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$: THEEFFECTOFKERRNONLINEARITY,DOPPLER …prr.hec.gov.pk/jspui/bitstream/123456789/2833/1/Hazrat... · 2018-07-23 · Hazrat Ali, Ziauddin and Iftikhar Ahmad \The e ect of Kerr non-linearity$
THE EFFECT OF KERR NONLINEARITY, DOPPLER

BROADENING AND SPONTANEOUS GENERATED

COHERENCE ON SLOW LIGHT PROPAGATION

by

Hazrat Ali

A Dissertation

Submitted To The University Of Malakand In Partial Fulfillment Of

Requirements For The Degree Of

DOCTOR OF PHILOSOPHY

In Physics

Major Professors: Prof. Dr. Iftikhar Ahmad

Asst. Prof. Dr. Ziauddin

DEPARTMENT OF PHYSICS UNIVERSITY OF MALAKAND,

CHAKDARA, PAKISTAN

2016

i

ii

Abstract

The influence of Kerr non-linearity, Doppler Broadening and spontaneous

generated coherence (SGC) is presented when a probe light pulse is incident

on dispersive atomic medium. We consider different atom-field configura-

tions, i.e., N -type electromagnetically induced transparency (EIT), Four-

level Λ-type and tripod atomic systems. Initially, we consider a four-level

N -type atomic medium and exploited the light pulse propagation through

the medium. It is found that the Kerr non-linearity and relaxation rate of

forbidden transition affect the dispersive properties of the atomic medium.

A more slow group velocity of light pulse propagation is achieved via in-

creasing the Kerr field. We also explored the influence of relaxation rate

of forbidden decay rate on dispersive properties of the atomic medium. By

increasing the atomic number density, the relaxation rate of forbidden de-

cay rate increases which leads to control the slow and fast light propagation

through the medium. Next, we consider a four-level Λ-type atomic medium

and investigated the influence of Kerr non-linearity and Doppler broadening

on the dispersive properties of the medium. It is found that the combined

effect of Kerr non-linearity and Doppler broadening influence the dispersive

properties of the atomic medium more sharply as compared to separate effect

of Kerr non-linearity or Doppler broadening. The combined effect of Kerr

non-linearity and Doppler broadening on light pulse propagation then leads

to a more slow group velocity through the medium. Further, we included the

SGC and Kerr non-linearity in four-level atomic system and study the light

pulse propagation through the medium. A very steep dispersion is achieved

via the combined effect of SGC and Kerr non-linearity. A steep dispersion

then leads to more slow group velocity through the medium. Next, we ex-

tended our studies to the propagation of light pulse propagation to four level

iii

tripod atomic medium via two Kerr nonlinear fields. We expect That a very

slow group velocity can be achieved, which in turns leads to stop or halt the

light pulse through the medium.

iv

Acknowledgements

I acknowledge the support of all those people who helped me directly or in-

directly in the completion of this dissertation. I am grateful to my supervisors

Dr. Iftikhar Ahmad and Dr. Ziauddin (CIIT, Islamabad) for their kind su-

pervision, discipline, guidance and encouragement during my research work.

I am very thankful to Dr. Joseph. H. Eberly for hosting me at the university

of Rochester, USA, and providing me the opportunity to learn from his expe-

riences. I am also thankful to all the faculty members of the Department of

Physics, graduate students of the Center for Computational Materials Science

and Joseph. H. Eberly group especially Mr. Saifullah, Dr. Shafiq Ahmad,

Dr. Imad, Dr. Zahid, Mr. Bilal, Mr. Sheraz, Mr. Banaras Khan, Mr.

Fawad Khan, Mr. Rashid Iqbal, Mr. Sajid Khan, Mr. Amin Khan, Mr. Gul

Rehman and Mr. Philippe Lewalle for facilitation and providing a friendly

working environment. I would like to say the words of thanks to my friends

Jan Muhammad and Shakir Khan for their cooperation in my research work.

I gratefully acknowledge the financial support of the IRSIP section of the

Higher Education Commission of Pakistan for my visit to the University of

Rochester. Most importantly, my deepest love and gratitude will go to my

parents and siblings for their love, patience and support throughout my stud-

ies. Special thanks to my wife for her moral support during the process of

the completion of my Ph. D. studies and love to my kids Sara Khan, Yaman

Khan and Abyan Khan.

Hazrat Ali

v

List of publications

This thesis consist of the following published papers

1. Hazrat Ali , Ziauddin, Iftikhar Ahmad “Control of wave propagation

and effect of Kerr nonlinearity on group index,” Commun. Theor. Phys. 60,

87-92 (2013).

2. Hazrat Ali , Ziauddin and Iftikhar Ahmad “The effect of Kerr non-

linearity and Doppler broadening on slow light propagation,” Laser Phys.

24, 025201(1-5) (2014).

3. Hazrat Ali , Iftikhar Ahmad and Ziauddin, “Control of Group Veloc-

ity via Spontaneous Generated Coherence and Kerr Nonlinearity,” Commun.

Theor. Phys. 62, 410-416 (2014).

Other publications

4. Bakht Amin Bacha, Iftikhar Ahmad, Arif Ullah and Hazrat Ali ,

“Superluminal propagation in a poly-chromatically driven gain assisted four-

level N-type atomic system,” Phys. Scr.. 88, 045402(1-7) (2013).

5. J. H. Eberly, Xiao-Feng Qian, Asma Al Qasimi, Hazrat Ali , M. A.

Alonso, R. Gutirrez-Cuevas, B. J. Little, J. C. Howell, Tanya Malhotra and

A N Vamivakas, “Quantum and classical opticsemerging links,” Phys. Scr..

91, 063003(1-9) (2016).

vi

vii

Contents

1 Introduction 1

2 Literature Review 7

2.1 Origin of Slow Light . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Slow Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Experimental Realization of Slow Light . . . . . . . . . . . . . 12

2.4.1 Coherent Population Trapping . . . . . . . . . . . . . . 12

2.4.2 Coherent Population Oscillations . . . . . . . . . . . . 12

2.4.3 Stimulated Brillouin Scattering . . . . . . . . . . . . . 13

2.4.4 Stopped Light . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 Electromagnetically Induced Transparency . . . . . . . . . . . 15

2.6 Nonlinear Optics . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6.1 Kerr Effect and EIT Kerr Non-linearity . . . . . . . . . 19

2.7 Doppler Broadening . . . . . . . . . . . . . . . . . . . . . . . . 22

2.8 Spontaneous Generated Coherence . . . . . . . . . . . . . . . 24

3 Calculation Details 27

3.1 Density Matrix Formalism and Rotating Wave Approximation 28

3.2 Liouville’s Equation . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Hamiltonian in the Rotating Wave Approximation . . . . . . . 32

viii

4 Results and Discussion 39

4.1 Control of Wave Propagation and Effect of Kerr Nonlinearity

on Group Index . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.1.3 Presentation of the results . . . . . . . . . . . . . . . . 44

4.2 Effect of Kerr Nonlinearity and Doppler Broadening on Slow

Light Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2.3 Results presentation . . . . . . . . . . . . . . . . . . . 58

4.3 Control of Group Velocity via Spontaneous Generated Coher-

ence and Kerr Nonlinearity . . . . . . . . . . . . . . . . . . . . 64

4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 65


4.4 Control of Group Velocity via Double Kerr Nonlinearity in

Four-Level Tripod Atomic System . . . . . . . . . . . . . . . . 79

4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 80


4.4.4 Control of Group Velocity via a Single Kerr Field . . . 83

4.4.5 Control of Group Velocity via Double Kerr Fields . . . 85

5 Conclusions 92

Appendices 96

ix

List of Figures

2.1 (Color online) presentation of EIT in λ type system . . . . . 16

2.2 diagram shows the intensity dependent refractive index (a)

strong laser beam affects its own propagation (b) strong laser

beam affects the propagation of weak pulse. . . . . . . . . . . 18

2.3 The energy eigen level which describing the existence of Kerr

nonlinearities in (a) N-type system, (b) Tripod atomic system

(c) and M-type system . . . . . . . . . . . . . . . . . . . . . . 20

3.1 The interaction of the off resonant probe field with two-level

atom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.1 (Color online) Schematics of the atom-field interaction. . . . . 42

4.2 (Color online) Plots of (a) real (solid) and imaginary (dashed)

parts of the susceptibility χ(k) (b) group index n(k)g versus

probe field detuning ∆p for Ωk = 0 and Γ = 0.002γ. The

inset in Fig. (b) is the group index ranging from -0.1γ to

0.1γ. The corresponding parameters are Ω1 = 2γ, γ3 = γ,

γ = 1MHz, ∆1 = 0, γ1 = 0.1γ, γ2 = 0.1γ and νp = 1000γ. . . 46

x



probe field detuning ∆p for Ωk = 0 and Γ = 2γ. The other

parameters are the same as in Fig. 4.2. . . . . . . . . . . . . . 47


parts of the susceptibility χ(k) for normal dispersion and (b)

group index n(k)g versus probe field detuning ∆p when Ωk = 1γ

and Γ = 2γ. The inset in (b) is the group index ranging from

−0.1γ to 0.1γ. The other parameters are the same as in Fig.

4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48


parts of the susceptibility χ(k) for anomalous dispersion and

(b) group index n(k)g versus probe field detuning ∆p when Ωk =

1γ and Γ = 2γ. The other parameters are the same as in Fig.

4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.6 (Color online) Plots of real parts of the susceptibility χ(k) for

(a) normal dispersion when Γ = 0.002γ and (b) for anomalous

dispersion versus probe field detuning ∆p when Γ = 2γ. The

other parameters are the same as in Fig. 4.2. . . . . . . . . . . 50

4.7 (Color online) Plots of group index n(k)g versus Kerr field Ωk/γ

(a) for normal dispersion when Γ = 0.002γ and ∆p = 0 (b) for

anomalous dispersion when Γ = 2γ and ∆p = 0, the remaining

parameters are the same as in Fig. 4.2. . . . . . . . . . . . . . 51

xi

4.8 (Color online) (a) Schematic of the atom-field interaction, we

choose the ground state hyperfine levels 5S1/2, F = 2, m =

2;F = 2, m = 0;F = 1, m = 0 of 87Rb atom for |a〉, |c〉and |d〉, 5P3/2, F

/ = 2, m = 1 for |b〉, respectively. (b) A

block diagram where the probe, control and driving fields are

propagating inside the medium. . . . . . . . . . . . . . . . . . 55

4.9 (Color online) Plots of real parts of susceptibilities (χ(0), χ(k),

χ(b) and χ(kb)) versus probe field detuning. The parameters

are γ = 1MHz, γ1 = γ,γ2 = 0.1γ, Γ1 = 0.002γ, Γ2 = 0.2γ,

∆1 = ∆2 = 0, Ω1 = 2γ, Ω2 = 3γ, D = 10kHz and β = γ. . . . 60

4.10 (Color online) Plots of group indexes (n(0)g , n

(k)g , n

(b)g and n

(kb)g )

versus probe field detuning ∆p, the parameters remains the

same as in Fig. 4.9. . . . . . . . . . . . . . . . . . . . . . . . . 61

4.11 (Color online) Plots of group index n(k)g and n

(kb)g versus Kerr

field Ω2 at ∆p = 0, (a) without Doppler broadening effect

(b) with Doppler broadening effect, the remaining parameters

remains the same as in Fig. 4.9 . . . . . . . . . . . . . . . . . 63

4.12 (Color online) (a) Schematics of the four-level N -type rubid-

ium atomic system (b) dipole moments of driving and probe

fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66


parts of the susceptibility χ(0) (b) group index n(0)g versus

probe field detuning ∆p for q = 0, γ = 1MHz, γ1 = γ2 =

γ3 = 1γ, Γ = 0.002γ,Ω01 = 4γ, and νp = 1000γ. . . . . . . . . . 72

xii


parts of the susceptibility χ(0) (b) group index n(0)g versus

probe field detuning ∆p for q = 0.99, the remaining parame-

ters remains the same as in Fig. 4.13. . . . . . . . . . . . . . . 73



probe field detuning ∆p for q = 0 and Ωk = 2γ, the remaining

parameters remains the same as in Fig. 4.13. . . . . . . . . . . 74



probe field detuning ∆p for q = 0.99 and Ωk = 2γ, the re-

maining parameters remains the same as in Fig. 4.13. . . . . . 75

4.17 (Color online) Plots of (a) group index versus Kerr field for

q = 0 and ∆p = 0 (b) group index versus Kerr field for ∆p = 0

and q = 0.99, the remaining parameters remains the same as

in Fig. 4.13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.18 (Color online) Schematics of theD1 line in the rubidium (87Rb)

four-level tripod atomic system . . . . . . . . . . . . . . . . . 80


parts of the susceptibility χ(k1) (b) group index n(k1)g versus

probe field detuning ∆p for Ωk2 = 1γ, Ωk1 = 0.5γ, γ41 = 0.1γ,

γ = 1MHz, ∆k1 = ∆k2 = 0, γ31 = 1γ, γ21 = 1γ and νp = 1000γ 84


parts of the susceptibility χ(k1) (b) group index n(k1)g versus

probe field detuning ∆p for Ωk1 = 1γ, the remaining parame-

ters remains the same as in Fig. 4.19 . . . . . . . . . . . . . . 86

xiii


parts of the susceptibility χ(k1k2) (b) group index n(k1k2)g versus

probe field detuning ∆p for Ωk1 = Ωk2 = 0, the remaining



parts of the susceptibility χ(k1k2) (b) group index n(k1k2)g versus

probe field detuning ∆p for Ωk1 = Ωk2 = 1γ, the remaining


4.23 (Color online) Plots of group index n(k1)g versus (a) single Kerr

field when Ωk2 = 1γ and∆p = 0 (b) and n(k1k2)g versus double

Kerr fields, the remaining parameters are the same as in Fig.

4.19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

xiv

Chapter 1

Introduction

Light-matter interaction has a long history and has played a key role in the

discovery of many physical processes such as reflection, refraction, absorb-

tion, scattering and dispersion. The subject, quantum mechanics explains

the structure and energy level of atoms and other microscopic particles in-

cluding sub atomic particles [1]. The physical processes such as black body

radiation, photoelectric effect and Compton effect can be explained using the

concept of quantized nature of light. The atom absorbs or emits one quan-

tum of energy (~ν), when electrons make a transition between energy eigen

states of the atom. The quantized theory of light is useful to study spectro-

scopic nature of various atoms. Besides, fully quantum mechanical theory

of light and matter interaction, semi- classical theory can be used to explain

many fascinating physical phenomena. In semi-classical theory, the matter

(atom) is treated as a quantized entity while the light is treated as a wave

(continuous entity) according to Maxwell theory of electromagnetism. Using

a semi-classical approach, one can obtain the refractive index of a medium as

a function of optical frequency and easily understand the principles behind

the optical nonlinearity [2].

A Laser is an intense, monochromatic, and coherent source of light, which

1

has contributed a lot to the fields of quantum optics and nonlinear optics.

Some of the exciting applications of single mode laser-matter(two-level atom)

interaction are Autler-Townes splitting, self-induced transparency, resonance

florescence and Ramsey fringes [3]. The interaction of an intense laser beam

with matter gives rise to some of the interesting nonlinear effects such as the

Kerr effect, parametric down conversion, Raman effect and higher harmonic

generation [2]. This kind of interaction has provided an opportunity to build

a high resolution spectroscopic detector, which is helpful for studying vari-

ous spectroscopic properties of light such as laser-induced florescence spec-

troscopy, time-resolved laser spectroscopy and saturation absorption spec-

troscopy [4]

In atom-field interaction, the optical properties of a quantum system can

be effectively controlled and modified by atomic coherence. The coherent in-

teraction of laser field with the atom and the interference due to spontaneous

emission leads to coherence in the atomic system. It has some of the remark-

able applications, which includes electromagnetically induced transparency

(EIT) and lasing without inversion which are predicted earlier in cascade con-

figuration [5, 6]. Zhu and two another have shown lasing without inversion in

V-type atomic system [7]. Lasing without inversion has been predicted and

observed in V-type atomic system, when spontaneous emission in the sys-

tem is suppressed [8]. Beside from electromagnetically induced transparency

(EIT) and lasing without inversion, another prominent and key application of

atomic coherence is the variation that can occur in dispersion of the medium,

which leads to subluminal or superluminal pulse propagation.

Slow light refers to the propagation of a light signal in a medium with

reduced group velocity. The velocity of light v = cn

inside the medium

depends on the refractive index of the medium. Control over the group index

2

leads to two important properties associated with the propagation of light

signals: slow and fast light. High positive group index means a delayed pulse,

which corresponds to slow light propagation inside the medium; negative

group index corresponds to fast light propagation in the medium.

High positive group index can be obtained near the sharp transmission at

the probe field frequency. The field which is associated with the propagation

of weak laser light under observation is known as probe field. Boyed [2]

showed that nonlinear effects can create a sharp transmission through the

atomic medium and also can be used to manufacture special photonic devices

[9]. Control over light pulses plays a central role in practical applications to

the communication systems [10]. Ultra-slow light pulses have been created

by several groups in different atomic media such as vapors [11, 12, 13, 14, 15,

16, 17, 18, 19], photonic crystals [9], solid materials [20, 21, 22, 23, 24, 7, 25],

optical fibers [26] and liquid crystal [27]. Sensing [28], interferometry [29, 30]

and optical communication [10] also require slow light pulse propagation.

A high potential device based on a slow light pulse requires large delay

bandwidth product, low absorption loss and small distortion [2, 31, 32]. Each

property has its own limitation, which vary from application to application.

A large dispersion is required for interferometer based on slow light while a

large delay bandwidth product rather than absolute delay is required for an

optical buffer. Fast tunability of the group velocity is important to increase

the quality of optical communications.

The nonlinear interaction of the optical field with matter modifies the

optical properties of a medium. Kerr non-linearity is an effect by which

the refractive index of the medium changes as a function of light intensity.

Phase conjugate mirror effects [33], four wave mixing [34, 35, 36, 37], Ra-

man scattering [38], number squeezing of the field [39], generating single

3

[40, 41] and correlated [42, 43, 44] photon are examples of EIT nonlinear

effects. All of these effects are prominent in situations with light having

high intensity. The nonlinear optics have found increased applications since

the invention of laser. It has also been noticed that nonlinearity plays a

key role in various optical processes such as quantum information science,

remote sensing, spectroscopy and optical communication. The importance

of Kerr nonlinearity was reinforced after its experimental demonstration by

Schmidt and Imamoglo in 1996 [45]. Both the refractive and absorptive

kerr nonlinearities are predicted and demonstrated in continuous wave (CW)

condition [46, 47, 48, 49, 50] while only absorptive Kerr nonlinearity has

also been shown [51, 52, 53]. Giant Kerr nonlinearity has also been sug-

gested and observed in M-type [54, 55] as well as in tripod [56, 57] atomic

system. The application of Kerr nonlinearity can be broadened to include

most optical phenomenon such as Entanglement [58, 59], quantum computing

[60, 61, 62, 63, 64], quantum gates [65, 66, 67] etc.

Doppler broadening effect is prominent when we are dealing with atoms in

thermal motion. The apparent change between the frequency of the optical

probe field and atomic transition frequency can bring a significant change in

the susceptibility of a medium. It is believed that the Doppler broadening

effect will modify the dispersion and absorption properties of the medium.

The enhancement in the positive group index delays the light pulse in the

medium and hence a slow light pulse can be achieved. Slowing down of light

pulse has noticed in Doppler broadening medium [68]. A larger gain has been

reported in a four level N-type Doppler broadening medium [69]. Switching

between sublumninal and superluminal light propagation has been reported

by Agarwal et al. [70].

Spontaneous emission decreases coherence in the system [71], however

4

spontaneous generated coherence (SGC) effect can produce an additional

coherence in the system. SGC occurs in degenerate or nearly degenerate level

systems, where the interference between the spontaneous emissions channels

is from the excited level to two closely spaced ground levels or from two closely

excited level to the ground levels. This gives rise to an additional coherence

in the system. SGC arises due to the interaction of the closely spaced level

with vacuum fluctuation and has diverse effects on the dynamics of a system.

The condition necessary for SGC are (1) closely spaced structure, and (2)

no dipole orthogonal dipole matrix elements. Using the SGC effect, probe

absorption can be changed into probe gain [72]. The enhancement of Kerr

nonlinearity via spontaneous generated coherence (SGC) was first reported

in 2006 [73].

Desirable manipulation over optical properties of a medium is important

as well as necessary for various practical applications in the optical world.

For instance a high speed modulator requires us to control the refractive

index of certain materials with an optical field. The focus of the present

work is to exploit various optical properties by individual as well as combine

application of Kerr nonlinear field, Doppler broadening and SGC effect. This

control could ultimately lead us to achieve slow light, guided and stopped

light.

The present thesis contains five chapters. The running chapter introduces

the problem of slow light pulses and their practical applications in the optical

world. The second chapter addresses some basic review of literature, which is

helpful in describing our problem of slow light. In chapter third, we describe

how Liouville’s equation and rotating wave approximation for the atomic

system can be derived. The Liouville’s equation is in turn used to find out the

susceptibility of the medium. The dispersion and absorption can be obtained

5

from the susceptibility. The susceptibility is related to the group index,

which is in turn used to find the group velocity of light. The calculations

of susceptibility in Doppler broadened media and SGC are included in this

chapter. The results and discussion of this study are presented in chapter

four. Finally conclusion of the thesis is presented in chapter five.

6

Chapter 2

Literature Review

2.1 Origin of Slow Light

A wave packet comprised of different frequency dependent parts, which may

travel with mutually distinct velocities. Brillouin inspected the periodic

group of compound harmonic oscillators and obtained the displacement for

each and every oscillator in the form of propagating waves [74]. Later on, it

was found that such solution is quite general i .e., individual waves of wave

packet do not generally propagate with the same speed, leads to the idea of

a dispersion relation. The group velocity of light pulse propagation differs

from the phase velocity. It may be greater, equal or smaller, or even its direc-

tion of propagation can be opposite to the phase velocity. It is obvious that

slow group velocity means slowed light propagation, however the information

travels at group velocity.

To inspect how the wave packet propagate slowly, we consider the addition

of two plane waves of the same amplitude and slightly different frequency ω

and wave vector k as

y(x, t) = Re[ei((k+∆k)x−(ω+∆ω)t) + ei((k−∆k)x−(ω−∆ω)t)] (2.1)

7

y(x, t) = 2cos(kx− ωt)cos(∆kx−∆ωt) (2.2)

The above equation is the product of two co-sinusoidal waves. The first

one describes the carrier wave with frequency ω and velocity vk

while the

other modulated one is the envelope wave having frequency ∆ω and velocity

∆v∆k

. The two waves propagate with different velocities. The envelope can

propagate slower than, faster than, or even opposite direction to the carrier

waves. The wave which has a collection of different frequencies instead of a

monochromatic wave and whose frequencies are closely packed around the

central frequency, will propagate with group velocity

vg =dω

dk. (2.3)

The wave number k in terms of the refractive index

k(ω) =n(ω)ω

c, (2.4)

where ω is the angular frequency and c is the speed of light in vacuum. The

relationship between group index and group velocity can be expressed as.

1

vg=dk(ω)

dω, (2.5)

1

vg=n(ω) + ω dn(ω)

dω

c, (2.6)

1

vg=ng

c. (2.7)

The above equations are used to study the group velocity of light relative

the speed of light inside the medium. It is clear from Eq. (2.6) that a large

group index may be obtained by either finding a material having greater

phase index n(ω), or large derivative of the phase index over the optical

frequency. The term dn(ω)dω

is know as the dispersion and contributes more to

slow propagation of light inside the medium. A steep slope of the dispersion

8

profile can be obtained either by a large change in the refractive index dn or

by a very small range of frequencies through which the changes occur.

2.2 Dispersion

The interaction of light pulse with material medium exhibits some of the in-

teresting optical phenomena such as dispersion, absorption etc. The process

by means of which the phase velocity undergoes such changes with respect

to its frequency is known as dispersion. The medium exhibiting this prop-

erty is known as a dispersive medium. If the refractive index of a material

depends on the frequency of interacting light, then the process is known as

material dispersion. The spreading of light through a prism, rainbow and

chromatic abberation in lenses, are famous examples of the material disper-

sion. In waveguide dispersion, the phase velocity of the wave changes with

frequency of light due to structural geometry. The wave guide dispersion can

be found in optical fibres as well as in photonic crystal. The various optical

phenomena become complex (having real and imaginary part), when there

is an optical loss or gain in the medium. Susceptibility, index of refraction

and propagation constant are the examples of complex quantities. The real

part of the susceptibility gives the dispersion profile of a medium, while the

imaginary part of the susceptibility describes absorption properties of the

medium.

Light interacting with a dispersive medium experiences two type of dis-

persions i.e., normal and anomalous dispersion. The phenomenon in which

shorter wavelength pulse moves slower than the longer wavelength pulse is

called normal dispersion, while on the other hand if the shorter wavelength

pulse arrives earlier than the longer wavelength, then the dispersion is known

as anomalous dispersion. The expression for the group index from Eqs. (2.6)

9

and (2.7) can be written as

ng = n(ω0) + ωdn(ω)

dω. (2.8)

The term n(ω0) is known as the phase index of the medium at resonance

frequency ω0, while the term dn(ω)dω

describes the optical dispersion of the

medium. When dn(ω)dω

= 0, it means that the group index of the medium is

independent of the optical frequency and remains constant with increasing or

decreasing frequency. In this case the medium has no dispersion. dn(ω)dω

> 0 is

that spectral region, where the refractive index increases with the change in

the optical frequency and is known as normal dispersion. Normal dispersion

usually occurs in transparent materials (glass, water) for the visible spectrum

as well as nearly infrared and ultraviolet spectrum. Anomalous dispersion

occurs in that spectral region, where refractive index of the medium decreases

with respect to the change in optical frequency i.e., dn(ω)dω

< 0. Generally this

kind of dispersion can be noticed in the medium which is not too opaque at

resonance frequency. For example, a prism doped with certain dyes can be

used to observe the anomalous dispersion.

2.3 Slow Light

We have discussed that how the optical medium can slow down a light pulse

propagation through a medium. Slow light has been demonstrated exper-

imentally in Bose-Einstein condensates [75] as well as in a ruby crystal at

room temperature [21]. There are various techniques used to obtain slow light

pulse in the medium. These techniques are divided in to two main categories

i.e, microscopic and macroscopic division. The increase in the group index

due to the interaction of light pulse at the atomic or molecular level can be

categorized as microscopic slow light. The group index ng is quite different

10

than refractive index n because the extra term ω dndω

contains in the group

index which brings an appreciable change in the medium and causes slow

propagation of pulses in the medium i.e., ω dndω> 0 −→ vg < c . Microscopic

slow light technique can be used in various high-teach potential applications

such as coherent population oscillations (CPO) [75, 10, 22, 76], stimulated

Raman scattering (SRS) [77], parametric amplification [78, 79], stimulated

Brillouin scattering (SBS) [26, 80] and spectral hole burning [81].

The interaction of a light pulse with structural geometry of matter leads

to change in the effective group velocity of light. However, the wavelength

of the light pulse used must be comparable to or greater than the size of the

element of matter. The light obtained in this process is known as macroscopic

slow light. Group delay could be a best choice to describe the propagation

of light pulse in the whole element or through one period of the periodic

structure. The period can be expressed as

tg =dϕ(ω)

dω. (2.9)

The term ϕ(ω) is known as phase, arises in the transfer function H(ω) =

A(ω)eiϕ. To understand macroscopic nature of slow light completely, we need

to know the difference between effective group index and effective refractive

index of the homogeneous medium.

nr,ef(ω) =ϕ(ω)c

ωL, (2.10)

ng,ef = nr,ef + ωdnr,efc

dω, (2.11)

here c is speed of light in vacuum, L is the length or period of the medium.

The macroscopic nature of slow light has a wide range of applications,

some of them are optical ring resonators [82, 83, 84], photonic band gap

structures [85, 86, 87, 88, 9, 89] and fibre grating structures [90].

11

2.4 Experimental Realization of Slow Light

This section describe various experimental techniques to achieve slow light

pulse propagation, how to slow down or stop and preserve light pulse in the

medium.

2.4.1 Coherent Population Trapping

A special technique, used to create transparency window in three level medium

is known is coherent population trapping (CPT) [91]. The basic idea of CPT

is to prepare an atom in the superposition states. Beside Stark or Zeeman

sub-levels, non allowed Raman transition states could be best choice to push

it in to superposition states [92]. The difference between the probe and cou-

pling laser is chosen the same to the frequency of the Raman transition of

the levels. It was observed by Alzetta et al. in 1976 [93]. A single multi

mode laser can also be used to produce superposition. The width of the

transparency window is independent of radiative transition while it can be

controlled by setting the de phasing rate between the ground states. CPT

technique have been used to carry out an initial experiment for slow light

[16, 11, 12].

2.4.2 Coherent Population Oscillations

Coherent population oscillations (CPO) is another useful mechanism, which

can be used to produced a transparency window in the medium. In this

process, a modulated coupling laser is used to excite atoms to the metastable

state at a given frequency. The modulated frequency of the probe laser is

made slightly different than the modulated frequency of the coupling laser.

If the delay between the modulated probe and coupling laser is small enough

12

through the medium, then it is possible that the medium can not absorb

photons from the probe field. It is due to fact that most of atoms are in the

metastable state. The atoms in the ground state oscillate between the probe

and coupling laser as is obvious from the name CPO. A high-intensity laser

field can be used to produce both probe and coupling light [21]. Schweinsberg

et al. [76] observed slow light of 25000 m/s in Erbium-doped fiber using CPO

mechanism.

2.4.3 Stimulated Brillouin Scattering

Stimulated Brillouin scattering (SBC) is another special mechanism devel-

oped by Kwang et al. [94] to achieve subluminal pulse propagation in an

optical fiber. SBC arises due to the interaction between the pump and Stokes

waves. Acoustic wave can be produced, when the frequency difference of the

two counter propagating waves become equal to the Brillouin shift of the

medium. Thus photons are scattered by the acoustic wave from higher to

lower frequency wave. The Stokes wave is slowed down and interestingly

it is not absorbed by the medium. This mechanism is not so effective in

controlling the light pulse as compared to CPT and CPO. A similar tech-

nique, which is more promising and effective than SBC is stimulated Raman

scattering (SRS). In this process phonons (vibrational mode in material) are

involved instead of acoustic waves.

2.4.4 Stopped Light

The speed of light in vacuum is 300 thousand kilometers per second, the

fastest thing ever known in the universe, was slowed and even stopped com-

pletely for some time. When a probe pulse interacts with the medium is

slowed down, then small portion of the energy is left over with the probe

13

pulse and some portion of the energy is stored in the form of holographic

imprint and most of the remaining energy is added to the coupling or control

field. The probe pulse gets back the energy from the atom and the control

field, when it leaves from the medium. Thus the probe pulse increases the

long lived state of the atom and it can be easily retrieved from the atomic

medium. The initial research done by Liu et al.[17] and Phillips et al. [95]

has opened new ways to stop light and they also shown that a light pulse

can be stored in the medium for a period more than a second [96].

A high steep dispersion of the pulse in a medium leads to slow light

propagation. The dispersion is related to the group index i.e., ng = n+ω ∂n∂ω

,

where n and ω are the phase index and angular frequency, respectively. A

high dispersive medium corresponds to high group index, which in turn leads

to more slowly propagating pulse inside the medium i.e, vg = cng

. Various

attempt have been made by several groups in achieving the slow light in

vapors [11, 14, 15, 16, 17, 18, 19] and solids [20, 21, 22, 23, 24, 7, 25].

Rubidium (Rb) atom has the ability to maintain a longer ground state

coherence time than the pulse width which allows us to preserve the light

pulse and recover it in the future. Signal and spontaneously produced pulses

are stored in the rubidium by four wave mixing mechanism. The pulses

generated in this processes are called number squeezed. This mechanism is

very useful in storing two correlated fields. Spatial mode information of light

pulse has been stored in hot atomic vapor to avoid diffusion [97]. Stopped

and stored light pulses may also be used in the remote sensing, imaging and

information processing.

14

2.5 Electromagnetically Induced Transparency

The phenomenon by which an opaque medium can be made transparent

by electromagnetic field is called electromagnetically induced transparency

(EIT). Consider three level lambda system having |a〉, |b〉 and |c〉 states as

shown in the Fig. 2.1. The transition between |a〉 ←→ |b〉 and |a〉 ←→ |c〉are dipole allowed and are driven by the resonant probe field having Rabi

frequency Ωp and resonant control field of frequency Ωc, respectively. If the

difference between the optical frequencies of the probe field and control field

is equal to the difference of the ground state transition frequency then there

appears a dark state, which is the coherent superposition of the ground states.

i.e,

|d〉 =Ωp|b〉+ Ωc|c〉

Ω. (2.12)

Where Ω =√

Ω2p + Ω2

c . The destructive interference occurs between the

probability amplitude for the transition |a〉 ←→ |b〉 and |a〉 ←→ |c〉, so the

dark state is decoupled from the excited state. When the electron is trapped

in the dark state, it cannot be excited to the state |a〉 and hence the medium

becomes transparent to the optical field. This phenomenon was first observed

by Harris in 1990 [98] and commonly known as electromagnetically induced

transparency. Soon after its discovery, EIT effect has also been noticed in

lead [99] and strontium [100] vapors. The narrow EIT window leads to steep

dispersion, which delays the optical pulse inside the medium [5, 14, 15]. Hau

et al. [16] reduced the velocity of signal pulse to 17m/s in ultra cold atomic

gases. It has been noticed that various nonlinear effect can be enhanced

through EIT medium [101].

EIT effect can be seen when the probe field is taken much smaller than

the control field (Ωc >> Ωp) and often this assumption makes the analyt-

15

-4 -2 0 2 4

0.2

0.4

0.6

0.8

1.0

-4 -2 0 2 4

0.0

0.2

0.4

0.6

0.8

1.0

Wc=0

Wc=2g

Figure 2.1: (Color online) presentation of EIT in λ type system

16

ical calculations very easy. EIT is used to attain gain without inversion

[8, 102, 103, 104]. The EIT effect has been used for raising the frequency

standards [105] and also in spectroscopy [106, 107]. Dark state polaritons

is another prominent characteristic, which exists in the EIT medium [108].

The quantum optical field is converted into quasi particles, when propagating

through the EIT medium. The quantum state and pulse shape of the trapped

quasi particles can be transfered to the ground state coherence. Dark state

polaritons are trapped only in a medium by turning off the control field adi-

abatically, and when the control field is turned on, then one can get retrieve

them from the medium.

The steep dispersion profile corresponds to narrow EIT window, which is

clear from the well known Kramer-Kronig relations. This kind of dispersion

results the slow pulse propagation near the EIT transparency window. The

EIT effect has also been used by several groups in 1999 to achieve the slow

group velocity inside the atomic medium [11, 12, 16]

2.6 Nonlinear Optics

Non linear interaction of light with matter is one of the leading optical mech-

anisms, which brought revolution in the world of optics. Nonlinear optics de-

scribes the behavior of light in a nonlinear medium. The strength of optical

field varies nonlinearly to the dielectric polarization in nonlinear media. The

nonlinearity in the medium can only be seen, when subjecting the medium

to light having enough high intensity (round about 108v/m ). Lasers can

provide such high intensities and the first nonlinear phenomenon i.e, sec-

ond harmonic generation [109] was described shortly after the discovery of

laser. Second harmonic generation mechanism can be produced only when

the electric field of the light pulse varies quadratically in the medium.

17

Figure 2.2: diagram shows the intensity dependent refractive index (a) strong

laser beam affects its own propagation (b) strong laser beam affects the

propagation of weak pulse.

18

The polarization of the medium plays a vital role to explain the mecha-

nism behind the nonlinear optical process. The dielectric polarization varies

with strength of the electric field of the interacting light pulse with medium.

The linear polarization of the system can be expressed as

P = ε0χ1E. (2.13)

Where χ1 is the linear susceptibility and ε0 is known as permittivity of free

space. The polarization in the nonlinear optics can be described in terms of

electric field strength as

P = ε0χ1E1 + ε0χ

2E2 + ε0χ3E3. (2.14)

The terms χ2 and χ3 are described as 2nd and 3rd order nonlinear suscep-

tibilities, respectively. The second term in the above equation is termed as

the second order polarization and can be used to explore the modified op-

tical properties which arise due to the quadratic interaction of the optical

field with the medium. The third term is known as the third order nonlinear

polarization, which is responsible for the phenomenon involving third order

interaction of the electric field of light pulse with the medium.

2.6.1 Kerr Effect and EIT Kerr Non-linearity

Kerr effect is a nonlinear optical phenomenon, which arises due to the inter-

action of an intense light beam with the medium (crystal, liquid and gases).

The concept behind the Kerr effect is the nonlinear polarization response

of the medium to the strength of electric field of the optical pulse, which

modifies the properties of the propagating pulse through the medium. The

phenomenon in which the group index of medium changes with the intensity

19

(b)

(a)

Wc Wc1

WcWc2

Wc Wc1

|e

(c)

Wc1

Figure 2.3: The energy eigen level which describing the existence of Kerr

nonlinearities in (a) N-type system, (b) Tripod atomic system (c) and M-

type system

of the light pulse is also known as Kerr non-linearity. The refractive index of

certain material medium can be expressed as

n = n1 + n2 < E2 >, (2.15)

the term n1 is the refractive index of the weak probe field and n2 is known as

the second order refractive index of the medium. Typically one evaluates the

enhancement of the refractive index of the medium with increasing strength

of the intensity of the optical field. The bracket denotes a time average. The

electric field of the optical pulse can be written as

E(t) = E(ω)e−iωt + E(ω)∗eiωt, (2.16)

which leads to

< E2 >= 2E(ω)E(ω)∗ = 2|E(ω)|2, (2.17)

20

substituting Eqs. (2.17) in (2.15), the refractive index of medium takes the

form

n = n1 + 2n2|E(ω)|2, (2.18)

Eqs. (2.15) and (2.18) are usually used for Kerr effect. The interaction of

an intense light beam with medium can also lead to nonlinear polarization,

which can be expressed as

Pnl(ω) = 3ε0χ3|E(ω)|2E(ω), (2.19)

and the total polarization of the medium takes the form as

Ptot(ω) = ε0χ1E(ω) + 3ε0χ

3|E(ω)|2E(ω) = ε0χeffE(ω), (2.20)

where χeff is known as the effective susceptibility and can be expressed as

χeff = χ1 + 3χ3E(ω), (2.21)

the general expression of the refractive index in terms of effective suscepti-

bility is given as

n =√

1 + χeff . (2.22)

Substituting Eqs. (2.18) and (2.21) in Eq. (2.22) and then simplifying the

analytical expression, the linear and nonlinear refractive indices can be cal-

culated as

n1 =√

1 + χ1, (2.23)

and

n2 =3χ3

4n1

. (2.24)

These refractive indices are calculated using a single laser field. The discus-

sion can be extended to two (weak and strong) laser beams [2]. The nonlinear

refractive index can be calculated as

n2X =

3χ3

2n1. (2.25)

21

The term n2X is known as cross coupling effect and almost twice of that

obtain for single laser beam.

The EIT Kerr effect was initially explained and suggested in 1996 [45].

An additional energy level and control field was added to lambda type sys-

tem, which takes the shape similar to N, see figure 2.3(a). The additional

field produces the Stark shift, which disturbs the EIT medium. The Stark

shift along with EIT dispersion brings a prominent changes in the optical

properties of the medium which, leads to cross phase modulation (XPM) of

the light pulse. There are two types of EIT Kerr effects i.e., refractive and

absorptive Kerr effects, which are both demonstrated in continuous wave con-

dition [42, 46, 49]. Harris et al. [110] presented the absorptive Kerr medium

and they found that instead of one photon the medium absorbs two photons .

Enhancement in refractive EIT Kerr nonlinearity has been observed by Pack

et al. in 2007 [111]. Several other schemes have also been proposed for EIT

Kerr nonlinearity such as M-type system [54, 55] shown in Fig. 2.3(b) and

tripod system [56, 57] shown in Fig. 2.3(c).

2.7 Doppler Broadening

Doppler effect is the change in the wavelength or frequency of light or sound

waves due to the relative motion of source and observer. A famous example

is that of the moving vehicle having siren passes near the stationary listener.

The listener catches the apparent change in pitch of the siren, whether the

vehicle moves towards or away from the listener. The wavelength get elon-

gated, as the vehicle moves away from the listener, and the listener receives

a lower pitch and vice versa. A similar process is observed with light waves.

A shift from red to blue light is observed, when red light source is moving

towards the observer while a shift from blue to red is observed, when blue

22

light source is moving away from the observer. The astronomer uses the

concept of Doppler shift to find out the velocity of stars and galaxies.

The study of Doppler effect can also be extended to the moving atoms

relative to the light pulse. The atoms in thermal motion experiences a small

variation in frequency, when the light is incident upon it. The shift in fre-

quency of light pulse not only modifies the dispersion and absorption proper-

ties of the medium but also brings a significant changes in the group index of

the medium. The Doppler broadening effect for the spectral line interacting

with thermal media was predicted and discovered in 1932 [112]. The effect

was calculated using conservation laws (energy and momentum) during the

emission of radiation. Dicke [113] used the broadening phenomenon, and

observed the narrowed spectral line of the thermal atom. It has been de-

scribed that how translational motion of the atoms changes due to the recoil

momentum of the radiation, which leads to the broadened phenomenon.

The Doppler broadening effect has been used in two level system to slow

down the light pulse propagating through it [68]. It was shown that the group

index of the system can be obtained of the order of 103. Fan et al. [69] have

studied the optical properties of the medium in four level N-type Doppler

broadening medium. They found that by changing the Doppler width, one

can bring a significant change in the absorption (gain) and dispersion prop-

erties of the medium. The coherent spectral hole burning in the absorption

line can be achieved in the Doppler broadened medium, when two laser field

(one is co-propagating and the other is counter-propagating) are applied si-

multaneously to the Λ type system [114]. This mechanism is used to obtain

the small group velocity of light pulse in the atomic vapors. Spatial dis-

persion due to Doppler broadening effect is used to halt and stop the light

pulse inside the medium [115]. The transmission coefficient of the light pulse

23

can be increased from 40 to 90 percent in inhomogeneous broadened media

[116]. The experiment [117] shows that broadening effect can be used to

obtain the ultra narrow EIT width for the helium atomic medium. Another

experiment carried out by Camacho et al. [13] that describes the broadening

and delay features of the propagating light pulses through the Lorentzian

medium. They showed that dispersive Doppler broadening effect dominates

the line shape through the double Lorentzian system and the shape of the

pulses remains undistorted for a given time delay. The results are presented

for various effects related to the motion of the atom in λ system [118] i.e.,

Doppler, Dicke and Ramsey effects. Doppler-Dicke effect for one and two

photons absorption is described. The Doppler effect for single photon in

buffer gas is broadened while the Dike narrowing effect is observed in the

two photon line.

2.8 Spontaneous Generated Coherence

It is generally believed that spontaneous emission limits the coherence process

[71], but the suitable use of spontaneous emission in the system leads to an

additional coherence and can enhance many optical properties of the medium.

Spontaneously generated coherence (SGC) is one of the proper mechanisms,

which is used to increase the coherence in the system. SGC can only be

active in that system, where the two degenerate eigen states are coupled to

a common exited or common ground eigen state [119, 120]. The SGC is

effective in the case, where two dipole moment arise due to the interaction

of optical field with the atomic system may not be orthogonal. Menon et al.

[121] studied the pump and probe field response in Λ type atomic system and

found that the system maintains both the CPT and EIT phenomena even in

the presence of SGC. However, the SGC not only brings a significant changes

24

in the time scale associated with CPT sate but also affects absorbtion and

dispersion lines of the system.

The light pulse propagation is studied in Λ-type atomic system with de-

generate lower eigen states [122]. The system was driven into squeezed vac-

uum (SV) and coherent field. It was found that the small absorption or gain

can be produced in the presence of both the SV and SGC. The subluminal

or superluminal pulse obtained in this process has much smaller distortion.

The careful analysis of the propagation of weak probe pulse has been carried

out in V-type system in the presence of SGC and incoherent field [123]. The

incoherent pump field increases the group velocity of the light pulse, while

the SGC effect acts as a knob for controlling the light pulse propagation from

subluminal to superluminal. It was also shown that the anomalous disper-

sion can be observed only, when the SGC is taken in to account. The relative

phase along with SGC effect has been studied in three level V-type atomic

system [124]. It was reported that the system is sensitive to the induced

interference effect, and that the group index of the medium can be changed

from positive to negative with increasing the strength of spontaneous quan-

tum interference. The effect of SGC on the absorption properties of rubidium

medium is demonstrated experimentally [125]. The rubidium medium con-

sists of four level (N-type and inverted Y-type) system. The broad and deep

transparency window in the system is reported. The relative phase in Y-type

system arises due to the coherence produced by SGC [72]. The relative phase

manipulates the absorption and dispersion properties of the medium. The

study of SGC in rubidium atoms are further extended to photon counting

statistics [126]. The rubidium atoms are trapped in coherent state, leads to

new transparency channel, which exists even if the strong probe field is used.

The ultra narrow peaks in the absorption profile is reported in the presence

25

of SGC.

26

Chapter 3

Calculation Details

The properties of weak prob field are described through out our studies. The

fields with low intensities are termed as weak fields. These kind of field, do

not change the optical properties of the medium under observation. The

atom absorbs photon from the probe beam and decays spontaneously back

to the original eigen state ang the emitted photon can be absorbed again.

However, if the atom decays spontaneously to some other off resonant eigen

state with probe beam, then the atom will not absorb any more photons. A

high intense beam excites sufficient number of atoms which decays to both

‘off’ and ‘on’ resonant eigen states. The absorption properties of the probe

beam in the medium are modified and hence probe beam will no longer be

considered weak. There are different analytical techniques used to explore

various properties of the medium but, here, we carry out our calculation

based on density matrix formalism.

27

|a

|b

Figure 3.1: The interaction of the off resonant probe field with two-level

atom.

3.1 Density Matrix Formalism and Rotating

Wave Approximation

Consider a single optical pulse propagating in x direction inside two level

medium as shown in the figure 3.1. The optical field in the carrier envelope

form can be written as.

E(x, t) = ε(x, t)ei(kx−ωt) + ε∗(x, t)e−i(kx−ωt), (3.1)

ε(x, t) is the field envelope function, ω is the angular frequency and k is

the wave number. The interaction between optical field is assumed to be off

resonant with the two-level medium, the optical angular frequency ω is tuned

close to the atomic transition frequency. The expansion of wave function in

terms of eigen basis |a〉 and |b〉 can be written as

|ψ(x, t)〉 = ca(t)|a〉+ cb(t)|b〉, (3.2)

where ca(t) and ca(t) are the amplitudes of time dependent probability of

eigen basis |a〉 and |b〉, respectively. The time evolution of the wave function

can be described by Shrodinger’s wave equation:

28

∂

∂t|ψ(x, t)〉 =

1

i~H|ψ(x, t)〉, (3.3)

where H is Hamiltonian and can be written for the two level system inter-

acting with optical field as:

H = ~ωa|a〉〈a|+ ~ωb|b〉〈b| − ~dab|a〉〈b| − ~dba|b〉〈a|, (3.4)

where, ~ωa and ~ωb are energies of the eigen states |a〉 and |b〉, respectively

and dab is the diploe matrix element, which results from the interaction of

the optical field with the system.

Single wave function cannot describe properly the real quantum mechan-

ical system and needs much general formulism to describe such a system.

There are several other physical phenomena such as phase changing collision

that exists in the ensemble of atoms, which can change the dipole moment

and leave the population of the atoms unaltered. To consider such physi-

cal phenomena into account, the density matrix can be the best choice to

describe the quantum mechanical system.

ρ =

ρaa ρab

ρba ρbb

, (3.5)

the column order of the density matrix in terms of energy eigen state is

|a〉 and |b〉. The density matrix depends on both space and time, but for

simplicity we use the notion i.e., ρab = ρab(x, t).

The density matrix of the pure state in terms of direct product of the

wave function can be written as

ρ = |ψ〉〈ψ|. (3.6)

After evaluating Eqs. (3.2) and (3.7), we obtain

29

ρ =

|ca|2 cac∗b

c∗acb |cb|2

. (3.7)

Here the diagonal elements ρaa = |ca|2 and ρbb = |cbb|2 describe the population

in the states |a〉 and |b〉, while the off-diagonal elements ρab = cac∗b and

ρba = c∗acb describe the coherence in the system. The scope of the density

matrix is wider as compared to the wave function. The wave function can

not describe mixed state but the density matrix approach can completely

analyze the behavior of the atom either in pure or mixed state. If there is no

coherence in the system i.e., ρab = cac∗b = 0 , then the density matrix could

be best approach to study the behavior of such systems.

ρ =

|ca|2 0

0 |cb|2

. (3.8)

The trace of the density matrix either for pure or mixed state is one i.e.,

ρaa + ρbb = 1.

3.2 Liouville’s Equation

Shrodinnger wave equation can only describe the evolution of a single pure

state and limited to ensemble of particles. A suitable approach is required

to examine the evolution of ensemble of particles. One of such approach is

Liouville’s equation, which gives the time evolution of density matrix. The

time evolution from initial state ψ(t1) to final state ψ(t2) can be expressed

as

|ψ(t2)〉 = U(t2, t1)|ψ(t1)〉 (3.9)

It is clear from the above equation that

30

U(t2 = t1, t1) = 1. (3.10)

The Shrodinger wave equation can be written as

i~ ˙|ψ〉 = H|ψ〉, (3.11)

putting Eq. (3.10) and its partial differentiation in Eq. (3.12), we get

i~[∂U(t2, t1)

∂t|ψ(t1)〉+U(t2, t1)

∂|ψ(t1)〉∂t

] = |ψ(t1)〉HU(t2, t1)+U(t2, t1)H|ψ(t1)〉,(3.12)

Comparing both sides of the equation, we have

∂U(t2, t1)

∂t=−i~U(t2, t1), (3.13)

The solution of the equation is

U(t2, t1) = e−i~

H(t2−t1), (3.14)

Let ρ be another function evolves as the state |ψ(t2)〉 and can be expressed

as

〈ψ(t2)|ρ|ψ(t2)〉 = 〈ψ(t1)|U †(t2, t1)ρU(t2, t1)|ψ(t1)〉, (3.15)

= 〈ψ(t1)|ρ0|ψ(t1)〉, (3.16)

=⇒ ρ0 = U †(t2, t1)ρU(t2, t1), (3.17)

after using the unitary operator, we have

ρ = U(t2, t1)ρ0U†(t2, t1), (3.18)

taking differential of both sides with respect to t, we obtain

31

∂ρ

∂t= U(t2, t1)ρ0

∂U †(t2, t1)

∂t+ U(t2, t1)

∂ρ0

∂tU †(t2, t1) +

∂U(t2, t1)

∂tρ0U

†(t2, t1),

(3.19)

∂ρ

∂t=

1

i~[Hρ− ρH] +

∂ρ

∂t, (3.20)

∂ρ

∂t=

1

i~[H, ρ] +

∂ρ

∂t. (3.21)

=⇒ ρ =1

i~[H, ρ] +

∂ρ

∂t, (3.22)

here,

∂ρ

∂t= −Γρ, (3.23)

the compact form of Liouville’s equation can be expressed as

ρ =1

i~[H, ρ]− Γρ. (3.24)

This equation is also known as von Neumann equation.

3.3 Hamiltonian in the Rotating Wave Ap-

proximation

The approximation in which the rapid oscillating terms are ignored is known

as rotating wave approximation. We are interested to find out the analytical

approximate solution of Hamiltonian, where it has no any rapid oscillating

term. We need a rotating frame to figure out the term, which could be

replaced by its zero average value. The unitary matrix can be used in such

rotating frame and is given as

32

U = eiωt

1 0

0 e−i(kx−ωt)

, (3.25)

the transformation of density can be expressed as

ρrw = UρU †, (3.26)

the subscript “rw” represents rotating wave. The unitary transformation

leaves the von Neumann equation unchanged, therefor

ρrw =1

i~[H

rw, ρrw]. (3.27)

The rotating wave modifies the Hamiltonian and can be expanded as

Hrw = UHU † + i~(∂U

∂t)U †, (3.28)

here, H is laser atom Hamiltonian with out approximation and its matrix

form can be deduced from Eq. (3.4) as

H =

~ω1 −d.E−d∗.E∗

~ω2

, (3.29)

putting Eqs. (3.26) and (3.30) in Eq. (3.29), we obtain the following matrix

Hrw =

0 −(dab.E)ei(kx−ωt)

−(dba.E)e−i(kx−ωt)~(ω2 − ω1 − ω)

. (3.30)

The term Eei(kx−ωt) of the right off diagonal elements converts to εe2i(kx−ωt)+

ε∗ after inserting the value of electric field from Eq. (3.1) and similarly the

term of other diagonal element get equal to εe−2i(kx−ωt) + ε. The rapid oscil-

lating term i.e., ±2iωt can be ignored according to the rotating wave approx-

imation. Here, the envelope function is assumed to fluctuate slowly than the

33

carrier wave. The Hamiltonian of the system in the rotating approximation

can be expressed as

Hrw =

0 −dab.ε∗

−dab.ε ~∆p

. (3.31)

The term ∆p = ~(ω2 − ω1 − ω) is known as detuning. The matrix can also

be written as

Hrw =

0 −~

2Ω∗

p

−~

2Ωp ~∆p

, (3.32)

the term Ωp is known as Rabi frequency of the probe field and is given as

Ωp(x, t) =2dab.ε(x, t)

~. (3.33)

The Rabi oscillations has a key role in describing the system interacting

with light field. It contains the information about the dipole moment and

envelope function.

The von Neumann equation and Hamiltonian in rotating wave approxi-

mation can be extended to many levels system. To find various dynamics of

the system we need to expand the density matrix equation. The expansion

takes the form as

ρnm =1

i~[〈n|Hrwρ|m〉 − 〈n|ρHrw|m〉]− 〈n|Γρ|m〉. (3.34)

Here n = 1, 2, 3.... and m = 1, 2, 3... represent the energy eigen states and

Γ is the decay from state n to m. j2 rate equations can be found for j-

level system. There are four rate equations for two level system and 9 rate

equations for 3- level system and so on. The rate equations have exponential

time factors i.e., e±(∆p,c1,c2...)t and can be eliminated using some assumptions.

34

The assumptions ρ = ρe±(∆p,c1,c2...)t, can leave the rate equation with no

time factor and transform the equations to the form ˙ρ = Bρ. According

to the weak probe approximation, we can take first order in the probe field

while all orders in control field. The coupling equations are chosen from the

rate equation and using the initial conditions, where the atoms are prepared

initially in the state ρoii = 1 and the populations is assumed to be zero in all

other states i.e., ρojj = 0 and ρo

ij = 0. Then the coupling equation can take

the form as:

˙ρ = Aρ + C. (3.35)

The solution of the equation can be written as

G(t) =

∫ t

−∞

e−A(t−t0)Cdto (3.36)

G(t) = −A−1C. (3.37)

Here, G(t) is the column matrix having density matrix elements, A is the

square matrix and C is the column matrix having constants. The density

matrix element ρij can be calculated from G(t) and is related to the polar-

ization of the medium as

P = 2Nµijρij. (3.38)

N is the atomic number density and µij is the dipole matrix element between

eigen states |i〉 and |j〉. The polarization of the medium is also related to the

electric susceptibility and can be written as

P = ε0χεp, (3.39)

35

where, ε0 is the permittivity of free space, χ is the susceptibility of the

medium and εp is the amplitude of the oscillating envelope field. The sus-

ceptibility of the medium can be calculated using Eqs. (3.39) and (3.40).

χ =2Nµij

ε0εpρij, (3.40)

where

χ = χ1 + iχ2, (3.41)

with χ1 and χ2 the real and imaginary parts of susceptibility, respectively.

The real part of the susceptibility describes the dispersion properties of the

probe field while the imaginary part gives the absorption or gain profile of

the medium. These optical properties depend on the initial preparation of

the atom and also depend that where the probe field is employed between

the eigen states. The group index of the medium directly depends on the

susceptibility and can be expressed mathematically as

ng = 1 + 2πχ1 + 2πωp∂χ1

∂∆p

, (3.42)

where ωp and ∆p are the frequency and detuning of the probe field, respec-

tively. The group velocity of light pulse propagation in the medium can be

calculated from the group index as

vg =c

ng

, (3.43)

and,

vg =c

1 + 2πχ1 + 2πωp∂χ1

∂∆p

. (3.44)

Kerr field, Doppler broadening and SGC are other coherent effects that

can manipulate the optical properties of the medium. Kerr nonlinearity

36

modifies the electric susceptibility of the medium which in turns alter the

absorbtion, dispersion and group index of the medium. The polarization in

terms of optical Kerr field can be expanded in power series as

Ptot(ω) = ε0χ1E(ω) + 3ε0χ

3|E(ω)|2E(ω) = ε0χeffE(ω), (3.45)

so,

χeff = χ1 + 3ε0χ3|E(ω)|2, (3.46)

the first term of Eq. (3.46) is the susceptibility of the medium in the absence

of the Kerr field while the second term arises due to Kerr nonlinearity. The

corresponding total ground index of the medium can be expressed as

ntg = n0

g +K|E(ω)|2. (3.47)

Doppler effect is useful to describe ensemble of atoms in thermal motion.

The relative motion of the atoms and the optical field modifies the optical

susceptibility of the medium and the optical detuning can be replaced by

∆j = ∆j ± kjv in the equation for susceptibility. kj is the wave vector for jth

mode. The modified optical susceptibility can be written as

χb =1

D√

2π

∫ ∞

−∞

χ(kv)e−(kv)2

2D2 d(kv). (3.48)

Here, D is the doppler width and may be defined as υ0

c

√

2kBT/m, where as

kB, is Bortzmann constant, T is temperature and m is molecular mass.

SGC is another effect, that can generate an additional coherence in the

system. This effect is effective in a system, where two degenerate eigen states

are coupled to common ground or common exited eigen states and the effect

may be defined as

37

q =−→µij · −→µjk

|−→µij · −→µjk|= cos θ. (3.49)

Here, −→µij is the dipole moment between |i〉 and |j〉 while −→µjk is the dipole

moment between |j〉 and |k〉. It arises due to the interference between the

two decay channels. θ is the angle between two dipole moments. The two

dipole moments are parallel, then the SGC effect will be maximum and when

the two dipole moments are orthogonal to each other, then the SGC effect

is zero. The modified optical fields in the presence of SGC effect can be

expressed as

ΩK = Ω0K sin θ. (3.50)

ΩK = Ω0K

√

1− q2. (3.51)

Here, Ω0K is the optical field in the absence of SGC and q represents the

quantum interference parameter.

38

Chapter 4

Results and Discussion

4.1 Control of Wave Propagation and Effect

of Kerr Nonlinearity on Group Index

4.1.1 Introduction

In the recent years, atom-field interaction has attracted great attention due to

its practical application in the high-tech photonic devices. The group velocity

of light in a medium can exceeds or fall behind than the speed of light in

vacuum. It has been reported for the first time that the group velocity of

light inside the atomic medium can exceed the speed of light in vacuum [74].

Fast light or superluminal pulse propagation has been observed in diffrerent

classes of atomic media [70, 127, 128, 129, 130, 131, 132]. Aside from ultra fast

light propagation, slow light propagation in various material media has been

extensively studied with in the framework of the electromagnetic induced

transparency (EIT) effect [98].

The slow light pulse propagation has been suggested in rubidium atoms

with minimum absorption under EIT condition [5]. The slow or halted light

pulse has been reported experimentally in cold cloud of sodium atoms using

39

the effect of EIT [17]. In another experiment, it has been described that the

spatial dispersion due to Doppler broadening effect is used to halt and stop

the light pulse inside the medium via EIT [115]. Ultra slow light pulses have

been created experimentally by several groups in different atomic media such

as vapors and solid materials, and can be used to store the information inside

it [11, 16, 17, 21].

Further, the idea behind the fast light or superluminal pulse propagation

corresponds to the negative group delay, has attracted many people due its

fundamental nature. Already established fact is that to preserve causality

no pulses could travel faster then light speed [133]. Therefore, several groups

have paid attention to study the group velocity of the pulses having greater

speed then light [70, 127, 128, 129, 130, 131, 132]. It has been observed

that the group velocity could not carry information, hence causality does

not violated [134]. Negative group delay has wide range of applications and

could be used properly to design an electronic circuits. these circuits includes

negative group delay synthesizer [135] and radio frequency circuit design

[136, 137], which are based on the topology of the negative group delay.

In the year 2003, Kang et al. [49] studied Kerr nonlinearity in four-level

rubidium Rb atoms and minimum absorbtion was found in the medium.

A large cross modulation was investigated under the condition of EIT by

Chen et al. [138]. The slow light propagation via kerr nonlinearity has

been demonstrated in N-type atomic medium using EIT Effect by Dey and

Agarwal [139]. They used the idea of an earlier work [45], where an enhanced

Kerr nonlinearity has been noticed using the concept of an EIT. Enhanced

kerr nonlinearity has been reported experimentally using an ultra cold gas of

sodium atoms [16] in Bose-Einstein condensate.

The Kerr nonlinearity has been used by Sheng et al. [140] for entan-

40

glement purification protocol (EPP). In this protocol, they has been used

the concept of Kerr nonlinearity and constructed a quantum nondemolation

detector. Similarly, in 2009 a scheme [141] has been presented for quantum

nondemolition detectors (QNDs) based on cross-Kerr nonlinearities and the

scheme has independent of the controlled-not gates. Recently, some other

schemes [142, 143] has also used for constructing the quantum nondemola-

tion measurement based on Kerr nonlinearity and obtained the entanglement

purification. Besides, the role of Kerr nonlinearity in entanglement purifica-

tion, it has also been used for entanglement concentration [144, 145].

The Kerr nonlinearity not only play an important role in nonlinear optical

processes [2], but have many applications, these include for example, quan-

tum information processing, quantum nondemolation measurement, quan-

tum state teleportation and quantum logic gates.

In 2006, a very important behavior between EIT and Raman gain process

has been studied theoretically [146] and experimentally [147]. It has been ob-

served that the EIT is only applicable for low atomic densities, and whenever

atomic densities are increased Raman gain process became dominant.

In this section, we follow the same idea as has been observed earlier

[146, 147] and study the effect of forbidden decay rate in four-level atomic

medium when light is propagating through that medium. The behavior

of light propagation in atomic medium is changing from normal (slow) to

anomalous (fast) dispersion by increasing the forbidden decay rate via in-

creasing the atomic density. We also study the effect of Kerr field on the

group index in a four-level atomic medium for normal and anomalous disper-

sion. Our scheme is based on the extension of three-level EIT Λ configuration

[148].

41

Figure 4.1: (Color online) Schematics of the atom-field interaction.

4.1.2 Model

The schematics of the atom field interaction are presented in Fig. 4.1. We

assume four-level atomic system each having energy levels |a〉, |b〉, |c〉 and

|d〉. An intense driving laser field E1 is applied between level |a〉 and |b〉 with

corresponding Rabi frequency Ω1. Similarly, we also apply a weak probe field

Ep and a strong Kerr nonlinear field Ek between |b〉 → |c〉 and |c〉 → |d〉 with

corresponding Rabi frequency Ωp and Ωk, respectively.

To calculate the linear optical susceptibility corresponding to the probe

light Ep, we follow the same approach as in [148]. The interaction picture

hamiltonian of the system in rotating wave and dipole approximation is given

by

V = −~/2(Ω1e−i∆1t|b〉〈a|+ Ωpe

−i∆pt|b〉〈c|+ Ωk|d〉〈c|+ cc) (4.1)

where ∆1 and ∆p are the corresponding driving and probe field detunings,

respectively. We assume that the control and Kerr fields are strong while

the probe field is weak field such that |Ω1| and |Ωk| >> |Ωp|. Now the

42

corresponding rate equations can be written as

ρbc = [i∆p − γ1 − γ2]ρbc + i/2Ω1ρac + i/2Ωp(ρcc − ρbb)

−i/2Ωkρbd, (4.2)

ρac = [−i(∆1 −∆p)− Γ]ρac + i/2Ω1ρbc − i/2Ωpρab

−i/2Ωkρad, (4.3)

ρbd = [−i∆p − γ1 − γ2 − γ3]ρbd + i/2Ω1ρad + i/2Ωpρcd

−i/2Ωkρbc, (4.4)

ρad = [−i(∆1 −∆p)− γ3]ρad + i/2Ω1ρbd − i/2Ωkρac, (4.5)

where γ1, γ2 and γ3 are the decay rates as shown in Fig. 4.1, whereas Γ is

the relaxation rate of forbidden decay rate from level |c〉 to level |a〉.To obtain the susceptibility of the medium, we should first find the density

matrix element ρbc. it can be obtain by considering weak probe field approx-

imation. we can take the probe field in the first order while the control and

kerr fields in all orders. Following weak field approximation we presume that

driving (E1) and control (Ek) fields are significantly stronger then probe field

( Ep), which means that |Ωp| is much weaker then |Ω1| and |Ωk|. The dielec-

tric susceptibility can be found analytically from the dielectric polarization

and density matrix [148], which can be expressed as

χ = β8i(1/4(Γ + i∆1 − i∆p)Ω

21 + A(C) + Ω2

k/4)

B, (4.6)

where

A = (γ1 + γ2 + γ3− i∆p)

C = (−iΓ + ∆1 −∆p)(iγ3 −∆1 + ∆p)

43

and β=N |℘bc|2

ε0~with N be the atomic density, ℘bc is the dipole matrix element

whereas the denominator B is given in the appendix1.

The influence of Kerr nonlinear field on the susceptibility can be analyzed

by the following relation

χ(k) = χ(0) + Ω2k

∂χ

∂Ω2k

|Ωk=0, (4.7)

where the first term on the right side gives the dielectric susceptibility with-

out Kerr field, i.e., Ωk = 0, while second part of the equation (4.7) is the

contribution of the kerr nonlinear field to the dielectric susceptibility. The

group index is defined as n(k)g = c/vg, where c and vg be the speed of light

and the group velocity, respectively, can therefore be calculated using the

expression

n(k)g = 1 + 2πRe[χ(k)] + 2πνpRe[

∂χ(k)

∂∆p], (4.8)

where νp is the frequency of probe field.

4.1.3 Presentation of the results

Here, we study the influence of relaxation rate of forbidden decay on the

dielectric susceptibility of the medium which can change the behavior of the

medium dramatically from normal to anomalous dispersion. We increase the

forbidden decay rate via increasing the number of atoms. We also show that

manipulation of the group index of the medium via Kerr nonlinearity affects

the fast and slow light behavior. Using Eq. (4.7), we calculate the real and

imaginary parts of the dielectric susceptibility χ(k) where as the group index

of the medium is calculated using Eq. (4.8).

Initially, we assume that Kerr nonlinear field is zero in the medium i.e.,

Ωk = 0, therefore the nonlinear term in Eq. (4.7) vanishes and χ(k) = χ(0).

We take the forbidden decay rate Γ = 0.002γ. We also consider that the

44

atoms are prepared in level |c〉 and the corresponding parameters are Ω1 =

2γ, γ3 = γ, γ = 1MHz, ∆1 = 0, γ1 = 0.1γ, γ2 = 0.1γ and νp = 1000γ.

4.2(a) shows curves of absorption and dispersion with probe field detuning.

We get a normal dispersion along with two absorption peaks which gives the

usual EIT process [2]. The absorption is almost zero at resonance point i.e.,

∆p = 0. At this point the medium get transparent to the probe field. In

order to study another optical property of the medium i.e., group index, we

show the curve between the group index n(k)g and probe field detuning ∆p,

as shown in Fig. 4.2(b). This shows that the group index is positive for the

normal dispersion.

Next, to study the influence of forbidden decay rate Γ, we consider

that the Kerr field is still zero and increase the forbidden decay rate from

Γ = 0.002γ to Γ = 2γ. The behavior of the wave propagation in the atomic

medium becomes change dramatically from normal to anomalous dispersion.

Here, again we plot the real and imaginary parts of the dielectric susceptibil-

ity χ(k) and group index n(k)g versus probe field detuning ∆p, see Fig. 4.3. In

this case we observe anomalous dispersion and negative group index as shown

in Fig. 4.3(a) and 4.3(b). From this we can establish that the behavior of

the wave propagation in atomic system can be controlled via forbidden decay

rate Γ. The forbidden decay rate Γ of the atomic system can be increased via

increasing the number of atoms. It is due to the fact that collision between

the atoms increases and enhances the forbidden decay rate.

Now, to study the influence of Kerr nonlinearity, we consider that the

Kerr field is not zero. Then, three-level EIT atomic system extends to four-

level EIT, we again plot the real and imaginary parts of the susceptibility χ(k)

and group index n(k)g versus probe field detuning, see Figs. 4.4 and 4.5. The

behavior of the wave propagation inside the four-level EIT atomic system

45

Figure 4.2: (Color online) Plots of (a) real (solid) and imaginary (dashed)

parts of the susceptibility χ(k) (b) group index n(k)g versus probe field detuning

∆p for Ωk = 0 and Γ = 0.002γ. The inset in Fig. (b) is the group index

ranging from -0.1γ to 0.1γ. The corresponding parameters are Ω1 = 2γ,

γ3 = γ, γ = 1MHz, ∆1 = 0, γ1 = 0.1γ, γ2 = 0.1γ and νp = 1000γ.

46



∆p for Ωk = 0 and Γ = 2γ. The other parameters are the same as in Fig.

4.2.

47


parts of the susceptibility χ(k) for normal dispersion and (b) group index n(k)g

versus probe field detuning ∆p when Ωk = 1γ and Γ = 2γ. The inset in (b)

is the group index ranging from −0.1γ to 0.1γ. The other parameters are

the same as in Fig. 4.2.

48


parts of the susceptibility χ(k) for anomalous dispersion and (b) group index

n(k)g versus probe field detuning ∆p when Ωk = 1γ and Γ = 2γ. The other

parameters are the same as in Fig. 4.2.

49

Figure 4.6: (Color online) Plots of real parts of the susceptibility χ(k) for (a)

normal dispersion when Γ = 0.002γ and (b) for anomalous dispersion versus

probe field detuning ∆p when Γ = 2γ. The other parameters are the same

as in Fig. 4.2.

50

Figure 4.7: (Color online) Plots of group index n(k)g versus Kerr field Ωk/γ

(a) for normal dispersion when Γ = 0.002γ and ∆p = 0 (b) for anomalous

dispersion when Γ = 2γ and ∆p = 0, the remaining parameters are the same

as in Fig. 4.2.

51

remains the same as compare to the case of three-level EIT atomic system.

We notice that the group indeces increases in four-level atomic system due

to the fact of Kerr field. We also compare the normal and anomalous disper-

sion of three- and four-level atomic systems and notice that the dispersion

increases in four-level atomic system, see the enlarge view of normal and

anomalous dispersion of three- (without Kerr field) and four-level (with Kerr

field) in Fig. 4.6(a) and (b), respectively. It is due to the fact that the Kerr

nonlinearity increases the group index.

Now we investigate the group index of the medium for different values of

the Kerr field, we display the graph between the group index n(k)g and Kerr

field Ωk/γ. We notice that the group index of the atomic medium becomes

more negative for the anomalous dispersion and more positive for the normal

dispersion with an increase in the strength of the Kerr field, as shown in Fig.

4.7. It is clear that by increasing the intensity of the Kerr field we can obtain

much slower pulse propagation of light inside the atomic medium. Therefore

it is obvious that the light pulse can be stopped or halted inside the medium

via Kerr field.

In this section, we investigated the influence of Kerr nonlinearity and

relaxation rate of forbidden transition on the propagation of light through

four-level N-type atomic medium. In the coming section, we have a plan to

study the effect of Kerr nonlinear field and Doppler Broadening on the light

pulse propagation by considering four-level Λ-type atomic medium.

52

4.2 Effect of Kerr Nonlinearity and Doppler

Broadening on Slow Light Propagation

4.2.1 Introduction

In the previous section, we have studied the control of light pulse propagation

through atomic medium via Kerr field as well as relaxation rate of forbidden

transition. The group velocity of light pulse propagation through the medium

has been reduced via Kerr nonlinearity. To achieve a more slow group velocity

through an atomic medium, next we incorporate Doppler broadening effect

along with Kerr nonlinearity in the four-level Λ-type system. The Doppler

broadening effect becomes more important for assembly of moving atoms.

Following this effect earlier in-homogeneously Doppler broadened medium

has been considered for slow light [115]. Similarly, the possibility of pro-

ducing slow light in an inhomogeneous Doppler broadened medium has been

investigated by Agarwal and Dey using two-level atomic system [68, 149]. In

2005, Baldit et al. [116] did an experiment by considering inhomogeneous

Doppler broadened medium consist of rare-earth-ion-doped crystal and ob-

served group delay of the order of 1.1 s. They explain the difference between

a Doppler broadened gaseous medium and a solid-state medium like rare-

earth-ion-doped crystal i.e., the susceptibilities are to be averaged over an

inhomogeneous distribution. Recently, a four-level N-type Raman gain con-

figuration [150] has been utilized and studied the enhancement of anomalous

dispersion with Doppler effect. They also studied that the constructive in-

terference became more strengthen via Doppler broadening effect.

In this section, we consider a four-level lambda-type atomic configuration

of 87Rb atoms and monitor the propagation of weak probe light inside the

medium. The atom-field interaction exhibits an EIT process and we study

53

the effect of a Kerr field and Doppler broadening on the amplitude of the

group index of the medium. In this section, we expect control over the group

index of an EIT process via a Kerr field and Doppler broadening effect which

leads to more slow group velocity inside the medium. The major motivation

for this work is the investigation of more slow group velocity with a very

high group index using the combined effect of Kerr non-linearity and Doppler

broadening.

4.2.2 Model

We consider a realistic four-level lambda atomic configuration (87Rb) having

energy-level |a〉, |b〉, |c〉 and |d〉. An intense driving laser field E1 is applied

between level |b〉 and |a〉 with the corresponding Rabi frequency Ω1. Sim-

ilarly, we also apply a weak probe field Ep and a microwave coupling field

E2 between |b〉 → |c〉 and |c〉 → |d〉 with corresponding Rabi frequency Ωp

and Ω2, respectively. To calculate the expression of optical susceptibility

corresponding to the probe light Ep, we follow the same approach as in [148].

The interaction picture Hamiltonian for system in rotating wave and dipole

approximation is given by

V = −~

2(Ωpe

−i∆pt|b〉〈c|+ Ω1e−i∆1t|b〉〈a|

+Ω2e−i∆2t|c〉〈d|+H.c) (4.9)

where ∆1 = ω1−ωba, ∆2 = ω2−ωcd and ∆p = ωp−ωbc are the corresponding

driving, control and probe field detunings, respectively. Now the required

54

Figure 4.8: (Color online) (a) Schematic of the atom-field interaction, we

choose the ground state hyperfine levels 5S1/2, F = 2, m = 2;F = 2, m =

0;F = 1, m = 0 of 87Rb atom for |a〉, |c〉 and |d〉, 5P3/2, F/ = 2, m = 1 for

|b〉, respectively. (b) A block diagram where the probe, control and driving

fields are propagating inside the medium.

55

density matrix equations can be written as

ρbc = [i∆p − γ1]ρbc +i

2Ω1ρac +

i

2Ωp(ρcc − ρbb)

− i2Ω2ρbd,

ρac = [−i(∆1 −∆p)− Γ1]ρac +i

2Ω1ρbc −

i

2Ωpρab

− i2Ω2ρad,

ρbd = [i(∆2 + ∆p)− γ2]ρbd +i

2Ω1ρad +

i

2Ωpρcd

− i2Ω2ρbc,

ρad = [−i(∆1 −∆2 −∆p)− Γ2]ρad +i

2Ω1ρbd

− i2Ω2ρac.

(4.10)

where γ1 and γ2 are the decay rates from level |b〉 to |c〉 and |d〉, respectively,

whereas Γ1 (collisional decay) and Γ2 are the decay rates from level |c〉 and

|d〉 to |a〉.The dielectric susceptibility of the medium can be calculated from Eq.

(4.10). First, we need to find the density matrix element ρbc, it can be found

by using certain approximation. The control and Kerr field are assumed to be

much stronger than the probe field and hence we consider probe field in the

first order while Control and Kerr fields are in all orders. We consider that

all the atoms are initially prepared in level |c〉. The dielectric susceptibility

can be found analytically from the dielectric polarization and density matrix

[148], which can be expressed as

χ = βi[iΩ1F + (CA + Ω2

2)D]

B, (4.11)

56

where

B = iΩ1[−Γ1∆pΩ1 − i∆1∆pΩ1 + i∆2pΩ1 − iΩ3

1 + iΩ1Ω22

−iΓ1Ω1γ1 + ∆1Ω1γ1 −∆pΩ1γ1] + A[iΩ2(−iΓ1Ω2

+∆1Ω2 −∆pΩ2) + (Ω21 + C)E]D

+iΩ2[iΩ21Ω2 − iΩ2(Ω

22 + ED)], .

C = Γ1 + i∆1 − i∆p, A = Γ2 + i∆1 − i∆2 − i∆p,

D = −i∆2 − i∆p + γ3, E = −i∆p + γ1,

F = −iΓ1Ω1 + ∆1Ω1 −∆pΩ1,

and β = N |℘bc|2

ε0~εpwith N be the atomic density, ℘bc is the dipole matrix

element. The group index of the medium which is defined as ng = c/vg,

where c and vg be the speed of light and group velocity, respectively, can

therefore be calculated using the expression

ng = 1 + 2πRe[χ] + 2πνpRe[∂χ

∂∆p], (4.12)

where νp is the frequency of the probe field.

The susceptibility of the medium via Kerr nonlinear field can be describe

by the following relation [139]

χ(k) = χ(0) + Ω22

∂χ

∂Ω22

|Ω2=0, (4.13)

Eq. (4.8) can be used to obtain the group index of the medium in presence

of Kerr field.

n(k)g = 1 + 2πRe[χ(k)] + 2πνpRe[

∂χ(k)

∂∆p

], (4.14)

57

Next, by considering the frequency shifts due to Doppler broadening effect

for an atom moving with velocity ~v, we replace ω1 by ω1 + kv, ω2 by ω2 + kv

and ωp by ωp + kv. After incorporating the Doppler broadening effect, we

can write the optical susceptibility (χ(b)) and the group index (n(b)g ) as

χ(b) =

∫ ∞

−∞

χ(0)(1

D√

2πe−(kv)2/2D2

)d(kv), (4.15)

and

n(b)g = 1 + 2πRe[χ(b)] + 2πνpRe[

∂χ(b)

∂∆p], (4.16)

respectively, where D is the Doppler width which may defined as D =

ν0

c

√

2kBT/m.

Similarly, we can write down the optical susceptibility χ(kb) and the group

index n(kb)g including both control (Kerr) and Doppler broadening effect

χ(kb) =

∫ ∞

−∞

χ(k)(1

D√

2πe−(kv)2/2D2

)d(kv), (4.17)

and

n(kb)g = 1 + 2πRe[χ(kb)] + 2πνpRe[

∂χ(kb)

∂∆p

]. (4.18)

4.2.3 Results presentation

We discuss the manipulation of group velocity using a four-level lambda-type

system using the concept of an EIT. Here, we study the effect of Kerr non-

linearity, Doppler broadening and the combined effect of Kerr nonlinearity

and Doppler broadening on group velocity inside the medium. We chose the

parameters as γ = 1MHz, γ1 = γ,γ2 = 0.1γ, Γ1 = 0.002γ, D = 10kHz,

Γ2 = 0.2γ, ∆1 = ∆2 = 0 and β = γ. The plots of real parts of susceptibilities

(χ(0), χ(k), χ(b) and χ(kb)) versus probe field detuning ∆p are shown in Fig.

4.9. Initially, we assume that there is no Kerr field involved in the system and

therefore the nonlinear term of Eq. (4.13) becomes zero, i.e., χ(k) = χ(0), then

58

the system behaves like a simple three-level EIT [148]. For the corresponding

condition we plot the susceptibility (χ(0)) versus probe field detuning that

show the normal dispersion (slow group velocity), see Fig. 4.9(a). Now we

switch on the Kerr field Ω2 and study the group velocity inside the medium.

Using Eq. (4.13) we again plot the real part of susceptibility χ(k) versus

probe field detuning for Ω2 = 3γ as shown in Fig. 4.9(a). The Kerr field

affect the normal dispersion which has been noticed earlier [139, 151]. The

slope increases via Kerr field that leads to more slow group velocity inside

the medium.

The analysis of slow group velocity inside the medium is discussed above

in Fig. 4.9 which is only reasonable when atoms are in stationary state.

It will be more constructive when atoms are considered in random motion

and investigated the light propagation inside the medium. Then the Doppler

broadening effect becomes important. To study the light propagation inside

the Doppler broadened medium, we consider three laser fields i.e., driving,

probe and Kerr pass through 87Rb vapor cell as shown in Fig. 4.8(b). Again

we focus on real part of susceptibility of the Doppler broadened atomic vapor

cell by considering that the Kerr field Ω2 = 0. We use Eq. (4.15) and plot

the susceptibility χ(b) versus probe field detuning, see Fig. 4.9(b). We notice

that the slope of normal dispersion increases with incorporating the Doppler

broadening effect, see the difference between Re[χ(0)](without Doppler broad-

ening effect) and Re[χ(b)](with Doppler broadening effect) in Fig.4.9(c)

In the above discussion we investigated the increase of slope for normal

dispersion that leads to more slow group velocity inside the medium via Kerr

non-linearity as well as Doppler broadening effect in two different processes.

The control of group velocity inside different atomic media have been no-

ticed earlier via Kerr nonlinearity [139, 151] and Doppler broadened medium

59

Figure 4.9: (Color online) Plots of real parts of susceptibilities (χ(0), χ(k),

χ(b) and χ(kb)) versus probe field detuning. The parameters are γ = 1MHz,

γ1 = γ,γ2 = 0.1γ, Γ1 = 0.002γ, Γ2 = 0.2γ, ∆1 = ∆2 = 0, Ω1 = 2γ, Ω2 = 3γ,

D = 10kHz and β = γ.60

Figure 4.10: (Color online) Plots of group indexes (n(0)g , n

(k)g , n

(b)g and n

(kb)g )

versus probe field detuning ∆p, the parameters remains the same as in Fig.

4.9. 61

[68, 149]. It will be more constructive to study the control of group velocity

inside the medium by incorporating the combined effect of Kerr nonlinearity

and Doppler broadening. Now we incorporate the combined effect of Kerr

nonlinearity and Doppler broadening and study the group velocity inside the

medium which is the major part of this work. We are expecting more and

more slow group velocity inside the medium via combined effect of Kerr non-

linearity and Doppler broadening. We use Eq. (4.17) and plot the real part

of susceptibility χ(kb) versus probe field detuning for Ω2 = 3γ, see Fig. 4.9(b).

For the combined effect of Kerr nonlinearity and Doppler broadening we in-

vestigate a very steep normal dispersion that leads to more and more slow

group velocity inside the medium. In Fig. 4.9(c), we plot all the real parts of

susceptibilities versus probe field detuning, from this plot we conclude that

the slope of normal dispersion for Re[χ(kb)] is very steep as compare to the

slopes of χ(0), χ(k) and χ(b).

There is a strong correlation between dispersion and group index of the

medium i.e., the group index increases with increase the dispersion of the

medium and vice versa. In the above analysis we show the enhancement

of normal dispersion using Kerr nonlinearity and Doppler broadening effect.

The group index also change with changing the dispersion of the medium,

so we plot n(0)g , n

(k)g , n

(b)g and n

(kb)g versus probe field detuning, see Fig. 4.10.

Due to the combined effect of Kerr nonlinearity and Doppler broadening we

notice that the group index n(kb)g is very large as compare to n

(0)g , n

(k)g and

n(b)g .

Now to study the behavior of group index of the atomic medium for

different choices of the intensity of Kerr field , we plot the group index n(k)g

and n(kb)g versus Kerr field Ω2. In Fig. 4.11(a), the plot shows the group

index n(k)g versus Kerr field, here we notice that the group index of the atomic

62

Figure 4.11: (Color online) Plots of group index n(k)g and n

(kb)g versus Kerr

field Ω2 at ∆p = 0, (a) without Doppler broadening effect (b) with Doppler

broadening effect, the remaining parameters remains the same as in Fig. 4.9

63

medium becomes more positive for the normal dispersion with an increase

in the strength of the Kerr field. In Fig. 4.11(b) we plot the group index

n(kb)g versus Kerr field and investigate similar behavior as we observed in Fig.

4.11(a). The group index n(kb)g of the atomic medium is more positive as

compare to the group index n(b)g , because in n

(kb)g both the Kerr and Doppler

broadening effects are now involve.

In this section, we have discussed the control over the optical properties

of the medium via Kerr nonlinearity and Doppler Broadening effect. In

the coming section, we have a plan to explore the control over the optical

properties of the EIT N-type medium in the presence of both Kerr field and

SGC effect.

4.3 Control of Group Velocity via Sponta-

neous Generated Coherence and Kerr Non-

linearity

4.3.1 Introduction

We have discussed earlier, the control over light pulse propagation through

various atomic media via Kerr nonlinearity and Doppler broadening. In

this section we study the control over light pulse propagation through the

medium in the presence of both Kerr and SGC effect. Spontaneous emission

usually minimize the coherence in the system while SGC is an extra control

parameter, which enhances the coherence in the system and plays a useful

role in many optical process. The enhancing Kerr nonlinearity via SGC is

reported in 2006 [73]. SGC depends on quantum interference of spontaneous

emissions between two channels. The quantum interference effect has been

64

noticed earlier in three-level Λ-type system, where the spontaneous emission

interfere from a single excited state to two lower closely spaced levels [119].

Similarly, in V-type atomic system the spontaneous emission interfere from

two closely spaced upper levels to a common ground state [120]. Actually, this

coherence based on nonorthogonality of the two transition dipole moments.

More recently, The control of group velocity has been noticed in different

atomic media via a Kerr field [151, 152, 153]. In their work, it has been

noticed that a Kerr nonlinearity enhanced the group index of the medium,

which leads to slow group velocity inside the medium. Now it will be more

constructive to study the control of group velocity using the collective effect

of SGC and Kerr field.

In this section, we study the light pulse propagation inside a medium

via Kerr nonlinearity and SGC. Each atom of a medium consist of N -type

atomic configuration of 85Rb atoms. A Kerr field enhance the group index

which leads to slow group velocity as noticed earlier [139, 151, 152, 153]. We

also investigate the individual effect of SGC on group velocity. The important

and major part of this article is to study the control of group velocity via

the collective effect of SGC and Kerr field.

4.3.2 Model

The energy-level configuration of the atom-field interaction are presented in

Fig. 4.12. We consider a realistic four-level N -type atomic system of rubid-

ium atoms (85Rb) each having energy levels |a〉, |b〉, |c〉 and |d〉. An intense

driving laser field E1 is applied between level |a〉 and |b〉 with corresponding

Rabi frequency Ω1. Similarly, a weak probe field Ep and a strong Kerr field

Ek are applied between |b〉 → |c〉 and |c〉 → |d〉 with corresponding Rabi fre-

quency Ωp and Ωk, respectively. Here, γ1 and γ2 are the spontaneous decay

65

Figure 4.12: (Color online) (a) Schematics of the four-level N -type rubidium

atomic system (b) dipole moments of driving and probe fields.

rates of the excited level |b〉 to the ground levels |a〉 and |c〉. For generation

of SGC the two lower levels |a〉 and |c〉 must be closely spaced, it is due to

the fact that the two transitions of the excited state interact with the same

vacuum mode.

The interaction picture Hamiltonian for system in rotating wave and

dipole approximation is given by

V = −~[(∆1 −∆p)|c〉〈c|+ ∆1|b〉〈b|+ (∆1 −∆p + ∆k)

|d〉〈d|+ Ω1|a〉〈b|+ Ωp|c〉〈b|+ Ωk|c〉〈d|], (4.19)

where ∆p, ∆1 and ∆k are the corresponding probe and driving field de-

tunings, respectively. We consider that the driving laser field E1 and Ek are

strong fields while the probe field Ep is a weak field which means |Ω1|and|Ωk| >>

66

|Ωp|. Now the corresponding rate equations can be written as

ρbc = [i∆p − γ1 − γ2]ρbc + iΩ1ρac + iΩp(ρcc − ρbb)

−iΩkρbd,

ρac = [−i(∆1 −∆p)− Γ1]ρac + iΩ1ρbc − iΩpρab

−iΩkρad + 2q√γ1γ2ρbb,

ρbd = i(∆p −∆k)ρbd + iΩ1ρad + iΩpρcd

−iΩkρbc,

ρad = [−i(∆1 −∆p + ∆k)− γ2]ρad + iΩ1ρbd

−iΩ2ρac,

ρbb = (−γ1 − γ2)ρbb + iΩ1ρab − iΩ1ρba,

ρab = (−i∆1 − γ1)ρab + iΩ1ρbb,

ρba = (i∆1 − γ1)ρba − iΩ1ρbb,

(4.20)

where γ1, γ2 and γ3 are the decay rates as shown in Fig.4.12 whereas Γ is the

forbidden decay rate between level |a〉 and |c〉.Here, the parameter q denotes the alignment of two dipole moments ~µba

and ~µbc. For the orientations of the atomic dipole moments ~µba and ~µbc the

effect of SGC is very sensitive. The parameter q may further be defined as

q = ~µba.~µbc/ |~µba.~µbc| = cosθ arises due to the quantum interference between

two decay channels |b〉 → |a〉 and |b〉 → |c〉, whereas θ is the angle between the

two dipole moments. The term q√γ1γ2 represents the quantum interference

resulting from the cross coupling between the spontaneous emission channels

|b〉 → |a〉 and |b〉 → |c〉. In fact, the parameter q represents the strength

of the interference in spontaneous emission. If the two dipole moments are

orthogonal to each other then q = 0, which clearly shows that there is no

67

quantum interference due to spontaneous emission. When the two dipole

moments are parallel to each other then the quantum interference is maximal

and q = 1. The control of alignment of two dipole moments depends on the

angle θ between them. The angle θ may be adjusted with the help of external

driving (E1) and probe (Ep) fields, see Fig. 4.12(b). It is due to the fact

that the probe and control fields do not interact with each other’s transitions

so that one must be perpendicular to the dipole moment coupled to the

other [121]. So the quantum interference effect can be adjusted by control

the alignments of two dipole moments. The driving (Ω1) and probe (Ωp)

fields associated to the angle θ and therefore we can write as Ω1 = Ω01sinθ =

Ω01

√

1− q2.

The polarization P of the medium depends on the electric field Ep and

can be calculated as:

P = χε0Ep; (4.21)

where P = 2Nµbcρbc and Ep = Ωp~/µcb. After simplification the expression

for the dielectric susceptibility χ for the given atom-field system as shown in

Fig. 4.12 can be written as

χ =2N |µbc|2~ε0Ωp

ρbc, (4.22)

where N represent density of the atomic medium while µbc and ρbc are the

dipole and off-diagonal matrices elements, respectively, for the corresponding

optical transition. It is clear form Eq. (4.22) that we should first find the

density matrix element ρbc. it can be obtain by considering weak probe field

approximation. we can take the probe field in the first order while the control

and kerr fields in all orders. Following weak field approximation we presume

that driving (E1) and control (Ek) fields are significantly stronger then probe

field ( Ep), which means that |Ωp| is much weaker then |Ω1| and |Ωk|. Here,

68

the zeroth-order solution of the probe field elements are equal to zero except

ρ0cc = 1, it is due to the fact that the atom are initially prepared in the

ground state |c〉. The above Eq. (4.20) can be solved easily following the

recipe discussed in the Appendix 2 to get ρbc with resonance condition i.e.,

∆1 = ∆k = 0

ρbc =[(iΓ + ∆p)(iγ3∆p + ∆2

p − Ω21)−∆pΩ

2k]Ωp

B, (4.23)

where

B = γ2γ3∆2p − iγ2∆

3p − iγ3∆

3p −∆4

p + iγ2∆pΩ21

+iγ3∆pΩ21 + 2∆2

pΩ21 − Ω4

1 + [iγ2∆p + iγ3∆p

+2(∆2p + Ω2

1)]Ω2k − Ω4

k + Γ[(γ1 + γ2 − i∆p)

(iγ3∆p + ∆2p − Ω2

1)− (γ3 − i∆p)Ω2k]

+γ1∆p[γ3∆p − i(∆2p − Ω2

1 − Ω2k)]. (4.24)

Using Eq. (4.22) and (4.23) we can write the optical susceptibility for the

atom-field interaction can be written as

χ = β(iΓ + ∆p)(iγ3∆p + ∆2

p − Ω21)−∆pΩ

2k

B, (4.25)

where β=2N |µbc|2

~ε0.

We consider Ωk is a Kerr field, then the effect of Kerr field Ωk on the

susceptibility can be studied by the following expression [139, 151, 152, 153].

χ(k) = χ(0) + Ω2kχ

(1), (4.26)

where χ(0) is the susceptibility of the medium without Kerr nonlinearity

and can be calculated as

69

χ(0) = − (Γ− i∆p)

γ1∆p + γ2∆p − i∆2p + Γ(iγ1 + iγ2 + ∆p) + iΩ2

1

.

(4.27)

The second part in Eq. (4.26) shows the nonlinear part of the suscepti-

bility via Kerr field Ωk, where χ(1) can be calculated as

χ(1) =∂χ

∂Ω2k

∣

∣

∣

∣

Ωk=0

(4.28)

χ(1) =Γ2(iγ3 + ∆p)−∆p(iγ3∆p + ∆2

p + 3Ω21) + 2Γ(γ3∆p − i(∆2

p + Ω21))

(iγ3∆p + ∆2p − Ω2

1)[Γ(γ1 + γ2 − i∆p)− iγ1∆p − iγ2∆p −∆2p + Ω2

1]2.

(4.29)

The group index of the medium in the presence of Kerr Field can be

calculated using the expression given in Eq. (4.8).


The enhancement of group index of a medium has been studied earlier [139,

151, 152, 153] via strength of Kerr nonlinearity. The enhancement of group

index of a medium then leads to slow group velocity inside the medium.

In the following we start our discussion by studying the control of group

velocity via Kerr nonlinearity and SGC when a light pulse is propagating

inside a medium. Initially, we study the control of group velocity inside a

medium via SGC and then by Kerr nonlinearity. Further, we consider the

collective effect of SGC and Kerr field on group velocity in four-level N -type

system. We also consider that the atoms are initially prepared in level |c〉and the corresponding parameters are γ = 1MHz, γ1 = γ2 = γ3 = 1γ,

Γ = 0.002γ, Ω01 = 4γ and νp = 1000γ.

70

We investigate the effect of SGC on group velocity when a light pulse is

propagating inside a medium. We study the group velocity in the absence

of Kerr field i.e., Ωk = 0, then the system becomes a simple three-level

Λ-configuration. In Fig. 4.13(a) we plot the real and imaginary parts of

susceptibility χ(0) versus probe field detuning without considering SGC i.e.,

q = 0. The plot shows a normal dispersion (subluminal behavior) which has

been noticed earlier for EIT process [49, 148]. In this plot we consider q = 0,

which means that there is no quantum interference between spontaneous

emission channels. We also plot the group index versus probe field detuning

and at resonance condition we calculate the group index n(0)g = 393. Next,

we increase the value of q from 0 to 0.99 and again we plot the susceptibility

χ(0) and group index n(0)g versus probe field detuning, see Fig. 4.14. A strong

quantum interference effect of the spontaneous emission channels occurs for

the maximum value of q. In this time we get a steep dispersion along with

narrow EIT window as shown in 4.14(a) which has been noticed earlier [73,

121]. As EIT window depends on stark splitting which is directly related to

the control or driving field. When the strength of the control field increases

the width of EIT window also increases and vice versa. Now it is obvious

from the relation Ω1 = Ω01

√

1− q2 that q can affect the control field Ω1.

If the value of q increases the control field decreases and the EIT window

also decreases. So at high value i.e, q = 0.99 the width of the EIT window

decreases. For the maximum value of q we also study the group index of the

medium and at resonance condition and calculate n(0)g = 20× 103.

Now we switch on a Kerr field Ωk and investigate the light pulse prop-

agation inside the medium. We consider initially the value of q = 0 which

shows that there is no quantum interference effect of the spontaneous emis-

sion channels. We plot the susceptibility χ(k) and group index n(k)g using

71


parts of the susceptibility χ(0) (b) group index n(0)g versus probe field detuning

∆p for q = 0, γ = 1MHz, γ1 = γ2 = γ3 = 1γ, Γ = 0.002γ,Ω01 = 4γ, and

νp = 1000γ.

72


parts of the susceptibility χ(0) (b) group index n(0)g versus probe field detuning

∆p for q = 0.99, the remaining parameters remains the same as in Fig. 4.13.

73



∆p for q = 0 and Ωk = 2γ, the remaining parameters remains the same as in

Fig. 4.13.

74



∆p for q = 0.99 and Ωk = 2γ, the remaining parameters remains the same as

in Fig. 4.13.

75

Figure 4.17: (Color online) Plots of (a) group index versus Kerr field for

q = 0 and ∆p = 0 (b) group index versus Kerr field for ∆p = 0 and q = 0.99,

the remaining parameters remains the same as in Fig. 4.13.

76

Eqs. (4.26) and (4.8), respectively. The dispersion as well as the group in-

dex increases with increasing the Kerr field, see Fig. 4.15. We consider the

Kerr field Ωk = 2γ and calculate the group index at resonance condition

i.e., n(k)g = 687. This clearly tells us that the Kerr field enhance the group

index which leads to slow group velocity. Similar effect of Kerr field on the

dispersion and group index has been investigated earlier in four-level N -type

atomic medium [151].

Further, it will be more appropriate to study the collective effect of SGC

and Kerr field on control of group velocity. As discussed previously that the

individual effect of SGC and Kerr field control the group velocity when a light

pulse is propagating inside the medium. In our proposed atomic configuration

both SGC and Kerr field are present then it is more constructive to study the

collective effect of these phenomenon on the slow light propagation. We are

expecting a more slow group velocity inside a medium as compared to the

individual effect of SGC and Kerr field. We consider that there is a strong

quantum interference effect i.e., q = 0.99 and the Kerr field Ωk = 2γ. We

plot the real and imaginary parts of susceptibility χ(k) and group index n(k)g

versus probe field detuning, see Fig. 4.16. Due to the collective effect of SGC

and Kerr field the normal dispersion (solid curve) becomes more and more

steep as compared to the individual effect, see Fig. 4.16(a). We also plot the

group index as shown in Fig. 4.16(b), at resonance condition we calculate

the group index i.e., n(k)g = 7× 105. This enhancement of group index then

leads to more slow group velocity inside the medium.

To study the enhancement of group index of the medium with increasing

the strength of the Kerr filed Ωk, we plot the group index n(k)g versus strength

of the Kerr field. In Fig. 4.17(a), the plot shows the group index n(k)g versus

strength of a Kerr field by considering no quantum interference effect present

77

i.e., q = 0, here we notice that the group index of the atomic medium becomes

more positive for the normal dispersion with an increase in the strength of

a Kerr field. Next, we consider a strong quantum interference effect in our

system i.e., q = 0.99 and plot the group index n(k)g of the medium versus the

strength of Kerr field and investigate similar behavior as we examined earlier

(in Fig. 4.17(a)), see Fig. 4.17(b). But at this time the group index becomes

more and more positive via the collective effect of SGC and Kerr field. Now

It is clear that by increasing the intensity of the Kerr nonlinear field we can

obtain much slower pulse propagation inside the medium. Obviously halted

or stopped pulses can be achieved via Kerr Field.

The slow light have potential applications, these include for example, slow

light devices are considered for enhancing other optical nonlinearities [154].

Slowing or stopping light is also used to achieve the long storage times to

perform quantum operations [17, 58]. Slow light could be used to enhance

the sensitivity of spectral interferometer which has been noticed by Shi and

co-workers in 2007 [155]. Similarly, slow light has been used in laboratory

settings to achieve true time delay to synchronize the radio frequency emitters

of a phased-array radar system [156, 157].

In this section, we have studied the influence of Kerr field and SGC effect

on the propagation of light pulse through N-type atomic medium. In the

forthcoming section, we have a scheme to study the effect of two Kerr fields

by considering tripod atomic medium.

78

4.4 Control of Group Velocity via Double Kerr

Nonlinearity in Four-Level Tripod Atomic

System

4.4.1 Introduction

In the last sections, we presented a detail study on the behavior of light prop-

agation via Kerr field, Doppler broadening and SGC effect through various

atomic media. The Kerr non-linearity, Doppler broadening and SGC affect

the light pulse propagation through the medium, which leads to slow group

velocity. To attain a more slow group velocity inside the medium, next we

consider two Kerr fields in a single atomic configuration of tripod atomic sys-

tem. In this section, we theoretically investigate the behavior of light pulse

propagation inside a medium consist of four-level tripod atomic configura-

tion. Each atom of a four-level tripod atomic medium consist of D1 line of

87Rb. A single Kerr field enhance the group index which leads to control of

group velocity as noticed earlier [139, 151, 152, 153]. Here, in this section

we expect a strong control of group velocity via double Kerr nonlinear fields

inside a four-level tripod atomic medium. The motivation comes from an

earlier experimental work where four-level tripod atomic medium has been

used to obtain large cross-phase modulation in a cold D1 line of rubidium

(87Rb) medium without implying high magnetic field [158]. In the article

[158] a trigger, probe and pump fields are considered. We follow the same

experimental model and consider trigger and pump fields as Kerr nonlinear

and investigate the control of group velocity via double Kerr nonlinear fields.

The control of group velocity using double Kerr fields then leads to more and

more slow group velocity inside the medium.

79

Figure 4.18: (Color online) Schematics of the D1 line in the rubidium (87Rb)

four-level tripod atomic system

4.4.2 Model

The schematics of the atom field interaction are presented in Fig.4.18. We

suggest an experimental model of four-level tripod atomic system [158] each

having energy level |1〉, |2〉, |3〉 and |4〉. Two intense driving laser fields Ek1

and Ek2 are applied between level |2〉 ↔ |4〉 and |3〉 ↔ |4〉 with corresponding

Rabi frequencies Ωk1 and Ωk2, respectively. Similarly, a weak probe field Ep

between |4〉 ↔ |1〉 is applied with corresponding Rabi frequency Ωp.

The interaction picture Hamiltonian for the system in rotating wave and

dipole approximation is given by

V = −~(Ωpe−i∆pt|4〉〈1|+ Ωk1e

−i∆k1t|4〉〈2|

+Ωk2e−i∆k2t|4〉〈3|+ cc), (4.30)

80

where ∆p, ∆k1 and ∆k2 are the corresponding probe and driving field detun-

ings, respectively. We consider that the driving laser field Ek1 and Ek2 are

strong fields while the probe field Ep is a weak field which means |Ωk1|and|Ωk2| >>|Ωp|. Now the corresponding rate equations can be written as

ρ41 = [i∆p − γ41]ρ41 + iΩp(ρ11 − ρ44) + iΩk1ρ21

+iΩk2ρ31,

ρ21 = [i(∆p −∆k1)− γ21]ρ21 + iΩk1ρ41 − iΩpρ24,

ρ31 = [i(∆p −∆k2)− γ31]ρ31 + iΩk2ρ41 − iΩpρ34,

ρ24 = [−i∆k1 − γ24]ρ24 + iΩk1(ρ44 − ρ22)− iΩpρ21

−iΩk2ρ23,

ρ34 = [−i∆k2 − γ34]ρ34 + iΩk2(ρ44 − ρ33)− iΩk1ρ32

−iΩpρ31,

ρ23 = [−i(∆k1 −∆k2)− γ23]ρ23 + iΩk1ρ43 − iΩk2ρ24.

(4.31)

where γ21, γ31 and γ41 are the decay rates from level |2〉, |3〉 and |4〉 to |1〉,respectively.

ρ41 can be calculated from the coupled rate equations given in Eq. (4.31)

using some approximations. Following the weak probe approximation, the

susceptibility of the medium can be obtain as:

χ = βi(γ21 + i∆k1 − i∆p)(γ31 + i∆k2 − i∆p)

B, (4.32)

where β=N |℘41|2

ε0~with N be the atomic density, ℘41 is the dipole matrix

element and B is given as:

B = [γ31 − i(−∆k2 + ∆p)][(γ21 + i∆k1 − i∆p)(γ41 − i∆p) + Ω2k1]

−iΩk2(iγ21Ωk2 −∆k1Ωk2 + ∆pΩk2), (4.33)

81

Here, we consider Ωk1 is a Kerr field while Ωk2 has no Kerr nonlinearity,

then the effect of Kerr field Ωk1 on the susceptibility can be studied by the

following expression [139, 151, 152, 153]

χ(k1) = χ(0) + Ω2k1

∂χ

∂Ω2k1

∣

∣

∣

∣

Ωk1=0

, (4.34)

where χ(0) is the optical susceptibility of the medium in the absence of Kerr

field, i.e., Ωk1 = 0. The second part in Eq. (4.34) gives the nonlinear behavior

of the optical susceptibility via Kerr field Ωk1. Now the group index of the

medium which is defined as n(k1)g = c/vg where c and vg be the speed of light

and the group velocity, respectively, can therefore be calculated using the

expression

n(k1)g = 1 + 2πRe[χ(k1)] + 2πνpRe[

∂χ(k1)

∂∆p], (4.35)

where νp is the frequency of the probe field.

Similarly, by considering Ωk2 is a Kerr field while Ωk1 has no Kerr nonlin-

earity, the effect of Kerr field Ωk2 on the susceptibility and group index can

be studied as

χ(k2) = χ(0) + Ω2k2

∂χ

∂Ω2k2

∣

∣

∣

∣

Ωk2=0

, (4.36)

and

n(k2)g = 1 + 2πRe[χ(k2)] + 2πνpRe[

∂χ(k2)

∂∆p], (4.37)

respectively.

When both Ωk1 and Ωk2 are considered as Kerr fields, which having same

strength equal to Ωk i.e., Ωk = Ωk1 = Ωk2 , then the collective effect of the

Kerr fields on the susceptibility and group index can be studied as

χ(k1k2) = χ(0) + Ω2k

∂χ

∂Ω2k

∣

∣

∣

∣

Ωk=0

, (4.38)

and

n(k1k2)g = 1 + 2πRe[χ(k1k2)] + 2πνpRe[

∂χ(k1k2)

∂∆p], (4.39)

82

respectively.


We know from an earlier study [139, 151, 152, 153] that the group index of a

medium increases via increasing the strength of a Kerr field. The enhance-

ment of group index then leads to slow group velocity inside the medium. A

single Kerr field is used to control the group velocity in the above study. In

the following we start our discussion by studying the control of group velocity

via Kerr nonlinearity when a light pulse is propagating inside a four-level tri-

pod atomic medium. We suggest an experimental model consist of four-level

tripod atomic medium with initial atomic state in the D1 line of 87Rb atom.

Initially, we study the control of group velocity inside a medium via a single

Kerr field. Then we consider double Kerr field in four-level tripod atomic

medium and study the control of group velocity. We also consider that the

atoms are initially prepared in level |1〉 and the corresponding parameters

are γ41 = 0.1γ, γ = 1MHz, ∆k1 = ∆k2 = 0, γ31 = γ21 = 1γ, and νp = 1000γ.

4.4.4 Control of Group Velocity via a Single Kerr Field

We study the effect of a single Kerr field on the group velocity when a light

pulse is propagating inside the medium. We consider that Ωk1 is a Kerr field

while Ωk2 = 1γ which has no Kerr nonlinearity. We use Eq.(4.34) and plot the

real and imaginary parts of the optical susceptibility χ(k1) where as the group

index of the medium is plotted using Eq. (4.35). Fig. (4.19)(a) presents real

and imaginary parts of the susceptibility χ(k1) versus probe field detuning

∆p for Ωk1 = 0.5γ. At resonance or round about resonance condition the

real part of the susceptibility gives the dispersion (normal) property while

the imaginary part gives the absorption property. We plot the group index

83


parts of the susceptibility χ(k1) (b) group index n(k1)g versus probe field de-

tuning ∆p for Ωk2 = 1γ, Ωk1 = 0.5γ, γ41 = 0.1γ, γ = 1MHz, ∆k1 = ∆k2 = 0,

γ31 = 1γ, γ21 = 1γ and νp = 1000γ

84

n(k1)g versus probe field detuning ∆p for the same parameters and calculate

the group index n(k1)g = 1299 of the medium at resonance condition ∆p = 0.

Next, we slightly change the Kerr field Ωk1 from 0.5γ to 1γ and remain all

the other parameters unchanged. We again plot the susceptibility χ(k1) and

the group index n(k1)g versus probe field detuning. We calculate the group

index again at resonance condition, the group index increases from 1299 to

5193 with increasing the Kerr field from 0.5γ to 1γ. We notice that the group

index increases approximately four times when the strength of the Kerr field

increases from 0.5γ to 1γ.

Now we study the control of group velocity inside the medium by con-

sidering Ωk2 is a Kerr field. The other field Ωk1 = 1γ which has no Kerr

nonlinearity while all other parameters remain unchanged. To investigate

the effect on group velocity via Kerr field Ωk2, we use Eq. (4.36) and Eq.

(4.37) for susceptibility and group index of the medium, respectively. We no-

tice that the susceptibility and group index remains the same as we observed

above for the case of Kerr field Ωk1.

4.4.5 Control of Group Velocity via Double Kerr Fields

The control of group velocity using a single Kerr field have been studied

earlier in different atomic media [139, 151, 152, 153]. We study the control

of group velocity via a single Kerr field using an experimental model of D1

line of 87Rb atom. We also study the individual effect of two Kerr fields Ωk1

and Ωk2 on group velocity inside a four-level atomic medium. Now it will

be more constructive to investigate the collective effect of two Kerr fields on

the control of group velocity inside a medium. Initially, we consider that the

two Kerr fields are zero, the second and third part of Eq. (4.38) vanishes

and then χ(k1k2) = χ(0). The four-level atomic system reduces to a simple

85


parts of the susceptibility χ(k1) (b) group index n(k1)g versus probe field de-

tuning ∆p for Ωk1 = 1γ, the remaining parameters remains the same as in

Fig. 4.19

86


parts of the susceptibility χ(k1k2) (b) group index n(k1k2)g versus probe field

detuning ∆p for Ωk1 = Ωk2 = 0, the remaining parameters remains the same

as in Fig. 4.19.

87

two-level and becomes an absorptive medium. We plot the susceptibility

χ(k1k2) and group index n(k1k2)g versus probe field detuning, see Fig. (2.21).

We observe a high absorption along with anomalous dispersion and negative

group velocity that leads to fast light propagation inside the medium. The

dispersion change dramatically from anomalous to normal when the two Kerr

fields is switched on simultaneously. The normal dispersion then leads to slow

group velocity inside the medium. Here, we also notice that the medium

turns into an amplifier exhibiting slow pulse propagation inside the medium.

The normal dispersion that we notice here is very steep by just considering

the two Kerr fields Ωk1 = Ωk1 = 1γ, see Fig. (4.22)(a). Further, we study

the group index of the medium for the same parameters and observe a very

high group index at resonance condition, see Fig.(4.22)(b). We investigate

a very strong control of the group velocity via collective effect of two Kerr

fields. In section A, we observed the group index at resonance condition i.e.,

n(k1)g = 5193 for a Kerr field Ωk1 = 1γ. The collective effect of two Kerr

fields enhance the group index to 2.5× 107 by considering Ωk1 = Ωk2 = 1γ.

The collective effect of two Kerr fields enhance the group index more than

4814 times as compare to a single Kerr field. To study the enhancement

of group index of the medium with increasing the strength of the Kerr filed

Ωk1, we plot the group index n(k1)g versus strength of single Kerr field. In Fig.

(4.23)(a), the plot shows the group index n(k1)g versus strength of single Kerr

field, here we notice that the group index of the atomic medium becomes

more positive for the normal dispersion with an increase in the strength of

a Kerr field. Next, we consider Ωk1 = Ωk2 and plot the group index n(k1k2)g

of the medium versus the strengths of two Kerr fields and investigate similar

behavior as we observed earlier (in Fig. 4.23(a)), see Fig. 4.23(b). But at

this time the group index becomes more and more positive via the collective

88


parts of the susceptibility χ(k1k2) (b) group index n(k1k2)g versus probe field

detuning ∆p for Ωk1 = Ωk2 = 1γ, the remaining parameters remains the same

as in Fig. 4.19.

89

effect of two Kerr fields.

In our proposed atomic system there is a strong control of the group

velocity via external Kerr fields. The light pulse can be slow down even

that one can stop the light pulse and store it as per their requirement. We

give a theoretical idea using a four-level experimental system of D1 line of

87Rb atoms. For the Kerr fields Ωk1 = Ωk2 = 10γ we notice that the group

velocity inside the medium reduces to vg = c/(2.5×109) = 0.12m/s, see Fig.

4.23(b). With increasing the strength of the Kerr fields the group velocity

decreases further. The slow light have potential applications, these include

for example, slow light devices are considered for enhancing other optical

nonlinearities [154]. Slowing or stopping light is also used to achieve the

long storage times to perform quantum operations [58, 17]. Slow light could

be used to enhance the sensitivity of spectral interferometer which has been

noticed by Shi and co-workers in 2007 [155]. similarly, slow light has been

used in laboratory settings to achieve true time delay to synchronize the

radio frequency emitters of a phased-array radar system [156, 157].

90

Figure 4.23: (Color online) Plots of group index n(k1)g versus (a) single Kerr

field when Ωk2 = 1γ and∆p = 0 (b) and n(k1k2)g versus double Kerr fields, the

remaining parameters are the same as in Fig. 4.19.

91

Chapter 5

Conclusions

In conclusion, we considered different atomic media and study the influence

of Kerr non-linearity, Doppler broadening and SGC on light pulse propaga-

tion. We theoretically investigated that the group velocity of light pulse is

reduced via Kerr non-linearity, Doppler broadening and SGC. The reduced

group velocity of light pulse then leads to slow light inside the medium. For

four-level N -type atomic medium we studied the influence of relaxation rate

of forbidden transition and Kerr non-linearity on light pulse propagation in-

side the medium. It is found that the relaxation rate of forbidden transition

change the behavior of light pulse propagation through the medium. The

group velocity change from positive to negative via changing the relaxation

rate of forbidden transition (Γ). In the same atomic medium we also studied

the the influence of Kerr non-linearity on light pulse propagation. The posi-

tive as well as negative group index increased via increasing the strength of

Kerr field. At Ωk = 10γ the group indeces are 5 × 104 and −1.6 × 105 for

normal and anomalous dispersion, respectively. It is found that the group

index for normal and anomalous dispersions can be increased via increasing

the strength of Kerr field. The increase of group index clearly shows that

by increasing the strength of Kerr field a more slow group velocity can be

92

achieved.

Next, we considered a four-level lambda-type configuration of 87Rb atoms

and study the weak pulse propagation inside the medium. As previously in-

vestigated that the Kerr field influenced the group index of the medium. In

the present scheme we incorporated the Kerr as well as the Doppler broaden-

ing to monitor the light pulse propagation through the medium. we expected

control over the group index of an EIT process via a Kerr field and Doppler

broadening which leads to a slower group velocity inside the medium. Ini-

tially, we studied the influence of Kerr field and Doppler broadening on light

pulse propagation through the medium separately. It is found that the group

index increased via Kerr field as well as by Doppler broadening. To achieve

a more slow group velocity inside the medium next we considered both the

Kerr field and Doppler broadening and studied the behavior of light pulse

propagation through the medium. To see a more clear picture of light pulse

propagation inside the medium when both Kerr field and Doppler broaden-

ing are considered. The behavior of group index of the medium for different

choices of intensity of Kerr field is plotted for two cases i.e., without Doppler

broadening and with Doppler broadening. We noticed that the group index

is 1.2 × 106 at Ωk = 10γ when there is no Doppler broadening. The group

index increased from 1.2 × 106 to 4.7 × 107 at Ωk = 10γ for the case when

Doppler broadening is considered. This clearly shows that the group velocity

decreases more with including both Kerr field and Doppler broadening.

As discussed earlier, that Kerr field and Doppler broadening affects the

group velocity of light through the medium. In the present scheme we stud-

ied the influence of the Kerr field along with SGC on the propagation of light

pulse through N -type atomic medium. The Kerr field enhances the group

index of the medium, which leads to slow group velocity. The separate effect

93

of SGC on the propagation of light pulse bring an appreciable change to the

transparency window. The narrow transparency window is found with in-

creasing SGC parameter. The SGC increases the group index of the medium

and we have observed 20×103 for maximum SGC. To attain more slow group

velocity, we incorporated both Kerr and SGC effects inside the medium. It is

found that the group index increase Via Kerr field along with SGC, which re-

sulted the slow group velocity through the medium. To inestigate more clear

picture of the propagation of light pulse through the medium, we plotted the

group index with Kerr field for two cases i.e., without SGC and with SGC.

The value of group index at Ωk = 10γ is found to be 8×103, when there is no

SGC effect involve in the medium. The value of group index increased from

8 × 103 to 1.8 × 107 at Ωk = 10γ, when SGC effect in the medium is taken

into account. It clearly shows that an ultra-slow group velocity can be found

by incorporating both Kerr field and SGC simultaneously in the medium.

Next, we considered an experimental model consist of four-level tripod-

configuration with initial atomic state in the D1 line of 87Rb atom and mon-

itor the propagation of weak pulse through the medium. Previously, we

discussed the light pulse propagation via Kerr field, Doppler broadening and

SGC, which have reduced the group velocity of light through various atomic

medium. A strong control over light pulse propagation can be obtained by

considering two Kerr fields in a single atomic medium. The single Kerr field

also enhances the group index of the this medium as discussed earlier for the

other atomic medium. It is found that a high group index of the medium

can be observed by taking two Kerr fields of the same strength. To present a

clear picture over the control of light pulse propagation, we plotted the group

index of the medium with single and two Kerr fields of the same strength.

The value of the group index is found to be 4.7 × 105 at Ωk1 or Ωk2 = 10γ.

94

The value of the group index of the medium increased from 4.7 × 105 to

2.7 × 109, when two Kerr Fields of the same strength i.e., Ωk1= Ωk2 = 10γ

are applied in the medium. The group velocity of light is reduced to 0.11

m/s in the presence of two Kerr fields. It clearly shows that the light pulse

can be halted or stopped by considering two Kerr fields in a single medium.

95

Appendices

96

Appendix1

B = Ω2k(4CD − Ω2

1 + Ω2k) + 16(γ3 + i∆1 − i∆p)(D(EC + Ω2

1/4) + 1/4EΩ2k)

+Ω21(4EC + F ).

Where

C = (γ1 + γ2 − i∆p),

D = (γ1 + γ2 + γ3 − i∆p),

E = (Γ + i∆1 − i∆p),

and

F = (Ω1 − Ω2)(Ω1 + Ω2).

97

Appendix2

Here, we consider the details of solving Eq. (4.20). We follow the same

method as has been used in [148, 159], saharai-four-level. We can write Eq.

(4.20) in the form as

R = −MR + C, (1)

where R, C and M are the column vectors and matrix, respectively, as given

below

R =(

ρbc ρac ρbd ρad ρbb ρab ρba

)T

,

C =(

iΩp 0 0 0 0 0 0)T

,

M =

−i∆p + γ1 + γ2 −iΩ1 iΩk 0 0 0 0

−iΩ1 i(∆1 − ∆p) + Γ 0 iΩk −2q√

γ1γ2 0 0

iΩk 0 −i(∆p − ∆k) −iΩ1 0 0 0

0 iΩk −iΩ1 i(∆1 − ∆p + ∆k) + γ3 0 0 0

0 0 0 0 γ1 + γ2 −iΩ1 iΩ1

0 0 0 0 −iΩ1 i∆1 + γ1 0

0 0 0 0 iΩ1 0 −i∆1 + γ1

.

Now the formal solution of such an equation can be written as

R(t) =

∫ t

−∞

e−M(t−t)Cdt = M−1C. (2)

We use Eq. 2 and get the solution for ρbc which is given in 4.23.

98

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