DEVELOPMENT OF A STRONTIUM MAGNETO-OPTICAL TRAP FOR
PROBING CASIMIR–POLDER POTENTIALS
by
PAUL J. MARTIN
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
March 2017
DISSERTATION APPROVAL PAGE
Student: Paul J. Martin
Title: Development of a Strontium Magneto-Optical Trap for Probing Casimir–Polder
Potentials
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 G. Raymer Chair
Daniel A. Steck Advisor
Benjamin J. McMorran Core Member
George V. Nazin Institutional Representative
and
Scott L. Pratt Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate
School.
Degree awarded March 2017
ii
c© 2017 Paul J. Martin
This work is licensed under a Creative Commons
Attribution-NonCommercial (United States) License.
iii
DISSERTATION ABSTRACT
Paul J. Martin
Doctor of Philosophy
Department of Physics
March 2017
Title: Development of a Strontium Magneto-Optical Trap for Probing Casimir–Polder
Potentials
In recent years, cold atoms have been the centerpiece of many remarkably
sensitive measurements, and much effort has been made to devise miniaturized
quantum sensors and quantum information processing devices. At small distances,
however, mechanical effects of the quantum vacuum begin to significantly impact
the behavior of the cold-atom systems. A better understanding of how surface
composition and geometry affect Casimir and Casimir–Polder potentials would benefit
future engineering of small-scale devices. Unfortunately, theoretical solutions are
limited and the number of experimental techniques that can accurately detect such
short-range forces is relatively small.
We believe the exemplary properties of atomic strontium—which have
enabled unprecedented frequency metrology in optical lattice clocks—make it an
ideal candidate for probing slight spectroscopic perturbations caused by vacuum
fluctuations. To that end, we have constructed a magneto-optical trap for strontium
to enable future study of atom–surface potentials, and the apparatus and proposed
detection scheme are discussed herein. Of special note is a passively stable external-
cavity diode laser we developed that is both affordable and competitive with high-end
commercial options.
iv
CURRICULUM VITAE
NAME OF AUTHOR: Paul J. Martin
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR
University of New Mexico, Albuquerque, NM
DEGREES AWARDED:
Doctor of Philosophy, Physics, 2017, University of Oregon
Master of Science, Physics, 2014, University of Oregon
Bachelor of Science, Physics, 2007, University of New Mexico
Bachelor of Science, Applied Mathematics, 2007, University of New Mexico
AREAS OF SPECIAL INTEREST:
Atomic, Molecular, and Optical Physics
Experimental Design and Control
PROFESSIONAL EXPERIENCE:
Graduate Teaching Fellow, University of Oregon, 2007–2011, 2013–2017
Graduate STEM Fellow in K–12 Education, NSF GK–12, University of Oregon,
2011–2013
Undergraduate Research Assistant, University of New Mexico, 2007–2008
Physics Tutor, Center for Academic Program Support at the University of New
Mexico, 2004–2007
GRANTS, AWARDS AND HONORS:
Weiser Senior Teaching Assistant Award, University of Oregon, 2015
Graduate STEM fellow in K-12 Education, NSF GK-12, University of Oregon,
2011-2013
Honorable Mention, National Science Foundation Graduate Research Fellowship
Program, 2008
Qualifying exam award for outstanding performance, University of Oregon, 2008
v
Eoin Grey Award for “Best graduating senior in physics” at the University of
New Mexico, 2007
Departmental and University Honors, University of New Mexico, 2007
Summa cum Laude, University of New Mexico, 2007
National Merit Scholarship Finalist, 2002
PUBLICATIONS:
Chris Seck, Paul J. Martin, Eryn C. Cook, Brian Odom, and Daniel A. Steck,
“Noise reduction of a Libbrecht–Hall-style current driver,” Review of Scientific
Instruments 87, 064703 (2016).
Jayampathi C. B. Kangara, Andrew J. Hachtel, Matthew C. Gillette, Jason T.
Barkeloo, Ethan Clements, Samir Bali, Brett E. Unks, Nicholas A. Proite,
Deniz D. Yavuz, Paul J. Martin, Jeremy J. Thorn, and Daniel A. Steck,
“Design and construction of cost-effective tapered amplifier systems for laser
cooling and trapping experiments,” American Journal of Physics 82, 805
(2014).
Eryn C. Cook, Paul J. Martin, Tobias L. Brown–Heft, Jeffrey C. Garman, and
Daniel A. Steck, “High-passive-stability diode-laser design for use in atomic-
physics experiments,” Review of Scientific Instruments 83, 043101 (2012).
Rob L. Cook, Paul J. Martin, and J. M. Geremia, “Optical coherent state
discrimination using a closed-loop quantum measurement,” Nature 446,
(2007).
vi
ACKNOWLEDGEMENTS
Work behind the contents of this dissertation took place over several years and
would not have been possible without the help of many people who assisted me both
professionally and personally.
First, I need to acknowledge the work presented here belongs equally to my good
friend and colleague Eryn Cook. We built the Sr apparatus together from scratch,
and this thesis could not have been realized without her efforts. Eryn exemplifies the
skeptical and inquisitive nature of a scientist, and I appreciate the objective clarity
she brings when we are attempting to understand a process or solve a problem. On
many occasions she prevented us from chasing a misconception—wasting time and
effort—before thinking something through more clearly.
My advisor, Dan Steck, is unique as a hybrid of experimentalist and theorist, and
he has been a wealth of knowledge and patience during this project. I thank him for
providing Eryn and me with the opportunity to take a new project and develop it into
our own. I admire Dan’s ability to fully research and focus on a topic (even outside
of physics), and especially appreciate his attention to detail. I hope I absorbed these
qualities in some way and can better emulate them in the future.
Other members of Steck Lab also positively impacted my experience (including
the lab dog, Maya!). The venerable first generation—Tao, Libby, and Jeremy—set a
high bar that we strove to meet in the Sr apparatus. Jonathan, Richard, and Wes also
were fun to work with, and we shared in many enjoyable non-physics conversations,
including discussions about heavy metal, woeful baseball teams, and Kaggle. I am
thankful for the help of many undergrads in the construction of the apparatus, but I
must explicitly thank Tobias Brown–Heft for his extraordinary contributions.
vii
The resources and expertise available at the UO TSA machine shop were
invaluable in my experience. We owe a lot to the efforts of Kris and John, but Jeffrey
in particular deserves special thanks. He masterfully crafted eight of our novel ECDLs
and his consultation improved the design considerably.
I have also enjoyed wonderful investment from my friends, church, and family.
Within the physics department I’d like to especially acknowledge—in alphabetical
order—the support from my friends: Tom Baldwin, Eryn Cook, Brian Hake, Jonathan
Mackrory, Erin Mondloch, Ryan Quitzow–James, Libby Schoene, Jake Searcy, Mike
Taormina, and Dash Vitullo. My friends who have little to do with physics are also
a deep well of encouragement: Stu Johnson, my “buddies,” and the “Sunday Suppa”
group. When my wife and daughter lived in Albuquerque during the final push, I was
effectively adopted by several friends—especially the ‘other’ Martins, the Kruses, D.
Bear, and Stu—and I am forever grateful for their love and hospitality.
Part friend, family, and new-parent and physics comrade, my brother Kyle
understands my life more than most, and I am thankful for our conversations. My
parents are my most stalwart supporters, and I am grateful for their love and sacrifices,
from driving across the country with me in a moving van (twice) to enduring an hour-
long physics thesis defense. My in-laws also provided prayers, love, and support, and
I am thankful for the ways they support my family.
Finally, I owe the deepest gratitude to my wonderful and beautiful wife, Analisa,
and our darling daughter, Elise. Analisa faithfully and patiently encouraged me, and
provided uplifting refuge during difficult times. I am grateful for the sacrifices she
made being married to a graduate student, and I have immensely enjoyed discovering
the difficulties and delights of sharing my life with someone. Elise is a blessing to our
family, and I am thankful for the unbridled joy she brings with a simple smile.
viii
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Precision Metrology with Cold Atoms . . . . . . . . . . . . . . . 1
1.2. Cold Strontium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3. Quantum Vacuum Effects . . . . . . . . . . . . . . . . . . . . . . 3
II. CASIMIR–POLDER POTENTIALS AND COLD STRONTIUM . . . . . 5
2.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1. Casimir–Polder Potential . . . . . . . . . . . . . . . . . . . 7
2.1.2. Previous Experimental Approaches . . . . . . . . . . . . . . 12
2.2. Strontium Atoms as Casimir–Polder Detectors . . . . . . . . . . 16
2.2.1. Optical Lattice Technique . . . . . . . . . . . . . . . . . . . 18
2.2.2. Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.3. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
III. NOVEL EXTERNAL-CAVITY DIODE LASER . . . . . . . . . . . . . . 23
3.1. Stabilizing Cavity Length . . . . . . . . . . . . . . . . . . . . . . 25
3.1.1. Diffraction Grating Arm . . . . . . . . . . . . . . . . . . . 26
3.1.2. Laser Diode Current Source . . . . . . . . . . . . . . . . . 28
3.1.3. Additional Measures . . . . . . . . . . . . . . . . . . . . . . 29
3.2. Sensitivity to Perturbations . . . . . . . . . . . . . . . . . . . . . 32
3.3. Linewidth Measurements . . . . . . . . . . . . . . . . . . . . . . 38
3.4. Construction and Assembly Details . . . . . . . . . . . . . . . . . 43
3.4.1. Customizing a Particular Wavelength . . . . . . . . . . . . 43
ix
Chapter Page
3.4.2. Final Assembly and Alignment . . . . . . . . . . . . . . . . 47
IV. EXPERIMENTAL APPARATUS . . . . . . . . . . . . . . . . . . . . . . 51
4.1. Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1.1. Hot Strontium Source . . . . . . . . . . . . . . . . . . . . . 53
4.1.1.1. Oven Design . . . . . . . . . . . . . . . . . . . . . . . 53
4.1.1.2. Molecular Flow Conductance . . . . . . . . . . . . . . 55
4.1.1.3. Operating Lifetime and Beam Flux . . . . . . . . . . 56
4.1.1.4. Atomic Beam Considerations . . . . . . . . . . . . . . 59
4.1.2. Chamber Design . . . . . . . . . . . . . . . . . . . . . . . . 60
4.1.3. Assembly and Bakeout . . . . . . . . . . . . . . . . . . . . 63
4.1.4. Magnetic Coils . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.5. Zeeman Slower . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2. Laser Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2.1. Tapered Amplifier . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.2. Second-Harmonic Generation . . . . . . . . . . . . . . . . . 84
4.2.3. Frequency Stabilization . . . . . . . . . . . . . . . . . . . . 88
4.2.3.1. Heat-Pipe Vapor Cell . . . . . . . . . . . . . . . . . . 88
4.2.3.2. Frequency-Modulation Spectroscopy . . . . . . . . . . 91
4.2.3.3. Helical Resonators . . . . . . . . . . . . . . . . . . . 95
4.2.4. Slave Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3. Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.3.1. Radio-Frequency Electronics . . . . . . . . . . . . . . . . . 99
4.3.2. Computer Control . . . . . . . . . . . . . . . . . . . . . . . 101
x
Chapter Page
V. TRAPPING AND COOLING STRONTIUM . . . . . . . . . . . . . . . . 103
5.1. Double-MOT Scheme . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2. Blue MOT Temperature Measurements . . . . . . . . . . . . . . 107
5.2.1. Imaging System . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2.2. Expansion Data and Temperature Estimate . . . . . . . . . 110
5.3. Cooling in a Red MOT . . . . . . . . . . . . . . . . . . . . . . . 116
5.3.1. Transfer Process . . . . . . . . . . . . . . . . . . . . . . . . 117
5.3.2. Current Progress . . . . . . . . . . . . . . . . . . . . . . . . 119
VI. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
APPENDICES
A. TRAPPING BASICS: MECHANICAL FORCES OF LIGHT . . . 126
A.1. Optical Molasses . . . . . . . . . . . . . . . . . . . . . . . . 128
A.2. Magneto-Optical Trap . . . . . . . . . . . . . . . . . . . . . 131
A.3. Dipole Force and Optical Lattices . . . . . . . . . . . . . . . 133
B. STRONTIUM DETAILS . . . . . . . . . . . . . . . . . . . . . . . 135
B.1. Dipole Matrix Elements . . . . . . . . . . . . . . . . . . . . 139
B.1.1. Example: Ground State Dipole Potential . . . . . . . . 141
B.2. Casimir–Polder Level Shifts . . . . . . . . . . . . . . . . . . 142
B.3. Tabulated Data . . . . . . . . . . . . . . . . . . . . . . . . . 145
C. DELAYED SELF-HETERODYNE METHOD . . . . . . . . . . . 149
C.1. Mathematical Framework . . . . . . . . . . . . . . . . . . . 150
C.2. Heterodyne Measurement . . . . . . . . . . . . . . . . . . . 153
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Chapter Page
C.3. Delayed Self-Heterodyne Measurement . . . . . . . . . . . . 155
C.3.1. White Noise . . . . . . . . . . . . . . . . . . . . . . . . 157
C.3.2. Adding 1/f Noise . . . . . . . . . . . . . . . . . . . . . 159
C.3.3. Laser Linewidth and Observation Time . . . . . . . . . 160
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
xii
LIST OF FIGURES
Figure Page
2.1. Feynman diagram of an atom in its ground state interacting with the
electromagnetic vacuum field . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2. Level shift of the ground state of strontium near a perfectly conducting
plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3. Trapping potentials of the optical lattice and the CP effect . . . . . . . . 20
2.4. Expected differential level shifts due to the Casimir–Polder potential
between strontium atoms and a perfect conductor . . . . . . . . . . . . . 21
3.1. Sketch of a Littrow-configuration ECDL . . . . . . . . . . . . . . . . . . 26
3.2. A rendering of the ECDL diffraction grating arm, which rotates via a
notch-hinge flexure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3. Laser susceptibility to noise in the current supply . . . . . . . . . . . . . 30
3.4. Assembly drawing of the novel ECDL, featuring labels of all
peripherals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5. Schematics for methods we use to analyze the laser’s susceptibility to
environmental perturbations . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.6. Laser response while driving the piezo at acoustic frequencies in the cavity
resonance spectroscopy setup . . . . . . . . . . . . . . . . . . . . . . . . 34
3.7. Response of a prototype laser under various driving amplitudes in cavity
resonance spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.8. Frequency-noise power spectral density (fSD) as a function of frequency,
measured with the setup in Fig. 3.5(b) from 0–20 kHz . . . . . . . . . . 37
3.9. Schematic depicting the delayed self-heterodyne measurement . . . . . . 39
3.10. Delayed self-heterodyne spectra of a 922 nm laser . . . . . . . . . . . . . 41
3.11. Delayed self-heterodyne spectra of long and short 689 nm lasers . . . . . 43
xiii
Figure Page
3.12. Wavelengths that can be realized with the novel ECDL design using stock
Richardson diffraction gratings between 300 nm and 1500 nm . . . . . . 44
3.13. Numerically determining the location of anamorphic prisms . . . . . . . 46
4.1. Schematic of the UHV chamber used for trapping and cooling strontium
atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2. Schematic of the strontium source. A crucible and collimation nozzle are
centered in a custom-length nipple and heated externally . . . . . . . . . 54
4.3. Vapor pressure of strontium . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4. The estimated lifetime of the effusive oven and atomic beam flux as a
function of temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5. Sketch of atomic beam passing through a capillary and constricted by
downstream nickel gaskets . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.6. Model used to estimate expected initial pressures in the vacuum
chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.7. Photograph illustrating support structure for the trapping chamber . . . 68
4.8. Photograph of the permanent-magnet Zeeman slower . . . . . . . . . . . 73
4.9. Magnetic field profile experienced by atoms as they pass through the
Zeeman slower and into the trapping region . . . . . . . . . . . . . . . . 75
4.10. Basic block diagram of lasers required in the cold strontium
experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.11. Details of the tapered amplifier (TA) chip and housing . . . . . . . . . . 79
4.12. Output power of tapered amplifiers as a function of drive current and seed
power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.13. Schematic showing how 461 nm blue light is prepared for interaction with
strontium atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.14. Photograph of a strontium heat-pipe vapor cell . . . . . . . . . . . . . . 89
4.15. Schematic of frequency-modulation spectroscopy used to stabilize the blue
461 nm and red 689 nm lasers to an atomic reference . . . . . . . . . . . 92
xiv
Figure Page
4.16. Error signal produced in FM spectroscoy of the 5s2 1S0 −→ 5s 5p 1P1 “blue
MOT” transition in strontium . . . . . . . . . . . . . . . . . . . . . . . . 93
4.17. Error signal produced in FM spectroscoy of the 5s2 1S0 −→ 5s 5p 3P1 “red
MOT” transition in strontium . . . . . . . . . . . . . . . . . . . . . . . . 94
4.18. Photographs of the helical resonating cans used to amplify RF signals for
the EOMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.19. Overview of electronics used to control RF devices . . . . . . . . . . . . 101
5.1. Schematic of the arrangement of MOT beams for both the top and bottom
traps, viewed from the atomic source . . . . . . . . . . . . . . . . . . . . 104
5.2. Photographs of cold strontium atoms . . . . . . . . . . . . . . . . . . . . 106
5.3. Control timing sequence used to trap and image the bottom MOT . . . 111
5.4. Example of attempt to subtract the image of the loaded MOT in the blue
temperature measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.5. Sum of pixel values along columns of CCD images, showing the difference
between signal and background images and shot-to-shot fluctuations in
MOT brightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.6. Series of images taken in an attempted temperature measurement of the
blue MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.7. Integrated pixel values of the series of subtracted images in the blue MOT
measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.8. Modulating the detuning of the red MOT beams to interact with atoms
at blue MOT temperatures . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.9. Control timing sequence used to test if the red laser is affecting the
atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
A.1. Net scattering force an atom feels from two beams traveling in opposite
directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
B.1. Relevant energy levels used in trapping and cooling of neutral 88Sr . . . 135
B.2. Vapor pressure of strontium from 300–700◦C . . . . . . . . . . . . . . . 137
xv
Figure Page
B.3. Isotopic shifts of the ‘blue MOT’ 5s2 1S0 −→ 5s 5p 1P1 and ‘red MOT’
5s2 1S0 −→ 5s 5p 3P1 transitions, relative to 88Sr . . . . . . . . . . . . . . 138
B.4. Expected level shifts of strontium atoms near a perfect conductor . . . . 145
C.1. Frequency spectrum of a laser with linewidth γ centered at frequency ω0
beaten against a single-frequency plane wave with frequency ωr . . . . . 153
C.2. Plot of analytical solutions of the photocurrent power spectrum in the
delayed self-heterodyne technique for white noise . . . . . . . . . . . . . 158
xvi
LIST OF TABLES
Table Page
4.1. Maximum temperatures of various vacuum chamber components . . . . 65
4.2. Magnet size, location, and strength in the permanent magnet Zeeman
slower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3. List of AOMs used in the experiment . . . . . . . . . . . . . . . . . . . . 100
A.1. Doppler and recoil temperatures for several atomic species . . . . . . . . 130
B.2. Reduced dipole matrix elements and total decay rates for transitions of
the form 1S0 −→1 P1, originating from the 5s2 ground state. . . . . . . . 146
B.3. Reduced dipole matrix elements and partial decay rates for transitions of
the form 5s 5p 3PJ −→ 5sNs 3SJ′ (N = 6–10). . . . . . . . . . . . . . . . 146
B.4. Reduced dipole matrix elements and partial decay rates for transitions of
the form 5s 5p 3PJ −→ Np2 3PJ′. . . . . . . . . . . . . . . . . . . . . . . 147
B.5. Reduced dipole matrix elements and partial decay rates for transitions of
the form 5s 5p 3PJ −→ 5sNd 3DJ′ (N = 4–6). . . . . . . . . . . . . . . . 147
B.6. Reduced dipole matrix elements and partial decay rates for transitions of
the form 5s 5p 3PJ −→ 5sNd 3DJ′ (N = 7–9). . . . . . . . . . . . . . . . 148
xvii
CHAPTER I
INTRODUCTION
Atoms are incredibly useful in that all atoms of the same kind are identical.
This is more than just a likeness in composition, however, since they share chemical
properties and energy spectra as well. This makes atoms nice quantum systems
because—unlike solid-state objects like quantum dots or nitrogen-vacancy centers in
diamond—they are in some sense immediately well-characterized and predictable.
This knowledge of how an atom ought to behave can be leveraged into learning
something about its interaction with the environment. Loosely speaking, more can
be learned over longer and more controlled observations, and reducing atomic motion
is a straightforward way to improve measurement results.
1.1. Precision Metrology with Cold Atoms
In the 1970s—not long after the advent of the laser—physicists became interested
in tuning coherent radiation to be resonant with atoms and affect their mechanical
motion [1]. Over the course of a couple of decades, the field of cold-atom physics
burgeoned into a high-profile and successful arena, and four Nobel Prizes have been
awarded for work related to laser-cooled atoms.1 Such attention was garnered not
just because the physics is fundamentally interesting, but also because the resulting
cold-atom media are immensely useful. State-of-the-art atomic beam clocks were
almost immediately surpassed by atomic fountain clocks [3], and cold samples of atoms
significantly decreased the momentum uncertainty in matter-wave interferometers [4].
1In particular, trapping and cooling of neutral atoms was developed in the 1980s, and the Nobel
Prize lecture of William Phillips should be mandatory reading for anyone remotely interested in the
mystery of cooling atoms by shining light at them [2].
1
Later, the realization of Bose–Einstein Condensates (BECs) [5] enabled long-range
coherent interactions, allowing studies of phenomena such as quantized vortices [6]
and Mott insulator phase transitions [7].
Metrology is enhanced in cold atomic samples because interaction times are
extended, position can be controlled, and slight changes to atomic behavior
are more readily detected. Highly sensitive inertial navigation systems [8, 9],
magnetometers [10], and electrometers [11] have all been made using cold atoms.
Contributions relevant to fundamental physics have also been made, such as
measurement of the fine structure constant [12] and searches for the electric
dipole moment [13]. Frequency metrology and timekeeping, in particular, have
long compelled developments in atomic physics, and recent advances have allowed
researchers to directly observe a gravitational red shift by raising an optical table
33 cm [14].
1.2. Cold Strontium
It was with frequency metrology in mind that work began in earnest to cool
atomic strontium around the turn of the millennium [15–17]. The groups of Hidetoshi
Katori (University of Tokyo) and Jun Ye (University of Colorado at Boulder)
pioneered development of optical lattice clocks [18, 19], which offer much lower
frequency instability δf/f than the microwave transition in a cesium clock. This
is because the clock transition is on a doubly-forbidden, ultranarrow transition in
the optical regime (where absolute frequency measurement is made possible through
optical frequency combs [20]). The results obtained have been astounding, with timing
accuracy better than one second out of the age of the universe [21]. Optical lattice
clock technology is quite mature, and work toward mobile strontium clocks has been
2
pursued, with one successfully built inside a car trailer [22] and plans to put one on
the International Space Station by 2023 [23].
Strontium has attracted interest aside from timekeeping, however. The spinless
ground state and low atom–atom interaction make it an ideal candidate for precision
sensing [17], and Guglielmo Tino’s group has proposed using it as a tool to search
for short-range corrections to Newtonian gravity [24, 25]. They have also compared
measurements of 87Sr and 88Sr in a test of the equivalence principle between fermions
and bosons, probing the possibility of spin–gravity coupling [26]. Additionally,
strontium’s narrow-band secondary cooling transition at 689 nm can produce sub-
recoil temperatures [27]. These cold traps have enabled BEC production without
evaporative cooling, a promising first step toward realizing a continuous atom
laser [28].
1.3. Quantum Vacuum Effects
Cold atoms are also appealing as a medium for quantum information processing
and quantum sensors, and much effort has been made to miniaturize devices to
enhance utility and scalability [29, 30]. The small distances involved, however,
mean that quantum vacuum effects become manifest. For example, the lifetimes
of neutral atom traps have been observed to decrease significantly for atom–surface
separation on the order of a few µm [31]. Another series of experiments demonstrated
strong coupling between single cesium atoms and microtoroidal resonators, but found
the Casimir–Polder (CP) potential became a significant contributor to coupling
dynamics [32].
A better understanding of geometries and materials in such quantum vacuum
effects could enable engineering of small-scale devices which mitigate negative effects,
3
or—perhaps preferably—inspire novel devices whose functionality are based on
Casimir and CP potentials. Examples of these include potentially using carbon
nanotubes as conductors in atom-chip devices [33] or using nanoscale photonic crystals
to create subwavelength vacuum lattices [34].
The contents of this thesis are based on the marriage of the above ideas:
the sensitivity of the optical transitions in ultracold neutral strontium make it a
reasonable medium for observing CP potentials spectroscopically. Chapter II first
explores the origins of mechanical forces due to the quantum vacuum and discusses
previous experimental work. After this, the prospects of measuring CP forces using
88Sr atoms in an optical lattice are discussed. Chapters III, IV, and V document
the extensive construction and initial loading of a magneto-optical trap (MOT) for
strontium, including development of a novel external-cavity diode laser (ECDL),
ultra-high vacuum (UHV) system, and stabilized light sources for interacting with
the atoms. The appendix includes a brief overview of cooling and trapping of neutral
atoms, a reference for some properties of strontium, and a discussion of measuring
laser linewidth using the delayed self-heterodyne technique.
4
CHAPTER II
CASIMIR–POLDER POTENTIALS AND COLD STRONTIUM
2.1. Background
Quantum electrodynamics (QED) is a remarkably successful and complete
field theory [35, 36], somewhat appropriately bearing the same moniker as quod
erat demonstrandum. It provides the mathematical framework to describe the
interaction of light and matter, and it has produced accurate calculations of the
Lamb shift, the fine-structure constant, and corrections to the electron’s magnetic
dipole moment [37, 38]. In this chapter, we will be concerned with interactions with
the electromagnetic vacuum that give rise to mechanical forces in so-called Casimir,
Casimir–Polder (CP), London, or van der Waals (vdW) physics.
To sketch out the basic phenomenon, we begin with a brief overview of QED.
The Hamiltonian for the electromagnetic field in the Coulomb gauge is
HF =
∫
d3r
[
Π2
2ǫ0
+
1
2
ǫ0c
2(∇×A)2
]
, (2.1)
where Π = −ǫ0E is the momentum conjugate to A, with E being the electric field
and and A the magnetic vector potential. By separating the spatial and temporal
components, this Hamiltonian is analogous to that of a quantum harmonic oscillator,
and a canonical quantization of the electromagnetic field can be performed [36, 39],
resulting in
HF =
∑
k,ζ
~c|k|
(
a†
k,ζak,ζ +
1
2
)
. (2.2)
5
The k and ζ indices represent wave vector and polarization states of spatial field modes
fk,ζ , which are populated by the field creation operator a
†
k,ζ (and correspondingly
depopulated by the conjugate annihilation operator). The classical field modes are
normalized, and the raising and lowering operators obey the commutation relations
∫
d3r fk,ζ(r) · f∗k′,ζ′(r) = δ3k,k′δζ,ζ′ (2.3)[
ak,ζ , a
†
k′,ζ′
]
= δ3
k,k′δζ,ζ′. (2.4)
The creation (and annihilation) operators work in the same way as in the standard
harmonic oscillator, but each applies to its own set of modes, including at zero
population. That is,
a†
k,ζ |0k,ζ〉 = |1k,ζ〉 (2.5)
applies for every field mode vacuum state, and the collective “electromagnetic
vacuum” consists of infinitely many unpopulated modes.1 Since the energies of each
oscillator mode are nonzero in the ground state, the terms “zero-point energy” or
“zero-point fluctuations” are often used to reference vacuum energy. A summation
of the contributions of these vacuum modes in Eq. (2.2) is obviously divergent, and
renormalization techniques are needed to make physical predictions.
In a seminal paper in 1948, Hendrick Casimir found that zero-point fluctuations
lead to mechanical forces between macroscopic bodies [40]. He considered two infinite,
parallel, planar conductors in vacuum, separated by a distance d. The bodies create
boundary conditions which define the allowed interior electromagnetic vacuum modes
1Spontaneous emission requires the machinery of QED to be understood as an atom coupling to
these unpopulated modes through stimulated emission.
6
fk,ζ and their associated frequencies ωk = k/c. To recover a physical quantity,
he compared the total vacuum energy—found by summing ~ωk/2 for the allowed
modes—to the energy when d → ∞. Increased separation results in a larger energy
because more modes are allowed at low frequencies. The result is then an attractive
potential, V (d)− V (∞) < 0, and Casimir found the distance scaling to be d−3. The
result can be alternately derived through a more physical interpretation involving
radiation pressure, where vacuum field modes can be considered to impart momentum
through “virtual photons” reflecting off a surface [41]. Here, regions interior and
exterior to the plates are considered, and again there is a discrepancy between the
allowed field modes in the two geometries. Consider for simplicity the wave vectors
normal to the planes. The lowest allowed frequency between the two planes is πc/d,
but vanishingly small frequencies can exist in the exterior region. The pressure of
the modes acting on the bodies is therefore unbalanced, and the plates are pushed
together.
2.1.1. Casimir–Polder Potential
Before his result involving macroscopic bodies, Casimir worked with Dirk Polder2
on the effect of retardation in London–vdW forces between atoms and molecules. In
particular, they were intrigued by observations by Verwey and Overbeek in colloidal
suspensions [42]. At long distances, the usual z−6 scaling (where z is the separation of
the particles) did not match measurements as well as a z−7 power law. By modeling
two atoms as polarizable particles interacting with quantized field modes, Casimir and
2Because of the authors of the seminal papers, quantum vacuum effects between macroscopic
bodies are called Casimir interactions, while Casimir–Polder (CP) refers to particle–particle or
particle–body interplay; broadly speaking, study of the interaction of dispersive objects with
electromagnetic fluctuations is referred to as vdW or Casimir physics, but this thesis will reserve
the term vdW for referencing near-field CP effects.
7
Polder confirmed the observed inter-particle potential scaling; they also performed
similar analysis of an atom–conductor system, finding that the attractive potential
shifted from the vdW z−3 behavior to a long-range potential that falls off more quickly
at z−4 [40]. Roughly speaking, the change in scaling occurs when the separation
becomes larger than the dominant atomic transition wavelength. This claim requires
some further explanation, but first it is helpful to develop some intuition involving
an atom’s interaction with the fluctuating vacuum.
Working in the QED formalism defined above, we can calculate the effect of the
quantized field interacting with an atom in its ground state and recover the basic form
of the CP potential. For an atom with multiple excited levels, the atomic Hamiltonian
is given by
HA =
∑
j
~ωj0|ej〉〈ej|, (2.6)
where ωj0 = ωj − ω0 = (Ej − E0)/~ is the frequency splitting between the jth
excited state |ej〉 and the ground state |g〉. The electromagnetic field Hamiltonian
HF is given by Eq. (2.2), and we consider the dipole approximation for the interaction
Hamiltonian, HAF = −d ·E. The quantized electromagnetic field is a sum over spatial
modes and their annihilation and creation operators [36, 39],
E(r) = −
∑
k,ζ
√
~ωk
2ǫ0
(
fk,ζ(r)ak,ζ + f
∗
k,ζ(r)a
†
k,ζ
)
. (2.7)
8
The atomic dipole operator d can be written in terms of atomic raising and lowering
operators. For an atom in the ground state,
d =
∑
j
(
dgejσj + d
∗
gej
σ†j
)
, (2.8)
where dgej := 〈g|d|ej〉 is the dipole matrix element connecting the ground and jth
excited state and σj := |g〉〈ej| is the atomic lowering operator.
The total Hamiltonian is thus H = HA+HF+HAF, and to find the CP shift due
to the interaction with the vacuum field we treat HAF as a perturbation. The first-
order correction 〈g|HAF|g〉 is zero because of the atom raising and lowering operators,
so we calculate to second order:
VCP =
∑
j
∑
k,ζ
|〈g, 0|HAF|ej, 1k,ζ〉|2
Eg,0 − Eej ,1k,ζ
. (2.9)
The interaction Hamiltonian has four terms which go as σja
†
k,ζ, σ
†
jak,ζ, σjak,ζ , and
σ†ja
†
k,ζ. The latter two of these combine to produce the only non-vanishing term in
the numerator of Eq. (2.9):
~ωk
2ǫ0
〈
g, 0
∣∣dgej · fk,ζ σjak,ζ ∣∣ ej , 1k,ζ〉〈ej , 1k,ζ ∣∣d∗gej · f∗k,ζ σ†ja†k,ζ ∣∣ g, 0〉. (2.10)
This is notable for a couple of reasons. First, these two combinations of atom
and field raising and lowering operators are usually discarded in the rotating-wave
approximation (RWA) because they are the counter-rotating terms [39]. They also
violate conservation of energy when considered individually—for example, σ†ja
†
k,ζ both
raises the atomic state and adds a photon to the field—however, when both sets of
operators are combined there is no net energy change. This process is sketched in the
9
|g, 0〉 |e, 1k,ζ〉 |g, 0〉
FIGURE 2.1. Feynman diagram of an atom in its ground state interacting with the
electromagnetic vacuum field. As the atom is excited a photon is created in field
mode fk,ζ. The photon is later annihilated as the atom drops to the ground state.
Feynman diagram in Fig. 2.1. The interpretation is that the atom ‘emits’ a virtual
photon into the vacuum mode before ‘absorbing’ it a short time later.
The energy difference in the denominator is E0 −Ej − ~ωk, so the CP potential
is then
VCP = −
∑
j
∑
k,ζ
(
~ωk
2ǫ0
) |dgej · fk,ζ |2
~ (ωj0 + ωk)
. (2.11)
From here, the calculation requires the explicit field modes for a given geometry.
Such calculations are difficult because fk,ζ are nontrivial for even relatively tame
geometries. Furthermore, Casimir and CP forces are non-additive [43] because the
addition of extra matter fundamentally alters the field modes.3 A physical (non-
infinite) potential is found by subtracting the CP shift the field induces on the atom
in free space (the limit where geometric structures influencing the field modes are far
away from the atom). Using Fig. 2.1 again for intuition, the physical CP potential
can loosely be thought of as the collection of second-order processes where the virtual
photons interact with surrounding objects—those that do not ‘see’ the geometric
influences will be canceled under renormalization. This idea is evident in numerical
3In fact, this is how some of the hallmark results can be obtained: the Lamb shift can be found
as the energy change in the free-space vacuum modes when a single atom is added; and modification
from adding a second atom results in the London and CP attractive potentials [36].
10
worldline techniques, which only count the energy contribution of Monte–Carlo paths
that intersect surfaces [44–46].
In the planar conductor geometry considered by Casimir and Polder (where the
atom is displaced by zzˆ from the conductor in the x-y plane), we insert half-space
quantization modes [36] into Eq. (2.11) and the sum can be carried out analytically.
For a spherically symmetric atom in the ground state, the result is [39]
VCP = − 1
4πǫ0
1
4π
∑
j
|〈g| zˆ · d |ej〉|2
[
∂2z
1
z
f(2ωj0z/c)
]
, (2.12)
where f(a) is one of the auxiliary functions of the sine and cosine integrals,
f(a) = sin aCi(a) + cos a
[π
2
− Si(a)
]
, and (2.13)
∂af(a) = −g(a) = cos aCi(a)− sin a
[π
2
− Si(a)
]
. (2.14)
In the short-distance limit, f(a) → π/2, and the resulting z−3 scaling is identical to
the simple case of an atomic dipole interacting with its image [36]. Even though the
atom is in its spherically symmetric ground state and 〈g|d|g〉 = 0, the second-order
dipole moment 〈g|d2|g〉 is non-vanishing. This means that fluctuations in the vacuum
field that give rise to the CP potential can equivalently be cast as fluctuations in the
atomic dipole, which is implied by the vertices in Fig. 2.1.
Equivalence of the CP result with the vdW scaling breaks down as the distance
between the atom and conductor is increased. Mathematically, this is clear from
the auxiliary functions decaying with increasing z. The retarded interaction of the
fluctuating dipole and its image provide a more physical explanation: If the oscillation
period of the fluctuating dipole is short compared to the amount of time it takes to
exchange information with its image, then the two become uncorrelated and the
11
attraction is reduced.4 In fact, the correct long-range potential can be derived in a
semiclassical treatment in this manner, but a normal ordering of the fields must
be invoked, thus implying the fundamentally quantum mechanical nature of the
effect [47]. We can also corroborate our intuition from the retarded dipole interaction
by considering the picture of an atom emitting virtual photons. An individual event
only contributes to the CP potential if the time period of the brief violation of
conservation of energy encompasses the time it takes the photon to leave the atom,
interact with the mirror, and return.
Thus the important timescale for an atomic transition is 2z/c = 2π/ω0j = λ0j/c,
which of course is implied by the argument of the auxiliary function in Eq. (2.12).
Also important, however, is the strength of the dipole moment connecting |g〉 to
|ej〉. For reference, the strongest, lowest-frequency resonance of an atom is typically
∼ 1 µm, so the crossover will occur at distances smaller than this. The ground-state
level shift for strontium—whose strong cycling transition is at 461 nm—is pictured in
Fig. 2.2 along with the near- and far-field approximations. At a few hundred nm, the
scaling behavior of the exact solution noticeably deviates from the z−4 long distance
behavior. Additionally, it is interesting that the vdW dipole-dipole energy does not
correctly predict the actual shift except at extremely small distances, so the classical
static intuition is really only useful for understanding the scaling.
2.1.2. Previous Experimental Approaches
At the end of his famous paper, Casimir estimates the size of the attractive
force he calculated. He then proceeds to significantly underestimate the impact
4Even if ignoring orientations of the dipoles, the magnitudes are uncorrelated and the product of
the dipole and its image is on average smaller than in the instantaneous case, 〈datom(t)dimage(t−τ)〉 ≤
〈datom(t)dimage(t)〉 = 〈d 2atom(t)〉].
12
atom-wall separation (nm)
0 500100 200 300 400
gr
ou
n
d
 s
ta
te
 l
ev
el
 s
h
if
t 
(M
H
z)
0
-3
-2
-1
near field
approx
far field
approx
CP exact
101 102 103
FIGURE 2.2. Level shift of the ground state of strontium near a perfectly conducting
plate. The inset shows the same curves on a logarithmic scale, extending the range
to 10 nm separation. Small- and long-distance limits of Eq. (2.12) illustrate the
transition that occurs between roughly 10 nm and 1000 nm.
of his work, musing, “This force may be interpreted as a zero point pressure of
electromagnetic waves. Although the effect is small, an experimental confirmation
seems not unfeasable [sic] and might be of a certain interest” [40]. Since then, the
field of Casimir and vdW physics has become vibrant and important, overlapping
with many disciplines, including atomic physics, condensed-matter systems, surface
chemistry, and structural biology. As such, this thesis will only provide selective,
brief overviews of landmark achievements; more complete discussion is detailed in
numerous books and review articles [48–52].
In practice, understanding Casimir and CP potentials for real bodies is quite
difficult, and for a field of physics that is seventy years old, analytic solutions are few
and far between. In the years following Casimir’s original result, Lifshitz generalized
the theory to accommodate arbitrary bodies at finite temperature [53, 54], but the
dielectric permittivity and magnetic permeability must be well-characterized at all
13
frequencies. Thus, writing down the field modes fk,ζ is not just a geometric problem,
but also one of materials, which builds additional model-based complications into
predictions regarding Casimir and CP forces. Luckily, experimental progress has
been steadily growing in the past three decades, which can be helpful in addressing
questions about appropriate theoretical models [55].
Remarkably, accurate demonstration of Casimir’s parallel plate prediction was
not achieved for fifty years. The reason for this long delay was the technical difficulty
in realizing a clean, charge-free system with parallel surfaces. In the end, experiments
became feasible using a sphere to approximate one of the bodies as an infinite
plane at the point of closest approach [56] (parallel plate experiments have since
been performed—albeit with relatively poor accuracy [57]). This technique opened
the door for sphere–plate experiments in micro- and nano-electromechanical systems
(MEMS and NEMS), including the use of atomic-force microscopy [58] and small-
scale torsional oscillators [59]. The latter became somewhat of a model system and
allowed investigation into reducing the Casimir force through thin-film metallic skin
depth effects [60] and nanostructured materials [61, 62]. Researchers in Ref. [63]
demonstrated a repulsive interaction when adding a layer of bromobenzene between
a silica surface and gold sphere (where the permittivities satisfy εg > εb > εs,
thus meeting the repulsive criteria found in Ref. [54]). The flurry of macroscopic
Casimir experiments have (not coincidentally) occurred alongside progress in MEMS
and NEMS technologies. As devices are miniaturized there is great interest in
leveraging understanding of the Casimir effect in engineering new devices and
mechanisms; circumventing negative effects of bodies irrevocably bonding together
through Casimir-induced “stiction” has long been a goal of research [64, 65], but
novel arrangements, like low-power Casimir-force switches [66], are also tantalizing.
14
In a recent proposal, the Casimir effect is used to control an optical switch based on
optomechanically-induced transparency (OMIT) [67].
While macroscopic Casimir experiments have largely been achieved by sensitively
measuring slight deflections of bodies which are brought close together, their CP
counterparts have had to be creative in extracting surface potentials through
dynamical interaction of atomic systems. One subclass of CP experiments uses atomic
beams traveling past surfaces [68–70]. The seminal result of Ref. [71] confirmed both
vdW and CP scaling by passing a beam of sodium atoms through a gold-coated groove.
By focusing an ionizing laser through the beam at the output (and then counting the
ions with an electron multiplier) they could monitor the ‘opacity’ of the channel as
a function of plate separation. Other hot-atom experiments have found success in
the vdW regime. Atoms passing through transmission gratings pick up a small phase
change which can be detected through atomic-beam interferometry [72, 73], and this
effect has been used to detect the increase in attraction from adding a thin gold layer
to a silicon-nitride grating [74].
Cold-atom approaches have also been developed. In a series of admirable
experiments, the Cornell group measured the influence of a surface on a Bose–
Einstein condensate (BEC) of 87Rb atoms [75]. By bringing a magnetically trapped,
cigar-shaped BEC near a surface, the researchers could accurately measure CP
perturbations to the harmonic potential. This system provided good signal from
6–12 µm, allowing them to study temperature dependence of the CP force [76].
They also investigated sensitivity to surface patch potentials (caused by rubidium
atoms adsorbed onto the surface), concluding that short-range investigations are likely
difficult in this system [77]. Another host of experiments use quantum reflections of
atoms off of CP surfaces potentials [78–83]. Such an experiment provided the first
15
measurement of CP forces in the intermediate distance regime [84] using a well-
characterized evanescent field at the surface of borosilicate glass to tune the effective
turnaround point for cold rubidium atoms.
Finally, a class of spectroscopic experiments have also been pursued. By using
a two-laser scheme to probe an excited transition (where the CP shifts are larger
than for the ground state), changes in the resonance have been probed in an atomic-
beam experiment [70] and in atomic vapor cells [85–87] in the vdW regime. Recently,
this technique has demonstrated an increase in the attraction between cesium and
sapphire with increasing surface temperature [88].
2.2. Strontium Atoms as Casimir–Polder Detectors
The atomic physics approaches outlined above have been impressive in their
ability to sense CP forces, particularly in the near-field vdW regime. Absent, however,
is a spectroscopic approach with atoms which are a controllable distance away from
the surface. Current spectroscopic techniques have some drawbacks. The results
in Ref. [88] rely on a surface-reflection measurement using a probe laser, which is
sensitive to shifts in the atomic energy levels in the vapor to a penetration depth
of roughly λprobe/2π ∼ 100 nm. The varying thickness of the “nanocell” used in
Ref. [87] allows some control of the distance of the atoms sampled, but the information
obtained by the surface reflection is ultimately model dependent. There is also some
extra uncertainty since the transition probabilities are not well characterized in these
excited state optical transitions [88].
A useful addition to the class of spectroscopic measurements of the CP
potential would therefore be a direct measurement of an atomic transition. In
particular, ultracold atoms with a well-characterized transition would provide detailed
16
observation of surface potentials without the need to assume a particular theoretical
comparison. Strontium is a good candidate for such a measurement because its
optical “clock” transitions are extremely narrow and well-characterized. The sub-
Hz accuracy obtained in Ref. [89] was limited by the interrogation laser’s linewidth,
and such frequency resolution would easily probe CP-sized shifts (compare to Fig. 2.4,
for example).
In addition to the narrow linewidth, strontium offers other benefits over alkali
atoms, especially in the variety of surfaces it can probe. Thermal excitations in
metallic surfaces have been shown to be a significant source of loss in atom traps,
since small fluctuations in the magnetic field can drive oscillations of atoms which are
resonant with the trap’s fundamental frequency. Ion traps are strongly susceptible
to this effect—where significant loss is incurred even tens of micrometers away of
a surface—and neutral atoms with spin are affected at the ∼µm scale [90]. This
was studied by the Cornell group shortly before they began work in their BEC–
surface CP experiments, where they were exploring the feasibility of microtraps of
87Rb involving current carrying wires [91]. In contrast to the 52S1/2 ground state of
87Rb, the nuclear-spin-free 88Sr has a spinless ground state that is decoupled from
the thermal fluctuations [90], making trapping more convenient in close proximity to
a metallic surface. A more considerable limitation on the Cornell BEC experiments,
however, was the effect of stray patches of charge from adsorbed atoms (“adatoms”)
on the surface. Roughly speaking, adatoms on a metal are slightly ionic because the
valence electron penetrates into the surface, creating a dipole-like separation—in the
case of Rb—of roughly 0.5 nm. These charges created potentials of 1/r2 for titanium
(conductor) and 1/r2.3 for silicon (semiconductor) [77], so the CP measurements had
to be carried out with fused silica and sapphire [91]. Strontium adatoms, on the
17
other hand, should be much less ionic in character because its ionization energy is
larger than the work function of most materials [77], opening the door to probing
both metals and insulators. Furthermore, a spectroscopic method of detection (in
contrast to the BEC center-of-mass method) does not require as great a number of
atoms, and surface contamination would be largely reduced.
Finally, cold strontium is a promising medium for probing CP potentials
because of extensive recent work on the optical lattice clock. In addition to
detailed spectroscopy knowledge, frequency-measurement systematics are extremely
well characterized [89] or have been eliminated altogether. Supreme among these is
the ability to circumvent spatial-dependent ac Stark shifts in an optical trap by using
the “magic wavelength” of the clock transition [92], which is central to the proposed
measurement scheme discussed below.
2.2.1. Optical Lattice Technique
Cold strontium is thus well-suited to use in CP measurements, but we need to be
able to confine it in a reasonably small cloud next to a test surface. A focused dipole
trap could be used to trap atoms and move them near the surface, but trap widths
are typically many µm in the radial direction and even larger axially [93]. An optical
lattice, on the other hand, provides steep changes in potentials that hold atoms in
very well-defined positions along the beam’s axis. If we orient the lattice normal to the
surface, then a single lattice site would provide good spatial resolution of the atom–
surface separation, but we would need to be able to sample various distances to make
a useful measurement. Forming the lattice by retro-reflecting a beam from the surface
is problematic because atoms can only be trapped at positions z = λL/4 + nλL/2. A
basic requirement of using cold strontium in such a configuration, therefore, is that
18
the test surface be at least somewhat transparent to the optical lattice wavelength.
This way one of the lattice’s beams comes from behind the surface, and the trapping
regions can be positioned at arbitrary z.
The frequency at which the test surface is transparent is a stringent condition,
however, because the wavelength of the optical lattice is not negotiable. In an optical
field, the total shift of an energy level in an atom is found by summing the ac Stark
shift from all dipole-allowed transitions connected to that state, and in general the
ground and excited states of a particular transition will not shift by the same amount.
Because of the varying intensity in a lattice potential, the differential shift depends
on position, which would complicate the search for a small frequency shift in a CP
measurement. Fortunately there are particular wavelengths at which the shifts of
the two levels are identical. Such magic wavelengths occur at 813.428(1) nm for the
5s2 1S0 −→ 5s5p 3P0 transition in 87Sr used in the optical lattice clocks [92], and at
914(1) nm for the 5s2 1S0 −→ 5s5p 3P1 intercombination line in 88Sr [94]. By operating
our trapping laser at a magic wavelength, the differential ac Stark shift is eliminated,
and all atoms within the trap will have the same resonant frequency for the transition
we want to interrogate.
To shift the location of the trapped atoms and probe different z, we propose
direct translation of the lattice via a linear air-bearing translation stage (Aerotech
ABL10100-LT), which was used in previous, one-way barrier experiments with
rubidium in our lab [95]. The resolution of this stage is 5 nm, and it has a repeatability
of 50 nm, meaning that atoms could be repositioned reasonably well as data are
recorded. Atoms can be loaded from a MOT far away from the surface before they
are moved into position for the CP measurement, which greatly reduces the risk
of unwanted adatom contamination on the test surface. Note that the nearest the
19
atom-wall separation (nm)
0 2000200 400 600 800 1000 1200 1400 1600 1800
gr
ou
n
d
 s
ta
te
 l
ev
el
 s
h
if
t 
(M
H
z)
0
-1
-0.8
-0.6
-0.4
-0.2
FIGURE 2.3. Trapping potentials of the optical lattice and the CP effect, plotted
both individually (dashed) and combined (solid green). Here the lattice has a depth
of 22 µK, corresponding to a power of 380 mW and a waist of 50 µm. At close range,
the strength of the CP potential overwhelms the confining potential of the lattice,
placing a lower limit on achievable atom–surface separation.
atoms can be brought to the surface before getting pulled out of the lattice is roughly
200 nm, as shown in Fig. 2.3 for a trapping depth of 22 µK.
2.2.2. Feasibility
The spectroscopic measurement is an interrogation of the frequency separation of
the ground and excited states, and we therefore are only sensitive to the differential
CP shift of the two states. The generalization of Eq. (2.12) for an excited state
is identical for vacuum-couplings to higher-lying states, but is modified for lower-
lying states. This is because in the perturbative calculation the dipole interaction
induces the atom to emit additional “real” photons which reflect off the surface and
are reabsorbed. The modification of Eq. (2.12) adds oscillatory terms which are
consistent with a dipole radiating near a mirror [39].
20
atom-wall separation (nm)
200 1000300 400 500 600 700 800 900
d
if
fe
re
n
ti
al
 l
ev
el
 s
h
if
t 
(k
H
z)
50
-200
-150
-100
-50
0
3P0  mJ = 0
3P1  mJ = 0
3P1  mJ = 1
3P2  mJ = 2
3P2  mJ = 1
3P2  mJ = 0
FIGURE 2.4. Expected differential level shifts between the ground state and various
excited states in strontium. Shifts of a particular level are computed by assuming
an infinite planar conductor and summing over dipole-allowed atomic transitions (see
Appendix B).
CP potentials for various energy levels in Sr are calculated by summing over
dipole-allowed transitions, and the differences in the ground- and excited-state shifts
are shown in Fig. 2.4. These are computed using the generalized version of Eq. (2.12)
derived in Ref. [39], and details about calculating the various dipole matrix elements
are discussed briefly in Appendix B. The differential shift is largest for the 3P2 excited
states, but at 200 nm—where the lattice trap begins to break down—the shift of the
3P1 state is roughly 50 kHz for magnetic sublevel mJ = 1. Such a shift can be resolved
by the 7.6-kHz-wide intercombination line, and a first generation of experiments
can interrogate the potential using the same laser used to make a red MOT (see
Chapters IV and V) and a magic wavelength lattice operating at 914 nm.
Probing the CP shift at a particular distance will require many repetitions of
loading and translating the lattice and performing a spectroscopic interrogation. Since
the absorption of the narrow transition is so small (Isat is only 3 µW/cm
2), the
21
interrogation procedure will consist of a π/2 pulse on the 689-nm line followed by a
measure of the ground state population using a resonant 461-nm beam. By varying
the detuning of the interrogating laser, the transition’s lineshape can be mapped out,
so shifts in the resonant frequency can be sensed well below the 7.6-kHz width of the
transition.
2.2.3. Outlook
Demonstrating the viability of this lattice-based approach would add diversity
to atom–surface CP experiments. In particular, this method would join the reflection
experiments of Ref. [84] as a measure of intermediate distances where the CP potential
departs from the far-field z−4 scaling. Moreover, the cold-strontium technique is
more amenable to a variety of surfaces than previous experiments, with the only
real requirement being that the test surface is somewhat transparent at the lattice
wavelength and that it does not significantly distort the beam.
The remainder of this thesis discusses the development of an apparatus to
provide a cold-strontium source for such CP experiments. The first proof-of-principle
experiment will be performed using a pyrex vacuum cell as the interaction surface.
In addition to investigating a variety of surfaces, subsequent evolution of the project
would add the ability to interrogate the true 3P0 clock transition, providing orders of
magnitude higher sensitivity to the CP potential.
22
CHAPTER III
NOVEL EXTERNAL-CAVITY DIODE LASER
Diode lasers, as semiconductor devices, benefit from manufacturing techniques
driving computing technology, making them fairly cheap to produce in large
quantities. They are also highly efficient and compact, greatly simplifying the
overhead needed to furnish a laser lab. They have become the lasers of choice in
modern atomic physics laboratories [96–99] because the linewidth can be made narrow
relative to an atomic transition and the frequency is easily tuned and controlled.
Discussion of a laser’s linewidth typically begins with the Schawlow–Townes
linewidth ∆νST, which is a result of phase fluctuations from spontaneous emission
into the laser mode [100]. In diode lasers, however, phase fluctuations in the gain
medium can alter the carrier density, causing fluctuations in both amplitude and
optical path length [101]. The result is a broadening of the laser linewidth via β,
a “cavity enhancement factor” [102, 103], which typically has values between 2 and
8 [101]. The modified Schawlow–Townes linewidth of a laser diode operating at
frequency ν is then
∆νLD = ∆νST(1 + β
2) =
hνgnsp (∆νC)
2
P0
αt
(
1 + β2
)
, (3.1)
where h is Planck’s constant, g is the gain, nsp is the number of spontaneous emission
events, P0 is the power in the mode, αt is the cavity loss, and ∆νC is the cavity
bandwidth [98]. Typically, linewidths of laser diodes are 20–50 MHz [97], and need
to be narrowed before interacting with atomic transitions. The enhancement factor
is an obvious source of broadening, but it is fundamentally determined by the diode’s
23
semiconductor composition and is thus difficult to alter. The linewidth can, however,
be narrowed by improving the quality factor Q of the lasing cavity. To lowest order,
Q =
νC
∆νC
= − l
∆l
(3.2)
in an optical cavity [104], and the laser’s linewidth is then inversely related to the
square of the length of the cavity. This means that the linewidth of a diode laser
(which has a cavity length of only a few hundred µm) can be narrowed considerably by
creating a much longer ‘external-cavity diode laser’ (ECDL), where a reflective optical
element overrides one of the diode’s output facets and acts as a cavity mirror [105,
106]. Extending the length too far does have drawbacks, however. Longer lasers are
more sensitive to mechanical vibrations (see Sec. 3.1), and significant reduction in the
cavity’s free spectral range makes mode hops between closely spaced cavity modes
more likely to occur.
Typically, a frequency-selective element is also used so the diode’s frequency
output is not only narrowed, but tunable as well. A simple and common way to
create a tunable ECDL is through the use of a diffraction grating in the Littrow
configuration, as pictured in Fig. 3.1.1 The laser’s two cavity mirrors are formed by
the diode’s rear facet and the grating. First-order diffracted light is fed back into the
laser-diode gain medium, allowing coarse tuning of the cavity’s resonant frequency
through rotation of the grating. Fine tuning is controlled with a piezoelectric device
1An alternative popular ECDL design uses a “cat’s-eye” configuration, which focuses the cavity
mode onto a mirror. An interference filter within the cavity can be tilted to coarsely select the
operating wavelength [107–109]. Such a design is attractive because mechanical stability of the
tuning and feedback elements are, in principle, not as paramount to frequency stability as in a
grating-feedback configuration. It can be argued, however, that the narrow linewidths achieved
in the cat’s-eye designs are a result of the long cavity lengths needed to fit the necessary optical
components.
24
placed behind the grating, and laser frequency is stabilized via electronic feedback to
the piezo. These grating-stabilized ECDLs are found in virtually all atomic physics
laboratories, and it is not uncommon that they are built in-house [110–113].
Constructing a new strontium experiment requires a variety of laser systems
(four in our case, including one for the particularly narrow 7.6 kHz intercombination
line, see Sec. 4.2), and we decided it was worth the effort to create a new Littrow-
configuration ECDL design. In particular, we wanted to surpass the passive stability
achieved by the previous generation of lasers built in our lab [95, 114, 115], and, if
possible, produce lasers which were competitive with high-end commercial products.
We have achieved these goals, published the results [116], and made our design freely
available to the atomic-physics community (including machine drawings and step-
by-step pictorial construction instructions [117]). Since our publication is fairly
comprehensive, the current chapter will not be concerned with reproducing that
content in its entirety, but will instead cover major highlights and fill in gaps which
add to the utility and discussion of the design.2
3.1. Stabilizing Cavity Length
As is apparent from combining Eqs. (3.1) and (3.2), the frequency stability of a
diode laser can be severely impacted by mechanical instability in the length of the
external cavity. A change in length can happen slowly (drifts in pressure, humidity,
or temperature) or on faster timescales (mechanical vibrations or current-induced
changes in the diode’s index of refraction). Whatever the cause, if a displacement is
large enough, stabilization electronics will eventually fail. Low-frequency resonances
are particularly concerning, because these can limit the frequency bandwidth of the
2Another useful resource is the undergraduate thesis of Erica Mason [118], who thoroughly
documented the process of constructing the laser without our assistance.
25
ag
FP diode laser
aspheric lens diffraction grating
output
FIGURE 3.1. Sketch of a Littrow-configuration ECDL. The output of a Fabry–Pe´rot
(FP) diode laser is collimated by an aspheric lens before being reflected by a diffraction
grating. The first-order mode is reflected back at angle αg into the laser diode, and
the zeroth-order is the usable output.
electronic servomechanism if no special compensation is designed into the controller.
In a Littrow-geometry ECDL, though, tuning is fundamentally achieved by physically
moving the diffraction grating, and a main challenge is simultaneously implementing
mobility while retaining rigidity.
Our approach is similar in spirit to that of Carl Wieman’s group [119]: by carving
the laser cavity out of a monolithic block of aluminum, the cavity length is better
isolated from external vibrations. Sensitivity to background noise is additionally
ameliorated (it is, of course, impossible to remove effects of all background noise,
since it diverges at low frequency [120]) by increasing the fundamental vibrational
resonance of the arm which moves the diffraction grating. Ideally, the resonance
should be higher than most lab acoustic noise, whether it is generated by cooling
fans, discoursing scientists, or clumsy graduate students.
3.1.1. Diffraction Grating Arm
A simple arm with a symmetrically notched hinge—formed by carving two
semicircular channels into a cantilever—can have a relatively high fundamental
26
frequency while allowing deflection from a piezoelectric stack (we use Noliac
SCMAP02/S2/A/5/5/10/60/10.6/1000, which has a blocking force of 1000 N). A
finite-element analysis [121] yields an empirical formula for the torsional constant of
the hinge,
κ =
Eht3
24KR
, (3.3)
where E is the elastic modulus of the material (68.9 GPa in 6061-T6 aluminum), h is
the height of the hinge, t is the minimum thickness of the hinge, R is the radius of the
semicircular channel, and the factor K is determined from the approximate formula
K = 0.565 t/R + 0.166. In the case of a rectangular bar (of length l and width w)
attached to such a hinge, the resonant frequency is
f =
1
2π
√
κ
I
=
1
2π
√
t3
2KRwl(w2 + 4l2)
. (3.4)
Note that the height of the hinge cancels out for objects whose moments of inertia
I are proportional to h (anything with a uniform cross section through the height
of the arm). We constructed a sample flexure using a 0.25′′ ball-end mill to carve
out a notch of thickness 0.09′′. For an aluminum (density 2770 kg/m3) arm of length
l = 1.25′′ and width w = 0.25′′, the calculated resonance frequency was f ∼ 1700 Hz,
consistent with the measured f ∼ 1520 Hz.
The final laser design uses a grating arm with dimensions similar to this simple
case. To increase the resonant frequency, however, much of the material in the arm
is removed, as shown in Fig. 3.2. Four 2-56 clearance holes are counterbored through
almost the entire height of the arm, allowing the arm to be attached to a support
base during machining. Eight additional holes are drilled through the arm and the
27
FIGURE 3.2. A rendering of the diffraction grating arm, which rotates via a
notch-hinge flexure at left. Mass is removed to increase the fundamental resonance
frequency.
front half is mostly removed. We also use a custom-sized grating (6 mm tall, 12 mm
long, and only 3 mm deep) to minimize the mass we add to the end of the pendulum.
Plucking the arm in a constructed laser body (with no grating attached) produces
a 2.05 kHz tone. The target Littrow grating angle is 45◦, but the arm is machined at
an angle of 43◦, and must be pushed from behind by piezo and coarse adjust screw
to be “loaded” into final position. The support from the piezo and coarse adjust
screw significantly increases the frequency of the fundamental resonance (see Sec. 3.2).
Care should be taken to avoid deflecting the arm too far past the desired operational
point and fatiguing the aluminum [118]. Using an expression for the maximum yield
stress [121], we estimate that angular deflections beyond ∼ 0.3◦ will begin to enter
this regime, so it is best to monitor the laser’s wavelength during initial alignment of
the cavity and have a reasonable expectation for the laser’s output direction. Note
that this is especially true when using a shim behind the grating (Sec. 3.4.1) because
the angle provided by the shim is somewhat difficult to control.
3.1.2. Laser Diode Current Source
As mentioned earlier, frequency fluctuations in a diode laser can be caused by
changes in charge-carrier density due to noise in the supply current. This effect turns
out to be significant, and—if unchecked—can actually be the dominant contributor
28
do a diode laser’s linewidth. While making preliminary stability measurements of a
prototype of the ECDL, we powered the diode using a homebuilt current source (based
on Ref. [122] and discussed in Ref. [123]; output current is driven by a pair of BUF634
op-amps which are regulated by a proportional–integral [PI] servo monitoring the
voltage dropping over a sense resistor). As shown in Fig. 3.3, the current supply has
an obvious noise bump around 400 kHz in its power spectrum, and this is imprinted
on the frequency noise spectrum of the laser and broadens its linewidth significantly.
Using a f3dB = 15 Hz low-pass filter on the circuit’s output corrects this behavior, but
hampers the practicality of the controller by disallowing current modulation. Instead,
it is best to use a low-noise circuit, such as a Libbrecht–Hall-style controller [124, 125],
which cleverly pairs a low-noise op-amp and metal-oxide-semiconductor field-effect
transistor (MOSFET) to regulate the current. Incidentally, the laser’s sensitivity to
current noise enabled our collaborators to uncover an issue with the Libbrecht–Hall
design operating near its current limit, and this led to a joint publication about simple
ways to make improvements [126].
3.1.3. Additional Measures
We make several efforts to further stabilize the length of the laser cavity, as
shown in the assembly drawing of our ECDL in Fig. 3.4. The main laser-cavity area
is carved from a 1′′-thick piece of 6061-T6 aluminum, and a separate block—which
holds a Thorlabs LT230P-B diode-collimation tube—is secured into the side to define
one end of the lasing cavity. A diffraction grating is glued to the notched-hinge
arm to complete the cavity. The main body has a top and bottom lid, and the
stack is actively temperature-controlled with two thermoelectric coolers (TECs) in
series. Temperature is monitored at the diode to have control over the gain curve,
29
frequency (Hz)
103 106104 105
fr
eq
u
en
cy
 n
oi
se
 (
ar
b
it
ra
ry
)
laser response
current supply
FIGURE 3.3. Laser susceptibility to noise in the current supply. The current
controller (blue) has a noise bump around 400 kHz. Laser frequency noise (green) is
converted to voltage fluctuations using an atomic absorption feature as in Fig. 3.5(b),
and exhibits a rise in its power spectral density matching that of the current controller.
but temperature control of the entire assembly stabilizes the path length of the cavity
(for 6061-T6, the coefficient of expansion is 23.6 µm/m/◦C, so a temperature change
of less than half a degree would result in a mode hop in a 2-cm-long, 780-nm laser,
even if the diode’s temperature was constant).
A hermetically sealed cavity limits effects of slow changes in the lab’s pressure
or humidity. We seal all joints and points of mechanical access, electrical inputs, and
optical outputs using Viton O-rings or a high-temperature vacuum epoxy (Epotek
353ND). A vacuum seal-off valve is machined into the front of the main body, where
a pump can be attached to evacuate the cavity or backfill it with an inert gas to
remove moisture, also enabling operation below the dew point. (We had initially
hoped that evacuating the cavity might improve acoustic stability, but the methods
used in Sec. 3.2 seem to indicate that this is not an important effect.)
30




Diffraction 
grating
Low-voltage 
piezo
Anamorphic
prism pair
Grating 
adjustment screw
Brewster 
window
Vacuum 
seal-off 
valve
Fiber coupler
Optical isolator
Diode 
collimation
tube
Protection 
circuit
D-sub 15 
connector
Thermoelectric coolers
O-ring
seal
Wire feed 
through
7.750"
3.125"
4.500"
Tilt adjust
screw holes
Temperature
sensors
Sapphire
discs
FIGURE 3.4. Assembly drawing of the novel ECDL, featuring labels of all peripherals.
Complete discussion is available in [116]. The cavity body is hermetically sealed using
O-rings and epoxy, and the beam exits through a microscope slide at Brewster’s angle
before beam-shaping, optical isolation, and fiber-coupling.
31
Vibration isolation is provided from below by viscoelastic (E-A-R Specialty
Composites, Isodamp C-1002-06)3 and aggregate (custom part from Castinite)
materials, and a molded, floppy silicon cover. The silicon cover, built in house [127],
additionally damps effects of fluctuations in lab temperature. Beam-shaping optics,
Faraday isolation, and fiber coupling are all integrated, and extreme isolation from
local environmental effects could be achieved by moving the ECDL to a sandbox in
an atmospherically controlled chamber, for example. The point of the novel design,
however, is to make such extreme measures unnecessary; while we have not carefully
characterized the long-time frequency stability of our laser, we have noticed that
the passive measures are quite effective: day-to-day drifts of a free-running laser are
typically less than a few linewidths of the 30-MHz-wide 5s2 1S0 −→ 5s 5p 1P1 cycling
transition in 88Sr.
3.2. Sensitivity to Perturbations
To measure the laser’s response to external perturbations, we use a couple of
techniques, pictured in Fig. 3.5. First, we actively drive the ECDL’s piezo—effectively
shaking the grating arm—and monitor light through a Fabry-Pe´rot (FP) cavity. We
offset the signal by half of the FP peak height, and slowly lock (100 ms integration
time) to the side of the resonance. Changes in frequency at the laser are transformed
into changes in amplitude of the error signal, Fig. 3.5(a). This technique is mainly
to search for acoustic resonances in the laser and we name it “cavity-resonance
spectroscopy.” Scope traces are produced using a Stanford Research Systems DS345
function generator to drive the piezo (0.03 Vp-p amplitude before passing through a
3There may be some benefit to cutting this material into smaller pieces, as implied in the
discussion about Sorbothane in Sec. 5.2.1.
32
photodiodelaser
PZT
0-20 kHz
FP cavity
slow lock
PZT
photodiode
laser
PZT
slow lock
l/2 l/4
Rb vapor cell
(a)
(b)
ND
pump
probePBS
feedback
erroroscilloscope
error spectrum analyzer
0 20kHz
0 100kHz
20 dB
FIGURE 3.5. Schematics for methods we use to analyze the laser’s susceptibility to
environmental perturbations. In (a), an active perturbation excites the laser’s piezo,
and we measure the amplitude of the frequency response by slow-locking a Fabry-
Pe´rot (FP) cavity to the laser and monitoring the error signal. In (b), the laser’s
passive frequency power spectral density is measured in a similar manner by locking
the laser’s frequency to an atomic resonance in 85Rb.
20.2 dB attenuator) from 0–20 kHz. Two hundred traces are squared and averaged
to produce the frequency-deviation data in Figs. 3.6 and 3.7.
In Fig. 3.6, we compare the novel ECDL with an existing “bronze” laser in
our laboratory (named for its 954 aluminum-bronze alloy [95, 114, 115]), which
is designed with a longer grating arm to optimize the pivot point and mechanical
feed-forward [98]. This makes the bronze laser much more susceptible to vibrational
perturbations than the new ECDL, especially at lower acoustic frequencies. Indeed,
a rough estimate of the fundamental frequency (as in Sec. 3.1.1) yields 360 Hz. To
compare the two lasers, the data are scaled to overlap the high-frequency tails, where
the response of the lasers should be the same. At low driving frequencies, the
33
piezo modulation frequency (kHz)
0 202 4 6 8 10 12 14 16 18
rm
s 
fr
eq
u
en
cy
 d
ev
ia
ti
on
(a
rb
it
ra
ry
 u
n
it
s)
0.5
0
bronze laser
with O-ring
new laser
without O-ring
new laser
0 205 10 15
100
10-3
10-1
10-2
FIGURE 3.6. Laser response while driving the piezo at acoustic frequencies in the
cavity resonance spectroscopy setup, Fig. 3.5(a). For comparison, an existing laser
made from 954 aluminum-bronze alloy is also analyzed. The inset shows the same
data on a logarithmic scale.
measured response is a result of the grating arm being tuned by the piezo rather
than being an actual mechanical resonance. So while both lasers operate at 780 nm,
the bronze laser has a smaller frequency deviation because the diffraction angles are
different (the bronze and new lasers use 1200 and 1800 lines/mm, respectively).
The difference in resonances is significant between the two lasers, and the log-
scale inset shows that lowest-frequency resonance in the new laser occurs around
12 kHz. The somewhat larger resonance at 14 kHz (which is the fundamental torsional
resonance, shifted from 2.05 kHz by the piezo supporting the end of the arm) is
dramatically damped by introducing a Viton O-ring underneath the grating arm.
The shear damping does not affect the 12 kHz feature, and it is likely that this
resonance is related to the lids. (The original prototype had 0.5′′-thick lids instead of
0.75′′, and we noticed that a similar resonance would shrink if the top lid was removed
34
piezo modulation frequency (kHz)
1 153 5 7 9 11 13
rm
s 
fr
eq
u
en
cy
 d
ev
ia
ti
on
(a
rb
it
ra
ry
 u
n
it
s)
0.06
0
medium perturbation
small perturbation
large perturbation
FIGURE 3.7. Response of a prototype laser under various driving amplitudes in
cavity resonance spectroscopy. These data were useful in diagnosing a loosely glued
diffraction grating (resonance just below 7 kHz), and were recorded in the following
order: medium, large, small.
or thickened.) We did not try other materials to dampen the arm motion, but it is
likely that Sorbothane would work well [128].
As implied in the preceding paragraph, cavity-resonance spectroscopy can be
useful in understanding and assuaging sensitivity to perturbations. As a concrete
illustration of this, in our prototype lasers we had occasionally seen small peaks in
the 3–7 kHz region which did not seem to have consistent location or height and were
unaffected by altering the lid or gluing masses onto the grating arm. The sequence
of data presented in Fig. 3.7 helped us to confirm these resonances were caused by
diffraction gratings not being glued in place firmly enough. We had originally tried
to use little epoxy to reduce the mass as much as possible, but when an undergrad
used larger quantities in a final production run, these resonances did not appear (to
see how much was used in this run, see the pictorial guide [117]). The hypothesis
was that the resonance of a shaking grating would behave in a nonlinear manner, and
35
driving the piezo with larger amplitudes would cause the grating resonance to grow
differently than the fundamental arm resonance. We tested this on a laser with no
O-ring damping, using drive amplitudes of first 0.6 Vp-p (“medium perturbation”),
then 3.0 Vp-p (“large perturbation”), and finally 0.03 Vp-p (“small perturbation”).
The change in the grating resonance (just below 7 kHz) is clear: the response for
the large perturbation is much more extreme than the medium, where the latter is
barely visible. Furthermore, after the violent shaking, the resonance is still larger
for small drive than it was for medium, implying that the grating has been shaken
loose. After applying more epoxy, we retested the small and medium amplitudes
and confirmed that the sensitivity near 7 kHz had vanished (we avoided the large
perturbation condition, afraid that we might once again loosen the grating).
A second method we used to characterize the laser’s response to perturbations
is a measurement of its frequency-noise spectral density (fSD). Here we converted
frequency fluctuations using an atomic resonance, which provides a narrower feature
than the FP cavity (and also eliminates coupling from vibration of FP mirrors to the
measured fSD). In this case, the laser itself was slowly locked (3 s integration time) to
the F = 3 −→ F ′ = 3, 4 hyperfine crossover of 85Rb using simple variant of saturated
absorption spectroscopy (SAS) [129], and the fluctuations of the error signal were
monitored on a digital spectrum analyzer (Tektronix RSA3408A). We did not worry
about subtracting off the Doppler-broadened background because absolute frequency
accuracy is not important in this measurement. The hyperfine splittings help convert
the error signal voltage to a frequency via a factor η, and we used this to transform
the power spectrum P (f) measured by the analyzer into an fSD Sν(f) for the laser
via Sν(f) = η
2RP (f), where R is the 50-Ω impedance of the spectrum analyzer.
36
frequency (kHz)
0 202 4 6 8 10 12 14 16 18
fr
eq
u
en
cy
 n
oi
se
 (
H
z2
/H
z)
109
104
105
106
107
108
new laser
bronze laser
background
10-1 101100
109
105
107
FIGURE 3.8. Frequency-noise power spectral density (fSD) as a function of frequency,
measured with the setup in Fig. 3.5(b) from 0–20 kHz. Inset is on a log-log scale up
to 10 kHz to highlight that many of the low-frequency noise peaks are associated with
pickup during acquisition since they are also in the background data.
Data for the frequency-noise power spectral densities are presented in Fig. 3.8,
including a trace taken with the light to the photodetector blocked to serve as
a baseline. Each trace represents several minutes of averaging performed by the
spectrum analyzer. The lasers were powered by a Libbrecht–Hall-style current
source from Vescent Photonics (D2-105) to minimize the supply’s contribution to
the frequency noise. The measurement was performed in a quiet environment: no
one spoke, the optical table was “floated,” and we turned off unnecessary equipment
such as a high-efficiency particulate arrestance (HEPA) filter fan. Even under such
quiet laboratory conditions, the difference between the new laser design and the older
bronze laser is stark. The resonances in the bronze laser appearing in Fig. 3.6 are all
visible here, highlighting the importance of eliminating susceptibility to low-frequency
perturbations in the new design. The fSD of the new laser sits below the bronze, save
for the small peak at 14 kHz, above which the two lasers are indistinguishable.
37
We also looked at the fSD measurements while actively perturbing the lasers
(clapping, tapping on the optical table, etc.) and observed that the new laser was
much less sensitive. In fact, after increasing the robustness of the feedback servo (by
adding a proportional stage and decreasing the integrating time constant), the new
laser was difficult to unlock, even via direct physical contact. This led us to make
similar fSD measurements under extreme, “calibrated” perturbation by an airhorn
(roughly 105 dBA in the vicinity of the lasers). The frequency excursions of the
bronze laser were larger than the crossover feature, so we instead had to lock onto the
side of the Doppler-broadened D2 line. The fSD in this case could not be averaged,
and a single trace from each laser are presented in [116].
3.3. Linewidth Measurements
The mean-frequency excursion ∆νrms, akin to the laser linewidth due to noise in
a particular frequency band, is found by integrating the fSD between f1 and f2 [130],
∆νrms =
√∫ f2
f1
S¯ν(f)df
= η
√
R
∫ f2
f1
P¯ (f)df. (3.5)
Here, S¯ν(f) and P¯ (f) are the the adjusted fSD and power spectral density, found
by subtracting the background signal. The lower frequency can be interpreted to
be related to the observation time f1 = 1/Tobs, which should be specified when
discussing a laser’s linewidth (see Sec. C.3.3). Integrating the data in Fig. 3.8 from
20 Hz to 100 kHz gives ∆νrms = 296(26) kHz for the bronze laser and 146(8) kHz
for the new laser over a 50 ms observation time. The difference in these measured
linewidths is entirely in the band below 10 kHz, and by more carefully controlling the
38
w0
w0 + W
fiber delay
AOM
W
to spectrum 
analyzer
polarizer
fast  
photodiode
l/2
PBS
laser
0
1
FIGURE 3.9. Schematic depicting the delayed self-heterodyne measurement. An
AOM operating at frequency Ω splits a laser into two paths of an interferometer.
One of the arms has a long fiber delay before the two beams are combined on a fast
photodiode (New Focus 1801-FS). The resultant spectrum, centered at Ω, contains
information about the laser’s linewidth (see Appendix C).
mechanical behavior of the laser at these frequencies, we have cut the linewidth in half.
Moreover, in the case of the airhorn, the integrated rms noises are ∆νrms = 40 MHz
and ∆νrms = 1 MHz for the bronze and new laser, respectively, demonstrating the
superior performance of the new laser under active perturbation.
Another method used to measure laser linewidth is the delayed self-heterodyne
technique [131, 132], discussed in more detail in Appendix C. In this process, depicted
in Fig. 3.9, a laser is interfered with itself, producing a beatnote on a photodiode (from
which linewidth information can be ascertained). An acousto-optic modulator (AOM)
shifts the beatnote spectrum away from dc for easy measurement on a radio-frequency
(RF) spectrum analyzer. One version of the laser is time-delayed by traveling through
a long fiber so that two different time periods can be compared. If the delay is
long compared to the coherence time of the laser, the two beams have uncorrelated
frequency noise and can be regarded as separate sources. In this case, extraction of
the laser’s full width at half maximum (FWHM) linewidth is straightforward, but a
more precise understanding is generally necessary.
39
Using the machinery of phase-fluctuation statistics [133], the laser’s frequency is
modeled to have both white- and 1/f -noise components:
Sω(ω) = γ +
k
|ω| . (3.6)
The white-noise parameter γ is the modified Schawlow-Townes linewidth, Eq. (3.1),
while the k parameter characterizes the strength of the 1/f noise expected in an
electronic device. The current power spectral density measured at the photodiode in
the self-heterodyne technique is
SI(ω) ∝
∫ ∞
−∞
dτe−i(ω−Ω)τ exp
(− γmin{|τ |, |td|} − δdriftτ/2− (δRBWτ)2/16 ln 2)×[
|τ + td|−k(τ+td)2/2pi |τ − td|−k(τ−td)2/2pi |τ |kτ2/pi |td|kt2d /pi
]
, (3.7)
where td is the delay time, Ω is the AOM frequency, and δRBW is the Gaussian
resolution bandwidth (RBW) of the spectrum analyzer. This equation is the same as
Eq. (C.20) with the addition of the δdrift term, which accounts for drift in Ω during
averages on long timescales by convolving the resulting spectrum with a Lorentzian
of width δdrift. Information about the observation time Tobs is contained in δRBW, and
this term explicitly cuts off the integral, which would be otherwise divergent for large
td [39].
We have three lengths of single-mode telecommunication fiber to use as delay
paths: 2.260 km, 4.415 km, and 4.457 km. These are combined to take a series of
self-heterodyne spectra at various delay times, such as those pictured in Fig. 3.10.
For short delays, coherent wiggles can be clearly seen in the wings of the spectrum,
and these are used to fit the time delay td in the 2.260 km fiber. This sets the light’s
wavelength-dependent propagation speed in the fiber, which is used to determine td
40
foo-ofm (kHz)
-1000 1000-800 -600 -400 -200 0 200 400 600 800
p
ow
er
 s
p
ec
tr
al
 d
en
si
ty
 (
d
B
m
) -35
-95
-40
-50
-60
-70
-80
-90
2.260 km
4.415 km 6.675 km
8.872 km
FIGURE 3.10. Delayed self-heterodyne spectra of a 922 nm laser, including both
data (green line) and numerical fitting (dashed blue line) described the text. The RF
frequency of the AOM is fm = Ω/2π = 80 MHz. Four fiber lengths are used, and
fitted spectral parameters are γ = 1.86(29) kHz and
√
k/2π = 27.0(15) kHz, yielding
an expected 32 kHz FWHM linewidth for Tobs = 100 µs.
for longer delays. The spike in the center of the spectrum is caused by remaining
coherence in the laser’s frequency fluctuations and by extra RF pickup of the AOM
signal by the spectrum analyzer.
We numerically fit the data using Eq. (3.7), taking as fitting parameters δRBW
and δdrift for the spike, and γ and k for the spectral tails. Amplitudes for the spike and
remaining spectrum, as well as a background baseline level, are also used to fit the
height of the spectra. The results of the fit (dashed blue line) are plotted with the data
(solid green line) for a 922 nm laser in Fig. 3.10, and the model is clearly sufficient to
characterize both the center and the tails of the spectra. (Often, linewidth analyses
only consist of fitting a Lorentzian to the spectrum and taking half of the resulting
FWHM as the linewidth, but these miss the Gaussian part caused by 1/f noise and
underestimate the actual width [116, 132].) After the γ and k parameters are known,
41
the overall FWHM linewidth can be computed for a specified rectangular-window
observation time Tobs [39].
We used the self-heterodyne technique to characterize several lasers we
made [116], but of special interest is an extra-long ECDL we constructed at 689 nm
for use in narrow-band cooling on strontium’s 5s2 1S0 −→ 5s5p 3P1 intercombination
line (which is less than 10 kHz wide). For comparison, we also constructed an
otherwise identical 689-nm laser in the standard “short” length depicted in Fig. 3.4.
As discussed previously, a diode laser’s linewidth is inversely related to the square of
the cavity length [see Eqs. (3.1) and (3.2)], so the linewidths of two ECDLs operating
at the same frequency with similar cavity length stability are related by
∆ν1 = ∆ν2
L22
L21
. (3.8)
For an ECDL, the total cavity length is the sum of the optical path length in the
diode itself and the extended cavity length: L = nd + Lext. Typically, nd ∼ 600 µm,
and the added lengths of our two cavities are 2.2 cm and 10.0 cm. We therefore
expect the linewidth of the long laser to be reduced by roughly a factor of 20. Self-
heterodyne spectra for the 2.260 km fiber for the short- and long-cavity lasers are
shown in Fig. 3.11. Clearly the short laser has a much smaller coherence time since
periodic structure is not visible in the tails of the spectrum. Averaging the fits of two
fiber lengths (because Rayleigh scattering at 689 nm is significant, longer delays are
not feasible), we find the noise parameters to be γ/2π = 93(36) kHz and
√
k/2π =
87.0(7) kHz for the short laser and γ/2π = 4(1) kHz and
√
k/2π = 22(1) kHz for the
long. The reduction in white-noise linewidth is consistent with the increase in cavity
length. Using these parameters the short- and long-laser’s FWHM are 254 kHz and
11.7 kHz, respectively, for Tobs = 100 µs.
42
foo-ofm (kHz)
-1000 1000-800 -600 -400 -200 0 200 400 600 800
p
ow
er
 s
p
ec
tr
al
 d
en
si
ty
 (
d
B
m
) -35
-95
-40
-50
-60
-70
-80
-90
long laser
short laser
FIGURE 3.11. Measured data (green lines) and numerical fits (dashed red lines) using
a 2.260-km delay fiber for two 689 nm lasers with different cavity lengths. Reduction
in linewidth of the long laser matches expectations (see text).
3.4. Construction and Assembly Details
Complete details of the laser’s peripherals are discussed in [116], and the pictures
posted on our website [117] are useful, but inexperienced users may require a little
more guidance to assemble the ECDL. This section provides some extra instruction
where necessary.
3.4.1. Customizing a Particular Wavelength
A primary goal in designing the new laser was to make it cheap to construct
and easily adaptable to many wavelengths. We made efforts to minimize the
number of orientations and tool changes required on a computer numerical controlled
(CNC) milling machine, and also used stocked commercial peripherals when possible.
Interactions with the machine shop are simplified by using a standard set of machined
pieces for any laser, with the operating wavelength being determined by a few key
43
 300  350  400  450  500  550  600  650  700  750  800  850  900
Wavelength (nm) Opposite grating shimArm deflection only
Grating shim and arm deflection
 900  950  1000  1050  1100  1150  1200  1250  1300  1350  1400  1450  1500
FIGURE 3.12. Wavelengths that can be realized with the novel ECDL design using stock Richardson diffraction gratings
between 300 nm and 1500 nm. Available wavelengths for ruled gratings are presented above those for holographic ones.
Red regions do not require any shimming of the grating, but blue and green require normal and reverse shimming,
respectively (where “normal” shimming is performed at the end of the grating arm). Details of choosing the correct
number of lines/mm for a particular wavelength are given in the text.
44
optical components—the laser diode, diffraction grating, anamorphic prism pair,
optical isolator, and fiber coupler. This also adds flexibility and utility to spare
or retired lasers without need of additional machine work.
Referring again to Fig. 3.4, the basic cavity geometry has light from the diode
incident on the diffraction grating at ∼ π/4 radians. The exact angle is determined
by the diffraction equation in the Littrow configuration, where αin = αout =: αg and
the diffraction order is m = 1:
λ =
d
m
(sinαin + sinαout)
= 2d sinαg. (3.9)
Here d is the separation between grooves in the grating, but values are typically
given in the number of lines/mm. Recall that the diffraction grating arm (Sec. 3.1.1)
is designed to be “loaded” into place by the piezo and fine adjust screw, but since
we want to avoid fatiguing the aluminum, large changes to this location should be
avoided. This means angles that π/4 − αg should be positive (to retain tension in
the flexure pivot and avoid “unloading” the arm) and relatively small. We have
successfully deflected the arm by π/4 − αg ≈ 2◦, but any angle larger than this
requires a shim behind the outer edge of the grating. Our custom-sized gratings have
a width of 12 mm, so this angle is converted to shim thickness (in thousandths of an
inch) via t = 1000(12/25.4) sin(π/4 − αg). We usually undersize the shim by about
0.005′′ to allow leniency for burrs or layers of epoxy adding extra thickness. Fig. 3.12
shows a range of wavelengths that can be achieved using our laser’s geometry and a
list of available ruled and holographic diffraction gratings from Newport’s Richardson
Gratings (these are gratings for which they have a “master” grating, which cost about
45
output aperture
a
a1
a2
Thorlabs  
PS871-B
custom-sized  
grating
780 nm  
1800 g/mm
qpi  = 16°  
q||   = 8.7°
914 nm  
1500 g/mm
qpi  = 24°  
q||   = 10°
922 nm  
1500 g/mm
qpi  = 24°  
q||   = 10°
689 nm  
2000 g/mm
qpi  = 19°  
q||   = 8.5°
994 nm  
1350 g/mm
qpi  = 26°  
q||   = 10°
852 nm  
1596 g/mm
qpi  = 30°  
q||   = 8.5°
p/4-ag
wpi
w||
FIGURE 3.13. Numerically determining the location of anamorphic prisms. The
grating angle αg is set by the Littrow condition for λ by Eq. (3.9), and α1, α2,
and a are calculated to enable fiber coupling at the output aperture. Footprints for
prism locations (which are to scale and can be directly printed from this document)
are shown for strontium-wavelength lasers as well as common rubidium and cesium
wavelengths. Diverging angles θ⊥ and θ‖ are taken from laser diode data sheets.
$10k to produce). Note that “normal” shimming is preferred over “reverse” based on
the location of the beam output from the laser body. (The idea of a reverse shim to
expand wavelength coverage came after we finalized and published the design, and we
anticipate that much of the green areas in the figure cannot be nicely circularized or
fiber-coupled without some extra machining or creativity.) Note also that the figure is
not representative of wavelengths covered by existing laser diodes, and that a suitable
source still needs to be attained.
The magnification of the prism pair, meanwhile, is determined by the desired
operational wavelength and the divergence angles of the laser diode, typically given
46
as FWHM angles θ⊥ and θ‖, where the orientation is relative to the diode junction
(for a TE diode, light is polarized parallel to the junction, along the minor axis of the
elliptical output). Fig. 3.13 shows a (not-to-scale) cartoon of the grating and prisms,
where the beam is circularized by transforming w⊥ into w‖ (while not affecting w‖ in
the orthogonal direction). We start with the position and orientation of the diffraction
grating for the given λ, assuming the grating has been pivoted and shimmed at the
standard location of the supporting arm. Next, three parameters—the angle of the
first prism α1, the angle of the second prism α2, and the distance between them a—
are adjusted numerically until the beam is circularized and passes normal to the laser
body through a fixed output point (for fiber coupling). A computer script generates an
outline footprint of the resultant prism locations which can be printed and glued into
the prism pocket for aid in alignment (we etch the outline on a thin piece of acrylic
using a laser cutter, which works nicely). The outlines in Fig. 3.13 for strontium,
rubidium, and cesium wavelengths are presented to scale and can be printed directly
for use in building lasers at these wavelengths. Additional wavelengths require the use
of the Octave script posted on our web site [117]. By default the beam is constrained
to pass through the center of the prisms, but they might need to be nudged to avoid
touching each other or the pocket boundary.
3.4.2. Final Assembly and Alignment
After a suitable geometry is chosen, the grating is epoxied in place using a
Torr-Seal equivalent (Duniway “Epoxi-Patch” EP-8034) as one of the final steps in
assembly. A machined spacer positions the grating against the top outer edge of the
arm, and a laser-cut jig holds the laser so the grating lays flat (we made a couple of
attempts to clamp the grating in place, but nothing worked as well as using gravity
47
to hold it in place, especially with the complication of sandwiching a small shim in
place). We use three dots of epoxy on the exterior of accessible corners of the grating-
arm interface and carefully slide the spacer out after the epoxy sets overnight. It’s a
good idea to refrain from putting epoxy underneath the grating: the grating should
be secure enough that it does not vibrate as in Fig. 3.7, but should also be easy to
break free if necessary.
To more easily align the laser diode, it is helpful to be able to lift the entire diode
can mount out of the laser body and move it to an external platform. This means
that wires connecting the protection circuit, diode, and temperature sensors should
be as long as possible while being short enough to fold nicely into the main cavity
pocket. First, the laser’s output is pointed at a far wall and the aspherical optic in the
collimation package is adjusted to minimize the spot size (ensuring the beam does not
go through a focus). Next, a polarization beamsplitting cube (PBS) is used to orient
the polarization. Since for our diodes we wanted our light to be vertically polarized,
we minimized transmission through the PBS by rotating the collimating tube inside
the mounting block. A 4-40 screw secures the tube in place, and it is helpful to
pre-tighten this somewhat with the tube sticking about 0.25′′ out of the front of the
block. The tube can then be rotated with fingers to controllably find the minimum,
at which point the screw is tightened a bit more. The tube is then pushed into the
mount until the front faces are flush—while making sure the minimum transmission
does not change—and the screw is fully tightened. We recommend this procedure
because using the wires at the rear to rotate the can be frustrating, not to mention
dangerous for the diode.
The aligned diode can mount is placed inside the main cavity body, loosely
attached using two 1/4′′ 8-32 screws, and the wires are carefully packed into the
48
available space behind the diode (the protection circuit should not be epoxied in place,
since removal of the diode may be necessary in the future). Standard procedures are
followed to align the cavity [110], with horizontal alignment performed via the screw
behind the grating and vertical alignment with two rounded 4-40 screws threaded
through the diode can block, pushing off the bottom lid. It is helpful to set the
current just above threshold and iterate horizontal and vertical degrees of freedom
while monitoring the wavelength on a wavemeter (such a step was crucial in prototype
versions of our laser, where the grating adjustment screw did not push in the center
of the arm and the horizontal and vertical degrees of freedom were coupled). When
the desired wavelength is reached, the grating is in the correct position, and threshold
should be minimized using the vertical tilt screws. Raising the diode (clockwise screw
motion) is easier than lowering, which requires force on the adjusting ball driver to
push the can downward. To rigidly secure the diode can mount to the main cavity
block, the 8-32 mounting screws are tightened while the tilt screws are removed
(the diode’s position is thus not over-constrained and is less susceptible to future
creep). This process is somewhat difficult, and iterative tightening and loosening by
small amounts is required. Even then, a discontinuous shift in alignment can happen
as tension in the tilt screws is released, and tightening often needs to be repeated
numerously in a somewhat stochastic process.
Finally, anamorphic prisms are installed using the printed outline as a guide. The
laser’s output is monitored about a meter away, and small changes are made to prism
locations and angles so that the beam is roughly round and—more importantly—
perpendicular to the laser body. An optical isolator is attached to the main cavity
body, and the laser is then ready to be coupled into fiber. We have had some difficulty
with the fiber couplers we use from Oz Optics (HPUC series). The locking screws are
49
not rounded and will cut slight grooves into the anodized laser-head adapter; upon
subsequent coupling and locking, these produce uncontrollable shifts in the position
of the coupling lens. We rounded our own screws (McMaster 92196A048) and inserted
small sapphire disks (Swiss Jewel W4.07) to circumvent this issue.
50
CHAPTER IV
EXPERIMENTAL APPARATUS
4.1. Vacuum System
Maintaining atoms in a MOT for an appreciable amount of time requires the
trap to be sufficiently isolated from collisions with background gas. Trap lifetime
is inversely related to the pressure [134], and one can actually use this to infer the
pressure through lifetime measurements [135, 136]. Because of this sensitivity to
background-gas collisions, an indispensable part of an atom-trapping apparatus is an
ultra-high vacuum (UHV) system. This section discusses the design and construction
of our UHV chamber, pictured in Fig. 4.1, and its associated peripherals. Vacuum
technology is a mature field of physics, and there are many sources which cover in
much greater detail the methods and components described here [137–139].
The chamber for our setup has two distinct regions: a high-pressure side (left)
featuring a hot-oven atomic-beam source, and a low-pressure side (right) where
cooling and trapping occurs. The pipe transitioning between the two regions acts
as a differential pumping tube and is where preliminary cooling of the atomic beam
takes place in a “Zeeman slower.” In the low-pressure region, atoms are trapped in a
“top MOT” at the center of a spherical octagon and can be delivered to a “bottom
MOT” inside a pyrex cell (ColdQuanta CQMC0006), which provides increased optical
access and isolation from the stream of hot strontium atoms coming from the oven.
51
StarCell 75 L/s
✶ ✁✂❤
✁✐✄✁☎
✈❛✄✈☎
❩☎☎✆❛✁ ✝✄✞✇☎❡
tt❛✁✟✆
✝✟s✄✆❛t✞✁ ✠✟✆✠
✠②❡☎✡ ✂☎✄✄
✇❛t☎❡✐✂✞✞✄☎♦
❆☛ ✂✞✄✝
☎✦✟✝✈☎ ☞❡ ✞✈☎✁
❡r❤t✐❛✁r✄☎
✈❛✄✈☎
❡r❤t✐❛✁r✄☎
✈❛✄✈☎
s☎✄✄✞✇✝
✺✺ ✌✍✝
✞✁ ✠✟✆✠
✼✺ ✌✍✝
✞✁ ✠✟✆✠
t✞ t✟❡s✞ ✠✟✆✠
✝❛✠✠❤❡☎
✇✁♦✞✇
✈☎✇✠✞❡t
✈☎✇✠✞❡t
✈☎✇✠✞❡t
❛t✞✆✂ s☎❛✆
✝❤✟tt☎❡
◆ r❛✝✎☎t
❛✠☎❡t✟❡☎✝
❤r❤✐✠❡☎✝✝✟❡☎✍✝✞✟❡✂☎ ❡☎r✞✁ ✄✞✇✐✠❡☎✝✝✟❡☎✍t❡❛✠✠✁r ❡☎r✞✁
FIGURE 4.1. Schematic of the UHV chamber used for trapping and cooling strontium atoms. Atoms originate from
the effusive oven (high pressure, far left), pass through a permanent-magnet Zeeman slower (center), and are trapped
in a spherical octagon and a vertically oriented pyrex cell (low pressure, right). A pair of right-angle valves allow a
turbomolecular pump to be connected to both sides of the chamber via temporary bellows.
52
4.1.1. Hot Strontium Source
4.1.1.1. Oven Design
A schematic drawing of our strontium oven is shown in Fig. 4.2. All internal
parts are machined from 316L stainless steel, as this is inert with Sr vapor [140].
Solid strontium is placed in a crucible with a circular output aperture of 0.25′′
diameter. Adjacent to the crucible, a collimating nozzle is made of a bundle of
capillary tubes, which are 1-cm long with nominal 203 µm inner diameter (ID) and
414 µm outer diameter (OD). Capillaries are cut from long hypodermic tubing (Small
Parts B000FN0TL2) using a narrow grinding wheel, and are then cleared of burrs
with a wire brush and individually inspected for a clear aperture. The nozzle and
crucible are sized to fit snugly in the center of a custom-length 12′′ nipple, which
has a smooth-bore, 0.620′′ ID interior. (We first ordered the custom-length nipple
from Kurt J. Lesker Company, but this tube was rolled and welded and had a 0.020′′
ridge on the inside. The UO machine shop had some pipe with higher quality interior
surface finish we used instead.) Four loose-fitting tubes (0.615′′ OD) slide into the
nipple and act as spacers to accurately position the location of the effusive oven.
Fortunately, we can achieve temperatures to create sufficient Sr vapor without
using electrical feedthroughs and heaters inside the vacuum chamber. The yellow
rectangles in Fig. 4.2 represent semi-cylindrical clamshell radiative heaters, which
heat the crucible and nozzle from the exterior. Two custom-length pairs of heaters
(Thermcraft RL106-S-L, 110 W) are imbedded in semi-cylindrical insulation housing
(Thermcraft VIP-2.5-10-0.75-2) and secured around the 0.75′′ OD oven pipe. Wired
in series, the two heaters around the nozzle are heated to be 50◦C warmer than
those around the crucible to counteract clogging from condensing Sr. The heaters are
53
Sr crucible
collimating 
nozzle
radiative heaters
insulation
loose-fitting  
spacers
to chamber
FIGURE 4.2. Schematic of the strontium source. A crucible and collimation
nozzle are centered in a custom-length nipple and heated externally. The nozzle’s
temperature is maintained to be 50◦C warmer than the crucible to prevent clogging.
connected to a Variac set to 38 V rms source, switched by solid-state relays (Omron
G3NA-220B) that are driven by temperature controllers (Love Controls TCS-4011)
which monitor the temperature using type-J thermocouples. We originally attempted
to switch heaters with the Love controller directly, but its internal relay failed (in the
closed position, effectively removing the ability to switch off the heater) after several
weeks.
Inspecting a photo of the completed nozzle reveals N = 184 capillaries in our
bundle, with an additional capillary that appears to be blocked at the input aperture.
We loaded a total mass of 3.21 g of ≥ 99% pure Sr (Sigma Aldrich 474746-25G) into
the crucible and assembled the oven inside a glove bag filled with argon. The crucible
is tapped on the rear end (left side in Fig. 4.2) for extraction when recharging the oven,
although we anticipate that the crucible will fuse to the nipple and be irremovable.
54
In this case, before reproducing a new set of the aforementioned custom components,
there are additional Sr oven solutions which are worth considering [141, 142].
4.1.1.2. Molecular Flow Conductance
Dynamics of molecular flow depends on the density of gas and geometry of
its container. Typically, the Knudsen parameter Kn = l/a defines two distinct
regimes (here, l is the mean-free path of the gas and a is the characteristic length
of the container, often the radius of a cylindrical pipe [137]). At high densities,
Kn < 0.01, the dominant interactions are inter-particle collisions of gas molecules,
and flow dynamics are that of a compressible, viscous fluid. Low densities, meanwhile,
are governed by particle-wall collisions, and for Kn > 1 flow can be more simply
determined from a statistical ensemble of non-interacting thermal particles. In the
geometry of our oven, strontium atoms entering a 203 µm ID capillary will be in the
latter regime (referred to as the molecular or effusive regime) for temperatures below
700◦C.
Velocities in the oven are given by the Maxwell-Boltzmannn distribution f(v),
and we can find the distribution of atoms moving in the z-direction by integrating
over the transverse components,
f(vz) =
∫
f(v)dvxdvy
= n
(
m
2πkBT
)1/2
exp
(
− mv
2
z
2kBT
)
. (4.1)
This is normalized to the number density n, so the flux of atoms with a particular
velocity passing through an aperture of area A is then b(vz) := Avzf(vz), where b(vz)
is the velocity distribution for a molecular beam with vz > 0. Integrating b(vz) over
55
temperature (Celsius)
300 700400 500 600
v
ap
or
 p
re
ss
u
re
 (
to
rr
)
100
10-7
10-6
10-5
10-4
10-3
10-2
10-1
FIGURE 4.3. Vapor pressure of strontium. A larger version is given in Appendix B.
all positive velocities gives the total atomic flux through the aperture. The result is
ΦA =
Apv√
2πmkBT
, (4.2)
where the number density n has been rewritten in terms of the vapor pressure
pv, assuming an ideal gas. The vapor pressure is determined using empirical data
tabulated by C.B. Alcock [143],
log10pv = 5.006 + 9.226−
8572
T
− 1.1926 log10T, (4.3)
and is plotted in Fig. 4.3.
4.1.1.3. Operating Lifetime and Beam Flux
Conductance through a pipe or duct is the product of the aperture flux of
Eq. (4.2) with a geometric transmission probability α. If we assume the nozzle has a
56
hexagonal close-packed arrangement of capillaries, then the gaps between capillaries
are approximately triangular and we can estimate the transmission probability
through these orifices [144]:
Φtotal =
pv√
2πmkBT
(NαA +Nα∆A∆) , (4.4)
where the symbols  and ∆ refer to capillaries and gaps, respectively. This is an
overestimate of the flux because the triangular gaps have an area which is nearly a
factor of three larger than the area in a close-packed arrangement. (Furthermore,
from the machine drawing of the nozzle, the full area of the aperture is measured
to be 27.5 mm2, while the combination of N = 184 capillaries and triangles covers
28.3 mm2.) However, it could also be an underestimate if there are large gaps between
capillaries (that is, the close-packed assumption is bad) or around the outside of the
bundle. Given that we loaded 3.21 g of strontium in our oven, we find the temperature-
dependent operating time plotted in Fig. 4.4. This assumes round-the-clock operation,
but we typically only run the oven while we are in the lab tending the experiment.
The flux of atoms which is delivered to the top MOT trapping region is influenced
by the geometry of the gaskets in our differential pumping tube, schematically shown
in Fig. 4.5. Atoms which pass uninhibited through a capillary spread at a full-angle
of about 2.3◦. We further collimate this beam by using two nickel gaskets placed
on either side of the all-metal angle valve which separates the high- and low-pressure
chambers. Atoms entering the trapping region thus have limited transverse velocities:
v⊥ < avz/d, where d is the distance between the beginning of the capillary and the
final aperture of radius a [145].
With these new boundary conditions, we integrate out the transverse velocities of
the Maxwell-Boltzmann distribution to find the velocity distribution for a collimated
57
oven temperature (Celsius)
450 600500 550
ov
en
 l
if
et
im
e 
(y
ea
rs
)
102
10-1
100 1013
101 1014
b
ea
m
 f
lu
x
 (
1/
s)
1015
1012
FIGURE 4.4. The estimated lifetime of the effusive oven under continuous operation
(blue) and atomic beam flux delivered to the top MOT trapping region (green) as a
function of temperature. We make conservative estimates for each (see text), so the
curves represent a lower-bound of expected values. Running at a cooler temperature
drastically improves how long we can operate before recharging the strontium crucible.
molecular beam,
bcol(vz) = Avzf(vz)
= 2πAvz
∫ avz/d
0
v⊥f(v) dv⊥
= An
(
m
2πkBT
)3/2
vz exp
(
− mv
2
z
2kBT
)[
1− exp
(
− mv
2
z
2kBT
a2
d2
)]
≈ Anπa
2
d2
(
m
2πkBT
)3/2
v3z exp
(
− mv
2
z
2kBT
)
, (4.5)
where A is again the area of the capillary and the final approximation assumes d≫ a.
Note that the mean, average, and rms velocities of bcol(vz) are skewed to be higher
than those from the typical Maxwell-Boltzmann distribution [146, 147]. The total
58
Capillary
a = 6.35 mm
d = 46.4 cm
200 mm
Ni Gaskets
FIGURE 4.5. Sketch of atomic beam passing through a capillary and constricted by
downstream nickel gaskets. Labeled dimensions are used to estimate the atomic flux
delivered into the trapping region.
collimated flux is then approximately
Φcol = N
∫ ∞
0
bcol(vz)dvz
= N
a2
d2
ΦA, (4.6)
where we have made a conservative estimate by considering only atoms passing
through the N = 184 capillaries (and ignored excess conduction through other gaps in
the nozzle). The atomic flux delivered to the trapping region is plotted as a function
of temperature in Fig. 4.4.
4.1.1.4. Atomic Beam Considerations
Because of the high vapor pressure, strontium atoms will readily stick to the
cold surfaces of the chamber. Anything not directly in the oven’s beam path is
unlikely to accumulate large numbers of atoms, meaning that most windows and
viewports will remain clear. We do, however, use a sapphire viewport (Larson VSZ-
150-F2, exterior surface AR-coated for 461 nm) at the entrance of the Zeeman slowing
beam to avoid the direct stream of strontium damaging a silica window. The atoms
59
will not react with sapphire [148], but will still deposit on the surface. To remove
the condensed atoms, we heat the sapphire window, which is extended from the
chamber on a 5′′-long nipple. We wrap a one-inch-wide silicone rubber heater (Omega
SRFG-108/10) around the 2.75-inch ConFlat (CF) joint, and heat it to 200◦C, near
the heater’s maximum operating temperature. An infrared thermometer measures
the temperature of the center of the window to be 175◦C. Assuming evaporation
into a perfect vacuum, the Hertz-Knudsen equation predicts the flux of atoms at
temperature T leaving an area A to be the same as the flux of atoms passing through
an orifice as in Eq. (4.2) [149]. In this limit, the evaporation rate is comparable to the
flux of the atomic beam. In practice, we have noticed that running the oven at 600◦C
for several hours without loading a top MOT has produced a film on the sapphire
window which takes a few days to evaporate.
To reduce strontium contamination on the sapphire window, we have an atomic-
beam shutter which uses a magnetic vacuum feedthrough (Kurt J. Lesker DS275VPS).
The shutter is actuated with timing pulleys and a stepper motor (ROB-09238, driven
by the “EasyDriver” stepper motor, ROB-12779, both ordered from SparkFun), and
can be controlled by the computer to be timed with the rest of the experiment.
However, since opening or closing the shutter takes a few hundred milliseconds and
creates substantial vibration, we typically close the shutter only when we do not
intend to load any atoms into the top MOT for several minutes.
4.1.2. Chamber Design
Optical access to the atomic beam is provided via Kovar-sealed glass viewports
(six of 0.75′′ clearance on a 1.33′′ CF flange [Larson VP-075-F1] and nine of 1.5′′
clearance on a 2.75′′ CF flange [Larson VP-150-F2], all AR-coated for 461 nm by
60
Spectrum Thin Films), most of which are attached to two spherical octagons (Kimball
Physics MCF450-SphOct-E2A8 and MCF600-SphOct-F2C8). On the high-pressure
side of the chamber, the smaller spherical octagon allows observation of the atomic
beam and the option to cool atoms transversely as they exit the oven. We currently do
not cool the beam in this manner, but this could be useful if a larger flux is necessary,
as is the case for many groups who use strontium [17, 19, 150]. In the low-pressure
spherical octagon, the six MOT cooling beams enter through the four ports diagonal
to the atomic beam (in the plane of the optical table), a viewport at the top of the
chamber, and the end of the pyrex cell (in the vertical direction). The only non-stock
parts used in the chamber are the aforementioned oven pieces, a pair of nickel gaskets
(discussed in more detail below), and the differential pumping tube (a 15-inch-long
1.33′′ CF nipple).
Vacuum is maintained in our chamber using two rebuilt Varian Starcell ion
pumps (Duniway RVIP-75-ST-M and RVIP-55-ST-M), a titanium sublimation pump
(Duniway TSP-275-003), and two all-metal right angle valves (VAT 54032-GE02)—
where we attach both sides of the chamber via temporary bellows to a turbomolecular
pump for initial evacuation during bakeout. To simplify achieving UHV, we minimized
the inner surface area and number of CF joints, and maximized conductance to the
pumps. The titanium sublimation pump (TSP) is positioned so it will not sputter into
any other pumps or onto any windows. We abstained from using dedicated pressure
gauges, and instead used the ion pumps to monitor chamber pressures. Ideally,
vacuum will only need to be broken when the strontium source is depleted. When
this happens, an inline angle valve (Varian 9515052) can seal off the low-pressure side
of the chamber to maintain vacuum during the refilling process.
61
oven
high-pressure 
chamber
low-pressure 
MOT chamber
nozzle ZS tube
0.2 L/s
C2
S2 = 75 L/s
C3
S3 = 1000 L/s
ion pump Ti-sub pump
QZ
Q2 Q3
QL = qA2
P1 < 10
-5 torr P2 < 10
-9 torrP0 ~ 10
-3 torr
QH = qA1
C1
S2 = 55 L/s
ion pump
Q1
0.15 L/s
Q0
FIGURE 4.6. Model used to estimate expected initial pressures in the vacuum
chamber. Conductances, in red, are calculated from chamber geometry, and
throughtputs QL,QH , and Q0 are estimated using expected outgassing rates in
stainless steel [151]. From these and the pumping rates, Si, pressures are calculated.
Before finalizing the design, we considered a model of the chamber with three
pressure regions, depicted in Fig. 4.6. Conductances of the nozzle, differential
pumping tube, and pipes leading to the pumps were calculated using Santeler’s
empirical formula [152], and outgassing rates Q = qA were estimated in the chamber
based on a post-bake value of q = 1014 torr-L/s/cm2 for stainless steel [151] over an
area of A1 ≈ 1400 cm2 and A2 ≈ 4400 cm2 in the high- and low-pressure sides,
respectively. Since the contents of the oven were not going to be baked before
assembly, we assumed a higher outgassing rate from hydrogen diffusing out of the
stainless steel [151], and calculated the expected initial rate for our oven and nozzle
operating at T = 550◦C (the outgassing area is roughly 160 cm2, counting the
capillaries). Pumping rates are 55 L/s, 75 L/s, and ∼ 1000 L/s for the two ion
pumps and TSP, respectively (the TSP pumps at 20 L/s/in2, which is multiplied
by the inner surface area of the 6′′ nipple). All these were combined in a system
of molecular throughput and conductance equations (combinations of Q = PS and
62
Q = C∆P , where Q is throughput, C is conductance, S is volumetric pumping rate,
and P is pressure [144]). The resulting estimate for pressure in the MOT region was
below 10−9 torr, and this value only decreases as hydrogen is depleted from the oven’s
steel. Thus the conductance in our differential pumping tube allows UHV operation
even though the pressure in the oven region is quite high.
4.1.3. Assembly and Bakeout
During preparation and assembly of the chamber, components were handled
with powder-free nitrile gloves and connected following standard procedure for CF
flanges [153]. Stock vacuum parts from vendors made from 304L or 316L stainless
steel were inspected for cleanliness and baked in air at 450◦C for 24 hours. High
temperatures increase the rate at which hydrogen—trapped monatomically inside
the metal during production—diffuses out of the metal [151], thereby reducing the
quantity which can outgas in the future. Additionally, by baking in air, an oxide layer
forms on the surface, which can act as a barrier to hydrogen recombining and leaving
the metal (with a caveat that the oxide layer will break down and increase outgassing
if the part is subsequently heated to 250◦C) [154]. After this pre-bake, parts were
cleaned with spectroscopically-pure acetone and methanol, dried with canned air,
and wrapped in UHV-grade aluminum foil. Custom-made parts, meanwhile, were
machined with sulfur- and silicon-free cutting fluid (we have used Trim-sol and Kool
Mist 78), and then sonicated in soapy water (Alcanox), deionized water, and isopropyl
alcohol before final cleaning in the same manner as the stock parts. We chose not
pre-bake the 316L stainless steel components of the oven since its normal operation
quickly reduces the hydrogen content anyway.
63
Because the two sides of the chamber are mated via a long 1.33′′ CF nipple, we
built the chamber on an 80-20 support frame to increase rigidity and mobility. After
separate construction, the two halves were joined by sliding the 80-20 pieces under
the high-pressure side until the CF flange at the end of the differential pumping tube
met the 6′′ spherical octagon. Preliminary alignment of the chamber was performed
before the contents of the oven were filled: by looking into the chamber through
the sapphire window and adjusting the position of the two halves of the chamber
on the 80-20 frame, we tried to make all circular features concentric—in particular,
the two nickel gasket apertures, light from the far end of the oven, and the sapphire
window. (Alignment was rechecked after bakeout with the oven activated to produce
blackbody glow, confirming line of sight between the window and oven.)
Parts exposed to atmosphere have water vapor adsorbed on the surface, so
achieving UHV requires an extended bakeout while evacuating with a turbomolecular
pump. The 80-20 support frame enabled us to relocate the chamber during bakeout
(this is necessary because we have space constraints with flammable materials above
our optical table). We wrapped heater tape around the chamber and contained the
heat using a combination of tented UHV aluminum foil and firebricks. We monitored
the temperature with 16 thermocouples and used variable transformers to drive the
heaters and brought the chamber to a target of 200◦C. Changes in temperature were
applied slowly (we aimed for less than 1◦C per minute) to suppress stress from thermal
expansion in CF joints and windows, and special care was taken to avoid overheating
temperature-sensitive components, listed in Table 4.1. Since the TSP cartridge is at
the center of a 6′′ nipple inside the chamber, it is cooler than the surrounding walls
during bakeout. Molecules outgassing from the walls adsorb on the filaments, only to
outgas later when the pump is activated. To prevent this, we heated each filament by
64
Item Manufacturer Part Number Limit (◦C)
Right-angle valves VAT 54032-GE02 300
Inline valve Varian 9515052 450
Kovar windows Larson Electronics VP-150-F2 400
Sapphire window Larson Electronics VSZ-150-F2 450
Pyrex cell ColdQuanta CQMC0006 300
Magnet wire MWS 40466 240
Epoxy EpoTek 353ND 225
Ion Pumps Varian Starcell 350
TABLE 4.1. Maximum temperatures of various vacuum chamber components. The
target temperature during bakeout is 200◦C, but the ion pumps should not be
switched on at this temperature since their operational max is 150◦C. The most
sensitive parts are the epoxy and polyimide-coated magnet wire used in the anti-
Helmholz coil.
applying 30 A of current (we have three filaments and can only apply current through
one at a time, so we switched every few hours).
After baking and pumping for several days, the pressure (measured near the
turbo pump) approached 10−7 torr, and we flashed each TSP filament at 40 A for
one minute. Several hours later, we cooled the chamber to a temperature that was
safe to turn on our ion pumps, keeping the rest of the chamber a bit hotter than
the pumps so that any outgassing was pumped away rather than adhering to the
walls of the chamber. Next, we heated our strontium oven to 400◦C (to outgas that
part of the chamber while the turbo pump was still attached) and monitored the
pressure using the ion pumps. (The first time we did this, the pressure rise was so
sharp we thought a leak had formed. After cooling to room temperature and pumping
overnight, however, the pressure dropped once more. Presumably, the rise in pressure
was initial outgassing from the untreated stainless steel in the oven or a pocket of
gas was released from behind the tightly fitting crucible. To assure ourselves that
everything was okay, we performed a short bakeout and observed that the pressure
65
did not rise as abruptly when turning on the oven again.) We then cooled the chamber
to room temperature and activated one of our TSP filaments for two minutes at 45 A.
Finally, we sealed off the chamber using the right-angle valves and moved it back onto
the optical table. Final readings of both ion pump controllers were off scale, meaning
the pressure was well below 10−9 torr. Several years after sealing the chamber and
regularly cycling the oven, the ion pumps measure pressures of about 1.0× 10−9 torr
and 6.5× 10−9 torr in the low- and high-pressure sides, respectively, with the oven at
500◦C (this is without engaging the TSP since the final bakeout).
In the future if vacuum needs to be broken (to refill the oven, for example), it
might be possible to avoid such extensive bakeout. First of all, the inline valve can
be closed to isolate the two halves of the chamber. The side to be opened can then
be filled with a dry gas (e.g. argon) to atmospheric pressure using a venting valve
in our turbo pump. The uppermost viewport can be replaced carefully by an argon
source (with little agitation, hopefully no moist air will enter the chamber), and slight
positive pressure applied after a lower seal is broken. If a bakeout of the high-pressure
side of the chamber is in the end necessary, it is reasonable to do this portion in place
and not move the entire chamber off the optical table as we did in the initial baking
process.
During the process of assembling our chamber, we had difficulty achieving a
reliable seal with nickel gaskets in a CF-flange assembly. (Since copper reacts with
strontium, we have to use nickel gaskets in areas which are exposed to high beam
flux [140]). Gaskets for smaller, 1.33′′ CF flanges were purchased from MDC and
sealed nicely, but larger, 2.75′′ CF gaskets for the beam constriction apertures—
blanks of Ni-200, purchased from Grass Manufacturing—were problematic. In the
first assembly, small leaks detected at one flange improved after baking out, and it
66
seemed as if Ni flowed on a longer timescale than Cu, and that heating the gasket
helped. However, after subsequent reassembly, leaks were still problematic, even
after bakeout. We eventually realized that the gaskets were work-hardened (during
the cold-rolling and stamping process), and that pressure from a CF knife edge only
exacerbated this effect. Inspecting used gaskets, we found that dimples in a gasket’s
surface were pushed in by the knife edge and that the metal did not flow as well
as copper. We settled on buying cold-rolled, water-cut gaskets which were slightly
oversized. To prepare them for use, we machined them to the appropriate CF size,
annealed in air at 1500 ◦F for 5 minutes, and then buffed off the resulting oxide layer
using scotch pads. The first two gaskets prepared in this manner sealed nicely, and
we have not had to break vacuum since. Incidentally (we discovered later) a Japanese
group investigated the sealing reliability issue and developed a “concave” gasket [155].
4.1.4. Magnetic Coils
While not directly a part of the vacuum system, some experimental devices had
to be determined during its conception. Because the top MOT is trapped inside
the 6′′ spherical octagon, we need a fairly large set of anti-Helmholtz (AH) coils
that have clearance for the four-way cross flange. The coil forms attach to a pair of
aluminum plates, which are connected to the spherical octagon on an exterior ring of
tapped holes and provide stability by securing to the 80-20 support frame. Fig. 4.7
shows the support plates and one AH coil (not in its final position) to illustrate the
geometry (note that the set of vacuum screws attaching the plate to the octagon in
this particular photograph have washers which are too large to fit in the counterbored
holes, but they ultimately do not protrude beyond the top of the plate). Clearly, the
AH coil needs to be in place before the chamber assembly is finished and baked, and
67
FIGURE 4.7. Photograph illustrating support structure for the 6′′ spherical octagon
and science cell. Aluminum plates are attached to the octagon and the 80-20 frame
(the screws are countersunk in the final configuration), and the top-MOT anti-
Helmholtz coils can clear the outer flange dimension.
we follow UHV procedure in machining and assembling the forms to mitigate possible
contamination of viewports during bakeout.
A coil form is created by carving a 0.5′′-wide, 0.75′′-deep channel into an 8′′-
outer-diameter aluminum annulus. A second channel for water-cooling is carved and
later sealed by welding a ring and threaded cooling-line coupling block into place.
The coils are wound using 20 AWG insulated magnet wire (single-build Polyimide,
MWS P/N 40466), and have 308 turns each (roughly 22 layers of 14 wraps). High-
temperature epoxy is painted on the wire as the coils is turned to secure it in place
and to assist in removing heat from the center of the bundle. Assuming all turns
are overlapped at the center of the bundle—giving an AH pair of radius 9.15 cm and
separation 11.25 cm—the axial field gradient is approximately ∂Bz(I) = 3.83 I G/cm,
68
where I is the applied current. The resistance of a coil, meanwhile, is estimated to
be 6.3 Ω (constructed coils end up agreeing with this estimate, and we measure total
resistance values of the two coils in series to vary between 12.0 Ω and 13.7 Ω because
their temperatures change with operating current).
The trapping region in a MOT is roughly defined by the location where the
Zeeman shift matches the laser’s detuning, and an atom at rest is therefore resonant
with the light. Setting the velocity equal to zero in the effective detuning, Eq. (A.3),
we find the MOT’s radius (along the axial direction of the AH coils) to be
RMOT =
~∆
µ˜ ∂Bz
, (4.7)
where µ˜ = µ′ − µ = µB(m′J′g′J′ −mJgJ) is the cooling transition’s effective magnetic
moment. Because the blue trapping transition in strontium has a relatively broad
30-MHz linewidth, a large field gradient is desirable to be able to form a tight MOT.
We use a Kepco ATE 100-10M power supply to drive the coils in series, and
the 100-V maximum corresponds to a maximum field gradient of roughly 28 G/cm.
The power dissipated in these coils is significant and requires water-cooling with
a recirculating refrigeration bath. For these, we use two 400 W Neslab chillers—
one for each coil—and isolate vibration on the cooling line by potting a few loops
inside a concrete-filled plastic tub. To prevent possible overheating (we would
be forced to break vacuum if the upper coil’s insulating coating degrades to the
point of shorting), we use a protection circuit which only enables current if signals
from two flow switches (McMaster-Carr 2371K4) indicate that cooling is enabled.
Originally, thermal switches (Digi-Key 480-3221-ND) were also in place, but a five-
minute epoxy holding them to the coil forms failed with temperature cycling and
they were not reinstalled. The Kepco supply, however, acts as a nice fail-safe since
69
it will automatically shut down if the load creeps beyond 100 V. This corresponds to
a current of roughly 7.4 A, and the temperature of the wire is measured to be 160◦C
with an infrared thermometer in this extreme limit (compare this temperature to the
wire’s maximum rated temperature of 240◦C given in Table 4.1). We typically load
a top MOT using 6.8 A, corresponding to a field gradient of about 26 G/cm.
Anti-Helmholtz coils for the bottom MOT are positioned around the pyrex cell
so that their longitudinal axes point in the north-south direction. We use a pair of
coils wrapped around a machined Delrin form, which was decommissioned from a
previous setup in the lab [123]. There are 216 turns per coil, and the coil diameter
and separation are roughly 40 mm and 50 mm, respectively. These provide a field
gradient of ∂Bz(I) = 24.6 I G/cm.
4.1.5. Zeeman Slower
If we try to load atoms directly from the thermal beam into a MOT, only a subset
will be trapped. The MOT’s capture velocity is the limit where the energy dissipated
by a trapping beam matches the atom’s kinetic energy, and can be approximated
by [156]
vc =
√
rcFrad
m
. (4.8)
Here rc is the trapping radius, typically given by the size of the MOT beams, and Frad
is the radiative scattering force. Assuming a trapping radius of 1 cm and maximum
force, Frad = ~kΓ/2 (which should, in principle, be smaller because the transverse
MOT beams are at an angle to the atomic beam), the capture velocity is about
80 m/s. If we integrate the expression for the collimated beam flux up to this capture
velocity [instead of to infinite velocities, as in Eq. (4.6)], the resulting flux of atoms
70
that can be trapped (given an oven operating temperature of 500◦C, for example) is
only 2.2× 1010 atoms/s, or about 0.2% of the total collimated flux.
To increase the loading rate, we can use a counter-propagating slowing beam
which interacts with the atoms as they travel from the oven to the trapping region.
This beam is red-detuned in the same manner as Doppler cooling, but some tactic
must be employed to enable continued interaction with the atoms as they slow and
fall out of resonance with the light. One option is chirping the laser frequency to tune
the laser and follow the shifting resonance [147, 157], and another alternative tunes
the Doppler shift through a creative series of reflections of the slowing beam [158].
The simplest method experimentally is to use the Zeeman effect to spatially tune
the atoms so they continually interact with a fixed laser frequency [2]. For a given
laser detuning and magnetic field, a particular velocity class of atoms will scatter
light and slow down. As the atoms proceed, the changing field decreases the velocity
at which they are resonant. Additionally, a new group of atoms (which previously
were not interacting with the light) will begin to be slowed as well. In this manner,
a Zeeman slower effectively sweeps through the velocity distribution of the atomic
beam, bunching atoms into the same velocity class.
Assuming the maximum scattering rate Fmax = ~kΓ/2, the minimum slower
length required to bring atoms to rest from velocity v is determined by basic kinematic
equations. To stop 88Sr atoms with v = 400 m/s, this distance is less than 10 cm,
which—because of the broad linewidth and higher-energy 461 nm photons—is much
shorter than for alkali atoms. In practice, however, the length of a slower needs to
be longer than this minimum distance, because it assumes perfect resonance. Small
imperfections in the magnetic field profile cause decreases in the scattering force,
and the atoms will not stay on the designed velocity profile, eventually becoming
71
unaffected by the slowing light. So to effectively slow atoms, the field should change
monotonically and meet the condition,
∣∣∣∣dB(z)dz
∣∣∣∣ < (~k)2Γµ˜ v(z) . (4.9)
This expression is found by writing a(z) = v(z) dv/dz and finding dv/dz by equating
the Zeeman and Doppler shifts [159].
The magnetic field is commonly made using coils wrapped around the axis of
the atomic beam in a tapered solenoid [160], which requires careful simulation and
construction. In this geometry, the slower is an enduring structure built around a
section of the vacuum chamber and needs to be bakeable. Furthermore, the assembly
must be water-cooled because high currents are required to produce the magnetic
field. These difficulties can be circumvented by using a permanent magnet Zeeman
slower, and we chose to follow the design of a transverse slower in Refs. [161] and [162].
Such a slower consumes no energy and is easily adjusted in the future. The design
can even be changed entirely if desirable; permanent-magnet Zeeman slowers have
become popular in the last few years, and have evolved since we chose our design (for
example, there are permanent magnet slowers producing longitudinal fields [163, 164],
slowers which cleverly mount the magnets [165, 166], and even a computer-controlled
slower which allows dynamical configuration for multiple atomic species [167]).
Fig. 4.8 shows our completed Zeeman slower. Stacks of magnetic dipoles are
glued in color-coded mounts and are positioned using threaded brass studs. The
surrounding box is made from nickel-plated cast iron, and is normally closed on
all sides. We use combinations of three different thicknesses of 5/8′′-diameter N42
neodymium magnets (K&J Magnetics DA1, DA2, and DAH1) to create individual
dipole stacks. Locations and dipole moments of the magnets are given in Table 4.2.
72
FIGURE 4.8. Photograph of the permanent-magnet, transverse-field Zeeman slower.
The atomic beam enters from the left and experiences a monotonically increasing
magnetic field. Magnet holders are color-anodized to indicate polarity (the field
points from red to blue)
(We chose to use cast iron to better mimic the results in Refs. [161] and [162], but a
mild steel would also work well and be much easier to machine.)
As evidenced by the color reversal, the magnetic field switches polarity in the
latter half of the slower. This zero-crossing feature allows the use of smaller magnetic
field magnitudes, but would be problematic for atoms with magnetic sublevels in the
ground state. Since the resonance condition matches the tuning of only one pair of
excited- and ground-state sublevels, an atom ending up in a different ground-state
sublevel after the degeneracy at zero field will no longer be slowed. A zero-crossing
slower works nicely for 88Sr, though, because S = L = J = I = 0 in the ground state.
Note that this slower has a transverse magnetic field, so the cooling laser does
not propagate along the atoms’ quantization axis. This means that in order to couple
73
z (mm) Stack thickness (in) |m| (A-m2) x (mm)
0 1/10 + 2(1/8) 1.85 31
21 1/10 + 2(1/8) 1.85 42
42 1/10 + 2(1/8) 1.85 44
62 1/10 + 2(1/8) 1.85 50
83 1/10 + 2(1/8) 1.85 58
104 1/10 + 1/16 0.86 51
125 1/16 0.33 55
146 1/16 0.33 62
163 no magnets
187 1/10 0.53 45
208 1/10 + 1/16 0.86 46
229 1/10 + 2(1/8) 1.85 40
257 1/10 + 2(1/8) 1.85 28
TABLE 4.2. Magnet size, location, and strength in the permanent magnet Zeeman
slower. Distances in the x-direction are measured from the axis to the center of the
stack of magnets, and atoms travel through the slower in the +z-direction. Magnetic
moments |m| = BrV/µ0 are calculated using Br = 1.32 T as the residual induction in
N42 magnets. Each position along the slower has two symmetric magnet stacks, so in
total we use fifty-six magnets. The cast-iron shield has boundaries at z = −20 mm,
z = 277 mm, and at x = ±80.5 mm.
from the mJ = 0 ground state to the m
′
J
= 1 excited state, the light needs to be
linearly polarized so that it can be decomposed into σ+ and σ− components along
the quantization axis. Because only one of these components interacts with the
appropriate transition, ultimately half of the laser power does not contribute to the
cooling process. The correct polarization of the laser is orthogonal to the direction
of the magnetic field, because the other orientation would only drive π-polarized
transitions (and the m′
J
= 0 excited state is not tuned by the slower).
The field was modeled based on the positions of the dipoles and their first-order
images in the magnetic shield, and measured at several locations (before the slower is
placed around the vacuum chamber) to confirm accuracy. A subset of these data are
shown in Fig. 4.9 along with the model which terminates at the shield boundary. For
74
❞✁✂✄☎✆✝ ✂✞✟✠✡☛✞ ✁☞✠✌✝✟ ✍✆✎✏
✲✑✒ ✻✒✒ ✑✒ ✷✒ ✸✒ ✹✒ ✺✒
♠
✓
✔
✕
✖
✗
✘
✙
✚
✘
✖
✛
✜
✢
✣
✓
✤
✥
✥
✦
✹✒✒
✲✹✒✒
✲✸✒✒
✲✷✒✒
✲✑✒✒
✒
✑✒✒
✷✒✒
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❝✧★❝✩★✧✪✫✬ ✭✮✫★✬✯✫✧✰✩✱✫✬ ✭✮✫★✬
❆✳ ✭✮✫★✬ ✧★✴✵✶ ✼✫✧✯
FIGURE 4.9. Magnetic field profile experienced by atoms as they pass through the
Zeeman slower and into the trapping region. Note that the two curves represent
different quantization axes: the Zeeman slower field is transverse to the atomic beam,
and the anti-Helmholtz (AH) field points along the atoms’ velocity.
reference, the field produced by the AH coils operating at I = 7 A is also plotted along
the atomic beam’s trajectory (elliptic integrals are used to calculate the off-axis AH
field [168], but the effects of the chamber or cast-iron shield are not considered). The
extreme field values are −300 G at the slower’s entrance and +330 G at its exit. For
a laser detuning of ∆ = −500 MHz, these correspond to capture and exit velocities
of 424 m/s and 18 m/s, respectively. The final velocity is well within the range that
can be caught by the MOT, and integrating the collimated velocity distribution up
to the slower’s capture velocity shows that 35% of atoms in the beam can now be
loaded from an oven at 500◦C—a substantial increase from 0.2% with no slower.
To conservatively estimate the required power, we assume that all atoms are
slowed from capture to exit velocity, corresponding to about 41300 recoil events apiece.
For 500◦C, the beam flux is 3.9× 1012, and 140 mW of laser power is required in this
estimate (including the fact that twice the power is needed for the transverse slower).
Of course, not all atoms require this many photon recoil events in the slowing process,
75
and we still observe decent loading in our MOT for less than 20 mW of power in our
Zeeman-slowing beam. Normally we use ∼ 45 mW in our beam, which would cool
1.3× 1012 atoms/s in the above assumption.
Transverse heating of the atoms is dissuaded by slightly focusing the slowing
beam [147, 161]. We use a pair of lenses (50.2 mm and 300 mm, Newport
KPX082AR.14 and KPX112AR.14, respectively) to expand the beam (which has
an initial waist of 0.38 mm) and focus it near the oven crucible. The beam requires
a periscope to reach the level of the atoms, and we initially set this up beside the
chamber to fine tune the position of the lenses. In particular, we checked that the
beam would not clip on the aperture where the differential pumping tube connects to
the trapping chamber (preventing unnecessary loss of slowing beam power) and that
the beam was roughly the same size as the bundle of collimating capillaries at the
oven’s output. Once the relative positions were determined, we shifted the periscope
and lenses into final position and directed the beam down the length of the chamber.
As mentioned previously, the atomic beam deposits strontium on the sapphire window
at the end of the chamber, and we overlapped the entry of the light with the visible
buildup. Loading of the upper MOT is fairly robust to the exact positioning of the
slowing beam, but we often check the pointing alignment using a camera looking into
the small spherical octagon.
4.2. Laser Systems
Trapping and cooling alkali atoms, like rubidium (780 nm) and cesium
(852 nm), is convenient because of widespread diode-laser technology at the
necessary wavelengths. Manipulation of strontium atoms, however, demands several
substantially different wavelengths which are unfortunately not as cheap or convenient
76
second-harmonic  
generation
922 nm 
ECDL
tapered 
amplifier
FM spectroscopy
PZT
locking signal
461 nm
689 nm 
ECDL
slave 
laser
FM spectroscopy
PZT
locking signal
689 nm
spatial 
filter
914 nm 
ECDL
tapered 
amplifier
914 nm
994 nm 
ECDL
497 nm
second-harmonic  
generation
FIGURE 4.10. Basic block diagram of lasers required in the cold strontium
experiment.
to realize. As a result, we have had to add several layers of complexity to create the
required light. A basic block diagram of the four lasers we have made is shown in
Fig. 4.10, and associated atomic transitions are labeled in Fig. B.1 in the appendix.
The strong cycling transition, which we call the “blue MOT” transition, is at 461 nm;
a secondary cooling transition, the 5s2 1S0 −→ 5s 5p 3P1 intercombination line we call
the “red MOT” transition, is at 689 nm; the “magic wavelength” for the red MOT
transition is at 914 nm (this is different than the 813 nm magic wavelength used in
the strontium clock, which operates on the doubly forbidden 5s2 1S0 −→ 5s 5p 1P0
transition [19]); and a repumping scheme operates at 497 nm [17]. While each laser
system is unique, there are several shared components and techniques that can be
discussed individually. The starting point of each laser is the novel ECDL, which has
been described in detail in Chapter III. In this section, light amplification, second-
harmonic generation (SHG), and frequency stabilization and control will be addressed.
77
4.2.1. Tapered Amplifier
In situations where ECDLs do not generate enough power it is common to
boost power by injection-locking a second diode laser [95, 169]. Here a ‘master’
seeds the lasing process in the injection-locked ‘slave,’ which can operate with higher
intracavity powers. However, the single-mode power that can be achieved is limited
by the geometry of the device. In particular, the transverse dimension of the gain
medium must be smaller than roughly the wavelength of the laser light, or higher-order
transverse modes will be supported within the cavity. To overcome this limitation in
applications which require even more power, a diode with a tapered gain region—
called a tapered amplifier (TA)—can be used [170–172] (often the term master-
oscillator power amplifier [MOPA] is also used to describe the setup). Like ECDLs,
TAs are becoming ubiquitous in atomic physics laboratories, and we worked with
collaborators to create a tutorial in how to successfully implement them [173].
Inside the TA device, illustrated in Fig. 4.11(c), seed light is coupled into the
single-mode rear facet, which has similar geometry to the active region of a laser
diode. After some length, L1, the gain medium begins to expand at roughly the
diffraction angle of the design wavelength (the spatio-temporal dynamics of the gain
medium are actually quite sensitive to the details of this expansion [174]). The
light adiabatically expands over a length L2, and is amplified due to the increased
area in the diode’s active region before exiting at the front facet. Higher-order
transverse modes threaten single-mode operation, and the use of diffraction expansion
rather than explicit waveguide structures helps suppress them. Additionally, cavity-
spoiling grooves deter resonances in the injection portion, and it is likely that a lack
of these geometric considerations added difficulty to early attempts in creating TA
diodes [172]. Note that, unlike laser diodes, the front- and rear-facet are not coated
78
removable block
aspheric  
lenses
(b)
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(c)(a)
seed input
TA chip
glass rods
output
wires
wing (-)
base (+)
TA chip
FIGURE 4.11. Details of the tapered amplifier (TA) chip and housing. (a) chip
purchased from Eagleyard Photonics with relevant parts labeled, (b) our homebuilt
housing for aligning optics, (c) schematic of the chip illustrating tapered gain medium,
(d) TA operating at 922 nm. Note that in all but (c) the input side is to the right.
Portions of this figure were adapted from Ref. [173]. Photographs by Dan Steck.
to be reflective: the TA is a single-pass device which can provide gain as high as
27 dB [175] and maintain the narrow-linewidth properties of a master laser [176].
We purchased TA chips from Eagleyard Photonics (EYP-TPA-0915-01500-3006-
CMT03-0000), which have dimensions w1 = 3 µm, w1 = 190 µm, and L1 + L2 =
2750 µm. The chips have a nominal output power of 1.5 W at 3 A operating
current and center wavelength of 915 nm, though the particular diodes we were
sent are instead centered at 905 nm. A 50 nm FWHM, however, means they are
able to amplify our master ECDLs operating at 914 nm and 922 nm. The TA chip,
pictured in Fig. 4.11(a), comes in a C-mount package; electrical current is applied
79
through the base (anode) and an isolated ‘wing’ (cathode), which is connected to the
semiconductor device with a series of small wires. The diode has TE polarization,
meaning that the electric field is in the plane of the junction—or parallel to the optical
table—and the diode’s elliptical output is oriented vertically.
TA diodes are susceptible to catastrophic failure [173], and we follow several
practices to reduce the chances that this occurs. First of all, as a semiconductor
device, the TA can be shocked electrostatically, and is handled with the same
grounding considerations as a regular laser diode [97]. It is also easy to accidentally
create high power in the single-mode region and burn out the chip. This can happen
from directly injecting too much seed power [177] or from a back reflection seeding
the TA in reverse. These situations are avoided by being aware of the seed power,
slightly tilting optics placed in the output beam, and by using a high degree of optical
isolation after the TA. Somewhat more subtle, however, is the problem of amplified
spontaneous emission (ASE). If the TA is powered on while it is not seeded, ASE
generates light which travels both forward and backward in the diode, and can also
cause damage at the interface between the tapered and single-mode regions. ASE is
not harmful when driving the TA at low current (and can actually be quite useful as
discussed below), but high current should not be used unless the presence of a seed
and proper alignment is confirmed [178]. We employ a seed-monitoring photodiode
(see Fig. 4.13[a]) and a relay to turn off the TA’s current controller if the seed light
is switched off or blocked [179].
We house the TAs in homebuilt structures, originally discussed in Ref. [177] and
pictured in Fig. 4.11(b). The walls and lid are laser-cut from Delrin and acrylic,
respectively, and the inside of the structure is sealed so that dust and debris do
not land on the high-power output of the TA chip. Small air gaps resulting from
80
the kerf of the laser cutter are sealed with epoxy around the walls, and the lid
and baseplate interfaces have a soft, silicon rubber gasket (McMaster 86435K452).
Brewster windows are attached at the input and output (again sealed with a silicon
gasket and oriented for light polarized parallel to the optical table). The main body is
made from oxygen-free high conducting (OFHC) copper since it is both the positive
electrical contact and the thermal reservoir for the C-mount diode, and it is mounted
on two TECs (Laird 56460-501) wired in series for temperature regulation. The
TECs are on an aluminum baseplate that has channels for optional water-cooling,
which has not been necessary. The negative electrical contact, meanwhile, is created
by clamping the chip’s wing between a piece of copper and a polycarbonate spacer.
We have thus avoided using a soldering iron in the vicinity of the chip. We control
the temperature and current of the TA using modified versions of controllers used in
our lab for ECDLs. The originals are described in previous theses [95, 123], and the
modified versions have been documented on our lab website [179].
Coupling and collimating aspheric lenses (Thorlabs A230TM-B, f = 4.51 mm)
are threaded into brass sleeves which are epoxied into place in a 120-degree ‘v-groove’
after alignment (see below). The output of a TA, however, is highly astigmatic (500–
700 µm in our diodes), and the output lens will only collimate the fast (vertical) axis.
The slow axis is refocused by the lens, and is separately collimated using a 75 mm
cylindrical lens (CVI RCX-30.0-20.0-38.1-C-633-1064) outside the housing (see figure
6 in Ref. [172] for an illustration of the output beam geometry). If a TA chip needs
to be changed, the output v-groove block can be removed to access the M2 screw
securing the TA on the main OFHC base. Hopefully, the replacement diode will have
a similar position as the original and alignment of the aspheric lenses will not have
to be repeated anew.
81
Initial positioning and alignment of the aspheric lenses was performed by
attaching the brass sleeves (with a five-minute epoxy) to posts hanging from a pair
of towers which could be positioned using a three-axis translation stage. To align the
seed laser into the TA, we adapted a procedure outlined in an Eagleyard application
note [180] (another alignment procedure is also discussed in Ref. [173]). First, we used
ASE straight out of the TA to roughly align the path along the row of holes where the
master ECDL fiber launcher would be placed and then collimated the ASE with the
input coupling aspheric lens (leaving it slightly diverging). We then put in the seed-
light fiber launcher and used steering mirrors to couple the ASE into an optical fiber
to ensure alignment. It was helpful to use a pair of anamorphic prisms to spatially
mode-match the Gaussian fiber output with the roughly elliptical output of the ASE.
(We magnified the vertical dimension with a 2:1 prism pair in the 922 nm TA setup
and a 3:1 pair for the 914 nm. That was what we had on hand at the time, and both
seem to work okay.) Next, we connected an ammeter to the TA, essentially using it as
a photodiode to maximize the current produced by the seed light. For a seed power of
7.5 mW, initial current values were around a few µA, and by moving the coupling lens
closer to the TA and using the steering mirrors—illustrated in Fig. 4.13(a)—current
was maximized to be nearly 2 mA. After this preliminary alignment, we connected
the current supply to the TA, switched it on, and finally made small changes in the
alignment of the seed light and coupling lens to maximize the output power. At this
point, the output aspheric lens was adjusted to minimize the vertical spread of the
beam in the far field, and the cylindrical lens was inserted and positioned to finish
the collimation. Note that the collimating lenses should be positioned while running
the TA at the desired operating current since the astigmatism changes with current-
dependent refractive index. With alignment optimized, we then secured the aspheric
82
current (A)
0 3.00.5 1.0 1.5 2.0 2.5
T
A
 o
u
tp
u
t 
p
o
w
er
 (
m
W
)
700
0
100
200
300
400
500
600
5 mW seed
7 mW seed
10 mW seed
seed power (mW)
0 204 8 12 16
1000
0
200
400
600
800
922 nm
914 nm
FIGURE 4.12. Output power of tapered amplifiers (TAs) as a function of drive
current (left) and seed power (right). Seed power data are taken with an operating
current of 2 A. Data for the left plot are taken with the 922 nm TA, which has lower
output power than the 914 nm TA, as evidenced by the right plot.
lenses in place by placing 3 mm diameter glass rods between the brass sleeves and
v-grooves and epoxying them in place (with Epotek 353ND). After allowing the epoxy
to set for 60 hours, we broke the translation-stage assembly free from the brass sleeves
(residue from the five-minute epoxy can be seen on the output lens in Fig. 4.11[b])
Output power from the TA depends on both the supplied current and the power
of the seed light, as shown in Fig. 4.12. These data were taken before the aspheric
lenses were epoxied in place and do not include any losses from sending the output
through the optical isolators shown in Fig. 4.13(a). As mentioned previously, too
much seed power could damage the TA (our chips are rated for a 50 mW maximum
seed), and we ideally would like to operate at levels where the output power begins
to saturate. For these diodes, we have found this to be around 15 mW. Note that
the output power for the 922 nm TA is significantly lower than for the 914 nm TA.
This is because, as mentioned previously, the TA’s gain curve is centered at 905 nm
83
and it is less efficient at amplifying the 922 nm light (the same inefficiency in the
master lasers explains why less seed power is available for that laser as well). We
were able to tune the temperature of the chip, shift the gain curve, and increase the
output power slightly. However, this required heating to several degrees above room
temperature—potentially shortening the lifespan of the TA—and we elected to accept
the reduced power output rather than operate in this regime. We typically run the
922 nm TA at 21.3◦C with 2 A of current, which results in ∼ 350 mW of power after
the optical isolators (when injected with about 10 mW of seed power). The 914 nm
TA, meanwhile, produces nearly 900 mW of power after the isolators with a seed
power of 13 mW and operating current of 2.9 A.
Finally, after setting up the TAs and working on the 461 nm doubling cavity,
we noticed that a small amount of ASE—even with the diode seeded—appeared to
couple back through the optical fiber and destabilize the 922 nm ECDL. We solved
the issue by adding a second optical isolator to the output of the master laser before
the fiber coupler.
4.2.2. Second-Harmonic Generation
At the time we started work on this project, laser diodes were not a viable option
at 461 nm or 497 nm, which led us to develop lasers based on frequency doubling near-
infrared ECDLs.1 Crystals with χ(2) nonlinearity can produce higher-order optical
harmonics when illuminated with a single pump laser [183]. Since the power output
of the second harmonic depends on the square of the power of the fundamental laser,
1As of this writing, 497 nm is still a difficult regime, but it is possible to find laser diodes which
can operate at 461 nm. Nichida’s NDB4216, for example, has a nominal wavelength of 450–460 nm
and is not absurdly expensive, priced at $3300. It has been shown that using two of these diodes in
a master-slave configuration can produce more than 100 mW of 461 nm power [181]. Additionally,
it is possible to use a master ECDL to seed a cheaper, multimode slave to generate high powers for
applications which are less sensitive to linewidth, such as Zeeman slowing [182].
84
PBS anamorphic  
prisms
safety 
monitor
Brewster window
TA chip
acrylic housing
4.5 mm  
aspheric lens
75 mm  
cylindrical lens
ECDL 
fiber
isolator isolator
oven
PPKTP
dichroic 200 mm 
mode-matching
cavity lock
PZT
feedback
PBS
FM spectroscopy
nL= n0 - 220 MHz
Second  
harmonic  
generation
0
-11
0
to Zeeman slower
280 MHz
180 MHz
to upper MOT
1
0to lower MOT
180 MHz
30 mW 45 mW
12 mW
10 mW
Master laser 
amplification
Frequency 
control
dual HR input coupler
25 mm 
mirrors
350 mW
10 mW
(b)
(a)
(c)
FIGURE 4.13. Schematic showing how 461 nm blue light is prepared for interaction
with strontium atoms. A 922 nm master ECDL is amplified in a tapered gain medium
(TA) before interacting with a periodically poled potassium titanyl phosphate crystal
(PPKTP) in a build-up cavity to generate frequency-doubled light. The 461 nm
light is sent to an atomic reference for frequency-stabilization of the master laser (see
Fig. 4.15), and frequency adjustments are made using several AOMs.
it is advantageous to focus a Gaussian beam at the center of a crystal in an optical
resonator [184, 185]. We chose to use a periodically poled potassium titanyl phosphate
(PPKTP) crystal inside a linear build-up cavity, essentially a hybrid of two previously
published SHG setups [186, 187]. The design will be discussed briefly here, but it is
a rather complicated part of our experiment and will be covered in detail in Eryn
Cook’s thesis [188].
Fig. 4.13(b) shows a simplified schematic of our SHG setup for the 461 nm laser.
Vertically polarized fundamental light comes from the 922 nm TA, and is coupled into
85
a linear cavity with a 200-mm mode-matching lens. The cavity mirrors have radius of
curvature of 25 mm and are separated by about 5 cm, producing a fundamental waist
of roughly 60 µm at the center of the 10-mm-long PPKTP crystal. The cavity’s rear
mirror is highly reflective for both wavelengths, while the input coupling mirror is AR-
coated at 461 nm and coated for ∼ 9% transmission at 922 nm for proper impedance
matching (all coatings were performed by Spectrum Thin Films). The result is a
Fabry–Pe´rot resonator at 922 nm and a double-pass configuration for the blue light,
which is extracted by a dichroic mirror after exiting the cavity. The PPKTP crystal
(5.5 µm poling period, manufactured by Raicol) is housed in a homebuilt OFHC
oven, which is temperature-stabilized to be 29.7◦C, and placed on a four-axis tilting
kinematic mount (New Focus 9071) for position adjustments. The crystal is angle-
polished (nominally 8 mrad) to allow the thickness of the final poling period to be
adjusted by translating the crystal. This provides a mechanism to optimize SHG
efficiency by controlling the phase relationship between the forward- and reverse-
traveling blue light, although we never saw as clear an effect of this translation as in
Ref. [186].
The length of the cavity is stabilized to be resonant with the fundamental by
monitoring 922 nm transmission and feeding back to a piezo behind the rear cavity
mirror. The mirror and piezo are rather massive, and the cavity is susceptible to being
unlocked by mechanical noise. To improve utility, we have modified a circuit [189] that
automatically reengages the lock, and our improved circuit is described in Ref. [188].
Locking the cavity has also been difficult because of thermal effects similar to what
was seen in Ref. [190], and we frequently need to reposition the crystal slightly to find a
location that both produces high blue power and remains locked. The dichroic mirror
is slightly reflective for 922 nm, and we spatially separate the second harmonic using
86
a Pellin–Broca prism (not pictured in Fig. 4.13). Since small changes in the cavity’s
alignment change the pointing direction of the output beam, a pair of steering mirrors
and irises are used to frequently realign the path through the Pellin–Broca. A well-
aligned cavity can produce powers over 150 mW (at times the conversion efficiency
has been over 50%, even with the poor spatial mode of the TA output limiting mode-
matching into the cavity), but the thermal effects seem to be more significant when
more blue light is being produced. We typically have 120–150 mW of useable blue
power to send to the experiment, and we have found that atoms are still efficiently
slowed and trapped with as little as 100 mW total power (this is the power measured
before the first beamsplitter after the cavity in Fig. 4.13, so it includes all necessary
power for stabilizing the laser and loading both MOTs).
Frequency-doubled light for the 497 nm laser, meanwhile, is created using
a periodically poled lithium niobate (PPLN) crystal (Covesion MSHG976-0.5-20,
mounted in an oven, PV20). Much less second-harmonic power is required at this
wavelength than the main cycling transition since the saturation intensity of the
5s 5p 3P2 −→ 5s 5d 3D2 transition is only 2.2 mW/cm2 (compared to 42.7 mW/cm2
for the 461 nm line). Additionally, we only need to send a single beam into the trap
to influence population and lifetime, and that beam does not need to be much larger
than the MOT (for example, 500 µW would be enough to saturate a MOT with 5 mm
diameter). Because of the low power requirements, we do not use a TA to amplify
the output of the 994 nm master ECDL, and we have constructed a linear cavity
using a spare set of mirrors which were coated for the 922 nm SHG process. So,
while certainly not optimized, the cavity is good enough to produce a small amount
of 497 nm light.
87
4.2.3. Frequency Stabilization
To ensure that a laser is tuned to an atomic transition, absorption is monitored
in a reference sample. Typically, Doppler-free lineshapes are created through a
process called saturated-absorption spectroscopy (SAS), in which an opposing pump-
probe scheme isolates a particular velocity class of the atomic sample [129, 191]. A
dispersive lineshape, convenient for electronic feedback, can be created by modulating
the frequency of the laser and using lock-in amplification to analyze the signal from
a photodiode (frequency modulation [FM] can be performed through electro- or
acousto-optic effects [192] or—in the case of laser diodes—through controlling the
applied current [123]). Alternatively, a clever use of light-induced birefringence can
create Doppler-free lineshapes in a process called polarization spectroscopy [193], and
combining this with a magnetic field (used to create a dispersive lineshape in the
dichroic-atomic-vapor laser lock [DAVLL] scheme [194]) yields lockable signals which
require no frequency modulation [195]. This Doppler-free DAVLL technique can be
used to stabilize lasers to strontium in a “see-through” hollow-cathode lamp [196],
but such an atomic reference is costly. We instead, use a different method to produce
strontium vapor samples and stabilize the lasers with FM spectroscopy.
4.2.3.1. Heat-Pipe Vapor Cell
As discussed in Sec. 4.1, strontium needs to be heated to several hundred Celsius
to create an appreciable vapor pressure and it reacts unfavorably with vacuum
viewports and copper gaskets. Thus, simple glass vapor cells, like those typically
used for alkali atoms, are not a viable option. Instead, we have made heat-pipe vapor
cells [197, 198] that use an inert buffer gas to prevent Sr vapor from reaching windows.
If the mean-free path of the buffer gas is small compared to the distance between a
88
heater insulation package
pressure gauge
pump-out valve
viewportSr crucible
FIGURE 4.14. Photograph of a completed heat-pipe vapor cell. Radiative heaters
at the center of an insulation package heat strontium contained in a nickel crucible,
and an argon buffer gas prevents strontium vapor from reaching the windows. For a
scale reference, the holes in the optical table are on a double-density 1′′ grid.
central heated region and the windows, the Sr gas is well-contained. One of these
devices is pictured in Fig. 4.14. Using an argon-filled glove bag as in Sec. 4.1.1.1,
several grams of Sr were loaded into a 10-cm-long nickel crucible which has a 0.25′′
hole in the base and end cap to pass light. A fine, nickel mesh is also placed inside the
crucible to encourage wicking back into the center, but this is not needed unless the Sr
is actually melted. The crucible was placed in a smooth-bore, 0.62′′ ID stainless steel
pipe, roughly centered between two viewports (Larson VPK-075-F1). One viewport
is attached to a 1.33′′ CF flange that was welded by the UO machine shop to be
tilted by 3◦ (right side in the picture) to suppress resonant reflections between the
two windows. Nickel gaskets are used in joints nearest the Sr sample (MDC 191060),
and we did not have the same troubles with these that we had with the gaskets in the
UHV chamber (Sec. 4.1.3). A compact vacuum valve (HPS CV16-C1C1-MKKCV)
and thermocouple sensor (Duniway DST-531) allow us to pump out the vapor cell
and monitor the pressure, respectively.
89
Heat is applied to the Sr sample using two semi-cylindrical ceramic heaters wired
in series (Thermcraft RL106-S-L, with insulation package VIP-2.5-8-0.75-2). The
signal from a type-J thermocouple is monitored by an on-off temperature controller
(Love Controls TCS-4011), which we originally used to directly switch voltage from a
Variac to drive the heaters. This simple setup was problematic since the relays inside
the Love controllers failed after running the heaters for about a week (they failed in
the closed position, and the temperature would run away to whatever maximum that
could be attained by the Variac voltage). Initially, we thought this was an issue of
switching inductive loads, which would cause arcing as the mechanical component
switched between leads, eventually welding the switch in place. We upgraded the
relays inside the Love controllers (to something with a higher load capacity that
would fit the same printed circuit board [PCB] footprint, Omron G2RL1EDC12) and
added transient-voltage suppressors (TVS) to limit the strength of the inductive kick.
This seemed to solve the issue, but the relays again failed after about three months
of operation. This time, however, we realized they had reached their rated maximum
number of switching events. The final iteration—which has worked without issue
for several years—used the Love controller to switch a 5-V dc signal that controls a
solid-state relay (SSR) with built-in varistor (Omron G3NA-220B-DC5-24), which in
turn switched the Variac voltage. This is the same temperature controlling scheme
used with the Sr oven in Sec. 4.1.1.1.
We attached a turbomolecular pump to evacuate the chamber and used the
venting valve to backfill with argon gas. Originally, we did not bake the assembly,
but after it was sealed for several months, outgassing caused the pressure to rise and
we performed a gentle bake at 150◦C for about a week. It was difficult to control the
pressure of the argon during backfilling, but we ended up with about 10 mTorr in
90
one heat pipe and 140 mTorr in the other. When we turned on the heaters, however,
the pressures rose due to outgassing (as high as 450 mTorr in one of the pipes) but
eventually fell to be about 25 mTorr in each chamber, presumably because the Sr
vapor was acting as a getter pump [199].
4.2.3.2. Frequency-Modulation Spectroscopy
A laser can be stabilized to an atomic reference in a rather simple SAS setup, such
as the one we used in Fig. 3.5(b). The pump-probe interaction creates Lamb dips [145]
on a Doppler-broadened background, and through FM techniques, the laser can be
locked to them. However, the presence of the Doppler-broadened background means
there is a slight offset in frequency between the locking point and the actual atomic
resonance. This by itself would not be so annoying, but the difference is coupled to
the laser’s power and to background light levels, meaning the actual frequency of the
laser would drift slightly in an uncontrolled manner. In SAS, the Doppler-broadened
background is commonly removed by using differential photodetection on two probe
beams [129], where only one of them has interacted with atoms affected by the pump.
This method is not easily realized in our setup because the heat pipe is long and
thin, making the necessary beam geometry difficult to achieve. Instead, we use a
collinear pump-probe geometry and an extra layer of lock-in detection to remove the
background. A diagram of the setup we use to stabilize the 461 nm and 689 nm lasers
is shown in Fig. 4.15.
We use an AOM to chop the pump beam at 175 kHz, thereby modulating the
saturation of the atoms. The probe’s absorption spectrum will always have the
same Doppler-broadened background, but the Lamb dips are only present half the
91
laser Sr heat pipe
l/4
AOM
0
1
500 mm
500 mm
EOMl/2
servo
PZT
pump probe
nL
nLnL+2WAOM
l/2
(double-passed)
175 kHz
f
f
WEOM
WAOM
FIGURE 4.15. Schematic of frequency-modulation spectroscopy used to stabilize the
blue 461 nm and red 689 nm lasers to an atomic reference. Values for the blue setup
are ΩAOM = 220 MHz and ΩEOM = 8.773 MHz, while they are ΩAOM = 80.5 MHz and
ΩEOM = 3.161 MHz for the red.
time. Lock-in detection at this frequency therefore isolates the non-broadened atomic
features, and all that remains is turning these into lockable signals.
We do not want to modulate the frequency of either of our lasers using the
master diode’s current (for the 922 nm master this would complicate locking the
SHG cavity, while we want the frequency of the 689 nm laser to be as stable as
possible to best interact with the narrow intercombination line), so we instead use
an electro-optic modulator (EOM) to modulate the phase of the probe. Driving the
EOM with frequency ΩEOM puts sidebands on the laser. Interference of the sidebands
and carrier at the photodetector create a signal, which—after locking in on frequency
ΩEOM and selecting the correct phase shift—when swept across a resonance produces
a dispersive lineshape [192].
92
piezo voltage (arbitrary units)
0 1
E
rr
or
 s
ig
n
al
 (
v
ol
ts
)
0.3
-0.3
-0.2
-0.1
0
0.1
0.2
86Sr
88Sr 87Sr
FIGURE 4.16. Error signal produced in FM spectroscoy of the 5s2 1S0 −→ 5s 5p 1P1
“blue MOT” transition in strontium. Both a single trace (light green) and averaged
signal (blue) are presented. The three individual hyperfine transitions in 87Sr cannot
be distinguished, but a feature caused by F ′ = 9/2 and F ′ = 11/2 is pointed out.
The AOM functions in the +1 order and is double-passed [95], shifting the
frequency of the pump to be 2ΩAOM larger than that of the probe. This means that
atoms that are resonant with both the pump and probe are traveling to the right with
a velocity corresponding to a Doppler shift of ΩAOM. The resonance condition is then
ν0 = νL +ΩAOM, and a locked laser will be red-detuned from the atomic transition by
ΩAOM. Thus, the 461 nm laser is red-detuned by 220 MHz (after frequency doubling),
and the 689 nm laser is red-detuned by 80.5 MHz.
Fig. 4.16 shows the error signal produced in the FM spectroscopy setup for the
461 nm laser. The largest feature is the 88Sr isotope, which is where we usually lock
the laser. 86Sr is visible as a smaller zero-crossing feature to which the laser can also
be locked. The 87Sr triplet produces both the highlighted shoulder and narrows the
right-side of the 88Sr feature. The signal-to-noise ratio is not fantastic because, as
shown in Ref. [200], the ratio of the sideband splitting to the linewidth of the atomic
93
piezo voltage (arbitrary units)
0 1
E
rr
or
 s
ig
n
al
 (
v
ol
ts
)
6
-6
-4
-2
0
2
4
86Sr
88Sr
FIGURE 4.17. Error signal produced in FM spectroscoy of the 5s2 1S0 −→ 5s 5p 3P1
“red MOT” transition in strontium. Both a single trace (light green) and averaged
signal (red) are presented.
transition is important. Because the blue MOT linewidth is so broad, we actually
had to increase the EOM drive frequency (which was not trivial, see Sec. 4.2.3.3) to
ΩEOM = 8.773 MHz to improve the error signal. Using a commercial low-noise fast
photodetector (New Focus 1801-FS) produces a significantly nicer signal than the
homebuilt photodiode used to record Fig. 4.16. The homebuilt photodiode circuit
is essentially a buffered transimpedance amplifier, and we can increase the gain at
the expense of the frequency bandwidth. A decent signal was attained by using a
Hamamatsu S5973 photodiode with a transimpedance gain of 40 V/mA (which is the
same as the commercial unit).
We made the same adjustment to the photodiode used in stabilizing the red
MOT laser, and the error signal produced in this case is shown in Fig. 4.17. The
ability to see two isotopes here allows us to estimate the width of the 88Sr feature.
We convert the error signal amplitude into frequency deviation and estimate that the
locked signal has rms frequency uncertainty of 250 kHz. This is likely caused by noise
94
in the detection and demodulation process and by pressure-broadening of the line in
the 420◦C heat-pipe oven.
4.2.3.3. Helical Resonators
We inherited a free-space EOM (Conoptics 350-50) and borrowed another
(Conoptics 350-52) to use as phase modulators, but do we not have the associated
high-voltage driving electronics. To generate modulation voltages that are large
enough to put significant power into the sidebands, we need to amplify an RF signal.
To do this, we have built resonating circuits similar to what ion-trapping groups often
use to make their trapping RF fields [201, 202]. These resonators are coaxial devices
whose core is wrapped in a helix, and yield high quality factors in a relatively small
size. Photographs of the helical resonators we built are presented in Fig. 4.18.
We used the empirical formulae from Ref. [203] to design a resonator that used
standard-sized copper pipe and a helix wrapped from AWG 23 magnet wire we
had on hand. Unlike Ref. [201], we are not restricted to build a resonator at a
particular frequency, which allows for this cost-saving simplification. We used a 5′′
length of 2′′ ID copper pipe, and a 4′′-long coil with 1.1′′ ID. The helix was wrapped
around a slotted-section of polyvinyl chloride (PVC) pipe, following the build-your-
own-inductor method of an amateur radio enthusiast [204]. The helix has 88 turns
which are separated by the thickness of the wire, and consistent spacing was achieved
by inserting notched, laser-cut ribs into the PVC slots before wrapping the coil.
These were epoxied in place to provide rigidity before the coil was removed from the
wrapping pipe. The helix was suspended within the copper pipe using nylon screws,
and electrical connections are made through 2′′-pipe end caps (McMaster 5520K49).
Coupling to the resonator can be made through antennae [202] (which also allows
95
input
outputhelical coil
laser-cut rib
connection to can
output
FIGURE 4.18. Photographs of the helical resonating cans used to amplify RF signals
for the EOMs. The left picture shows the inner-coil of the coaxial resonator, which
is wound by hand with the help of a laser-cut spacing tool. Inputs and outputs
are connected via BNC jacks at the bottom and top of the vertically-standing can,
respectively.
fine tuning impedance matching), but we simply tapped and grounded the helix at
the about the third turn to inductively couple primary and secondary coils which
were connected to the input and output, respectively. To realize and maintain a
high-Q resonator, we attempted to limit oxidation and therefore maximize the can’s
conductance (in principle, OFHC copper would work even better than the stock pipe
we used, but this would be a custom, costly part). We polished the pipe with Brasso,
painted the inside with a thin layer of Q-dope, and sealed the completed resonator
with clear nail polish.
Based on the geometry, the resonant frequency of the unloaded amplifier should
be about 30 MHz and has a Q of 550, but when loading with a capacitive EOM
these should be smaller. Using the Conoptics 350-50 EOM and a directional coupler
(Minicircuits ZDC-10-1), we measured f1 = 3.2 MHz and Q1 = 180 for the first
resonator and f2 = 3.16 MHz and Q2 = 220 for the second. As mentioned earlier,
we realized that we had to increase the resonant frequency of the first can, and we
96
rewrapped the helix, skipping two notches in our laser-cut ribs (so that the separation
between coils was five times the thickness of the wire). With the number of turns
in the coil reduced to 30, we measured the altered (loaded) resonator values to be
f1 = 8.76 MHz and Q1 = 50. We confirmed that sidebands were produced with
the amplified EOM signal by interfering the pump and probe on a photodiode and
looking at the RF spectrum centered around 2ΩAOM. The amplitude of the RF source
is increased until we see second-order sidebands begin to appear.
4.2.4. Slave Laser
To amplify the 689 nm, red-MOT light, we injection-lock a slave diode laser
(Hitachi HL6738MG) in a homebuilt housing [95]. We use the rejection port of an
optical isolator to seed the slave diode [205], and monitor both the master and slave
on an FP cavity to confirm if the slave is locked. The diode’s temperature was tuned
to bring its free-running wavelength close to the stabilized master’s, which enabled
locking over a broader range of diode current. Output from the slave is passed through
an AOM to provide fine control of the laser’s detuning when it interacts with atoms
in the MOT.
The master ECDL in this case is the special, long laser we constructed, which
we power using the commercial low-noise current supply from Vescent photonics.
In the locking circuit, we use low-gain proportional and slow integral feedback to
avoid injecting noise into the laser. We have monitored the output of the slave
in a delayed self-heterodyne measurement and confirmed that the linewidth is not
noticeably increased by the seeding process. While we have not at this point fit
the data to characterize the linewidth, the self-heterodyne traces are qualitatively
similar to those of the long ECDL (see Fig. 3.11), and we measured the HWHW on
97
a spectrum analyzer (which as discussed in Appendix C gives the upper bound of
the linewidth) to be less than 40 kHz with Tobs = 1 ms (the spectrum analyzer has a
1 kHz Gaussian resolution bandwidth).
4.3. Electronics
Most of the electronics used to operate the experiment are homebuilt. Each
ECDL and slave laser has its own dedicated current and temperature controller [95,
122, 123], and the TAs use slightly modified versions [179]. Another refashioned
version of the laser current driver is used to control current in a magnetic coil [123].
This uses a push-pull arrangement of two OPA549 chips (from Texas Instruments)
and features a thermal-protection section that can disable current if a monitored
temperature reaches a user-defined threshold. An analog and digital input can set
the current target and disable output, respectively, allowing computer control of the
magnetic fields used in the experiment. We use these to control current in pairs
of coils, wired in series, including the field-canceling Helmholtz (HH) coils and the
bottom-MOT AH coils.
We also have homemade analog feedback and control electronics, packaged
together in what we refer to as a laser “lockbox.” These include a phase-
adjustable coupled-oscillator circuit to generate modulation signals at 175 kHz;
gain, demodulation, and filtering stages for a photodiode input signal; low-frequency
proportional and integral feedback to the tuning piezo (which can also be manually
tuned or ramped with an onboard triangle-wave function generator); and optional
high-frequency feedback to the laser current with a feed-forward stage. The degree of
front-panel adjustability is high, but we have occasionally made a few custom changes
to the general design typically used in the lab. In the lockboxes which stabilize the
98
ECDLs to the heat pipes, we have used a comparator to convert the sinusoidal 175 kHz
modulation into the TTL pulse train shown in Fig. 4.15. The lockboxes controlling
the SHG cavities, meanwhile, have been modified to modulate at 68 kHz, which is
the resonant frequency we measured for the piezo/mirror combination in these setups.
The small modulation signal is added to the overall signal applied to the piezo, and
we needed to drive on resonance to increase the effects of the modulation and create
a nice error signal. Finally, in the lockbox for the 922 nm master ECDL, we have
added an input which allows the SHG cavity’s automatic relocking circuit [188] to
temporarily disable the integral feedback. This prevents the controller from driving
the laser frequency too far from resonance while the blue cavity restabilizes.
4.3.1. Radio-Frequency Electronics
To drive our electro- and acousto-optic devices, we have built our own custom
PCB wrappers for commercial surface-mount RF components. These are less costly
than packaged solutions, and they also allow us to parallelize and organize our RF-
electronics chains in a clean and compact manner, as seen in Fig. 4.19(c).
The sequence of electronics we use to drive an AOM is given in Fig. 4.19(a),
and the frequencies, powers, and modulators we use are listed in Table 4.3. We
begin with a function generator, which uses a voltage-controlled oscillator (VCO)
(Minicircuits JTOS-100+, JTOS-200+, or JTOS-300+, depending on the frequency)
and a voltage-controlled attenuator (VCA) (Minicircuits RVA-2500+) to produce a
customizable RF signal.2 The frequency and amplitude can be set with manual
adjustment knobs or controlled by analog signals controlled by a computer. The
2Frequencies of the two AOMs used in the red MOT beam path are set by direct digital
synthesizers since this gives us better control of the frequency, important for interacting with the
narrow, intercombination linewidth.
99
Model Purpose ΩAOM (MHz) PRF (W) η (%)
NeosTech. N15200-0.67-KD blue lock 220 0.58 20*
Isomet 1250C-829 Zeeman 280 1.45 50
Crys.Tech. 3200S top MOT 180 0.92 63
Crys.Tech. 3200-121 bottom MOT 180 0.39 47
Isomet 1205C-2 red lock 80.5 0.32 2*
Isomet 1205C-2 red MOT various 1.25 38
Brimrose TFM-80-30-.800 lattice 80 0.94 80
TABLE 4.3. List of AOMs currently in use in the experiment, including the
manufacturer and part number, purpose, operating RF power PRF and frequency
ΩAOM (MHz), and efficiency η. The asterisk denotes double-pass efficiency.
RF signal can then be switched (Minicircuits MSWA-2-20) with a digital signal (for
example, this is where the TTL pulse in the FM-spectroscopy setup chops the AOM)
before it is sent on to a preamplification stage (Minicircuits ERA-3SM) and an RF
power amplifier. The power amplifier is an SOT-115J package CATV amplifier in
a shielded box (Compac R51160-100-0) with appropriate thermal management. We
use two different CATV amplifiers: the MHW1345 component from Freescale has
34.5 dB gain and 10–200 MHz bandwidth, and the CGD1044H from NXP has less
gain at 25 dB gain but a wider 40–1000 MHz bandwidth. Like Minicircuits power
amplifiers, they are sensitive to RF reflections and can be destroyed if powered on
while an input or output is disconnected. Unfortunately, this particular package
seems to be obsolete and we have purchased large quantities of each CATV amplifier
to have a lifetime supply.
The electronics chain for the EOMs is shown in Fig. 4.19(b). The frequencies we
use are ΩEOM = 8.773 MHz for the blue laser and ΩEOM = 3.161 MHz for the red, and
sources for these are the direct digital synthesizer (DDS) controllers, described below
in Sec. 4.3.2. Note that most of the components are not pictured in Fig. 4.19(c). Out
of the DDS, the small-amplitude signal is split to a homebuilt phase shifter (ϕ) and
100
preamplifiers
rf function  
generators
rf switch
analog out
digital out
direct digital 
synthesizers
func gen
analog 
output
digital 
output
switch preamp power amp
freqamp
AOM
rf power 
amplifiers
(a) (c)
(b)
EOM
direct digital 
synthesizer amp
ampf
from PD
LO
helical resonator
FIGURE 4.19. Overview of electronics used to control RF devices in the experiment.
Block diagrams are shown for (a) AOM signals and (b) EOM signals, and a
photograph of an electronics rack (c) shows homebuilt packaging of electronics.
amplifier (amp). The phase shifting circuit uses two first-order all-pass filters [206] to
shift the phase by more than 180◦, but the op-amps rail if too large a signal is used.
The amplification stages used in the EOM sequence consist of a pair of cascaded
inverting LM7171 op-amps, which have a unity gain bandwidth of 200 MHz and a
high slew rate of 4100 V/µs. The amplifiers increase the DDS signal before sending
it on to the helical resonator (with an amplitude as large as 20 V peak-to-peak), and
also increase the phase-shifted signal used as the local oscillator (LO) in the mixer
(Minicircuits ZP-10514+) used to demodulate the photodiode signal.
4.3.2. Computer Control
We interface the experiment with a computer for precise timing of electronic
signals. Designs for hardware that produce digital and analog outputs were created
by Todd Meyrath when he helped develop a parallel bus control system as a graduate
101
student in Mark Raizen’s group at The University of Texas at Austin [168]. The
digital output channels can each drive a 50-Ω load, and we have built two boards for
a total of 32 channels. The analog boards use DAC7744 chips by Texas Instruments
and produce output voltages between ±10 V with 16-bit accuracy. Each channel here
has an output driven by a BUF634 and can drive a maximum of 250 mA. We have
built two of these boards, totaling 16 analog channels. Finally, we have six DDS
boards, each built around an AD9852 from Analog Devices. Fully digitized signals
can be created up to 135 MHz. In principle, arbitrary waveforms can be created, but
our software does not currently allow for reprogramming the DDS in time with the
rest of the experiment. We can, however, program two frequencies and then switch
or ramp between the two via frequency-shift keying (FSK). Timing of the switching
is controlled with an external digital signal which is well-timed.
Computer control is enabled using software communication with open-source
embedded ethernet devices [207]. The project is discussed in detail in Ref. [208].
A computer programs a ‘box’ via the ethernet-enabled controller, and an interface
board loads data into first in, first out (FIFO) memory so they can be read out to
the parallel bus hardware. The analog output box is designated to be the ‘master,’
and one of its channels sends a start pulse to other boxes, marking the beginning of
an experimental run. Subsequent synchronization is maintained through the use of a
shared clock signal at 10 MHz, derived from a rubidium standard.
102
CHAPTER V
TRAPPING AND COOLING STRONTIUM
With all the components described in the previous two chapters, we can now
discuss our progress in trapping and cooling strontium with our apparatus. The
geometry of the cooling setup is pictured in Fig. 5.1. Cooling beams are delivered to
the vacuum chamber with free-space optics, and the top and bottom six-beam MOTs
are each created via retroreflection of three beams. Top-MOT beams are split from
a single beam, which has a waist of 0.55 mm and is detuned to the red of 461 nm
resonance by 40 MHz. The transverse beams are expanded with telescopes using
50.2 mm and 500 mm lenses (Newport KPX082AR.14 and KPX211AR.14) to largely
fill the 1.5′′ clear aperture viewports. These beams are raised to the level of the top
MOT with periscopes and directed through the spherical octagon using mirrors on an
elevated breadboard, attached to the vacuum chamber’s 80-20 support frame. The
axial beam for the top MOT is delivered through the bottom of the pyrex cell, and
is thus smaller, expanded by a telescope using 100 mm and 300 mm lenses (Newport
KPX094AR.14 and KPX112AR.14).
Beams for the bottom MOT—also detuned by 40 MHz—are sized to fill as much
of the pyrex cell as possible without clipping. All three retroreflected beams are
expanded with 24.5 mm and 100 mm achromat lenses (Newport PAC022AR.14 and
PAC052AR.14) from an initial waist of 0.69 mm. As shown in Fig. 5.1, the AH coils
are placed on either side of the hanging pyrex cell, and the transverse MOT beams
are directed diagonally through the cell. The imaging camera (not pictured) views
atoms in the lower MOT from the left side of the illustration.
103
l/4
l/4
upper MOT
lower MOT
AH coil frame
5.0 in
upper MOT 
transverse beams
upper MOT 
axial beam
2.15 in 5.125 in
1 in 
optical post
choose 
upper/lower
AH fields
24.6 G/cm 
per Amp
3.83 G/cm 
per Amp
FIGURE 5.1. Schematic of the arrangement of MOT beams for both the top and
bottom traps, viewed from the high-pressure side of the vacuum chamber. The
transverse top-MOT beams are in the plane of the optical table, and the axial beam
is oriented vertically. Beams for the bottom MOT point into the page, with the
transverse beams reflected diagonally through the pyrex cell and the axial beam
passing through the lower AH coils on either side of the cell. The inset shows the
orientation of magnetic quadrupoles, with approximate field gradients for each AH
pair.
We use polarizing beamsplitter (PBS) cubes (Photop Technologies BPS0201)
and dual-wavelength λ/2 waveplates (also from Photop Technologies, but with no
part number) to divide each of the top- and bottom-MOT beams. We have found
that when trying to reflect vertical polarization that is a small fraction of the overall
power, there is significant residual horizontal polarization also reflected, and that the
beam needs to be cleaned up with an additional polarizer. We use equal power in
each MOT’s transverse beams, which each have twice the power of the axial beam.
104
Circular polarization is created using dual λ/4 waveplates (Photop Technologies, λ/4
at both 461 nm and 689 nm; ⊘38.1 mm for the top-MOT transverse beams, and
⊘25.4 mm for the rest).
Repumping light at 497 nm is combined on a dichroic beamsplitter (Semrock
LM01-480-25) with the upward, axial top-MOT beam. Thus, it passes through both
trapping regions, reducing the power budget. The 689 nm light is combined with the
lower-MOT beams before splitting and expansion, also using a dichroic beamsplitter
(Semrock FF495-Di03-25x36). The red light’s frequency is controlled with a single-
passed AOM, so it is steered slightly when we make adjustments to the detuning (a
frequency change of 10 MHz deflects the beam by only a couple of millimeters at the
MOT). Finally, we have used a dichroic beamsplitter in the beam path for the top
MOT in case we want to use red light there as well.
5.1. Double-MOT Scheme
We have successfully cooled atoms in both the top and bottom regions, and
photographs of our MOTs are shown in Fig. 5.2. The top MOT is loaded directly from
the slowed atomic beam and loads large numbers of atoms quickly (for comparison, we
can load a MOT in several tens of milliseconds, but the one-way-barrier experiments
in rubidium in our lab used a 20-s loading time). By tuning the piezo of the 922 nm
master laser, we are able to trap all four naturally occurring isotopes of strontium,
but we lock the laser to the 88Sr resonance and work exclusively with this isotope.
Our first observation of cold atoms occurred when the top-MOT axial beams had
incorrect circular polarization. Rather than confining the atoms in the top MOT, the
vertical beams were ejecting them (see the middle picture in Fig. 5.2), and we saw a
faint fluorescent beam of atoms inside the pyrex cell. This happened because we did
105
2D MOT 
ejection jets
Zeeman  
beam
top  
MOT
bottom 
MOT
FIGURE 5.2. Pictures of cold strontium atoms: top MOT (left), top MOT in the
bottom-MOT-loading configuration (center), and bottom MOT (right). These photos
were taken with the atomic source oven operating at 600◦C. Photos by Dan Steck,
Paul Martin, and Eryn Cook with a Canon EOS 6D, Canon EOS REBEL T5i, and
Sony DSC-RX100M3, respectively.
not account for an inversion in handedness from the reflection off the dielectric mirror
below the pyrex cell (we have since switched to metallic mirrors for MOT beams after
their polarizations are circularized [Newport 10D20ER.1]). Observing this ejected, 2D
MOT was serendipitous, however, in that it gave us an efficient method to load the
bottom MOT. We had planned on cooling atoms in the spherical octagon, releasing
them, and allowing them to fall into the pyrex cell. This would be inefficient because
of the high temperature of the blue MOT; a small cloud of atoms at 1 mK would
spread to over 3 cm before entering the constricted entrance of the pyrex cell (which
is only about ⊘1.5 cm). By ejecting atoms from the 2D MOT, we are able to load
more atoms in a continuous fashion.
We are able to switch between loading a top or bottom MOT by rotating a
λ/4 waveplate to change the vertical beam’s polarization from confining to ejecting,
respectively. Notice from Fig. 5.1 that the bottom trapping region is fairly close to
the top MOT’s lower AH coil. This means that when the top MOT coils have high
currents (∼6.8 A for loading a top MOT) the zero-crossing of the magnetic field in the
lower region is shifted downward significantly. We therefore must also decrease the
106
current in the top AH coils when loading a bottom MOT. Typically, we use a current
of around 2 A (roughly 8 G/cm, resulting in a 2D MOT that is much more diffuse
than pictured in Fig. 5.2) and use z-axis Helmholtz (HH) coils to shift the magnetic
zero to be at the center of the lower AH coil pair. The relative orientation of the
top- and lower-AH coils is important because the vertical beam should be polarized
to eject atoms from the top MOT but not the bottom.
We find that the repump significantly increases the MOT’s brightness and
lifetime, as expected from previous observations in strontium [16, 17]. Since we only
have a small power available, we keep the beam compact to have maximum impact
on the atoms. The brightness of the bottom MOT is fairly sensitive to the alignment
of the repumping beam. We carefully overlap it with the vertical MOT beam, then
misalign it slightly to maximize the brightness we observe with a security camera
(EverFocus EQH5102). The misalignment is likely helpful because the repump is
most effective when it overlaps the bottom MOT, which does not necessarily form
within the vertical ejection beam. We do not stabilize the 497 nm light to an atomic
transition, but rather use the brightness of the atoms to frequently confirm that the
light is resonant. In the future if more power is produced at 497 nm (either by getting
more power from the master 994 nm ECDL or improving the reflectivity of the mirrors
in the SHG cavity), the light could be actively stabilized to the transition using the
optogalvanic technique [209, 210].
5.2. Blue MOT Temperature Measurements
To measure the temperature of the atoms in the MOT we can release the trapped
cloud, allow it to ballistically expand, and then image it using molasses beams to
‘freeze’ the atoms in place [211, 212]. This is somewhat simpler than the original
107
methods used to measure the temperature of optical molasses because MOTs have the
additional benefit of spatial confinement.1 After releasing and allowing the atoms to
evolve for tdelay, the measured spatial distribution of the cloud will be the convolution
of the MOT’s initial spatial distribution and a second spatial distribution—created
by expansion over tdelay due to the initial velocity distribution. Assuming the
atoms exhibit a Maxwell–Boltzmann distribution and a Gaussian spatial profile, the
resulting expanded cloud will also be Gaussian, with width
σx(tdelay) =
√
σ2x0 + (σpxtdelay/m)
2. (5.1)
Here, σpx =
√
mkBT is the width of the atoms’ momentum distribution before release.
By varying tdelay, the hyperbolic form of Eq. (5.1) can be fit to find the temperature
of the cloud (for large values of tdelay, the initial size of the MOT is negligible and a
linear fit suffices).
The duration of the imaging pulse is important because the cloud will still be
expanding—albeit diffusively—under the influence of molasses beams. While imaging
atoms from a cesium MOT, experimenters in Ref. [211] varied the temporal extent of
the freezing molasses. They observed that significant diffusion did not occur for times
less than about 20 ms. Since the scattering rate in strontium is roughly a factor of 6
larger than that in cesium, we should use an imaging pulse of only a few milliseconds
to avoid artificial broadening of the expanded cloud width.
1The “release-and-recapture” method uses the same experimental mechanics as the freezing
method we use, but it can have large uncertainties due to sensitivity to molasses spatial
distribution [213]. The “time-of-flight” method, on the other hand, is less geometrically sensitive,
but requires an extra (resonant) laser—used to probe the absorption of released atoms as they fall
and pass through it [2].
108
5.2.1. Imaging System
We take fluorescence images using a charge-coupled device (CCD) camera
(“MicroLine” camera from Finger Lakes Instrumentation—we call it the “FLI
camera”) which was used in previous experiments and is also described Ref. [123].
We use a Micro-Nikkor 55 mm f/3.5 lens from Nikon at its closest focus and block
background light (especially 689 nm and 914 nm light which could scatter into the
camera) using a color filter (Schott BG40, part number FGB37 from Thorlabs). The
colored glass is not AR-coated and transmits roughly 89% of 461 nm light, meaning
the quantum efficiency of our camera (which has a Kodak KAF-0402ME sensor) is
about 45%. We used a scaled target reticle (Edmund NT39-460) and measured a size
conversion factor for our images to be ∼ 23.75 µm/pixel.
The shuttering system in our CCD camera is problematic in a number of ways. In
the past the shutter blades would often stick or break, requiring frequent cleaning and
repair. A small change to the actuating mechanism [123] seems to have helped (we
have only had to replace one shutter blade, and have not noticed significant sticking
since). Additionally, in its original use, the mechanical vibrations produced by the
camera could cause the “bronze” lasers in the rubidium experiments to come unlocked.
Previous graduate students resolved this issue by mounting the camera on a large,
heavy damping plate that was connected to a second base with a strip of 0.125′′-thick
Sorbothane sandwiched in between. On our optical table, however, the SHG cavities
are quite sensitive to vibrations and the damping solution unfortunately was not as
effective. We learned through an informal conversation with Luis Orozco—who has
used Sorbothane to reduce vibrations in cavity QED systems [128]—that Sorbothane
is much more effective at damping vibrations when it is cut into smaller pieces.
The reduced size allows for more transverse expansion of the viscoelastic material,
109
which is the mechanism for extracting vibrational energy. The ratio of the loaded to
unloaded surface area (called the “shape factor” in Sorbothane’s Engineering Design
Guide [214]) should be between 0.3 and 1 for good damping, and the strip we were
using had a ratio over 10. We cut up small 0.25′′ square pieces (shape factor of 0.5),
and the improvement was dramatic: before, the disturbance of taking a picture was
so severe that the SHG cavities would often come unlocked, but now the response of
the error signal is greatly reduced.
Temperature measurements of the blue MOT are difficult given our camera
shutter’s finite opening time. By varying the timing of an exposure pulse in relation
to the camera trigger pulse, we determined the timing of the shutter response. The
observed intensity as a function of time was roughly an error function, as suggested
by the camera profile in Fig. 5.3. The camera shutter begins to open 10 ms after the
triggering pulse and is fully open about 20 ms after the trigger. This means that we
cannot image the expanded cloud for tdelay < 10 ms without also partly imaging the
MOT before it is released. Unfortunately, since 88Sr does not experience sub-Doppler
cooling effects, the temperature of the blue MOT is quite hot (Tdoppler = 770 µK),
and 10 ms is too long to wait and still have an appreciable signal. (The cesium atoms
in Ref. [211], for comparison, were on the order of 10 µK and were measured with
expansion times of 15 ms ≤ tdelay ≤ 40 ms.)
5.2.2. Expansion Data and Temperature Estimate
Despite the issues of the slow shutter, we attempted to make a temperature
measurement by including the partial image of the loaded MOT in our background
signal. The experimental timing sequence used in this measurement is shown in
Fig. 5.3, with timing referenced to the release of the trapped atoms. AH fields and
110
top MOT 
beams
top MOT 
AH fields
bottom MOT 
beams
bottom MOT 
AH fields
camera
0-50-100-150-200 50
time (ms)
trigger
shutter 
fully open
illumination 
pulse
37 G/cm
54 G/cm
8 G/cm
drop 
event
FIGURE 5.3. Control timing sequence used to trap and image the bottom MOT
during temperature measurements. The delay between triggering the CCD camera
and the aperture fully opening is roughly 20 ms. The time between the drop event
and the illumination pulse tdelay is varied between 0.25 ms and 2.00 ms in Fig. 5.6.
MOT beams were initially turned off to dump atoms before loading a bottom MOT
for 100 ms. After this, we turned off the top-MOT cooling beams and magnetic
fields and adjusted the lower HH fields while increasing the AH field gradient to
54 G/cm to tighten the trap. These changes spatially shift the MOT, and we waited
for 50 ms to let magnetic field transients decay. Finally, we released the bottom MOT
by inhibiting the AH fields while switching off the cooling beams with an AOM. The
camera is triggered so that its shutter is fully open at the time the atoms are released.
A 2-ms-long illumination pulse images the atoms after tdelay, which we vary between
0.25 ms and 2.00 ms. A “background” sequence is identical, but does not have an
illumination pulse.
We had hoped that subtraction of the image of the loaded MOT would
provide distributions that could be fit with Gaussians to extract the temperature.
111
signal background subtracted
MOT reflections
cell scatter
FIGURE 5.4. CCD images for a drop delay of 50 ms using 100 repetitions of the
timing sequence in Fig. 5.3. Some sources of light pollution are identified and can be
seen to persist in the subtracted image.
Unfortunately, variations in MOT brightness make this difficult. These variations
have several underlying causes, including fluctuations in the number of atoms loaded
in the MOT (from unstable power of the repumping beam), slight changes in the
timing of the opening of the camera shutter (exposing the MOT for an uncontrolled
amount of time), and changes in the intensity of the MOT beams. Even with
the improved Sorbothane damping, the blue laser can have small changes to its
intensity as the shutter opens, and the output power of the SHG cavity can also
have long-term drifts. We attempted to remedy these fluctuations by using a pulse-
stabilizer circuit [215] that was built for absorption imaging in the previous rubidium
experiments in our lab [123]. This circuit compares an analog voltage to the integrated
power it receives from a picked-off portion of the bottom-MOT beams. When the
two are equal, a digital output is sent to the RF switch controlling the beam’s
AOM. We set the analog voltage to correspond to a roughly 2 ms pulse length, but
the pulse-stabilizer circuit ultimately controls the timing, ensuring that a consistent
illumination energy is used from run to run.
112
pixel
0 35050 100 150 200 250 300
co
lu
m
n
 s
u
m
 (
ar
b
it
ra
ry
 u
n
it
s)
signal
background
pixel
0 35050 100 150 200 250 300
co
lu
m
n
 s
u
m
 (
ar
b
it
ra
ry
 u
n
it
s)
50 ms
75 ms
100 ms
125 ms
FIGURE 5.5. Sum of pixel values along columns of the signal and background images
in Fig. 5.4 are shown on the left. The signal has a broader tail than the background
due to expansion of the atomic cloud during the delay time. Similar sums of the
background images are shown on the right for several delay times. The peak of the
background images varies, showing the volatility in the brightness of the MOT during
these measurements.
An example of the image subtraction process is shown in Figs. 5.4 and 5.5 for
tdelay = 50 ms. We alternate between “signal” and “background” sequences and
average together 100 images of each before subtracting. Pixel values can be integrated
along the rows or columns of an image to produce a distribution from which the width
can be extracted. It is clear from the signal’s broad tails in the left plot in Fig. 5.5
that a larger Gaussian shape is hidden by the bright MOT. Ideally, the central MOT
image can be removed perfectly, but the variation in MOT brightness (even with
100 images averaged) is problematic. This is illustrated in the right plot in Fig. 5.5,
where averaged background images have been integrated for several delay times (the
delay time serves simply as a label, of course, since the background images have no
illumination pulse). Since the variation of the peak amplitude of the background
113
MOT is on the same order as the amplitude of the expanded cloud’s distribution (in
the left plot), this clearly is problematic.
Subtracted images for a sequence of expansion times are pictured in Fig. 5.6 and
the integrated distributions are shown in Fig. 5.7. CCD images have been individually
normalized before applying a color scaling, so the size in the subtracted image is
not necessarily representative of the width of the expanding cloud. The problems
of fluctuations in MOT brightness is clear in these images: the amplitude of the
integrated distribution at tdelay = 0.25 ms is much smaller than at 0.50 ms or 0.75 ms.
It is also apparent that the atoms expand rapidly, and images after about 1.50 ms
are not useful. Moreover, the contributions of extra scatter on the pyrex cell are
significant with the small subtracted signals, complicating the prospect of fitting
data. Ultimately, however, we will be performing temperature measurements on
much colder atoms in a red MOT, and these complications will be avoided.
To make a very rough estimate of the temperature, we can use the size of
the unexpanded MOT and estimates of the restoring force. From the equipartition
theorem, the size of the MOT in one dimension is related to the temperature via
kBT = κz
2
rms, where κ is defined in Eq. (A.8) [147]. The FWHM of the MOT is
typically between 22 and 25 pixels, and the axial bottom-MOT beam (which confines
the atoms roughly in the horizontal direction as viewed by the camera) has a power of
∼ 2 mW over approximately a square centimeter. This corresponds to a temperature
range of 3–4 mK, which is reasonable, given blue MOT temperatures measured by
other groups using 88Sr [16, 17].
114
0.25 ms 0.50 ms 0.75 ms 1.00 ms
1.25 ms 1.50 ms 1.75 ms 2.00 ms
FIGURE 5.6. Series of images taken in an attempted temperature measurement of
the blue MOT. For each delay time, the signal image (average of 100 runs) is shown
above the subtracted result. Each image is normalized to itself, so false color levels
are not necessarily comparable.
115
pixel
0 35050 100 150 200 250 300
ro
w
 s
u
m
 (
ar
b
it
ra
ry
 u
n
it
s)
0.25 ms
0.50 ms
0.75 ms
1.00 ms
1.25 ms
1.50 ms
1.75 ms
2.00 ms
pixel
0 35050 100 150 200 250 300
co
lu
m
n
 s
u
m
 (
ar
b
it
ra
ry
 u
n
it
s)
0.25 ms
0.50 ms
0.75 ms
1.00 ms
1.25 ms
1.50 ms
1.75 ms
2.00 ms
FIGURE 5.7. Integrated pixel values of the series of subtracted images shown in
Fig. 5.6. Pixel values are independently totaled horizontally (left) and vertically
(right). Contributions of scatter in the cell can be seen as increases in the summed
value at small and large row number and as a feature near column 100.
5.3. Cooling in a Red MOT
Narrow-band cooling is interesting because it can be a regime where single
photon scattering events are important in MOT dynamics. Usually atomic cooling is
performed on a broad, cycling transition where Γ ≫ ωrecoil. In this limit, individual
recoil events do not significantly change the effective detuning δeff, and a semiclassical
treatment of the field involving scattering rates suffices. Here atoms in molasses
beams can be considered to be in Brownian motion, leading to the concept of the
Doppler-cooling temperature limit [147]. The recoil frequency shift of the 7.6-kHz-
wide intercombination line in 88Sr, however, is 4.7 kHz (Γ/ωrecoil = 1.6), and a
quantized model of the field is needed to develop a Doppler-cooling theory [216].
In this regime, atomic temperatures are much smaller than the traditional Doppler
limit, even reaching values below the single-photon recoil level [27].
116
If a transition is too narrow, however, then a traditional MOT operating on the
line is infeasible. Calcium, for instance, has a 408-Hz-wide 1S0 −→3 P1 transition
at 657 nm. This linewidth is so narrow that the maximum scattering force is not
much larger than gravity (~kΓ/2mg = 1.5), and atoms literally fall out of the MOT.
(Despite this issue, cold 40Ca samples have been created by quenching the excited
state to artificially decrease the lifetime [217] and by using an alternate narrow-
band transition connected to the long-lived 3P2 state [218].) Strontium is convenient
because the intercombination line is narrow, yet broad enough that the radiation
pressure can overcome the gravitational force (~kΓ/2mg = 16). Images of a red MOT
show the atoms sagging to the bottom of the trapping region, where the position-
dependent force is maximum [27].
The narrow transition is also quite sensitive to power broadening because the
saturation intensity is so small (3 µW). This means the effective linewidth Γeff =
Γ
√
1 + I/Isat can be increased significantly, entering the broadband cooling regime
once more. Cooling-beam intensity and detuning thus have a profound impact on
final temperatures reached in the red MOT, and this has been explored in depth by
the Ye Group [19, 27, 89, 219]. We plan to use intensities for our 689 nm laser that
are large enough to capture many atoms at several µK.
5.3.1. Transfer Process
Cooling in a red MOT is complicated by the narrow transition because atoms
will only be resonant in a small velocity window. At 1 mK, for instance, the average
atomic speed is 0.5 m/s, but a velocity change of only 0.005 m/s gives a Doppler shift
equal to the transition’s linewidh—meaning that most atoms are out of resonance with
the laser. In order to combat this effect, the red molasses beams can be frequency-
117
detuning
-3 0-2.5 -2 -1.5 -1 -0.5
Doppler profile, 5 mK
Doppler profile, 2 mK
Dmod
D1
FIGURE 5.8. Modulating the detuning of the red MOT beams to interact with
atoms at blue MOT temperatures. Doppler-broadened distributions of atoms are
shown for temperatures of 2 and 5 mK. The narrow laser—pictured here with a
7.5 kHz linewidth—will only interact with a small number of atoms, so to cool the
entire population it is artificially broadened by ramping the detuning between ∆1 and
∆1 +∆mod.
chirped [220] or artificially broadened [15] to address and cool more atoms. This idea
is illustrated in Fig. 5.8, where a narrow cooling laser is shown alongside the frequency
spread of Doppler-broadened distributions of atoms at 2 and 5 mK. By ramping the
laser’s detuning between ∆1 + ∆mod and ∆1, the narrow lineshape sweeps across a
significant fraction of the thermal atoms, increasing the transfer efficiency from blue
MOT to red.
Additionally, for a given magnetic field gradient, the boundaries of the blue
and red trapping regions are also quite different. Using Eq. (4.7) again as intuition,
we can estimate the expected size discrepancy. The detuning of the blue MOT is
∆blue = 40 MHz, and the largest detuning in Fig. 5.8 is ∆red = ∆1 +∆mod = 2 MHz
(note that the boundary of the red MOT is ramped along with the frequency). The
effective magnetic moments for the two transitions, meanwhile, are µ˜blue = µB and
µ˜red = 3µB/2. Thus, at its largest extent the red MOT will be a factor of 30 smaller
in the same magnetic field. This again means that many of the atoms will be lost
unless we also significantly decrease the magnetic field gradient to encompass the
atoms within the red MOT’s trapping region.
118
As a starting point, we have followed the procedure of the Katori and Ye
Groups [15, 27], where transfer efficiency from the blue to red MOTs can be as high
as 50% [19]. After loading atoms in a lower blue MOT (by following the sequence
of events for t < 0 in Fig. 5.3, we simultaneously turn off the blue light, lower the
magnetic field gradient from 54 G/cm to 3 G/cm, and turn on modulated red light.
After the atoms are cooled for roughly 10 ms, we ramp the magnetic field gradient
up to 10 G/cm over 50 ms. This serves to compress the red MOT, which can then be
cooled further with the unmodulated red laser operating at detuning ∆1. We achieve
modulation of the red light through the ramped FSK ability of the DDS boxes. In
the computer we can set the final detuning ∆1 , the modulation depth ∆mod, and
the frequency at which the laser sweeps back and forth between the two extremes.
The computer controls a digital signal which ramps the DDS frequency up and down
linearly.
5.3.2. Current Progress
At the end of the described sequence, we turn off the AH field and pulse on the
blue imaging light to observe the atoms. Unfortunately, we do not see any evidence
that we have successfully loaded a red MOT, and are still debugging the process.
We have observed that shining the modulated red laser into a blue MOT reduces the
brightness of the atoms, which can be seen in both our security camera monitor and in
the FLI CCD images (this is true even if the light is blue detuned). This presumably
happens because the laser is driving population to the 3P1 state, reducing the number
of atoms that can fluoresce at 461 nm. The red light is therefore successfully locked
to the 689 nm transition.
119
bottom MOT 
blue beams
bottom MOT 
AH field
bottom MOT 
red beams
camera
0 50
time (ms)
100 150 200 250
3 G/cm
10 G/cm
54 G/cm
recapture 
event
recapture 
event
transfer 
event
FIGURE 5.9. Control timing sequence used to test if the red laser is affecting the
atoms. Atoms are released from the blue trap at t = 0 and allowed to interact with
modulated red light for some time before the blue trap is reengaged. Two testing
scenarios are shown: recapture event (a) occurs after ramping the AH field from
3 G/cm to 10 G/cm (thick lines), and recapture event (b) occurs after leaving the
AH field low for long timescales (thin lines).
Additionally, we have tried to recapture remaining atoms in a blue MOT after
interaction with the red light. Two timing sequences are shown in Fig. 5.9: the first
(a) goes through the cooling sequence described above (save for ceasing modulation
in the final cooling stage) before recapture, while the second (b) leaves the field at
a low value, allowing the modulated red light to interact with the atoms for some
time before recapture. We have found that we can recapture some of the atoms
in each case, evidence that the red light is inhibiting some atoms from leaving the
trapping region. We find that we can recapture a larger fraction of the atoms for
smaller values of ∆1 (on the order of 100 kHz) and moderate modulation depths of
∆mod ∼ 1 MHz. (The signal is greatly reduced both when the modulation is disabled
and when ∆mod > 2 MHz. We have typically ramped the frequency between the
120
extreme values at a rate of 25 kHz, and we might need to increase this rate if we
want to modulate over a broader range of frequencies.) For parameters which yield
large recapture fractions, we find that the scenario in (b) can load atoms more than
500 ms after the release event.
We currently overlap the lower blue and red MOT beams on a dichroic before
using polarization optics to split the beams into three paths. This assumes that the
dual waveplates are actually λ/4 for both wavelengths, like we specified in our order
from Casix. The Ye Group, however has waveplates which are λ/4 for 461 nm and
3λ/4 for 689 nm, calling into question the retardance of our waveplates.2 We have
tested the waveplates by alternating the polarity of magnetic fields during the red-
laser-interaction times of the timing sequences in Fig. 5.9. We have found that atoms
are recaptured for each polarity, but that consistently more are recaptured when the
sign of the magnetic field matches what we use in the blue MOT. Thus, we think our
waveplates are indeed λ/4 for each wavelength.
So, given that the laser is resonant with the atoms and that we are using the
correct field gradient polarity, why are we unable to load the red MOT? It’s possible
that background magnetic fields are not nulled well enough, and the zero-crossing of
the AH field is displaced significantly far from where the blue MOT forms. Atoms in a
red MOT that are shifted away from the focal plane of the FLI camera would then be
harder to see unless ‘recaptured’ by again turning on the high field gradient of the blue
MOT. Another possible issue is related to the geometry of our setup. The orientation
of the axis of the AH coils in Ref. [27] is perpendicular to the optical table, and
the vertical trapping beam directly opposes gravity. Our vertically-oriented beams,
2Investigating further, we found that the material shipped with the waveplates indicated that
they were indeed λ/4 at each wavelength, but the vendor—after we recently asked directly—said
they were 3λ/4 at 689 nm.
121
however, are diagonally positioned (as shown in Fig. 5.1), so the radiative force is
reduced by
√
2 (having two beams does not double the upward force since they are
well above saturation intensity). This means the maximum scattering force is now
only 11.3 times the force of gravity, and it’s possible that atoms are not well spatially
confined. For either of these, a potential debugging tool is to attempt to load the red
MOT in the spherical octagon, where we have both better control over the magnetic
fields (the HH coils here are closer to optimal spacing) and a trapping beam directly
opposing gravity. Alternatively, we could add an additional red MOT beam, oriented
vertically to test the latter possibility in the lower trapping region.
122
CHAPTER VI
CONCLUSION
This dissertation has detailed the development of a strontium MOT for future use
in measurements of atom–surface interactions. Strontium is a promising candidate for
such measurements because, as an alkaline earth element, it offers spectroscopically
narrow spin-flip transitions which have helped enable unprecedented timekeeping in
strontium optical lattice clocks. These narrow transitions provide ample resolution to
make direct measurements of CP-induced level shifts when atoms are near a surface.
A secondary benefit of using strontium is the ability to confine atoms in a magic
wavelength optical lattice that eliminates light-shift systematics while providing good
spatial resolution of atom–surface separation. We propose using test surfaces which
are partially transparent at the magic wavelength so that one of the lattice beams
comes from behind the surface. In this configuration the locations of the lattice
minima are not fixed by the boundary of the surface, like they would be in a reflected
lattice scheme. Strontium atoms could therefore be loaded far away from the test
surface and controllably translated into position for measurement of the CP shift. In
such a lattice scheme, atoms could be brought as close as 200 nm before the trapping
potential is overwhelmed by the CP effect, and this technique would join Ref. [84] as
a rare experimental investigation into intermediate-range CP potentials.
The experimental apparatus needed for such measurements requires extensive
development and construction. Our lab embraces a cost-effective, do-it-yourself
approach, and during this nascent project we have produced several complex laser
systems, numerous control electronics, and a UHV system with a hot-strontium
source. We have been fortunate to be able to share some of our achievements
123
in a pedagogical and open-source fashion [116, 126, 173], and our passively stable
ECDL design has been adopted by many groups in the atomic physics community
(in addition to some labs pursuing their own machining solutions, the University of
Oregon Technical Science Administration machine shop has shipped more than fifteen
sets of components to groups from around the world). With our stabilized 461 nm
laser, we can slow strontium atoms from an atomic beam, load them into a primary
MOT, and transfer them to a secondary MOT that is more isolated from background
gas collisions and offers better optical access.
Atoms in this blue MOT are still quite warm, however, and further narrow-
line cooling on the 689 nm intercombination line is necessary before they can be
efficiently loaded into an optical lattice. We have been able to confirm recently that
our stabilized 689 nm laser is resonant with the atoms: We can release the blue MOT,
switch on a red molasses, and recapture many of the atoms a few hundred milliseconds
later. Unfortunately, we have not observed evidence that we are successfully loading
a red MOT, and addressing this issue is the next step in the continued maturation of
the apparatus.
Once methods for loading a red MOT have been developed, proof-of-principle
experiments can begin to be pursued. For simplicity, these will probe the relatively
broad intercombination line and use the side of the hanging pyrex cell as a test
surface. Short-term work that needs to be done includes integration of the air-bearing
translation stage, alignment of the optical lattice, and possibly some narrowing of the
689 nm interrogation laser by stabilizing it to a high-finesse optical cavity. Impact
of these initial measurements will be significant because relatively few schemes exist
for measuring vacuum-induced potentials, and additional experimental avenues are
valuable.
124
Moreover, the apparatus can be improved after initial measurements to enable
more ambitious CP investigations. First, much finer resolution can be obtained by
probing an ultranarrow clock transition instead of the intercombination line. This
will involve building a pair of new lasers—one to serve as the optical lattice at the
new transition’s magic wavelength, and one that is extremely narrow to probe the
CP shifts. Additionally, investigations of test surface geometry and composition
can be pursued since the lattice technique only requires some transparency at the
magic wavelength. In contrast to alkali atoms, 88Sr presents high flexibility in surface
choice because it has no nuclear spin and has a higher ionization energy (for metallic
surfaces these characteristics reduce trap losses and ionic character of adsorbed atoms,
respectively).
A better understanding of CP potentials could impact engineering of small-scale
cold-atom sensors and quantum information processing devices. We believe that cold
strontium can be a useful resource for probing atom–surface interactions in novel
regimes, and the apparatus we have developed is an important first step in realizing
such an investigative tool.
125
APPENDIX A
TRAPPING BASICS: MECHANICAL FORCES OF LIGHT
Light can influence the motion of an atom through radiation pressure. Consider
an isolated atom that is illuminated by a monochromatic plane wave, nearly resonant
with the atom. Both absorption and emission of a photon changes an atom’s
momentum by ~k, where k is the magnitude of the wavevector, k = 2π/λ = |k|. The
direction of the momentum change is fixed for absorption but random for emission.
A single absorption–emission scattering event is therefore not very meaningful, but
after many such interactions, the momenta from emitted photons have no net effect.
Thus, the light source can “push” an atom in the same direction as k.
In a semiclassical treatment, we consider a two-level atom to be at rest while
interacting with a classical plane-wave light source via the dipole potential [39, 147].
The mean radiation force is a quantum of momentum times the rate of scattering
events,
〈Frad〉 = ~kRscatt = ~kΓρee. (A.1)
The scattering rate is expressed in terms of ρee, the excited state population, and
Γ, the rate of decay from |e〉 to |g〉 (also called the linewidth). The excited state
population is found by solving the optical Bloch equations under the rotating wave
approximation (RWA) [39], and the radiation pressure force becomes
〈Frad〉 = ~kΓ
2
s0
1 + s0 + (2∆/Γ)2
. (A.2)
126
Here, s0 is the on-resonant saturation parameter, s0 = I/Isat = 2|Ω|2/Γ2, where Isat
is the saturation intensity and the Rabi frequency Ω is the rate at which resonant
light causes the atom’s population to oscillate between the ground and excited state.
The detuning is the mismatch of the laser’s frequency with the atomic resonance,
∆ = ωL − ω0, and light is said to be “red detuned” for ∆ < 0 and “blue detuned”
for ∆ > 0. As we might expect, the force is a Lorentzian function of ωL, and the
maximum amplitude of Fmax = ~kΓ/2 for high intensity resonant light (for large
intensities, the atom is equally likely to be in the ground and excited states and
ρee = 1/2).
In general, however, a laser of frequency ω0 will not produce the maximal
scattering force because the atom’s resonance can be shifted away from ω0. It is
better to consider an effective detuning in Eq. (A.2),
∆eff = (ωL +∆Doppler)− (ω0 +∆Zeeman)
= ∆ +∆Doppler −∆Zeeman, (A.3)
which includes a shift of the laser frequency due to the velocity-dependent Doppler
effect, ∆Doppler = −k · v, and a shift of the atomic resonance by the magnetically-
induced Zeeman effect, ∆Zeeman = µ˜ · B/~ (where µ˜ is the difference in the atom’s
excited- and ground-state magnetic moments). For simplicity, we will restrict our
discussion to one dimension and examine the influence of the detuning modifications
individually.
127
A.1. Optical Molasses
The Doppler shift in 1D has a magnitude of kv, with the sign being determined by
the relative directions of the laser and atom. If the laser opposes atomic motion, the
frequency is shifted to the blue, ωL+kv. Thus, a laser that is red-detuned is more likely
to scatter photons for atoms traveling toward it, extracting energy in a dissipative
process. This is the basic idea of an “optical molasses,” in which counter-propagating,
red-detuned lasers damp atomic motion and reduce their kinetic temperature [213].
In our 1D picture, consider two beams, one with k > 0 and the other with k < 0.
Their combined force on an atom of velocity v is
Fmolasses =
~kΓ
2
(
s0
1 + s0 + [2(∆− kv)/Γ]2 −
s0
1 + s0 + [2(∆ + kv)/Γ]2
)
. (A.4)
This force is shown for low illumination intensity (s0 ≪ 1) in Fig. A.1, where the
x-axis is the atom’s velocity. Atoms which move to the right (left) with a positive
(negative) velocity are shifted into resonance with the left-going (right-going) beam,
and are correspondingly slowed. The detuning of the lasers moves two Lorentzians
away from the origin, and for ∆ = −Γ/2, the momentum transfer is uniform over a
maximum range of velocities. In practice, however, there is some power broadening
of the transition because s0 is not negligible, and it is reasonable to use a larger
detuning. The figure shows lineshapes using ∆ = 1.25 Γ, which is what we use for
our blue strontium trap, described in Chapter V.
In the limit of kv ≪ Γ, Eq. (A.4) reduces to
Fmolasses = 4~k
2s0
2∆/Γ
[1 + s0 + (2∆/Γ)2]
2 v =: −βv. (A.5)
128
velocity (Gl/2p) or position (G/m' $Bz)
-5 5-4 -3 -2 -1 0 1 2 3 4
sc
at
te
ri
n
g 
fo
rc
e
right-going beam
left-going beam
total
D/G
FIGURE A.1. Net scattering force of two beams traveling in opposite directions. The
x-axis can either be considered to be velocity (corresponding to viscous damping) or
position (harmonic restoring force), as discussed in the text.
This limit corresponds to atomic velocities which are ‘contained’ within the two
Lorentzian lineshapes, and since ∆ < 0 for red-detuned light, the coefficient of v
is strictly negative. The positive parameter β is defined to emphasize the linear
dependence of the force on velocity, clearly consistent with viscous damping (as
implied by the term ‘molasses’). This cannot damp an atom’s velocity all the way to
zero, however, since the mechanism for removing energy relies on scattering photons
in a random direction. This acts as a heating process—where the atom experiences
a random walk in velocity, mediated by photon recoils—which balances the cooling
in the molasses at a point called the Doppler cooling limit. For ∆ = −Γ/2 the
temperature value is [221],
TDoppler =
~Γ
2kB
. (A.6)
129
λ (nm) Γ/2π (MHz) TDoppler (µK) Trecoil (µK)
23Na 589 10 235 2.4
87Rb 780 6 146 0.362
133Cs 852 5.2 125 0.198
88Sr 461 32 770 1.0
88Sr 689 0.0075 0.180 0.460
TABLE A.1. Doppler and recoil temperatures for several atomic species. Also
included are the cooling transition’s wavelength and linewidth. The Doppler
temperature for the 689 nm transition in 88Sr is not well defined, as suggested by
the fact that it is smaller than the recoil temperature.
This defines one temperature of significance in cold-atom physics, and when optical
molasses were first formed, researchers were surprised to find the atoms were
substantially cooler than this [221].1Another useful temperature reference is defined
by the velocity of a single recoil event, and is called the recoil temperature,
Trecoil =
~
2k2
kBm
. (A.7)
Example Doppler and recoil temperatures are listed in Table A.1 for several atomic
species. Of special note for this thesis are the temperatures for the two transitions in
88Sr. Because of the large linewidth, Doppler-cooling on the 461 nm transition results
in a fairly hot sample of atoms. The 689 nm transition, on the other hand, is so narrow
the kv ≪ Γ assumption in Eq. (A.5) does not apply and Doppler temperature is not
a well-defined concept. In fact, the transition is so narrow, the change in frequency
due to a single recoil is 4.8 kHz, on the same scale as the linewidth. The process of
cooling in this regime are clearly a bit strange, and will be discussed more in Sec. 5.3
1Since mechanisms for sub-Doppler cooling do not apply to atoms with a spinless ground state,
they are outside the scope of this thesis, but nice descriptions can be found in Refs. [222] and [2].
130
A.2. Magneto-Optical Trap
If we ignore velocity, and instead focus on the 1D Zeeman shift in Eq. (A.3), the
mathematical analysis used above is similar except the magnitude of the frequency
shift is replaced by kv → µ˜Bz/~. Again, µ˜ is effective magnetic moment of the
transition, which is found by taking the difference of the ground and excited state
moments, µ˜ = µ′ − µ = µB(m′J′g′J′ − mJgJ). Here, µB is the Bohr magneton, mJ
is the magnetic quantum number of the atom (whose quantization axis is oriented
with the magnetic field along the z-direction), and gJ is the level’s Lande´ g-factor.
Throughout this document, the prime shorthand is used to indicate the excited state.
To simplify discussion a little, let’s assume thatmJ = 0 and−1 < mJ′ < 1 (which
is actually true for 88Sr), so the frequency shift is then ∆Zeeman = µBm
′
J′
g′
J′
Bz/~.
Spatially tuning the magnetic field can thus create a potential landscape where atoms
are shifted into resonance with a laser and feel an increased scattering force. We can
use this idea to create a 1D “trap” that uses the light force to confine atoms in a
particular region. Using Fig. A.1 again for intuition, consider a positive magnetic
field gradient with zero field at the origin, Bz = (∂Bz)z (where shorthand for the
derivative is used, ∂Bz := ∂B/∂z). When an atom has position z > 0, we want it
to scatter light from the left-going beam to force it back toward the origin. This
happens by ensuring the red-detuned laser light is coupled to mJ′ = −1 to bring ∆eff
closer to zero. In contrast, an atom with position z < 0 should become resonant
with a laser coupling to a sublevel mJ′ = +1. Note that the lasers interact with
different sublevels, which has the added benefit of increasing ∆eff when a laser would
be pushing the atoms away from z = 0.
131
The total force on an atom from the two laser beams is again the sum of two
Lorentzians,
Ftrap =
~kΓ
2
(
s0
1 + s0 + [2(∆− µBg′J′Bz/~)/Γ]2
− s0
1 + s0 + [2(∆ + µBg′J′Bz/~)/Γ]
2
)
≈ 4~k2s0 2∆/Γ
[1 + s0 + (2∆/Γ)2]
2
µBg
′
J′
∂Bz
~k
z
= − κz, (A.8)
where the approximation is made for ∆Zeeman ≪ ∆ (again, the linear region near
z = 0 contained between the two Lorentzians in Fig. A.1). This is a linear restoring
force with spring constant κ. Thus the red-detuned lasers used in Doppler cooling
can be extended to create a spatially confining potential by adding a linear magnetic
field gradient and using circularly polarized light which couples to the appropriate
magnetic sublevels. In the example above, the left-going and right-going beams should
have polarizations of σ− and σ+, respectively. Note that these polarizations are with
respect to the atom’s quantization axis, however, and that the two beams will actually
both have right-hand circular polarization in this case.
This is the simple 1D version of the magneto-optical trap (MOT), which is
generalized to three dimensions by using six laser beams and anti-Helmholtz (AH)
coils [223]. The lasers are typically derived from the same source, and thus have the
same detuning, but their polarization requires some special attention. The magnetic
quadrupole field created by the AH coils enters through the axis of the coil pair and
exits uniformly in the lateral space between the two (or vice-versa, depending on the
direction of current). Because ∇·B = 0, the axial field gradient has the opposite sign
of the transverse gradients and twice the magnitude. This means that if the lateral
132
confining beams are right-hand circularly polarized, then the axial beams must be
left-hand circular.
A.3. Dipole Force and Optical Lattices
Thus far we have only considered actual absorption and emission events. This is
not complete, however, because an off-resonant light field can create mechanical effects
without directly driving excitations. In the two-level atom, the dipole interaction in
the RWA produces an ac Stark shift of the ground state [39],
V =
~|Ω|2
4∆
. (A.9)
This is called the dipole potential, which—in contrast to the radiative pressure force—
is largely conservative. Confinement of an atom in the ground state is therefore
possible by creating gradients in intensity of a red-detuned beam, with deeper trap
depths provided by higher intensity fields. Since detuning of the trapping beam
is much larger than the atomic linewidth, relatively few photons are scattered and
heating effects are small. These traps have long lifetimes and are useful for holding
pre-cooled atoms during the course of an experiment.
One-dimensional traps are created by focusing a single beam to a small waist
(often called a dipole trap) or by using two counter-propagating beams in a standing-
wave configuration (called an optical lattice) [93]. An optical lattice can be created
by overlapping two dipole traps via reflection, and because of interference, the lattice
depth is four times greater than a single dipole trap with the same laser power. The
resulting potential is modulated in space by half the laser’s wavelength, and has an
133
the intensity envelope of a Gaussian beam [104],
V (r, z) = 4V0 cos
2(kz)
w20
w2(z)
exp
[
− 2r
2
w2(z)
]
, (A.10)
where w(z) = w0
√
1 + (z/z0)2, and w0 and z0 are the minimum waist and Rayleigh
length, respectively. The potential of each lattice site can be approximated as
harmonic in both r and z, and the frequencies at the center of the lattice are [224]
fz =
2
λ
√
2V0
m
(A.11)
fr =
2
πw0
√
V0
m
. (A.12)
Note that the spatial confinement in a lattice can be quite narrow in the z-direction.
For instance, consider 1-µK 88Sr atoms that are in a 10-µK-deep lattice at 914 nm.
A rough estimate of the half width of the cloud of atoms is given by equating the
harmonic and thermal energies, mω2zz
2 = kBT , and for these parameters 2z ∼ 30 nm.
134
APPENDIX B
STRONTIUM DETAILS
This appendix presents some basic details about strontium which are used in
calculations and discussions within the body of the thesis. The energy levels which
are relevant to the trapping and cooling schemes we use are shown in Fig. B.1. A
strong transition at 461 nm provides primary cooling in the Zeeman slowing and blue
MOT stages, but the atoms have some probability to decay to the J = 1 and J = 2
fine structure states of the 5s 5p 3PJ manifold. Unfortunately, the J = 2 level is quite
long-lived, and atoms are lost to the blue cooling process if they are not pumped
out of that state. There are several schemes in which to do this [225], and we use a
turquoise beam at 497 nm to couple to the 5s 5d 3D2 state.
914 nm 
magic  
wavelength
5s 5d 3DJ
J = 3
J = 2,  g = 7/6
J = 1
J = 2,  g = 3/2
J = 1,  g = 3/2
J = 0
5s 5p 3PJ
5s 4d 1D2
5s 5p 1P1
497 nm 
2.3 MHz 
repump
689 nm 
7.6 kHz 
red MOT
461 nm 
32 MHz 
blue MOT
g = 1
5s2     1S0
FIGURE B.1. Relevant energy levels used in trapping and cooling of neutral 88Sr,
including transition wavelengths, linewidths, and Lande´ g-factors. Also included is
the 914 nm magic wavelength for the 5s2 1S0 −→ 5s 5p 3P1 intercombination line.
135
Isotope Abundance (%) Mass (amu) I F ′ ∆ν461 (MHz) ∆ν689 (MHz)
88Sr 82.58 87.905612 0 1 0 0
7/2 -9.7 1352.0
87Sr 9.86 85.909260 9/2 9/2 -68.9 221.7
11/2 -51.9 -1241.5
86Sr 7.00 86.908877 0 1 -124.8 -163.8
84Sr 0.56 83.913425 0 1 -270.8 —
TABLE B.1. Principal isotope data for strontium [226], including abundance, mass,
and isotopic shifts of the 5s2 1S0 −→ 5s 5p 1P1 and 5s2 1S0 −→ 5s 5p 3P1 transitions—
labeled ∆ν461 and ∆ν689, respectively. Shifts are presented relative to
88Sr, and are
taken from Ref. [227] for the 461 nm transition and Refs. [228] and [229] for the
689 nm intercombination line. These data are presented visually in Fig. B.3.
Since Sr is an alkaline earth element, it has triplet states where the two valence
electrons have aligned spins. The 3PJ manifold has two transitions to the ground state
which are doubly forbidden (they are forbidden both by total angular momentum
consideration and because S 6= S ′), offering extremely narrow linewidths that are
exploited in the strontium optical lattice clocks in 87Sr [19]. The semi-forbidden
J = 1 intercombination line used for secondary cooling is only 7.6-kHz wide, and can
cool atoms to sub-recoil temperatures [27].
The vapor pressure of Sr is significantly lower than that of alkali atoms, and is
plotted in torr in Fig. B.2 using empirical data tabulated by C.B. Alcock [143],
log10pv = 5.006 + 9.226−
8572
T
− 1.1926 log10T. (B.1)
There are four naturally occurring isotopes of strontium [226]. Some of their
properties are detailed in Table B.1, and frequency shifts of the 461-nm and 689-
nm transitions are shown in Fig. B.3
136
temperature (Celsius)
300 700400 500 600
v
ap
or
 p
re
ss
u
re
 (
to
rr
)
100
10-7
10-6
10-5
10-4
10-3
10-2
10-1
FIGURE B.2. Vapor pressure of strontium using empirical data from Ref. [143].
137
-400 100-300 -200 -100 0
46
1 
n
m
 a
b
so
rp
ti
on
0
84Sr 86Sr
87Sr
7/2
11/2
9/2
88Sr
detuning from 88Sr (MHz)
-1500 1500-1200 -900 -600 -300 0 300 600 900 1200
68
9 
n
m
 a
b
so
rp
ti
on
0
86Sr87Sr 11/2 87Sr 9/2 87Sr 7/2
88Sr
FIGURE B.3. Isotopic shifts of the blue MOT 5s2 1S0 −→ 5s 5p 1P1 and red MOT
5s2 1S0 −→ 5s 5p 3P1 transitions (top and bottom plots, respectively). Frequency
differences are presented relative to 88Sr, and to illustrate the absorption profile,
lineshapes are plotted as Lorentzians whose heights are weighted by the abundance
of each isotope. For the transition at 461 nm, a realistic curve (thick blue) uses
linewidths of 32 MHz, but a version with linewidths of 2 MHz is also included (thin
green) to more clearly show the individual isotopes. At 689 nm, meanwhile, the peaks
shown are much broader than the actual transition (10 MHz compared to 7.6 kHz)
for aesthetic appeal.
138
B.1. Dipole Matrix Elements
Calculations of quantities involving the Rabi frequency Ω(r) require the ability
to compute dipole matrix elements. In general a sum over several transitions is
necessary, and this Appendix lays some groundwork for performing calculations for
strontium. The Rabi frequency is given by the dipole interaction between the atom
and the electric field,
Ωq =
−〈F mF |erq|F ′ m′F 〉Eq
~
=
−〈J mJ |erq|J ′ m′J〉Eq
~
, (B.2)
where subscript q indicates the polarization of light, and the primed and unprimed
quantities are associated with the excited and ground states, respectively. The second
equality is true for 88Sr since it has no nuclear spin, and thus F = J + I = J. It is
helpful to rewrite the matrix element using the Wigner-Eckart theorem [39]:
〈J mJ |erq|J ′ m′J〉 = 〈J‖er‖J ′〉(−1)J
′−1+mJ
√
2J + 1

 J ′ 1 J
m′
J
q −mJ

 , (B.3)
where the object in parentheses is the the Wigner 3-j symbol. The result is a
factorization into a geometric term and a reduced matrix element which no longer
depends on the magnetic quantum number. Since the dipole does not interact with
the spin of the electron, we can further simplify by decomposing into a reduced matrix
139
element involving just the orbital quantum number L,
〈J‖er‖J ′〉 ≡ 〈L S J‖er‖L′ S ′ J ′〉
= 〈L‖er‖L′〉δSS′(−1)J ′+L+1+S
√
(2J ′ + 1)(2L+ 1)


L L′ 1
J ′ J S

 ,
(B.4)
where the object in brackets this time is the Wigner 6-j symbol. Dipole matrix
elements are computed using these reduced matrix elements and partial lifetimes of
fine structure transitions [230],
1
τJ′J
= ΓJ′J =
ω3
J′J
3πǫ0~c3
2J + 1
2J ′ + 1
|〈J‖er‖J ′〉|2. (B.5)
For strontium, available quantities are typically the overall lifetimes of excited state
fine structure manifolds decaying into lower state manifolds. This is a sum over all
possible decay paths,
1
τL′L
= ΓL′L =
1
2S ′ + 1
∑
J ′,J
ΓJ′J, (B.6)
where the factor of 2S ′ + 1 is necessary because considering all J ′ → J possibilities
essentially multiplies the L′ → L decay rate by the number of J ′ states in the excited
manifold. Plugging in the expression for 〈J‖er‖J ′〉 in Equation B.5, the known decay
rate is thus
ΓL′L =
|〈L‖er‖L′〉|2
3πǫ0~c3
1
2S ′ + 1
∑
J ′,J
ω3
J′J
(2J + 1)(2L+ 1)


L L′ 1
J ′ J S


2
. (B.7)
This makes calculating potentials for arbitrary states somewhat involved, since all
ωJ′J frequency splittings must be used to first find |〈L‖er‖L′〉|2 before determining
140
the individual ΓJ′J or |〈J‖er‖J ′〉|2.1 Such calculations are necessary to estimate
the expected magic wavelengths for the 5s2 1S0 −→ 5s 5p 3P0 clock and 5s2 1S0 −→
5s 5p 3P1 narrow-line cooling transitions [19, 89, 231], and to calculate the expected
Casimir–Polder shifts presented in Chapter II. Note that the normalization convention
used in this document is that of Brink and Satchler [232], which is different than what
is used in the referenced magic wavelength calculations. The two reduced matrix
elements are related by 〈L‖er‖L′〉 = √2L+ 1〈L‖er‖L′〉, with the matrix element on
the right-hand-side being from this document.
B.1.1. Example: Ground State Dipole Potential
In the special case of transitions connected to strontium’s ground state, the dipole
matrix elements can be found immediately from the excited state lifetimes. Dipole
interactions only couple the 5s2 1S0 ground state to higher-lying
1P1 singlet states,
meaning there are only single J ′ = 1→ J = 0 decay paths and therefore ΓJ′J = ΓL′L.
For linearly polarized light, the dipole matrix elements are then
|〈J mJ |er0|J ′ m′J〉|2 =
1
3
|〈J‖er‖J ′〉|2 (B.8)
= 3πǫ0~c
3ΓL′L
ω3
J′J
. (B.9)
As an example calculation, these can be used to find the dipole potential the atom
experiences when it’s in the ground state. If the trapping laser is tuned very far
from resonance, the RWA no longer applies. The dipole potential includes a counter-
1An alternative way to think about this [19, 89, 231] is combining the set of splittings ωJ′J and
branching ratios to find the “center of mass” frequency between fine structure manifolds ωL′L and
then applying the L analog of Eq. (B.5)
141
rotating term (the Bloch–Siegert shift [233]),
V (r) = −~Ω(r)
2
4
(
1
ωL + ω0
− 1
ωL − ω0
)
, (B.10)
where Ω(r) is the Rabi frequency, ω0 is the atomic resonance, and ωL is the laser
frequency. The total ground state shift due to a linearly polarized field E0 is thus a
sum over states satisfying Eq. (B.9):
V (r) = −3πǫ0c
3
4
E20
∑
k
Γk
ω3k
(
1
ωL + ωk
− 1
ωL − ωk
)
. (B.11)
Here ωk and Γk are the frequency splitting and decay rate from the kth excited state to
the ground state, and we can compute the sum using the data presented in Table B.2.
If we assume a Gaussian beam with waist parameter w0 and power P , the maximum
potential is
V (r) = − 3c
2P
w20
∑
k
Γk
ω3k
(
1
ωL + ωk
− 1
ωL − ωk
)
=
P
w20
· (5.0160× 10−37 [m2 · s]) , (B.12)
where we have assumed the laser to be operating at the 914 nm magic wavelength of
the 5s2 1S0 −→ 5s 5p 3P1 intercombination line.
B.2. Casimir–Polder Level Shifts
In the case of an atom near a perfectly conducting plane, the atom–field
interaction can be evaluated analytically to second order in perturbation theory. For
142
an arbitrary atomic level α the resulting CP shift is [39]
Vα =
∑
j
sgn(ωjα)|ωjα|3
4πǫ0πc3
[(
d2j,‖
2
− d2j,z
)
1
z′j
[
f(z′j)−Θ(ωαj)π cos z′j
]
−
(
d2j,‖
2
+ d2j,z
)[
1
z′2j
+
(
2
z′3j
− 1
z′j
)[
f(z′j)−Θ(ωαj)π cos z′j
]]
−
(
d2j,‖
2
+ d2j,z
)
2
z′2j
[
g(z′j)−Θ(ωαj)π sin z′j
] ]
. (B.13)
Here ωjα := ωj − ωα = −ωαj is the level splitting, sgn( ) is the signum function, and
Θ( ) is the Heaviside step function. Again, f and g are auxiliary function of the sine
and cosine integrals [Eq. (2.13)], and a rescaled position z′j = 2ωjαzj/c is also used.
When ωjα > 0 the level is coupled to higher-lying states, and the result is the same
as Eq. (2.12) if the atom is assumed to be spherically symmetric. When ωjα < 0,
however, the coupling is to lower-lying states and the resulting oscillatory potential
(in z) resembles that of a classical dipole [39].
The perpendicular and parallel dipole matrix elements are written compactly as
d2j,‖ and d
2
j,z, respectively, with
d2j,z = |〈α|zˆ · d|j〉|2 , and (B.14)
d2j,‖ = |〈α|xˆ · d|j〉|2 + |〈α|yˆ · d|j〉|2.
For fine structure states, the Wigner 3-j symbols in Eq. (B.3) can be evaluated, and
the dipole matrix elements are then written as mJ-dependent factors multiplying
reduced dipole matrix elements:
143
d2j,z = |〈J,mJ |er0|J ′, mJ〉|2
= |〈J‖er‖J ′〉|2 ×


1/3, J = 0→ J ′ = 1
1−m2
J
, J = 1→ J ′ = 0
m2
J
/2, J = 1→ J ′ = 1
(4−m2
J
)/10, J = 1→ J ′ = 2
(4−m2
J
)/6, J = 2→ J ′ = 1
m2
J
/6, J = 2→ J ′ = 2
(9−m2
J
)/21, J = 2→ J ′ = 3
, (B.15)
d2j,‖ = |〈J,mJ |er+1|J ′, mJ〉|2 + |〈J,mJ|er−1|J ′, mJ〉|2
= |〈J‖er‖J ′〉|2 ×


2/3, J = 0→ J ′ = 1
m2
J
, J = 1→ J ′ = 0
1−m2
J
/2, J = 1→ J ′ = 1
(6 +m2
J
)/10, J = 1→ J ′ = 2
(2 +m2
J
)/6, J = 2→ J ′ = 1
1−m2
J
/6, J = 2→ J ′ = 2
(2 +m2
J
)/21, J = 2→ J ′ = 3
. (B.16)
We use the various ΓJ′J from the data tables in Sec. B.3 to calculate the reduced
dipole matrix elements and evaluate the CP level shift for strontium’s ground state
and each of the magnetic sublevels in the 5s 5p 3PJ triplet state. In addition to the
tabulated transitions, we include the intercombination line when appropriate, where
ΓJ′J = 7.6 kHz. The results are plotted in Fig. B.4, and the the oscillatory behavior
of the 3P1 excited state is clearly visible for mJ = 1.
144
atom-wall separation (nm)
0 2000200 400 600 800 1000 1200 1400 1600 1800
le
v
el
 s
h
if
t 
(k
H
z)
5
-20
-15
-10
-5
0
3P0
3P1  mJ = 1
3P1  mJ = 0
3P2  mJ = 2
3P2  mJ = 1
3P2  mJ = 0
1S0
FIGURE B.4. Expected level shifts of strontium atoms near a perfect conductor,
evaluated for the 5s2 1S0 ground and 5s 5p
3PJ triplet states. Differences between the
excited- and ground-state shifts are plotted in Chapter II.
B.3. Tabulated Data
The following tables give spectroscopic data of various transitions connected to
strontium’s 5s2 1S0 ground and 5s 5p
3PJ triplet states. Wavelength and ΓL′L decay
rate data are taken from Ref. [231], which compiled data from several sources [198,
234–236]. We use Eq. (B.7) to compute the reduced dipole matrix elements
|〈L||er||L′〉|2 as well as partial decay rates ΓJ′J to enable more straightforward
calculation of expected shifts of individual fine-structure energy levels.
145
Exc. Orbital λ (nm) ΓL′L (Mrad/s) source |〈L||er||L′〉|2 (C2 ·m2) ΓL′L/2pi (MHz)
5s 5p 460.8618 190.01 expt [235] 1.9795×10−57 30.2410
5s 6p 293.2685 1.87 expt [198] 5.0201×10−60 0.2976
5s 7p 257.0238 5.32 expt [198] 9.6141×10−60 0.8467
5s 8p 242.8826 14.9 thry [236] 2.2722×10−59 2.3714
4d 5p 242.8100 12 thry [236] 1.8283×10−59 1.9099
5s 9p 235.5027 11.6 thry [236] 1.6126×10−59 1.8462
5s 10p 230.7980 7.6 thry [236] 9.9446×10−60 1.2096
5s 11p 227.5921 4.88 thry [236] 6.1230×10−60 0.7767
TABLE B.2. Reduced dipole matrix elements and total decay rates for transitions of
the form 1S0 −→1 P1, originating from the 5s2 ground state.
N J J ′ λ (nm) ΓL′L (Mrad/s) source |〈L||er||L′〉|2 (C2 ·m2) ΓJ′J/2pi (MHz)
0 1 679.2890 1.6263
6 1 1 688.0208 85 expt [94] 3.4090×10−58 4.6956
2 1 707.2020 7.2063
0 1 432.7657 0.2231
7 1 1 436.2933 12 expt [237] 1.2093×10−59 0.6532
2 1 386.6546 1.0335
0 1 378.1588 0.1519
8 1 1 380.8495 8.22 thry [236] 5.4922×10−60 0.4460
2 1 386.6546 0.7104
0 1 355.4459 0.0835
9 1 1 357.8221 4.53 thry [236] 2.5068×10−60 0.2455
2 1 362.9417 0.3920
0 1 343.5236 0.0510
10 1 1 345.7426 2.77 thry [236] 1.3818×10−60 0.1500
2 1 350.5201 0.2399
TABLE B.3. Reduced dipole matrix elements and partial decay rates for transitions
of the form 5s 5p 3PJ −→ 5sNs 3SJ′ (N = 6–10).
146
N J J ′ λ (nm) ΓL′L (Mrad/s) source |〈L||er||L′〉|2 (C2 ·m2) ΓJ′J/2pi (MHz)
0 0 – –
0 1 483.3459 6.64104
0 2 – –
1 0 474.3244 18.8283
5 1 1 478.5654 120 expt [237] 4.7394×10−58 4.8495
1 2 487.7674 5.0432
2 0 – –
2 1 472.3595 7.6337
2 2 481.3222 14.3001
TABLE B.4. Reduced dipole matrix elements and partial decay rates for transitions
of the form 5s 5p 3PJ −→ Np2 3PJ′.
N J J ′ λ (nm) ΓL′L (Mrad/s) source |〈L||er||L′〉|2 (C2 ·m2) ΓJ′J/2pi (MHz)
0 1 2603.1254 0.0441
0 2 – –
0 3 – –
1 1 2736.1990 0.0285
4 1 2 2692.1930 0.412 expt [238] 5.2050×10−58 0.0538
1 3 – –
2 1 3067.0208 0.0013
2 2 3011.8377 0.0128
2 3 2923.3748 0.0561
0 1 483.3388 5.6352
0 2 – –
0 3 – –
1 1 487.7432 4.1129
5 1 2 487.3849 61 expt [239] 4.2552×10−58 7.4196
1 3 – –
2 1 497.3051 0.2587
2 2 496.9326 2.3334
2 3 496.3643 9.3656
0 1 394.1915 2.2589
0 2 – –
0 3 – –
1 1 397.1161 1.6570
6 1 2 397.0383 24.62 expt [238] 9.2527×10−59 2.9843
1 3 – –
2 1 403.4318 0.1054
2 2 403.3514 0.9488
2 3 403.1512 3.8008
TABLE B.5. Reduced dipole matrix elements and partial decay rates for transitions
of the form 5s 5p 3PJ −→ 5sNd 3DJ′ (N = 4–6).
147
N J J ′ λ (nm) ΓL′L (Mrad/s) source |〈L||er||L′〉|2 (C2 ·m2) ΓJ′J/2pi (MHz)
0 1 363.0175 1.2993
0 2 – –
0 3 – –
1 1 365.4964 0.9548
7 1 2 365.4307 14.2 thry [236] 4.1568×10−59 1.7196
1 3 – –
2 1 370.8396 0.0609
2 2 370.7719 0.5488
2 3 370.6953 2.1965
0 1 347.8360 0.7777
0 2 – –
0 3 – –
1 1 350.1113 0.5720
8 1 2 350.0670 8.51 thry [236] 2.1887×10−59 1.0299
1 3 – –
2 1 355.0111 0.0366
2 2 354.9656 0.3293
2 3 354.9094 1.3178
0 1 339.0948 0.5032
0 2 – –
0 3 – –
1 1 341.2568 0.3702
9 1 2 341.2568 5.51 thry [236] 1.3122×10−59 0.6665
1 3 – –
2 1 345.9102 0.0237
2 2 345.9102 0.2133
2 3 345.8528 0.8537
TABLE B.6. Reduced dipole matrix elements and partial decay rates for transitions
of the form 5s 5p 3PJ −→ 5sNd 3DJ′ (N = 7–9).
148
APPENDIX C
DELAYED SELF-HETERODYNE METHOD
Optical frequency sources with narrow linewidths can be difficult to characterize.
There are a few simple methods which are either impractical or do not work very well
for narrow linewidths:
– An optical spectrum analyzer—a simple diffraction grating, for instance—
measures a broad range of optical frequencies. The resolution is typically no
better than a few GHz, and lasers with small δf/f end up looking like a δ-
function in frequency.
– It is possible to convert optical frequency fluctuations into a more reasonable
quantity by passing the laser light through a high-Q optical resonator. The
linewidth of the discriminating cavity, however, needs to be narrow relative
to the laser (which can be costly), and this measurement cannot distinguish
between noise on the laser and noise which perturbs the length of the resonator.
– Another approach is convert the laser spectrum from optical to radio frequency
domains through heterodyne spectroscopy. This allows the narrow linewidth
to be measured directly by high-resolution RF spectrum analyzers, but can
be cumbersome because it requires a second laser that has similar frequency
(different by at most a few GHz) and bandwidth as the target laser.
An alternative and more practical option is to beat the laser in question against
a time-delayed, frequency-shifted version of itself. This “delayed self-heterodyne”
technique is very similar to the final point above, except a narrow reference is not
required.
149
Since it is a fairly common trick [240–244], this Appendix is meant as an overview
to give intuition to well-established theoretical work [131, 132, 245]. First, a simple
model [133] is shown to give rise to the expected Lorentzian line shape due to random
phase noise. Next, the heterodyne and self-heterodyne techniques are discussed in the
case of purely white noise. Finally, 1/f noise is added and details are given regarding
how to extract a laser’s linewidth from the output of a photodetector.
C.1. Mathematical Framework
In this section we hold off on discussing how we measure a laser’s linewidth and
instead develop some mathematical tools to help us analyze the methods discussed
later. A laser’s linewidth can be attributed to many processes, like collisional
broadening and spontaneous emission, but we will use a simple model that lumps all
these together into a single, randomized phase. That is, assuming a monochromatic
plane wave, the laser’s electric field is given by
EL(t) = E0e
i[ω0t+φ(t)], (C.1)
where φ(t) is a stochastic, random variable. We assume no amplitude noise and
ignore shot noise throughout this analysis because they do not change the resulting
lineshape, which is our primary concern. We also will be using two-sided frequency
functions, taking the positive frequencies to be the relevant quantity [130].
The (two-sided) power spectrum of the laser’s electric field, according to the
Wiener-Khinchin theorem, is given by the Fourier transform of the autocorrelation
150
function
SE(ω) =
∫ +∞
−∞
CE(τ)e
−iωτdτ
=
∫ +∞
−∞
〈
E∗(t)E(t + τ)
〉
e−iωτdτ. (C.2)
Plugging in Eq. (C.1) and defining the quantity ∆φ(t, τ) := φ(t + τ) − φ(t), known
as the phase fluctuation or jitter, we find for the autocorrelation function
〈
E∗(t)E(t + τ)
〉
= eiω0τ
〈
ei∆φ(t,τ)
〉
= eiω0τe−
1
2
〈∆φ2(t,τ)〉. (C.3)
The simplification has been made by using the integral representation of the
expectation value, completing the square in the exponential, and using the fact
that ∆φ is a Gaussian random variable of mean value zero. The mean-square phase
fluctuation can be broken down into correlation functions of phase,
〈
∆φ2(t, τ)〉 = 〈φ2(t+ τ)〉+ 〈φ2(t)〉 − 〈φ(t+ τ)φ(t)〉 − 〈φ(t)φ(t+ τ)〉
= Cφ(0) + Cφ(0)− Cφ(−τ)− Cφ(τ), (C.4)
and we can again make use of the Wiener-Khinchin theorem—this time the inverse.
That is, the autocorrelation, Ci(τ), of a quantity i is the inverse Fourier transform of
its power spectral density, Si(ω). Thus the mean-square phase fluctuation is expressed
151
in terms of a phase-fluctuation spectrum,
〈
∆φ2(t, τ)〉 = 1
2π
∫ +∞
−∞
{[
Sφ(ω) + Sφ(ω)
]
eiω0 − Sφ(ω)(−ω)eiωτ − Sφ(ω)eiωτ
}
dω
=
1
2π
∫ +∞
−∞
Sφ(ω)
[
2− e−iωτ − eiωτ ]dω
=
1
π
∫ +∞
−∞
Sφ(ω)
[
1− cos(ωτ)
]
dω
=
2
π
∫ +∞
−∞
Sφ(ω) sin
2 (ωτ/2) dω. (C.5)
Now, ω = dφ/dt implies Sφ(ω) = Sω(ω)/ω
2 [130] (write Sω(ω) as a time-averaged
function of φ and integrate by parts), which allows us to work with frequency-
fluctuation spectra. In the case where Sω(ω) is independent of frequency—that is
the frequency jitter is a white-noise process—the result is
〈
∆φ2(t, τ)
〉
=
2Sω
π
∫ ∞
−∞
1
ω2
sin2 (ωτ/2)dω
=
Sωτ
2
2π
∫ ∞
−∞
(
sin(ωτ/2)
ωτ/2
)2
dω
= Sω|τ |
=: γ|τ |. (C.6)
We have used integral properties of the sinc function and noticed that
〈
∆φ2(t, τ)〉 ≥ 0.
After plugging into Eq. (C.3) and taking the Fourier transform, the resultant power
spectrum of the electric field is a Lorentzian distribution centered at ω0 with full
width at half maximum (FWHM) of γ:
SE(ω) =
E0
2
π
γ/2
(γ/2)2 + (ω − ω0)2 . (C.7)
152
γ γ
ω0ω0 − ωr ωr0
FIGURE C.1. Frequency spectrum of a laser with linewidth γ centered at frequency
ω0 (blue) beaten against a single-frequency plane wave (black) with frequency ωr.
The resulting beat note (red) is centered at ω0 − ωr and also has linewidth γ.
Note that we have adopted the definition Sω = γ rather than Sω = 2γ (as in [131, 245])
because we want γ to represent the FWHM laser linewidth.
So, assuming a white-noise process and modeling it using a stochastic phase,
we recover the expected result and generate some useful identities in the process. In
particular, we will again use the simplification in Eq. (C.3) and the result of Eq. (C.5).
C.2. Heterodyne Measurement
As a useful introduction, consider a special case of heterodyne spectroscopy. Here
the target laser is interfered with a perfect laser that has zero linewidth under this
model. That is, the reference laser is monochromatic and its power spectrum SE(ω)
is a delta function in frequency space. We expect the resulting beat signal to have
the same line shape as the target laser, but centered at the frequency difference of
the two lasers (see Fig. C.1).
153
If the two electric field amplitudes of the two lasers are related by a factor of α,
the combined field is
Etotal(t) = EL(t) + Eref(t)
= E0e
i[ω0t+φ(t)] + αE0e
iωrt. (C.8)
This field is measured on a photodetector, which produces a signal current
proportional to the square of the field. The resulting power spectrum is then the
Fourier transform of the current autocorrelation function CI(τ). Ignoring shot noise
and dropping constants of proportionality, the autocorrelation function is
CI(τ) = 〈I∗(t)I(t+ τ)〉
∝ 〈E∗(t)E(t)E∗(t+ τ)E(t + τ)〉
=
〈 (
E0e
−i(ω0t+φ(t)) + αE0e
−iωrt
) (
E0e
i(ω0t+φ(t)) + αE0e
iωrt
)×
(
E0e
−i(ω0(t+τ)+φ(t+τ)) + αE0e
−iωr(t+τ)
) (
E0e
i(ω0(t+τ)+φ(t+τ)) + αE0e
iωr(t+τ)
) 〉
.
(C.9)
Of the sixteen terms, many turn out to be zero under a time average, 〈eixt〉 = 0.
The result is four frequency-independent terms and two others, which happen to be
complex conjugates,
CI(τ) ∝ E04(α2 + 1)2 + E04α2
[
e−i(ω0−ωr)τ
〈
e−i∆φ(t,τ)
〉
+ ei(ω0−ωr)τ
〈
ei∆φ(t,τ)
〉]
= E0
4(α2 + 1)2 + 2E0
4α2e〈∆
2φ(t,τ)〉/2 cos [(ω0 − ωr)τ ] . (C.10)
We have again used ∆φ = φ(t + τ) − φ(t) and applied the identity from Eq. (C.3).
Defining Ω = ω0 − ωr, again assuming Sω(ω) = γ, and evaluating the Fourier
154
transform, the measured current spectrum becomes
SI(ω) ∝ 2πE04(α2 + 1)2δ(ω) + 2E0
4α2
π
[
γ/2
(γ/2)2 + (ω − Ω)2 +
γ/2
(γ/2)2 + (ω + Ω)2
]
.
(C.11)
This spectrum is simply a delta function at zero frequency and two Lorentzians,
centered at ±Ω. In principle, there should be a frequency-independent term that
comes from shot noise, but since that does not alter the shape of the power
spectrum, it can be ignored. Note that if two identical lasers are used in such a
heterodyne measurment, the derivation changes slightly, but the resulting spectrum
has a linewidth of 2γ, with each laser contributing equally to the total noise [39].
C.3. Delayed Self-Heterodyne Measurement
More practical than the heterodyne technique, the delayed self-heterodyne setup
requires only a single laser. As pictured in Fig. 3.9, a laser is interfered with a time-
delayed and frequency-shifted version of itself (the frequency shift makes the resultant
RF spectrum easier to measure by shifting it away from dc). This is easily achieved
by coupling the zeroth order of an AOM into a long fiber and combining the output
with the AOM’s first-order beam on a beamsplitter. Note that we can delay either
order of the AOM, which operates at a frequency Ω, but choose the zeroth for power
considerations in a lossy fiber. Without a delay, phase fluctuations in the two beams
are entirely correlated, and the measured power spectral density gives no information
about the laser’s linewidth. As the delay time td = Lfiber/cfiber increases, however, the
two beams become more independent, and the result converges toward that of the
heterodyne measurement.
155
The previous sections explicitly demonstrated the method used to calculate the
photocurrent’s power spectral density, but providing in-depth calculations for the
self-heterodyne technique is beyond the scope of this appendix. What follows will
outline the results presented in the literature [131, 132], with the details from the
previous pages adding clarity to the process (for a more detailed discussion—which
also uses the more experimentally relevant one-sided spectral density functions—see
the second chapter of Dan Steck’s Quantum Optics notes [39]).
The model for the electric field is the same as before, but now we combine the
laser with itself,
Etotal(t) = EL(t− td) + αEL(t)eiΩt (C.12)
= E0e
i[ω0(t−td)+φ(t−td)] + αE0e
i[(ω0+Ω)t+φ(t)], (C.13)
where α is again the ratio of amplitudes between the two fields and Ω is the frequency
difference (the operation frequency of the AOM). Again ignoring shot noise, the
autocorrelation function on the detector is
CI(τ) ∝ 〈E∗(t)E(t)E∗(t+ τ)E(t + τ)〉
= E0
4(α2 + 1)2 + 2E0
4α2 cos (Ωτ) exp
(
− [∆φ(t, τ) + ∆φ(t− td, τ)]
2
2
)
,
(C.14)
156
where again many terms have averaged to zero. The exponential term can be rewritten
in in a useful way [131] which allows the identity of Eq. (C.5) to be repeatedly applied:
− [∆φ(t, τ) + ∆φ(t + td, τ)]
2
2
= −∆φ2(τ)−∆φ2(td) + ∆φ
2(τ + td)
2
+
∆φ2(τ − td)
2
=
4
π
∫ ∞
−∞
Sω(ω)
ω2
sin2
(ωτ
2
)
sin2
(
ωtd
2
)
dω. (C.15)
Thus, after dropping terms with no frequency dependence, we obtain
SI(ω) ∝
∫ ∞
−∞
dτe−iωτ cos (Ωτ) exp
[
4
π
∫ ∞
−∞
Sω(ω
′)
ω′2
sin2
(
ω′τ
2
)
sin2
(
ω′td
2
)
dω′
]
.
(C.16)
as the resulting power spectral density of the photodiode current.
C.3.1. White Noise
In general, the Fourier transform must be performed numerically, but the case
of white noise Sω(ω
′) = γ has an analytical result [245]:
SI(ω) ∝
∫ ∞
−∞
dτe−iωτ cos (Ωτ) exp
(− γmin{|τ |, |td|})
= exp(−t˜)δ(γω˜) + 1
γ
1
1 + ω˜2
[
1− exp(−t˜)
(
1
ω˜
sin
(
ω˜t˜
)
+ cos
(
ω˜t˜
))]
. (C.17)
Here ω˜ = (ω−Ω)/γ is the frequency difference measured in linewidths, and t˜ = tdγ =
td/τc expresses the time delay in terms of the coherence time τc of the laser. The
spectrum consists of a delta function and a Lorentzian centered at Ω. The Lorentzian
is broadened by an oscillatory term, whose contribution is exponentially suppressed
with increasing delay times. The interpretation is that the delta function comes from
sampling when the two beams are still perfectly correlated, and the oscillatory piece
157
w~
-30 30-20 -10 0 10 20
S
I 
(w
)
10-3
10-2
10-1
100
td = 0.5tc
td = tc
td = 2tc
td = 3tc
td → ∞delay
0.5tc
1tc
2tc
3tc
 ∞ 
FWHM
11.61g
6.08g
3.39g
2.57g
2g
FIGURE C.2. Plot of analytical solutions of the photocurrent power spectrum in the
delayed self-heterodyne technique for Sω(ω) = γ (white noise), see Eq. (C.17). Spectra
are plotted for a variety of values of time delay, expressed in terms of the coherence
length of the laser, τc = 1/γ, and corresponding full width at half maximum (FWHM)
are given. Periodicity of the wiggles in the wings of the spectra are determined by
the length of the delay fiber.
represents residual coherence when the delay is not long compared to the coherence
time. The result is plotted (without the delta-function) in Fig. C.2 for several delay
times. For short delays, the broadening and scalloping of the Lorentzian is significant,
but as the length of the fiber delay is increased, the oscillations become shallower and
more closely spaced until they disappear entirely. As td →∞, the spectrum’s FWHM
linewidth converges to 2γ, which was the heterodyne result. From the perspective
of making a quick measurement in the lab, note that the HWHM gives an upper
bound of the white-noise parameter γ, regardless of the length of delay fiber (but also
notice how bad this estimate gets for delay times less than τc). It is also possible to
estimate the length of a (shorter) fiber using the periodic structure of the wings of the
spectrum. If the linewidth of a laser is quite narrow, however, eliminating coherent
behavior is impractical and the lineshape should be fit to determine γ. (For example,
158
if γ = 1 kHz, the coherence length Lc = cfiber/γ = 200 km, which will yield extremely
low output powers due to Rayleigh scattering if the laser does not operate at telecom
wavelengths).
C.3.2. Adding 1/f Noise
For a gas-medium source (such as a He–Ne laser), it is a good approximation to
include only white noise in modeling the laser’s linewidth. Semiconductor devices,
however, exhibit the 1/f or “flicker” noise inherent in all electronics, and we need to
include an additional term in the frequency spectral noise density [246]:
Sω(ω) = γ +
k
|ω| . (C.18)
The 1/f component dominates laser linewidth, especially on long timescales since
the noise diverges for small frequencies. Under the delayed self-heterodyne setup, we
plug in the frequency spectral density and evaluate the integral in the exponent of
Eq. (C.16) [132],
SI(ω) ∝
∫ ∞
−∞
dτe−iωτ cos (Ωτ) exp
(− γmin{|τ |, |td|})×[
|τ + td|−k(τ+td)2/2pi |τ − td|−k(τ−td)2/2pi |τ |kτ2/pi |td|kt2d /pi|
]
. (C.19)
At this point numerical methods must be used to compute the Fourier transform and
obtain the current power spectral density.
159
C.3.3. Laser Linewidth and Observation Time
Unfortunately, as td → ∞ the integrand in Eq. (C.19) diverges for large
τ . This means that in the heterodyne case, the 1/f -noise component causes the
laser to have infinite linewidth (which is quickly clear when trying to evaluate the
integral in Eq. (C.5) with ω2|ω| in the denominator). Because of divergent noise
at low frequencies, the laser’s free-running frequency is unbounded and can—over
long enough times—run away without returning to its initial value. This is why
stating a laser’s linewidth should always be accompanied with a specification of the
observation time over which it is monitored.
In the case of Eq. (C.19), experimental constraints already fix the divergence
in some sense. A spectrum analyzer’s resolution bandwidth (RBW) determines the
smallest frequency that can be resolved, therefore cutting off timescales above the
instrument’s acquisition time. As an example, a Gaussian RBW implies that the
total spectrum measured on the analyzer is the convolution of SI(ω) with a Gaussian
of FWHM δRBW, and Eq. (C.19) is then modified to be
SI(ω) ∝
∫ ∞
−∞
dτe−i(ω−Ω)τ exp
(− γmin{|τ |, |td|} − (δRBWτ)2/16 ln 2)×[
|τ + td|−k(τ+td)2/2pi |τ − td|−k(τ−td)2/2pi |τ |kτ2/pi |td|kt2d /pi
]
, (C.20)
where we have for convenience also expanded the cos(Ωτ) term and dropped the
mirror image of the spectral feature, centered at ω = −Ω (this is allowed if Ω ≫ γ
and the tail of one peak does not contribute to the other). This result is representative
of what is measured in the lab, and we can use Eq. (C.20) to characterize the γ and
k parameters of a laser [116].
160
Declaring the linewidth of the laser with 1/f noise is another matter, however.
Stating that the linewidth is infinite, while thought-provoking, is not very useful. The
laser’s unbounded frequency means that the time-averaged phase jitter 〈∆φ(τ, t)〉 6= 0
for finite times and the identity in Eq. (C.3) is not true (it should really involve
the variance, reading 〈e±ix〉 = e−Var(x)/2). Statistical averages involving ∆φ(t, τ)
are instead calculated over a specified time Tobs [247], which alters the integrand in
Eq. (C.5). Photocurrent power spectral densities can be calculated with these new
statistics, and the FWHM linewidth can in turn be calculated for known γ and k
parameters at arbitrary Tobs [39].
161
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