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

Bibliography

[1] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg. Atom-Photon Interac-

tions: Basic Processes and Applications. Wiley-Interscience, 1992.

[2] R.W. Boyd. Nonlinear Optics. Academic Press, 2003.

[3] K.-J. Boller, A. Imamoglu, and S.E. Harris. Observation of electromagnetically

induced transparency. Phys. Rev. Lett., 66(20):2593 – 6, 1991.

[4] S.E. Harris, J.E. Field, and A. Imamoglu. Nonlinear optical processes using

electromagnetically induced transparency. Phys. Rev. Lett., 64(10):1107 – 10,

1990.

[5] B. Lounis and C. Cohen-Tannoudji. Coherent population trapping and fano

profiles. J. Phys. II France, 2:579–592, 1992.

[6] G. Grynberg, M. Pinard, and P. Mandel. Amplification without population

inversion in a v three-level system: a physical interpretation. Physical Review

A., 54(1):776 –, 1996.

[7] D.J. Fulton, S. Shepherd, R.R. Moseley, B.D. Sinclair, and M.H. Dunn.

Continuous-wave electromagnetically induced transparency: a comparison of v,

lambda , and cascade systems. Phys. Rev. A, 52(3):2302 – 11, 1995.

167

168

[8] J. Gea-Banacloche, Yong-Qing Li, Shao-Zheng Jin, and Min Xiao. Electromag-

netically induced transparency in ladder-type inhomogeneously broadened me-

dia: theory and experiment. Phys. Rev. A, 51(1):576 – 84, 1995.

[9] L.Allen and J.H.Eberly. Optical Resonance and Two-Level Atoms. Wiley, 1975.

[10] S. Baluschev, N. Leinfellner, E.A. Korsunsky, and L. Windholz. Electromagnet-

ically induced transparency in a sodium vapour cell. Eur. Phys. J. D, 2(1):5 –

10, 1998/05/.

[11] W.D. Phillips. Laser cooling and trapping of neutral atoms. In Proceeding of

the Euria Fermi Summer School, Corso CXVIII, Editrice Compositori, 1992.

[12] G. Alzetta, A. Gozzini, L. Moi, and G. Orriols. An experimental method for

the observation of rf transitions and laser beat resonances in oriented na vapour.

Nuovo Cimento B, 36B(1):5 – 20, 1976.

[13] R.M. Whitley and Jr. Stroud, C.R. Double optical resonance. Physical Review

A, 14(4):1498 – 513, 1976.

[14] E. Arimondo and G. Orriols. Nonabsorbing atomic coherences by coherent two-

photon transitions in a three-level optical pumping. Lettere al Nuovo Cimento,

17(10):333 – 8, 1976.

[15] H.R. Gray, R.M. Whitley, and Jr. Stroud, C.R. Coherent trapping of atomic

populations. Optics Letters, 3(6):218 – 20, 1978.

[16] E. Arimondo. Progress in Optics. North-Holland, 1996.

169

[17] P M Radmore and P L Knight. Population trapping and dispersion in a three-

level system. J. Phys. B, 15(4):561–573, 1982.

[18] Yong-qing Li and Min Xiao. Electromagnetically induced transparency in a three-

level lambda-type system in rubidium atoms. Physical Review A, 51(4):2703 –,

1995.

[19] J.E. Field, K.H. Hahn, and S.E. Harris. Observation of electromagnetically in-

duced transparency in collisionally broadened lead vapor. Physical Review Let-

ters, 67(22):3062 – 3065, 1991.

[20] Yong qing Li, Shao zheng Jin, and Min Xiao. Observation of an electromag-

netically induced change of absorption in multilevel rubidium atoms. Physical

Review A, 51(3):1754 – 7, 1995.

[21] J.P. Marangos. Electromagnetically induced transparency. J. Mod. Opt.,

45(3):471 – 503, 1998.

[22] S.E. Harris. Electromagnetically induced transparency. Phys. Today, 50(7):36 –

42, 1997.

[23] M. Fleischhauer, A. Imamoglu, and J.P. Marangos. Electromagnetically induced

transparency: optics in coherent media. Reviews of Modern Physics, 77(2):633

– 73, 2005.

[24] U. Fano. Effects of configuration interaction on intensities and phase shifts. Phys.

Rev., 124:1866–1878, 1961.

170

[25] S.E. Harris, J.E. Field, and A. Kasapi. Dispersive properties of electromagneti-

cally induced transparency. Physical Review A (Atomic, Molecular, and Optical

Physics), 46(1):29 – 32, 1992/07/01.

[26] Min Xiao, Yong-Qing Li, Shao-Zheng Jin, and J. Gea-Banacloche. Measurement

of dispersive properties of electromagnetically induced transparency in rubidium

atoms. Physical Review Letters, 74(5):666 – 9, 1995.

[27] A. Kasapi, Maneesh Jain, G.Y. Yin, and S.E. Harris. Electromagneti-

cally induced transparency: propagation dynamics. Physical Review Letters,

74(13):2447, 1995.

[28] Z. Dutton L.V. Hau, S.E. Harris and C.H. Behroozi. Light speed reduction to

17 metres per second in an ultracold atomic gas. Nature, 397:594–598, 1999.

[29] M.M. Kash, V.A. Sautenkov, A.S. Zibrov, L. Hollberg, G.R. Welch, M.D. Lukin,

Y. Rostovtsev, E.S. Fry, and M.O. Scully. Ultraslow group velocity and enhanced

nonlinear optical effects in a coherently driven hot atomic gas. Physical Review

Letters, 82(26):5229 – 32, 1999.

[30] M.D. Lukin and A. Imamoglu. Nonlinear optics and quantum entanglement of

ultraslow single photons. Physical Review Letters, 84(7):1419 – 22, 2000.

[31] U. Leonhardt and P. Piwnicki. Ultrahigh sensitivity of slow-light gyroscope.

Physical Review A, 62(5):055801 – 1, 2000.

171

[32] Chien Liu, Z. Dutton, C.H. Behroozi, and L.V. Hau. Observation of coherent op-

tical information storage in an atomic medium using halted light pulses. Nature,

409(6819):490 – 3, 2001.

[33] Aephraim M. Steinberg and Raymond Y. Chiao. Tunneling delay times in one

and two dimensions. Physical Review A, 49(5 pt A):3283, 1994.

[34] A. Kuzmich L.J. Wang and A. Dogariu. Gain-assisted superluminal light prop-

agation. Nature, 406:277, 2000.

[35] L.J. Wang, A. Kuzmich, and A. Dogariu. Erratum: Gain-assisted superluminal

light propagation (nature (2000) 406 (277-279)). Nature, 411(6840):974, 2001.

correction on the 2000 paper.

[36] R.Y. Chiao. Amazing Light.

[37] R. Y. Chiao and A. M. Steinberg. Tunneling times and superluminality. Progress

in Optics, (37):347, 1997. Review on fast light and causality.

[38] R.W. Boyd and D. J. Gauthier. Progress in Optics, volume 43, chapter ‘Slow’

and ‘Fast’ light, pages 497–530. Elsevier, Amsterdam, 2002.

[39] D. Marcuse. Proc. IEEE, 51:849, 1963.

[40] H.K. Holt. Gain without population inversion in two-level atoms. Phys. Rev. A,

16(3):1136 – 40, 1977.

172

[41] V.G. Arkhipkin and Yu.I. Heller. Radiation amplification without population

inversion at transitions to autoionizing states. Phys. Lett., 98A(1-2):12 – 14,

1983.

[42] S.E. Harris. Lasers without inversion: interference of lifetime broadened reso-

nances. Phys. Rev. Lett., 62:1033 – 1036, 1989.

[43] S.E. Harris and J.J. Macklin. Lasers without inversion: single-atom transient

response. Phys. Rev. A, 40(7):4135 – 7, 1989.

[44] A. Imamoglu and S.E. Harris. Lasers without inversion: interference of dressed

lifetime-broadened states. Opt. Lett. (USA), 14(24):1344 – 6, 1989.

[45] A. Imamoglu. Interference of radiatively broadened resonances. Phys. Rev. A,

40(5):2835 – 8, 1989.

[46] L.M. Narducci, H.M. Doss, P. Ru, M.O. Scully, S.Y. Zhu, and C. Keitel. A

simple model of a laser without inversion. Opt. Commun., 81(6):379 – 84, 1991.

[47] L.M. Narducci, M.O. Scully, C.H. Keitel, S.-Y. Zhu, and H.M. Doss. Physical

origin of the gain in a four-level model of a raman driven laser without inversion.

Opt. Commun., 86(3-4):324 – 32, 1991.

[48] K. Ichimura, K. Yamamoto, and N. Gemma. Evidence for electromagnetically

induced transparency in a solid medium. Physical Review A, 58(5):4116 – 20,

1998.

173

[49] Shaozheng Jin, Yongqing Li, and Min Xiao. Hyperfine spectroscopy of highly-

excited atomic states based on atomic coherence. Optics Communications, 119(1-

2):90 – 6, 1995.

[50] R. Moseley and M. Dunn. Lasing turned upside down. Phys. World, 8(1):30 –

4, 1995.

[51] R.I. Thompson, B.P. Stoicheff, G.Z. Zhang, and K. Hakuta. Nonlinear generation

of 103 nm radiation with electromagnetically-induced transparency in atomic

hydrogen. Quantum Optics, 6(4):349 – 58, 1994.

[52] R.R. Moseley, S. Shepherd, D.J. Fulton, B.D. Sinclair, and M.H. Dunn. Inter-

ference between excitation routes in resonant sum-frequency mixing. Physical

Review A, 50(5):4339 – 49, 1994.

[53] Ying Wu, J. Saldana, and Yifu Zhu. Large enhancement of four-wave mixing by

suppression of photon absorption from electromagnetically induced transparency.

Phys. Rev. A, 67(1):13811 – 1, 2003.

[54] B.S. Ham, M.S. Shahriar, and P.R. Hemmer. Enhanced nondegenerate four-

wave mixing owing to electromagnetically induced transparency in a spectral

hole-burning crystal. Optics Letters, 22(15):1138 – 40, 1997.

[55] S. H. Autler and C. H. Townes. Stark effect in rapidly varying fields. Phys. Rev.,

100:703–722, 1955.

[56] C. Cohen-Tannoudji. Amazing Light.

174

[57] C. Delsart and J. C. Keller. Optical autler-townes effect in doppler-broadened

three-level systems. Journal de Physique (Paris), 39(4):350 – 360, 1978.

[58] Changjiang Wei, Dieter Suter, Andrew S.M. Windsor, and Neil B. Manson. ac

stark effect in a doubly driven three-level atom. Physical Review A. Atomic,

Molecular, and Optical Physics, 58(3):2310, 1998.

[59] Shi-Yao Zhu, L.M. Narducci, and M.O. Scully. Quantum-mechanical interference

effects in the spontaneous-emission spectrum of a driven atom. Phys. Rev. A,

52(6):4791 – 802, 1995.

[60] G.S. Agarwal. Nature of the quantum interference in electromagnetic-field-

induced control of absorption. Phys. Rev. A, 55(3):2467 – 70, 1997.

[61] Jr. Bentley, C.L. and J. Liu. Lwi in a driven lambda three-level atom and effects

of the probe laser on eit. Opt. Commun., 169(1-6):289 – 99, 1999.

[62] A.S. Zibrov, M.D. Lukin, D.E. Nikonov, L. Hollberg, M.O. Scully, V.L. Velichan-

sky, and H.G. Robinson. Experimental demonstration of laser oscillation without

population inversion via quantum interference in rb. Phys. Rev. Lett., 75(8):1499

– 502, 1995.

[63] S. Shepherd, D.J. Fulton, and M.H. Dunn. Wavelength dependence of coherently

induced transparency in a doppler-broadened cascade medium. Physical Review

A, 54(6):5394 – 9, 1996.

175

[64] L.M. Narducci, M.O. Scully, G.-L. Oppo, P. Ru, and J.R. Tredicce. Spontaneous

emission and absorption properties of a driven three-level system. Phys. Rev. A,

42(3):1630 – 49, 1990.

[65] A.S. Manka, H.M. Doss, L.M. Narducci, P. Ru, and G.-L. Oppo. Spontaneous

emission and absorption properties of a driven three-level system. ii. the lambda

and cascade models. Phys. Rev. A, 43(7):3748 – 63, 1991.

[66] L. M. Narducci. Introduction to laser physics, phys 750-508, Drexel University,

2002. (unpublished).

[67] H.J. Carmichael. Statistical Methods in Quantum Optics 1, page 27. Springer,

1999.

[68] B.R. Mollow. Power spectrum of light scattered by two-level systems. Phys.

Rev., 188(5):1969 – 75, 1969.

[69] B.R. Mollow. Absorption and emission line-shape functions for driven atoms.

Phys. Rev. A, 5(3):1522 – 7, 1972.

[70] P. L. Patterson. Evidence of the existence of an he3+ ion. Journal of Chemical

Physics, 48:3625, 1968.

[71] C. Cohen-Tannoudji and S. Haroche. Absorption and scattering of optical pho-

tons by an atom interacting with radiofrequency photons. J. Phys. (France),

30(2-3):153 – 68, 1969.

176

[72] C. Cohen-Tannoudji and S. Reynaud. Dressed-atom description of resonance

fluorescence and absorption spectra of a multi-level atom in an intense laser

beam. J. Phys. B, 10(3):345 – 63, 1977.

[73] C. Cohen-Tannoudji and S. Reynaud. Simultaneous saturation of two atomic

transitions sharing a common level. J. Phys. B, 10(12):2311 – 31, 1977.

[74] G. Grynberg and C. Cohen-Tannoudji. Central resonance of the mollow ab-

sorption spectrum: physical origin of gain without population inversion. Opt.

Commun., 96(1-3):150 – 63, 1993.

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.