slow light, stopped light and guided light in hot rubidium vapor

Slow Light, Stopped Light and Guided Light in Hot

Rubidium Vapor Using Off-resonant Interactions

by

Praveen Kumar Vudya Setu

Submitted in Partial Fulfillment

of the

Requirements for the Degree

Doctor of Philosophy

Supervised by

Professor John C. Howell

Department of Physics and Astronomy

Arts, Sciences and EngineeringSchool of Arts and Sciences

University of RochesterRochester, New York

2011

ii

Dedicated to my parents.

iii

Curriculum Vitae

The author was born in Kurnool, Andhra Pradesh, India on 4 January, 1983.

He attended the Indian Institute of Technology, Kharagpur from 2000 to 2005

and obtained his Bachelor of Science and Master of Science degrees in Physics.

He came to the University of Rochester in the Fall of 2005 for graduate studies

in Physics and received a Master of Arts degree in 2007. He pursued his doctoral

research in atomic physics and quantum optics under the supervision of Professor

John C. Howell.

iv

Publications

“A double Lorentzian atomic prism”, P. K. Vudyasetu, S. M. Bloch, D. J.

Starling, J. S. Choi and J. C. Howell, Physical Review Letters (Submitted).

“Rapidly reconfigurable slow-light system based on off-resonant Raman ab-

sorption”, P. K. Vudyasetu, R. M. Camacho and J. C. Howell, Physical Review A

82, 053807 (2010).

“Interferometric weak value deflections: Quantum and classical treatments”,

J. C. Howell, D. J. Starling, P. Ben Dixon, P. K. Vudyasetu and A. N. Jordan,

Physical Review A 81, 033813 (2010).

“All Optical Waveguiding in a Coherent Atomic Rubidium Vapor”, P. K.

Vudyasetu, D. J. Starling and J. C. Howell, Physical Review Letters 102, 123602

(2009).

“Four-wave-mixing stopped light in hot atomic rubidium vapour”, R. M. Ca-

macho, P. K. Vudyasetu and J. C. Howell, Nature Photonics 3, 103 (2009).

“Storage and Retrieval of Multimode Transverse Images in Hot Atomic Rubid-

ium Vapor”, P. K. Vudyasetu, R. M. Camacho and J. C. Howell, Physical Review

Letters 100, 123903 (2008).

“Slow-Light Fourier Transform Interferometer”, Z. Shi, R. W. Boyd, R. M.

Camacho, P. K. Vudyasetu and J. C. Howell. Physical Review Letters 99, 240801

(2007).

v

Conference Presentations

“Four Wave Mixing (FWM) and Electromagnetically Induced Transparency

(EIT) Based Coherent Image Storage in Hot Atomic Vapors”, P. K. Vudyasetu,

R. M. Camacho, and J. C. Howell, CLEO/QELS 2008 paper: QThB3 (oral pre-

sentation).

“Storage and Retrieval of Images in Hot Atomic Rubidium Vapor”, P. K.

Vudyasetu, R. M. Camacho, and J. C. Howell, Slow and Fast Light (SL) 2008

paper: SWD4 (oral presentation).

“Storing and Manipulating Multimode Transverse Images in Hot Atomic Va-

pors”, P. K. Vudyasetu, D. J. Starling, R. M. Camacho, and J. C. Howell, Laser

Science (LS) 2008 paper: LWD3 (oral presentation).

“Fast Reconfigurable Slow Light System based on Off-resonant Raman Ab-

sorption Scheme”, P. K. Vudyasetu, R. M. Camacho, and J. C. Howell, Frontiers

in Optics (FiO) 2010 paper: FThN5 (oral presentation).

vi

Acknowledgments

This thesis would not have been possible without the help of many individuals.

First and foremost, I would like to thank my thesis supervisor, Prof. John C.

Howell, for his invaluable guidance through the highs and lows of research over

past 5 years. His enthusiasm for research is infectious and made working in the

laboratory more enjoyable. I cherish his patience and his commitment towards

the progress of my graduate education, which has culminated in this thesis.

I wish to acknowledge my mentors who helped me acquire key research skills.

I would like to thank Michael V. Pack for teaching me optics experimental skills.

He was the man with answers for whatever questions I had, and I learned a great

deal from him about EIT experiments. He was very generous in helping me with

MATLAB and in lending me his books. I enjoyed working with Ryan M. Camacho

on several projects. He taught me the value of good presentations of research ideas

and I have learned a lot about scientific writing from him. He was always there

with help and kind words of wisdom whenever I needed it.

I would like to thank my lab-mates for all their help. I was very fortunate

to work with Curtis J. Broadbent, David J. Starling, Gregory Armstrong, Steven

M. Bloch and Joseph Choi on various projects, who were all fun to work with

and I have learned something or the other from each of them. I, also, had an

vii

opportunity to work with Zhimin Shi from Prof. Boyd’s group and I thank him

for the discussions on slow light devices.

I would like to extend my sincere gratitude to the university community for

all the warmth and terrific hospitality without which traveling from a far away

country for studies would not have been so wonderful. In particular, I would like

to thank Barbara Warren, Shirley Brignall, Connie Hendricks, Michie Brown and

Ali DeLeon of the Physics department and the staff at the International Services

Office, with whom I have interacted on numerous occasions and they were always

helpful.

I, also, would like to acknowledge the support from the funding agencies

DARPA DSO Slow Light and NSF.

Finally, I would like to thank my family and friends for being supportive and

encouraging. I thank my parents for all the sacrifices they made to ensure my

success.

viii

Abstract

This thesis presents the applications of some of the coherent processes in a

three-level atomic system, to control spatial and temporal properties of a signal

pulse. We use two Raman absorption resonances in rubidium vapor separated by a

few MHz to achieve a rapidly tunable slow-light system. We control the slow-light

characteristics all-optically by tuning the frequency and power of a coupling beam.

A dual absorption slow-light system is known to cause less pulse broadening than

a single transmission resonance system, and thus, a tunable double absorption

system is advantageous. We use a four-wave mixing process to demonstrate pulse

storage in rubidium vapor for times much greater than the pulse width. We

demonstrate storage of both the temporal and spatial profile of the pulse. We

overcome the diffusion of spatial information during the storage in warm atomic

vapor by storing the Fourier transform of the image instead of an image with a

flat phase. The Raman absorption resonance is also used to control the transverse

refractive index profile of the signal beam. The refractive index of the signal

interacting with a coupling beam in a Raman process is dependent on the coupling

beam intensity. We use a first order Laguerre-Gaussian (LG01) coupling beam

to create a waveguide like transverse refractive index profile. We demonstrate

propagation of a focused signal beam for lengths much greater than the Rayleigh

length. Finally, we demonstrate a dual absorption atomic prism, which is capable

ix

of spatially separating spectral lines that are 50 MHz apart and which can precisely

measure frequency fluctuations. This simple prism is a valuable spectral filtering

tool for a variety of atomic experiments.

x

Table of Contents

Curriculum Vitae iii

Acknowledgments vi

Abstract viii

List of Figures xii

Foreword 1

1 Background 3

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Interactions in a Lambda System . . . . . . . . . . . . . . . . . . 7

1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Double Raman Absorption Slow Light 27

2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 33

xi

2.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Stopped Light Using Four Wave Mixing 44

3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46



3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4 All Optical Waveguiding Using Raman Absorption 61

4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61



4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 Slow Light Prism Spectrometer 78

5.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79



5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6 Conclusions 89

Bibliography 94

A Derivation of Susceptibility for a Lambda System 109

xii

List of Figures

1.1 Various configurations for the interaction of three atomic energy

levels and two optical fields. . . . . . . . . . . . . . . . . . . . . . 5

1.2 Three level Λ system . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Lines shapes for various detunings . . . . . . . . . . . . . . . . . . 10

1.4 Four wave mixing in double-Λ system . . . . . . . . . . . . . . . . 14

2.1 Dependence of C1 and C2 on ∆ . . . . . . . . . . . . . . . . . . . 29

2.2 (a) & (b) The energy level diagram for vapor cells VC1 and VC2

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3 The outline of the setup used in the experiment. . . . . . . . . . . 34

2.4 Variation of absorption line shapes with coupling beam power. . . 36

2.5 Variation of fit parameters with coupling beam power. . . . . . . 36

2.6 Transmission of the signal through a vapor cell in the presence of

two coupling beams differing in frequency.Green curve is a refer-

ence absorption with single coupling beam frequency. The cou-

pling beams are separated in frequency by 1 MHz and 2 MHz for

the transmission curves in red and blue respectively. . . . . . . . . 37

xiii

2.7 The signal pulse out of a vapor cell with two coupling beams dif-

fering in frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.8 The transmission profile of the probe corresponding to the observed

delay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.9 Reference pulse and the delayed pulse. . . . . . . . . . . . . . . . 39

2.10 Measurement of turn-on and turn-off times of Raman absorption. 41

2.11 Beating in the cw signal beam resulting from coupling beam fre-

quency change. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.12 The experimental observation of slow light switching. . . . . . . . 42

3.1 (a) Experimental setup. (b) Four wave mixing energy levels. (c)

Representation of the synchronized timing of signal (dashed red),

delayed signal (blue) and coupling (black) beams. . . . . . . . . . 50

3.2 Measured (asterisks) and theoretical (solid) steady state signal (blue)

and idler (red) intensities as a function of signal detuning. . . . . 52

3.3 Demonstration of four-wave mixing slow light and stored light . . 54

3.4 Idler pulse peak power as a function of coupling input power. . . . 55

3.5 CCD camera capture of the signal intensity profile at the object

plane and at the vapor cell (Fourier plane). . . . . . . . . . . . . . 56

3.6 Input signal profile (a) and the time evolution of measured (b) and

calculated (c) transverse images. . . . . . . . . . . . . . . . . . . 57

3.7 Theoretical time evolution of stored ground state coherence of Rb

atoms. The inset shows a close up of the time evolution near zero

crossover points of electric field amplitude. . . . . . . . . . . . . . 58

xiv

4.1 Beam propagation as a function of κ . . . . . . . . . . . . . . . . 63

4.2 Beam propagation as a function of coupling rabi frequency . . . . 66

4.3 Beam propagation as a function of Raman detuning . . . . . . . . 66

4.4 The experimental schematic for all optical waveguiding using atomic

rubidium vapor. The focusing scheme for control beam (black) and

signal (gray) is shown in the inset . . . . . . . . . . . . . . . . . . 69

4.5 The experimental plot of the variation in the transmission of signal

versus Raman detuning. . . . . . . . . . . . . . . . . . . . . . . . 70

4.6 The plot of refractive index of the signal, tuned -1.5 MHz away

from Raman resonance, as a function of coupling beam power. . . 71

4.7 (a) shows the spatial variation of refractive index. (b) shows the

plot of refractive index versus position along one of the axes. . . . 71

4.8 The Snap shots of the signal beam profile at the back of the vapor

cell with the coupling beam off (a) and on (b). Beam profiles along

the longer axis of the beams at the front face of the cell (c) and at

the back face of the cell(d). . . . . . . . . . . . . . . . . . . . . . 72

4.9 Plots of the signal beam size at the back face of the vapor cell versus

the control beam power . . . . . . . . . . . . . . . . . . . . . . . . 74

4.10 Plots of the signal beam size at the back face of the vapor cell versus

the Raman detuning . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.11 The plot of output signal power versus Raman detuning. . . . . . 75

4.12 The plot of output signal power versus input signal power. The

plot is nearly linear, the slope of the linear fit to data is 0.43. . . . 75

xv

5.1 The rubidium vapor cell can be approximated as a dispersing prism.

For our setup θp ≈ 79o, θ1 ≈ 20o and thus the geometrical param-

eter A is about 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2 A schematic of the experimental setup. . . . . . . . . . . . . . . . 81

5.3 The camera images for different modulation frequencies. . . . . . 83

5.4 One dimensional intensity scans for different modulation frequencies. 84

5.5 Plot of deflection as a function of frequency. Circles represent the

experimental data and solid line is the linear fit. The slope of the

line is 1.95 µm/MHz. . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.6 (a) Four wave mixing beams are split at the prism and imaged at

the camera. (b) The one dimensional scan of the intensity profile. 86

1

Foreword

All the research included in this thesis was done under supervision of my thesis

advisor Prof. John C. Howell. In addition, several graduate students collaborated

with me on projects which resulted in this work. Their individual contribution to

this thesis is elaborated in the following paragraphs.

The research in chapter 2 was performed in collaboration with Dr. Ryan M.

Camacho and Prof. John C. Howell. Ryan and I worked on experiments which

resulted in figures 2.8, 2.9 and 2.12. Ryan created all three of these figures. Rest of

the work, both experimental and analytical, was done by me with guidance from

both Ryan and Prof. Howell. Most of this work was published in Physical Review

A 82, 053807 (2010), which I wrote with the help of Ryan and Prof. Howell.

The research in chapter 3 was performed in collaboration with Dr. Ryan M.

Camacho and Prof. John C. Howell. Ryan and I worked on experiments which

resulted in all the experimental plots. Ryan created figures 3.1(b), 3.2, 3.3 and

3.4 and the rest of the figures were created by me. This chapter was adapted

from two published journal papers, Nature Photonics 3, 103 (2009) and Physical

Review Letters 100, 123903 (2008). The Nature Photonics paper was written by

Ryan with inputs from me and Prof. Howell and the PRL paper was written by

me with inputs from Ryan and Prof. Howell.

2

The research in chapter 4 was performed in collaboration with David J. Starling

and Prof. John C. Howell. David and I worked on experiments which resulted in

all the experimental plots. I created all the figures in this chapter. Most of this

work was published in Physical Review Letters 102, 123602 (2009), which I wrote

with the help of David and Prof. Howell.

The research in chapter 5 was performed in collaboration with Steven M.

Bloch, David J. Starling, Joseph S. Choi and Prof. Howell. Steve worked on the

experimental setup with the help of Joe. I worked on experiments with Steve,

David and Joe which resulted in all the experimental plots. I created all the

figures in this chapter. I wrote a paper based on this work with the help of Steve,

David, Joe and Prof. Howell, which we submitted to Physical Review Letters and

is under review.

3

1 Background

1.1 Introduction

The study of interactions between optical fields and matter has a long history

and has resulted in the discovery of many physical processes. The optical prop-

erties of matter such as absorption, reflection, refraction, scattering, dispersion

etc., can all be explained by the properties of the atoms and molecules with which

the light is interacting. The advent of quantum mechanics resulted in the under-

standing of atomic structure and electronic energy levels inside an atom [Schiff,

1968]. The quantum nature of light was used to explain physical processes like

black body radiation, photoelectric effect, etc. It was postulated by Bohr that an

electron jumps from one electronic state of an atom to another electronic state

by absorbing or emitting one quantum of energy hν. The quantum theory also

enabled us to understand the spectroscopic properties of various elements. A

thorough understanding of matter-light interactions requires us to quantize both

the optical field and matter. However, one can explain many interesting phenom-

ena in a semi-classical regime wherein we quantize just the matter and consider

4

the optical field as a classical electromagnetic wave satisfying Maxwell’s equa-

tions. Following this approach one can derive the refractive index of a material

as a function of light frequency, obtain the absorption coefficient and understand

many nonlinear optical processes [Boyd, 2003].

The invention of laser provided optical physicists a powerful, highly coherent,

nearly monochromatic source which opened up the fields of quantum optics and

nonlinear optics. A single mode laser interacting with a two-level atom leads to

several interesting phenomena like resonance florescence, Autler-Townes splitting,

Ramsey fringes, self-induced transparency etc [Allen and Eberly, 1987]. A power-

ful laser can also lead to nonlinear effects like parametric down conversion, higher

harmonic generation, Kerr effect, Raman effect etc [Boyd, 2003]. It also proved a

highly effective spectroscopic tool enabling several techniques like saturation ab-

sorption spectroscopy, time-resolved laser spectroscopy, laser-induced florescence

spectroscopy etc [Demtroder, 2002].

While the coherent interactions of an optical field with a two-level atom lead us

to many applications, the interactions of multiple optical frequencies with multiple

levels of atom result in an even richer variety of physical phenomenon. In such

a system, the optical properties of the medium can be altered by one or many

optical fields. The simplest of such a multi-level system is a three-level atom

interacting with two optical frequencies where the optical interaction between two

of the energy levels of the atom is forbidden. The three-level system can be in the

form of a cascade, a “V” or a “Λ” shaped energy diagram as shown in 1.1.

The three-level system has been used to demonstrate various phenomena.

Atomic cascade systems have been used to demonstrate selective reflection of

the probe beam in a pump-probe scheme [Schuller et al., 1993; Amy-Klein et al.,

5

Cascade/

LadderV Λ

E1

E2

E1 E2 E2E1

1

2

3

1

2

3

1

2

3

Figure 1.1: Various configurations for the interaction of three atomic energy levelsand two optical fields.

1995]. Effects like electromagnetically induced transparency (EIT) and lasing

without inversion have also been demonstrated in a three-level cascade atomic

system [Xiao et al., 1995; Sellin et al., 1996]. Based on the polarization interfer-

ence between one photon and two photon processes in such a system, ultrafast

modulation spectroscopy has been demonstrated in sodium vapor [Fu et al., 1995].

Lasing without inversion has also been predicted in a V type three-level system

[Zhu, 1992]. It was shown that spontaneous emission can be canceled in a V type

system, resulting in inversion without lasing [Scully et al., 1989]. Comparison of

electromagnetically induced transparency in cascade, V and Λ types systems has

been carried out by Fulton et al. [Fulton et al., 1995].

This thesis focuses on the applications of interactions in three-level lambda

system for slow light and stopped light. It was shown by Gray et al. [Gray et al.,

1978] that the population of the three-level lambda system can be trapped in

the lower ground states, an effect known in literature as “coherent population

trapping” (CPT). One can use magnetic sub-levels or hyperfine levels and an

excited state of alkali atoms to form a lambda system. The three-level lambda

6

system has been studied extensively in the context of electromagnetically induced

transparency (EIT) [Fleischhauer et al., 2005]. The narrow resonance feature of

EIT results in steep dispersion which has been used to obtain ultra-slow group

velocities [Harris et al., 1992; Hau et al., 1999]. The ground state atomic coherence

induced by two optical fields is long lived and this property has been exploited to

store optical information [Phillips et al., 2001]. Using appropriate pulse shapes of

the two optical fields, one can coherently transfer populations between two ground

states by stimulated Raman adiabatic process (STIRAP) [Bergmann et al., 1998].

Coherent preparation of the atomic lambda system also proved helpful in en-

hancing several nonlinear processes in the atomic systems. Harris et al. have

shown that a four-wave mixing process can be enhanced by electromagnetically

induced transparency [Harris et al., 1990]. Preparation of the lambda system in a

maximally coherent state can enhance the frequency conversion efficiencies [Jain

et al., 1996]. Hemmer et al. demonstrated an efficient phase conjugation scheme

using coherent population trapping [Hemmer et al., 1995]. Schmidt and Imamoglu

proposed a scheme for a giant cross phase modulation making use of the steep

dispersion resulting from electromagnetically induced transparency [Schmidt and

Imamoglu, 1996]. This giant Kerr effect means that one can achieve large phase

shift in one optical mode using only few photons of another optical mode. In fact

it is predicted by Lukin and Imamoglu that two slowly propagating photons can

induce large phase shift on each other [Lukin and Imamoglu, 2000].

This chapter is organized as follows. In section 1.2, we introduce a lambda

system and briefly discuss some of the interactions relevant to this thesis like

electromagnetically induced transparency, Raman absorption and four-wave mix-

ing. In section 1.3 we provide the motivation behind the thesis. We introduce

7

1

2

3

∆

δ

Γ/2 Γ/2

γ

Signal Coupling

Figure 1.2: Three level Λ system

the concepts of slow light stopped light and all-optical waveguide and discuss the

advantages of using off-resonant interactions in rubidium for these problems. In

section 1.4 we present an outline for the thesis.

1.2 Interactions in a Lambda System

In this section we discuss some of the consequences of coherent interactions of

two optical fields with three-level lambda system. We first introduce a lambda

atomic system. The energy level diagram of a lambda system is shown in Fig.

1.2. The two lower energy levels are optically coupled to an excited state by the

“signal” and the “coupling” beams. The dipole interaction between the two lower

energy levels is forbidden. The excited state life time is given by 1/Γ while the

ground state decoherence rate is given by γ. In a typical lambda system, Γ ≫ γ.

The signal of frequency ωs is detuned from 1 → 2 transition frequency ω12

by the single photon detuning ∆. The coupling frequency, ωc, is detuned from

3 → 2 transition frequency ω32 by ∆− δ where δ is the two photon detuning. The

two photon effects are usually observed when δ is of the order of γ. The Rabi

8

frequencies of signal and coupling beams are denoted by Ωs and Ωc respectively.

For the remainder of the thesis we assume that Ωs ≪ Ωc. We also assume that

the atom is initially in the state |1〉. Under these assumptions the susceptibility

at the signal frequency is given by

χ(∆, δ,Ωc) = βδ − iγ

(δ − iγ)(∆− iΓ/2)− |Ωc|2/4, (1.1)

where β = Nµ2/hǫ0 where N is the number density and µ is the transition dipole

moment. The above equation has been derived in the appendix A. The above

equation for susceptibility can be used to explain many physical phenomena in

this thesis and is a good starting point for the discussion that follows in this

section. We note that in the limit of Ωc = 0, we obtain the usual susceptibility

for a two level system.

The real and imaginary parts of the susceptibility, χ′ and χ′′, can be written

as,

χ′ = β∆

∆2 + Γ2/4

[

1 +

(

2δ0 −|Ωc|24∆

)

δ′

δ′2 + γ′2− 2γ0

γ′

δ′2 + γ′2

]

, (1.2)

χ′′ = βΓ/2

∆2 + Γ2/4

[

1 + 2δ0δ′

δ′2 + γ′2−

(

2γ0 −|Ωc|22Γ

)

γ′

δ′2 + γ′2

]

. (1.3)

In the above, δ0 = |Ωc|2∆/(4∆2 + Γ2), γ0 = |Ωc|2Γ/(8∆2 + 2Γ2), δ′ = δ − δ0

and γ′ = γ + γ0. The first terms in each of the above equations are the real

and the imaginary parts of the susceptibility χ0 when the coupling beam is off.

The second and the third terms have dispersive and absorptive functional forms.

The single photon detuning ∆ determines how each term contribute to the total

susceptibility.

We can observe many qualitative features of signal transmission from the above

9

equations. There is a broad absorptive feature of width Γ/2 resulting from the

signal interaction with 1 → 2 transition. Within that feature there is another

narrow feature of width γ′ centered at δ = δ0. The narrow feature resulting from

two photon interaction can be transmissive or absorptive depending on ∆.

Consider the case where ∆ = 0. The real and imaginary parts of the suscep-

tibility at signal frequency for this case are given as,

χ′(∆ = 0) = −β|Ωc|2Γ2

δ

δ2 + (γ + |Ωc|2/2Γ)2, (1.4)

χ′′(∆ = 0) =2β

Γ

[

1− |Ωc|22Γ

γ + |Ωc|2/2Γδ2 + (γ + |Ωc|2/2Γ)2

]

. (1.5)

The first term in the imaginary part of the susceptibility corresponds to absorption

of signal in the absence of the coupling beam. The coupling beam results in

decrease in absorption. The result is a Lorentzian transparency profile in the

background of absorption with width given by γ + |Ωc|2/2Γ. Here |Ωc|2/2Γ is

power broadening term and in the case where power broadening is not dominant,

the width of transparency is given by the ground state decoherence rate γ. This

coupling beam induced transparency is known in literature as electromagnetically

induced transparency (EIT). Choosing appropriate coupling beam intensity, it is

possible to achieve zero absorption for δ = 0.

Now consider the case where ∆ >> Γ. In this case the susceptibility can

be approximated as zero for the signal frequency with no coupling beam. With

the coupling beam, the real and imaginary parts of the susceptibility at signal

10

−100 0 100

0.5

1

0−100 0 100

−1

0

1

−100 0 1000

2

4

x 10−3

−100 0 100

−2

0

2

x 10−3

−5 0 50

3

x 10−5

−5 0 5

−7

−6

−5

x 10−5

Real(χ)Imag(χ)

Frequency (KHz) Frequency (KHz)

∆ = 0

∆ = 100xΓ/2

∆ = Γ/2

(a)

(f)

(e)

(d)

(c)

(b)

Figure 1.3: Lines shapes for various detunings

frequency are approximately given by

χ′(∆ >> Γ) = β|Ωc|24∆2

δ′

δ′2 + γ′2, (1.6)

χ′′(∆ >> Γ) = β|Ωc|24∆2

γ′

δ′2 + γ′2. (1.7)

The absorptive line profile is approximately Lorentzian, with a width of γ′ and the

strength proportional to |Ωc|2/∆2. The line center is determined by the frequency

of the coupling beam, and the width and the depth of absorption are determined

by the intensity of the coupling beam. Thus, by changing the properties of the

coupling beam one can alter the properties of the absorption. We call this coupling

beam induced absorption as Raman absorption.

11

The imaginary and real parts of susceptibility for various single photon de-

tunings are plotted in Fig. 1.3. The plot of imaginary part of susceptibility as a

function of δ show that for ∆ = 0, we have a Lorentzian transmission feature, for

∆ = Γ/2 the transmission profile line shape is dispersive and for ∆ ≫ Γ/2, we

have a Lorentzian absorption.

1.2.1 Electromagnetically Induced Transparency

We have seen in the previous discussion that for ∆ = 0, the signal experi-

ences electromagnetically induced transparency. EIT was investigated first by

Harris in 1990 [Harris et al., 1990]. It was shown that EIT can enhance nonlinear

susceptibility while reducing signal absorption. EIT was soon demonstrated in

strontium vapor [Boller et al., 1991] and in lead vapor [Field et al., 1991]. It was

pointed out that the narrow transmission feature of EIT is also accompanied by

steep linear dispersion which results in optical pulse delays [Harris et al., 1992;

Xiao et al., 1995; Kasapi et al., 1995]. It was also shown that signal and coupling

pulses of arbitrary shape evolve into a matched pulse shape after propagating cer-

tain characteristic distance through EIT medium [Harris, 1993; Harris, 1994]. Hau

et. al demonstrated ultra slow propagation (17 m/s) of signal pulses in an ultra

cold atomic gas [Hau et al., 1999]. Enhancement of various nonlinear processed by

EIT has been investigated by various groups (for review, see [Fleischhauer et al.,

2005]).

In order to have qualitative understanding of EIT, consider the three-level

lambda system shown in Fig. 1.2. The coherence ρ12 can be written in the form

12

of a geometric series as follows.

ρ12 =Ωs

2

1

∆− iΓ/2× (1.8)

[

1 +

(

Ωc

2

1

δ − iγ

Ω∗c

2

1

∆− iΓ/2

)

+

(

Ωc

2

1

δ − iγ

Ω∗c

2

1

∆− iΓ/2

)2

+ ...

]

.

The first term in the above represents the probability for the transition 2 → 1. The

subsequent terms represents the probability for the transitions 2 → 3 → 2 → 1,

2 → 3 → 2 → 3 → 2 → 1, and so on. The combination of all the terms for ∆ = 0

and δ = 0 results in the atomic coherence,

ρ12 =iΩs

Γ

(

1− |Ωc|2/2Γγ + |Ω|2/2Γ

)

. (1.9)

For γ << |Ω|2/2Γ, ρ12 = 0. Thus, for the resonant lambda system, the polariza-

tion at the signal frequency is zero when Raman resonance condition is satisfied.

This means that the signal will propagate as if the medium is absent.

This effect can also be understood in a dressed state picture. It has been

shown that, in an appropriate basis, the ground states can be written in terms of

a “bright-state” and a “dark-state” [Fleischhauer and Manka, 1996]. In this basis,

only bright state is coupled to the excited state by an effective Rabi frequency

while the dark state is sees no interaction. If the medium is prepared in such a

dark state, the signal is completely decoupled from atoms and hence propagates

without any losses.

One characteristic feature of EIT is the existence of dark state polaritons

[Fleischhauer and Lukin, 2000]. It was shown that the quantum signal field in the

EIT medium propagates as quasi-particles and that these particles, when trapped

in the medium, transfer their pulse shape and quantum state to the atomic ground

13

state coherence. Trapping of the dark state polariton occurs when the coupling

beam is adiabatically turned off. One can retrieve the signal by turning on the

coupling beam at a later time. Pulse storage was first demonstrated in a cold

atomic cloud by Liu et. al [Liu et al., 2001], in warm atomic vapor by Phillips et.

al [Phillips et al., 2001] and in a solid by Turukhin et. al [Turukhin et al., 2001].

The storage of quantum states of light was demonstrated by Van Der Wal et. al

[van der Wal et al., 2003].

1.2.2 Raman Absorption

For the far detuned case, the system can be treated like a two-level system

[Gerry and Eberly, 1990]. If Ωs and Ωc are Rabi frequencies of signal and coupling

fields respectively, the effective Rabi frequency coupling level 1 and level 3 for far

detuned case is ΩsΩc/∆. We observe Lorentzian absorption line shapes similar

to two-level atoms and it has been predicted that the three-level system produce

similar non-classical effects as a two-level atom.

The Raman absorption resonance in an off-resonant lambda system received

considerable attention recently in the context of slow light and fast light research.

Mikhailov et al. have reported pulse advancements of about 300 µs when the signal

pulse is tuned to the center of Raman absorption (However, the signal pulse is

attenuated because of absorption) [Mikhailov et al., 2004]. Knappe et al. have

studied signal line shapes for the off-resonant lambda system in terms of simple

parameters [Knappe et al., 2003]. More recently these absorption were used to

achieve gradient echo memory (GEM) [Hetet et al., 2008]. In this scheme the

Raman resonance is broadened using a magnetic field that varies linearly along

the light propagation direction and reversing that gradient results in photon echo.

14

1

2

3

δ

Γ/2 Γ/2

γ

Coupling

Signal Idler

∆1∆2

Figure 1.4: Four wave mixing in double-Λ system

Reim et al. demonstrate a memory capable of more than a GHz signal bandwidth

using Raman resonance [Reim et al., 2010].

1.2.3 Four-wave Mixing

Consider the double lambda system shown in Fig. 1.4. The coupling beam in

the lambda system discussed in the previous sections can also act on the ground

state |1〉 and scatter idler photons. The four-wave mixing process (FWM) grows

with propagation distance and at high optical densities it can dominate the other

processes discussed previously. The transmission line shape of the signal under

FWM conditions look like a single Lorentzian transmission, like EIT, and the

dispersion within the gain profile can give rise to slow light.

Four-wave mixing in the double lambda configuration has been studied by

various research groups. Generation of squeezed states of light using four-wave

mixing was proposed by Reid and Walls [Reid and Walls, 1985]. It was soon

demonstrated in sodium vapor by Slusher et. al [Slusher et al., 1985]. Four wave

mixing has also been studied in optical fibers and many applications like frequency

15

conversion and amplification are based on degenerate four-wave mixing [P., 2001].

More recently, four-wave mixing in rubidium vapor was used and strong relative

intensity squeezing was achieved [McCormick et al., 2007; McCormick et al., 2008].

It was also shown that the signal and idler pulsed propagate slowly in such a

medium [Boyer et al., 2007]. Quantum mechanical aspects of FWM were also

demonstrated [Marino et al., 2009; Pooser et al., 2009a; Pooser et al., 2009b].

1.3 Motivation

Controlling optical properties of a material is desirable for many applications.

For example, the control of refractive index of some crystals by the electric field

is an enabler for high speed optical modulators. This work deals with all-optical

control of temporal and spatial propagation of optical pulses in rubidium vapor.

By optically tuning the dispersion characteristics of rubidium vapor we can control

the group velocity of an optical signal and the transverse shape of the beam as it

propagates through the medium. This optical control has been applied to achieve

slow light, stopped light and guided light.

1.3.1 Slow Light

The field of slow light research deals with the control of the group velocity

of an optical pulse through various means. In a dispersive medium the group

velocity is given by

vg =c

n0 + ν dndν

, (1.10)

where n0 is the refractive index at the central frequency of the pulse, ν0. We call

the denominator of the right hand side of the above equation as group index ng.

16

For highly dispersive material we can approximate ng as ν dndν. Controlling ng by

various means, we can control the group velocity of the pulse. For positive ng,

the pulse is delayed and hence we call it slow light. Similarly, for negative values

of ng, we will have fast light.

One can obtain very large group indices in the vicinity of a sharp transmission

at the signal frequency. Such sharp transmissions can be achieved using variety of

nonlinear effects in atomic media [Boyd, 2003] or by fabricating specially designed

photonic structures [Baba, 2008]. In this work we deal with the former case, that

is, obtaining slow-light by tailoring the refractive index of the medium. From

the Kramers-Kronig relations we know that sharp change in the refractive index

within a frequency range is accompanied by sharp change in the transmission

properties at those frequencies. Thus narrow spectral features result in slow or

fast light.

The ability to control the group velocity of a light pulse is central to the

practical realizations of all optical communication systems [Ku et al., 2002]. After

an initial demonstration of ultra slow group velocities of light pulses by Hau et al.

[Hau et al., 1999], several groups have demonstrated slow-light in a wide variety of

media like atomic vapors [Kash et al., 1999; Budker et al., 1999; Camacho et al.,

2006; Harris et al., 1992; Kasapi et al., 1995; Budker et al., 1999; Hau et al., 1999;

Liu et al., 2001; Yang et al., 2007; Boyer et al., 2007], solid materials [Turukhin

et al., 2001; Bigelow et al., 2003a; Bigelow et al., 2003b; Khurgin, 2005; Gehring

et al., 2006; Zhu and Gauthier, 2006a; Shumakher et al., 2006], optical fibers

[Okawachi et al., 2005], liquid crystals [Residori et al., 2008], photonic crystals

[Baba, 2008] etc. Slow-light has applications in many diverse fields like optical

communications [Ku et al., 2002], interferometry [Shi et al., 2007; Shi and Boyd,

17

2008], sensing [Shahriar et al., 2007], etc.

An ideal slow light system must have large delay-bandwidth product with fast

tunability, large bandwidth, low distortion and low absorptive losses [Boyd, 2003;

Milonni, 2005; Khurgin, 2010]. The relative importance of each of the charac-

teristic varies with the application. For example, very large dispersion is more

important for a slow light based interferometer than very wide bandwidth. In

contrast, for an optical buffer, the large delay-bandwidth product is more impor-

tant than large absolute delay.

For the purposes of optical communication systems and other applications,

rapid tunability of the group velocity of a light pulse is advantageous. For example,

one could use such a system to synchronize a pulse train by tailoring the group

velocity of each pulse in the pulse train. Such an application demands the group

velocity switching times to be less than the signal pulse width. We can also use it

to get single photons on demand, by slowing down the single photons and turning

off the delay at appropriate time. For applications such as slow light enhanced

Fourier transform interferometry [Shi et al., 2007], the time required to determine

the spectrum of an unknown signal is directly proportional to the group velocity

switching time.

Camacho et al. [Camacho et al., 2006] have shown that one can achieve multi-

ple pulse delays by operating between two Lorentzian absorption resonances. Such

a double Lorentzian system is shown to dramatically reduce absorptive broaden-

ing and cancel the dispersive broadening, and thus is advantageous over a single

Lorentzian system like an electromagnetically induced transparency (EIT) based

system. Slow-light is primarily limited by absorptive broadening in addition to

absorptive losses in such a double Lorentzian system. The depth and line-width of

18

each of the resonance along with the separation between two resonances determine

the system bandwidth, the group delay and the absorptive losses.

Camacho et al. use naturally occurring Rubidium or Cesium resonances to

achieve double absorption system. In such a system, line-width and the depth of

each of the resonance is fixed and the slow light tuning is achieved primarily by

changing the number density of the system. By using Raman absorption lines,

one can have optical control over the properties of the absorption line and thus an

optical control over the delay and bandwidth. The double absorption slow light

is also demonstrated in fibers using anti-stokes absorption resonances by Zhu and

Gauthier [Zhu and Gauthier, 2006a].

Double Raman absorption is also useful for applications requiring rapid change

of group index. A slow light based Fourier transform interferometer, for example,

requires a scan of group index of the slow light medium and the interferometer

output is recorded as function of group index which is then Fourier transformed

to obtain the signal spectrum. Shi et al [Shi et al., 2007] demonstrated such an

interferometer using naturally occurring double absorption resonance slow light

system. A rapidly tunable slow light system lessens the spectral determination

time and the steeper refractive index gradient in Raman absorption based system

improves the spectral resolution by at least two orders of magnitude.

1.3.2 Stopped Light

“Stopped light” usually refers to the interconversion of electromagnetic fields

and long-lived atomic coherences. It allows for recording coherent signals for later

retrieval even at very low light levels. The initial work by Liu et al.[Liu et al.,

2001] and Phillips et al. [Phillips et al., 2001] stimulated additional research with

19

a recent demonstration of storage times in excess of one second [Longdell et al.,

2005]. Stopped light may be useful for applications in remote sensing, image

processing, and quantum information.

Typically, the pulses used in stopped light experiments are several kilometers

long in free space. However, because the stopped light medium usually ranges

from a few tens of microns up to several centimeters, slow light is first used to

spatially compress the optical pulses inside the medium. Slow light is achieved

when a steep linear dispersion is obtained in a medium, leading to a large group

index ng = n + ω ∂n∂ω

and a small group velocity vg = c/ng, where n is the phase

index and ω is the angular frequency. Slow group velocities have been achieved in

a variety of media, including atomic vapors [Harris et al., 1992; Kasapi et al., 1995;

Budker et al., 1999; Hau et al., 1999; Liu et al., 2001; Yang et al., 2007; Boyer

et al., 2007] and solids [Bigelow et al., 2003a; Bigelow et al., 2003b; Khurgin, 2005;

Gehring et al., 2006; Zhu and Gauthier, 2006b; Okawachi et al., 2006; Shumakher

et al., 2006].

The ability of rubidium to preserve the ground state coherence for times much

longer than the pulse width enables us to store the signal and retrieve it at a

later time. The motivation here is to store the signal information like spatial and

temporal characteristics and not to store the energy of the pulse. We employ a

four-wave mixing process in rubidium (we elaborate more about the process later)

to store the signal as well as spontaneously generated idler pulses. The signal and

idler beams produced in such a process are known to be number squeezed. Storing

such fields is fundamentally interesting and may be useful in applications where

two correlated fields need to be stored. In addition, we demonstrate a scheme in

which we store the spatial mode information of the signal beam in warm vapor

20

overcoming diffusion. We believe that storing images may have applications in

image processing, quantum information and remote sensing.

1.3.3 Guided Light

An all optical waveguide refers to a waveguide whose transverse refractive

index profile is set by the interaction of an optical control beam with the medium.

The properties of the waveguide can be altered or modified to fit a particular

experimental requirement simply by changing the properties of the control beam.

Several schemes have been proposed to achieve all optical wave guiding and there

have been some experimental realizations. Moseley et al. have proposed [Moseley

et al., 1996] and realized [Moseley et al., 1995] focusing and defocusing of the

signal beam using electromagnetically induced focusing. There are several other

papers which address Raman focusing [Walker et al., 2002; Yavuz et al., 2003;

Shverdin et al., 2004; Proite et al., 2008]. Truscott et al. have achieved optical

wave guiding [Truscott et al., 1999] and their scheme is analyzed in detail by

Kapoor et al. [Kapoor and Agarwal, 2000] and Anderson et al. [Andersen et al.,

2001]. In their scheme the control beam modifies the refractive index of the

medium by pumping the ground state rubidium atoms to an excited state. There

are several other schemes proposed for waveguiding [Shpaisman et al., 2005], and

more recently image guiding [Firstenberg et al., 2009], using electromagnetically

induced transparency in lambda and double lambda systems.

Nonlinear optical properties of various gases within an optical waveguide struc-

ture have received considerable recent attention [Benabid et al., 2002; Ghosh et al.,

2006]. The use of an optical waveguide enables one to have high intensities at

lower optical powers over distances much greater than the diffraction length. The

21

combination of higher intensities, longer interaction lengths and higher atomic

densities within the mode volume results in efficient nonlinear processes [Ben-

abid et al., 2002]. Benabid et al. demonstrated an efficient stimulated Raman

scattering process in a hollow-core photonic crystal fiber filled with hydrogen gas

requiring two orders of magnitude less control beam power than any other pre-

viously reported experiments [Benabid et al., 2002]. Efficient nonlinear processes

have been demonstrated by injecting rubidium vapor in such a waveguide [Ghosh

et al., 2006] and also by using the small optical mode area of a tapered-nano-fiber

(TNF) placed in rubidium vapor [Spillane et al., 2008].

Previous proposals and demonstrations of an all optical waveguide use the

sharp refractive index variation near an EIT line. This means that the signal is at

the atomic resonance frequency. This limits the application of such a waveguide

because of the signal losses due to absorption. If the signal is guided in the core of

the donut shaped control beam, it is advantageous to have the signal in the trans-

parent region. Off-resonant Raman absorption discussed in the previous section

can be used to create an all optical wave guide. The dependence of signal refrac-

tive index on the control beam intensity enables us to control the spatial mode of

the signal. One can choose a spatial mode of the coupling beam which results in a

quadratic dependence of signal refractive index on the radial coordinate. Such a

quadratic index medium is known to confine the optical mode within small radial

bounds. By changing the frequency or power of the coupling beam we can tune

the size of the signal mode propagating in the medium.

Propagation of the focused signal beam for lengths greater than the diffrac-

tion length can lead to efficient nonlinear processes which depend on the signal

intensity. For example, a weak pulse of light can produce enhanced AC-Stark

22

shift in the atoms which may be useful for the single photon non-demolition mea-

surement schemes. One can also obtain enhanced nonlinear optical frequency

conversion using the compressed mode inside the waveguide.

1.3.4 Rubidium Prism

Prisms have played a fundamental role in our understanding [Newton, 1704]

and manipulation of light. The dispersing power of the prism is a function of the

material’s frequency-dependent index of refraction. The faster the index changes

as a function of frequency, the better the dispersing power. For example, in the

visible region of the spectrum, BK7, a popular prism material, has dn/dν ≈

5×10−17 Hz−1. We recently showed using precision deflection measurements that

the frequency sensitivity of a BK7 prism could reach approximately 100 kHz/√Hz

using about 1 mW of light [Starling et al., 2010]. The low dispersing power of

these standard prisms limits their usefulness to systems that exhibit large spectral

changes or for broadband applications. Gratings can improve the dispersing power

by a couple orders of magnitude, but even this is too small for many applications.

There has been an ever increasing demand for spectral resolution and disper-

sion. Common techniques for high resolution spectroscopy include Fourier trans-

form interferometry [Hariharan, 2003] and high finesse cavities [Fortier et al.,

2006]. However, these systems are single mode and must be scanned to determine

other frequencies in the system. Optical frequency combs are used as efficient

frequency counters and find applications in high precision optical spectroscopy

[Udem et al., 2002]. In addition, the teeth of the frequency comb can be used

in “Direct Frequency Comb Spectroscopy” (DFCS) [Stowe et al., 2008]. However

since the teeth are closely spaced in frequency, one needs a highly dispersive el-

23

ement to separate individual teeth spatially. Bartels et al. [Bartels et al., 2009]

have overcome this by making the comb less spectrally dense and making use

of regular dispersing elements. With high dispersion one can spatially separate

dense optical frequency combs. A spatially dispersing element can also be used as

an efficient frequency filter for applications like entanglement filtering in quantum

information [Okamoto et al., 2009].

The primary advantage of using prisms is their ability to unambiguously de-

termine spectral lines [Demtroder, 2002]. The dispersing power of a prism can be

improved by using light frequencies near a material resonance (see for example

[Wood, 1988; Marlow, 1967]). For example, Finkelstein et al. showed, using the

resonance enhancement of dispersion (of a single absorption resonance), that a

mercury vapor prism could resolve the Raman lines of CO2 [Finkelstein et al.,

1998]. Previous experiments have focused on a single resonance, typically a single

absorption [Lin et al., 1996; Finkelstein et al., 1998; Zheltikov et al., 2000] or sin-

gle transmission resonance [Sautenkov et al., 2010]. The difficulty in working with

single absorption resonances is the nonlinear dispersion, the strong absorption and

strong frequency-dependent absorption, and the inability to resolve many spectral

lines.

The steep dispersion in a slow light system can be used to achieve highly disper-

sive prism that can be used as an experimental tool for off-resonant experiments

in rubidium vapor. Experiments involving several optical frequencies in the same

optical mode require a spectral filter. We present a rubidium vapor prism which is

capable of spatially resolving closely spaced frequencies. In addition, it is capable

of measuring frequency fluctuations with an accuracy better than 1 part in 1013.

This rubidium prism is capable of separating signal, idler and coupling beams in a

24

four-wave mixing process or signal and coupling beams in an off-resonant Raman

process.

1.4 Thesis Outline

In this chapter, we have reviewed some of research work centered around pro-

cesses in a three-level lambda system. In the following chapters we discuss the

experimental results that are central to this thesis.

In chapter 2, we present a slow light system based on dual Raman absorption

resonances in warm rubidium vapor. Each Raman absorption resonance is pro-

duced by a coupling beam in an off-resonant Λ system. This system combines

all-optical control of the Raman absorption and the low dispersion broadening

properties of the double Lorentzian absorption slow light. The bandwidth, group

delay and central frequency of the slow-light system can all be tuned dynami-

cally by changing the properties of the coupling beam. We demonstrate multiple

pulse delays with low distortion and show that such a system has fast switching

dynamics and thus fast reconfiguration rates.

In chapter 3, we demonstrate a four-wave mixing based storage of signal pulses

in warm rubidium vapor for times much longer than the pulse width. We first

show the results which demonstrate storage of temporal information of a single

transverse mode pulse. The signal is stored in and retrieved from the long-lived

ground state atomic coherences. We show that the signal and spontaneously

generated idler can be stored simultaneously in the medium and can be retrieved

by just turning off and turning on the coupling beam. We also show that the

retrieved signal pulse power varies linearly with the retrieval beam demonstrating

25

that a coherently prepared vapor generate signal and idler more efficiently than an

unprepared medium. We also present an experimental realization of the storage

of multiple transverse modes of the signal in the rubidium vapor. We show that

an image impressed onto a 500 ns pulse can be stored and retrieved up to 30 µs

later. The primary limitation in storing multiple transverse modes in hot vapor

is atomic diffusion. The image storage is made robust to diffusion by storing the

Fourier transform of the image.

In chapter 4, we present experimental results demonstrating the signal beam

propagation with a small spot size over several diffraction lengths. We use the

off-resonant coherent Raman absorption in warm atomic rubidium vapor to create

a medium whose refractive index changes along the radial direction quadratically.

Such a medium is known to guide light and we demonstrate such a waveguide

whose properties are controlled by a low power Laguerre Gaussian coupling laser

beam. We first present the basic theoretical predictions of a lossless quadratic

index medium and then show that the spot size of the signal as it propagates in

the medium can be controlled using a coupling beam. We investigate the behavior

of the signal beam size as we vary coupling beam parameters. We also show that

the coupling efficiency of the signal beam into the waveguide varies linearly with

the signal power.

In chapter 5, we present a tool for off-resonant atomic experiments. We present

an atomic prism spectrometer with five orders of magnitude greater dispersing

power than a standard glass prism. The prism utilizes the steep linear dispersion

between two strongly absorbing hyperfine atomic resonances of rubidium. We

show that the number of resolvable spectral lines is proportional to the slow light

delay-bandwidth product. We resolve spectral lines 50 MHz apart and realize

26

a spectral sensitivity of 20 Hz/√Hz via precision deflection measurements. To

demonstrate the practicality of this setup, we spatially separate collinear pump,

signal and idler beams resulting from a four-wave mixing process. We believe this

prism will have applications in quantum information, in spatially separating sev-

eral teeth of an optical frequency comb and as a spectral filter in atomic resonance

experiments.

In chapter 6, we make some concluding remarks with suggestions for the pos-

sible future work.

In Appendix A, we present a simple theoretical model that predicts some of

the key features of the transmission of the signal beam through a lambda system

in the presence of a coupling beam. We derive the susceptibility of the signal beam

in the presence of coupling beam using density matrix equations and discuss the

role of single photon detuning in the shape of signal transmission line profile.

27

2 Double Raman Absorption

Slow Light

In this chapter we present a fast, optically tunable double absorption slow

light system where the absorption resonance for the signal is artificially created

by a coupling beam in such a three-level off-resonant Λ system. We show that

the properties of each of the absorption lines can be changed optically by tuning

the properties of the coupling beams which create them and consequently acheive

optical tunability of slow light.

The resonant Λ system has been studied extensively in the context of EIT

[Fleischhauer et al., 2005]. The Raman absorption line produced by an off-

resonant Λ system has been studied and used to produce fast-light [Mikhailov

et al., 2004; Knappe et al., 2003]. Such a process has also been used by Reim et

al. to store very weak signal pulses [Reim et al., 2010]. Proite et al. create a gain

line and an absorption line using two coupling lasers in order to obtain enhanced

refractive index [Proite et al., 2008]. Here we propose to use two absorption lines

in order to realize a double Lorentzian slow light system. We also study the tran-

sient response to the switching of the coupling beam properties and show that

one can dynamically and rapidly tune the group delays. This system combines

28

all optical control of the Raman absorption and the low dispersion broadening

properties of the double Lorentzian absorption slow light. The bandwidth, group

delay and central frequency of the slow-light system can all be tuned dynami-

cally by changing the properties of the control beam. We demonstrate multiple

pulse delays with low distortion and show that such a system has fast switching

dynamics and thus fast reconfiguration rates.

2.1 Theory

The susceptibility for a three-level lambda system has been derived in the

Appendix A. The real and imaginary parts of susceptibility at the signal frequency

can be written as

χ′(∆, δ,Ωc) = χ′0 + C1

δ′

δ′2 + γ′2− C2

γ′

δ′2 + γ′2, (2.1)

χ′′(∆, δ,Ωc) = χ′′0 + C2

δ′

δ′2 + γ′2+ C1

γ′

δ′2 + γ′2. (2.2)

The effective detuning and the effective line width are given by δ′ = δ−δ0 and

γ′ = γ+γ0 respectively where δ0 = |Ωc|2∆/(4∆2+Γ2) and γ0 = |Ωc|2Γ/(8∆2+2Γ2).

The constants C1 = β |Ωc|2

4∆2−Γ2/4

(∆2+Γ2/4)2and C2 = β |Ωc|2

4∆Γ

(∆2+Γ2/4)2.

Fig. 2.1 illustrates the dependence of C1 and C2 on ∆. Solid line in the plot

shows 2C1/Γ, dashed line shows 2C2/Γ and dotted line shows normalized χ′′0. C2

is zero for ∆ = 0, is maximum at ∆ = Γ/2√3 and falls back to zero for large

∆. C1 is negative for ∆ < Γ/2 and positive for ∆ > Γ/2. We also see that

for ∆ >> Γ/2, C2 approaches zero faster than C1 and this is the region we are

interested in to produced an Lorentzian-like absorption line.

29

0 1 2 3 4−6

−4

−2

0

2

∆/Γ

2C1/Γ

2C2/Γ

n’’/n’’max

Figure 2.1: Dependence of C1 and C2 on ∆

3

∆ ∆

Γ/2 Γ/2 Γ/2 Γ/2

γγ

δ1 δ2

Ωc1 Ωc2Ωs

VC1 VC2

(a) (b)

Ωs

22

1 1

3

Figure 2.2: (a) & (b) The energy level diagram for vapor cells VC1 and VC2respectively.

30

Now consider the case where the signal has to propagate through two identi-

cally prepared vapor cells VC1 and VC2 as shown in Fig. 2.2. The effective suscep-

tibility at the signal frequency is given by the sum of susceptibilities in each of the

cells. The control beams in the two cells only differ by frequency 2δc. The effec-

tive susceptibility is given by χeff (∆, δ,Ωc, δc) = χ(∆, δ−δc,Ωc)+χ(∆, δ+ δc,Ωc)

where χ(∆, δ,Ωc) is given by Eqn. (1). Expanding the refractive index around

δ = |Ωc|2∆/(4∆2+Γ2), we obtain the following expressions for real and imaginary

parts of the refractive index.

n′ = n′0 −

C1

δc

[

δ′

δc+

(

δ′

δc

)3

+ ...

]

− C2γ′

δ2c

[

1 + 3

(

δ′

δc

)2

+ ...

]

, (2.3)

n′′ = n′′0 −

C2

δc

[

δ′

δc+

(

δ′

δc

)3

+ ...

]

+C1γ

′

δ2c

[

1 + 3

(

δ′

δc

)2

+ ...

]

. (2.4)

For the off-resonant case, the absorption line shape can be approximated to a

Lorentzian and thus we can obtain a double Lorentzian system. From the above

equations, we can derive the group index, defined as ng = n′0 + ωsdn/dω, to be

ng = n′0+ωsC1/δ

2c . The relevant slow light parameters for a propagation distance

L, the group delay τd and absorption αL are given as

τd =βωsL

c

|Ωc|24∆2

1

δ2c, (2.5)

αL =βωsL

c

( |Ωc|24∆2

)2Γ

2δ2c. (2.6)

Following the approach of Camacho et al. [Camacho et al., 2006], we can

write the dispersive and absorptive broadening for a pulse with Gaussian envelop

31

exp(−t2/(2T 20 )) after propagation through the medium of length L as

T 2a = T 2

0 +12ωsLC1γ

′

cδ4c= T 2

0 +12τdγ

′

δ2c, (2.7)

T 2d = T 2

0 +

(

3ωsC1L

δ4c cT20

)2

= T 20 +

(

3τdδ2cT

20

)2

. (2.8)

We define the bandwidth, B, of the system as the bandwidth of the pulse

for which the absorptive broadening is equal to two. Thus the bandwidth of the

system is given as B = δc/(2√αL). Also, from above equations we note that

τd = αL/γ0 and hence the time delay for one absorption length is simply the

inverse of the power broadened line width of the Raman absorption resonance.

The delay-bandwidth product for one absorption length is thus given by

τd ×B =δc2γ0

. (2.9)

The bandwidth of the system can be tuned simply by changing the frequency

difference between two control beams and the delay-bandwidth product can be

changed by changing the line width, γ0, of the Raman resonance. γ0 can be

tailored by changing the single photon detuning, ∆, or by changing the power of

the control beams. Thus we obtain all optical control over the bandwidth and the

delay bandwidth product.

In order to study the time response of the system, we need to solve the dynamic

equations for atomic coherence. In the weak perturbation regime for the probe,

we write the following equation for the ground state coherence ρ21.

32

ρ21 + i(∆ + δ − iΓ/2− iγ) ˙ρ21

−[

(∆− iΓ/2)(δ − iγ)− |Ωc|2/4]

ρ21 +Ωs

2(δ − iγ) = 0 (2.10)

We first solve the above equation for the case where the control beam is turned

on at time t = 0. The solution is of the form

ρ21(t) =

[

λ−

λ+ − λ−eλ+t − λ+

λ+ − λ−eλ−

t

]

(ρ(ss)21 − ρ

(0)21 ) + ρ

(ss)21 , (2.11)

where ρ(ss)21 is the steady state coherence and ρ

(0)21 is the steady state coherence

without the control beam on. The constants λ+ and λ− for large ∆ are given by

λ+ = i|Ωc|24∆

− |Ωc|2Γ8∆2

, (2.12)

λ− = −i

(

∆+|Ωc|24∆

)

− Γ

2

(

1− |Ωc|24∆2

)

. (2.13)

For |Ωc|2

4∆2 ≪ 1 we can see that exp(λ−t) is rapidly decaying compared to exp(λ+t)

Hence the characteristic transient time for turning on the slow-light medium is

given by reciprocal of real part of λ+.

τ (on) =8∆2

|Ωc|2Γ= 1/γ0. (2.14)

In order to calculate the turn off time, we consider the case where the coherence

is ρ(ss)21 for t < 0 and pump is turned off at t = 0. The time evolution of the

33

coherence for t > 0 is given by

ρ21(t) =(

ρ(ss)21 − ρ

(0)21

)

e−(i∆+Γ/2)t + ρ(0)21 . (2.15)

The characteristic turn off time in this case is just the excited state life time.

τ (off) = 2/Γ. (2.16)

Lastly, for the case where the frequency of the control beam is changed without

changing its intensity, we state without explicitly writing the equations that the

characteristic time for the transients to decay is same as τ (on). The coherence also

has a beat note during this transient time oscillating at the difference frequency

of the initial and final control beams.

We see that the turn on time τ on is the inverse of the Raman absorption line

width and the turn off time is the inverse of the homogeneous line width of the

excited state. Hence τ off ≪ τ on. τ on is also equal to the time delay obtained for

one absorption length. Thus for a system with a delay-bandwidth product of one

we can change the slow light parameters at the bit rate of the pulse stream and we

can turn off the slow light within the time much less than the pulse width. Such

fast dynamics makes this scheme an attractive candidate for pulse synchronization

techniques.

2.2 Experimental Setup

We demonstrate the dual Raman absorption slow light using warm atomic ru-

bidium vapor. The outline of the experimental setup is depicted in Fig. 2.3. The

34

DLaser

795nm

λ/2

PB1 PB2 PB3 PB4 PB5

AO

1

AO2 AO3

VC1 VC2 PD

C1 C2

S

BS

Figure 2.3: The outline of the setup used in the experiment.

795 nm laser comprises of a narrow line-width tunable diode laser followed by a

tapered amplifier to obtain high power, narrow line-width laser beam. The fre-

quency of the laser is tuned to the required frequency of the coupling beam. Signal

“S” is obtained by frequency shifting part of the the laser beam by 3.035GHz by

double passing it through a tunable 1.5GHz acousto-optic modulator (AOM1).

The other part of the laser beam is further split and sent through two 80 MHz

AOMs resulting in two coupling beams “C1” and “C2”. The frequency difference

between C1 and C2 is controlled by changing the RF frequency fed to these AOMs.

The coupling beams C1 and C2 are combined with the signal at the polarizing

beam splitters PB2 and PB4 in front of the vapor cells VC1 and VC2 respectively.

C1 is filtered from signal at PB3 before VC2 and C2 is filtered at PB5 placed after

the vapor cell VC2. The signal is measured at the photo detector PD.

VC1 and VC2 are identical 5 cm long vapor cells placed inside hollow mu

metal tubes to block stray magnetic fields. The vapor cells are heated by strip

heaters and the current passing through the strip heaters is identical for each

cell to ensure similar temperatures for both vapor cells. Each cell along with the

mu metal tubing and the heaters is placed inside a teflon tube enclosed by anti-

reflection coated windows at each end. The vapor cells contain both rubidium

35

isotopes in their natural abundance. In addition we also have 20 torr neon in each

cell which acts as a buffer gas. The temperature of each vapor cell is about 80 C

resulting in a number density of about 1012 cm−3 in each cell.

2.3 Experimental Results

2.3.1 Single Absorption Characteristics

Fig. 2.4 shows the experimental observations of coupling beam induced ab-

sorption line shapes. We measure the signal transmission through one of the vapor

cells shown in Fig. 2.3. The two photon Raman detuning is changed by changing

the modulation frequency at the AOM1. The observed transmission as a func-

tion of Raman detuning is then divided by transmission with the coupling beam

off to obtain normalized transmission. The natural logarithm of the normalized

transmission gives the absorption coefficient times the length of propagation, αL.

Fig. 2.4 shows the absorption lines for three different coupling powers 3 mW

(squares), 6 mW (circles), 10 mW (diamonds). The solid lines show the fitted line

for the experimental data shown by markers. We fit the experimental data to the

function given by Eqn. 2.4 with C1,C2,δ0,γ0 and n′′0 as fit parameters.

Fig. 2.5(a) shows the variation of fit parameters C1 (squares), C2 (circles)

and Fig. 2.5(b) show the variation of δ0 (circles) and γ0 (squares) with coupling

beam power. From the definitions, we expect a linear dependence of C1 and C2

with coupling beam intensity. There is a clear linear dependence upto 10 mW.

Beyond 10 mW, four-wave mixing begins to affect the line shape and so we observe

nonlinear behavior beyond 10 mW. We can also see that the absorption and the

line width increase linearly with the increasing coupling power. At low coupling

36

−0.2 −0.1 0 0.1 0.2

0

1

2

3

4

Raman Detuning (MHz)

αL

Figure 2.4: Variation of absorption line shapes with coupling beam power.

0 5 10 15 200

2

4

0 5 10 15 200

20

40

60

Coupling Power (mW)

Fre

qu

en

cy (

kH

z)

(a)

(b)

Figure 2.5: Variation of fit parameters with coupling beam power.

37

−2 −1 0 1 20

0.2

0.4

0.6

0.8

1

1.2


Tra

nsm

issio

n (

arb

. u

nits)

Figure 2.6: Transmission of the signal through a vapor cell in the presence of twocoupling beams differing in frequency.Green curve is a reference absorption withsingle coupling beam frequency. The coupling beams are separated in frequencyby 1 MHz and 2 MHz for the transmission curves in red and blue respectively.

beam powers where the we can ignore power broadening, the line width is relatively

constant. This gives the ground state decoherence rate γ for our system. Thus

we can tune the absorption properties just by changing the coupling beam power.

We can also change the center of absorption just by changing the frequency of the

coupling beam by tuning the AOM.

2.3.2 Demonstration of Slow Light

In order to obtain double absorption slow light we first introduce two coupling

beams at two frequencies and the signal into just one vapor cell. Fig. 2.6 shows

the observation of line shapes. The green curve shows the line shape with just one

coupling beam frequency. The red curve shows the transmission with two coupling

38

−40 −20 0 20 40

0

0.05

0.1

0.15

0.2

Time(ms)

Am

plit

ud

e (

Arb

. U

nits)

Figure 2.7: The signal pulse out of a vapor cell with two coupling beams differingin frequency.

beam frequencies separated by 1 MHz and the blue curve shows the transmission

with frequency separation of 2 MHz. The powers in each of the coupling beam

are equal. We see that having two coupling beam frequencies in one vapor cell

reduces the depth of absorption in each of the absorption compared to having just

one frequency. This is due to competing nonlinear effects producing frequencies

at the multiples of the coupling frequency difference. We can see evidence of this

effect in the red curve where we have small absorptions on either sides of the

double absorption feature we expect. A signal pulse tuned to a frequency between

two absorption resonances exhibit beating at the difference frequency of the two

coupling beams at the output. Fig. 2.7 demonstrate this effect. Gray curve is the

reference pulse and black curve is the pulse shape at the output of the cell.

We need to use two vapor cells with a single coupling beam in each to overcome

the above mentioned problems. The transmission profile for this case is shown in

39

Tra

nsm

issio

n (

arb

. u

nits)

−15 −10 −5 0 5 10 150

10

20

30

40

50

Frequency (MHz)

Figure 2.8: The transmission profile of the probe corresponding to the observeddelay.

−0.4 −0.2 0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

µs)

Inte

nsity

Time (

Figure 2.9: Reference pulse and the delayed pulse.

40

Fig. 2.8 with 3 MHz of frequency difference between two coupling beams. Fig

2.9 shows the experimental observation of the delay. The frequency difference

between two coupling beams in VC1 and VC2 is 3 MHz. The alignment of the

coupling and signal beams is adjusted for each vapor cell to obtain the Raman

absorption dip. The signal pulse with full width at half maximum (FWHM) of

about 179µs is tuned to the center of transparency in between the absorption dips.

We observe a delay of about 374ns with the coupling beams on. This corresponds

to a delay bandwidth product of about 2 (bandwidth here is measured as FWHM

width of the pulse). The FWHM of the output pulse is about 262ns and thus a

broadening of factor 1.4. Thus we demonstrate slow group velocity with relatively

less pulse broadening using the two coupling induced absorption dips.

2.3.3 Slow Light Switching

First we will measure transient times for the change in absorption of the signal

because of the changes in coupling beam properties. Fig. 2.10 shows the exper-

imental observations of switching times. In order to measure turn off time, the

signal pulse, tuned to the center of Raman absorption, is sent through the medium

with the coupling beam on and as the pulse is propagating inside the medium,

coupling beam is turned off. In the Fig 2.10, dotted curve on the top shows the

reference pulse and on the bottom shows the output pulse in the presence of cou-

pling induced absorption. The dashed line shows the coupling level. The solid

line on the top shows the turn-off characteristics of the absorption and on the

bottom the turn-on characteristics. Upper solid curve shows that the pulse has

lower transmission before turning off the coupling beam because of Raman ab-

sorption. After the coupling beam is turned off, the signal transmission increases

41

−20 −10 0 10 20

Time(µs)

Figure 2.10: Measurement of turn-on and turn-off times of Raman absorption.

to match with the dotted line within 50ns and the rise time is limited in this case

by the rise time of the AOM used to switch coupling beam. The turn on time is

measured by having the coupling beam initially off and turning it on as the signal

is inside the medium as shown in the lower curve of Fig. 2.10. The turn on time

is measured to be about 0.75 µs.

The transients for the change in frequency of the coupling beam has similar

time constant and we see modulation in the signal during transition time. Fig.

2.11 shows the transient characteristics of the signal transmission for the change

in coupling beam frequency. Coupling beam frequency is initially tuned to Raman

resonance so that signal experiences absorption. Then we change the frequency

of the coupling beam so that Raman detuning is far off resonance where the sig-

nal experiences no absorption. During the transition time from being absorptive

to being transmissive, the signal experiences beating in its amplitude with a fre-

quency equal to the difference in the coupling beam frequency.

Fig. 2.12 shows the experimental observation of slow light switching. The

42

−30 −20 −10 0 10 20

0

0.1

0.2

0.3

0.4

0.5

0.6

Time (µs)

Tra

nsm

issio

n (

arb

. u

nits)

Figure 2.11: Beating in the cw signal beam resulting from coupling beam fre-quency change.

2 4 6 80

0.02

0.04

0.06

0.08

0.1

0.12

Tra

nsm

issio

n(a

rb. u

nits)

Time(µs)

Figure 2.12: The experimental observation of slow light switching.

43

dotted line is the input pulse without any delay and the broken line is the output

pulse with delay. The black line demonstrates the slow light switching. As the

pulse is compressed inside the vapor cell, we turnoff the control fields as shown by

the black dashed line. We turn off the slow propagation of the pulse all optically

within 50ns demonstrating high speed switching of slow light.

2.4 Summary

We have presented a double absorption slow light system based on the Raman

absorption dip. The strength, the width and the center of the absorption can be

modified all optically by changing the properties of the coupling beam. In addition

we can even tune the number density and the single photon detuning to further

modify the properties of the slow light system. The wide range of independent

controls makes this slow light system versatile. However the limitation to this

versatility is imposed by the competing nonlinear four-wave mixing process which

dominates at high atomic number densities. This limitation is dealt to some

extent in this paper by careful alignment of the probe and coupling beams in

order to have a phase matching favoring absorption and suppressing four-wave

mixing. Also we avoid the problem of multi-wave mixing resulting from having

two coupling beams and the probe beam in one vapor cell by using two vapor

cells.

44

3 Stopped Light Using Four

Wave Mixing

In this chapter we present experiments that demonstrate the ability to store

light using two Λ resonances simultaneously, and hence record information from

two distinct optical frequencies for later retrieval. The four-wave mixing process

discussed in chapter 1 is used to obtain the double lambda system. The signal

beam and spontaneously generated idler beam are stored simultaneously in warm

rubidium vapor. As in EIT based slow light, light pulses are mapped onto a

coherent polarization (spin) wave in an atomic ensemble, and then retrieved by

converting the spin wave back into an optical pulse. The four-wave mixing gain

on Raman resonance, and absorption away from Raman resonance results in large

pulse compression allowing for storage of a complete waveform. In addition, the

parametric process generates a frequency-shifted idler pulse that may also be

stored and retrieved simultaneously with the signal pulse.

Many experiments have been performed where optical information is stored

in one lambda transition and retrieved with another [Zibrov et al., 2002; Wang

et al., 2005; Chen et al., 2006]. In these experiments, a collective atomic spin

wave is coherently prepared by injecting both write beams and stokes beams

45

simultaneously, and then anti-Stokes beam is read out using a read beam whose

frequency is chosen to complete a four-wave mixing process. In the present scheme,

the signal and idler generated in the four-wave mixing are known to be correlated.

The storage of such fields may have important implications in image processing,

quantum information, and remote sensing, where two correlated fields may need

to be preserved for later use. If macroscopic fields are stored, one may imagine

retrieving only small fractions of the coherence multiple times, allowing for the

production of weak correlated fields at regular intervals.

Recently there has been some interest in extending the ideas of single trans-

verse mode slow and stopped light to multiple transverse modes where the atomic

medium delays and stores the spatial profiles of the stored pulses [Jain et al., 1995;

Camacho et al., 2007; Pugatch et al., 2007; Zhao et al., 2008; Shuker et al., 2008].

One of the difficulties of using hot vapors as the storage medium for stopped light

experiments is the diffusion of atoms during storage period. Recently, Pugatch et

al. [Pugatch et al., 2007] showed that an optical vortex with a phase singularity

in the transverse spatial profile can be stored in an atomic medium despite strong

diffusion. Subsequently there has been some theoretical work done to make image

storage in hot vapors robust to diffusion [Zhao et al., 2008].

We demonstrate the storage and retrieval of a transverse image in a hot

atomic vapor using a combination of electromagnetically induced transparency

(EIT)[Harris et al., 1990] and four-wave mixing (FWM) techniques. We overcome

the adverse effects of diffusion by storing the Fourier transform of the image in

the stopped light medium rather than the image itself. While the optical phase

in the object plane is constant, the phase in the Fourier plane varies in such a

way that atoms with opposite phase destructively interfere during diffusion [Zhao

46

et al., 2008]. This is similar to the storage of the Laguerre-Gauss beams [Pugatch

et al., 2007] under diffusion where atoms on opposite sides of the phase singularity

have relative phases of π and destructively interfere.

3.1 Theory

From Appendix A, we know the linear susceptibility at the signal frequency

for a lambda system. The linear polarization at signal frequency can be written

as,

P (1)s = Nµ12

Ωs

2

∆R

∆s∆R − Ω2c

4

(3.1)

The quantities ∆s = ∆s − iΓ/2 and ∆R = ∆s − ∆c − iγ are the complex single

photon and two photon (Raman) detunings where Γ and γ represent the transverse

excited and longitudinal ground-state decay rates respectively, N is the atomic

number density, and Ωj = Ej · µj/h represents the Rabi frequency induced by

electric field amplitdue Ej via the dipole matrix element µj.

The nonlinear polarization at the signal frequency is proportional to the idler

field and is responsible for four-wave mixing:

P (NL)s =

Nµ12

8

ΩiΩ2c

∆1∆2∆3 − Ω2c∆1/4

, (3.2)

where ∆1 = ∆c+ω12− iΓ/2 ,∆2 = ∆c+ω12−∆i− iγ, and ∆3 = 2∆c+ω12−∆i−

iΓ/2 are the complex one, two and three photon detunings along this excitation

pathway.

The total steady state polarization oscillating at the signal frequency is then

Ps = P(1)s +P

(NL)s . A similar result may be derived for the polarization oscillating

47

at the idler frequency:

P(NL)i =

Nµ32

8

ΩsΩ2c

∆s∆R∆1 − Ω2c∆1/4

, (3.3)

which, except for being proportional to Ωs instead of Ωi, is identical to the non-

linear contribution (P(NL)s ) to the polarization at the signal frequency when the

idler beam is generated in a parametric process, which guarantees the equivalence

of ∆s and ∆3 as well as ∆R and ∆2.

The coupled amplitude equations for signal and idler for perfect phase match-

ing can thus be written as:

∂Ωs

∂z= gsΩi − αsΩs, (3.4)

∂Ωi

∂z= giΩs. (3.5)

The gain coefficients gs and gi can be written as:

gs =ωs

2cnsℑ(χFWM(ωs))|Ec|2, (3.6)

gi =ωi

2cniℑ(χFWM(ωi))|Ec|2. (3.7)

Solving the coupled amplitude equations we obtain the following solution for Ωs(z)

and Ωi(z):

Ωs(z) = Ωs(0)

[

cosh(ξz)− αs

2ξsinh(ξz)

]

e−αsz/2, (3.8)

Ωi(z) = Ωs(0)gigsξ

sinh(ξz)e−αsz/2, (3.9)

where ξ =√

α2s/4− gigs

48

The steady state coherence between ground states |1〉 and |3〉 can be written

as:

ρ13 =ΩsΩc

∆s∆R − Ω2c

+ΩiΩc

∆1∆2 − Ω2c∆1

∆3

. (3.10)

We note that the ground state coherence set up in the medium has two distinct

terms, one proportional to signal field amplitude and one proportional to the idler

field amplitude, and that each contribution identifies a unique excitation pathway.

As we adiabatically turn off the coupling beam, all coherences associated with the

excited state decay within excited state lifetime which is very fast. The ground

state coherence decays much slower. When we turn on the coupling beam at a

later time before ρ13 decays we resume the gain process and obtain signal and

idler.

We now model storing a transverse spatial profile of the signal in the atomic

vapor. Consider the system shown in Fig. 3.1. Suppose we have a 4f imaging

system such that the vapor cell is placed in between two lenses as shown in the

figure. In this way the transverse intensity profile at the vapor cell is the Fourier

transform of the object. The second lens images the object onto a camera.

We adopt the diffusion model of Pugatch et al. to simulate image diffusion

in the Fourier plane. We first determine the Fourier transform of the field in the

back focal plane of L1. Let Eo be the field at the object plane which is also the

front focal plane of a spherical lens of focal length f . At the back focal plane of

the lens, the field Ef is given by the Fourier transform of the field in the object

plane:

Ef = F (Eo) =1√λf

∫ ∞

−∞

Eo exp(−i2π

λfξu)dξ, (3.11)

where ξ and u are the coordinates of object plane and Fourier transform plane

49

respectively. The ground state atomic coherence at the time of storage is given

by ρ13 = gΩs, where g is the nonlinear coupling coefficient and Ωs is the Rabi

frequency of the signal field. The time evolution of the atomic coherence is given

by

∂ρ13(u, t)

∂t= D

∂2ρ13∂u2

− ρ13Tc

(3.12)

where Tc is ground state decoherence time of the coherence ρ13, and D is the

diffusion coefficient of the atoms. Assuming a constant pump intensity along the

transverse dimension of the cell, the only spatial dependence of the ground state

coherence comes from the signal field amplitude.

The field at the image plane, Ei, is recovered by inserting Eq. (1) into the

diffusion equation [ Eq. (2)] and taking another Fourier transform, which upon

integration gives

∂

∂tEi(x, t) = −(

1

Td

+1

Tc

)Ei(x, t), (3.13)

with a solution given by

Ei(x, t) = Ei(x, 0) exp

[

−t(1

Td+

1

Tc)

]

, (3.14)

where Td = λ2f2

(2π)2Dx2 is the diffusion time constant and x is the coordinate in the

image plane.

There are two features worth noting in Eq. 3.14. First, each spatial point in

the image decays exponentially in time, with a time constant given by 1/Td +

1/Tc. This means that dark areas of the image remain dark for appreciable times

compared to the temporal pulse length. Second, since Td falls off as 1/x2, the

central portion of the image has maximum storage time. We can increase the

diffusion time Td by making the image smaller or making the focal length of the

50

T T

Input Signal

Pump Pulse

Retrieved Signal

1 2

Delayed Signal

ObjectRb

Vapor CellPBS PBS Image

Camera

Signal

3.035 GHz Acousto-Optic

Modulator

Laser 795 nm

Pump

L1 λ

2

L2

(a)

(c)

Ωs

Ωi

ΩC

ΩC

|1

|3

|2∆s ∆c

∆ i

(b)

Figure 3.1: (a) Experimental setup. (b) Four wave mixing energy levels. (c)Representation of the synchronized timing of signal (dashed red), delayed signal(blue) and coupling (black) beams.

imaging lens larger. In either case, the Fourier transformed spatial profile at

the vapor cell would be larger, requiring a correspondingly larger coupling beam

diameter and vapor cell.


A schematic of the experimental setup is shown in Fig. 3.1(a). An external

cavity diode laser followed by an amplifier is the source for coupling beam. The

signal is generated by frequency shifting part of the coupling by 3.035 Ghz using

51

an acousto-optic modulator. The full-width at half maximum signal and coupling

field intensity spatial profiles for pulse storage experiments were approximately

500 µm and 300 µm respectively. For image storage experiments, the 1/e2 beam

diameter of the coupling is approximately 4 mm and that of signal approximately

1 mm at the object plane. The powers of coupling and signal beams are approx-

imately 12 mW and 300 µW respectively. The coupling and signal beams are

orthogonally polarized and are combined using a polarizing beam splitter (PBS)

before the cell. The coupling is filtered from the signal using another PBS after

the cell. We use the D1 transitions of 85Rb to create a Λ configuration. The cou-

pling is detuned by 700 MHz to the blue of the optical transition connecting the

F = 2 ground state to the F ′ = (2, 3) excited states, and the signal is set 3.035

GHz (the ground state hyperfine splitting) to the red of the coupling.

A 12.5 cm long Rb vapor cell with natural isotopic abundance containing 20

torr neon buffer gas is placed in a magnetically shielded oven and heated to ap-

proximately 180 C, yielding a number density of approximately 1013 atoms/cm3.

A 4f imaging system is used to image the object on to the camera as well as place

the Fourier plane at the cell. It consists of two lenses, L1 and L2, each of focal

length of f = 500 mm separated by a distance 2f . The object is placed at the

front focal plane of L1 and the image is obtained at the back focal plane of L2.

The vapor cell is placed at the back focal plane of L1 (the Fourier plane). The

diameter of the cell is 1 cm and the transverse diameter of the signal beam is

chosen such that the profile of Fourier image fits in the cell. The coupling beam

is orthogonally polarized to the signal to filter out the coupling. In addition to

polarization filtering, we also performed a temporal filtering correlation measure-

ment using a 100 MHz detector (3 dB roll off). A 25 µm slit is placed in the focal

52

−0.1 −0.05 0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

1

Ω (

arb

. u

nits)

s 2

δ (MHz)r

Figure 3.2: Measured (asterisks) and theoretical (solid) steady state signal (blue)and idler (red) intensities as a function of signal detuning.

plane and a bucket detector is placed behind the slit. The position of the slit is

scanned in the image plane and the temporal intensity profile of the retrieved light

pulse hitting the bucket detector is recorded is measured on a 1.5 GHz scope.

Our slow light scheme is based on a combination of EIT and FWM in a Λ

system which consists of two lower energy levels coupled to a common higher

energy level of the atom by two electromagnetic fields. The relevant energy levels

of 85Rb are shown in Fig. 1(b). To obtain a highly transparent region which

also exhibits steep dispersion, the pump and signal lasers are detuned several

hundred MHz from the zero velocity class in a Doppler broadened vapor. The

signal experiences both FWM gain and EIT when its frequency is tuned to the

two-photon Raman resonance. As a note, optical alignment, buffer gas pressures,

laser detunings etc. all affect the transmission and dispersion. The rapid change

in the transmission profile near Raman resonance leads to a steep dispersion and

slow group velocity. Once the signal has been compressed inside the cell, we turn

53

off the pump, storing the image. The transverse spatial profile of the signal is

mapped onto the long-lived ground state coherence of the 85Rb atoms. The signal

is retrieved by turning on the pump at a later time. The timing of the pulses is

illustrated in Fig. 1(c). The retrieved signal intensity falls off exponentially with

storage time due to diffusion and decoherence.


To demonstrate pulse storage and retrieval we take the object and lenses out

of the setup. The signal beam is pulsed using a separate AOM (not shown) and

the coupling beam is turned off while the pulse is compressed within the cell and

then turned on at a later time. A plot of the measured signal and idler intensities

as a function of signal detuning around Raman resonance is shown in Fig. 3.2

for the case of 3 mW coupling field intensity and 50 µW of signal field intensity

before entering the cell. Accompanying the experimental data are theory plots,

generated by inserting Eqs. 3.8 and 3.9. We used the parameters Ωs(0) = 1,

z = 10 cm, N = 2 × 1011 atoms/cm3, Γ = 36 Mhz, γ = 10 Khz, Ωc = 7 Mhz,

∆c = 700 Mhz, ∆i = 2∆c + ω12 − ∆s and have included 600 Mhz of doppler

broadening for theory plots. The theory curves agree well with the data except

for a frequency offset of approximately 0.25 Mhz to the blue (not shown), which

may be due to buffer gas suppression of linear stark shifts [Nagel et al., 1999].

Figure 3.3 shows the delay of a 1.4 µs signal pulse and the simultaneous gener-

ation of a idler pulse (a), as well the storage and retrieval of both signal and idler

pulses 20 (b) and 120 (c) µs later. Part (a) of the figure shows an example of an

incident signal pulse (black), delayed signal pulse (blue) and generated idler pulse

(red) for the case when the coupling field (3 mW) remains on for the duration of

54

0

1

2

3

4

5P

ea

k In

ten

sity (

% o

f in

pu

t p

uls

e)

0 10 20 30

0 10 120 130

x 1/20 x 3

0 10 20 30

(a)

(b)

(c)

Time (µs)

Figure 3.3: Demonstration of four-wave mixing slow light and stored light

the measurement. In parts (b) and (c) of the figure, the coupling field (dashed

line) is turned off approximately 500 ns after the peak of the signal pulse enters

the medium, and then turned back on 20 µs and 120 µs later, respectively. The

retrieved pulse waveforms correspond to that fraction and shape of each pulse

that was in the medium when the coupling field was turned off, demonstrating

that the stored coherence contains information about each waveform that may be

separately retrieved

The maximum storage time observed was approximately 500 µs. We note that

the entire signal pulse waveform is retrieved and is broadened temporally by a

factor of two. Also of note is that only a fraction of the idler pulse is recovered,

indicating that each pulse retains its individual waveform during storage and

retrieval. This means that the retrieval process is not simply incoherent scattering,

55

101

102

103

100

101

102

Pump Retrieval Power (µW)

Sig

na

l P

ow

er

(µW

)

Figure 3.4: Idler pulse peak power as a function of coupling input power.

as might be expected by storing the signal pulse in the absence of four-wave mixing

gain and then retrieving near a four-wave mixing resonance. If that were the case,

then one would expect similar pulse shapes for both the signal and idler beams.

It is also of note that the intensity of the retrieved pulses varies linearly with

coupling retrieval power, as shown for the idler beam (the signature of four-wave

mixing) in Fig. 3.4. The pulses were stored using approximately 3.5 mW of

coupling power, and then retrieved using progressively smaller coupling powers.

Such a scheme may prove useful in cases where low background noise is needed for

precision low light level measurements using optical nonlinearities, since the strong

beams used to prepare the media are turned off long before the weak scattering

beams enter the media. It has recently be shown [McCormick et al., 2007], for

example, that -7.1 dB of relative intensity squeezing may be achieved in a similar

system before storage, and the present scheme may be beneficial for exploring

the entanglement properties between the stored signal and idler fields and in the

generation of narrowband entangled pairs of photons.

56

Image plane Fourier plane

−2

−1

0

1

2

Po

sitio

n (

mm

)

Figure 3.5: CCD camera capture of the signal intensity profile at the object planeand at the vapor cell (Fourier plane).

Having demonstrated temporal pulse storage, we now present the storage of

transverse spatial profile of the signal. We now use the 4f imaging system to

achieve this task. The intensity profile of the signal at the object plane and its

Fourier transform at the vapor cell are shown in 3.5. Figure 3.6 shows the input

image (a), as well as the retrieved (b) and calculated (c) image profiles for several

storage times. The theory plots are generated by using Eq. 3.14 to propagate the

measured input image with a diffusion coefficient of 10 cm2/s. The object used

is an amplitude mask containing a 5 bar test pattern. We note that the image

contrast remains high even for the longer storage times, even though the wings of

the image decay faster than the central part, as predicted earlier.

The physical mechanism responsible for preserving the image can be under-

stood in terms of the phase distribution of the stored optical wavefront and the

diffusion of the atoms (shown graphically in Fig. 3.7). Each atom in the field

acquires the local coherence set by the signal and pump fields. As the atoms dif-

57

Input Image

1

Position (mm)−1 0 1

0

1

(a)

−1 0 1 −1 0 1

Position (mm) Position (mm)

1 µs

Retrieved Images

TheoryExperiment Time

4 µs

16 µs

32 µs

(b) (c)

Figure 3.6: Input signal profile (a) and the time evolution of measured (b) andcalculated (c) transverse images.

58

−1 0 1

−1

0

1

0

Reference10 µs30 µs

Position (mm)

Am

pli

tud

e

Figure 3.7: Theoretical time evolution of stored ground state coherence of Rbatoms. The inset shows a close up of the time evolution near zero crossover pointsof electric field amplitude.

fuse in the Fourier plane, atoms of opposite phase tend to destructively interfere

preserving the high contrast. This is similar to the topological stability of stored

Laguerre-Gauss beams as demonstrated by Pugatch et al. and in good agreement

with the theoretical predictions by Zhao et al.

We note that our theoretical model does not account for the finite numerical

aperture of the 4f imaging system. The size of the numerical aperture in our

system is set by the size of the pump beam at the cell, which has a 1/e2 intensity

diameter of approximately 4 mm. Since this is larger than the spatial extent of all

relevant features in the Fourier transformed image (see Fig. 2), this approximation

is valid. In addition, we have assumed that all modes of spatial diffusion for the

59

prepared atoms remain within the pump beam diameter, so that higher order

modes which exit from and then return to the pump beam during storage do not

cause interference [Xiao et al., 2006]. Since the diffusion length for the longest

storage time is given by√Dt ≈ 170 µm, and the closest relevant feature in the

Fourier transform of the image is farther than 500 µm from the edge of the pump

beam, this is also a reasonable assumption. As a note, we also performed numerical

integration over the relevant finite dimensions of our experiment which produced

negligible errors.

Figure 3.7 shows the evolution of the ground state coherence under the con-

ditions of decoherence and diffusion. The inset of the figure shows a section of

plot containing zero crossover points. We see that the zeros of ρ13 are unchanged,

though the amplitude on either side of zero crossovers decreases with time. As the

atoms with positive and negative phases have equal probability of reaching the

zero crossover point, the retrieved fields from such atoms at those points tend to

destructively interfere, maintaining a zero in the field amplitude. At other points,

the same interference process results in a decrease in amplitude while maintaining

the field profile.

3.4 Summary

In summary, we demonstrated pulse storage and retrieval in rubidium vapor

using four-wave mixing process. We have shown that the four-wave mixing gain

enables us to achieve complete pulse localization within the warm rubidium vapor

cell and allows us to store full wave form. We also demonstrated that the four-wave

mixing efficiency can be improved by coherently preparing the medium ahead of

time using storage.

60

We also demonstrated slowing and storage of an arbitrary transverse image

in a hot atomic vapor, and shown that the retrieved image is robust to atomic

diffusion. We achieved this by storing the Fourier transform of the image instead

of an image with flat phase front. This remarkable feature allows the coherent

storage of spatial information even in doppler broadened media with large diffusion

constants.

61

4 All Optical Waveguiding

Using Raman Absorption

Off-resonant Raman absorption discussed in chapter 2 in the context of slow

light can be used to create an all optical wave guide. The refractive index depen-

dence on the coupling beam power has been exploited to reduce diffraction of the

focused signal beam. We show that the signal beam propagates with a small spot

size over several diffraction lengths. This all optical waveguide is imprinted by a

low power Laguerre Gaussian coupling laser beam. The refractive index at the

annulus of the donut control beam is lower than that at the core for signal fre-

quencies tuned to the blue of Raman resonance. We also show that the coupling

efficiency of the signal beam into the waveguide varies linearly with the signal

power.

4.1 Theory

We first review some of the properties of a Gaussian beam propagating in a

medium whose refractive index decreases quadratically with the radial coordinate

as we move away from the optical axis. The electric field distribution of a Gaussian

62

mode propagating in the z direction is given by

E(r, z) = E0ω0

ωexp

[

−i(kz + φ)− ikr2/2q]

, (4.1)

where φ(z) is the phase shift due to the geometry of the beam and q is the complex

parameter describing the Gaussian beam. q is defined as

1

q=

1

R− i

λ

πω2. (4.2)

R in the above equation is the radius of curvature of the phase front, ω(z) is

the radius of the beam at a location z and ω0 is the beam radius at the beam

waist. As the beam propagates in free space from the beam waist, the spot size

and the radius of curvature vary according to the following equations:

ω2(z) = ω20

[

1 +

(

λz

πω20

)2]

, (4.3)

R = z

[

1 +

(

πω20

λz

)2]

. (4.4)

We have a much simpler transformation relation for the beam parameter q and

it is given by,

q2 = q1 + z, (4.5)

where q1 and q2 are the beam parameters at two spatial locations separated by

the distance z along the propagation direction.

The transformation relation when the beam passes a converging thin lens of

focal length f is given by,

1

q2=

1

q1− 1

f. (4.6)

63

0 2 4 6 8 100

50

100

150

200

250

300

κ = 0

κ = 10

κ = 20

κ = 30

κ = 50

z (cm)

Sig

na

l S

ize

(m

icro

ns)

Figure 4.1: Beam propagation as a function of κ

It is known that in the media where the refractive index varies quadratically

with the transverse position, the transformation for the beam parameter can be

written in terms of an ABCD matrix as,

q2 =Aq1 +B

Cq1 +D. (4.7)

It has been shown by Yariv and Yeh [Yariv and Yeh, 1978] that for the lossless

quadratic index medium, whose refractive index is given by n = n0[1− (κ2/2)r2],

64

the transformation relation is given by,

q(z) =cos(κz)q0 + sin(κz)/κ

−sin(κz)κq0 + cos(κz). (4.8)

Fig. 4.1 shows the change in signal beam size for various values of κ. We

show the change for a propagation distance of 10 cm. The initial width of the

beam is assumed to be 100 µm. κ = 0 corresponds to the case of the free space

propagation. We can see that the beam expands to more than 2.5 times its initial

size in free space. As the value of κ increases, we see that the divergence of

the beam is reduced. For higher values of κ the signal size exhibits oscillatory

behavior. For each κ, the mode is confined to a finite transverse size or, in other

words, the beam is guided within that transverse cross-section. If we can control

the properties of κ, we have a control over the properties of the waveguide.

Next we show that in the Raman system, we can control κ all optically by the

power, size and frequency of the control beam. Consider the coupling beam in the

Laguerre-Gaussian LG-01 mode (donut mode) whose intensity can be written as,

Ic(r) =2P

πw2c

r2

w2c

exp(−2r2

w2c

), (4.9)

where P is the coupling beam power and wc is its width.

When r << wc, we can approximate the above expression as,

Ic(r) =2P

π

r2

w4c

. (4.10)

From above, the square of the coupling beam Rabi frequency can be written

65

as,

|Ωc|2 =4µ2

ch2ǫ0

P

πw4c

r2. (4.11)

Using the above coupling beam rabi frequency, the off-resonant refractive index

discussed in chapter 2 can be written as,

n = 1 +β

2

|Ωc|24∆2

δ

δ2 + γ2= 1 +

β

2

µ2

ch2ǫ0

P

πw4c

δ

∆2(δ2 + γ2)r2. (4.12)

Note that we ignored power broadening and the AC stark effect in the above

equation because the region of interest is the center of the donut beam where the

coupling beam intensity is very small. We can see from the above equation that

we have a medium with a refractive index that changes quadratically with the

radial coordinate.

From Eqn. 4.12, we can write κ2 as,

κ2 = −βµ2

ch2ǫ0

P

πw4c

δ

∆2(δ2 + γ2). (4.13)

We can see from the above equation that for negative values of δ, κ is real

and hence we have an oscillatory solution discussed in the previous section. For

positive values of δ, the refractive index at the core of the LG-01 beam is lower

than at the annulus and hence we have a diverging solution. We also note that the

value of κ2 is proportional to P and hence higher values of coupling beam power

results in better mode confinement. Also note that the value of κ2 is inversely

proportional to w4c and hence the waveguiding is strongly dependent on the size

of the coupling beam.

Fig. 4.2 shows the dependence of output signal size on coupling beam power.

66

0 10 20 30 40 500

100

200

300

400

500

600

700

800

900

1000

100 microns

75 microns

50 microns

25 microns

Coupling Power (mW)

Ou

tpu

t S

ign

al S

ize

(m

icro

ns)

Input Signal Size

Figure 4.2: Beam propagation as a function of coupling rabi frequency

−5 0 5

x 105

0

500

1000

1500

2000

2500

100 microns

75 microns

50 microns

25 microns


Ou

tpu

t S

ign

al S

ize

(m

icro

ns)

Input Signal Size

Figure 4.3: Beam propagation as a function of Raman detuning

67

Each of the curve is plotted for different input signal beam widths. Eqns. 4.8 and

4.13 are used to simulate the beam size after propagating a distance of 10 cm. We

use the coupling beam width of 800 µm, β of 105 and a single photon detuning of

1.5 GHz for the simulation. We also assumed that δ = −γ = 100 KHz. We can

observe that the behavior of the signal output size is also dependent on its initial

size. Smaller initial size results in faster oscillations in its size.

We can also see from Eqn. 4.13 that κ2 has a dispersive relation with the

Raman detuning. Fig. 4.3 depicts this behavior. The output signal size is plotted

against the Raman detuning for various input signal sizes. The coupling beam

power is assumed to be 10 mW. We see that signal size is smaller for negative δ

and larger for positive δ. The size approaches the free space value for large δ on

either side of zero Raman detuning. The change is faster for smaller input signal

beam size.


The experimental setup shown in Fig. 4.4 consists of a 795 nm external cavity

tunable diode laser followed by a tapered amplifier. The beam is split in two at

a 50/50 beam splitter. One beam acts as the signal after frequency shifting it by

about 3.035 GHz to the red by double passing it through a 1.5 GHz acousto-optic

modulator. The other beam acts as the coupling beam which is sent through a

spatial filter in order to clean up its mode and is followed by a charge one spiral

phase plate, resulting in a first order Laguerre Gaussian beam. The orthogonally

polarized coupling and signal are then combined at a polarizing beam splitter.

We use a configuration where the coupling focuses at the back face and the

68

signal focuses into the core of the donut coupling beam at the front face of the

vapor cell. We use the configuration of focusing coupling beam rather than a

collimated beam because we want to show that the signal beam follows the size

of the coupling beam. The transmission properties of the beams and two photon

characteristics are observed by taking off the spiral phase plate and the lenses L1

and L2. The coupling is filtered at another polarizing beam splitter after the cell.

We image the back face of the cell with a 4f imaging system to determine the

size of the signal. The anti-reflection coated vapor cell is 5 cm long and contains

a natural abundance of rubidium isotopes with a 20 torr neon buffer gas. The

vapor cell is placed inside a magnetically shielded oven and is maintained at a

temperature of about 80 C which results in number densities of approximately

1012 cm−3. We have a positive single photon detuning, ∆, of about 500 MHz.


The plot showing the transmission of the signal beam as a function of Raman

detuning is shown in Fig. 4.5. The coupling and the signal are tuned to be

about 500 MHz to the blue of Rb85 F = 2 to F′ = (2,3) and F = 3 to F′ =

(2,3) D1 transitions respectively. Both coupling and signal are collimated and are

co-propagating. The line shape is similar to what we obtained in chapter 2.

The dispersion of the medium is obtained by applying Kramers Kronig rela-

tions on the observed transmission profile. We obtain dispersion profile in this

manner for various coupling beam powers. Fig. 4.6 shows the plot of the varia-

tion of the refractive index versus coupling beam power at a fixed signal frequency

with δ = -1.5 MHz. We can see that the refractive index decreases with increasing

coupling beam intensity. We choose the signal frequency to be close to Raman

69

0 0

Laser

795 nm

AOM

λ/2

PBS PBS

Rb

Signal

Coupling

0

0

DCamera

2ff f

Spiral

Phase

Plate

L1L2

L3 L4

50/50

BS

Figure 4.4: The experimental schematic for all optical waveguiding using atomicrubidium vapor. The focusing scheme for control beam (black) and signal (gray)is shown in the inset

70

Raman Detuning (MHz)-3 -2 -1 0 1 2 3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Tra

nsm

issio

n (

AU

)

Figure 4.5: The experimental plot of the variation in the transmission of signalversus Raman detuning.

resonance such that there is good contrast in refractive index for higher and lower

coupling beam powers and away from the absorption dip shown in Fig 4.5. We

found that we have optimum guiding at a δ = -2 MHz.

Fig. 4.7(a) shows the refractive index profile along the transverse plane. White

indicates higher refractive index and black, along the ring of the donut, is lower

refractive index. The intensity profile at the front of the vapor cell is captured

with a camera and Eqn. 4.12 is used to obtain the refractive index. Fig. 4.7(b)

shows the index profile along one of the axes of the beam. We see that we have a

refractive index contrast of about 10−5 between maximum and minimum refractive

indices. We use a single photon detuning of 500 MHz, a Raman detuning of -1

71

0 5 10 15 20 25 30−16

−14

−12

−10

−8

−6

−4

−2

0

2x 10 −7

Coupling Beam Power (mW)

Refractive Index - 1

Figure 4.6: The plot of refractive index of the signal, tuned -1.5 MHz away fromRaman resonance, as a function of coupling beam power.

-792 -342 108 558

9.6

9.65

9.7

9.75

x 10−4

Position (micro meters)

Re

fra

ctive

In

de

x -

1

(c) (b)(a)

Figure 4.7: (a) shows the spatial variation of refractive index. (b) shows the plotof refractive index versus position along one of the axes.

72

−500 0 500

−50

0

50

100

150

−500 0 500

−50

0

50

100

150

Inte

nsity (

AU

)

Position(microns) Position(microns)

(a) (b)

(c) (d)

(a) (b)

Figure 4.8: The Snap shots of the signal beam profile at the back of the vapor cellwith the coupling beam off (a) and on (b). Beam profiles along the longer axis ofthe beams at the front face of the cell (c) and at the back face of the cell(d).

MHz and 30 mW coupling beam power in Eqn. (4.12) to obtain the refractive

index profiles from a camera snap shot of intensity profile.

Note that the integrated intensity of the black curve is approximately 43%

of the gray curve, indicating a good coupling of power into the waveguide. The

coupling beam power is 18 mW. The coupling beam intensity in (c) and (d) is

normalized to fit in the figure while the signal intensities in (d) are relative.

Fig. 4.8 shows the main result of the experiment. The coupling beam is

converging along the cell and has a focus at the back face of the cell. The snap

73

shots of the signal beam with and without the coupling beam at the back face

of the vapor cell are shown in Fig. 4.8(a) and Fig. 4.8(b). The signal is focused

into the “core” of the coupling beam at the front face as shown in Fig. 4.8(c).

The black dashed line is the measured coupling beam intensity profile, the black

dotted line is the measured signal intensity and the solid black line is the gaussian

fit to the measured signal intensity. The gaussian width of the signal is 56 µm. In

the absence of the coupling beam, the signal beam diverges along the length of the

cell as shown by the gray curve in Fig. 4.8(d). In Fig. 4.8(d), the black dashed

line is the measured coupling beam profile, gray and black dotted lines are the

measured signal beam profiles when the coupling is off and on respectively and

the solid gray and black lines are gaussian fits to the dotted lines. The gaussian

widths of gray and black curves are 102 µm and 35 µm respectively. When the

coupling is on, the signal is guided along its core and so the signal beam diameter

is smaller at the back face of the cell. For example, the Gaussian width of the

signal at the back face is 35 µm with an 18 mW input coupling (black curve of

Fig. 4.8(d)). The integrated intensity of the black curve is approximately 43%

of the gray curve implying that there is a good coupling of the signal power into

the waveguide. Note that the peak intensity of the black curve is more than that

of the gray curve. The coupling beam intensity in Fig. 4.8(c) and Fig. 4.8(d)

is normalized to fit in the figure while the signal intensities in Fig. 4.8(d) are

relative. The laser beams are slightly elliptical and so the axis mentioned above

is along the longer axis.

We saw in Figs. 4.2 and 4.3 the variation of the output signal beam size with

the coupling beam power and the two photon Raman detuning. Fig. 4.9 and Fig.

4.10 shows the experimental observation of the dependence. We see from Fig. 4.9

74

0 5 10 15 2030

40

50

60

70

80

90

100

Coupling Power (mW)

Sig

na

l B

ea

m W

idth

(m

icro

ns)

Figure 4.9: Plots of the signal beam size at the back face of the vapor cell versusthe control beam power

−5 0 530

40

50

60

70

80

90

100

110

120

Sig

na

l B

ea

m S

ize

(m

icro

ns)


Figure 4.10: Plots of the signal beam size at the back face of the vapor cell versusthe Raman detuning

75

−5 0 540

60

80

100

120

140

160

180

200

220

Sig

na

l P

ow

er

(AU

)

Frequency (MHz)

Figure 4.11: The plot of output signal power versus Raman detuning.

50 100 150 200 250 300 350 400 450 500

40

60

80

100

120

140

160

180

200

220

Input Signal Power (micro Watts)

Ou

tpu

t S

ign

al P

ow

er

(mic

ro W

atts)

Figure 4.12: The plot of output signal power versus input signal power. The plotis nearly linear, the slope of the linear fit to data is 0.43.

76

the oscillatory behavior predicted in the theory. The gray curve beam size along

the horizontal and the black curve shows the beam size along the vertical. Fig.

4.10 shows the expected dispersive looking plot of the output beam size versus

Raman detuning. We have waveguiding when the Raman detuning is negative

and divergence for positive detuning. It also shows that we can have waveguiding

for a range of frequencies over a bandwidth of few MHz. This means that we can

potentially guide optical pulses with bandwidths of a few megahertz. Fig 4.11

shows the output powers for different frequencies. We see that in the frequency

range where we have good waveguiding, we also have relatively good transmission

of the signal power.

In order to verify that the waveguide is the result of the Raman absorption and

not due to other nonlinear effects like four-wave mixing, we measure the output

signal power for various input signal powers as shown in Fig. 4.12. We see that the

plot is linear and hence the waveguiding effect is linear in signal power. This also

means that the waveguiding effect is not due to self focusing effect. Instead it is

due to coupling beam dependent focusing. The slope of the linear fit is about 0.43,

which means that we couple about 43% of input signal power into the waveguide.

This implies that we can expect to guide light of very low power signal without

significant loss.

4.4 Summary

In summary, we use the intensity dependent refractive index resulting from a

Raman transition in a Λ system to create an all optical waveguide. We are able to

transmit about 43% of the power along the waveguide, for lengths much greater

than the diffraction length, using a low power control beam.

77

This all optical waveguide can be used to achieve efficient nonlinear processes

at very low light levels. For example, in the case of naturally abundant rubidium,

we can use one isotope to guide the signal and the other isotope as a medium for

the nonlinear processes such as two photon absorption, Stark shift, Kerr effect, etc.

Lukin and Imamoglu [Lukin and Imamoglu, 2000] suggested the use of rubidium

isotopes for two simultaneous, independent nonlinear processes to achieve large

Kerr nonlinearities. One can also optimize the waveguide to increase the band-

width and allow for multiple frequency waveguiding. Finally, one can use this

waveguide as a building block for an all optical beam-coupler and beam-splitter,

similar to solid-state waveguide devices.

78

5 Slow Light Prism

Spectrometer

In this chapter, we present a rubidium vapor prism spectrometer that operates

in the transparent region between two strongly absorbing resonances with six

orders of magnitude greater dispersing power than a standard glass prism. Unlike

the earlier proposals which utilize enhanced refraction in the absorption region,

this scheme utilizes the steep linear dispersion in the transparent region between

two resonances. Such a transparent region also gives rise to slow light and has

been studied in various systems recently [Khurgin, 2010]. We show that the

number of resolvable spectral lines in such a system is proportional to slow light

delay-bandwidth product. The delay bandwidth product for double absorption

slow light in rubidium has been shown to be nearly 100 [Camacho et al., 2006;

Camacho et al., 2007] and thus is advantageous over electromagnetically induced

transparency (EIT) [Fleischhauer et al., 2005] based slow light prisms [Sautenkov

et al., 2010] where the delay bandwidth products are typically less than 1. Our

slow light prism is capable of measuring optical frequency fluctuations with a

precision of 20 Hz/√Hz and can spatially resolve frequencies of about 50 MHz.

79

θ1

θp

θ2 θ3

θ4

x0

n(ω)Rb Prism

Figure 5.1: The rubidium vapor cell can be approximated as a dispersing prism.For our setup θp ≈ 79o, θ1 ≈ 20o and thus the geometrical parameter A is about2.

5.1 Theory

First we compute the dispersing power of the double Lorentzian prism then

show that the number of resolvable lines between the resonances is proportional

to the delay bandwidth product. Consider a double absorption slow light medium

[Camacho et al., 2006] with an angled interface with air as shown in Fig. 5.1.

Assuming that the index of refraction of air is unity, the change in the direction of

the beam at the interface is small and sin(θ1) ≈ θ1, we can obtain the exit angle of

the ray propagating through the prism as, sin(θ4) = n(ν) sin(θp− θ1/n(ν)), where

θp is the apex angle of the prism and θ1 is the angle made by the ray with the

normal of the first surface of the prism as shown in Fig 5.1. For n(ν) ≈ 1, the

angular dispersion of frequencies can be written as

dθ4dν

= Aλτ

L, (5.1)

where the geometrical parameter A = tan(θp−θ1)+θ1. In the above equation, we

assumed a steep linear dispersion and replaced dn/dν with ng/ν where ng is the

group index. For a medium of length L and a group delay of τ , ng/ν = λτ/L where

λ is the wavelength. The group delay in such a system is approximately given by

80

τ = αL/Γ where α is the absorption coefficient at the center of the transparency

and Γ is the full width at half maximum of each absorption [Camacho et al., 2006].

We now compute the number of spatially resolvable frequencies. We place a

lens of focal length f near the exit face of the prism, the displacement of the beam

in the focal plane is ∆y = ∆θ(ν)f . For a beam with the Gaussian diameter of D

before the lens, the Fourier transform limited diameter of the beam at the detector

is given by, d = 4λπ

fD. In order to estimate the spatial resolution, we calculate the

amount of frequency shift needed for one beam waist displacement of the beam

at the detector. Setting ∆y = d, we find

δν =d

fA

L

λτ. (5.2)

Suppose ∆ν is the bandwidth of the prism, the maximum deflection for the system

is given by,

∆ymax = fλAτ∆ν

L. (5.3)

We see that the maximum deflection depends on the delay-bandwidth product

over unit length. We are interested to have ∆ymax/d >> 1 in order to spatially

separate large number of frequency components.

In order to maximize ∆ymax/d we need a slow light system with large delay

bandwidth product like a double Lorentzian absorption system. The bandwidth

of the system is governed by the separation between the two absorptions and

the delay is dependent on the optical depth. The hyperfine absorption lines in

alkali metals can provide the required double absorption resonances. Our demon-

stration in this paper uses the dispersion between two such rubidium absorption

resonances.

81

Laser 780 nm λ/2

Fiber EOM

LensHeated

Rb Vapor cell Camera

Figure 5.2: A schematic of the experimental setup.

The simplified model discussed above can predict our experimental results.

However for more accuracy, we need to consider the effect of centroid shift due

to differential absorption across the transverse cross-section of the beam in the

prism. The centroid of the Gaussian beam is shifted by αL0w2/4x0, where α is

the absorption coefficient, L0 is the propagation distance for the centroid when

there is no absorption, w is the Gaussian beam spot size (D/2) and x0 is the

distance of the beam from vertex of the prism. We note that L and τ in all the

equations correspond to the length of propagation and delay for the centroid of

the exit beam.


The schematic of the experimental setup is shown in Fig. 5.2. A narrow

line width external cavity diode laser at 780 nm is tuned between the two Rb85

hyperfine resonances of the D2 line and coupled into a fiber electro-optic modulator

(EOM) which is driven by an arbitrary waveform generator. A λ/2 wave plate is

used to control the efficiency of the sideband creation in the EOM. The light is

82

then focused through an angled hot vapor cell onto a camera a distance 38 cm

away. The vapor cell, 7.5 cm long with angled windows, has rubidium with natural

isotopic abundance. The light enters the vapor cell near the edge of one of the

windows and exits through the curved surface as shown in Fig. 5.2. This creates

an effective prism of length 6 mm along the center of the exit beam. The vapor cell

is placed approximately 3 cm away from the focusing lens. The Gaussian spot size

of the beam at the vapor cell is about 1.6 mm and the beam is at approximately

3.2 mm from the apex of the prism. The deflection is observed on the camera

through a coarse change of the laser frequency or by modulation of the EOM. A

removable 50/50 beam splitter is put in the path of the beam after the vapor cell

to monitor the optical depth of the cell.


As we change the frequency of the laser within the transparent region between

the two Rb85 hyperfine resonance frequencies, we see a shift in the position of the

beam at the camera. The effective focal length of the focusing lens is chosen such

that the displacement of the beam for maximum possible frequency shift is within

the active area of the camera. We note that the displacement of the beam as well

as its focal spot size increase for longer focal distance. The temperature of the

vapor cell is tuned such that we have about 25% transmission at the center of the

transparent region. Even though the deflection of the beam increases for higher

temperatures of the vapor cell, the effective bandwidth of the system decreases

due to increased absorption. At our working temperature the bandwidth of our

system is about 1 GHz. The Gaussian diameter of the focal spot at the camera is

about 90 µm.

83

ν = 0 MHz

ν = 50 MHz

ν = 100 MHz

ν = 300 MHz

ν = 500 MHz

Figure 5.3: The camera images for different modulation frequencies.

The frequency dependent deflection is quantified by first turning off the EOM

and tuning the frequency of the laser to the center of the transparency. Turning on

the EOM results in frequency side bands. Different frequency bands in the signal

are spatially separated after the rubidium vapor cell and the resultant spatial

distribution of intensities is recorded at the camera. Fig. 5.3 shows the camera

images for different RF modulation frequencies. The central spot in each of the

images is the zeroth order (unmodulated) frequency followed by the first order

and second order side bands. Frequency sidebands up to second order are clearly

visible for the modulation frequency of 200 MHz. The first order side bands are

visible up to the modulation frequency of 550 MHz. The intensity dependence

on the position for different modulation frequencies is shown in the Fig. 5.4.

Even though the phase modulation efficiency of the EOM is constant for the RF

frequencies we used, the frequency dependent absorption causes the change in

84

0 0.5-0.5 1-1 1.5-1.5

Position (mm)

Figure 5.4: One dimensional intensity scans for different modulation frequencies.

85

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

0 200 400 600-200-400-600

Def

lect

ion (

mm

)

Frequency (MHz)

DataLinear Fit

Figure 5.5: Plot of deflection as a function of frequency. Circles represent theexperimental data and solid line is the linear fit. The slope of the line is 1.95µm/MHz.

relative intensities. One can obtain the exact spectral information of the input

signal by correcting for the frequency dependent losses at the vapor cell. We also

note that for the modulation frequencies less than 50 MHz, the sidebands are not

well separated in space.

Fig. 5.5 shows that the displacement changes linearly with the modulation

frequency for the frequency range of about 1100 MHz. The circles show the

experimental data and the solid line its linear fit. From the slope of the linear

fit, we deduce that the displacement at the camera per unit frequency change is

about 1.95 µm/MHz. This is a relatively large displacement for a small change in

frequency. The maximum displacement of the beam ymax is measured to be 2.145

mm for a frequency range of 1100 MHz. This implies that dn/dν for the prism

is about 2.7 × 10−12 Hz−1. Using Eq. (5.3), for a delay of 26 ns and propagation

length of 6± 1 mm, we expect a maximum deviation of 2.6± 0.4 mm. From Eq.

86

0-0.2-0.4-0.6 0.2 0.4 0.6

Position (mm)

SignalIdler

Pump

Figure 5.6: (a) Four wave mixing beams are split at the prism and imaged at thecamera. (b) The one dimensional scan of the intensity profile.

(5.2), using the measured focal spot size, we expect a spatial frequency resolution

δν of 37±6 MHz. These predictions are close to what we observed experimentally.

The sensitivity of the spectrometer is quantified using a similar configuration

to Fig. 5.2. Instead of a CCD camera we use a position sensitive quadrant detec-

tor. We use the EOM at a modulation frequency of 1 GHz and then adjust the

frequency of the laser such that one of the first order sidebands is in the transpar-

ent region between the two Rb85 hyperfine resonances. The central peak and the

other sidebands are in a highly absorbent region. The sideband frequency is then

modulated using an external arbitrary waveform generator. For this measurement

we modulated the frequency of the sideband by 10 kHz at an external rate of 60

kHz. The signal from the quadrant detector is low pass filtered at 100 kHz and

high pass filtered at 30 kHz. The power of the beam after passing through the

87

cell was roughly 300 µW. The sensitivity of the spectrometer is found to be ap-

proximately 20 Hz/√Hz, which corresponds to a signal to noise ratio of 1. This

is almost four orders of magnitude improvement over the standard glass prism

technique [Starling et al., 2010]. The theoretical limit for a shot noise limited

system is less than 1 Hz/√Hz, but we were limited by turbulent noise and other

technical noise.

To demonstrate the practical utility of this rubidium prism, we show that we

can spatially separate different frequencies resulting from a collinear four-wave

mixing process in Rb85. The signal and idler beams resulting from the four-wave

mixing process in rubidium have been shown to be number squeezed and are ex-

pected to be entangled. Thus, four-wave mixing in rubidium is a potential source

for narrow-band entangled photons. However, the residual pump noise in the sig-

nal and idler beams and the spontaneous emission noise in hot atomic vapor make

such experiments challenging. We can use the rubidium prism to separate differ-

ent frequency components spatially and reduce the amount of crosstalk between

the beams.

The signal beam is to the blue of the F = 3 → F ′ = 2, 3 Rb85 transition

and the pump beam, separated by 3.035 GHz in frequency from the signal beam,

is to the blue of the F = 2 → F ′ = 2, 3 Rb85 transition. Signal, idler and pump

beams at the output of a rubidium vapor cell are coupled into a fiber and at the

output of the fiber we place our rubidium prism. Signal and pump beams fall

in the highly dispersive frequency regions of the rubidium prism and hence are

deflected. The magnitude of deflection is different for signal and pump because

of differing refractive index of rubidium at those frequencies. The idler beam

which is about 6 GHz to the blue of signal beam is far away from any resonance

88

frequencies and hence it does not experience much deflection. Fig. 5.6(a) shows

the image at the detector and Fig. 5.6(b) its horizontal cross-sectional plot. The

central spot is the signal beam and the left and right spots are of pump and idler,

respectively. We thus see that each of the frequencies are well separated from one

another. We believe this prism will be an invaluable tool for spectral filtering in

atomic experiments.

5.4 Summary

To summarize, we demonstrated a highly dispersive atomic prism with a spatial

frequency resolution of 50 MHz. We show that the number of spatially resolv-

able spots is related to the slow light delay-bandwidth product, which makes a

double absorption system ideal for the prism. In addition, we made a precision

measurement of frequency fluctuations with a sensitivity of 20 Hz/√Hz which is

approximately a measurement of 1 part in 1013 using a very simple setup and thus

this prism may have applications in frequency metrology. We also demonstrated

a spatial separation of the pump, signal and idler beams from a four-wave mixing

process in rubidium. Number squeezed signal and idler beams have applications

in quantum metrology and hence this prism can be used to reduce the noise in

the signal and idler beams resulting from the crosstalk between the beams.

89

6 Conclusions

We have presented many experimental results in the preceding chapters re-

garding the optical control of temporal and spatial propagation of light. We used

Raman absorptions to obtain slow group velocities for the signal pulse. We used a

four-wave mixing process to store signal pulses in the atomic medium and we have

demonstrated a technique to store transverse information of pulse in vapor which

is robust to diffusion. We used coupling beam intensity dependent refractive index

to control the signal beam size as it propagates in rubidium vapor. Finally, we

demonstrated an atomic prism which can be used as a spectral filter in atomic

experiments. In this chapter, we summarize the results and provide an outlook

for further research.

In chapter 2, we demonstrated a slow light system based on dual Raman

resonances. Slow light based on dual absorption resonances are known to cause

low dispersive and absorptive broadening on signal pulses compared to a single

transmission resonance based slow light. We have shown that the bandwidth

of such a system is proportional to the frequency separation of the two Raman

absorptions and the group index is proportional to the strength of absorptions.

90

The turn-on time of the slow light is inversely proportional to the width of the

absorption and the turn-off time is proportional to the excited state life time. All

of these properties, except the excited state life time, can be controlled by the

coupling beam, and thus we have an ultra-tunable slow light system. Competing

nonlinear processes like four-wave mixing set a limit on the maximum achievable

strength of Raman absorption and thus set a limit for tunable range. We presented

experimental results which show the dependence of Raman absorption line shapes

on coupling beam powers. We have demonstrated experimentally a delay of more

than two pulse widths and slow light tuning times faster than a pulse width.

Tunable dual resonance slow light systems can have many applications. The

actual design parameters may vary according to application. For example, an

optical buffer may require large bandwidth and fast tuning times. In such a design,

we choose broad absorption resonances separated in frequency by the required

bandwidth. In interferometric and other sensing applications, we need large group

indices and in such designs we need two strong and narrow resonances which are

very close in frequency. Slow light based Fourier transform interferometer has

been demonstrated by Shi et al. [Shi et al., 2007]. The interferometer output as a

function of group index is first obtained and the signal spectrum is then retrieved

by Fourier transforming the output. The speed with which we determine the

spectrum can be dramatically improved by using optical tuning of the slow light.

The spectral resolution can also be improved by two orders of magnitude.

In chapter 3, we demonstrated four-wave mixing based stopped light. The

signal pulse and spontaneously emitted idler pulse are coherently mapped onto

the rubidium ground state coherence. Signal and idler pulses can be retrieved

before the ground state decay time, which in our case is shown to be about 120 µs.

91

We show that the retrieval efficiency is linear in retrieval beam power. The four-

wave mixing storage is interesting because of the quantum mechanical properties

of the signal and idler photons. More work needs to be done to explore the

possibility of enhanced correlated photon generation. This system is capable of

both amplifying and slowing the signal pulses and thus has applications for optical

communications.

We also demonstrate storage of multiple transverse modes of signal by storing

the Fourier transformed image instead of a flat phase image. This way the storage

is more robust to diffusion. We demonstrate the storage of a test image and show

that the dark areas of the image remain dark and the retrieved image is attenuated

more near the edges of the image than at the center. The central idea of this

scheme is that the atomic coherences carry phase information of the signal and

that oppositely phased coherences destructively interfere during diffusion.

The image storage scheme differs from other holographic storage devices. Even

though the image storage times for the atomic scheme is much less, the holographic

storage require much greater signal powers. The storage and retrieval of image

storage is also much faster. The scheme discussed here can potentially store

the transverse profile of a single photon and thus may be useful for quantum

information. More work needs to be done to demonstrate this possibility. This

is also useful for remote sensing and Lidar applications where fast storage and

retrieval of a reference image is required.

In chapter 4, we demonstrated waveguiding of the signal beam using Raman

absorption. The refractive index of the signal beam is shown to be dependent on

the coupling beam intensity. The first order Laguerre-Gauss coupling beam is used

to obtain quadratic dependence of the refractive index of the signal on the radial

92

coordinate. The signal beam size propagating inside the medium can be changed

by changing the power or frequency of the coupling beam. We experimentally

demonstrate waveguiding and also show the dependence of the output signal size

on coupling beam parameters.

The next step in this work is to demonstrate an improvement of the efficiencies

of the nonlinear processes. For example, we may use a sample with two isotopic

species, like in the case of rubidium, to achieve waveguiding using one species at

frequencies useful for nonlinear processes in another species. For instance, Raman

absorption resonance can be created for Rb87 at a frequency which is resonant to

Rb85 transition. We can then demonstrate degenerate EIT at a very low coupling

and signal beam powers.

In chapter 5, a sensitive atomic prism has been presented. We have shown

that the prism is capable of spatially separating spectral lines 50 MHz apart. This

property can be used to filter different optical frequencies in atomic experiments

involving many optical frequencies. For example, signal and coupling beams in

EIT experiments can be spatially separated after the vapor cell using this prism

instead of a polarizing beam splitter. This allows for exploring an EIT regime in

which signal and coupling beams have parallel polarizations. We demonstrated

that the signal, idler and coupling beams resulting from the four-wave mixing

process can be filtered using this prism. We believe that we can obtain more

spectral purity using this filter when compared to a Fabry-Perot etalon which we

used in chapter 3.

We also measured frequency fluctuations with an accuracy of 20 Hz/√Hz. Ac-

curate measurement of frequencies opens the door for many precision experiments.

For example, low powered optical signals can be measured in a nondestructive way

93

by measuring very small induced AC-stark shifts. One can also use this prism to

measure magnetic fields by converting magnetic field dependent resonances to

observable optical deflections. There is a possibility of tuning the properties of

the prism using an additional pump beam which can deplete the ground state

population to cause a decrease in refractive index and an increase in available

bandwidth.

To summarize, off-resonant interactions in rubidium discussed in this thesis

are very interesting and have potential use in fields like optical communications,

quantum information, precision measurements and sensing. We demonstrated

slow light, stopped light and guided light in this thesis and these demonstrations

lead us to more interesting experiments. Finally, the atomic prism presented in

this thesis can serve as a valuable tool for many atomic experiments.

94

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109

A Derivation of Susceptibility

for a Lambda System

In this appendix we derive susceptibility at the signal frequency in a three-

level atom in a Λ configuration. In a Λ system, two lower energy atomic states are

coupled optically with an exicted state. Optical coupling between two lower energy

states is forbidden. The resulting simultaneous interaction of the atom with two

optical modes has some interesting properties. For example, the transmission and

the refractive index of one optical mode is altered by the properties of the other

mode. We call the optical mode of our interest as signal beam and the optical

mode that controls the properties of the signal beam as control beam.

Consider the three-level Λ system shown in Fig. 1.2. The levels 1 and 2 are

coupled by the probe field Es

2e−i(ωst−kz) + c.c. The control field coupling states 2

and 3 can be written as Ec

2e−i(ωct−kz)+ c.c. The Rabi frequencies of signal and the

coupling beams are defined as. Ωs,c(t) = (µEs,c/h)e−i(ωs,ct−kz) = Ωs,ce

−i(ωs,ct−kz).

The detunings are defined as ∆ = ∆s = (ω2 − ω1)− ωs, ∆c = (ω2 − ω3)− ωc and

δ = ∆s −∆c. The Hamiltonian for this system is given by

110

H = hω1 |1〉〈1|+ hω2 |2〉〈2| + hω1 |3〉〈3| −hΩs(t)

2|2〉〈1| − hΩ∗

s(t)

2|1〉〈2|

− hΩc(t)

2|2〉〈3| − hΩ∗

c(t)

2|3〉〈2| . (A.1)

Let the state of the system is given by |Ψ〉 = a |1〉+ b |2〉+ c |3〉. The evolution of

the state follows the Schrodinger’s equation, ih ∂∂t|Ψ〉 = H |Ψ〉. We can write this

in the matrix notation as follows:

ih∂

∂t

a

b

c

= h

ω1 −Ω∗s(t)/2 0

−Ωs(t)/2 ω2 −Ωc(t)/2

0 −Ω∗c(t)/2 ω3

a

b

c

. (A.2)

Using the transformation,

a

b

c

=

e−iω1t 0 0

0 e−iω2t 0

0 0 e−iω3t

a′

b′

c′

, (A.3)

we get,

∂

∂t

a′

b′

c′

= −i

0 −Ω∗se

−i∆st/2 0

−Ωsei∆st/2 0 −Ωce

i∆ct/2

0 −Ω∗ce

−i∆ct/2 0

a′

b′

c′

(A.4)

111

Using another transformation,

a′

b′

c′

=

1 0 0

0 ei∆t 0

0 0 eiδt

a′′

b′′

c′′

, (A.5)

we get,

ih∂

∂t

a′′

b′′

c′′

= −h

0 Ω∗s/2 0

Ωs/2 −∆ Ωc/2

0 Ωc/2 −δ

a′′

b′′

c′′

(A.6)

By comparing the equations A.2 and A.6, the effective Hamiltonian for this

system in this rotating frame is given by

Heff = −h

0 Ω∗s/2 0

Ωs/2 −∆ Ωc/2

0 Ω∗c/2 −δ

. (A.7)

Assuming that we have a closed system with the total spontaneous decay rate

from the excited state given by Γ and the ground state decoherence rate given by

γ, the decoherence matrix D is given by

D =

−Γρ22/2 Γρ12/2 γρ13

Γρ21/2 Γρ22 Γρ23/2

γρ31 Γρ32/2 −Γρ22/2

. (A.8)

Using the evolution equation for the density matrix, ρ = − ih[Heff , ρ]−D, the

equations of motion for the density matrix elements are given by,

112

ρ11 =Γ

2ρ22 − ℑΩ∗

sρ21 (A.9)

ρ22 = −Γρ22 + ℑΩ∗cρ23 + Ω∗

sρ21 (A.10)

ρ33 =Γ

2ρ22 − ℑΩ∗

cρ23 (A.11)

ρ23 = −[

i∆c +Γ

2

]

ρ23 + iΩc

2(ρ33 − ρ22) + i

Ωs

2ρ13 (A.12)

ρ21 = −[

i∆s +Γ

2

]

ρ21 + iΩs

2(ρ11 − ρ22) + i

Ωc

2ρ31 (A.13)

ρ31 = −(iδ + γ)ρ31 + iΩ∗

c

2ρ21 − i

Ωs

2ρ32 (A.14)

In the above equations ℑ(ρ) denotes the imaginary part of ρ. Assuming that

ρ11 ≈ 1 and ρ22 = ρ33 ≈ 0, the steady-state perturbative solution for ρ21 up to

first order in Ωs is,

ρ(ss)21 =

Ωs

2

δ − iγ

(δ − iγ)(∆− iΓ/2)− |Ωc|2/4. (A.15)

The susceptibility at the signal frequency is thus given by,

χ(∆, δ,Ωc) = βδ − iγ

(δ − iγ)(∆− iΓ/2)− |Ωc|2/4. (A.16)

where β = Nµ2/hǫ0 where N is the number density and µ is the transition

dipole moment.

113

The real and imaginary parts of the susceptibility can be written as,

χ′(∆, δ,Ωc) = β∆

∆2 + Γ2/4

[

1 +

(

2δ0 −|Ωc|24∆

)

δ′

δ′2 + γ′2− 2γ0

γ′

δ′2 + γ′2

]

,

χ′′(∆, δ,Ωc) = βΓ/2

∆2 + Γ2/4

[

1 + 2δ0δ′

δ′2 + γ′2−(

2γ0 −|Ωc|22Γ

)

γ′

δ′2 + γ′2

]

.

(A.17)

In the above, δ0 = |Ωc|2∆/(4∆2 + Γ2), γ0 = |Ωc|2Γ/(8∆2 + 2Γ2), δ′ = δ − δ0

and γ′ = γ + γ0. The first terms in each of the above equations are the real and

imaginary parts of the susceptibility χ0 when the coupling beam is off. The second

and the third terms have dispersive and absorptive functional forms. The single

photon detuning ∆ determines how each term contribute to the total susceptibility.

slow light, stopped light and guided light in hot rubidium vapor

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