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Interference Between Competing Pathways in the Interaction of
Three-Level Atoms and Radiation
A Thesis
Submitted to the Faculty
of
Drexel University
by
Tony Y. Abi-Salloum
in partial fulfillment of the
requirements for the degree
of
Doctor of Philosophy
June 2006
c©Copyright 2006
Tony Y. Abi-Salloum. All Rights Reserved.
ii
Dedication
On this preliminary page I embed the name: Souheila Wakim, my mom, to whom
I dedicate this work. I would not have been able to make it through these difficult five
years without the cheerful and soothing spirit that my mom has always surrounded
me with. Bhebbik ya ahla mama.
iii
Acknowledgments
I take the opportunity here to thank all the people who supported me in one way
or another during this entire pleasurable Ph.D. journey.
I am genuinely honored for having been advised by the knowledgeable, intuitive,
patient, humble, and world-wide known physicist, Dr. Lorenzo Narducci. He believed
in my capabilities, introduced me to his field of research, and gave me a chance to
enjoy a real and pure physics research experience. Dr. Narducci also taught me how
to handle both teaching and research work simultaneously. Dr. Narducci is and will
always be my ideal lecturer and researcher.
All faculty members, staff, and students of the physics department helped to ease
my years of work at Drexel. I am particularly grateful to the continuous and amazing
support of the department head, Dr. Michel Vallieres. I am also touched by the
precious friendship of my colleagues Tatjana Miletic and Fiona Hoyle. The countless
help I received from the staff Jacqueline Sampson, Janice Murray, Laura DAngelo,
and Lisa Ferrara is overwhelming.
Alexis Finger has been a teaching inspiration, an academic orientation consultant,
and a wonderful friend, since the Spring of 2001.
iv
My family, mom, brothers and sisters, are my everlasting unconditional support.
My godfather, Tony R. Salloum, encouraged me to pursue my studies in the U.S.
and my brother Michel Abi-Salloum supported my plans and challenging trip. My
uncle Dr. Naji Wakim illuminated my path and shared with me keys of success in
the academic field.
My relatives in the U.S. always surrounded me by their love and warmth. Aunt
Mary Salloum, and aunt Najat Wakim welcomed me to their homes. My cousin
Richard Salloum complemented my Physics knowledge with his alternative, holis-
tic, and metaphysical ideas. Management and re-focusing conversations were always
helpful when held with my cousin George Abi-Salloum. My cousins Armand Sal-
loum, Bernard Abi-Salloum, Gilbert Abi-Salloum, and Rabih Awad, as well as my
uncle Ibrahim Wakim, and Charles Antoun were always there when needed.
I may not have the right words to express my deep gratitude to my precious buddy
Elie Zainoun. He was my cousin and my friend who shared my highs and cheered me
up during my lows. Elie has been my personal nutritionist, always following up my
health issues. His constant technical support had a strong impact on my work’s pro-
fessionalism and appearance. Elie has been a truly thoughtful and supportive brother.
Table of Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
1 Introduction 1
1.1 Three-Level Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 A Survey of the Literature . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Two-Level System 30
2.1 The Field Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Atomic Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.1 The Atomic Master Equation . . . . . . . . . . . . . . . . . . 37
2.3 Connection Between Macroscopic and Microscopic Variables . . . . . 39
2.3.1 The Low Saturation Limit . . . . . . . . . . . . . . . . . . . . 43
2.4 Discussion of The Results . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Perturbative Technique 48
3.1 Cascade-EIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
v
vi
3.1.1 Derivation of the Master Equation: An Outline . . . . . . . . 49
3.1.2 Perturbative Solution of the Atomic Equations: . . . . . . . . 55
3.2 Cascade-AT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3 Discussion of the Results . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3.1 Absorption Coefficients . . . . . . . . . . . . . . . . . . . . . . 60
3.3.2 Index of Refraction . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3.3 Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 Two-Time Atomic Correlation Functions and the Regression Theo-
rem 66
4.1 Cascade-EIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.1.1 Atomic Equations and Steady State Solutions . . . . . . . . . 69
4.1.2 Emission Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1.3 Probe’s Absorption . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2 Cascade-AT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2.1 Density Matrix Elements in Steady State . . . . . . . . . . . . 84
4.2.2 Emission Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2.3 Probe’s Absorption . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Secular Limit 89
5.1 Manifolds and Dressed States . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Transition Decay Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 Atomic Equations in the Secular Limit . . . . . . . . . . . . . . . . . 101
vii
5.5 Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.6 Absorption of the Probe Field . . . . . . . . . . . . . . . . . . . . . . 112
5.6.1 Cascade-EIT Configuration . . . . . . . . . . . . . . . . . . . 113
5.6.2 Cascade-AT Configuration . . . . . . . . . . . . . . . . . . . . 114
5.6.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6 EIT and AT Effects as Scattering Processes 118
6.1 Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.1.1 Transition and Probability Amplitudes . . . . . . . . . . . . . 120
6.1.2 Resolvent Operator . . . . . . . . . . . . . . . . . . . . . . . . 126
6.2 Bare States Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2.1 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.2.2 Low Saturation Limit . . . . . . . . . . . . . . . . . . . . . . . 144
6.3 Dressed States Picture . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.3.1 Cascade-AT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.3.2 Cascade-EIT . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7 Conclusions and Prospects for Future Work 162
Bibliography 167
A Scattering Technique 177
List of Figures
1.1 Three-level systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.2 Atomic three energy levels . . . . . . . . . . . . . . . . . . . . . . . . 24
1.3 Abs. coeff. of the field E1 as a function of the carrier frequency ω1:
E2=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4 Abs. coeff.t of the field E1 as a function of the carrier frequency ω1:
E2 6=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5 Rubidium’s partial energy diagram: 85Rb . . . . . . . . . . . . . . . . 25
1.6 Sodium’s partial energy diagram . . . . . . . . . . . . . . . . . . . . . 26
1.7 Cascade-EIT configuration . . . . . . . . . . . . . . . . . . . . . . . . 27
1.8 Cascade-AT configuration . . . . . . . . . . . . . . . . . . . . . . . . 27
1.9 Coherent Population Trapping in a lambda system . . . . . . . . . . 28
1.10 Electromagnetically Induced transparency in a lambda system . . . . 28
1.11 Autler-Townes split in the absorption line . . . . . . . . . . . . . . . 29
1.12 Autler-Townes effect in a Cascade-AT system . . . . . . . . . . . . . 29
2.1 Absorption line for a two-level system . . . . . . . . . . . . . . . . . . 47
2.2 Dispersion line for a two-level system . . . . . . . . . . . . . . . . . . 47
viii
ix
3.1 Absorption lines for the Cascade-EIT and Cascade-AT configurations:
Resonant and strong coupling field . . . . . . . . . . . . . . . . . . . 64
3.2 Absorption lines for the Cascade-EIT and Cascade-AT configurations:
Resonant coupling field in the low saturation limit . . . . . . . . . . . 64
3.3 Absorption’s maxima separation for the Cascade-EIT and Cascade-AT
configurations vs Ωc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 Dispersion lines for the Cascade-EIT and Cascade-AT configurations:
Resonant and strong coupling field . . . . . . . . . . . . . . . . . . . 65
4.1 Cascade-EIT configuration with W12 excitation . . . . . . . . . . . . 88
4.2 Cascade-AT configuration with W23 excitation . . . . . . . . . . . . . 88
5.1 Cascade configuration . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2 Transition decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3 Absorption line for the Cascade-EIT configuration compared between
the Secular and Perturbative technique . . . . . . . . . . . . . . . . . 117
5.4 Absorption line for the Cascade-AT configuration compared between
the Secular and Perturbative technique . . . . . . . . . . . . . . . . . 117
6.1 Resonances of the Cascade-EIT config. in the bare states picture . . . 160
6.2 Resonances of the Cascade-AT config. in the dressed states picture . 160
6.3 Resonances of the Cascade-EIT config. in the dressed states picture . 161
7.1 Ranges of the coupling field strength used in the different techniques 166
x
Abstract
Interference Between Competing Pathways
in the Interaction of Three-Level Atoms and Radiation
Tony Y. Abi-Salloum
Lorenzo M. Narducci
The appearance of quantum interference in the microscopic world is one of the
deepest mysteries at the very root of quantum mechanics. When light interacts with
atoms, it can induce transitions by way of distinct but indistinguishable pathways and
yield unexpected and often counter-intuitive results. Electromagnetically Induced
Transparency (EIT), Coherent Population Trapping (CPT), and Lasing Without In-
version (LWI) are modern examples of phenomena where the traditional rules that
govern absorption and dispersion undergo major revisions.
Spectacular consequences of this new state of affairs include the ability of a light
beam to propagate through a normally absorbing medium with little or no absorp-
tion, or to undergo amplification even if the active medium is not prepared in a state
of population inversion. Light signals have been shown to travel with strongly sublu-
minal group velocities of only a handful of meters per second and, even, to propagate
at speed greater than the ordinary speed of light, but without violation of causality
or of the established rules that control the transfer of optical information through
space.
Three-level atoms can interact with two coherent electromagnetic fields accord-
ing to three different systems, known as Cascade, Lambda and Vee systems. The
propagation of a weak probe field can be deeply affected by the presence of a second
xi
stronger beam, the so-called coupling field. Transparency windows, splitting of emis-
sion and absorption lines and enormously enhanced dispersion have been documented
in numerous experiments.
In this thesis we explore the physical origin of the transparency induced in these
systems by the simultaneous interplay of the coupling and probe fields. We focus our
attention on the Cascade system and study its two configurations, Cascade-EIT and
Cascade-AT. We develop a variety of complementary approaches for the description
of these two configurations, some semiclassical and others fully quantum mechani-
cal. We prove the existence of quantum interference in Electromagnetically Induced
Transparency and also the surprising absence of interference in a closely related phe-
nomenon, known as the Autler-Townes effect.
Finally we complement the traditional machinery of theoretical quantum optics
with techniques borrowed from quantum scattering theory, and offer what we believe
is the most convincing physical evidence for the appearance, or for the absence, of
quantum interference effects.
Chapter 1: Introduction
The main objective of this thesis is to investigate the phenomenon of quantum in-
terference in optical processes where three-level atoms, or molecules, interact with
two resonant, or nearly resonant, electromagnetic fields. When the atoms are driven
by the fields, they undergo transitions from an initial to a final state. In some in-
stances, these transitions can occur by way of multiple independent pathways, each
characterized by a quantum mechanical complex amplitude. Because the transition
probability is given by the squared magnitude of the total amplitude, the occurrence
of a physical process is often accompanied by the interference of the various ampli-
tudes, leading to the enhancement or suppression of the independent contributions.
Thus, more specifically, our eventual objective is to calculate the multiple quantum
mechanical amplitudes and to identify the interference processes that are responsible
for the appearance of specific final states.
The generic physical setting of interest to this work involves electromagnetic waves
propagating through a medium of atoms or molecules, although we will typically call
“atoms” the elementary constituents. As the wave propagates, it affects the state of
the atoms and, in turn, undergoes modifications of its own which can then be detected.
1
2
A well known and very common consequence of the interaction of an electromagnetic
wave with a passive medium1 is the attenuation of the beam along the direction of
propagation, although under special conditions, for example if the medium is pre-
pared in a state of population inversion, a propagating beam can be amplified.
The attenuation of the wave is the consequence of two main effects: a transfer of
energy and momentum to the medium, and the scattering of radiation by the atoms.
The latter can be elastic, such as Rayleigh scattering [1], or inelastic (for example,
Raman [1] and Brillouin [2] scattering), while the former can be usually thought
of as the removal of electromagnetic energy from the field, followed by the creation
of various forms of excitation in the medium. The energy stored in the medium is
eventually re-emitted by spontaneous emission in the surrounding space [1] or redis-
tributed through the constituent atoms by non-radiative processes with an increase
in the temperature of the environment.
Absorption is accompanied by dispersion. At the most elementary level, we as-
sociate dispersion with the change in phase velocity of the various monochromatic
components of light that occurs when a wave enters the medium, for example from
vacuum. This effect can be observed most readily if a light beam with a broad spectral
distribution crosses the boundary of the medium with a non-zero angle of incidence.
1We call a medium “passive” when the atomic populations are distributed according to a thermal,or Boltzmann distribution. By contrast, an “active” medium is characterized by strong non-thermalpopulation distributions induced by various forms of pumping processes.
3
In fact, Snell’s law predicts that individual frequency components will undergo refrac-
tion at different angles. To be more precise, this is only an indirect manifestation of
the different phase velocities associated with the individual monochromatic compo-
nents. A more direct consequence of dispersion is the change in the temporal profile
of a light pulse upon propagation. As the different monochromatic components travel
through the medium with their own phase velocities, they get out of step from each
other and the light pulse undergoes a modifications of its initial temporal profile.
A quantitative description of these phenomena was provided by Lorentz who sim-
ulated the individual atoms as elementary dipole oscillators which are set in motion
by the electric field of the traveling wave. Lorentz’s model has the great virtue of ex-
plaining the basic features of linear absorption and dispersion phenomena. However,
it is unable to describe other more recently discovered effects which are intimately
relevant to this thesis.
Our work is concerned primarily with Electromagnetically Induced Transparency
(EIT), an effect that was discovered approximately 16 years ago [3, 4]. While later
chapters will provide a detailed theoretical description of EIT, it is appropriate at
this point to summarize the essential features of this phenomenon so that we may
emphasize the role that it plays in connection with the emergence of quantum inter-
ference effects.
Consider a collection of atoms whose three energy levels of interest are shown
4
schematically in figure 1.2. If a coherent, tunable field E1 is nearly resonant with the
transition 1→3 and if the field’s carrier frequency, ω1, is scanned continuously from a
red-detuned to a blue-detuned configuration, at first the field is weakly absorbed. As
the field frequency is tuned closer to the atomic transition frequency, ω31, the field is
absorbed progressively more, with the maximum absorption taking place under ideal
resonance conditions. If, now, the field frequency continues to increase, the absorption
coefficient decreases and eventually vanishes in a strongly blue-detuned configuration.
There is nothing very surprising about this qualitative description which becomes an
especially good fit to the true behavior of the system if the field E1 is weak enough
that the number of excited atoms in level 3 is always a small fraction of the ground
state population in level 1. We call this the weak saturation condition which we cover
in subsection 2.3.1
We hold the quantitative and more precise description of this process for later
sections of this work and summarize what we just said by stating that the power
absorption coefficient of the field E1, as a function of the carrier frequency ω1, has
the approximate form shown in figure 1.3.
Next we want to repeat the measurement of the power absorption coefficient of
the field E1 but, now, we first turn on a second field E2 (the so-called coupling field)
whose carrier frequency matches the atomic transition frequency ω32. We also assume
that the second field to be considerably stronger than the first in the sense that, on
resonance (ω2 = ω32), this field can saturate (or nearly saturate) the atomic transition
5
2→3. A scan of the power absorption coefficient of field E1 as a function of the carrier
frequency ω1 is shown in figure 1.4.
At first, the appearance of a minimum in the power absorption coefficient of the
probe field when a coupling field is allowed to interact with a different transition is
quite counter-intuitive. The implication is that, as widely verified experimentally, the
attenuation length of the probe field can be increased considerably, relative to the
situation when the coupling field is absent. Thus, the medium may be made “trans-
parent” for the probe, even in the presence of a strongly allowed transition at the
frequency ω31. This effect is called Electromagnetically Induced Transparency (EIT).
This is not the only unexpected consequence of EIT. The dispersion spectrum of the
probe also changes drastically in the presence of the coupling field, as we will discuss
in this work, and the group velocity of a probe pulse may undergo a dramatic reduc-
tion from the expected value c/n (where n is the index of refraction of the medium
at the frequency ω31) or even, under appropriate conditions, an increase which may
lead to values of the group velocity larger than c/n. This so-called “superluminal”
propagation has been shown convincingly not to conflict with our traditional under-
standing of the way light behaves under propagation.
A very closely related phenomenon to the EIT is Coherent Population Trapping
(CPT) [5]. An understanding of the characteristics of CPT can be done by recon-
sidering the atomic system (Fig. 1.2) studied for the EIT case, but this time we
monitor the fluorescence out of the excited level, level 3, instead of the absorption
6
of the probe field. In the absence of the coupling field, E2, the fluorescence follows
the behavior of the probe absorption coefficient (Fig. 1.3). Thus, the total fluores-
cence intensity increases when the probe’s carrier frequency approaches the atomic
transition frequency ω31 at which the fluorescence is maximum. When the coupling
field is turned on, a dip in the fluorescence spectral profile, i.e. a reduction of the
fluorescence, develops in the neighborhood of ω1 = ω31 very similar to the one ex-
perienced by the absorption (Fig. 1.4). The naming of this phenomenon, CPT, is
associated with the intuitive understanding that coherent effects can cause trapping
of the population in a coherent supperposition of two levels which, in turn, causes
the reduction in fluorescence from the third.
In addition to the Electromagnetically Induced Transparency (EIT) and Coherent
Population Trapping (CPT) phenomena, which have been associated with quantum
interference, other effects originate from the interaction of a three-level atom with two
optical fields. Lasing Without Inversion (LWI) [6] for example, is a consequence of the
EIT and CPT effects. A general review of these phenomena is provided in section 1.2.
Another effect that is featured by three-level systems is the Autler-Townes (dy-
namic Stark-Shift) effect which is reviewed in the next section. Even though the AT
effect resembles the EIT phenomenon in that they both reduce the absorption of a
probe field for appropriate values of the carrier frequency, AT is not associated with
interference. This important difference between the two effects is investigated further
in the following chapters of this thesis.
7
In this introductory chapter we set up the foundations for the upcoming chapters.
We introduce first three different systems that can be engineered when a three-level
atom interacts with two fields. Out of these three systems, we study in detail one
of them because this leads to the two configurations of interest to this work. A
brief review of the existing literature of the interaction of light with matter is then
presented. This review is followed by an evaluation of some of the loose ends of the
existing theories and a critical review of certain fundamental assumptions. Before
moving into the core part of this work, an explicit layout of the chapters supporting
their contents is presented.
1.1 Three-Level Systems
Three-level atoms, or molecules, interact with two nearly-resonant coherent fields
forming three different systems (as shown in Fig.1.1). Each field connects a separate
transition, but the two transitions share a common energy level. This thesis considers
the first type of these systems, the so-called cascade system.
We are interested in the case where one of the fields is allowed to have an arbitrary
strength, while the second is weak enough not to affect the atom appreciably, in a
sense that will be made more precise in subsection 2.3.1. According to the established
terminology, the first field is usually referred to as the coupling field, while the second
is known as the probe.
8
Of course, three-level atoms are idealizations but we can find reasonable approxi-
mations among the hydrogenic atoms and even alkali atoms and diatomic molecules
such as dimers of lithium, potassium and sodium. Before discussing the three main
systems of experimental significance it will be instructive to survey the lowest energy
levels of the alkali atoms rubidium and sodium, taken as examples
The 85Rb atom and its isotope, 87Rb, are commonly used for the purpose of cre-
ating three-level systems. Figure 1.5 shows a simplified energy diagram of 85Rb. An
electron in the level n=5 can have five different values of the orbital angular momen-
tum, ~L. The corresponding quantum number is L, where 0 ≤ L < 4. The intrinsic
angular momentum, or spin, of the electron interacts with the orbital angular momen-
tum and originates the fine structure. through the spin-orbit interaction. The total
electronic angular momentum ~J , carries the quantum number J, given by J = |L±S|,
where S is the quantum number of the total spin angular momentum, ~S, of all elec-
trons in the outer shell. In the case of interest here, one electron exists in the outer
shell of the 85Rb atom, which leads to S=1/2. The diagram (Fig. 1.5) includes all
levels from the ground state up to the third orbital, L=2 (D). The hyperfine struc-
ture is a result of the interaction of the total electron angular momentum, ~J , with
the nuclear angular momentum, ~I, I=5/2 for 85Rb. The total angular momentum,
~F , has the quantum number F, |J − I| ≤ F ≤ |J + I|, labeling the hyperfine states.
When a magnetic field is turned on, each degenerate hyperfine level splits into 2F+1
levels each with a quantum number mF , where −F ≤ mF ≤ F . We use the n(2S+1)LJ
9
notation to denote the fine energy levels.
We note that the excitation and decay processes of electrons can take place be-
tween levels characterized by ∆L = ±1 and ∆mJ = 0,±1. These conditions lead to
the requirements that ∆F = 0,±1.and ∆mF = 0,±1
Following the atomic selection rules, if two fields couple two different transitions
sharing a common energy level, three different systems can be formed. The first
of these, the cascade system (Fig.1.1.a), which is the system of interest in this the-
sis, can be achieved by coupling the transitions 52S1/2 − 52P3/2 and 52P3/2 − 52D5/2
[7, 8] of the 85Rb atom (Fig. 1.5). Fulton [7] also used the 85Rb atom to cre-
ate two other configurations. Coupling the transitions 52S1/2(F = 2) − 52P1/2 and
52S1/2(F = 3)− 52P1/2 Fulton created what is known as the lambda, Λ, system (Fig.
1.1.b). Fulton also formed the third system, V (Fig. 1.1.c) by connecting the transi-
tions 52S1/2(F = 3)− 52P3/2(F = 4) and 52S1/2(F = 3)− 52P1/2(F = 2).
Similar observations can also be carried out with sodium atoms. Reviewing the
partial diagram of the Sodium atom (Fig.1.6) [9] one can see how the Lambda system,
for example can be achieved by coupling the transitions 32S1/2(F = 1)−32P1/2(F = 2)
and 32S1/2(F = 2)− 32P1/2(F = 2) [10].
Either one of the two fields of every system showed in Fig. 1.1 can act as the probe
or the coupling field, depending on their respective strengths, a parameter which is,
10
in principle, under the control of the experimentalist. This fact leads to two different
configurations for every one of the three systems.
When the field which acts on the transition 1-2 of the cascade model is acting as
the probe field and the field connecting levels 2 and 3 functions as the coupling field,
we call the system Cascade-EIT (Fig. 1.7) in recognition of the fact that this con-
figuration is appropriate for demonstrating EIT behaviors. Switching the strengths
of the fields leads to another cascade configuration which we denote as Cascade-AT
(Fig. 1.8). This is the configuration that we have selected for our in-depth analysis
of the similarities and, specially, differences between the EIT and The AT effects.
For uniformity, we define the detuning parameters of the fields from exact reso-
nance to be positive when the frequency of the field is larger than the atomic transition
frequency, ωij, where ωij = (Ej −Ei)/~. The symbols Ej and Ei denote the energies
of the upper and lower levels coupled by the field. Hence, we define the detuning
parameters for the different cascade configurations in the following manner:
Cascade− EIT : δp = ωp − ω21, (1.1)
δc = ωc − ω32. (1.2)
Cascade− AT : δp = ωp − ω32, (1.3)
δc = ωc − ω21. (1.4)
Ordinarily, the atomic transition of interest is broadened not only by spontaneous
11
emission but also, and usually to a much greater extent, by the thermal motion of
the atoms, collisions and other external perturbations. For simplicity, we confine
ourselves only to the spontaneous emission broadening, a situation that has become
accessible nowadays with the use of atomic samples trapped and cooled in a magneto
optical trap [11]. The polarization decay rates are described by a full quantum theory
of spontaneous emission for an arbitrary n-level atom yielding, in the collision free
case, to the decay rates γij given by
γij =1
2
n∑
k
(Wik + Wjk), (1.5)
where Wij denotes the population decay rate from level i to level j.
All decay rates take the same form for our different systems. Equation 1.5 leads
to
γ12 =1
2W21, (1.6a)
γ23 =1
2(W21 + W32 + W31) , (1.6b)
γ31 =1
2(W32 + W31) . (1.6c)
We note that in chapter 4 in addition to the incoherent spontaneous decays, some
incoherent excitations are also considered. These additional rates will cause only mi-
nor modifications in the set of equations (Eq. 1.6).
12
It is important here to point out that in our following work we always express
the equations in terms of dimensionless units. The natural unit of frequency for our
problem is W21 so that W21 = 1 and the other decay rates must be interpreted as
the ratios of Wij/W21. For convenience of notation we use no special symbols for the
scaled decay rates and for the scaled Rabi frequencies which are directly proportional
to the driving field amplitudes, as we will make more precise at the appropriate point.
1.2 A Survey of the Literature
When three-level atoms interact with two electromagnetic fields they can display
non-linear optical behaviors which have been of great interest to researches in the
quantum optics and atomic physics communities. In this section we review the main
phenomena displayed by these systems, in so far as they relate to our work, their
applications, and existing theoretical descriptions.
A decrease in the fluorescence of a three-level Sodium atom was detected in 1976
by Alzetta et al. [12]. This experiment provided the first demonstration of Coherent
Population Trapping (CPT), which we introduced earlier in this chapter. Theoretical
explanations [13, 14, 15] of the trapping phenomena have appeared in the literature.
A broad topical review was written in 1996 by Arimondo [16] who explained the
trapping phenomenon with an in-depth study of the lambda system (Fig. 1.1b). The
two ground states |1 > and |2 > of the lambda system are coherently mixed by the
13
fields forming the so-called coupled state, |C >, and the non-coupled state, |NC >.
These states |C > and |NC > are respectively coupled and non-coupled to the upper
excited state, |3 >. Even though the non-coupled state is not coupled to the excited
state, it can get populated in two indirect ways. First by the relaxation rate Γ12,
which is almost zero in this case, and second by Raman detuning, δR, which is the
consequence of two photon stimulated absorption and emission respectively of the two
fields E1 and E2. Thus, atoms can be transfered to the non-coupled state and remain
trapped there causing the reduction in emission out of the upper level, 3. The differ-
ent couplings between the three states |3 >, |C >, and |NC > are shown in figure 1.9.
Another perspective of the physical origin of the trapping process was given by
Lounis and Cohen-Tannoudji [5] who studied the scattering process of one probe
photon. The authors showed that different scattering pathways interfere, causing the
reduction in absorption and consistently in the emission. CPT in the cascade and
lambda systems were also studied in ref. [17]. We will adopt in chapter 6 this theo-
retical perspective for our own investigations of quantum interference effects.
In 1990 the term “Electromagnetically Induced Transparency (E.I.T.)” was in-
troduced by Harris et al. [4] whose theoretical work showed an enhancement in the
third-order susceptibility and simultaneously an enhanced transparency in a collection
of Cascade-EIT-like systems. The first experiment confirming the EIT phenomenon
was done in 1991 by Boller, Imamoglu, and Harris [3] in a Strontium lambda system
(Fig. 1.1b). Boller and co-workers stated that the transparency may be interpreted
14
as a combination of the Stark effect and of another interference phenomenon under-
stood at the level of a special set of states called the dressed states. These dressed
states, which are eigenstates of to the interacting atom and fields system, are coherent
superpositions of the atomic bare states mixed by the fields. The proposed concept
is that the coupling field mixes the two levels 2 and 3 creating the two dressed states
A and B (Fig. 1.10). A dressed atom in the dressed level 1’ can become excited to
either one of the two dressed states, A and B. These two possible excitation pathways
interfere, causing a reduction in the probe absorption. Of special note is also reference
[18] were EIT was shown in a lambda system. We already denoted in the previous
section one of the cascade configurations as Cascade-EIT. This name, at least at this
early point in the thesis, recognizes the experiments [19, 20, 8] which showed EIT in
this specific system. Many reviews [16, 21, 22, 23] have been written about EIT.
A physical phenomenon of relevance in the literature of EIT is known as the Fano
profile. In 1961 Fano [24] studied the interference between two excitation channels
that couple a discrete level to a continuum. Two ionization pathways, one direct and
another one proceeding through an intermediate autoionised state, interfere leading
to a zero transition probability and thus a reduction of the ionization probability. The
underlying physical phenomenon responsible for EIT has been generally assumed to
be similar to the physical origin of the Fano profile.
EIT and CPT originate from closely related physical phenomena. EIT is associ-
ated with the reduction in the absorption of the probe field due to the interference
15
between different excitation pathways, while CPT is characterized by the reduction
in spontaneous emission due to the trapping of the population in the non-coupled
state.
In the case of EIT, and in the range of the frequency of the probe where the
absorption is reduced, the dispersion of the medium can be made very steep and to
acquire positive or negative signs. Due to the fact that dispersion is related to the
group velocity of the wave (this relationship will be discussed in chapter 2), light can
be made to propagate “slow” or “fast”. This effect was first studied by Harris and
co-workers [25] who demonstrated reduction in the speed of light by a factor of 250
relative to its speed in the nearly vacuum environment of a Pb vapor cell. Other
early experiments demonstrating reduction in the speed of light [26, 27, 28, 29] have
also been presented. Very recent works have suggested the use of “slow” light for
the production of quantum entanglement [30], and for an improved design of a gyro-
scope [31]. Furthermore, even the possibility of “stopping” light was suggested in ref.
[32]. In addition to the “slow” light, in 1994, “fast” light was also proposed [33] and
demonstrated [34] in 2000 (although a correction was provided by the authors [35] at
a later time) where vg = −c/310. The “fast” light in this case does not violate the
causality principle, as addressed for example in references [36, 37]. In 2002, Boyd [38]
wrote an extensive review dedicated to the topic of “slow” and “fast” light.
Another interesting consequence of atomic coherence is Lasing Without Inversion
(LWI) and the closely related Amplification Without Inversion (AWI). This phe-
16
nomenon was suggested in 1963 by Marcuse [39] and followed by a later study in
1977 by Holt [40]. In 1983, Arkhipkin and Heller [41] showed that the physical origin
of AWI is closely connected with the Fano interference. The authors claimed that the
probability of absorption can be made very small even when the emission probability
is different than zero. In 1989, Harris and co-workers [42, 43, 44] related the LWI
phenomenon to quantum interference. These authors showed that excitations exhibit
a Fano-like destructive interference, which causes a reduction in the absorption of the
field. In 1989, Imamoglu [45] extended the work of Harris by studying a standard
cascade system having its upper levels (2 and 3 in Fig. 1.1a) decaying to one com-
mon level. Other significant contributions were presented in 1991 by Narducci and
collaborators [46, 47] who emphasized the role of the atomic coherence in the LWI
phenomenon. In 1996, using techniques borrowed from quantum scattering theory,
Grynberg and his colleagues [6] showed that interfering pathways in a V system can
lead to amplification without inversion. In general, LWI requires, in addition to the
three-level system, a pump mechanism from the ground state to the upper state of
the lasing process. It is known that short-wavelength laser radiation is very hard to
achieve because of the rapid increase in the spontaneous emission rate of energetic
transitions relative to their optical counterparts. Thus, the LWI proposal circum-
vents the technical difficulties associated with the production of inverted population
between highly energetic transitions.
Bringing the applications of EIT closer to useful devices, the phenomenon has
been investigated in solid media. The challenge in this case is provided by the large
17
dephasing rates that are common in condensed matter systems. The first experiment
demonstrating EIT in solids was carried out in 1998 by Ichimura and his colleagues
[48].
Different applications benefit from the characteristics of EIT. The features of
reduction in absorption and fluorescence were used to study the spectroscopy of hy-
perfine levels in 85Rb atoms [49, 50]. Four Wave Mixing (FWM) usually suffer from
having one of the emitted fields strongly reabsorbed. This absorption problem can
be solved with the help of EIT [51, 52, 53] even in solid media [54].
A phenomenon quite unlike most of what we have surveyed this far is the so-
called dynamic Stark effect. In 1955 Autler and Townes published a paper entitled
“Stark Effect in Rapidly Varying Fields” [55]. The authors studied theoretically and
experimentally the effect of an RF field on the absorption line of a gas consisting of a
collection of molecules. The absorption line of the RF field splits into two components
creating a dip (reduction in absorption) at the atomic frequency similar to what is
shown in figure 1.11, where ω is the angular frequency of the field. Figure 1.12 shows
the model that was offered by Cohen-Tannoudji [56] in explaining the AT effect. A
coupling field acts on the transition 1-2 of a three-level cascade system (Fig. 1.12a).
Dressed by the field, the atom acquires the energy level structure shown in part b of
the figure, where N is the number of photons in the coupling field mode. Due to the
interaction between the coupling field and the atom, the two states |1, N + 1 > and
|2, N > become mixed and generate the two dressed states |a(N) > and |b(N) >. The
18
dressed states, in return, open the path for two transitions (represented by dashed
arrows in Fig. 1.12c) which lead to the split in the absorption line detected by Autler
and Townes. The split corresponds to the energy difference between the dressed states
which is equal to ~Ω, where the effective Rabi frequency Ω is defined as Ω =√
Ωc + δc.
We note that on resonance, δc = 0, the split becomes linear with respect to the cou-
pling field strength, a characteristic that we will use later in this thesis to identify the
existence of AT effect. The AT effect is used in three-level cascade systems [57] and
four-level systems [58]. We note that the system studied by Cohen-Tannoudji and
presented here is a simplified version (no probe field) of what we call in this work the
Cascade-AT configuration.
1.3 Motivation
The many experimental observations carried out with three-level atoms driven by two
fields are generally well established and understood. However some of the effects still
lack detailed understanding. For example, in both phenomena EIT and AT, a probe
field whose frequency is varied in the vicinity of the unperturbed atomic resonance
undergoes a drop in absorption (i.e. increased transmission), and this raises the nat-
ural question: are these phenomena fundamentally different so that one can clarify
the use of the different terms EIT and AT? Is there more to these effects beyond the
decreased absorption?
19
After reviewing the literature which we covered in the previous section, one learns
that unlike in the AT case [56], it has been suggested [19] that the EIT is an ef-
fect of quantum interference origin. This suggestion has been broadly accepted by
the quantum optics community which assumed that EIT is related to the Fano profile.
This matter of existence or absence of quantum interference effects was investi-
gated in three-level systems by different researchers. In 1995, Zhu, Narducci, and
Scully [59] studied simplified models of the Cascade-EIT and V system. The authors
showed that unlike the V system, the Cascade-EIT configuration exhibits interference
in the emission. In 1997, Agarwal [60] studied the interference in the four configura-
tions Cascade-EIT, Cascade-AT, lambda, and vee. The author claims the existence
of destructive interference in the Cascade-EIT and lambda systems, and constructive
interference in the Cascade-AT and vee systems. What is also important is the theory
of simultaneous existence of the EIT and AT effects in the lambda [3, 61], vee [62],
and Cascade-EIT [63] systems.
We investigate in this thesis the question of EIT vs AT by studying two of their
corresponding systems which we denoted by Cascade-EIT (Fig. 1.7) and Cascade-AT
(Fig. 1.8). What is interesting is that a simple change in the strengths of the fields
(use a weaker coupling field and turn it into the probe field and vice versa) turns the
Cascade-EIT configuration, for example, into the Cascade-AT one.
A specific aspect that sets these two cascade configurations appart and which has
20
been demonstrated in refs. [64, 65, 7] is that one needs a stronger coupling field to
create a transparency window, i.e. a reduction in the absorption, in the Cascade-AT
and V systems than in the Cascade-EIT and lambda systems.
We will argue in this work that EIT and AT absorption spectra originate, in fact,
from different underlying physical processes. We identify different interfering path-
ways in the Cascade-EIT model. We also clarify the absence of interference in the
Cascade-AT model. These results originate from our analysis of the problem with the
help of different techniques and under different limits. In the end, our conclusions are
unified and made more rigorous by the adoption of the scattering technique pioneered
by Lounis and Cohen-Tannoudji [5].
1.4 Layout
Before studying in detail the two cascade configurations of interest we provide first,
in chapter 2, a review of the well-established theory of absorption and dispersion in
quantum mechanical two-level systems, the prototypical atomic absorbers. We also
reveal the relationship between the atomic coherence and the absorption and disper-
sion, which will be used in chapters 3, and 4.
In chapter 3 we restate in greater details the major points that are to be addressed
in this thesis. We emphasize the observed similarities and differences between the two
21
cascade configurations and construct the atomic equations in the semi-classical pic-
ture, when quantized atoms interact with classical fields. We solve the equations
perturbatively with respect to the probe field, which is considered to be very weak
relative to the coupling field. We note that the perturbative technique imposes no re-
strictions on the strength of the coupling field. Through an analysis of the absorption
spectra for different values of the coupling field, we reveal the existence of different
situations. In the strong field limit, the absorption lines corresponding to the two
cascade configurations show similar dips which reflect reductions in the absorption
coefficient. This similarity does not hold in the weak coupling field regime. A reduc-
tion in the absorption persists in the Cascade-EIT case but not in its counterpart,
the Cascade-AT configuration, which shows no reduction in absorption. Studying the
apparent difference in one regime and the similarity in another, requires the use of
a variety of techniques which are valid over different ranges of the strength of the
coupling field.
In chapter 4 we introduce appropriate two-time atomic correlation functions and
the so-called regression theorem which we use to derive the absorption and fluo-
rescence spectra. With no constraints over the strength of the coupling field, we
reproduce the same analytical expressions of the absorption spectra found in chapter
3, Perturbative Technique. This match in the absorption spectra confirms the con-
sistency of our descriptions under the same limits regardless of the adopted technique.
The strong field limit is explored in chapter 5 where we confirm the similarity
22
between the behaviors of the two cascade configurations which was already discussed
in chapter 3. We derive the atomic equations for a fully quantized system, atom with
fields, in the so-called secular limit (introduced in chapter 5) with the help of the
master equation. The derived spectra are the sum of well separated Lorentzian-like
lines, which do not overlap in the secular limit. In the strong field regime and with
the help of the secular limit, we prove the absence of any interference phenomena
within the approximate solution of the problem.
We discuss the probability of absorbing a photon from the probe field mode in
chapter 6. We study the Cascade-EIT configuration in both bare and dressed (atom
dressed by the fields) pictures, and compare the respective results. As it turns out,
we discuss the Cascade-AT configuration only in the dressed state picture because of
technical limitations that will be discussed in chapter 6. The calculated transition
amplitude, which is associated with the process of absorption of interest, shows the
existence of interference in the Cascade-EIT configuration regardless to the strength
of the coupling field. We also show in the weak coupling field limit the absence of any
type of interference effect in the Cascade-AT configuration. The excitation pathways
of both configurations are studied and presented by appropriate diagrams.
The assumptions and limits which are adopted in every chapter make the results
valid over different ranges of the strength of the coupling field. Chapter 7 concludes
by relating the results found in the different techniques, and also by reemphasizing
the existence of interfering excitation pathways in the Cacasde-EIT configuration and
23
their absence in the Cascade-AT one. We also present in chapter 7 a general outline
of our future work.
Figure 1.1: Three-level systems: a) Cascade (Ξ), b) Lambda (Λ), and c) V.
24
Figure 1.2: Atomic three energy levels
0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ω1
Pow
er A
bs. C
oeff.
ω31
Figure 1.3: Abs. coeff. of the field E1 as a function of the carrier frequency ω1: E2=0
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
ω1
P
ower
Abs
. Coe
ff.
ω31
Figure 1.4: Abs. coeff. of the field E1 as a function of the carrier frequency ω1: E2 6=0
25
Figure 1.5: Rubidium’s partial energy diagram: 85Rb
26
Figure 1.6: Sodium’s partial energy diagram
27
Figure 1.7: Cascade-EIT configuration
Figure 1.8: Cascade-AT configuration
28
Figure 1.9: Coherent Population Trapping in a lambda system
Figure 1.10: Electromagnetically Induced transparency in a lambda system
29
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.01
0.02
0.03
0.04
0.05
0.06
0.07
ω
P
ower
Abs
. Coe
ff.
Figure 1.11: Autler-Townes split in the absorption line
Figure 1.12: Autler-Townes effect in a Cascade-AT system
Chapter 2: Two-Level System
In this chapter we describe the propagation of a quasi-monochromatic, polarized clas-
sical wave through a collection of two-level atoms [66]. Because the field is described
by the Maxwell wave equation and the atoms by a suitable density operator, obeying
the Liouville equation, this approach is usually called semi-classical. The atoms are
assumed to form a dilute gas, so that direct atom-atom interaction (e.g. dipole-dipole
interactions) can be neglected.
Our objective is to calculate the field absorption coefficient and the dispersion
spectrum under steady state conditions. We start by finding the field equation in the
quasi-monochromatic approximation which is known as the reduced field equation.
We then derive in the rotating-wave approximation the coherent part of the atomic
equations of motion following from the Liouville-von Neumann equation. These equa-
tions are completed by phenomenologically adding the appropriate incoherent decay
terms. After establishing the relation between the microscopic atomic coherences (the
off-diagonal density matrix elements) and the macroscopic polarization, we derive the
Maxwell-Bloch equations. In steady state we calculate the polarization, which after
the introduction of the low saturation limit becomes related to the absorption and
30
31
dispersion coefficients experienced by the field. We present at the end of this chapter
a discussion of the derived absorption and dispersion coefficients.
2.1 The Field Equation
As we know, the effect of the magnetic field of an electromagnetic wave is generally
negligible, as compared to the effect of the electric field on atoms. Thus, we neglect
the magnetic part of the wave and consider only the electric field component, which
can be represented in the rather general form
E(z, t) = Eo(z, t)e−i(kz−ωt) + Eo∗(z, t)ei(kz−ωt). (2.1)
In equation 2.1, Eo(z, t) is a complex envelope function describing the field varia-
tion in space and time as it propagates, while ω is the carrier frequency of the field,
and k = ω/c is the vacuum wave number.
When the electromagnetic wave is quasi-monochromatic, the electric field ampli-
tude Eo(z, t) and its complex conjugate are slowly varying in both space and time.
In the presence of the electric field, a polarization appears in the medium, and its
general form is:
P (z, t) = Nµi[P o(z, t)e−i(kz−ωt) − P o∗(z, t)ei(kz−ωt)
], (2.2)
32
where N is the number of atoms per unit volume, µ is the modulus of the atomic
transition dipole moment, and P o(z, t) is a dimensionless complex envelope function.
According to Maxwell’s equations, the polarization acts as the driving source of
the field, and its origin can be traced to the atoms in the medium. Obviously, we will
need to find an explicit relation between the polarization envelope P o(z, t) and the
microscopic atomic variables.
Our first step is to take advantage of the quasi-monochromatic approximation
which, as already mentioned, implies that the space-time variations of the field enve-
lope are slow as compared to those of the field carrier. More precisely we require
∣∣∣∣∂Eo(z, t)
∂z
∣∣∣∣ << k |Eo| ,∣∣∣∣∂Eo(z, t)
∂t
∣∣∣∣ << ω |Eo| , (2.3)
and for consistence we also require:
∣∣∣∣∂P o(z, t)
∂t
∣∣∣∣ << ω |P o| . (2.4)
In this way, the wave equation for the full Maxwell field
∂2E(z, t)
∂z2− 1
c2
∂2E(z, t)
∂t2= µo
∂2P (z, t)
∂t2, (2.5)
33
reduces to the much simpler equation for the field and polarization envelopes:
c∂Eo(z, t)
∂z+
∂Eo(z, t)
∂t= −α P o(z, t), (2.6)
where
α =N µ ω
2 εo
. (2.7)
2.2 Atomic Equations of Motion
We consider now the medium at a microscopic scale and study the atoms from a
quantum mechanical point of view. When dealing with a mixture of states, which
are brought about by the incoherent interaction of the atoms with the external en-
vironment (collision, spontaneous emission, etc.) the Liouville equation provides the
appropriate description. This equation is given by
i~dρ
dt= [H(t), ρ] , (2.8)
where ρ is the density operator of a single atom and H(t) is the total Hamiltonian,
H(t) = H0 + H1(t). (2.9)
In equation 2.9, H0 represents the unperturbed contribution in the absence of the
applied electric field, i.e,
H0 = E2 |2 >< 2|, (2.10)
34
and E2 = ~ω21 is the energy of the atomic excited state |2 >, measured relative to
the ground state |1 >, which is considered as the origin of the energy axis; ω21 is the
atomic transition frequency. The interaction Hamiltonian in the dipole approximation
has the well-known form
H1(t) = −p E(z, t), (2.11)
where p is the atomic dipole operator in the direction of polarization of the field and
E(z,t) is the field, evaluated at the position of the atom. As already discussed in
section 2.1, a convenient representation for the electric field is
E(z, t) = Eo(z, t)e−i(kz−ωt) + Eo∗(z, t)ei(kz−ωt)
= Eo(z, t)eiωt + Eo∗(z, t)e−iωt, (2.12)
where the new amplitudes Eo and Eo∗ are introduced for notational convenience.
The dipole moment operator for a two-level system has the form
p = µ (|1 >< 2| + |2 >< 1|), (2.13)
where µ is the modulus of the dipole moment connecting the states of interest. The
free time evolution of the projectors |1 >< 2| and |2 >< 1|, in the absence of the
35
applied field, is given by
(|1 >< 2|)t ' e−iω21t, (2.14)
(|1 >< 2|)t ' eiω21t. (2.15)
It is obvious that the Hamiltonian H1(t) (Eq. 2.11) contains rapidly varying
terms, oscillating at frequencies ±(ω21+ω), and slowly varying terms whose harmonic
variation is of the type ±(ω21−ω). Because the average evolution of the atom occurs
over a much longer time scale than (ω21 + ω)−1, to an excellent approximation it is
possible to ignore the rapidly varying terms in the interaction Hamiltonian (this is
usually called the rotating-wave approximation), so that
H1∼= −µEo(z, t)eiωt|1 >< 2| − µEo∗(z, t)e−iωt |2 >< 1|. (2.16)
The remaining explicit time-dependence in the Hamiltonian can be removed with
the help of a small trick. We begin by writing the total Hamiltonian in the form
H = E2|2 >< 2| +[−µEo(z, t)eiωt|1 >< 2| − µEo∗(z, t)e−iωt|2 >< 1|]
≡ HA + HB, (2.17)
36
where we define HA and HB function of an unknown parameter ωM as
HA = ~ωM |2 >< 2|, (2.18)
HB = ~(ω21 − ωM)|2 >< 2| − (µEo(z, t)eiωt|1 >< 2|+ c.c.
). (2.19)
Next, we transform the Liouville equation (Eq. 2.8) to the interaction represen-
tation, and finally select ωM in such a way that the interaction Hamiltonian in the
interaction representation does not contain any explicit dependence on time. To be
more specific, we define ρ (the density operator in the interaction picture) as
ρ = U ρ U−1, (2.20)
where
U = ei~HAt = eiωM t|2><2|. (2.21)
With this transformation the Liouville equation
i~dρ
dt= [HA + HB(t), ρ] , (2.22)
becomes
i~dρ
dt=
[HB(t), ρ
], (2.23)
37
where
HB(t) = U HB(t) U−1
= ~(ω21 − ωM)|2 >< 2| − µ(Eo(z, t)|1 >< 2|ei(ω−ωM )t|2><2| + c.c.). (2.24)
Now, if we select ωM = ω, the interaction Hamiltonian becomes explicitly time
independent, i.e,
HB(t) = ~(ω21−ω) |2 >< 2| − µ (Eo(z, t) |1 >< 2| − Eo∗(z, t) |2 >< 1|) . (2.25)
It’s now a trivial matter to construct the equations of motion for the matrix
elements ρij of the density operator in the interaction picture. The result is
˙ρ11 = iµ
~(Eo(z, t)ρ21(t)− Eo∗(z, t)ρ12(t)) , (2.26a)
˙ρ12 = iµ
~Eo(z, t) [ρ22(t)− ρ11(t)] − i(ω − ω21)ρ12(t), (2.26b)
˙ρ21 = −iµ
~Eo∗(z, t) [ρ22(t)− ρ11(t)] + i(ω − ω21)ρ21(t), (2.26c)
˙ρ22 = −iµ
~(Eo(z, t)ρ21(t)− Eo∗(z, t)ρ12(t)) . (2.26d)
2.2.1 The Atomic Master Equation
The previous set of equations (Eqs. 2.26) describe the reversible (coherent) dynamics
of the atom while it interacts with the external field. In reality, an atom is also subject
38
to many other effects, usually random in nature, such as collisions with other atoms,
interaction with the external thermal environment, and spontaneous emission, just
to mention the most important ones. For our purpose, mentioned in section 1.1, we
focus only on the effects of spontaneous emission.
A rigorous description of this process [67] requires a fully quantum mechanical
theory of the electromagnetic field which is beyond the scope of the present discussion.
Remarkably, however, the essential physical consequences of spontaneous emission can
be captured by the inclusion of phenomenological relaxation terms in equation 2.26,
i.e., by the addition of the damping terms
(˙ρ11
)incoh
= W21 ρ22, (2.27a)
(˙ρ12
)incoh
= −γ12 ρ12, (2.27b)
(˙ρ21
)incoh
= −γ21 ρ21, (2.27c)
(˙ρ22
)incoh
= −W21 ρ22. (2.27d)
Physically, the inclusion of the terms ±W21ρ22 in equations 2.26a and 2.26d, re-
spectively, simulates the effect of spontaneous emission at a rate W21, while the terms
−γ12ρ12 and −γ21ρ21 describe how spontaneous emission (a random, incoherent pro-
cess) destroys the coherence which may have been established within the atom at the
beginning of the evolution. Equation 1.5, γij = 1/2∑n
k(Wik +Wjk), in this case yields
γ12 = γ21 =1
2W21. (2.28)
39
For simplicity of notation, we let ρij = Rij and the final form of the atomic
equation becomes
R11 = iµ
~(Eo(z, t)R21(t)− Eo∗(z, t)R12(t)) + W21R22, (2.29a)
R12 = iµ
~Eo(z, t) [R22(t)−R11(t)] − i(ω − ω21)R12(t) − γ12R12(t), (2.29b)
R21 = −iµ
~Eo∗(z, t) [R22(t)−R11(t)] + i(ω − ω21)R21(t) − γ21R21, (2.29c)
R22 = −iµ
~(Eo(z, t)R21(t)− Eo∗(z, t)R12(t)) − W21R22. (2.29d)
2.3 Connection Between Macroscopic and Microscopic Vari-
ables
The average dipole moment of one atom is given by
< p > = Tr(ρp) = µ(ρ12 + ρ21). (2.30)
Thus the macroscopic polarization for an ensemble of N atoms per unit volume is
P = N < p > = Nµ(ρ12 + ρ21). (2.31)
An important byproduct of equations 2.30 and 2.31 is that they provide a concrete
meaning for the off-diagonal elements of the density matrix: ρ12 and ρ21 are directly
related to the average dipole moment per atom.
40
Because the source of the field equation (Eq. 2.6) is the macroscopic polarization
envelope and the atomic equations involve microscopic variables, we must find a link
between the two. This is readily established if we compare equation 2.2 with equation
2.31, i.e.,
i P o(z, t)e−i(kz−ωt) = − i ρ12. (2.32)
Changing back from ρij to Rij and in view of the relation ρ12 = R12 eiωt, we have
P o(z, t) = − i R12 eikz. (2.33)
With the help of equation 2.29b we now have
∂P o(z, t)
∂t= − i
(−i
µ
~Eo(z, t) D − iδR12(t) − γ21R12(t)
)eikz, (2.34)
where we defined :
δ = ω − ω21 (the field detuning) (2.35)
D = R11 − R22 (the population difference) (2.36)
Finally using equation 2.33 and the equation Eo(z, t) = Eoe−ikz we have
∂P o(z, t)
∂t= −µ
~Eo(z, t) D − (γ12 + iδ)P o. (2.37)
41
The equation of motion for the population difference follows from equations 2.37,
2.29a and 2.29d. The result of a few simple manipulations is
dD
dt= 2
µ
~(Eo(z, t)P o∗ + Eo∗(z, t)P o) + W21(1−D). (2.38)
The three equations 2.6, 2.37, and 2.38 form the so-called Maxwell-Bloch equations
c∂Eo(z, t)
∂z+
∂Eo(z, t)
∂t= −α P o(z, t) (2.39a)
∂P o(z, t)
∂t= −µ
~Eo(z, t) D − (γ12 + iδ)P o (2.39b)
dD
dt= 2
µ
~(Eo(z, t)P o∗ + Eo∗(z, t)P o) + W21(1−D)
(2.39c)
In steady state, the Maxwell-Bloch equations become
c∂Eo(z, t)
∂z= −α P o(z, t), (2.40a)
0 = −µ
~Eo(z, t) D − (γ12 + iδ)P o, (2.40b)
0 = 2µ
~(Eo(z, t)P o∗ + Eo∗(z, t)P o) + W21(1−D). (2.40c)
Now we can easily calculate the steady state atomic variables P o(z, t → ∞) and
D(z, t →∞) in terms of the steady state field envelope Eo(z, t →∞) from equations
42
2.40b and 2.40c, with the result
P o(z, t →∞) = − W21 Ω
2 (1 + iδ)D, (2.41)
D(z, t →∞) =1 + δ2
1 + δ2 + Ω2, (2.42)
where we have defined
δ =δ
γ12
, (2.43a)
W21 =W21
γ12
, (2.43b)
Ω =Ω
γ12
, (2.43c)
where Ω is the Rabi frequency of the field defined as
Ω =2 µ Eo
~. (2.44)
With the help of equation 2.42, the steady state polarization envelope takes the
form
P o(z, t →∞) = − W21Ω
2
1 − i δ
1 + δ2 + Ω2. (2.45)
The polarization is a nonlinear function of the Rabi frequency and has the same
direction as the field. It’s often convenient to express equation 2.45 in the form
P o = εo χ Ω, (2.46)
43
where the complex susceptibility, χ, which is defined as
χ(z, t →∞) = −W21
2 εo
1 − i δ
1 + δ2 + Ω2, (2.47)
depends on both the field intensity and the detuning parameter. The intensity de-
pendence implies that we are not dealing with a linear medium.
2.3.1 The Low Saturation Limit
In the weak field limit where
Ω ¿ δ, (2.48)
or Ω ¿ γ12, (2.49)
the susceptibility (Eq. 2.47) reduces to
χ(z, t →∞) ≈ −W21
2 εo
1 − i δ
1 + δ2. (2.50)
In this case equation 2.40a implies
Eo(z, t →∞) = Eo(0)e−αχz, (2.51)
what is known as Beer-Lambert law.
In the weak field limit the populations difference, D = R11−R22 (Eq. 2.36), given
44
in steady state by equation 2.42,
D(z, t →∞) =1 + δ2
1 + δ2 + Ω2, (2.52)
is approximately equal to unity. In this case, most of the population exist in the
ground level 1. This is why the two conditions, equations 2.48 and 2.49, lead to what
is known as the low saturation limit.
After expressing χ in terms of real and imaginary parts, χ = χR + iχI , equation
2.51 can be rewritten in the form
Eo(z, t →∞) = Eo(0)e−αχRze−iαχIz, (2.53)
so the complete Maxwell field now takes the form
E(z, t →∞) = Eo(0)e−i((αχI+k)z − ωt) e−αχRz. (2.54)
Equation 2.54 can be interpreted in the following way: The field enters the medium
with an initial amplitude Eo(0) and then it is attenuated (or amplified, depending on
the sign of χR) by the factor e−αχRz. Furthermore its vacuum wave number k is
modified and in the medium it becomes αχI + k, which is frequency dependent. As
we know from elementary electromagnetic theory, αχR can be interpreted as the ab-
sorption coefficient, while −αχI + k is proportional to n(ω) − 1, where n(ω) is the
index of refraction.
45
It is important at this point to notice that the absorption coefficient, αχR, is
proportional to the real part of χ, which is in turn proportional to the real part of P o
by the equation P o = εoχΩ (Eq. 2.46). Now, by the use of equation 2.33, P o = −iR12,
we fined that the absorption coefficient is proportional to the negative imaginary part
of R12. Similar analysis yields that n(ω) − 1 is proportional to the real part of R12.
These last two pieces of information will be very useful in the development of the
following chapters of this thesis.
2.4 Discussion of The Results
In the previous section, αχR and −αχI + k were calculated and related respectively
to the absorption coefficient and index of refraction in the low saturation limit (i.e.
Ω ¿ γ12). We study in this section, as in chapters 3 and 4, the imaginary and real
parts of the appropriate atomic coherence, R12 in this case. The negative imagi-
nary part of R12 will be referred to as the absorption coefficient while the real part as
the dispersion coefficient, which is the rate of change of the index of refraction, dn/dω.
Figure 2.1 shows how the absorption coefficient varies with respect to the field
detuning from the atomic transition frequency. The absorption line has a Lorentzian
shape, whose maximum is controlled by the decay rate W21 (hence by the square of
the atomic dipole moment), while the width at half-max. is controlled by the polar-
ization relaxation rate γ12. The maximum is located at δ = 0, corresponding to a
46
situation where the field is resonant with the atom. The further we move away from
the resonance, the less absorption is experienced by the field.
In figure 2.2 we study the behavior of the dispersion coefficient (essentially the
behavior of the index of refraction) inside the medium and in the neighborhood of the
atomic resonance. To be specific, figure 2.2 displays [n(ω) − 1] as a function of the
detuning parameter δ, where n(ω) is the index of refraction. Away from the resonance,
n(ω) increases as a function of increasing frequency, but in the vicinity of δ = 0,
instead, it decreases. This behavior is usually referred to as anomalous dispersion.
This slope, dn/dω, is directly related to the group velocity by the equation vg = c/ng
where the group refractive index, ng, is defined by the equation ng = n + ω dn/dω
[38]. Note that the anomalous dispersion region correspond to a negative slope of
the index of refraction. This point should be kept in mind for later discussion. Just
as the absorption, dispersive effects become less important away from resonance and
eventually vanish for frequencies that are greatly removed from the atomic resonance
frequency (of course, one must keep in mind that the real atom have innumerable other
resonances whose effects add up together). In this case the wave passes through the
medium without appreciable absorption or dispersion effects.
47
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
δ
Abs
orpt
ion
Coe
ffici
ent
Figure 2.1: Absorption line for a two-level system: Ω = 0.5
−10 −8 −6 −4 −2 0 2 4 6 8 10−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
δ
Dis
pers
ion
Coe
ffici
ent
Figure 2.2: Dispersion line for a two-level system: Ω = 0.5
Chapter 3: Perturbative Technique
The physical settings of interest, the two cascade configurations, involve a strong cou-
pling field which greatly affects the original unperturbed state of the atom. In this
arrangement our intention is to explore the stationary configuration of the medium
without further modifying its proprieties. This is done by sending a second field (the
probe) through the atoms, which is weak enough to be affected by the medium but
incapable of altering its steady state to any appreciable amount. This situation is
just right for the adoption of perturbative handling of the steady state.
We start this chapter by studying the Cascade-EIT configuration in detail in
section 3.1 where we generalize the derivation of the atomic equations done in the
previous chapter to the system of interest, and complement it by introducing the per-
turbative technique. The end results of the Cascade-AT configuration are presented
in section 3.2 and then studied in comparison with the results of the Cascade-EIT
configuration in section 3.3.
This chapter, specifically section 3.3, defines the problem of this thesis. Varying
the strength of the coupling field, we study different plots of the absorption coefficient
48
49
and other variables. We reveal the differences and similarities between the Cascade-
EIT and Cascade-AT configurations and set up well defined points to be explored in
the following chapters.
3.1 Cascade-EIT
Following the same procedure described in detail in chapter 2, but now generalized to
account for the fact that the atoms interact simultaneously with two fields, we derive
the Master equation for the Cascade-EIT configuration (Fig. 1.7) in subsection 3.1.1.
The atomic equations of motion are then solved in subsection 3.1.2 perturbatively to
the lowest order in the strength of the probe field.
3.1.1 Derivation of the Master Equation: An Outline
The probe and the coupling fields are represented by the plane waves
Ep(z, t) = Eop (z, t)e−i(kpz−ωpt) + Eo∗
p (z, t)ei(kpz−ωpt)
= Eop(z, t)e
iωpt + Eo∗p (z, t)e−iωpt, (3.1)
Ec(z, t) = Eoc (z, t)e−i(kcz−ωct) + Eo∗
c (z, t)ei(kcz−ωct)
= Eoc (z, t)e
iωct + E∗oc (z, t)e−iωct, (3.2)
50
and their envelopes obey the two wave equations
c∂Eo
p (z, t)
∂z+
∂Eop (z, t)
∂t= −αp P o
p (z, t), (3.3)
c∂Eo
c (z, t)
∂z+
∂Eoc (z, t)
∂t= −αc P o
c (z, t), (3.4)
where
αp =N µ ωp
2 ε0
, (3.5)
αc =N µ ωc
2 ε0
. (3.6)
Equations 3.3 and 3.4 are not independent of each other because the respective
source terms P op and P o
c are linked by the atomic dynamics. If the coupling field is
sufficiently intense, it is likely to suffer only negligible attenuation and dispersion as
it propagates through the medium. In this case, which will be the setting of greatest
interest in this work, the probe field in steady state will be affected mainly by the
coupling field at the input part of the medium.
At the same time, the atom is described by its density operator
ρ =
ρ11 ρ12 ρ13
ρ21 ρ22 ρ23
ρ31 ρ32 ρ33
, (3.7)
which in the absence of the fields, evolves only under the action of the unperturbed
51
Hamiltonian H0 (using E1 as the origin of the energy axis) of the form
H0 = E2|2 >< 2| + E3|3 >< 3|. (3.8)
In the presence of the fields, the interaction Hamiltonian, H1, is given by
H1(t) = −p1Ep(z, t) − p2Ec(z, t)
= −µ12(|1 >< 2|+ |2 >< 1|) (Eop(z, t) eiωpt + h.a.) −
µ23(|2 >< 3|+ |3 >< 2|)(Eoc (z, t) eiωct + h.a.), (3.9)
which upon using the rotating wave approximation to eliminate the high frequency
components, reduces to
H1(t) = − (µ12 |2 >< 1| Eo∗
p (z, t)e−iωpt + h.a.) −
(µ23 |3 >< 2| Eo∗
c (z, t)e−iωct + h.a.). (3.10)
In order to remove the explicit time dependence from the Liouville equation (Eq.
2.8), we transform the equations to the interaction picture, after defining the new
density operator, ρ, as
ρ = U ρ U−1, (3.11)
where
U = ei~HAt = exp [i (ωpt |2 >< 2| + (ωp + ωc)t |3 >< 3|)] . (3.12)
52
The new interaction Hamiltonian in the interaction picture, HB, is
HB(t) = U HB(t) U−1
= −~δp|2 >< 2| − ~(δp + δc)|3 >< 3| −
µ23 (Eoc (z, t)|3 >< 2|+ h.a.)− µ12
(Eo
p(z, t)|2 >< 1|+ h.a.). (3.13)
The density operator, ρ, satisfies the Liouville equation
i~dρ
dt=
[HB(t), ρ
]. (3.14)
After using equation 3.14 explicitly in terms of the matrix elements ρij we obtain
a description of the reversible (coherent) atomic dynamics. As already mentioned,
many processes of a random character introduce irreversible behavior. We consider
only the effects of spontaneous emission which can be simulated by the inclusion
of phenomenological relaxation terms. At this point, after the change in notation
ρij → Rij, we obtain the following atomic equations,
R12 = −i
µ12E
o∗p (z, t)
~(R11 −R22) + δpR12 +
µ23Eoc (z, t)
~R13
− γ12R12, (3.15a)
R13 = −i
−µ12E
o∗p (z, t)
~R23 + (δp + δc)R13 +
µ23Eo∗c (z, t)
~R12
− γ13R13, (3.15b)
R22 = −i
−µ23E
o∗c (z, t)
~R32 −
µ12Eop(z, t)
~R12 + c.c.
−W21R22 + W32R33,(3.15c)
R23 = −i
µ23E
o∗c (z, t)
~(R22 −R33)− µ12Ep(z, t)
~R13 + δcR23
− γ23R23, (3.15d)
R33 = −i
−µ23E
oc (z, t)
~R23 +
µ23Eo∗c (z, t)
~R32
− (W32 + W31)R33, (3.15e)
53
where R11 = 1 − R22 − R33, Rij = R∗ji, and the polarization decay rates, γij, are
given by the set of equations 1.6.
At this point it becomes convenient to exhibit explicitly the z dependence through
the definitions
Eop(z, t) = Eo
p eikpz , Eo∗p (z, t) = Eo∗
p eikpz,
Eoc (z, t) = Eo
c eikcz , Eo∗c (z, t) = Eo∗
c eikcz,
and then to absorb it into the dynamical variables with the help of a final change of
variables, as follows
→ Sii = Rii, (3.16a)
Eop(z, t) R12 = Eo
p eikpz R12 → S12 = eikpz R12, (3.16b)
Eo∗p (z, t) R21 = Eo
p e−ikpz R21 → S21 = e−ikpz R21, (3.16c)
Eoc (z, t) R23 = Eo
c eikcz R23 → S23 = eikcz R23, (3.16d)
Eo∗c (z, t) R32 = Eo∗
c e−ikcz R32 → S32 = e−ikcz R32, (3.16e)
→ S13 = ei(kp+kc)z R13. (3.16f)
It is also convenient to define the probe and coupling Rabi frequencies according
54
to
Ωp =2 µ12 Eo
p
~, (3.17a)
Ωc =2 µ23 Eo
c
~, (3.17b)
and to introduce the complex rates
Γ12 = γ12 + iδp, (3.18a)
Γ23 = γ23 + iδc, (3.18b)
Γ13 = γ13 + i(δp + δc). (3.18c)
With these definitions, the final form of the atomic equations of motion is
S12 = −Γ12S12 + iΩ∗
p
2(S22 − S11) − i
Ωc
2S13, (3.19a)
S13 = −Γ13S13 + iΩ∗
p
2S23 − i
Ω∗c
2S12, (3.19b)
S21 = −Γ∗12S21 − iΩp
2(S22 − S11) + i
Ω∗c
2S31, (3.19c)
S31 = −Γ∗13S31 − iΩp
2S32 + i
Ωc
2S21, (3.19d)
S23 = −Γ23S23 + iΩ∗
c
2(S33 − S22) + i
Ωp
2S13, (3.19e)
S32 = −Γ∗23S32 − iΩc
2(S33 − S22) − i
Ω∗p
2S31, (3.19f)
S22 = iΩ∗
c
2S32 − i
Ωc
2S23 − i
Ω∗p
2S21 + i
Ωp
2S12 −W21S22 + W32S33, (3.19g)
S33 = iΩc
2S23 − i
Ω∗c
2S32 − (W31 + W32)S33, (3.19h)
55
where
S11 = 1 − S22 − S33. (3.20)
3.1.2 Perturbative Solution of the Atomic Equations:
As mentioned in section 2.3, the absorption and dispersion coefficients are respectively
proportional to the negative imaginary and real parts of the appropriate off-diagonal
density matrix element. In this case, the coherence of interest is ρ12 (S12) because
the probe acts on transition 1-2. We also need to calculate the populations of levels
2 and 3, ρ22 (S22) and ρ33 (S33), which are directly related to the total fluorescence
of these levels.
We begin by casting the eight atomic equations in the vector form
~ϕ = L0~ϕ + iΩp
2L1 ~ϕ + i
Ωp
2I, (3.21)
where the matrices L0 and L1 are independent of Ωp and where we have defined the
components of ~ϕ as follows
S12 → ϕ1, S13 → ϕ2, S21 → ϕ3, S31 → ϕ4,
S23 → ϕ5, S32 → ϕ6, S22 → ϕ7, S33 → ϕ8.
56
The explicit form of the matrices is
L0 =
−Γ12 −iΩc
20 0 0 0 0 0
−iΩ∗c2
−Γ13 0 0 0 0 0 0
0 0 −Γ∗12 iΩ∗c2
0 0 0 0
0 0 iΩc
2−Γ∗13 0 0 0 0
0 0 0 0 −Γ23 0 −iΩ∗c2
iΩ∗c2
0 0 0 0 0 −Γ∗23iΩc
2−iΩc
2
0 0 0 0 −iΩc
2iΩ∗c
2−W21 W32
0 0 0 0 iΩc
2−iΩ∗c
20 −(W31 + W32)
,
L1 =
0 0 0 0 0 0 2 1
0 0 0 0 1 0 0 0
0 0 0 0 0 0 −2 −1
0 0 0 0 0 −1 0 0
0 1 0 0 0 0 0 0
0 0 0 −1 0 0 0 0
1 0 −1 0 0 0 0 0
0 0 0 0 0 0 0 0
, I =
−1
0
1
0
0
0
0
0
.
Next, we approximate the vector ~ϕ by a second order expansion in the small
57
parameter Ωp, i.e,
~ϕ = ~ϕ(0) + Ωp ~ϕ(1) + Ω2p ~ϕ(2). (3.22)
In steady state, when ~ϕ = 0, equations 3.21 and 3.22 yield the following results
in increasing order of perturbation
Zeroth order : L0 ~ϕ(0) = 0, (3.23a)
First order : L0 ~ϕ(1) + i ~I + i L1 ~ϕ(0) = 0, (3.23b)
Second order : L0 ~ϕ(2) + i L1 ~ϕ(1) = 0. (3.23c)
The variables of interest are ϕ1 (S12), ϕ7 (S22) and ϕ8 (S33). Upon solving the
zeroth order equation 3.23a, we find
~ϕ(o) = 0, (3.24)
while the first order, equation 3.23b, yields
ϕ(1)1 = −i
γ13 + i(δp + δc)|Ωc|2
4+ [γ12 + iδp] [γ13 + i(δp + δc)]
, (3.25a)
ϕ(1)7 = 0, (3.25b)
ϕ(1)8 = 0. (3.25c)
58
The second order, equation 3.23c, after some algebraic manipulations gives
ϕ(2)1 = 0, (3.26a)
ϕ(2)7 =
2(CEIT + W32 + W31)AbsEITp − W31P
EIT
D, (3.26b)
ϕ(2)8 =
2CEITAbsEITp + W21P
EIT
D, (3.26c)
where
D = CEIT(W31 + W21) + W21(W32 + W31), (3.27a)
CEIT = 2γ23
γ223 + δ2
c
|Ωc|24
, (3.27b)
PEIT = 2<[ |Ωc|2/4[γ23 + iδc] (|Ωc|2/4 + [γ12 + iδp] [γ13 + i(δp + δc)])
], (3.27c)
AbsEITp = −=
[ϕ
(1)1
]. (3.27d)
In summary, the required atomic parameters are given by
S(1)12 = −i
γ13 + i(δp + δc)
|Ωc|2/4 + [γ12 + iδp] [γ13 + i(δp + δc)], (3.28)
S(2)22 =
2(CEIT + W32 + W31)AbsEITp − W31P
EIT
D, (3.29)
S(2)33 =
2CEITAbsEITp + W21P
EIT
D. (3.30)
A detailed interpretation of these results is given in section 3.3.
59
3.2 Cascade-AT
The work that must be done for the Cascade-AT configuration (Fig. 1.8) is very
similar to what we previously presented for the Cascade-EIT system. In the Cascade-
AT case the probe field couples levels 2 and 3. Again we solve the different orders of
perturbation for the density matrix elements S23, S22, and S33, and find that
S(0)22 =
CAT
W21 + 2CAT, (3.31)
S(1)23 =
−i
W21 + 2CAT
1
γ12 + iδc
[γ12 + iδc][γ13 + i(δp + δc)]CAT + W21
|Ωc|24
[γ13 + i(δp + δc)][γ23 + iδp] + |Ωc|24
, (3.32)
S(2)22 = − 2
W32 + W31
W31 + CAT
W21 + 2CATAbsAT
p , (3.33)
S(2)33 =
2
W32 + W31
AbsATp , (3.34)
where CAT = 2γ12
γ212 + δ2
c
|Ωc|24
, (3.35)
AbsATp = −=
[S
(1)23
]. (3.36)
3.3 Discussion of the Results
This section studies the absorption and dispersion coefficients, and the fluorescence
spectra of the upper levels, 2 and 3. Each figure in this section is a comparison of two
plots. Each plot corresponds to one of the two Cascade configurations. These figures
bring to our attention the similarities between the two systems in some regimes and
the differences in some other regimes.
60
3.3.1 Absorption Coefficients
In this subsection we study the absorption lines spectra of the two different cascade
configurations in a variety of cases. AbsEITp (Eq. 3.27d) and AbsAT
p (Eq. 3.36) are
plotted simultaneously in figures 3.1 and 3.2 for comparison.
In the resonant and strong coupling field case, δc = 0 and Ωc À γij, the two ab-
sorption lines look very similar (Fig. 3.1). They are made out of two Lorentzian-like
bands separated by a dip. The dip is referred to as the transparency window, which
reflects the reduced absorption experienced by the probe. The similarity of the two
absorption spectra might tempt us to assume that the two configurations, in general,
exhibit the same physics phenomena. Figure 3.2 indicates that this conclusion is
wrong.
Figure 3.2 is a plot of the absorption coefficients for a resonant weak coupling
field. In this case, the coupling field, which is always stronger than the probe field,
is weak in comparison with the atomic decay rates. The transparency in the absorp-
tion line of the probe field in the Cascade-EIT configuration persists, contrary to the
one of the Cascade-EIT configuration. What can be learned here is that a physical
phenomenon is initiated only in the Cascade-EIT configuration by a weak field.
The previously discussed difference and similarity can also be revealed by studying
the separation between the two maxima of the two peaks in the resonant coupling
61
field case. Figure 3.3 is a plot of the separation between the maxima as a function
of the coupling’s Rabi frequency, Ωc. Unlike the Cascade-EIT configuration, in the
weak field regime the Cascade-AT configuration shows no splitting in the absorption
line. Both separation lines become linear in Ωc in the very strong field regime. This
linear dependence is a signature of the AT effect, which we introduced in Chapter 1.
The presented discussion in this subsection shows that two phenomena exist in
the Cascade-EIT configuration. The first phenomenon which is triggered by a weak
coupling field, is absent in the Cascade-AT configuration, and is unknown at this
point. The other phenomenon which is the AT effect, is originated in the strong
coupling field regime in both cascade configurations. Apparently, in the Cascade-EIT
case, the AT effect dominates the first phenomenon which makes the absorption line
look very similar to the one displayed by the Cascade-AT configuration.
3.3.2 Index of Refraction
The reduction of the absorption opens the opportunity to study the dispersion felt
by the field while passing through the medium. Figure 3.4 studies the dispersion
coefficient, with a plot of n(ω)− 1 as a function of the detuning of the probe field, δp.
An important feature of the index of refraction is its slope as a function of frequency.
Because transparency is maximum near δp = 0, we are specially interested in the
slope near the origin, which we discussed in section 2.4. The slope in this case varies
with Ωc and can have a positive or a negative value. If dn/dω is large and positive,
62
vg < c, a situation which is usually referred to as “slow” light. If dn/dω is large
and negative, vg > c, corresponding to what is currently known as “fast” light. In
some cases, dn/dω can be negative enough to make the group index negative, which
corresponds to a negative group velocity. A negative group velocity is observed by the
emergence of a new peak from the medium at the other side before the initial peak
enters the medium from the first side. The existence of a negative group velocity was
verified experimentally with the use of EIT. The probe pulse was advanced by 62 ns,
corresponding to vg = −c/310 [34].
3.3.3 Fluorescence
We offer in this section a detailed study of the relationship between the absorption
of the probe field and the total fluorescence of each one of the two excited levels in
the two different configurations.
The total fluorescence spectra of levels 2 and 3 are respectively proportional to the
populations ρ22 (S22) and ρ33 (S33). For the Cascade-EIT (Cascade-AT) configuration
the populations are given by equations 3.29 and 3.30 (3.33 and 3.34). These equations
are functions of the variables CEIT (CAT) and PEIT, which are of equal magnitude.
Unlike the PEIT, the CEIT (CAT) is δp independent.
Let us consider first the Cascade-EIT configuration case. In the equation of S(2)22
(Eq. 3.29), PEIT has W31 as a multiplication factor. In general, the 3-1 transition
63
is dipole forbidden. Hence, W31 is negligible (we use W31 = 0.01 for the considered
Rubidium atomic levels). In this case the fluorescence of level 2 is “almost” pro-
portional to the absorption of the probe field. We note that the extra contribution,
W31PEIT, in the fluorescence is responsible of the Lasing Without Inversion (LWI)
phenomenon. On the other hand, S(2)33 (Eq. 3.30) has PEIT multiplied by W21. Hence
the fluorescence of level 3 is very different from the probe’s absorption.
Now we study the fluorescence spectra of the Cascade-AT configuration. In the
absence of the probe field, the population of level 2, S(0)22 (Eq. 3.31), is not equal
to zero when Ωc 6= 0. This probe’s field detuning independent term, S(0)22 , can be
interpreted as a constant background fluorescence. When the probe field is turned
on, the population of level 2 decreases as electrons get excited from level 2 to level 3.
This is why S(2)22 (Eq. 3.33) is negative, and directly proportional to the absorption
coefficient. The total fluorescence of level 2, S22 = S(0)22 + S
(2)22 , is positive definite.
The fluorescence of level 3 is directly proportional to the absorption coefficient.
64
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
δp
Abs
orpt
ion
Coe
ffici
ent
Cascade−EITCascade−AT (scaled)
Figure 3.1: Absorption lines for the Cascade-EIT and Cascade-AT configurations:W32 = 0.2, W31 = 0.01, δc = 0, and Ωc = 1.5.
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
δp
Abs
orpt
ion
Coe
ffici
ent
Cascade−EITCascade−AT (scaled)
Figure 3.2: Absorption lines for the Cascade-EIT and Cascade-AT configurations:W32 = 0.2, W31 = 0.01, δc = 0, and Ωc = 0.4.
65
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
1
2
3
4
5
6
Ωc
Max
ima
Sep
arat
ion
Cascade−EITCascade−AT
Figure 3.3: Absorption’s Maxima Separation: W32 = 0.2, W31 = 0.01, and δc = 0.
−10 −8 −6 −4 −2 0 2 4 6 8 10−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
δp
Dis
pers
ion
Coe
ffici
ent
Cascade−EITCascade−AT (scaled)
Figure 3.4: Dispersion lines for the Cascade-EIT and Cascade-AT configurations:W32 = 0.2, W31 = 0.01, δc = 0, and Ωc = 1.5.
Chapter 4: Two-Time Atomic Correlation
Functions and the Regression Theorem
Driven by the electric fields of the radiation, the atoms develop oscillating atomic
dipoles, which in turn emit electromagnetic fields related to the atomic dipole mo-
ments. What is also known is that the dipole moment operator is proportional to the
polarization operator. The Weiner-Khintchine theorem relates the two-time correla-
tion function of any statistically stationary variable to its spectral density by way of a
Fourier transform. This is how the emission spectrum is related to the inverse Fourier
transform of the two-time correlation function of the polarization. This relation is at
the core of our work in this chapter.
In 1969 [68] Mollow calculated the power spectrum of light scattered by two-level
systems using the previously mentioned relationship between the emitted spectral
density and the two-time polarization correlation function. The regression theorem,
which will be discussed in subsection 4.1.2, is also used. In 1972 [69] Mollow used the
same technique for a three-level system. In addition to the emission spectra, Mollow
worked out also the absorption spectrum of a weak field (probe) by evaluating the
perturbative work done by the probe on the existing field-atom system. Mollow re-
66
67
lated the absorption of the probe field to an appropriate two-time atomic correlation
function which we derive in section 4.1.3
As in the perturbative technique, presented in the previous chapter, the calcu-
lations carried out in this chapter following the lines of Mollow’s work are valid for
any coupling’s field strength. Using a totally different technique, we re-derive in this
chapter the same analytical expressions found for the absorption spectrum by the per-
turbative technique. This fact builds a strong connection between the two approaches.
The additional benefit of this technique over the perturbative one is in the cal-
culated emission spectra. Beside finding the populations of the excited levels, we
evaluate the power spectrum of the emitted light. This power spectrum is a func-
tion of the emission frequency for fixed values of the detunings of the coupling and
probe fields. This is unlike the total fluorescence (populations) found in the perturba-
tive technique where the populations are function of the detuning of the probe field.
In this work the populations and emission power spectra are evaluated before the
inclusion of the probe field. A non-zero solution requires the addition of an appropri-
ate incoherent pump rate over the transition on which the probe acts. For a closer
comparison with the results of the perturbative technique, the incoherent excitation
rate is eliminated by setting it equal to zero when the probe’s absorption is calculated.
In this chapter, as in the previous one, we start by introducing the technique by
studying explicitly the Cascade-EIT case. The setting for the Cascade-AT configura-
68
tion is outlined in section 4.2. The fluorescence spectra are derived but not analyzed
in detail because they are not directly relevant for the work of this thesis. The ab-
sorption spectra are also not studied in detail because they are identical to the ones
derived and analyzed in the previous chapter (Perturbative Technique).
4.1 Cascade-EIT
Figure 4.1 shows the energy diagram of the Cascade-EIT configuration. The probe
field is represented by a dashed arrow to indicate its absence in the initial system
(coupling field + atom) which we study to find the fluorescence spectra. Due to the
absence of the probe field, we need to introduce an incoherent excitation rate, W12,
from level 1 to level 2.
We start this section by deriving the atomic equations describing the atom, lo-
cated at z=0, interacting with the coupling field. Only resonant and quasi-resonant
terms are retained because terms oscillating at larger frequency introduce negligible
contributions, on average. We then calculate the density matrix elements in steady
state. The emission spectrum of the light radiated during the atomic transition 2-
1 is evaluated in subsection 4.1.2 and used in subsection 4.1.3 where we derive the
absorption of the probe, which will be added as a perturbation to the pre-existing
coupling field plus atom system.
69
4.1.1 Atomic Equations and Steady State Solutions
As we did in chapters 2 and 3 we derive in this subsection the atomic equations using
the Louiville equation (Eq. 2.8), i~dρdt
= [H0 + Hc(t), ρ].
The unperturbed Hamiltonian, H0 (Eq. 3.8), of the atom can be cast in its gen-
eral form, H0 = ~∑
k ωk1|k >< k|, where the atomic frequency, ωk1, is defined as
ωk1 = 1/~(Ek − E1). The coupling field interacts with the atom according to the
interaction Hamiltonian Hc(t) = −p2Ec(z, t).
The Louiville equation leads to the general atomic equations for the matrix ele-
ments ρij
dρij
dt= −iωijρij − i
∑
k
[ − λjk
(Eoc (z, t)eiωct + Eo∗
c (z, t)e−iωct)ρkj
+ λkj
(Eoc (z, t)eiωct + Eo∗
c (z, t)e−iωct)ρik], (4.1)
where
λjk =µjk
~. (4.2)
After adding phenomenologically the incoherent decay terms to equation 4.1 we
70
write the two general atomic equations for the populations and polarizations
(d
dt+
∑m
Wjm
)ρjj −
∑m
Wmjρmm = i(Eo
c (z, t)eiωct + Eo∗c (z, t)e−iωct
)×∑m
(λjmρmj − λmjρjm) , (4.3)
(d
dt+ γjk + iωjk
)ρjk = i
(Eoc (z, t)eiωct + Eo∗
c (z, t)e−iωct)×
[(λjkρkk − λjkρjj) +
∑
m6=k,j
(λjmρmk − λmkρjm)
].
(4.4)
The polarization decay rates, γij, in this case are different from the ones defined
in chapter 1 (Eqs. 1.6). Including the incoherent pump rate, W12, equation 1.5 gives
γ12 =1
2(W21 + W12) , (4.5a)
γ23 =1
2(W21 + W32 + W31) , (4.5b)
γ31 =1
2(W32 + W31 + W12) . (4.5c)
For j=2 and k=1 equation 4.4 leads to
(d
dt+ γ21 + iω21
)ρ21 = i
(Eoc (z, t)eiωct + Eo∗
c (z, t)e−iωct)
[λ21 (ρ11 − ρ22) + (λ23ρ31 − λ31ρ23)] . (4.6)
The l.h.s. of the equation oscillates with the frequency −ω21. We study all terms
on the r.h.s. of the equation and only keep the resonant and quasi-resonant ones.
71
The term e±iωctρii oscillates at the frequency ±ωc and must be omitted. The angular
frequencies ω32 ± ωc are both very different from ω21 so we ignore the terms e±iωctρ23
where the density operator element ρ23 oscillates at the frequency ω32. Same way we
drop the term e−iωctρ31 of oscillating frequency −(ωc +ω31). The only remaining term
on the r.h.s. of equation 4.6 is eiωctρ31 of frequency of oscillation ωc−ω31 = −ω21 +δc.
This term is resonant with l.h.s. of the equation when δc = 0 and quasi-resonant in
other cases. After dropping all the fast oscillating terms, equation 4.6 becomes
(d
dt+ γ21 + iω21
)ρ21 = iλ23Eo
c (z, t)eiωct ρ31. (4.7)
The atomic equations for all the other density matrix elements can also be de-
rived from equations 4.4 and 4.3, and then reduced by keeping only the resonant and
quasi-resonant terms.
With the help of the following identifications
ρii = ρii, (4.8a)
ρ32 = ρ32e−iωct, (4.8b)
ρ21 = ρ21e−iω21t, (4.8c)
ρ31 = ρ31e−iωcte−iω21t, (4.8d)
72
the reduced atomic equations take the form
d
dtρ21 + γ21ρ21 = iλ23Eo
c (z, t) ρ31, (4.9a)
d
dtρ31 + (γ31 − iδc) ρ31 = iλ∗23Eo∗
c (z, t)ρ21, (4.9b)
d
dtρ32 + (γ32 − iδc) ρ32 = iλ∗23Eo∗
c (z, t) (ρ22 − ρ33) , (4.9c)
d
dtρ33 + (W31 + W32) ρ33 = i [λ∗23Eo∗
c (z, t)ρ23 − λ23Eoc (z, t)ρ32] , (4.9d)
d
dtρ22 + W21ρ22 −W12ρ11 −W32ρ33 = i [−λ∗23Eo∗
c (z, t)ρ23 + λ23Eoc (z, t)ρ32] , (4.9e)
d
dtρ11 −W21ρ22 −W31ρ33 = 0, (4.9f)
where δc = ωc − ω32.
After eliminating ρ11 with the help of the conservation condition of the trace,
1 = ρ11 + ρ22 + ρ33, the steady state solutions are
ρ22 =W12(W31 + W32 + CEIT)
D, (4.10a)
ρ33 =W12C
EIT
D, (4.10b)
ρ32 = iEo∗
c (z, t)λ∗23
γ23 − iδc
W12(W31 + W32)
D, (4.10c)
73
where
CEIT = 2γ23
γ223 + δ2
c
Ω2c
4, (4.11a)
D = (W12 + W21)(W31 + W32) + (2W12 + W21 + W31) CEIT, (4.11b)
Ω2c = 4|Eo
c (z, t)λ23|2. (4.11c)
We note that the term CEIT (Eq. 4.11a) and the Rabi frequency Ωc (Eq. 4.11c)
are identical to the corresponding terms defined in the perturbative technique (Eqs.
3.27b and 3.17b ). The denominator D (Eq. 4.11b) becomes equal to the one (Eq.
3.27a) defined in the previous chapter after setting the incoherent excitation rate W12
to zero.
4.1.2 Emission Spectrum
In this subsection we derive the emission spectrum of the light emitted during the
transition between the states |2 > and |1 >. This spectrum, which we will not an-
alyze because it is irrelevant for the work of this thesis, will be needed in the next
subsection where the absorption of the probe field is evaluated and studied.
As mentioned in the introduction of this chapter, the power spectrum of the
scattered field is the inverse Fourier transform of the two time atomic polarization
correlation function. Because we are dealing with a statistical system, we use the
average of the polarization’s correlation function and define the emission spectral
density as
74
∫ +∞
−∞dt eiνt < P (−)P (+)(t) >, (4.12)
where the atomic polarization operators are given by
P (−) = µ21a†21 + µ32a
†32, (4.13)
P (+)(t) = µ21a21(t) + µ32a32(t). (4.14)
The atomic operators, aij and a†ij, are defined as
aij = |j >< i|, (4.15a)
a†ij = |i >< j|, (4.15b)
so that the density matrix elements can also be expressed in the form
ρij = < aij > = Tr (ρ|j >< i|) , (4.16a)
ρii = < a†ijaij > = Tr (ρ|i >< j|j >< i|) . (4.16b)
Thus the spectral density, ge(ν), of the radiation emitted at the frequency ν during
the atomic transition between states |2 > and |1 > is given by
ge(ν) =
∫ +∞
−∞dt eiνtge(t), (4.17)
75
where the corresponding two-time atomic correlation function, ge(t), is
ge(t) = < a†21a21(t) > . (4.18)
If we integrate ge(ν) over all emission frequencies, ν, we obtain the total intensity
of the emitted radiation during the atomic transition |2 > - |1 > to be
1
2π
∫ +∞
−∞dν ge(ν) = ρ22, (4.19)
as expected.
Regression Theorem
Unlike the density matrix elements, ρij =< aij > and ρii =< a†ijaij >, two-time
correlation functions like ge(t), ge(t) =< a†21a21(t) >, can not be calculated using
the Louiville equation. This is one instance when the Regression theorem plays an
especially useful role.
The Regression theorem states that if M, Q, and N are members of a complete
set of system operators Sµ, and if one-time expectation values can be expressed in
the form
< M(t) > =∑
µ
Oµ(t, t′) < Sµ(t′) >, t′ < t, (4.20)
where Oµ(t, t′) are complex functions of time, then two-time expectation values take
76
the form
< Q(t′)M(t)N(t′) > =∑
µ
Oµ(t, t′) < Q(t′)Sµ(t′)N(t′) >, (4.21)
under the assumption that the system is Markovian. This condition implies that the
expectation values of every observable at a given time depend only on the state of
the system at that time and not on its previous history.
The Regression theorem implies that the correlation function < a†21a21(t) > be-
haves in time just as the single-time average < a21(t) >= ρ21 which, in turn, is coupled
to ρ31 according to equations 4.9a and 4.9b. The matrix element ρ21 evolves in time
according to the formal solution
ρ21(t) = U21;21 ρ21(0) + U21;31 ρ31(0), (4.22)
which, in turn, implies
< a21(t) > = U21;21 < a21(0) > +U21;31 < a31(0) > . (4.23)
According to the Regression theorem we can then derive the following rules of
correspondence:
< a21(t) > → < a†21a21(t) >, (4.24a)
< a21(0) > → < a†21a21 >, (4.24b)
< a31(0) > → < a†21a31 >, (4.24c)
77
which, when substituted in equation 4.23, lead to
< a†21(0)a21(t) > = U21;21(t) < a†21a21 > +U21;31(t) < a†21a31 >, (4.25)
ge(t) = U21;21(t)ρ22 + U21;31(t)ρ32. (4.26)
The Fourier transform of ge(t), ge(ν), is related to the Laplace transform, ge(−iν),
by the relation [68]
ge(ν) = 2<[ge(−iν)], (4.27)
where the Laplace transform of ge(t), ge(−iν), is given by
ge(−iν) = U21;21(−iν)ρ22 + U21;31(−iν)ρ32. (4.28)
Upon solving equations 4.9a and 4.9b in Laplace space and casting the solution
of ρ21 in the form of equation 4.22 we find that
U21;21(−iν) =γ13 − i(∆ν + δc)
f(−iν), (4.29a)
U21;31(−iν) = iEo
c λ23
f(−iν), (4.29b)
where
f(−iν) = [γ12 − i∆ν][γ13 − i(∆ν + δc)] +Ω2
c
4, (4.30a)
∆ν = ν − ω21. (4.30b)
78
The emission spectrum, which we do not need to study, is proportional to the real
part of ge(−iν) (Eq. 4.28). The functions ge(−iν) (Eq. 4.28), and U21;21 (Eqs 4.29)
are needed also in the next subsection where we derive the absorption spectrum of
the probe field.
4.1.3 Probe’s Absorption
The absorption spectrum of a probe field can be calculated by studying the pertur-
bative work done by a weak field on a pre-existing coupling field plus atom system.
Following Mollow [69], we consider a probe field of the form
Ep(z, t) = Eop (t)eiωpt + Eo∗
p (t)e−iωpt. (4.31)
The total Hamiltonian of the full system can be written as
H = (H0 + Hc(t)) + Hp(t), (4.32)
where H0 is the unperturbed Hamiltonian of the atom, and Hc (Hp) is the interaction
Hamiltonian of the coupling (probe) field.
The Louiville equation in the interaction picture is
∂ρ′(t)∂t
= − i
~[H ′
p(t), ρ′(t)
], (4.33)
79
where H ′p(t), the probe field-atom interaction Hamiltonian in the interaction picture,
is given by
H ′p(t) = U(t)Hp(t)U
−1(t), (4.34)
and where U(t), the unitary transformation operator, is defined as
U(t) = exp
[i
~
∫ t
−∞dt′(H0 + Hc(t
′))]
. (4.35)
The density operator in the interaction picture, ρ′, differs from its steady state
value only by the small contribution δρ′(t) created by the weak probe field, i.e.
ρ′(t) = ρss + δρ′(t). (4.36)
To first order in the perturbation, equations 4.33 and 4.36 lead to
∂
∂tδρ′(t) = − i
~[H ′
p(t), ρss
], (4.37)
whose solution is
δρ′(t) = − i
~
∫ t
−∞dt′
[H ′
p(t′), ρss
]. (4.38)
The rate of work done, W ′(t), by the perturbation field on the system [69, 70]
is the derivative of the Hamiltonian averaged over the perturbative correction of the
density matrix, δρ′(t), caused by the probe field
W ′(t) = Tr
(∂H ′(t)
∂tδρ′(t)
). (4.39)
80
Substituting equation 4.38 into equation 4.39 leads to
W ′ =1
i~
∫ t
−∞dt′Tr
ρ
[∂H ′
p(t)
∂t,H ′
p(t′)]
. (4.40)
In the previous chapter, and in the rotating wave approximation, the interaction
Hamiltonian describing the action of the coupling and probe fields on an atom was
shown to have the form H1(t) = − (µ12 |2 >< 1| Eo∗
p (z, t)e−iωpt + h.a.)−
(µ23 |3 >< 2| Eo∗c (z, t)e−iωct + h.a.) (Eq. 3.10). Similarly here, the interaction part
of the atom-probe field Hamiltonian in the resonant approximation is
H ′p(t
′) = −~λ∗21Eo∗p (t′)a†21 − ~λ21Eo
p (t′)a21, (4.41)
where
λ21 = λ12 =µ21
~. (4.42)
After taking the derivative of H ′p(t) we obtain
∂H ′p(t)
∂t= −i~ωp
[λ∗21Eo∗
p (t)e−iωpta21(t)− λ21Eop (t)eiωpta†21(t)
], (4.43)
so that equations 4.41, and 4.43 yield the commutator
1
i~
[∂H ′
p(t)
∂t,H ′
p(t′)]
= ~ν|λ21Eop |2
eiν(t−t′)[a21(t), a
†21(t
′)]− e−iν(t−t′)[a†21(t), a21(t′)]
.
(4.44)
Because in steady state the system must reach a stationary configuration, the
81
trace depends only on t − t′ = τ and not, separately, on t or t′. , In this case, and
after substitution of equation 4.44 into equation 4.40 we find that
W ′(τ) = ~ν|λ21Eop |2
∫ 0
−∞(−dτ)
eiντTr
(ρ[a21(τ), a†21]
)− e−iντTr
(ρ[a†21(τ), a21]
).
(4.45)
After changing the variable of integration form τ to −τ in the second integral,
and using the fact that in steady state the correlation functions depend only on τ ,
i.e. [a21, a†21(−τ)] = [a21(τ), a†21] equation 4.45 reduces to
W ′(τ) = ~ν|λ21Eop |2
∫ ∞
−∞dτeiντTr
(ρ[a21(τ), a†21]
), (4.46)
or
W ′ = ~ωp|λ21Eop |2 ga(ωp). (4.47)
where ga(ωp) is the Fourier Transform of the atomic correlation function, ga(τ),
ga(τ) = < [a21(τ), a†21] >,
= < a21(τ)a†21 > − < a†21(0)a21(τ) > . (4.48)
After setting < a21(τ)a†12 >= gd(τ) to represent the total absorption spectrum,
we find that the net absorption, ga, can now be written in the form
ga(ωp) = gd(ωp)− ge(ωp), (4.49)
82
where ge(ωp) is the fluorescence spectrum derived in the previous subsection.
Using the Regression theorem and along the lines of the derivation of ge(−iν), we
find that
gd(−iωp) = U21;21(−iωp) ρ11, (4.50)
which leads to
ga(−iωp) = U21;21(−iωp) (ρ11 − ρ22)− U21;31(−iωp) ρ32. (4.51)
In the absence of the incoherent pumping process, whose rate is W12, the steady
state solutions of the density matrix elements (Eqs. 4.10a-c) reduce to
ρ22 = 0, (4.52a)
ρ33 = 0, (4.52b)
ρ32 = 0. (4.52c)
which leads to ρ11 = 1.
In this case, equation 4.51 becomes
ga(−iωp) = U21;21(−iωp), (4.53)
83
and after using equation 4.29a we obtain
ga(−iωp) =γ13 − i(δp + δc)
[γ12 − iδp][γ13 − i(δp + δc)] + Ω2c
4
, (4.54)
where
δp = ωp − ω21. (4.55)
We can conclude by inspection that the real part of ga(−iωp), which is related to
the net absorption spectrum by the relation ga(ωp) = 2< [ga(−iωp)], is equal to the
negative imaginary part of S(1)12 , which is the absorption of the probe field, given by
equation 3.28. This exact match of the absorption spectra (apart from a multiplication
factor) unifies the derivation presented in this chapter with the perturbative technique
discussed in the previous chapter.
4.2 Cascade-AT
In the Cascade-AT configuration case, and when the probe field is missing, the inco-
herent pump rate, W23, is added (Fig. 4.2). In the following subsections we sketch
the results of interest for the fluorescence and probe absorption spectra.
84
4.2.1 Density Matrix Elements in Steady State
The fluorescence and absorption spectra derived in this section are assigned by the
following steady state density matrix elements
ρ22 =(W31 + W32)C
AT
D, (4.56a)
ρ33 =W23C
AT
D, (4.56b)
ρ21 = iEo∗
c (z, t)λ∗12
γ12 − iδc
(W21 + W23)(W31 + W32)−W32W23
D, (4.56c)
where
CAT = 2γ12
γ212 + δ2
c
Ω2c
4, (4.57a)
D = CAT(2W31 + 2W32 + W23) + W21(W31 + W32) + W31W23, (4.57b)
and where
Ω2c = 4|Eo
c (z, t)λ12|2, (4.58a)
δc = ω − ω21. (4.58b)
4.2.2 Emission Spectrum
The emission power spectrum, ge(ν), associated with the atomic transition |3 > -
|2 >, is the inverse Fourier transform of the two time correlation function, ge(t),
85
defined as
ge(t) = < a†32a32(t) >, (4.59)
whose Laplace transform is given by
ge(−iν) = U32;32(−iν) ρ33. (4.60)
After solving the coupled atomic equations of ρ32 and ρ31 in Laplace space, we
find the two following functions:
U32;32(−iν) =γ13 − i(∆ν + δc)
f(−iν), (4.61a)
U32;31(−iν) = −iε∗0λ12
f(−iν), (4.61b)
where
f(−iν) = [γ23 − i∆ν][γ13 − i(∆ν + δc)] +Ω2
c
4, (4.62)
and where
∆ν = ν − ω32, (4.63)
Upon substituting equations 4.61a and 4.56b into equation 4.60 we obtain
ge(−iν) =γ13 − i(∆ν + δc)
f(−iν)
W23CAT
D. (4.64)
86
4.2.3 Probe’s Absorption
The Laplace transform of the total probe’s absorption correlation function gd =
< a32(τ)a†23 > is given by
gd(−iωp) = U32;32(−iωp) ρ22 + U32;31(−iωp) ρ21, (4.65)
leading to the net absorption spectrum
ga(−iωp) = U32;32(−iωp) (ρ22 − ρ33) + U23;31(−iωp) ρ21. (4.66)
In the absence of the incoherent pump field the set of equations 4.56 give
ρ22 =CAT
W21 + 2C, (4.67a)
ρ33 = 0, (4.67b)
ρ21 = iEo∗
c (z, t)λ∗12
γ21 − iδc
W21
W21 + 2CAT, (4.67c)
where
CAT = 2γ12
γ212 + δ2
c
Ω2c
4. (4.68)
Thus equations 4.67, 4.61, and 4.66 lead to
ga(−iωp) =1
W21 + 2CAT
1
[γ12 − iδc]
[γ12 − iδc][γ13 − i(δωp + δc)]CAT + W21
Ω2c
4
[γ23 − iδωp][γ13 − i(δωp + δc)] + Ω2c
4
,
(4.69)
87
where
δωp = ωp − ω32. (4.70)
It is important to observe that in this case also <[ga(−iνp)] = −=[S(1)23 ], where
S23 is given by equation 3.32. Once again the two techniques described in this chapter
and the previous one, Perturbative Technique, lead to the same analytical expressions
for the absorption spectra.
88
Figure 4.1: Cascade-EIT configuration with W12 excitation
Figure 4.2: Cascade-AT configuration with W23 excitation
Chapter 5: Secular Limit
We showed in chapter 3, Perturbative Technique, that in the strong coupling field
limit and on resonance the two different cascade configurations display very similar-
looking absorption spectra. The origin of this similarity is clarified analytically in
this chapter, and the absence of quantum interference will be verified with the help
of a technique which is valid only in the strong field limit.
In 1969, Cohen-Tannoudji and Haroche introduced what is now called the dressed-
atom approach to describe the interaction of an atom with radiofrequency photons
[71] in which both the atom and the field are quantized. Dressed by the field, the
atomic unperturbed states turn into what are known as dressed states. As a con-
ceptually similar example let us consider the H2 molecule. The two hydrogen atoms
forming the molecule are dressed by the electrostatic interaction between them. The
energy level diagram of the dressed system, two hydrogen atoms coupled by the field,
is totally different from the energy levels of the individual atoms.
In 1977, using the dressed-atom approach, and in the secular limit (the effective
Rabi frequency of the system, which is related to the strengths of the different used
89
90
fields, is much greater than the atomic decay rates), Cohen-Tannoudji and Serge Rey-
naud [72, 73] studied dressed multi-level atoms.
The tools adopted in this chapter for the derivation of the fluorescence and ab-
sorption spectra are the same as we discussed in the previous chapter, i.e. two-time
atomic correlation functions and the regression theorem. The fields are represented
by single-mode quantum operators. The atomic equations are derived from a suitable
master equation which includes both reversible and irreversible contributions to the
evolution of the atomic dynamical variables.
Due to the mathematical complication of the problem (8x8 coupled first order dif-
ferential atomic equations), the master equation is solved in the secular limit which
leads to the elimination of the non-secular (non-resonant) terms in the atomic equa-
tions. This elimination process is similar to the one used in section 4.1.1 where the
non-resonant terms where eliminated. Setting the coupling field on resonance leaves
no ambiguities in distinguishing the secular and non-secular terms. We use resonant
fields through out the chapter.
We start this chapter by the general description of the dressed atom. We define
different manifolds, derive their corresponding dressed states, and calculate the tran-
sition decay rates between the manifolds. The decay rates help understanding the
cascade radiative decays which are associated with the emission spectra. In section
5.3 the master equation is solved and the atomic equations are derived. After deriving
91
the atomic equations we find the explicit time dependence and some of the steady
state solutions of the variables of interest. The fluorescence spectra emitted during
both transitions 2-1 and 3-2 are derived in section 5.5. We finish the chapter by
deriving and discussing the absorption spectra of the probe field in the two cascade
configurations. The derived absorption spectra in the strong coupling field regime are
compared numerically to the spectra found in the previous chapters, 3 and 4.
5.1 Manifolds and Dressed States
In this section and in most of the following ones we study the general cascade system
(Fig. 5.1) made of the atom and two fields. We note here that the considered cascade
system has no specific configuration, i.e. it can be either Cascade-EIT or Cascade-AT,
because the strengths of the fields are arbitrary. A weak field (probe) will be added
when needed with the simultaneous turn off of the corresponding field in the last two
sections of this chapter.
The manifolds form a two-dimensional lattice (Fig. 5.2) that the dressed atom de-
scends through while emitting photons. Studying the manifolds helps us understand
the spontaneous emission spectrum which is the consequence of the transition decays
between different manifolds.
If we turn off the atom-fields interaction the manifolds consist of the unperturbed
92
states of the system which are the Cartesian products of the atomic states with the
states of the fields. The states |1, N1, N2 >, |2, N1− 1, N2 >, and |3, N1− 1, N2− 1 >
form the manifold E(N1, N2), where N1 and N2 are respectively the numbers of pho-
tons in the fields Ω1 and Ω2 (Fig. 5.1).
On resonance, ω1 = ω12 and ω2 = ω23, where ω1 and ω2 are the frequencies of the
fields, the unperturbed manifold E(N1, N2) is three-fold degenerate with an energy
E0(N1,N2) = ~ω1N1 + ω2N2. On the vertical dimension (Fig: 5.2) two manifolds of
quantum numbers (N1, N2) and (N1 − 1, N2) are separated by the energy ~ω1. On
the other dimension (horizontal), an energy of ~ω2 separates the manifolds E(N1, N2)
and E(N1, N2 − 1).
The unperturbed Hamiltonian of the system is
H0 = ~ω21a22 + ~ω31a33 + ~ω1σ+1 σ1 + ~ω2σ
+2 σ2, (5.1)
where σ+1 and σ1 (σ+
2 and σ2) are the creation and annihilation field operators, and
where aii is the atomic operator defined as aii = |i >< i|.
The laser fields and the atom interact according to the interaction Hamiltonian
H1 = −DE⊥, (5.2)
93
where the atomic dipole operator is defined as
D = µ12(|2 >< 1|+ |1 >< 2|) + µ23(|3 >< 2|+ |2 >< 3|), (5.3)
and the field, E⊥, is given by
E⊥ = E⊥,1 + E⊥,2, (5.4)
=
√~ω1
2εoV(σ1 + σ†1) +
√~ω2
2εoV(σ2 + σ†2). (5.5)
Avoiding modifications in the spontaneous emission of the atom we consider a
large cavity volume, V, and large numbers of photons in the fields, i.e. the ratio of
Ni (i=1, 2) over V remains finite when V tends to infinity. This condition keeps the
spontaneous emission unaffected during the interaction of the atom with the fields.
In this case, the disappearance of photons from the fields by absorption followed by
the appearance of a new photon in one of the vacuum modes by spontaneous emission
would not reduce the number of photon in the fields dramatically so the absorption
and emission processes are effected.
Define the coupling constants
g1 = −µ12
√~ω1
2εoV, (5.6)
g2 = −µ23
√~ω2
2εoV. (5.7)
94
In the rotating wave approximation, after eliminating of the fast oscillating terms,
the interaction Hamiltonian (Eq. 5.2) takes the form
H1 = ~g1
(σ1a12 + a†12σ
+1
)+ ~g2
(σ2a23 + a†23σ
+2
). (5.8)
For instance, the first term of the Hamiltonian, a12σ1, corresponds to the process
of one photon annihilation out of the first field, and the excitation of an electron from
level 1 to level 2.
After solving the characteristic equation of the interaction Hamiltonian, which
corresponds to diagonalizing the matrix H1, we find the three eigenvalues
E1a = ~G, (5.9a)
E1b = 0, (5.9b)
E1c = −~G. (5.9c)
These eigenvalues lead to their corresponding eigenstates, which form the manifold
95
E(N1, N2), and which are given by
|a(N1, N2) > =1√2(sinθ|1, N1, N2 > +|2, N1 − 1, N2 > +
cosθ|3, N1 − 1, N2 − 1 >), (5.10a)
|b(N1, N2) > =1√2(cosθ|1, N1, N2 > −sinθ|3, N1 − 1, N2 − 1 >), (5.10b)
|c(N1, N2) > =1√2(−sinθ|1, N1, N2 > +|2, N1 − 1, N2 > −
cosθ|3, N1 − 1, N2 − 1 >), (5.10c)
where
sinθ =g1
√N1
G, (5.11a)
cosθ =g2
√N2
G, (5.11b)
and where G is the effective Rabi frequency, defined as
G =√
g21N1 + g2
2N2. (5.12)
The eigenstate |b(N1, N2) > is at the same energy level E0(N1,N2), E0
(N1,N2) =
~ω1N1 +ω2N2, of the unperturbed states. The two other dressed states, |a(N1, N2) >
and |c(N1, N2) > are respectively shifted from E0(N1,N2) by energies equal to ~G and
−~G. In the Secular limit, G >> γij, the dressed states are well separated from each
other, which eliminates the possibility of an overlap in their linewidths. This previous
comment supports the absence of interference in the strong field limit and it will be
96
clarified further later in this chapter when calculating the emission and absorption
spectra.
5.2 Transition Decay Rates
Beside the need for the transition decay rates in the derivations of the emission and
absorption spectra, these rates are essential for an understanding of the dynamics of
the dressed atom. The decay lines show us the various transitions that the dressed
atom follows between the manifolds.
An atom in the manifold E(N1, N2) level |α(N1, N2) > decays to the manifold
E(N1 − 1, N2) level |β(N1 − 1, N2) > via a 2→1 transition with the transition decay
rate
Γαβ = | < β(N1 − 1, N2)|D|α(N1, N2) > |2. (5.13)
Defining the appropriate atomic polarization operator, P (+) = |1 >< 2|, Γαβ can
be written as
Γαβ = µ212
(P (+)
)2
βα, (5.14)
where
(P (+)
)βα
= < β(N ′1 − 1, N ′
2)|1 >< 2|α(N ′1, N
′2) > . (5.15)
97
The atomic polarization operator, P (+) (Eq. 5.15), has the elements
(P (+)
)ab
=(P (+)
)bb
=(P (+)
)cb
= 0, (5.16a)
(P (+)
)aa
=(P (+)
)ac
=sinθ
2, (5.16b)
(P (+)
)ba
=(P (+)
)bc
=cosθ
2, (5.16c)
(P (+)
)ca
=(P (+)
)cc
= −sinθ
2, (5.16d)
which substituted into equation 5.14 lead to
Γba = Γbb = Γbc = 0, (5.17a)
Γaa = Γac = Γca = Γcc = µ212
sin2θ
4, (5.17b)
Γab = Γcb = µ212
cos2θ
2. (5.17c)
The decay rates between the manifolds E(N1, N2) and E(N1, N2 − 1) by a 3→2
transition corresponding to transitions between the states |α(N1, N2) > and
|β(N1, N2 − 1) > are defined as
Γ′αβ = | < β(N1, N2 − 1)|D|α(N1, N2) > |2, (5.18)
98
and are given by
Γ′ba = Γ′bb = Γ′bc = 0, (5.19a)
Γ′aa = Γ′ac = Γ′ca = Γ′cc = d′2cos2θ
4, (5.19b)
Γ′ab = Γ′cb = d′2sin2θ
2. (5.19c)
As mentioned in the previous section the manifolds form a two-dimensional cell
(Fig 5.2). The decay rates, Γαβ, are associated with a“downward” decay from the
manifold E(N1, N2) to the manifold E(N1 − 1, N2). Following the calculated decay
rates (Eqs. 5.17) we draw six decay lines which lead to the corresponding five differ-
ent spontaneous emission frequencies ν = ω21, ω21 ±G, and ω21 ± 2G. The two fine
dotted lines represent transitions with the same energy ~ω21. These five emission lines
will be discussed in detail in section 5.5 where the fluorescence spectra are calculated.
The decay rates Γ′αβ (Eqs. 5.19) can also be studied leading to the“right side” decay
lines in figure 5.2. The corresponding emission lines, ν = ω32, ω32±G, and ω32± 2G,
will also be studied in section 5.5.
We notice that in both transitions, right and down, no decays take place out of
the state |b(N1, N2 >. The assumption that if a dressed atom decays onto the dressed
state |b(N1, N2 > it will not leave that state is wrong. It will eventually oscillate
between this state and the other ones in the same manifold by stimulated emission
and absorption and then decays out of one of the states |a(N1, N2 > or |c(N1, N2 >
99
to another manifold.
5.3 Master Equation
In the previous chapters, we used the Louiville equation to derive the coherent part
of the atomic equations and then added an incoherent part phenomenologically. This
technique is not applicable in the dressed state picture where the damping terms are
very complicated. In this section, we introduce the master equation which contains
both the coherent and incoherent parts and which will be used in the next section to
derive the full atomic equations.
The master equation, which describes the spontaneous emission, has the general
form
dρ
dt= − i
~[H, ρ]coh + Λincoh, (5.20)
where the irreversible term, Λincoh, is given by
Λincoh =∑ij
ajiρa†ji(Ajiij + A∗
jiij)− ajjρAjiij − ρajjA∗jiij
. (5.21)
The complex rate constants, Ajiij, are related to the population decay rates, Wij,
by the relation
Wij = Ajiij + A∗jiij. (5.22)
These complex rate constants are also related to the polarization decay rates, γij,
100
by the equation
γij =∑
k
<(Aikki + A∗jkkj), (5.23)
which combined with equation 5.22 leads to the relation γij =1
2
∑k(Wik +Wjk) (Eq.
1.5) introduced in chapter 1.
The frequency shifts, ∆Ωij, are defined as
∆Ωij = −∑
k
=(Ajkkj + A∗ikki), (5.24)
and are set equal to zero due to following the assumption that the elastic collisions
are negligible.
The following relation
<Aijji = <(
Ajiijexp(~ωij
kT)
), (5.25)
leads to
Wji = exp
[−~ωij
kT
]Wij, (5.26)
which shows that the upward thermal excitations are much smaller than the down-
ward ones in the optical regime. Hence we ignore the upward transitions due to
thermal excitations. We also ignore the pure phase relaxation effects (i.e. set Aiiii=0).
101
After writting Λincoh explicitly and using all the relationships and approximations
mentioned above, we find
Λincoh =
|1 >< 2|ρ|2 >< 1|W21 + |2 >< 3|ρ|3 >< 2|W32 + |1 >< 3|ρ|3 >< 1|W31 −1
2W21 (|2 >< 2|ρ + ρ|2 >< 2|)− 1
2(W32 + W31) (|2 >< 2|ρ + ρ|2 >< 2|) . (5.27)
5.4 Atomic Equations in the Secular Limit
Before solving the master equation, we study the degeneracies of the energy separa-
tions, ~ωαβ, between the dressed states of the same manifold. The frequencies, ωαβ,
represent the oscillation frequency of the atomic density matrix elements, ραβ.
The set of equations 5.9 lead to the energy separations
~ωac = 2~G, (5.28a)
~ωca = −2~G, (5.28b)
~ωab = ~ωbc = ~G, (5.28c)
~ωba = ~ωcb = −~G, (5.28d)
from which we observe that the two frequencies ωab and ωbc (ωba and ωcb) are degen-
erate. The other frequencies of oscillation, which are different by a factor of G (Eq.
6.26), are non-resonant in the secular limit.
102
When solving the master equation for the density matrix elements, we only retain
the secular terms. For example, the atomic equation of the coherence ρab will only
contain terms of ρab and ρbc. This is unlike ρac and ρca which are independent of all
the other coherences. The populations, which have no oscillating frequency, can be
coupled to each other but not to the coherences.
Projecting the master equation onto the dressed state basis, specifically onto the
states < α(N1 − p,N2 − q)| and |β(N1, N2) > we find the atomic equations for the
density matrix ρp,qαβ(N1, N2) defined as
ρp,qαβ(N1, N2) = < α(N1 − p,N2 − q)|ρ|β(N1, N2) >, (5.29)
which after summing over N1 and N2 reduces to
ρp,qαβ =
∑N1,N2
ρp,qαβ(N1, N2). (5.30)
103
In the Secular limit, and after defining the constants
λ1 = W21sin2(θ)
4+ W32
cos2(θ)
4, (5.31a)
λ2 = W211
2+ W32
cos2(θ)
2, (5.31b)
µ1 = W32sin2(θ)
2, (5.31c)
µ2 = W21cos2(θ)
2, (5.31d)
ν1 = W31sin2(θ) cos2(θ)
4, (5.31e)
ν2 = W31cos2(θ)
2, (5.31f)
ν3 = W31cos4(θ)
2, (5.31g)
ν4 = W31sin4(θ)
2, (5.31h)
ν5 = W31sin2(θ)
2, (5.31i)
104
we find the set of atomic equations:
ρp,qaa = −i(−pω1 − qω2)ρ
p,qaa +
(λ1 − λ2 + ν1 − ν2)ρp,qaa + (µ1 + ν4)ρ
p,qbb + (λ1 + ν1)ρ
p,qcc , (5.32a)
ρp,qbb = −i(−pω1 − qω2)ρ
p,qbb +
(4ν1 − 2µ1 − 2ν5)ρp,qbb + (µ2 + ν3)ρ
p,qaa + (µ2 + ν3)ρ
p,qcc , (5.32b)
ρp,qcc = −i(−pω1 − qω2)ρ
p,qcc +
(λ1 − λ2 + ν1 − ν2)ρp,qcc + (λ1 + ν1)ρ
p,qaa + (µ1 + ν4)ρ
p,qbb , (5.32c)
ρp,qac = −i(2G− pω1 − qω2)ρ
p,qac − (λ1 + λ2 − ν1 + ν2)ρ
p,qac , (5.32d)
ρp,qca = −i(−2G− pω1 − qω2)ρ
p,qca − (λ1 + λ2 − ν1 + ν2)ρ
p,qca , (5.32e)
ρp,qab = −i(G− pω1 − qω2)ρ
p,qab −
(λ2
2+ µ1 +
ν2
2+ ν5 + 2ν1
)ρp,q
ab + 2ν1ρp,qbc , (5.32f)
ρp,qbc = −i(G− pω1 − qω2)ρ
p,qbc −
(λ2
2+ µ1 +
ν2
2+ ν5 + 2ν1
)ρp,q
bc + 2ν1ρp,qab , (5.32g)
ρp,qba = −i(−G− pω1 − qω2)ρ
p,qba −
(λ2
2+ µ1 +
ν2
2+ ν5 + 2ν1
)ρp,q
ba + 2ν1ρp,qcb , (5.32h)
ρp,qcb = −i(−G− pω1 − qω2)ρ
p,qcb −
(λ2
2+ µ1 +
ν2
2+ ν5 + 2ν1
)ρp,q
cb + 2ν1ρp,qba . (5.32i)
105
The explicit time dependent solutions of most of the density matrix elements
will be needed when calculating the fluorescence and absorption spectra. Solving for
ρac(t) and ρca(t) is straightforward but the rest of the elements are best solved with
the help of Fourier transform techniques. Due to the mathematical complication of
solving separately for ρp,qaa (t) and ρp,q
cc (t) we solve for the sum of the two elements
which is what will be needed. The explicit time dependent solutions of the variables
of interest are
ρp,qaa (t) + ρp,q
cc (t) =
(ρp,q
aa (0) + ρp,qcc (0)− 2(µ1 + ν4)
2µ1 + µ2 + ν3 + 2ν4
)e−(2µ1+µ2+ν3+2ν4)t +
2(µ1 + ν4)
2µ1 + µ2 + ν3 + 2ν4
, (5.33a)
ρp,qbb (t) =
(ρp,q
bb (0)− µ2 + ν3)
2µ1 + µ2 + ν3 + 2ν4
)e−(2µ1+µ2+ν3+2ν4)t +
µ2 + ν3
2µ1 + µ2 + ν3 + 2ν4
, (5.33b)
ρp,qac (t) = exp [−i(2G− pω1 − qω2)t] e−(λ1+λ2−ν1+ν2)tρp,q
ac (0), (5.33c)
ρp,qca (t) = exp [−i(−2G− pω1 − qω2)t] e
−(λ1+λ2−ν1+ν2)tρp,qca (0), (5.33d)
ρp,qab (t) = exp [−i(G− pω1 − qω2)t] (L1ρ
p,qab (0) + L2ρ
p,qbc (0)) , (5.33e)
ρp,qbc (t) = exp [−i(G− pω1 − qω2)t] (L2ρ
p,qab (0) + L1ρ
p,qbc (0)) , (5.33f)
ρp,qba (t) = exp [−i(−G− pω1 − qω2)t] (L1ρ
p,qba (0) + L2ρ
p,qcb (0)) , (5.33g)
ρp,qbc (t) = exp [−i(−G− pω1 − qω2)t] (L2ρ
p,qba (0) + L1ρ
p,qcb (0)) , (5.33h)
106
where
2 L1 = exp
[−(
λ2
2+ µ1 +
ν2
2+ ν5)t
]+ exp
[−(
λ2
2+ µ1 +
ν2
2+ ν5 + 4ν1)t
],(5.34a)
2 L2 = exp
[−(
λ2
2+ µ1 +
ν2
2+ ν5)t
]− exp
[−(
λ2
2+ µ1 +
ν2
2+ ν5 + 4ν1)t
].(5.34b)
After defining the steady state populations as ρ0,0αα = Πα, we find the only needed
solutions which are of orders p=q=0 given by
Πa = Πc =µ1 + ν4
µ2 + 2µ1 + ν3 + 2ν4
, (5.35a)
Πb =µ2 + ν3
µ2 + 2µ1 + ν3 + 2ν4
. (5.35b)
5.5 Fluorescence
The fluorescence spectra are calculated using the same technique discussed in the
previous chapter. Hence, the emission spectrum is the inverse Fourier transform of
the two-time atomic correlation function, ge(τ), which leads to
ge(ν) = Re
[∫ ∞
0
ge(τ)e−iντdτ
], (5.36)
where
ge(τ) = < P (−)(τ)P (+)(0) > . (5.37)
We start by solving for the fluorescence spectrum of the light emitted during the
107
transition 2-1. In this case the corresponding atomic polarization operators are
P (−) = |2 >< 1|,
=∑α,β
N1,N2
(P (−)
)α,β|α(N1, N2) >< β(N1 − 1, N2)|, (5.38)
P (+) = |1 >< 2|,
=∑γ,ϕ
N1,N2
(P (+)
)γ,ϕ|γ(N ′
1 − 1, N ′2) >< ϕ(N ′
1, N′2)|. (5.39)
The average of the negative part of the atomic polarization operator is
< P (−)(τ) > = Tr(ρP (−)
),
=∑
α,β
(P (−)
)α,β
ρ1,0β,α(τ). (5.40)
With the help of the Regression theorem, we establish the rules of correspondence
< P (−)(τ) > −→ < P (−)(τ)P (+)(0) > ≡ ge(τ), (5.41)
ρ1,0β,α −→ (
P (−))
βαΠβ. (5.42)
After applying the previous substitutions and with the help of the set of equations
5.33 which give the explicit time dependent solutions of ρp,qβ,α(τ) function of the initial
108
conditions ρp,qβ,α, and for p=1 and q=0 equation 5.40 leads to
ge(τ) ≡ < P (−)(τ)P (+)(0) >, (5.43)
=sin2θ
2Πa eiω1τ e−(λ2+ν2)τ +
sin2θ
4Πa
(ei(ω21+2G)τ + ei(ω21−2G)τ
)e−(λ1+λ2−ν1+ν2)τ +
cos2θ
4Πa
(ei(ω21+G)τ + ei(ω21−G)τ
)e−(
λ22
+µ1+4ν1+ν22
+ν5)τ +
cos2θ
4Πa
(ei(ω21+G)τ + ei(ω21−G)τ
)e−(
λ22
+µ1+ν22
+ν5)τ . (5.44)
The previous equation of the correlation function ge(τ) can be written in the
general form
ge(τ) =∑
(weighting factor) ei(central frequency)τ e−(linewidth)τ , (5.45)
which after substitution into the equation of the emission spectrum (Eq. 5.36) leads
to
ge(ν) =∑ 1
π(weight factor)
linewidth
(ν − ωcentral)2 + (linewidth)2. (5.46)
Each term of this sum is a Lorentzian made of well defined weight factor, linewidth,
and central frequency. We cast the different components of the Lorentzians which
make the spectrum in the following table:
109
Weight Factor Central Frequency Linewidth
sin2θ
2Πa ω21 λ2 + ν2
sin2θ
4Πa ω21 ± 2G λ1 + λ2 − ν1 + ν2
cos2θ
4Πa ω21 ±G
λ2
2+ µ1 + 4ν1 +
ν2
2+ ν5
cos2θ
4Πa ω21 ±G
λ2
2+ µ1 +
ν2
2+ ν5
The spectral line is made of three Lorentzians centered at the frequencies
ν = ω21, ±2G and two more lines centered at ν = ω21 ± G which are superposi-
tions of two Lorentians with different linewidths. In the Secular limit, G >> γij,
the lines which make this spectrum are well separated. The separations between the
central positions of these different parts of the spectrum are much greater than their
linewidths which are combinations of the natural linewidths γij. These five emission
lines are the ones mentioned earlier in section 5.2 and represented in figure 5.2 by the
“downward” decay arrows.
In a similar way we can calculate the fluorescence spectrum associated with the
3-2 decay. In this case we use the atomic polarization operators P (−) = |3 >< 2|, and
110
P (+) = |2 >< 3| which lead to the correlation function
ge(τ) =cos2θ
2Πa eiω32τ e−(λ2+ν2)τ +
cos2θ
4Πa
(ei(ω32+2G)τ + ei(ω32−2G)τ
)e−(λ1+λ2−ν1+ν2)τ +
sin2θ
4Πb
(ei(ω32+G)τ + ei(ω32−G)τ
)e−(
λ22
+µ1+4ν1+ν22
+ν5)τ +
sin2θ
4Πb
(ei(ω32+G)τ + ei(ω32−G)τ
)e−(
λ22
+µ1+ν22
+ν5)τ . (5.47)
As before, the emission spectrum is a sum of Lorentzian lines as summarized in
the following table
Weight Factor Central Frequency Linewidth
cos2θ
2Πa ω32 λ2 + ν2
cos2θ
4Πa ω32 ± 2G λ1 + λ2 − ν1 + ν2
sin2θ
4Πb ω32 ±G
λ2
2+ µ1 + 4ν1 +
ν2
2+ ν5
sin2θ
4Πb ω32 ±G
λ2
2+ µ1 +
ν2
2+ ν5
These are the five expected emission lines centered at the frequencies ν = ω32, ω32±
G, and ω32 ± 2G which we represent in figure 5.2 by the “right side” decay arrows.
The two derived emission spectra in this section can be tested by turning either
one of the fields off. We start with the trivial setting where the first field is eliminated,
111
g1 = 0. In this case the sets of equations 5.11 and 5.35 reduce to
sin θ = 0, (5.48a)
cos θ = 1, (5.48b)
Πa = Πc = 0, (5.48c)
Πb = 1. (5.48d)
Under these conditions, all weight factors in both tables are zero. As expected, there
is no emission from any transition due to the trapping of the population in level 1.
On the other hand, when the second field is turned off, i.e. g2 = 0, we have
sin θ = 1, (5.49a)
cos θ = 0, (5.49b)
Πa = Πc =1
2, (5.49c)
Πb = 0, (5.49d)
and, emission takes place only out of the 2-1 transition. The calculated spectrum
in this case is the well known three-triplet fluorescence spectrum of a two-level atom
driven by a strong field. The Lorentzian lines of the spectrum summarized in the
following table match analytically, on resonance, the ones derived by Cohen-Tannoudji
112
and co-authors [1]
Weight Factor Central Frequency Linewidth
1/4 ω21 γ21
1/8 ω21 ± Ω13
2γ21
where Ω1 is the first field’s Rabi frequency defined as
Ω1 = 2g1
√N1. (5.50)
5.6 Absorption of the Probe Field
We calculate in this section the absorption spectrum of a probe field as the rate of
work done by this weak field on the pre-existing strong field (coupling) plus atom
system. In the Cascade-EIT (Cascade-AT) configuration the field Ω1 (Ω2) will be
turned off. Comments about the two absorption spectra will be given at the end of
the section.
As proved in the previous chapter (Chp. 4), the absorption spectrum is given by
ga(ωp) =
∫ ∞
−∞dτeiωpτga(τ), (5.51)
113
where ga(τ) is the average of the commutator defined as
ga(τ) = <[P (+), P (−)(τ)
]>,
= < P (+)P (−)(τ) > − < P (−)(τ)P (+) >,
= gd(τ)− ge(τ). (5.52)
5.6.1 Cascade-EIT Configuration
In the Cascade-EIT case, the lower field is eliminated, g1 = 0, and the corresponding
atomic polarization operators are P (+) = |1 >< 2| and P (−) = |2 >< 1|.
As mentioned in the previous section, setting g1 = 0 leads to ge(τ) = 0 which in
turn leads to ga(τ) = gd(τ). Along the lines of the work presented for the calculations
of ge(τ) in the previous section, we find that
ga(τ) =1
2exp
[i
(ω21 ± Ω2
2
)τ
]exp
[−
(γ32
2
)τ], (5.53)
which when substituted into equation 5.51 leads to the absorption spectrum presented
in the following table
Weight Factor Central Frequency Linewidth
1/2 ω21 ± 1
2Ω2
1
2γ32
where Ω2 is the Rabi frequency of the second field and is related to the coupling
114
constant, g2 (Eq.5.7), by the equation
Ω2 = 2g2
√N2. (5.54)
5.6.2 Cascade-AT Configuration
After solving for the absorption spectrum of the probe field in the Cascade-AT config-
uration case, where the appropriate atomic polarization operators are P (+) = |2 >< 3|
and P (−) = |3 >< 2|, we find that the spectrum is made of the following two
Lorentzians
Weight Factor Central Frequency Linewidth
1/4 ω32 ± 1
2Ω1 γ32 − γ21
2
5.6.3 Comments
It is apparent that the absorption spectra for both configurations have the same struc-
ture. Each spectrum is the sum of two non-interfering Lorentzian lines. The separa-
tion between the maxima is linear with the coupling field (field 2 for the Cascade-EIT
configuration and field 1 for the Cascade-AT configuration). This linear behavior in
the strong field limit, which is associated with the Autler-Townes (Stark shift) effect,
was introduced in the introduction chapter, and then shown in figure 3.3 in chapter 3.
All fluorescence and absorption spectra calculated in the last two sections match
115
analytically with the ones calculated by L. Narducci et. al. [65]. Analytical results
derived by the authors with the help of an approximated matrix inversion technique
was numerically matched with exact matrix inversion code.
In addition to this analytical match, the derived absorption spectra of the probe
field in the Cascade-EIT and Cascade-AT case match numerically with the ones de-
rived in chapter 3 (the imaginary parts of equations 3.28 and 3.32), and chapter 4 (the
real parts of equations 4.54 and 4.69). This numerical match is shown in figures 5.3
and 5.4 for the Cascade-EIT and Cascade-AT respectively. We note that the stronger
the coupling field the better the match is.
116
Figure 5.1: Cascade configuration
Figure 5.2: Transition decays
117
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
δp
Abs
orpt
ion
Coe
ffici
ent
PerturbativeSecular (scaled)
Figure 5.3: Absorption line for the Cascade-EIT configuration compared between theSecular and Perturbative technique: W32 = 0.2, W31 = 0.01, δc = 0, and Ωc = 5
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
δp
Abs
orpt
ion
Coe
ffici
ent
PerturbativeSecular (scaled)
Figure 5.4: Absorption line for the Cascade-AT configuration compared between theSecular and Perturbative technique: W32 = 0.2, W31 = 0.01, δc = 0, and Ωc = 5
Chapter 6: EIT and AT Effects as Scattering
Processes
Three techniques have been used so far in this thesis for the study of the cascade con-
figurations, Cascade-EIT and Cascade-AT. We have already established that quan-
tum interference operates only in the Cascade-EIT configuration. In addition, we also
proved in the chapter entitled Secular Limit that the interference effects are washed
out in the strong field regime.
In the secular limit chapter we also learned that an advantage of dealing with
quantized fields is the ability to identify the pathways followed by the atom when
absorbing and emitting photons. In this chapter we treat again the fields quantum
mechanically and, as a result, we are able to trace the different intermediate steps
taken by the atom in going from one state to another.
A different way to understand the absorption of the probe field can be achieved
by answering the question of how likely it is for a photon to be absorbed out of the
probe field. Obviously, when, as a function of the probe frequency, the absorption
coefficient decreases (for example), a photon from the probe beam has a smaller prob-
118
119
ability of being absorbed. Even in a simpler form, the question can be stated as what
is the probability for the system to go from a state to another by absorbing one probe
photon.
Following Lounis and Cohen-Tannoudji [5] we interpret the absorption of a probe
photon as a scattering process induced by the atom while interacting with several
pump photons. We calculate the corresponding scattering amplitude and, if they
exist we identify the multiple physical paths followed by the system as it evolves to
its final state. The existence of multiple paths in the transition amplitude results in
quantum interference effects exhibited by the transition probability. In 1993, Gryn-
berg and Cohen-Tannoudji [74], using the same scattering technique [5], traced the
physical origin of gain in the central resonance of the Mollow absorption spectrum.
The scattering technique in the dressed states picture was also used to study the Vee
system [6].
In the first section of this chapter, section 6.1, we survey the mathematical tools
which are needed to calculate the probability that a system evolves from a given initial
to a final state. For some important technical reasons discussed in the introduction
of section 6.2, only the Cascade-EIT configuration can be studied in the bare states
picture. It is in this same section where we show the existence of interference between
two excitation-emission pathways in the process of absorption of a probe photon in the
Cascade-EIT case. The results are simplified by considering the low saturation limit.
Section 6.3 studies the scattering of a probe photon in the dressed states picture only
120
in the low saturation limit due to physical requirements of the technique which will
be discussed in the introduction of the section. Both configurations are studied. The
expected absence of interfering pathways in the Cascade-AT case is verified. Also,
the results found in the Cascade-EIT case are consistent with the ones found in the
bare state picture.
6.1 Technique
In this section we derive and study the general elements of the scattering technique
that will be used in the two following sections. The transition and probability am-
plitudes, which are related to the probability, are defined and calculated non pertur-
batively in the first subsection and then rederived in the second subsection after the
introduction of an operator called the Resolvent.
6.1.1 Transition and Probability Amplitudes
A system in an initial state |i > at a time ti has a probability Pfi(tf , ti) of performing
a transition to a final state |f > at a later time tf . The system evolves between the
two times, ti and tf , according to the unitary transformation
|f > = U(tf , ti)|i >, (6.1)
where the evolution operator, U(tf , ti), is defined as
U(tf , ti) = e−iH(tf−ti)/~, (6.2)
121
and depends on the total Hamiltonian H.
After substituting equation 6.1 into the Schrodinger equation for the final state,
given by
i~d
dtf|f > = H|f >, (6.3)
we find that the evolution operater obeys the integral equation
U(tf , ti) = Uo(tf , ti) +1
i~
∫ tf
ti
dtUo(tf , t)V U(t, ti), (6.4)
where the unperturbed evolution operator, Uo(tf , ti), is defined as
Uo(tf , ti) = e−iHo(tf−ti)/~. (6.5)
By successive iterations, equation 6.4 yields
U(tf , ti) = Uo(tf , ti) +∞∑
n=1
U (n)(tf , ti), (6.6)
where for tf > τn > ... > τ1,
U (n)(tf , ti) =
(1
i~
)n ∫ tf
ti
dτn...dτ1 e−iHo(tf−τn)/~V (τn)...V (τ1)e−iHo(τ1−ti)/~. (6.7)
In order to eliminate the explicit time dependence in the equations, we switch
122
from the Schrodinger to an interaction picture defined by the transformation
U = eiHotf /~U(tf , ti)e−iHoti/~. (6.8)
The n-th contribution to the evolution operator in equation 6.7 takes the form
U (n)(tf , ti) =
(1
i~
)n ∫ti
tf dτn...dτ1 V (τn)...V (τ1), (6.9)
after being transformed into the interaction picture. In equation 6.9 V is the inter-
action Hamiltonian in the interaction picture.
The probability amplitude that the system will go from the initial state |i > to
the final state |f > is defined as
Sfi ≡ Ufi = Ufi = < f |U(tf , ti)|i >, (6.10)
which, can also be expressed in the form
Sfi =∞∑
n=0
S(n)fi , (6.11)
with the help of equation 6.6.
The transition probability, P , is equal to the modulus squared of the the proba-
123
bility amplitude, Sfi,
P = |Sfi|2. (6.12)
After substituting equation 6.9 into equation 6.10 we can identify the various
orders of the probability amplitude. The zeroth order is given by
S(0)fi = δfi, (6.13)
and is equal to zero in the usual case of interest where the initial and final states are
different.
For n=1, the first order probability amplitude is given by
S(1)fi = < f |U (1)(tf , ti)|i >, (6.14)
which can also be written explicitly in the form
S(1)fi = < f | 1
i~
∫ tf
ti
dτ1 e−iHoτ1/~V (τ1)e−iHoτ1/~ |i >, (6.15)
and eventually reduces to
S(1)fi =
1
i~
∫ tf
ti
dτ1 Vfiei(Ef−Ei)τ1)/~. (6.16)
If we select the origin of time so that ti = −T/2 and tf = T/2 where T is the
124
duration of the interaction, we obtain
S(1)fi = −2πiδ(T )(Ef − Ei)Vfi, (6.17)
where the diffraction function, δ(T )(Ef −Ei), has a maximum amplitude of T/2π~ at
Ef = Ei and a width of 4π~/T , and is defined as
δ(T )(Ef − Ei) =1
π
sin(Ef − Ei)T/2~Ef − Ei
. (6.18)
This approximate δ-function represents the conservation of energy with an uncer-
tainty ~/T due to the finite interaction time.
The second order of the probability amplitude has the explicit form
S(2)fi = < f |
(1
i~
)2 ∫ T/2
−T/2
dτ2dτ1 e−iHo(tf−τ2)/~V (τ2)V (τ1)e−iHo(τ1−ti)/~ |i >, (6.19)
which by inserting an identity operator between V (τ2) and V (τ1) leads to
S(2)fi =
(1
i~
)2 ∫dτ2dτ1
∑
k
VfkVkiei(Ef−Ek)τ2/~ ei(Ek−Ei)τ1/~. (6.20)
After eliminating the restriction τ2 > τ1 by multiplying by the Heaviside function
Θ(τ2 − τ1) (=1 for τ2 > −τ1 > 0 ; =0 for τ2 − τ1 < 0) and after using the identity
eiE(τ2−τ1)/~Θ(τ2 − τ1) = limη→0+
− 1
2πi
∫ +∞
−∞
e−iE(τ2−τ1)/~
E + iη − Ek
dE, (6.21)
125
equation 6.20 reduces to
S(2)fi = −2πiδ(T )(Ef − Ei) lim
η→0+
∑
k
VfkVki
Ei − Ek + iη. (6.22)
On the one hand, the function∑
k(VfkVki)/(Ei − Ek + iη) varies very rapidly
when the state |k > is in the neighborhood of the initial state, and Ek ≈ Ei. On the
other hand, the delta function δ(T )(Ef −Ei) requires that the difference between the
energies of the initial and final states stays within the limit ~/T .
Thus to second order we have
Sfi = δfi − 2πiδ(T )(Ef − Ei)
[Vfi + lim
η→0+
∑
k
VfkVki
Ei − Ek + iη
]+ Θ(V 3). (6.23)
After defining the transition amplitude, Tfi, as
Tfi = < f |V |i > + < f |V 1
Ei −Ho + iηV |i > +
< f |V 1
Ei −Ho + iηV
1
Ei −Ho + iηV |i > + ... (6.24)
the probability amplitude is given by
Sfi ' δfi − 2πiδ(T )(Ef − Ei) limη→0+
Tfi. (6.25)
126
6.1.2 Resolvent Operator
The transition amplitude (Eq. 6.24) is a perturbative expansion in orders of the
interaction Hamiltonian, V, and depends on the term 1/(Ei−Ho + iη). The appear-
ance of this term suggests to the introduction of a resolvent operator of the total
Hamiltonian, H, in the form
G(z) =1
z −H. (6.26)
After using the identity
1
A=
1
B+
1
B(B − A)
1
A, (6.27)
the Resolvent (Eq. 6.26) can also be cast into the form
G(z) = Go(z) + Go(z)V G(z), (6.28)
where
Go(z) =1
z −Ho
, (6.29)
is the unperturbed Resolvent.
By successive iteration equation 6.28 yields
G(z) = Go(z) + Go(z)V Go(z) + Go(z)V Go(z)V Go(z) + ... (6.30)
127
With the help of equation 6.29, the transition matrix (Eq. 6.24) leads to
Tfi = < f |V |i > + < f |V Go(Ei+iη)V |i > + < f |V Go(Ei+iη)V Go(Ei+iη)V |i > +...
(6.31)
which after using equation 6.30 reduces to
Tfi = < f |V |i > + < f |V G(Ei + iη)V |i > . (6.32)
Define the limit of the resolvent, G, as
G+(Ei + iη) = limη→0+
G(Ei + iη). (6.33)
With the help of equation 6.32, the probability amplitude (Eq. 6.25) written as a
function of G+ takes the form
Sfi = δfi − 2πiδ(T )(Ef − Ei) [< f |V |i > + < f |V G+(Ei + iη)V |i >] , (6.34)
which in the cases of interest where |f > 6= |i > reduces to
Sfi = −2πiδ(T )(Ef − Ei) [< f |V |i > + < f |V G+(Ei + iη)V |i >] . (6.35)
The probability Pfi (Eq. 6.12) in this case is given by
Pfi = 4π2 [δ(T )(Ef − Ei)]2 | < f |V |i > + < f |V G+(Ei + iη)V |i > |2, (6.36)
128
which implies that
Pfi = π2 [δ(T )(Ef − Ei)]2 |Vfi +
∑
k,l
VfkVli < k|G+(Ei + iη)|l > |2. (6.37)
The states |k > and |l > are discrete intermediate states of the scattering process
between the initial and final states, |i > and |f >. The system is not observed at the
intermediate states, but the the probability amplitude between the initial and final
states is the sum over all probability amplitudes of all the intermediate states. The
corresponding equation is given by
< f |U(tf , ti)|i > =∑
k
< f |U(tf , tk)|k >< k|U(tk, ti)|i > . (6.38)
The matrix element < k|G(Ei + iη)|l > of the resolvent operator, G, is the matrix
element of G projected onto the two subspaces that the states |k > and |l > belong
to. For example, if the two intermediate states |k > and |l > belong to the same
subspace Eo given by Eo = |ϕ1 >, ... , |ϕn >, we have
Gkl(Ei + iη) =< k|PG(Ei + iη)P |l >, (6.39)
where the projector P is defined as
P =n∑i
|ϕi >< ϕi|. (6.40)
The other possible case is when only one of the intermediate states, |l > for
129
example, belongs to the subspace Eo while the second state, |k >, belongs to the
supplementary subspace of Eo, Lo. In this case we have
Gkl(Ei + iη) =< k|QG(Ei + iη)P |l >, (6.41)
where Q is the supplementary projector of P defined as
Q = 1− P. (6.42)
In order to find explicit expressions for PG(z)P and QG(z)P, we start by writing
the original definition of the resolvent operator (Eq. 6.26) in the form
(z −H)G(z) = 1. (6.43)
After multiplying the previous equation by the projection operator P from the left
and from the right, and inserting the identity P + Q = 1 between (z-H) and G(z) we
find
P (z −H)PG(z)P + P (z −H)QG(z)P = P. (6.44)
After using the identities PHoQ = 0, PP=P, QQ=Q, and PQ=0 we obtain
P (z −H)P [PG(z)P ]− PV Q[QG(z)P ] = P. (6.45)
130
A similar analysis leads to
−QV P [PG(z)P ] + Q(z −H)Q[QG(z)P ] = 0. (6.46)
The two coupled equations 6.45 and 6.46 lead explicit expressions for PG(z)P and
QG(z)P given by
PG(z)P =P
z − PHoP − P
(V + V
Q
Z −QHQV
)P
, (6.47)
QG(z)P =Q
z −QHQV
P
z − PHoP − P
(V + V
Q
Z −QHQV
)P
. (6.48)
We now define the level shift operator, R(z), as
R(z) = V + VQ
z −QHQV, (6.49)
(the properties and significance of this operator will be described in the next section).
With the help of the indentity, 6.27, equation 6.49 can also be cast into the form of
a perturbative expansion in powers of V
R(z) = V + VQ
z −QHoQV + V
Q
z −QHoQV
Q
z −QHoQV + ... (6.50)
After writting equations 6.47 and 6.48 as functions of the level-shift operator, R
131
(Eq. 6.49), we obtain
PG(z)P =P
z − PHoP − PRP, (6.51)
QG(z)P =Q
z −QHQV PG(z)P. (6.52)
6.2 Bare States Picture
The probability that a system will evolve from an initial to a final state is independent
of the picture in which the calculations are carried out. The different pictures only
effect the intermediate states associated with the pathways followed by the system as
it evolves from the initial to the final state. Some pictures, the dressed states picture
for example, pose specific restrictions to the application of the scattering technique.
This fact will be addressed further in section 6.3 where the dressed states picture will
be used.
In this section we adopt the bare states picture seeking an understanding of the
possible pathways followed by the atom through intermediate bare states in the pro-
cess of scattering one probe photon. We consider the physically realistic situation
where the system, which is the bare atom in this case, is initially in a quasi-stable
state. At the end of the scattering process the system must also exist in a quasi-stable
state. If this were not the case, the evolution would not yet be over.
In a cascade system the excited atomic states |2 > and |3 > are not stable be-
132
cause of their spontaneous decay out of these states. This leaves the ground state,
|1 >, as the only bare atomic state which can be used as an initial and final state
for the scattering process. In the Cascade-EIT configuration, unlike the Cascade-AT
configuration, the state |1 > is coupled by a weak field (probe) to state |2 >. This
fact keeps the state |1 > stable. The situation is quite different in the Cascade-AT
case where state |1 > is coupled to state |2 > by a strong field (coupling). The fact
that the strong coupling field forces the atom to oscillate between the two states |1 >
and |2 > with a large Rabi frequency makes the state 1 > unstable.
The problem of not having a bare stable state in the Cascade-AT configuration
eliminates the possibility of applying the scattering technique on this configuration in
the bare states picture. We study in this section only the Cascade-EIT configuration.
In the next section, instead, we show that one of the dressed states is quasi-stable
and this allows the study of the Cascade-AT configuration in the dressed state picture.
The initial setting of the Cascade-EIT system of interest involves the atom in the
atomic ground state |1 > interacting simultaneously with the coupling field, having
Nc photons in its mode, and with one probe photon. Thus, the initial state has the
form
|i > = |1; (1)p, (Nc)c, (0)j > . (6.53)
Considering only one photon in the probe field corresponds to the simplest scat-
tering process, where one probe photon is absorbed followed at a later time by the
133
emission of a new photon in one of the vacuum modes. The case where NP (NP > 1)
photons exist in the probe field is examined in appendix A, where we show that the
inclusion of more than one probe photon is associated with higher orders of interac-
tion which are not of interest in this work.
Based on the scattering process and the requirement that the state be quasi stable,
the final state corresponds to the situation where the atom is in its ground state with
Nc photons in the coupling field and no photons in the probe field. This state is given
by
|f > = |1; (0)p, (Nc)c, (1)ω > . (6.54)
The interaction of the atom with two single mode fields was discussed in the pre-
vious chapter (see equation 5.8). Here, the interaction Hamiltonian, V, also includes
the interaction of the atom with the infinite number of modes (j) of the vacuum
leading to
V = ~gp
(a12σp + a†12σ
+p
)+ ~gc
(a23σc + a†23σ
+c
)+
∑j
[gj
(a12σj + a†12σ
+j
)+ g′j
(a23σj + a†23σ
+j
)]. (6.55)
The coupling constants, g, are related to the respective Rabi frequencies by the
134
following equations
Ωp = 2gp
√Np, (6.56a)
Ωc = 2gc
√Nc, (6.56b)
Ωj = 2gj = 2
(−µ12
√~ωj
2εoL3
), (6.56c)
Ω′j = 2g′j = 2
(−µ23
√~ωj
2εoL3
). (6.56d)
The interaction Hamiltonian, acting on the initial and final states gives
V |i > = ~Ωp
2|ϕ2 >, (6.57)
V |f > = ~Ω
2|ϕ2 >, (6.58)
where the discrete state |ϕ2 > is defined as
|ϕ2 > = |2; (0)p, (Nc)c, (0)j > . (6.59)
With the help of equation 6.57, which leads to
< f |V |i > =Ωp
2< f |ϕ2 > = 0, (6.60)
the transition amplitude, Tfi (Eq. 6.32), reduces to
Tfi =~2ΩΩp
4< ϕ2|G(Ei + iη)|ϕ2 > . (6.61)
135
After selecting the energy of the state |ϕ2 > as the energy reference, Eϕ2 = 0, and
after defining the state |ϕ3 > as
|ϕ3 > = |3; (0)p, (Nc − 1)c, (0)j >, (6.62)
we find that the states |i > and |ϕ3 > have the energies
Ei = ~δp, (6.63a)
Eϕ3 = −~δc, (6.63b)
and are quasi-degenerate with the state |ϕ2 > (the quasi-degenerate results from the
smallness of the detuning parameters relative to all the other frequencies of the prob-
lem).
For the same reasons discussed in the previous chapter (see section 5.1), we take
here the infinite volume limit, L →∞. Unlike the coupling between the states |ϕ3 >
and |ϕ2 >,
< ϕ2|V |ϕ3 > = ~Ωc
2
L→∞−→× 0, (6.64)
which remains finite, the coupling between the two states |ϕ2 > and |i >,
< ϕ2|V |i > = ~Ωp
2
L→∞−→ 0, (6.65)
vanishes. This fact eliminates the initial state |i > from the subspace of the state
136
|ϕ2 >.
In this case the matrix element < ϕ2|G(Ei+iη)|ϕ2 > is the element of the operator
G projected onto the subspace Eo defined as
Eo = |ϕ2 >, |ϕ3 > . (6.66)
We define the matrix PG(Ei + iη)P in the form
PG(Ei + iη)P =
G22(Ei + iη) G23(Ei + iη)
G32(Ei + iη) G33(Ei + iη)
, (6.67)
which with the help of equation 6.51 is identified as the inverse of the matrix
(PG(Ei + iη)P )−1 =
Ei + iη − Eϕ2 −R22(Ei + iη) −R23(Ei + iη)
−R32(Ei + iη) Ei + iη − Eϕ3 −R33(Ei + iη)
.
(6.68)
After using the set of equations 6.63 the limit of the projection of the resolvent
operator onto the subspace Eo takes the form
limη→0+
PG+(Ei−iη)P =~D
δp − limη→0+ R22(Ei − iη) − limη→0+ R23(Ei − iη)
− limη→0+ R32(Ei − iη) δp + δc − limη→0+ R33(Ei − iη)
,
(6.69)
where D is the determinant of the (PG(z)P )−1 matrix (Eq. 6.68) in the limit of
η → 0+.
137
To second order in the interaction Hamiltonian, the level-shift operator (Eq. 6.50)
is given by
R(Ei − iη) = V + VQ
Ei − iη −QHoQV, (6.70)
which leads to the diagonal matrix element
R22(Ei − iη) = < ϕ2|V |ϕ2 > + < ϕ2|V Q
Ei − iη −QHoQV |ϕ2 > . (6.71)
The action of the interaction Hamiltonian, V, on the state |ϕ2 > gives
V |ϕ2 > = ~Ωc
2|ϕ3 > +
∑j
~Ωj
2|ϕ1,ωj
>, (6.72)
where
|ϕ1,ωj> = |1; (0)p, (Nc)c, (1)j > . (6.73)
After substituting equation 6.72 into equation 6.71 we realize that the first term
of the level-shift matrix element is zero in this case. Because the operator Q does not
contain the projector |ϕ3 >< ϕ3|, the product QV |ϕ2 > reduces to∑
j ~Ωj
2|ϕ1,ωj
>.
In this case, equation 6.71 reduces to
R22(Ei − iη) =∑
j
~2Ω2j
4< ϕ1,ωj
| 1
Ei − iη −QHoQ|ϕ1,ωj
>, (6.74)
138
which implies that
R22(Ei − iη) =∑
j
~2Ω2j
4
1
Ei − iη − Eϕ1,ωj
. (6.75)
After applying the identity
1
x + iη=
x
x2 + η2− iη
x2 + η2= P
(1
x
)− iπδ(x), (6.76)
on the term 1/(Ei − iη − Eϕ1,ωj), equation 6.75 leads to
R22(Ei + iη) = ~(
∆2(Ei)− iΓ2(Ei)
2
), (6.77)
where
∆2(Ei) =1
~P
(∑j
~2Ω2j
4
1
Ei − iη − Eϕ1,ωj
), (6.78)
and
Γ2(Ei) =2π
~∑
j
~2Ω2j
4
1
Ei − iη − Eϕ1,ωj
δ(Ei − Eϕ1,ωj). (6.79)
After substituting Ei by Eϕ2 (Ei = ~δp ' Eϕ2 = 0), and Eϕ1,ωjby Eϕ1 − ~ωj
equations 6.78 and 6.79 reduce to
limη→0+
∆2(E2) =1
~P
(∑j
~2Ω2j
4
1
Eϕ2 − Eϕ1 − ~ωj
), (6.80)
limη→0+
Γ2(E2) =2π
~∑
j
~2Ω2j
4
1
Eϕ2 − Eϕ1 − ~ωj
δ(Eϕ2 − Eϕ1 − ~ωj). (6.81)
139
The terms ∆2(E2) and Γ2(E2) respectively correspond to the energy shift of state
|ϕ2 > due to the coupling of the atom with the field and the rate of spontaneous
emission out of state |ϕ2 > to all lower energy levels. The term Γ2(E2), for all levels
a (with a < b), is defined as
Γb(Eb) =∑
a
Wba (6.82)
We define new energies equal to the old ones plus the shifts, ∆2(E2) and ∆3(E3),
which are approximately equal, E3 = ~δc ' Eϕ2 = 0. Thus, equation 6.77 reduces to
limη→0+
R22(Ei + iη) = −i~W21
2. (6.83)
Along the same lines of the calculations leading to the level-shift matrix element
R22, we also find that
limη→0+
R33(Ei + iη) = −i~W32 + W31
2. (6.84)
The off-diagonal matrix element of R (Eq. 6.70), R32, is given by
R32(Ei − iη) = < ϕ3|V |ϕ2 > + < ϕ3|V Q
Ei − iη −QHoQV |ϕ2 > . (6.85)
After subsituting equation 6.72 into equation 6.85 we obtain
R32(Ei − iη) = ~Ωc
2+ < ϕ3|V Q
Ei − iη −QHoQ
(~Ωc
2|ϕ3 > +
∑j
~Ωj
2|ϕ1,j >
).
(6.86)
140
The action of the interaction Hamiltonian on the state < ϕ3| leads to
< ϕ3|V = ~Ωc
2< ϕ2|+
∑i
~Ωi
2|ϕ2,i >, (6.87)
where
|ϕ2,i > = |2; (0)p, (Nc − 1)c, (1)j > (6.88)
After substituting equation 6.87 into equation 6.86 we find that the second term
is equal to zero. In this case the limit η → 0+ of equation 6.86 yields
limη→0+
R32(Ei − iη) = ~Ωc
2. (6.89)
The two off-diagonal matrix elements limη→0+ R32(Ei−iη) and limη→0+ R23(Ei−iη)
are equal because the coupling Rabi frequency, Ωc, is real.
With the help of equations 6.83, 6.84, and 6.89, equation 6.69 reduces to
limη→0+
PG+(Ei − iη)P =~D
δp + iW21
2−Ωc
2
−Ωc
2δp + δc + i
W32 + W31
2
, (6.90)
where the determinant, D, is given by
D = ~2 (δp + δc + iγ13) (δp + iγ12)− ~2 Ω2c
4. (6.91)
141
After substituting equation 6.90 into equation 6.61 we obtain
limη→0+
Tfi =ΩΩp
4
~ (δp + δc + iγ13)
(δp + δc + iγ13) (δp + iγ12)− Ω2c
4
. (6.92)
We can see by inspection that the previously derived equation for the transition
amplitude, Tfi , is equal to the complex conjugate of the first order perturbative so-
lution of the coherence ρ12 (Eq. 3.28) derived in chapter 3, Perturbative Technique.
With the help of the fact that =(S
(1)12
)∝ AbsEIT
p , the equation
−= (T +fi
) ∝ =(S
(1)12
), (6.93)
leads to
−= (T +fi
) ∝ AbsEITp . (6.94)
This previous relation (Eq. 6.94) shows that the absorption coefficient of the
probe field is directly proportional to the imaginary part of the transition amplitude
of the scattering process of one probe photon.
6.2.1 Resonances
The determinant D (Eq.6.91) of the matrix (PGP )1 can be written explicitly in terms
of the eigenvalues of the matrix, i.e.
D = (δp − ZII)(δp − ZIII), (6.95)
142
where the eigenvalue, ZII and ZIII are given by
2ZII = −(δc + iγ23) +√
(δc + iγ13 − iγ12)2 + Ω2c , (6.96a)
2ZIII = −(δc + iγ23)−√
(δc + iγ13 − iγ12)2 + Ω2c . (6.96b)
After writing the transition amplitude (Eq. 6.92) in terms of the eigenvalues we
obtain
Tfi =~2ΩΩp
4(ZII − ZIII)
(ZIII + δc + iγ13
δp − ZII
− ZIII + δc + iγ13
δp − ZIII
). (6.97)
Because the transition amplitude is the sum of two complex numbers, the scatter-
ing process is characterized by an interference between two possible evolution path-
ways followed by the atom during the scattering process. Each of the two complex
numbers is associated with an intermediate state, or a sequence of intermediate states.
We denote these pathways as 1st and 2nd resonance an, thus, write equation 6.97 in
the form
Tfi =~2ΩΩp
4(ZII − ZIII)
(1st Resonance + 2nd Resonance
). (6.98)
When the coupling field is turned off, Ωc = 0, the eigenvalues, ZII and ZIII ,
approach their unperturbed values
ZII → −iγ12, (6.99a)
ZIII → −δc − iγ13. (6.99b)
143
Consistently, we expect that the eigenstates ¯|ϕ2 > and ¯|ϕ3 >, corresponding to
the eigenvalues ZII and ZIII , approach the unperturbed states |ϕ2 > and |ϕ3 > in
the absence of the coupling field.
After setting
ZII = −iγ12 − δ′c + iγ′, (6.100)
and using the conservation of the trace we obtain
ZIII = −δc − iγ13 + δ′c − iγ′. (6.101)
The corrections to the energy and radiative broadening of levels 2 and 3 due to
the existence of the coupling field can be understood in the following way. The term
−δ′c (δ′c) represents the light shift of level 2 (3). Similarly, the term -γ′ (γ′) is the
radiative correction of the unperturbed level 2 (3).
The first resonance is centered at δp = <(ZII) = −δ′c, which implies that ~ωp =
~ω21−~δ′c, which is the optical resonance between level 1 and the shifted level 2. This
optical resonance has a width of γ12−γ′ which approaches γ12 when Ωc tends to zero.
The second resonance is centered at δp = <(ZIII) = −δc + δ′c which is equivalent
to ~ωp + ~ωc = ~ω31 + ~δ′c. This resonance corresponds to the Raman resonance
condition between the light-shifted level 3 and level 1 and has a linewidth of γ13 + γ′.
144
6.2.2 Low Saturation Limit
In this section we consider the low saturation limit (see section 2.4) in which the level
shifts and the linewidth corrections acquire more transparent forms. Moreover, the
pathways corresponding to the different resonances become obvious.
In the low saturation limit equation 6.96b reduces to
ZIII = −δc − iγ13 − Ω2c/4
δc + i(γ13 − γ12), (6.102)
which after comparison with equation 6.101 leads to
δ′ = −δcΩ2
c/4
δ2c + (γ13 − γ12)2
, (6.103a)
γ′ = −(γ13 − γ12)Ω2
c/4
δ2c + (γ13 − γ12)2
. (6.103b)
After substituting the set of equations 6.103 for δ′ and γ′ into the equation of the
transition matrix (Eq. 6.97) we obtain
Tfi =~ΩΩp
4
(1
δp − ZII
+
[− Ωc/2
δc + iγ13 − iγ12
]21
δp − ZIII
)(6.104)
where the two terms between the parentheses correspond to the first and second res-
onances.
145
The first resonance is given explicitly by
1st Resonance =~ΩΩp
4
1
δp + δ′c + i(γ12 − γ′), (6.105)
where γ′ << γ12. We also consider the limit δ′c << δp which yields the final result
1st Resonance ≈ ~Ω2
1
~(δp + iγ12)
~Ωp
2. (6.106)
The first resonance is the product of these factors. Starting from the right hand
side, the factor ~Ωp/2 describes the absorption process of the probe photon. This
absorption process leaves the atom in the state 2 which has an energy ~δp and a
radiative decay γ12. The last factor of the first resonance, ~Ω/2, is associated with
the emission of a photon of frequency ω into one of the vacuum modes. Thus, the
pathway of the scattering process corresponding to the first resonance can be sketched
graphically as shown in figure 6.1.
The second resonance has the form
2nt Resonance =~ΩΩp
4
[− Ωc/2
δc + iγ13 − iγ12
]21
δp + δc − δ′c + i(γ13 + γ′). (6.107)
This contribution becomes especially large when δp ≈ −δc + δ′c leading to the
approximation δc ≈ −δp + δ′c ≈ −δp. After neglecting the radiative correction γ′,
146
γ′ << γ13, equation 6.107 reduces to
2nd Resonance ≈ ~Ω2
1
~(δp + iγ12 − iγ13)×
~Ωc
2
1
~(δc + δc + iγ13)
~Ωc
2×
1
~(δc + iγ12 − iγ13)
~Ωp
2. (6.108)
The second resonance (Fig. 6.1) corresponds to the absorption of the probe pho-
ton followed by simultaneous absorption and emission of coupling field photons and
ending with the spontaneous emission of one vacuum photon.
The transition amplitude (Eq. 6.104), after all of the approximations discussed
previously, reduces to
Tfi ≈ ~Ω2
1
~(δp + iγ12)
~Ωp
2+
~Ω2
1
~(δc + iγ12 − iγ13)
~Ωc
2
1
~(δp + δc + iγ13)
~Ωc
2
1
~(δc + iγ12 − iγ13)
~Ωp
2.
(6.109)
6.3 Dressed States Picture
We already mentioned in the introduction of the previous section that quasistable
bare states do not exist in the Cascade-AT configuration. This makes the scattering
technique unsuitable for the analysis of the scattering process. This difficulty can be
removed in the dressed states picture, and in particular in the low saturation limit,
147
as we are going to show in this section.
As our first step we describe the Cascade-AT configuration in the dressed states
picture and prove the absence of interference in the low saturation limit. In subsection
6.3.2 we study the Cascade-EIT case and reproduce the results found in the bare state
picture (the two pictures, of course, must be equivalent to each other).
6.3.1 Cascade-AT
A three-level cascade atom interacting with a coupling field that acts on its two lower
states, 1 > and |2 >, is described by the total Hamiltonian
H = ~ω21a22 + ~ω31a33 + ~ωcσ+c σc + ~gc
(σca12 + a†12σ
+c
), (6.110)
where we set the energy of the ground state as the energy reference, E1 = 0.
We split the total Hamiltonian into an unperturbed part, Ho, defined as
Ho = ~ω31a33 + ~ωc(a22 + σ+c σc), (6.111)
and an interaction part, Vc, given by
Vc = −~δca22 + ~gc
(σca12 + a†12σ
+c
). (6.112)
148
We define the manifolds E(Nc) which, in the absence of the atom-field interaction,
consists of the unperturbed states |1, Nc >, |2, Nc − 1 >, and |3, Nc − 1 >. These
unperturbed states are Cartesian products of the atomic bare states, |1 >, |2 >, and
|3 >, and of field states, |Nc > and |Nc − 1 >.
After diagonalizing the interaction Hamiltonian we find the two eigenvalues
Ea = −~δc
2+~2
√δ2c + Ω2
c , (6.113a)
Eb = −~δc
2− ~
2
√δ2c + Ω2
c , (6.113b)
and the corresponding eigenstates
|a(Nc) > = cos θ|1, Nc > + sin θ|2, Nc − 1 >, (6.114a)
|b(Nc) > = cos θ|2, Nc − 1 > − sin θ|1, Nc >, (6.114b)
where
sin θ =Ωc√
Ω2c +
(δc +
√Ω2
c + δ2c
)2, (6.115a)
cos θ =δc +
√δ2c + Ω2
c√Ω2
c +(δc +
√Ω2
c + δ2c
)2. (6.115b)
The interaction of the atom with the coupling field, expands the manifold E(Nc)
to include the three eigenstates |a(Nc) >, |b(Nc) >, and |3(Nc) >, where |3(Nc) >≡
149
|3, Nc − 1 >.
The two eigenstates, equations 6.114a and 6.114b, are mixtures of the two unper-
turbed states |1, Nc > and |2, Nc−1 >. In order to make one of these two eigenstates
quasi-stable we take the low saturation limit where
Ωc
δc
<< 1. (6.116)
In this case, equations 6.115a and 6.115b take the approximate form
sin θ ≈ Ωc
2δc
, (6.117a)
cos θ ≈ 1 (6.117b)
and the dressed states (Eqs. 6.114a and 6.114b) become
|a(Nc) > = |1, Nc > +Ωc
2δc
|2, Nc − 1 >, (6.118a)
|b(Nc) > = |2, Nc − 1 > −Ωc
2δc
|1, Nc > . (6.118b)
In the low saturation limit the eigenvalues (Eqs. 6.113a and 6.113b) of the dressed
states reduce to
Ea = 0, (6.119a)
Eb = −δc. (6.119b)
150
The dressed state |a(Nc) > (Eq. 6.118a) is made of the sum of the atomic bare
state |1 > and a correction term due to the coupling of the atom with the field. Be-
cause the correction term is small in the low saturation limit, the spontaneous decay
rate out of level |a(Nc) > is approximately equal to that of the ground state, |1 >,
i.e. the state is quasi-stable.
The same argument holds for the other dressed state, |b(Nc) >, which approaches
to the atomic state |2 > in the absence of the coupling field. We note that the total
decay rates 6.82 of the two dressed states are given by
Γa = 0, (6.120a)
Γb = W21. (6.120b)
In this case the dressed state |a(Nc) > is quasi-stable and can be used as the
initial and final state of the scattering process. The process begins with one photon
in the probe field and ends with one photon in the vacuum and no photons in the
probe field. Thus, we define the following initial and final states
|i > = |a(Nc), (1)p, (0)j >, (6.121a)
|f > = |a(Nc), (0)p, (1)ω > . (6.121b)
151
The total interaction Hamiltonian is comprised of several contributions:
V = H0 + Vc + Vp + Vv, (6.122)
where Vp is the atom-probe field interaction Hamiltonian defined as
Vp = ~gp
(a23σp + a†23σ
+p
), (6.123)
Vv is the atom-vacuum interaction, given by
Vv =∑
j
[gj
(a12σj + a†12σ
+j
)+ g′j
(a23σj + a†23σ
+j
)], (6.124)
and Vc is the atom-coupling field interaction Hamiltonian given by equation 6.112,
Vc = −~δca22 + ~gc
(σca12 + a†12σ
+c
). (6.125)
The the interaction Hamiltonian acting on the initial and final states, equations
6.121a and 6.121b, gives
V |i > = Ea|a(Nc), (1)p, (0)j > +~Ωp
2a23|a(Nc), (0)p, (0)j > +
~Ω1
2a†12|a(Nc), (1)p, (1)ω1 >, (6.126a)
V |f > = Ea|a(Nc), (0)p, (1)ω > +~Ω2
a12|a(Nc), (0)p, (0)j > +
~Ω′
2a23|a(Nc), (0)p, (0)j > +
~Ω2
2a†12|a(Nc), (0)p, (1)ω,ω2 > . (6.126b)
152
After defining the projector matrix elements (aij)αβ as
(aij)αβ = < α(Nc)|(aij)|β(Nc) >, (6.127)
and with the help of equation 6.119a for the approximated eigenvalue of the dressed
state |a(Nc) >, the set of equations 6.126 reduce to
V |i > =~Ωp
2(a23)3a|3(Nc), (0)p, (0)j > +
~Ω1
2(a†12)1a|1(Nc), (1)p, (1)ω1 >, (6.128a)
V |f > =~Ω2
(a12)1a|1(Nc), (0)p, (0)j > +
~Ω′
2(a23)3a|3(Nc), (0)p, (0)j > +
~Ω2
2(a†12)1a|1(Nc), (0)p, (1)ω,ω2 > . (6.128b)
Next, with the help of equations (Eq. 6.128), the transition amplitude, Tfi (Eq.
153
6.32), yields
Tfi(Ei − iη) =
~2ΩΩp
4(a12)1a(a23)3a < 1(Nc), (0)p, (0)j|G(Ei − iη)|3(Nc), (0)p, (0)j > +
~2ΩΩ1
4(a12)1a(a
†12)1a < 1(Nc), (0)p, (0)j|G(Ei − iη)|1(Nc), (1)p, (1)ω1 > +
~2Ω′Ωp
4(a23)3a(a23)3a < 3(Nc), (0)p, (0)j|G(Ei − iη)|3(Nc), (0)p, (0)j > +
~2Ω′Ω1
4(a23)3a(a
†12)1a < 3(Nc), (0)p, (0)j|G(Ei − iη)|1(Nc), (1)p, (1)ω1 > +
~2Ω2Ωp
4(a†12)1a(a23)3a < 1(Nc), (0)p, (1)ω,ω2|G(Ei − iη)|3(Nc), (0)p, (0)j > +
~2Ω2Ω1
4(a†12)1a(a
†12)1a < 1(Nc), (0)p, (1)ω,ω2|G(Ei − iη)|1(Nc), (1)p, (1)ω1 > .
(6.129)
All terms except for the third correspond to two states which belong to comple-
mentary subspaces. As shown in appendix A, every element, except for third one,
vanishes. Thus, the transition amplitude reduces to
Tfi(Ei − iη) =~2Ω′Ωp
4(a23)3a(a23)3aG33(Ei − iη), (6.130)
where we defined the Resolvent matrix element G33(Ei − iη) as
G33(Ei − iη) = < 3(Nc), (0)p, (0)j|G(Ei − iη)|3(Nc), (0)p, (0)j > . (6.131)
The intermediate state |3(Nc), (0)p, (0)j > belongs to its own one dimensional
space. In this case, the corresponding form of equation 6.69 in a one dimensional
154
space is
limη→0+
(PGP )33(Ei − iη) =1
Ei − iη −R33
. (6.132)
The matrix element R33 of the level-shift operator is given by equation 6.84 which
after substitution into equation 6.132 leads to
limη→0+
G33(Ei − iη) =1
~δp + ~δc + i~W32 + W31
2
. (6.133)
After substituting equation 6.133 into equation 6.130 and with the help of the
matrix element of the projector a23 which is given by
(a23)3a = < 3(Nc)|a23|a(Nc) > =Ωc
2δc
, (6.134)
we obtain
Tfi(Ei − iη) =Ωc
2δc
~Ω′
2
1
~ (δp + δc + iγ31)
~Ωp
2
Ωc
2δc
, (6.135)
where we replaced (W32 + W31)/2 with the polarization decay rate γ31.
The transition amplitude (Eq. 6.135) consists of only one term corresponding
to a single resonance. There is only one complex number associated with the only
pathway followed by the dressed atom as it evolves from the initial state (Eq. 6.121a)
to the final state (Eq. 6.121b). This fact shows the absence of interference in the
absorption process of a probe photon in the Cascade-AT case. The process described
by this transition amplitude is shown by figure 6.2 where we have taken the detuning
155
of the coupling field to be negative, a choice leading which leads to a higher energy
level for the state B, B≡ |b(Nc) >, relative to the state A, A ≡ |a(Nc) >. In addition
to the stimulated absorption and emission of photons out and into the mode of the
coupling field within the dressed state ≡ |a(Nc) >, the scattering process involves
the absorption of a probe photon followed by the emission of a photon in the vacuum
field via the intermediate state |3(Nc), (0)p, (0)j >.
6.3.2 Cascade-EIT
In this subsection for completeness we sketch the calculations of the Cascade-EIT
configuration using the dressed states picture. In this case the interaction Hamiltonian
is comprised of the new contributions
Vc = −~δca33 + ~gc
(σca23 + a†23σ
+c
), (6.136)
Vp = ~gp
(a12σp + a†12σ
+p
), (6.137)
while the interaction part describing the atom-vacuum coupling remains the same.
The dressed states of the Cascade-EIT system are
|a(Nc) > = |2, Nc > +Ωc
2δc
|3, Nc − 1 >, (6.138a)
|b(Nc) > = |3, Nc − 1 > −Ωc
2δc
|2, Nc > . (6.138b)
156
The initial and final states are the same as defined in the bare states picture,
|i > = |1(Nc), (1)p, (0)j >, (6.139a)
|f > = |1(Nc), (0)p, (1)ω >, (6.139b)
where here we defined the state |1(Nc) >≡ |1; Nc >.
The action of the interaction Hamiltonian on the initial and final states yields
V |i > =~Ωp
2a12|1(Nc); (0)p, (0)j >, (6.140a)
V |f > =~Ω2
a†12|1(Nc); (0)p, (0)j > . (6.140b)
Next, we define the projector matrix elements
(a12)a1 = 1, (6.141a)
(a12)b1 = −Ωc
2δc
, (6.141b)
with
(a12)αβ = (a†12)βα, (6.142)
157
and the set of equations 6.140 became
Tfi =~2ΩΩp
4[
(a†12)1a(a12)a1 < a(Nc); (0)p, (0)j|G(Ei + iη)|a(Nc); (0)p, (0)j > +
(a†12)1a(a12)b1 < a(Nc); (0)p, (0)j|G(Ei + iη)|b(Nc); (0)p, (0)j > +
(a†12)1b(a12)b1 < b(Nc); (0)p, (0)j|G(Ei + iη)|b(Nc); (0)p, (0)j > +
(a†12)1b(a12)a1 < b(Nc); (0)p, (0)j|G(Ei + iη)|a(Nc); (0)p, (0)j > ]. (6.143)
With the help of the expansion of the Resolvent operator (Eq. 6.30) in powers of
the interaction Hamiltonian, we can see that the second and fourth terms of equation
6.143 are zero, so that the transition amplitude (Eq. 6.143) reduces to
Tfi =~2ΩΩp
4[
(a†12)1a(a12)a1 < a(Nc); (0)p, (0)j|G(Ei + iη)|a(Nc); (0)p, (0)j > +
(a†12)1b(a12)b1 < b(Nc); (0)p, (0)j|G(Ei + iη)|b(Nc); (0)p, (0)j >]. (6.144)
In the low saturation limit (Eq. 6.141) we can also write
Tfi =~2ΩΩp
4[ < a(Nc); (0)p, (0)j|G(Ei + iη)|a(Nc); (0)p, (0)j > +
Ω2c
4δ2c
< b(Nc); (0)p, (0)j|G(Ei + iη)|b(Nc); (0)p, (0)j >]. (6.145)
The Resolvent matrix elements corresponding to the intermediate states
|a(Nc); (0)p, (0)j > and |b(Nc); (0)p, (0)j >, which belong to two one-dimensional sub-
158
spaces, are given by
Gaa(Ei + iη) =1
Ei − Ea − i~W21
2
, (6.146a)
Gbb(Ei + iη) =1
Ei − Eb − i~W31 + W32
2
, (6.146b)
where
Ea = 0, (6.147a)
Eb = −δc, (6.147b)
Ei = δp + δc. (6.147c)
Thus, after substituting equations 6.146 into equation 6.145 we obtain
Tfi = ΩΩp
(1
δp + iγ12
+Ω2
c
4δ2c
1
δp + δc + iγ13
), (6.148)
which can be rewritten in the form
Tfi = Ω1
δp + iγ12
Ωp + ΩΩc
2δc
1
δp + δc + iγ31
Ωc
2δc
Ωp. (6.149)
The transition amplitude (Eq. 6.149) found here is the sum of two terms which are
associated with two resonances. Figure 6.3 shows the two resonances in the dressed
states picture. The first resonance corresponds to the excitation of the dressed atom
from level |1(Nc) > to level |a(Nc) > (A≡ |a(Nc) >) followed by the decay to the
159
ground state by spontaneous emission of one photon into the vacuum. The second
term corresponds to a process which is very similar to the first but where the dressed
atom gets excited to the dressed state |b(Nc) > (A≡ |b(Nc) >) not |a(Nc) >.
The transition amplitude (Eq. 6.109),
Tfi ≈ ~Ω2
1
~(δp + iγ12)
~Ωp
2+
~Ω2
1
~(δc + iγ12 − iγ13)
~Ωc
2
1
~(δp + δc + iγ13)
~Ωc
2
1
~(δc + iγ12 − iγ13)
~Ωp
2,
(6.150)
calculated in the bare states picture in the low saturation limit has the same structure
as the amplitude ( Eq. 6.149) derived in the dressed states picture under the same
conditions. The physical interpretations given for the two resonances, 1st Resonance
and 2nd Resonance, at the end of subsection 6.2.2 are consistent with the previous
discussion given for the transition amplitude (Eq. 6.149).
160
Figure 6.1: Resonances of the Cascade-EIT configuration in the bare states picture
Figure 6.2: Resonances of the Cascade-AT configuration in the dressed states picture
161
Figure 6.3: Resonances of the Cascade-EIT configuration in the dressed states picture
Chapter 7: Conclusions and Prospects for Future
Work
We conclude this dissertation by reviewing first the relationships between the absorp-
tion spectra found using different techniques. After showing the uniformity of the
results, we summarize the conclusions that were learned through this work leading to
a broad and detailed understanding of the Cascade-EIT and Cascade-AT configura-
tions. Some possible future projects will be outlined at the end of the chapter.
Figure 7.1 shows the relative ranges of the coupling’s Rabi frequency over which
the adopted techniques are valid. The perturbative technique (Chap. 3), two-time
atomic correlation functions and the regression theorem (Chap. 4), and the scattering
process in the bare states picture (Sec. 6.2) are valid over all strengths of the coupling
field. In chapter 4 we derived the absorption spectra (the real parts of equations 4.54
and 4.69) of the two cascade configurations which have the same analytical expres-
sions of the absorption spectra (the imaginary parts of equations 3.28 and 3.32) found
in chapter 3. We calculated in section 6.2 the transition amplitude (Eq. 6.92) of the
scattering of one probe photon in the Cascade-EIT case. The negative imaginary
part of the derived transition amplitude matches analytically the calculated absorp-
162
163
tion spectra in chapter 3 (Eq. 3.28) and chapter 4 (Eq. 4.54).
In the strong field regime and using the secular limit (Chap. 5), we showed that
the calculated absorption spectra match numerically (Fig. 5.3 and 5.4) the spectra
derived in chapters 3 and 4, which also in the Cascade-EIT case match the negative
imaginary part of the transition amplitude which was calculated in section 6.2. This
numerical match is represented in figure 7.1 by a dashed arrow.
The derived transition amplitude (Eq. 6.92) for the Cascade-EIT configuration
in section 6.2 was approximated in the low saturation limit. The approximate result
(Eq. 6.109) approach analytically (represented by a solid arrow in Fig. 7.1) the am-
plitude (6.149) found in the Dressed states picture in the low saturation limit.
Each of these complementary techniques gave a piece of information about the
problem under study, leading to a complete understanding of the two cascade con-
figurations. With the help of the perturbative technique, for example, we analyzed
the problem and pointed out two main cases. In the strong coupling field regime we
noted that the two cascade configurations exhibit similar behaviors. This similarity
was shown in figure 3.1 which displays the absorption spectral lines corresponding to
the two configurations of interest. In figure 3.3, we studied the separations between
the two maxima of the absorption lines. We showed that in the strong coupling field
regime and for both configurations, the separation is linear with respect to the cou-
pling’s Rabi frequency. This behavior was associated with the AT effect which was
164
assumed to be the cause of the detected reduction in absorption. In the weak coupling
field regime a different situation was revealed. In this case, the plots of the absorp-
tion lines (Fig. 3.2) showed that the two configurations are different. A persisting
reduction in the absorption of the probe field in the Cascade-EIT which was assumed
to be of a quantum interference origin, did not appear in the Cascade-AT case.
The matter of similarity between the two configurations was investigated with the
help of the Secular limit in chapter 5. We proved that in the secular limit (strong
coupling field regime) both models show no interference effects. The derived spectra
are the sum of well separated Lorentzian-like lines. These non-interfering pieces of
the spectra eliminate the existence of any type of interference. We also learned in
chapter 5 that the interference effects, if they exist, need to be explored in the weak
field regime.
With the help of the scattering technique, which we introduced in chapter 6, we
studied the Cascade-EIT configuration in the bare and dressed states pictures. The
Cascade-AT configuration was studied only in the dressed states picture for reasons
that were stated in section 6.3. In the Cascade-EIT case we showed that the transi-
tion amplitude is the sum of two complex numbers associated with two resonances.
These resonances correspond to interfering scattering pathways which we described
in the bare states picture (Fig. 6.1), and in the dressed states picture (Fig. 6.3) in
the weak field regime. The transition amplitude corresponding to the Cascade-AT
case was derived in the low saturation limit, and in the dressed states picture. The
165
calculated amplitude was associated with one resonance presented in figure 6.2. The
existence of only one possible scattering pathway eliminates the possibility of inter-
ference effects in the Cascade-AT configuration.
In summary, we hypothesized in chapter 3 the existence of interference effects
in the Cascade-EIT configuration. This hypothesis was proved in chapter 6 where
two interfering resonances were studied in the bare and dressed states pictures. This
quantum interference effect emerges in the weak field regime but is dominated by the
Autler-Townes effect in the strong field regime (chapter 5: Secular limit). No inter-
ference effects are exhibited by the Cascade-AT setting. This absence is confirmed
with the help of the scattering technique by the existence of only one resonance. In
the strong field regime the Cascade-AT configuration exhibits the AT effect studied
in chapter 5.
Having explored and understood the different aspects of the adopted techniques
in this thesis, we would like to use these techniques to study new systems of interest.
A type of interesting system for example is the four level systems which are proved
experimentally to display Lasing Without Inversion.
Another future work of interest is the inclusion of the Doppler broadening into
the picture. The majority of the EIT experiments are done in a Doppler broadened
medium. One of the things that we would like to investigate is the claim of some
experimentalists that in some cases the Doppler broadening increases the generated
166
coherence in the system under study and, by inference, the appearance of interference
effects.
Figure 7.1: Ranges of the coupling field strength used in the different techniques
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Appendix A: Scattering Technique
In the case where the Probe field contains Np (Np > 1) photons, the initial and final
states (Eqs. 6.53 and 6.54) are given by
|i > = |1; (Np)p, (Nc)c, (0)j >, (A.1)
|f > = |1; (Np − 1)p, (Nc)c, (1)ω > . (A.2)
The action of the interaction Hamiltonian on the initial and final states gives
V |i > = ~Ωp
2|ϕ2 >, (A.3)
V |f > = ~Ω
2|ϕ2 > +~
Ωp
2|ϕ′2 >, (A.4)
where we have defined the states
|ϕ2 > = |2; (Np − 1)p, (Nc)c, (0)j >, (A.5)
|ϕ′2 > = |2; (Np − 2)p, (Nc)c, (1)ω > . (A.6)
177
178
The transition amplitude, Tfi (Eq. 6.32), reduces to
T +fi =
~Ωp
2
(~Ω2
< ϕ2|G+(Ei + iη)|ϕ2 > +~Ωp
2< ϕ′2|G+(Ei + iη)|ϕ2 >
). (A.7)
The consideration of Np (Np > 1) photons in the probe field creates one additional
term < ϕ′2|G+(Ei + iη)|ϕ2 >, in the transition amplitude (Eq. 6.61) relative to the
case where the probe field carries only one photon in its mode.
The two intermediate states |ϕ2 > and |ϕ′2 > do not belong to the same subspace.
The state |ϕ2 > belongs to the subspace Eo (Eq. 6.66), while |ϕ′2 > belongs to the
complementary subspace Lo.
After substituting equation 6.52 into equation 6.41, where |k > and |l > corre-
spond to |ϕ′2 > and |ϕ2 >, we obtain
< ϕ′2|G+(Ei + iη)|ϕ2 > = < ϕ′2|1
Ei + iη −QHQV (|ϕ2 >< ϕ2|G|ϕ2 > +
|ϕ3 >< ϕ3|G|ϕ3 >). (A.8)
After expanding the term 1/(Ei +iη−QHQ)V into powers of V using the identity
6.27, and following the line of calculations leading to equation 6.84 we obtain
< ϕ′2|1
Ei + iη −QHQ=
1
Ei + iη − Eϕ′2 − i~W21/2< ϕ′2|, (A.9)
179
which substituted into equation A.8 leads to
< ϕ′2|G+(Ei + iη)|ϕ2 > =1
Ei + iη − Eϕ′2 − i~W21/2(< ϕ′2|V |ϕ2 >< ϕ2|G|ϕ2 > +
< ϕ′2|V |ϕ3 >< ϕ3|G|ϕ3 >). (A.10)
The two intermediate states |ϕ′2 > and |ϕ2 > (|ϕ′2 > and |ϕ3 >) are connected
by two sequential processes, the absorption of a probe photon and the spontaneous
emission of a photon of frequency ω into one of the vacuum modes (absorption of a
coupling photon and a spontaneous emission of a vacuum photon). This fact leads to
< ϕ′2|V |ϕ2 > = 0, (A.11a)
< ϕ′2|V |ϕ3 > = 0. (A.11b)
After substituting the matrix elements (Eqs A.11a and ??) of the interaction
Hamiltonian, V, into equation A.10 we obtain
< ϕ′2|G+(Ei + iη)|ϕ2 > = 0. (A.12)
To second order, the transition amplitude (Eq. A.7) in the case of Np (Np > 1)
photons in the mode of the probe field reduces to the transition amplitude (Eq. 6.61)
calculated in the case where the probe field contains only one photon.
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