HEMISPHERICAL OPTICAL MICROCAVITY FOR CAVITY-QED STRONG
COUPLING
by
JUSTIN MICHIO HANNIGAN
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
December 2009
II
University of Oregon Graduate School
Confirmation of Approval and Acceptance of Dissertation prepared by:
Justin Hannigan
Title:
"Optical Microcavities for Cavity-QED Strong Coupling"
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:
Hailin Wang, Chairperson, Physics
Michael Raymer, Advisor, Physics
Jens Noeckel, Member, Physics
Richard Taylor, Member, Physics
Andrew Marcus, Outside Member, Chemistry
and Richard Linton, Vice President for Research and Graduate Studies/Dean of the Graduate
School for the University of Oregon.
December 12, 2009
Original approval signatures are on file with the Graduate School and the University of Oregon
Libraries.
© 2009 Justin Michio Hannigan
iii
IV
An Abstract of the Dissertation of
Justin Michio Hannigan for the degree of
in the Department of Physics to be taken
Doctor of Philosophy
November, 2009
Title: OPTICAL MICROCAVITIES FOR CAVITY-QED STRONG
COUPLING
Approved:
Professor M. G. Raymer
This thesis reports on progress made toward realizing strong cavity quantum
electrodynamics coupling in a novel micro-cavity operating close to the hemispherical
limit. Micro-cavities are ubiquitous wherever the aim is observing strong interactions
in the low-energy limit.
The cavity used in this work boasts a novel combination of properties. It
utilizes a curved mirror with radius in the range of 40 - 60 11m that exhibits high
reflectivity over a large solid angle and is capable of producing a diffraction limited
mode waist in the approach to the hemispherical limit. This small waist implies a
correspondingly small effective mode volume due to concentration of the field into a
small transverse distance.
The cavity assembled for this investigation possesses suitably low loss (suitably
low linewidth) to observe vacuum Rabi splitting under suitable conditions.
According to best estimates for the relevant system parameters, this system should
vbe capable of displaying strong coupling. The dipole coupling strength, cavity loss
and quantum dot dephasing rates are estimated to be, respectively, g = 35j..leV, K =
30j..leV, and I = 15j..leV.
A survey of two different distributed Bragg reflector (DBR) samples was carried
out. Four different probe lasers were used to measure transmission spectra for the
coupled cavity-QED system.
The system initially failed to display strong coupling due to the available lasers
being too far from the design wavelength of the spacer layer, corresponding to
a loss of field strength at the location of the quantum dots. Unfortunately, the
only available lasers capable of probing the design wavelength of the spacer layer
had technical problems that prevented us from obtaining clean spectra. Both a
Ti:Ab03 and a diode laser were used to measure transmission over the design
wavelength range.
The cavity used here has many promising features and should be capable of
displaying strong coupling. It is believed that with a laser system centered at the
design wavelength and possessing low enough linewidth and single-mode operation
across a wide wavelength range strong coupling should be observable in this system.
CURRICULUM VITAE
NAME OF AUTHOR: Justin Michio Hannigan
PLACE OF BIRTH: Hila, Hawaii
DATE OF BIRTH: July 18, 1977
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon
University of Hawaii, Hila
DEGREES AWARDED:
Doctor of Philosophy in Physics, 2009, University of Oregon
Bachelor of Science in Physics and Mathematics, 2000, University of
Oregon
AREAS OF SPECIAL INTEREST:
Quantum Optics
Quantum Information
Semiconductor Optics
PROFESSIONAL EXPERIENCE:
Graduate Research Assistant,
University of Oregon, 2000 - 2009
Graduate Teaching Assistant,
University of Oregon, 2006 - 2007
Physics and Mathematics Thtor,
Academic Learning Services,
University of Oregon, 1998 - 2000
PUBLICATIONS:
G. Cui, J. M. Hannigan, R. Loeckenhoff, F. M. Matinaga, M. G.
Raymer, S. Bhongale, M. Holland, S. Mosor, S. Chatterjee, H. M.
Gibbs, and G. Khitrova. A hemispherical, high-solid-angle optical
micro-cavity for cavity- QED studies. Opt. Express, 14(6):22892299,
2006.
VI
vii
ACKNOWLEDGEMENTS
First and foremost I would like to acknowledge my adviser, Michael Raymer.
He has seen my academic journey progress from the point when I first entered
. .
the University of Oregon as an undergraduate to the completion of my studies
as a graduate student. While I consider that many things could have gone better
during my graduate career, Mike has proven himself as a steadfast and dedicated
adviser. I have truly learned a lot from him, both as a physicist and as a human.
I consider myself fortunate to have been able to work on this project, despite the
lack of positive results. I have struggled and learned a lot during this time at the
University of Oregon.
I thank Hyatt Gibbs and Galina Khitrova for their work in designing, fabricating
and characterizing the semiconductor structures used in this work as well as for
providing surface characterization for the micro-mirrors that we produced. This
work would not have been possible without their hard work and expertise.
Murray Holland should also be acknowledged for providing early calculations
that provided estimates of mode volume for the microcavity modes as well as
helping secure funding for the project.
Of course, I must acknowledge my colleague of many years in this work,
Guoqiang Cui. Our time together working on this project led to a close relationship
viii
and I truly consider him to be one of my closest friends. Our trajectories have
diverged, but I am sure they will cross again in the future.
I must also acknowledge David Foster, my predecessor on this project. He is
responsible for many engineering details, including design of the tripod structure
and the vacuum chamber. His attention to detail provided a good example for me
when pursuing further design work.
It was also a great experience to work with Franklin Matinaga. He came to
work with us from Universidade Federal de Minas Gerais on two separate occasions.
His experience was of great help in obtaining good signal quality in the cavity-
modified photoluminescence experiments.
I must acknowledge the Technical Science Administration professionals who
have provided excellent support in many aspects of design and construction.
Thanks to David Senkovich, Kris Johnson, John Boosinger, Jeffrey Garman and
Cliff Dax for helpful guidance and support over the years.
Thanks also to my entire committee (Michael Raymer, HaHin Wang, Jens
Nackel, Richard Taylor and Andrew Marcus) for supporting me.
Thanks to Andrew Cook for sharing in adventures with me.
Finally, I must acknowledge my parents (Stephen and Jeanne Hannigan) for
encouraging curiosity and allowing intellectual freedom. I hope to provide such an
environment for my daughter, Lillian.
DEDICATION
ix
To my daughter, Lillian Haruko Hannigan-Thompson: I dedicate this work to you.
Believe in yourself and live your dreams. I love you.
TABLE OF CONTENTS
Chapter
I. INTRODUCTION
1.1 Introduction
1.2 Context . .
1.3 Motivation
1.4 Outline
II. THEORY ...
2.1 Overview
2.2 Theoretical Considerations for IFQDs
2.3 The Jaynes-Cummings Hamiltonian
2.4 Diagonalization - Normal Modes ..
2.5 Linewidth, Dissipation and Open Systems
2.6 Numerical Studies of Non-paraxial Modes
III. THE HEMISPHERICAL MICROCAVITY ..,
3.1 Flat Semiconductor DBR Containing QDs .
3.2 Large Solid-Angle Micro-mirrors ..
3.3 Mounted Micro-mirrors .
3.4 The Near-hemispherical Microcavity
3.5 Tripod Support System . . .
3.6 Cryostat and Cold-Finger ..
3.7 Ultra-high Vacuum Chamber
3.8 Summary . . . . . . . .
IV. EXPERIMENTS AND DATA.
4.1 Introduction
4.2 Input Beams
4.3 Output Beams
4.4 Mode Matching .
4.5 Cavity-Modified Photoluminescence
x
Page
1
1
1
4
12
15
15
15
24
55
70
93
100
100
108
115
119
122
129
130
132
135
135
135
138
141
145
Chapter
4.6 Using the Newport Laser as Probe . .
4.7 Using the New Focus Laser as Probe.
4.8 Sacher LiON as Probe. . . .
V. CONCLUSIONS AND OUTLOOK.
5.1 Conclusions .
5.2 Improvements to the Apparatus
5.3 Outlook .
APPENDICES
Xl
Page
146
152
154
160
160
165
166
A. TIME-DEPENDENCE OF EXPECTATION VALUES . . . . . . . . .. 168
B. QUANTIZATION OF A PARAXIAL RESONATOR.
B.1 Classical Field of the Resonator .
C. ACRONYMS . . . . . . . . . . . : . . .
177
177
194
BIBLIOGRAPHY ..... . . . . . . . . . . . . . . . . . . . . . . . . . . .. 199
Xll
LIST OF FIGURES
Figure
1.1 A QCPG scheme based on trion-tr'ion interactions.
Page
6
2.13 'Ifansmission spectrum ..
2.10 Energies of bare and dressed states
2.11 Rabi flopping . . . . . . .
2.12 Quantum dynamical map
9
17
18
20
22
23
26
37
61
63
65
69
73
92
94
Bulk GaAs band structure near r . .
Rotation from bare states into dressed states.
2-level system energies .
Selection rules for QDs in two bases.
Effect of interaction on energy levels; normal mode splitting
Selection rules - asymmetric QD .
2.8
2.9
2.3 Selection rules - axisymmetric QD .
2.1
2.2
2.5
2.6
1.2 Entangled photon pairs from a biexciton coupling to cavity.
2.4 Anisotropic exchange splitting ...
2.7 ElM field energy levels.
2.14 FDTD calculation of micro-cavity modes.
xiii
Figure
2.15 Gaussian to V-mode transition
Page
96
3.12 Flexure mount for X-Y translation
3.4 Micro-PL for CAT96 and CAT97
3.5 Nano-PL from CAT97 .
3.3 Reflection spectrum for CAT96 and CAT97
97
101
103
105
106
107
110
112
114
116
118
120
124
126
128
131
133
134
137
DBR surface roughness .
Micro-mirror substrate preparation
Micro-mirror sphericity and surface roughness
3.1
3.6
3.7 Procedure for removing glass chips with PDMS films
3.2 DBR reflectivity as a function of distance
2.16 Amplitude at y = 0 and polarization for z = 0 .
3.8
3.9 Micro-mirror T(O): Setup and data .
3.10 Twyman-Green interferometer for mounting micro-mirrors
3.11 Cavity stability parameters . . . .
3.13 Schematic of sample column ....
3.16 Schematic: ultra-high vacuum (DRV) system
3.15 Schematic of cooling system . . . . . . . . . .
3.14 Ridden line drawing of the support structure
3.17 Photograph of the support tripod and cooling system .
4.1 Laser sources filtered by single-mode fiber . . . . . . .
xiv
4.16 Sacher vs. New Focus spectra.
Figure
4.15 Sacher - double peaks ..
4.13 Sacher single peaks ....
Page
139
141
143
144
146
147
148
149
150
151
153
154
155
155
156
157
159
Detection configuration for passive cavity control
Detection setup for active control of cavity length
Lens system schematic .
4.2
4.3
4.4
4.8 Cavity-modified PL - long cavity
4.9 Polarized PL from intracavity IFQDs .
4.10 A missing peak .
4.5 Effect of the lens system
4.6 Cavity-modified PL - setup
4.7 Cavity-modified PL - short cavity
4.12 Doublets except on the fundamental
4.11 Narrow absorption tuned through modes
4.17 Length scans with Sacher laser at 765 nm
4.18 Length scans for fixed probe at various locations on the sample
4.14 Sacher - strange features.
LIST OF TABLES
xv
Table
3.3 Relevant degrees of freedom and their role in the experiment.
3.1
3.2
Properties of borosilicate microcaps .
Micro-cavity parameters . . . . . . .
Page
109
120
123
1CHAPTER I
INTRODUCTION
1.1 Introduction
This dissertation reports on the effectiveness of near-hemispherical microcavities
for achieving strong coupling between a single exciton bound to an interface
fluctuation quantum dot and a single photon occupying a single mode of the
electromagnetic field. The near hemispherical nature of the optical cavity provides
for small mode volumes and exhibits large solid angle, non-paraxial modes.
1.2 Context
This research takes place in the subfield of quantum optics referred to as cavity
quantum electrodynamics (CQED). Generally, CQED involves the use of optical
resonators (i.e. resonant cavities) to modify the nature of the interactions between
the electromagnetic field and a material system.
One of the first predictions of what is now commonly viewed as a CQED
effect (the Purcell effect) did not involve a cavity at all. It was a prediction made
regarding nuclear magnetic resonance (NMR) systems. In a 1946 abstract, E. M.
2Purcell predicted that nuclear spins coupled to a resonant electronic circuit would
experience an enhanced spontaneous emission rate [52]. The emitter in Purcell's
scheme was a nuclear spin in a magnetic field while in contemporary CQED it is
quite often an atom. Of course, a two-level system may be used as an idealized
model for both a simple atom with one valence electron and a nuclear spin in a DC
magnetic field. Consequently, we naturally find many common features in NMR
and quantum optics.
Much later, in 1981, inhibited spontaneous emission was also predicted [36].
Both of these phenomena (inhibition and enhancement of spontaneous emission)
occur in a regime well described by perturbation theory. Both correspond to a
general modification of the emission rate due to modification of the density of
available output states and henceforth I will refer to both phenomena as simply
the Purcell effect, in accordance with common parlance. the Purcell effect can be
understood as a straightforward application of Fermi's golden rule. The Purcell
effect was observed first in superconducting (niobium) microwave cavities [25].
Other experiments with a variety of atoms and cavity configurations followed
[32, 29, 43, 30, 46].
The Purcell effect arises due to an incoherent interaction between the electric
field inside of the cavity and the optical emitter. Changes in lifetime or energy
shifts [30] may be observed in this regime but not phase dependent phenomena. In
3order to observe quantum interference the dynamics must enter a regime of strong
coupling in which the two systems evolve in time coherently.
Roughly, the strong coupling regime corresponds to situations in which the
intracavity mode coupling to the material emitter is greater than any other
(environmental) interactions. A consequence of this is that the differential equations
for the time evolution of the system exhibit complex eigenvalues and the interacting
systems behave analogously to classical, coupled oscillators: The time evolution
can concisely be described in terms of normal-modes, which are each marked by a
shared frequency.
The rationale for using microcavities in CQED becomes obvious when the
relationship between electromagnetic energy and volume is considered:
1 J ( 2 1 2)[; = '2 dr foE + /-10 B .
v
If energy is held constant, then fields of arbitrary intensity (squared modulus)
can be produced by confining the energy to a small enough volume (i.e. to keep
the value of the integral constant while integrating over a tiny volume the values
of E 2 and B 2 have to increase by an appropriate amount over that volume). This
explains the ubiquity of microcavities or even nanocavities in experiments that
seek to observe strong coupling. In order to observe a coherent coupling between
interacting systems it is necessary to make the interaction channel between the
4two systems stronger than any other interactions with external degrees of freedom
(i.e. losses). The interaction may be made stronger by confining the field to a small
volume.
. 1.3 Motivation
This work shares many of the same motivations as the atomic CQED experiments.
Namely, that strong coupling in CQED leads to inter-system coherence and
entanglement and this can be used as a resource in the field of quantum information.
Two obvious connections are to quantum computing and quantum key distribution.
A pair of quantum dots coupled to a microcavity has been shown to be a
potential physical system for instantiation of a quantum logic gate [14, 48]. In
this scheme, dopants provide for excess electrons or holes (depending on whether
the dopant is n-type or p-type). These excess charge carriers end up confined to
the quantum dots and the ground state of the dots will be a degenerate pair of
states corresponding to the two components of spin along a particular axis. A qubit
encoded in the spin of this trapped particle will enjoy much longer coherence times
than those based on properties of excitons.
This spin may be used as a qubit. Quantum conditional logic may be obtained
via dynamics that involve charged exciton (trion) states. For the case of an n-
doped sample, in which the trapped spin belongs to an electron, the transition
to a trion state is allowed only for the case that the second electron's spin is
------------------~~
opposite to the first electron's spin state -~ otherwise the Pauli exclusion principle
prohibits the transition. This mechanism provides for a way to build a (two qubit)
quantum controlled phase gate, an important building block in quantum computer
architecture.
If neighboring quantum dots are brought into the trion state the exchange
interaction will dominate the unitary time evolution and there will be a collective
phase accumulation for the two dots. Only one particular input state will be
excited to a trion-trion state by a given (circularly polarized) 1r- pulse. This
produces a conditional phase shift controlled by the logical values of the input
qubits.
5
10) 11)
-~±=F
F1~_ Ft
10)
It-
10) 11)
l±=t- 11 F
FtJ l
11) 11)
.1t=t ~tf-
.j=--=t _Fl
e i ¢/I) e i ¢ll)
11 F 1t=t
J lJ l
6
FIGURE 1.1: A QCPG scheme based on trion-trion interactions [48]. A pair of quantum
dots is depicted on each side of the dotted line. The pair on the left side is in an input
state 101) while the pair on the right side is in the input state 111). A 7f-pulse of right-
hand circularly polarized light is applied in both cases. The quantum dot whose spin
initially corresponds to a logical value of 10) cannot be excited to a trion state because of
Pauli blocking. Each of the other dots allows the transition to a IX-) trion state. The
pair of quantum dots on the right hand side now accumulates phase due to an exciton-
exciton exchange interaction. Finally, another 7f-pulse is used to bring the quantum dots
back to the initial state. The QCPG has mapped 111) :---+ ei ¢ 111), while mapping any
other input state back to itself.
7After unitary time evolution to build up phase on the pair of trions, another
pulse is used to return both dots to the ground state. On an appropriate time scale,
the phase due to unitary time evolution will be small on each input qubit state
except for 111). In the limit that this phase accumulation is negligible, it may be
seen that the entire process, beginning and ending with the quantum dots in their
ground state, may be expressed in the ordered basis {lOa), 101),110), Ill)} of qubits
as
1 a a a
a 1 a a
QCPG=
a a 1 a
The technique of molecular-beam epitaxy will enable one to grow two quantum
wells (QWs) with a several-nanometer separation, with a large enough barrier
potential to prevent electron tunneling, where interface fluctuation quantum dots
(IFQDs) formed in each QW are each doped with an excess electronl .
Another important application of such strongly coupled cavity-QD systems is
the deterministic generation of single-photons [38, 35, 45, 13] or of polarization-
entangled photon pairs on demand [55].
The cavity design should also lend itself to applications in atomic quantum
optics [42] as well as semiconductor optics [33] and seems ideally suited to test
IThe scheme, as presented, relies on circularly polarized optical transitions, so use of IFQDs
would necessitate the use of a strong magnetic field to satisfy this.
certain recent predictions about non-paraxial modes [19, 18] that will be discussed
in the next chapter.
The scheme by Stace, Milburn and Barnes utilizes decay from the biexciton
state of an asymmetric quantum dot [55], such as an IFQD. Whereas the two
photon cascade emitted from the biexciton state of asymmetric quantum dots
(QDs) is generally entangled in both frequency and polarization, a properly
designed cavity acts to erase the "which path" information and causes the pair of
emitted photons to be entangled in polarization only. This is advantageous, from
an experimental point of view, because polarizations can be manipulated with
linear optics.
The cavity also acts to collect the biexciton emission into a pair of adjacent,
longitudinal modes with the same transverse, spatial profile. Thus, the biexciton
emission may be collected (primarily) into a single spatial mode, governed by
the geometry of the optical cavity. This provides much higher efficiency than
attempting to collect light emitted into free space.
The achievement of either a deterministic single-photon source [38, 35, 45, 11]
or a source of entangled photons would help to enable an optical scheme for
performing quantum computations - linear optical quantum computing (LOQC)
[37J. In turn, a single-photon source is a valuable resource for schemes such as
Bennett and Brassard, 1984 (BB84), which rely on a source of single photons.
8
9W3
Ivac}----''---_'---__-'--_
(a)
0)4 0)2
FSR
Frequency
(b)
I~
" J,~
0)1 0)3
FIGURE 1.2: Entangled photon pairs from a biexciton coupling to cavity. This diagram
depicts the proposed setup in [55]. The energy levels of the ground state biexciton
and the ground state excitons in an asymmetric quantum dot are shown in (a). The
frequencies of quantum dot transitions are shown in blue. The frequencies of two different
cavity modes separated by one FSR are shown in red. The anisotropy splitting is labeled
by 1:1 and is on the order of tens of f.leV, while the biexciton binding energy (labeled
by E)is on the order of several meV. The biexciton emission spectrum is shown in (b)
in blue. The red trace shows the location of the cavity modes with respect to the QD
transitions.
The ability to produce entangled photon pairs enables techniques such as
quantum teleportation [60] or entanglement swapping [47]. The ability to supply
entangled photon pairs rather than single photons is also known to reduce the
resource requirements in LOQC.
Such sources have wide applications in the emerging field of quantum information
science [4]. This is particularly true for quantum cryptography, in which an
essential element of secure quantum key distribution (QKD) is an optical source
emitting a train of pulses that contain one and only one photon [2]. For example,
a source having zero probability for generating two or more photons in a pulse
and greater than 20% probability of generating one photon would lead to a great
advance in QKD in daylight through the atmosphere [31, 54, 49].
10
Solid-state emitters have certain distinguishing advantages over atoms and
molecules. In particular, their lack of mobility removes the requirement for robust
trapping and cooling. The population of emitters in a particular solid sample has
to do with structural peculiarities of the crystal (as opposed to what one might
call structural peculiarities of free space). Due to the rigid lattice of a crystal, the
structural defects have well-defined positions.
Accordingly, when dealing with solid state emitters the number of emitters in a
particular volume of the crystal is fixed 2. In contrast, an atomic beam intersecting
a cavity will typically exhibit shot noise. It is exceedingly difficult to keep one and
only one atom at a well-defined position inside of a micro- cavity. Using structural
.defects such as QDs takes care of this problem, though at the cost of well-defined,
consistent properties. Delivering single atoms to a microcavity requires exacting
technical expertise. From an experimental standpoint, absorbing heat from a
crystal into a cold finger is trivial by comparison.
The emitters of choice for this work are interface fluctuation quantum dots
(IFQDs). They make a good choice for CQED experiments due to very large dipole
moments [5, 44]. In fact two of three first indications of single quantum dot (SQD)
CQED strong coupling [61, 53, 50] involved IFQDs; [53] employed self-assembled
InAs QDs.
2This assumes stable operating conditions. The number of available bound states depends
on a particle's energy compared to the binding energy of a given potential. Factors that affect
either of these two quantities will therefore have an effect on the number of transitions that can be
observed. If the particle's energy is high enough no bound states are accessible and there will be
no observable emitter.
11
The primary difference in this work, compared with those just cited, is the
type of cavity. While successful efforts have used microdiscs, micropillars and
photonic crystal nanocavities, this work combines an IFQD with an external,
near-hemispherical cavity. External cavities provide certain advantages, such as
the ability to scan the waist of the cavity mode in search of suitable emitters.
Independent detuning of cavity frequency compared to emitter frequency is also
possible, in contrast to monolithic structures where temperature tuning is often
the only technique available. Temperature tuning can be undesirable because in
addition to thermal expansion, temperature also determines the population of
phonons present, a chief source of dissipation and dephasing in solid-state systems.
The external cavity allows a given emitter to be positioned at a region of
maximum field strength while also providing for independent tuning of cavity
frequency to match the emitter's transition frequency. This also provides for clearer
examination of temperature dependence of the coupled system since the detuning
may be held fixed while tuning the temperature.
The near-hemispherical geometry of the cavity provides for a very small (close
to the diffraction limit) mode waist. This means a small mode volume for a given
cavity length. A mirror with a radius of curvature of 50 !-lm should yield a mode
volume of less than about 50 1-lID3 - a rough estimate based on an approximate
spot size at the flat mirror multiplied by the length of the optical cavity.
12
As the cavity is shortened, the radius of the mode waist increases and the
divergence angle decreases. Eventually, the paraxial limit is reached and the mode
may be treated as a Gaussian beam. In this situation we would expect a mode
volume of %w5 .L, as calculated in appendix B. The non-paraxial nature of the
modes under the near-hemispherical conditions) investigated here defies a treatment
in terms of Gaussian beams.
The external microcavity system we have built also provides a nearly ideal
environment for assessing certain predictions like mixing of nearly degenerate
modes as predicted in [19]. Mode mixing is attended by non-simple polarization
distribution - at times characterized by mixtures of circular and linear polarization
at different transverse locations within a given mode.
1.4 Outline
Here, a brief sketch of the structure of the succeeding chapters is provided.
Chapter 2 will cover theoretical considerations for this experiment. First, a
description of the IFQDs is provided, along with a justification for treating them as
a two-level system. Following this, the two levels of interest will be used to express
a Hamiltonian in terms of pseudo-spin operators.
The quantization of the field follows this. Due to the absence of analytic
solutions for the near-hemispherical cavity's non-paraxial modes, the particular
form of the quantized field will not be treated explicitly. A quantization of the
13
paraxial Gaussian fundamental mode is carried out in appendix B. This provides
guidance for the relationship between the classical fields and the quantum operators
invoked in the theoretical analysis and helps make clear the relationship between
the parameter ~ff that appears in the quantum field operators and the amplitude
of the field at the position of interest.
A treatment of the dipole interaction follows this and the pieces are assembled
into the familiar Jaynes-Cummings Hamiltonian.
The remainder of the chapter involves methods for including dissipation via
interactions with unmeasurable reservoirs and coupling to an input and output
mode so that a steady-state transmission spectrum can be calculated. The chapter
ends with some numerical results pertaining to non-paraxial mode properties for
cavities with configurations similar to ours. These results come from both Murray
Holland's group at University of Colorado and Jens Nockel's group here at the
University of Oregon.
Chapter 3 will discuss the design and construction of the microcavity system.
It will account for the equipment and designs used and remark about suitability
for achieving the stated goals. It will briefly describe the processing of curved
micromirrors and the basic design of the semiconductor DBRs used to form the
cavity boundaries. This work was done in collaboration with Guoqiang Cui in the
Raymer Lab and Hyatt Gibbs and Galina Khitrova at the University of Arizona,
and was published in [12].
14
Chapter 4 will discuss the experimental work. It begins with a discussion of the
observed modes and steps taken to mode-match into the desired mode.
The next topic is intra-cavity photoluminescence (PL) of the dots, to observe
the presence of IFQDs and determine the orientation of the crystal axes by
measuring polarized PL.
Finally, an array of transmission measurements under various conditions will
be presented. Data was taken for two different DBR samples. Each sample was
measured at various transverse positions, various cavity lengths and with various
ranges of probe wavelength (covered by three different probe lasers).
Chapter 5 will summarize the results of the experiment and assess possible
improvements to this work. Ideas for future research and directions will also be
discussed.
,....--------------------
15
CHAPTER II
THEORY
2.1 Overview
I
This chapter's goals are to provide a theoretical framework for analyzing the
data obtained in the experiment. At first, some theory about the quantum dots
used in this work will be presented. Later sections will provide a theory for the
interaction of a two-level system with a single mode of the electromagnetic field.
In particular, the Hamiltonian for an analog closed system is developed. Next,
the Hamiltonian will be diagonalized (i.e. transformed to a dressed-state basis)
to provide for basic predictions of vacuum Rabi splitting. Finally, coupling to
electromagnetic and vibrational reservoirs will be included to provide a more
realistic account of the expected dynamics.
2.2 Theoretical Considerations for IFQDs
The purpose of this section is to provide a theoretical description of the emitters
to be used in this work. The emitter used here will be described as a two-level
system. The two energy levels correspond to vacuum and a ground-state exciton
16
in the exciton basis. To justify this description, a brief account of the properties of
these dots will be provided. The emitters of interest in this work are IFQDs. They
are shallow (weak lateral confinement), anisotropic GaAs/AlxGal_xAs quantum
dots that have large lateral extent and thus exhibit a large dipole moment, making
them attractive emitters for CQED strong coupling.
2.2.1 Confinement and Light Holes
Quantum dots are spatially ,localized regions of semiconductor with lower
energy than their surroundings. They are characterized by confinement in all three
dimensions and because of this they may be regarded as quasi-zero-dimensional
systems.
Excitations in undoped semiconductor systems result from the promotion of an
electron in a valence band to the conduction band. In the neighborhood of the r-
point, where Ikl = 0, there are two degenerate valence bands in bulk GaAs. The
two valence bands have band edges with different curvatures, leading to different
effective mass for the holes left by promoting an electron to the conduction band.
Holes in the higher-curvature band have a larger smaller effective mass and are
called heavy holes, while those in the lower-curvature band have smaller effective
mass and are called light holes.
In confined systems, such as quantum wells and QDs, the energy of the light
hole is much higher than the heavy hole and it becomes decoupled. As mentioned
in [23], for well widths narrower than around 12 nm the light-hole exciton overlaps
17
E
CB
k
HH Band
LH Band
FIGURE 2.1: Bulk GaAs band structure near r. The two degenerate valence bands are
shown for bulk GaAs. Another split-off band is not shown because it occurs at a lower
energy; adapted from [15].
the continuum states of the heavy-hole exciton. Also, calculations of lateral barrier
height for narrow GaAs IFQDs are roughly of the order of 16 meV [58], whereas
the light-hole exciton is generally tens of meV higher in energy than the heavy-
hole exciton l . This means that for the IFQDs used in this work, which occur in
QWs with widths of 3.86 nm, we can consider the exciton ground states as being
almost purely bound states of electrons and heavy holes. It suggests that mixing
between heavy-holes and light holes only occurs for continuum states of heavy holes
(i.e. holes that are no longer bound by the shallow lateral potential due to the
monolayer fluctuations). Therefore, we assume that the exciton ground states are
IThe barrier height is actually smaller than the thermal energy at room temperature, which is
about 26 meV. This means that at room temperature excitons have higher energy than the lateral
potential and there are effectively no quantum dots.
18
I-~) I~) 1-2) 1-1) 11) 12)
--.- --.-,
, /
0"+ ,/ 0"- (J_ (J+/ ,
/ ,
I-~) / , I~) Ivac)
(a) The electron-hole basis (b) The exciton basis
FIGURE 2.2: Selection rules for QDs in two bases. Selection rules show dipole-
allowed transitions in a description involving electrons and heavy holes (a) or
excitons (b). The solid arrows indicate dipole-allowed transitions.
bound states of a heavy hole and an electron. There are four possible such bound
states and there exists an underlying four-fold degeneracy [1, 51, 40].
2.2.2 Selection Rules and the Exchange Interaction
The four possible pairings of the electron with the heavy hole may be written
as 1-3/2}h l-l/2}e' 1-3/2}h l+l/2}e' 1+3/2h l-l/2}e and 1+3/2}h l+l/2}e in the
electron-hole basis.
Alternatively, it is possible to label these combinations in terms of total z-
component of spin and talk about excitons rather than electrons and holes. In this
case, the same four bound states may be labeled, respectively, 1-2}, I-I}, I+l} and 1+2}.
The /+1} and I-I} states are referred to as the spin-up and spin-down bright
excitons. In turn, 1+2} and 1-2} will be the spin-up and spin-down dark excitons.
One of the two standard approaches for dealing with the exchange interaction is
the Ivchenko-Pikus representation in which the state of the hole IJz = ±3/2} is
represented by a pseudospin, ISh,z = =fl/2}.
19
The Hamiltonian for the electron-hole exchange interaction may then be
parametrized in terms of the observed energy splittings of an anisotropic quantum
dot, as in [21]:
(2.1)
where 00 is the energy difference between the dark and bright excitons, Ob is the
energy splitting of the bright exciton due to the long range exchange interaction,
and Od is the energy splitting of the dark exciton states. The matrix in the ordered
basis of 11), 1-1), 12) and 1-2) is:
00 Ob 0 0
~ 1 Ob 00 0 0[Hex] = 2" (2.2)
0 0 -00 Od
0 0 Od -00
This immediately indicates the decoupling of the 1±1) excitons from the
1±2) excitons. The eigenstates are again determined by finding the vectors that
diagonalize the Hamiltonian.
20
Symmetric Quantum Dots
In spherical (or axisymmetric) dots there is rotational invariance in 2 (or 1)
angles. The azimuthal invariance of the Hamiltonian implies a conserved quantity.
Noether's theorem implies a conservation of angular momentum in this c~e. The
selection rules for such dots are depicted in figure 2.3.
~
...1........
1+,-)
Eexc 0"+ 0"_
1 1-) 1+)
r 0"+Eexc 0"_
Ivac)
FIGURE 2.3: Selection rules - axisymmetric QD. The dipole-allowed selection rules for
an axisymmetric QD are shown
The degeneracy between the dark and bright excitons is lifted by the short-
range exchange interaction. The long-range interaction is negligible due to
transverse symmetry and 6b ~ O. The bright excitons, thus occupy a degenerate
doublet. The exciton states are eigenstates of total angular momentum and the
angular momentum projection along the z- axis. The 1+1) and 1-1) excitons
are generated by absorption of a left-hand circular polarized photon and a right-
hand circular polarized photon, respectively. They must, respectively, involve an
invocation of angular momentum ladder operators applied to the vacuum state.
21
More concretely, as stated in [2:1.], the Bloch functions in a quantum well have
symmetries like this:
1+1) = (IX + iY)h j)(ls)e 1)
1-1) = (IX - iYh l)(ls)e j).
This represents the bright excitons as bound states of two spin-l/2 particles
aligned anti-parallel to each other. The excess angular momentum appears
as an orbital angular momentum portion of the heavy-hole pseudospin in this
representation. The heavy-hole kets in the above expressions are immediately
reminiscent of the form of angular momentum ladder operators and also of
circularly polarized light [34].
Asymmetric Quantum Dots
(2.3)
(2.4)
For asymmetric (anisotropic) QDs (such as IFQDs) the rotational invariance
is destroyed and angular momentum is no longer a good quantum number. The
new selection rules will be described briefly. The long-range exchange interaction
acquires a finite effect and leads to eigenstates which are totally symmetric and
anti-symmetric combinations of the angular momentum eigenstates.
No Interaction
11>,1-1>
11> - 1-1>
Exchange Interaction
"""-- 11>+ 1-1>
22
FIGURE 2.4: Anisotropic exchange splitting. This figure shows the level diagram with
and without the exchange interaction for spin-l excitons.
Working with the 1±1) manifold diagonalize the matrix in the usual way. In this
situation (h is nonzero.
::::} >.2 - 2>'80+ (85 - 8D = 0
::::} >. = 280 ± J485- 4 (85 - 8n
2
ei¢
The eigenvectors corresponding to >. = 80± 81 are Iv) = 2 (11) =f 1-1)).
Taking the positive and negative combinations of (2.3) and (2.4) provides for
ei ¢
IX) = 2 (11) + 1-1))
ei ¢
IY) = -i- (1 1) -1-1))
2
The directions X and Y tend to correspond with the [110] and [110], respectively
for GaAs IFQDs. The excitons are dipole active along these directions.
---------_. ----------
23
/vac)
IXX=YY)
X)
- ......... ... ...................
i ~~ ~~
exc 1ry 1rx
·~~l ... IX) l' l'· . I·.. t.......... j 1,' •••• j~
exc
1ry 1rx
,r ,r
E
E
FIGURE 2.5: Selection rules - asymmetric QD. This figure shows the linear polarized
optical transitions connecting the excitons to the biexciton state and the vacuum state.
The bright-exciton splitting is shown here as 6b• The exciton binding energy is labeled as
~.
Taking the above together with (2.3) and (2.4) it is evident that the spin states
of the electron are mixed together and also that the polarization of the absorbed
or emitted photons must be linear. Experimentally observed splittings are well
described by this theory, as shown in [22, 39, 24].
So, due to the confinement and shallowness of these dots, the excitons may be
thought of as combinations of only the heavy hole and the electron. Short-range
exchange interactions split the possible exciton states into dark states and bright
states. Due to anisotropy, the degeneracy of the bright doublet is lifted and the
selection rules for these transitions require linear polarization of the light field.
24
Going forward from here, a simple two-level system with dipole moment
pointing along the [110] direction of the crystal will be used to model the pair
of states corresponding to the vacuum state and the ground-state exciton of an
IFQD. Because the experiment seeks to operate in the single excitation limit, it
is not necessary to distinguish carefully between whether the excitons have bosonic,
fermionic or mixed characteristics: Differences between fermions and excitons only
become evident when multiple excitations are confronted and effects like exclusion
or bunching have a chance to manifest.
2.3 The Jaynes-Cummings Hamiltonian
In this section, the basic Jaynes-Cummings Hamiltonian is developed. This
system consists of a single, quantized mode of the electromagnetic field coupled
via electric dipole interaction to a single two-level system. The Hamiltonians for
the two non-interacting systems will be developed followed by the interaction
Hamiltonian. Finally, the Hamiltonian will be written in a simplified form.
2.3.1 Two-level Hamiltonian
The Hamiltonian for the two energy levels will be expressed in terms of pseudo-
spin operators (Pauli matrices).
Consider a two-level system with energies Ea and Eb. Let the difference of the
two energies be given by
(2.5)
25
The Hilbert space is spanned by two states labeled by la) and Ib). In this
case, the lower level Ib) corresponds to the vacuum state of an IFQD and the la)
corresponds to the ground state exciton. We may calculate the matrix elements of
the 2-level Hamiltonian by exploiting the two energy eigenstates,
1t2-1evel la) = Ea la)
1t2-1evellb) = Eb Ib) .
This allows us to write down the atomic Hamiltonian by calculating its matrix
elements. Meanwhile, completeness of our basis ensures that
la) (al + Ib) (bl = 1.
This expression for the identity permits the following description of the 2-level
Hamiltonian. Inserting two complete sets we obtain:
'H2-1evel = L li)(il 1t2-level Ij)(jl
i,jE{a,b}
= L li)(jl (il 1t2-level jj)
i,jE{a,b}
8ij
,.-"-..
= L li)(jl (i Ij) Ej
i,jE{a,b}
= L li)(il Ei ·
iE{a,b}
(2.6)
(2.7)
(2.8)
(2.9)
26
The Hamiltonian may then be written explicitly in the basis of energy eigenstates
as:
H2-1evel = Ea la) (al + Eb Ib) (bl· (2.10)
This is a completely general Hamiltonian for a two-level system. It merely consists.
of projection operators into the two possible states with the corresponding energies
as coefficients.
a) This general two-level system is mathematically
equivalent to a non-relativistic spin-1/2 particle
(i.e. the dynamics may be formulated in terms
of Pauli pseudospin operators). As an aside: A
concrete mapping to the Pauli matrices is obtained
b) by choosing a particular map of our state vectors
into an orthogonal pair of column vectors. This
given energy difference
FIGURE 2.6: 2-level system map generates a set of 2 x 2 matrices which are
energies. This shows the two identically the Pauli matrices. For the moment,
energy levels in question with it is not necessary to choose such a map, so the
matter will be left for later. Suffice it to say that the
dynamics may suitably be described by operators in the group SU(2).
I
I
!
27
Without loss of generality, we may define a set of operators by taking outer
products of our basis states:
s+ - la) (bl
s_ -Ib) (al
az - la) (al - Ib) (bl·
The two-level Hamiltonian can be simplified significantly from (2.10). By
applying an appropriate additive identity, the above may be cast into a form
consistent with the Pauli description.
(2.11)
(2.12)
(2.13)
H2-1evel = Ea la)(al + Eb Ib)(bl
1 1
= 2(Ea la)(a/ + Eb Ib)(bl) + 2 (Ea la)(al + Eb Ib)(bl)
1 1
- 2(Eb la)(al - Ea Ib)(bl) + 2 (Eb la)(al - Ea Ib)(bl) (2.14)
1 1
= 2 (Ea - Eb) (\a)(a\ - Ib)(bl) + 2 (Ea + Eb) (Ia)(a\ + Ib)(b\)
1 1 ~
= 2 (Ea - Eb) az + 2 (Ea + Eb) Jr.
On the fourth line (2.13) and (2.8) have been used.
To study the dynamics of the system it is sufficient to look at how the energy
scales. The constant energy term (proportional to the identity operator) plays no
role in determining the dynamics of the system. It is safe to ignore it for now. It
can always be added at a later point if it becomes convenient to do so. At this
28
point, retain the part proportional to the Pauli operator. Using (2.5) gives the
following atomic contribution to the energy:
(2.15)
2.3.2 Field Hamiltonian
Appendix B reviews the theory of a paraxial cavity. In particular, the forms
of the Hamiltonian and the quantized electric field operator are found, as well as
the mode volume for a paraxial Gaussian mode of the cavity. The treatment of
the field in that appendix rests heavily on the paraxial approximation, whereas a
hemispherical cavity has a diffraction-limited spot-size and large solid angle, and so
is not described well by that approximation.
Departing from the paraxial regime means that approximations like sin () ~
(), tan () ~ (), and cos () ~ 1 are no longer justified. In other words, it becomes
necessary to go beyond first-order in the theory. The standard trick of separation of
transverse and longitudinal variables fails because we can no longer make the above
approximations.
The lack of an analytical framework for characterizing the solutions to Maxwell's
equations under such conditions pushes research in the direction of numerical
solutions. Numerical investigations into cavities of similar geometry to the ones
considered here [20] have led to predictions of novel modes whose behavior depends
29
intimately on the vector nature of light [18] (i.e. the spatial distribution of the
amplitude of the modes is not independent of the spatial distribution of the
polarization of the modes).
Interestingly, such studies have also yielded predictions of optical spin-orbit
coupling due to mode mixing induced by polarization-dependent phase shifts (upon
reflection from layers in a Bragg stack) that extend arbitrarily far into the paraxial
regime [19].
Nevertheless, while non-paraxiality may lead to certain novel behaviors not
easily described by analytic methods, it does not affect the expression for the
energy density of an electromagnetic field; nor does it change the fact that E(r, t)
lies in the plane of the QW. We proceed by quantizing the classical electromagnetic
Hamiltonian.
Classical Field
It is possible, for the case of paraxial modes (see appendix B) to carry out
the necessary integration of the mode function explicitly to determine the mode
volume. In this case, without an analytic representation of the mode function, the
mode volume will not be explicitly determined.
The plan of attack here is to write down the classical energy and show that it
(as in the paraxial regime) has the form of a harmonic oscillator. This classical field
30
will then be quantized by canonical quantization. The quantized electric field will
be used later to write down the interaction Hamiltonian2•
(2.16)
Of course, the fields must obey the source-free, vacuum Maxwell's equations.
Thus, each of the fields must satisfy a wave equation.
(2.17)
(2.18)
The time dependent part of the equations may easily be separated from the
vectorial, space dependent part. Working with the electric field, define it to be a
product of a vectorial function of a position vector with a scalar function of time.
Let
E(r, t) = R(r) . T(t).
2In this section boldface denotes vectors in JR3 , caret denotes a unit vector in JR3.
(2.19)
31
Then,
Now scalar multiply both sides from the left by R(r)/IR(r)I' T(t) (where R denotes
a spatial unit vector pointing along R). This provides a separated harmonic
equation for the scalar time-dependent function and a seemingly non-separable
vector equation.
The time dependent function is, therefore, a harmonic function of time and has
solutions as given in (B.5). In the absence of a resonator the eigenvalues would
generally have a continuous spectrum. In cavity-QED, we may assume the presence
of a stable resonant cavity so that the eigenvalue spectrum becomes sharply
peaked. It is well known that self- consistency of the modal field in a resonant
cavity leads to sharply peaked density of states and the eigenvalues (w) become
bunched about certain discrete values called the resonant frequencies of the cavity.
32
Additionally, with the small length of the cavities considered in this work, the
mode spacing can become quite large and it is not difficult to tune a probe beam to
become resonant with only one cavity mode. Henceforth, discussion will concern
only one mode at a time. Define v = kc. Then each mode will be labeled by a
particular value of v and it is reasonable to write
(2.20)
Armed with (2.20) return to the wave equation for the electric field (2.17).
Since the time dependent function has been found take this opportunity to change
notation, switching from R(r) for the vectorial, spatially dependent factor to the
clearer E(r). This provides
(2.21)
The vector Helmholtz equation on the last line of (2.21) can be written more
simply due to the source free Gauss's Law.
= \7 x (\7 x E(r)) + k2E(r) = 0
33
This can plainly be satisfied provided the following expression for the curl of the
field holds:
\7 x E(r) = ±ikE(r).
Now the application of Faraday's Law is straightforward.
a ~
at B(r, t) = -\7 x E(r, t) = ~ikk x E(r, t)
(2.22)
(2.23)
Differentiate (2.23) with respect to time to achieve the magnetic field's second
time derivative.
B(r, t) = ~ik k x E(r, t) = -v2B(r, t)
1 ,.
=? B(r, t) = -2(~ik k x E(r, t)
v
Now, that the relationship between the fields has been determined the electromagnetic
energy of the intracavity field may be written down. The electromagnetic energy, as
given by (2.16), is
1 J ( 2 1 k 2 I ~ 1 2 .)H ="2 d3r EO IE(r)1 T 2 (t) + 110 v4 k x E(r) T 2 (t)
v
= ~Jd3r (IE(r)12T Ct) + :21k x E(r)1 2t 2(t))
v
= ~ (1'2(t) + :~)Jd'rIE(r)I'
v
(2.24)
34
Write this in terms of the effective mode volume. Suppose that the field has a
uniform intensity everywhere. The effective mode volume is defined so that when
(2.25)~ff' IE(ro) 12 = Jd3r IE(r) 12
V
it multiplies the uniform intensity, the result is the same as when integrating the
varying intensity of the real field over the cavity volume.
Combine (2.25) with (2.24) to produce
(
T'2)EO 2 2H ="2 T (t) + -;Ji . ~ff' IE(ro)1 .
Finally, define a new dynamical variable, which is just a scaled version of T(t).
Let
q(t) = VEO;ff IE(ro) IT(t). (2.26)
For future consideration, E(ro) may be regarded as the product of a scalar
amplitude, a (possibly) position-dependent vector field and a normalized mode
function. That is,
(2.27)
with the obvious generalization:
E(r) = Eou(r)c(r). (2.28)
35
Due to (2.26) the classical energy is written as follows:
This is the energy of a classical, harmonic oscillator with unit mass with
canonical coordinate q. In this case p = q so that
Quantization of the field
(2.29)
Now, the problem is quantized according to canonical quantization by simply
replacing the quantities q, p and H with operators q,p and it. The operators q and
p obey the Heisenberg uncertainty principle:
[q,p] = in
It is usually convenient to express the dynamics in terms of an alternate set of
operators: q, p ---+ a, at. Let the new basis be:
1
a = ~ (vq+ip)
y2nv
1at = -- (vq - ip) .
V2nv
(2.30)
(2.31)
36
This gives, for the inverse transformation:
The new aoperators obey the following commutation relationship:
[a, at] = 1.
Now we can use the above to re-write the Hamiltonian in the familiar form
(2.32)
(2.33)
(2.34)
using the creation (annihilation) operators. These operators generate the familiar
ladder of states beginning at the ground state and proceeding, in units of nv, to
infinity. From (B.26) the Hamiltonian is:
_ 1 nv [(A At)2 (A At)2]
- -- a+a - a-a
2 2
= n; [2 (aat + ata) ]
= n; [(1 + 20,to,) ]
= nv(ata+~).
37
As with the atomic Hamiltonian drop the constant energy term. This gives the
form of the Hamiltonian that will be used in the rest of the analysis.
(2.35)
~-
nv
T
In+ 1)
In)
12)
11)
10)
Finally, write down the electric field operator
since it is relevant to the discussion in section
2.3.3, where the interaction of a dipole with the
electric field is to be considered. Since the QDs
are stationary we can consider the position of
the dipole to be fixed at a particular location
(ro) and thus the effective mode volume can be
considered constant.
By (2.27), IE(ro) I - Eou(ro) and thus, by
FIGURE 2.7: ElM field energy
levels. This shows the ladder of
states of th~ quantum harmonic
oscillator.
(2.26),
ff2trq = Eou(ro)T.EO Veff
This permits (2.28) to be expressed in terms
of q as
j5;V 2E(r,t) = trq'c,EO veff
38
Now that the field is expressed in terms of the canonical variable q, the
quantized version of the classical field can be given in one of the following forms:
E
A
( ) _ j;f2 A _ {;!fV (~ ~t)r - -v;:qe - 2 v;: a + a e.
EO elf EO elf
Time evolution may be considered to apply to operators or vectors (states) in
quantum mechanics, so explicit time dependence has been dropped here.
2.3.3 Interaction Hamiltonian
(2.36)
Under the assumption that the field is almost constant across the dimension
of the emitter the interaction is that of a dipole interacting with the electric field.
A constant field will polarize a quantum dot containing an exciton by pulling the
hole and electron in opposite directions. The IFQDs have lateral size of the order of
A./15 (around 50 nm). This is the so-called dipole approximation.
The energy of an electric dipole interacting with an electric field is:
V=-/-L·E. (2.37)
In order to quantize this energy equation look at the atomic dipole operator.
We already have a quantized form for the electric field from the previous subsection.
Expand the dipole operator in the basis of atomic operators by once again inserting
two complete sets. The dipole operator is expressed as: it = e· x. This has an
39
odd parity, so it will have non-vanishing matrix elements only between states of
different parity.
/L = L li)(il PIj)(jl
i,jE{a,b}
= L li)(jl (il PIj)
i,jE{a,b}
= L li)(jl (il ex Ij)
i,jE{a,b}
= /Lab /a)(bl + /Lba Ib)(al
Hermiticity of the dipole operator requires that the nonzero dipole matrix
elements be related by a complex conjugate.
P= /Lab la)(bl + /Lba IbXal
pt = /L':w Ib)(al + /Lba la)(bl
~ ~ t */L = /L {:} /Lab = /Lba'
(2.38)
(2.39)
Using the results of (2.38) and (2.39) and writing the complex dipole matrix
elements in polar form, the dipole operator may be written as:
40
A suitable change of basis will render the dipole matrix elements real. Suppose
that
(a'i it la') = (b'l it Ib') = 0 (a'i it Ib') = (b'l it la') = J1
where la') and Ib') are related to la) and Ib) by a change of basis 1':
, ~ i<l>
la) = T la) = e T la)
~ -i<l>
Ib') = T Ib) = e-2 Ib).
Notice that l' is a unitary operator. Its matrix will be diagonal so that complex
conjugation has the same effect as a Hermitian adjoint; to produce an inverse
operator 1't = 1'-1.
Now just write down the matrix elements of it in the primed basis.
(a'lP,jb') = (al (e- iiI>/2p,e-iiI>/2) Ib)
= (al P, Ib) e-iiI> = J1
(b'l it la') = (bl (eiiI>/2iteiiI>/2) la)
= (bl it la) eiiI> = J1.
41
Equivalently, one may leave the two energy eigenstates fixed and regard the
transformation as applying to the dipole operator (i.e. a unitary or similarity
transformation) .
Now that it has been shown that a change of basis can produce real matrix
elements as desired, just relabel the basis vectors so that a and b correspond to the
basis states that produce real matrix elements in the dipole matrix. This gives us
the resulting expression for the dipole operator:
(2.40)
Now multiply with the electric field operator which is given in (2.36). The resulting
interaction Hamiltonian is given by the following equation:
'HInt = -jJ,. E (r, t)
= -JjVf~~u(r) (8++8_) (a+at)cos<p. (2.41)
Here cos <p is due to the dot product between the field and the dipole unit vectors.
Let the coefficient of the operators be equal to n· g.
ng = JjEocos <p
g = JjVn~V cos<p. (2.42)
42
The new parameter 9 is the coupling constant and is the characteristic frequency
for the interaction. The interaction Hamiltonian now appears as follows:
(2.43)
Rotating Wave Approximation
To obtain a simplified form for the Hamiltonian, we briefly examine the time
evolution of the terms in the interaction Hamiltonian in the interaction picture.
In the interaction picture, the time evolution of operators is generated by the free
Hamiltonian, while the interaction generates time evolution for the state vector.
The free Hamiltonian is:
(2.44)
The time evolution for operators is governed by a Heisenberg equation. A
general operator obeys the following equation.
;. 1 [A A]0= in 0,110 . (2.45)
Or, more explicitly,
. 1 [ A]a = ih a,l£o
1
= ih [a, hv (ata)]
= -iv (aata - ataa)
= -iv (aat - ata) a
= -iv [a, at] a
= -iva,
which has the immediate solution:
a(t) = a(O)e-ivt .
Taking the Hermitian conjugate of equation (2.46) gives the evolution of the
creation operator.
Similarly, calculate the time evolution for the atomic operators.
43
(2.46)
(2.47)
44
Taking the adjoint provides us with the equation of motion for the raising operator.
From (2.43) we see that the terms have the following time evolution.
at(t)s+(t) = at(O)S+(O)eiCW+V)t
a(t)S_(t) = a(O)S_(O)e-iCw+V)t
at(t)S_(t) = at(O)S_(O)e-iCw-v)t
a(t)s+(t) = a(O)s+(O)eiCw-V)t
(2.48)
(2.49)
(2.50)
(2.51)
(2.52)
Normally, under the rotating wave approximation (RWA) we boost to a rotating
frame and then take the DC component. In this situation where we consider the
product of two operators and under the condition that the atomic transition and
the cavity mode are close to resonance we really just need to take the low frequency
components; high frequency components will quickly average to zero. Dropping
terms corresponding to (2.49) and (2.50) while keeping terms given by (2.51) and
(2.52) gives us the following simplified interaction Hamiltonian.
(2.53)
45
2.3.4 Total Hamiltonian
The dynamics unfold in a composite Hilbert space. The composite space
is simply the tensor product of the two independent Hilbert spaces. First, the
formalism for a composite Hilbert space is presented following the treatment of
[6]. Then, the final form of the operators in the composite Hilbert space will be
derived. It will be seen that simply expanding the Hamiltonian in terms of the
basis vectors over the composite space automatically produces operators of the
correct dimensionality.
Composite Hilbert Space
The Hilbert space is composite. Taking the tensor product of the two independent
Hilbert spaces produces a new Hilbert space whose dimension is the product of the
dimensions of the constituent Hilbert spaces.
(2.54)
Then, given fixed orthonormal bases {1<p}A)) } and {I <P3B )) } in SjA and .fJB
respectively, a general state in .fJ may be written:
1?jJ) = L I<p}A)) ® I<P3B )).
i,j
(2.55)
46
It then follows that an operator over this composite space has the following
expansion:
(2.56)
In the context of the current problem our basis consists of {I¢field) @ I¢]-level) }.
This set of vectors spans the composite Hilbert space. Then, a unified Hamiltonian
may be constructed by expanding in terms of these vectors. Let Ii, m) = 1m) @ Ii).
The Hamiltonian is composed of an pseudospin part, a field part, and a part
that overlaps both spaces. Exploit the completeness of the states and insert the
corresponding identity operators:
H = H2-level + HField + HInt
00
:::;. H = :L :L Ii, m)(j, nl (i, ml H 2-levellj, n)
i,jE{b,a} m,n=O
00
+ :L :L Ii, m)(j, nl (i, ml HField Ij, n)
i,jE{b,a} m,n=O
00
+ :L :L Ii, m)(j, nl (i, ml HAF Ij, n).
i,jE{b,a} m,n=O
(2.57)
47
It is now time to calculate the appropriate matrix elements in the composite
basis.
A 1(i, ml1t2-level Ii, n) = '2fu.u (il (ml (/a)(al - Ib)(bl) In) Ii)
1
= '2fu.u (m In) ((i la)(a Ii) - (i Ib)(b Ii))
= n:; 6mn (c5ia6aj - 6ibc5bj )
(i, ml HField Ii, n) = (il (ml fu.u .n In) Ii)
(i, ml HInt Ii, n) = -11,g (i, ml (8+ .a+ 8_ .at) Ii, n)
= -11,g (i, ml ( la)(bl .a+ Ib)(al .at) Ii, n)
= -11,g[(i, m la) (bl a\j, n) + (i, m \b) (al at \j, n)]
= -11,g[y'n (i, m la) (b Ii, n - 1) + vn" + 1 (i, m Ib)
x (ali,n+1)]
(2.58)
(2.59)
- - ---- -- ---- ----
48
Substitution of the results (2.58), (2.59) and (2.60) into (2.57) gives the
following expression for the Hamiltonian.
H=
00
i,jE{b,a} m,n=O
00
- 2: 2: Ii, m)(j, nl ng[yn. biabbjbm,n-l + v'n + 1· bibbajbm,n+l]
i,jE{b,a} m,n=O
Rewriting the outerproduct Ii, m)(j, nl = Im)(nl Q9li)(jI, as suggested by (2.56) yields
the expression below.
H=
00
i,jE{b,a} m,n=O
00
- 2: 2: Im)(nl Q9li)(j1 ng[yn· biabbjbm,n-l + v'n + 1· bibbajbm,n+l]
i,jE{b,a} m,n=O
The interaction reduces to the following expression:
HInt = -ng 2: (In)(n + 11 Q9~+ In + l)(nl Q9~)vn+1. (2.61)
n=O A A
s+ s_
49
We utilize the following result:
L li)(jl Oij = L li)(il = Ih-level
i,jE{a,b} iE{a,b}
(2.62)
00 00L Im)(nl (jm~ = L In)(nl = ITField'
m,n=O n=O
(2.63)
This leads to the following simplified form for the Hamiltonian operator over the
composite Hilbert space.
H = HField Q9 IT2-level + ITField Q9 H2-level + HInt (2.64)
It is now possible to write out explicit matrix representations of these operators
by choosing an appropriate map between kets and column vectors. For the present
work it makes sense to use
50
and
1
o
o
1
o
o
o
{IO) , 11) , 12) , ... , In) , ... } -+
o 1
o ,...
1
} ...
These column vectors can be suitably used to represent the dynamics of
the problem at hand. Matrices provide a representation that can convey visual
information about interactions and can allow visual identification of non-interacting
subspaces. This map of kets into column vectors allows the various pieces of the
Hamiltonian to be written in matrix form. First write down the matrix for the 2-
level system and the field. Combine these to get the free Hamiltonian (with no
interactions). Then write down the interaction Hamiltonian and combine it with
the free Hamiltonian to produce the full Hamiltonian for the Jaynes-Cummings
system.
In order to produce matrices that span both spaces, employ a direct (or
Kronecker) product between operators in each Hilbert space. The direct product
51
is defined such that the components of a matrix C = A ® B are given by
where rn = p(i - 1) + k, n = p(j - 1) + land where"B is a p x p matrix. This can
be succinctly displayed by supposing that A is a 2 x 2 matrix. "Then we can directly
write
By writing the outer product Ii, rn)(j, nl as Irn)(nl ® li)(jl a representation has
already been chosen. The 2-level Hilbert space stands to the right in the direct
product. This means that the operators for the composite Hilbert space will tend
to contain 2 x 2 blocks, as will be shown shortly.
52
The matrix for the 2-level Hamiltonian over the extended Hilbert space is given
by the matrix elements in (2.58):
-1
1
-1
1
-1
1
(2.65)
The matrix for the field Hamiltonian can be written down using the matrix
elements written in (2.59)
o
o
A A nv
1iField 0 n2-level = 2
1
1
2
2
53
The matrix for the free Hamiltonian may then be obtained by addition. The result
is:
'Ho = Ii
As expected, the Hamiltonian of the non-interacting system is given by a
diagonal matrix. The bare states, consisting of products of definite states of the
two-level system and definite states of the field are the normal modes.
Next, write down the matrix for the interaction Hamiltonian using the matrix
elements from (2.60).
54
'HInt = -fig
(::) (::) (::)
(: :) (~ :)
(::) (:~)
(: :) (::) (:~)
(::)
(: :)
(~ :)
55
Finally, the complete Jaynes·-Cummings Hamiltonian may be expressed as a
matrix. It takes the following form:
w
2
w
2
-g
-g
w
v- -
2
w
v+-2
w
nv+-2
-gvn+!
-gvn+! (n + l)v + ~
2.4 Diagonalization - Normal Modes
Having obtained the matrix for the Hamiltonian, it is now time to turn to
the matter of diagonalization - of finding the eigenvectors and using them to
transform the Hamiltonian into a diagonal matrix. Equivalently, we wish to expand
the Hamiltonian in terms of the normal modes of the system. The normal modes
are independent of each other and undergo simple time evolution at a single
56
frequency. As in classical mechanics, they provide a mathematically convenient
and conceptually simple basis for, expanding the dynamics of systems of coupled
oscillators.
2.4.1 Minimal Decoupled Subspace
As in classical mechanics off-diagonal elements in the Hamiltonian matrix
signify coupling between the various degrees of freedom. The Hamiltonian is block
diagonal, consisting of 2 x 2 blocks along the major diagonal. It is for this reason
that the Hilbert space was decomposed into the current ordering for the direct
product. By writing the atomic Hilbert space to the right we are provided with
a matrix containing 2 x2 blocks. The block diagonal structure means that each
2 x 2 block represents an independent subspace which is not coupled to the rest of
the Hilbert space. As indicated previously, this is only a choice of representation
and doesn't change any intrinsic properties of the vector space (e.g., which states
couple to which other states). However, the particular choice of representation can
greatly simplify calculation and interpretation. In practice it is generally easier to
identify non-interacting subspaces when their terms are grouped together in blocks
in matrix representations.
The Hamiltonian given by the nth block is:
(2.66)
57
where..6. - W - l/ is called the detuning and OR =2gvn + 1 is .called the Rabi
frequency. This implies that each subspace is spanned by two quantum states:
la, n) and Ib, n + 1). Based on the structure of the Hamiltonian over the entire
Hilbert space we see that the kets in the combined Hilbert space map naturally
map to a normally ordered two-dimensional basis in the following way:
2.4.2 Eigenvalues and Eigenvectors
This reduced subspace is manageable and may be easily diagonalized. The
first matrix in (2.66) is proportional to the identity matrix so it will remain
diagonal under any suitable principal axis transformation. We need only find the
eigenvectors of the second matrix.
First label the matrix to be diagonalized as n'. The factor of n/2 has been
dropped, for now.
det (0' - AI) =0.
..6. - A -OR
*
=0
-OR -..6. - A
(2.67)
58
:::} (A - ~) (~ + A) - n~ = 0
(2.68)
Now to generate the appropriate eigenvectors, apply each of the eigenvalues to
the eigenvalue equation -- 0'· v = Av.
(2.69)
Additionally, the change of basis should preserve the normalization. Invoke the
normalization condition for the eigenvectors.
V= (a,b)
(2.70)
59
Using (2.69) and (2.70) gives a set of normalized eigenvectors. For A = -Dn :
iL = VU':J.. - Dn )2 + D~
DR
Meanwhile, a similar calculation for A = Dn gives the other independent
eigenvector.
V(!j,. - Dn )2 + D~
~-Dn
Consistent with the normalization condition we can make a trigonometric substitution
for the components of the eigenvectors.
60
(2.71)
(2.72)
The transformation matrix which diagonalizes the Hamiltonian may now be
written as:
[
COS On - sin on]
p= .
sin On cos On
In other words, the diagonalized Hamiltonian is obtained from the original
Hamiltonian via a similarity transformation (a unitary transformation).
H' = P-lHP
::::}H
H·y =Ey
PH/P-l.y =Ey
H/P-l.y = EP-l.y
::::} y' =p.y
(2.73)
61
Thus, the inverse matrix of P provides for the change of basis from the bare states
into the dressed states (the new eigenvectors). For example, The v+ vector is
transformed by p-l as follows:
sin on] [cos en] _ [1]
cos On sm On 0
The map between dressed states and column vectors in the dressed state basis is
therefore given by
{In'+)d'In'-)d} = {[:], [:]}.
In, -) Ib,n + 1)
la,n)
In,-) Ib, n + 1)
In,+)
la,n)
FIGURE 2.8: Rotation from bare states into dressed states. This diagram shows a
geometric representation of the change of basis from the bare states into the dressed
states in terms of the mixing angle On. The graph on the left shows the dressed states
in terms of bare basis vectors. The graph on the right shows the bare states in terms of
dressed basis vectors.
62
The eigenstates of the "free" Hamiltonian are referred to as bare states. It
is said that the interaction dresses the states and the new basis, in which the
Hamiltonian is diagonal is referred to as the dressed state basis. Alternatively,
(2.74)
the dressed states are sometimes referred to as normal modes since the situation
is analogous to a classical system of coupled oscillators, in which it becomes
advantageous to talk about collective motions of the coupled oscillators rather than
insisting on describing the motions of each oscillator separately.
2.4.3 Normal Mode Splitting
Acting on each of the eigenvectors with the Hamiltonian over our chosen
manifold, (2.66), we obtain the eigenvalues of the Hamiltonian.
The eigenvalues of the complete Hamiltonian are the energies of the dressed
states. They are simply:
1 Ii
En ± = liv (n + -) ± -On.
, 2 2 (2.75)
63
The lifting of the degenerate eigenvalues of the free Hamiltonian by the
inclusion of the interaction is called normal-mode splitting. At zero detuning (i.e.
on resonance), the energy now exhibits a split spectrum as shown in figure 2.9.
In, + } =cost}nla, n}
- sint} Ib, n+l}
n
No interaction
With interaction
In, -} =sint}n la, n}
+ cost} Ib, n+l }
n
FIGURE 2.9: Effect of interaction on energy levels; normal mode splitting. The energy
levels of the energy eigenstates are shown in the case of inclusion of the interaction (right)
and sans interaction (left). The lifting of the degeneracy by the interaction is sometimes
called normal mode splitting.
Consider varying the cavity length so that the resonance fequency of the intra-
cavity field changes. If we plot the energies of the dressed and bare states we can
see that the dressed states become asymptotic to the bare states in the limit of
large detunings. For the n = a manifold the bare states correspond to a vacuum
state of the field with an excited state (ground-state exciton) of the quantum dot or
a single excitation in the field and a vacuum state of the quantum dot.
The eigenvalues for the n = a manifold are Eo,± = ~ [v ± J(w - V)2 + n~]. In
the limit of large .6. we can ignore the factor of n~ and evaluate the square root as
64
having the value of ±..6.. Equivalently,
Eo,± = @(v ± J..6.2 + n~)
_ Ii [ .1A2.~
- 2" v ± v il~V 1 +~J
~ @[v ± W (1 +~ ] for 1..6.1 » InRI
Ii~ 2" [v ± Iw - vi]·
This approximation is only valid asymptotically. In each of those limits we have
I!!:. (v + w - v) for v « wEo = 2,+ Ii2" (v + v- w for) v » w
{
liw
- for v« w
_ 2
- liv - m; fow» w
Similarly, the other set of eigenvalues may be described, asymptotically as
Iliv - liw for v « wEo- = 2, liw2 for v» w
By including the constant energy terms that were omitted previously, the energ
will be uniformly shifted up by a factor of liw /2. This makes the lower energy
contribution by the two-level system, corresponding to the quantum dot vacuum
state, become zero and the upper state take on a value of liw. This is entirely
65
Energy
...
Energy of upper
dressed state
Energy of lower
dressed state
v
Energy of single intra-cavity
photon = fJv
Energy of ground-state
exciton = fJw
FIGURE 2.10: Energies of bare and dressed states. The energies for both bare states
and both dressed states are shown. It can be seen that each of the dressed states
asymptotically approaches a different bare state on either side of the resonance.
optional and doesn't affect any dynamical predictions of the system. However, it
does enable a slightly simpler labeling of the energy diagram. In this case we may
associate the asymptotic bare energies as corresponding to purely one or the other
of the bare systems. In other words, the asymptotic bare state energies can be
thought of as either an excited quantum dot with vacuum in the cavity field, or a
vacuum state of the quantum dot with a single excitation in the cavity field.
2.4.4 Rabi Oscillations
The advantage of transforming into the dressed-state (diagonalized) basis
becomes clear when we look at the time evolution of the system. Because the
dressed states are eigenstates of the Hamiltonian, they are each necessarily also
eigenstates of the time-evolution operator. For example,
In, +(t)) = (; (t, to) I\II (to)) = e-i</>(t) In, +(to)) ,
because the time-evolution operator may produce a time-dependent eigenvalue,
but leaves an eigenstate of the Hamiltonian unchanged. For a time-independent
Hamiltonian the form of the time-evolution operator is:
A [-iH (t - to)]U (t, to) = exp n .
This gives the following time evolution for the quantum state.
\1/J (t)) = exp [-iHt/n] 11/J (0))
00
= L L exp [-iEnmt/n] In, m)(n, ml1/J (0))
n=O mE{+,-}
66
67
The eigenvalues of the Hamiltonian and the associated eigenvectors have been
determined, so the time evolution is trivial.
0]) [cn+ (0)]
o Cn - (0)
(2.76)
In order to see the so-called Rabi oscillations we transform back into the bare
state basis. To evolve the bare states forward in time the time-evolution operator
must be cast into the bare-state basis.
P IVJ (t))D = PUD.p-l P IVJ (O))D
IVJ (t)) B = PUDP-1 IVJ (0)) B
[
Ca,n (t) ] [.2n + 1 ] A [e~mnt
= exp 1. vt· P
2
Cb,n+l (t) 0
Using (2.73) and performing the necessary matrix multiplication we obtain the
following result for the time evolution of the bare states.
[
Ca,n (t) ] A [ Ca,n (0) ]
= [UB(t)] .
Cb,n+l (t) Cb,n+l (0)
68
where
A [.2n±1 ] [cos (¥) + i cos 2Bn sin (¥)[UB(t)] = e t 2 vt .
i sin 2Bn sin (¥)
i sin 2Bn sin (¥) 1
cos (n2't) - i cos 2Bn sin (n2't)J
Suppose the initial state of the system is Ca,n(O) = 1, Cb,n+l = O. Also, suppose
that the mixing angle Bn = 7f / 4 (i.e. the detuning parameter ~ = 0). Then the
state at a time t will be given by:
[
Ca,n (t) ] [ .2n + 1 ]
= exp 'l vt·
2
Cb,n+l(t)
So, the probability to find the system in either of the bare states varies
sinusoidally. This periodic switching between maximal probability and zero
probability is referred to as Rabi flopping.
2.4.5 The Single Quantum Manifold
Now consider the situation in the n = 0 manifold. The bare states that span
this manifold are: Ib, 1) and la, 0). The representation of the change of basis as
a rotation is convenient. In the case of resonance between the cavity mode and
the transition between the two levels, the effective mixing angle is 7f / 4. The time
69
(~ =0)
FIGURE 2.11: Rabi flopping. This is a plot of the probabilities of measuring the system
in either of the two bare states as a function of time, assuming zero detuning. Each
27r
probability oscillates with a period of D
n
'
evolution, in this case, is simply given by
Correspondingly, the quantum state exhibits maximal entanglement the
two Schmidt numbers have equal magnitude. This is commonly referred to as a
Schrodinger cat state.
The normal-mode splitting, in this case, is commonly referred to as vacuum
Rabi splitting and the oscillations of the state vectors as vacuum Rabi oscillations.
An example of Rabi oscillations in a semiclassical context are oscillations in the
state of a two-level atom driven by a resonant optical field. The frequency of the
oscillations (flopping) is proportional to the amplitude of the applied electric field.
70
The Rabi flopping in this case cannot be explained semiclassically. In the case
of an initially excited two-level system, the electromagnetic field is in a vacuum
state.· It may roughly be said that the excited atom undergoes emission that's
stimulated by the vacuum field of the cavity mode and for this reason the splitting
of the energy levels and the oscillations in the probabilities of measuring either of
the bare states are referred to, respectively, as vacuum Rabi splitting and vacuum
Rabi oscillations.
Obviously, this phenomenology will apply to any two level system interacting
with a single mode of the electromagnetic field.
2.5 Linewidth, Dissipation and Open Systems
The preceding treatment is sufficient for explaining the appearance of certain
features in experimental studies, but fails to properly describe most real world
systems that one might hope to measure. The reason for this is the rather idealized
closed system. The situation parallels that of a driven, undamped oscillator in
classical mechanics. The spectrum consists of a set of delta function peaks.
Realistic oscillators exhibit damping, which may arise in the context of classical
mechanics as (velocity dependent) dissipative forces, or in quantum mechanics
as dissipative couplings to external degrees of freedom. They both correspond to
interactions that carry energy out of the system of interest and lead to damped
responses, characterized by finite amplitudes and linewidths for oscillators.
---------------------~-~---
71
More precisely, a realistic model of a quantum optical system should allow for
interactions with unmonitored degrees of freedom, which may lead to dissipation
and dephasing (decoherence). Dissipation corresponds to reduction of probabilities
(diagonal components of the density matrix). This also, necessarily reduces the size
of the off-diagonal coherence terms. This is termed dissipation-driven decoherence.
Dephasing corresponds to decay of the off-diagonal terms and the approach to
a classical mixture (also retreat from a quantum pure state). Decoherence that
proceeds without associated dissipation may be called pure dephasing.
It is, therefore, necessary to employ a model that includes couplings to external
degrees of freedom in order to account for real-world phenomenology. For example,
the mode of the optical cavity is connected to other modes of the electro-magnetic
field via scattering and transmission. The two-level emitter, which in these
experiments is an interface fluctuation quantum dot, may also couple to non- cavity
modes of the field via spontaneous emission (especially in the case of an open-sided
cavity as used here). In the case of a single atom, the primary interaction channels
involve radiative couplings. In the case of a QD, a number of other channels may
also be involved. The QD may still emit radiatively to either the cavity mode or
non-cavity modes. However, non- radiative interaction channels are also available,
such as tunneling of the exciton to neighboring QDs or interactions with phonons.
Since the interest here is merely in providing a proper account of the expected
spectrum of the transmission of a weak probe beam, care will not be taken to
72
distinguish each possible reservoir in detail. A primary contribution to the overall
dissipation and dephasing properties of quantum dots does appear to be due to
interactions with phonons, however [57, 16].
We will continue development of the theory in an ad-hoc manner, first including
the effects of reservoir couplings, then introducing a driving field so that a transmission
spectrum may be derived.
2.5.1 Reservoirs
Time evolution for closed systems in quantum mechanics is represented as a
one- parameter unitary group. Among other things, this says that for each time
evolution operator propagating the system into the future there exists an inverse
operator carrying the system back in time. Ignoring the measurement/ collapse
problem (which breaks the assumption of a closed system anyway), the dynamics
are time-reversal invariant. So far, the model has been closed and the Hamiltonian
is the generator of time evolution (i.e U(t, to) = exp[iH(t - to)] ).
For many systems studied in quantum optics the assumption of a closed system
is not appropriate. Spontaneous emission is a good example of behavior that is not
time-reversal invariant and therefore not well described by models involving closed
systems.
A successful method of dealing with this is to include interactions between the
open system of interest and a reservoir (e.g. an infinite set of oscillators). This
leads to non-unitary time-evolution that exhibits an arrow of time. In the case of
73
unitary time "evolution the eigenstates of the Hamiltonian are also eigenstates of
the time evolution operator. A closed system prepared in an energy eigenstate will
remain in that state forever.
For the case of an open-system coupled to a reservoir, however, the open system
is free to emit energy into the reservoir (or vice versa) until they reach equilibrium
(i.e. the likelihood for the open system to emit a quantum of energy into the
reservoir is equal to the likelihood of it absorbing a quantum of energy from the
reservoir) .
unitary
p(O) = Ps(O) Q9 PR __e_v_ol_u_tio_n_~>p(t) = U(t, 0) [Ps(O) Q9 PR]Ut(t, 0)
ThR1 1ThR
ps(O) ) ps(t) = V(t) . Ps(O)dynamical
map
FIGURE 2.12: Quantum dynamical map. A commutative diagram shows the action of a
dynamical map V(t), reproduced from [6].
The commutative diagram shown in figure 2.12 says that time evolution and
taking the partial trace over the reservoir are commutative. That is, it doesn't
matter in which order we take these two operations. We may calculate the time
evolution over the closed system S EEl R then trace out the reservoir degrees of
freedom to find the state of the open sub- system at a later time. Alternatively,
and of great practical importance, it is possible to trace out the reservoir in the
beginning and calculate the (non- unitary) time evolution of the open sub-system
74
(8) purely in terms of states and operators that are defined in .f)s. As is indicated
in the diagram, the evolution of 8 is given by a dynamical map (V).
The expression at the upper left corner of the diagram in figure 2.12 presupposes
that the combined system of the subsystem of interest together with the reservoir is
factorizable at t = O. In other words, there are no correlation, initially.
Commutativity requires that
Ps(t) = V(t)ps(O) - trR {U(t, O)[ps(O) 0 PR]Ut(t, O)} .
If it is assumed that the reservoir is large enough that its state does not depend
on the interactions with the subsystem (which in this case can be considered to
contain only a single quantum of energy), then Vt > 0, V(t) : ns ---t ns. That is, it
defines a map from ns into itself for time translations into the future (but not the
past). It is clear from the commutative diagram that the dynamical map V must
involve only operators in ns.
2.5.2 Approximations
In pursuit of a formulation of the dynamics of the open-system in terms of a
master equation in the Lindblad form, it is necessary to invoke some approximations.
Three approximations are generally needed to obtain a quantum master equation in
the Lindblad form [6, 7]:
75
1. The Born approximation involves only working to second order in the terms
that couple the system to the reservoir. This is justified if the couplings are
weak.
2. The Markoy approximation requires the existence of two widely separated
time-scales. The time-scale over which the system is to be examined must
greatly exceed the time-scale over which correlations in the reservoir persist.
The Markov approximation allows for a description that is local in time, it is
equivalent to taking the two-time correlation function for reservoir operators
to be a delta-function.
where tilde indicates operators in the interaction picture.
3. The rotating-wave approximation was made already when deriving the
Jaynes-Cummings Hamiltonian. It is necessary to make the same approximation
for the couplings between the system and reservoir in order to derive a master
equation in the Lindblad form from a Born-Markov master equation.
If each of these approximations may be justified, then the quantum dynamical
map V(t) may be described as an element of a quantum dynamical semigroup.
(2.77)
76
Here, t1 and t 2 label changes in time. The dynamical map for the time evolution
of the open subsystem depends only on the total time elapsed between the initial
state and the final state. It does not depend on the particular times involved.
The dynamical map may be said to be independent of history, consistent with the
Markov approximation.
Under these conditions, it may be assumed that the dynamical map V,
belonging to a semi-group, may be formulated in terms of an infinitesimal generator
of the semi-group. That is,
V(t) = exp[L:t].
Then, in terms of the generator L: of the semigroup, we may write
d A f" A
dtPS = J..,Ps· (2.78)
Since we are assuming that the Born-Markov approximation and the RWA hold,
we may assert that the generator, L:, is in the Lindblad form. In this case, we may
write (2.78) as:
(2.79)
77
The term weighted by IP is a phenomenological pure dephasing term as
discussed in [8].
The time evolution of expectation values can then be calculated in the following
way:
+ K,(2apat - atap - pata6
IP (A AA A)oA)+ 2 o-zpO-z - p .
(2.80)
The coupling to the reservoirs has the effect of coupling the single quantum
manifold to the vacuum state, 10, b). The density matrix for the scenario under
consideration may thus be expanded in a basis consisting of three states. In other
words, coupling to reservoirs causes the decoupling between the subspaces of the
closed system model to no longer be rigorously observed. The Hamiltonian now
contains terms that couple these blocks to each other and provide for transition to
lower energy manifolds. Starting off in the single quantum manifold, the Hilbert
space for the open subsystem is now spanned by three states:
1
° °
{10, b) ,11, b), 10, a)} -+ ° , 1 , °
° °
1
78
The quantum state in this 3-state basis can be written as
1'1/;) = cliO, b) + c211, b) + c310, a) for Ci E C.
In turn the density matrix over the open subsystem looks like
where due to the orthonormality of the basis we can consider the matrix to be
(2.81)
a sum over the 9 outer products of the basis vectors (i.e. an expansion over the
matrix elements). For example, the term CIC;, due to
010
° ° °
is one such matrix element.
It turns out that each of these matrix elements may be interpreted as the
expectation value of an operator defined on the subspace. For example, the matrix
element depicted above is associated with an overlap between the states 10, b) and
11, b). We may demonstrate that this corresponds directly to the expectation value
79
of the annihilation operator. The expectation values of higher moments are zero in
the restricted subspace.
The five3 independent components of the density matrix may each be written
down similarly in terms of expectation values of operators.
[p] =
1 - (ata) - (Eh!L)
(at)
(a) (EL)
By specifying an initial condition in this Hilbert space, one may formulate a
well-posed initial value problem. This is a necessary course of action if one wishes
to study the emissive properties of the CQED system. Though the goal here is to
obtain a transmission spectrum rather than a spontaneous emission spectrum, it
will be worthwhile to pause here to calculate the mean- value equations for the a
and S_ operators. These may be calculated as shown in (2.80).
The calculations are worked out in appendix A. As can be seen from (A.6) and
(A.7), the annihilation operator and the lowering operator for the quantum dot are
3Hermiticity reduces the number of independent components of the matrix to 6 (from 9).
Normalization (tr[p] 1) makes one of the diagonal matrix elements linearly dependent on the
other two.
coupled. The pair of equations may be written in matrix form as follows:
80
ig ] l(&) ]
--(f + iw) (8_)
(2.82)
81
The general solution of a linear, homogeneous system of differential equations
(y = M· y) is given by y = AVI exp[A1t] + BV2 exp[A2t] + C, where Ai and Vi are
eigenvalues and eigenvectors generated by M. The eigenvalues are
K, + r .w + 1/ 1vi . .2A± = --- - 't-- ± - (K, - r - 't~) - 4g2
2 2 2 (2.83)
For certain values of K" rand g, the time evolution may be seen to be governed
by a complex eigenvalue and thus corresponds to damped harmonic motion. So,
without solving the problem in detail, it's already evident that the behavior of the
undriven system corresponds to either damped harmonic time evolution or to pure
exponential decay, depending on whether the eigenvalues are complex or purely
real, respectively.
2.5.3 Input-Output Formalism
The experiment that we wish to model is the transmission spectrum of a weak
probe beam through the coupled cavityjemitter system. In pursuit of this, we now
expand upon the damped Jaynes-Cummings Hamiltonian considered so far. The
intracavity field will be connected to the probe field via transmission. Similarly, the
intracavity field drives a pair of output fields via transmission.
First consider the coupling to an input probe field. The interaction that
transfers energy from one mode to the other is transmission at the input mirror.
Input and output both involve interactions with reservoir modes. The interaction
82
Hamiltonian that couples the intracavity mode to the input probe field and to
the output field that will be measured by a detector is the same Hamiltonian that
generates dissipation and dephasing.
(2.84)
It is appropriate to consider the reservoir modes as traveling waves obeying
periodic boundary conditions along one dimension. These traveling modes thus
have field operators whose amplitudes depend on the volume of a box with length
L'. The box defines a natural length scale for the density of states. In particular,
a wave entering the box on one end will end up in the same state at the other side
of the box. The reciprocal of this crossing time defines a FSR of l" where L' is the
length of the box (we can take lim later). This is the frequency difference between
L'->oo
reservoir modes and leads directly to a density of states (over angular frequency),
which will be used later:
L'
g(w) = 21rc. (2.85)
We consider a reservoir mode coupled to one mirror as the input probe beam.
Assume that one mode with angular frequency WL is in a coherent state, with all
other modes in the vacuum state. In this situation, the one input mode can be
merged into the Hilbert space, leaving the rest of the vacuum modes to contribute
83
via the Lindblad form. The interaction with the probe beam can then be written as
(2.86)
Due to the linearity of the trace, the contribution to the rate equations can be
calculated separately. The Hilbert space now may be considered to be
The fact that the input field is a coherent state means that the mean value of the
envelope of the probe field is the displacement parameter. This allows a simple
calculation of the probe field contribution to the time evolution of the intracavity
field.
(2.87)
84
where (3 is the displacement parameter of the coherent state produced by the probe
laser.
We are now in a position to determine the steady state solution of the mean
field inside of the cavity. By including (2.87) in the damped rate equations given in
(2.82), we obtain a pair of rate equations for the driven CQED system.
The resulting rate equations for the mean values of the field and quantum dot
polarization operators are
(2.88)
This is a non-homogeneous system of first order differential equations. It is possible
to solve these equations directly and get the time dependent solutions. This would
be necessary for describing experiments sensitive to transient behavior. In this case,
where we probe the CQED system with a continuous-wave (CW) beam, we are only
interested in the steady state behavior.
2.5.3.1 Steady-state solutions
In order to find a steady state solution, transform into a frame rotating at the
frequency of the driving field. This permits a static solution in terms of the slowly
varying envelope of the input field. The rate equations for the mean field and mean
85
. polarization can be found by using the chain rule:
(2.89)
Using (2.89) with (2.88) we obtain rate equations in the rotating frame.
To find the steady state solutions, set the left-hand side of (2.88) equal to zero
- i.e. the rate of change of the dynamical quantities vanishes. This converts the
system of ordinary differential equations (ODEs) into a linear system of algebraic
equations:
This can be solved by substitution or Gaussian elimination. The resulting steady-
state solution for the mean field is
(2.92)
86
In order to calculate the transmission spectrum, the output field generated by
the input field needs to be determined.
2.5.4 Transmission Spectrum
To calculate the transmission spectrum. the output field needs to be compared
to the input field. The output field couples to the intracavity field in the same
way as the input field, via (2.84). When treating the input field, it was possible to
simplify the analysis considerably by specifying that the input field was vacuum
except for a single mode that coupled to the intracavity mode. Now that we have
found the steady state cavity mean field, it is necessary to permit interaction with
many output modes. In order to treat the output modes, we will use a Heisenberg
picture approach to find the time evolution of the output reservoir mode operators
in terms of the intracavity mode operators. This subsection follows the treatment
in [9] closely.
The self energy for the reservoir modes is simply
HR = nL KkO,tfk + K*o'il
k
The time evolution of the operator Tk may then be calculated by evaluating its
commutator with the Hamiltonian:
87
One of the terms in th€ sum has been eliminated because [fk , fk'l = O. FUrther,
because [fk , ft,] = 6kk" the sum may be truncate~ to include only the term k' = k.
This first-order non-homogeneous differential equation can be solved by the method
of integrating factors. Rewrite the above result with the non-homogeneous driving
term on one side, so that it has the form y' + P(x)y = Q(x).
Proceed by multiplying through by an appropriate integrating factor that will
allow the left-hand side to be expressed as a complete derivative. The integrating
factor will be of the form JP(x)dx. In this case, the integrating factor is eiwkt •
Multiplying the above equation by this integrating factor, we obtain
88
where in the last line, we have used a = &'e-iwct with We denoting the frequency of
the cavity mode. Solving for Tk(t) we obtain
Tk(t) = Ce-iwkt - iK~ it &'ei(wk-wC)t' . e-iWktdt'
= Tk(O)e- iwkt _ iK~eiwct it &'ei(Wk-WC)(t'-t)dt'.
In the last line the integration constant has been given the more physical label of
The reservoir field is
E(z, t) = eo L V2~~' Tk(t)· exp [i (~k z+ </>(z))] ,
k 0
which naturally separates into two pieces; one part due to the free reservoir field,
the other due to the intracavity field. In this case, we measure the transmitted
field, so the initial reservoir fields are all vacuum and don't contribute to the
expectation value of the field. We need only consider the field emanating from
thecavity:
(2.93)
89
It is convenient to replace the sum over k with an integral over the density
of states. This may be regarded as conversion of a Fourier series into a Fourier
transform or as counting of states by integrating the density of states rather than
adding up k (i.e. w/c)values. In any case, we use (2.85) to replace the sum over k
with an int~gral -- L -+ Jdwg(w) = 2~CJdw:
k
E(+)(z t) = -ie ~e-i[Wc(t-Z/C)-</>(z)]
R' 0V~
X 2~c100 dwVWK,*(W) It dt'~(t')ei(Wk-WC)(t'-Hz/c). (2.94)
To go further, some approximations must be made. In particular, invoke the
slowly varying envelope approximation, which says that ~ is almost constant
on the time scale of an optical period (27f/ we). Furthermore, the exponential
term oscillates 'quickly' for frequencies away from We, leading to cancellation
on frequency intervals away from We. Armed with this argument, we set VW ~
..;we and K,*(w) ~ K,*(we). This allows (2.94) to be rewritten as
E~+)(z, t) = -ieoJ2~~C f{e-i[Wc(t-z/C)-</>(Z)]K,*(wc)
x [It dt'~(t') 2~100 dwei(Wk-WC) (t'-Hz/c) ]
= -ieoJ nwe (iJe-i[wc(t-Z/C)-</>(Z)}K,*(we)
2EOAc V~
x [It dt'~(t')f5(t' - t + z/c)].
90
For negative values of z, the delta-function selects a value of t' outside of the
range of integration. The field operator for reservoir modes emanating from the
cavity is
'j-ieov nwAc (iJK,*(wc)a(t - z/c) ct > z > 0Ek+) (z, t) = 0 21:0 C V~ (2.95)
z<o
Now, the interaction constant K,*(wc) may be determined by imposing a
constraint. In steady state, we require that
Combining (2.95) with (2.92), we obtain the steady-state mean field emitted
by the pumped CQED system. Because the intracavity field is proportional to the
input field, we replace Wc with WL.
91
Finally, the transmission coefficient is
2
A
(Eprobe)
2
j$,~
T . 12K, _----.,.-_--'-:(r_+_i(-'-w..,...,.-----:--wL_)-,-)__---,--1
2
g2 + (K, + i(v - WL)) (r + i(w - WL)) ·(2.96)
2.5.4.1 Transmission as a Function of Probe Energy and Cavity-QD Detuning
Figure 2.13 shows (2.96) plotted as a function of probe frequency and cavity-
QD detuning. It depicts an aerial view of a 3-dimensional graph. The detuning
is plotted in units of l-!eV while the probe energy is plotted in units of eV. The
detuning parameter is ~ = W - v. The graph is oriented so that the mode energy
increases toward the rear of the graph. The plot was generated using 9 = 351-18V,
K, = 30 l-!eV, and r = 'Y1 = 15 I-18V. The value for the coupling constant (g)
corresponds to a mode volume of 50 I-!m3with a modest electric dipole moment of
31 Debye. Cavity decay rate (K,) corresponds to observed finesse and decay rate for
the IFQDs is based on observed free-space linewidths. Figure 2.13 is an obvious
generalization of figure 2.10, which showed the avoided crossing for the closed-
system Jaynes-Cummings energies.
92
1.6232
1.6230
1.6228
r
-I
I
I
'.
.---- -
.-
-I~
-
.
--I~-I-+--'--f--~
I I
\\.__r;-~~'
i~~ 4;d~1:'
-T, ,-l.-A b~r. tIt r r l
-",,:,..-' - I I I I
_l_;--I
1--T-1--+---I----ir----:--+--t--1----: - --.f--1--;...I-+-1--t--jJ~-I.-I- --I'~~ .. - !-,-~l_L 1_
--
f---r--+-;-- ,-
1---1-
-100 a 100
FIGURE 2.13: Transmission spectrum, This plot shows an aerial view of a 3-
dimensional plot of the transmission coefficient, as calculated in (2,96), plotted against
cavity-QD detuning (x-axis) and probe energy (y-axis), The choice of parameters in this
calculation is g = 35 ~eV, K. = 30 ~eV; and r = 15 ~eV, corresponding to best estimates
based on anticipated mode volume and observed finesse, The probe energy is plotted in
units of eV, while the detuning is in units of ~eV,
93
2.6 Numerical Studies of Non-paraxial Modes
As mentioned previously, the large angular spread of modes in a near- hemispherical
cavity violate the conditions for making a paraxial approximation. In order
to ascertain the effective mode volume for this experiment, it is necessary to
characterize the modes and identify the strength of the electric field at the location
of the QDs and this requires the use of various numerical methods.
Through collaboration with the theory group of Murray Holland at the
University of Colorado, as well as the theory group of Jens Nackel at the University
of Oregon, we have determined quantitatively the coupling strength between a
QD and the mode of our external microcavity. Calculating the mode function
is nontrivial, since the field for such a high-numerical aperture (NA) cavity is
non-paraxial. As a consequence, this field is not separable into a simple pair of
polarization components. Whereas in paraxial systems the polarization may be
considered to be roughly independent of the spatial mode, this is not necessarily
the case for non-paraxial systems. The non-paraxial field is also not separable into
longitudinal and transverse modes.
The computations deal carefully with the structure of the DBR while treating
the curved mirror as a simple conductor. While both mirrors consist of Bragg
stacks, the field should strike the curved mirror everywhere at normal incidence,
so the effects of angle-dependent phase-shifts should be negligible.
94
FIGURE 2.14: FDTD calculation of micro-ca.vity modes. Calculations confirm that the
mode waist is on the order of a wavelength near the hemispherical limit.
An example of the finite-difference time-domain (FDTD) method, showing the
calculated energy density of the mode in the V = 0 plane is shown in figure 2.6.
The calculations show that even in the presence of angle-dependent phase shifts,
the mode waist in the non-paraxial regime is smaller than one wavelength.
The hybrid method mentioned earlier [20] has produced certain novel predictions.
It relies on numerical calculation of E and H values in a plane (say y = 0) and
azimuthal integration to generate the behavior in other planes.
;..rear the hemispherical limit, qualititatively new types of modes may be
identified. One such mode, a p-polarized "V" mode [18], appears to adiabatically
continue from the Gaussian fundamental as the cavity length is varied. This
method has also yielded predictions of optical spin-orbit coupling due to mode
mixing of nearly degenerate pairs of higher-order transverse modes induced by
95
polarization-dependent phase shifts (upon reflection from layers in a Bragg stack)
that extend into the paraxial regime [19].
Calculations for a 60 I-tm cavity suggest that working slightly short of the
hemispherical limit, where the modes are making the transition from paraxial to
non-paraxial, may be beneficial. It appears that a Gaussian mode under paraxial
conditions transforms into a p-polarized "V" - mode as the cavity length approaches
the radius of curvature of the cavity (L ~ R).
A Gaussian beam has transverse spatial distribution and angular distribution
(i.e. distribution of wave vector directions) that both obey Gaussian distributions.
Figure 2.6 shows the growth of a p-polarized lobe as described in [18] for increasing
cavity length. The parameter Zl = R - L measures the amount by which the
length of the cavity is shorter than hemispherical. It can be regarded as a sort of
paraxiality parameter in the sense that the modes become more and more paraxial
for monotonically increasing Zl'
Calculations of the field strength for various non-paraxial modes suggest that
the field strength for this set of modes has a local maximum at a cavity length
shorter than hemispherical. It is also found that of the low order (Iml = 1) modes,
the maximum of the field strength lies on the optical axis (p = 0).
Figure 2.16a shows intensity contours for the field in the plane y = 0 in a
cavity matching our cavity configuration. The mode plotted is similar to a paraxial
Gaussian mode but with a non-Gaussian angular distribution of the p-polarized
96
"S intensity - \ S intensity--\
>; P intensity - -
.>; \
P intensity - -
~ \I-<
"
I-< cO
"
..., I-<
:.s ..., "~ :.s \I-< \
>, ~ \
..., >, \
'en ..., \>:: 'enQ) >:: \...,
Q) \>:: ...,
...... >:: \
...... \
\
"0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90
Angle (degrees) Angle (degrees)
(a) Zl = O,5!!ID (b) Zl = O.43!!ID
S intensity-- /' S intensity>; P intensity - - . >; I \ P intensity~ ...- .... \ ~ I \
I-<
...- \ I-< I \..., -_ .... ...,:.s \ :.s I \I-< I-<~ \ ~ I \
>, \ >, I \
..., \ ..., I \'en "en>:: \ >:: I \Q) \ Q)..., ..., I \>:: \ >::...... ...... / \
\ / \
\ ...- \
\ .... / \
0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90
Angle (degrees) Angle (degrees)
(c) Zl =O.425!!ID (d) Zl = O.40!!ID
FIGURE 2.15: Gaussian to V-mode transition [17J and [18J. The sequence of plots
shows the modification of angular distribution of the lowest order mode as paraxiality
of the system is reduced. The angular distribution begins as a Gaussian (a) when
R - L Zl = O.40l-l.m. The angular distribution becomes non-Gaussian in (b) and
(c) and eventually the central lobe disappears, leaving a V-mode in (d).
97
1
o
-36
x [11m]
(a) Mode contours for y = 0
o
X (~m)
(b) Polarization vectors for z = 0
FIGURE 2.16: Amplitude at y = 0 and polarization for z = O. In (a), a contour plot
shows the amplitude of the field in the plane y = O. The calculation assumes a mirror with
radius of curvature 60 Ilffi, as used in the experiment, and a cavity length of 59.575 !-Lm,
corresponding to 2.15c. Polarization vectors in the plane z=O are shown in (b).
98
(i.e. transverse magnetic (TM)) component. This mode is the one plotted in figure
2.15c.
Figure 2.16b shows the waist of the mode and demonstrates a couple of things:
• The polarization if? linear and the amplitude maximum at radial polar.
coordinate p = O.
• The polarization is slightly non-uniform, with components along the y-axis for
points along the ±45° directions.
As a result of these investigations we have a good idea of the electric field
strength at the location of the quantum dots. Fortuitously, the simulations also
indicate that the mode with the largest amplitude is a continuation of the paraxial
Gaussian mode, at least for the NA supported by the current set of mirrors.
Another mode, called VTE (V-mode with transverse-electric polarization) has
even higher amplitude on the optical axis but is inaccessible to the current optics
because the angular spectrum extends to angles not covered by the current mirrors.
For the hybrid Gaussian-VTM mode, the polarization on the optical axis is
linear and this may be easily oriented to align with the dipole moment of the
IFQDs.
For cavity lengths very close to hemispherical the amplitude of the field on
the optical axis actually decreases, suggesting an advantage to working slightly
shorter than hemisphericaL A further advantage is that higher-order modes
99
remain spectrally distinct and the situation is better described in terms of a
single mode of the field interacting with a two-level system. At the hemispherical
limit the transverse modes with even parity all become degenerate, and treating
this situation as a two-level system coupled to a single mode of the electric field
becomes less justified. Even though a probe beam can be mode-matched so that
it couples to only a single intracavity mode, the QD will still have a set of cavity
modes with which to interact and the situation would be better modeled by a set of
modes with different coupling constants.
100
CHAPTER III
THE HEMISPHERICAL MICROCAVITY
In this chapter the various components of the apparatus will be described.
Discussion will begin with the microcavity and proceed outward to the larger scale
support structure and auxiliary components. Design, construction and assembly
considerations will be discussed. The results of characterization measurements,
where applicable, will also be provided.
3.1 Flat Semiconductor DBR Containing ODs
One of the reflective surfaces bounding the resonant cavity is a semiconductor
DBR containing GaAs QDs. These structures were designed, fabricated and
characterized by our collaborators, Hyatt Gibbs and Galina Khitrova, at the
University of Arizona.
3.1.1 Semiconductor DBR
Semiconductor planar DBR mirror with exceptionally good surface smoothness
and high reflectivity can be grown by molecular beam epitaxy (MBE) techniques
[56]. The surface roughness on transverse length scales relevant for our needs (rv
1~m) competes favorably with that of the best super-polished dielectric mirrors
101
10"
.-...
N
E
c
E 10"
E
-o
(f)
(L 10.5
--Arizona DBR with IFQDs
10-6 --Super Dielectric mirror
from CalTech
10 100 1000 10000
Spatial Frequency (1/mm)
FIGURE 3.1: DBR surface roughness compared with SuperDielectric mirrors. This
plot shows the spatial power spectral density for a University of Arizona semiconductor
DBR containing IFQDs. A reference SuperDielectric mirror from CalTech is shown for
comparison.
of the type used in atomic CQED experiments. Figure 3.1 shows a comparison of
the two kinds of mirrors - the MBE-grown DBR and a super-polished dielectric
mirror (made by the Kimble group at Caltech). The figure plots the power spectral
density (PSD) of surface roughness versus transverse spatial frequency) measured
with a Wyko interferometer. It is seen that the planar semiconductor mirror has far
larger roughness for low spatial frequencies) while the commercial dielectric mirror
is slightly rougher at spatial frequencies a.bove 1000 mm -1; the region of interest for
our cavity, since the mode waist has a radius of less tha.n 1~m.
102
The DBR in the simulation in figure 3.2 is composed of alternating ~ optical
thickness layers of AlAs and AlxGal_xAs. The simulation is of a sample with mole
fraction x = 0.24 and a stack formula of (LH)30LHHWHHC where L (H) represents
a quarter wave optical thickness layer of the low (high) index of refraction material.
W represents a 4 nm quantum well and C represents a thin capping layer of GaAs
to discourage oxidation of the the spacer material, which is Alo.24Gao.76As. The
second stack adds an anti-reflection coatingl .
Both DBRs used in this work were grown on a GaAs substrate by MBE. Both
samples were grown with a stack formula of LH22 LHHWHHC. One sample involves
an alloy of Alo.2sGao.7sAs for H. The other sample has an alloy of Alo.24Gao.76As
acting as H. Wand C denote layers of GaAs; a QW of 3.86 nm thickness and a
cap-layer of 3.28 nm thickness, respectively.
Several different mirror structures were grown for use in the cavity-QED study.
Several bare DBRs, a DBR with spacer layer but no QDs, and a couple of DBRs
containing IFQDs. Each of these structures was grown on a substrate of GaAs via
MBE.
While the properties of a stack generally depend on the entire arrangement
of layers (indices of refraction, absorption, layer thickness, etc.), in this situation
the behavior can roughly be factored into two distinct pieces. A highly reflective
IThe simplest form of this is a >../4 thick layer of a material with a lower index of refraction.
Optimal performance is achieved when the index of refraction is the geometric mean of the two
materials forming an interface.
103
AR-coating o. top o. a f~ spacer
no AR-co ating
1llllU'lI fUlJl.nl lJlllruUUULUUU lfUl
I ~ ~II'"II! lJ 1,/ 'I/'V\i\/,,,ev,,~ _\,I,
iUlJUUUU1JlfJuLfUUUUUlnJlJlJIJl1UlJUUUl
-I+---+,·~~'I!~IW""~~~ __~__
10
--
8.
:::3
ro 6.....-.
~
~
(J) 4c:
(1)
....,
c: 2
0
10
-- 8
:::3
.
CO
....... 6~
.....
(J)
4c:
(1)
.....
c:
2
0
-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
P sition Jlm]
FIGURE 3.2: DBR reflectivity as a function of distance. The red line shows the index of
refraction of the structure. The black line shows the electric-field intensity (in arbitrary
units). This figure shows the comparison of a DBR with lambda spacer and a DBR with
lambda spacer and an anti-reflection coating. This figure illustrates the basic design,
though it does not represent the particular stack formula for the sample measured in the
accompanying discussion.
104
stack and a spacer layer of thickness equal to one wavelength of optical path length
containing a rough quantum well at its center.
The high-reflector portion of the stack corresponds to (LH)22L. Its terminal
layers are L to provide for proper inclusion of a spacer layer made of the H material
(AlAs). The portion corresponding toHHWHH is a one wavelength spacer layer
with a quantum well at its center. This structure will be discussed in the next
subsection.
The decay of the transmitted field is evident in both plots. The formula that
includes an anti-reflection coating actually has faster attenuation, but the intensity
at the location of the quantum dot is reduced. Because the aim is to maximize
the coupling constant, g, it is important to maximize the field amplitude at the
location of the quantum dot. The samples used in this work do not include an anti-
reflection coating for this reason.
3.1.2 Spacer-Layer Containing Quantum Dots
As mentioned in the discussion in chapter 2, the GaAs QDs that we use are
interface fluctuation quantum dots. They are formed through the influence of
monolayer-thick interface fluctuation during the MBE-growth of a quantum well
(QW), creating elliptically shaped regions about 50-100nm across [63, 22].
In order to integrate the emitters into a monolithic mirror structure, the DBR
is grown with a quantum well located at the center of a one wavelength (of optical
path length) thick spacer layer. The bounding layers of the quantum well are grown
105
100
90
~ 80
.0 70:~
..., 60.(.)
~ 50&!
40
30
20 ..., I I I
650 675 700 725 750 775 800 825 850 875 900
Wavelength (nm)
(a) CAT96
100
90
~ 80
>.
:~ 70
...,
fil 60
E;:1&! 50
40
30
20 ..........-'-............~L.....o..-'- ............~L.....o..-'- ............~.L-o._
650 675 700 725 750 775 800 825 850 875 900
Wavelength (nm)
(b) CAT97
FIGURE 3.3: Reflection spectrum for CAT96 and CAT97. The spectrum for both of the
DBRs used in this work are shown. The dip near the center of the stop-band is due to
the low-finesse etalon mode.
with two minute growth interruptions. This produces surface irregularities in the
quantum well. The purpose of this spacer layer is to enforce that light of the design
. wavelength will have an antinode occupying the location of the quantum well, as
indicated in figure 3.2. Of course, a layer of material of thickness equal to one
wavelength of optical path length can be thought of as a Fabry-Perot etalon. Due
to the small length of the etalon only one mode is present within the wavelength
range probed by this experiment. The reflection spectrum from the mirror exhibits
a dip due to the presence of this etalon mode. Free-space measurements taken at
room temperature show a reflection coefficient of only about 80%. This dramatic
drop is due to buildup of energy in the etalon mode and absorption by the GaAs
QW at the antinode. When the semiconductor structure is cooled the etalon
resonance shifts to higher energies (shorter length scales), allowing for high-finesse
operation of the cavity.
106
CAT96 CAT97
05 05
..0 ..D>-<@0.4 ell 0.4
'---'
>-, >,
+-'
"'""
O.:l'w 0.3
'00>:::
Q) s::
+-' Q>::: 0.2 "'"" 02I-< s::
....:J
-(l..
.....:l0.1 P-. 01
730 7GO 770 790 810 830 850 730 750 770 790 810 830 8GO
Wavelength (nrn) Wavelength (nm)
(<1) CAT96 (b) CAT97
FIGURE 3.4: II/Iicro-PL for CAT96 and CAT97.
The reflectivity of the DBRs is shown in figure 3.3. The theoretical peak is R =
98.97% for both mirrors.
Micro-PL spectra have also been recorded for both, giving a good idea of the
distribution of sizes of the quantum dots. Figure 3.4b shows that the DBR labeled
as CAT97 exhibits a broader distribution, as well as a contribution near 805 nm
from an unidentified source (possibly some impurities in the substrate).
The areal density of the IFQDs is such that tens to hundreds of them are
expected to be found within an area corresponding to the waist of a cavity mode.
This rules out spatial selection as a viable method of choosing an individual QD.
However, nano-PL measurements indicate that certain positions on a DBR exhibit
features consistent with emission from spectrally isolated IFQD.
107
760
Wavelength (nm)
750 740
-+-.,....- -~~,..A____.. .-------..,~.... ..-._~_~~~_~
300l,~~~-----....~~~~~ ~
,.. ..--~ ........... .. -~~--~.~.-
_____________~__...J~~. ~
~~~.~-~~~
_____._v.- .~J.J~..-'\."JL/'---'''_"o~~_
1.63 1.64 1.65
Energy (eV)
1.66 1.67 1.68
FIGURE 3.5: Nano-PL from CAT97, Each horizontal trace corresponds to nano-PL
collected from a different spot on the sample. Each trace is produced by translating the
spot by 300 nm and measuring the PL for a particular integration time, The size of the
spot is between 0.8 ~m and 0.44 ~m, as produced by a focused beam from an objective
coupled to the surface through a solid-immersion lens, The two circled peaks are good
candidates for single, spectrally-isolated quantum dots,
108
3.2 Large Solid-Angle Micro-mirrors
The cavity is nominally a 50 f..tm radius of curvature hemispherical cavity. The
mode volume for this cavity is of the order of 50 f..tm3.
One of the technical achievements of this experiment is the successful construction
of small radius of curvature micro-mirrors. They are constructed of borosilicate
glass through a process that will be described below.
3.2.1 Micro-mirror Fabrication
The substrates for the micro-mirrors are the surfaces of tiny bubbles of nitrogen
formed in molten glass. The process is outlined below:
A. Place 50 micro-capillary tubes in a graphite crucible
B. Bake under dry nitrogen at llOO°C
c. Polish front side with diamond disc to 6f..tm surface roughness to open some
dimples
D. Inspect with a microscope to verify the presence of suitably sized bubbles
polished to the correct depth
E. Polish backside of sample to a thickness of 150~170f..tm
F. Fine polish backside with abrasive slurries to achieve optical finish
G. Coat surface with high-reflectivity dielectric coating
109
The bulk of production was done in our lab: Willamette Hall, Room 242;
University of Oregon. The bubble formation (glass fusing) was carried out using
tube furnaces located in the basement of Klamath Hall at the University of Oregon.
The coatings were produced by Spectrum Thin Films Corp ..
The micro-capillary tubes are commercially available Drummond Mieroeaps. The
glass is manufactured by Kimble Glass. Manufacturer data is given in the table
below.
Glass Type Working Soften Anneal Strain p (glee) n
Pt. Pt. Pt. Pt.
N-51A Boro 1140 785 570 530 2.33 1.49
TABLE 3.1: Properties of borosilicate microcaps as reported by Kimble Glass.
Approximately fifty of these tubes are placed in a graphite crucible, which is
inserted into a fused-quartz tube for baking. The fused-quartz tube is placed into
a programmable tube furnace. A baking program, which consists of a ramp up to
lloooe followed by steady baking at temperature and a slow ramp down (several
degrees per minute) was devised, which optimized the bubble size and density for
our purposes. By adjusting the baking parameters it is possible to produce samples
with bubbles in the range of several tens of microns up to a few hundred microns.
After cooling, the samples are returned to our lab for further preparation. At
this stage, the sample is a rounded chunk of glass containing voids of various sizes.
The sample is bonded to an aluminum plate with CrystalBond 509 wax (from SPI
(a) (b) (c)
110
(d) (e)
FIGURE 3.6: Micro-mirror substrate preparation. Borosilicate micro-capillary tubes
are placed in a graphite crucible (a) and heated under a nitrogen atmosphere in a tube
furnace to produce a lump of glass containing bubbles of various sizes (b). This piece of
bubbled glass is ground down on both sides to produce a slipof glass of suitable thickness
and at least one exposed bubble of suitable radius of curvature and depth (c). The slip of
glass has been rotated so that the flat surface faces the page in (d). A photograph shows
the surface surrounding one such bubble in (e). The photograph was taken under 40x
magnification and shows a bubble with, approximately, a 60llm radius of curvature.
111
Supplies) and ground at a lapping station on a rough diamond wheel (grit size =
15 !im).
When roughly half of the sample has been polished away, the wheel is switched
to a 6 !im-grit wheel to reduce the roughness of the surface of the polished surface
and to more finely control how much material is removed. At this time, the
substrate is inspected under a microscope for candidate bubbles of an appropriate
radius of curvature. Polishing proceeds with periodic pauses to check the depth of
the remaining bubble surface.
Surfactants are introduced in the water stream to help reduce the chance of
glass chips coming into close contact with the bubble surface and adhering due to
van der Waals interactions. We have observed that introducing a small amount of
dish detergent into the water stream reduces the relative frequency of such glass
chips considerably. One may hypothesize that this is due to formation of bilayer
and micelle boundaries separating bits of glass from the bubble surface.
Once the depth of the remaining bubble is equal to about half of the radius of
curvature of a bubble of suitable size, the sample is inspected for glass chips. We
discovered that elastomers may be used to remove some glass chips that remain.
The effectiveness of this procedure depends on the shapes of the contaminating
glass chips. We used a two-component formulation -liquid PDMS mixed with a
cross-linking agent (Sylgard 184 from Dow Corning) - to produce an elastomeric
mold of the substrate. Once the elastomer had cured it was carefully pulled away
112
(c) (d)
FIGURE 3.7: Procedure for removing glass chips with PDMS films. A view of a bubble
surface with glass chip contaminants is shown in (a). First, some PDMS and cross-
linking agent are mixed in the prescribed quantities. This is applied to the surface of the
glass substrate containing the open bubbles we wish to clean (b). After the elastomer has
cured, it is peeled from the surface (c), lifting some glass pieces away with it. In (d) the
surface is shown with one of the contaminating glass chips removed.
from the mirror substrate. This process proved to reduce the number of remaining
glass particles by about 1/2.
After this process is completed, the substrate is flipped over and mounted to an
aluminum disc with Crystalbond 509 wax. The back side is then polished down in
a similar fashion. When the thickness is on the order of 150-170 I-lm, a polishing
pad is mounted on the lapping station and polishing continues, using abrasive
(colloidal silica) slurries. The process terminates with the use of a slurry with a
colloidal suspension of 0.05 I-lm diameter silica particles, leaving a high-quality
optical finish.
It is of interest that the mirror surfaces, themselves, are not polished. Optical
finishes are produced naturally as the hot glass seeks equilibrium. This is important
113
since a good technique for polishing small spherical voids is unknown at this time.
There do exist certain schemes for creating spherical voids by wet-etching through
apertures but these are expected to produce surfaces with much greater roughness
[28].
The rough surface surrounding the dimple is shown in figure 3.6e. This is the
surrounding glass substrate which has been polished by the 6 I-lm grit diamond
wheel. The actual surface of the dimple is out of focus due to the depth of focus of
the microscope objective. It is important not to apply abrasive slurries to this side
of the glass slip, otherwise abrasives will degrade the surface quality of the mirror
substrate.
Due to the tiny ratio between the radius of a bubble and the radius of Earth,
the magnitude of the tidal force at any point on the bubble's surface will be small.
The equilibrium shape of the bubbles is expected to be very close to spherical
for this reason. The sphericity has been measured with a Wyko interferometer,
see figure 3.8a. At the bottom of a dimple, in a circle of 15 I-lm diameter, the
deviations from spherical were found to be less than 10 nm across the measured
region.
3.2.2 Micro-mirror Coating Design
As is clear from the analysis in the theory portion of this dissertation it is
important to have high reflectivity (low I'i:). Moreover, in order to be able to
communicate with the device we need the transmission rate to be much larger than
114
30
-12.5 --w.V ·s.O (1.0 5.0
!
10.0 14.7 mn
o -'r-----,---r----r
o 10 20 30 LIm
(a) lVIicro-mirror sphericity
le+04
r--------
-- Bubble #1
I
Bubble #2
-- Bubble #3
-- Bubble #4
-- Bubble #51-- SuperDielectric I
0.01 "l
.-
N
S
~
S
S
--0
CfJ
~ le-07
Ie-DB
le-O
1 10 100 le+03
Spatial Frequency (mm- l )
(b) Surface roughness of several (5) micro-mirror substrates compared with the same
SuperDielectric mirror as the DBR in figure 3.1.
FIGURE 3.8: Micro-mirror sphericity and surface roughness. This plot shows the
deviations of a micro-mirror from spherical (a) and surface roughness (b), as measured by
a Wyko interferometer.
115
absorption or scattering rates. Briefly, we require that R » T » A, S. Moreover,
there is a requirement that this hold over a wide range of wave-vector directions, so
that high finesse operation can be achieved near the hemispherical limit.
The mirror coatings were produced by Spectrum Thin Films Corporation.
The coatings specifications called for greater than 99.5% reflectivity across a
frequency range of 740 - 810 nm. A coating deposited by a directional source is
expected to exhibit cosine variations in the layer thicknesses. For large angles
(away from the center of the concave surface) the thinning of the layers will lead to
shifting of the stop-band toward shorter wavelengths. In the interest of maintaining
performance out to high angles, it is reasonable to specify a longer wavelength
for the center of the stop-band compared with a flat mirror to account for layer
thinning at higher angles.
To characterize the performance of the coatings at points away from the center
of the concave surface, measurements of the transmission through the mirror
coating were made for meridional beams with varying angles with respect to the
substrate normal.
Figure 3.9b indicates that the reflectivity is large over a large solid angle.
3.3 Mounted Micro-mirrors
To ease handling and coupling into the cavity, the slip of glass containing the
coated micro-mirror is mounted directly to an immersion objective. Norland 88
116
~ 2.0
'---'
40-20 0 20
Angle (Deg.)
(b) T( 8) data
-40
Substrate containing
. .
mIcro-mIrror
(a) Setup for measuring micro-mirror T(8)
FIGURE 3.9: Micro-mirror T(B): Setup and data. In (a) the geometry for measuring
transmission is shown. In particular, it. defines the angle fl. To collect this data, the
beam was focused to a spot and incident at normal incidence to the mirror surface. Each
data point was recorded by changing the angle between the mirror substrate normal and
the optical axis of the probe beam while maintaining normal incidence with the mirror
surface. In (b) the transmission coefficent is plotted for various points along the surface of
a micro-mirror. The data shows that transmission remains less than 0.5% for a half-angle
greater than 40 degrees.
117
optical adhesive has an appropriate index of refraction to be used as a kind of rigid
immersion fluid. It also has low vapor pressure and is suitable for inclusion in a
DRVenvironment. It is only rated for operation between -60°C and 90°C, yet we
have found it to provide good adhesion properties through repeated cycling between
room temperature and an operating temperature of about 16 K.
The correct position (working distance) of the micro-mirror is determined with
the aid of a Twyman-Green interferometer. This allows reliable positioning of the
curved surface to make the job of mode matching easier. Once the correct position
has been determined the adhesive is hardened2 by illuminating under an ultraviolet
lamp. The result is a microscope objective with a highly reflective spherical surface
matched to the wavefronts produced when a collimated beam is introduced to the
objective's aperture.
This objective can simply be threaded into an appropriate mount and pumped
with a collimated beam. If the beam has the correct trajectory then the wavefronts
will automatically match the curved mirror. Mode matching is now primarily an
issue of steering a collimated beam so that it follows the correct trajectory into the
objective.
The mounting process is illustrated in the attending diagram.
2Care should be taken to compensate for the change in the index of refraction of the optical
adhesive as it hardens.
118
Monitor
Beam
splitter
,,
• 3-axis
Micro-mirror translation
~ '~t~_...t =r--:;i sage
,r-\""'- Optical adhesive
(~)
jlmmersion
, , objective (NA= 1.3)
I
CCD camera
o 00
FIGURE 3.10: Twyman-Green interferometer for mounting micro-mirrors. The diagram
shows the alignment setup for the micro-mirrors. The interference pattern is collected
by a CCD and displayed on a monitor. Proper positioning is achieved when the output
interference pattern corresponds to plane wave interference.
119
3.4 The Near-hemispherical Microcavity
The result of moving the mounted micro-mirror within a suitable distance of the
DBR containing QDs is a unique half-monolithic, hemispherical micro-cavity for
semiconductor cavity-QED. For mirror separations such that L ~ R, the mirror
surfaces define a stable resonator. The stability parameters for the resonator are
L
91 = 1--R1
L
92 = 1 - R
2
'
where 0 ~ 9192 ~ 1 for a stable resonator. This cavity operates with 92 = 1
due to the flat mirror. At hemispherical, L = Rl, so 91 = O. For decreasing
cavity length, 91 should increase monotonically and the cavity stability parameters
approach those of a planar cavity. The shape and solid angle of the micro-mirrors
used here prevent L « R and an upper bound exists for 91. In practice, since the
depth of the mirror is roughly half of the radius of curvature, L ~ R/2, so that
o ~ 91 ~ 1/2. The range of possible cavity configurations forms a line segment in
the first quadrant of the plane parametrized by the cavity stability parameters.
The cavity parameters are in a novel range, tabulated in table 3.2. Cavity-
length is limited by the minimum separation allowed by the geometry of the curved
mirror.
This cavity design contains two unique features:
120
FIGURE 3.11: Cavity stability parameters. The black line segment indicates the range
of possible stability parameters for the micro-cavities described here.
Mode Parameter Symbol Quantity
Cavity length (Il-m) L 40 - 60
Finesse F 200a
Waist radius (Il-m) Wo ~1
Divergence angle (deg.) ()D ±40
Flnumber f/# ~1
TABLE 3.2: Micro-cavity parameters. Properties of modes supported by a near-
hemispherical micro-cavity.
aThis should be amenable to increase by an order of magnitude by using higher reflectivity
coatings.
121
1. The use of a concave micro-mirror with high-reflectivity over a large solid-
angle.
2. The use of an integrated DBR mirror containing the QD sample in an
external-cavity configuration. .
The 40 - 60ll-m curved-mirror substrate has a high degree of sphericity and an
excellent surface quality, enabling the application of a custom-designed, multilayer,
dielectric coating with 99.5% reflectivity over a high solid angle. Such large solid
angle is unique compared with, for example, a recently reported half-monolithic
micro-cavity design [59].
The radius of the mode waist, located at the planar mirror, is denoted woo Since
the QD is to be placed in this waist, this radius should be minimized in order to
maximize the coupling between the QD and the field. The angular half-width of
the cavity mode is Be. Diffraction dictates that the smaller Wo is made, the larger
Be becomes. When Wo equals one optical wavelength, the angle Be is roughly 40
degrees.
The effective mode volume ~ff, which depends on the location of the QD,
is defined as a spatial integral of the field intensity, normalized to unity at the
location of the QD. For example, if the transverse spatial distribution of the field
amplitude can be described as a (paraxial) Gaussian function (or equivalently, if
the angular distribution of the mode is Gaussian) with l/e contours that define a
122
spot size w(z) at a position z along the cavity axis, then the effective mode volume
will be ~ff = 7r~5L, as determined in appendix B.
3.5 Thipod Support System
The mirrors of our cavity must be positioned with respect to each other. For
this purpose a tripod system has been devised which will allow positioning of the
fiat semiconductor DBR with respect to the curved micro-mirror. This structure
must allow for coarse as well as fine positioning of the sample mirror. The design
must also allow the sample to be cooled to cryogenic temperatures. To avoid
cooling the entire experiment this mirror must be thermally isolated from the rest
of the experiment.
The requirements for the support system include:
A. Control over translations and rotations
B. Long range of travel in z-direction
C. Low vapor-pressure materials to comply with DRV requirements
D. Remotely operable [via computer]
E. Minimal vibrational coupling to environment
F. Suitable thermal isolation to allow cooling of a minimal amount of material
123
Operation Role
Lateral translations (along i and j) Lateral translations provide a means to
include different populations of quantum
dots in the optical cavity.
'Ifanslation along k 'Ifanslation along k are necessary for
controlling the cavity length, since k
roughly points along the optical axis of
the cavity. It is also necessary to correct
for thermal expansion during cooling,
warming.
Pitch and yaw Pitch and yaw provide for angular
pointing of the cavity's optical axis.
In principle, they provide a way to align
the input optical axis with the cavity's
optical axis. These degrees of freedom
play a role in mode matching.
Roll Roll provides a way to rotate the dipole
moment of the IFQDs with respect to the
polarization axis of the electric field.
TABLE 3.3: This table lists the relevant degrees of freedom for the cavity along with a
description of the role they play in the experiment.
To address item A, the various degrees of freedom and the role they play in this
experiment will be enumerated.
The support structure is designed to manipulate the flat semiconductor mirror
with respect to a fixed curved mirror. 'Ifanslations along k provide for control
over the length of the cavity. Cavity length must be controlled precisely as it
directly affects the detuning between cavity modes and a dot. Additionally, the
large temperature range experienced by parts of the structure when cryogens are
introduced means that the ability to translate the mirror over large distances is
124
Piezo-driven Translation Stage
P iezo cartidges
50-micron stroke
Sample-column
attaches here
FIGURE 3.12: Flexure mount for X-V translation. The structure used to allow X-
Y translations of the DBR is shown. The optical axis of the transmitted beam passes
through the hole at the center of the stage.
required: The position of the fiat mirror changes by more than 100 ~m as the
structure is cooled down from room temperature to about 15 K. Actuation of a
tripod provides for translations as well as pitch and yaw. In practice roll (rotations
of the fiat mirror around the z-axis) turns out to be unnecessary. While it would
allow for orientation of the quantum dot dipole moments, it is much easier to
simply rotate the polarization of the electric field using waveplates.
By placing the fiat mirror in the pivot plane (the plane that includes the
support bearings) the tilt becomes decoupled from translation. This comes at the
expense of the lateral translation. Lateral translation capability is restored by the
inclusion of a simple flexure stage that allows lateral movement of a central flange
with respect to the center of the tripod's base. The face of this structure is shown
in figure 3.12. The face of the structure attaches to the base of the tripod and
contains threaded holes (not shown) that allow the sample column to be bolted to
the center portion of the structure.
125
Because of a lack of position readout on each leg of the tripod we did not
actively control for pitch and yaw. The tripod essentially was used as a piston and
changes in the pitch and yaw were dealt with by adjusting the trajectory of the
input beam.
A series of flanges are mounted to a set of tube components. This provides a
modular way of attaching components together. Extending from the bottom of the
tripod's base, the following components are bolted together:
1. Thbe-shaped piezoelectric transducer
2. Vespel (polyimide) tube for thermal isolation
3. High purity aluminum sample holder
4. Sapphire disk with semiconductor DBR attached
Moving from the end of the list backward, the semiconductor DBR was attached
to a sapphire disk oriented with the c-axis normal to the surface to minimize the
effects of birefringence. Sapphire was chosen for its combination of transparency
and large thermal conductivity at low temperatures. This was attached to a
high purity aluminum sample holder with Epo-Tek T7110 thermally conductive
epoxy from Epoxy Technologies. Aluminum offers a favorable combination of
low density and high conductivity. This aluminum sample holder provides the
attachment points for the cold finger. This is discussed in further detail in section
3.6.
126
A Vespel (polyimide) tube supports a large thermal gradient and allows the
tripod, support structure, and piezo element to remain at room temperature
while the sample and holder are brought to cryogenic temperatures. This stage is
bolted to a tube-shaped piezoelectric transducer. The transducer is a multilayer
Lead-zirconate-tantalate (PZT) stack operated at low voltage (O-150V), which is
responsible for the controlling the cavity length on small length scales.
PZT tube actuator
Vespel tube
Not to cale
AI flanges
99.9999% purity eu
path to cold finger
FIGURE 3.13: Schematic of sample column: This shows a schematic representation
of the column that contains the sample holder and attaches to the base of the tripod.
High-purity copper wires connect the sample holder to the cold finger.
The Invar tripod is supported on a kinematic mount consisting of a set of
ruby bearings that contact a cone, a groove and a plane (respectively). This
properly constrains the position and orientation of the tripod and allows it to be
127
manipulated by adjusting the heights of the support bearings. Burleigh Inchworm
UHVL motors were chosen to control the tripod positions because of a few factors:
1. They are capable of long travel when compared to other piezo based actuators.
2. They do not require any lubricants to operate properly. Thus, they are well
suited to a DRV environment.
3. They can support a modest load even without any drive power. By design,
they clamp down when voltage is absent.
The design of this structure was performed on a computer running TurboCAD.
The alloy Invar36 was selected for the bulk of the apparatus due to its extremely
low coefficient of thermal expansion. Parts were made of Invar36 where possible to
avoid bi-metallic junctions and improve the stability of the system during baking.
An isometric drawing of the assembled mechanical support structure is shown
in figure 3.5. This includes a set of three Burleigh UHVL Inchworm stages designed
for DRV operation. A damping structure consisting of two plates with matching
"V"-grooves separated by 0.5" diameter Viton cylinders is visible, near the bottom
of the drawing. This structure proved to be instrumental in getting the vibrations
of the cavity to a tolerable level.
A photograph taken from the main port of the vacuum chamber is shown in
figure 3.17a. The cold-finger and radiation shield are visible to the right of the
cavity and support tripod.
128
FIGURE 3.14: Hidden-line drawing of the support structure. The figure shows an
isometric line drawing of the support structure. The drawing was produced by TurboCAD.
129
3.6 Cryostat and Cold-Finger
As is mentioned in footnote 1, the existence of the quantum dots as well as
the optimization of their individual coherence times depends upon attainment of
a suitably low temperature. Heat is pumped out via a cold-finger attached to the
aluminum sample holder with a collection of high purity (99.9999% pure) copper
wires.
The cold-finger brings the proximity of the cold reservoir a short distance from
the DBR.
The temperature is maintained by cold helium pumped out of a helium dewar,
through a transfer tube (Oxford LLT) and into an Oxford Ultrastat - a low-
vibration, continuous-flow cryostat. Cold helium gas is drawn back through a
return pathway in the transfer tube, which helps to keep the helium cold while it
is sent to the cryostat. The liquid helium absorbs heat from a heat transfer block
at the end of the cryostat. The temperature of the heat transfer block can be
controlled by throttling the flow rate of the helium, which provides a coarse level
of control, or by using a heater attached to the block. The controller for the system
reads a temperature sensor mounted to the transfer block and can drive the heater
with a proportional-integral-derivative (PID) circuit to control the temperature to
within ±O.1 K. The temperature at the sample holder is measured by a 4-probe
measurement on a diode (Lakeshore DT-470) bolted to the sample holder.
130
One of the difficulties we faced is that a pathway used for heat conduction can
also conduct vibrations into the structure that it cools. In this apparatus, only the
DBR is cooled, while the curved micro-mirror and surrounding structures remain at
room temperature. The cold finger connects only to the aluminum sample holder,
to which the semiconductor DBR is mounted. The length of the optical cavity
is, therefore, sensitive to vibrations in the direction of the cavity's optical axis
that propagate to the cavity from the cryostat. In order to stabilize the cavity,
it was necessary to mount the cryostat with a vertical orientation. This provided
for a slightly longer path for the cold finger, but it effectively decoupled the main
vibrational modes of the cryostat from changes in the cavity length.
A schematic representation of the cooling system is shown in figure 3.15. Before
installation in the UHV chamber, all copper parts were polished and treated with
an acetic acid solution to remove any residual oxide or hydroxyl layers [10] in order
to improve thermal conductivity at the metallic interfaces.
3.7 Ultra-high Vacuum Chamber
The low temperature of the sample necessitates an UHV environment. Atmospheric
components will adsorb onto cold surfaces, spoiling surface quality of the cold
mirror. For this reason, cryostats generally operate in high or ultra-high vacuums.
The main chamber is pumped out by a Varian Turbo-V 70 turbopump backed
by a Varian SD-90 mechanical pump. This arrangement is capable of bringing the
131
Exit window
~
Radiation shield
U O?jectiv~ withIDlCro-IDlITOr
Transfer tube
Cryostat
Thermal
block __
FIGURE 3.15: Schematic of cooling system. The sample column is depicted on the
left side above the objective with mounted micro-mirror. The support tripod has been
omitted. The walls of the DRV chamber are shown, along with the feeclthrough for the
cryostat and the attached cold finger. A radiation shield of polished, OFRC copper
surrounds the cold finger and the cooled sample holder to reduce heating by blackbody
radiation.
132
chamber pressure down to a pressure of the order of about 10-6 Torr. To proceed
beyond a high vacuum to an ultra-high vacuum the turbopump and roughing pump
are isolated from the chamber with a pneumatic-actuated inline valve. Finally,
a gate valve is opened to expose an ion-pump to the main chamber. The base
pressure in this configuration is several nTorr.
The layout of the vacuum system is indicated in the following diagram:
3.8 Summary
The apparatus may be roughly considered to be the preceding materials
and the associated electronics and control software. We may consider this to
constitute the environment in which the experiment takes place. We have identified
experimentally important degrees of freedom and addressed how they are to be
controlled in the experiments.
The laser sources, and beam-control and detection systems will be described in
the next chapter, since the configurations of these are subject to change depending
on which measurements need to be made. Except for the use of two different
DBRs, the components described in this chapter are not subject to change and may
be roughly considered to constitute the apparatus that we seek to manipulate and
whose optical properties we wish to measure.
133
Gate
valve
Turbo
Pump
Valve
Flexible
steel hose
Valve
Vacuum
Cham er
Ion
Pump
Mechanical
roughing pump
Exhaust D HVand UHV
D Low Vacuum
D Atmospheric
FIGURE 3.16: Schematic: UHV system. This block diagram shows the layout of
the ultra-high vacuum system.
134
(b)
FIGURE 3.17: Photograph of the support tripod and cooling system taken from the
main port of the vacuum chamber (a) and a magnified view (b).
135
CHAPTER IV
EXPERIMENTS AND DATA
4.1 Introduction
In this section, account will be made of efforts to observe vacuum Rabi
splitting in this system. A description of both the input-beam preparation and
the collection/detection system will first be provided. Later, we examine some data
from each of three lasers used as a probe for the microcavity system.
4.2 Input Beams
Here we briefly describe how the lasers are prepared for entry into the CQED
system.
4.2.1 Laser Sources
In order to measure the transmission spectrum for the coupled cavity-QD
system, two different diode lasers were used. One of them, a Velocity 6312
tunable diode-laser from New Focus, was capable of emission over a range from
771 - 789 nm and was used as a reference laser to fix the length of the optical
cavity. Because it could be tuned to energies lower than either the ground state
136
excitons of the IFQDs or the continuum of QW states, it can be used without fear
of generating excess excitations in the CQED system.
In the initial runs, a tunable diode laser from Newport (a 1040A motor-driven
Littman-Metcalf) was used as a probe. It was capable of emission across a range
from 748 nm to about 759 nm. Later, a Ti:Ab03 and another diode laser (LiON
by Sacher Lasertechnik) were used to investigate the system in the vicinity of
the design wavelength of the spacer-layer. The Ti:AI2 0 3 laser was determined to
have an inhomogeneous linewidth in excess of 2 GHz as measured with an optical
spectrum analyzer (a scanning Fabry-Perot). Because the output of this laser was
found to be spectrally broad due to the presence of multiple modes, data from
these scans will not be included here. These scans did not yield any useful data.
In an effort to provide clean mode matching from free-space modes into micro-
cavity modes, both lasers (probe and reference) were coupled through a length
of polarization-maintaining single-mode fiber (SMF). The output from the fiber
should correspond closely to a single Gaussian mode.
By preparing both probe and reference beams with orthogonal polarizations
they can be efficiently combined onto a common path with a polarizing beam
splitter (PBS). Both beams are then launched into a polarization-maintaining
SMF. This maintains the purity of the polarizations (along with relative orientations)
and provides a common optical axis for each beam as each one couples to a free-
space mode at the other end of the fiber.
137
Mirror fFif'E.~=::::;
Anamorphic
prism pair.I~j\;m;j"""",---
Reference laser
FIGURE 4.1: Laser sources filtered by single-mode fiber. Probe and reference lasers are
prepared with orthogonal polarizations and combined on a PBS. Both beams then enter a
single-mode fiber.
Because both the reference laser and the probe laser are diode lasers whose
, beams emerge emerge with large aspect ratios, anamorphic prism-pairs are used,
as indicated in figure 4.1, to correct beam shape and improve coupling into the
optical fiber. Each beam is passed through a Faraday isolator to prevent retro-
reflected light from entering the laser cavity. An acousto-optic modulator (AOM)
is inserted into each beam to allow for lock-in detection at the output. The probe
AOM operates at 80 MHz, the reference AOM at 100 MHz.
Examining the beams that emerge from the SMF, we verify that the beams
have smooth Gaussian spatial distributions and well-defined, linear polarization.
The probe beam is polarized vertically (V), with respect to the table and the
reference beam is polarized horizontally.
138
These beams are both roughly collimated. Because a microscope objective is
used to collect the output from the SMF, the focal length for the two beams is
slightly different due to dispersion and different indices of refraction for the two
beams.
A periscope brings the combined beam to the correct height and a pair of
'steering mirrors is used to send it through an optical window in the side of the
DRV chaber. From here it strikes a mirror at 45° and propagates vertically,
approximately along the cavity optical axis defined by the orientation of the flat
mirror. The beam now enters the Zeiss Plan-Neofluar immersion objective with
integrated micro-mirror.
4.3 Output Beams
The emission from the cavity in the forward direction is collected by an aspheric
collection lens. Aligning the collection lens is a delicate matter because it must be
aligned while the mirror separation is small enough for a stable resonator to exist
so that there are modes to collimate.
The collimated modes exit the DRV chamber through an anti-reflection-coated
window. The cavity emission, which contains both probe and reference beams, is
steered through a pair of irises that defines the optical axis of the collection system.
The probe and reference beams are then separated in one of two ways, depending
on whether active stabilization is required or not.
139
?I'ljn~
Iris
~Vlicro-cavj ty
IIris
Probe
DoubiC'l bearn
(f = l()() rnm).
Dillmction
grating f\
-- 7\
~ -·Z--.7~,.
..-!' '-
,:;.
7
(1~~/A
~ \ Del,ector,;
Aspheric " -- '
collection ) " ~f. Doublet
lens ~ U\ (f = YO 111111)
Reference
beam
FIGURE 4.2: Detection setup for active control of cavity length. The emitted light is
collected by an aspheric lens and steered through a pair of irises. The probe and reference
beams are separated by reflection from a diffraction grating (with 1800 lines/mm).
Doublets are used to image the grating to photo-detectors locat.ed in each first-order
diffracted beam. The solid lines indicate the short wavelength end of the respective
tuning ranges. The dashed line indicates the long wavelength end, The doublets image
the surface of the grating to the detectors, regardless of which wavelength of light is used
for the reference or probe.
4.3.1 Active Configuration
1£ active stabilization is used then a grating must be used to separate the two
beams based on wavelength, A blazed grating separates the two beams and each
one is sent to a separate detector.
Doublet lenses are used in each first-order diffracted beam to image the surface
of the diffraction grating to the surface of a photo-detector. These doublets have
sufficient numerical aperture to accept angular variations in the diffracted beams
due to changes of the laser wavelengths.
In this configuration, we implement a form of the Pound-Drever-Hall method of
laser stabilization [3]. The version of this method used here involves slow dithering
140
(at 5 kHz) of the laser frequency around the target wavelength. The frequency
modulation is produced by dithering the laser frequency by several GHz via a
sinusoidal signal applied to the piezo element attached to the laser cavity's end-
mirror.
An error curve is produced by mixing the cavity transmission signal with the
sinusoidal signal used to drive the piezo. This is done using a lock-in amplifier.
By applying appropriate gain and phase shifts to the error curve, a control
signal is produced and fed to the piezo actuator that controls cavity length. A
circuit was built up using a board designed by Dan Steck (Univesity of Oregon) for
the purpose of driving laser diodes. This circuit contains proportional and integral
response stages and permits stabilization against thermal drift and low-frequency
noise over long time scales.
4.3.2 Passive Configuration
There is a loss of efficiency from collecting the light from a single diffraction
order. (Typically, one can expect to get at most half of the energy in the first
diffracted order.) If active stabilization is not needed, then efficiency can be
improved by using a PBS to separate the two polarization components. In this
configuration, there is too much cross-talk to run both beams simultaneously. The
reference laser is used to set the cavity length at the beginning of a scan and then
it is blocked while the scan is carried out.
141
PBSAspheric
collectioll0/ Ic_'ns_'--- DOllblel'~t. (r=,oo,"m~
~f\eferencc
Micro-em'ity
(I~/
Prol'l'
lJCfUll
DOllblel
(f = 70 mill)
bram
FIGURE 4.3: Detection configuration for passive cavity control. A PBS separates the
transmitted light into two orthogonal polarization components. This gives more detail
about the polarization properties of the mode, but does not permit the reference beam to
occupy the cavity while data is being collected.
Once vibration issues had been properly dealt 'Nith by inclusion of Vi ton
supports, the cavity was found to be stable enough on the time scale of a scan, so
this configuration was used to collect all data after July 2, 2007.
4.4 Mode Matching
From experience with paraxial modes it is expected that higher order modes will
distribute the energy across a larger radial distance and provide a larger effective
spot size and consequently, a larger efi'ective volume. Numerically, modes with
, higher total angular momentum (necessarily higher order modes) have turned out
to have smaller field strength than the on-axis strength of the VTM6 mode, In
accordance with the findings in section 2.6, the mode exhibiting a small spot and
non-Gaussian angular spectrum should give the best coupling strength.
142
At the hemispherical limit, the modes sort themselves into two groups, according
to two-dimensional parity inversion symmetry, each separated from the next by half
of the free spectral range. In order to be able to probe one mode alone, even when
the cavity length is such that two or more modes align at the same frequency, it is
important to be able to exercise some control over the mode matching from free-
space traveling modes into modes of the micro-cavity.
Angular misalignment of the input beam will tend to put energy into odd
modes. Consider that the phase front of the input beam will exhibit a gradient
when compared with the equipotential imposed by the surface of the mirror. I.e.
the phase at the equipotential surface will have an advanced lobe and a retarded
lobe.
Translational misalignments of the input beam with respect to the cavity optical
axis will tend to drive combinations of the lowest order odd-modes along with the
fundamental.
To lowest order, driving the odd modes can be eliminated by placing the input
beam onto the correct trajectory; centered properly and angularly aligned to be
collinear with the cavity optical axis. This can be accomplished with a pair of
steering mirrors.
Not including the fundamental, which simply has a constant phase across a
certain region of the input mirror, the lowest order even modes correspond to errors
in the Rayleigh range of the input beam. The input beam should be the correct
143
II = 500mm h = -200mm
D = d1 + d2 = 300mm
FIGURE 4.4: Lens system schematic. The total distance between L1 and L3 may
remain fixed. Different configurations may be achieved by modifying, for example, d1 .
Gaussian to match the curvatures at the two mirror surfaces that form the cavity,
thus it should have a Rayleigh range that matches the cavity.
Running some calculations of ray transfer matrices led us to a configuration
of lenses that allowed a range of modification of curvature without substantially
altering the diameter of the beam at the location of the cavity. The arrangement of
lenses is shown in figure 4.4.
The effect of the lens system is shown for three different configurations in figure
4.5. Figure 4.5d shows the reduction of the transmission spectrum to a single
mode output. Conveniently, the reference laser also experiences effectively single-
mode throughput in an overlapping range of configurations (i.e. we see single-mode
throughput for d1 = 90mm).
144
10 8
7
8
6
.e 6 .e 5
~ ~ 4
~ 4 ~
III "00 3c: c:
~ .$2 E 2
0
0
-2 -1 ~~
0 2 3 4 5 6 7 8 0 2 3 4 5 6 7 8
Time[s) Time[s)
(a) (b) d1 = 30mm
IV-I
.',1, .1 ,,;.,
"
2
8
7
6
.e 5
~ 4~
III 3c:
.$
E 2
1
0
-1
o 345
Time[s)
(c) d1 = 50mm
6 7 8
10
8
.e 6
~
~ 4
III
c:
.$
E 2
0
-2
0 2 3 4 5 6 7 8
Time[s)
(d) dl = 90mm
IV-I
~ Ii.,. ... .L w.,
2
10 IV -I
8
.e 6
.....,.
-e
~ ~
~ 4 ~
III "00
c: c:
.$ .$
E 2 E
..,li
'"'0
-2
0 2 3 4 5 6 7 8
Time [s]
(e) d1 = 150mm
10
9
8
7
6
5
4
3
2
1
o
-1
o 345
Time [s)
(f) d1 = 170mm
6 7 8
FIGURE 4.5: Effect of the lens system. The effect of the lens system on the Newport
laser is shown for 5 different configurations. For reference, (a) shows the transmission
spectrum without lenses. All scans cover a range of 750 - 758 nm of wavelength. For a
fairly broad range of values of d1 around 90 mm the throughput is effectively single-mode
(i.e. the higher modes do not rise above the noise floor.
145
4.5 Cavity-Modified Photoluminescence
In order to check the population and distribution of dots on the sample, we
measured the cavity-modified PL. We pumped the dots with a red diode laser
at 658 nm coupled through the SMF shown in figure 4.1. Th~ pump laser was
inserted into the probe arm with a collimating lens and an anamorphic prism
pair toimprove coupling into the SMF. The SMF is not single-mode for the
pump wavelength. Nonetheless, it provides good collinearity of the pump with the
reference laser path.
The cavity mirrors are not high reflectors at the wavelength of the pump. The
free pump beam emerges from the other side of the cavity and is separated from
the PL light by a colored-glass filter that transmits light longer than CHECK THIS
700 nmCHECK THIS. This allows the PL to pass unimpeded, while absorbing the
transmitted pump light.
In order to analyze the polarization properties of the PL, a polarizer was placed
in the path of the beam; just before the fiber in figure 4.6. The peak signal was
recorded for a set of polarizer angles to determine the orientation corresponding
to maximum oscillator strength. This provides an in situ method of verifying the
orientation of the dipole moment for the IFQDs. The result of this investigation is
shown in figure 4.9.
146
Spectrometer
___~Mirror
I:.- -T - - - ~~
He-Ne Laser I I
I I
I_I Immersion objective
A
6 L fil ~ Micro-cavityt ,~ ow-pass tel' .f-.~
LN2 cooled CCD ~ i l r
. ~ ~ • _- .:- _-~~ ~~~ .:- 1-~Iirror
Smgle-mode fiber (;/ ~
FIGURE 4.6: Cavity-modified PL - setup. This figure shows the setup for measuring
cavity-modified PL. A red diode laser pumps the QDs, producing luminescence. Resonant
luminescence will be collected into cavity modes and transmit through both cavity
mirrors. PL is collected from the side of the cavity bounded by the semiconductor DBR.
After transmission through a low-pass filter that screens out the pump light, the PL is
coupled through a single-mode fiber and sent to a spectrometer where the frequency
components of the PL are resolved and integrated on a liquid-nitrogen-cooled CCD.
The signals are peaked around a vertical polarization (900 ). This verifies that
the primary dipole moment of the IPQDs points in a direction corresponding to
vertical polarization of the input beam.
4.6 Using the Newport Laser as Probe
Under the original scheme, the New Pocus laser acts as a reference and the
Newport laser is tuned between 748 nm and 760 nm to look for vacuum Rabi-
splitting.
Figure 4.10 shows the complete absorption of a mode near the hemispherical
limit under a small change in the cavity length. This indicates that the absorption
6000 ,------------------------------,
........
~
--J
783
-_. - - ._----
778
--II -----
768 773
Wavelength (nm)
763
I!-~o -1-1--------,--------r----r-----...,.-------lL...:...::::~~
758
1000 --Hl--H.lt-H----'l- ++--l.
5000
Short Cavity PL
~ 3000
'iii
c
~
-
-= 2000
..J
0.
~
~ 4000
~
.Q
~
"'-
FIGURE 4.7: Cavity-modified PL - short cavity. This plot shows PL emitted by intracavity IFQDs. The cavity has been
shortened so that the transverse mode spacing is larger than the linewidth of the spectrometer. The QDs are pumped with a
red diode laser at 658 nm. The pump power is 318 ~W.
4000
Wavelength (nm)
f-'
~
00
""""""'';'", ~""'"""""""""II. of/l"l
770760750
3500
1500
I i I I1000 I I iii I i 780 790
--:- 3000
-0
!....
«
--
.q 2500
C/)
c
Cl)
......
c
--' 2000
Q..
FIGURE 4.8: Cavity-modified PL -long cavity. Compa.rcc! with figm(~ 4.7, the spectrum here is much simpler. All emission
comes out in peaks spaced by half of the cavity FSR.
149
2500
1500
Ii I
I II I i;
! II Ii 1\ II A)0UULA.,JUW l..JlJL.....-.
750 755 760 765 770 775 780
Wavelength (nm)
(b) 30°
4000
.--...
.0 3500I-l
ell
----->, 3000...,
.~
Q) 2500...,
l=1
......
~ 20000..
1500
Wavelength (nm)
(d) 90°
4000
750 755 760 765 770 775 780
~ 3500
-----
0:; 2000
.Q 3000
.~
Q)
...,
l=1
I-<
4000
..ci 3500I-l
ell
----->, 3000:';:;
00
l=1
Q) 2500...,
l=1
......
~ 2000
0..
1500 I I I I~V.....Jc...A.,/I_Jl-v LL........,
750 7·55 760 765 770 775 780
Wavelength(nm)
(a) 0°
4000
750 755 760 765 770 775 780
Wavelength(nm)
(c) 60°
1500
~ 20000..
.--...
.0~ 3500
----->,
:';:; 3000
00
l=1
$ 2500
l=1
I-<
FIGURE 4.9: Polarized PL from intracavity IFQDs. This sequence of PL spectra shows
the effect of rotating a polarizer introduced into the beam measured by the spectrometer
in figure 4.6. The polarizer angle is measured with respect to the table horizontal.
150
2 4 6 8 10 12 14 16
(b) One P-ltiilik[sl
0.014 ,--,----,----------.-..,.------r------r=fT=='ll
0.012
0.01
....., 0.008
-e
~ 0.006
::-
'jg 0.004
2
.s 0.002
o
-0.002
-0.004 '-'---'---'--'----'-'----"------'-----"----'----
2 4 6 8 10 12 14 16 0
(a) Two }lliltIk1.SI
0.014 r---r---,--------,---,-----,----;:::::::F,====n
0.012
0.01
£ 0.008
~ 0.006
~
'jg 0.004
2
.s 0.002
o
-0.002
-0.004 '-----'-'---_-'-----------"--'------'-'--------'-_-'-'-'-~'--'------.J
o
FIGURE 4.10: A missing peak. Two scans at the same location on the sample but
different cavity lengths. Both scans cover a wavelength range of 748 - 763 nm at 450 nW.
The longer wavelength peak in (b) has been completely absorbed.
spectrum is non-uniform and it should be possible to locate regions in which only
one IFQD interacts with the mode.
The cavity in figure 4.10 has L ~ R. We have also seen narrow absorption
features interact with groups of modes in a slightly shorter cavity. In certain places,
where narrow absorption features overlap with modes of a shorter cavity, we can see
similar behavior in more detail.
Figure 4.11 shows the presence of a narrow absorption feature as revealed by
scanning a group of modes through the spectral region containing the absorption
via incremental changes in the cavity length. In this set of scans we can see the
gradual absorption of the fundamental and a gradual increase in the intensity of
the neighboring higher-order mode. While situations like this were encountered on
a few occasions, we were never able to observe any splitting of the fundamental.
151
(a) (e)
(b)
(c)
(f)
(h)(d)
753 754 755 756 757 758759'760 753 754 755 756 757 758 759 760
I I
Probe Wavelength [nm]
FIGURE 4.11: Narrow absorption tuned through modes. A set of scans with
incremental increase of cavity length is shown in (a) - (h). The dashed lines indicate
the spectral location of the absorption feature.
152
4.7 Using the New Focus Laser as Probe
The spectrum of micro-PL from the IFQDs (figure 3.4) shows a long, inhomogeneously
broadened tail out to about 775 nm. This indicates a small population of dots with
longer lengths, suggesting a better possibility for finding spectrally isolated dots,
compared to probing near the maximum of the inhomogeneously broadened PL
signal. Additionally, the lower density of dots in this spectral range is expected
to improve dissipation induced dephasing rates. The idea is that lower abundance
of neighboring dots at or below the energy of a given, excited dot should lead
to reduced non-radiative transition rates from a Fermi's Golden Rule type of
argument. However, moving to longer wavelengths also reduces the coupling
strength between the cavity mode and the dots due to movement of the antinode
out of the plane of the QW.
4.7.1 Splittings in Transverse Modes
Some of the data collected using the longer wavelength New Focus laser
exhibited splittings in the higher-order transverse modes. These modes are already
known to be non-degenerate, so this is not intrinsically new behavior. However,
there are still some interesting features in this batch of data.
There appears to be a splitting of the higher-order transverse modes that is
not easily seen with the other probe lasers. This splitting does not appear to have
153
0.18
0.16
0.14
0.12
-e 0.1
.!!!.
.~ 0.08
c:
~ 0.06
0.04
0.02
0
1 ~ . .1
~ ~
0.12
0.1
0.08
:ci
~ 0.06
~
c: 0.04~
0.02
0
I~ ............... 1
11~
'.
k
20 40 60 80 100
Time[s]
(a)
I~ ............. 1
N
j \....L",,,,,,,,, <;ki~~
55 60 65 70
Time[s]
(d)
1008040 60
Time[s]
(b)
20
-0.02
o
0.12
0.1
0.08
€ 0.06
.!!!.
~
c: 0.04$
.E
0.02
0
-0.02
50807570
Time[s]
(c)
65
-0.02
o
0.18
0.16
0.14
0.12
€ 0.1
.!!!.
f;- 0.08
Ul
c:
$ 0.06
.E
0.04
0.02
0 ."
-0.02
60
FIGURE 4.12: Doublets except on the fundamental. Long range scans are shown in (a)
and (b). The doublets are clearly evident on the N=2 modes in (c) and (d), which show
portions of the same scans in finer detail. The laser scan rate is 1 A/s.
154
0.12 0.12 0.14
0.1 0.1 0.12
0.08 0.08 0.1
.., ..,
~ € -e 0.080.06 .!!!. 0.06 .!!!.
f f :; ~ 0.060.04 0.04 I~ 0.04
0.02 0.02 0.02
-0.02
0123456
(atme [5)
-0.02 '---"~~----,:-~~-:---,~--.J
8 9 10 0 1 2 3 4 5 6 7 8 9 10(bYime [5)
-0.02 ,---"~~~~~~~--.J
o 1 2 3 4 5 6 7 8 9 10(CYime[S)
FIGURE 4.13: Sacher single peaks. Data from the early part of the run on
November 17, 2008.
anything to do with interactions with QDs due to its appearance at many different
locations and cavity lengths.
4.8 Sacher LiON as Probe
The Sacher laser lases over more than 10 nm of wavelength, centered at 765
nm. It appeared to offer the best chance to see strong coupling in this experiment.
Initial scans with the Sacher diode laser exhibited single peaks, reminiscent of a
lot of data taken with the other lasers.
Data from the end of the run started to exhibit interesting features.
Beyond this date, all wavelength scans exhibited double peaks on every mode
order. Representative scans are shown in figure 4.8
In order to ensure that the laser was lasing in a single longitudinal mode, an
optical spectrum analyzer was used to measure the spectrum of the Sacher laser.
The spectrum didn't show any evidence of multiple frequency components and this
0.012 r-......~~~~~FiT===n
0,01
0.008
0.006
0.004
0.002
6 8 10 12 14 16 18 -20
TIme lsi
(a)
0.Q1
0.008
0.006
i'
.!!. 0.004
~ 0.002~
,0.002
-0.004 L--~~~.
o 2 4 6 8 10 12 14 16 18 20
Timels}
(b)
155
0.05 r-~-~~~~~---.""",==n
0.045
0.04
0.035
i o~~~
._! 0.02
-:: 0.Q15
0.01
0.005 o<nc- 1/"'"".,.\1"',,,,,
o
-0.005
o 2 4 6 8 10 12 14 16 18 20
Time[s]
(e)
FIGURE 4.14: Sacher strange features. The data began to exhibit some non-Lorentzian
lineshapes.
0.05
0.04
i' 0.03
!l!-
1 0.02
~ 0.01
-0.01 L---~~~~~_~~~~--'
o 2 4 6 8 10 '2 14 16 18 20
(atme [s)
0.03
0.025
0.02
i'
!l!- 0.015
f 0.01~
0.005
2 4 6 8 10 12 '4 16 ,8 20(b) Time Is)
0.5 ~
0.4
i' 0.3
!l!-
~ 0.2
~ 0.1 ~,
,0.1 L-~~~~_~~~~~e....-J
o 2 4 6 8 10 12 14 '6 18 20
( eTmelS)
FIGURE 4.15: Sacher - double peaks. Now the data shows double peaks persistently,
across multiple transverse orders.
0.025
0.02
0.015
.......
.0
....
~ 0.01
>-
+-'
'en 0.005c(J)
+-'
c
0
771 771.5 772
Wavelength [nm]
(a) Sacher
0.03
l"""";'" 0.025
..0
s-
ea
........ 0.02~
'00
c 0.015(J.)
.....
c
0.01
0.005
o
771 771 .5 772
Wavelength [nm]
(b) New Focus
156
FIGURE 4.16: Sacher vs. New Focus spectra. The two scans overlap and are carried
out at the same cavity length and transverse location on the DBR.
led to a temporary suspicion that the double peaks were associated with some kind
of cavity effect.
Comparison of overlapping portions of scans carried out with the New Focus
laser as probe and the Sacher laser as probe demonstrated that the persistent
doublets were due to the Sacher laser, not a property of the CQED system.
4.8.1 Length Scans
The final attempt to observe vacuum Rabi splitting employed the Sacher laser
as a fixed-wavelength probe. Rather than scanning the probe energy, transmission
as a function of varying cavity-QD detuning was measured. This did, indeed
produce single, Lorentzian-like peaks.
157
0.025
V
0.02 H
0.015
~
..ci
...
~ 0.01
.c:-
'00
0.005c:Q)
-E
0
-0.005
-0.01
0 2 4 6 8 10 12 14 16 18 20
Time [5]
(a)
0.035
0.03
0.025
:e 0.02
~ 0.015
>.
-'00 0.01c:
Q)
-E 0.005
0
-0.005
-0.01
0 2 4 6 8 10 12 14 16 18 20
Time [5]
(b)
FIGURE 4.17: Length scans with Sacher laser at 765 nm. Two scans are shown: with
the mode-matching optics inserted (a) and without (b).
158
Figure 4.17 shows scans at the same probe wavelength with and without the
mode-matching optics. The peaks show no evidence of the doublets evident when
tuning the probe wavelength.
Some scans did show doublets even for the fixed probe wavelength under certain
conditions.
Unfortunately, as shown in figure 4.18, these doublets appear to be associated
with certain unstable wavelengths of the probe laser. TIming the wavelength
slightly causes the doublets to disappear, while large translations across the surface
do not have an effect.
159
0.01 0.01
0.008 0.008
0.006 0.006
'C 'C
.a
-e~ 0.004 ~ 0.004
.c .c
.iii
0.002
'iIi
0002c cQJ QJ
C C
0 0
-0.002 -0002
-0.004 -0.004
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [5] Time [5]
(a) (52.0,17.8) (b) (52.0,19.0)
0.012 0.01
0.01 0.008
II0.008 0.006
-e 'C0.006 -e
tIl~ ~ 0.004.c 0.004 .c'iii 'iIic c 0.002
OJ 0.002 2 IC c
0 0
-0.002 -0.002
-0.004 -0.004
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
Time [5] Time Is]
(c) (52.0,28.0) (d) (52.0,43.5)
0.015
0.01
0.005
o
-0 .005 L---,---,-_~--,-~_,--~~_,----,
o 2 4 6 8 10 12 14 16 18 20
Time [5]
(e) (52.0,43.5)
FIGURE 4.18: Length scans for fixed probe at various locations on the sample. The
probe wavelength is fixed at ..\ 765.17 nm. The coordinates that label each subfigure
indicate the voltage applied to the X-Y piezos. Each volt corresponds to roughly 300 nm
of translation. Doublets are evident in each of (a), (b),(c) and (d); each of them separated
by at. least. several IFQD l(~llgths. The final scan (c) is takcll at. the same location as (d)
but for a probe wavelength of /\ = 765.19 nm.
160
CHAPTER V
CONCLUSIONS AND OUTLOOK
In this chapter we seek to identify the accomplishments made in pursuit of our
goals and to try to take stock of the difficulties that prevented the ultimate goals
from being realized. We will also point out improvements to the apparatus that
may help others wishing to pursue studies with a similar apparatus.
5.1 Conclusions
We successfully designed and fabricated micro-mirrors exhibiting high reflectivity
over a large solid angle. The bubbles used to form the substrates for these mirrors
range in size from 30 ~up to around 200 ~ in radius. The highly curved mirrors
offer high reflectivity over a large solid angle. We achieved mirrors with a half-angle
of 60°. The mirror coatings were measured to have R ~ 99.5% out to about a 40°
half angle. These micro-mirrors can readily be paired with a wide array of other
mirror geometries. When paired with planar mirrors they can readily support non-
paraxial modes. The mirror substrates can be ground to retain an arbitrary portion
of a sphere, though at this time it's not known what the limits are for currently
available coating techniques. Techniques like atomic layer epitaxy (ALE) offer
highly conformal coatings, but tend to have lower surface quality that exhibits
161
higher scattering losses than conventional coating techniques. In any case, the solid
angle supported here already provides for attractive confinement of the transverse
field and could be used with a planar dielectric mirror to investigate predictions of
non-paraxial mode properties such as described in [19, 18].
We also assembled a 5-axis support structure, capable of translation in 3
dimensions as well as pitch and yaw control. As described in chapter 3, roll
is experimentally unimportant due to freedom (and ease) of controlling the
polarization axis of the incident field.
We attached one side of the cavity to a cold-finger using high-purity copper
wires. With high-purity copper components and an OFHC-copper radiation shield,
we succeeded in cooling the sample to 15 - 17 K.
We were able to cut down the vibrations in the cavity to be on the order of an
angstrom on short time scales, as evidenced by the amount of noise on the peaks in
transmission scans. The linewidth of the mode in terms of cavity length depends on
the probe wavelength. For a probe at 765 nm, the FSR in terms of cavity length is
f).L ~ 383 nm. The linewidth is then 6L = f).L/F = 38230~m = 1.9nm. Noise at
the top of the transmission peaks is small compared to the peak height, indicating
small fluctuations in cavity length during the time scale for the laser to cross a
given peak. The Lorentzian appearance of the transmission peaks indicates that
the amount of noise introduced by fluctuations of cavity length are small, at least
on the time scale it takes for the probe to traverse the resonance. For fluctuations
162
in cavity length of bx = 0.15 x b£ = 0.3 nm we would expect to see fluctuations of
about 2.2% in the amplitude at the peak of the resonance (assuming a Lorentzian
lineshape) .
We were also able to collect PL from the IFQDs in situ. Analyzing the polarization
of the PL allowed us to verify the axis corresponding to maximum oscillator
strength and align the polarization of the probe along this axis.
Scans made with the Newport acting as probe generally showed broader
linewidth, consistent with coupling to more absorbers. At longer cavity lengths
we could occasionally see places with selective absorption of a single mode, as in
figure 4.10, or increases in the transmission of certain modes in the vicinity of an
absorber, as in figure 4.11. This indicates that the spatial resolution is high enough
to observe highly non-uniform absorption features. However, no indication of a
splitting was ever observed for the fundamental.
The New Focus laser exhibits superior tuning stability but worse passive
stability (e.g. against thermal drift) compared with the Newport laser, which was
selected to be used as a probe laser. A more ideal scenario would have allowed the
Newport laser to act as the reference laser and the New Focus to provide mode-hop-
free tuning. However, availability of laser diodes for the respective lasers made our
chosen configuration the only possibility.
Using the New Focus laser as a probe did reveal some interesting features,
such as a splitting in the higher-order modes. Again, as for the Newport laser, we
163
never saw any evidence of a splitting for the fundamental. The persistence of the
splittings in the higher-order modes across many cavity lengths and positions on
the sample suggest that these effects have something to do with the mode structure
of the resonator than with an interaction with QDs. In general, the scans using
this laser as probe exhibited lower absorption but also fewer features like selective
absorption of individual modes.
The initial data taken with the Sacher laser showed single peaks. Subsequent
data indicated that the split peaks have to do with scanning behavior of the
Sacher laser. This is further suggested by the ability to see single peaks for scans
of cavity length with a fixed probe wavelength. Unfortunately, the confirmation
that the laser scans were unreliable came after a lot of time spent ruling out other
exlanations.
A set of cavity-length scans was collected for various positions on the sample
and various wavelengths close to 765 nm. This did uncover doublets, but they
turned out not to depend on the position on the sample. The immediate disappearance
of the split peaks upon a small change in the probe wavelength indicates that
this too was due to an artifact of the probe output. It is certainly possible that
something could have been uncovered with further, extensive length-scans after
careful characterization of which wavelengths produce the doublet behavior.
A better alternative, requiring both time and money would be to get the laser
repaired. Unfortunately, due to time constraints, by the time the doublets had been
164
definitively traced to the Sacher laser there was not enough time to return it for
repair.
Given the magnitude of the electric dipole moment, approximate mode volume
and cavity and QD dephasing rates, we have a system that should be capable of
strong coupling. Of course, the potential to observe signatures of strong coupling
was not realized in this work and some attention should be given to possible
reasons for this.
It should be acknowledged that there is quite a large parameter space to search
through. In order to find a single dot, we must find the right location on the
sample such that a single quantum dot is within the mode waist and has a size
different enough from the others that it can be considered spectrally isolated from
its neighbors. Furthermore, the design of the spacer layer means that not just any
isolated dot will do. The dot should have an energy corresponding to the etalon
mode for the spacer or else the cavity mode will have an antinode away from the
position of the quantum dot when it is resonant with the exciton transition - i.e.
the goal is to find a spectrally isolated dot that matches the spacer-layer design
wavelength. With a maximum lateral translation range of 50 11m, it is also possible
that a much larger translation is necessary to find a region with optimal overlap of
spectrally isolated dots with the spacer-layer resonance.
165
5.2 Improvements to the Apparatus
The current apparatus supports lateral scanning over an area of 250 !-.l.m2 ,
limited by the stroke length of piezo actuators. It would not be difficult to
adjust this design to allow for much increased scanning ranges. Another major
improvement would be the inclusion of position sensors on the tripod legs and on
the lateral actuators. This would give a much better control over the orientation of
the cavity and would also provide the ability to move to specific locations on the
sample. Due to hysteresis in the piezo actuators as well as large thermal expansions
each time the sample is cooled and reheated, we were not able to reliably return to
the same point on the sample on successive runs.
A way to read out the positions of each leg of the tripod would allow it to be
used to effectively control the yaw and pitch. In the absence of information about
the particular leg positions, the tripod was essentially used as a piston without
regard to the angular orientation of the DBR. In practice this was not a huge
problem because the input beam could be steered to align with cavity optical axis
(whose direction is defined by the pitch and yaw values). Nonetheless, with the
benefit of hindsight, a "next-generation" apparatus would include digital encoders
for each tripod leg.
The dominant source of vibrations in the current incarnation of the apparatus
is helium boiling inside of the cryostat. Newer cryostat and transfer tube designs
166
are capable of keeping the helium liquid inside of the cryostat. The passive stability
of the cavity would improve if the turbulence associated with the boilling helium
could be moved out of the cryostat and into the return path of the transfer tube.
5.3 Outlook
The properties of this cavity make it a good candidate for investigating non-
paraxial modes. If small flat mirrors could be built, such as coated (optical)
fiber-tips, then truly hemispherical micro-mirrors could be used to form a stable
cavity. This isn't possible with current mirrors because the lateral dimension
of the flat mirrors is orders of magnitude larger than that of the curved micro-
mirror. If mirror coatings could be produced with high reflectivity over the entire
hemispherical surface, then a cavity formed with a small-diameter flat-mirror could
probe short, paraxial cavities all the way out to a truly hemispherical cavity. While
the mirror geometry described in this dissertation support non-paraxial modes,
there are predicted modes with angular spectrum exceeding what these mirrors can
support. If the aim is to probe non-paraxial optics, then it would seem worthwhile
to attempt to produce a curved mirror with even larger numerical aperture.
Of course, a curved micro-mirror attached to an immersion objective as used
in this work could be used to investigate any variety of emitter located near
the surface of a flat mirror. There appears to be some interest recently in using
diamond color-centers [41, 62] and it should be possible to integrate these into a
167
monolithic mirror structure in a similar fashion to the integration of IFQDs into the
DBRs used in this work.
Additionally, the proposal [55] to use biexciton emission from an anisotropic
quantum dot as a source of polarization-entangled photons seems readily attainable
for a cavity similar to the cavity described here. In that scheme it is desirable
to match the FSR of the cavity to the biexciton shift and the linewidth to the
anisotropy splitting. The range of bubble sizes produced by the technique described
in this dissertation covers the biexciton shift for GaAs QDs (both self-assembled
dots and IFQDs). A linewidth of a few tens of J..leV would suffice and for a given
FSR, is easily accessible with conventional coating techniques.
Although signatures of strong-coupling eluded us, significant achievements were
made in contructing new and novel micro-cavities that have applications involving
cavity-QED, quantum information science and non-paraxial optics.
168
APPENDIX A
TIME-DEPENDENCE OF EXPECTATION VALUES
The Hamiltonian for an open subsystem consisting of a two-level system
interacting via a dipole interaction with a single mode of the electric field is given
by (2.64):
The time-evolution of the density operator is determined by a master equation
in the Lindblad form provided the Born-Markov and rotating wave approximations
are justified (2.79).
"/P(A AA 1)+ 2 {jzp{jz - .
(A.1)
Then the time evolution of operators may be found by multiplying both sides
of (A.1) by the operator under consideration and taking the trace over the open
subsystem. The time evolution of a general operator is thus,
169
In this appendix the goal is to compute the expectation values of the operators
a, fL, ata, fh fL and atfL. This provides results needed for calculating spectra in
chapter 2.
The differential equations for the single-time averages (expectation values)
depends on the trace. A couple of properties of the trace will be exploited in this
pursuit.
Invariance under cyclic
tr{ABC} = tr{CAB} = tr{BCA} permutation of the order of its (A.2)
arguments.
tr {A + B} = tr {A} + tr {B} The trace is a linear map. (A.3)
Because of these properties and linearity of the commutator, each term in the
system Hamiltonian will yield a null contribution for those cases where the time
evolution is being calculated for an operator that commutes with said term. In
other words, suppose that [IV, 0] = O. Then
tr { [IV, p] O} = tr { IVpO - pIVO}
= tr {pOIV } - tr {pIV0}
= tr {pIVO} - tr {PIVO}
=0
170
Because of this, begin by omitting each commutator term in the master
equation whose system operators commute with the operator whose expecation
value is being calculated. Also, notice that the dissipative terms enclosed by
parentheses each evaluate to zero if multiplied by a commuting operator (an
operator that commutes with all operators other than the density operator) and
traced over.
Now write down some relevant commutators for use in the following calculations:
(A.4a)
(A.4b)
(A.4c)
Since we work in the 3-state basis of {10, b) , 11, b) , 10, an, it follows that any
product of two raising (creation) or lowering (annihilation) operator corresponds to
a null operator in this Hilbert space.
A final result before determining the dynamical equations for the expectation
values: Find two useful triple products of the 2-level operators.
(A.5a)
(A.5b)
First calculate the time evolution for the field operator, a:
(it) = tr{ -iv [ata,p] a+ig [ats_,p] a+ig [as+,p] a}
{ 'Y A A A A A ) }+ tr 2" 2S_pS - -p - pS+S_ a
+ tr {rL (2apat - atap - pata) a}
+~
= tr{ - ivp [a, at] a+ igp [a, at] s_ + rLP [at, a] a}.
'-v-' '-v-' "-v-"'
=1 =1 =-1
The result is
~(a) = -(rL + iv)(a) + ig(S_) .
Next, find the ODE governing the time evolution of the two-level lowering
operator:
(8_) = tr{ - i~ [az,p] S_ +ig [as+,p] S_
'Y A A AA AA A
+ 2"(2S_pS+ - S+S_p - pS+S_ )S_
'YP(A AA A)S}+"2 azpaz- p -
23_ -fjz
~ ,..-A--...
= tr{ - i~P[s_, az] +igpa [s_, s+]
3_
~
+ ~(2PS+~ - pS_S+S_-~)
-3_
---:-- A
'Yp ( A A S A AS) }+"2 paz _az-p -
171
(A.6)
172
The observant reader will notice that one of the expectation values involved here
is (ao-z ). The 3-state basis offers a way to simplify this expression. First, the
expectation value of the annihilation operator may be written down as
(a) = ('Ij!1 a1'Ij!) = ('Ij!1 a(COb 10, b) + Cl,b 11, b) + COa 10, a))
= Cl,b ('Ij! 10, b) .
Meanwhile the desired expectation value is
('Ij!1 ao-z 1'Ij!) = ('Ij!1 a(la)(al - Ib)(bl)(cOb 10, b) + Clb 11, b) + COa 10, a))
= -Clb ('Ij! 10, b) .
Direct comparison leads to the conclusion that in the context of the Hilbert space
spanned by the current 3-state basis, (ao-z ) = -(a). This yields the corresponding,
simplified differential equation governing the time evolution of the lowering
operator's expectation value.
(A.7)
The equations for (At) and (8+) can be obtained by simply taking the Hermitian
adjoint of (A.6) and (A.7), respectively. Now we turn to the matter of calculating
173
the projection operators (ata and 8+8_) for the excited states of the field and the
two-level system.
=tr{ -il/_)+ig[,o8_(ataat-~]
+ ig[,o8+~ - aata)] - 2fl;,oat(1- ata)a}
= tr{ ig,08_at - ig,08+a - 2fl;,oata}
We see that the number operator evolves in time according to
where h.c. represents the Hermitian conjugate.
The next task is to determine the differential equation governing the time
(A.8)
evolution for the projection operator into the excited state of the two-level system. 1
dAA W AA [A t A JAAdt (8+8_) = tr{ - i"2 [o-z, ,0] 8+8_ + ig 0,8+ + a 8_, ,0 8+8_
'"Y A A AA AA AA
+ 2(28_,08+ - 8+8_,0 - ,08+8_)8+8_
1We can evaluate the operator 8+8_ by writing it in terms of the states of the two-level
system: 8+8- = ja)(b Ib)(al = \a)(aj
174
/P (A AA A) A A }+"2 (5Zp{5Z - P 8+8_
= tr{ - i~p(8+8_&z - &z8+8_) +igp(~+ a8+8_8+
- at8_8+8_ -*)-~p(8+8_8+8_ + 8+8_8+8_)
/P A A A A
+ "2P(&z8+8_&z - 8+8_)}
= {-i~p~+igp(a8+-at8_)
-,p8+8_ - /Pp~)},
2
which provides the final form of the differential equation for 8+8_,
(A.9)
The final independent expectation value that needs to be solved for is at8_.
+ ~(28_p8+ - 8+8_p - p8+8_ )at8_
+ /P (A AA A) At8 }"2 (5zp{5z - p a -
= tr{ - i~ pat [8_, &z] - illpat [at, a] 8_ + igp(ata8_8+ - aat8+8_)
- K,p(.a±-a;ta+ ataat)8_
175
Because o-zfLo-z = (Ia)(al - Ib)(bl) Ib)(al (Ia)(al - Ib)(bl) = -Ib)(al = -fL, this can
be written as
d t~ { t~ t~ t ~ ~dt (a 8_) = tr - iwpa 8_ + ivpa 8_ + igp(-a ao-z - 8+8_)
- fl,p( at -.atat1i) tL
I' ~~t8~
- -pa _
2
+ 'YP(-2at8 )}2 -
= -iw(at8_) +iv(at8_) -ig((atao-z) + (8+8_))
- fl,(at8_) - 'l(at8_) - 'Yp(at8_)
2
As before, the expectation value (at 0,0-z) can be simplified in the 3-state Hilbert
space.
('ljJ1 ata j'IjJ) = ('ljJ1 ata(COb 10, b) + Clb 11, b) + COa 10, a)) = Clb ('IjJ 11, b) ;
Using this result, the differential equation governing the time evolution of
(at S_) can be written in simplified form as
176
(A.10)
,------------------
177
APPENDIX B
QUANTIZATION OF A PARAXIAL RESONATOR
In this appendix, the problem of determining the fields and mode-volume inside
of a stable, paraxial resonator will be considered. Quantization may be performed
by canonical quantization once the Hamiltonian has been written in the form of a
classical, canonical coordinate and its time-derivative: H(q, q). However, this step is
carried out in chapter 2 and will not be repeated here.
The basic strategy is to specify the electric and magnetic fields satisfying
the Helmholtz equation and subject to the boundary conditions of a resonator
whose surfaces are sections of spheres. Once these have been specified then the
classical Hamiltonian may be calculated by integrating the electrodynamic energy
density over the volume of the intra-cavity fields. This turns out to consist of
some position-independent factors multiplied against the integral of the mode
intensity over the volume of the cavity (or unit cell in the case of periodic boundary
conditions) .
B.l Classical Field of the Resonator
The resonator contains no sources, so any electric or magnetic fields must satisfy
the vacuum Maxwell's equations. In particular, we seek solutions to the vacuum
178
wave equation.
In Cartesian coordinates, the vector Laplacian can be written in a simple form.
(B.1)
Due to (B.1) we can write the wave equation as a set of individual scalar
equations in Cartesian coordinates. In this case, we consider a paraxial mode of a
resonator bounded by two spherical, reflective boundaries. The transverse nature
of the field tells us that the polarization vectors for the fields will lie in spherical
surfaces in the neighborhood of the optical axis. The immediate task is to write
this in terms of Cartesian coordinates so that the simplification allowed by B.1 can
be exploited.
Due to a coordinate singularity in spherical coordinates1 it is desirable to use a
modified set of spherical coordinates to describe the orientation of the polarization
vectors. The singularity can be nicely avoided by choosing the polar angle to be
zero along the x-axis rather than the z-axis. Due to this change we now have
sin e ~ 7f/2 and ¢ ~ 7f /2 close to the optical axis. Additionally, the transformation
1Points on the z-axis correspond to (z, 0, ¢) - the azimuthal coordinate is completely
undefined. Two dimensional polar coordinates the origin has an analogous coordinate singularity.
The origin occurs at p = 0 while ¢ is undefined.
179
from spherical basis vectors to Cartesian basis vectors takes place as follows:
r
(j
cos (j sin (j cos ¢ sin (j sin ¢
- sin (j cos (j cos ¢ cos (j sin ¢
x
y
o -sin¢ cos¢" z
Thus, we have (j = - sin (jx + cos (j cos ¢f) + cos (j sin ¢i ~ -x + i!: .1Vf + i!: iyz & z
in the vicinity of the optical axis but away from z = O. This provides us with a
relationship between the amplitudes of the different electric field vector's Cartesian
components that satisfies the condition of lying within a spherical surface and near
the optical axis.
This constraint allows the wave equation to be written as the following:
_v2Eo(r, t) -0;Eo(r, t)
1 (B.2)0
c2 0
V2~Eo(r, t) ~o;Eo(r, t)
z z
Each component obeys a scalar wave equation. This can be attacked by the
usual separation of variables. Suppose that Eo(r, t) = R(r) .T(t). Then the equality
of the first row requires that
- V 2 [R(r) .T(t)] = - ~ 0; [R(r) .T(t)]
c
V 2R(r) ~(t)
=} =--
R(r) T(t)
180
Proceed by noticing that the only way for the equation to hold is if each side is,
separately, equal to the same constant. Rewrite as two equations and utilize a
clever choice of separation constant.
\72R(r)
= _k2
=} R(r)
T(t)
= _k2
c2T(t)
This yields
(B.3)
and
(BA)
Here, (B.3) is the Helmholtz equation, while (BA) provides for a harmonic time-
dependent part. Formally, this can be demonstrated by converting the second order
ODE into a system of two first-order ODEs and solving the characteristic equation
to find the eigenvalues. The general solution to equation (BA) can then be written
as
(B.5)
where A± are the two eigenvalues. The characteristic equation trivially leads to two
imaginary eigenvalues, A± = ±v, where v = kc. Inserting these values into (B.5)
181
provides the solution in this case.
(B.6)
The main goal here is to solve the Helmholtz equation (B.3) as that provides
for the spatial characteristics of the field. Since we consider here the standing wave
modes of an optical resonator we can appropriately make a quasi-monochromatic
or slowly-varying envelope approximation. The slowly varying envelope means that
the spectrum for the field is narrow around the wavenumber of the plane wave. The
spread of wavenumbers present in a Fourier decomposition of the envelope is small
and therefore the corresponding length scale over which changes occur along the
optical axis is long compared to the wavelength. That is, suppose that
R(r) = A(r)eikz .
The slowly varying envelope approximation constrains the variations of the
envelope function, A, so that
(B.7)
(B.8)
182
When (E.8) is inserted into (B.3) it permits the following,
=0.
It is now clear that R(r) will be a solution of the Helmholtz equation (B.3)
provided that the envelope function obeys a paraxial Helmholtz equation:
\7~A(r) + 2ik a~~r) = o.
Taking a small sideways step, it's also possible to assert that a spherical
wave is a well known solution to the Helmholtz equation whose wavefronts are
(B.9)
spherical surfaces. Invoking the paraxial approximation, the corresponding wave
is a parabolic wave.
183
- ----r==:;======:;:=~
r JX2+ y2 + Z2
AeikZVl+x2z+,j'2
This provides an example that the paraxial approximation already contains the
essential content of the slowly varying envelope approximation.
The equation determining Ez just turns out to be obtained from the equation
for Ex by a simple substitution:
\/2 (:: . E ( )) = ...::..- ;]2 Eo(r, t)
v 0 r, t 2 8 2
Z zc t
::::} T(t)\72 (:: . R(r)) = xR(r) 82T(t)
z zc2 8t2
Z 2 (X ) 1 82T(t)
::::} xR(r) \7 -;. R(r) = c2T(t)' 8t2 .
Necessarily, the same equation is obtained (B.4) for the time dependence as
before. Meanwhile, the spatial equation is
(B.I0)
184
which, not surprisingly, is just the equation obtained for Ex by taking R -7 iE.R.
z
Since we already have solved for R(r) it is straightforward to verify that it remains
a solution to the Helmholtz equation when multiplied by x/z.
So, a paraboloidal field of the following form obeys the boundary condition for a
single spherical equipotential surface z #- 0:
(B.ll)
However, as pointed out previously, this field runs into problems at z = o.
The field's amplitude is undefined in the entire plane! The oscillatory part of the
envelope also begins to oscillate arbitrarily quickly under approach of z = O. To
avoid these difficulties and permit cavities that include a flat surface at z = 0-
such as the cavity employed in the current experimental work -- it suffices to move
the pole away from the real axis by taking z -7 q(z) = Z - iZR. The resulting
solution is the Gaussian beam. Its wavefronts are flat at z = 0 while retaining
spherical wavefronts everywhere (flat corresponds to a circle of infinite radius of
curvature). Additionally, the curvature of a wavefront intersecting the optical axis
at z = z' can be tuned by selecting a different ZR.
The q-parameter is often written in terms of more physically meaningfully
quantities. This provides a representation for the beam in terms of understandable
physical quantities, such as the radius of the beam (as measured to the distance
185
from the optical axis where the intensity drops to a value of l/e2 compared to the
value on the axis), the radius of curvature of the wavefronts, and a residual phase
shift (the Gouy phase).
The Gaussian envelope is:
A ( ) = Ao [ik (x
2+ y2)] .
r ( ) exp ( ) ,q Z 2q Z (B.12)
where q(z) is the complex q parameter, Z - izR • A representation of the beam
written in terms of more physical properties of the beam can be obtained by
defining the following:
1 z . ZR 1 A
q(z) = Z2 + zA + '/, Z2 + zA = R(z) + i nW2 (z)'
where the following definitions have been made:
(B.13)
(B.14)
w(z) is the cylindrical radius of the l/e2 contour, Wo is the radius at the waist
-located at z = 0 - and R(z) is the radius of curvature of the wavefronts at
distance along the optical axis given by r. By applying (B.13) and (B.14) to (B.12)
186
a representation allowing more direct physical interpretation is obtained.
I-'lL
ZR .F
1 +i c:r
iAo Wo [. ( z )]= - .-- exp -~ arctan -
ZR w(z) ZR ·F,
where F has been used temporarily to stand in for the appropriate exponential
factors. The final step follows from the identity [26}:
1 1 + i~aretan~ = -In--.
2i 1 - i~
Without loss of generality, the constant factors in the front of the expression
may be absorbed into AD, since it is an unspecified complex number to be determined
by boundary conditions. Define the resulting phase factor arising from the above
identity as ((z) = arctan L. This finally leaves the Gaussian envelope in its
ZR
familiar form.
187
(B.15)
Absorption of a factor of 1/ZR into the coefficient Ao appears consistent when
consid(i)ring the units of Ao in (B.12) in comparison to those of Ao in (B.15). At
this point the electric field inside of the cavity has been determined. Recalling
that the general solution permits combinations of both eigenvalues, take this
opportunity to replace eikz with cos(kz) to reflect a priori knowledge that the cavity
mode contains waves traveling in both directions due to reflections at the mirror
surfaces.
E(r, t) = A1T(t) ( -5; + Z _XiZR Z) w~:) e- wq~z) exp [ik2~;Z) - i((Z)] cos(kz)
(B.16)
Having found the electric field it is straightforward to calculate the corresponding
magnetic field. The corresponding B field can be determined by using Maxwell's
equations. Begin by invoking Faraday's Law. In differential form and 81 units it has
the following form:
Calculating the curl of the electric field is straight-forward. After differentiating
and invoking the paraxial approximation to eliminate terms of greater than first
188
order in xlz or ylz, the following expression remains:
o
OtB(r, t) = A(r)T(t) lk sin(kz)
ky
q(z) cos(kz)
Next, differentiate once more and satisfy the remaining Maxwell-Ampere
equation by invoking the wave equation.
o
B (r) = A(r)T(t) k sin(kz)
ky
q(z) cos(kz)
(B.17)
The magnetic field obeys an identical wave equation. This leads to the same
separation of variables resulting in harmonic time dependence and a Helmholtz
equation, permitting V'2B = -k2B.
1 .. 2
-B = V' B
c2 (B.18)
189
Use the result of (B.18) with (B.17). This gives the magnetic field.
o
B ( ) = _ A(r)T(t)r,t 2
lJ
k sin(kz)
iky
q(z) cos(kz)
(B.19)
Having obtained the form of the magnetic field we turn to the matter of
calculating the energy in the field. Express the electrodynamic energy as the
volume integral of the energy density using the fields determined previously.
From classical theory [27], the energy density may be written in the following
way:
1 ( 2 1 2)U = - EO lEI + -IBI .
2 Mo
(B.20)
The total energy depends upon the intensities of the electric and magnetic fields
integrated over a characteristic volume. Because the energy depends on the square
of the modulus of the fields we can neglect the longitudinal component of each of
the fields: After squaring, the longitudinal electric and magnetic field components
will be, respectively, 0 ((~?) and 0 ((~?).
190
H = ~Jd3r (EOE2+ :0B 2)
V
= ~Jd3r IA (r)1 2 [EOT2(t) cos2(kz) +~ EO~Ot2(t)k2 Sin2 (kz)]2 ~ v
v
Notice that the integral is essentially over the squared modulus of the resonator
mode function, R (r). To proceed we calculate the mode volume for the Gaussian
beam. Both terms are a Gaussian envelope applied to a standing wave which is
integrated over many wavelengths. Calculation will be done for the electric-field
mode function, but the reader should note that an identical calculation carries
through for the magnetic field.
The mode function for the electric field is:
Taking the modulus square of this we have:
2 [2 2 ]2 2 W o P 2IR (r) I = AoW 2 (z) exp - W2 (z) cos (kz).
(B.21)
(B.22)
---~._------ -----
191
Once again, the mode volume is the integral of the mode-intensity over the cavity
volume. This may now be written as:
J 3 2 2J W5 [2p2 ] 2d r IR (r)1 = Ao dT W 2(Z) exp - W2 (z) cos (kz)
v v
2 2 J27r
J
OO
J
L 1 [2p2 ] 2
= Aowo pdpd¢dz W2 (z) exp - W2 (z) cos (kz)
o 0 0
2 2 JL 1 2) JOO [2 p2 ]
= 21rAoWo dZ W2 (z) cos (kz pdpexp -W2 (z)
o 0
The p integral can be performed directly via substitution. Let ~ = - ~~2Z)' Then
d~ = - W;~z) dp. This allows us to re-write the integral in the following way:
L -00j d3r IR(r)1 2 = 21rA2w2jdz 1 cos2(kz) j W2(z)d~e~o 0 W2 (z) -4
v 0 0
L
= iA6w6J dz cos2 (kz)
o
The above result is actually an approximate upper bound due to integration
over an interval of [0,00] in p. In the case of an extremely paraxial limit this
approximation seems justified as this is consistent with a mirror with very large
radius of curvature. The above calculation is for a region of space bounded by two
fiat mirrors enclosing a Gaussian beam.
192
The effective mode volume is a volume that, when multiplied against the
squared modulus of the mode function evaluated at the location of the emitter,
provides the same numerical result as integrating the modulus of the mode function
over the actual volume of the cavity. It is defined by:
Jd3rIR(r)1 2 = IR(r2-level)12 ·Veff •
V
(B.23)
Take the position of the atom to be at the cylindrical coordinates (0, ¢, 0). When
inserted into (B.22) this provides us with IR (r2-level) 12 = IAoI2 • Inserting this into
(B.23) provides the effective mode volume:
V
eff
= 1fW5L .
4
(B.24)
Armed with this insight, it is possible to write down the total energy in a more
suggestive way.
1 I 12 ( 2) 1· 2 )H = 2EO~ff Ao T (t + l/2T (t) (B.25)
Without loss of generality, we can scale the coefficient and the amplitude of T(t).
Scale the coefficient so that:
193
VEOVeff .Define the scaled dynamical variable as q(t) - 2v2 T(t). This leaves the
Hamiltonian in the following familiar form:
This is the energy of a classical harmonic oscillator with unit mass, such that p = q
(B.26)
194
APPENDIX C
ACRONYMS
AIGaAs AlxGal-xAs. An alloy with a molar fraction of Al (Ga) of x (l-x).
ALE atomic layer epitaxy. A coating technique that uses se1£- limiting chemical
reactions to allow epitaxial growth. It provides a conformal coating but tends to
suffer from higher scattering than traditional coating techniques.
ADM acousto-optic modulator. A device that uses Bragg scattering off of a moving
grating comprised of pressure waves (sound) in a crystal. Diffracted beams
exhibit frequency shifts as well as angular shifts and the diffracted orders may be
modulated by controlling the driving signal into the crystal.
CCD charge-coupled device. A device operated by manipulating charge packets
held in capacitive bins.
CQED cavity quantum electrodynamics. A subfield of QED in which a resonator is
used to modify the nature of matter-field interactions.
8884 Bennett and Brassard, 1984. The first quantum key distribution protocol.
Published by Bennett and Brassard in 1984, its security rests on the No-Cloning
theorem.
--------- ---------
195
CW continuous-wave. A term used to describe lasers operating with continuous
output.
DBR distributed Bragg reflector. Quarter-wavelength optical thickness
semicnd]lctor stack; a high reflector.
FDTD finite-difference time-domain. A method for numerically solving PDEs on a
grid.
FSR free spectral range. The frequency difference between two adjacent
longitudinal modes of the same transverse order in a resonator. In a
hemispherical cavity FSR = .-::..., where £ is the distance between the mirrors.
2£
IFQD interface fluctuation quantum dot. A quantum dot formed from fluctuations
of the interfaces of a quantum well.
lOQC linear optical quantum computing. A scheme for performing quantum
computations that relies on single photons, linear optics and projective
measurements.
MBE molecular beam epitaxy. A process in which molecular beams are used to
deposit materiallayer-by-Iayer.
MOT magneto-optical trap. A method of trapping atoms using both a tapered
magnetic field and counter-propagating laser beams to achieve a high degree of
cooling.
196
NA numerical aperture. A measure of the acceptance angle of an optical system.
NA= nsinO, where n denotes index of refraction.
NMR nuclear magnetic resonance. Phenomena relating to resonant interactions
between the m~gneticmoment of a nuclear spin and an (electro-) lIlagnetic field
(usually in the RF band).
ODE ordinary differential equation. An equation involving only functions of a
single independent variable and its derivatives.
OFHC oxygen-free, high-conductivity. An acronym used to describe copper
specified as having a negligible content of oxides and hydroxyl groups. This
provides for good conductive properties - both thermal and electrical.
PBS polarizing beam splitter: An optical device in which (for an assumed
orientation of the device) vertically polarized light gets reflected and horizontally
polarized light gets transmitted.
PDMS poly(dimethyl-siloxane). An organic silicone compound with elastomeric
properties; commonly used to create molds for nano replication.
PID proportional-integral-derivative. A term describing a general class of feedback
circuits capable of applying a control signal generated from some system output
via proportionality, rate of change and integrated output.
197
Pl photoluminescence. A form of spectroscopy utilizing optical excitation and
measurement of the resulting emission spectrum.
PSD power spectral density. A measure of the "intensity" of a time series per unit
frequency. In the cont~xt of surface measurements, it corresponds to the modulus
square of height variations per unit spatial frequency.
PZT Lead-zirconate-tantalate. A widely used piezo-electric ceramic.
QCPG quantum controlled phase gate. A two-qubit quantum gate that applies a
phase factor to each input provided the two qubit input state is 111). All other
input states map back to themselves.
QD quantum dot. A zero dimensional object or a three dimensional potential well.
Typically, it consists of a region of semiconductor surrounded on all sides by a
material with a larger band gap.
QKD quantum key distribution. Any of various schemes utilizing properties of
quantum systems to guarantee secure distribution of random bit-strings (i.e.
one-time pad).
QW quantum well. A two dimensional structure exhibiting confinement in one
dimension. It is generally formed by a layer of semiconductor. surrounded by
another semiconductor with a higher band gap.
198
RWA rotating wave approximation. An approximation corresponding to moving to
rotating coordinates.
SMF single-mode fiber. An optical fiber capable of supporting only a single mode
at a specified wavelength.
SQD single quantum dot. A solitary semiconductor quantum dot.
TE transverse electric. A field whose electric field is everywhere perpendicular to
the optical axis.
TM transverse magnetic. A field whose magnetic field is everywhere perpendicular
to the optical axis.
TSA Technical Science Administration. Offers professional technological expertise
and service to the scientific community of the University of Oregon.
UHV ultra-high vacuum. Pressure range of roughly: P ~ 10-8 Torr.
199
BIBLIOGRAPHY
[1] G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki. Exciton binding energy
in quantum wells. Phys. Rev. B, 26(4):1974-1979, Aug 1982.
[2] C. H. Bennett, G. Brassard, and A. Eckert. Quantum cryptography. Scientific
American, 267:50, 1992.
[3] E. D. Black. An introduction to pound-drever-halllaser frequency
stabilization. American Journal of Physics, 69(1):79-87, 2001.
[4] D. Bouwmeester, A. Ekert, and A. Zeilinger, editors. The physics of
quantum information: quantum cryptography, quantum teleportation, quantum
computation. Springer-Verlag, London, UK, 2000.
[5] R. Bratschitsch and A. Leitenstorfer. Quantum dots: Artificial atoms for
quantum optics. Nature Materials, 5:855-856, Nov 2006. 10.1038/nmatI768.
[6] H.-P. Breuer and F. Petruccione. The Theory of Open Quantum Systems.
Oxford University Press, 2002.
[7] H. J. Carmichael. Statistical Methods in Quantum Optics 1. Springer, 1999.
[8] H. J. Carmichael. Statistical Methods in Quantum Optics 1, chapter 1.5, pages
20 - 26. Springer, 1999.
[9] H. J. Carmichael. Statistical Methods in Quantum Optics 1, chapter 7.3.
Springer, 1999.
[10] K. L. Chavez and D. W. Hess. A novel method of etching copper oxide using
acetic acid. Journal of The Electrochemical Society, 148(11):G640-G643, 2001.
[11] G. Cui. An external optical micro-cavity strongly coupled to optical centers for
efficient single-photon sources. PhD thesis, University of Oregon, 2008.
[12] G. Cui, J. M. Hannigan, R. Loeckenhoff, F. M. Matinaga, M. G. Raymer,
S. Bhongale, M. Holland, S. Mosor, S. Chatterjee, H. M. Gibbs, and
G. Khitrova. A hemispherical, high-solid-angle optical micro-cavity for cavity-
qed studies. Opt. Express, 14(6):2289-2299, 2006.
[13] G. Cui and M. G. Raymer. Quantum efficiency of single-photon sources in the
cavity-qed strong-coupling regime. Opt. Express, 13,:9660~9665, 2005.
200
[14] P. Domokos, J. M. Raimond, M. Brune, and S. Haroche. Simple cavity-
qed two-bit universal quantum logic gate: The principle and expected
performances. Phys. Rev. A, 52(5):3554-3559, Nov 1995.
[15] M. 1. Dyakonov. Spin Physics in Semiconductors, chapter 1, page 11. Springer,
2008.
[16] X. Fan. Pure dephasing induced by exciton phonon interactions in narrow gaas
quantum wells. Solid State Communications, 108:857-861, Nov. 1998.
[17] D. H. Foster. Focused modes in the 60 \-lm cavity. Private communication.
[18] D. H. Foster, A. K. Cook, and J. U. Nacke!. Goos-hiinchen induced vector
eigenmodes in a dome cavity. Opt. Lett., 32(12):1764-1766, 2007.
[19] D. H. Foster and J. U. Nacke!. Bragg-induced orbital angular-momentum
mixing in paraxial high-finesse cavities. Optics Letters, 29(23):2788-2790, 2004.
[20] D. H. Foster and J. U. Nacke!. Methods for 3-d vector microcavity problems
involving a planar dielectric mirror. Optics Communications, 234(1-6):351 -
383,2004.
[21] D. Gammon, A. Efros, J. Tischler, A. Bracker, V. Korenev, and 1. Merkulov.
Electronic and nuclear spin in the optical spectra of semiconductor quantum
dots. In T. Takagahara, editor, Quantum Coherence Correlation and
Decoherence in Semiconductor Nanostructures, pages 207 - 280. Academic
Press, San Diego, 2003.
[22] D. Gammon, E. S. Snow, B. V. Shanabrook, D. S. Katzer, and D. Park. Fine
structure splitting in the optical spectra of single gaas quantum dots. Physical
Review Letters, 76(16):3005-3008, 1996.
[23] A. V. Gopal and A. S. Vengurlekar. Well-width dependence of light-hole
exciton dephasing in gaas quantum wells. Phys. Rev. B, 62(7):4624-4629, Aug
2000.
[24] S. V. Goupalov. Anisotropy-induced exchange splitting of exciton radiative
doublet in cdse nanocrystals. Physical Review B (Condensed Matter and
Materials Physics), 74(11):113305, 2006.
[25] P. Goy, J. M. Raimond, M. Gross, and S. Haroche. Observation of cavity-
enhanced single-atom spontaneous emission. Physical Review Letters,
50(24):1903-1906, 1983.
[26] 1. S. Gradshteyn and 1. M. Ryzhik. Table of Integrals, Series, and Products,
chapter 1. Academic Press, 5 edition, January 1994.
201
[27] D. J. Griffiths. Introduction to Electrodynamics. Prentice-Hall, Inc., 2nd
edition, 1989.
[28] C.-H. Han and E.-S. Kim. Study of self-limiting etching behavior in wet
isotropic etching of silicon. Japanese Journal of Applied Physics, 37(Part 1,
No. 12B):6939-6941, 1998.
[29] D. J. Heinzen, J. J. Childs, J. E. Thomas, and M. S. Feld. Enhanced and
inhibited visible spontaneous emission by atoms 'in a confocal resonator. Phys.
Rev. Lett., 58(13):1320-1323, Mar 1987.
[30] D. J. Heinzen and M. S. Feld. Vacuum radiative level shift and spontaneous-
emission linewidth of an atom in an optical resonator. Phys. Rev. Lett.,
59(23):2623-2626, Dec 1987.
[31] R. J. Hughes, J. E. Nordholt, D. Derkacs, and C. G. Peterson. Practical free-
space quantum key distribution over 10 km in daylight and at night. New
Journal of Physics, 4:43, 2002.
[32] R. G. Hulet, E. S. Hilfer, and D. Kleppner. Inhibited spontaneous emission by
a rydberg atom. Phys. Rev. Lett., 55(20):2137-2140, Nov 1985.
[33] A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss,
M. Sherwin, and A. Small. Quantum information processing using quantum
dot spins and cavity qed. Phys. Rev. Lett., 83(20):4204-4207, Nov 1999.
[34] E. L. Ivchenko. Optical Spectroscopy of Semiconductor Nanostructures, chapter
5.5, page 267. Alpha Science International Ltd., 2005.
[35] A. Kiraz, C. Reese, B. Gayral, L. Zhang, W. V. Schoenfeld, B. D. Gerardot,
P. M. Petroff, E. L. Hu, and A. Imamoglu. Cavity-quantum electrodynamics
with quantum dots. Journal of Optics B: Quantum and Semiclassical Optics,
5(2):129-137, 2003.
[36] D. Kleppner. Inhibited spontaneous emission. Physical Review Letters,
47(4):233-236, 1981.
[37] E. Knill, R. Laflamme, and G. J. Milburn. A scheme for efficient quantum
computation with linear optics. Nature, 409(6816):46-52, Jan 2001.
[38] A. Kuhn, M. Hennrich, and G. Rempe. Deterministic single-photon source for
distributed quantum networking. Phys. Rev. Lett., 89(6):067901, Jul 2002.
[39] V. D. Kulakovskii, G. Bacher, R. Weigand, T. Kiimmell, A. Forchel,
E. Borovitskaya, K. Leonardi, and D. Hommel. Fine structure of biexciton
emission in symmetric and asymmetric cdsejznse single quantum dots. Phys.
Rev. Lett., 82(8):1780-1783, Feb 1999.
202
[40] A. Larionov, M. Bayer, J. Hvam, and K. Soerensen. Collective behavior of a
spin-aligned gas of interwell excitons in double quantum wells. JETP Letters,
81(3):108-111, Feb 2005.
[41] M. Larsson, K. N. Dinyari, and H. Wang. Composite optical microcavity of
diamond nanopillar and silica microsphere. Nano Letters, 9(4):1447-1450,
2009.
[42] H. Mabuchi and A. C. Doherty. Cavity quantum electrodynamics: Coherence
in context. Science, 298(5597):1372-1377, 2002.
[43] F. D. Martini, G. Innocenti, G. R. Jacobovitz, and P. Mataloni. Anomalous
spontaneous emission time in a microscopic optical cavity. Phys. Rev. Lett.,
59(26):2955-2958, Dec 1987.
[44] P. L. McEuen. Quantum physics: Artificial atoms: New boxes for electrons.
Science, 278(5344):1729~1730, 1997.
[45] J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. Buck, A. Kuzmich, and
H. J. Kimble. Deterministic generation of single photons from one atom
trapped in a cavity. Science, 303(5666):1992-1994, 2004.
[46] S. E. Morin, C. C. Yu, and T. W. Mossberg. Strong atom-cavity coupling over
large volumes and the observation of subnatural intracavity atomic linewidths.
Phys. Rev. Lett., 73(11):1489-1492, Sep 1994.
[47] J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger. Experimental
entanglement swapping: Entangling photons that never interacted. Phys. Rev.
Lett., 80(18):3891-3894, May 1998.
[48] E. Pazy, E. Biolatti, T. Calarco, 1. D'Amico, P. Zanardi, F. Rossi, and
P. Zoller. Spin-based optical quantum computation via pauli blocking in
semiconductor quantum dots. EPL (Europhysics Letters), 62(2):175-181, 2003.
[49] C.-Z. Peng, T. Yang, X.-H. Bao, J. Zhang, X.-M. Jin, F.-Y. Feng, B. Yang,
J. Yang, J. Yin, Q. Zhang, N. Li, B.-L. Tian, and J.-W. Pan. Experimental
free-space distribution of entangled photon pairs over 13 km: Towards satellite-
based global quantum communication. Phys. Rev. Lett., 94(15):150501, Apr
2005.
[50] E. Peter, P. Senellart, D. Martrou, A. Lemaitre, J. Hours, J. M. Gerard, and
J. Bloch. Exciton-photon strong-coupling regime for a single quantum dot
embedded in a microcavity. Phys. Rev. Lett., 95(6):067401, Aug 2005.
203
[51] U. Pohl, A. Schliwa, R. Seguin, S. Rodt, K. Potschke, and D. Bimberg.
Advances in Solid State Physics, chapter Size-Tunable Exchange Interaction
in InAs/GaAs Quantum Dots, pages 45-58. Springer Berlin, 2007.
[52] E. M. Purcell. Spontaneous transition probabilities in radio-frequency
spectroscopy. Physical Review, 69:681 [A], 1946.
[53] J. P. Reithmaier, G. Sek, A. Loffier, C. Hofmann, S. Kuhn, S. Reitzenstein,
L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel. Strong
coupling in a single quantum dot-semiconductor microcavity system. Nature,
432,:197-200, 2004.
[54] K. Resch, M. Lindenthal, B. Blauensteiner, H. Bohm, A. Fedrizzi,
C. Kurtsiefer, A. Poppe, T. Schmitt-Manderbach, M. Taraba, R. Ursin,
P. Walther, H. Weier, H. Weinfurter, and A. Zeilinger. Distributing
entanglement and single photons through an intra-city, free-space quantum
channel. Opt. Express, 13(1):202-209, 2005.
[55] T. M. Stace, G. J. Milburn, and C. H. W. Barnes. Entangled two-photon
source using biexciton emission of an asymmetric quantum dot in a cavity.
Physical Review B (Condensed Matter and Materials Physics), 67(8):085317,
2003.
[56] R. P. Stanley, R. Houdre, U. Oesterle, M. Gailhanou, and M. Ilegems.
Ultrahigh finesse microcavity with distributed bragg reflectors. Applied Physics
Letters, 65(15):1883-1885, 1994.
[57] T. Takagahara. Theory of exciton dephasing in semiconductor quantum dots.
Physical Review B (Condensed Matter and Materials Physics), 60(4):2638-
2652, 1999.
[58] A. Thdinhardt, C. Ell, G. Khitrova, and H. M. Gibbs. Relation between dipole
moment and radiative lifetime in interface fluctuation quantum dots. Phys.
Rev. B, 65(3):035327, Jan 2002.
[59] M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir,
E. Kukharenka, and M. Kraft. Microfabricated high-finesse optical cavity with
open access and small volume, 2005.
[60] R. Ursin, T. Jennewein, M. Aspelmeyer, R. Kaltenbaek, M. Lindenthal,
P. Walther, and A. Zeilinger. Communications: Quantum teleportation across
the danube. Nature, 430(7002):849-849, Aug 2004.
[61] T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper,
C. Ell, O. B. Shchekin, and D. G. Deppe. Vacuum rabi splitting with a single
quantum dot in a photonic crystal nanocavity. Nature, 432,:200-203, 2004.
204
[62] Y. Zhang and M. Loncar. Submicrometer diameter micropillar cavities with
high quality factor and ultrasmall mode volume. Opt. Lett., 34(7):902-904,
2009.
[63] A. Zrenner, L. V. Butov, M. Hagn, G. Abstreiter, G. B6hm, and G. Weimann.
Quantum dots formed by interface fluctuations in alas/gaas coupled quantum
well structures. Phys. Rev. Lett., 72(21):3382-3385, May 1994.