ∣ ∣δL| ∣Ω̃∣
F~ (v) = −[ ∣ ] vẑ. (2.72)
∣∣
∣∣2 2
1 + 2 Ω̃∣ + 4δ2L
Compared to the V-atom in the Equation 2.71, the damping coefficient β for the
extended two-level atom is slightly larger.
While the damping of the motion should drive the atom to rest, this damping
force is balanced by the random emission of photons by the atom. Each photon
absorption, which damps the motion is followed by an emission, which gives the atom
a momentum bump of magnitude ~k in the opposite direction of the photon direction.
With a detuning ∆ = −Γ/2, to maximum β for both models in equations 2.71 and
2.72, analyzing the diffusion of the atomic velocity distribution [2, 25] results in a
minimum atomic energy of
~Γ
Umin = . (2.73)
4
34
With the 1D equipartion theorem, this results in what is called the Doppler
temperature, the lower limit on temperatures for atoms in an optical molasses.
~Γ
TD = . (2.74)
2kB
2.5 Magneto-Optical Traps
As noted above, atoms in optical molasses are still free to diffuse [68] without
being confined to any one location. Confining the atom to make an actual trap can
be done with magnetic fields.
2.5.1 Magnetic Confinement
Placing an atom into a region with a spatially variying magnetic field will give
the excited states of the atoms a spatially varying Zeeman shift. In particular, adding
a linear magnetic field with gradient −B′z (here, assume B′z > 0) shifts the excited
state energies (for the V-atom) by
µBgF
∆ = (−B′B z) , (2.75)~ z
following the convention from Equation 2.59. The spatial energy shifts of the |±〉
states are shown in Figure 2.5. From this Figure, an atom located at z > 0 has its
|+〉 excited state energy shifted down in energy. This reduces the detuning of the laser
from resonance, ∆L (shown in Figure by the thin, red line), improving the atomic
interaction with σ+ light. With σ+ light traveling form the z > 0 direction, photons
absorbed from that field push the atom back toward z = 0. The same analysis holds
true for an atom located at z < 0 with the |−〉 level and the σ− field. This results
35
z = 0
|i
L
|+ i
 +
|g i
FIGURE 2.5. Magneto-Optical Trap Level Diagram. An atom displaced from z = 0
has one excited state energy Zeeman shifted closer to frequency of the red detuned
laser. Correctly matching the polarization of a laser to the direction of the magnetic
field applies a restoring force on the atom towards z = 0.
in the magnetic field imposing a restoring force from the lasers onto the atom. This
behavior is the position-space analog to damping in optical molasses.
This interaction is quantified following similar steps to those in Section 2.4.1.
The detuning of the atoms from resonance follow the equations
∆+ = ω+ − ω0 = ω − (ω0 −∆B) = ∆L + ∆B
(2.76)
∆− = ω− − ω0 = ω − (ω0 + ∆B) = ∆L −∆B,
where it is important to note that the energy differences here are the result in shifts
of the excited state frequencies, rather than shifts in the laser frequencies as was the
case for optical molasses. The force on an atom is then
F ss(z) = [−~kΓρ(z)+,+ + ~kΓρ(z)−,−] ẑ, (2.77)
36
where the explicit dependence on position comes from the detuning energy shifts in
Equation 2.76. This equation has exactly the form of Equation 2.68 in position-space
rather than velocity space. The graph of this function will be the same as the red
curve in the graph of Figure 2.4a with the horizontal axis being position in units of
~∆L/µ ′BgFBz. This graph agrees with the restoring force interpretation presented
above. A change of sign for the magnetic field (B~ (z) = +B′zzẑ) requires flipping the
polarization of the two beams to create a trap again.
Typical MOT parameters, for our single-atom MOT, have B′z ≈ 241 G/cm.
Following the gF derivation in [62] to use gF ≈ 1.33, the distance scale is 135µm.
Atoms in our MOT have typical displacements from the center of the MOT on the
order of 10 µm, again allowing examination of small position displacements (“small”
being defined so atomic Zeeman shifts are much smaller than the detuning of the
laser). A graph of this is shown Figure 2.4c. Again, this force is very close to linear
so that we can expand the equation to get a restoring force F~ = F~0ẑ − κzẑ with the
same value for F0 as Equation 2.69 and
{
2 2
′ | | × (|Ω̃ |
2
m +|Ω̃ 2 2 2p| )(|Ω̃m| −[|Ω̃p| ) +4(|Ω̃m|2+|Ω̃p|2)κ = 8kµBgFBz δL ]2 +[1+4δ2 +2|Ω̃ |2+2|Ω̃ |2] 16δ2L m p L+(2+|Ω̃ |2m +|Ω̃ 2p| )} (2.78)
4(|Ω̃ |2m +|Ω̃p|2)(1+[ 4δ2L)+16|Ω̃ 2m| |Ω̃p|2 ]
2 .
[1+4δ2 +2|Ω̃ |2+2|Ω̃ |2] 16δ2 +(2+|Ω̃ |2+|Ω̃ |2L m p L m p )
Returning briefly to the purely magnetic trapping of Section 2.3.1, the magnetic
trapping force of Equation 2.53 for the V-atom was
F~ ss ′Mag. Trap = −µBgFBz [ρ(z)+,+ − ρ(z)−,−] ẑ. (2.79)
This force is identical in form to Equation 2.77 but with scaling µ g B′B F z. For the
typically MOT parameters above, the magnetic trapping force is of order 10−23. This
37
is orders of magnitude smaller than the MOT force scaling ~kΓ, which is of order
10−20.
As seen with the |0〉 energy level, without electric fields to excited the atom,
those excited states will not be populated. Without these fields, then, the V-atom
will have no magnetic trapping as Fg = 0. For the full
87Rb atom, however, the ground
states do have angular momentum. Thus, there can still be magnetic trapping for
87Rb without near-resonant electric fields. In addition, the |Fg = 1,mg〉 states have
gF = −1/2, which results in atoms in this ground state being repelled from the
minimum of the magnetic field magnitude rather than trapped [63, 69].
For the magnetic confinement equation, in the case where Ω− 6= Ω+, the F0 term
is non-zero and shifts the “center” of the MOT – the position where the restoring
force is zero. A similar effect occurs with a background magnetic field B~0 = B0ẑ to
give a total field of
B~ (z) = −B′zzẑ +B0ẑ. (2.80)
This shifts the location where B~ = 0, again moving the center of the MOT and
modifying the force equation
( )
~ ~ − B0F = Fz,0 κ z +
B′
ẑ. (2.81)
z
From Equation 2.78, κ can be written as κ = κ̃B′z. Then the force equation becomes
F~ = F~z,0 − κzẑ −B0κ̃ẑ. (2.82)
38
Therefore, a non-zero background field of magnitude B0 = Fz,0/κ̃ can cancel the
offsetting force due to beam imbalance to return a purely linear restoring force F~ =
−κzẑ. This can also be used to cancel gravitational forces on the MOT.
Comparing the equation for restoring constant κ to Equation 2.70, the optical
molasses damping constant, we have
µ ′BgFB
κ = β z , (2.83)
~k
which is the same as the equation for the extended two-level atom [25]. Therefore,
the extended two-level atom solution has the same overly strong assumptions for the
restoring force strength as it does for the damping force.
Taking the case with balanced electric fields, the restoring force becomes F =
−κV-atomz with
∣∣ ∣∣2
16kµ ′BgFBz |δL| ∣Ω̃∣
κV-atom = [ ∣ ∣ ] [ ∣ ∣ ] . (2.84)∣ ∣2 ∣ ∣2∣ ∣ 2 2 ∣ ∣ ∣
∣ ∣∣4
1 + 4 Ω̃ + 4δL 1 + 4δL + 2 Ω̃ + ∣Ω̃∣
2.5.2 Finally, a MOT
Combing the effects of the atomic motion-based Doppler shift creating a damping
force and the magnetic field creating a spatially dependent restoring force, the force
on an atom in the MOT is that of a damped, harmonic oscillator
F = −βz − κv. (2.85)
Here, the F0 forces from MOT beam imbalance have been suppressed.
39
With such a force, an atom in a MOT should behave as in a harmonic potential,
U = 1κz2. As above, the damping of the atomic motion by the MOT lasers will not
2
force the atom to rest because of the random emission of photons. Thus the atom
should have an average energy due to motion that has the form 〈U〉 = 1κ〈z2〉. With
2
the equipartion theorem [25], in one dimension the atomic position should follow
〈z2〉 kBT= . (2.86)
κ
With this relation, measurements of the size of a magneto-optical trap can give a
measure of the trap’s temperature as done in many of the temperature techniques
discussed in Chapter I. Early MOT experiments expected temperatures close to
the Doppler temperature, Equation 2.74, but experiments measured clearly lower
temperatures [30, 70].
2.5.3 Sub-Doppler Cooling
Atomic temperatures below the Doppler temperature are a result of polarization
changes seen by the atom moving in an optical molasses [64, 71]. In both cases, this
enhanced cooling only appears for atoms with multiple ground states, such as the one
shown in Figure 2.6, so we’re getting a bit ahead of ourselves for the discussion in
Chapter V. For optical molasses with counter-propagating linearly polarized electric
fields, alluded to in the discussion of the extended two-level atom in Section 2.4.1,
the sub-Doppler cooling mechanism is named the Sisyphus effect.
In this linear field arrangement, the polarizations for the two counter-propagating
fields created by MOT beams are are right angles. Their interference creates two
potentials, which underpin the force of Equation 2.29, that the atoms move through.
40
FIGURE 2.6. Atom with Mutliple Ground States
One potential oscillates as sin(kx) where k is the wavenumber of the light and
corresponds to the σ+ polarized light. The other potential oscillates as sin(kx + π)
and corresponds to the σ− polarized light.
Atoms primarily interacting with the σ+ light, such as atoms oscillating between
the outer two levels on the right side of Figure 2.6, will follow that light’s potential
energy curve, increasing and decreasing speed as it moves. This is shown as the
leftmost atom (black dot) in Figure 2.7. However, on occasion, when the atom is
at the peak of the potential, with the smallest kinetic energy, the atom can absorb
a σ− photon (center atom in Figure 2.7). When it does so, it moves onto the σ−
curve, which is at its lowest point in the potential energy curve (show as the arrow
for the center atom). The kinetic energy of the atom here does not change (except
for a small change due to photon emission recoil), but it now is at a potential energy
minimum (rightmost atom in the Figure). Repeating this process lowers the overall
mechanical energy of the atom and results in temperatures lower than predicted by
Doppler cooling.
For optical molasses with circular polarizations, such as the MOT described
in 2.5, the sub-Doppler cooling mechanism arrises from an additional enhanced
41

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U-
U+
FIGURE 2.7. Sisyphus Cooling. Atoms (left) interacting with only σ+ light follow
the potential energy curve U+. If an atom (center) absorbs a σ− photon while at the
top of the U+ curve, it will drop to the U− curve (right), loosing mechanical energy
in the process.
scattering rate from a beam that an atom is moving towards. In this arrangement,
the light’s linear polarization direction rotates around the beams’ propagation axis.
In the reference frame of the atom seeing a fixed polarization direction, the changing
electric field direction appears as a magnetic field that Zeeman shifts the atom’s
energy levels. This shifting induces pumping between different ground states of the
atom leading to preferential excitation by the light field the atom is moving towards
[71, 72]. This preferential interaction slows the atoms more quickly than Doppler
theory predicts giving a lower atomic speed and temperature. As our MOT is based
on circular beams, this is the sub-Doppler cooling mechanism expected for our MOT.
This cooling, polarization gradient cooling, only occurs with atoms that have
multiple ground states. The V-atom has just one by construction, while the full
87Rb atom has many. The added cooling from polarization gradient cooling is clearly
demonstrated in the much larger slope, due to enhanced restoring constant κ, near
the origin of Figure 5.2 for the full atom (blue) as opposed to the V-atom (red).
Evoking the equipartition theorem to calculate the temperature still holds for the
42
sub-Doppler cooling, although there is not an analytic solution for κ for the full 87Rb
atom in a MOT. In any case, measuring temperature via the equipartition theorem, as
was developed for Equation 2.86, the temperature is inversely proportional to κ. The
evidently larger value for κ for the 87Rb atom then reveals the cooler-than-Doppler
temperature.
These two sub-Doppler cooling mechanisms do not rely on the presence of the
magnetic field for the MOT. They originate entirely from the counter-propagating
electric fields and thus also appear for atoms in an optical molasses. The magnetic
field for the MOT is, again, strictly responsible for the position-dependent trapping
of the atoms near z = 0.
43
CHAPTER III
EXPERIMENTAL SETUP
This chapter will discuss the experimental apparatus for the experiments
discussed in chapters IV and VI. The experiment design implements a dual-MOT
setup in which one MOT is singularly used to increase load rates into the second
MOT through radiation pressure. The second MOT is where our experiments are
performed. For moving from a many-atom MOT to a single-atom MOT, changes
were made primarily to the second MOT.
In this chapter, the vacuum chamber and laser system will be first discussed.
These systems were designed and built by prior students and are discussed in detail
in their theses [73–76]. The systems are discussed in brief, focusing on changes made
to the system in order to allow for single-atom trapping. Larger changes made to the
experimental apparatus are discussed after, looking at high-gradient magnetic field
systems and a single-photon detection apparatus.
3.1 Vacuum
Our vacuum chamber is segmented into two chambers, a “high pressure” chamber
and a “low pressure” chamber, as shown in Figure 3.1. The “high” and “low”
designations refer to their relative pressures as both sides fall into the ultra-high
vacuuum (UHV) regime. The high pressure chamber originally had a pressure of
∼ 10−8 torr and the low pressure chamber had a pressure of ∼ 10−10 torr [73]. The
high pressure chamber contains the first MOT and the low pressure chamber contains
the second MOT. An important feature of the low pressure chamber is an 8” long,
44
30mm square rectangular cell made of fused silica. This glass cell is installed to stick
outward along the long axis of the optical table, giving broad optical access.
The two chambers are connected by a differential pumping tube. The diameter
of this narrow, tapered tube grows at an angle of about 3◦ from the high pressure
chamber to the low pressure chamber [75]. The narrowness of the tube reduces the
probability for atoms to transit its length between chambers. The taper allows the
atomic beam of atoms to expand as it is sent between the first MOT and the second
MOT (see Section 3.2 for a discussion of the two MOT design).
As a way to improve the chances to capture just a single atom, we decided to
reduce the already low background pressure. This was done by closing the valve
leading to the rubidium source for the past many years. The background vapor has
gradually adsorbed onto the inner chamber or filtered out through an ion pump
(Varian VacIon Plus Starcell 75), lowering the background pressure below levels
measurable by the pump (10−8 torr). The loading rate of MOTs with a small number
of atoms can be used to estimate the background pressure of atoms using the loading
rate of single atoms [6, 77]. Our permanent magnet MOT discussed in Section 3.3.1,
with lasers detuned by about −2Γ (Γ is the natural line width of Rubidium 87, about
5.75 MHz [62]), has a loading rate of 0.0058 atoms/sec. This suggests the main
vacuum chamber has a pressure on the order of 10−15 torr and background rubidium
density of 40 atoms/cm3.
After an unplanned rebaking of the vacuum system (see Section 3.1.2) the
background pressure was kept low by severely limiting the initial atomic rubidium
vapor. In the initial build for a large MOT, the system was flooded with rubidium
vapor by heating a 1 g rubidium source contained in the system [73]. After the
rebake, the rubidium source was reopened via valve and vapor was allowed to reenter
45
Thorlabs 1.5"x8" post (x4, $52 ea.)
Thorlabs 1.5"x8" post (x4, $52 ea.) with PB4 base adaptor (x2, $12 ea.)
with PB4 base adaptor (x2, $12 ea.) mount with PF175 clamping fork (x2, $16 ea.)
mount with PF175 clamping fork (x2, $16 ea.) fasten to ion pump with C1498 post clamps (x4, 28 ea.)
fasten to ion pump with C1498 post clamps (x4, 28 ea.) also need M6x10mm socket head cap screws (18-8 SS)
also need M6x10mm socket head cap screws (18-8 SS)
Varian Vacion Plus Starcell 75
Varian Vacion Plus Starcell 75 M6 (x4)
M6 (x4)
Varian Vacion Plus Starcell 20
Varian Vacion Plus Starcell 20
4mm Gap 4mm Gap
Kimball MCF450-SS20400-A ($1030)
Kimball MCF450-SS20400-A ($1030) Kimball MCF450-SC60000-C ($1700)
Kimball MCF450-SC60000-C ($1700)
Differential pumping tube, 2° taper, 0.05 l/s conductance
Differential pumping tube, 2° taper, 0.05 l/s conductance
HPS 100885000 4.5" CF blank (x2, $? ea.)
HPS 100885000 4.5" CF blank (x2, $? ea.)
HPS 100883000 2.75" CF blank ,
HPS 100883000 2.75" CF blank , 1/4-20 blind tapped hole added ($?)
1/4-20 blind tapped hole added ($?)
Newport
New Focus 9952 pedestal riser,
New Focus 9952 pedestal riser, with standard post and holder ($18)
with standard post and holder ($18)
Thorlabs
Side View
Main Mount Mirrors Mounted
Vacuum Chamber Small holes are 8-32 through holes Coil Arms Mounted Coils Mounted
  face Large holes go around bolts
Front View
Top View
Vacuum Chamber Reflections
Coil Mounting Plate (a) Differential Pumping Tube
Pyramid MOT 

mirrors
Front View Side View
MWS 23AWG magnet wire, "heavy" Kapton coating (240°C continuous rating) ,
Ø0.0253" overall, 20.4 Ω/1000 ft., 625 ft/lb.)
75mm focal distance
2" diameter
Top View
ŷ
n=1.45358
x̂ MOT1 Anti-Helmholtz Coils
3mm beam size
(b) Low Pressure Chamber High Pressure Chamber
Top View
Fused Silica Cell
End View
Side View
MOT2 Anti-Helmholtz Coils

(water cooled)
ẑ
Vacuum Chamber Coil Arms
Mirrors Mounted
  face
Main Mount Coil Arms Mounted x̂
Small holes are 8-32 through holes Coils Mounted
Large holes go around bolts
FIGURE 3.1. Experimental Diagram of Vacuum Chamber. (a) Top-down view. (b)
Side view. The two chambers are joined by a differential pumping tube. The high
pressure chamber contains MOT 1. The low pressure chamber contains MOT 2 within
the fused-silica cell and is where experiments are done. The red dash-dot line shows
the beam path of the MOT 1 laser through the chamber. The solid-red lines show the
beam paths for the second MOT. Only vacuum chamber elements important for the
discussion in chapters IV and VI are shown. Missing from this diagram are pumps
and titanium getters, the rubidium source, and mirrors for a high powered fiber laser
[73].
46
the vacuum. Without heating the source, only a small amount of rubidium enters the
system and allowed us to capture single atoms again.
3.1.1 Light Induced Atomic Desorption
Significantly lowering the background vapor of rubidium helps the chances to
capture just one atom, but it also limits the chances to capture any atoms. To briefly
improve loading rates, we implemented a light-induce atomic desorption (LIAD)
system.
Shining bright, off-resonant light into a vacuum chamber whose walls have been
coated with rubidium can desorb the rubidium from the walls to increase the vapor
pressure in the chamber [11, 78]. This has been observed for not just rubidium
but other atomic vapors [79–81] and even for molecules [82]. For many surfaces,
this desorption has been explained as the light breaking ionic bonding between alkali
atoms and silicon-oxygen chains in the surface [82]. This effect has also been observed
to occur for atoms adsorbed onto stainless steel, although the physical mechanism for
this is less well understood [81, 83].
A number of groups have used LIAD to increase atomic counts in MOTs [11, 81,
84, 85]. LIAD as a method to load traps has the benefit of allow for a low background
vapor pressure but still allow for quick loading of the trap by temporarily increasing
the vapor pressure. A low vapor pressure is important for single-atom MOTs to limit
loading rates from the background and to increase MOT lifetimes by limit background
atom collisions. A MOT loaded with LIAD demonstrates those effects [86, 87].
We built and implemented an LIAD system based on the work of [81] using three
1W blue power LEDs (Newark P/N 51R2234). These were arranged symmetrically
around a used copper vacuum gasket and the gasket was mounted outside of the first
47
MOT access window as shown in Figure 3.2. The LEDs were powered with the circuit
shown in Figure 3.3. The double transistor design was implemented to use a high
current PNP transistor (Mouser P/N 511-BD238) we had on hand. The addition of
the NPN transistor functions as NOT gate to reverse TTL logic appropriately for the
PNP transistor. The parallel resistor design is for safe power dissipation while the
circuit is running.
For a many-atom MOT, our LIAD setup gives an atom number increase in the
second MOT of around 10%. This is significantly below what was seen in [81] as
that experiment loaded the MOT directly from the desorbed gas. Our tests did not
measure the atom number increase in a MOT loaded directly from the desorbed gas
(our first MOT), but from a MOT that was loaded by this (potentially) enlarged
MOT. Our experimental setup doesn’t allow for measurement of the atom number in
the first MOT. A LIAD setup is not used near the second MOT as the bright lights
of the LIAD could damage the single-photon counting avalanche photodiode used to
detect single atoms in the second MOT.
3.1.2 Table-Top Bakeout
An unplanned breaking of the vacuum system (do not push on the fused-silica
cell!) required a rebaking of the system to achieve UHV pressures. The baking process
vaporizes gasses which have deposited onto the vacuum chamber and allow them to
be removed from the vacuum via pump. Because the system should still be relatively
clean even after filling with air, a lower temperature bake was done compared to the
original construction [73]. When the system was originally built, a power failure also
broke vacuum near the end of the bake. An additional final bake, to only 150◦C, done
48
FIGURE 3.2. Installed LIAD system. +T12hVree blue +L1E2VDs are attached to the reverse
of the copper coil seen in left image, which is also the dark, black ring in the right
image. The fiber in the foreground10ckaΩrries trapping and rep5uxm15p0Ωlight for MOT 1.
+12VTTL 
Control +12V 1kΩ
10kΩ 5x150Ω
50Ω
TTL 
Control 1kΩ
Ω
50Ω 05
1
x
5
V
FIGURE 3.3. LED circuit for LIAD setup 21
+
Ω
k
1
V
49 21
+
Ω
Ω k
0 0
5 1
l
o  
tr L
n T
o T
C
+12V
+12V
10kΩ 5x150Ω
TTL 
Control 1kΩ
50Ω
over night was sufficient to return to UHV pressures. For this rebake, a similar low
temperature bake was sufficient to return to UHV pressures.
Because the vacuum chamber was already in place on the optical table and
positions of experimental equipment were set relative to it, we decided to attempt
a bake of the chamber in place instead of moving it away from the table. This
was done following the “many heater” method described by Birnbaum [88]. After
removing sensitive experimental equipment around the vacuum system, heater tape
(P/N Omega SRT101-060 and similar) was wrapped around various vacuum chamber
parts and thermocouples were positioned around the chamber. Around the tip of most
of the thermocouples, a short sleeve of fiberglass (harvested from around broken heater
tapes) was placed to electrically insulate the exposed ends of the thermocouples from
being grounded to the vacuum chamber. The thermocouples were positioned so that
temperature gradients were easy to detect. For example, a series of thermocouples
were located at the window of the first MOT, one one each side of the differential
pumping tube, and one on each side of the (newly replaced) fused-silica cell. This
allows us to monitor the temperature gradient along the full length of the chamber,
making sure a relatively even temperature was kept along that axis.
After positioning the heater tapes and themocouples, the chamber was covered in
many layers of aluminum foil. Windows were covered by wrapping oil-free aluminum
foil (All Foils UHV aluminum foil) around the window flange and its bolts, creating
a heightened ring of foil around the rim of the window, as shown in Figure 3.4a.
More foil was then placed over the ring and ends were stretched to wrap around the
chamber holding the foil over the top of the window in place and leaving a gap of
air between the foil and the window. This was to prevent foil scratching the glass
and provide an insulating pocket of air to keep high temperatures. Around the fused-
50
silica cell, foil wrapped fire bricks were stacked to a few inches below the cell. A
single heater tape was laid over the top of the firebricks under the cell. Four very
long pieces of UHV aluminum foil were folded into long, stiff strips which were bent
into arches that stretched over the cell and tucked under the bricks on each side of
the cell. Similar long, folded pieces of UHV foil were wrapped between the arches,
building the framework of an oven around the cell (affectionately referred to as The
Barn). This is shown in Figure 3.4b. A layer of UHV foil was wrapped around the
sides of The Barn and folded onto the top and a layer of UHV foil was wrapped
over the top. The UHV foil covering windows and the rest of the vacuum chamber,
including ion pumps and pump hoses, were wrapped in many layers of kitchen quality
aluminum foil. The fully wrapped vacuum chamber is shown in Figure 3.4c.
The vacuum chamber was heated slowly with the heater tapes to temperatures
around 120◦C over a few days. The 2nd vacuum chamber temperature increase was
particularly slow due to its size and limited surface area where heating tape could be
wrapped, which slowed the process considerably as the entire system was attempted
to be kept close to the same temperature. The ion pumps were heated much higher
than the other portions of the chamber (to 180◦C and 170◦C for the larger and
smaller ion pump respectively). This was to clean off material that had built up
inside the ion pumps. The small ion pump had not worked for years as collected
material had shorted the high voltage difference across the pump. Particularly
warm or cool spots of the chamber were adjusted by removing and adding aluminum
foil, respectively. To help visualize the temperatures, the thermocouple values were
plotted spatially in three dimensions with their locations corresponding to the (ideal)
position of the thermocouples. A few additional temperatures were estimated based
on the temperature measurements of nearby thermocouples. Lines between these
51
(a) (b)
(c) (d)
FIGURE 3.4. (Re)baking the vacuum chamber. (a) Protected window coverings. (b)
Oven constructed around the fused-silica cell. (c) Entire vacuum system wrapped
and baking. (d) Positional temperature monitoring graph.
52
points were colored as relative temperature of the extreme points from an estimate
of the temperature at the center of the large MOT chamber. One such of these
graphs is shown in Figure 3.4d. The graph was particularly helpful in visualizing the
temperatures and potential strong gradients in the system. This graph, along with
graphs of the individual thermocouple temperatures over time, were used to adjust
the various heater tape voltages to bring the chamber up to temperature.
Prior to baking but after replacing the fused silica cell and pumping out the
system, the lowest pressure measured by the turbo pump (BOC Edwards P/N EXT
70H 24V) was 3.2× 10−8 torr. After baking, the pressure had reduced to 6.4× 10−9
torr. This is comparable to a final pressure of 8.3 × 10−9 torr achieved during the
initial baking of the system. This difference is primarily due to a small leak in the
home-made gasket for mounting the fused-silica cell that was present during the initial
bake. Tightening the flange for the cell closed the leak after ending the bake sealed
it, but limited the pressure during the bake. No such leak was present before or after
the rebake. Pressure readings with the ion pumps measured post-baking pressures of
< 4×10−10 torr and 3×10−9 torr on the low- and high-pressure sides of the chamber,
respectively.
3.2 Lasers
The laser system for our MOT has been largely unchanged from previous
experiments [76]. All of the lasers for the experiment are homemade external-cavity
diode lasers with outputs of around 100 mW near 780 nm, the transition wavelength
for 87Rb.
The main MOT trapping laser consists of a master laser injection locked to two
slave lasers, discussed in detail in [76]. The beam path for the MOT trapping laser
53
Trapping Glass
 PBSPlate
Master 
Laser
/2
/2 Glass

Plate
isolator
AOM
MOT 1 
Slave /4
Laser
/2
AOM
Differential

Photodiodes
/2 /4 To Fabry-Perot Cavity
AOM
MOT2 Trapping

To MOT1
  Laser

Fiber
 MOT 2 
To 
  Fiber CouplerCoupler Slave 
Fabry-Perot

 Cavity Laser
FIGURE 3.5. Beam Path of MOT Trapping Lasers
Glass

Plate
Repump 
Laser MOT2 Repump Laser

 Fiber Coupler
/2
isolator
PBS
Glass

Plate /2 MOT1

PB  Fiber 
S Coupler
From MOT1

Slave Laser
Differential

Photodiodes
FIGURE 3.6. Beam Path of MOT Repumping Lasers
54
isolator
isolator
Rb Vapor Cell
Rb Vapor Cell
AOM
PBS
AOM
PBS
is shown in Figure 3.5. The two slave lasers reproduce the frequency of the input
light and effectively increase the total MOT laser power. A portion of the light from
a slave laser is sent to a Fabry-Perot cavity. Monitoring the output of the cavity
verifies successful injection of the maser laser as only a single peak will appear in the
cavity signal. The remainder of the slave lasers’ outputs are coupled to optical fibers
for use in the MOTs. One slave laser is used to operate each of the two MOTs.
The repumping beam, needing much less power than the trapping laser, does not
seed slave lasers. This beam is split and directly coupled into two fibers, one for each
MOT. The beam path for the repump trapping laser is shown in Figure 3.6.
3.2.1 MOT 1 Laser
For the first MOT, light from the repump laser and light from one of the slave
lasers are combined via a beamsplitter before being coupled into a 5 µm core diameter
optical fiber (Oz Optics P/Ns PMJ3A3A-850-5/125-3-x-1, where x corresponds to
various fiber lengths). The fiber emits light freely into space (there is no collimation
lens) before reaching a 3-inch lens which collimates the now greatly expanded beam.
The fiber and lens are visible in pictures of the LIAD, Figure 3.2. The now collimated
light travels into the vacuum chamber through a large window on the right side of
the high pressure chamber as marked with the green dash-dot line shown in the
MOT chamber diagram (Figure 3.1). Inside the chamber, the beam reaches a set of
“pyramid mirrors” to form the MOT.
The pyramid mirrors are shown in Figure 3.7. These are a set of four Pyrex
pieces coated to reflect light at 780nm and are each aligned at 45◦ degree inclines. As
the collimated light reaches the pyramid mirrors, it is reflected perpendicular to the
original beam. As shown in Figure 3.7a, the reflected beams counter-propagate from
55
(a) (b)
FIGURE 3.7. Pyramid MOT Mirrors. (a) Diagram of arrangement of pyramid MOT
mirrors with beam behavior shown. (b) Picture of built pyramid MOT mirrors.
opposing mirrors. The four sets of beams reflected from the pyramid mirrors, together
with large anti-Helmholtz coils mounted on the exterior of the vacuum chamber form
a 2-dimensional MOT at the center of the pyramid. They are trapped on the plane
that is perpendicular to the MOT 1 beam axis.
As shown in Figure 3.7, the mirror pieces are cut to leave a hole at their center
to allow atoms to escape from the 2-dimensional MOT [89, 90]. The hole aligns
with the small-diameter opening of the differential pumping tube. The portion of the
MOT light that travels directly along the center of the pyramid will travel through
hole and along the differential pumping tube. This is shown as the green dash-dot
line in Figure 3.1. Before entering the tube, this light intersect the 2-dimensional
MOT. The light applies radiation pressure (see Section 2.2.3) to the cooled atoms
and pushes them along its path, through the differential pumping tube and into
the high pressure chamber.The fused-silica cell is attached opposite the differential
pumping tube so that the light, and atoms, traveling through the tube will continue
on along the length of the cell.
The second MOT’s center is located about 5-3/8” from the end of the fused-
silica cell and is positioned a quarter inch below the beam path for the first MOT
56
laser through the differential pumping tube. The atoms pushed by this light travel
approximately the full length of the cell while still inside the beam [74]. However,
some atoms sink out of this path and can fall into the range of the second MOT.
This drooping is caused by gravity and atomic motion perpendicular to the beam
axis (this motion is the same that necessitates the taper in the differential pumping
tube). Thus, we can load atoms into the second MOT from the first by capturing
atoms dropping out of the beam. We have seen that using the first MOT to load the
second MOT increases the loading rate of the permanent magnet MOT (see Section
3.3.1) by a factor of 4.
No changes were made to the first MOT laser in preparation for single-atom
experiments.
3.2.2 MOT 2 Laser
Unlike the MOT 1 laser, the trapping and repump beams for the second MOT
are coupled into separate fibers. These two fibers are the inputs of a fiber beam
splitter array (custom design from Canadian Instrumentation and Research) which
combines the two inputs and splits the light to 6 fibers, the 6 fibers for our MOT.
The fiber array was designed so that each output beam received equal power, but
is not perfect. The lowest power beam has around two-thirds of the power of the
highest power beam. The output beams, as used in the experiment, are matched so
counterpropagating beams have about equal power to limit effects of beam imbalances
(see Section 2.4.1). An imbalance ratio between counterpropagating beams is defined
as
P+ − P−
w = , (3.1)
P+ + P−
57
where P+ and P+ are the powers in the higher powered beam and the lower powered
beam respectively [91, 92]. This definition is convenient as P+ = 〈P 〉(1 + w) and
P− = 〈P 〉(1 − w) where 〈P 〉 is the average power of the counterpropagating beams.
With this definition, the MOT beam pairs have imbalance ratios of w = 0.0367,
w = 0.0034 and w = 0.115. The largest imbalance ratio was chosen for the two
beams traveling vertically up and down, along the axis of the MOT anti-Helmholtz
coils. This large beam imbalance has implications for the discussion in Chapter VI
and is resolved as in 3.2.4.
Originally, the six fibers are connected to a stack of components that contain a
fiber collimator (Thorlabs P/N F810FC-780) and a quarter-wave plate (Casix P/N
WPL1225-lambda/4-780nm-M) [73]. The collimators produce a 6.6 mm diameter
beam (the 1/e2 power diameter) and the wave plates convert the linearly polarized
input light to circular light. The waveplate is mounted on the face of a 0.594” diameter
aluminum tube. The tube (and thus waveplates) are rotated around the collimator
to give correct light polarization to create the MOT.
Four of the fibers launch light parallel to the surface of our optical table,
intersecting the fused silica cell at 45◦. Because the beams enter non-normal to
the cell, the beams experience a shift in propagation direction due to refraction. The
fiber launchers are positioned on the table to account for this, resulting in opposing
beams that counterpropagate as desired. These beam paths are the red lines shown
in Figure 3.1a.The final two beams are the vertical beams discussed above. These
beams intersect normal to the cell and do not experience refraction. These are the
red lines seen in Figure 3.1b.
Two changes were required for the second MOT lasers when moving to the single
atom experiments. Changing the anti-Helmholz coils to permanent magnets required
58
repositioning the fiber launcher that was below the experimental cell. This launcher
was moved to propagate light parallel to the table and reflect to vertical off an extra
pyramid mirror. The reflection reversed the polarization of the light, so the wave
plate in the fiber launcher had to be adjusted to account for this.
The second change added 25 cm irises (Thorlabs P/N IN25) between the fiber
launcher and experimental cell. The irises are closed to typical diameters of 3 mm.
The irises allowed us to reduce the diameter of the MOT beams, which has three
benefits. One, the clipped beams reduce the total amount of light reaching the
experimental cell. This reduced background scattered light during an experiment
- an important feature as single-atom detection is done by photon counting (Section
3.4). Second, the smaller beams reduce the loading rate of the MOT. Although the
size of trapping region of the MOT is dominated by the magnetic-field gradient of
the anti-Helmholtz coils, the counterpropagating beams outside this region create an
optical molasses which slows atoms without trapping them [1–3]. These slowed atoms
can wander into the trapping region, which increases the loading rate of the MOT.
Lastly, clipping the beams with irises, as opposed to replacing the collimator with one
that produces a small diameter beam, allows easy adjustment of the beam size without
altering the beam intensity at the center of the MOT. Atoms in the MOT should be
trapped very close to the intersection of the centers of the six beams. Clipping the
tails of beams will not affect the intensity at center, keeping the parameters affecting
atomic fluorescence and motion the same, while allowing us to change the loading
rates and reduce the photon background.
59
3.2.3 Trapping Beam Detuning Control
The beam paths for the trapping and repump lasers was shown in figures 3.5
and 3.6, but a detailed description of them was left for prior students’ dissertations,
particularly that of Tao Li [75]. However, due to the importance of laser detuning
to MOT performance, a detailed view of the frequency control elements for the two
MOT beams is below.
The frequency of the MOT trapping beam is controlled by the the beam diagram
shown in Figure 3.8. This diagram shows only pieces of the overall beam path (Figure
3.5) that are necessary for frequency control. Most notably absent is a split in the
beam path at point 3 to seed both slave lasers. The slave lasers only affect the
beam power before going to a final AOM and then the fiber couplers. The output
frequency of the laser is locked via a home designed lock-in amplifier [75] whose
input comes from a doppler-free spectral measurement of the hyperfine structure of
rubidium via saturation absorption [93]. In this method, two lasers counter propagate
through an atomic sample. Close to resonance, atoms moving in opposite directions
absorb photons from opposite beams. This is very similar to the enhanced absorption
from beams that lead to the optical molasses in Section 2.4.1, although here the
beam polarization is linear. Measuring the output from each beam would reveal an
absorption signal that is doppler broadened. because of atomic motion. However, if
one beam has a much higher power, atoms whose speed is nearly zero will interact
much more strongly with that beam. These atoms will be saturated (see Equation
2.63 and its discussion) by the stronger beam and will absorb little light from the
weaker beam. A photodiode signal from the weaker beam would not show absorb at
frequencies very close to the atomic resonances of the atom. Measuring this signal,
60
PBS
/4
From 

Laser 2
3 +2f
To Saturated   

      Absorption 1 +fI AOM
4
AOM 2f To Fiber        
SA Coupler
/4
FIGURE 3.8. MOT Frequency Controlling Beam Path. Numerically labeled points
along the beam path are reference points for the discussion in the text.
and subtracting away the doppler signal from an identical unsaturated beam, can
reveal a non-doppler broadened spectrum for the atomic sample.
One such spectrum is shown in Figure 3.9 and is measured (effectively) at the
beam path point 1 as shown in Figure 3.8. The spectrum shows 6 peaks, only three
of which correspond directly to hyperfine transitions of the D line of 872 Rb. These
peaks are the peaks labeled A, D, and F in the spectrum and correspond to optical
transitions. Peak A corresponds to the transitions |Fg = 2〉 → |Fe = 3〉, which is
the MOT trapping transition. Peak D corresponds to the |Fg = 2〉 → |Fe = 2〉
transition. And peak F corresponds to the |Fg = 2〉 → |Fe = 1〉 transition. Using the
|Fg = 2〉 → |Fe = 3〉 transition as a reference frequency, the peaks are located at the
frequencies given in the spectrum (frequencies from Steck [62]). The other three peaks
are “crossover” peaks originating from moving atoms whose speeds causes resonant
light frequencies for one atomic transition to become doppler shifted into resonance
with a different transition. In this case, the frequency of these peaks are the average
of the two different resonant transitions, as shown in the spectrum [48].
61
AOM
PBS
0MHz -133.325 MHz -211.798 M-H26z6.650 MHz -345.124 MHz -423.597 MHz
0 MHz -133.325 MHz -211.798 M-H26z6.650 MHz -345.124 MHz -423.597 MHz
0 MHz 0 MHz -211.798 MHz --221111..779988  MM-HH26zz6.-625101 .M79H8z M--2H266z66..665500  MMHHzz --422636..569570  MMHHzz -423.597 MHz
0 MHz -211.798 MHz -211.798 M-H26z6.650 MHz -266.650 MHz -423.597 MHz
0 MHz -133.325 MHz -211.798 M-H26z6.650 MHz -345.124 MHz -423.597 MHz
0 MHz -133.325 MHz -211.798 M-H26z6.650 MHz -345.124 MHz -423.597 MHz
A B C D E F
A B C D E F
FIGURE 3.9. 87Rb D-2 line spectrum. This spectrum shows decays to the |Fg = 2〉
state. Frequencies listed are peak locations relative to the decay energy from the
|Fe = 3〉 state, peak A.
A A B CB D C ED FE F
A B C D E F
A B C D E F
The laser is locked to peak B in Figure 3.9, giving the light at the point 1 in
A B C D E F
Figure 3.8 a frequency of ∆ = −133.326 MHz, where the frequency is referenced to
the |Fg = 2〉 → |Fe = 3〉 transition. Prior to reaching this point, the beam double
passes through an acousto-optic modulator (AOM), the AOM at the bottom left of
Figure 3.9. Because the lasers are locked via lock-in detection, their measurement
requires an added modulation. Adding the modulation directly the output laser is
not feasible as a MOT needs very stable laser frequencies. Instead, this AOM has
its input frequency modulated between 143 MHz and 147 MHz at a rate of 200 kHz.
Taking the average of the two input frequencies as a single frequency of the AOM,
fSA, the laser has a frequency of ∆ = −133.326 MHz − 2fSA before entering this
AOM. The AOMs can be used to give either a positive or negative shift in frequency
depending on angle of entry of the laser beam. For the remainder of this discussion,
these positive or negative shifts are taken by adding or subtracting the frequencies
62
input into an AOM. This frequency is the same frequency at position 2 in Figure 3.8
and must be the frequency of the light as it leaves the laser.
The portion of the light that is not used for frequency locking double passes
through an AOM with a single frequency. This AOM, unlike the other two in this
setup, has a tunable frequency using our home build experimental control system
ZOINKS [94]. If this AOM has a frequency f, after double passing through it, the
light has frequency ∆ = −133.326 MHz − 2fSA + 2f . This is the light frequency at
position 3. This is also the position where the light is split into two paths, and each
path is injection locked to a slave laser. This is not shown in the figure as it only
increases beam power. The two light paths, after being split, both follow the rest of
the path to point 4 shown in Figure 3.8, although they go to different AOMs with
the same set frequency.
After having the power increased by the slave laser and before entering the 6-way
fiber splitter array, the light (for both MOTs) passes through a final AOM whose use
is to control the overall beam intensity and shutter the beam when it is not wanted
in the experiment. This AOM has a frequency fI , giving a final laser frequency of
∆ = −133.326 MHz− 2fSA + 2f + fI , (3.2)
where again ∆ is the detuning of the light from the |Fg = 2〉 → |Fe = 3〉 transition,
as this was the reference frequency in Figure 3.9. This is the frequency at position 4
of Figure 3.8 and is the frequency of the light that creates our the MOT.
Taking typical values of fSA = 72.5 MHz and fI = 79.782 MHz, the detuning
follows
∆ = −198.544 MHz + 2f
63
The AOMs used have a frequency range of 80 MHz to 100 MHz giving usable a MOT
laser detuning in the range of about -36 MHz to +2MHz. The MOT is typically ran
with detuning between −2Γ and −Γ.
The repump beam for the magneto-optical trap has a much simpler frequency
control setup. There is only one AOM involved in shifting the frequency, as seen in
Figure 3.6. The repump beam is locked to the same cross-over peak as the trapping
beam. This peak is detuned from resonance of the |Fg = 1〉 → |Fe = 2〉 transition
by −78.474 MHz. There are no frequency changes made between the repump laser
output and the repump saturation absorption setup, so that the laser output is
detuned this same amount. Before traveling to fiber couplers, the beam travels
through a single AOM that applies a shift of +fR to the beam. This provides a
repump beam detuning of
∆R = −78.474 MHz + fR (3.3)
where ∆R is the detuning from the |Fg = 1〉 → |Fe = 2〉 repumping transition.
3.2.4 Recircularizing and Balancing the MOT2 beams
The fiber array for the MOT2 lasers are (linear) polarization maintaining fibers.
As our design for the MOT requires circular polarization, previous experiments
mounted quarter-wave plates to the output fiber launcher for each of the six beams
for the 2nd MOT [73]. As noted Section 3.2.2, the power output of these beams
was not particularly well balanced. This imbalance shifts the loading location of the
MOT, which was not desirable for our position measurements of a single atom. To
64
(a) (b)
λ/4
P1 P2
FIGURE 3.10. Fiber Polarization Mount. (a) Schematic drawing of entire mount.
One polarizer (P1) is mounted to be aligned with the polarization axis of the fiber.
The quarter-wave plate is glued in placed aligned with the fast axis of the 2nd polarizer
(P2). (b) Picture of built mount. The colors of each piece roughly correspond to the
colors in image (a). Black tape covers set screws which hold the polarizations in
place. They are to prevent accidental unmounting of the polarizers while adjusting
the mounts.
resolve this, the quarter-wave plate mount was modified to allow for control of the
power output of each beam as shown in Figure 3.10.
A new mount was made that holds a 1/2” sheet polarizer (Laser Components
P/N 11006083) mounted in a modified 1/2” to 1” optics adaptor (Thorlabs P/N
AD1T). This polarizer is labeled P1 in the Figure 3.10a. On top of this, the original
quarter-wave plate mount was installed after being modified to hold a second 1/2”
polarizer (labeled P2) inside of it (the quarter-wave plate is glued on the output
face of the mount, just as noted in Section 3.2.2). The 2nd polarizer is aligned so
that its fast axis aligns with the axis of the quarter-wave plate, allowing for any
light that travels through the polarizer to be circularized by the quarter-wave plate.
The first polarizer is aligned to the polarization axis of the fiber and acts as a filter
to limit polarization angle noise in the fiber. Rotating the 2nd mount, with the
2nd polarizer and quarter-wave plate, adjusts the power output through the mount
without changing the circularization of the beam. The entire two-polarizer-quarter-
wave-plate mount system is shown in Figure 3.10.
65
The power output through a polarizer goes as the cosine2 of the angle between
the input light polarization and the polarizer axis (see Appendix B). While this is
parameter we hope to use to control individual MOT beam power, any fluctuations in
the polarization angle of the input light will cause fluctuations in the beam power. We
measured polarization angle fluctuations of around δ ◦n = 2 from the light through
the optical fibers for the 2nd MOT. For beams where we want very little (if any)
attenuation (the angle being close to 0), this noise is not a problem, but for fibers
were large attenuation was needed to balance the beam power, this can cause large
power fluctuations. As shown in Appendix B, aligning the first polarizer to the ideal
axis of the fiber, limits power fluctuations output through the three-optic system to
order δ2n, compared to order δn when only a single polarizer is used.
3.3 Magnetic Fields
Extending the lab’s previous experiments [95, 96] to the single atom regime
required increasing the field gradient to trap single atoms. We accomplished this
in two ways, first with a matched pair of permanent magnetic rings to create a
quadrupole magnetic field. This did successfully allow us to trap single atoms, but
the permanent magnetic field was not ideal for experimenting, so we designed and
built a high-current, water cooled pair of electro-magnets.
3.3.1 Permanent Magnet Anti-Helmholtz Coils
The first attempt to greatly increase the anti-Helmholtz magnetic field gradient
used a pair of matched permanent ring magnets with their magnetic poles on the
66
Property Value
Inner Radius, R1 0.5in
Outer Radius, R2 1in
Magnet Thickness, L 0.25in
Magnet Separation, s 2.25in
Maximum Magnetic Field, Bmax 1468G
TABLE 3.1. Permanent Ring Magnet Parameters
flat-faces of the magnets. (K&J Magnetics P/N RY0X04). The measurements of the
magnets are in Table 3.1
With the origin at the center of the (hole in the) magnet and the z-axis running
on the central axis of the magnet (coordinate system shown in Figure 3.11), the field
along the z-axis is given by
 −1
B (z) = B  L Lz z,max √( ) − √2 ( )
 ×
L 2 L 2+R +R2
 2 2 2 1( ) ( )
L L
√ z − z −2 2( ) − √ +2 ( )2
z − L +R21 z − L +R22 2 2
( ) ( ) 
z + L L−√ 2( ) + √
z +
2
2 ( )
 , (3.4)
2
z + L +R2 L1 z + +R
2
2 2 2
where Bz,max is the magnetic field z-component at the origin, R1 and R2 are the inner
and outer radii of the ring magnet, and L is the thickness of the magnet.
This can be derived one of two ways. First, by integrating the magnetic field of
a magnetic dipole oriented in the ẑ direction [97] over the volume of the permanent
magnet (Figure 3.11a). This integral, in cylindrical coordinates, is:
∫ ∫ ( )
µ R2 L/2 ′ 2 20m 2(z − z ) − ρ
Bz(z) = ρdρ dz
′ .
2 R1 −L/2 [(z − z′)2 + ρ2]5/2
67
I
m̂
I
(a) (b)
FIGURE 3.11. Permanent Magnet Magnetic Field Calculation Methods. (a)
Calculation method that integrates a magnetic dipole field over the volume of the
magnet. (b) Calculation method that treats the permanent magnet as opposite-
direction sheets of current on the inner and outer walls of the magnet.
The second method is to assume the magnetic is made up of a sheet of current with
height L and radius R1, and a second sheet of current with height L and radius R2
with the same current magnitude but in the opposite direction of the inner-sheet
(Figure 3.11b). Both of these derivation methods require replacing the unknown
magnetization, m̂, or unknown current, I, the appropriate equation involving the
magnetic field at the origin, Bz,max.
The paired permanent magnets, separated by distance s, have a total field
Btot(z) = Bz (z + s/2)−Bz (z − s/2) (3.5)
The ±s/2 terms shift the magnets by half the separation, redefining the origin to be
at the mid-point between the magnets. This quadrupole field using the permanent
magnet measurements in Table 3.1 is plotted in Figure 3.12 along with measured
values for the magnetic field for our magnets. Near the center of the coils, the total
68
BHGL
200
100
zHcmL
-1.5 -1.0 -0.5 0.5 1.0 1.5
-100
-200
FIGURE 3.12. Permanent Magnet Quadrupole Field
magnetic field is nearly linear2 with a (theoretical) gradient of B′z = 163 G/cm and a
measured gradient of B′z = 165 G/cm.
The permanent magnet MOT was mounted around the experimental cell cased
in laser-cut acrylic. A first designed shown in 3.13a used clear acrylic (McMaster-Car
P/N 8560K354) and did allow for our first trap of single atoms, but the design suffered
from noise issues. Reflections of the lasers off the experimental cell scattered off the
acrylic and to the avalanche-photodiode used to detect since atoms. The the design
also could flex, which changed the scatter and shifted background fluorescence levels.
A second design used black acrylic (McMaster-Carr P/N 8505K92). Stiffer supports
solved the flexing issue and built in Wood’s horns [98, 99] solved scatter issues. The
horns were made by bending and clamping closed metal tubing (McMaster-Carr P/N
2Btot(z) of Equation 3.5 is an odd function and the magnitude of the z
3 component of the field
within 100µm of the center is an order of magnitude smaller than the linear component using the
magnet parameters in Table 3.1. These justify a linear assumption.
69
(a) (b)
FIGURE 3.13. Single Atom MOT Magnetic Field Designs. (a) First design with
permanent magnets. (b) Water-cooled electromagnets installed in the experiment.
5177K69 and P/N 8955K141). The inside of the tubes were painted black (Krylon
Black indoor/outdoor primer).
3.3.2 Electromagetic Anti-Helmholtz Coils
Electromagnetic anti-Helmholtz coils are preferred for our experiment as they can
be easily turned off to do away with external magnetic fields. They were designed to
closely match the fields of the permanent magnets. These installed MOT coils are
shown in 1.1b and in 3.13b.
The MOT coils were designed looking at two factors – the magnetic field gradient
at the center of the coils and the power dissipated in the coils. Both of these are
predominately set by the current in the coils and by their shape. For two single loops
of wire with radius r and resistivity ρ that are separated by a distance s, the magnetic
field gradient at the midpoint between them and the power dissipation by them are
70
given by the equations
∣
∂B ∣∣ −3µ0I r
2s
∂z ∣ = (3.6)z=0 2 [(s/2)2 + r2]5/2
P = 4πrρI2. (3.7)
The magnetic field equation can be found from Ampere’s Law and the power equation
comes from the power dissipated by two resistors in series (having the coils in series
is preferable for our experiment so that the current in each is the same). Clearly, the
magnetic field gradient scales linearly with current in the coils, while the power scales
as a square of the current. Reducing the size of the coils, through decreasing the
radius of the coils, increases the magnetic field gradient and it will reduce the power
dissipated, both things we wanted to occur. Numerically calculating field gradients
and power use with multiple loops of wire with different radii and separation between
coils was used to decide on general design for the coils.
Additionally, the thickness of the wires had be taking into account two ways.
First, the thickness controlled the radius of the next layer of wires outward from the
center of a single coil. Second, the loops were assumed to be closely packed together,
with each outward layer of coils nestled in the “gap” between two coils of this previous
layer. This negatively affects the design by increase the separation of the coils and by
reducing the number of vertical stacks of wire in each even numbered layer of wires. It
improves the design by reducing the radius of the wires for every subsequent outward
layer, reducing the overall size of the coil of wires, and it makes wrapping the coil of
wires easier as the wires will naturally fall into this gap when wrapped tightly.
71
The final design examined various gauges of wire, the number of outward layers
of wire and the number of vertically stacks of wire. The calculations lead to a design
with 270 individual loops of 20 gauge wire, arranged with 20 outward layers arranged
with 14/13 vertical stacks of wire in odd/even outward layers. The inner-most radius
of the loops of wire was 0.5” and the separation of the two facing sides of the coils was
4cm. Relative to input current, this design should produce a magnetic field gradient
of B′z/I = 27.4 G/A·cm and each coil should have a resistance of 2.2 Ω. When
measured after being built, the designed coils created a gradient of B′z/I = 26.88
G/A·cm and their combined resistance is 4.75 Ω.
The coils are powered with a Kepco ATE75-15 power supply which can output
up to 15 A. The power supply is programmed using a PC-12 adaptor allowing for
external current control and in fast mode, which allows fast altering of the current
and voltage output [100]. Running the experiment in fast mode is desired for trapping
only single atoms (see Section 4.1) and for making our position measurements (see
Section 6.2). Setting a voltage between 0 V and 1 V across pins 15 and 30 of the
PC-12 will set the output of the power supply between 0 A and 15 A.
3.3.3 Water Cooling
Because of the high power dissipation of the electromagnets, their mounting
brackets were designed to allow water cooling to occur on the surfaces of the brackets
opposite the experimental chamber, as shown with blue rectangles in Figure 3.1b.
Each coil has a 1/8” tall and half-inch wide channel inside the coil. Each channel
runs 300◦ around the coil with a water intake and output on either end. Water
lines are connected to 3/8” Swagelok brass tube fittings (Portland Valve Fitting P/N
B-600-6). The water lines are made of rigid polyethylene tubing (McMaster-Carr
72
P/N 50375K47) as visible in Figure 3.13b. The water lines are attached to a water
chiller (Neslab RTE-100) which can pump about 3 gallons/minute. The water chiller
has a 1.3 gallon tank and can cool up to 350 W when operated at 20◦C. The water
temperature is typically set around 15◦C. This temperature is chosen to improve
cooling of the coils, according to Equation C.10, and so that touching the below-room
temperature coils will be cool to the touch to verify they are being chilled correctly.
The water chiller does not seem to have standard pipe fittings, so we adapted 1/4”
compression tube fittings (brass sleeve fitting McMaster-Carr P/N 50385K72 and
brass nut McMaster-Carr P/N 50385K62) for a short stretch of 1/4” tubing. This
was expanded (Portland Valve Fitting P/N B-600-1-4) to match the larger 3/8” tubing
of the rest of the water line.
There are three additional elements in the water line between the chiller and
coils. First, just after leaving the chiller, there is a vibration damper that prevents
vibrations created by the chiller from reaching the experiment. The damper is made
of a long length of tubing that is coiled in 3 circles of about 1’ diameter. The coils
are inside of a plastic trashcan and the can was filled with cement. Second, there is
a flowmeter (McMaster-Carr P/N 4351K37) attached to the water lines that lets us
control the water flow between 0.3 gallon/minute and 3 gallons/minute. The water
line then travels to the MOT coils, in series, before turning through the vibration
damper. Lastly, the water runs through a low-flow switch (McMaster P/N 2371K4)
that allows us to monitor the flow of water through the tubing. The water then
returns to the chiller. The complete plumbing diagram for water cooling the MOT
coils is shown in Figure 3.14.
To verify the coils will not over heat in our experiment, we analyzed the heating
and cooling rates of the coils. This analysis is shown in C and reveals the temperature
73
Flowmeter
Flow Switch
Vibration Damper
MOT coils
RTE100 
Chiller
FIGURE 3.14. Plumbing Diagram for water cooling the MOT coils
of the coils as a function of time is
( )
β β
T (t) = Tw + e
γt − , (C.10)
γ γ
where Tw is the set-temperature of the water and parameters γ and β are complicated
combinations of parameters given in Equations C.8 and C.9. To keep γ < 0, there
is a maximum driving current (Equation C.11) that the coils can cool. For our coils,
temperature settings, and chiller, (see Table C.1) this current is Ilim = 61A, well
above our designed operating current. Because resistance is tied so closely to the coil
temperature, the resistance will also raise as the temperature does according to the
equation
[ ( )]
R(t) = R′0 1 +Rm 1− e−|γ|t , (C.12)
where R′0 is an effective initial resistance, Rm is a maximum resistance. This form is
nice as it allows simple measurement of γ through monitoring the resistance of the
coils. At current of 9A we measure γ = −0.017 ± 0.11/s. Equation C.8 predicts a
value of γ = −0.23/s using experimental parameters in Table C.1.
74
+5V Thermal
Cutoff 1
Control
Voltage 
1kΩ Input
Thermal
Cutoff 2
1kΩ
Flow 
  Switch 1kΩ Control
Voltage 
Output
FIGURE 3.15. Water Cooled Coils Protection Circuit
The likely cause of this mismatch is over simplification in the calculation for the
rate flows into the water from the coils. This derivation assumed the length of the
cooling channel, L, is much larger than a characteristic channel diameter, DH (see
Appendix C.3). Our design has L = 19DH , which is only sort of longer. As discussed
in Appendix C.3, this means there is an inward temperature gradient from the surface
of the cooling channel walls, reducing the efficiency of convection into the water from
the walls of the channel. Even with this overestimate of the cooling, the coils remain
cool to the touch even after operating at high currents for an extended period.
3.3.4 Protection
Because of the high power output of the MOT coils and the potential for over
heating, we installed a temperature protection circuit as shown in Figure 3.15. This
circuits uses TTL logic to trigger a relay (Mouser P/N 528-107-1) that allows the
control voltage from the ZOINKS system to reach the coil power supply. The TTL
signals run through three BNC outputs. Two these BNC connections are attached
75
to thermal cutoffs (Mouser P/N 667-EYP-2BN082) which can be attached to the top
and bottom coils. The cutoffs act as fuses, blowing out if they, and therefore the coils,
reach 82◦C. The third is attached to the low-flow switch which is attached in-line for
the MOT coils water chiller and gives an open circuit if the flow rate falls below 0.1
gallons/minute. These three guarantee the power supply will only run if the coils are
being cooled and if the coils are not overheating.
Care is taken as to not leave the coils running without being monitored. The
power supply and the water chiller are both turned off over night. We have had
mistakes, however. At least once, the temperature control of the chiller was turned
down well below room temperature and left on overnight. Water condensed on the
cooled coils, which spread onto the experimental cell, dripping from above and wicking
from below. The coils have also been left running at high currents for an extended
period, allowing them to overheat. This caused a residue to form on the experimental
cell. In both cases, removing the coils and carefully cleaning the experimental cell
with acetone solved the problems.
3.3.5 Measuring Magnetic Fields
The Kepco power supply has a very obvious noise source of around 27 kHz.
Eliminating this noise is done with an interchangeable capacitor to create a simple
low-pass filter. This is the capacitor shown in Figure 3.16. Typically, a high voltage 1
µ F capacitor is used (Mouser P/N 5984-100V1-F). Forcing oscillations of the current
necessitates careful monitoring of the current as the oscillation amplitudes will be
heavily limited by the filter.
The current sent to the MOT trapping coils is monitored in 3 ways. Before
going to the coils, the output current of the power supply is sent to the circuit shown
76
From
Power Ammeter
Supply Monitor
Sense Resistor
+ Monitor
CLN50
RS INA128 +
- TL071
-
Hall Sensor
To 
Monitor 50!
MOT
coils
FIGURE 3.16. MOT Coils Current Monitoring Circuit. Red and blue coloring for
the current labels the current path before reaching the MOT coils and after returning
from the MOT coils, respectively.
in Figure 3.16. Before reaching the coils, the current passes through a dual-throw
switch which allows for (or bypasses) a direct ammeter measurement. The current
then travels through a 0.01 Ω sense resistor (Mouser P/N 684-SR20-0.01), whose
voltage output can be monitored. After traveling through the coils, the return current
travels through a Hall sensor [101] (FW Bell CLN-50, Newark P/N 83F2355), before
returning to the power supply.
3.3.6 Helmholtz Coils
The MOT theory in Section 2.5 reveals that the center of the trap is located
where the magnetic field vanishes. Because of the Earth’s magnetic field, this does not
correspond to the mid-point between the anti-Helmholtz coils. This can be managed
by using three sets of Helmholtz coils, a pair of matching coils separated by some
distance with current running in the same direction, as sketched in Figure 3.17.
77
I
Ly
S
Lx
FIGURE 3.17. Helmholtz Coil Layout. This sketch shows the dimensions given in
Equation 3.8 for the magnetic field at the center of the Helmholtz coils, marked by
the x.
While the magnetic field cancels at the center of Helmholtz coils, the magnetic field
along the axis of Helmholtz coils is (closer to) uniform. The Helmholtz coils can be
arranged on the faces of a box centered around the midpoint of the anti-Helmholtz
coils. Manipulating the current in each coil pair creates a (nearly) uniform magnetic
field pointing in any direction, which can be used to cancel the Earth’s magnetic field
or move the center of the MOT away from the midpoint of the anti-Helmholtz coils.
The prior experiments built rectangular Helmholtz coils around each MOT. The
coils for the second MOT are visible in 3.13b as the red and white wire structures
built around the chamber. These coils are formed with loops of 10-strand ribbon
cable connected so current runs through each strand in series. If one pair of matched
coils having N turns and dimensions Lx and Ly are separated by distance S (as shown
in Figure 3.17), the magnetic field (relative to the coil current, I) points along the at
the center of the coils is given by
Bcenter 4µ0N √ LxLy 2S
2 + L2 2(x + Ly= ) (3.8)
I π S2 + L2 2x + Ly (S
2 + L2x) S
2 + L2y
78
Helmholtz Coil Lx, in Ly, in S, in N B/I, G/A
First MOT, x-direction 13.4 13.4 8 6× 10 1.7
First MOT, y-direction 12 9 6.5 3× 10 1.7
First MOT, z-direction 12.4 10.8 6.2 3× 10 2.6
Second MOT, x-direction 10 4.5 6 6× 10 3.4
Second MOT, y-direction 5.5 7 10 10× 10 2.6
Second MOT, z-direction 6 10 4.5 4× 10 3.2
TABLE 3.2. Helmholtz Coil Field Gradients. Lengths and turn counts are from
Schoene [73]. The turn number is written as the number of loops of ribbon cable
with the number of wires in the cable. Field gradients calculated from 3.8. Note that
for the first MOT, the directions labels are different from those defined by Schoene,
who referenced directions relative to the axes for each set of anti-Helmoholtz coils.
Here, the directions are labeled universally, so that the first and second MOTs use
the same direction. These directions are as labeled in 3.1. Translations between the
two direction systems can be done by remapping the directions in figure 2.14 onto
the directions as labeled in figure 2.21 from Schoene.
calculated simply using Ampere’s Law. The measurements and ribbon-cable loops
numbers for the experiment are shown in Table 3.2 taken from Schoene [73]. These
are repeated here because there is a disagreement with the magnetic field values found
by Schoene for the second MOT.
The Helmholtz coils can be used to move the center of the MOT where where
the magnetic field vanishes following Equation 2.81. By changing the current in
the Helmholtz coils and imaging the location where the MOT loads, we are able to
calculate the field generated by the coils. Because of our experimental geometry,
we can only do this for the x- and z-direction (the y-direction is perpendicular to
the camera imaging plane). Doing so gives that the Helmholtz coils in the x- and
z-direction generate fields given by Bx/I = 2.73± 0.05 G/A and Bz/I = 2.91± 0.17
G/A. While still different from values calculated in Table 3.2, the results are closer
to the values in Table 3.2 than the values given in [73].
79
3.4 Photon Collection
To detect single atoms, we built and implemented a single-photon counting
system. The key component is a single-photon resolving avalanche-photodiode
(APD). Light from atoms is collected by a series of lenses and fiber coupled to the
APD. The photons measured by the APD are transmitted as TTL pulses to a field-
programable gate array (FPGA) system which counts the arrived pulses. Photon
counts, and time tagged photon arrival, were sent to lab computers over ethernet.
Later experiments to image the position of the atom require a single photon
resolving CCD camera. After careful study by Jeremy Thorn [74], we purchased and
installed a Hamamatsu C9100-13 Electron-multiplied CCD camera. An off-the-shelf
Cannon lens is attached to the camera, although Matt Briel [102] designed a lens
system for the camera to optimize off-axis position precision for future single-atom
experiments.
The camera is installed along the axis of the APD’s lens imaging system, but
on the opposite side of the fused silica cell. In Figure 3.13b, the camera is visible at
the top of the image and the tube housing the APD lenses is visible at the bottom.
This camera and lens tube location allows careful positioning of the APD lenses.
The camera is first moved as to focus on the MOT. Laser light is coupled backward
through the lens system and shined it onto the camera, showing the focal spot of the
lens system in the camera’s imaging plane. The lenses can be moved with translation
stages until the imaged focal spot overlaps the loading position of the MOT as imaged
with the camera.
80
FIGURE 3.18. Single Photon Collect Lens System
3.4.1 APD
The APD (PerkinElmer P/N SPCM-AQRH-12-FC) used has a dark-count rate3
of <500 photons/sec (measured rate of 321 photons/sec). The APD ordered has an
FC/PC fiber coupler attached.
Light from the atoms is collected by a system of lens shown in Figure 3.18. This
lens system was based on a similar system in use at the University of Texas at Austin
[103, 104] and originally designed by Wolfgang Alt [105, 106]. Our lens system lacked
the complex aberration compensation of the Alt design but did attempt to match
magnification ratios. Removing the aberration compensation just reduces the overall
efficiency of the photon collection system.
Our lens system consists of a two 50 mm diameter lenses with focal lengths of
75 mm (Edmund Optics, P/N NT69-507) and 150 mm (Edmund Optics, P/N NT69-
510). The 50 mm diameter is to increase the collection solid angle of the lens tube
(0.388 steradian) and the focal lengths are to mimic the 2:1 magnification of the Alt
3A phone call with PerkinElmer revealed that all of their APDs are identical in fabrication.
Dark count rates are measured after the APDs are built. The APDs are then classified and priced
accordingly. There is no design or fabrication difference between APDs of different dark count rates.
81
design. The 150 µm pinhole (Thorlabs P/N P150s) is to limit background scatter.
The matched 25 mm lens with focal lengths of 25 mm (Edmund Optics P/N NT65-
524) create a 1:1 system to focus into the FC fiber coupler (Thorlabs P/N SM1FC)
at the end of the lens system. The system is built into 1” and 2” Thorlab lens tubes
(SM1 and SM2 product line). The 2” and 1” lens tube sections are joined with an
adaptor (Thorlabs P/N SM2A6). The fiber used is a custom 105 µm diameter patch
cable from Thorlabs (fiber P/N AFS105/125Y, double FC/APC couplers, 2 meters).
Because of the mismatch between lens tube diameter and lens diameter, each lens
has a few narrow, thin (0.020” thickness for the 50 mm lenses and 0.012” thickness for
the 25 mm lenses) brass sheets of around 1” in length between the lens and the tube.
These shims were roughly equally spaced around the lens to limit their movement
while securing the lens in place. I would not recommend using lenses and lens tubes
with slightly different diameters.
Just prior to the pinhole, a small photodiode and relay are placed with wires
leading outside the lens tube. The photodiode connects to the APD protection
circuit (Section 3.4.3) to collect background light. The relay is also connected to
the protection circuit and, when triggered, covers the pinhole in order to block light
to the APD.
The light collection efficiency of each step between an atom and APD is shown in
Table 3.3. Using the photon scatter rate for an atom (using Equations 2.28 and 2.27),
with typical MOT parameters of detuning ∆ ∼ −Γ, anti-Helmholtz coil current of
9A, and laser power of 1.1 mW , the photon count rate by the APD should be around
44,000 photons/sec. This falls far below measured values (a maximum of 6,5000
photons/sec). This mismatch between anticipated count rates and measured count
rates is consistent with other single atom experiments [20, 77, 106].
82
Contribution Efficiency
Spatial Collection 0.3889/4π
Transmission through experimental cell (reflection) 0.9325
Transmission through lens tube (reflection) 0.9900
Fiber Coupling 0.22
Transmission through fiber (reflection) 0.9329
Transmission to APD chip (reflections) 0.9176
APD Quantum Efficiency 0.66
Total Efficiency, e 3.551× 10−3
TABLE 3.3. Photon Collection Lens System Collection Efficiency
3.4.2 FPGA
The FPGA used in the experiment is a Terasic Cyclone II chip built into an
Altera DE2 Development Board. The development board was chosen as it already
contains many inputs and outputs, has an ethernet port, and can be configured with
part of the FPGA operating as a NIOS II microprocessor [107]. The DE2 also has a
large user base, providing a large selection of programs and uses that have been freely
published online.
The DE2 board has two large banks of male header pins. We designed a small
female plug that interfaced a few BNC plugs to the board’s header. Each BNC input
contained a 3/2 voltage divider shown in Figure 3.19 as the DE2 boards use a 3.3
V TTL logic. Shorting the 20Ω resistor and replacing the 30 Ω resistor with a 50Ω
resistor will give standard 50 Ω terminated TTL inputs. The BNCs were used to
connect FPGA to the APD and to the ZOINKS experiment control system. The
board also contains a large selection of switches, which were configured to control
timing for photon counting. The FPGA system does not have direct start and stop
control. Instead, it has a run and reset trigger.
83
+ +
TL072a TL072a
- - + +
TL072b TL072b
1000! 1000!
- -
1000! 1000! Out to Out to
1000! 1000!
MOD MOD     power     power 
  IN   IN supply supply
12000 Ω0! 10300 0Ω!
BNC 

Input
1000! 1000!
DC DC Header 

   IN    IN Output
FIGURE 3.19. BNC-Header Adaptor for DE2 FPGA
Photon counting is simple input-pulse counting, implemented with very similar
VHDL programing to the photon counting done by Mark Beck’s group [108, 109],
and data processing is done via the NIOS II microprocessor. This microprocessor is
not a separate chip, but a portion of the FPGA chip that is configured to function as
a microprocessor. The NIOS II, running code written in C, manages data from the
photon counter and manually writes UDP ethernet packets for an onboard DM9000A
ethernet chip. This chip routes the bare packets to our control computer.
We use two separate implementations of the FPGA board. The first simply
counts photons for a predetermined time (set by switches), binning the photon count
data. Timing is done with an onboard 50 MHz oscillator. After recording for the set
time, data are sent to the computer. This occurs for a predetermined number of bins
(set by switches). This schematic is shown in Figure 3.20. This implementation is
used for the bayesian atom counting algorithm discussed in Chapter IV.
The second implementation does not count photons, but records the arrival time
of photons (again, using the 50 MHz oscillator). This is done in VHDL and the arrival
times are sent to the microprocessor. The microprocessor numbers the photons and
stores the arrival times for successive photons. As the data is recored, ten of these
arrival times are bundled together and written as one UDP packet before being set to
the computer. The bundles are numbered to check for dropped packets and thus lost
84
Reset
Run


NIOS Processor
Pulse 
 Number Bin
Counting
Write UDP 
APD 
 Packet DM9000A
Pulses
To Control 
Binning Time 
 Number of Bins
 Computer
(Switches) (Switches)
FIGURE 3.20. FPGA implementation for counting photons. Both the pulse counting
and the NIOS processor are programmed onto the FPGA chip. The other elements are
external connections on the DE2 board that interface with the FPGA. The “Number
Bin” and “Write UDP Packet” blocks are segments of code written in C and running
on the microprocessor.
photon data. Additionally, on the control-computer side of the experiment, a manual
delay must be built in at the end of an experiment to check that all of the photon
arrival data has arrived from the APD. With photon arrivals on the order of tens of
thousands per second, an additional 30 s (after an experimental time of around 100
s) is needed so that the APD’s backlog of photon arrival times can be bundled and
sent to the computer. The schematic for the APD implementation is shown in Figure
3.21. This implementation is used for the spectral experiments discussed in Chapter
VI.
The DE2 board does have a dedicated clock input, so a higher speed oscillator
could be used to record timing of photons [107]. Our APD has a maximum photon
count rate of 20 MHz making the need for higher resolution on the FPGA unnecessary
for our experiments. Additionally, rubidium has a decay rate of Γ = 38.1 MHz, so
85
Reset
Run


NIOS Processor
Number 
Record  Photon
Arrival  Store 

Time Photons
APD 
 Write UDP 
Pulses Packet DM9000A
Data Recording Time
 To Control 
(Switches) Computer
FIGURE 3.21. FPGA implementation for timing photons
when measuring light from a single atom, we should not see photon counts faster
that this [62]. While testing the FPGA for our counting algorithms, it was able to
function properly at rates up to 30 MHz using a function generator.
Because the ethernet packets are sent bare, there is no communication between
the FPGA and the control computers. This allows for the possibility for dropped
packets. Luckily at our lowish count rates (well under 100 kphotons/sec), we haven’t
noticed this occurring, but it is possible. Solving this could be done by designing
a TCP/IP stack for the FPGA to deal with communications between the computer
and FPGA. This would allow the computer to report back to the FPGA that it
received the packet successfully. It could allow additional signals to be sent to the
FPGA, such as timing settings (rather than using switches). Alternatively, there is
an implementation of µLinux that has been run on the NIOS II microprocessor. This
contains its own TCP/IP stack that would handle the communication for us. We were
unable to compile and execute this OS successfully. Instead, packets are numbered
when being sent over ethernet and checking for dropped packets is done by making
86
sure all successive packets are received. Dropping a packet, then, can only be detected
and not corrected. This occurs very rarely, but when it does the experimental run
with lost data is thrown away.
3.4.3 APD Protection
Photon arrival rates higher than around 30 MHz risk damaging the APD. To
prevent this a protection circuit was designed and built (not shown due to complexity).
The circuit has two input measurement systems which trigger two safety measures.
Once the protection mechanisms are triggered, the system can only be reset manually
via a momentary switch. This requires a user to notice the problem and fix it before
restarting an experiment.
Prior to the pinhole located in the APD’s lens system, there is a photodiode
and an electric relay as shown in Figures 3.18 and 3.22a. The photodiode measures
light detected in the tube and is connected to the protection circuit. If the
photodiode voltage output corresponds to measuring 30 Mphotons/second, it triggers
the protection mechanisms. Because the photodiode is offset from the beam path of
light focused to the APD, it can only detect broad, bright sources such as room lights.
The TTL output of the APD is also used as a protection mechanism. A copy of
this data is put through a double-resistor RC circuit, shown in 3.22b. The double-
resistor design allows voltage growth across the capacitor to have a much longer time
constant than the voltage decay. Thus, the circuit behaves as an integrator for the
TTL signal. Modeling the circuit, an input count rate of 18 Mcounts/sec gives an
output voltage of 0.39 V after 1 ms. The circuit measures the voltage across the
capacitor and triggers the protection mechanisms if the voltage is above 0.39 V. This
circuit does cause some issues at count rates lower than this as the voltage across the
87
(a) (b)
photodiode
APD 3.3kΩ Integrated 
Data Count-rate

input Output
1μC
Shutter 100kΩ
(c)
FIGURE 3.22. APD Protection Mechanisms. (a) Background light monitoring
pinhole and protection shutter. (b) Integrating double-resistor RC circuit. (c) APD
protection box
capacitor can eventually reach the cutoff voltage. For example, at 5 Mphotons/sec,
the cutoff voltage is reached after 4.2 ms, at 1 Mphotons/sec the cutoff is reached after
23.5 ms. At 100 kphotons/sec, however, the voltage across the capacitor saturates
to 0.22 V. Most of our experiments have photon count rates on just the order of 10
kphotons/sec, allowing the protection to only trigger if there is something a major
issue. This circuit element protects against direct coupling of light into the APD. This
could be triggered things such as by loading a large MOT of hundreds of thousands
of atoms or misaligned or scattered MOT beams that could couple into the APD.
The APD is gated to turn on/off with a TTL input. One of the protection
mechanism resets the TTL to turn off the APD. The ZOINKS system ideally
controls the gating of the APD, but its control is routed through the protection
circuit, allowing interruptions of the on signal. The protection circuit triggers this
88
interruption if the the APD count rate is too high or if the photodiode measures too
much background light to safely use the APD. The second protection mechanism is
a shutter installed along with the photodiode in the APD lens system, as was shown
in Figure 3.22a. The shutter is a black-painted copper flag that moves in front of the
pinhole. This blocks light from reaching the fiber coupler to the APD. The flag is
attached to a T90 relay (Mouser P/N 655-T90N1D12-12) which triggers with a signal
from the protection circuit.
89
CHAPTER IV
OUR SINGLE ATOM MOT
Single-atom MOTs typically rely on three mechanisms to greatly reduce the
atomic loading rate to the single atom-level, all three of which have been used in our
experiment.
1. Reducing the MOT beam diameter, typically on the scale of a few mm, reduces
the volume of the optical molasses region from which atoms are trapped in the
MOT [6, 110]. This was done simply with off-the-shelf irises in front of the
MOT beam fiber launchers (see Section 3.2.2).
2. Reducing the background gas pressure for the atomic species to be trapped
limits the number of atoms which can “wander” into the trapping region [6, 18,
77]. This is also beneficial to increase the lifetime of the trap as background
atoms colliding with those in the trap is the major atomic loss mechanism.
This was done originally by having the experiment’s rubidium source closed for
years while continuing to pump the vacuum and, after rebaking the MOT, only
leaving the rubidium source open for a short period after the rebake.
3. Increasing the magnetic field gradient reduces the radius of the trap (defined as
the distance were the zeeman shift of the energy levels equals the laser detuning)
[7, 11, 20]. This again limits the “volume” inside of which atoms can be trapped
inside the MOT. We used both permanent and water-cooled electromagnets to
create fields on order of 220 G/cm (see Section 3.3).
All these methods reduce MOT numbers strictly by reducing the probability of
trapping atoms rather than any direct manipulation of atoms in the trap. These
90
methods can be used, however, to control the number of atoms in the MOT by
increasing and decreasing loading rates as needed [10, 111].
This chapter looks more closely our single atom MOT, focusing on the detection
methods to verify there is only a single atom. After detecting a single atom during
an experimental run, we can trigger our system to reducing loading rates for atoms
in the MOT and perform an experiment. To detect the single atom, we developed an
algorithm based on bayesian statistics, discussed in the final section.
4.1 Detecting A Single Atom MOT
In large MOTs, the number of atoms is estimate from either the intensity of light
emitted from the atoms [5] or by the reduction of light due to absorption of a light
source that travels through the atoms [95, 96, 112]. With a small number of atoms, on
the order of 10, it is possible to precisely measure the number based on the fluorescence
from the atoms [7, 9, 20]. Each atom will emit photons at (nearly) the same rate,
so fluorescence from the atoms should appear in discrete jumps corresponding to
the number of atoms in the MOT. With a larger number of atoms, this technique
is limited by factors such as the spatial extent of a larger MOT, motion of atoms
within the MOT, and the large number photon counting limitations. With a very
large number of atoms reabsorption of emitted photons by other atoms in the MOT
[113, 114] is also an appreciable effect that prevents photon counting utility as a direct
way to measure atom number.
91
 40000 a) 6 atoms
5 atoms
 30000 4 atoms
3 atoms
 20000 2 atoms
1 atom
 10000 0 atoms
 0  100  200  300  400  500
Time, sec
 24000 b)
2 atoms
 20000
 16000 1 atom
 12000
0 atoms
 8000
 24000  0 c)  20  40  60  80  100
Time, sec
 21000
1 atoms
 18000
 15000 0 atoms
 12000
 0  20  40  60  80  100
Time, sec
FIGURE 4.1. Sample Photon Collection Data. (a) Long data run recorded with
permanent magnets. (b) Data run with anti-Helmholtz coils with 4 A current. (c)
Data run with anti-Helmholtz coils with 9 A current. Graphs (b) and (c) share
horizontal axes. Red line is recorded data, black line is linear estimate of fluorescence
at a given atom number. Atom number dashed lines estimated from tracking jumps
in data.
92
Photon countrate, photons/sec Photon countrate, photons/sec Photon countrate, photons/sec
4.1.1 Detection : Fluorescence Jumps
For our small atom MOT, a few fluorescence records are shown in Figure 4.1. In
each of these, and in general for the experiments discussed in this dissertation, the
MOT lasers are shifted to frequencies above the atomic resonance to prevent loading
atoms into the MOT for a few seconds. This gives a measure of the background
fluorescence rate for an individual experimental run. This has two direct benefits.
First, it gives a background level for the bayesian algorithm discussed in the next
section. Second, it allows for estimation of the MOT beam power for an individual
data run without measuring the power directly. While beam power is not measured
directly, a set of data runs with known MOT beam powers can be done without
loading atoms into the MOT to correlate background scatter rates measured from
the APD to MOT beam powers. This is useful for modeling the behavior and
temperature of atoms in the MOT (see Chapters VI). After the few seconds for
background estimation, the laser detuning is reset to its desired value to allow atoms
to load in the MOT. Sharp changes from the background are signals of atoms entering
and leaving the MOT. These jumps are on the order of a few thousand photons/sec,
generally counted for 100ms, which are the red curves in the three sample graphs.
The jump size is impacted by MOT laser intensity and detuning as evident by the
atomic fluorescence rate in Equations 2.27 and 2.28, but a much more significant
impact on the jump size is alignment of the APD lens system with the MOT location
location.
Comparisons between the three graphs in Figure 4.1 show the direct influence
of the first MOT. For the graph in Figure 4.1a, the first MOT is left on during the
entire data run and this one data run has an atom loading rate of 0.034 atoms/s.
For Figure 4.1b and Figure 4.1c, the first MOT is turned off once a single atom has
93
detected (using the bayesian algorithm triggering discussed in Section 4.2.7). In (b),
there is a much weaker magnetic field in the second MOT (once one-atom has been
detected in the MOT), but the loading rate is greatly reduced, just 1 atom in about
seconds). In (c), the magnetic field of the second mot very closely matches that of
the permanent magnet MOT in (a), but no additional atoms load into the MOT as
they had in (a). This is directly the result of turning off the first MOT.
The discrete jumps in fluorescence are made more clear in histograms of the count
rates, as commonly done in single atom experiments [6, 24, 115, 116]. The histograms
for the graphs in Figure 4.1 are shown in Figure 4.2. In these graphs, the measured
background rate has been subtracted from each data point before the histogram was
created. The discrete separation between atom number counts is obvious here.
The broader width of the 1-atom peak in Figure 4.2b is tied to the weaker
confinement of the atom. Because the MOT κ coefficient is proportional to the
magnetic field gradient (Equation 2.78), the smaller current allows for a weaker
trapping potential. This allows the atom to explore a larger region of space, altering
the coupling between the atom and the APD’s detector area. This is more obvious
in the 2-atom peak where the motion of both particles greatly increases the width of
the (tiny and hard to see) peak.
4.1.2 Atom Counting by Fluorescence Jumps
In the graphs of Figure 4.1, the the atomic number fluorescence levels (dashed
lines) are calculated from the data after the experimental run completes. For these
values, a running average and standard deviation for fluorescence rate is made until
a subsequent data point differs from the running average by a threshold defined by
a multiple of the standard deviation (typically 5 − 10σ). This suggests a change
94
 140 a)
 120
 100
 80
 60
 40
 20
 10 0
b) 0  5000  10000  15000  20000  25000  30000  35000
 80
 60
 40
 20
 10 0
c)  0  5000  10000  15000  20000  25000  30000
 80
 60
 40
 20
 0
 0  5000  10000  15000  20000  25000  30000  35000
Photon countrate, photons/sec
FIGURE 4.2. Histograms of sample photon collection data. Graphs correspond to
data runs in Figure 4.1. Horizontal axis of each graph shifted by average zero atom
fluorescence rate for that data run. All graphs share horizontal axes and fluorescence
bin size of 150 photons/sec.
95
in atomic number at which point a new running average and standard deviation
are made. These average values are the black horizontal lines plotted in the graphs
and give an average fluorescence rate for the given number of atoms at that time.
The average rates plotted as dashed lines in each graph are the average of correctly
numbered black line averages. The number of atoms in the MOT is approximated by
tracking the positive and negative jumps in fluorescence.
This atom counting method does have faults which are not present in the bayesian
algorithm outlined in the next section. Drifts in fluorescence rate, especially with
multiple atoms in the MOT, can also be interpreted as a spurious fluctuation in the
atom number. Loading events with multiple atoms tends to result in very incorrect
atom number approximations. For example, initially loading two atoms during a
single fluorescence record only registers as a single atom. Subsequent individual
atom losses from the trap result in an atom count of -1. Additionally, the counting
algorithm does include the possibility for double-atom loss events, discussed briefly
in Chapter I. This can cause similar misestimates of the atom number as a single-loss
event could register as a double atom loss if the threshold for atoms loading and
leaving the MOT is not carefully set. Even with these issues, the method gives a
fair approximation of the atom number, especially for data runs with large jumps in
fluorescence. This method is typically used to “seed” fluorescence rate estimates for
subsequent data runs using the bayesian algorithm described below.
Evident in both Figures 4.1b and c, the initial count rates reported by the APD
occasionally do not match the background levels. These levels, instead, more closely
match the ending fluorescence levels from the prior data run. This is a result of the
programming of the FPGA (see Section 3.4.2). Because the counting is done directly
by the FPGA and controlling the number of binned data runs is done by software
96
running on the NIOS processor, there is a mismatch in when to count data. When the
final data point of one run is fully counted, the FPGA immediately begins counting
the next data point until the NIOS software tells it to stop. This results in occasional
too-high count rates for the first data point of the next data run. This was a larger
problem initially as the high count rate overlapped with multiple data points. Adding
the “reset” signal to the FPGA (as shown in Figure 3.20), reduced this to just a single
packet. This could be fixed by implementing the TCP/IP stack for the FPGA, but
for the data recorded in this thesis, the very first data point is simply ignored when
calculating the background fluorescence rate.
The atom number estimating method above produces standard deviations for
the fluorescence level for each number of atoms. The variance of the fluorescence for
each atom number in the graphs of Figure 4.1 is shown in Figure 4.3 where each data
point corresponds to a different number of atoms. It is clear that the variance of
these peaks grows faster than the average. The background is nearly poissonian (the
initial point of each graph lies nearly on top of the black poissonian line). This is
as expected since individual photon arrivals are un-related. For entirely single-atom
sources, sub-poissonian photon statistics could be expected [117, 118], however the
large background scatter swamps sub-Poisson statistics. On the other hand, super-
Poisson statistics can be expected for single atoms [119, 120] when the driving laser
detuning is larger than an atom’s decay rate. Our experiments use detunings close to
the half-width-half-max, which would allowing for Poisson fluorescence distributions
to return, as noted by others [121]. In general, atomic motion tips the scales to
expected Poisson statistics with a time-dependent average as the atomic motion will
alter the coupling between the atom and the APD through the lens system. This
effect should be more pronounced when there are multiple atoms which can move
97
 20000
Graph a
Graph b
Graph c
 15000
 10000
 5000
 0
 1000  1500  2000  2500  3000  3500  4000  4500
Average Fluorescence, photons/sec
FIGURE 4.3. Variances of sample photon collection data. Graphs refer to those in
Figure 4.1. Also plotted is a black line corresponding to a Poisson distribution.
around, which is what is observed as the graph of Figure 4.3b deviates from the black
Poissonian line.
4.1.3 Detection : Atomic Pictures
Further evidence of small numbers of atoms is revealed when imaging the MOT
with our CCD camera. Four data runs with varying numbers of atoms are shown
in Figure 4.4. In each data run, a picture was taken at time t=90 s. As clear from
the recorded APD data in graph (a), these four data runs had 0, 1, 2 and 3 atoms
in the MOT at the time the picture was taken. These (background subtracted and
cropped) images are shown as a series in (b), with outline colors corresponding to the
graph colors in (a) and ordered in terms of increasing atom number. These pictures
are gray-scale normalized such that the highest intensity pixel is white and the lowest
intensity pixel is black. In the first image in series (b), the white dots correspond
to random background fluctuations. The remained of these images show a clear
bright spot where the MOT loads. Because each of these pictures are normalized
98
Fluorescence Variance, (photons/sec)2
 35000
 30000 a)
 25000
 20000
 15000
 10000
 5000
 0  10  20  30  40  50  60  70  80  90  100
Time, sec
b) c)
d)
els
 Pix
tica
l
Horizontal V
er
 Pixels
FIGURE 4.4. Atom counting with CCD camera. (a) Recorded fluorescence signal
over 4 data runs. A picture of the MOT was taken at 90s for each data run. (b)
Individually normalized MOT pictures. (c) Picture intensities normalized on same
scale. (d) Raw picture pixel count data. Image and graph outline colors in b-d
correspond to graphs with same color in a.
99
Pixel Count, a.u.
Photon countrate, photons/sec
individually, it is hard to see much relation between then. In the series of pictures
in (c). the four MOT pictures are normalized on the same scale. Clearly, increasing
brightness occurs with higher number of atoms in the MOT. To further this, the
background-subtracted, but un-normalized, pixel data is plotted in the graphs of (d).
There is a clear increase in pixel intensity that corresponds to a larger number of
atoms in the MOT. The large increase in pixel intensity is sufficient that it can be
used, rather than fluorescence rates, by the bayesian algorithm discussed below.
4.2 Bayesian Algorithm
Many other single atom MOT experiments count atoms in the MOT by
comparing the fluorescence level from the MOT to the known discrete steps in
fluorescence contributed from each atom, very similar to the method described in
Section 4.1.2. In many cases of these cases, including ours, the discrete steps
are smaller than expected signaling unknown efficiency loses in their experiment
[6, 20, 77, 106]. We have expanded on this approach implemented an atom counting
algorithm based on bayesian estimation.
4.2.1 Atom-Number Probability Distribution
The probability for there to be n atoms in the MOT at time ti is defined as
P i(n). Of interest is how this probability will evolve as fluorescence measurements
are made. Probability evolution follows Bayes’ rule [122],
p(x| ∫ p(x)p(y|x)y) = , (4.1)
p(x)p(y|x)dx
x
100
where p(x) is the initial probability distribution for a parameter x, p(x|y) is the
probability distribution after measuring a value y for the parameter, and p(y|x) tells
the likelihood to have measured y when the parameter was actually valued as x. The
experiment records a fluorescence measurement at time i from a prior fluorescence
distribution and updates that distribution at time ti+1. Comparing to the general
form of Bayes’ rule we can define P i(n) ≡ p(x) as the initial atom-number probability
distribution of fluorescence before a measurement, P (yi|n) ≡ p(y|x) as the likelihood
to have gotten measurement y with n-atoms in the MOT, and P (yi|n) ≡ p(y|x) as the
updated atom-number probability distribution after fluorescence measurement result
y. The integral over possible measurement values x becomes a sum over the number
of atoms in the MOT. In this way, we have
i
i+1 | ∑P (n)P (yi|n)P (n y) = ,
n P
i(n)P (yi|n)
or writing the evolution of P i(n) in terms of a differential change between times i and
i+ 1, P i+1(n) = P i(n) + dP (n), we have
[ | ]P (yi n)
dP (n) = ∑ | − 1 P
i(n). (4.2)
P in (n)P (yi n)
With distributions P i(n) as being what is evolved in time, the most important term
here is the likelihood function P (yi|n), which gives the likelihood that the photon
measurement at time i came from a state with n atoms in the MOT.
101
4.2.2 Fluorescence Distribution
At a single time i, define the probability distribution for the fluorescence, x, from
a MOT with n atoms as
[ ]
1 −[x− (nr +B)]2
Fln(x, r) = √ exp , (4.3)
2πσ2 2σ2n n
where r is the single atom fluorescence rate and B is the background fluorescence
rate. Because photon arrivals are independent, the probability should be Poisson-
distributed. The Poisson-distribution of the light arrival from an atom is further
justified in Section 6.1 where the arrival rates of the atoms are important. However,
looking at count rates, the noise isn’t quite Poissonian as the atom is free to move
slightly. This movement changes r. As seen in Figure 4.3, experimental measurements
of σn for both n = 0 (i.e. for the background fluorescence) and n > 0 have variances
a bit larger than the mean, justifying treating the count rates as Gaussian rather
than strictly Poissonian. In practice, following the suggestion of Section 4.3, σn is
treated as nearly Poissonian in the form σ2n = 〈nr + B〉 (1 + ζ0 + nζn), where ζ0 is a
small non-Poissonian contribution to the background and ζn is a increase in variance
due to atomic motion. Typically used values for the non-Poissonian contributions are
ζ0 = 0.08 and ζn = 0.2.
Motion of the atom changes coupling of atomic fluorescence into the APD, noise
in laser power, magnetic fields, and polarization all change the photon emission rate.
Thus, r is not constant, but is assumed to also have a gaussian probability distribution
P (r) [ ]
1 −(r −R)2
Fl1-at(r) = √ exp , (4.4)
2πσ2 2σ2R R
where R is the average emission rate.
102
Combining these two probabilities gives the probability distribution for n atoms
if we measure a photon rate of x with an average background rate, B, and average
single-atom photon rate, R,
∫ −∞
Fln(x) = Fln(x, r)Fl1-at(r)dr
−∞ [ ]
√ 1 −[x− (nR +B)]
2
= exp . (4.5)
2π (n2σ2 + σ2) 2 (n2σ2 2r n r + σn)
A few plots of the fluorescence probability from n atoms in the MOT are plotted
in Figure 4.5 with a single FPGA measured data run. Each peak along the y-axis is
the distribution, Fln(x), for atom numbers n = 0 to n = 5. The plotted probabilities
are the final distributions calculated from the last measurement of the data run. The
values for R, B, σR, and σn that make these distributions are calculated following
the method in Section 4.2.6 from the data shown in the figure.
4.2.3 Noisy Measurements
The measurements take are noisy fluorescence measurements from the
experiment. This noise is not fluctuations in the background signal or in the
fluorescence rate from the atom, those are built into the fluorescence distributions
Fln(x). Instead, this noise is from measurement errors—dark counts from the APD
and possible counting mistakes at the FPGA. Making a measurement, y, produces a
sampled value x from the distribution P in(x) plus some noise, written as ζ (note that
a noiseless measurement would then sample x values directly from Fln(x)). So the
measurement outcome has a value
y = x+ ζ. (4.6)
103
 4500
Probability Distributions
data
 4000
 3500
 3000
 2500
 2000
 1500
 1000
 0  200  400  600  800  1000
FIGURE 4.5. Sample data for bayesian fluorescence number estimation. Vertical
axis is in units of photons/100 ms and horizontal axis is in units of recorded FPGA
packets. Note the increasing distribution width for larger values of n. This comes
from the factor of n2 in the variance of Equation 4.5. This overlapping fluorescence
distributions for n and n + 1 atoms shows the limitations of photon counting to
estimate atom number at a large number of atoms.
104
To treat this noise, it is assumed that values for the noise follows a Gaussian
distribution [ ]
1 −ζ2
p(ζ) = √ exp , (4.7)
2
2πσ2 2σζζ
so that the probability to have had a noise value of ζ = y − x is
[ ]
− √ 1 −(y − x)
2
p(ζ = y x) = exp . (4.8)
2
2πσ2 2σζζ
The probability for the noise to take this value is the same as the probability to have
measured y when the sampled fluorescence value is x.
For the Bayesian evolution, the likelihood to measure y with n atoms in the MOT
is needed. This is the the probability for the noisy measurement to have sampled a
value x and averaged over the probability for the fluorescence to have value x with n
atoms in the MOT as below.
∫
P (yi|n) = p(ζ = yi − x)Fln(x)dx
[ ]
√ 1 − [yi − (nR +B)]
2
= ( ) exp ( ) (4.9)2 2 2 2
2π n2σ2 + σ2 + σ2 2 n σR + σn + σR n ζζ
This is the desired likelihood function for the bayesian estimation of the number
of atoms in the MOT. It tells the probability that when a fluorescence rate yi is
measured, there are n atoms in the MOT, given that there is
1. an average photon emission rate per atom of R and with standard deviation of
σR,
2. average fluorescence background B,
105
3. background (plus atomic motion) fluorescence rates with standard deviation σn,
which are slightly super-Poissonian, and
4. systematic noise with variance σζ .
4.2.4 Keeping P (n) > 0
From Bayes’ rule, if P in(x) = 0, for any time ti, then for all later times, the
probability to have n atoms in the MOT will always be zero. This can be resolved
numerically one of two ways. First, manually setting Pi(n) =  for some fixed small
value  at each time step if Pi(n) < . Second, using a loading-rate method. This
method includes, in the probability evolution, atomic loading and loss terms for the
MOT. With the loading rate method, the probability to have n atoms in the MOT
evolves as
dP i(n)
= −nΓP i(n) + (n+ 1)ΓP i(n+ 1)− LP i(n) + LP i(n− 1), (4.10)
dt
where Γ is the rate that an atom is lost from MOT, and L is the loading rate of atoms
into the MOT. The first term represents any one of the n atoms leaving the MOT.
The 2nd term represents any one of the atom leaving a MOT that used to have n+ 1
atoms. The 3rd term represents at atom loading into the MOT from the background
gas (to create a MOT with n+1 atoms). The 4th term represents an additional atom
loading from a MOT with n − 1 atoms. This loading-rate equation is identical to
loading-rate model analysis done for small numbers of atoms in a MOT [20].
Using this method numerically requires a maximum number of atoms, Nmax to
be set. Doing this adds a cut-off term Θ(Nmax−n) to the 3rd term so that an addition
106
atom “cannot” load if n = Nmax. This definition also allows
n=∑Nmax dP (n)
= 0,
dt
n=0
so that the loading-rate method appropriately conserves probability.
Tests with both methods give the same predictions for N(t) under a variety of
other parameters. In practice, we use the loading-rate method with typical parameters
L = 0.006, Γ = 0.003, and Nmax = 8. These values are measured atomic loading and
loss rates from the experiment.
4.2.5 Number Estimation
Combining the Bayesian evolution with the loading-rate differential gives an
overall evolution for the atom-number probability
[ ]
dP i+1(n) = −nΓP
i(n) + (n+ 1)ΓP
i(n+ 1)− LP i(n) + LP i(n− 1) dt+
 P (yi|n) ∑ − 1P
i(n). (4.11)
P (yi|n)
n
Again, this equation conserves probability when summed over n. It also solves the
issues of any P (n) → 0 as the probability for there to be n atoms in the MOT
will be increased slightly by the loading rate portion (first term) of the differential.
This probability evolution could be interpreted as a deterministic “Hamiltonian” like
evolution at all times, punctuated with the noisy measurements at times ti, similar
to the stochastic evolution of a system in quantum measurement theory [123–125].
107
This algorithm is simple to implement in real time while we record fluorescence
from the MOT and estimate the number of atoms in it. To determine the number
of atoms in, we typically assume the n with the largest probability is the correct
number of atoms in the MOT. This generally works well as often the average single-
atom fluorescence rate is much larger than the width of its fluorescence distribution,
R σR.
In some cases, such as poor alignment of the APD lens system with the MOT
center or cases where atomic position distributions are large, just taking the largest
probability as the number of atoms in the MOT causes problems. The main error seen
is constant fluctuations in the estimate of the number of atoms in the MOT as two
values for Pi(n) are close to 0.5 (typically for n = 0 and n = 1). In such cases, assume
that states with n > 1 remain essentially unpopulated and initially P i(0) = 0.51, so
that there are believed to be no atoms in the MOT. Updating the probabilities after
measurement could give P i+1(1) = 0.51 so there is now believed to be one atom in the
MOT. Another update gives P i+2(0) = 0.51, so the state again returns to there being
zero atoms in the MOT. This can repeat often if measurements of the fluorescence
tend to stay in the “middle” between the peaks of the likelihood functions for zero
atoms or one atom. This most often happens when the initial assumption about the
single-atom fluorescence rate, R, is larger than the actual rate in the experiment.
Other than using a more realistic single-atom fluorescence rate, this can be
solved via more complicated assumptions about when the number of atoms in the
MOT changes. One method is to require the largest probability at ti+1 to be
above some threshold value (larger than 0.5) before determining the atom number
changed from time ti. A second method could require the largest probability
be above the 2nd largest probability by some determined factor, limiting jumps
108
between two states with probabilities close to 0.5. A third method would require
that the maximum probability remain the maximum over a number of fluorescence
measurements, avoiding the possibility of fast fluctuations in number. A final method
is to reduce the possibility of such oscillations by updating the background and single-
atom fluorescence rate as data were being recorded, described in detail below.
4.2.6 Background and single-atom fluorescence estimation
Mean values for single-atom fluorescence rates are often 5 or 6 times larger than
the standard deviation of the background signal (this fact is occasionally used after
data is recorded to locate times when a given number of atoms are in the MOT,
as noted in the atom counting method in Section 4.1.2). Because of this, in many
cases P i(n) is very close to unity. For example, for the data shown in Figure 4.1c,
at times when n=0 gives the largest probability, the mean value of P (0) is 0.9986±
2.2× 10−4, and at times when n=1 is the largest probability, the mean value of P (1)
is 0.9979 ± 1.4 × 10−2. With good alignment of the imaging system with the MOT
center, these are not uncommon values. With P (n) ≈ 1 for some n, we can leave n
fixed in Equation 4.5 and use the fluorescence measurements to update values for R
or B rather than update predictions for n.
Taking n = 0, Equation 4.5 gives
[ ]
√ 1 − (x−B)
2
Fl0(x) = exp .
2πσ2 2σ20 n
Because the noise in a measurement is assumed to be Gaussian, updated values for the
background mean value and variance after a measurement can be written analytically
109
[122]. Following this, a measurement of yi allows updating of B and σ
2
0 as
B σ2i ζ + y
2
iσ0,i
Bi+1 =
σ20,i + σ
2
ζ
(4.12)
σ2 2
2 0,i
σζ
σ0,i+1 = ,σ2 20,i + σζ
where, again, σζ is systematic noise in the measurement. These updates are easy to
make and include in subsequent measurements and probability calculations.
When there is an estimate of n > 0 atoms in the MOT, instead of updating the
background and its variance, the atomic fluorescence rate and its variance is updated
using the n-atom fluorescence probability Equation 4.5. The background level and its
variance complicate the calculation for updating R and σR. This calculation is done
in Appendix D and the conclusions are
R 2iσζ + (y
2
i −B)nσR,i
Ri+1 =
n2σ2 2R,i + σζ
2 2 (D.3)σ σ
2 Ri ζσR,i+1 = .n2σ2 2R,i + σζ
These two updating methods are effectively just a Bayesian filter [122] for the
background (when n = 0) and for the fluorescence rate (when n > 0).
4.2.7 Bayesian Algorithm
The full Bayesian algorithm is sketched schematically in Figure 4.6. At each
time step, ti, a fluorescence rate, yi, is measured. From this, calculate the likelihood
functions for each atom number according to Equation 4.9. These are then used to
update the atom-number probability distribution with Equation 4.2. The number of
110
Record Fluorescence Rate
Calculate likelihood functions,P (yi|n)
Evolve atom number distributions  
with Bayesian evolution
P i+1(n) = P i(n) + dP (n)
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Estimate number of atoms in MOT, Ni+1
Ni+1 = 0 Ni+1 > 0
Ni+1  Ntrig
Ni+1 < Ntrig Trigger 
Experiment
Update Update
Bi+1 & 2 i+1 2B,i+1 R & R,i+1
Evolve atom number distributions  
with deterministic evolution
i+1
P i+1 i+1
dP (n)
(n) = P (n) + dt
dt
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FIGURE 4.6. Bayesian algorithm flow chart. Initial background fluorescence
measurement and algorithm ending elements are not shown.
111
atoms in the MOT is updated according to any of the methods discussed in Section
4.2.5.
If the algorithm is set to trigger an experiment based on atom number, this
number is checked against the triggering atom number, Ntrig. If the number of atoms
meets or surpasses that value, whatever experiment is to be done is triggered. It is
also possible to trigger an experiment manually after a given time. The experiment is
a predetermined order of commands and runs “in the background” while the Bayesian
algorithm continues.
Based on the predicted number of atoms in the MOT and the measured
fluorescence, the background fluorescence parameters (with Equations 4.12) or the
atomic fluorescence parameters (with Equations D.3) are updated. Because the
atomic number estimate is made before updating rates, Equations D.3 should be
modified to use the number of atoms assumed to be in the trap when the data was
recorded. Thus the values N should become Ni+1 in the equations. This completes
the number probability update due to the measurement.
During the time before the next measurement, the probability is evolved
according to the deterministic evolution of Equation 4.10. The time variable dt is
the time between measurements. Alternatively, the probabilities could be checked as
to not fall below the pre-determined minimum  as described in Section 4.2.4. This
prepares the probabilities for updating on the next measurement.
Starting and stopping of the algorithm are done based on the information from
the FPGA. The algorithm does not start on the first data from the FPGA. Instead,
all experiments are designed with a few second “dead time” where the MOT trapping
laser detuning is shifted above resonance. This guarantees no atoms load in the MOT.
The data recorded by the FPGA is only background fluorescence. The algorithm
112
knows when the dead time ends and once it receives photon data whose time stamp
matches the end of the dead time, it can calculate initial background fluorescence
and then begins the Bayesian evolution. The algorithm is ended once the FPGA
transmits its final count rate. The final FPGA time-data packet is followed by a
series of packets that just read ’‘STOP” rather than photon data for this reason.
113
CHAPTER V
ATOMIC FORCES IN A MOT
In this chapter the MOT theory in Chapter II is expanded to more closely model
the behavior of a real 87Rb atom. The atom is expanded from the two-level or V-atom
to the full D2 level structure, in Figure 2.1. The magnetic and optical fields, along
with the atom’s position and velocity, are expanded to three dimensions, requiring
6 optical fields. The discussion in Section 5.1 briefly discusses modifications to the
Hamiltonians in Chapter II to change to the full atom in 3D. Section 5.3 looks at
these changes in the dynamics of the atom, but internally as the evolution of its
density matrix and externally as the MOT trapping force on the atom. After laying
the groundwork for the simulation, a discovered atomic loss mechanism is discussed
along with a resolution to agree with established MOT theory.
5.1 3D and 87Rb Hamiltonians
5.1.1 Atomic Hamiltonian
The free atomic Hamiltonian closely matches that of Equation 2.57 with a few
small changes. The detuning is defined relative to the ground state energy rather
than the excited state energies. This is done as a separate electric field is needed
to excite atoms from each of the two ground states. The |Fg = 2〉 ground state is
defined with (detuned) energy ∆M and the |Fg = 1〉 ground state is defined with
energy ∆R. Here, the M subscript is in reference to the “MOT laser”—the laser field
for the MOT trapping transition (see Section 3.2 and Figure 2.1). The R subscript
114
Energy Level With Repump Without Repump
|Fe = 3〉 0.35 5.6×10−7
|Fe = 2〉 2.9×10−4 4.6×10−10
|Fe = 1〉 3.6×10−5 5.8×10−11
|F = 0〉 2.7×10−8 2.6×10−14e
|Fg = 2〉 0.65 1.0×10−6
|Fg = 1〉 8.3×10−4 1
TABLE 5.1. Repumping field and populations of 87Rb D2 energy levels. Values are
steady-state populations (summed over magnetic sublevels) for energy levels. Repump
field powers, relative to trapping field power, are 10−1 and 10−10. Values calculated
with ∆M = −Γ, ∆R = 0.5 MHz, trapping beam intensity 10Isat, and without Zeeman
shifts of the magnetic sublevels. These values were calculated as described in Section
5.3.1.
is in reference to the “repumping laser”. With these, the free atomic Hamiltonian is
∑3 ∑Fe
HA = ~ ∆Fe |Fe;me〉〈Fe;me|+
Fe=0 me=−Fe
∑2
~ (∆M + ∆Fe=3) |Fg = 2;mF 〉〈Fg = 2;mF |+ (5.1)
mF=−2
∑1
~ (∆R + ∆Fe=2) |Fg = 1;mF 〉〈Fg = 2;mF | ,
mF=−1
where the terms ∆Fe give the (relative) energy differences between the various excited
states. The two ground states also have different energies, but because their energy
is defined relative to the excited state energy (through the detunings), the energy
difference between the two grounds must match the energy shift of the excited state
to which they are coupled. These energy differences are associated with the angular
momenta of the atom and are defined in [48, Eq. 7.134].
For hyperfine splitting, transitions which do not change the total angular
momentum are allowed, which makes excitations |Fg = 2〉 → |Fe = 2〉 possible. The
likelihood of this excitation is small as it is detuned from the MOT trapping field
115
by 267 MHz (peak D in Figure 3.9). Even with such a large detuning, excitations
that do occur can then decay to the |G;F = 1〉 ground state. Excitations out of
this ground state are detuned by 6.8 GHz from the MOT trapping transition [62],
effectively making this a dark (inescapable) state—atoms that fall into this state will
remain there. The repumping field is thus required to excite atoms out of this state
back into the MOT transition by coupling |Fg = 1〉 back to the |Fe = 2〉 (as shown
in Figure 2.1). These atoms could then return to the |Fg = 2〉 state and the MOT
transition. This result is shown numerically in Table 5.1. The values listed are the
steady-state level populations (summed over magnetic mF sublevels). Populations on
the left are with a repump field (10% of the power of the trapping field) and values
on the right are without the repump field. The decay of atoms into the inescapable
|Fg = 1〉 state without a repumping field is clear.
5.1.2 Atom-Magnetic Field Hamiltonian
In one dimension, the magnetic field for the MOT was assumed to be linear with
gradient −B′z, which agrees with the magnetic field strength for permanent magnets
in Figure 3.12 and for anti-Helmholtz electromagnets in Equation 3.6 near the center
of the MOT. In a full 3D theory, the magnetic field at any point in space for anti-
Helmholtz coils, with their axis defined as a z-axis, will be
∫ [2π
((z+s/2) z+s/2~ µ I cos θx̂+( ) sin θŷ+(1− x cos θ− y sin θ0 R R R R ) )ẑB(x, y, z) = dθ2πR 2 2 2 3/2
0 1+( x ) + y +R (R) (
z+s/2) −2 x cos θ−2 y sin θR R R ] (5.2)
((z−s/2) ( z−s/2− cos θx̂+ ) sin θŷ+(1− x cos θ− y si)n θ)ẑR R R R 3/2 ,
1+( x
2 2
) z−s/2
2
+( y ) +( ) −2 x cos θ−2 y sin θR R R R R
where R is the radius of the coils, s is the separation between coils and the origin is
along the axis exactly inbetween the two coils. As long as the atom stays close to the
116
origin, the magnetic field is nearly linear along the axis of the coils with a (negative)
gradient of ∣
′ ∣ 3Bz ≡ −
∂Bz ∣ 3µ0IsR∣ = . (5.3)∂z ~r=0 [R2 + (s/2)2]5/2
In order for the divergence of the magnetic field to vanish, the x- and y-direction
contributions to the divergence must cancel the contribution from the z-direction.
Additionally, the x- and y-direction magnitudes must be equal as the magnetic field
should be symmetric around the z-axis. These, together with the linear assumption
of the z-direction magnetic field, require that the magnetic field have the form
(
~ ′ x y
)
B (~r) = Bz x̂+ ŷ − zẑ . (5.4)2 2
This can be checked simply with the full form of the field in Equation 5.2. Looking
at the field along the x-axis is
∫
~ µ
2π s
0I cos θx̂+
s sin θŷ
B(x, y = 0, z = 0) = dθ( R R( ) ( ) ) (5.5)2πR 2 3/20 2
1 + x + s/2 − 2 x cos θ
R R R
The y-component clearly integrates to zero so that the field is just along the x-
direction with magnitude
 [ ] [ ] 
2
µ Is (R + x
2 + (s/2)2)E −4xR K −4xR
0 (x−R)2+(s/2)2 (x−R)2+(s/2)2
Bx(x) = √ − √  ,
πx (x−R)2 + (s/2)2 ((x+R)2 + (s/2)2) (x−R)2 + (s/2)2
(5.6)
117
where K[k] and E[k] are complete elliptical integrals. Near x = 0, the field is nearly
linear with a slope of
′ dBx 3µ0IsR
3
Bx = lim = , (5.7)
x→0 dx 2 [R2 + (s/2)2]5/2
which is half the z-direction gradient with the opposite sign. Thus the form of
equation 5.4 is valid near the origin.
Do note, in our simulations, the z-direction gradient value, B′z that is used is
the measured value from the constructed water-cooled anti-Helmholtz coils. As noted
in Section 3.3.1, this values was also numerically calculated, but must summed over
many coil pairs with different radii Ri and separations si to account for many layered
loops of wire in our coils.
In writing the form of the atom-magnetic field coupling Hamiltonian, Equation
2.49, it was assumed that the atomic dipole moment µ~ stayed aligned with the
magnetic field. This allowed the Hamiltonian to be written just as the Zeeman shifts
of the energy levels. This assumption is kept here, giving the Hamiltonian
√ [
B′ ∑ ∑Ĥ = µ z x2 + y2z B + 4z2 g2 Fe me Feme |Fe;me〉〈Fe;me|+
∑ ∑ ] (5.8)
gFg F m mg |Fg;mg〉〈Fg;m | ,g g g
where the gF values are as described in Section 2.3. When considering the motion of
the atom, this assumption requires that the direction of the atomic dipole changes
along with the magnetic field. This can be particularly burdensome when passing
through the origin as the magnetic field direction changes abruptly (from +ẑ to −ẑ
when traveling along the z-axis, for example). Allowing for the atom to do so also
assumes that the motional times scale of the atom is much slower than the time
118
scale for the atomic dipole moment to precess and align with the magnetic field, the
adiabatic limit [63].
5.1.3 Atom-Electric Field Hamiltonian
With the full level structure of rubidium, there are many transitions between the
various magnetic sublevels. Rather than writing individual transitions independently,
and with thoughts towards the polarization of the experimental light fields, they can
be grouped by linear and circular transitions that change mF by ±1 or 0. In this way,
the lowering operators are written as in
∑
Σ̂q = s (Fe, Fg,me) |Jg,me + q〉〈Je,me| , (5.9)
Fg ,Fg ,me
with coefficients ([48, Eq . 7.407])
√
s(F , F ,m ) = (−1)Fe+Jg+1+Ie g e (2Fe + 1)(2Jg + 1)×
 
  (5.10) J J 1 
〈 e gFg,me + q|Fe,me; 1, q〉 
,
F F g e I
where each term in the sum lowers the atom from the |Je,me〉 state to the |Jg,mg =
me+q〉 state. Do note that under this definition, transitions with q = +1 increase the
mF sub-levels of the atom, which correspond to σ− transitions as used in Chapter II.
With these lowering operators, the atom-field interaction Hamiltonian can be defined
in the same form as Equation 2.56, with appropriate care due to the σ±,0 and Σq=∓1,0
119
Propagation direction Polarization P[ olarization V] ec√tor
+z σ− [ 1, −i, 0 /] √2
−z σ+ [ −1, −i, ] 0√ / 2
+x σ+ [ 0, i, 1 /] √2
−x σ− [ 0, −i, 1 ] /√2
+y σ+ [ −i, 0, 1] √/ 2
−y σ− i, 0, 1 / 2
TABLE 5.2. MOT Beam Circular Polarizations. These are defined so atoms along an
axis are pushed towards the origin as shown in Figure 1.1. Note the change flipping
of polarizations between the x- and y-directions compared to the z-direction, which
arrises from the change in the sign of the magnetic field gradient in the x- and y-
directions. The polarization vectors are in cartesian coordinates. Labels of σ± are in
reference to the polarization seen by the atom. In the frame propagating with each
beam, the polarizations for opposing beams is identical (see Section 2.4).
relationship. This Hamiltonian is
~∑[ ]
ĤAF = Ω
∗Σ̂ + Ω Σ̂†q q q q . (5.11)2
q
The optical field for the MOT is made of six lasers as described in Chapter I. A single
beam has the form of Equation 2.14 with the E~+ component (in the rotating atom
frame) as
E0̂
E~+ (~r) = e−iφ i
~
e k·~r, (5.12)
2
where ~k is the beam’s propagation direction, ̂ is the beam polarization, and φ is the
beam phase. In the MOT, the beam polarizations are circular and their direction
is closely linked the magnetic field along an axis, as discussed in Section 2.5.1. The
required polarizations are given in Table 5.2 in terms of their circular polarizations.
These polarizations are defined so that atoms located along the beam axis feels a net
force pushing them toward the origin.
120
The net electric field E~T is the sum of the individual fields
∑
E~+T (~r) = E
~+
i (~r) , (5.13)
i
where the sum is over each MOT beam present in the trap. To sum the fields, the
polarizations need to be written in a common (lab-based) Cartesian basis, which are
the polarization vectors listed in the right column of Table 5.2. This gives the field in
the cartesian basis, which then needs to changed to the linear and circular basis as to
write the field Hamiltonian as in Equation 5.11. In this basis, the Rabi frequencies
are
   
 Ωq=−1   −√
1 √i 0
   2 2 
~   〈Jg = 1/2|d|J3 = 3/2〉Ω  

~
z =  Ωq=0  = ~  0 0 1 ET , (5.14)   
Ω √1 iq=+1 √ 02 2
where 〈Jg = 1/2|d|J3 = 3/2〉 is the D2 dipole transition matrix element for 87Rb [62].
This basis has the angular momentum quantization axis in the z-direction. To
change to a basis where the quantization axis is along the magnetic field direction, the
angular momentum vector must be rotated. From Rose [126], rotating an operator
that changes total angular momentum by 1 to a basis with a different quantization
121
axis is done with a rotation operator4 whose matrix elements are
[ √
x
(1) −imγ −im′
∑
α (−1) (1 +m)!(1−m)!(1 +m′)!(1−m′)!Dm′,m(αβγ) = e e − ′ − − ′ − ×(1 m x)!(1 +m x)!(x+m m)!x!
x
( )2+m−m′−2x( ) ′ ]m −m+2x
β β
cos − sin , (5.15)
2 2
where the angles α, β, and γ correspond to the standard Euler angles. Because only
the direction of the z-axis is important to the rotation (the orientation of the x- and
y- axis does not matter), we have γ = 0. With possible values m = −1, 0, 1, the
(correctly indexed) rotation matrix is the 3× 3 matrix :
 
1eiα (1 + cos β) −√1 sin β
1e−iα (1− cos β)
 2 2 2 



 
R(α, β) =  √1 eiα sin β cos β −√
1 e−iα sin β  . (5.16) 2 2  


1eiα (1− cos β) √1 sin β 1e−iα (1 + cos β)
2 2 2
This rotation matrix is identical to an operation that converts the circular basis
polarizations to the Cartesian basis, rotates the Cartesian vectors with classical
rotations around the z-axis by (polar) angle α and then around the y-axis by
(azimuthal) angle β, then converts back to the circular basis.
For each position in the MOT, the magnetic field has spherical coordinate angles
φB and θB (corresponding to azimuthal and polar angles respectively). With the
4This operator, based on historic derivation by Wigner, has indices ordered different from common
∑matrix defintions. So the matrix elements are for the rotation operator, R, given by R|1,m〉 =
(1)
Dm′,m|1,m′〉.
m′
122
atom aligned to the magnetic field, it “sees” an electric field with polarization vector
Ω~ B = R (φB, θ ) Ω~B z (5.17)
The circular and linear components of this vector are the Rabi frequencies Ωq used
for the atom-field coupling Hamiltonian 5.11.
5.2 Matching Simulation to Experiments
Because of a lingering mismatch between experimental measurements and
numeric results, the simulated MOT has had a large number of additional features
included to better approximate the experiment. Some of these are discussed in brief
here.
5.2.1 MOT Beam Power
The most obvious issue present in the MOT beams is imbalance in beam power,
as noted in Section 3.2.2. This is corrected easily by providing each MOT beam its
own field strength E0 in Equation 5.12. In the simulation, a global laser beam power
P (in milliwatts) is defined and each beam has a power ratio factor ri relative to this
power. With these, the field strength for each MOT beam is
√
Ei
2 (Pri) 20
0 = × , (5.18)πw2 0c
where the first term is the central intensity for a Gaussian beam and the second term
relates beam intensity to field strength [48]. In this equation, w is the MOT beam
123
waist (measured in cm) and the second term has an extra factor of 10 to convert from
lab-measurement-units for beam intensity in mW/cm2 to MKS units.
In our experiment, the MOT beams have a gaussian beam (intensity) profile
rather than being pure plane waves as written above. The beam waist (measured
as the 1/e2 power radius) is 0.33 mm. For a single atom that remains within tens
of microns of the MOT beams, the field intensity seen by the atom should be fairly
uniform and close to the peak intensity. However, there is a small change in intensity
which can be taken into account. For an atom located at ~r and a MOT beam
propagating in the direction ~k and originating from position ~b0, the (square of the)
distance from the axis of the beam and the atomic position is
2 ∣∣∣ ~ − ∣
∣2 [( ) ]2
d = λb0 ~r∣ − λ~b0 − ~r · ~k , (5.19)
where λ is the laser wavelength. This has been calculated by looking at two points
along the MOT beam separated by one wavelength and basic point-line distance
formulae. With this, the atom seems a field strength of
i → i −d2/w2E0 E0e . (5.20)
Note that this is the correct formula as the beam waist is the 1/e2 radius (rather than
its variance) and the field strength is proportional to the square root of the intensity.
MOT beam power measurements were done outside of vacuum (obviously), but
the experimental cell was not anti-reflection coated. As such, the horizontal and
vertical MOT beams inside the cell will have different powers. As seen in the vacuum
system in Figure 3.1, the vertical beams enter the cell at nearly normal incidence,
so that the beam is almost entirely polarized in the x-y plane, which is S-polarized
124
to the surface. However, if this beam is not exactly vertical, light polarized along
the z-direction is P-polarized. For the horizontal beams (which enter the cell at
nearly 45◦), the light polarization component in the x-y plane is P-polarized and
light polarizaed along the z-direction is S-polarized. For the appropriate wavelength
and material, the field strength for each (Cartesian) polarization direction is reduced
√ √
by a factor of either 1− 2RS or 1−RP , where RS and RP are the S-polarization
and P-polarization reflection coefficients, respectively. The factor of 2 accounts for
reflections on the outer and inner surface of the experiment cell walls. In the case
where RS =6 RP , this will shift the beams out of purely circular polarization inside
the cell.
5.2.2 MOT Beam Direction
For an ideal MOT, the laser pairs are exactly counter-propagating and are normal
to beams in other directions. This is, of course, not the case in physical MOTs.
Instead, it is common while building a MOT to adjust the beam directions very
slightly until the MOT “looks” good—that is to say it appears approximately round
when imaged and there is a high atom number in the MOT. As discussed in great
detail in Section 5.4 below, this adjusts the interference pattern of the lasers to
minimize pathways for atoms to escape from the MOT. This is done numerically
by rotating each beam’s propagation vector slightly with a classical rotation matrix.
This rotation must also be applied to a MOT beam’s Cartesian polarization vector,
although done with an angular momentum rotation matrix as above. Based on our
experiment’s MOT beam alignment system, the misalignment of our MOT beams is
no more than about half a degree, putting an upper limit on the angular displacement
of a beam in the simulation.
125
Similarly, it is possible that an entire MOT beam is displaced from its ideal
launching position. For example, a horizontal MOT beam could be a little too low,
but angling its beam upward with a slight tilt would still have the beam strike the
center of the MOT. This could still load a MOT without much difficulty. In the
simulation then, each beam has a three dimensional positional offset vector to deal
with this. It is important to note that this only comes into account when using a
beam’s gaussian profile, where the offset is the vector~b0 discussed above. If the beams
are plane waves there is no intensity dependence on the transverse dimension. Only
a beam’s propagation direction matters, not their displacement from ideal launching
position.
5.2.3 MOT Beam Polarization
As discussed in Section 3.2.4, the experimental MOT beam polarizations are
elliptical rather than circular. How much the beams are elliptical can be found by
measuring the power (of the elliptical beam) through a polarizer. With perfectly
circular light, the power through the polarizer will be constant for all angles through
the polarizer. For elliptical light, the power will maximize at some angle and minimize
at 90◦ from that angle.
For Cartesian axes a and b, normal to the beam propagation direction, the
difference between the two powers in each polarization can be quantified as
(√ )
Pa
γ −1a = cos , (5.21)
Pa + Pb
where the angle is defined relative to the a-axis and the values Pa and Pb are the
(measured) powers of the polarization in the two directions. Note that if the beam
126
is polarized completely along the a-direction, Pb = 0 and thus γ = 0. Similarly when
the beam is linearly along the b-direction if Pa = 0 and γ = π . If the beams are
balanced then γ = π/2. Incorporating this into the polarization vectors in Table 5.2,
√
the 1/ 2 is replaced by cos γa for the polarization component the a-direction and
sin γa for the polarization the b-direction. Each of the six beams will have its own
value for γa and the direction a can be defined as either of the two directions normal
the propagation direction.
5.2.4 Magnetic Fields
It is also possible to have a more complex formula for the magnetic field. Rather
than the linearized form of Equation 5.4, a full form of the field at all positions from
anti-Helmholtz coils could be used. Additionally, a background magnetic field B~back
could be present either from the Earth, the Helmholtz coils discussed in Section 3.3.6,
miscellaneous equipment in the lab, or the lab next door. In this case, the magnetic
field is
(x y )
B~ (~r) = B′z x̂+ ŷ − zẑ +B~0, (5.22)2 2
where, as above, B′z is the field gradient along the axis of the MOT magnetic field
coils (or permanent magnets). Dealing with this magnetic field is straight forward as
its magnitude is easily calculated for use in the atom-magnetic field Hamiltonian and
the angles to use for the polarization rotation matrix in Equation 5.16 are calculated
from this field.
The experiment sits on an optical table, whose top is a large conducting slab. As
such, currents in the anti-Helmholtz coils will produce mirror images in the conductor.
While this effect is small, it could play a larger role when magnetic field is modulated.
127
Including these effects just calls for including another term in the total magnetic field
for another pair of anti-Helmholtz coils (with current in the opposite direction) whose
placement is below the surface of the table a distance equal to the height of the real
anti-Helmholtz coils above the table.
5.3 3D and 87Rb Calculations
5.3.1 Atomic Equation of Motion
Spontaneous emission is handled identically to the |Fg = 0〉 → |Fe = 0〉 case
with the appropriate forms of Σq as the lowering operators. That is, the differential
equation governing the evolution of the atomic density matrix is
d − i
[ ] [ ] [ ] [ ]
ρ = ĤA + ĤAF + Ĥz, ρ +ΓL Σ̂q=−1 ρ+ΓL Σ̂q=0 ρ+ΓL Σ̂q=+1 ρ (5.23)
dt ~
where the Hamiltonians are defined in Equations 5.1, 5.8, and 5.11, and the Lindblad
superoperator was defined in Equation 2.6.
With the 24 magnetic sublevels of the D2 transition for
87Rb, the density matrix
is 24× 24, probably not analytically solvable in 3D, but it can be done numerically.
We implement this by reforming the density matrix into a 24 × 1 vector ρv and the
Hamiltonian and Lindblad superoperators into an appropriate 24 × 24 matrix M.
With these, the equation of motion simply becomes
d
ρ =Mρv. (5.24)
dt
Steady state solutions for this equation are found via LU decomponsition to invert
the matrix M [127]. This is implemented through the LAPACK library [128].
128
The populations for the MOT trapping transition excited states are plotted in
Figure 5.1a as a function of position from the origin in a 1D MOT. The preference
for populating the outer most Zeeman sublevel is a result of these levels having only
a single ground state to which they can decay. The growth (or much smaller decay)
of the states with me > 0 states result from these states having their energy levels
red shifted (recall B(z > 0) < 0 as shown in Figure 2.5). The rapid growth (or decay
for me < 0) of populations in the outer most magnetic sublevels result from their
having the largest Zeeman shifts, which most quickly moves these levels into (or out
of) resonance with the electric fields. Additionally, as seen in other mutli-level atoms
with ground state Zeeman shifts, at small magnetic detunings a two-photon process
couples neighboring ground states [64].
As a test for success of the numeric simulation, the atom can be returned to the
|Fg = 0〉 → |Fe = 1〉 model and the steady state populations were identical to the
analytic results in Appendix A. For comparison between these two atomic models,
the (MOT trapping) excited energy level steady state populations in a 1D MOT for
the V-atom and the full rubidium atom are plotted in Figure 5.1b. For the rubidium
atoms, the populations are summed over Zeeman sublevels for clarity in the graph.
5.3.2 Force Formalism
The force on the atom is calculated exactly as in Equation 2.29. The effect on
the magnetic field Hamiltonian is exactly
∇~ ĤzĤz = [xx̂+ yŷ + 4zẑ] . (5.25)
x2 + y2 + 4z2
129
 0.1  0.12
 0.09 (a) (b)
 0.1
 0.08
 0.07 |me = +3 i  0.08
 0.06 |me = +2 i
|me = +1 i me > 0 V-atom 0.05
| i  0.06m = 0 me = 0 87e Rb
 0.04 |me = 1 i me < 0
 0.04
 0.03 |me = 2 i
|me = 3 i
 0.02
 0.02
 0.01
 0  0
 0  10  20  30  40  50  0  10  20  30  40  50
Position, μm Position, μm
FIGURE 5.1. Excited state populations for 87Rb and V-atom. Calculation done
for a 1D MOT along the +z-axis. (a) Populations for 87Rb excited states. (b)
Populations for both 87Rb and V-atom model. The populations for 87Rb are summed
over magnetic sub-levels for clarity.
Noting that along the z-axis, the strength of Hz is twice as large as along the x- or
y-axis, the force that results from this should also be twice as strong along the z-axis.
This is expected as the magnetic field gradient is twice as large along the z-axis. If
the system is treated as having no background magnetic field, this equation for the
gradient of the atom-magnetic field Hamiltonian does not change.
The effect on the atom-field Hamiltonian follows the derivation of the optical
molasses force. The gradient of Hamiltonian becomes just a gradient of the electric
field propagation term e−i
~k·~r, adding a factor −i~k to each term in the sum of the
total electric field in Equation 5.13. Because the electric field is already a vector,
the gradient here is a tensor which is most clear when written in terms of individual
directions. For the (pre-rotated) Rabi frequency vector of Equation 5.14, the gradient
130
Excited State Populations
Excited State Populations
in the ` direction is
 
1
 −√ √
i 0
2 2
∂ 〈Jg = 1/2|d|

J3 = 3/2〉  ( )
Ω~ = 
∂` z ~  0 0 1
 −ikE~+` + ikE~−` . (5.26) 
√1 √i 0
2 2
This vector must then be rotated to the atomic dipole reference frame. In this view,
the polarization rotation matrix is assumed to not change with position to first order.
However, because the magnetic field changes with position, its direction will also
change with position. This would change the rotation matrix as well, but this effect
is ignored to first order, when the change in field orientation is small with respect to
the change in the field polarization.
5.3.3 Steady State Force
To find the force in the s-direction for the atom located at position ~r, calculate
the steady state of the atomic density matrix, ρss(~r), as above. The gradient of the
Hamiltonian in the `-direction is then calculated and is given by
∂ ∂ ∂
Ĥ = Ĥz +R (φB, θ ~B) Ω`. (5.27)
∂` ∂` ∂`
Calculating the expectation value of the force is done by tracing this operator over
the steady state density matrix:
[( ) ]
〈 ∂F ss`〉 = Tr Ĥ ρ . (5.28)
∂`
131
The form of Equation 5.27 does include both the purely magnetic trapping of Section
2.3.1 and the MOT magnetic confinement of Section 2.5.1. The magnetic trapping
derived from the gradient of the magnetic Hamiltonian and the magnetic confinement
derived from the gradient of the atom-field Hamiltonian. Both derive their the spatial
dependence from the spacial dependence of ρss.
For comparison, a one-dimensional force in steady state for the full 87Rb atom
and the V-atom is plotted in Figure 5.2 both with a trapping field gradient of 242G/cm
(a current of 9A in our magnetic field coils). A few features are clear. First, near the
center of the trap, the force on the full 87Rb atom is larger. A linear fit of the force
full rubidium force data near z = 0 gives a force of F~ (z) = − [3.68× 10−16Rb N/m] zẑ,
an order of magnitude larger than the V-atom force F~V-at(z) = − [4.3× 10−17N/m] zẑ
with the balanced restoring constant from Equation 2.84.
This enhancement to the trapping force is a result of having multiple ground state
magnetic sublevels for the full atom [48]. For ground state sublevels where the energy
is redshifted due to the magnetic field, a two-photon absorption process can couple
neighboring ground state levels with slightly different detunings from resonance [64].
The effect is to enhance coupling to the MOT beam that the atom is closer to, leading
to a larger force from this beam5. This effect “turns off” at larger Zeeman shifts when
the ground state magnetic sublevels shift further apart in energy. This gives rise to a
second feature, the change in slope of the force for the full 87Rb atom around 20 µm
for these parameters. In general, the width of the two-photon feature is
AsΓ2~
δz = ′ | | (5.29)4µBgFgBz ∆
5For slowly moving atoms, a similar effect occurs in momentum-space (with Doppler shifts playing
a similar role to the detuning) and is the polarization gradient cooling discussed in Section 2.5.3
[72].
132
 0.25
4 040
 0.2 3 030
2 020
 0.15 1 010
0 0
 0.1 -1-010
-2-020
 0.05 -3-030
-4-040
-4-400 --2200  0  20  40Positi0on, um 20 40
 0 Position, μm
-0.05
-0.1
-0.15
-0.2
-0.25
-3 -2 -1  0 "  1 #  2  3
PositPioosnit,io un,n sictasl eodf ~ |L|z µ g @BB F @z
FIGURE 5.2. 1D forces on 87Rb and V-atom. Red curves show V-atom results, blue
curves show Full 87Rb results. Inset shows forces in scaled to region near where the
atom in the MOT is. Axes for main graph and inset are scaled identically to figures
2.4(a) and (c) respectively.
133
ForcFeor,c eu,n sictasle odFf [~k]
Force, 10-22 N
Force, 10^-22
where s is the saturation parameter defined in Equation 2.38 and A is a numerical
factor that depends on the structure of the atom [64].
The final feature is a larger trapping region for the complete 87Rb atom. This
results from the smaller Zeeman shifts of the inner magnetic sublevels. Once magnetic
field grows to a large enough magnitude, the |me = ±3〉 energy levels are blue shifted
out of resonance with the MOT lasers. However, the |me < 3〉 energy levels are still
red-detuned of the electric field, letting them continue to interact with the beams.
This extends the region where trapping can occur.
5.3.4 Velocity
The atomic velocity is not taken into account in our calculation, meaning that
there is no doppler shifting of the MOT beams. This could be done, and requires
restructuring the atomic hamiltonian of Equation 5.1, as each MOT beam will have a
different detuning due to the atomic motion. recalculation as the detunings in must
written separately for each beam. Without including velocity in the calculation, there
are also no sub-doppler effects that may arise. The sub-Doppler effects on the atom
can be taken into account simply by limiting the maximum energy (temperature) of
the atom when calculating the probability distribution from the potential energy of
the atom.
5.4 Escape Channels
Even the results graphed in Figure 5.1 is relatively simple as it represents a just
a 1D system. Expanding to three dimensions as laid out above and can lead to an
134
 60
(a)
 40
 20
 0
-20
-40
-60
 0  2  4  6  8  10  12  14
 400 (b) r, µm
 200
 0
-200
-400
 0  2  4  6  8  10  12  14
r, µm
FIGURE 5.3. 3D MOT forces for an assortment of beam phases. Calculation done
for a 3D MOT along the +z-axis. Figure (a) shows the forces and figure (b) shows the
resulting potential energy from these forces. The potential is found by numerically
integrating the force along the positive z-axis.
unanticipated result. Figure 5.3 shows (a) the force and (b) the potential energy along
the positive z-axis for a random selection of phases for the six 3-D MOT beams.
From the figure, it is clear that different arrangements of the MOT beam phases
can greatly change the force on an atom. Mostly clearly, unlike in the 1D case, the
force oscillates. This is a result of interference of the MOT beams creating an optical
lattice inside the MOT (see the brief discussion in Section 1.3). Additionally, from
the potential energy graphs, there are some phase arrangements where the force on
an atom pushes it from the center of the trap. These are cases where the potential
energies decrease as the atom travels outward from the center of the trap (the green
135
Potential Energy, µK
Force, 10-22 N
and red graphs). While not shown, in some of these cases the potential energy does
eventually “turn around” becoming positive again. In such cases, this effectively just
moves the “center” of the MOT away from the location where the magnetic field
vanishes. This is similar to the beam imbalance or background magnetic field as
discussed in Chapter II, both which also displace the center of the MOT.
In other cases, the potential does not “turn around” but instead continues to
decrease. In these cases, atoms which found themselves along these paths could
potentially escape from the trap. While shown below for just along the positive z-
axis, these paths can arise in many directions and typically form narrow channels
along which the force is not restorative. These channels are narrow, on the order of
the light wavelength, and they grow wider further from the MOT center. Such non-
restorative forces in MOTs [92, 129] have been observed before. In optical lattices,
similar non-cooling forces can appear in momentum space [130]
The origin of these channels can be seen by more carefully consider the electric
fields in the MOT. First, consider only the two z-beams. These beams have (positive-
rotating) fields, as defined in Equation 5.12,
   
 −1  
1
 
~ (+) E√0  −  −iφz− +ikz ~ (+)Ez− =  i  e e and E E√0 2 2 z+ = 2 2  −  e
−iφ
i z+e
−ikz. (5.30)
   
0 0
136
Note that the field sub-scripts refer to the direction the beams come from, no the
direction the beams propagate in. The total field is then
 
−e−iφ +ikz −iφ z−e + e z+e
−ikz
 E0  ~ (+)E = √  −ie−iφz−e+ikz − ie−iφz+e−ikz T ,2 2  
0
and, with Equation 5.14, the Rabi frequency is
 
e−iφz−e+ikz
〈  ~ d〉E0 Ω =  0 

z .
2~  
e−iφz+e−ikz
Taking a position on the negative z-axis where B~ ||ẑ (for simplicity), the rotation
matrix of Equation 5.16 is the identity, R (φBθB) = 1, so that Ω~ = Ω~ z. The Rabi
frequency vector has magnitude
 
∣ ∣ 〈 〉 
1
∣ ∣ d E0  

∣Ω~ ∣ =  0  , (5.31)2~  
1
as is expected. The electric field (made by the two beams) is equal parts σ− and σ+
light. This is the normal assumption made when working with a 1-D MOT. When
looking at forces, we need the negative gradient of each component of Ω:
 ( ) 
−ik e−iφ z−e
+ikz

−∇~ ~ 〈d〉E0
 
Ω =  0  ẑ.2~  ( ) 
+ik e−iφz+e−ikz
137
Since Ωq=−1 is associated with σ+ polarization, this equation suggests σ+ light is
responsible for forces in the negative-z direction. This is exactly is what was expected
as defined in Table 5.2. Similarly, σ− light (associated with Ωq=+1) is responsible for
forces in the positive-z direction. Optical molasses, discussed in Section 2.4.1, takes
advantage of these different force directions by increasing coupling to the beam that
is counter-propagating to atomic motion in order to cool the atom. MOTs take
advantage of this by using Zeeman shifts to increase coupling to the beam which will
push the atom towards the center of the trap. The gradient equation above is exactly
the situation plotted in Figure 5.2.
The phase dependence can be seen by adding just a single additional MOT beam.
Say a beam propagating in the −x direction (from the +x direction). This beam has
field
 
 0 
~ (+) E0Ex+ = √ 
2 2  
 e−iφx+e−ikxi .

−1
With the two z-beams in Equation 5.30, the total field is
 
−e−iφz−e+ikz + e
−iφz+e−ikz
E0 

~ (+)E = √  −ie−iφz−e+ikz − ie−iφ
−ikx
z+e−ikz + ie−iφx+eT
2 2  
 .

−e−iφx+e−ikx
138
The Rabi frequency vector (with the magnetic field along z still) is
 
e−iφz−e+ikz −
1e−iφx+e−ikx
2
〈 〉  d E0
Ω~ = 

~ −√
1 e−iφx+e+ikx 
2  2 
.
e−iφz+e−ikz − 1e−iφx+e−ikx
2
Looking at this vector in terms of forces, we have gradient
 ( )   ( ) 
+ik e−iφx+e−ikx   −ik e
−iφz−e+ikz
〈d〉E0  √ ( )  〈d〉

E  −∇~ Ω~ =  −i 2k e−iφ −ikx 
0
x+e  x̂+  0  ẑ (5.32)4~  ( )  2~  ( ) 
+ik e−iφx+e−ikx +ik e−iφz+e−ikz
The terms associated with forces in the z-direction do not change, which is good.
The z-forces should still be controlled by the two z-beams. For the x-direction, forces
arise from all three polarizations. This is to be expected as the field in the x-direction
beam has has a component of each polarization (in the circular basis). Note that the
circular polarization components result in forces in the positive x-direction—opposite
the propagation direction of the beam.
From the gradient, it appears thethat force in the z-direction should be the
same as when there is no off-axis MOT beams. However, because Ω~ is different,
the steady-state atomic density matrix will be different. Thus the average force, as
calculated from Equation 5.28, will be different. From Equations 2.27, A.1 and A.2,
the excited state populations are generally proportional to the square of the exciting
Rabi frequency component (just |Ω|2 for the two-level atom and |Ω |2and |Ω |2− + for
the two excited states in the V-atom). Assuming this generally holds for the more
139
complex multi-level atom,6 we can look at the square of the Rabi vector:
 
5
∣ ∣  + cos [kz − kx− (φz− + φ4 x+)] 
∣∣~ ∣∣
2 〈d〉2E2  
Ω = 0 
~2  1/2
 . (5.33)
4  
5 + cos [kz + kx+ (φ
4 z+
+ φx+)]
The linear component has a fixed value, but the two circular polarizations depend on
position as well as the phase relationships between the various beams. This leads to
a complicated force in the x- and z-direction as a function of position. Of specific
interest, are locations where the oscillating terms have larger negative values. In
these cases, the σ+ or σ− components can be less than their associated values in
the two-beam case above, Equation 5.31. This would tend to decrease the excited
state populations, and with the same form for the z-component of −∇~ Ω, and would
reduce the overall force in the z-direction. Under some phase arrangements, this could
change the sign of the force, resulting in a force that pushes the atom away from the
center of the MOT, as seen in Figure 5.3.
As an illustrative example, the specific case for φz− = φz+ = φx+ = 0 is shown
in Figure 5.4. In this figure, the graphs for both the 1D two-vertical beam MOT and
the 3-beam field arrangement just discussed is shown. Graphs (a) and (b), showing
the (normalized) polarization components and the Rabi frequency magnitudes, agree
with the calculations in Equation 5.33, where in all four sets of graphs the σ− and
σ+ components are equal. In graph (d), which shows the components of the force on
the atom, the total force in the x-direction is always negative as is expected because
the beam is propagating in this direction.
6Looking at Equation 5.11, the average force is a sum over 〈Σ̂q〉ss, weighted by the gradient of
the appropriate component of Ω~ . For the analytically solved atoms, 〈Σ̂ 〉 ssq ss = ρqq.
140
From graph (c), which shows the expectation values of the atomic lowering
operators, the Σq=0 lowering operator always has a larger expectation value than
either the Σq=−1 and Σq=−1 lowering operators. Compared to the forces in graph
(d), 〈Σq=0〉 is minimized at positions where Fx = 0. These are also locations where
the light is split evenly between circular and linear polarizations, as shown in graphs
(a). This balance and cancel the two directions for the x-direction force as noted in
Equation 5.32.
Because the two vertical beams essentially form a one-dimensional MOT, the
force is (nearly) linear as expected and results in a quadratic potential. However,
the addition of the third beam produces strong oscillations in z-direction force.The
z-direction force does go to zero where 〈Σq=−1〉 = 〈Σq=+1〉 (graph (c)), suggesting
the atom experiences forces from the counter-propagating σ+ and σ− beams equally.
While not clear from the z-direction force itself, the potential curve (calculated just
from integrating the force) does show MOT trapping as from the 1D case, overlaid
with deeper potential energy wells associated with the spatial polarization oscillations
as shown in graphs (a). Detangling these two potentials is necessary for estimating
the MOT temperature.
Adding additional beams further complicates the equation for Ω. The complex
position relationship is not of specific interest as calculations for the temperature will
integrate over position. However, the phase relationship between the six beams as
evident above will greatly change the Rabi frequencies, the steady state populations
and the force on the atom. This does includes situations where the overall force
is away from the center of the MOT as shown in Figure 5.3. In these cases, the
polarization arrangement creates a state where one of the 〈Σ〉q values moves the
atom away from the center of the MOT. For example, for the beam arrangement
141
Two vertical beams Two vertical beams & negative-X beam
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1/4
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0
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 0  2  4  6  8  10  12  14  0  2  4  6  8  10  12  14
r, µm r, µm
(b) 1.0
0.8
0.6
0.4
0.2
|⌦+| |⌦0| |⌦|
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0.20 r, µm r, µm
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0.15
0.10
0.05
0
 0  2  4  6  8  10  12  14  0  2  4  6  8  10  12  14
x10-24 x10-22(d) 60 r, µm r, µm
Fx    Fy   Fz
30
0
-30
-60
 0  2  4  6  8  10  12  14  0  2  4  6  8  10  12  14
(e) 50 r, µm r, µm
40
30
20
10
0
 0  2  4  6  8  10  12  14  0  2  4  6  8  10  12  14
r, µm r, µm
FIGURE 5.4. 3D MOT phase dependence. The left graphs show results for the
MOT with just two vertical beams. The right graphs show the results for the MOT
with an additional beam in the negative x-direction. In all graphs, the ho√rizontal
axis is positions along the positive-z axis. Calculations done with Ω = Γ/ 2 and
|B′z| = 1 G/cm. Graphs (a) and (b) show the normalized polarization components
and Rabi frequency amplitude, respectively for the three circular polarizations as
defined in equations 2.55. Figure (c) shows the three lowering operators’ (Equation
5.9) expectation values in steady state. Figure (d) shows the force in the x-, y- and
z-direction. Note that the two graphs have different scalings. Figure (e) shows the
potential energy along the positive-z axis.
142
Lowering Operator
 Rabi Frequency Magnitude, 

Potential Energy, μK Scaled Force, Newtons Expectation Value relative to Γ Normalized Polarizations
leading to Equation 5.32 if the beam phases (and possibly the MOT Zeeman shifts)
are such that when z > 0 we have 〈Σ̂−〉ss > 〈Σ̂+〉ss, then the resulting force along
the z-direction would push the atom towards in the positive-z direction—away from
the center of the MOT. These phase arrangements are the ones which lead to the
anti-trapping forces in Figure 5.3.
5.5 Recovering Potential
In addition to anti-trapping, the force curves shown in Figure 5.3 have an
additional problem. The forces, in general, are non-conservative. Integrating the
force along another path (besides along the z-axis as calculated in Figure 5.3b) result
in different values for the potential energy. In such a circumstance, it is impossible
to define a potential energy of the form
F~ = −∇~ U. (5.34)
Rectifying this can be done one of three ways. First, the Helmholtz theorem states
any vector field, ~v (~r), can be written as
~v (~r) = ~v|| (~r) + ~v⊥ (~r) , (5.35)
where ∇~ × ~v|| = 0 and ∇~ · ~v⊥ (~r) = 0. This first irrotational (curl-less) term is
conservative, so that applying the Helmholtz theorem to the force we can define a
potential as ∫
U = − F~ || (~r) · d~r. (5.36)
143
The α component of irrotational force can be computed directly frosm the full force
[48] as
∫ [ ∫ ]
|| 3 ′ 1 3 kαkβ ~ ~′Fα (~r) = d r d k e
ik·(~r−r ) F ~′β(r ).
(2π)3 k2
In a more numerically accessible form, this is
[ ]
|| −1 kαFα (~r) = F B , (5.37)k2
where
( )
B = ~k · F̃ ~k ,
and where
( )
F̃ ~k = F [Fx (~r)] x̂+ F [Fy (~r)] ŷ + F [Fz (~r)] ẑ
is the Fourier transform of the force field.
These two transforms, to find the fourier transforme of the force and the inverse
transform to get the irrotational component, are three-dimensional. This requires
numeric solutions for the force over a large grid that encompasses the region in which
the atom exists. The region should be at least large enough such that the probability
for an atom to be outside of the grid is small. Similarly, the position-space spacing of
the grid should be smaller than the wavelength of the lasers to allow for oscillations in
the results. For a fair sized MOT (approximate 50 µm in diameter) and square grid
spacing λ/10, this is about 263 million grid points, each which has 3 components to the
force. While the calculation for the irrotational force is not challenging, calculating
the atomic density matrix in steady state at each point is numerically intensive (but
definitely possible).
144
The second approach, much simpler numerically, is to calculate the force in a
smaller region, or even in one dimension, for many phases of the six MOT beams.
The forces are averaged together before calculating the potential energy. As our
experimental lasers are not phase controlled, drift in phase, and potentially even
change beam direction, an atom will see a variety of phase relationships between the
beams. Averaging data over many data runs should approximate the numeric phase
averaging.
The third approach, which we have not investigated numerically but mimics
the experimental process to make “good” MOTs discussed above, is to adjust beam
direction and phases until the escape channels vanish in the simulated results.
145
CHAPTER VI
POSITION AND TEMPERATURE MEASUREMENTS
Soon after loading a single atom MOT with our water-cooled electromagnetic
anti-Helmholtz coils, pure curiosity lead us to check on the power spectrum of
photons collected from our single atom. A very clear peak at around 21 kHz was
visible and, soon after, an oscillation of the same frequency was seen in the anti-
Helmholtz coil current. While it isn’t revolutionary that an oscillation in one property
(magnetic field strength) results in oscillations in properties that depend on it (atomic
fluorescence rates through the Zeeman-shifted detuning), because the magnetic field
strength varies with position, the strength of the field oscillation also has a positional
dependence. This should encode some atomic position information about into the
fluorescence oscillation strength, allowing us to learn about motion of the atom in the
MOT, as well as it is temperature, from the light that it emits.
The theory and data analysis technique are laid out at the beginning of the
chapter. Following that, measurements of the atomic fluorescence oscillation in a
variety of experimental contexts are shown and discussed. To close the chapter, two
interesting additional effects are examined.
6.1 Theory
Following Equation 2.28, the rate that a multilevel atom scatters photons is
∑
R = Γ ρei,e , (6.1)i
i
where the sum is over all of the possible excited states of the atom. These excited
state populations depend strongly on the detuning of the electric field from atomic
146
resonance. This is directly evident in the steady state equations for the two-level atom
2.27 or the V-atom excited states A.1 and A.2, which all depend on the detuning from
resonance. For demonstration, the two-level population and the single-excitation
field excited-state population for the V-atom (Equation 2.61), can be written in a
generalized form as
∣∣ ∣∣2∣Ω̃∣
ρe,e = ∣∣ ∣∣2 (6.2)
1 + 4δ2 + 2 ∣Ω̃∣
where δi and Ω̃i are the detuning from resonance and Rabi frequency for light coupled
to the excited state, both scaled by the spontaneous decay rate Γ. Obviously, a smaller
detuning from resonance results in a higher population. As was done in Chapter II,
the detuning can be written in terms of the laser detuning plus a Zeeman shift (see
Equations 2.76) as
δi = δL,i +miδB,
where mi is the magnetic quantum number for the excited state, and again the
detunings are scaled by Γ, e.g. δ ≡ ∆/Γ. When the Zeeman detuning δB is much
smaller than the laser detuning δL,i, which is true for an atoms in a MOT as discussed
in Section 2.5, the population can be expanded in terms of small Zeeman detuning.
In one dimension, with Equation 2.75 defining the Zeeman shift’s frequency, thus
becomes
∣∣ ∣∣2 ∣∣ ∣ | | ∣∣ ∣
∣2
Ω̃i 8µBgFmi δL,i Ω̃i∣
ρei,e =i ∣∣ ∣∣2 + ( ∣ ∣ ) B
′
zz. (6.3)
2 ∣ ∣ ∣ ∣2
2
1 + 4δL,i + 2 Ω̃i Γ~ 1 + 4δ2L,i + 2 ∣Ω̃i∣
Interestingly, this form shows a clear breakdown of treating the atom in a MOT
as either a two level atom or as the extended two-level atom described in Equation
2.4.1. For the two level atom’s single excited state, m = 0 and there is no Zeeman
147
shift, and hence no position/magnetic field dependence in the fluorescence. For the
extended two level atom, m± = ±1 and when summing over both states to get a
total scatter rate, the position and magnetic field dependence vanishes due to the
linear dependence on mi. Of course, an expansion to order δ
2
B would return the
position dependence as terms of m2i would appear. Checking for linear or quadratic
dependence of the fluorescence rate on the position/magnetic field would inform us
as to if the extended-two level atom is a fair model for out experiment (hint: it is
not, see below Section 6.3.1).
6.1.1 Fluorescence Oscillations
In any case, the magnetic field dependence of the excited state population is clear
from Equation 6.3. As noted above, in a MOT, this dependence is weak and unlikely
to be detectable in our MOT without much improved efficiency in photon collection.
However, introducing oscillations, as done by accident above, should clearly reveal
the dependence on the magnetic field. In one dimension again, an oscillation in the
magnetic field of the MOT can be written as
B~ (z, t) = −B′z(1 +  cos(2πft))z (6.4)
where  is typically a small value,  . 0.15. To examine this analytically, we will
turn to the V-atom model in Section 2.4 which had steady state populations7 given
by Equations A.1, A.2 , and A.3. With the magnetic field as defined above, the
7While it is true that the magnetic field is changing in time, steady state populations can still
be used as the timescale for the internal evolution of the atom to the steady state is very different
than the magnetic field oscillation period. The internal atomic dynamics timescale on the order of
1/Γ = 27 ns [48] while the timescale for the field oscillations is on the order of 100 µs or longer
(frequencies for driving oscillations in the experiment are typically on the order of 100 Hz to 1 kHz).
With the much longer time scale for the magnetic field, the atom basically sees a constant field
148
magnetic field detuning frequency (scaled by atomic decay rate) is
µB
δB(t) = B
′
zz [1 +  cos(2πft)] (6.5)~Γ
(see the derivation of Equation 2.59). Here, we have imposed |m±| = 1 and positive
or negative shifts in frequency for the different excited state levels are written as
δ±(t) = δL ± δB(t). With this, the fluorescence is (by expansion around small )
F ≈ 〈Fl〉+ dF cos(2πft), (6.6)
with
( ∣ ∣ )
〈Fl〉 = Γ ρss ∣ + ρss ∣+,+ −,− and (6.7)
( ∣=0 =0
∂ρss ss
∣ ) ′
+,+ ∣ ∂ρ−,− ∣ µBB z
dF = ∣ z
∂δ ∣ +
∣∣ . (6.8)
B =0 ∂δB =0 ~
Thus, the fluorescence oscillates with the same frequency and (unwritten) phase as the
magnetic field, around an average value that matches the non-oscillating field value,
and with an amplitude that is proportional to both position/magnetic field and the
driving frequency. While the values for m± have been suppressed here, it is clear
from the form of δB that the linearity of dF with respect to  requires linearity with
respect to m±. Then, when experimentally measuring values for dF , linear scaling
with respect to  will rule out the extended two-level atom model as noted above.
while evolving to steady state and we can assume the atom is always in its steady state value for
the magnetic field at time t.
149
It is not particularly enlightening, but the derivatives of the excited state
populations for the V-atom are given by
[ ]
∂ρss− − 1 ∂N, = ρ′ − ρss
∂δB N [ −− −−∂δB ] (6.9)
∂ρss 1 ∂N+,+ = ρ′ − ρss ,
∂δB N ++ ++∂δB
with
′ s(1− w) { [ ] }ρ−− = 16s (2δB,0 + δ 2L) + 64δB,0 1 + 4δ2 B,0 + 32δL [1 + 4δB,0 (3δB,0 + δL)]
′ s(1 + w) { [ ] }ρ++ = 16s (2δ 2B,0 − δL) + 64δB,0 1 + 4δB,0 − 32δL [1 + 4δB,0 (3δB,0 − δL)]2
∂N ( ) [ ] [ ]
= 8δB,0 2 + s+ 8δ
2 2 2
∂δ B,0
4 + 5s+ 16δB,0 − 24swδL 2 + s+ 24δB,0 +
B [ ]
128δ2LδB,0 s− 16δ2B,0 − 64δ3L [sw − 8δLδB,0]
where δB,0 is the scaled magnetic field detuning (Equation 6.5) with  = 0. In these
equations, the two Rabi frequencies have been written in terms of beam imbalance
ratio w (as defined in Equation 3.1), saturation parameter s as defined in Equation
2.38, and here s is calculated with the average power of the two beams). In these
forms, the (decay rate scaled) Rabi frequencies are
∣∣ ∣2∣ ∣Ω̃±∣
s
= (1± w) , (6.10)
2
which simplify the form of the steady state equations greatly.
There is another piece to note briefly here. The magnetic field underlies the
mechanism that traps atoms in the MOT through a harmonic restoring force described
in detail in Section 2.5.1. The restoring force strength κ, as shown in Equation 2.84,
is proportional to the magnetic field gradient. Modulating the field gradient through
150
the current in the anti-Helmholtz coils then also modulates the value for κ. In doing
so, it is possible to excite an additional resonance in the atomic motion, called a
parametric resonance [131]. With a well damped atom, (i.e. large damping constant
β), a small oscillation parameter , and driving the oscillations far from resonance (or
rather far from twice the resonant frequency, see Appendix F), the influence of the
field oscillations on the atomic motion is negligible and our theoretical framework is
still valid. These resonances are discussed in much more detail in Section 6.5.
6.1.2 Position Averaging
The form of the equation for the fluorescence oscillation amplitude dF shows
a dependence on position that is difficult to get at experimentally. Instead, it may
be beneficial to look at spatially averaged values for this amplitude based on the
temperature and potential energy seen by the atom. Following the damped harmonic
oscillator formalism of Section 2.5.2, this potential will be
1
U(z) = κz2, (6.11)
2
where κ is the restoring force, given for the V-atom by Equation 2.84.
From this equation, it is clear that modulating the trapping strength will oscillate
the potential energy of the atom. In doing so, it is possible to heat the atom and thus
expand grow the position distribution for the atom [39]. At high frequencies (relative
to a characteristic frequency fc = κ/2πβ, where β is the damping coefficient for that
atomic motion), the oscillations are too fast for the atom to respond and it experiences
the average non-modulated trap. At low frequencies, the particle’s position variance
−1/2
grows by a factor of (1− 2) above its non-modulated value. The characteristic
151
frequency, with the form of the trapping strength of Equation 2.83 is
µ g B′B F
f = zc ,
2π~k
where the only experimentally controlled value is the MOT magnetic field gradient B′z.
For our large field MOTs this frequency is on the order of tens of kHz. Experiments
are typically modulated at frequencies of hundreds of Hz to a few kHz, putting the
experiment deep into the low frequency range. Here, the modulation is slow enough
to impact the motion of the atom and increase its positional variance. However, with
small oscillation amplitudes, the growth is only on the order of a few percent of the
non-oscillatory variance, which should not affect our calculations seriously.
In this “small” frequency limit, and defining an effective temperature value T ′
as just the variance of the position distribution as via the equipartition theorem, the
average value for the fluorescence oscillation amplitude for the atom is
√ ∫ [ ]
〈 〉 κ κz
2
dF = dF (z) exp − ′ dz. (6.12)2πkBT 2kBT
Compared to the “true” temperature of the atom, the effective temperature is
√
T ′ = T/ 1 + 2, just a few percent higher than the “true” temperature. This small
difference will be ignored in the remainder of the calculations in this chapter. The
form of dF (z) in Equation 6.8 appears to be an odd function with respect to z (so
that the average is just zero), but the derivatives of the excited state populations are
also spatially dependent through the oscillation-free Zeeman detuning, δB,0.
152
Looking ahead to extracting this oscillation from experimental data, the RMS
oscillation amplitude is
[√ ∫ [ ] ]1/2
κ 2 − κz
2
dFRMS =
2πk T ′
dF (z) exp dz] , (6.13)
B 2kBT
or, more generally,
( ∫ [ ] )1/2
U(z)
dF 2RMS = A dF (z) exp − ′ dz , (6.14)kBT
where A is some constant to normalize the position distribution. With the form of
dF (z) for the V-atom, both dF ′RMS and 〈dF 〉 should be proportional to Bz, the
amplitude of the field oscillation.
Again, looking ahead to the measurement, rewriting the fluorescence of the atom
as
F = 〈Fl〉 [1 +m cos(2πft)] , (6.15)
then the dimensionless amplitude for the fluorescence oscillation can be calculated as
either
〈 〉 ∫ [ ]〈 〉 dF A U(z)m = 〈 〉 = 〈 〉 dF (z) exp − ′ dz or (6.16)Fl F l k
[ [B
T
] ]
1/2 ∫ 1/2dFRMS A 2 −U(z)mRMS = 〈 〉 = 〈 dF (z) exp dz] . (6.17)Fl F l〉 k T ′B
The dimensionless form for the fluorescence amplitude has at least two benefits.
First, it is easily comparable to the similar dimensionless amplitude for the driving
oscillations in the anti-Helmholtz coils, . Second, when comparing to experimental
results, photon collection efficiency factors vanish. Rather than fitting data to
153
an exact number of photons per second for the oscillation amplitude, which is
complicated by the unknown factor limiting collection efficiency (see Section 3.4.1),
the fit is relative to the average count rate, which has the same unknown efficiency
factor. The two collection factors multiply both photon rates (average and oscillation
amplitude) and thus cancel in the final calculation.
6.1.3 Numeric Calculations
The RMS form of the dimensionless fluorescence amplitude, m, is written in
a generic form so that any potential energy can be used, including for a three
dimensional system with z → ~r and dz → dV . In the case of the one-dimensional
V-atom, there is analytic form for the potential and for dF (z) as described above,
although it is (likely) there is no analytic form for the integral.
For the full rubidium atom calculation found discuss in Chapter V, there is no
analytic form for the potential, but it is possible to numerically find a solution for a
given temperature. The potential energy is found as done in Section 5.5—integrating
the irrotational component of force (in one dimension, the entire force is irrotational)
and the force is found as described in Section 5.3.3.
For dF (z) (or dF (~r) if in three dimensions), there is also no closed form solution
as there is for the the V-atom; rather, this can be done by assuming the fluorescence
oscillates at a constant frequency and finding the amplitude from the difference in its
extreme values, i.e. taking half the difference in the average fluorescence from when
the magnetic field has its largest value (B(z) = (1 + )B′zz) and its smallest value
(B(z) = (1− )B′zz) :
Fl+(z)− Fl−(z)
dFRb(z) = . (6.18)
2
154
This is exactly equivalent to taking a numeric derivative of the fluorescence rate with
respect to . In deriving Equation 6.6, the general form for the small  expansion is
∣∣
F (z) ≈ dF (z)F (z)| ∣=0 +  .d ∣=0
Ignoring the cosine term (remember that the atom is assumed to be in steady state
at all times, so the cosine is effectively a constant and only its extremes are of interest
for finding the amplitude), comparing to Equation 6.6 dF must be
∣ ( )
dF (z) ∣
dF =  ∣ Fl+(z)− Fl−(z)=  ,
d ∣=0 2
where the derivative is written numerically to first order in . This simplifies to
exactly the form of dFRb in Equation 6.18.
The average fluorescence rate is found numerically exactly as in Equation 6.1—
summing the steady state excited state populations and weighting by the atomic
decay rate Γ.
These three numeric calculations for the potential (Ui), fluorescence rate (〈F 〉i)
and fluorescence oscillation amplitude (dFi) can be done for all one dimensional
positions zi or for a multidimensional grid. Then, the two averages for the fluorescence
155
oscillation amplitude of Equations 6.16 and 6.17 are calculated as
1 ∑
exp [−Ui/kBT ] dFi∆z
A
〈m〉(T ) = i (6.19)
1 ∑
exp [−Ui/kBT ] 〈Fl〉i∆z
A
i
√
1 ∑
exp [−Ui/kBT ] dF 2i ∆zA
i
mRMS(T ) = ∑ (6.20)1
exp [−Ui/kBT ] 〈Fl〉i∆z
A
i
where ∆z is the spacing of points in the numeric calculation and A is a factor to
∑
normalize the potential, A = i exp [−Ui/kBT ] ∆z. These equations are written
fully, without simplification, because while it seems trivial to simplify factors of A
and ∆z, it is not always so clear while coding (lol!). In multiple dimensions, a similar
form can be used most easily where the label i refers to points in a multi-dimensional
grid, ∆z factors are dropped, and A is recalculated appropriately (also without the
spacing factor ∆z).
Experimental measurements of either average, mRMS or 〈m〉, can then be used
to fit a temperature for an atom to these expressions, with an appropriate model to
calculate U , dF 2i i , and 〈Fl〉i.
6.2 Analysis of Photon Arrivals
Extracting oscillation information can be done particularly well by looking at
the spectrum of photon arrivals. The spectrum can be numerically calculated directly
from the record of photon arrivals by an autocorrelation measurement of the photon
arrivals as per the Wiener-Khinchin theorem [42]. It is numerically simpler to bin
photon arrival data (creating a list of photon counts per timer) and calculate a power
156
spectrum directly with a fourier transform as is
∣∣∣ ∣
∣2
f̃(f)∣
S(f) = , (6.21)
Tmax
where f̃(f) is the Fourifer transform of the binned photon time data [48]. This is
implemented, for this dissertation, via Octave to do the Fourier transforming. The
Octave implementation returns both positive and negative frequency components of
the spectrum, but in a non-intuitive order common in numeric Fourier transform
algorithms. The code snippet below takes in photon count times in the array data
and produces an appropriate power spectrum with the array indices ordered from most
negative frequency to largest frequency for natural graphing of the power spectrum.
Spec=fftshift(fft(ifftshift(data))) * dt;
PowSpec=real(spec.*conj(Spec))./tMax;
In this code snippet, dt is the for one photon bin and tMax is the maximum time
for the data. This clearly indicates two times must be defined. The first is a time
to bin photon counts, ∆t (dt in the code snippet). This sets a time step and gives a
maximum frequency that will calculated in the spectrum as fmax = 1/2∆t. Selecting
a shorter bin time allows for higher frequencies to be revealed in the spectrum,
but produces larger data sets to analyze impacting calculation time. Second is the
maximum time to collect data, Tmax (tMax in the code snippet). This gives a frequency
spacing for the spectrum as ∆f = 1/2Tmax. Selecting a longer maximum time gives a
higher resolution spectrum, but limits the number of spectra that can be calculated
and averaged together to reduce noise.
In Figure 6.1a, one data run with 100 ms binned photon data (the binning time
used to trigger the bayesian algorithm while collecting data) is shown. In Figure
157
6.1b, spectra for a few different maximum times are shown. In Figure 6.1c, spectra
with a few different maximum frequencies (and thus different bin times) are shown
for this same data. Figure 6.1d, the photon oscillation amplitudes for the spectra in
(b) and (c) are shown, calculated using the method described below (Equation 6.28).
In graphs (b-d) here, the spectra and oscillation amplitudes were calculated from the
photon arrival time collected from the data run in (a).
In practice, the bin time is chosen to give maximum frequencies of 25 kHz to
sample multiple harmonics (see Section 6.6) of the typical few kHz driving frequencies.
With useable data collection times of around 90 s per experiment, maximum times
for a spectra are typically around a 5 or 6 seconds to allow for many spectra to be
averaged for a single photon’s experimental lifetime.
6.2.1 Oscillation Calculation
From the spectra, information about the average fluorescence, amplitude of the
oscillation and phase can be extracted. The measured fluorescence is assumed to be
Poissonian distributed8. Defining the background average photon rate to be B and
the atomic fluorescence rate to oscillate with frequency fα
α(t) = α0 [1 +mα cos(2πfαt)] , (6.22)
so that the Poissonian sampled fluorescence signal is Flm(t) = B +α(t). As noted in
Appendix E, the average rate of photon collections, 〈Flm〉 = B+α0, (Equation E.2),
8As revealed in Figure 4.3, the fluorescence measured from the experiment is super-poissonian.
The calculation in Appendix E there assumes a gaussian distribution a variance larger than the
mean. The calculation there shows that even with the variance many times larger, there is little
change in the power spectrum.
158
 35 (a)
 32.5
 30
 27.5
 25
 0  10  20  30  40  50  60  70  80  90  100
Time, sec
 350 (b)
 300
Tmax=5.0s Tmax=9.5s
 250
Tmax=6.5s
 200
 150 Tmax=3.5s
 100 Tmax=1.5s
 50
 0
3001 3001 3001 3001 3001
Frequency, Hz
 350 (c)
 300
 250 fmax=10kHz fmax=25kHz fmax=42kHz fmax=60kHz
 200 fmax=5kHz
 150
 100
 50
 0
3001 3001 3001 3001 3001
Frequency, Hz
fFmmaax,x ,k kHz
 0  10  20  30  40  50  60  70
 0.2 (d)
 0.15
 0.1
 0.05
 0
 0  1  2  3  4  5  6  7  8  9  10
TTmax,,max   sseecc
FIGURE 6.1. Sampled oscillation spectrum. (a) Fluorescence rate as a function of
time. An atom appears in the MOT around 10seconds. (b) Averaged single atom
spectra for a variety of maximum times. (c) Averaged single atom spectra for a
variety of one photon bin times (or maximum spectrum frequencies). In both b and
c, only ±2 Hz is shown around the driving frequency, 3001 Hz. (d) Average values
for unit-less oscillation amplitude mα calculated from the spectra in (b) and (c). For
this data run, the anti-Helmholtz current modulation amplitude was 0.058.
159
FFluluooreresscceennccee A Ammppliltituuddee,,  mmαα PPoowweerr  SSppeeccttrruumm,,  kkpphhoottoonnss//sseecc PPoowweerr  SSppeeccttrruumm,,  kkpphhoottoonnss//sseecc FFlulouroersecsecnecnec eR raatete, ,k kpphhoototonnss/s/seecc
can be found from the tails of the power spectrum as
S(f →∞) = 〈Flm〉. (6.23)
This appears clearly in the spectra shown in Figure 6.1(b) and (c), which all have
similar backgrounds to the peak near 3001 Hz. When this is done numerically, an
average of a power spectrum’s tail is used rather than just the spectrum value for
fmax. The average rate can also be gotten directly from the photon arrival data
without needing to take a spectrum. With ni photons in each time bin, the average
rate is
1 ∑〈Flm〉 = ni. (6.24)
Tmax i
This, of course, is the total background rather than just the single atom fluorescence
rate. To get the single atom fluorescence rate, the background fluorescence rate just
needs to be known. As was noted in the single-atom detection method Section 4.1, our
experimental runs always begin with a few seconds with blue detuned laser frequencies
so that no atoms can load into the MOT. Thus, we have a built in measurement of the
no-atom fluorescence, B. Then the single atom fluorescence rate is α0 = 〈Flm〉 − B.
This rate can be measured entirely without a spectrum and is necessary for the
Bayesian algorithm in Chapter IV.
For the oscillation amplitude, as calculated in E and in [132], there should be a
peak in the power spectrum, calculated over a maximum time Tmax, whose amplitude
is
α2T α2 2max
S (f ) = 〈Fl 〉+ 0 m2 + η 0mαα m
4 α 4〈 , (E.12)Fl〉
where η is defined in Equation E.3, a parameter that describes how much larger the
standard deviation of the fluorescence is compared to the average fluorescence (i.e.
160
how close to Poissonian the signal is). The dependence of the peak height on Tmax
is revealed in the spectra in Figure 6.1b, with a few qualifications. First, the spacing
of the spectral points changes with Tmax. With wider spacing, the peak gets spread
out which over emphasizes the peak height visually. Second, with Tmax being integer
seconds, there are spectral points that align exactly with integer frequencies, as such
as the peak at Tmax = 5s in the figure. This peak appears higher than the peak
at Tmax = 6.5 s because the peak is shifted over a bit and widened to align with
the spectral points. This lowers the peak heights when the maximum time is not an
integer. The lowering is compensated for by the integration method described below
for calculating the peak height (see Equation 6.26) and does not have an impact the
calculation of the oscillation amplitude as shown in Figure 6.1d.
Assuming a perfectly Poissonian distribution or (η = 0), as noted in Appendix
E, when there is a large background fluorescence (〈Flm〉  η), the amplitude of the
oscillation is then √
2 S (fα)− 〈Flm〉
mα = 〈Flm〉 −
. (6.25)
B Tmax
In this form, all values are directly measurable. The peak height, or as we’ll see
the area under the peak height, is S(fα). The background fluorescence rate, B,
is measured from the data run before any atoms were loaded into the trap. The
measured, experimental fluorescence rate, 〈Flm〉, can either come from the tails of
the spectrum (as noted in Appendix E) or just from the raw photon arrival data
(Equation 6.24).
The power spectral height of Equation E.12 assumed that we had f = fα,
the externally driven magnetic field frequency. This is an implicit delta function
in the general spectrum, however the experimental data are certainly not a delta
function, see Figures 6.1b and c. Instead, due to the numeric calculation, the delta
161
function is spread over many frequencies. To get the original peak height back, we
can numerically integrate over the peak. This produces
∫ k∑=N
S(fα) = S(f)δ(f − fα)df ≈ S(fα + k∆f)∆f, (6.26)
k=−N
where the value N is a defined window size to integrate over, typically done so that
the window around the peak is ±1 Hz, i.e. N = 1/∆f . In this numeric language
then, directly from the power spectrum it is possible to calculate the square of the
fluorescence modulation amplitude as
k∑=N
dF 2 = α2m2
4
m 0 α = [S(f + k∆f)− S(f =∞)] . (6.27)T dt2 αmax
k=−N
Note that a factor of ∆f has been lost here in order to return the integrated area of
Equation 6.26 to a delta-function peak height. There is an additional factor of 1/dt2
present in this equation. The spectrum is calculated from data are in raw photon
count numbers, so that the amplitude measured is in terms of just photon number2.
The factor is to return the amplitude to units of photons/s. This is done to compare
directly to the atomic fluorescence rate defined in Equations 6.1, which is given in
terms of the atomic decay rate Γ. Additionally, rescaling amplitudes to photons/s
gives a systematic unit for comparison between analysis with different values of dt
(the blue data in Figure 6.1d). Because the value for dF 2m can be pulled directly out
of the experimental data, this implies that the RMS form of the analytic fluorescence
amplitude (mRMS of Equation 6.20) should be used to for calculating the temperature
of the atom. For this comparison, the measured dimensionaless oscillation amplitude
is then √ √
dF 2m dF
2
mα = =
m
α0 〈
(6.28)
Flm〉 −B
162
where again α0 is the average fluorescence rate from the atom (which can/must be
found separately as noted above).
The phase for the fluorescence oscillation comes directly from the spectra as well.
Take, for example, the cosine function with frequency f0 and phase φ. The Fourier
transform is
1 [ ]
f̃(f) = e−iφδ(f + f0) + e
+iφδ(f − f0) (6.29)
2
Taking the tangent of the real and imaginary parts of this wave gives
( )
Im[f̃ ] sinφ [−δ(f + f0) + δ(f − f0)]
tan = − = tanφ (6.30)Re[f̃ ] cosφ [δ(f + f0) + δ(f f0)]
when evaluated at f = f0. Then, if the complex Fourier transform of an oscillating
time signal is written as
∣∣∣ ∣
∣
f̃(f) = f̃(f)∣ eiφ, (6.31)
the phase φ corresponds to the phase of the underlying oscillation and can be found
just by taking the inverse tangent of the real and imaginary parts of the transformed
function. Doing this numerically with the binned photon data returns the appropriate
phase, offset by factors of π that are related to the total time and spacing between
bins.
As noted above, choosing shorter segments of time to create power spectra
from limits the resolution of the spectra, but it allows for more averaging of data
for a clearer signal. This averaging is particularly important in light of Equations
6.25 and 6.27 as the background fluorescence is subtracted. For individual data
runs, particularly those with small MOT coil current amplitudes, , small peaks in
a spectrum or a noisy background and produce negative values for dF 2 and thus
imaginary amplitudes for the fluorescence oscillation, mα. This is resolved by taking
163
just the real part of the calculated dF 2 and averaging it over many spectra before
calculating the fluorescence amplitude. The averages are calculated as below.
1. Photon arrival times are recorded with the FGPA system described in Section
3.4.2.
2. Using the bayesian algorithm to predict times when a single atom was in the
MOT (see Section 4.2), photons collected during these times are counted in bins
of time length ∆t, as above. The size of the bin is determined by the maximum
desired frequency of the spectrum.
3. For a defined maximum collection time, Tmax, the appropriate number of bins
(Nbin = Tmax/∆t) are spliced out of the data.
4. The spliced data are used to calculate a spectrum. Total background
fluorescence 〈Flm〉 (Equations 6.24 or 6.23), the square of the fluorescence
oscillation amplitude dF 2m (Equation 6.27) and oscillation phase φ (Equation
6.30) are calculated from the spectrum, or photon counts, for a number of
harmonics of the known driving frequency of the MOT coils.
5. Splicing and calculations are done with consecutive numbers of bins until
remaining number of bins is smaller than Nbin.
6. Average spectra, fluorescence background and square of the fluorescence
oscillation amplitudes together. Note that averaging the square of the
fluorescence oscillation amplitude and taking a square root calculates exactly
the RMS of the fluorescence oscillation amplitude.
Together with a measurement of the non-atom background, B, from each
data run, the three measured averaged values for dF 2m, 〈Flm〉 and B calculate the
164
dimensionless oscillation amplitude of Equation 6.28. When these three measured
value have error σdF , σFl, and σB, respectively, the error in the calculated oscillation
amplitude σm is
2 2 2
2 σdF dFm (σFl + σ
2 )
σm = ( )2 + ( )B4 . (6.32)
4 〈Fl 2m〉 −B dFm 〈Flm〉 −B
This can be done for individual data runs and thus specific atoms, as was done for the
data in Figure 6.1. With a particular MOT model, these data can be fit to Equation
6.17 or Equation 6.20 to find a temperature of the atom while it is still in the MOT,
as is done in other atomic temperature measurements [133]. The individual atom,
then, can be used for other experiments.
It is also possible to average the spectra and amplitudes for many data runs
together with specific MOT system parameters. This does produce an approximate
atomic temperature for an atom in the MOT with those system parameters, rather
than an in-situ measure of the temperature of a specific atom. However, as is done
with the other temperature measurements discussed in Chapter I, it is assumed that
atomic temperatures are primarily a function of the MOT system parameters.
6.3 Measurements
Fluorescence measurements were previewed briefly in Figure 6.1, which showed
the fluorescence peak in a spectrum measured from a single atom. Some complete
spectra are shown in Figure 6.2 for a variety of driving frequencies, but all with
a driving anti-Helmholtz current modulation amplitude of  = 0.025. Some
clear features can be noted about these spectra that are universal to single-atom
fluorescence spectra measured in our experiments. The tail of the low-frequency peak
165
 1e+06
101Hz 467Hz 953Hz 1499Hz 2711Hz
 100000
 10000
 1000
 100
 10
0.1 1 10 100 1000 5000
Frequnecy, Hz
FIGURE 6.2. Assorted measured spectra. Single-atom fluorescence spectra with a
variety of driving frequencies. All spectra done with a driving current amplitude of
 = 0.025.
scales as 1/f 2, which is not surprising. There is a wide peak in the spectra around
27 Hz. This peak originates from an unknown magnetic field oscillation that appears
ubiquitous in the building around the lab. It can be seen even on simple loops of
wire connected to a spectrum analyzer. The frequency is not fixed, but drifts slightly
causing the widened peak in the spectrum. There are peaks at 60 Hz because of
course there are. As calculated in Appendix E, the spectral tails equal the average
measured fluorescence rate. Finally, although the current modulation amplitude is
the same for all the peaks in Figure 6.2, the peak heights, and thus the fluorescence
oscillation magnitudes, change with frequency. This fact is analyzed in more detail
in Section 6.3.2 below.
To check the developed temperature measurement technique, comparisons to
known methods should be done. Figure 6.3 shows temperature measurements for a
single atom in our high gradient MOT with the release-recapture method described in
Section 1.4 [4, 16]. In the figure, two sets of data and three simulations with different
atomic temperatures is shown. The red data are done with the magnetic field turned
166
Power Spectrum, kphotons/s
off with the MOT lasers, while the blue data leaves the magnetic field on during the
test. At short times (below about 6 ms) the recapture rate is quite similar in both
cases. This is due to slow ramp down of the anti-Helmholtz coil current while turning
off the field.9 The field persists during this time and magnetically traps the atom (see
Section 2.3.1). At long times, the two cases diverge as the field is no long trapping
atoms. With the influence of the magnetic field taken into account, the long-time
tail for the recapture rate should be used to estimate the temperature. Thus, the
atom is likely around 160 µK, surprisingly above the Doppler temperature. From the
data and simulation, it is expected that the atom has temperatures around 160 µK,
which should be the “target” for verifying our temperature measure technique. The
measurements and simulations in Figure 6.3 may be discussed in more detail in the
dissertation of Erickson [134], but serve as a fine comparison for this work.
6.3.1 The Linearity of Fluorescence Amplitude
The most pressing measurement is verifying the scaling of the fluorescence
oscillation amplitude to the driving current modulation amplitude. As noted in
Section 6.1, linear scaling of the fluorescence amplitude would rule out the extended
two-level model to describe the atom in the MOT, as this model would predict
fluorescence modulation only at higher orders of the current modulation amplitude.
The measured fluorescence amplitude as a function of current oscillation
magnitude for two different driving frequencies are shown in Figure 6.4. Red data
9This ramp-down is not caused by the RC circuit created by the anti-Helmholtz coil and the
filtering capacitor discussed in Section 3.3.5. These have an RC time constant of τ=4.6 µs, much
faster than the ramp-down time. This is also much slower than the ramp-down time due to internal
capacitors in the current supply, which have capacitance 1.5 µF as programed for our power supply
[100]. This time delay, instead, may be a result of self-inductance in our MOT coils or eddy currents
in the various conducting elements around the experiment.
167
 1 B-Field OffB-field On
Simulated T=30K
Simulated T=90K
Simulated T=160K
 0.8
 0.6
 0.4
 0.2
 0
 0  2  4  6  8  10  12
Release time, ms
FIGURE 6.3. Single-atom temperatures with release-recapture method. Measured
data in red done while ramping down MOT magnetic field on release. Data in blue
have the MOT magnetic field left on during the measurements.
f =1154Hz f =1154Hz
 0.8 (af0=)3001Hz (s
0caled 4x)  0.8 (bf
0
0=)3001Hz (scaled 4x)
 0.6  0.6
 0.4  0.4
 0.2  0.2
f0=1154Hz
 0  0.8 f0=3001Hz (scaled 4x)  0
 0  0.02  0.04  0.06  0.08  0.1  0  0.02  0.04  0.06  0.08  0.1
Driving Current Amplitude, ε Driving Current Amplitude, ε
 0.6
FIGURE 6.4. Fluorescence amplitude scaling with current amplitude. The same data
appear in both graphs, but is fit (the curves in each graph) with a power law in (a)
 0.4
and linearly in (b).
 0.2
168
 0
 0  0.02  0.04  0.06  0.08  0.1
Driving Current Amplitude, ε
Fluorescence Amplitude, mα
Recapture Probability
Fluorescence Amplitude, mα
Fluorescence Amplitude, mα
are recorded at a frequency of 1154 Hz and the blue data are recorded at a frequency
of 3001 Hz. It is important to note that the blue data have been (vertically) scaled
by a factor of 4 so that it is behavior is clear on the graph. In Figure 6.4a, the lines
are power law fits. The low frequency data scales as m ∝ 0.82±0.05α and the high
frequency data scales as m ∝ 0.81±0.03α . While not quite linear, the scaling is still
far from 2. Thus we can rule out the extended two-level atom as a model for our
experiment. Linear fits to the data are shown in Figure 6.4. Here, the lower frequency
data have a slope of 5.2± 0.55 and the higher frequency data have an unscaled slope
of 1.0± 0.03.
The large difference in slope (and larger fluorescence amplitude in general) for
the two measurements in Figure 6.4 corresponds to the different frequencies of the two
data sets. This is analyzed in Section 6.3.2 in more detail, but we’ll note here that
the higher frequency measurement is a better model to calculate temperature. The
lower frequency data can excite mechanical resonances in the atom causing additional
motion in the atom, as noted in Section 6.1.1. The higher frequency oscillations avoid
these resonances and should reveal the behavior of the atom without influence of the
magnetic field oscillations.
Figure 6.5 shows simulated results for the slope of the fluorescence amplitude
compared to the magnetic field modulation amplitude for a variety of models. Here,
the slope is plotted as a function of atomic temperature following Equation 6.20. In
Figure 6.5a, the three lines are all calculations for a 1D MOT for the V-atom, full
87Rb atom, and a Jg = 1 → Je = 2 atom (referred to as J1→2 through the rest of
this work). The J1→2 atom was used in some of our simulations for comparison to an
analytic derivation of the spring constant for this atom in the MOT [92]. The MOT
169
 0.016  0.016
(a) (b)
 0.014  0.014
 0.012  0.012
 0.01  0.01
 0.008  0.008
 0.006  0.006
 0.004  0.004
 0.002  0.002
V-atom MOT beam phase integration
87Rb atom Additional MOT oscillation
Jg=1 -> Je=2 atom (20x scaled) Effective 3D distriubtion
 0  0
 0  100  200  300  400  500  0  100  200  300  400  500
Temperature, µK Temperature, µK
FIGURE 6.5. Simulated 1D modulation slopes. Values are slopes of fluorescence
amplitudes to driving current modulation amplitudes (i.e., the slopes of the linear fits
in Figure 6.4). The two graphs have the same vertical scale.
spring constant in our simulation does match this analytic form and thus this atom
model has been used to examine some results of our simulation.
With the larger total angular momentum of the excited state for the J1→2 atom
larger than the V-atom’s, the outermost excited state energy level for the J1→2 has
a larger Zeeman splitting than that of the V-atom. This would imply the J1→2 atom
should have a larger amplitude in the fluorescence modulation, as it is driven by
oscillations in the Zeeman shift of the atoms energy levels. However, in calculating
the fluorescence amplitude for the V-atom, the simulation added an additional factor
of 3 to its excited state Zeeman splitting by changing gF to 3gF . This was to better
mimic the behavior of the outmost energy levels of the full 87Rb atom and leads the
larger fluorescence amplitude for the V-atom compared to the J1→2 atom in Figure
6.5a.
All three of these 1D simulations clearly show slopes that are significantly less
than the slope of order 1 from the data in Figure 6.4. Thus in Figure 6.5b, simulations
are of the 3D MOT with the full 87Rb atom as described in Chapter V. For all of
170
Fluorescence Amplitude Slope
Fluorescence Amplitude Slope
these, as noted in Section 5.5, the calculation is averaged over many phases of the
MOT beams. This direct averaging is the red curve in figure (b). While its results are
larger than the 1D results, and particularly true for low (reasonable experimental)
temperatures, they are still much less than the measured slopes .
Recalling the effective temperature discussed in Section 6.1.2 and with prior
knowledge of additional atomic motion as seen in Section 6.5, we can add an additional
“forced” modulation of the atom to our calculation. This is done by assuming that the
atom’s entire position distribution oscillates around the peak position with amplitude,
A. This amplitude is small enough to not change the coupling of light from the atom
into the APD detection system (unlike the oscillations observed in Section 6.4). With
this “forced” oscillation model, it is assumed that the modulation frequency of the
magnetic field is much faster than this oscillation frequency of the atom in the trap.
This is generally true for most of our measurements and certainly for the data in
Figure 6.4, where the relevant position oscillations are on the order of 100 Hz (see
section 6.3.2). With this slow position oscillation, the atom would then see the full
modulation of the magnetic field at each position, so that the average fluorescence
amplitude (and thus its slope) would just be its value at each position averaged over
one period of the atom’s motion. For a fluorescence amplitude that depends on
position, mα(a), then the average rate is
∫ A
〈mα〉 = mα(a)P (a)da (6.33)
−A
where P (a) is the probability to be at position a for an object oscillating with
amplitude A: √
1 1
P (a) = − . (6.34)π A2 a2
171
The results of these slow atomic oscillations are shown as the green data of Figure
6.5b.
While the calculations above for the “3D” MOT are only looking at the forces
(and thus the potential energy) along one axis, the larger 3D environment can
be modeled more directly with an effective 3D distribution. A majority of our
calculations, and indeed all of those seen above in this section, have been done along
one MOT beam axis. This gives a position distribution along one axis rather than
a true 3D distribution. If we assume that the atom’s position distribution is small
in the other directions (a fair assumption with a strong confining force due to the
high-gradient fields), we can map an effective 3D solution onto our 1D calculation.
With a well confined atom (and exactly true for a 1D MOT), the field strength largely
dictates the probability to be at each position along an axis. With the 3D mapping,
we re-weight the 1D position probabilities by the number of points in the 3D with
the same magnetic field magnitude. As points in the 3D move away from the axis,
the field magnitude grows and thus the higher magnetic-field tails of the distribution
are move heavily weighted. This effectively turns the 1D position distribution into
a (local) magnetic field magnitude distribution for the atom. Renormalizing this
distribution with its heavier tails should result in a higher fluorescence oscillation
as the field oscillation is also higher in the tails. While this seems artificial, it does
mimic the behavior of the 3D atom. The atom explores regions off-axis that have
a stronger magnetic field and stronger oscillation in the field than points on the
axis. This numeric mapping method takes these into account without calculating the
(potentially non-conservative) force off-axis. The blue data in Figure 6.5b shows this
effective 3D distribution with a off-axis grid size of 15 µm.
172
It is still evident that simulations are not matching the measured experimental
results. Thus, calculations for approximate MOT temperatures will be dropped from
the remaining calculations and functional behavior will be the focus.
6.3.2 Frequency Dependence of Fluorescence Amplitude
As revealed in Figures 6.2 and 6.4, the amplitude of fluorescence oscillations
changes with frequency. In Figure 6.6, the amplitude as a function of driving
frequency is plotted. Here all the driving currents have modulation amplitude10
 = 0.025. In this graph, also shown are fits to the low frequency response (f < 500)
and the high frequency response (f > 700). The low frequency signal scales as a
power law f 0.55±0.02. The high frequency tails scale with a power law as f−1.51±0.11
(or with exponential decay that has decay constant (8.30± 0.36)× 10−4 Hz−1).
These two power law scalings closely match the scaling of a Lorentzian
distribution weighted by f 1/2. For a particle with a small modulation to its trapping
frequency ω0, its position variance should have a Lorentzian shape as a function of
the driving frequency [39, 135]. Since the fluorescence amplitude depends closely
on this position variance (see Section 6.1.2), it is expected that the fluorescence
amplitude follows this same shape at high frequencies. As discussed above, for a
fast modulation of the fluorescence rate, at all positions the atom sees the whole
range of fluorescence rate values so that its fluorescence rate can just be averaged
10As a practical note, the modulation in anti-Helmholtz coil current was introduced by adding a
signal from a function generator to the control signal that went to the coil power supply. Due to
both the MOT coil low-pass filtering capacitor and the power supply’s own frequency response, a
number of amplitudes for the function generator signal have to be tested and the current response in
the coils measured with a Hall probe for each frequency. The amplitude of the coil current, A, was
calculated by comparisons to the measured current variance σ2, which are related by A2 = 2σ2 for a
sinusoidal oscillation (Table 6.2 gives the relationship between amplitude and variance for different
waveforms). The amplitudes for the function generator signal were adjusted until the measured
current amplitude reached the desired value.
173
 0.5
Low Frequency Power Law
High Frequency Power Law
 0.4 High Frequency Exponential DecayMeasured Spectrum
 0.3
 0.2
 0.1
 0
 0  500  1000  1500  2000  2500  3000  3500  4000
Driving Frequency, Hz
FIGURE 6.6. Fluorescence amplitude spectrum. Along with the data (blue) low-
frequency and high-frequency fits to the data are shown.
over positions—producing the same frequency distribution for the fluorescence. At
lower frequencies, the spectrum is weighted by f 1/2. Therefore, the spectrum can
then be written as √
a f/πΓ
S(f) =
(f − , (6.35)2f 2 2z) + Γ /4
where Γ is the (Lorentzian) FWHM, a is a scaling factor, and 2fz is the (Lorentzian)
peak frequency, which is twice the frequency of the (undamped) atomic motion in the
√
MOT [39, 136]. The atomic motion has frequency 2πfz = ω0,MOT = κ/m where κ
is the MOT spring constant.
There is an additional trapping constant associated with magnetic trapping in
the MOT. As discussed in Section 2.5.1, the magnetic trapping force is of the form
F~mag = −κmagzẑ, (6.36)
with
µBg
′
fB
κ zmag = κ~ MOT
. (6.37)
kΓ
174
Fluoresence Amplitude, mα
With this, the magnetic trapping frequency is
√
µBgFB′
ω z0,mag = ω0,MOT . (6.38)~k
The data shown in the spectrum of Figure 6.6 were recorded with MOT laster
detuning ∆ = −4.8 MHz, magnetic field gradient B′L z = 215 G/cm, and laser Rabi
frequency |Ω| = 0.575Γ, where Γ is the atomic decay rate. With the (laser power
balanced) V-atom solutions for κ and β in Equations 2.84 and 2.71, the MOT has
a trapping frequency of f0,V-at, MOT = 1056 Hz and the magnetic trap has frequency
f0,V-at, Mag = 77 Hz. It is also good to look at the frequency for the full
87Rb atom
calculation, as the spring constant for the full atom had a much stronger trapping
force. Using the scaling between the full atom and the V-atom confinement forces
from fitting the simulated data in Figure 5.2, the MOT has a trapping frequency of
f0,87Rb-MOT = 3094 Hz and the magnetic-trapping frequency is F0,87Rb-Mag = 228 Hz.
All of these values, of course, are for the vertical axis of the MOT. In the horizontal
direction, the magnetic field and thus the trapping strength, is reduced by half. These,
√
then, give an additional frequencies scaled by 2 compared to the z-axis frequencies.
In principle, then, the spectrum of Figure 6.6 could be made of four Lorentzian
√
peaks, all scaled by f . However, due to the high MOT frequencies, it is more likely
that the magnetic trapping results in the peaks shown in Figure 6.6. This gives a
spectrum of the form
[ ]
√ Γz/2π Γxy/2π
S(f) = a f − + √ . (6.39)(f 2f )2 + Γ2z /4 (f − 2 2f )2 2z z + Γxy/4
The two spectrum formulae of Equations 6.35 and 6.39 are plotted along with the
fluorescence data in Figure 6.7. The green curve in the figure is a fit to the the single
175
 0.5
Single Lorentzian Fit
Double Lorentzian Fit
 0.4 Bad Double Lorentzian FitMeasured Spectrum
 0.3
 0.2
 0.1
 0
 0  500  1000  1500  2000  2500  3000  3500  4000
Driving Frequency, Hz
FIGURE 6.7. Lorentzian fluorescence amplitude spectrum. The green curve is a fit
to the single Lorentzian spectrum of Equation 6.35 and the red curve is a fit to the
double Lorentzian spectrum of Equation 6.39. The black curve is a fit to the double
Lorentzian spectrum without the group of 5 “low” values on the interior of the curve,
hence its “Bad” labeling.
Lorentzian spectrum of Equation 6.35 with fit values a = 42 ± 0.3, Γ = 1442 ± 11,
and fz = 100 ± 3 Hz. The red curve is a fit to the double Lorentzian spectrum of
Equation 6.39 with fit values a = 20 ± 0.1, Γz = 2608 ± 68, Γxy = 834 ± 14, and
fz = 94 ± 1 Hz. The black curve, which should be treated as suspect, is a fit of
the double Lorentzian spectrum without the 5 “low” data points in the interior of
the spectrum. Without these, the fit parameters are a = 21 ± 0.2, Γz = 1153 ± 88,
Γxy = 1500 ± 105, and fz = 143 ± 5 Hz. While this is not an appropriate way to
analyze data, it does show that the spectrum equation is pretty spot-on to (most of)
the data. All of these frequencies are the appropriate order for the magnetic trapping
frequency.
176
Fluoresence Amplitude, mα
 0.09 700
4A 5A 6A 7A 8A 9A
(a) fα=2711Hz, ε=0.03fα=6007Hz, ε=0.07 400 (b)
 0.08
 0.07 100
 0.06
20
2711 2711 2711 2711 2711 2711
 0.05 Frequency, Hz
700
(c) 4A 5A 6A 7A 8A 9A 0.04 400
 0.03
100
 0.02
 0.01
 100  120  140  160  180  200  220  240  260 20 6007 6007 6007 6007 6007 6007
Magnetic Field Gradient, G/cm Frequency, Hz
FIGURE 6.8. Fluorescence amplitude scaling with field gradient. (a) Calculated
fluorescence amplitudes. (b) Low-frequency (red data) fluorescence spectra. (c) High
frequency (blue data) fluorescence spectra. In both spectra (b) and (c), graphs are
labeled as their desired anti-Helmholtz currents while horizontal positions in (a) are
in terms of measured field gradients.
6.3.3 MOT Size: Trapping Strength
The “size” of the MOT can be scaled by changing the trapping strength, κ for
the MOT. This is done easily by changing the anti-Helmholtz coil gradient. This is
done for two data runs in Figure 6.8. In Figure 6.8a, two very different relationships
are seen. The blue data shows clear independence of the fluorescence amplitude at
different DC field gradients. This could be expected as the current amplitude  is
constant for all field gradients. With the weaker trapping of the field, the atom may
explore further distances from the center of the trap, but the (relative to DC) size of
the oscillation is the same everywhere.
The red data in Figure 6.8a show a linear relationship between the gradient and
the fluorescence. This could be expected as at every position the total oscillation of
the magnetic field magnitude (in Gauss) is larger for the higher field gradient. So
177
Fluorescence Amplitude, mα
Power Spectrum, kphotons/sec Power Spectrum, kphotons/sec
 0.0003
40 µK
120 µK
160 µK
300 µK
 0.00025
 0.0002
 0.00015
 0.0001
 5e-05
 0
 0  50  100  150  200  250
Magnetic Field Gradient, G/cm
FIGURE 6.9. Simulated fluorescence amplitude scaling with field gradient. They are
linear.
even while the atom explores shorter distances from the center of the trap with the
higher gradient, the change in magnetic field strength is larger in these regions.
These two different scalings is also reflected directly in the spectra of Figures 6.8b
and 6.8c. In Figure 6.8b, the peak heights clearly grow with increasing DC current,
while in 6.8c, they are relatively constant. So which is correct? Apparently the linear
results. In Figure 6.9, a simulation for the 1D 87Rb atom shows the fluorescence
amplitude as a function of the DC anti-Helmholtz current at a variety of temperatures
in µK. Clearly the results are linear. The two experiments shown in Figure 6.8
have similar MOT parameters—their only clear difference is the frequency of the
oscillations. It is unclear why the two give very different results.
178
Fluorescence Amplitude, mα
6.3.4 MOT Size: Detuning
It is also possible to scale the size of the MOT through the detuning of the lasers.
As can be seen from the “traditional” view of the atomic Zeeman shifts in the MOT
of Figure 2.5, there is some distance from the center of the MOT where the Zeeman
shifts cause the lasers become blue detuned. From Equation 2.75, this happens at a
z-direction “radius”
~∆L
zrad = (6.40)
µBgFm ′FBz
and twice this distance from the MOT center for the x- and y- directions (due to the
halved magnetic field gradient in these directions). This blue detuning as a function
of position is also responsible for the “turn off” of the enhanced trapping of the atoms
with multiple ground states discussed in Section 5.3.3. For our high magnetic field
gradient MOT, B′z = 215 G/cm (an anti-Helmholtz current of 8 A to match the data
in Figure 6.10) and lasers detuned by ∆L = −Γ and the outer most excite state for
the 87Rb atom (mF = ±3), this radius is 48 µm, which is significantly larger than an
atom’s position distribution at MOT temperatures, as seen both in simulations and
images of the atom (see the single-atom fluorescence profile in Figure 6.15a, which
has standard deviation of around 15 µm).
This is examined in two ways in Figure 6.10. The red data shows the fluorescence
oscillation strength as a function of laser detuning with a fixed magnetic field gradient
of 215 G/cm. The blue data show the fluorescence amplitude as a function of
detuning with a constant MOT radius as defined in Equation 6.40. Figure 6.10a shows
measured experimental data and Figure 6.10b shows simulated results. A few things
are noticeable. First, the vertical scale is very different between the two, but that
is understandable based on the discussion in Section 6.3.1. Second, while the shape
179
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(a) Fixed CurrentFixed Radius (b)
 18
 0.35
 16
 14
 0.3
 12
 0.25  10
 8
 0.2
 6
 4
 0.15
 2
Fixed Current
Fixed Radius
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 2  4  6  8  10  12  2  4  6  8  10  12
Detuning, MHz Detuning, MHz
FIGURE 6.10. Fluorescence amplitude as a function of laser detuning. (a) Measured
Data and (b) 1D simulated data for 87Rb with a temperature of 190 µK. Red data
in both graphs changes detuning with keeping the anti-Helmholtz current fixed. The
blue data in both graphs changes the anti-Helmholtz current along with the detuning
to keep the MOT “radius” in the z-direction constant as in Equation 6.40.
of the fixed radius graphs roughly agree, the fixed current graphs are quite different.
Third, the strong dependence of the amplitude on the detuning near ∆L = −Γ could
play an important role in noisy signals. Our experiments are done with detunings right
around −Γ. As seen in Equation 3.2, the detuning of the laser light is controlled by
the frequency of 3 AOMs. Poor calibration between the experimental control system
and the AOM’s output frequency could cause systematic errors in the detuning.
6.3.5 Position Changes: Background Fields
Another clear check on the positional dependence of the fluorescence oscillations
can be done by shifting the center of the MOT (defined as where B~ = 0) using
a background magnetic field. In being shifted away from the center of the anti-
Helmholtz coils, the atom should see higher total magnetic field modulations. This
agrees with the measured and simulated results of Section 6.3.3 which showed the
higher field gradient gave larger fluorescence amplitudes. Here, rather than higher
180
Fluorescence Amplitude, mα
Fluorescence Amplitude, mα
gradients producing larger total fields for the atom to see, shifting the center of the
MOT with a uniform background field puts the atom at locations with a higher field
gradient.
Doing this test requires a change to the Bayesian algorithm as described in
Section 4.2.7. Rather than triggering the experiment based on measuring a single
atom, the experiment must be triggered manually after some set time. Because there
is a background field, increasing the magnetic field gradient (to limit loading in the
MOT) causes the “center” of the MOT to shift, as discussed in Section 2.5.1. Thus,
to collect light from the single atom, the APD must be focused on the location where
the atom will appear after the gradient is increased, rather than the location where
the atom started (when there were low magnetic field gradients). Without the APD
focused on the center of the MOT, an average loading time must be “guessed” to
ramp up the magnetic fields and hope that a single atom has arrived in the trap.
Normally, a time of 10-20 s works well for just a single atom to load into the MOT.
This matches will with the loading rate of 0.006 atoms/100ms used in the Bayesian
algorithm (see Section 4.2.4).
In Figure 6.11a, the slope of the fluorescence oscillation amplitude (compared
to the current amplitude) is plotted as a function of the DC background magnetic
field. This field was applied along the lab x-axis (parallel to the face of the imaging
camera). Here, the slope of the fluorescence amplitude was chosen to limit noise that
may arise in individual dI settings. The shape of the graph is not too surprising. It
is symmetric, which is to expected as the anti-Helmholtz magnetic field magnitude
is symmetric. It is a little surprising that the graph is not centered on B = 0, but
this can be easily explained as a result of an additional background field and/or an
181
 2.5  2.5
(a) (b) Balanced Beams (35x scaled, -0.2 G shift)5% Unbalanced (35x scaled, -0.3 G shift)
1D MOT (120x scaled, -0.2 G shift)
Measurement
 2  2
 1.5  1.5
 1  1
 0.5  0.5
 0  0
-0.8 -0.6 -0.4 -0.2  0  0.2  0.4 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4
Background Field, G Background Field, G
FIGURE 6.11. Fluorescence amplitude with background magnetic fields (a) Measured
Data. (b) Scaled and shifted simulations for different models for the MOT.
imbalance in the MOT beams. While considerable effort is put into canceling these
(see Section 3.2.4 in particular), the cancelation must not have been complete.
In Figure 6.11b, three MOT models have been used to attempt to match the
measured data, with varying success. Each model, however, still requires a large
overall scaling factor as well as a manual shift of data center to match the experimental
data. The scalings and shifts are given in the figure key. The 3D theory matches
the experimental results as closely as possible. It does use all six MOT beams and
calculates the force along an axis that is in between the beams, rather than along the
axis of a beam. As discussed in Section 5.2.1, the beams enter the experimental cell
at 45◦ while the camera images the MOT along the length of the cell. Thus when
moving the atoms horizontally, they do not move down the axis of a beam. All six
MOT beams are used in the 3D calculations, with MOT beam phases averaged over.
The 3D calculations do include an addition atom position oscillation of 5 µm, but does
not include the off-axis 3D probability estimation. Because the this estimator maps
3D position onto the magnetic field magnitude, the large background field quickly
swamps the change in field magnitude for atoms that are off-axis. In fact, this tends
182
Fluorescence Amplitude Slope
Fluorescence Amplitude Slope
to narrow the atomic distribution rather than broaden it, which was the goal of
implementing the 3D mapping.
From the theory graphs in Figure 6.11b, there is still a numeric factor of 35
between the measured experimental data and the best theory we have developed.
The increasingly complex simulations discussed in Section 6.3.1 have reduced this
scaling factor from around 350 between the data in Figure 6.4 and the 1-D V-atom
simulation in Figure 6.5 (at 190 µK) to “just” 35 here, but this gap persists.
There is also a large background field offset that does not match well with
the experiment. In the experimental data, the “minimum” of the slope graph is
around -0.2 G, but is still close to zero for the simulated data. In Figure 6.11b, the
simulation results are is shifted manually by 0.2G to 0.38G to have them overlap with
the measured data. These additional shifts could be explained by imbalance in the
experimental MOT beam powers. Despite careful balancing of their beam powers with
the new fiber launcher systems discussed in Section 3.2.4, a lasting power imbalance of
5% is reasonable (although still quite large). This imbalance is simulated as the pink
curve in Figure 6.11b, but only shifts the MOT center by around 0.1G. An imbalance
of 10-20% could result in the observed shift in the data, but this imbalance is much
larger than in the experiment.
A lingering background magnetic field would also be a surprise source for the
central shift, as the “default” currents for the Earth canceling Helmholtz coils are set
by keeping the atom in one place when ramping the MOT coil magnetic field. This
ensures any Earth or lab based background field is canceled, prior to adding a known
background field for the experiment. While this technique to cancel background fields
could also impose a background field that cancels the MOT beam power imbalances
183
(see Equation 2.82), such a large imposed field of the order 0.2G as seen in the data
would require the same 10-20% imbalance as discussed previously.
6.4 Multiple Atom Fluorescence Amplitudes
Another reasonable experiment would be to look at the modulation from more
than one atom. In this case, there are n atoms that are emitting fluorescence at the
modulated rate. These should all have the same phase as the current modulation,
but would have different amplitudes based on the position of the different atoms in
the MOT. For n atoms in the MOT, the spectrum calculation of Appendix E would
change in Equations E.1 where the measured average fluorescence rate and its variance
would become ∑
fl(t) = 〈Fl〉 [1 + [( i i) cos(2πfαt)]
(6.41)
σ2 = σ2 2B + nσα,
where 〈Fl〉 is the average fluorescence rate, i is the oscillation amplitude for MOT
atom i, σ2B is the background fluorescence variance and σ
2
α is the fluorescence variance
for one atom. As shown in Figure 4.3, the deviation from a Poisson distribution is
small for multiple atoms in the tightly confined MOT. So with a large fluorescence
rate, it is still valid to assume the fluorescence is Poisson distributed just as discussed
for a single atom in the appendix and Section 6.2.1. Then the only difference between
the single-atom calculation the multiple atom calculation is an effective fluorescence
amplitude
∑
′ (~r, t) = i (~r, t) , (6.42)
i
where their specific time and position dependence is shown here. With a single
atom, the time dependence is managed by averaging over data runs and the spatial
dependence is what provided for the position and temperature measurements. With
184
 10000
0 atoms 1 atoms 2 atoms 3 atoms
 1000
 100
 10
1497 1501 1497 1501 1497 1501 1497 1501
Frequency, Hz
FIGURE 6.12. Spectra of multiple atoms in a MOT. Magnetic field driven at 1499
Hz with modulation amplitude  = 0.025.
multiple atoms, these averages are challenging as the atoms move independently and
possibly interact (although the interaction cross-section for a few atoms in the MOT
is small [15]). Even if the averaging was straightforward, it would be impossible to
distinguish between photons from each atom so their overall oscillation amplitude
would still appear as just one amplitude. With this, the amplitude from multiple
atoms does not appear as a useful tool for measuring temperatures. Despite its lack
of utility, spectral peaks from multiple numbers of atoms are graphed in Figure 6.12.
6.5 Parametric Resonances
As we’ve seen in Chapter II, atoms in MOTs and objects confined by dipole forces
behave as damped, harmonic oscillators. In such systems, modulating the strength
of the restoring force can bring about new resonances as noted in Section 6.1. These
resonances can caused forces oscillations called parametric oscillations and have been
studied some in optical systems for atoms in large MOTs [136–138] and for beads in
a dipole trap [39].
185
Power Spectrum, kphotons/s
Under parametric oscillation, the equation of motion for the location of an atom
in the MOT (in 1D) is given by
mz̈ = −βż − κ [1 +  cos(ωt)] z, (6.43)
where  is the “strength” of the oscillation and ω is a parametric driving frequency,
√
not necessarily the same as the natural frequency of the oscillator, ω0 = κ/m. As
calculated in Appendix F, additional excitation modes of the oscillator can be induced
if the oscillator strength is above a threshold strength given by
4β2 4β2
2TH = = , (F.8)m2ω20 mκ
and the parametric frequency occurs in the region
√ √
− ω02ω 2 − 2 ω00 TH < ω < 2ω0 + 2 − 2TH. (F.7)2 2
Comparing the equation for TH to the known MOT relationship between κ and β in
Equation 2.83, it simplifies to
2 4β~kTH, MOT = ′ . (6.44)mgFµBBz
Additionally, we could look for parametric excitation in the magnetic trapping force.
As discussed above the magnetic trapping force has a trapping strength related to the
MOT strength according to Equation 6.37. With this the magnetic trapping could
186
Natural Frequency, ω0/2π Threshold Strength
MOT (V-atom) 897 Hz 0.051
Magnetic Trapping (V-atom) 70 Hz 0.65
MOT (87Rb) 2624 Hz 0.017
Magnetic Trapping (87Rb) 205 Hz 0.22
TABLE 6.1. High-gradient MOT parametric resonance conditions. Experimental
parameters have typical MOT values as noted in the text.
also experience parametric excitation with threshold strength
2 4β~2k2ΓTH, mag = . (6.45)
mg2Fµ
2 ′ 2
B (Bz)
For both MOT magnetic confinement and magnetic trapping, the natural frequency
and threshold strengths are given in Table 6.1 using typical (high-gradient) MOT
values δ̃L = −1, |Ω| = 0.5Γ and B′z = 242 G/cm. This table also shows approximate
values for the full 87Rb atom, using the the scaling between the full atom and V-atom
MOT confinement force from fitting the simulated data in Figure 5.2.
While the solution to Equation 6.43 predicts exponential growth of the atomic
position, including higher order terms of the position and velocity (z3 and v3) in the
expansion for the force Equations 2.77 and 2.68, allows for stable oscillations
z(t) = R cos(ωt+ φ), (6.46)
where the amplitude is given by
 ( ) √ ( ) 2
2 16 2ωR = 4 − 2 − 2 2 2 2 2ω 
2 2 ω TH
+  (4 + TH)− 4TH − 1
3A (4 +  ωTH) 0 0
(6.47)
and Aκ/m is the coefficient for the force term proportional to −z3 [138].
187
These oscillations are clear when looking at pictures of the atom, such as those
in Figure 6.13a. These pictures are the average of a number of data runs with one
atom and were taken with an exposure time of 150 ms. The magnetic fields had
a parametric frequency of 1154 Hz, so that the pictures reveal the atom’s position
distribution over 173 oscillation periods. The pictures clearly show the double-peaks
as would be expected for the position distribution of an oscillating particle. Even
without averaging over many pictures the double-peak shape appears, as shown in
Figure 6.16a. The graphs in Figure 6.13b were found by summing pixel intensity
across the rows of the pictures in (a). The slight tilt, around 4◦ was ignored for the
integration. These figures clearly show the double-peaks as would be expected for the
position distribution of an oscillating particle. The oscillation amplitudes measured
in Figure 6.13c compare the distance between peaks.
As discussed in Section 2.5.1, the center of a MOT tracks the location where the
magnetic field vanishes. If there is a background magnetic field, B0, in 1D, the field
is given by Equation 2.80. If the magnetic field gradient oscillates, the location where
the magnetic field vanishes also oscillates according to
−B0 ≈ −B0 B0z0(t) = ′ ′ + ′  cos(ωt) (6.48)Bz [1 +  cos(ωt)] Bz Bz
by solving Equation 2.81 in the equal laser field strength regime. If the oscillations
present in Figure 6.13a were due to the movement of the center of the MOT, a linear
fit of the graph in Figure 6.13b would correspond to the ratio B /B′0 z. This fit gives a
value of B0/B
′
z = 551±53 µm. This data was recorded with a magnetic field gradient
of 219 G/cm, which would require a background field of B0 = 12± 1 G. This is much
larger than the actual background field. Such large oscillations are best explained,
then, by a parametric resonance.
188
(a)
 100
(b) (c)
 75
 50
 0
 0.04  25
 0.08
 0.12
-8 -6 -4 -2  0  2  4  6  8  0.16  0  0  0.04  0.08  0.12  0.16
Modulation Amplitude, ε
Pixel Row Number Modulation Amplitude, ε
FIGURE 6.13. Images of Single Atom Parametric Resonance. (a) CCD exposures
of one atom in a parametric MOT with increase modulator strength. (b) Pixel
intensity of images integrated across a row (normalized to a position distribution).
(c) Oscillation amplitudes of an atom oscillating in a parametric MOT. The curve is
a power law fit to the data: A = 7961.21p .
189
Modulation Amplitude, ε
PPoossitiitoino Onsc ilAlatmion pAmliptluituddee, µ,m μm
For a particle oscillating with amplitude A, the probability to be at a position a
is given by Equation 6.34). We can assume our camera sees a gaussian distribution
for light detected from a single atom located at position a. The variance of this
distribution, σ2, is largely to displacement of the atom from the camera’s focal plane,
but there is also a contribution due to atom motion during an exposure. If the
exposure time is much longer than the period of oscillation for the atom, the light
distribution seen by the atom is given by
∫
1 A
[− ]√ (z − a)
2
√ daL(z) = exp (6.49)
π 2πσ2 −A 2σ
2 A2 − a2
or in dimensionless variables relative to σ, we have
∫ 1 [ ′ ′ ′ ]2 ′
L(z′
1 −(z − A a ) da
) = √ exp √ . (6.50)
π 2πσ2 −1 2 1− a′2
This does not have an analytic form, but is plotted in Figure 6.14. If the oscillation
amplitude is small relative to the standard deviation, the resulting distribution looks
like a single peak. At larger amplitudes, distinct peaks at the edges of the oscillation
can be resolved. These are both observable in Figure 6.13b.
Making one additional change, we can fit Equation 6.49 using the data from
6.13b. This change adds an offset to the integrable value a:
∫ [ ]
1 A −(z − (a− a0))2 da
L(z) = √ exp √ . (6.51)
π 2πσ2 −A 2σ
2
A2 − (a− a )20
We can fit this equation for for values of σ, A, and a0 using the distributions in Figure
6.13b. The data to fit is pixel data, which is the integrated fluorescence over region of
size 15 µm. To fit this appropriately, the fitting algorithm integrates a high-resolution
190
LL(z')0)
0.4
L̃(z = 0)
A0
1.0
= 0.1 0.8
A0 = 1 0.3 0.6
0.4
A0 = 2
A0
0.2
= 4 A0
0 0.2
2 4 6 8 10
A = 8
0.1
z'0
-10 -5 5 10 z
FIGURE 6.14. Light Distribution for Parametric Oscillating Atom. Numeric
solutions for Equation 6.50 with listed values for the scaled oscillation amplitudes
A′ = A/σ. Inset shows plots of Equation 6.53, the normalized light amplitude at the
center of the MOT. Both graphs have unit variance.
position spacing, then sums over a 15 µ region to generate information for effective
pixels before comparing to the data. A few plots of the fitted (higher-resolution)
distributions for some atom images are reproduced in Figure 6.15 along with graphs
of the resulting fitted values for the data in Figure 6.13.
The fit for the detected light distribution standard deviation σ (Figure 6.15d)
is consistently between 15 µm and 30 µm, which corresponds to a width of 1 and 2
pixels. This is consistent with single-atom photographs being a few pixels in size.
In Figure 6.15e, the oscillating atom’s position amplitude grows as the
modulation amplitude increases, which is expected. The values are also close to
the values measured amplitudes in Figure 6.13c, but a little larger. This is expected
from the graphs in Figure 6.14, where the oscillation amplitudes A′ (which were
measured in Figure 6.15e) are larger than the peak of the light distribution (which
191
were measured in 6.14c). This is due to the hard cutoff of the oscillation probability
distribution above a = ±A in Equation 6.34.
The center offsets in Figure 6.15f are all much smaller than a pixel, which is
expected from the images. The large noise in the fit data for small oscillations
are a result of the widening oscillation amplitude closely mimicking an increase of
the detected light distribution variance. The shared impact on the resulting light
distribution makes fitting the two values difficult.
At the center of the oscillation, Equation 6.50 does have an analytic solution
given by ( )
A′2I [ ′ ]0 2
L(z′ = 0) = √ 4 −Aexp (6.52)
2πσ2 4
where I0 is the 0-th order modified Bessel function of the first kind [139]. We can
normalize this to the condition of no oscillation (A = 0) to give a useful equation
(
A′
) [
2 −A′ ]2
L̃(z′ = 0) = I0 exp , (6.53)
4 4
which just represents the center intensity (relative to no oscillations) for the atom.
This equation is plotted as a function the oscillation amplitude in the inset of Figure
6.14. This is an expected shape as the atom spends less of its time close to the center
of the MOT when there is a large oscillation amplitude. We can verify this one of two
ways. First, by examining the light intensity in the center pixel for each atom image
in Figure 6.13a. Here, equation 6.51 is integrated over the size of one pixel. This is
shown in Figure 6.16a.
The second way is by looking at the average fluorescence for a single atom from
the APD data. This fluorescence is shown in Figure 6.16b. This graph shows the
drop of intensity directly. At around t = 31 s, there is a spike in fluorescence as the
192
(a) (d)
Magnetic Field Modulation Amplitude 0.0062
 0.5  30
 0.45
 25
 0.4
 0.35
 20
 0.3
 0.25  15
 0.2
 10
 0.15
 0.1
 5
 0.05
 0  0
 0  0.02  0.04  0.06  0.08  0.1  0.12  0.14  0.16
P(ixebls (1)5 µm) B Modluat(ione Am)plitude, ε
Magnetic Field Modulation Amplitude 0.055
 0.16  90
 0.14  80
 70
 0.12
 60
 0.1
 50
 0.08
 40
 0.06
 30
 0.04  20
 0.02  10
 0  0  0  0.02  0.04  0.06  0.08  0.1  0.12  0.14  0.16
Pixels (15 µm) B Modluation Amplitude, ε
Magnetic Field (Mcodula)tion Amplitude 0.12 (f)
 0.1  0.5
 0.09
 0
 0.08
 0.07
-0.5
 0.06
 0.05 -1
 0.04
 0.03 -1.5
 0.02
-2
 0.01
 0 -2.5
 0  0.02  0.04  0.06  0.08  0.1  0.12  0.14  0.16
Pixels (15 µm) B Modluation Amplitude, ε
Pixels B Modulation Amplitude, ε
FIGURE 6.15. Fitted Light Distributions for Parametric Oscillating Atoms. All
graphs are results fits for Equation 6.51 (a-c) Plots of fitted pixel data for modulation
amplitudes  =0.0062, 0.055, 0.12 respectively. Vertical lines show the ”boundaries”
of pixels. Black lines are fits, red lines are data. (d) Fitted values for light distribution
standard deviation for data in Figure 6.13. (e) Fitted values for position amplitude
for data in Figure 6.13. Line is a power law fit : A = 1960.546. (f) Fitted values for
center offset for data in Figure 6.13. Graphs (d-f) share a common horizontal scale.
193
Pixel Brightness (normalized) Pixel Brightness (normalized) Pixel Brightness (normalized)
Pixel Brightness (normalized) Pixel Brightness (normalized) Pixel Brightness (normalized)
Light Distribution

CenteCre nOter Offffssete, υmt, μm Position Amplitude, μmPosition Amplitude, υm StLaighnt Ddistraiubrtidon  SDtandeardv Dieaviattiioon, nυm, μm
atom enters the MOT. This triggers the bayesian algorithm to reduce the trapping
region and introduce the magnetic field oscillation. Immediately, the atom beings
oscillating and the fluorescence drops as the atom spends less time in the focus of
the APD lens system. This graph is representative of each data run that produced
the single atom images in Figure 6.13a. Here, the intensity is plotted as a function
of A/σ, the position oscillation amplitude relative to the light distribution standard
deviation. Also shown is a plot of Equation 6.54 using average values from the prior
fits (σ = 16.9 µm and a0 = −0.22 µm) for comparison.
Equation 6.53 looks at just the value at z′ = 0, which we cannot measure.
Instead, the values in Figure 6.16c are from a region of space equal to the focus
size of the APD lens system. To calculate this accurate, we integrate Equation 6.51
over some small region, again normalizing to the case of no oscillations. This gives
equation
√ [ ]∫ ∫ −(z−(a−a0))2
( [ ]2/π
3 s A exp 2
F̃ l = [ ]) 2σdz da√ , (6.54)
Erf s√+a0 + Erf s√−a0 −s −A A22σ 2σ − (a− a )20
where the integration region represents the width of the APD lens system focus.
The error functions appear from the normalization, an integral from −s → s over a
gaussian of variance σ and centered at a− a0.
This APD measured signal is plotted in Figure 6.16c, with the integration region
defined as the radius of the APD lens system focal spot size. Represents a fit of the
Equation 6.54. The longer tail in the APD data compared to the pixel intensity arises
from the larger focal size compared to pixel size.
194
(a)  1.2
 1
 0.8
 0.6
 0.4
 0.2
 0
 0  1  2  3  4  5  6  7
RelativeR Oelastivcei lOlasctiilolatnio nA Ammpplitulidteu, dA/eσ, A / σ
(b)  22
 20
 18
 16
 14
 12
 0  10  20  30  40  50  60  70  80  90  100
TimTimee, , sseecc
(c)  1.2
 1
 0.8
 0.6
 0.4
 0.2
 0
 0  1  2  3  4  5  6  7
RelativeR Oelastivcei lOlasctiilolantio nA Ammpplitluitdue,d Ae/σ, A / σ
FIGURE 6.16. Light intensity loss due to parametric oscillations. (a) Normalized
intensity of center pixel for parametrically oscillating atom. (b) Graph of APD
measured intensity for one data run with magnetic field modulation amplitude
 = 0.149. A single picture of this atom is also shown. (c) Normalized intensity of
APD signal for a parametrically oscillating atom. Lines are calculation of Equation
6.54 with average fit values for A, σ, and a0.
195
APD Signal Normalized Intensity PhotPohnot oRn aRtaete, , kkpphhoototnos/nssec/sec Central Pixel Normalized Intensity 
3-Dimensional Parametric Resonances
As discussed in Section 5.1.2, the magnetic field created by the anti-Helmholtz
coils have the form
( )
B~ (~r) = B′
x y
z x̂+ ŷ − zẑ , (5.4)2 2
where B′z is the magnitude of the linear gradient along the MOT axis. Recalling
Equation 2.78, the restoring spring-constant, κ along some axis in a MOT is
proportional to the magnetic field gradient along that axis. From these two, we
must have that the restoring spring-constant along the x- and y-directions in the
MOT is half that of along the z-direction. Writing the equation of motion for an
atom in a MOT, we then have
κ κ
m~̈r = −β~̇r − κzẑ − xx̂− yŷ (6.55)
2 2
where κ is defined as in Equation 2.78 for the V-atom or from the potential recovery
method in Section 5.5. In prior parametric resonance experiments for atoms in a
MOT, the z-direction MOT trapping beams were modulated to cause parametric
oscillations for just piece of this equation [136–138]. However, modulating the MOT
magnetic field also imposes parametric conditions on the x- and y-directions:
κ κ
m~̈r = −β~̇r − κ [1 +  cos(ωt)] zẑ − [1 +  cos(ωt)]xx̂− [1 +  cos(ωt)] yŷ (6.56)
2 2
The parametric oscillations along the z-direction were discussed and observed above.
However, because of the differing strength of the restoring spring-constant along the x-
and y-directions, there will be additional resonances that can appear which can cause
large oscillations perpendicular to the anti-Helmholtz coil axis. Because these occurs
196
in a plane that is normal to the camera’s imaging plane, these will be challenging to
image.
The parametric oscillations in x- and y-direction can be analyzed using the same
method as outlined for the 1D parametric resonance in Appendix F with a simple
change to Equations F.3 to make them
√
ω → 2ω0 + ζ
(6.57)
ν → √1 ω + 1ζ,
2 0 2
√
where ω0 is the oscillation frequency along the z-direction (i.e., ω0 = κ/m). The
resulting parametric resonance region is given by
√ √ √ √
2ω − √ω0 ω2 − 2 0 2 20 ⊥H < ω < 2ω0 + √  − ⊥H, (6.58)
2 2
where the threshold strength to excite x- and y-direction oscillations is
2 2β
2
⊥H = . (6.59)ω20m
2
This formalism reveals that the additional excitations in the x-y plane should appear
at a different frequency from the excitation along the z-direction and at a smaller
threshold strength. It is instructive look at these in terms of x- and y- frequencies:
√ √
2ωxy − ωxy 2 − 2⊥H < ω < 2ωxy + ωxy 2 − 2⊥H (6.60)
and
2
2
β
⊥H = . (6.61)ω2 m2x,y
197
From this form, the parametric resonance region is still centered on twice the natural
frequency in the x-y plane—which would be expected. Comparing to the threshold
strength along the z-axis, TH of Equation F.8, in the x-y plane the threshold strength
is half as large. This is easily explained by noting that the weaker restoring force in
the x-y plane requires a smaller perturbing strength () to overcome. Additionally,
the width of the parametric resonance region in the x-y plane is twice that of the z-
axis as revealed in Equation F.7. This is also easily explained by the weaker restoring
force allowing parametric excitation over a wider range of frequencies. This last piece
could help explain the broad peak of the spectrum in Figure 6.6.
6.6 Non-Sinusoidal Waveforms
The oscillation measurements described above focus on sinusoidal modulations.
The same analysis should hold for any periodic modulation of the MOT magnetic
field. Focusing on three common waveforms, all with fundamental frequency f1 and
amplitude A, the Fourier series for
– square waves contain only odd harmonics with amplitudes A/n,
– triangle waves contain only odd harmonics with amplitudes A/n2, and
– sawtooth waves contain all harmonics with amplitudes A/n,
where n denotes the index of the harmonic [140]. If the current in the anti-Helmholtz
coils is modulated with different waveforms, evidence of it should be present in
fluorescence spectra of a single atom.
Figure 6.17 shows the power spectra form an atom for a sine wave and the three
waveforms above. In each of these, the driving frequency was 401 Hz and the anti-
Helmholtz current modulation amplitude was 0.04. Some things stand out clearly in
198
 1000
Sine Current 1 Fluorescence
1/2
1/4
 100
0
 1  2  3  4  5  6  7  8
HHaarrmmoonnici cN uNmubemr, bner, n
 10
 1000  0  401  802  1203  1604  2005  2406  2807  3208
Frequency, Hz
Square Current 1 Fluorescence 
1/n 
1/n2.5
1/3
 100 1/51/7
0
 1  2  3  4  5  6  7  8
HaHarmrmoonnicic N uNmubmer,b ner, n
 10
 0  401  802  1203  1604  2005  2406  2807  3208
 1000 Frequency, Hz
Triangle Current 1 Fluorescence 
1/n2 
1/n3.5
 100 1/9
1/25
0
 1  2  3  4  5  6  7  8
HHaarrmmoonnici cN uNmubemr, bner, n
 10
 0  401  802  1203  1604  2005  2406  2807  3208
 1000 Frequency, Hz
Sawtooth Current 1 Fluorescence 
1/n 
1/n2.5
1/2
1/3
1/4
 100 1/5
0
 1  2  3  4  5  6  7  8
HaHarmrmoonnicic N uNmubmer,b ner, n
 10
 0  401  802  1203  1604  2005  2406  2807  3208
Frequency, Hz
FIGURE 6.17. Power spectra for different waveforms. Inset graphs show the
amplitudes of peaks at each harmonic, relative to the first. Solid lines show the
expected Fourier series amplitude and dashed line show the expected amplitude scaled
by a factor of the harmonic number, n−3/2.
199
Power Spectrum, kphotons/sec Power Spectrum, kphotons/sec Power Spectrum, kphotons/sec Power Spectrum, kphotons/sec
HarmonHiacrm Aomnicp Alimtuplditued e( r(erellaattiivvee) ) HarmHoanrmico nAicm Amplpiltituuddee ( r(erleatlivaet)ive) HarmHoanrmico nAicm Amplpiltituuddee ( r(erleatlivaet)ive) HarmHoarnmicon Aic mAmpplilittuuddee ( re(rlaetilvaet)ive)
these. First, the spectra do generally show the correct harmonics for each waveform.
Second, the fundamentals in some fo the graphs show sidebands who differ from the
peak by around ±27 Hz. This results from mixing with the unknown 27 Hz magnetic
field oscillation. Lastly, the triangle wave shows almost no harmonics above the first,
although willful examination of the spectra may show a peak at the third harmonic
frequency. The lack of higher harmonic peaks for the triangle wave is directly related
its much faster amplitude decay, n−2.
The inset graph for each waveform in Figure 6.17 shows the amplitude for the
first 8 harmonics calculated from the power spectra for the fluorescence (red data)
and the first 6 harmonics from a power spectrum of the anti-Helmholtz current (blue
data). The current signal is measured with an oscilloscope and the CLN-50 Hall
sensor (see Section 3.3.5), so the maximum harmonic measurable for the current is
limited by the time resolution of the oscilloscope. For clarity, these have been scaled
to the amplitude of the fundamental. The solid black line shows the appropriate
scaling of the Fourier series amplitudes as a function of the harmonic number n and
the dashed black line shows the appropriate scaling with an additional factor of n−3/2.
The amplitudes for the current modulation are close to their “correct” scaling.
For the square wave, the amplitudes of the odd harmonics scale as n−0.92±0.06. For
the triangle wave, the amplitudes of the odd harmonics scale as n−1.81±0.12. For the
sawtooth wave, the amplitudes of the harmonics scale as n−0.84±0.09.
On the other hand, the fluorescence modulation amplitudes scale closer to the
values when an additional factor for n−3/2 is included. This factor arrises from the
non-uniform frequency response of the fluorescence as shown in the power spectrum
of Figure 6.6, as its high frequency tail went as f−3/2. For the square wave, the
200
Waveform Function (one period) Variance
( )
Sinusoid y(t) = A sin 2πt σ2 = A2/2
T
Square y(t) = −AΘ(−t) + Aθ(t) σ2 = A2
Sawtooth y(t) = 2At σ2 = A2/3
T
[( ) ( ) ]
Triangle y(t) = 4A t+ T Θ(−t)− t− T Θ(t) σ2 = A2/3
T 4 4
TABLE 6.2. Variance compared to amplitude for common waveforms. Each waveform
has a period T and amplitude A.
odd harmonics scale as n−2.27±0.19. For the triangle wave, the odd harmonics scale as
n2.89±0.29. For the sawtooth wave, the harmonics scale as n−2.27±0.13.
The calculations for the current amplitudes from their (unshown) spectra do
not derive from Equation 6.28, which applies only for the oscillating average of a
Poissonian sampled fluorescence. The current is measured directly as noted above
and should follow the function
I(t) = I0 [1 +  sin(2πfαt+ φ)] . (6.62)
This gives rise to Equation 6.4, as the magnetic field gradient is directly proportional
to the current. The power spectrum of this current is
I22 I22
P (f) = I2δ(f) + 0 δ(f − f ) + 0I 0 α δ(f + fα). (6.63)4 4
201
When calculating the spectrum numerically, the amplitudes of the two delta functions
are found with with the same area-under the curve calculation as in Equation 6.26.
Doing this for the peak corresponding to each harmonic n gives the current harmonic
amplitudes n for the inset graphs in Figure 6.17.
The overall oscillation amplitude for these different waveforms can also be
calculated directly from the current measurement via its the variance, as was done for
the sinusoidal waves in Section 6.3.2. For the four waveforms of interest, the variance,
σ2, as a function of the amplitude A is given in Table 6.2. Each of these waveforms,
y(t), have an average value of zero when averaged over a period, so calculating the
variance is given by ∫ T/2
σ2 = y2(t)dt. (6.64)
−T/2
This method, of course, cannot extract the amplitudes of the Fourier series for
the waves, but it can easily find the overall wave amplitude, particularly useful for
calibrating amplitudes at different modulation frequencies.
202
CHAPTER VII
CONCLUSIONS
As discussed in Section 6.3.5, there remains a disagreement between experimental
measurements and theoretical predictions for the amplitude of fluorescence
modulations. At temperatures similar to the measured atomic temperature of
160 µK using the release-recapture method, our MOT model predicts fluorescence
amplitudes around 35 times smaller than our measured results. As noted in Section
6.1.3, selecting an appropriate model for the atom in the MOT is important for
our measurement. The measurement results were sensitive enough to rule out the
extended two-level atom model that is common in the literature [25]. Our best model,
a full D2 level structure for
87Rb in a 3D MOT with both added small oscillations
and off-axis atomic probabilities, still fell short of the measured predictions but is a
noticeable improvement over simpler models of both the atom and its environment.
The disagreement could, of course, be from either the data analysis rather than
the model. The analysis has been throughly checked against simulated fluorescence
signals with a known modulation amplitude, mα, and has been shown to correctly
extract its value via Equations 6.27 and 6.28. This fluorescence simulation randomly
sampled photon arrivals from an oscillating average fluorescence rate, producing
data that closely mimics the experimental data without any reference to a source
of photons. It did not simulate the behavior of the atom and tie that to fluorescence
measurements. This check was purely to verify the analysis of Section 6.2 was
appropriate. With confidence that our analysis measures fluorescence amplitudes
appropriately, the disagreement must lie in the theoretical model.
The continued improvement of the theoretical model to approach the measured
result (see, for example, Figure 6.5) gives a hint that there is potentially a missing
203
complication of the MOT that has yet to be included in the simulation. Based on the
magnetic field offset between measured amplitudes and theoretical ones seen in Figure
6.11, one potential route is to more closely model the magnetic field of the coils. At
the most basic level this would require using the full magnetic field of Equation 5.2.
We have exclusively used the linear description in Equation 5.4 which should be very
accurate near the center of the MOT coils. Shifting the MOT further from the center,
with either a background field or imbalanced MOT lasers, could move the atom to a
location where this description breaks down. However, the full field equation and the
linearized field differ in magnitude by less than 1.5% out to 500 µm from the center
of the trap in the z-direction and and around half that for displacements in the x-y
plane. This should comfortably cover the range of positions the atom would explore
even with large background fields.
Along a similar and potentially more valuable addition would take into account
known defects in the MOT coils. In the process of building the water cooled MOT
coils, one coil became tilted a few degrees so that the two are not perfectly coaxial.
This is largely responsible for the 4◦ tilt of the atom’s oscillations seen in Figure
6.13. While the effect does not prevent the MOT from loading, it could be partially
responsible for added noise in fluorescence from a single atom seen in Figure 4.3. The
blue data in this figure was recorded with the (second generation) permanent magnets
while the others are done with the electromagnets (without added modulation). The
permanent magnet MOT shows very close to Poissonian growth of the variance with
atom number while the two data sets for the electromagnets show added noise in
the fluorescence variance. This appears as a general feature of our single atom
electromagnet MOT—the variance in the fluorescence signal from a single atom is
larger than for its permanent magnet equivalent. This could arise from a more
204
complex magnetic field arrangement from the electromagnet coils due to their tilt.
We have investigated solving this tilt by manually rotating the coil to compensate
for the tilt (although this still causes the coils to have parallel axes rather than be
coaxial), but it did not have a noticeable impact on the fluorescence amplitudes.
Additionally, preliminary theoretical investigations of the coil defect has done just by
rotating the magnetic field of one coil slightly, which these showed very little impact
on the theoretical results. This test was done just along one MOT axis and with
an otherwise idealized MOT. How these defects interplay with other magnetic fields,
induced currents, or non-ideal MOT beams could explain some, but likely not all, of
the gap between experiments and the MOT model.
As mentioned briefly in Section 5.2.4, the presence of conducting materials
around our experiment could create image currents that impact the magnetic field
seen by the atom. With an oscillating magnetic field as in our experiments, causes
additional issues as the changing magnetic field will induce currents in the surrounding
material. The eddy currents would then add a time-dependent background field to
the atom, shifting the center the MOT, as shown in Equation 2.81. Of particular
note, the Helmholtz coils around experiment which help cancel the Earth’s magnetic
field will have oscillating currents induced in them. Because the induced currents
act to oppose the change in field, their impact would be to lessen the modulation in
the magnetic field and thus reduce the fluorescence oscillation. While the induced
currents are small, they have been measured directly by monitoring the voltage across
the coils during experimental runs. While these currents should be equal and opposite
for opposing Helmholtz coils, the induced current measured in one of the Helmholtz
coils was significantly lower than the current in the coil opposite it. This coil is the
closest to the vacuum system and has the vacuum flange that mounts the experiment
205
cell in its interior. Because there is this second conducing path for an induced current
(around the flange), the oscillating magnetic flux through the coils is reduced and thus
its induced current is smaller. The magnetic field this makes, then, does not cancel
the field from its opposite coil and leaves some residual induced field. This field will
be much smaller than the field from the MOT trapping coils, but does have a spatial
dependence and would impact the magnetic oscillations seen by the atom as it moves
around the trap. While this effect from the Helmholtz coils will likely be insignificant,
the measurable influence of eddy currents in the flange hints that they do influence
the system and may cause added motion to the atom.
Most experimental measurements have been done in the regime of fast magnetic
field modulation—where its frequency is must larger than the motional frequency of
atoms in the MOT.11 In this regime it was possible to ignore the motion of the atom
in the trap as discussed in Section 6.1.2 and even outside this regime the impact
on the position distribution of the atom is small [39]. As discussed Section 6.3.2
there are multiple relevant frequency scales (around hundreds of hertz for magnetic
trapping and kilohertz for the MOT) which complicate this high frequency regime.
The importance of the magnetic trapping to the shape of the amplitude-frequency
spectrum in Figure 6.7 as well as the parametric resonances seen in Section 6.5 further
suggest that motion of the atom in the trap may play a more significant role that the
present theoretical model assumes.
Extra motion of the atom has only been analyzed in our experiment through the
“forced” oscillations of the atom’s position (see Section 6.3.1). More careful analysis
of the motion of the atom could be done in a few ways. The numeric methods of
11The key exceptions to this being the frequency response measurements in Section 6.3.2 and the
non-sinusoidal waveform measurements in Section 6.6, where a lower frequency was needed so that
multiple higher harmonics would be visible in the spectra.
206
Chapter V could be altered to include velocity of the atom. To do so, the detuning of
each laser must be modified with a doppler shift so that the detuning for the i laser
field become
∆L,i → ∆ ~L,i + ki · ~v, (7.1)
where ~ki is the propagation direction of the field. This change would have to be
implemented for each beam and for both the trapping and repumping laser fields. In
this way, the wavefunction would be for both position and momentum and more
thorough analysis of motional dependence could be done. Similarly, the theory
could be reformed into a Wigner function formalism which reflects both position and
momentum distributions. This has been for many atomic models (in one dimension)
to examine sub-doppler cooling for atoms with many energy levels in a MOT [64, 72].
Doing so could reveal an overlooked physical mechanism for the measured, larger
than anticipated, fluorescence amplitude, much as Sisyphus and polarization gradient
cooling were unpredicted prior to the first MOT temperature measurements [71].
207
APPENDIX A
HOW A FG = 0→ FE = 1 ATOM BECOMES A V-ATOM
The derivation for the equations of motion for the Fg = 0 → Fe = 1 atom
is sketched in Section 2.5. The equation of motion for the evolution of the atom’s
density matrix is given in Equation 2.60. The individual element equations are in
Section A.2.
If there is no electric field which couples the ground state to an excited state,
then the Rabi frequency for that state is zero. For the atom discussed in Section 2.5,
there is no field coupling |0〉 to |g〉, which gives Ω0 = 0.
These equations will be analyzed in the steady state. Taking a peak forward
with that in mind, Equation A.5c drives ρ0,0 → 0 when Ω0 = 0. With this, combining
Equations A.5f, A.5a, and A.5g produce ρg,0 = ρ−,0 = ρ+,0 = 0 and combining
Equations A.5b, A.5d, and A.5e produce ρg,0 = ρ−,0 = ρ+,0 = 0. Thus, in the steady
state all density matrix elements associated with the |E;m = 0〉 state go to zero in
the steady state.
This effectively decouples this state from the rest of the atom, making the atom
behave as a V-atom: an atom with a ground state and two excited states. A level
diagram for such an atom is shown in Figure 2.2c. The internal dynamics of this
atom then evolve under the equations in Section A.1, where we’ve also let ∆0 → 0 for
the nonexistent field’s detuning. In the steady state, the V-atom equations produce
excited state populations
[
2
ρss = |Ω−|
2
−,− (|Ω 2 2 2 2 2 2N −| + |Ω+| ) + 4 (|Ω−| + |Ω+| ) + 16|Ω+| (δ− − δ+) +
( ) ( )] (A.1)
8 (|Ω |2− δ+ − |Ω 2+| δ−) (δ 2− − δ+) + 4 1 + 4δ+ 1 + (δ 2− − δ+)
208
and
[
|Ω |2ρss = ++,+ (|
2
Ω 2−| + |Ω 2+| ) + 4 (|Ω |2 + |Ω |2− + ) + 16|Ω 2−| (δ− − δ+)2 +N
( ) ( )] (A.2)
8 (|Ω |2− δ+ − |Ω+|2δ−) (δ− − δ+) + 4 1 + 4δ2− 1 + (δ− − δ )2+ ,
where Ω̃± = Ω± /Γ, δ± = ∆±/Γ, and the normalization factor is
| |2 | |2 3
[ ]
N = 2 ( Ω− + Ω+ ) + (|Ω |2− + | 2Ω 2+| ) 9 + 4 (δ− + δ 2+)
[ ]
+4 (|Ω−|2 + |Ω 2+| ) 1 + (δ− − δ+)2 +
[ ( ) ( ) ]
4|Ω |2 2− 4δ (2δ − δ ) δ2 2+ − + − − − δ+ + 3 δ2 + δ2− + + 2δ2+ +
[ ( ) ( ) ]
4|Ω |2 2 + 4δ (2δ − δ ) δ2 − δ2 + 3 δ2 + δ2 2− + − + − + − + + 2δ− +
(A.3)
− 220 [|Ω |2+ δ 2− + |Ω−| δ+] + 20|Ω 2 2+| |Ω−| [δ− − δ+]2 +
8 [|Ω |2δ + |Ω |2− − + δ 2+] [|Ω−| δ+ + |Ω 2+| δ−] +
[ ] [ ]
16 1 + (δ 2 2 2− − δ+) |Ω+| δ− + |Ω 2 2−| δ+ +
[ ] [ ] [ ]
4 1 + 4δ2− 1 + 4δ
2 2
+ 1 + (δ− − δ+)
The ground state population is given by ρss = 1− ρss − ρssg,g +,+ −,−.
209
A.1 V-atom Equation of Motion
( ∗ )ρ̇ i−,− = −Γρ−,− − Ω2 −ρg,− − Ω−ρ−,g
( )
ρ̇ i ∗−,+ = − [Γ + i (∆− −∆+ − 2∆B)] ρ−,+ − Ω2 −ρg,+ − Ω+ρ−,g
[ ]
ρ̇ = − Γ−,g + i (∆ −∆ ) ρ + i− B −,g Ω ρ + iΩ−2 2 + −,+ (ρ2 −,− − ρg,g)
( )
ρ̇+,− = − [Γ− i (∆− −∆+ − 2∆ i ∗B)] ρ+,− + Ω−ρ+,g − Ω+ρ2 g,−
( )
ρ̇+,+ = −Γρ − i+,+ Ω ρ ∗2 + g,+ − Ω (A.4)+ρ+,g
[ ]
ρ̇ Γ+,g = − + i (∆+ + ∆B) ρ+,g + iΩ ρ + iΩ+ (ρ2 2 − +,− 2 +,+ − ρg,g)
[ ]
− Γ − iΩ
∗
ρ̇g,+ = i (∆+ + ∆ ) ρ
i ∗ +
2 B g,+
− Ω−ρ−,+ − (ρ2 2 +,+ − ρg,g)
[ ] ∗
ρ̇ Γg,− = − − iΩi (∆ −∆ ) ρ − iΩ∗ ρ − − (ρ − ρ )2 − B g,− 2 + +,− 2 −,− g,g
( ) ( )
ρ̇g,g = Γ (ρ−,− + ρ ) +
i Ω ρ ∗ i ∗+,+ 2 − g,− − Ω−ρ−,g + Ω+ρg,+ − Ω+ρ2 +,g
210
A.2 Jg = 0→ Je = 1 Equation of Motion
i ( ∗ )ρ̇−,− =− Γρ−,− − Ω−ρg,− − Ω−ρ−,g2
ρ̇−,0 =− [Γ + i (∆− −∆B −∆0)] ρ−,0 −
i i
Ω ∗−ρg,0 + Ω
2 2 0
ρ−,g (A.5a)
i ( )
ρ̇−,+ =− [Γ + i (∆− −∆+ − 2∆B)] ρ−,+ − Ω−ρg,+ − Ω∗+ρ−,g
[ ] 2
Γ i i iΩ−
ρ̇−,g =− + i (∆− −∆B) ρ−,g + Ω+ρ−,+ + Ω0ρ−,0 + (ρ−,− − ρg,g)
2 2 2 2
i i
ρ̇ ∗0,− =− [Γ + i (∆0 −∆− + ∆B)] ρ0,− + Ω−ρ0,g − Ω0ρg,− (A.5b)2 2
i
ρ̇0,0 =− Γρ − (Ω ρ ∗0,0 0 g,0 − Ω0ρ0,g) (A.5c)2
− − − iρ̇ ∗ i0,+ = [Γ + i (∆0 ∆+ ∆B)] ρ0,+ + Ω ρ0,g − Ω0ρg,+ (A.5d)
[ ] 2
+ 2
− Γ i i iΩ0ρ̇0,g = + i∆0 ρ0,g + Ω−ρ0,− + Ω+ρ0,+ − (ρg,g − ρ0,0) (A.5e)
2 2 2 2
i i
ρ̇+,− =− [Γ− i (∆− −∆+ − 2∆B)] ρ+,− − Ω+ρg,− + Ω∗−ρ+,g2 2
i i
ρ̇+,0 =− [Γ + i (∆+ + ∆B −∆0)] ρ+,0 − Ω+ρg,0 + Ω∗0ρ+,g (A.5f)2 2
− − i
( )
ρ̇+,+ = Γρ+,+ Ω+ρg,+ − Ω∗+ρ+,g
[ 2 ]
Γ i i iΩ+
ρ̇+,g =− + i (∆+ + ∆B) ρ+,g + Ω−ρ+,− + Ω0ρ+,0 − (ρg,g − ρ+,+)
[ 2 ] 2 2 2
Γ i ∗ i ∗ iΩ
∗
ρ̇ 0g,0 =− − i∆0 ρg,0 − Ω−ρ−,0 − Ω+ρ+,0 + (ρg,g − ρ0,0) (A.5g)
[ 2 2] 2 2
− Γ − i i iΩ
∗
ρ̇g,+ = i (∆+ + ∆
∗ ∗ +
B) ρg,+ − Ω−ρ−,+ − Ω0ρ0,+ + (ρg,g − ρ+,+)
[ 2 ] 2 2 2∗
ρ̇g,− =−
Γ − i i iΩi (∆− −∆B) ρg,− − Ω∗+ρ+,− − Ω∗0ρ −0,− + (ρg,g − ρ−,−)2 2 2 2
i ( )
ρ̇ ∗g,g =Γ (ρ0,0 + ρ−,− + ρ+,+)− Ω−ρ−,g − Ω−ρg,− +2
i (− Ω∗
) i ∗
+ρ+,g − Ω+ρg,+ − (Ω0ρ0,g − Ω0ρg,0)2 2
211
APPENDIX B
DUAL POLARIZER NOISE REDUCTION
Let the output of the fiber have electric field E~ = EÊ with polarization angle
α+ δn(t) where δn(t) is a small amount of noise (on the order of a few degrees). This
field, along with its relation to polarizers are shown in Figure B.1.
B.1 One Polarizer
As shown in Figure B.1a, one polarizer has polarization vector D̂out at angle γ.
After passing through this polarizer, the electric field becomes
( )
E~ ~out = E · D̂out D̂out = E cos [γ − α− δn(t)] D̂out.
The power output is then
P 2out = Pin cos [γ − α− δn(t)] .
Doing a series expansion around small angles δn(t) ≈ 0, the power output is
[ ]
P 1 polout = Pin cos
2 [γ − α]+2Pin cos(γ−α) sin(γ−α)δn(t)+Pin sin2(γ − α)− cos2(γ − α) δ2n(t).
In general, the noise in the power output of the beam is of order δn(t), except in
limiting cases where γ − α = nπ .
2
212
E~ = EÊ D̂out E~
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FIGURE B.1. Laser power control with polarizers. (a) Electric field names (above
beam) and field polarization directions (below beams) before and after one polarizer.
(b) Electric field names and field directions before, between, and after two polarizer.
B.2 Two Polarizers
As shown in Figure B.1b, assume there is a middle polarizer between the fiber
and output polarizer. The middle polarizer has polarization vector D̂mid at angle β.
Then after the middle polarizer the electric field is
( )
E~ ~mid = E · D̂mid D̂mid = E cos [β − α− δn(t)] D̂mid.
After the output polarizer, the electric field is
( )
E~ ~out = Emid · D̂out D̂out = E cos [β − α− δn(t)] cos[γ − β]D̂out.
where, again, γ is the angle of the output polarization vector D̂out. The power output
is
P 2out = Pin cos [β − α− δ (t)] cos2n [γ − β].
213
Doing another expansion for small noise, the power is
Pout = Pin cos
2 [β − α] cos2 [γ − β] + 2Pin cos(β − α) sin(β − α) cos2 [γ − β] δn(t) +
[ ]
P cos2in [γ − β] sin2(β − α)− cos2(β − α) δ2n(t).
If the middle polarizer if aligned with the ideal polarization axis of the fiber β = α,
this becomes
P 2 pol 2out = Pin cos [γ − α]− Pin cos2 [γ − α] δ2n(t).
Therefore, for the 2 polarization setup, the noise is reduced to order δ2n(t).
214
APPENDIX C
MOT COIL WATER COOLING RATE
Due to large current used in the experimental anti-Helmholtz coils, they are
designed to be water-cooled to prevent overheating. This cooling method is analyzed
in detail here.
C.1 Heat Equation
In considering the heating of the coils, the only input source of energy is the power
dissipated by the current, P = I2R. This heating is dissipated by 5 mechanisms:
– Heating the water via conduction, Qwater
– Power radiated from coils, Prad
– Heating the aluminum coil mount, QAl
– Heating the copper wires, QCu
– Heating the air around the coils via conduction, Qair
Equating the input and output powers gives,
I2
d d d d
R = Qwater + Prad + QAl + QCu + Qair (C.1)
dt dt dt dt
A few simplifications should be made. Assume the copper and aluminum of the coils
have the same temperature. Both the convection to the air and the radiated power
are much smaller than the convection to the water and heating of the coils, so both of
these terms can be dropped12. The heating rate to the water, Qwater, is derived below
12Including conduction to the air changes the growth rate of the temperature, the variable γ in
as defined in equation C.8, by less than 1%. The radiative power dissipated, even though it scales
as T 4, is a few orders of magnitude less than other loss mechanisms.
215
in Section C.2. The heating of the aluminum and copper follows from introductory
thermodynamics as
QAl = mAlCAl∆T (C.2)
QCu = mCuCCu∆T , (C.3)
respectively. Additionally, the resistance of the coils will change as the coils change
temperature as,
R(T ) = R0 [1 + α(T − T0)] (C.4)
where R0 and T0 are a reference resistance and temperature respectively.
Combing these equations with C.1 gives
( [ ])
I2
−hcA ∂T
R0 [1 + α(T − T0)] = ṁwCw(T − Tw) 1− exp + (mC)eff , (C.5)
ṁwCw ∂t
where ṁw is the water mass flow rate through the cooling channel and (mC)eff is an
effective value for the combined heating of the aluminum mount and copper wires. If
the have the same temperature at all times, then this equals
(mC)eff = mAlCAl +mCuCCu. (C.6)
Because of the large contact area between the wires and mount and their relatively
small volume, it is safe to make this equal-temperature assumption.
The power equation has the form
∂T
= γT + β (C.7)
∂t
216
with
( [ ])
I2R α− ṁC 1− exp −hcA0 w ṁCw
γ = (C.8)
(mC)eff ( [ ])
I2R (1− αT ) + ṁC T 1− exp −hcA0 0 w w ṁCw
β = (C.9)
(mC)eff
The differential equation is solved by
( )
β β
T (t) = Tw + e
γt − (C.10)
γ γ
This result is problematic if γ > 0. The limiting case where γ = 0 (i.e. when the
heating just balances the cooling) gives a steady state temperature of Tw when the
current is √ ( [ ])
ṁCw
Ilim = 1−
−hcA
exp . (C.11)
R0α ṁCw
Currents above this will heat the coils indefinitely (γ > 0) according to the model.
This indefinitely heating results from ignoring radiative and thermal conduction to
the air. Currents below Ilim will result in a steady temperature of Tf = β/ |γ|. As
noted in the text, currents used in the experiment are far below Ilim.
With the solution for T (t), the resistance of the wires as a function of time
becomes
′ [ ( −| | )]R(t) = R0 1 +Rm 1− e γ t , (C.12)
where R′0 is the chilled coil resistance
R′0 = R0 [1− α(T0 − Tw)] (C.13)
217
Parameter Value
Aluminum support volume 3.52× 10−5 m3
Copper wire volume 1.73× 10−5 m3
Water flow rate, ṁw [141]
13 3.43 litre / min = 0.217 kg / s
Water chiller set temperature, Tw 15
◦C
Cooling channel Length, L 9.7× 10−2 m
Cooling channel surface area, A 3.29× 10−2 m2
TABLE C.1. Water-cooled MOT coil parameters. Variable names reference equations
C.2, C.3, and C.5.
T0 is a reference temperature used in defining the “normal” resistance of the wires,
R0, and Rm is a maximum change of resistance
I2R0 ∣
∣ ( [− ])∣∣ ṁCw hcA ∣
−1
R = ∣m | | = ∣α− 1− exp ∣ . (C.14)γ (mC) 2eff I R ṁCw
As noted in the text, measuring the resistance of the anti-Helmholtz coils as a function
of time is straightforward and provides a clear method to find a value for γ.
C.2 Water Cooling Rate Derivation
A closed channel through which water flows can be used to regulate the
temperature of the bulk medium. The rate that heat flows from the bulk into the
water and leaves the medium is calculated here, following the general formalism of
[142].
Divide the length of the channel into a small segment dx, as shown in figure C.1.
The mass of water, dm, that flows through this channel in time dt will absorb heat,
13We cannot locate a manual for our chiller, RTE100. The RTE101 has the same specifications
as our chiller and the plumbing and circuit diagrams match the innards of our chiller, leading us to
trust the RTE101 manual.
218
L
W
H
dx
FIGURE C.1. Water cooling channel dimensions
dq, from the walls of the channel according to
dq = q̇dt, (C.15)
where q̇ is the rate of heat transfer. This rate is from conduction from the walls into
the fluid, which follows
q̇ = hc(Ts − Tf )dA, (C.16)
where dA is the surface area of the fluid that is in contact with walls of the channel,
hc is the conduction coefficient (discussed below in Section C.3), and Ts is the (fixed)
temperature of the walls of the channel and Tf is the temperature of the fluid. If the
perimeter of the volume of water has length P, then the surface area is just dA = Pdx.
Assume the heat is absorbed uniformly throughout the fluid (there is no
temperature gradient from the surface into the bulk of the liquid), so that the absorbed
heat will warm the water by an amount dT as
dq = dmCdTf , (C.17)
219
where C is the specific heat of the water. Combining equations C.15 through C.17
gives
hc(Ts −
dm
Tf )Pdx = CdTf . (C.18)
dt
In this equation dm/dt is just the mass flow rate through the fluid, ṁ. The total
change in the temperature from the input, Tin, to the output, Tout, is found by
integrating along the total length of the channel, L.
∫ ∫
h P L Toutputc dTf
dx =
ṁC 0 T T − T[ ] input s f−hcPL Ts − Tout
exp = − (C.19)ṁC Ts Tin
Now, consider the total heat absorbed by the water in the channel,
Q = mC(Tout − Tin). (C.20)
This occurs at a rate
[ − ]Ts Tout
Q̇ = ṁC(Ts − Tin) 1− − . (C.21)Ts Tin
Using the result of equation C.19 gives an equation for the rate that heat is absorbed
into the water from the bulk of
[ [− ]]hcA
Q̇ = ṁC(Ts − Tin) 1− exp , (C.22)
ṁC
where A = PL is the total surface area of the water-flow channel and Ts is the
temperature of the walls of the water-flow channel.
220
C.3 Conduction Coefficient
The conduction coefficient, hc can be related to the Nusselt Number, Nu a ratio
between the thermal conduction of heat from surface into fluid and the thermal
convection of heat into into the fluid. This is,
hcLc
Nu = (C.23)
k
where Lc is a characteristic length of a flow and k is the thermal conductivity of
the fluid. For a long rectangular channel with a width that is 4-times the height
and a uniform temperature of the channel, the Nusselt number is 4.439 [142]. The
characteristic length for a flow through a long tube is
cross-sectional area 2HW
Lc = 4 = (C.24)
perimeter of cross-sectional area H +W
where H and W are as shown in figure C.1. This gives a conduction coefficient of
kNu(H +W )
hc = (C.25)
HW
Relating this to our experiment, for water, k = 0.6098 W / mK [143], and our channel
has H = 1/8′′ and W = 1/2′′. These give the conduction coefficient hc = 5.3 ×
102 W / K m2.
221
APPENDIX D
BAYESIAN EVOLUTION DERIVATION
Equation 4.4 gives the probability distribution for fluorescence rate (variable x)
from a single atom, Fl1-at(x). With a known number of atoms in the MOT, the
measurement of the fluorescence from n atoms in the MOT can be used to update
information about the fluorescence rate average, R, and standard deviation, σR, from
one atom with Bayes theorem.
With a noisy measurement, y as in section 4.2.3, the noise in a measurement
must have value
ζ = y − (B + nr) , (D.1)
where B is the (assumed constant) background fluorescence rate, n the number of
atoms in a MOT, and r is the single-atom fluorescence rate. The probability for the
noise to have this value is
[ ]
− √ 1 − (y − (B + nr))
2
p (ζ = y B + nr) = exp ,
2πσ2 2σ
2
ζ
ζ
which is similar to Equation 4.8 with multiple atoms in the MOT. Then, Bayes’
theorem just says that the single atom signal fluorescence evolves according to
Fl1-at(r)p (ζ = y −B + nr)
Fl1-at(r)→ ∫ ∞
Fl1-at(r)p (ζ = y −B + nr) dr
−∞
222
The normalization function is
∫ ∞ 1
Fl1-at(r)p (ζ = y −B + nr) dr = ×
−∞ 2πσRσzeta
∫ ∞ [ [ ]− ](r −R)2 − (y − (B +Nr))2
exp exp dr,
2
−∞ 2σR 2σ
2
ζ
which integrates to
[ ]
2
√ 1 − (y( ) exp (
− (B +NR))) . (D.2)
2 2 2
2π σ2N2 + σ2 2 σRN + σζR ζ
Thus, the single-atom fluorescence probability evolves according to
[ ] [ ]
→ √ 1 − (r −R)
2 − (y − (B + nr))2
Fl1-at(r) exp exp ×
2 2 2σ2 2σ 2σ
2π R
σζ R ζ
(σ2 2 2
[ r
n +σζ) ]
− (y(− (B + nR)))
2
exp
2 σ2 n2 + σ2R ζ  [ ] 2
− − (Rσ
2
ζ+ynσ
2
R−Bnσ
2
 R)r 2 21  n2σ +σ √ × R ζ = √ exp σ2 σ2  .σ2 σ2
2π R ζ 2
R ζ 
2 2 2
n2σ2 +σ2
n σR+σζ
R ζ
This is, of course, just a Gaussian. Starting at i, and evolving to i+1 while measuring
data point yi+1, the average and variance evolve as:
R σ2i ζ+(yi−B)nσ
2
R R,ii+1 = n2σ2R,i+σ2ζ (D.3)
σ2 2
σ2 = R
σ
i ζ
R,i+1 n2
.
σ2 2R,i+σζ
It is good to note that if n = 0, then R 2 2i+1 = Ri and σR,i+1 = σR,i. This should
be the case as with no atoms in the MOT, no information can be gained about the
223
fluorescence rate from a single atom. For this reason, when there are no atoms in the
MOT, our algorithm instead updates the background fluorescence rate average and
standard deviation as discussed in Section 4.2.6.
224
APPENDIX E
GAUSSIAN SAMPLED OSCILLATION AMPLITUDE
Photons are measured by the APD as both a background rate, β, which is
Gaussian-distributed with average rate B and a fluorescence rate from the atom,
α which is Gaussian distributed with variance σα and whose average oscillates in
time as described in the text:
α(t) = α0 [1 +mα cos(2πfαt)] . (6.22)
These two random values are sampled together, so that their total rate f and rate
variance σ2 are just the sum of the two,
fl(t) = 〈Fl〉 [1 +  cos(2πfαt)]
(E.1)
σ2 = σ2 + σ2B α,
with average total background fluorescence and (dimensionless) total fluorescence
oscillation amplitude defined as
〈Fl〉 = B + α0, and
(E.2)
 = α0mα/ (B + α0) .
For single-atom fluorescence, as shown in Figure 4.3, the fluorescence is relatively
close to a Poisson distribution and can be written after time T as
σ2(T ) = 〈fl(T )〉 (1 + η) , (E.3)
225
where η is a parameter that compares the variance to the mean rate. Because of the
Gaussian assumption for the photon rates, as time progresses both the variance and
the mean for the fluorescence distribution grows linearly. Thus, the parameter η is
constant over all times. The variances as measured in Figure 4.3 should then hold
for estimates of the oscillation parameter (m) for the atom. Photon arrivals between
time t = 0 and t = T produce a pulse chain,
∑
p(t) = δ(t− ti) (E.4)
i
which has a power spectrum [132]
1 [ { } ]
S(f) = 〈fl(T )〉+ 〈fl2(T )〉 − 〈fl(T )〉 〈ei2πf(tc−tc′ )〉 , (E.5)
T
where the brackets donate statistical averages and the exponentials result from Fourier
transforms of the photon arrivals [144]. With a variance defined relative to the average
rate, it is possible to write
〈fl2〉 = σ2 − 〈fl〉2 = 〈fl〉+ 〈fl〉η − 〈fl〉2, (E.6)
which simplifies the power spectrum to
1 [ { } ]
S(f) = 〈fl(T )〉+ 〈fl(T )〉2 + η〈fl(T )〉 〈ei2πf(tc−tc′ )〉 . (E.7)
T
226
To calculate the statistical average of the exponential, integrate over both tc and tc′
with both exponentials normalized by average photon rate. This produces
∣∣∫ ∣∣T 2
〈 1ei2πf(tc−tc′ )〉 = ∣〈 〉 ∣ f(t)e
−i2πftdt∣ , (E.8)
f(T ) 2 ∣0
so that the spectrum becomes
[ ∣ ∣ ] ∣∣∫ ∣∣∫
2
T
1 T
2 ∣ f(t)e−i2πftdt∣ ∣
∣
S(f) = 〈f(T )〉+ ∣ f(t)e−
∣
i2πft η 0dt∣∣ + 〈 〉 . (E.9)T 0 T f(T )
The first term is identical to the Poisson-distributed rate calculated by Matzner and
Bar-Gad [132], while the second term corresponds to the Gaussian modification made
here. The added noise from the Gaussian-distributed fluorescence then increases the
spectral power over the Poisson-distributed signal. The average fluorescence is given
by ∫ T
〈f(T )〉 = f(t)dt = 〈Fl〉T [1 +  sinc(2πfαT )] . (E.10)
0
In the limit of small  and at f = fα, the last term is
∣∣∫ ∣2∣ T ∣η fl(t)e−i2πftdt∣ [0 2
〈 〉 = η〈Fl〉× + sinc (πfαT ) +  sinc (πfαT ) +T fl(T ) 4 ]
2 sinc(2πfαT ) {1 + 4 sinc(πfαT ) + sinc (2πfαT )} .4
(E.11)
All of the oscillating terms decay rapidly at high frequencies (or long times), so they
can be dropped. Inserting the Poisson-distributed spectrum, gives a final form for
the power spectrum at f = fα
〈Fl〉2T2 2〈Fl〉
S (f = fα) = 〈Fl〉+ + η . (E.12)
4 4
227
Similarly, far from the driving frequency, fα, the spectrum becomes just
S(f →∞) = 〈Fl〉, (E.13)
which is just the average measured fluorescence rate (from the background and an
atom). Now, from the power spectrum from the APD, the oscillation amplitude can
be measured as √
4 [S (f = fα)− 〈Fl〉]
 = 〈 〉 (E.14)Fl 2T + η〈Fl〉
Just setting η = 0 returns the Poisson-distributed result of Matzner and Bar-Gad
[132] for the oscillation parameter mp. Comparing these results gives
√
〈Fl〉T
 = 〈 〉 mp. (E.15)Fl T + η
The quantity 〈Fl〉T is the total number of photons counted in time T without any
oscillations. This value should be much larger than the added super-Poissonian noise,
η. Using the simpler Poisson result of Matzner and Bar-Gad is justified.
The atom’s fluorescence is responsible for the oscillations. Then writing the
Poisson form of  (setting η = 0) in terms of the atomic oscillation amplitude gives
√
2 S (f = fα)− (B + α0)
mα = . (E.16)
α0 T
228
APPENDIX F
PARAMETRIC RESONANCE DERIVATION
A parametric resonantor is one where the value for the restoring spring constant
oscillates [131]. This means that the resonant frequency also oscillates. . Taking
the simplest equation for a damped, harmonic oscillator and allowing it to become
parametric gives the differential equation for its position, z
β κ
z̈ + ż + z [1 + cos(ωt)] = 0 (F.1)
m m
where m is the mass of the oscillator, κ is the restoring force spring constant, β is
the damping coefficient, and ω is the oscillation frequency of the parametric spring
constant. This frequency is not necessarily the same as the constant oscillation
√
frequency, ω0 = κ/m. Assume that the differential equation can be solved by
an equation z(t) given by
z(t) = a(t) cos(νt) + b(t) sin(νt)
[ ]
ż(t) = [ȧ(t) + νb(t)] cos(νt) + ḃ(t)− νa(t) sin(νt) (F.2)
[ ] [ ]
z̈(t) = ä(t) + 2νḃ(t)− ν2a(t) cos(νt) + b̈(t)− 2νȧ(t)− ν2b(t) sin(νt)
for some frequency ν. Putting these equations into the differential equation F.1 gives
[ ]
0 = ä+ 2νḃ− 2 βν a+ ȧ+ ω20νb+ ω20a+ ω20a cosωt cos(νt) +
[ m ]
b̈− β β2νȧ− ν2b+ ḃ− νa+ ω20b+ ω20b cosωt sin(νt).m m
229
Replacing the two frequencies with the natural frequency of the oscillator and a small
deviation frequency ζ:
ω → 2ω0 + ζ
(F.3)
ν → ω + 10 ζ,2
and noting that
cos(ωt) cos(νt) = cos [(ω + ν)t] + cos [(ω − ν)t] = cos [3νt] + cos [νt]
cos(ωt) sin(νt) = sin [(ω + ν)t]− sin [(ω − ν)t] = sin [3νt]− sin [νt] ,
the differential equation becomes
[ ]
β β
0 = ä+ 2νḃ− ν2a+ ȧ+ νb+ ω2 2
m m 0
a+ ω0a cos(νt) +
[ ]
b̈− β β2νȧ− ν2b+ ḃ− νa+ ω2 20b− ω0b sin(νt),m m
after dropping the quickly rotating 3ν terms. Further, assume that
ȧ ∼ ζa
(F.4)
ḃ ∼ ζb
230
Dropping terms that scale as the very small ζ2, the following changes are made to
the differential equations:
ä ≈ 0
b̈ ≈ 0
ν2 ≈ ω0(ω0 + ζ)
νȧ ≈ ω0ȧ
νḃ ≈ ω0ḃ.
These changes give a final differential equation of the form
[ ]
− a βȧ βνb0 = 2ḃ ζa+ ω0 + + cos(νt) +
[ 2 ω0m ω0m ]
− b βνa βḃ2ȧ+ ζb+ ω0 + − sin(νt).
2 ω0m ω0m
Under the assumption of small damping, β ∼ ζ, the differential equation can be
simplified to
[ ]
a β
0 = 2ḃ− ζa+ ω0 + b cos(νt) +
[ 2 m ]
− b β2ȧ+ ζb+ ω0 + a sin(νt).
2 m
This is solved only when the coefficients for both cos(νt) and sin(νt) vanish. Forcing
both to vanish produces coupled differential equations for ȧ and ḃ that follow the
231
vector equation
   ( )  
 a   β 1d ω + ζ a  − 2m 4
0 2  
=
dt ( )   . (F.5)
b 1ω ζ β
4 0
− b
2 2m
Assuming an eigenvalue solution, the equations of motion are
a(t) = a e−λt0
b(t) = b e−λt0
with eigenvalue √
β 1 ζ2
λ = ∓ 2ω20 − . (F.6)2m 16 4
A non-parametric resonator will be damped to z = 0. This solution will behave
similarly unless λ < 0. This exponential growth of the oscillator amplitude occurs if
2
ζ2
1
< 2ω20 −
β
.
4 m2
In terms of the frequencies, this parametric resonance is excited for frequencies
√ √
2ω − ω0 2 − 2 ω00 TH < ω < 2ω0 + 2 − 2TH (F.7)2 2
if the parametric strength is above a threshold strength
2 4β
2
TH = . (F.8)m2ω20
232
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