implementations of slow light using double-resonances

Implementations of Slow Lightusing Double-Resonances

by

Ryan M. Camacho

Submitted in Partial Fulfillmentof the

Requirements for the DegreeDoctor of Philosophy

Supervised byProfessor John C. Howell

Department of Physics and AstronomyThe College

Arts and Sciences

University of RochesterRochester, New York

2008

ii

For Melinda, David, Joseph, James, and Mark

iii

Curriculum Vitae

The author was born in Santa Clara, California on 12 February, 1979. He

attended Brigham Young University as a Heritage Scholar beginning in 1997 and

obtained his Bachelor of Science in Physics in 2003. He came to the University of

Rochester in the Fall of 2003 and began graduate studies in physics, receiving his

Master of Arts in 2005. He pursued his doctoral research in atomic physics and

quantum optics under the supervision of Professor John C. Howell.

CURRICULUM VITAE iv

Publications

Ryan M. Camacho and John C. Howell, “Slow Light in Atomic Vapors”, BookChapter, Eds. Jacob Khurgin and Rodney Tucker (to appear in 2008)

Curtis J. Broadbent, Ryan M. Camacho, Ran Xin, and John C. Howell,“Preservation of Energy-Time entanglement in a Slow Light Medium”, PhysicalReview Letters 100, 133602 (2008).

Praveen K. Vudyasetu, Ryan M. Camacho, and John C. Howell, “Storageand Retrieval of Multimode Transverse Images in Hot Atomic Rubidium Vapor”,Physical Review Letters 100, 123903 (2008).

Zhimin Shi, Robert W. Boyd, Ryan M. Camacho, Praveen K. Vudyasetu,and John C. Howell, “A Slow-Light Fourier Transform Interferometer”, PhysicalReview Letters 99, 240801 (2007).

M.V. Pack, R.M. Camacho, and J.C. Howell, “Transients of the electromag-netically induced transparency-enhanced refractive Kerr nonlinearity”, PhysicalReview A 76, 033835 (2007).

M.V. Pack, R.M. Camacho, and J.C. Howell, “Electromagnetically inducedtransparency lineshapes for large probe fields and optically thick media”, PhysicalReview A 76, 013801 (2007).

Ryan M. Camacho, Michael V. Pack, John C. Howell, Aaron Schweinsberg,Robert W. Boyd, “Wide-Bandwidth, Tunable, Multiple-Pulse-Width Optical De-lays Using Slow Light in Cesium Vapor”, Physical Review Letters 98, 153601(2007).

Ryan M. Camacho, Curtis Broadbent, Irfan Ali Khan and John C. Howell,“All-Optical Delay of Images using Slow Light”, Physical Review Letters 98,043902 (2007).

Ryan M. Camacho, Michael V. Pack, and John C. Howell, “Slow light withlarge fractional delays by spectral hole-burning in rubidium vapor”, Physical Re-view A 74, 033801 (2006).

M. V. Pack, R. M. Camacho, and J. C. Howell, “Transients of theelectromagnetically-induced-transparency-enhanced refractive Kerr nonlinearity:Theory”, Physical Review A 74, 013812 (2006).

Ryan M. Camacho, Michael V. Pack, and John C. Howell, “Low-distortion slowlight using two absorption resonances”, Physical Review A 73, 063812 (2006).

CURRICULUM VITAE v

Conference Presentations

Ryan M. Camacho, Praveen K. Setu, John C. Howell, “Coherence length mea-surement of a laser using slow light”, SPIE/Photonics West, San Jose, CA **In-vited** (January, 2008)

Ryan M. Camacho, Michael V. Pack, Curtis J. Broadbent, Irfan Ali-Khan,John C. Howell, Aaron Schweinsberg, Robert W. Boyd, “Implementations ofDouble-Resonance Slow Light”, OSA Annual Meeting, San Jose, CA (Septem-ber 2007)

R. M. Camacho, A. Schweinsberg, M. V. Pack, J. C. Howell, and R.W. Boyd,“Large optical pulse delays in cesium vapor”, ICQI, Rochester, NY (June 2007)

R. Camacho, C. Broadbent, I. Ali Khan, and J.C. Howell, “All Optical Delayof Images Using Slow Light”, CLEO/QELS, Baltimore, MD (May 2007)

R. M. Camacho, M. V. Pack, R. W. Boyd, J. C. Howell, “Large Fractional De-lays in a Hot Vapor”, SPIE/Photonics West, San Jose, CA **Invited** (January,2007)

Ryan Camacho, Michael Pack, John Howell, “Slow Light Near Two AbsorbingResonances”, OSA Annual Meeting, Rochester, NY (October 2006)

Aaron Schweinsberg, Ryan M. Camacho, Michael V. Pack, Robert W. Boyd,John C. Howell, “Tunable Slow Light in Cesium Vapor”, OSA Annual Meeting,Rochester, NY (October 2006)

Ryan Camacho, Michael V. Pack, John C. Howell “Large Fractional PulseDelays in a Hot Rubidium Vapor”, OSA Slow and Fast Light Topical Meeting,Washington, DC (July 2006)

Michael V. Pack, Ryan M. Camacho, John C. Howell, “Transients and RiseTimes of the Refractive EIT-Kerr Nonlinearity”, OSA Slow and Fast Light TopicalMeeting, Washington, DC (July 2006)

Michael V. Pack, Ryan M. Camacho, John C. Howell, “Low Light Level Switch-ing”, SPIE/Photonics West, San Jose, CA **Invited** (January, 2005)

vi

Acknowledgements

Graduate diplomas are quantized in the sense that they are awarded to single

individuals at a time. Nonetheless, there is demonstrable evidence that the work

reported here owes its completion to a collective ensemble. I take a moment now

to acknowledge some of the many who could rightly claim a share of my degree

were it divisible.

First and foremost, to my wife Melinda—I love you! You have sacrificed for

my success more than I. Your support, encouragement, and sheer bravery during

our sojourn in Rochester I will never forget. To my boys—David, Joseph, James,

and Mark—I appreciate your spirit and optimism for life; you make all hard work

worthy. To my parents, Mom and Dad, your support has meant more to me than

you know. I can trace most of the good decisions I’ve made in life to lessons you

have taught me.

For my academic development, I am indebted primarily to Professor John

Howell. In addition to being an excellent scientist, his enthusiasm for great exper-

iments is unmatched. I have been extremely fortunate for his careful mentoring

in the laboratory, as well as his belief that great research can coexist with happy

ACKNOWLEDGEMENTS vii

graduate students. His view that my success was his success contributed to his

meeting me in the lab at 5 am on a Saturday morning and flying back from sab-

batical in Italy for 2 days to help me finish an experiment. I am also grateful for

interaction with Professor Robert Boyd, with whom I had the chance to collabo-

rate on various experiments. His insights on how to turn undeveloped ideas into

great experiments has greatly shaped my views on scientific research.

All of the experiments that I have carried out have been done with the help of

my fellow students. I was fortunate to have Michael Pack as my first lab partner.

In addition to being a great pleasure to work with, he is among the brightest

people I have ever known. He more than anyone encouraged me to develop an eye

for detail. Michael can point out key weaknesses in an idea faster than anyone I

know, and taught me to be critical of my own work first so that others wouldn’t

have to do me the favor later on. The “Pack’isms” will likely live on for some time

in the group...“It’s easy to achieve the optimum, but hard to do better than the

optimum.” Irfan Ali Khan and Curtis Broadbent were the entanglement gurus that

patiently taught me the ways of single-photon counting and black shower curtains.

I was extremely fortunate to have their expertise on many occasions. From the

Boyd group I had the chance to work with two students: Aaron Schweinsberg,

and Zhimin Shi. I worked with Aaron on the Cesium slow light experiments.

He is an excellent experimenter and kept a very careful lab notebook. He also

has great grammatical prowess in paper-writing (our paper together is one of my

ACKNOWLEDGEMENTS viii

finest compositions). Zhimin Shi and I completed the fastest experiment of my

entire graduate career (three weeks from start to finish), after which he wrote

up the results and submitted the manuscript. It was perhaps the least painful

Physical Review Letter a graduate student ever published. Lastly, I have had

the opportunity to work with Praveen Vudyasetu, who was instrumental in the

completion of the results reported in the last two chapters of this dissertation. In

addition to being a good physicist, Praveen is a good friend who I would trust

with anything.

I would also like to thank Barbara, Ali, Connie, Michie, Sondra, Shirley, and

Janet, who managed a great portion of the non-academic part of my academic

life. They are extremely proficient. Barbara especially has gone out of her way

on my behalf, and as the face of the department during recruiting weekend made

a contribution to my decision to come to Rochester.

ix

Abstract

Many practical applications of slow optical pulse propagation (slow light) re-

quire that a pulse be delayed by many times its temporal width, or equivalently

that the product of the pulse’s delay and bandwidth greatly exceed unity. Im-

plementations in optical buffering, quantum memories, remote sensing, interfer-

ometry, and chip-scale optical circuits, for example, would be enabled by large

delay-bandwidth products. While in the last decade researchers have succeeded

in slowing optical pulses by up to a factor of 107, for a variety of reasons they have

not made exceptional progress in demonstrating large delay-bandwidth products.

The present work investigates the slow propagation of optical pulses whose carrier

frequency lies between two strongly absorbing Lorentzian resonances in an atomic

vapor and how this system mitigates the delay-bandwidth problem.

My studies show that owing to the symmetry of the double-resonance config-

uration, pulse distortion is reduced in such a way that multiple pulse delays are

possible. Detailed theoretical and experimental investigations are undertaken to

understand the physical origins of this effect, revealing an important interplay

between absorption and dispersion which allows for large delay-bandwidth prod-

ABSTRACT x

ucts. Specifically, I report on two proof-of-principle experiments demonstrating

tens of pulse delays using the incoherent ground-state hyperfine resonances in

85Rb and Cs, and develop a concise model based on dispersion theory using Feyn-

man diagrams. Several implementations of the scheme are then demonstrated

experimentally, including applications to imaging, interferometry, and quantum

entanglement. It is shown that double-resonances may also be prepared in a co-

herent fashion, allowing for tunable resonance parameters, fast switching, and

applications in nonlinear optics. Finally, it is shown that coherent preparation of

an atomic vapor may extend the the results on slow images to much longer time

scales, where atomic diffusion begins to dominate image distortion. A 4f imaging

method using analog Fourier optics is shown to mitigate this complication. My

work concludes with a summary of the principle results and a discussion of their

implications.

xi

Table of Contents

1 Introduction 1

1.1 Overview of Slow Light . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Slow Light in Atomic Vapors . . . . . . . . . . . . . . . . . . . . . 4

1.3 First Experiments in Slow Light . . . . . . . . . . . . . . . . . . . 5

1.4 Electromagnetically Induced Transparency . . . . . . . . . . . . . 7

1.5 Two-Level Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.6 Dispersion Management . . . . . . . . . . . . . . . . . . . . . . . 17

1.7 Conclusion and Dissertation Outline . . . . . . . . . . . . . . . . 23

2 Slow Light Between Two Ideal Absorbing Resonances 25

2.1 Slow Light and Lorentzian Resonances . . . . . . . . . . . . . . . 25

2.2 Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Ideal Double-Lorentzian . . . . . . . . . . . . . . . . . . . . . . . 28

2.4 Proof-of-Principle Experiment . . . . . . . . . . . . . . . . . . . . 32

2.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 38

CONTENTS xii

3 Slow Light between Two Absorbing Resonances–General Model 39

3.1 General Double-Lorentzian Model . . . . . . . . . . . . . . . . . . 39

3.2 Dispersion and Absorption . . . . . . . . . . . . . . . . . . . . . . 42

3.3 Proof-of-Principle Experiment . . . . . . . . . . . . . . . . . . . . 45

4 Implementations of Double-Resonance Slow Light 53

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 All Optical Delay of Images Using Slow Light . . . . . . . . . . . 54

4.3 Other Implementations of Double-Resonance Slow Light . . . . . 65

5 Slow and Stopped Light in Coherently Prepared Media 70

5.1 Introduction: Coherent vs. Incoherent Atomic Ensembles . . . . . 70

5.2 Feynman Diagram Model for a Coherent Four-Level System . . . 71

5.3 Coherently Prepared Double-Lorentzian . . . . . . . . . . . . . . . 75

5.4 Stopped Light in a Coherently Prepared Four-Level System . . . . 80

6 Stopped Images in Coherently Prepared Media 89

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.3 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7 Conclusions 102

CONTENTS xiii

Bibliography 107

A Feynman Diagrams 117

B Refractive vs. Dispersive Contributions to the Speed of Light in

Dielectrics 121

B.1 Electromagnetic Waves and the Group Index . . . . . . . . . . . . 121

B.2 Lorentz Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . 123

C Franson Interferometry 126

D Preservation of Energy-Time Entanglement in a Slow Light

Medium 137

E A Slow Light Fourier Transform Interferometer 142

xiv

List of Figures

Figure Title Page

1.1 Feynman diagrams for EIT . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Pulse delays in Pb EIT . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Pulse delays in Na BEC . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Pulse delays in two-level atoms . . . . . . . . . . . . . . . . . . . 15

1.5 Pulse delays using double-resonance . . . . . . . . . . . . . . . . . 22

2.1 Experimental schematic for double-resonance delays in Rb Vapor 32

2.2 Amplitude and phase in slow-light interferometer . . . . . . . . . 33

2.3 Pulse delays using Rb double-resonance . . . . . . . . . . . . . . . 36

2.4 Pulse broadening in Rb double-resonance . . . . . . . . . . . . . . 37

3.1 Cs ground-state hyperfine D1 optical resonances . . . . . . . . . . 43

3.2 Experimental schematic for delay in Cs double-resonance . . . . . 46

3.3 High bandwidth pulse delays using Cs double-resonance . . . . . . 47

3.4 Moderate bandwidth pulse delays using Cs double-resonance . . . 49

LIST OF FIGURES xv

3.5 Tunability of delay in Cs double-resonance . . . . . . . . . . . . . 51

3.6 Reconfiguration rates of Cs double-resonance . . . . . . . . . . . . 51

4.1 Experimental setup for the delay of transverse images . . . . . . . 56

4.2 Interference of a delayed image with a slightly diverging local os-

cillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3 Low-light-level 1-D image delays . . . . . . . . . . . . . . . . . . . 61

4.4 Low-light-level 2-D image delays . . . . . . . . . . . . . . . . . . . 63

4.5 Histogram of photon arrival times for delayed images . . . . . . . 63

4.6 Preservation of energy-time entanglement using slow light . . . . . 66

4.7 Output intensity of the slow-light FT interferometer . . . . . . . . 68

5.1 Experimental schematic and energy level-diagram for signal and

idler pulse storage . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2 Feynman diagrams for a four-level system . . . . . . . . . . . . . 74

5.3 Experimental setup for generating coherent double-resonance in Rb 76

5.4 Measured transmission profiles of coherent double absorption reso-

nances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.5 Pulse delays in a coherently prepared double-resonance . . . . . . 77

5.6 Transmission profile of coherent double-resonance used for pulse

delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.7 Fast switching using a coherently prepared resonance . . . . . . . 79

5.8 CW transmission near four-wave mixing resonance in Rb . . . . . 85

xvi

5.9 Delay and storage of signal and idler pulses using four-wave mixing 86

5.10 Linear retrieval of signal and idler beams with high efficiency . . . 87

6.1 Experimental schematic for image storage . . . . . . . . . . . . . 92

6.2 Measured image profiles in image and Fourier planes . . . . . . . 95

6.3 Measured and calculated evolution of stored images . . . . . . . . 97

6.4 Theoretical evolution of stored ground-state coherence . . . . . . 98

C.1 Schematic of a Franson interferometer with entangled photon

source S, and two distant detectors. . . . . . . . . . . . . . . . . . 127

1

Chapter 1

Introduction

1.1 Overview of Slow Light

It has long been known that a wave packet composed of many frequency compo-

nents may travel at a very different velocity than that of any of the individual

waves making up the packet. In his study of a periodic array of coupled harmonic

oscillators, for example, Leon Brillouin found solutions for the displacement of

each oscillator in the form of waves propagating along the array [1]. These waves

could be added together to form solutions with a dramatically different propa-

gation velocity (called group velocity) than that of any one wave solution. A

necessary characteristic of such solutions, later found to be quite general, is that

the individual waves must not all propagate at the same speed—they require some

dispersion relation. It has been shown more recently, however, in the context of

optical pulses that even when individual wave velocities differ from the mean by

1.1. OVERVIEW OF SLOW LIGHT 2

no more than a factor of 10−2, the velocity of the wave packet may differ by a

factor of 107.

Such possibilities for large differences in the group and phase velocities of light

pulses has stimulated fundamental and applied research in recent years, both of

which continue to be areas of active investigation.

To motivate the origin of slow group velocity, consider the sum of two plane

waves having the same amplitude but each with a slightly different frequency ω

and wavenumber k:

S(x, t) = <[ei((k+∆k)x−(ω+∆ω)t) + ei((k−∆k)x−(ω−∆ω)t)

]

= 2 cos(kx− ωt) cos(∆kx−∆ωt) (1.1)

The resulting superposition is the product of two sinusoidal oscillations. The first

(the carrier wave) describes a wave with frequency ω and velocity ω/k, while the

second (the envelope) denotes a wave with frequency ∆ω and velocity ∆ω/∆k.

When ω/k is nominally different than ∆ω/∆k, the envelope will propagate with

a different velocity than the carrier. The envelope may travel slower than, faster

than, or even opposite to the carrier wave. If instead of a pair of monochromatic

waves we have a collection of waves whose frequencies are closely packed around

the carrier frequency ω (a wave packet), the resulting envelope will propagate at

1.1. OVERVIEW OF SLOW LIGHT 3

the group velocity

vg =dω

dk. (1.2)

Since the wavenumber k may be written in terms of a frequency-dependent index

of refraction n(ω) and the speed of light in vacuum c,

k(ω) =n(ω)ω

c, (1.3)

we may express the group velocity in terms of the frequency-dependent refractive

index:

1

vg

=d

dωk(ω)

=n(ω) + ω d

dωn(ω)

c

=ng

c. (1.4)

We have introduced the group index ng, which is analogous to the phase index

except that it tracks the relative speed of an envelope relative to the speed of

light (with the important constraint that the envelope not be too distorted during

propagation), rather than the speed of points of constant phase. By inspection

of Eq. (1.4), we may see that one may obtain a large group index by finding a

material with either a large phase index n(ω), or a large derivative in the phase

index with respect to optical frequency. While it is difficult to find materials with

1.2. SLOW LIGHT IN ATOMIC VAPORS 4

refractive indices greater than 3, it is routine in laboratories to produce group

indices on the order of 106.

1.2 Slow Light in Atomic Vapors

While the the theoretical foundations of slow light have been well known for many

years, the experimental investigation of slow light did not begin until relatively

recently. From the beginning, atomic vapors have played an important role in

carrying out these experiments. Many of the preliminary studies of optical group

velocities were performed using atomic vapors, and atomic vapors continue to be

used in both fundamental and applied slow light research.

In the remainder of this chapter, we review the essential physics necessary to

understand slow and fast group velocities in atomic systems, and attempt to trace

the experimental observation of slow light from its origins, with a specific emphasis

on work that has been done in atomic vapors. In doing so, we adopt a less common

but unifying treatment, concentrating on the resonant interactions specific to

each slow light medium using double-sided Feynman diagrams1[2]. We take this

approach for two reasons: first, because it offers a pedagogical and systematic

way to visualize the varied atomic resonances leading to slow light. Second, this

treatment is less well-known and may provide insights not normally encountered

in the usual non-graphical density matrix formalism. A more traditional approach

as well as an excellent review of slow light up to 2002 has been made by Boyd

1.3. FIRST EXPERIMENTS IN SLOW LIGHT 5

and Gauthier [3] as well as Milonni [4]. Milonni also published a book in 2005

including chapters on slow, fast, and left-handed light [5].

Manifestations of slow light in atomic vapors may be broadly organized ac-

cording to the type of optical resonances used to obtain the necessary dispersion.

In the simplest case, slow group velocities may be obtained using a single laser

in a two-level atomic system. Of recent interest has been the study of slow light

in a three level system, most commonly in the configuration of electromagneti-

cally induced tranparency (EIT). Slow light has also been observed in four-level

systems, hole-burning schemes, and a variety of others, and in principle almost

any optical resonance may be used. While the number of ways to achieve slow

light is numerous, here we choose a few representative systems to illustrate how

one may make simple predictions about any one of them given the appropriate

Hamiltonian.

1.3 First Experiments in Slow Light

The first experimental studies of slow light were performed in the context of non-

linear optics; namely, the self-induced transparency (SIT) effect discovered by

McCall and Hahn in 1967 [6]. McCall and Hahn reported on an experiment in

which they passed pulses generated in a ruby laser through cooled rods of ruby

and observed appreciable group delay in addition to the SIT effect. That same

1See Appendix A for derivations leading to Feynman diagrams

1.3. FIRST EXPERIMENTS IN SLOW LIGHT 6

year Patel and Slusher demonstrated a similar effect using gaseous SF6 as the

delay medium [7]. The first study of slow light due to SIT in an atomic vapor was

performed in 1968 by Bradley et al., [8], who measured time delays in potassium

vapor. Interestingly, the experiment also included time-delay measurements away

from SIT resonance as a control experiment. This resulted in the first measure-

ments of slow light which in the linear regime. Slusher and Gibbs then reported

much more careful studies of SIT and slow light in an atomic vapor in 1972, in

which a two-level transition on the D1 line of Rb was studied [9]. Since all of

these studies relied on the nonlinearity generated by the pulse passing through

the medium, the group delay was dependent on the width, energy, and hence area

of the pulse. In fact, some researchers reported observations of group delay as an

indication of SIT, though it was quickly pointed out by Courtens and Szoke [10]

that this was not strictly the case.

When a transparency effect (EIT) was discovered that was linear in the input

pulse by Harris and coworkers [11], it was natural that the corresponding pulse

propagation dynamics should also be studied in relation to the new transparency

mechanism, which was taken up theoretically by Harris in 1992 [12]. The first

experimental studies of linear slow light, however, were carried out much earlier

by Grischkowsky [13]. Grischkowsky showed that even far away from resonance, a

pulse may experience a significant reduction in group velocity, and made contact

with the earlier discussions of SIT by describing his results in terms of an adia-

1.4. ELECTROMAGNETICALLY INDUCED TRANSPARENCY 7

badic following effect. We will return to this class of slow light experiments after

discussing slow light due to EIT.

1.4 Electromagnetically Induced Transparency

We begin with a discussion of a three level system in the Λ configuration (see Fig.

1.1) with two fields: a weak signal beam Es and a strong coupling beam Ec. The

well-known Hamiltonian describing the atom-field interaction in the absence of

damping may be written in the rotating frame as [14]

−~2

0 Ωs 0

Ωs −2∆1 Ωc

0 Ωc −2(∆1 −∆3)

, (1.5)

where Ωs and Ωc are the Rabi frequencies induced by the signal and coupling fields

respectively, and ∆1 and ∆3 represent the signal and coupling field detunings

from optical resonance. When the two photon detuning is zero (∆1 = ∆3), we

obtain eigenvalues of 0, ~/2(∆1 ± ΩN), where we have defined a normalizing

Rabi frequency ΩN ≡√

Ω2c + Ω2

s. The vanishing energy eigenvalue corresponds

to the case in which no atom field coupling exists and has an eigenvector of

|−〉 =Ωc

ΩN

|1〉 − Ωs

ΩN

|3〉. (1.6)


|1 1|

ωs

2|

|3

ωc

2|

ωc

|1

2|

|3

ωc

ωc

|1 1|

ωs

2|

|3

ωc

2|

ωc

|1

|1 1|

ωs

2||1

+ + + ...

ωs

ωc

|1

|3

|2

ωs

ωc

|1

|3

|2

ωs

|1

|3

|2

= + + + ...ωs

|1

|3

|2

ωc

Figure 1.1: Energy level diagrams with corresponding Feynman diagrams illustrating how in-terference between various atomic absorption pathways leads to EIT.

When the system is in this eigenstate, no absorption takes place and hence no

spontaenous emission occurs. For this reason, atoms prepared in this eigenstate

are said to be in a dark state, invisible to radiation at the signal frequency. We wish

to explore the dispersive properties of such a three-level system prepared in the

neighborhood of this dark eigenstate, and so must examine the atom field coupling

in the vicinity of two-photon resonance and include damping. The steady state

polarization nearly resonant with the signal frequency may be found by summing

the polarizations induced by all possible excitation pathways from state |1〉 to

state |3〉, conveniently represented using double-sided Feynman diagrams [2], as


shown in Fig. 1.1:

Ps = 2Nµ12Ωs

2

1

∆s

∞∑

n=0

rn

= Nµ12Ωs1

∆s − Ω2c

4∆R

(1.7)

where the summation over r = Ω2c/(4∆s∆R) accounts for the repeated emission

and absorption of a coupling photon. 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/~

again represents the Rabi frequency induced by electric field amplitude Ej via the

dipole matrix element µj.

We may obtain approximate expressions for the group delay and pulse broad-

ening by performing a series expansion of the steady state polarization res-

onant with the signal frequency [Eq. (1.7)] around zero two photon detun-

ing (∆R = 0). Assuming the coupling field to be on resonance (∆c = 0),

the real and imaginary parts of the index of refraction at the signal frequency

(ns =√

1 + χs ≈ 1 + Ps/2Es) may be written

n′s ≈ 1 +2Nµ2

12

~ε0

∆s

Ω2c

(1.8)

n′′s ≈ 2Nµ212

~ε0

(γ

Ω2c

+2Γ∆2

s

Ω4c

). (1.9)


While these expressions are sufficient to characterize the essential delay and broad-

ening features of an optical pulse, we take a moment to rewrite them in terms of

the optical depth of the atomic vapor, an experimentally easy quantity to mea-

sure. In the absence of the coupling field, the polarization at the signal frequency

is just the first term in Eq. (1.7):

Ps0 =Nµ12Ωs

2∆s

≈ 2Nµ12Ωs

(∆s

Γ2+ i

1

2Γ

), (1.10)

and we may easily write the linear optical absorption coefficients 2n′′ω/c in the

absence and presence of the coupling field, which we call α and β respectively:

α =Nωµ2

12

cε0~Γ

β =4Nωµ2

12γ

cε0~Ω2c

=4γΓ

Ω2c

α, (1.11)

allowing us to write the real and imaginary parts of the index of refraction as

n′s ≈ 1 +c

2ω

β

γ∆s (1.12)

n′′s ≈ c

2ω

[β +

2Γβ

γΩ2c

∆2s

]. (1.13)

A pulse which does not undergo significant distortion will travel at the group

velocity given by

vg =c

n′s + ωsdn′sdωs

≈ c

ωsdn′sdωs

(1.14)


where we have assumed that the dispersive term ωsdn′s/dωs is much larger than

the phase index n′s. Taking the appropriate derivatives we obtain

tg =L

vg

≈ βL

2γ=

2ΓαL

Ω2c

. (1.15)

The temporal broadening of an optical pulse passing through a slow light

medium originates from two sources, frequently classified as absorptive and dis-

persive broadening. Absorptive broadening occurs when the individual frequencies

making up the pulse waveform are absorbed at different levels in the medium. This

may be studied by treating the imaginary part of the index of refraction as an

optical filter of the form

S(∆s) = exp

[−βL

(1 +

2Γ

γΩ2c

∆2p

)]. (1.16)

When the input pulse is a bandwidth limited gaussian of temporal width T0 cen-

tered on Raman resonance, the spectrum of the output pulse is given by the

product of the input pulse spectrum Ain(∆s) and the filter:

Aout(∆s) = Ain(∆s)S(∆s) ∝ exp

[−βL−∆2

p

(T 2

0 +2Γ

γΩ2c

βL

)](1.17)


Figure 1.2: The first demonstration of optical pulse delay using EIT. Reprinted with permissionfrom M. M. Kash et al., “Ultraslow group velocity and enhanced nonlinear optical effects in acoherently driven hot atomic gas,” Physical Review Letters 82, 5229 (1999). Copyright (1999)by the American Physical Society.

which gives an output pulse of width

Tout =

√T 2

0 +2Γ

γΩ2c

βL =

√T 2

0 +8Γ2

Ω4c

αL. (1.18)

Dispersive broadening may be treated by considering the difference in group

delay for a pulse centered at ∆s = 0 and a pulse centered at ∆s = 1/T0. For

the case of EIT, however, and all other Lorentzian lineshapes, pulse broadening is

almost entirely due to frequency-dependent absorption and dispersive broadening

may be neglected.

This and other results from the above model may be verified by comparison to

the many slow light experiments that have be done using EIT in atomic vapors.

The group delay of pulses in an EIT system was studied first by Kasapi et al.

[15] and then later by many others [14, 16–18]. The results of Kasapi et al. are


shown in Figure 1.2. The experiment was performed in lead vapor, in which

the authors measure an EIT linewidth 2γ of approximately 107 rad/s. The peak

pulse transmission T may be read from the figure and used to approximate the

optical depth on EIT resonance as βL = −ln(T ) ≈ 0.25 for the pulse delayed by

approximately 23 ns, giving a group delay of τg = βL/2γ = 25 ns, in agreement

with the data. Another noteworthy demonstration of slow light in atomic vapors

using EIT was performed in 1999 by Hau et al.[19] In their experiment they input

a 2.5 µs optical pulse into a condensed cloud of sodium atoms with a spontaneous

decay rate of Γ = 61.3× 106 rad/s, and calculated optical depth of αL = 63 and

Rabi frequency Ωc = 0.56Γ. When these numbers are used in Eqs. (1.15) and

(1.18) for the pulse delay and broadening, we obtain

tg =2ΓαL

Ω2c

=2 · 63

(0.56)2Γ= 6.6 µs

Tout =

√T 2

0 +8Γ2

Ω4c

αL =

√(2.5 µs)2 +

8 · 63

(0.56)4Γ2= 2.8 µs

which agrees reasonably well with the experimental data reported (The actual

delay reported was 7.05 µs, and the broadening, while hard to quantify from the

figure, appears to be in line with the prediction). An equivalent classical treatment

of Hau et al.’s results has also been made by McDonald [20].

Shortly following the publication of Hau et al.’s results in 1999, many other

experiments were reported studying slow light in coherently prepared atomic va-

1.5. TWO-LEVEL SYSTEMS 14

Figure 1.3: Delay of a 2.5 µ s pulse in an low temperature cloud of sodium atoms. Reprintedwith permission from Macmillan Publishers Ltd: L. V. Hau, S. E. Harris, Z. Dutton, and C. H.Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature397, copyright (1999).

pors. Kash et al., for example, demonstrated that very slow group velocities could

be obtained without cooling the atomic sample [21], and Budker et al. reported

on slow light in the closely related phenomena of nonlinear Faraday rotation[22].

Since 1999, hundreds of experimental and theoretical papers have been published

on the subject of slow and fast group velocities, with a recent emphasis on possible

applications of slow light [23–29]. It should be noted that EIT, like SIT, may also

be studied in the nonlinear regime, in which matched pulse solutions can be found

that have coupled slow group velocities as well as a number of other interesting

properties [30, 31].


Figure 1.4: (Relative delays of σ+ and σ− polarized light pulses tuned away from optical reso-nance. Each division represents 5 ns. Reprinted with permission from D. Grischkowsky, “Adi-abatic following and slow optical pulse propagation in rubidium vapor,” Physical Review A 7,2096 (1973). Copyright (1973) by the American Physical Society.

1.5 Two-Level Systems

While EIT provided the first dramatic reduction of group velocity, the earliest

experimental studies of linear slow light (in which the input pulse has no effect on

the group velocity) were not performed using EIT, but rather using the naturally

occurring resonances in atomic vapors. We may understand such experiments in

the context of linear dispersion theory. For an optical field interacting with a two-

level system in the linear regime, we may use Eq. (1.10) to obtain an approximate

group velocity:


n ≈ 1 +c

2ωs

(2α∆s

Γ+ iα

)(1.19)

yielding

τg =L

vg

≈Lωs

dn′sd∆s

c=

αL

Γ(1.20)

The first slow light studies in this context appear to be those of Grischkowsky,

whose experimental results are shown in Fig. 1.4. In his experiment, he measured

the difference in propagation times for left-handed (σ−) and right-handed (σ+)

circularly polarized light pulses passing through an atomic vapor of Rb atoms. A

magnetic field was used to shift the Zeeman sublevels of the relevant transitions

such that the σ− pulse was much closer to resonance than the σ+ pulse, resulting in

a greater optical depth αL for the σ− pulse. Figure 1.4(a) shows an interferogram

of the pulses used in parts (b) and (c) to demonstrate that the optical linewidths

of the pulses are sufficiently narrow, part (b) shows the temporal profile of the

linearly polarized input pulse, and part (c) shows the output pulse resolved into

σ− and σ+ components. Each division corresponds to 5 ns in the figure. The

difference in optical depth between σ+ and σ− pulses from the figure appears to

be about αL = 1, implying a difference in delay of αL/Γ = 27 ns, in agreement

with the figure (where we have used Γ = 38.1 Mrad/s for Rb). One important

feature of these results is the complete separation of the pulses in time by many

1.6. DISPERSION MANAGEMENT 17

pulse widths with minimal distortion, the first experimental demonstration of a

delay-bandwidth product much greater than unity, a subject we address in the

next section.

Another line of study in the linear regime was taken up by Chu and Wong

[32, 33] in 1982 and later by Segard and Macke in 1985 [34]. Chu and Wong

measured the group velocity of a weak pulse on resonance in an alkali vapor with

the purpose of demonstrating the group velocity as a distinct measurable quantity

from the velocity of energy propagation. They showed that group velocity may be

either greater than c or less than c in the vicinity of optical resonance. The results

had been predicted by Garret and McCumber and can also be directly derived

from the well know results of Sommerfield and Brillouin [1].

1.6 Dispersion Management

Many implementations of slow light require a pulse to be delayed by many times its

temporal width, or equivalently the maximizing of the delay-bandwidth product

of the pulse. To examine in a qualitative way how one might accomplish this,

consider the dispersive term ωdn/dω in the denominator of the group velocity. In

order to reduce the group velocity, one must make this dispersive term large, which

requires a large slope of the frequency-dependent index of refraction. A steep slope

requires either a large change in the nominal value of the refractive index (i.e. large

dn) or a very small frequency range over which a modest change occurs (i.e. small


dω). Among the more compelling reasons EIT has been useful for fundamental

demonstrations of slow light is the extremely narrow resonances ( < 10 Hz) that

can be obtained in an EIT medium, which lead to a correspondingly small dω

and hence slow group velocities. However, narrow resonances can accommodate

only limited bandwidth, making a large delay-bandwidth product difficult in EIT

based systems. The few researchers that have managed delay-bandwidth products

greater than unity in EIT media have done so by creating samples with very large

differences in optical depth on and off resonance, thereby making the dn term large

enough for multiple pulse delays. Excellent discussions regarding the maximum

delay-bandwidth products attainable in EIT systems may be found in the work

of Boyd et al. [35] and Matsko et al. [36], who come to different but entirely

consistent conclusions. Miller [37, 38] as well as Tucker et al. [24] have also

discussed delay-bandwidth limitations in a slightly more general context.

Alternative approaches have been suggested to overcome these delay-

bandwidth limitations in atomic vapors, including channelization schemes [39, 40]

and ways to change the shape of the optical resonance in order to accommodate

larger bandwidths [41]. The ideal resonance shape was discovered many years ago

in the context of microwave physics to be a square transmission filter (in fact most

RF filters purchased quote a group delay among the specifications). It also hap-

pens that among the least ideal shapes is a single Lorentzian near resonance, such


as found in most EIT systems, since absorptive broadening limits the bandwidth

of the input pulse to approximately the linewidth of the Lorentzian.

Far away from resonance, however, Lorentzian lineshapes have quite different

dispersion characteristics. For example, by simple argument one may show that

for off-resonant optical interaction with an absorbing Lorentz oscillator in dilute

media, the dispersive term in the group index ωdn/dω is always larger than the

phase term n(ω) (less unity). For a single Lorentz oscillator with a complex

polarization given by Eq. (1.10) the dispersive term in the group index ω dndω

is

ωdn

dω∝ ω

∆2 − Γ2

(∆2 + Γ2)2. (1.21)

Subracting the phase index term n from the dispersive term ωdn/dω we find the

difference D between the two to be

D =ω(∆2 − Γ2) + ∆(∆2 + Γ2)

(∆2 + Γ2)2, (1.22)

which is always positive for off-resonant interaction (∆2 > Γ2) at optical frequen-

cies (ω À ∆). Thus, the slowing of light off resonance is mostly due to dispersive

effects even for nearly monocrhomatic fields.2

Working away from resonance also allows one to manage second order disper-

2See Appendix B for a more complete disucssion of this point.


sion more effectively and increase the delay-bandwidth product for a given pulse

broadening, such as seen in the above results of Grischkowsky.

To reduce pulse broadening further, several researchers have considered the

use of two resonances simultaneously. The first proposal for such as system ap-

pears to have been made by Steinberg and Chiao, who in 1994 suggested using

two widely spaced inverted resonances to observe superluminal group velocities

with little second order dispersion [42]. They noted that there exists a point be-

tween the two resonances in which the index of refraction has no second order

term, thereby reducing dispersive broadening. In 2000, Wang et al. used two

gain lines in a similar configuration to measure group velocities greater than the

speed of light [43, 44]. In 2003 Macke and Segard performed a careful study of

such negative group velocities using double gain resonances, and also suggested

that one may obtain pulse advancement by the proper spacing of two absorb-

ing lines[45]. The first time the reciprocal arrangement to that of Steinberg and

Chiao was investigated appears to have been in 2003, when Tanaka et al. used two

widely spaced absorbing resonances to measure optical delays with little distortion

[46]. This configuration has since been considered in a variety of theoretical and

experimental studies [29, 47–50].

We briefly outline the noteworthy features of this system by considering the

case of two absorbing Lorentzian lines of equal width separated by a spectral dis-


tance much larger than their widths. The susceptibility of the double-Lorentzian

is given by

χ =Nµ2

~ε0

(1

∆1

+1

∆2

), (1.23)

where ∆i represents the complex optical detuning ∆i − iΓ/2 from an optical

resonance centered at ωi.

Making the change of variables ω = (ω1 + ω2)/2+ δ and ω0 = (ω2 − ω1)/2 and

assuming that the pulse bandwidth does not exceed the spectral region between

the resonances, we may write the real and imaginary parts of the refractive index

as

n′ ≈ 1 +Aω2

0

δ +Aω4

0

δ3 (1.24)

n′′ ≈ fracAΓ2ω20 +

3AΓ

2ω40

δ2, (1.25)

where the power series are expanded about δ = 0 and A = Nµ2/~ε0. We first

note the absence of a second order term in the real part of the index of refraction,

indicating the absence of second order group velocity dispersion. It can also be

seen that dn′/dδ = dn′′Γ/2, which can be used to obtain a simple form for the

group velocity. Combining this result with αm = 2n′′ω/c, where αm is the optical

intensity absorption coefficient of the medium at the pulse carrier frequency, one


0 10 20 30 40 95 105 115

0

10

20

30

40

50

60

70

80

90

100

Time (ns)

Tra

ns

mis

sio

n (

%)

Theory

x10

Figure 1.5: Multiple pulse delays with little distortion using double resoncnes in Rb vapor. Thetheory plot is taken from Eq. (1.26).

obtains an approximate group velocity Γ/αm, leading to a group delay of

τg ≈αmL

Γ(1.26)

In the context of delay-bandwidth products this result is useful, since it pre-

dicts that the delay is independent of the spectral separation between the two

resonances and only depends on the absorption away from resonance and the

linewidth of the resonances. Also, owing to the shape of the transparency region

where delay takes place (it looks much more like a square filter than a single

Lorentzian) pulse broadening is dominated by third order dispersion rather than

second order absorption, allowing for much larger delay-bandwidth products [29].

An example of the use of two widely spaced absorbing Lorentzians to achieve

large delay-bandwidth products is shown in Fig. 1.5, in which optical pulses were

1.7. CONCLUSION AND DISSERTATION OUTLINE 23

delayed between the two ground-state hyperfine resonances in 85Rb. In addition to

the pulse delays, a plot of Eq. (1.26) is shown, where Γ = 36.1 Mrad/s. While the

delays are numerically much smaller than those achievable in EIT systems, there

is in principle no reason that the double-resonance technique could be applied to

produce longer delays.

1.7 Conclusion and Dissertation Outline

Experiments involving slow light in atomic vapors have produced the principle

results which continue to motivate slow light research. Most of the experimental

and theoretical results that continue to appear in the literature may be traced

to the early studies of slow light in vapors. In this chapter, we have briefly

reviewed the basic physics leading to the prediction of slow group velocities in

a few model systems. We have also given a brief overview of efforts that have

been made to minimize group velocity dispersion in slow light systems, with an

emphasis on the general principles necessary to understand the field. We have

not treated all possible variations of resonances that may be used to obtain slow

light in atomic systems, but have discussed a few representative systems using the

same general model built on Feynman diagrams. It is hoped that theory for other

types of systems (i.e. four-wave mixing schemes, hole burning, etc) may be easily

derived from the same framework once the basics outlined in this chapter are well

understood.

1.7. CONCLUSION AND DISSERTATION OUTLINE 24

In the remaining chapters of this dissertation, the basic concepts outlined in

the introduction will be given a more detailed treatment. The emphasis will of

course be on my own work in experimentally studying various slow light sys-

tems. Chapters 2-3 develop the double-resonance model and report on experi-

ments demonstrating its validity. Chapter 4 discusses several applications of slow

light using the double-resonance technique. The slowing of images receives a de-

tailed treatment, and two applications to interferometry and entanglement are

briefly discussed. In chapters 5-6, I discuss coherent implementations of the the

double resonance technique and a coherent four-wave mixing scheme that allows

for transverse image storage. The dissertation concludes with a summary of my

slow light research and the general implications of the results.

25

Chapter 2

Slow Light Between Two IdealAbsorbing Resonances

2.1 Slow Light and Lorentzian Resonances

While most researchers have used coherently prepared media (e.g. EIT) to produce

slow light, it is well known that slow light may be achieved in any medium with

frequency-dependent absorption. As discussed in the introduction, this includes

media consisting of two-level atoms and multilevel systems regardless of whether

they are in prepared in a coherent fashion. In this chapter, we discuss an incoherent

scheme to produce slow group velocities and manage its associated dispersion;

namely, the use of two widely spaced ideal Lorentzian absorbing resonances.

Most optical resonances have lorentzian lineshipes, and hence the means

for obtaining slow group velocities have usually involved a Lorentzian trans-

parency or gain resonance: electro-magnetically induced transparency (EIT)

[15, 19, 21, 22, 51], coherent population oscillations (CPO) [52–55], stimulated

2.2. PRACTICAL CONSIDERATIONS 26

Brillouin scattering (SBS) [56–58], stimulated Raman scattering (SRS) [59, 60]

etc..

As explained in the introduction, however, Lorentzian lineshapes do not man-

age dispersion well on resonance, but the spectral region between two resonances

does. This chapter considers both the group delay and pulse broadening of a

double-Lorentzian system using a simple model, as well as report on an experi-

ment demonstrating the ability to delay a pulse by multiple times its temporal

width with little distortion. The results of this chapter will be extended in the

next chapter, in which we discuss a slightly more general double-absortion system,

its temporal dynamics, and a corresponding experimental demonstration. In the

context of this dissertation, it is important to note that most of the applications

of slow light on which we report make use of the double-absorption technique

discussed in this and the following chapter. Hence, this chapter may be viewed as

both a fundamental study of slow light as well as a demonstration of a useful tool

to be used in later applications. As such, we first briefly indroduce the context

in which the optical delay of a pulse may be useful, and build the theory with an

emphasis in addressing concrete implementations.

2.2 Practical Considerations

There is currently considerable practical interest in developing all-optical delay

lines that can tunably delay short pulses by much longer than the pulse duration.

2.2. PRACTICAL CONSIDERATIONS 27

Slow light has long been considered a possible mechanism for constructing such

a delay line. Most commonly, the steep linear dispersion associated with a single

gain or transparency resonance provides the group delay. Most early work on delay

lines used the dispersion associated with electromagnetically induced transparency

[15, 19, 21, 22, 51, 61], but recently other resonances have been explored, including

coherent population oscillations [52–55], stimulated Brillouin scattering [56–58,

62], stimulated Raman scattering [59, 60], and spectral hole-burning [63].

As mentioned in the introduction, in addition to single-resonance systems,

double-gain resonances have been used for pulse advancement [43, 45, 64–67] and

delay [68]. Widely spaced gain peaks create a region of anomalous dispersion,

resulting in pulse advancement. When the spacing between the gain peaks is

small, a region of normal dispersion is created, resulting in pulse delay. Pulse

advancement is also possible by the proper spacing of two absorbing resonances

[45]. The possibility of pulse delay between two absorbing resonances has also

received some attention [13, 46, 48, 50, 69].

Ideally, an optical delay line would delay high bandwidth pulses by many pulse

lengths in a short propagation distance without introducing appreciable pulse dis-

tortion and be able to tune the delay continuously with a fast reconfiguration rate.

Minimal pulse absorption is also desirable, but not necessary because absorption

can be compensated through amplification.

2.3. IDEAL DOUBLE-LORENTZIAN 28

2.3 Ideal Double-Lorentzian

The susceptibility of the ideal double-Lorentzian is given by

χ =Nµ2

~ε0

(1

∆1

+1

∆2

)

= A(

1

ω1 − ω − iΓ2

+1

ω2 − ω − iΓ2

), (2.1)

where ∆i represents the complex optical detuning ∆i− iΓ/2 from an optical reso-

nance centered at ωi, and A = Nµ2/~ε0 is the strength of the susceptibility and Γ

is the full-width at half-maximum (FWHM) of each of the Lorentzian lineshapes.

Making the change of variables ω = (ω1 + ω2)/2 + δ and ω0 = (ω2 − ω1)/2 and

assuming the far detuned limit (i.e. ω0 À Γ), we may neglect the half-width

term in the denominator. We further assume the pulse frequencies to lie within

the range |δ| ¿ ω0, the pulse bandwidth to be larger than the Lorentzian half-

width, Γ/2, and χ ¿ 1. The real and imaginary parts of the refractive index

n = n′ + in′′ ≈ 1 + χ/2 may then be written as

n′ ≈ 1 +A2

(1

δ + ω0

+1

δ − ω0

)

≈ 1 +Aω2

0

δ +Aω4

0

δ3 (2.2)


and

n′′ ≈ AΓ

(1

(δ + ω0)2+

1

(δ − ω0)2

)

≈ AΓ

2ω20

+ 3AΓ

2ω40

δ2, (2.3)

where the power series are expanded about δ = 0.

The optical depth αL = 2ωLn′′/c (here L is the interaction length and α is

the intensity coefficient) at the midpoint between the Lorentzians is found to be

αmL = ωLAΓ/cω20 which implies ∂n′/∂δ|δ=0 = cαm/Γω. The group velocity is

then given by

vg ≈c

ω ∂n′∂δ

=Γ

αm

, (2.4)

and the group delay is given by

tg =L

vg

≈ αmL

Γ. (2.5)

The dispersive and absorptive broadenings in the small-pulse-bandwidth

limit (i.e., pulse bandwidth is much smaller than the spectral distance between

Lorentzians) are dominated by the second terms in the power series expansions

of the real and imaginary parts respectively. The absorptive broadening is due to

the spectrally dependent absorption in the wings of the pulse spectrum. In the

small-pulse-bandwidth limit the absorption can be approximated by a Gaussian


shaped spectral filter plus a constant absorption:

S(δ) = exp[−α(δ)L] ≈ exp[−αmL− 3δ2αmL/ω20] (2.6)

When the input pulse is a bandwidth-limited Gaussian, we find that in the fre-

quency domain the output pulse is the product of the spectral filter and the input

pulse spectrum Ain(δ):

Aout(δ) = Ain(δ)S(δ)

∝ exp[−αmL− δ2

(T 2

0 +3αmL

ω20

)],

where T0 is the half-width at 1/e intensity of the pulse. Thus, accounting for only

absorptive broadening the temporal half-width after traversing the medium is

Ta =

√T 2

0 +3αmL

ω20

. (2.7)

The temporal broadening due to dispersion is [70]

Td =

√T 2

0 +

(β2L

2T0

)2

+1

8

(β3L

2T 20

)2

(2.8)

The total pulse broadening is found by replacing T0 in eq. 2.8 with Ta from


eq. 2.7:

Ttot =

√T 2

0 +3αmL

ω20 ln 2

+3αmL

2Γω20 ln 2

(T 2

0 + 3αmLω2

0 ln 2

) . (2.9)

We focus on the case where ω0 À 1/T0 À Γ and Ta/T0 ≤ 2, corresponding to

our experimental parameters. For this case the dispersive broadening dominates

(i.e. the second term on the right hand side of eq. 2.8 contributes most to the

broadening). However, the quadratic absorption is still significant since it reduces

the effects of dispersive broadening by most strongly absorbing those frequencies

which experience the largest dispersion (i.e. frequency wings of the pulse). For

the parameters considered in this chapter, pulse broadening is less with both ab-

sorptive and dispersive broadening included than for dispersive broadening alone.

In single-Lorentzian systems, absorption is the dominant broadening mechanism

and this relationship between broadening mechanisms is not significant.

Although in hot Rb vapor the resonances experience strong inhomogeneous

Doppler broadening, in the far-wing limit the Rb resonances are essentially

Lorentzian and the double-Lorentzian formalism is a very good approximation.

The 85Rb D2 hyperfine resonances are separated by approximately 3 GHz, so

the Gaussian Doppler broadening of approximately 500 MHz has little effect on

the absorptive behavior. Also, collisional broadening was not significant for the

temperatures used in this work.

2.4. PROOF-OF-PRINCIPLE EXPERIMENT 32

Figure 2.1: Experimental schematic. A signal laser passes through a heated rubidium vaporcell and is either measuring directly using a fast detector, or after interference on a balancedphotoreceiver.

2.4 Proof-of-Principle Experiment

Experimental Setup

A diagram of the experimental setup is shown in Fig. 2.1. A narrowband (300

kHz) diode laser at 780 nm generates a beam of light tuned halfway between

the 85Rb D2 hyperfine resonances, which is fiber coupled into a fast electro-optic

modulator (EOM). An arbitrary waveform generator (AWG) drives the EOM,

producing light pulses with a duration of approximately 2 ns FWHM. The light

pulses then pass through a 10 cm glass cell containing rubidium in natural isotopic

abundance. The cell is heated with electronic strip heaters and enclosed in a cylin-

drical oven with anti-reflection coated windows. The pulse is then incident upon


−800 −600 −400 −200 0 200 400 600 800

0

10

20

30

Tra

ns

mis

sio

n (

%)

−800 −600 −400 −200 0 200 400 600 800

0

-0.5

0

0.5

1

Detuning (MHz)

Dif

f. S

ign

al

(arb

. u

nit

s)

(a)

(b)

Figure 2.2: (a) Signal transmission versus signal detuning and (b) difference-signal from thebalanced photoreceiver with each fringe corresponding to a 2π phase shift. The height of thefringes is in arbitrary units. Both transmission and phase data were taken with a 10 cm Rbvapor cell at approximately 130 C (corresponds to a group delay of 26 ns).

a 1 GHz avalanche photo-diode (APD) and recorded on a 1.5 GHz oscilloscope

triggered by the AWG.

A Mach-Zehnder interferometer was also used with a balanced photoreciever

in order to make continuous wave (CW) measurements of the transmission and

phase delay as a function of frequency. The difference-signal from the balanced

photoreciever provides phase information while transmission data is obtained by

blocking one of the photoreciever photodiodes. The beam splitter preceding the

vapor cell is polarizing to allow for easy balancing of the interferometer arms,

and the beam splitter immediately following the vapor cell is polarizing to allow

switching between the fast APD and CW balanced detection.


Results

Figure 2.2 shows (a) absorption and (b) phase spectroscopy scans for the transmis-

sion window resulting in a measured 26 ns pulse delay. The transmission window

has a width of approximately 1 GHz which is sufficient acceptance bandwidth for

the 2 ns pulses used in this experiment. The interference fringes were obtained by

sweeping the laser frequency and monitoring the intensity difference at the two

output ports of a Mach-Zehnder interferometer (see Fig. 2.1).

It is straightforward to predict the group delay from the absorption scan or

measure it directly using the interference fringes. From the absorption data, we

may extract the optical depth and calculate the group delay via Eq. (2.5), giving

approximately 26 ns for absorption data in Fig. 2.2a, in good agreement with

the measured delay. In contrast, from the interference fringes we may extract the

group delay directly:

tg =L

vg

=Lω ∂n′

∂δ

c≈ ∆φ

∆δ=

∆N

∆f, (2.10)

where ∆N is the number of fringes in a frequency range ∆f . For the resonance

shown in Fig. 2.2b there are approximately 25 fringes per GHz, giving a predicted

optical delay of 25/1GHz =25 ns, also in good agreement with measured values.

We note that the maximum delay-bandwidth product of a dispersive medium is


approximately given by the maximum number of interference fringes that can be

obtained within the acceptance bandwidth.

The above result (Fig. 2.2) is interesting, and was to the author’s knowledge

the first experimental result demonstrating the practical intersection of the fields

of slow light and interferometry. The dense fringe oscillation pattern shows that

the presence of the slow light medium has rendered the interferometer much more

sensitve to changes in the laser frequency (by a factor of ng to be exact). In addi-

tion, with the help of an interferometer, a pulsed laser is not required to character-

ize the group delay and broadening characteristics of a slow light medium—only

a tunable continuous wave laser is required. We will consider other interesting

aspects of slow light interferometry in the next chapter in relation to precision

measurements and quantum entanglement. It is worth noting at this point, how-

ever, that none of the practical results in interferometry, imaging, or quantum

entanglement reported in this thesis would have been possible without a slow

light medium capable of delay-bandwidth products much greater than unity.

Figure 2.3 shows signal pulse transmission and delay for various cell tempera-

tures, plotted in units of percent transmission. Using a 2.4 ns long pulse (FWHM)

and a single 10 cm vapor cell and varying the temperature between 90 C and 140

C we were able to tune between 8 ns and 36 ns of delay. We note that several

pulse delays are obtainable with greater than 1/e peak transmission. In order to

achieve 106 ns of delay with a delay-bandwidth product of 50 and a broadening


0 10 20 30 40 95 105 115

0

10

20

30

40

50

60

70

80

90

100

Time (ns)

Tra

ns

mis

sio

n (

%)

Theory

x10

Figure 2.3: Pulse delay at various optical depths. On the left, 2.4 ns pulses are passed througha 10 cm vapor cell and the delay is tuned by changing temperature. On the right, a 2.1 ns pulseis passed through four 10 cm cells and delayed 106 ns (50 fractional pulse delays).

of approximately 40% we used a 2.1 ns (FWHM) pulse incident on a series of four

10 cm vapor cells all heated to approximately 130 C. The theoretical prediction

of transmission as a function of group delay [Eq. (2.5)] is also plotted using the

Rb D2 homogeneous linewidth Γ = 2π×6.07 MHz from [71]. The discrepancy be-

tween the measured pulse intensities and the theoretical pulse energies can largely

be attributed to pulse broadening spreading the pulse energy over a larger time

resulting in lower peak intensities.

Figure 2.4 compares the fractional broadening of the delayed pulses shown in

Fig. 2.3 to the predicted values calculated using Eqs. 2.7-2.9. Shown in Fig. 2.4


10−3

10−2

10−1

100

0

0.2

0.4

0.6

0.8

1

1.2

Transmission

Fra

cti

on

al B

road

en

ing w/ absorptive correction

no absorptive correction

w/ chirp

measured

Figure 2.4: Fractional pulse broadening vs. natural log of transmission. Fractional broadening isdefined as the fractional increase pulse duration at FWHM (A value of 0 means no broadening).Due to absorption, the actual broadening is less than that predicted by the dominant dispersiveterm, even though absorptive broadening is negligible.

are the measured broadening values, the predicted total broadening without ab-

sorptive corrections,(Ta+Td−T0)/T0, the total predicted broadening, (Ttot−T0)/T0

and the total predicted broadening with a chirp-like like correction. As predicted

by Eq. (2.9), the data show that the quadratic absorption decreases the broaden-

ing due to dispersion. Also, for small optical depths the pulse width compresses

before broadening, which may be modeled by assuming a small negative chirp on

the input pulse. We do not know the origin of the chirp, but we found that by

including a small second order chirp in the theory we obtained a very good fit to

the data.

2.5. CONCLUDING REMARKS 38

2.5 Concluding Remarks

In conclusion, we have discussed the delay and broadening characteristics for

pulses propagating through a double-Lorentzian medium (i.e. a medium with two

widely spaced absorbing Lorentzian resonances). For many slow-light applica-

tions, absorptive double-Lorentzian systems seem to be better suited than gain-

like single-Lorentzian systems. Since the spacing between the two Lorentzians can

be arbitrarily large, the usable bandwidth may be proportionately large, though

practical considerations may limit the separation. Also, in contrast to single

Lorenztians, the double-Lorentzian lineshape is dominated by dispersive broad-

ening and not absorptive broadening, resulting in less pulse distortion for a given

delay. While the method of tuning the delay in the present experiment was slow

(increasing the temperature of the vapor cell), there may be ways to to achieve

fast reconfiguration rates. Some possibilities may be to drive a large number of

atoms to saturation with a strong auxiliary beam, or make use of light induced

desorption [72] of Rb to optically change the atomic number density. We consider

the first of these possibilities in the more general discussion of double-Lorentzians

in the next chapter.

39

Chapter 3

Slow Light between TwoAbsorbing Resonances–General

Model

3.1 General Double-Lorentzian Model

In this chapter we extend the results of the previous chapter to include the possi-

bility of two absorbing resonances of differing optical depths and investigate the

temporal characteristics of the group velocity. Again, the motivation is the the

possibility for practical implementations, which we also address. We also discuss

another proof-of-principle experiment in the context of the generalized model,

which was performed using a different resonant medium and allowed for shorter

pulses to be delayed. In brief, we report the tunable delay of a 1.6-GHz-bandwidth

pulse by up to 25 pulse widths and the tunable delay of a 600-MHz-bandwidth

pulse by up to 80 pulse widths by making use of a double absorption resonance in

3.1. GENERAL DOUBLE-LORENTZIAN MODEL 40

cesium. Furthermore, we show that the delay can be tuned with a reconfiguration

time of 100’s of nanoseconds.

In a medium with two Lorentzian absorption resonances whose oscillator

strengths differ, as illustrated in Figure 3.1, the complex index of refraction can

be approximated as

n(δ) = 1− A2

(g1

δ + ∆+ + iΓ2

+g2

δ −∆− + iΓ2

)(3.1)

where Γ is the homogeneous linewidth [full-width at half-maximum (FWHM) ],

g1 and g2 account for the possibility of different strengths for the two resonances,

δ = ω − ω0 − ∆ is the detuning from peak transmission, ω0 = (ω1 + ω2)/2, ω1

(ω2) is the resonance frequency for transition 1 (transition 2), ∆± = ω21 ± ∆,

ω21 = (ω2 − ω1)/2, and

∆ =g

1/31 − g

1/32

g1/31 + g

1/32

ω21. (3.2)

For example, alkali atoms have two hyperfine levels associated with their electronic

ground-state, leading to two closely spaced absorption resonances. For a vapor

of alkali atoms, the detunings satisfy ∆+ ≈ ∆− À Γ, and the strength of the

resonance is given in SI units by A = N |µ|2/[ε0~(g1 + g2)], where µ is the effective

far-detuned dipole moment [73], and g1 and g2 are proportional to the degeneracies

of the hyperfine levels.

Eq. (3.1) is also applicable for inhomogeneously broadened lines, such as

3.1. GENERAL DOUBLE-LORENTZIAN MODEL 41

Doppler broadened atomic vapors, if the detunings ∆− and ∆+ are greater than

the inhomogeneous linewidth by an order of magnitude or more. This result holds

because the homogenous Lorentzian lineshape has long wings while the inhomoge-

neous lineshape decreases exponentially. For cesium vapor near room temperature

the inhomogeneous broadening corrections to Eq. (3.1) near δ = 0 are less than

1%.

By expanding Eq. (3.1) about the point δ = 0, we find that the real part n′

and imaginary part n′′ of the index of refraction are given by

n′(δ) ≈ 1 + K0 + K1Aω2

21

δ + K3Aω4

21

δ3 (3.3a)

n′′(δ) ≈ K1AΓ

2ω221

+ 3K3AΓ

2ω421

δ2, (3.3b)

where

Ki =

(g

1/31 + g

1/32

2

)i+1 (g

(2−i)/31 + (−1)i+1g

(2−i)/32

), (3.4)

and where we have assumed that n − 1 ¿ 1 in keeping only the first few terms

in the expansion. Note that for the special case in which the two resonances are

of equal strength (i.e. g1 = g2 = g), the coefficients are given by Ki = 2g for i

odd and Ki = 0 for i even. For cesium, which has g1 = 7/16 and g2 = 9/16, the

error introduced by assuming g1 = g2 is approximately 0.5%. For this reason, we

3.2. DISPERSION AND ABSORPTION 42

make the simplifying assumption g1 = g2 = 1/2 while discussing the experimental

results below, though the general result was used in all computer modeling and

fits to the data.

Pulse propagation can be described in terms of various orders of dispersion,

which can be determined through use of Eq. (3.3a) as

βj =1

c

djωn′(ω)

dωj

∣∣∣∣ω=ω0+∆

, (3.5)

Thus the group velocity is given by vg = 1/β1, and the group-velocity dispersion

(GVD) and third-order-dispersion (TOD) are given respectively by β2 and β3.

The absence of second-order (first-order) frequency dependence in Eq. (3.3a) (Eq.

(3.3b)) means that near δ = 0 the GVD (absorption) is minimized regardless of

possible differences between g1 and g2. Thus, between two absorption resonances,

which can be described by Eq. (3.1), the maximum transparency is accompanied

by a minimum in GVD.

3.2 Dispersion and Absorption

We next develop a simple model to provide an understanding of the role of disper-

sion and absorption on pulse broadening. We provisionally define the pulse width

as the square root of the variance of the temporal pulse shape. For an unchirped

Gaussian pulse, i.e. E(0, t) = E0 exp (−t2/2T 20 ), the pulse width defined in this


0

50

100

Tra

nsm

issi

on

(%

)

-5 0 5-1

0

1x 10

-3

Signal Detuning (GHz)

n-1 v /

cg

0

0.1

(b)

(a)

Figure 3.1: (a) CW signal transmission (asterisks–measured, solid–fit ) overlayed with the spec-trum (dashed) of a 275 ps pulse and (b) index of refraction (solid) and group velocity (dashed),all versus signal detuning for cesium at approximately 114 C. All theory curves taken from Eq.(3.1) with A = 4× 105 rad/s, g1 = 7/16 and g2 = 9/16. High-fidelity optical delay is observedfor light pulses passing through the nearly transparent window between the two resonances.

way is simply T0. The pulse width after propagating through a distance L of

dispersive medium is then given to third-order in δ by [70]

T 2d = T 2

0 +

(2β2L

T0

)2

+

(β3L

2T 20

)2

(3.6)

where T0 is the initial pulse width. In the case of cesium, where ω0 ≈ 2π×3.5×1014

rad/s and ω21 ≈ π × 9.2× 109 rad/s, β2 can be neglected and Eq. (3.6) simplifies

to

T 2d ≈ T 2

0 +

(3τd

ω221T

20

)2

, (3.7)


where β3 has been calculated using Eqs. (3.5) and (3.3) and where τd ≈ α0L/Γ

is the pulse delay and α0L = 2ω0n′′L/c is the optical depth at the pulse carrier

frequency ω0. We further note that the change in pulse width due to absorption

only can be approximated as [35, 69]

T 2a = T 2

0 +3Γτd

ω221

, (3.8)

so long as (Ta/T0 − 1) < 1.

The fractional broadening due to dispersion, defined as Td/T0 − 1, scales as

1/T 30 , while the broadening due to absorption scales as 1/T0. In the present study,

τd ≈ 10−8 s, ω21 ≈ 1011 rad/s, T0 ≈ 10−10 s, and Γ ≈ 107 rad/s, indicating that

dispersion is the dominant form of broadening by about three orders of magnitude,

and the absorptive contribution to broadening can be ignored.

Experimentally it is much easier to quantify pulse widths in terms of their

FWHM rather than in terms of their variance as we have done in Eqs. (3.6) -

(3.8). For the remainder of this chapter, we will quote pulse widths in terms of

their FWHM.


3.3 Proof-of-Principle Experiment

Experimental Setup

Our experimental setup is shown in Fig. 3.2. The signal laser is a CW diode

laser with a wavelength of 852 nm. The signal frequency is tuned to obtain maxi-

mum transmission between the two Cs D2 hyperfine resonances and is pulsed at a

pulse repetition frequency (PRF) of 100 kHz using a fast electro-optic modulator

(EOM). The signal beam is collimated to a diameter of 3 mm, and two different

pulse widths are used, 275 ps or 740 ps FWHM, with a peak intensity of less

than 10 mW/cm2. The pulses then pass through a heated 10-cm-long glass cell

containing atomic cesium vapor. The 275 ps pulses are measured using a 7.5 GHz

silicon photodiode, and the 740 ps pulses are measured with a 1 GHz avalanche

photodiode. All electrical signals are recorded with a 30 GHz sampling oscillo-

scope triggered by the pulse generator. The pump beams were turned off except

during the experiments reported in Figs. 3.5 and 3.6.

CW Results

Figure 3.1(a) shows the transmission of a CW optical beam as a function of

frequency near the two cesium hyperfine resonances, overlayed with the spectrum

of a 275 ps Gaussian pulse. The data points are measured values and the solid

line fits these points to the real part of Eq. (3.1). The entire pulse spectrum


Signal Pumps

Figure 3.2: Experimental schematic. A signal pulse passes through a heated cesium vaporcell. Two pump beams combine on a beamsplitter and counter-propagate through the vapor, toprovide tunable delay of the signal pulse.


0 2 4 6 80

0.2

0.4

0.6

0.8

1

Time (ns)

Pu

lse

Inte

nsi

ty

TheoryExperiment

Increasing

Temperature

Air

Figure 3.3: Pulse shapes of 275 ps input pulses transmitted through a cesium vapor cell. Delaysa large as 25 pulse widths are observed. The temperature range from 90 C to 120 C

lies well within the relatively flat transmission window between the resonances,

resulting in little pulse distortion by absorption. Figure 3.1(b) shows the index of

refraction and frequency-dependent group velocity associated with the absorption

shown in Fig. 3.1(a), taken from the imaginary part of Eq. (3.1). We note that, in

the region of the pulse spectrum, the curvature of the frequency-dependent group

velocity is greater than that of the absorption, suggesting that group velocity

dispersion is the dominant form of pulse distortion. This is not the case for single-

Lorentzian systems, where the spectral variation of absorption is the dominant

form of distortion [3]. While most slow light experiments have worked by making

highly dispersive regions transparent, we have worked where a highly transparent

region is dispersive.


Pulse Delay Results

As shown above, the delay of a pulse is proportional to the optical depth of the

vapor. Figure 3.3 shows that we can control the delay by changing the temperature

(and thus optical depth) of the Cs cell. Using a 10 cm cell, and varying the

temperature between approximately 90 C and 120 C, we were able to tune the

delay of a 275 ps pulse between 1.8 ns and 6.8 ns. The theory curves in Fig. 3.3

were obtained using I(x, t) = n′(0)cε0|E(z, t)|2/2 where the electric field is given

by

E(z, t) =E0T0 exp [−i(ω0 + ∆)t]√

2π×

∫ ∞

−∞dδexp

[i

(ωn(δ)

cz − δt

)− δ2T 2

0

2

], (3.9)

and where we have used Eq. (3.1) for the index of refraction. The atomic density

N has been chosen separately to fit each measured pulse. We note that a pulse

may be delayed by many pulse widths relative to free-space propagation with little

broadening.

Longer pulses lead to delay with reduced pulse broadening because pulse broad-

ening is approximately proportional to 1/T 30 [see Eq. (3.7)]. To study the larger

fractional delays enabled by this effect, we used longer 740 ps input pulses for

which the dispersive broadening is significantly reduced. Figures 3.4(a) and 3.4(b)

show the delay and broadening of a 740 ps pulse after passing through a sequence


0 10 20 30 40 50 60

0.2

0.4

0.6

0

Time (ns)

Puls

e I

nte

nsity

0 20 40 60 800

0.2

0.4

0.6

Fractional Delay

F

ractio

na

l

Bro

ad

en

ing

Airx1/2

X 10 X 500

(a)

(b)

Figure 3.4: (a) Output pulse shapes and (b) fractional broadening as functions of fractionaldelay for a 740 ps input pulse. Fractional delay is defined as (τd/T0) and fractional broadeningis defined as (T − T0)/T0.

of three 10 cm cesium vapor cells. The plots correspond to a temperature range

of approximately 110 C to 160 C. Even though the pulse experiences strong

absorption at large delays, the fractional broadening of the pulse FWHM remains

relatively low. In both Figs. 3.3 and 3.4 dispersion, not absorption, is primarily

responsible for the decrease in peak intensity.

Tuning the Delay

In addition to temperature tuning, the optical depth can be changed much more

rapidly by optically pumping the atoms into the excited state using two pump

lasers. As shown in Fig. 3.2 each pump laser is resonant with one of the D2


transitions in order to saturate the atoms without optical pumping from one

hyperfine level to the other. The power of each pump beam is approximately

30 mW, and both pump beams are focused at the cell center. The signal beam

overlaps the pump beams and is also focused to a 100 µm beam diameter. The

pump beams are turned on and off using an 80 MHz AOM with a 100 ns rise/fall

time. Being on resonance with the D2 transitions, the pump fields experience

significant absorption (αL ∼ 300), and are entirely absorbed in spite of being well

above the saturation intensity.

With the pump beams on, the decreases in effective ground-state atomic den-

sity leads to smaller delay. Figure 3.5 shows a delayed pulse waveform consisting

of two 275 ps input pulses separated by 1 ns, with the pump on and off. We note

that pump fields create no noticeable change in the waveform shape or amplitude.

Also, we measured that the change in delay is essentially proportional to the pump

power.

In Fig. 3.6 the measured signal delay is shown as a function of the difference

between arrival time ts of the signal at the cell and the turn-on time tp of the pump.

The rise and fall times lie in the range 300-600 ns and vary slightly depending on

the relative detunings of the pumps.

In summary, we have observed large tunable fractional time delays of high-

bandwidth pulses with fast reconfigurations rates and low distortion by tuning

the laser frequency between the two ground-state hyperfine resonances of a hot


Time (ns)

Inte

nsi

ty

4 6 8

Pump On

1.0 ns

Pump Off

Figure 3.5: Pulse output waveforms with auxiliary pump beams on (dotted) and off (solid). Two275 ps input pulses separated by 1 ns are delayed by approximately 5.3 ns without pumping,but only 4.3 ns with pumping (a change of one bit slot) with little change in pulse shape.

5.6

6

6.4

De

lay

(n

s)

0 30t -t (µs)

Pump On

Pump Off

244S P

262 28

Figure 3.6: Pulse delay versus time following pump turn-on and turn-off, showing the reconfig-uration time for optically tuning the pulse delay. The two pump beams are tuned to separatecesium hyperfine resonances and are switched on at the time origin and switched off 24 µs later.


atomic cesium vapor cell. We have shown that in such a medium dispersion is the

dominant form of broadening, and we have characterized the delay, broadening,

and reconfiguration rates of the delayed pulses. These results are the foundation

for the experiments reported in later chapters utilizing slow light in practical

implementations.

53

Chapter 4

Implementations ofDouble-Resonance Slow Light

4.1 Overview

The number of researchers investigating slow light phenomena has grown sub-

stantially in recent years. This growth has resulted both because of fundamental

interest in slow group velocities, and the possibility for practical applications. Re-

cent applications include atomic linewidth measurements [74], enhanced nonlinear

effects[21, 75], atomic memories [76–78], chip-scale photonic circuits[79], improved

gyroscopes [80], and many others.

Most practical applications require the delay of a pulse by much longer than

the pulse duration, which has been enabled in atomic vapors by double-Lorentzian

absorption resonances. In the discussion following Fig. 2.2, it was noted that in

the large delay-bandwidth product regime, the intersection between slow light

and interferometry has the potential for practical application. In this chapter,

4.2. ALL OPTICAL DELAY OF IMAGES USING SLOW LIGHT 54

we discuss the usefulness of double-resonance slow light in constructing a delay

line for transverse images. We also comment on the usefulness of slow light in

two interferometric schemes, one classical and the other quantum mechanical in

nature.

4.2 All Optical Delay of Images Using Slow

Light

Introduction

All-optical methods for delaying images may have great potential in image pro-

cessing, holography, optical pattern correlation, remote sensing, and quantum

information. For example, in many digital image processing applications, the am-

plitude and phase information of images must be preserved. Electronic conversion

of optical images requires relatively intense optical fields and information is lost in

analog to digital conversions. Alternatively, one could use a long free-space delay

line; however, diffraction and physical space limitations impose serious restrictions

on such a system. A small all-optical buffer in which the phase and amplitudes

of the image are preserved would solve these problems.

In this section, we report on a series of several experiments showing that two-

dimensional images can be delayed while preserving the amplitude and phase

information of the images. The images are delayed using the double-resonance


system described in the previous chapters. This system has several noteworthy

characteristics in relation to the delay of images. First, it requires no additional

laser beams to prepare the slow light medium. This results in low background

noise and a high signal-to-noise ratio in the delayed image, even at very low

light levels. Also, the transverse images can be delayed by many times the pulse

length without affecting the phase stability of the image. This is demonstrated by

interfering the images with a pulsed local oscillator and monitoring the interference

pattern. The interference stability has almost no dependence on fluctuations in

the group velocity in the slow light system, but only on the phase velocity, which

is unaffected by a slow light medium. This property leads to stable and high fringe

visibility when the delayed image interferes with a local oscillator even if the slow

light medium has moderate thermal instabilities.

To the author’s knowledge, the only previous studies of transverse images in

a slow light medium were performed by Harris’ group [15, 81]. In the Cs system

used here, the group velocity is the same in all directions. Also, this system has

relatively low loss and minimal broadening of the pulse.

Experimental Setup

Consider the experimental setup represented in Fig. 4.2. Light pulses with a

duration of 2 ns full-width at half-maximum (FWHM), repeating every 7 ns, are

used in the experiment. The pulses are generated by passing a CW laser beam


Cesium Vapor Cell

Mask

Lens

Lens

Camera

Beamsplitter

Beamsplitter

Mirror

Mirror

Optional

Beam Block

Scanning Optical

Fibre

Figure 4.1: Experimental setup for the delay of transverse images. Light pulses of 2 ns durationare incident on a 50:50 beamsplitter. The transmitted pulses then pass through an amplitudemask and a 4f imaging system. The transmitted and reflected pulses are recombined at another50:50 beamsplitter. The transmitted part traverses a path approximately 5 feet shorter thanthe reflected path, and arrives at the second beamsplitter about 5 ns sooner than the reflectedpulse, preventing interference between the two pulses. The temperature of cesium vapor canthen be adjusted to give 5 ns of delay, resulting in interference. In the low-light-level experiment,the pulses are attenuated such that each pulse contains on average less than one photon andthe reflected path is blocked. A scanning optical fiber is used to collect the photons in theimage plane and the photon arrival times recorded using a photon counter with time-to-digitalconverter.


through a fiber-coupled, high-speed electro-optic modulator (DC to 16 Gb/s). The

laser frequency is set halfway between the two hyperfine ground-state resonances

of the D2 lines (852 nm wavelength) in Cesium. The pulses enter an unbalanced

Mach-Zehnder interferometer with a free-space path mismatch of 5 ns. The pulses

propagating in the long path are reference pulses (local oscillator) that are made

to interfere with the pulses exiting from the slow light medium in the short path.

In the short path, the pulses impinge on an amplitude mask, and are called image

pulses. A 4.5-lines-per-millimeter test pattern was used as the mask. The hot Cs

vapor is in the middle of a 4f imaging system, which consists of two identical 150

mm lenses. In a 4f imaging system the object is placed a focal length (150 mm)

in front of the first lens, the distance between the two lenses is two focal lengths

(300 mm), and the image is produced in the back focal plane of the second lens.

The 4f system was used to eliminate the quadratic phase in the image plane.

The group delays in the cell are varied by changing the vapor pressure through

temperature control of the cell.

The image and reference pulses interfere via the second 50/50 beam splitter.

One of the mirrors in the long path has a piezo-actuated mount allowing for

precision translations of the mirror. Movements on the order of a few nm are

possible allowing for control of the relative phase of the reference and image pulses

at the beamsplitter. By translating the mirror through a phase shift of π radians, it

is possible to measure the fringe visibility. The interference images were measured


on a CCD camera run in continuous mode (a CCD camera capable of gating pulses

in a 2 ns pulse window was unavailable for the present study).

Another experiment performed examined image delays for the case of weak

input pulses. The experimental apparatus for the weak light fields experiment

is considerably different from the macroscopic light fields experiment discussed

above. Pulses of light of 4 ns FWHM duration, repeating every 330 ns, are

created in the same way as the macroscopic light fields. However, the pulses are

attenuated so that, on average, there is less than 1 photon per pulse impinging on

the amplitude mask. The long arm of the Mach-Zehnder interferometer is blocked,

leaving only a straightforward 4f imaging system and the slow light medium. To

recreate the image, a scanning multimode fiber with a 62 µm diameter core is

used to collect the photons in the image plane. The multimode fiber is coupled

to a single-photon counting module with 300 ps detector jitter (Perkin Elmer

SPCM). The electronic signal from the detector is sent to a 16 ps resolution time-

to-digital converter and is time-stamped. The multimode fiber is continuously

scanned using computer-controlled translation stages with 20 nm resolution. The

position of the the translation stages is recorded as a function of time. The clock of

the computer-controlled translation stages is synchronized with that of the time-

to-digital converter using the electronic pulse that is driving the electro-optic

modulator. Thus, the 2-dimensional image is reproduced by binning the photon

detection events into the 2-dimensional positions at which they were detected.


position (mm)(a)

-2 -1 0 1 2position (mm)

(b)

-2 -1 0 1 2position (mm)

(c)

-2 -1 0 1 2

po

siti

on

(m

m)

-2

-1

0

1

2

Figure 4.2: Interference of a delayed image with a slightly diverging local oscillator. (a) An image(a black pattern of bars and a numeral) delayed by 5 ns interferes with a reference beam andproduces a ring pattern superimposed with the image. In the central dark spot, the two beamsdestructively interfere and cancel one another except in the image, which remains relativelybright. In the ring surrounding the central spot, the two beams constructively interfere andadd to create a bright ring except in the image, which remains relatively dim. The succeedingrings alternate between constructive and destructive interference. (b) and (c) show the samesuperposition of the two beams, but in the absence of slow light. In (b), the wavelength of thelaser is tuned outside of the dispersive region and in (c) the cesium cell is removed. In bothcases, no interference between the beams can be seen.

Background counts (e.g., light from the room or detector dark counts) in the image

are significantly reduced by only accepting time-binned data centered around the

relative delay of the image, within a time window that is determined by the

parameters of the pulse. This can be done in postprocessing of the image by

looking at the temporal histogram of arrival times. The interferometer was not

used in the weak fields experiment for the practical reason that the relatively large

interferometer was not phase stable for the entire scan duration.


Results

Consider the results for the macroscopic image interference shown in Fig. 4.2. In

Fig. 4.2(a), the Cs cell temperature is set to give 5 ns of delay, which matches

the arrival time of the image pulses at the second beamsplitter to that of the

reference pulses. The situation in which both pulses arrive at the beamsplitter

simultaneously will be referred to as “temporally matched”. The intensity along

the two paths is balanced for maximum interference. The phase of the local os-

cillator is set to give a dark fringe in the center of the image. Several π radians

of phase shift across the image can be observed. The only regions that do not

experience interference are the image points of the dark patterns of the ampli-

tude mask. Since there is no light in the delayed image at those points, the local

oscillator creates a constant background where the dark regions of the amplitude

mask are imaged. Hence, at the center of the dark fringe the inverse image is

created. An interference visibility of 90% ± 1% was observed for the temporally

matched pulse regime. The pulses from the two arms of the Mach-Zehnder in-

terferometer are then misaligned in time, so as to arrive at the beamsplitter at

different times. This is accomplished by either removing the cell or by tuning the

delay of the pulse. In both cases the observed visibility dropped, as seen in Figs.

2 b) and c), respectively. The images show the same number of phase shifts as

the temporally matched pulses but the interference visibility (after balancing the

intensity in each arm) is 6% for cell removal and 15% for delay tuning, far lower


0

50

100

Co

un

ts

0 0.2 0.4 0.6 0.80

500

1000

Position (mm)

Co

un

ts

0 5 100

500

1000

Time (ns)

Co

un

ts

(a)

(b)

(c)

Figure 4.3: (a) Delayed and (b) non-delayed one-dimensional low-light-level image with (c)accompanying histograms of photon arrival times. Each pulse contains, on average, 0.5 photonsbefore striking the image mask.

than the 90% visibility for the temporally matched case. As a note, there is always

a small amount of CW light leaking through the electro-optic modulator, which

has a 100:1 extinction ratio. The CW light is the primary culprit in giving the

nonvanishing interference visibility when the pulses are temporally mismatched.

The amount of CW light can be much greater than 1% of the total light since it

is constantly “on”, which can lead to a much larger integrated CW signal. In the

interference experiment, the CW background is about 5%. The CW background

light can be removed by using a camera that is able to gate around a 2 ns window

in a fashion similar to that of the low light-level experiment.

The experimental results for the weak field images are shown in figures 4.3

through 4.5. Figs. 4.3(a) and (b) show a delayed and non-delayed one-dimensional

image (a bar test pattern) where each pulse impinging on the image mask contains,


on average, 0.5 photons. The images are measured by scanning an optical fiber

in a line across the image plane for a total duration of 36 seconds. A histogram

of the photon arrival times is made for each incremented position of the fiber (an

effective pixel) as it scans across the image (shown in Fig. 4.3(c)). The measured

image is the convolution of the image with the fiber core. For these scans, the

laser frequency is set halfway between the optical resonances and the temperature

of the cell is set to give 9 ns of delay (shown in red). The process is repeated but

with the laser frequency tuned far from either resonance, which gives almost no

delay (shown in blue). Approximately 99% of extraneous counts from background

light and detector dark counts are removed by constructing the images using only

those photons which arrive in a 4 ns time window (out of the entire 330 ns window)

centered on the middle of the pulse arrival time distribution. An analysis of the

undesirable counts led to an estimate of approximately 2 extraneous counts per

spatial bin shown in Fig. 4.3, which is in good agreement with the image noise.

Figure 4.4 shows the delay of a two-dimensional image comprised of the letters

“UR” representing the researchers’ institution. In this part of the experiment, each

pulse contains, on average, 0.8 photons before arriving at the image mask. The

image is constructed by raster scanning a fiber across the image plane in a total

time of approximately 48 seconds. The time-binned filtering technique described

above was also used to remove background counts from the two-dimensional im-

ages. A histogram of the photon arrival times for the two dimensional images of


0 0.2 0.4 0.6 0.8 1.0

0

0.1

0.2

0.3

0.4

0.5

po

siti

on

(m

m)

position (mm)0 0.2 0.4 0.6 0.8 1.0

position (mm)(a) (b)

Figure 4.4: False color representation of a (a) Delayed and (b) non-delayed two-dimensional low-light-level image. An optical fiber was raster-scanned across a two dimensional image consistingof the letters “UR”. Though attenuated, the delayed imaged shows similar image fidelity andresolution to the non-delayed image. Each pulse contains, on average, 0.8 photons before strikingthe image mask.

−2 0 2 4 6 80

200

400

600

800

1000

Time (ns)

Cou

nts

Figure 4.5: Histogram of photon arrival times showing the delayed (red) and non-delayed (blue)two-dimensional image shown in Fig. 4.4


Fig. 4.4 is shown in Fig. 4.5., showing the delayed image pulses. Even though

every photon used to construct the image is delayed by approximately 3 ns, the

image is preserved with high fidelity.

Discussion and Concluding Remarks

A few comments about the results are in order. First, the propagation through

the medium is a classical effect, meaning that its behavior does not change in

going from classical fields to quantum fields. A formal demonstration of the

preservation of quantum fields was not undertaken in the present study. However,

the preservation of amplitude and phase as well as the low noise characteristics

imply that this system can be an integral part of quantum image buffering. The

development of a highly multimode quantum image buffer is a much different

goal than that of the preservation of two state systems that have been recently

studied (qubits)[51, 61, 82–85]. Second, the homogeneous linewidth of the cesium

atoms ultimately determines the upper limit in the absolute delay of our slow light

system. However, much narrower resonances could achieve a much larger upper

limit of the delay at the expense of the usable signal bandwidth.

In conclusion, we have shown in this section is that a transverse image can

be delayed in a slow light buffer. The buffer is shown to be able to delay the

image by many pulse widths, while also preserving its amplitude and phase char-

acteristics. The image is interfered with a pulsed local oscillator. When the local

4.3. OTHER IMPLEMENTATIONS OF DOUBLE-RESONANCESLOW LIGHT 65

oscillator and image pulse are temporally overlapped, high visibility fringes with

90% visibility are observed, demonstrating the preservation of phase information

after a 5 ns pulse delay. When the local oscillator and image pulse are temporally

misaligned, low visibility fringes are observed, demonstrating the pulsed nature of

the imaging system. The slow light system is then used to delay an image using

weak-light pulses. Pulses with less than one photon on average are used to image

an amplitude mask. The image of the mask is reproduced with high fidelity and

low noise, demonstrating 9 ns pulse delays of images at very weak light levels.

That both the phase and the amplitude remain intact under delay even at the

single photon level suggest that the double-resonance technique for slow light may

have useful applications in quantum information.

4.3 Other Implementations of Double-

Resonance Slow Light

Delay of Energy-time Entangled Photons

While the image pulses of the previous section were shown to have on average

fewer than one photon per pulse, the photons used were weak coherent states and

no attempt was made to demonstrate any quantum characteristics. In order to

test whether a true quantum state could be preserved under delay, we replaced the

pulsed laser source with a source of down-converted entangled photons. Instead


Figure 4.6: Interference Fringes. a) Predicted (solid) and experimentally measured Fransoninterference fringes corresponding to slowing of the idler photon (circles), and signal photon(triangle). The dotted curve shows the predicted interference fringes in the absence of theslow-light medium. The predicted fringes were calculated with T = 210C.

of a Mach-Zhender interferometer, which is ideally suited for demonstrations of

classical interference, we employed a Franson interferometer, which is capable of

measureing quantum interference1.

A heated Rb vapor cell was placed in one arm of the Franson interferometer,

and one of the pair of the entangled photons (the idler) was made to have band-

widths lying between the two strongly absorbing Rb D1 and D2 lines, such that it

would be slowed when passing through the Franson interferometer. The observed

shift in the quantum interference fringes as a result of the delay is shown in Fig.

4.6

1See Appendix C for an introduction to Franson interferometry


The well-known criteria for demonstrating entanglement between the photons

is a measured Fringe visibility of greater than 1/√

2 = 70.7% While the delayed

photon experiences significant broadening for large delays, the fringe visibility

remains sufficient to demonstrate the preservation of entanglement2.

Fourier Transform Interferometry

Another interferometric scheme in which double-resonance slow light can be shown

to be of great utility is that of Fourier transform interferometry. It was first

shown by Peter Fellgett[86] that the spectrum of co-propagating light waves may

be measured by passing the light through an interferometer and measuring the

output of one port of the interferometer while increasing the length of one of its

arms. He showed that the measured intensity corresponds exactly to the Fourier

transform of the input spectrum. Interestingly, the technique was not viewed

as having widespread utility at the time, owing to the difficulty in performing

the Fourier transform by hand. With the advent of high speed computers, the

technique is now widely used in modern laboratories.

It can be shown3that the the resolving power of a Fourier transform interferom-

eter is proportional to the allowable change in the optical path mismatch between

the two arms of the interferometer. As demonstrated in previous chapters, when

a tunable slow light medium is placed in one of the paths, the fringe spacing of

2See Appendix D for complete experimental details


Figure 4.7: Output intensity of the slow-light FT interferometer as a function of the group delayτg for an input field of two sharp spectral lines separated by 80 MHz.

the interference pattern at the output becomes compressed, demonstrating the

steep frequency dependence of the interference. When used in the configuration

of Fourier transform interferometry, this increased sensitivity to the frequency

of the input light directly yields an increased resolving power. Furthermore, if

the group velocity in the medium is tunable, Fourier transform may be gener-

ated without moving parts, another advantage over traditional Fourier transform

interferometry.

The results of an experiment using slow light to enhance Fourier transform

interferometry are shown in Fig. 4.7. In the experiment. A beam consisting of

two frequencies with a spectral separation of 80 MHz was passed through a slow

light Fourier transform interferometer constructed in our laboratory. The output

shown in the figure clearly shows a carrier frequency modulated by a beat note of


80 MHz. A complete description of the experimental apparatus and results may

be found in Appendix D.

3See Appendix E for a complete discussion of Fourier transform interferometry with andwithout slow light enhancement

70

Chapter 5

Slow and Stopped Light inCoherently Prepared Media

5.1 Introduction: Coherent vs. Incoherent

Atomic Ensembles

The experimental investigation of double-Lorentzian slow light described in the

previous chapters was made in a collection of incoherent Lorentz oscillators—the

atoms in the atomic ensemble had no definite phase relationship to one another.

In addition, there was little control over the width or spacing of the two resnonces,

since these quantities were determined by the lifetime of the excited state and the

ground-state hyperfine splitting of the atoms respectively, parameters over which

an experimenter has little control under most circumstances. While the delay

and broadening characteristics depend only on the shape of optical resonance, it

is interesting physically to ask what does change when the oscillators oscillate

5.2. FEYNMAN DIAGRAM MODEL FOR A COHERENTFOUR-LEVEL SYSTEM 71

in unison. It would also be of practical interest to devise a system in which the

spacing and the resonance width of two Lorentzians is completely tunable.

5.2 Feynman Diagram Model for a Coherent

Four-Level System

To undertake such a study, we consider a four-level system which may be thought

of a generalization of the EIT system discussed in the introduction. Consider the

energy level diagram shown in Fig. 5.1(b). We assume that all electron population

resides initially in ground-state |1〉. Again using Feynman diagrams, the steady

state polarization resonant with the signal frequency may be found by summing

the polarizations induced by all possible excitation pathways from state |1〉 to

state |2〉, as shown in Fig. 5.2. Such paths may be grouped into two categories:

those which begin with excitation by a signal photon (as shown in Fig. 5.2b), and

those that begin with excitation by a coupling photon (as shown in Fig. 5.2c).

In the first case, the induced polarization is linear in the signal and identical

to that of electromagnetically induced transparency (EIT) in the lambda config-

uration:

P (1)s = Nµ12

Ωs

∆s

∞∑

n=0

rn

= Nµ12Ωs

∆s − Ω2c

∆R

(5.1)


Laser

Acousto-Optic

Modulator

Polarizing

Beam Splitter

Rubidium

Vapor Cell

50/50

Beam Splitter

λ

2

Etalon

Detector

(a)

(b)

Ωs

Ωi

ΩC

ΩC

|1

|3

|2∆s ∆c

∆ i

Etalon

Figure 5.1: (a) Experimental schematic and (b) energy level diagram. A warm Rb vapor cell isprepared using a strong coupling field. A signal pulse passing through the vapor is compressedwhile simultaneously generating a frequency-shifted idler pulse via four-wave mixing. Bothsignal and idler pulses are stored in a long lived polarization wave by turning off the couplingfield immediately after compression. They are then released by turning on the coupling field,with each retrieved pulse retaining its own waveform.

where the summation over r = Ω2c/(∆s∆R) accounts for the repeated emission

and absorption of a coupling photon. 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/~

represents the Rabi frequency induced by electric field amplitdue Ej via the dipole

matrix element µj.

In the second case, the signal polarization is proportional to the idler field and

is responsible for nonlinear four-wave mixing:

P (NL)s = Nµ12

ΩiΩ2c

∆1∆2∆3 − Ω2c∆1

, (5.2)


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

at the idler frequency:

P(NL)i = Nµ12

ΩsΩ2c

∆s∆R∆1 − Ω2c∆1

, (5.3)

which, except for being proportional to Ωs instead of Ωi, is identical to the nonlin-

ear 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 steady state polarization induced between

ground-states |1〉 and |3〉 may be found in a similar manner:

P13 = Nµ13

ΩsΩc

∆s∆R − Ω2c

+ΩiΩc

∆1∆2 − Ω2c∆1

∆3

. (5.4)

We note that the ground-state polarization 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.

We also note that in the limit of vanishing two-photon detuning on each excitation


|1 1|

ωs

2|

|3

ωc

2|

ωc

|1

2|

|3

ωc

ωc

|1 1|

ωs

2|

|3

ωc

2|

ωc

|1

|1 1|

ωs

2||1

+ + + ...

ωs ωc

|1

|3

|2

ωs ωc

|1

|3

|2

ωs

|1

|3

|2

= + + + ...ωs

|1

|3

|2

ωc

ωs

|1

|3

|2

ωc

= +

|1

|3

|2

ωc

ωc

ωi

ωs

|1

|3

|2

ωc

ωc

ωi

(a)

(b)

(c)

|1

|3

|2

ωc

ωc

ωi

2|

|3

ωc

ωc

|1 1|

ωc

2|

|3

ωi

2|

ωc

|1

+ + ...

ωc

|1

|3

|2

= + ...+

ωc

|1

|3

|2

ωc

ωi

+

|1

|3

|2

ωc

ωi ωc

ωc

ωi

|1 1|

ωc

2|

|3

ωi

2|

ωc

|1

Figure 5.2: (Double sided Feynman diagrams showing the generated signal polarization. (a)The polarization oscillating at the signal frequency may be separated into two parts: [b (blue)]excitations that couple states |1〉 and |2〉 via the signal field leading to electromagneticallyinduced transparency, and [c (red)] excitations that couple states |1〉 and |2〉 via the idler fieldleading to four wave mixing gain or absorption. Interference among atomic excitation pathwaysleading to coherent preparation is shown by summing one photon, three photon, five photon,etc excitation pathways.

5.3. COHERENTLY PREPARED DOUBLE-LORENTZIAN 75

pathway, and negligible ground-state decoherence, the ground-state polarization

reduces to −Nµ13(Ωs +Ωi∆1/∆3)/Ωc, indicating that the ground-state coherence

induced by the idler field can be enhanced relative to that of the signal field by a

ratio of the one and three photon detunings, or may even have the opposite sign

and destructively interfere for negative three photon detunings.

In the remainder of this chapter, we will describe experiments performed under

two different regimes within the above model. The first is the case of a large

detuning of the coupling field from single photon resonance. This results in the

the absorption of signal photons and the emission of idler photons, but with low

gain. The second case will be with a smaller detuning for the coupling field, which

results in large gain on both the signal and the idler.

5.3 Coherently Prepared Double-Lorentzian

When the coupling field is far detuned from single photon resonance, and the

the signal field is tuned near two-photon resonance, coherent absorption of the

signal takes place, resulting in an absorption profile in the vicinity of two-photon

resonance. In order to make two Lorentzian absorption resonances for slow light,

one may prepare two separate ensembles and pass a pulse through them in series.

The experimental arrangement for such a system is shown in Fig. 5.3.

Examples of three experimentally measured transmission profiles of the signal

beam in the vicinity of two-photon resonance are shown in Fig. 5.4. In Fig. 5.4(a),


LaserTapered

AmpliferOptical

Isolator

1.5 GHz

AOMSpherical

Mirror

Rb Vapor Cell

Rb Vapor Cell

200 MHz

AOM

200 MHz

AOM

80 MHz

AOM

Figure 5.3: Experimental setup for obtaining two Lorentzian absorption resonances of tunablewidths and spacing.

-0 .5 0 0 .50

1

2

Tra

nsm

issi

on

-5 0 50

1

2

Tra

nsm

issi

on

-5 0 50

0 .5

1

Frequency (MHz)

Tra

nsm

issi

on

(a)

(b)

(c)

Figure 5.4: Measured transmission profiles of double absorption resonances.


Figure 5.5: Pulse delays in a coherently prepared double-resonance

the coupling fields are approximately 500 mW each and separated in frequency

by approximately 500 KHz. In Fig. 5.4(b) the separation between the coupling

field frequencies remains the same, but the power of each has been increased

to approximately 1 mW. In Fig 5.4(c), the separation of the coupling fields is

approximately 2 MHz and the coupling field power is approximately 4 mW for

each beam. These plots demonstrate the ability to easily tune both the width and

spacing of the resonances, allowing for corresponding tunability of slow pulses in

a slow light medium. It is worthy of note that the absorption on resonance can be

made very large, meaning that the medium has been rendered almost completely

opaque in an otherwise transparent region. This leads to the possibility of large

pulse delays.

Figure 5.5 is an example of the delay of a 200 ns pulse whose carrier fre-


Figure 5.6: CW transmission profile corresponding to delays in Fig. 5.5

quency has been tuned between two coherently prepared absorbing resonances.

For reference, the transmission profile of the resonances is shown in Fig. 5.6.

Because all atoms are prepared coherently by the pump, new effects can

be demonstrated that are not possible in incoherent double-Lorentzian systems.

Among these new effects is the possibility to switch the pump beam on or off

at a given time and thus change the absorbing properties of the entire ensemble

simultaneously. This allows for a type of pulse synchronization, as shown in Fig.

5.7. In Fig 5.7, the blue curve is the undelayed pulse, the red curve is the delayed

pulse, and the black curve is an example of a “synchronized” pulse. The the

black curve was taken by first preparing the double resonance system using the

coupling beam and then switching off the pump (show as a dashed line) after the

pulse had partially passed through the medium. As can be seen, the pulse resume

its non-delayed form very quickly. This could be useful in the syncrhonization


−400 −200 0 200 400 6000

0.2

0.4

0.6

0.8

1

Time (ns)

Inte

nsity

(ar

b. u

nits

)

Figure 5.7: Fast switching using a coherently prepared resonance

in the arrival of data packets, if a tunable delay was required before the pulse

amplitude reached a certain threshold value.

The possibility of coherently controlling the slow light parameters in the system

leads to other interesting effects as well, including the storage and retrieval of light

pulses and enhanced nonlinear optics, which we now discuss.

5.4. STOPPED LIGHT IN A COHERENTLY PREPAREDFOUR-LEVEL SYSTEM 80

5.4 Stopped Light in a Coherently Prepared

Four-Level System

Introduction

We next turn to the regime in which the coupling beam is much closer to resonance,

allowing for gain on both signal and idler fields. The ability to coherently prepare a

long-lived ground-state polarization still exists, so we may say that we are working

at the interesting intersection between the fields of coherent atom optics and

nonlinear optics. It is well known that coherently prepared media may be used to

enhance nonlinear optical interactions [11, 21, 87–92] and may also serve as atomic

memories for light storage [51, 82]. Recently there has been much theoretical

interest[93–98] in combining these two processes with the goal of enhancing optical

nonlinearities, pulse storage, or both. Recent experiments have demonstrated the

use of pulse storage to achieve large cross-Kerr nonlinearities [99] and the use of

optical nonlinearities to improve the readout in atomic memories [76].

In the field of atomic memories, many experiments have been performed

demonstrating the ability to prepare a collective atomic spin wave using a strong

off-resonant or weak resonant “write” beam, and then read out the coherence

at a later time using a resonant “read” beam [76, 78, 100–107]. In these tech-

niques, the signature of coherent atomic preparation is the detection of a Stokes

photon spontaneously emitted after the absorption of a write photon. An anti-


Stokes photon may then be deterministically generated by injecting a weak read

pulse, thereby inducing the completion of a parametric four-wave mixing interac-

tion. Other experiments have coherently prepared a collective atomic spin wave

by injecting both write and stokes beams simultaneously, and then reading out

an anti-Stokes beam using a read beam whose frequency is chosen to complete a

four-wave mixing process [108–110].

In all such experiments, the atomic ensemble stores information using a sin-

gle Λ resonance, and hence information about the waveform at a single optical

frequency. Likewise, the readout of the coherence stimulates emission on a single

(perhaps different) Λ resonance, and hence at a single optical frequency. The

present experiment demonstrates the ability to both write and read using two Λ

resonances simultaneously, and hence record information from two distinct optical

frequencies for later retrieval. The storage two fields may have important implica-

tions 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.

Here we report on our contribution to the field, which is the simultaneous

coherent storage of a 1.4 µs optical signal pulse and a frequency shifted idler pulse

in a hot atomic rubidium vapor, and show that the two pulses maintain different

waveforms throughout storage and retrieval. It is shown that the intensity of the


retrieved pulses varies linearly with the intensity of the retrieving coupling beam,

even at low light levels.

The present scheme is based on four-wave mixing using two input fields in

a three-level atomic ensemble, and is closely related to proposals involving reso-

nant double-lambda EIT systems. As in previous experiments, 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. However,

the present scheme has some distinct differences. The most important is that

pulse information transfer to the medium is accompanied by parametric gain on

Raman resonance, and absorption away from Raman resonance. This 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.

Experiment

The experimental setup is outlined in Fig 5.1(a). The frequency of a continuous

wave (CW) coupling laser is set approximately 700 Mhz to the blue of the optical

transition connecting the F = 2 ground-state to the F ′ = (2, 3) excited states of

the D1 line of 85Rb at 795 nm (since the exited state splitting is only 120 Mhz

and the doppler broadened linewidth is approximately 500 Mhz, we may regard

the excited states F ′ = (2, 3) as a single energy level). This coupling beam is then


sent through an acousto-optic modulator (AOM), which generates a deflected

beam oscillating at the signal frequency 3.0357 Ghz (the ground-state hyperfine

splitting, ω12) to the red of the coupling field, placing the signal frequency near the

midpoint between the F = 2 → F ′ and the F = 3 → F ′ optical transitions. The

coupling and signal beams are then made to have opposite linear polarizations,

recombined, and sent through a heated, magnetically shielded rubidium vapor cell

of natural isotopic abundance containing 20 torr of neon buffer gas, generating a

third beam (idler) 3.0357 Ghz to the blue of the coupling field. The output beam

is then split using a 50/50 beamsplitter and solid etalons are used to frequency

select the signal and idler beams before their intensities are measured on two

separate detectors. To demonstrate pulse storage and retrieval, 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. All experiments were performed using a 12.5 cm Rb vapor cell heated to

approximately 180 C. The full-width at half maximum signal and coupling field

intensity spatial profiles were approximately 500 µm and 300 µm respectively.

A plot of the measured signal and idler intensities as a function of signal

detuning around Raman resonance is shown in Fig. 5.8 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.

1–3 into the the solutions of the slowly varying phase-matched envelope equations


with no initial idler field:

Ωs(z) = Ωs(0)

[cosh(ξz)− αs

ξsinh(ξz)

]e−αsz (5.5a)

Ωi(z) = Ωs(0)

[gi

ξsinh(ξz)

]e−αsz (5.5b)

where ξ =√

α2s + 4gsgi, αs = =[P

(1)s ]ks/2Es is the linear absorption coefficient for

the signal, gs = =[P(NL)s ]ks/2Ei is the nonlinear gain coefficient for the signal, and

gi = =[P(NL)i ]ki/2Es is the nonlinear gain coefficient for the idler. For the theory

plots shown in Fig. 5.8, we have 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. 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 [111].

Figure 5.9 shows the delay of a 1.4 µs signal pulse and the simultaneous gen-

eration of a idler pulse (a), as well the storage and retrieval of both signal and

idler pulses 20 (b) and 120 (c) µs later. The maximum storage time observed

was approximately 500 µs. We note that the entire signal pulse waveform is re-

trieved 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, as might be expected by storing the


−0.1 −0.05 0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

1

Ω

(arb

. un

its)

p 2

∆ (MHz)p

Figure 5.8: Measured (asterisks) and theoretical (solid) steady state signal (blue) and idler (red)intensities as a function of signal detuning. The theory plots are generated from Eqs. 1-3,5 withparameters given in the text.

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. 5.10. 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 [112], for example, that -7.1 dB of


0

1

2

3

4

5

Pe

ak

Inte

nsi

ty (

% 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 5.9: Signal and idler pulse delay (a) and storage and retrieval [(b) and (c)]. Part (a)shows an example of an incident signal pulse (black), delayed signal pulse (blue) and generatedidler pulse (red) for the case when the coupling field (3 mW) remains on for the duration of themeasurement. In parts (b) and (c), the coupling field (dashed line) is turned off approximately500 ns after the peak of the signal pulse enters the medium, and then turned back on 20 µs and120 µs later, respectively. The retrieved pulse waveforms correspond to that fraction and shapeof each pulse that was in the medium when the coupling field was turned off, demonstratingthat the stored coherence contains information about each waveform that may be separatelyretrieved.


101

102

103

100

101

102

Pump Retrieval Power (µW)

Sig

na

l Po

we

r (µ

W)

Figure 5.10: Idler pulse peak power as a function of coupling input power.

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.

Concluding Remarks

In conclusion, we have reported on an experiment in which a signal beam and

a generated frequency-shifted idler beam were coherently stored in a hot Rb va-

por, and later retrieved. The compression of the signal beam was enhanced by

four wave mixing, resulting in the storage of the entire signal waveform for later

retrieval. In addition, coherent scattering from the stored coherence was shown


to generate both signal and idler beams with linear efficiency, even at low light

levels, while maintaining distinct signal and idler pulse waveforms.

89

Chapter 6

Stopped Images in CoherentlyPrepared Media

6.1 Introduction

In the preceeding chapter it was demonstrated that coherently prepared media

may be used to prepare double-Lorentzian absorption resonances with tunable

spacings and widths. It was also shown that because the dipole moments of all

atoms in the atomic ensemble oscillate in unison, a collective atomic coherence can

be stored in the medium which contains information about input light pulses, and

that this information may be recovered at a later time by scattering additional

light from the prepared coherence.

Condsidering these results in connection with the applications of slow light

discussed in chapter 4, it is natural to ask whether the image delay methods may be

be extended by using coherently prepared ensembles rather than incoherent ones.

In this chapter we demonstrate that indeed stopped light techniques may be used

6.1. INTRODUCTION 90

to store and retrieve the transverse spatial profile of a beam (an image) provided

that a method is devised to overcome the adverse effects of atomic diffusion. A

proof-of-principle experiment is discussed in which 1-d images are stored for up

to 30 µs and then retrieved.

As discussed, stopped light allows for the recording of coherent signals for later

retrieval even at very low light levels. The first demonstrations of stopped light in

its present context may be found in the initial work by Liu et al.[51] and Phillips

et al. [82]. This then stimulated additional research with a recent demonstration

of storage times in excess of one second [113]. 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, the stopped light medium usually ranges from a few

tens of microns up to several centimeters, so slow light is first used to spatially

compress the optical pulses inside the medium. Slow light is achieved when a

steep linear dispersion can be obtained in a medium, leading to a large group

index ng = n + ω ∂n∂ω

and 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 [12, 15, 19, 22, 51, 114, 115] and

solids [52, 53, 116–120].

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

6.2. EXPERIMENT 91

medium delays and stores the spatial profiles of the stored pulses [81, 121–124].

As mentioned above, 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. [122] 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 [123].

Here we demonstrate the storage and retrieval of a transverse image in a hot

atomic vapor using a combination of electromagnetically induced transparency

(EIT)[11] 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 [123]. This is similar

to the storage of the Laguerre-Gauss beams [122] under diffusion where atoms on

opposite sides of the phase singularity have relative phases of π and destructively

interfere.

6.2 Experiment

A schematic of the experimental setup is shown in Fig. 6.2(a). A 12.5 cm long

Rb vapor cell with natural isotopic abundance containing 20 torr neon buffer

6.2. EXPERIMENT 92

Ωs

Ωp

Ωp

Ωi

T T

Input Signal

Pump Pulse

Retrieved Signal

1 2

Delayed Signal

ObjectRb Vapor Cell

PBS PBS Image

Camera

Signal

3.035 GHz Acousto-Optic

Modulator

Laser 795 nm

Pump

L1 l

2

L2

(a)

(b)

(c)

2

1

3

Figure 6.1: (a)An external cavity diode laser followed by an amplifier is the source for pumpbeam. The signal is generated by frequency shifting part of the pump by 3.035 Ghz using anacousto-optic modulator. The 1/e2 beam diameter of the pump is approximately 4 mm and thatof signal approximately 1 mm at the object plane. The powers of pump and signal beams areapproximately 12 mW and 300 µW respectively. The pump and signal beams are orthogonallypolarized and are combined using a polarizing beam splitter (PBS) before the cell. The pumpis filtered from the signal using another PBS after the cell. The object, lenses L1 and L2, andthe camera form a 4f imaging system.(b) We use the D1 transitions of 85Rb to create a Λconfiguration. The pump is detuned by 700 MHz to the blue of the optical transition connectingthe F = 2 ground-state to the F ′ = (2, 3) excited states, and the signal is set 3.035 GHz (theground-state hyperfine splitting) to the red of the pump. (c) Representation of the synchronizedtiming of signal (dashed red), delayed signal (blue) and pump (black) beams. Once a signalpulse is inside the medium, the pump beam is turned off at time T1, storing the image. Thepump beam is then turned back on at time T2, retrieving the stored image pulse (solid red).

6.2. EXPERIMENT 93

gas is placed in a magnetically shielded oven and heated to approximately 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 pump beam is orthogonally

polarized to the signal to filter out the pump. In addition to polarization filtering,

we also performed a temporal filtering correlation measurement using a 100 MHz

detector (3 dB roll off). A 25 µm slit is placed in the focal 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. 6.2(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

6.2. EXPERIMENT 94

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

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. 6.2(c). The intensity profile of the signal at the object plane and

its Fourier transform at the vapor cell are shown in Fig. 6.2. The retrieved signal

intensity falls off exponentially with storage time due to diffusion and decoherence.

Figure 6.2 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. (6.4) to propagate the measured input image shown in Fig. 6.2(a).

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 by

diffusion theory. The physical mechanism responsible for preserving the image can

be understood in terms of the phase distribution of the stored optical wavefront

and the diffusion of the atoms (shown graphically in Fig. 6.2). Each atom in

the field acquires the local coherence set by the signal and pump fields. As the

6.2. EXPERIMENT 95

image plane Fourier plane

−2

−1

0

1

2

po

siti

on

(m

m)

Figure 6.2: CCD camera capture of the signal intensity profile at the object plane and at thevapor cell (Fourier plane).

6.3. THEORETICAL MODEL 96

atoms diffuse 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.

6.3 Theoretical Model

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(−i

2π

λfξu)dξ, (6.1)

where ξ and u are the coordinates of object plane and Fourier transform plane

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− ρ13

Tc

(6.2)


−1 0 1 −1 0 1

Position (mm) Position (mm)

1

Position (mm)−1 0 1

0

Input Image

1 ms

Retrieved Images

TheoryExperiment time

4 ms

16 ms

32 ms

1

(a)

(b) (c)

Figure 6.3: Input signal profile (a) and the time evolution of measured (b) and calculated (c)transverse images. The theory plots are generated using Eq. (6.4) with a diffusion coefficient of10 cm2/s).


−1 0 1−1

0

1

0

reference

10 µs

30 µs

Position (mm)

Am

pli

tud

e

Figure 6.4: Theoretical time evolution of stored ground-state coherence of Rb atoms. The insetshows a close up of the time evolution near zero crossover points of electric field amplitude.


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.

Figure 6.2 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.

The field at the image plane, Ei, is recovered by inserting Eq. (6.1) into the

diffusion equation [ Eq. (6.2)] and taking another Fourier transform, which upon

integration gives

∂

∂tEi(x, t) = −(

1

Td

+1

Tc

)Ei(x, t), (6.3)

with a solution given by

Ei(x, t) = Ei(x, 0) exp

[−t(

1

Td

+1

Tc

)

], (6.4)

6.4. DISCUSSION 100

where Td = λ2f2

(2π)2Dx2 is the diffusion time constant and x is the coordinate in the

image plane.

6.4 Discussion

We note that by integrating over the spatial coordinate u at the cell from negative

infinity to positive infinity we have not accounted 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. 6.2), this approximation is

valid. In addition, we have assumed that all modes of spatial diffusion for the

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 [125]. 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.

There are two features worth noting in Eq. (6.4). 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

6.4. DISCUSSION 101

compared to the temporal pulse length. Second, since Td falls off like 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

imaging lens larger. In either case, the Fourier transformed spatial profile at the

vapor cell would be larger, requiring a correspondingly larger pump beam diameter

and vapor cell.

In summary, we have slowed and stored an arbitrary transverse image in a hot

atomic vapor, and shown that the retrieved image is robust to atomic diffusion.

This remarkable feature allows the coherent storage of spatial information even in

doppler broadened media with large diffusion constants.

102

Chapter 7

Conclusions

The study of slow group velocities has progressed beyond fundamental demon-

strations to practical implementations. That progress, however, has been largely

dependent on the ability to delay optical pulses by times much longer than the

pulse duration. This simple criteria has proven rather intractable experimentally

for a number of years, especially if one also makes the practical requirement that

the pulse not be overly distorted. In this dissertation, we have investigated the use

of two widely spaced absorption resonances as a means to overcome these practical

limitations, and have reported on several experimental demonstrations. Further-

more, having found a solution to the delay-bandwidth problem, we have discussed

the use of the double-resonance technique to demonstrate a variety of new and

useful implementations. These include applications in imaging, interferometry,

and quantum entanglement.

103

The origin of the success of the double-resonance technique lies in the shape of

the transparency window between the resonances. Qualitatively, the transmission

spectrum has a nearly rectangular shape, which turns out to be important for

two reasons. The first has to do with absorption. Nearly all frequencies lying

between the two resonances are absorbed in equal amounts, meaning that the

relative spectral content of the output field will be similar to that of the input

field. If this were not the case, and some frequencies were absorbed more than

others, the bandwidth of the pulse would decrease and cause temporal broad-

ening. The second reason has to do with dispersion. Owing to the shape of the

transmission resonance and the symmetry of the configuration, the first correction

to the frequency-dependent index of refraction happens to be third order, rather

than second order as in single-resonance systems. This leads to a smaller variance

in the phase velocities of the individual frequencies away from the desired linear

distribution around the pulse carrier frequency, leading to less distortion of the

constructive interference responsible for the pulse shape. In fact, these two effects

actually work in tandem to reduce distortion even further than either individu-

ally. It turns out that those frequencies most likely to contribute to third order

dispersion are also those most strongly absorbed by the double-Lorentzian—the

offending frequencies are simply absorbed and thus cannot contribute to dispersive

broadening.

These remarkable features lead to the ability to slow optical pulses with a wide

104

variety of bandwidths and carrier frequencies without distorting the pulse. The

first demonstration discussed was performed in a nearly ideal double-resonance

structure of Rb. The hyperfine splitting in Rb leads to two ground-state energy

levels with similar coupling strengths to the first manifold of excited states. The

second demonstration used Cs, which has two nearly ideal absorbing resonances

for the same reason as Rb, but a much larger energy separation between the

resonances. This allowed for much higher bandwidth (temporally shorter) pulses

to be delayed and also demonstrated the application of the technique to systems

with different spectral and temporal characters.

Depending on one’s background, perhaps the most interesting element of the

double-Lorentzian technique is simply that it allowed for the intersection of the

field of slow light with other areas of optical science, and so a large portion of this

dissertation has been devoted to discussing specific examples of this crossover. In

the case of optical imaging, for example, it was shown that the phase and the

amplitude of transverse images may be preserved even when the group velocity of

the pulses carrying the images is slowed by several orders of magnitude. Without

complete temporal separation of a delayed image from a non-delayed reference,

it would be very difficult to distinguish which aspects of the image are preserved

under the delay. Likewise, in the case of Fourier transform interferometry, the

number of fringes obtained during the measurement of the Fourier spectrum is

proportional to the delay-bandwidth product of the slow light medium in the

105

spectral region of interest. These two subjects as well as a demonstration of

energy-time entanglement form the basis of the experimental demonstration that

the double-resonance slow light technique may have practical applications to other

fields.

Lastly, motivated by the historical relationship of the slow light community

with nonlinear optics and coherently prepared media, we have studied scenarios

in which a double-absorption resonance may be prepared in a coherent fashion,

and whether coherence and nonlinear effects may be used as a resource to extend

applications of slow light. It was found that by using a four-level system, two

absorption resonances may be prepared coherently with tunable linewidths and

separation. When the reconfiguration rates of the resonances were investigated, we

found that coherent preparation allowed for both fast switching of the resonance

(and hence pulse delays), and the preparation of a ground-state coherence which

stored information regarding optical interaction with all four levels. This coherent

memory was shown to have the ability to store both single longitudinal optical

modes as well as transverse modes. There is a basic difficulty, however, in storing

transverse modes of an optical field in warm vapors: the atoms carrying the

information move. In order to overcome the deleterious effects of atomic diffusion

on image storage, we implemented a scheme in which the the Fourier transforms

of the images were stored rather than the images themselves. This technique led

to much longer storage times and proved interesting in its own right.

106

As a matter of record and perhaps utility for those who wish to undertake

further research in this field (e.g. future new graduate students trying to read

this dissertation), a few comments may be appropriate. First, I consider among

the most useful parts of this thesis to be the calculational technique employed

to derive the coherent lineshapes. I spent hours solving density matrix equations

(which are at last indispensable) before discovering the diagrammatic technique

employed here, which saved much time and provided a picture to go along with

the words “atomic interference pathways”. Second, I recommend highly the sum-

maries written by Daniel Steck [73] of the alkali line data, which include derivations

and tables of the most important atomic parameters. I referred to it more often

than any other single reference throughout my graduate career.

107

Bibliography

[1] L. Brillouin, Wave Propagation and Group Velocity (Academic Press, NewYork, 1960).

[2] J. J. Su and I. A. Yu, “The study of coherence-induced phenomena usingdouble-sided feynman diagrams,” Chinese Journal of Physics 41, 627 (2003).

[3] R. W. Boyd and D. J. Gauthier, Progress in Optics (Elsevier, Radarweg 29,Amsterdam 1043 NX, 2002), pp. 497–530.

[4] P. W. Milonni, “Controlling the speed of light pulses,” Journal of PhysicsB-Atomic Molecular and Optical Physics 35, R31 (2002).

[5] P. W. Milonni, Fast Light, Slow Light, and Left-Handed Light (Institute ofPhysics, Bristol and Philadelphia, 2005).

[6] S. L. McCall and E. L. Hahn, “Self-induced transparency by pulsed coherentlight,” Physical Review Letters 18, 908 (1967).

[7] C. K. N. Patel and R. E. Slusher, “Self-induced transparency in gases.,”Physical Review Letters 19, 1019 (1967).

[8] D. J. Bradley, G. M. Gale, and P. D. Smith, “Self-induced transparency anddispersion delays in potassium vapour,” Nature 225, 719 (1970).

[9] R. E. Slusher and H. M. Gibbs, “Self-induced transparency in atomic ru-bidium,” Physical Review A 5, 1634 (1972).

[10] E. Courtens and A. Szoke, “On consequences of the different phase andsignal velocities in self-induced transparency,” Physics Letters A 28, 296(1968).

[11] S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processesusing electromagnetically induced transparency,” Physical Review Letters64, 1107 (1990).

[12] S. E. Harris, J. E. Field, and A. Kasapi, “Dispersive properties of electro-magnetically induced transparency,” Physical Review A 46, R29 (1992).

BIBLIOGRAPHY 108

[13] D. Grischkowsky, “Adiabatic following and slow optical pulse propagationin rubidium vapor,” Physical Review A 7, 2096 (1973).

[14] M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically in-duced transparency: Optics in coherent media,” Reviews of Modern Physics77, 633 (2005).

[15] A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, “Electromagnetically in-duced transparency - propagation dynamics,” Physical Review Letters 74,2447 (1995).

[16] M. Xiao, Y. Q. Li, S. Z. Jin, and J. Geabanacloche, “Measurement of dis-persive properties of electromagnetically induced transparency in rubidiumatoms,” Physical Review Letters 74, 666 (1995).

[17] O. Schmidt, R. Wynands, Z. Hussein, and D. Meschede, “Steep dispersionand group velocity below c/3000 in coherent population trapping,” PhysicalReview A 53, R27 (1996).

[18] J. P. Marangos, “Topical review electromagnetically induced transparency,”Journal of Modern Optics 45, 471 (1998).

[19] L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed re-duction to 17 metres per second in an ultracold atomic gas,” Nature 397,594 (1999).

[20] K. T. McDonald, “Slow light,” American Journal of Physics 68, 293 (2000).

[21] M. M. Kash et al., “Ultraslow group velocity and enhanced nonlinear opticaleffects in a coherently driven hot atomic gas,” Physical Review Letters 82,5229 (1999).

[22] D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, “Nonlinearmagneto-optics and reduced group velocity of light in atomic vapor with slowground state relaxation,” Physical Review Letters 83, 1767 (1999).

[23] A. B. Matsko, O. Kocharovskaya, Y. Rostovtsev, G. R. Welch, A. S. Zi-brov, and M. O. Scully, Slow, ultraslow, stored, and frozen light, Vol. 46 ofAdvances in Atomic Molecular and Optical Physics (Elsevier, Radarweg 29,Amsterdam 1043 NX, 2001), pp. 191–242.

[24] R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers:Capabilities and fundamental limitations,” Journal of Lightwave Technology23, 4046 (2005).

[25] G. T. Purves, C. S. Adams, and I. G. Hughes, “Sagnac interferometry in aslow-light medium,” Physical Review A 74, 023805 (2006).

BIBLIOGRAPHY 109

[26] Z. Shi, R. W. Boyd, R. M. Camacho, P. K. Vudyasetu, and J. C. Howell,“Slow-light fourier transform interferometer,” Physical Review Letters 99,240801 (2007).

[27] Z. M. Shi, R. W. Boyd, D. J. Gauthier, and C. C. Dudley, “Enhancingthe spectral sensitivity of interferometers using slow-light media,” OpticsLetters 32, 915 (2007).

[28] G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of atunable-bandwidth white-light interferometer using anomalous dispersionin atomic vapor,” Physical Review Letters 99, 133601 (2007).

[29] R. M. Camacho, M. V. Pack, J. C. Howell, A. Schweinsberg, R. W. Boyd,and Aq, “Wide-bandwidth, tunable, multiple-pulse-width optical delays us-ing slow light in cesium vapor,” Physical Review Letters 98, 153601 (2007).

[30] S. E. Harris, “Normal-modes for electromagnetically induced transparency,”Physical Review Letters 72, 52 (1994).

[31] J. H. Eberly, M. L. Pons, and H. R. Haq, “Dressed-field pulses in an ab-sorbing medium,” Physical Review Letters 72, 56 (1994).

[32] S. Chu and S. Wong, “Linear pulse-propagation in an absorbing medium,”Physical Review Letters 48, 738 (1982).

[33] S. Chu and S. Wong, “Pulse-propagation in an absorbing medium - reply,”Physical Review Letters 49, 1293 (1982).

[34] B. Segard and B. Macke, “Observation of negative velocity pulse-propagation,” Physics Letters A 109, 213 (1985).

[35] R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximumtime delay achievable on propagation through a slow-light medium,” Phys-ical Review A 71, 023801 (2005).

[36] A. B. Matsko, D. V. Strekalov, and L. Maleki, “On the dynamic range ofoptical delay lines based on coherent atomic media,” Optics Express 13,2210 (2005).

[37] D. A. B. Miller, “Fundamental limit for optical components,” Journal of theOptical Society of America B-Optical Physics 24, A1 (2007).

[38] D. A. B. Miller, “Fundamental limit to linear one-dimensional slow lightstructures,” Physical Review Letters 99, 203903 (2007).

BIBLIOGRAPHY 110

[39] Z. Dutton, M. Bashkansky, M. Steiner, and J. Reintjes, “Analysis and op-timization of channelization architecture for wideband slow light in atomicvapors,” Optics Express 14, 4978 (2006).

[40] M. Bashkansky, Z. Dutton, F. K. Fatemi, J. Reintjes, and M. Steiner,“Demonstration of bicolor slow-light channelization in rubidium vapor,”Physical Review A 75, 021401 (2007).

[41] R. M. Camacho, M. V. Pack, J. C. Howell, and Zp, “Slow light with largefractional delays by spectral hole-burning in rubidium vapor,” Physical Re-view A 74, 033801 (2006).

[42] A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal prop-agation in a medium with a gain doublet,” Physical Review A 49, 2071(1994).

[43] L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal lightpropagation,” Nature 406, 277 (2000).

[44] K. T. McDonald, “Negative group velocity,” American Journal of Physics69, 607 (2001).

[45] B. Macke and B. Segard, “Propagation of light-pulses at a negative group-velocity,” European Physical Journal D 23, 125 (2003).

[46] H. Tanaka et al., “Propagation of optical pulses in a resonantly absorbingmedium: Observation of negative velocity in rb vapor,” Physical Review A68, 053801 (2003).

[47] R. M. Camacho, M. V. Pack, J. C. Howell, and Wi, “Low-distortion slowlight using two absorption resonances,” Physical Review A 73, 4 (2006).

[48] B. Macke and B. Segard, “Pulse normalization in slow-light media,” PhysicalReview A 73, 043802 (2006).

[49] R. M. Camacho, C. J. Broadbent, I. Ali-Khan, and J. C. Howell, “All-opticaldelay of images using slow light,” Physical Review Letters 98, 043902 (2007).

[50] Z. Zhu and D. J. Gauthier, “Nearly transparent sbs slow light in an opticalfiber,” Optics Express 14, 7238 (2006).

[51] C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherentoptical information storage in an atomic medium using halted light pulses,”Nature 409, 490 (2001).

BIBLIOGRAPHY 111

[52] M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslowlight propagation in a ruby crystal at room temperature,” Physical ReviewLetters 90, 113903 (2003).

[53] M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slowlight propagation in a room-temperature solid,” Science 301, 200 (2003).

[54] X. Zhao, P. Palinginis, B. Pesala, C. Chang-Hasnain, and P. Hemmer, “Tun-able ultraslow light in vertical-cavity surface-emitting laser amplifier,” Op-tics Express 93, 7899 (2005).

[55] P. Palinginis, F. Sedgwick, S. Crankshaw, M. Moewe, and C. J. Chang-Hasnain, “Room temperature slow light in a quantum-well waveguide viacoherent population oscillation,” Optics Express 13, 9909 (2005).

[56] Y. Okawachi et al., “Tunable all-optical delays via brillouin slow light in anoptical fiber,” Physical Review Letters 94, 153902 (2005).

[57] M. Gonzalez-Herraez, K.-Y. Song, and L. Thavenaz, “Optically controlledslow and fast light in optical fibers using stimulated brillouin scattering,”Appl. Phys. Lett. 87, 081113 (2005).

[58] M. Gonzalez-Herraez, K.-Y. Song, and L. Thevenaz, “Arbitrary-bandwidthbrillouin slow light in optical fibers,” Optics Express 14, 1400 (2006).

[59] D. Dahan and G. Eisenstein, “Tunable all optical delay via slow and fastlight propagation in a raman assisted fiber optical parametric amplifier: aroute to all optical buffering,” Optics Express 13, 6234 (2005).

[60] J. E. Sharping, Y. Okawachi, and A. L. Gaeta, “Wide bandwidth slow lightusing a raman fiber amplifier,” Optics Express 13, 6092 (2005).

[61] A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. A. Musser, B. S.Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulsesin a solid,” Physical Review Letters 88, 023602 (2002).

[62] K. Y. Song, M. G. Herraez, and L. Thevenaz, “Long optically controlleddelays in optical fibers,” Optics Letters 30, 1782 (2005).

[63] R. M. Camacho, M. V. Pack, and J. C. Howell, “Slow light with large frac-tional delays by spectral hole-burning in rubidium vapor,” Physical ReviewA 74, 033801 (2006).

[64] R. Y. Chiao, “Superluminal (but causal) propogation of wave packets intransparent media with inverted atomic populations,” Physical Review A48, R34 (1993).

BIBLIOGRAPHY 112

[65] A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous disper-sion and superluminal light-pulse propagation at a negative group velocity,”Physical Review A 63, 053806 (2001).

[66] M. D. Stenner, D. J. Gauthier, and M. A. Neifeld, “The speed of informationin a fast-light optical medium,” Nature 425, 695 (2003).

[67] G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherentmanipulation of the raman gain process,” Physical Review A 70, 023802(2004).

[68] M. D. Stenner, M. A. Neifeld, Z. Zhu, A. M. C. Dawes, and D. J. Gauthier,“Distortion management in slow-light pulse delay,” Optics Express 13, 9995(2005).

[69] R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow lightusing two absorption resonances,” Physical Review A 73, 063812 (2006).

[70] G. P. Agrawal, in Nonlinear Fiber Optics (Academic Press, Radarweg 29,Amsterdam 1043 NX, 1995), p. 79.

[71] U. Volz and H. Schmoranzer, “Precision lifetime measurements on alkaliatoms and on helium by beam- gas-laser spectroscopy,” Physica Scripta T65, 48 .

[72] E. B. Alexandrov et al., “Light-induced desorption of alkali-metal atomsfrom paraffin coating,” Physical Review A 66, 042903 (2002).

[73] D. A. Steck, Cesium D line data, Technical Report No. LA-UR-03-7943, LosAlamos National Laboratory (2003), http://steck.us/alkalidata/.

[74] A. Kasapi, G. Y. Yin, M. Jain, and S. E. Harris, “Measurement of lorentzianlinewidths by pulse propagation delay,” Physical Review A 53, 4547 (1996).

[75] S. Chen, Y.-A. Chen, T. Strassel, Z.-S. Yuan, and B. Zhao, “Deterministicand storable single-photon source based on a quantum memory,” PhysicalReview Letters 97, 173004 (2006).

[76] C. H. van der Wal et al., “Atomic memory for correlated photon states,”Science 301, 196 (2003).

[77] B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, “Ex-perimental demonstration of quantum memory for light,” Nature 432, 482(2004).

BIBLIOGRAPHY 113

[78] M. D. Eisaman, L. Childress, A. Andre, F. Massou, A. S. Zibrov, and M. D.Lukin, “Shaping quantum pulses of light via coherent atomic memory,”Physical Review Letters 93, 233602 (2004).

[79] Q. F. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit ina photonic structure,” Nature Physics 3, 406 (2007).

[80] M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal, M. Messall, and K. Salit,“Ultrahigh enhancement in absolute and relative rotation sensing using fastand slow light,” Physical Review A 75, 053807 (2007).

[81] M. Jain, A. J. Merriam, A. Kasapi, G. Y. Yin, and S. E. Harris, “Eliminationof optical self-focusing by population trapping,” Physical Review Letters 75,4385 (1995).

[82] D. F. Phillips, A. Fleischhauer, A. Mair, and R. L. Walsworth, “Storage oflight in atomic vapor,” Physical Review Letters 86, 783 (2001).

[83] O. Kocharovskaya, Y. Rostovtsev, and M. O. Scully, “Stopping light via hotatoms,” Physical Review Letters 86, 628 (2001).

[84] M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in anatomic medium,” Nature 426, 638 (2003).

[85] M. F. Yanik, W. Suh, Z. Wang, and S. Fan, Physical Review Letters 93,233903 (2004).

[86] P. Fellgett, “Multi-channel spectrometry,” Journal of the Optical Society ofAmerica 42, 872 (1952).

[87] P. R. Hemmer, D. P. Katz, J. Donoghue, M. Croningolomb, M. S. Shahriar,and P. Kumar, “Efficient low-intensity optical-phase conjugation based oncoherent population trapping in sodium,” Optics Letters 20, 982 (1995).

[88] Y. Q. Li and M. Xiao, “Enhancement of nondegenerate four-wave mixingbased on electromagnetically induced transparency in rubidium atoms,” Op-tics Letters 21, 1064 (1996).

[89] B. L. Lu, W. H. Burkett, and M. Xiao, “Nondegenerate four-wave mix-ing in a double-lambda system under the influence of coherent populationtrapping,” Optics Letters 23, 804 (1998).

[90] M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entangle-ment of ultraslow single photons,” Physical Review Letters 84, 1419 (2000).

BIBLIOGRAPHY 114

[91] M. Katsuragawa, J. Q. Liang, F. L. Kien, and K. Hakuta, “Efficient fre-quency conversion of incoherent fluorescent light,” Physical Review A 65,025801 (2002).

[92] B. K. Dutta and P. K. Mahapatra, “Nonlinear optical effects in a doublydriven four-level atom,” Physica Scripta 75, 345 (2007).

[93] A. Raczynski and J. Zaremba, “Controlled light storage in a double lambdasystem,” Optics Communications 209, 149 (2002).

[94] A. Raczynski, J. Zaremba, and S. Zielinska-Kaniasty, “Electromagneticallyinduced transparency and storing of a pair of pulses of light,” PhysicalReview A 69, 043801 (2004).

[95] A. Joshi and M. Xiao, “Generalized dark-state polaritons for photon memoryin multilevel atomic media,” Physical Review A 71, 041801 (2005).

[96] Z. Li, L. S. Xu, and K. G. Wang, “The dark-state polaritons of a double-lambda atomic ensemble,” Physics Letters A 346, 269 (2005).

[97] A. Raczynski, M. Rzepecka, J. Zaremba, and S. Zielinska-Kaniasty, “Po-lariton picture of light propagation and storing in a tripod system,” OpticsCommunications 260, 73 (2006).

[98] E. Yomba, “New generalized hyperbolic functions to find new coupled ul-traslow optical soliton pairs in a cold three-state double-lambda system,”Physica Scripta 76, 8 (2007).

[99] Y. F. Chen, C. Y. Wang, S. H. Wang, and I. A. Yu, “Low-light-level cross-phase-modulation based on stored light pulses,” Physical Review Letters96, 043603 (2006).

[100] A. Kuzmich et al., “Generation of nonclassical photon pairs for scalablequantum communication with atomic ensembles,” Nature 423, 731 (2003).

[101] C. W. Chou, S. V. Polyakov, A. Kuzmich, and H. J. Kimble, “Single-photongeneration from stored excitation in an atomic ensemble,” Physical ReviewLetters 92, 213601 (2004).

[102] D. N. Matsukevich and A. Kuzmich, “Quantum state transfer between mat-ter and light,” Science 306, 663 (2004).

[103] M. D. Eisaman, A. Andre, F. Massou, M. Fleischhauer, A. S. Zibrov,and M. D. Lukin, “Electromagnetically induced transparency with tunablesingle-photon pulses,” Nature 438, 837 (2005).

BIBLIOGRAPHY 115

[104] T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, S. Y. Lan, T. A. B.Kennedy, and A. Kuzmich, “Storage and retrieval of single photons trans-mitted between remote quantum memories,” Nature 438, 833 (2005).

[105] S. A. Chen et al., “Deterministic and storable single-photon source basedon a quantum memory,” Physical Review Letters 97, 173004 (2006).

[106] J. K. Thompson, J. Simon, H. Loh, and V. Vuletic, “A high-brightnesssource of narrowband, identical-photon pairs,” Science 313, 74 (2006).

[107] J. Simon, H. Tanji, J. K. Thompson, and V. Vuletic, “Interfacing collectiveatomic excitations and single photons,” Physical Review Letters 98, 183601(2007).

[108] A. S. Zibrov, A. B. Matsko, O. Kocharovskaya, Y. V. Rostovtsev, G. R.Welch, and M. O. Scully, “Transporting and time reversing light via atomiccoherence,” Physical Review Letters 88, 103601 (2002).

[109] B. Wang, S. Li, H. Wu, H. Chang, H. Wang, and M. Xiao, “Controlledrelease of stored optical pulses in an atomic ensemble into two separatephotonic channels,” Physical Review A (Atomic, Molecular, and OpticalPhysics) 72, 043801 (2005).

[110] Y. F. Chen, P. C. Kuan, S. H. Wang, C. Y. Wang, and I. A. Yu, “Ma-nipulating the retrieved frequency and polarization of stored light pulses,”Optics Letters 31, 3511 (2006).

[111] A. Nagel, S. Brandt, D. Meschede, and R. Wynands, “Light shift of coherentpopulation trapping resonances,” Europhysics Letters 48, 385 (1999).

[112] C. F. McCormick, A. M. Marino, V. Boyer, and P. Lett, “Relative-intensitysqueezing at audio frequencies using four-wave mixing in an atomic vapor,”arXiv:quant-ph/0703111v1 (2007).

[113] J. J. Longdell, E. Fraval, M. J. Sellars, and N. B. Manson, “Stopped lightwith storage times greater than one second using electromagnetically in-duced transparency in a solid,” Physical Review Letters 95, 063601 (2005).

[114] W. G. Yang, D. B. Conkey, B. Wu, D. L. Yin, A. R. Hawkins, and H.Schmidt, “Atomic spectroscopy on a chip,” Nature Photonics 1, 331 (2007).

[115] V. Boyer, C. F. McCormick, E. Arimondo, and P. D. Lett, “Ultraslow propa-gation of matched pulses by four-wave mixing in an atomic vapor,” PhysicalReview Letters 99, 143601 (2007).

BIBLIOGRAPHY 116

[116] J. B. Khurgin, “Optical buffers based on slow light in electromagneticallyinduced transparent media and coupled resonator structures: comparativeanalysis,” Journal of the Optical Society of America B-Optical Physics 22,1062 (2005).

[117] G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd,“Observation of backward pulse propagation through a medium with a neg-ative group velocity,” Science 312, 895 (2006).

[118] Z. M. Zhu and D. J. Gauthier, “Nearly transparent sbs slow light in anoptical fiber,” Optics Express 14, 7238 (2006).

[119] Y. Okawachi, M. A. Foster, J. E. Sharping, A. L. Gaeta, Q. F. Xu, and M.Lipson, “All-optical slow-light on a photonic chip,” Optics Express 14, 2317(2006).

[120] E. Shumakher, N. Orbach, A. Nevet, D. Dahan, and G. Eisenstein, “Onthe balance between delay, bandwidth and signal distortion in slow lightsystems based on stimulated brillouin scattering in optical fibers,” OpticsExpress 14, 5877 (2006).

[121] R. M. Camacho, C. J. Broadbent, I. Ali-Khan, and J. C. Howell, “All-opticaldelay of images using slow light,” Physical Review Letters 98, 043902 (2007).

[122] R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, “Topolog-ical stability of stored optical vortices,” Physical Review Letters 98, 203601(2007).

[123] L. Zhao, T. Wang, Y. Xiao, and S. F. Yelin, “Image storage in hot vapors,”arxiv.org/abs/0710.4144v1 (2007).

[124] M. Shuker, O. Firstenberg, R. Pugatch, A. Ben-Kish, A. Ron, and N. David-son, “Angular dependence of dicke-narrowed electromagnetically inducedtransparency resonances,” Physical Review A 76, 023813 (2007).

[125] Y. Xiao, I. Novikova, D. F. Phillips, and R. L. Walsworth, “Diffusion-induced ramsey narrowing,” Physical Review Letters 96, 043601 (2006).

117

Appendix A

Feynman Diagrams

This appendix is meant to review the derivation leading to the Feynman dia-

grams used in the main text. As applied to nonlinear optics, Feynman diagrams

represent a calculational technique for arriving at the various orders of the non-

linear polarization. The physics of the diagrams has its origins in time-dependent

perturbation theory. We begin with the Schrodinger equation in the rotating

frame (interaction picture):

i~d

dtψI(t) = HI(t)ψI(t) (A.1)

where ψI and HI are the state vector and perturbing hamiltonian as seen in the

rotating frame. If we look for an interaction propagator UI(t, t0) that operates on

the state vector ψI(t0) in such a way that the resulting state vector is ψI(t), i.e.

ψI(t) = UI(t, t0)ψI(t0), (A.2)

118

we find that owing to Eq. (A.1) the propagator must obey

i~d

dtUI(t, t0) = HI(t)UI(t, t0), (A.3)

a standard result from perturbation theory which has a solution

UI(t, t0) = I − i

~

∫ t

t0

HI(t′)UI(t

′, t0)dt′, (A.4)

where I is the identity operator. We obtain an approximate solution by iteration:

UI(t, t0) ≈I − i

~

∫ t

t0

HI(t′)

[I − i

~

∫ t′

t0

HI(t′′)UI(t

′′, t0)

]

≈I − i

~

∫ t

t0

HI(t′) +

(−i

~

)2 ∫ t

t0

dt′∫ t′

t0

dt′′HI(t′)HI(t

′′) + . . . (A.5)

which is known as the Dyson series. We now switch back to the Schrodinger

picture and consider the probability of a transition from an initial state |ψi〉 to a

final state |ψf〉:

119

〈ψf |U(t, t0)|ψi〉 =δfie−iE0

i (t−t0)/~

+−i

~

∫ t

t0

dt′e−iE0f (t−t′)/~〈ψf |HI |ψi〉e−iE0

i (t′−t0)/~

+

(−i

~

)2 ∫ t

t0

dt′∫ t′

t0

dt′′∑

n

e−iE0f (t−t′)/~〈ψf |HI |ψn〉

× e−iE0n(t′−t′′)/~〈ψn|HI |ψi〉e−iE0

i (t′′−t0)/~ + . . . (A.6)

The interpretation of this series is best made by reading each term from right

to left. In the zeroth order term, the Dirac delta function guarantees that the

final state vector is the same as the initial state vector and evolution from t0 to

t is just that of a free particle (i.e. e−iEt/~). In the first order term, the state

vector ψi undergoes free evolution from time t0 until t′, at which time it has an

instantaneous interaction governed by the interaction Hamiltonian HI , causing a

transition from ψi to ψf . After the interaction, the new state vector ψf evolves

under free evolution from time t′ to time t. The second order term has a similar

interpretation: the state vector ψi evolves freely until time t′′, when the interaction

Hamiltonian causes a transition to a new state ψn, which can be any intermediate

state of the system. The state vector then evolves freely until time t′, when it is

taken from the intermediate state ψn to the final state ψf , and again undergoes free

evolution until time t. Because there may be more than one intermediate state,

the second order term includes a summation over all possible “two-transition”

120

routes to the final state (hence the summation over n). The integration over time

accounts for the possibility that the interaction causing the transition may occur

at any time.

Each diagram in a series of Feynman diagrams represents a possible pathway

for the state vector to evolve from its initial to final state. We may recover

the graphs used in this dissertation by identifying the interaction Hamiltonian

connecting any two states as

〈ψm|HI |ψn〉 = −E · µmn ≡ ~Ωmne−i(ωLt−iΓmn) (A.7)

where E is the electric field amplitude of a laser oscillating at a frequency ωL, con-

necting states ψm and ψn. We have phenomenoligically allowed for damping (i.e.

spontaneous emission and decoherence processes) by the inclusion of a imaginary

decay term Γmn. To enter the steady-state regime we allow the upper limit of all

integrals to extend to infinity. This yields the simple formulation used in the text

in which each possible transition is represented by a time-independent Rabi fre-

quency term, and each free evolution from one transition to the next is represented

by an inverse accumulated detuning 1/∆n (the result of time integration).

121

Appendix B

Refractive vs. DispersiveContributions to the Speed of

Light in Dielectrics

B.1 Electromagnetic Waves and the Group In-

dex

Assuming no free charges or currents, monochromatic electromagnetic waves travel

through linear dielectrics (i.e. D = εE) according to Maxwell’s equations:

∇ · E = 0 (B.1)

∇× E = −∂B

∂t(B.2)

∇ ·B = 0 (B.3)

∇×B = µε∂E

∂t, (B.4)

B.1. ELECTROMAGNETIC WAVES AND THE GROUP INDEX122

where we have defined ε = ε0(1 + χ). Taking the curl Eq A.2 and Eq. A.4 and

then using the fact that the divergence of both B and E is zero we arrive at the

usual wave equations for the electromagnetic fields:

∇2E = µε∂2E

∂t2(B.5)

∇2B = µε∂2B

∂t2. (B.6)

These wave equations of course have solutions of sinusoidal waves propagating

at a velocity

v =1√εµ

=c

n, (B.7)

where we have used the usual definition of c = 1/√

ε0µ0 and defined the index of

refraction n as the ratio of the speed of propagation of a monochromatic wave in

material to its speed in vacuum:

n ≡√

εµ

ε0µ0

=√

1 + χ

√µ

µ0

. (B.8)

We address the class of materials in which the electric susceptibility is much

greater than the magnetic susceptibility, and so let√

µ/µ0 = 1. The frequency

B.2. LORENTZ OSCILLATORS 123

dependent group index is then given by

ng = n(ω) + ωdn(ω)

dω

=√

1 + χ(ω) +ω

2√

1 + χ(ω)

dχ(ω)

dω

= n +ω

2n

dχ

dω(B.9)

where we have dropped the explicit ω dependence in the last equation for com-

pactness.

B.2 Lorentz Oscillators

Let us consider now the case a of Lorentz oscillator, with a complex susceptibility

given by

χ =A

∆− iΓ

= A[

∆

∆2 + Γ2+ i

Γ

∆2 + Γ2

], (B.10)

where we defined Γ in this case to be the half-width of the Lorentzian in this

appendix for notational simplicity (it is defined as the full-width in the main text).

Noting that d/dω = −d/d∆, the derivative of the real part of the susceptibility is

dχ

dω= A ∆2 − Γ2

(∆2 + Γ2)2. (B.11)


We may use this result to make a comparison between the magnitudes of the

linear part n and the dispersive part (ωdχ/dω)/2n of the group index. Taking the

difference the two quantities D = (ωdχ/dω)/2n− n we obtain

D =An

(ω

2

dχ

dω− (1 + χ)

)

=A2n

(ω(∆2 − Γ2)−∆(∆2 + Γ2)

(∆2 + Γ2)2

)− A

n. (B.12)

The important point to notice is that

A2n

(ω(∆2 − Γ2)−∆(∆2 + Γ2)

(∆2 + Γ2)2

)> 0 (B.13)

for off-resonant interaction (∆2 > Γ2) at optical frequencies (ω À |∆|). We may

now ask which term is the primary contributor to the delay of a pulse, the phase

delay L/c(n−1) or the group delay L/(cωdn/dω)? If D > −1 then the dispersive

term dominates and it can be said that light is propagating slowly mostly due

to dispersion. This is the case when second term in Eq. B.12, A/n, is less than

unity, corresponding to blue-detuned light (∆ < 0). For the case of red detuned

light, a comparison of the first and second terms in Eq. B.12 is necessary. The

condition that D > −1 for red detuned light may be approximated as

An

(1− ω

2∆2

)< 1. (B.14)


One example where this is clearly satisfied is when A < n and ω < ∆2 corre-

sponding to a dilute sample far from resonance. Either one of these constraints

may be relaxed if stricter bounds are placed upon the other. For the case of alkali

vapors, for example, the subject of this dissertation, µ ≈ 10−29C ·m, meaning that

A ≈ N · 2 × 10−11 s. Hence, very high number densities are needed in order for

the phase term to exceed the group term when red-detuned from resonance.

In summary, the optical delay of light passing through a medium composed

of Lorentz oscillators is always mostly due to dispersion (not refraction), when

the the frequency of the light is tuned to the blue side of all oscillators. This is

the case even for nearly monocrhomatic light, a surprising non-trivial result. One

would not a priori expect dispersive delay to dominate refractive delay even when

there are very few waves to disperse.

126

Appendix C

Franson Interferometry

This appendix serves as an primer on Franson interferometry for those who

may be unfamiliar with the basic concepts. It also includes a derivation for the

coincidence rate one may expect to observe using a Franson interferometer both

with and without a slow light medium in one arm. A more detailed description

of the actual apparatus used for the experiment reported in this thesis may be

found in Appendix D.

Consider the Franson interferometer shown in Fig. C.1. The two photon

wavefunction |ψ(x1, x2, t)〉 has the form

|ψ(x1, x2, t)〉 = (C.1)

∫ ∞

−∞

∫ ∞

−∞dω1dω2g(ω1, ω2)a

†k1

(x1, t)a†k2

(x2, t)|0〉.

Let us first examine the meaning of this wavefunction. The basis states (eign-

states) making up the wavefunction are a†k1(x1, t)a

†k2

(x2, t)|0〉. Each state contains

127

Figure C.1: Schematic of a Franson interferometer with entangled photon source S, and twodistant detectors.

just two monochromatic frequencies, ω1 and ω2, one in each of two spatial modes

having a wavenumber given by ki = nωi/c. Each basis state then has an eigen-

value ~(ω1 + ω2). The two creation operators serve to create a monochromatic

field with energy ~ωi in each mode, as well as define that field’s space and time

dependence in that mode. The explicit space and time dependence are actually

redundant in this wavefunction, since by definition a single frequency component

of the electromagnetic field must take the form A cos(kixi − ωit), so by labeling

the creation operators with a wavenumber ki we have already uniquely determined

the space-time dependence. Notice that the two creation operators are functions

of one frequency only, and hence each eigenstate of the system is factorable and

not entangled.

To represent the actual state of the system, as always, we generate a linear

combination of the basis vectors. The function g(ω1, ω2) serves to weight the

128

amplitude of each basis state in the linear combination. However, since g(ω1, ω2)

is a function of both ω1 and ω2, the two spatial modes are now coupled in frequency

and hence energy (i.e. they are entangled in energy). More explicity, g(ω1, ω2)

serves to couple the probabilities of measuring energies in each mode; |g(ω1, ω2)|2

tells me the probability of measuring an energy of ~ω2 in mode k2 conditional on

the measurement of energy ~ω1 in mode k1 and vice versa.

We would like measure some quantities at the detectors as a function of

time. To do this, we must either propagate my wavefunction [Eq. (C.1)]in time

(Schodinger picture), or choose an operator to represent what we wish to measure

and propagate it in time (Heisenberg Picture). We will do the latter, and try to

find the time-dependent electric field operator for each mode at the detectors.

The electric field operator for a mode ki can be written as a sum of field

operators for positive and negative frequency components:

Eki(xi, ti) = E+

ki+ E−

ki, (C.2)

E+ki

(xi, ti) = akie−i(kixi−ωiti) (C.3a)

E−ki

(xi, ti) = a†kiei(kixi−ωiti) (C.3b)

where we have left out some unimportant constants.

129

Notice that each mode ki has its own space and time coordinates, so the

operators representing the field in that mode are written in terms of that mode’s

space-time coordinates. To find the value of the operator at some physical point

x, we would first find the actual distance xi to that point along the optical path

of some mode ki, and plug that value into the operator. Since often we would like

to find the expectation value of products of operators with different space-time

coordinates (i.e. the product of the field at a detector located at x1 at time t1

with that of the field at a different detector located at x2 at time t2), it is useful

to let each mode have its own space and time coordinates.

We can find the value of the operators at each detector by modeling each 50/50

beamsplitter as a scattering matrix with two field operator inputs and two field

operator outputs. For the positive frequency electric field operators,

E+aout

E+bout

=

1 i

i 1

E+ain

E+bin

. (C.4)

Propagation through space may be modeled similarly with a matrix of the

form

E+aout

E+bout

=

e−i(kaxa−ωiti) 0

0 e−i(kbxb−ωiti)

E+ain

E+bin

(C.5)

130

Let us try first the lower part of the Franson interferometer. At the source,

the positive frequency electric field operator is

E+0 (0, 0) = ak1 (C.6)

The light must propagate through a beamsplitter, then through two arms of

an interferometer, and then through another beamsplitter. The total scattering

matrix for positive frequency components in the lower arm of the Franson may be

written

S+1 =

1 i

i 1

e−i(kaxa−ω1t1) 0

0 e−i(kbxb−ω1t1)

1 i

i 1

(C.7)

= eiω1t1

e−ikaxa − e−ikbxb i(e−ikaxa + e−ikbxb

)

i(e−ikaxa + e−ikbxb

)e−ikbxb − e−ikaxa

The wavenumber ka is that of free space, so kaxa = ω1xa/c. The wavenuber

kb has both a free space component and a dispersive component:

kbxb =ω1

c[xb + (n(ω1)− 1)L] (C.8)

=ω1

c(xb +

1

2χ(ω1)L)

where χ(ω1) is the frequency dependent susceptibility of the dispersive medium

of length L.

131

By symmetry, the scattering matrix of the upper arm of the Franson interfer-

ometer is identical except for the absence of any dispersive component:

S+2 = eiω2t2

e−ikcxc − e−ikdxd i(e−ikcxc + e−ikdxd

)

i(e−ikcxc + e−ikdxd

)e−ikdxd − e−ikcxc

(C.9)

where in this case kc = kd = ω2/c.

The field operators at the two detectors are then

E+1 (t1) = S

(1,1)+1 ak1 (C.10a)

E+2 (t2) = S

(1,1)+2 ak2 , (C.10b)

where by S(1,1)+1 we mean the upper-left hand element in the S+

1 scattering matrix.

The second order coherence function is then proportional to

< ψ(0)|E−1 (t1)E

−2 (t2)E

+2 (t2)E

+1 (t1)|ψ(0) > (C.11)

= < ψ(0)|S(1,1)−1 a†k1

S(1,1)−2 a†k2

S(1,1)+2 ak2S

(1,1)+1 ak1|ψ(0) >,

132

Let’s write out the last half of this expectation value explicitly:

E+2 (t2)E

+1 (t1)|ψ(0) > (C.12)

=

∫ ∞

−∞

∫ ∞

−∞dω1dω2S

(1,1)+2 ak2S

(1,1)+1 ak1g(ω1, ω2)a

†k1

a†k2|0〉

=

∫ ∞

−∞

∫ ∞

−∞dω1dω2e

i(ω1t1+ω2t2)(e−i

ω1xac − e−i

ω1c

(xb+12χ(ω1)L)

)

×(e−i

ω2xcc − e−i

ω2xdc

)g(ω1, ω2)|0〉.

In the case of parametric downconversion, we may approximate ω1 = ωp − ε

and ω2 = ωp + ε, so that we can substitute and obtain a single integral:

E+2 (t + τ)E+

1 (t)|ψ(0) >=

∫ ∞

−∞dεei((ωp−ε)t+(ωp+ε)(t+τ))

(e−i

(ωp−ε)xac − e−i

(ωp−ε)

c(xb+

12χ((ωp−ε))L)

)

×(e−i

(ωp+ε)xcc − e−i

(ωp+ε)xdc

)g(ε)|0〉. (C.13)

Eq. (C.13) can then be multiplied by its complex conjugate to obtain the

probability for obtaining coincident counts at detector 1 at time t and detector 2

at time t + τ . We may break this integral up into four separate integrals. The

first, which represents the coherence between the two short arms, will take the

form

133

∫ ∞

−∞dεei(ωp(2t+τ)+ετ)e−i(ωp(xa+xc

c)+ε(xc−xa

c))g(ε)

= e−σ2(xa−xcc

−τ)2e2iωp(t+τ−xac

) (C.14)

where we have assumed that

g(ε) =1

2σ√

πe− (ωp−ε)2

(2σ)2 . (C.15)

Notice that this contribution rapidly goes to zero as the difference in the path

lengths of the short arms get much longer than cτ , so that if we make the long

arms much shorter than the short arms, we may neglect the contributions from the

middle two integrals (long-short and short long). Notice that this also assumes

that the group delay in the dispersive medium is also less than τac. The final

(long-long) integral will then take the form

∫ ∞

−∞dεei(ωp(2t+τ)+ετ)e−i(ωp(

xb+xd+12 χL)

c)+ε(

xd−xb− 12 χL

c))g(ε) (C.16)

=

∫ ∞

−∞dεei(ωp(2t+τ−tbd−τχ)+ε(τ−τbd+τχ))e

− (ωp−ε)2

(2σ)2

where we have used the notation τχ = 12χL/c to represent the amount of phase de-

lay a particular frequency will undergo as a result of passing through the medium.

This integral cannot be evaluated unless a specific χ(ε) is specified. Let us mo-

134

mentarily set τχ to zero and evaluate the integrals to see if we can recover the

traditional franson fringes. In that case the second integral becomes

e−σ2(xb−xd

c−τ)2e2iωp(t+τ−xb

c) (C.17)

and so we have

E+2 (t + τ)E+

1 (t)|ψ(0) > (C.18)

= e−σ2(xa−xcc

−τ)2e2iωp(t+τ−xac

) + e−σ2(xb−xd

c−τ)2e2iωp(t+τ−xb

c)

We obtain total coincident counting rate by multiplying this by its complex

conjugate, which gives

e−2σ2(τ−τac)2 + e−2σ2(τ−τbd)2 + e−σ2(τ−τac)2e−σ2(τ−τbd)22 cos(2ωpτab) (C.19)

where have used the notation τac = (xc − xa)/c (this is the time difference of

the two short paths) and τbd = (xd − xb)/c (this is the time difference of the two

long paths and τab = (xb − xa)/c (this is the time difference between long/short

in one of the arms).

The detectors cannot detect only at times t and t + τ , but will in actuality

135

have some finite detection time. If we assume that the detection time ∆τ is long

enough that differences in the short paths and the long paths make no difference

(i.e. τac ¿ ∆τ and τbd ¿ ∆τ , then we may integrate over all detection times of

the second detector. The (normalized) coincidence rate then becomes

R =

√π

2σ

(2 + 2e−

σ2

2(τac−τbd)2 cos(2ωpτab)

)(C.20)

This looks correct. Now let us go back to the case in which we have a dispersive

medium in one arm of the lower part of the Franson. Since time dependence is ab-

sent in the final result, we may also remove it and write the long-long contribution

to E†2(t + τ)E†

1(t)|ψ(0)〉 as

L = e(iωp(τ−xb+xdc

))

∫ ∞

−∞dεe(iε(τ+

xb−xd+12 χL

c)−iωp

12 χL

c)g(ε) (C.21)

and the short-short may be written

S = e−σ2(xa−xcc

−τ)2e2iωp(τ−xcc

) (C.22)

and the second order coherence function will then be proportional to

g2 ∝ (L + S)∗(L + S) (C.23)

= L∗L + L∗S + S∗L + S∗S (C.24)

136

Let’s take these terms one at a time. Let us assume that it is one of the short

arms that moves during the measurement, and the relative difference between the

long arms stays fixed. (i.e xb − xd = 0)First the product of the two long-long

integrals:

L∗L =1

4πσ2

∫ ∞

−∞

∫ ∞

−∞dεdε′e(i(ε−ε′)τe−i(ωp−ε)τχ(ε)ei(ωp−ε′)τχ(ε′)e

(ωp−ε)2+(ωp−ε′)2(2σ)2 (C.25)

The important thing to note here is that there is no dependence in L∗L on the

movement of the short arms, and so this term is a constant. The smaller τχ the

more closely this constant approaches unity.

Now for the product of the two short-short terms:

S∗S = e−2σ2(τac−τ)2 (C.26)

This is the same result as before.

Lastly we have the interference between the short-short and long-long integrals,

which give both the envelope and oscillations inside the envelope. This is the

important term, but also the one that is most difficult to perform analytically.

When performed numerically, it yields a coincidence counting rate with a shifted

envelope, as one might expect.

137

Appendix D

Preservation of Energy-TimeEntanglement in a Slow Light

Medium

The following journal article is reprinted with permission from Curtis J. Broad-

bent, Ryan M. Camacho, Ran Xin, and John C. Howell, “Preservation of Energy-

Time Entanglement in a Slow Light Medium”, Physical Review Letters 100,

133602 (2008). Copyright (2008) by the American Physical Society.

Preservation of Energy-Time Entanglement in a Slow Light Medium

Curtis J. Broadbent, Ryan M. Camacho, Ran Xin, and John C. HowellDepartment of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA

(Received 5 December 2007; published 4 April 2008; corrected 7 April 2008)

We demonstrate the preservation of entanglement of an energy-time entangled biphoton through a slowlight medium. Using the D1 and D2 fine structure resonances of Rubidium, we delay one photon of the1.5 THz biphoton by 1:3 correlation lengths and measure the fourth order correlation fringes. After thegroup delay the fringe visibility is reduced from 97:0 4:4% to 80:0 4:8%, but is still sufficient toviolate a Bell inequality. We show that temporal broadening is the primary mechanism for reducing thefringe visibility and that smaller bandwidths lead to greatly reduced broadening.

DOI: 10.1103/PhysRevLett.100.133602 PACS numbers: 42.50.p, 03.67.Bg

Temporal control of the properties of qubits is a signifi-cant challenge in the quest for practical quantum informa-tion processing (QIP). For photons, or ‘‘flying qubits’’,their temporary storage in a quantum memory can facilitatemany applications (e.g., linear optics quantum computing[1]). Quantum memories have varying design requirementsdepending on the application. These range from the moststringent requirements of on-demand retrieval of storedqubits to simpler passive but tunable temporary qubitbuffers. Here we investigate entanglement preservation ofan energy-time entangled biphoton in a double resonanceslow light medium. We show that it can be used as apassive, tunable qubit buffer.

Different types of quantum memories for photons havebeen investigated, each with its own strengths. Cyclicalfree-space memories for photonic qubits using Pockel cellshave been demonstrated experimentally [2,3]. Cyclicalfiber-based memories have also been proposed [4]. Bothare well suited to QIP applications with a master clockcycle. Memories using Raman transitions and dark-statepolaritons [5] in atomic ensembles have been extensivelystudied [6–10]. While extremely versatile, these types ofmemories often suffer from the difficulty of efficientlyisolating and detecting single photons in the presence ofstrong coupling beams.

In this Letter we report on the buffering of a singlephoton from an entangled pair in a slow light mediumand show that entanglement is preserved. We use the finestructure resonances of Rubidium to delay the amplitude ofa single photon with a correlation bandwidth of 1.5 THz by1:3 correlation lengths. We observe a post-selectedfourth order correlation fringe visibility of 80:0 4:8%(reduced from 97:0 4:4%) for an energy-time entangledbiphoton. This exceeds the value of 70.7% needed toviolate a Bell inequality and demonstrates the preservationof entanglement. Additionally, we demonstrate that broad-ening is the primary mechanism responsible for the reduc-tion of fringe visibility.

The fourth order correlation fringes are created with aFranson interferometer, a useful tool in the analysis ofenergy-time entanglement [11–13]. It consists of two

Michelson interferometers with highly unbalanced armlengths so that no single photon interference occurs (seeFig. 1). Inserting a photon from an energy-time entangledbiphoton into each Michelson interferometer and thenscanning one of the four arms results in interference fringesin the detected coincidences. The fringes arise due tospectral correlations between the entangled photons. Thevisibility of these interference fringes can be used to verifythe presence of entanglement [14,15]. By inserting a slowlight medium into one arm of one of the interferometers,we delay the photon probability amplitude traveling in thatpath. The delay can be measured by observing a shift in thelocation of the Franson fringes. Additionally, the reductionof entanglement can be observed via the reduction of fringevisibility and the elongation of the fringe envelope due togroup velocity dispersion of the photons in the slow lightmedium.

The energy-time entangled pair is generated with spon-taneous parametric down-conversion (SPDC). The unnor-malized SPDC quantum state for an extremely narrowband pump is given by

ji /Zd!fsfi!a

y1 !a

y2 !p !j0i; (1)

where ! describes the phase-matching constraints forcollinear down-conversion [16], and the functions fs!and fi! represent any filtering or dispersion (e.g., in

Beam splitter

Rubidium Vapor Cell Scanning Mirror

Single Photon Detector

Mirror δx

FIG. 1 (color online). Interferometric setup. Two unbalancedMichelson interferometers are used to measure the delay andpreservation of entanglement of a photon traversing the heatedRb vapor cell slow light medium.

PRL 100, 133602 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending4 APRIL 2008

0031-9007=08=100(13)=133602(4) 133602-1 © 2008 The American Physical Society

fiber) encountered by the signal and idler photons beforeentering the Franson interferometer.

By evolving the creation operators of the biphoton statethrough the Franson interferometer we arrive at

jFi /Zd!fsfi!e2i!p!t12 e2i!p!t22

e2i!t11 e2i!t21tn!!

ay1 !ay2 !p !j0i; (2)

where n! is the Rb index of refraction, tjk refers to thefree-space optical path length of the jth arm of the kthMichelson interferometer, and LRb=c indicates thelength of the Rb vapor cell. t x=c refers to the trans-lation of the mirror in the interferometer arm containingthe slow light medium as shown in Fig. 1.

In the Franson interferometer, the interference fringesarise from photons detected simultaneously. The interfer-ence is a result of the relative phase between the biphotonprobability amplitudes which represent the short-short [ss]and long-long [ll] interferometer paths. Consequently, wepost-select the simultaneous coincidences (by ignoring theshort-long and long-short cross terms) to arrive at

jFi /Zd!fsfi!1 e2if!tn1!ptg

ay1 !ay2 !p !j0i; (3)

where we have taken tj1 tj2 for simplicity, and wheret t2k t1k is the common path mismatch. When firstintroduced, the functions fs! and fi! represented onlythe dispersion and filtering prior to the interferometer;however, we note that the form of Eq. (3) indicates thatthey may also be taken to include any dispersion or filteringwhich occurs after the interferometer before detection.

Since the experimental coincidence window, 3 ns, ismuch larger than the biphoton correlation time, b ’685 fs, and smaller than twice the common path mismatch,2t ’ 6 ns, the ss=ll coincidence rate R is given by

R /ZZ

dt0dt00jh0jE2 t00E1 t

0jFij2: (4)

We have assumed that the detectors have a flat spectralresponse so that the field operators are defined as

E j t /Zd!jei!jtaj!j: (5)

The coincidence rate can then be reduced to

R /Zd!jfs!j

2jfi!j22!!; (6)

where

! 1 e4ni!! 2e2ni!!

cos2f!t nr 1 !ptg; (7)

with nr! Ren!, and ni! Imn!. Thephase-matching function ! sinc~ngs ~ngi!!s0Lc=2c, is given in terms of the crystal length,Lc, and the signal and idler crystal group indices, ~ngs and~ngi. In the case where the idler is sent through the slow lightmedium Eq. (6) still applies except that ! is replaced by!p !.

Examination of Eq. (6) and (7) reveals some remarkableaspects of the Franson interferometer. First, the visibilityand fringe peak of Franson interference fringes are inde-pendent of any dispersion experienced outside of the in-terferometer (e.g., in optical fiber). Only the absolute valueof fs! and fi! enter into Eq. (6). Consequently, in thefollowing, fs! and fi! will only represent spectralfiltering of the signal and idler photons. Secondly, as shownin Eq. (7), all orders of dispersion and attenuation intro-duced by the slow light medium contribute to the final formof the interference fringes.

Evaluation of the integral in Eq. (6) can be accomplishednumerically with a suitable model for the index of the Rbvapor, n!. Including collisional broadening, the Rb va-por index of refraction can be modeled by

n! 1X2

j1

Njjj2

20@

X4

k1

gjk!!jk ij c

;

(8)

where N and c are the number density and colli-sional broadening full-width-half-maximum (FWHM)linewidth at temperature T. j, gjk, !jk, and 2j cor-respond to the far detuned effective dipole moment,the relative peak strength, the resonance frequency,and the homogeneous FWHM linewidth, for the hyper-fine resonances of the j f1; 2g ! fD1; D2g linesof 87Rb and 85Rb (k f1; 2; 3; 4g ! fj87Rb; F 1i;j87Rb; F 2i; j85Rb; F 2i; j85Rb; F 3ig). In the pres-ent experiment, we use a pump laser centered at pump atp 388 nm and generate signal photons with a centerwavelength of s0 785 nm and idler photons centered ati0 767:2 nm. By filtering the signal photons with a3 nm filter we reduce the biphoton bandwidth to1.5 THz. Actual line shapes are more accurately modeledby a Voigt profile, but since all photons are very far detunedfrom resonance (for example, the signal photon is detunedfrom the D2 resonances by2:5 THz whereas the FWHMof the doppler valley is 0:7 GHz) we ignore the rapidlydecaying doppler contribution in the index model. Thecollisional broadening width is given by cT 20vrmsTNT where, 0 ’ 2 1014 cm2 is theRb-Rb spin exchange cross section. All calculations pre-sented in this Letter include collisional broadening.

Using the design spectrum of the 3 nm signal filter,fs!, we have calculated the Franson fringes for threescenarios: (a) LRb 0 cm, (b) LRb 10 cm with the idlerphoton (i0 767:2 nm) sent through the interferometer


133602-2

with the slow light medium, and (c) LRb 10 cm with thesignal photon (s0 785 nm) sent through the interfer-ometer with the slow light medium. Though far outside theresonances, the idler photon will experience some delaywhen sent through the slow light medium due to the non-constant index of refraction of the Rb vapor at the idlerwavelength. The fringe envelopes were calculated withT 210 C and can be seen in Fig. 2.

By angle tuning 3 mm of BBO cut for type-II collinearSPDC we generate photons at the signal and idler wave-lengths using a 100 mW cw pump laser at p 388 nmwhich has a bandwidth of 2 MHz. The signal and idlerphotons are separated at a polarizing beam splitter andcoupled into single mode optical fibers (SMF). As previ-ously mentioned, a 3 nm bandpass filter, fs!, centered at785 nm filters the signal photons before coupling into fiber.The idler photons are not filtered, fi! 1. Signal andidler photons then enter the fiber coupled Franson interfer-ometer which has a common path mismatch of t 3 ns.The long arm of interferometer 1 contains a 10 cm pyrexRb vapor cell heated to 200 C. The path length of thelong arm in interferometer 1 is scanned using a 10 nmresolution translation stage. To reproduce the envelope ofthe fringes the stage is scanned with 50 nm steps over1 m at intervals of 20 m. Near the center of the inter-ference fringes the scanning is continuous for 20 m witha 50 nm step size. The photons are detected with Perkin-

Elmer SPCM detectors (timing jitter 600 ps) for 4 s ateach location of the translation stage.

After the cell has been heated to a temperature of200 C, scenarios (b) and (c) discussed above are mea-sured. Scenario (a) is not measured experimentally becauseremoval of the pyrex vapor cell significantly alters themeasured optical path length. In scenario (b) the idlerphoton is sent to the interferometer with the slow lightmedium and the fringes are measured. Then the fibers arethen switched to measure scenario (c), where the signalphoton centered at 785 nm is sent into interferometer withthe slow light medium. Only polarization adjustment isused to rebalance the ss=ll count rates in the interferome-ter. The entire experiment is completed in120 min . Anyturbulence during that time will only affect the phase oversubmicron length scales; thus, the fringe envelope is mea-sured correctly even in the presence of interferometricinstabilities.

Figure 2 also shows the measured Franson interferencefringes. The visibility of the fringes for scenario (b), whenthe idler photon is sent through the interferometer contain-ing the slow light medium, is 97:0 4:4%. The biphotonFWHM correlation length is b ’ 685 fs as set by the 3 nmbandwidth of the 785 nm filter in the signal photon’s beampath. The absolute peak-to-peak delay of the fringes withthe slowed idler, scenario (b), relative to the fringes withthe slowed signal, scenario (c), is td ’ 900 fs, giving afractional delay of fd td=b 1:3. The fractional broad-ening, fb, can be estimated by relating the FWHM of theinterference fringes in scenario (b) and (c), which givesfb 1:4. The fringes resulting from sending the signalphoton through the interferometer with the slow lightmedium, scenario (c), exhibit a visibility of 80:0 4:8%,demonstrating the preservation of entanglement.

The reduction of entanglement can be attributed to dis-persive pulse broadening that arises as a result of the non-zero second and third order terms in an expansion of Rbindex of refraction. The signal photons are significantlydetuned from atomic resonances (2:5 THz), as notedabove, causing absorption and absorptive broadening tobe negligible. Dispersive pulse broadening can be drasti-cally reduced by centering the pulse at the location wherethe second order terms in the index vanish, and by reducingthe pulse bandwidth [17]. Though the fractional delaydecreases linearly with the bandwidth, broadening fallsoff cubically, ultimately allowing larger absolute as wellas fractional delays for narrow bandwidth pulses.

The largest absolute delay obtainable is limited by thecreation of Rb2 molecules which occurs at number den-sities upwards of N 1016 cm3, corresponding to T 350 C. A high percentage of Rb2 molecules changes theindex profile and significantly increases the absorption anddistortion of the delayed photon. With the signal wave-length at the zero second order dispersion wavelength,s0 788:4 nm, we calculate a maximum attainable delay

0 450 900 1350 18000

0.2

0.4

0.6

0.8

1

Nor

mal

ized

Coi

ncid

ence

s (a

rb. u

nits

)

δx/c (fs)

0

0.5

1

0

0.5

1

FIG. 2 (color online). Normalized interference data. The dot-ted and solid lines trace the calculated interference fringe enve-lopes for scenarios (a), (b), and (c), as noted in the text. Thenormalized measured coincidences for scenarios (b) and (c) areshown by the triangles and circles, respectively. Inset plots areclose up views of the measured fringes near the center of eachfringe envelope. The tick marks on the horizontal axes of theinset plots indicate the expected fringe period of p=c. Themaximum and minimum coincidence counts (M, m) for scenar-ios (b) and (c) are (998,15) and (631,70), respectively. Signal andidler singles rates (s, i) for scenarios (b) and (c) were (12,28) and(12,25) kHz, respectively.


133602-3

of roughly 60 ps at T 350 C. The fractional delay fd isconsequently limited by the bandwidth of the biphoton andthe loss of entanglement due to third order dispersivebroadening. A fractional delay of fd ’ 60 can be achievedwith a bandwidth of 2 nm; the visibility of the calculatedFranson interference fringes in this case falls to 71.8% at350 C as shown in Fig. 3(b). With bandwidths narrowerthan 1 nm, the visibility remains higher than 98% at350 C as can be seen in Fig. 3(a).

The delay mechanism demonstrated in this Letter maybe applied to any double resonance structures and corre-sponding biphoton bandwidths. The 200 MHz biphotondiscussed in Ref. [18] should allow delays on the order of100 ns (fractional delays of 20) with relatively smallpulse broadening as discussed in Ref. [17]. The cell tem-perature need only be 140 C in this case; however, theloss would be quite large due to an increased optical depthresulting from the narrow 6.8 GHz hyperfine splitting.Regardless, such loss will be approximately spectrallyflat for the narrow band photons leaving the entanglementlargely unaffected.

While generating energy-time entangled biphotons witha cw pump laser may be of limited use for many QIPapplications requiring well defined single photon wavepackets or other types of entanglement (for example, po-larization or transverse position-momentum entangle-ment), properly engineered double absorption resonanceslow light media can still be expected to work effectively.

These types of slow light media have been shown to slowweak coherent light from a pulsed source, as well as topreserve the transverse wave vectors in a simple opticalimage [19]. Spectrally dependant Faraday rotation whichwould reduce polarization entanglement can be eliminatedby passive and/or active magnetic shielding of the vaporcell.

In summary, we have demonstrated the tunable delay ofone member of a pair of entangled photons relative to theother while preserving their entanglement. We sent anensemble of 1.5 THz photons from energy-time entangledpairs into a hot Rubidium vapor and observed a shift of thefourth order correlation fringes by1:3 correlation lengthsusing the Rb D1 and D2 fine structure resonances. The80:0 4:8% visibility of the slowed fourth order correla-tion fringes violated a Bell inequality value by nearly 2standard deviations, demonstrating the preservation of en-tanglement of the biphoton and preservation of the quan-tum state of the slowed photon. We examined thelimitations of this particular method of slow light, calcu-lating an absolute delay of 60 ps and a fractional delay offd ’ 60 for a 1 THz photon, while still exceeding the Bellinequality value for the visibility of 70.7%.

This work was supported by DARPA DSO Slow Light,MURI Quantum Imaging, and by a DOD PECASE.

[1] E. Knill, R. Laflamme, and G. J. Milburn, Nature(London) 409, 46 (2001).

[2] T. B. Pittman, B. C. Jacobs, and J. D. Franson, Phys.Rev. A 66, 042303 (2002).

[3] T. B. Pittman and J. D. Franson, Phys. Rev. A 66, 062302(2002).

[4] P. M. Leung and T. C. Ralph, Phys. Rev. A 74, 022311(2006).

[5] M. D. Lukin, Rev. Mod. Phys. 75, 457 (2003).[6] D. N. Matsukevich and A. Kuzmich, Science 306, 663

(2004).[7] A. V. Gorshkov et al., Phys. Rev. Lett. 98, 123601 (2007).[8] J. Nunn et al., Phys. Rev. A 75, 011401(R) (2007).[9] S. Chen et al., Phys. Rev. Lett. 97, 173004 (2006).

[10] M. D. Eisaman et al., Nature (London) 438, 837 (2005).[11] J. D. Franson, Phys. Rev. Lett. 62, 2205 (1989).[12] P. G. Kwiat, A. M. Steinberg, and R. Y. Chiao, Phys.

Rev. A 47, R2472 (1993).[13] J. Brendel et al., Phys. Rev. Lett. 82, 2594 (1999).[14] J. S. Bell, Physics 1, 195 (1964).[15] J. D. Franson, Phys. Rev. A 61, 012105 (1999).[16] R. Erdmann et al., Phys. Rev. A 62, 053810 (2000).[17] R. M. Camacho, M. V. Pack, and J. C. Howell, Phys.

Rev. A 73, 063812 (2006).[18] Y. J. Lu and Z. Y. Ou, Phys. Rev. A 62, 033804 (2000).[19] R. M. Camacho, C. J. Broadbent, I. Ali-Khan, and J. C.

Howell, Phys. Rev. Lett. 98, 043902 (2007).

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 600.5

0.6

0.7

0.8

0.9

1

δx/c (ps)

Nor

mal

ized

Frin

ge E

nvel

ope

(arb

. uni

ts)

a)

b)

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Fringe Visibility

FIG. 3. Calculated interference fringe envelopes. Calculatednormalized fringe envelopes vs delay for (a) 1 nm and(b) 2 nm for temperatures T 278, 304, 320, 332, 342,and 350 C as compared to free-space fringes (left most enve-lope). The signal wavelength is at the zero second order disper-sion wavelength, s0 788:4 nm. The dashed lines correspondto the Bell inequality visibility (right axis) of 70.7% as discussedin the text. The points indicate the visibility (right axis) of thecorresponding envelope.


133602-4

142

Appendix E

A Slow-Light Fourier TransformInterferometer

The following journal article is reprinted with permission from Zhimin Shi,

Robert W. Boyd, Ryan M. Camacho, Praveen K. Vudyasetu, and John C. Howell,

“A Slow-Light Fourier Transform Interferometer”, Physical Review Letters 99,

240801 (2007). Copyright (2007) by the American Physical Society.

Slow-Light Fourier Transform Interferometer

Zhimin Shi* and Robert W. BoydThe Institute of Optics, University of Rochester, Rochester, New York 14627 USA

Ryan M. Camacho, Praveen K. Vudyasetu, and John C. HowellDepartment of Physics and Astronomy, University of Rochester, Rochester, New York 14627 USA

(Received 11 July 2007; published 11 December 2007)

We describe a new type of Fourier transform (FT) interferometer in which the tunable optical delaybetween the two arms is realized by using a continuously variable slow-light medium instead of a movingarm as in a conventional setup. The spectral resolution of such a FT interferometer exceeds that of aconventional setup of comparable size by a factor equal to the maximum group index of the slow-lightmedium. The scheme is experimentally demonstrated by using a rubidium atomic vapor cell as the tunableslow-light medium, and the spectral resolution is enhanced by a factor of approximately 100.

DOI: 10.1103/PhysRevLett.99.240801 PACS numbers: 07.60.Ly, 07.60.Rd, 42.62.Fi

Recently, there has been considerable interest in thedevelopment of slow- and fast-light techniques [1] to con-trol the propagation velocity of light pulses through amaterial system. While early work concentrated on tech-niques to realize very large [2,3], very small [4], or evennegative [5] group indices, recent work has been aimed atdeveloping practical applications of slow-light methods[6–11]. In addition to applications in optical communica-tion systems [1], it was recently shown that slow light canalso be used to enhance the spectral sensitivity of certaintypes of interferometers [12–14].

Fourier transform (FT) interferometry [15] is a powerfultechnique that has an intrinsically high signal-to-noise ratio(SNR) and can have high resolving power. These proper-ties have led to its many applications in biomedical engi-neering, metrology [16], astronomy, etc. A conventionalFT interferometer [see Fig. 1(a)] is typically comprised of afixed arm and a moving arm, both of which contain non-dispersive media (typically air) with refractive index n.The length of the moving arm can be changed to achieve avariable optical delay time (ODT) nL=c, where L isthe length difference between the two arms, and c is thespeed of light in vacuum. To resolve the spectrum of aninput optical field with center frequency , needs to betuned from zero to a maximum value max with a step sizecomparable to 1=. The spectral resolution is given bymin 1=2max [15]. To achieve a high spectral reso-lution, one needs a large device size [typically with theorder of c=2nmin] and a large number of data acquis-ition steps [determined by =2min] for eachmeasurement.

In this Letter, we propose and demonstrate a new type ofFT interferometer that uses a continuously tunable slow-light medium to realize a tunable group delay between thetwo arms [see Fig. 1(b)]. We first develop the theory for theideal case in which the slow-light medium has a uniformgroup index ng (defined by ng n dn=d) and thus nogroup velocity dispersion over the frequency range of

interest. The frequency dependence of the refractive indexof such an ideal tunable slow-light medium in the vicinityof a reference frequency 0 is given by

n n0 n0g

0

0; (1)

where 0 0 is the frequency detuning and n0g ng n is the relative group index. We assume that for sucha medium n0g can be varied continuously, for example, bychanging the number density of an atomic vapor, from zeroto a maximum value n0g;max. Note that 0 is a referencefrequency chosen such that n0 remains constant as n0g istuned. We consider a Mach-Zehnder (MZ) interferometerwith such a tunable slow-light medium of length L in onearm and a nondispersive reference medium of length L2

and refractive index n2 in the other arm. For simplicity, welet I jEj2. When the input field has multiple fre-quency components, the output intensity at each of the twoports of such a MZ interferometer (see Fig. 2) is given by

Iout; 1

4

ZIinje

ikn0n0g0=0L eikn2L2 j2d; (2)

where k 2=c is the wave number at frequency in

detector

input field

moving arm

fixed arm

detector

input field

fixed arms

δ0.5 L

0.5Lmax

)b()a(

tunable slow-light medium

beamsplitter

beamsplitter

FIG. 1 (color online). Schematic diagrams of (a) a conven-tional FT interferometer with one moving arm and one fixedarm; (b) a FT interferometer with a tunable slow-light medium inone of the two fixed arms.

PRL 99, 240801 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending14 DECEMBER 2007

0031-9007=07=99(24)=240801(4) 240801-1 © 2007 The American Physical Society

vacuum. Note that, in practice, both arms also containother nondispersive media such as beam splitters and air.However, the optical path lengths contributed from thesemedia are assumed to be balanced between the two armsand therefore are not shown in Eq. (2). When the two armsare balanced such that n2L2 n0L, one can rewriteEq. (2) as follows:

Iout; 1

4

ZIinje

ikn0g0=0L 1j2d

1

2Iin

1

2

ZIin cosk

n0g0

0Ld: (3)

By subtracting the two outputs, and approximating k by k0,one obtains the following relation:

I0out Iout; Iout; ZIin cos2

n0g0

cLd; (4)

where I0out is the modified output that can be directlymeasured by using a balanced homodyne detection method[17]. Note that, for a pulse with center frequency near 0,the relative delay between the two arms of the interfer-ometer is given by

g ngL

cn2L2

c ng n0

Lcn0gL

c: (5)

We assume that the incident field contains only fre-quency components that are larger than 0, as is in thecase of the experiment shown below. In this way, one canobtain the following inverse Fourier transform relation:

I0outg Z 11

Iin0 0 cos20gd0

ReZ 11

Iin0 0ei20gd0

; (6)

where Refg denotes the real part. Thus, one can retrieve theinput spectrum by applying a Fourier transform to theoutput intensity scan as a function of g and taking onlythe result with positive detuning 0 > 0. Note that expres-sion (6) is similar to that of a conventional FT interferome-

ter [e.g., Eq. (11.4) in Ref. [15] ], except that in the presentcase the Fourier conjugate pair is the detuning 0 and thegroup delay g instead of the absolute frequency and theODT .

In the ideal case in which the slow-light medium islossless, the spectral resolution of such a slow-lightFT interferometer is limited by the largest achievable groupdelay g;max to 1=2g;max c=2n0g;maxL. Sincen0g;max can be very large when a suitable slow-light mediumis used, the spectral resolution of the slow-light FT inter-ferometer can be enhanced by the significant factor ofn0g;max with respect to that of a conventional setup.Alternatively, for a specified spectral resolution , thedevice size can be decreased by a factor of n0g;max.

In many real situations, however, a slow-light mediumintroduces some frequency-dependent loss. In a tunableslow-light medium of the sort we consider, the ratio be-tween the absorption coefficient and the relative groupindex n0g at each frequency 0 0 remains constant as n0gis scanned. Thus, for an input of an infinitely sharp spectralline at 0 0, the magnitude of the output I0out willdecrease exponentially as n0g (i.e., g) becomes larger.The Fourier transform of such a decay will result in aLorentzian-shape spectral line centered at 0 0 withan effective linewidth of c=4n0g. Thus, the overall spec-tral resolution near 0 0 is given by

0 0 max

c2n0g;maxL

;c0

0

2n0g

: (7)

The total spectral range of such a FT interferometer isgiven by c=2n0gLwhere n0g is the step size of thechange in n0g. Note that our slow-light FT interferometerdoes not require any moving arms, which is advantageousin certain situations in which vibration and translationerrors of a moving arm could introduce measurementerrors and decrease the SNR.

The theory of Eqs. (1)–(6) can be extended to the moregeneral case in which the slow-light medium has an arbi-trary frequency dependence of the refractive index near 0

in the form of n n0 n0g=0f

0, where f0describes the normalized dispersion function near 0. Insuch a case, one can replace 0 by f0 in Eqs. (2)–(6) andobtain the following inverse FT relation:

I0outg0 ReZ 11

Iin0 0e

i2f0g0d0; (8)

where g0 is the group delay of a pulse centered at 0.Note that g0 can be determined from the group delay ofa pulse centered at any known frequency 0 throughthe relation g0 n0g0g0 0=n0g0 0. TheFourier transform of I0outg0 gives first the spec-trum Iin as a function of f0. When each value of f0corresponds to a unique value of 0 0 within the spec-tral range of interest, one can then map out the inputspectrum Iin0 0 from Iinf. The spectral resolution

polarization beam splitter (PBS)

PBS

PBS

AOM

beam splitter (BS)

BS

λ/2

λ/2

λ/2

-

MZM

Laser

Rb cell

heater + controller

AWGI-

I+

Iout=I+-I-'

FIG. 2 (color online). Experimental setup of the FT interfer-ometer using a rubidium vapor cell as the slow-light medium.


240801-2

near frequency 0 0 is given by 0 0

maxfc=2n0g;max0 0L; c0 0=2n0g0

0g, where n0g;max0 0 and 0 0 are the maxi-mum relative group index and the absorption coefficient ofthe medium at frequency 0 0, respectively.

We have constructed a slow-light FT interferometerpossessing a MZ geometry to demonstrate the propertiesof our proposed scheme (see Fig. 2). A 10-cm-long rubid-ium vapor cell is used as the slow-light medium. Thetunability of the group index is realized by controllingthe temperature and thereby the atomic number densityof the cell. A tunable continuous wave (cw) diode laseroperating at approximately 780 nm with a linewidth ofapproximately 100 kHz is used as the primary source. Anacousto-optic modulator (AOM) is used to produce a sec-ond cw field whose frequency is 80 MHz lower than theprimary field. The two fields are combined and used as theinput field. Balanced homodyne detection is used to mea-sure the output intensity I0out. For monitoring purposes, apart of the primary laser is directed through a Mach-Zehnder modulator (MZM), which is driven by an arbitrarywaveform generator (AWG) to produce a pulse train with4-ns pulse duration. By measuring the group delay gexperienced by such pulses of known frequency in prop-agating through the vapor cell, the value of n0g at theprimary frequency is obtained.

The refractive index of a rubidium vapor near the D2

transition lines can be approximated as [18,19]

n 1A2

X4

j1

gj j i

; (9)

where the four terms in the summation correspond to thefour major hyperfine transitions of the rubidium D2 lines[see Fig. 3], gj and j are the relative peak strength and thefrequency center of the jth resonance, respectively, 6 MHz is the homogeneously broadened linewidth of theRb resonances, and A is a coefficient determined by theatomic number density and the dipole transition moments.The frequency spacing between the centers of neighboringresonances from low to high frequencies are 1.22 GHz,3.035 GHz, and 2.58 GHz, respectively. The natural abun-dances of 87Rb and 85Rb are 28% and 72%, respectively;therefore, the relative peak strengths among the fourresonances are g1:g2:g3:g4 5=8 0:28:7=12 0:72:5=12 0:72:3=8 0:28. The transmission as afunction of detuning 0 through the vapor cell at a tem-perature of approximately 100 C is plotted in Fig. 3(a).The thick and thin curves show the measured data and thetheory of Eq. (9) (with A 1:14 106 Hz), respectively.The reference frequency 0 is chosen between the reso-nances of the 85Rb F 2! F0 and 87Rb F 1! F0

transitions so that n0 1 according to Eq. (9). Notethat the theory curve for the absorption, which is based onthe use of Eq. (9), fits the data very well in the wings of thelines but not near the resonances themselves. This is be-

cause Eq. (9) ignores the influence of Doppler broadeningand the resulting Gaussian line shape. Since Gaussian lineshapes decay much more rapidly in the wings than doLorentzian line shapes, Eq. (9) accurately describes theatomic response at the frequencies (the gray region inFig. 3) at which our measurements were performed. Thecalculated corresponding refractive index n, and groupindex ng are plotted in Figs. 3(b) and 3(c), respectively,as functions of 0.

The primary frequency of the input field is chosen to beapproximately 1.79 GHz higher than the reference fre-quency 0 so that the dispersion model of Eq. (9) can beused. At room temperature, the vapor pressure is practi-cally zero so that n0g 0. As the temperature rises, theatomic number density increases [i.e., A in Eq. (9) in-creases] and therefore n0g increases. In the experiment,the pulse delay g and the output intensity I0out are mea-sured simultaneously as the vapor cell is heated from roomtemperature to approximately 120 C in a time of approxi-mately 1 min. The maximum group delay is approximately40 ns, which corresponds to n0g;max 120 at the primaryfrequency.

Figure 4 shows the experimental data for the outputintensity I0out as a function of the group delay g at theprimary frequency. The interference pattern clearly showsthe beating between the two closely spaced spectral lines.Note that the envelope of I0out is not at a maximum when n0gapproaches zero, which is probably because the phasedifference (e.g., due to the coatings on the surfaces ofvarious optical elements in our setup) between the two

ν

FIG. 3 (color online). (a) Transmission, (b) refractive index n,and (c) group index ng of the 10-cm-long rubidium vapor cell atthe temperature of approximately 100 C as functions of detun-ing 0. The thick curve is the measured transmission, and the thincurves are the fitted theory using Eq. (9). The inset shows theenergy levels of the 87Rb and 85Rb D2 transitions, and the grayregion is the frequency region over which the spectral measure-ment are performed.


240801-3

arms is not the same for different frequency components ofthe input field even when n0g 0.

The input spectrum is retrieved through the FT relationof Eq. (8) and the mapping process described above. Theresult is plotted as the solid line in Fig. 5. The dotted line isthe actual input spectrum, and the dashed line shows thesimulated result which is obtained from the calculated I0out

using the actual input spectrum, the rubidium model ofEq. (9) and the assumption of a balanced, noise-free inter-ferometer. One sees that the experimental result accuratelyresolves the position and the profile of the input field. Thespectral resolution demonstrated in the experiment is ap-proximately 15 MHz. This value agrees with the simula-tion result and is limited by the absorption of our slow-lightmedium. In contrast, a conventional setup with an opticalpath length difference between the two arms limited to10 cm could produce a spectral resolution no better thanapproximately 1.5 GHz. Thus, through use of slow-lightmethods, we have enhanced the resolution by a factorapproximately equal to the maximum group index (100)of our slow-light medium.

In summary, we have proposed and experimentally dem-onstrated a new type of Fourier transform interferometerthat has two fixed arms with a tunable slow-light mediumin one arm. We have shown that in such a FT interferometerthe spectrum of the input field and the modified outputintensity as a function of group delay form a Fourier trans-form pair. Since the maximum group delay through a slow-light medium can be very large under proper conditions,such a slow-light FT interferometer can provide very highspectral resolution. Moreover, a slow-light FT interferome-ter might be expected to outperform a conventional FTinterferometer by eliminating instabilities and positioningerrors associated with the moving arm of a conventionaldevice.

This work was supported by the DARPA/DSO SlowLight program and by the NSF.

*[email protected][1] R. W. Boyd and D. J. Gauthier, in Progress in Optics,

edited by E. Wolf (Elsevier Science, Amsterdam, 2002),Vol. 43, pp. 497–530.

[2] S. E. Harris and L. V. Hau, Phys. Rev. Lett. 82, 4611(1999).

[3] M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Phys.Rev. Lett. 90, 113903 (2003).

[4] M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Science301, 200 (2003).

[5] G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski,and R. W. Boyd, Science 312, 895 (2006).

[6] Y. Okawachi et al., Phys. Rev. Lett. 94, 153902 (2005).[7] M. Herraez, K. Y. Song, and L. Thevenaz, Appl. Phys.

Lett. 87, 081113 (2005).[8] P. Ku, C. Chang-Hasnain, and S. Chuang, Electron. Lett.

38, 1581 (2002).[9] A. Schweinsberg, N. N. Lepeshkin, M. S. Bigelow, R. W.

Boyd, and S. Jarabo, Europhys. Lett. 73, 218 (2006).[10] R. M. Camacho, M. V. Pack, and J. C. Howell, Phys.

Rev. A 74, 033801 (2006).[11] R. M. Camacho, M. V. Pack, J. C. Howell, A. Schweins-

berg, and R. W. Boyd, Phys. Rev. Lett. 98, 153601 (2007).[12] S. M. Shahriar et al., in Quantum Electronics and Laser

Science Conference (QELS), Technical Digest (OpticalSociety of America, Washington, DC, 2005), paperJWB97.

[13] G. T. Purves, C. S. Adams, and I. G. Hughes, Phys. Rev. A74, 023805 (2006).

[14] Z. Shi, R. W. Boyd, D. J. Gauthier, and C. C. Dudley, Opt.Lett. 32, 915 (2007).

[15] P. Hariharan, Optical Interferometry (Elsevier Science,Amsterdam, 2003), 2nd ed., Chap. 11, pp. 173–187.

[16] A. Gomez-Iglesias, D. O’Brien, L. O’Faolain, A. Miller,and T. F. Krauss, Appl. Phys. Lett. 90, 261107 (2007).

[17] N. Walker and J. Carroll, Electron. Lett. 20, 981 (1984).[18] A. Banerjee, D. Das, and V. Natarajan, Opt. Lett. 28, 1579

(2003).[19] D. A. Steck, http://steck.us/alkalidata.

1.5 1.6 1.7 1.8 1.9 2.0 2.10

0.5

1

rela

tive

inte

nsity

(a.

u.)

experimentsimulationactual input

detuning ν' (GHz)

FIG. 5 (color online). Retrieved spectrum of the input fieldusing experimental data (solid line) and simulated data (dashedline) and the actual spectrum of the input field (dotted line). Theresolution of a conventional FT interferometer of the same sizewould be approximately 100 times worse.

τ

FIG. 4 (color online). Output intensity of the slow-light FTinterferometer as a function of the group delay g for an inputfield of two sharp spectral lines separated by 80 MHz.


240801-4

implementations of slow light using double-resonances

Documents