Dynamics of Electromagnetically Induced Transparency Optical Kerr ...

Dynamics of Electromagnetically 

Induced Transparency Optical 

Kerr nonlinearities 

by 

Michael Vernon Pack 

Submitted in Partial Fulfillment 

of the 

Requirements for the Degree 

Doctor of Philosophy 

Supervised by 

Professor John C. Howell 

Department of Physics and Astronomy 

The College 

Arts and Sciences 

University of Rochester 

Rochester, New York 

2007

To my wife, 

Janene Kaufman Pack, 

for all of her love and support. 

ii

iii 

Curriculum Vitae 

The author was born in Logan, Utah on the 23rd December, 1974. He graduated 

from Holbrook High School in 1993. He attended Brigham Young University 

from 1996 untill 2001 and graduated with a BS degree in physics and electrical 

engineering and an MS degree in electrical engineering. 

He worked at Sandia 

National Laboratory in Albuquerque, NM from 2001 to 2003. He came to the 

University of Rochester in the fall of 2003, and received a MA degree in physics in 

2004. Under the supervision of Dr. John C. Howell, he has performed his doctoral 

research in the fields of quantum optics and atomic, and optical physics.

CURRICULUM VITAE 

iv 

Publications 

M.V. Pack, P. K. Setu, and J.C. Howell, “Using slow light to improve quantum 

non-demolition measurements using electromagnetically-induced-transparencyenhanced 

Kerr nonlinearities” (submitted to Phys. Rev. A) 

M.V. Pack, R.M. Camacho, and J.C. Howell, “Transients of the 

electromagnetically-induced-transparency-enhanced refractive Kerr nonlinearity: 

Experiment” Phys. Rev. A (in press) 

M.V. Pack, R.M. Camacho, and J.C. Howell, “Electromagnetically induced 

transparency lineshapes for large probe fields and optically thick media” Phys. 

Rev. A (in press) 

R. M. Camacho, M. V. Pack, J. C. Howell, A. Schweinsberg, R. W. Boyd, 

“Wide-bandwidth, tunable, multiple-pulse-width optical delays using slow light 

in cesium vapor” Phys. Rev. Lett. 98, 153601 (2007). 

R. M. Camacho, M. V. Pack, and J. C. Howell, “Slow light with large fractional 

delays by spectral hole-burning in rubidium vapor” Phys. Rev. A 74, 033801 

(2006). 

M. V. Pack, R. M. Camacho, and J. C. Howell, “Transients of the 


Theory” Phys. Rev. A 74, 013812 (2006). 

R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light 

using two absorption resonances” Phys. Rev. A 73, 063812 (2006). 

M. V. Pack, D. J. Armstrong, and A. V. Smith, “Measurement of the χ (2) 

tensor of GdCa 4 O(BO 3 ) 3 and YCa 4 O(BO 3 ) 3 crystals” J. Opt. Soc. Am. B 22, 

417 (2005). 

C. D. Nordquist, A. Muyshondt, M. V. Pack, P. S. Finnegan, C. W. Dyck, 

I. C. Reines, G. M. Kraus, T. A. Plut, G. R. Sloan, C. L. Goldsmith, and C. 

T. Sullivan, “An X-band and Ku-band RF MEMS switched coplanar strip filter” 

IEEE Microw. Wireless Compon. Lett, 14, 425 (2004). 

M. V. Pack, D. J. Armstrong, A. V. Smith, “Second harmonic generation with 

focused beams in a pair of walkoff-compensating crystals” Opt. Comm. 221, 211 

(2003). 

M. V. Pack, D. J. Armstrong, and A. V. Smith, “Measurement of the χ (2)


v 

tensor of KTiOPO 4 , KTiOAsO 4 , KTiOPO 4 and KTiOAsO 4 crystals” Appl. Opt. 

43, 3319 (2004). 

M. V. Pack, D. J. Armstrong, and A. V. Smith, “Measurement of the χ (2) 

tensor of the potassium niobate crystal” J. Opt. Soc. Am. B 20, 2109 (2003). 

D. J. Armstrong, M. V. Pack, and A. V. Smith, “Instrument and method 

for measuring second-order nonlinear optical tensors” Rev. Sci. Inst. 74, 3250 

(2003). 

Michael V. Pack, Darrell J. Armstrong, Arlee V. Smith, Michael E. 

Amiet, “Second harmonic generation with focused beams in a pair of walkoffcompensating 

crystals” Opt. Comm. 221, 211 (2003). 

J. Peatross and M. V. Pack , “Visual introduction to Gaussian beams using a 

single lens as an interferometer” Am. J. Phys. 69, 1169 (2001).


vi 

Presentations 

M. V. Pack, R. M Camacho, and J.C. Howell, “Large fractional delays in hot 

atomic vapors” SPIE Photonics West, San Jose, CA **Invited** (January 2007) 

M. V. Pack, R. M Camacho, and J.C. Howell, “Low-light-level all optical 

switching” Frontier in Optics, Rochester, NY (Oct 2006) 

M. V. Pack, R. M Camacho, and J.C. Howell, “Dynamics of the EIT Kerr 

nonlinearity” Slow light OSA, Wasington, DC (June 2006) 


switching” CLEO/QELS, Baltimore, MD (May 2005) 


switching” SPIE Photonics West, San Jose, CA **Invited** (January 2005) 

M. V. Pack and D. Armstrong and A. V. Smith, “Measurements of the second 

order nonlinear tensors for KTiOPO 4 , KTiOAsO 4 , KTiOPO 4 , KTiOAsO 4 , KDP, 

and KNBO 4 ” OSA Annual Meeting, Tucsun, AZ (Oct. 2003) 

M. V. Pack and D. Armstrong and A. V. Smith, “The D-factory: Measuring 

the second order nonlinear tensor of crystals” CLEO/QELS, Baltimore, MD (May 

2003)

vii 

Acknowledgements 

There are many people who have made my time at the University of Rochester 

both enjoyable and enlightening, and I would like to thank them. 

First, I am thankful to my advisor John Howell for the things I have learned 

from him and the pleasant working environment he creates. 

I lucked into an 

advisor who’s philosophy is “happy cows give good milk”, and who’s genuine 

enthusiasm for physics and life is not exceeded by anyone I have met. This enthusiasm 

is witnessed by John’s inability to stay out of the lab in Rochester while he 

was on sabbatical in Italy. While the technical skills I have acquired at Rochester 

are important, the other aspects of research such as teamwork, public relations, 

and how to search out and formulate new research problems are the real gems I 

am taking with me. In my experience, John is singularly adept at teaching these 

skills, and I am thankful to have learned them from him. 

I have worked with a number of graduate students while at Rochester, and 

have benefited by our exchange of ideas and tricks of the trade. I have worked 

predominantly with Ryan Camacho with Ryan emphasizing the slow-light work, 

and me emphasizing the EIT Kerr nonlinearity work. In our frequent discussions

ACKNOWLEDGEMENTS 

viii 

of physics, Ryan has helped me sharpen my understanding and arguments by not 

ceding any argument until my ideas were fully developed. Ryan’s insightful questions 

have often launched our research in new directions. I am also thankful for 

similar interactions with other members of our research group including Praveen 

Setu, Irfan Ali-Khan, Curtis Broadbent, and Ben Dixon. 

I have also enjoyed collaborations with Kevin Wright and Nick Bigelow, as well 

as Aarron Schweinsberg and Bob Boyd. Each research group has had something 

unique to offer, and I am glad to have been able to work with three of the best. 

Finally, I would never have been in a position to start much less finish this 

thesis except for the support of my parents and my wife. My parents provided 

me with the opportunities which made this possible, and Janene, my wife, has 

helped me realized my goals with her encouragement and support. Janene has 

done the little things that have made many late nights at the lab and writing 

papers possible, and she has also done the big things such as proof reading this 

thesis. 

Thank you all.

ix 

Abstract 

This thesis presents theoretical derivations and experimental demonstrations 

of Kerr optical nonlinearities and their dynamics. The Kerr nonlinearity studied is 

based on electromagnetically induced transparency (EIT), and is known for having 

an unusually large cross-phase modulation (XPM) [1]. Also, quantum nondemolition 

(QND) measurements of photon number using the EIT Kerr nonlinearity are 

discussed. 

We first discuss the EIT line shapes in hot atomic vapors both with and without 

buffer gasses, and for the specific case of EIT line shapes on the D 1 line of 

rubidium. Using a Bloch-vector formalism to describe EIT, we derive new expressions 

describing the dynamical evolution of the EIT Kerr nonlinearity and the 

rise-time of the EIT Kerr effect. 

The theory of the EIT Kerr dynamics is confirmed by experiments using hot 

rubidium vapor in a buffer gas. These experimental observations confirm that the 

rise-time of the EIT Kerr effect is linearly proportional to the optical thickness of 

the EIT medium and inversely proportional to the width of the EIT transparency 

resonance.

ABSTRACT 

x 

We find that the EIT Kerr effect is relatively slow, and that its rise-time is 

inversely proportional to the EIT Kerr susceptibility. The implications of the slow 

rise-time to QND measurements is investigated, and it is found that by matching 

the group velocities of the probe and signal optical fields, the signal to noise ratio 

for QND measurements is improved. Predictions for the EIT Kerr dynamics with 

slow group velocity for the signal field were confirmed experimentally in rubidium 

vapor with group velocities of about 10 4 m/s for both probe and signal optical 

fields.

xi 

Table of Contents 

1 Introduction 1 

1.1 Introduction to Electromagnetically Induced Transparency . . . . 3 

1.2 EIT enhanced nonlinear optics . . . . . . . . . . . . . . . . . . . . 6 

1.3 EIT Kerr nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . 7 

1.4 Other research related to EIT Kerr nonlinearities . . . . . . . . . 10 

1.5 Applications of the EIT Kerr nonlinearity . . . . . . . . . . . . . 16 

1.5.1 Number resolving photon counters . . . . . . . . . . . . . 17 

1.5.2 Quantum state preparation . . . . . . . . . . . . . . . . . 17 

1.5.3 Quantum logic gates . . . . . . . . . . . . . . . . . . . . . 18 

1.6 Outline of dissertation . . . . . . . . . . . . . . . . . . . . . . . . 19 

2 Theory of Electromagnetically induced transparency 23 

2.1 Back of the envelope theory of EIT . . . . . . . . . . . . . . . . . 24 

2.2 Perturbative solution for EIT . . . . . . . . . . . . . . . . . . . . 28 

2.3 Bloch-vector solution for EIT . . . . . . . . . . . . . . . . . . . . 39

CONTENTS 

xii 

2.4 Transient Solution for EIT . . . . . . . . . . . . . . . . . . . . . . 43 

2.5 Doppler broadening . . . . . . . . . . . . . . . . . . . . . . . . . . 46 

2.6 Coherence preserving collisions in 

buffer gasses and wall coatings . . . . . . . . . . . . . . . . . . . . 52 

2.7 Magnetic sublevels and hyper-fine levels . . . . . . . . . . . . . . 56 

2.8 Miscellaneous experimental considerations . . . . . . . . . . . . . 62 

3 Theory of the dynamics of the EIT Kerr nonlinearity 67 

3.1 The EIT Kerr nonlinearity . . . . . . . . . . . . . . . . . . . . . . 69 

3.1.1 EIT Kerr dynamics . . . . . . . . . . . . . . . . . . . . . . 72 

3.1.2 Rise-times in optically thin media . . . . . . . . . . . . . . 74 

3.1.3 Rise times in optically thick media . . . . . . . . . . . . . 76 

3.2 QND measurements & Kerr nonlinearities . . . . . . . . . . . . . 80 

3.2.1 QND measurement and SNR . . . . . . . . . . . . . . . . . 81 

3.2.2 SNR for EIT Kerr Measurements . . . . . . . . . . . . . . 88 

4 EIT Kerr dynamics Experiment 93 

4.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 

4.2 CW measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 99 

4.3 Transients Measurements . . . . . . . . . . . . . . . . . . . . . . . 105 

4.4 Discussion of Measurements . . . . . . . . . . . . . . . . . . . . . 111 

5 EIT Kerr nonlinearity with slow signal group velocity 116

CONTENTS 

xiii 

5.1 Slow Light EIT Kerr nonlinearity . . . . . . . . . . . . . . . . . . 117 

5.2 Experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . 126 

5.3 Slow signal EIT Kerr measurements . . . . . . . . . . . . . . . . . 131 

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 

6 Conclusions and perspective 140 

Bibliography 147 

A Appendix 1: EIT on the rubidium D1 line 158 

A.1 Choice of Polarizations . . . . . . . . . . . . . . . . . . . . . . . . 158 

A.2 Transition matrix elements . . . . . . . . . . . . . . . . . . . . . . 160 

A.3 EIT in rubidium 85 . . . . . . . . . . . . . . . . . . . . . . . . . . 161 

A.4 important parameters for rubidium . . . . . . . . . . . . . . . . . 167 

A.5 EIT in rubidium 87 . . . . . . . . . . . . . . . . . . . . . . . . . . 168

xiv 

List of Tables 

Table Title Page 

4.1 Experimental parameters for EIT Kerr measurements . . . . . . . 98 

A.1 Comparison of EIT for several choices of polarization . . . . . . . 171 

A.2 Dipole matrix elements for Rb 85 D 1 line with F = 2 . . . . . . . . 171 



A.5 Dipole matrix elements for Rb 87 D 1 line with F = 2 . . . . . . . . 172

xv 

List of Figures 

Figure Title Page 

1.1 Level diagram–EIT Λ system . . . . . . . . . . . . . . . . . . . . 3 

1.2 Three-level EIT systems–Λ, V , and ladder . . . . . . . . . . . . . 5 

1.3 Various EIT Kerr systems . . . . . . . . . . . . . . . . . . . . . . 9 

1.4 Autler-Townes splitting–dressed state picture . . . . . . . . . . . . 11 

1.5 Nonlinear magneto-optical rotation . . . . . . . . . . . . . . . . . 14 

2.1 Transformation to bright-dark basis . . . . . . . . . . . . . . . . . 25 

2.2 Level diagram–three level atom . . . . . . . . . . . . . . . . . . . 29 

2.3 Transformation to dressed state picture . . . . . . . . . . . . . . . 33 

2.4 Probe absorption–large single-photon detuning . . . . . . . . . . . 35 

2.5 Doppler integration . . . . . . . . . . . . . . . . . . . . . . . . . . 36 

2.6 Error bounds–Perturbative calculation . . . . . . . . . . . . . . . 39 

2.7 Transformation from GSC to susceptibility . . . . . . . . . . . . . 46 

2.8 EIT line shapes for various circumstances . . . . . . . . . . . . . . 51

LIST OF FIGURES 

xvi 

2.9 EIT linewidth VS rate of VCCPC . . . . . . . . . . . . . . . . . . 55 

2.10 Clebsch-Gordon coefficients for Rb 8 5 D 1 line . . . . . . . . . . . . 58 

2.11 Bright Rabi frequencies Rb 8 5 . . . . . . . . . . . . . . . . . . . . 60 

2.12 Average optical pumping rate R ave . . . . . . . . . . . . . . . . . 61 

2.13 Four wave mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 

2.14 Visual definition of EIT terms . . . . . . . . . . . . . . . . . . . . 66 

3.1 Level diagram four-level atom . . . . . . . . . . . . . . . . . . . . 68 

3.2 Susceptibility dynamics . . . . . . . . . . . . . . . . . . . . . . . . 73 

3.3 Rise time VS Raman detuning theory . . . . . . . . . . . . . . . . 74 

3.4 EIT Kerr rise time (thick media . . . . . . . . . . . . . . . . . . . 79 

3.5 Simple QND diagram . . . . . . . . . . . . . . . . . . . . . . . . . 82 

4.1 Schematic of EIT Kerr dynamics of experiments . . . . . . . . . . 94 

4.2 Level diagram EIT Kerr dynamics experiment . . . . . . . . . . . 97 

4.3 Probe absorption measured (T=58 ◦ C) . . . . . . . . . . . . . . . 100 

4.4 Measured Stark shift due to signal . . . . . . . . . . . . . . . . . . 101 

4.5 Measured Stark shift due to coupling/probe . . . . . . . . . . . . 102 

4.6 Raw data homodyne signal–CW . . . . . . . . . . . . . . . . . . . 103 

4.7 Processed data homodyne signal–CW . . . . . . . . . . . . . . . . 104 

4.8 Raw data homodyne signal–transient . . . . . . . . . . . . . . . . 105 

4.9 Processed data homodyne signal–transient . . . . . . . . . . . . . 106 

4.10 GSC dynamics–measured . . . . . . . . . . . . . . . . . . . . . . . 108

LIST OF FIGURES 

xvii 

4.11 Rise and fall times (T=58 ◦ C) . . . . . . . . . . . . . . . . . . . . 109 

4.12 Rise and fall times (different Temperatures) . . . . . . . . . . . . 110 

4.13 Rise and fall times VS optical depth . . . . . . . . . . . . . . . . 111 

4.14 Comparison of theory and measured–GSC dynamics . . . . . . . . 112 

5.1 Theory for XPM with slow signal group velocity . . . . . . . . . . 120 

5.2 Diagram of slow signal experiment . . . . . . . . . . . . . . . . . 127 

5.3 Doppler valleys of Rb 85 and Rb 87 D 1 line . . . . . . . . . . . . . . 129 

5.4 Beam areas for slow signal experiment . . . . . . . . . . . . . . . 130 

5.5 Probe absorption for EIT in slow signal . . . . . . . . . . . . . . . 131 

5.6 Signal absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 

5.7 Probe phase for signal pulses of various duration . . . . . . . . . . 134 

5.8 Probe phase with slow and fast signal pulses . . . . . . . . . . . . 135 

5.9 Probe phase for various signal group velocities–theory . . . . . . . 136 

5.10 Width of probe phase pulse VS signal group velocity . . . . . . . 138 

6.1 EIT Kerr experiment using an optical cavity . . . . . . . . . . . . 143 


A.2 Dark states in rubidium 85 . . . . . . . . . . . . . . . . . . . . . . 164 

A.3 Dark states in rubidium 85 . . . . . . . . . . . . . . . . . . . . . . 165 

A.4 Rubidium 85 D lines hyperfine splittings . . . . . . . . . . . . . . 167 

A.5 Rubidium 87 D lines hyperfine splittings . . . . . . . . . . . . . . 168

xviii 


A.7 Dark states in rubidium 87 . . . . . . . . . . . . . . . . . . . . . . 170

1 

Chapter 1 

Introduction 

The field of nonlinear optics explores how the interaction between light and 

and a medium modifies the medium’s optical characteristics. For example, the 

index of refraction can change as a function of the intensity of the light, which 

is known as the Kerr nonlinearity. Other nonlinear effects include generation of 

harmonic frequencies, frequency mixing between multiple fields (e.g. 

sum and 

difference frequency generation), and phase conjugate mirror effects. Generally, 

these nonlinear effects are only significant for optical fields with large intensities. 

Since the invention of the laser, nonlinear optics has become an integral part of 

optical communications, remote sensing, spectroscopy, and quantum information 

science. 

Over the past decade, there has been increasing interest in realizing low-lightlevel 

nonlinear optics using coherently prepared media (CPM). By decreasing 

the intensity required to achieve nonlinear optical effects, many new applications 

become possible. One intriguing application is cross-phase modulation (XPM),

2 

where the Kerr nonlinearity is used to perform a quantum nondemolition (QND) 

measurement of photon number. For a long time the largest Kerr nonlinearities 

were still quite small (χ (3) ∼ 10 −22 m 2 /V 2 ) limiting QND measurements to intense 

fields with large numbers of photons. These QND measurements are of greatest 

interest for small numbers of photons where individual number states can be 

resolved. 

Using electromagnetically induced transparency (EIT), Schmidt and Imamoğlu 

[1] showed that it is theoretically possible to achieve Kerr nonlinearities that are 

ten orders of magnitude larger than conventional Kerr nonlinearities. Their analysis 

concluded, “This shows that our scheme makes possible conditional phase shifts 

of the order of π with single photons, which should be beneficial for quantum nondemolition 

measurements of weak signals and quantum logic gate operation....The 

principal result here is that one can obtain arbitrarily large XPM phase shifts of 

the probe field by arbitrarily weak signal fields.” 

[1] This conclusion is overly 

optimistic because it ignores the rise-time of the EIT Kerr effect. However, it provides 

a reasonable glimpse into the possibilities offered by low-light-level nonlinear 

optics. 

The focus of this work is to better understand the limitations and potential of 

the EIT enhanced Kerr nonlinearity by exploring the dynamics and rise-time of 

the EIT Kerr effect.

1.1. INTRODUCTION TO ELECTROMAGNETICALLY 

INDUCED TRANSPARENCY 3 

γ 

2 

3 

γ 

2 

2 

ΩP 

ΩC 

1 

Figure 1.1: Simple level diagram of an EIT Λ system with probe Ω p and coupling Ω c fields in 

single-photon resonance and Raman resonance. 

1.1 Introduction to Electromagnetically Induced 

Transparency 

Consider the three-level system shown in Fig. 1.1 with three electronic states |1〉, 

|2〉, and |3〉, and two dipole allowed transitions |2〉 ↔ |3〉 and |1〉 ↔ |3〉, that 

are resonant with a probe optical field with Rabi frequency Ω p and a coupling 

optical field with Rabi frequency Ω c . If the system is in Raman resonance (i.e. 

the frequency difference between the optical fields ω p −ω c is equal to the frequency 

difference between ground state energies ω 12 ≡ (E 1 − E 2 )/), there is coherent 

superposition of ground states called the dark state that is defined as 

|−〉 ≡ Ω p|1〉 − Ω c |2〉 

, (1.1) 

Ω



where Ω ≡ √ Ω 2 c + Ω 2 p is the bright Rabi frequency. This dark state is decoupled 

from the excited state because probability amplitude for the electronic transition 

|2〉 → |3〉 destructively interferes with probability amplitude for the transition 

|1〉 → |3〉. Once in the dark state an electron is “trapped” and cannot be excited 

to state |3〉 making the medium transparent to the optical fields. 

There is also a superposition of ground states orthogonal to the dark state 

called the bright state |+〉 which is coupled to the excited state with Rabi frequency 

Ω. Once an atom is excited from the bright state to level |3〉 it spontaneously 

decays into either the bright or dark states with equal probability. This 

results in coherent preparation and eventual trapping of atoms in this dark state. 

This phenomenon was first observed in the late 1970’s by Alzetta et al. [2, 3], 

and was given the name of coherent population trapping (CPT) by Gray et al. 

[4]. This phenomenon is also commonly referred to as electromagnetically induced 

transparency (EIT), which is a term first coined by Harris in 1990 [5]. 

EIT can be found in many different types of electronic level structures with 

different number of levels [6]. Figure 1.2 shows examples of each of these systems 

with with level |2〉 being the “excited” state, and the dark state given by Eq. (1.1). 

EIT typically implies the assumption that the probe field is much smaller than 

the coupling field (Ω c ≫ Ω p ). Although this assumption is often convenient for 

simplifying the mathematics, it is not necessary. In this work we discuss Λ-type 

EIT systems exclusively.



3 

2 

1 

Ω C 

1 

Ω 

2 

P 

Ω 

1 

C 

Ω P 

3 

Ω C 

Ω P 

2 

3 

(a) (b) (c) 

Figure 1.2: Level diagrams for three different typed os EIT systems; (a) Λ, (b) V , and (c) ladder. 

Both the Λ and V systems will be Doppler free when the coupling and probe fields copropagate. 

The ladder system is Doppler free when the coupling and probe fields counter propagate. 

In the past 15 years EIT have developed into a very active research area. 

Much of this work in the early to mid 1990’s centered on using EIT for lasing 

without inversion (LWI). In LWI, EIT suppresses stimulated absorption but not 

stimulated emission such that when population is injected into the excited state 

gain is achieved without inversion [7–11]. EIT has also been applied to various 

precession measurement tasks such as magnetometery [12–15], spectroscopy [16, 

17], and improving frequency standards [18]. The value of EIT for precession 

measurement is due to the fact that the transparency resonance can be very 

narrow {e.g. an EIT linewidth with a full width at half maximum (FWHM) of 

20-100 of Hz is not uncommon [19, 20]}, making EIT a very sensitive meter for 

the Raman detuning and any environmental changes which affect it. 

The narrow transparency resonance of EIT has a correspondingly steep dis-

1.2. EIT ENHANCED NONLINEAR OPTICS 6 

persion curves as dictated by the Kramer-Krönig relations. This steep dispersion 

results in slow group velocities for pulses with frequencies near the EIT transparency 

resonance. In 1999 several groups used the steep dispersion of EIT to 

observe the slowing of light by seven orders of magnitude, reducing the group velocity 

to 10’s of meters per second [21–23]. Using similar EIT slow-light techniques 

several groups “stopped” a light pulse ∗ , and holographically stored the light for a 

short time before coherently retrieving the pulse [24–26]. In recent work, Longdell 

et al. have achieved storage times of up to ten second for light pulses in EIT media 

[27]. The incorporation of EIT and a CPM have been used to create single 

photon states, store these photons, and then retrieve them while preserving their 

quantum properties [28]. 

1.2 EIT enhanced nonlinear optics 

Apart from the EIT Kerr nonlinearity, which we will discuss separately, there 

have been many advances in using EIT to provide more efficient low-light-level 

nonlinear optical media. The field of EIT enhanced nonlinear optics began in the 

mid 1990’s only a few years after the first experiments in EIT. In 1995 Hemmer et 

∗ In stopped light there are no photons stored in the medium, only a holographic imprint of 

the pulse stored in the coherence of the medium. The term stopped light really refers to stopping 

a dark state polarition (a quasi-particle consisting of a superposition of photon and exciton and 

enabled by the dark state of EIT). The dark state polariton is stopped by adiabatically turning 

off the cw coupling field, which continuously slows down the probe field until it “stops”. When 

the coupling field goes completely to zero such that the dark state polariton is stationary, there 

are no photons in dark state polariton, which consists entirely of the coherence between ground 

states. Thus, “stopped” light is essentially the same as a hologram.

1.3. EIT KERR NONLINEARITY 7 

al. demonstrated that EIT could be used to realize efficient phase conjugate effects 

[29], and in 1996 several groups demonstrated efficient frequency conversion 

via EIT [30, 31]. Other subjects that have been studied in connection with EIT 

include coherent Raman scattering [32], four-wave mixing [33–36], generating single 

photons [28, 37], generating correlated photons [38–40], and number squeezing 

of optical fields [41]. 

1.3 EIT Kerr nonlinearity 

The EIT Kerr nonlinearity was originally proposed by Schmidt and Imamoğlu 

in 1996 [1]. The Schmidt-Imamoğlu proposal adds a fourth energy level and an 

additional optical field, labeled the signal field, to the Λ EIT system in Fig. 1.1, 

transforming the Λ into the shape of an N [see Fig 1.2(a)]. 

The signal field 

induces an AC Stark shift † that perturbs the ground states perturbing of the EIT 

system away from Raman resonance. The combination of Stark shift and steep 

dispersion due to EIT result in a significant change in the index of refraction, 

and the resulting XPM of the optical fields is the EIT Kerr nonlinearity. The 

advantage of the EIT Kerr effect is that for continuous wave (cw) fields, it can 

be arbitrarily large by either increasing the optical thickness of the EIT medium 

or by decreasing the EIT linewidth making the dispersion steeper. For pulsed 

† The AC Stark shift is also known as the light shift and the dynamic Stark shift.


applications, which are the focus of this work, the dynamics of the EIT Kerr 

nonlinearity limit the XPM [42]. 

Similar to the refractive Kerr nonlinearity, Harris and Yamamoto proposed a 

scheme for an absorptive EIT Kerr nonlinearity in which the EIT medium “absorbs 

two photons, but does not absorb one photon.” [43]. Although there are 

technically two EIT Kerr nonlinearities (i.e. absorptive and refractive), in this 

work we typically refer to the refractive EIT Kerr nonlinearity as the EIT Kerr 

nonlinearity. While we always explicitly refer to the absorptive EIT Kerr nonlinearity 

as being absorptive. 

While both EIT Kerr nonlinearities have been observed in cw conditions [44– 

48], only the dynamics of the absorptive Kerr nonlinearity have been observed 

and characterized [49–51]. The experiments described in this work are the first 

observations of the dynamics of the refractive EIT Kerr nonlinearity. 

Several other schemes for EIT Kerr nonlinearities followed the original 

Schmidt-Imamoğlu proposal. 

A few of these EIT Kerr schemes are shown in 

Fig. 1.3. The original four-level N systems proposed by Schmidt and Imamoğlu is 

shown in Fig. 1.3(a) [1]. Matsko et al. suggested that M-type systems like the one 

shown in Fig. 1.3(b) also exhibit large EIT Kerr nonlinearities [52, 53]. The tripod 

scheme shown in Fig. 1.3(c) was proposed by Petrosyan and Malakyan [54, 55]. 

Harris and Hau showed that the nonlinear interaction between probe and signal 

fields in an N system is limited by temporal walk-off. The temporal walk-off


(a) 

(b) 

Ω S 

ΩS 

Ω C 2 

Ω 

P 

Ω 

C 

Ω Ω 

P C 1 

(c) 

(d) 

Ω P 

Ω C 

Ω Ω C 

1 

P 

Ω S 

Ω S 

Ω C2 

Ω S 

Atomic 

Species #1 

Atomic 

Species #2 

Figure 1.3: Level diagrams for various systems exhibiting EIT Kerr nonlinearities; (a) N system, 

(b) M system, (c) a tripod system, and (d) the matched group velocity system.

1.4. OTHER RESEARCH RELATED TO EIT KERR 

NONLINEARITIES 10 

is the result of the different group velocities of the probe and signal pulses. The 

probe pulse propagates with a slow group velocity due to EIT ‡ . While, the signal’s 

group velocity is close to c the speed of light in vacuum [56]. To overcome 

this temporal walk-off and increase the interaction length several research groups 

proposed double Λ schemes, where the signal field would also be part of an EIT 

system and propagate with a slow group velocity [57, 58]. Figure 1.3(d) shows a 

diagram for the double Λ proposal by Lukin and Imamoğlu, in which the signal 

EIT is obtained using a second atomic species [57]. Petrosyan and Kurizki have 

also proposed a slightly more elegant double Λ scheme that requires only a single 

atomic system [58]. 

1.4 Other research related to EIT Kerr nonlinearities 

In addition to the research discussed above there has been a tremendous amount of 

related work that provides valuable context for our research. Due to the sheer volume 

of work related to quantum coherence and coherent transients in multi-level 

systems, our review cannot be comprehensive. We focus on the most important 

historical developments and those experiments that are closely related to the EIT 

Kerr effect. 

‡ The coupling field is assumed to be cw.



2 

ΩP 

3 

Undressed 

Picture 

ΩC 

1 

hΩ C 

2 

Ω P 

Dressed State 

Picture 

3' 

1' 

Figure 1.4: Transformation to the dressed state picture in which the atom is dressed by the 

coupling field. In the dressed state picture the Autler-Townes splitting of the excited state 

becomes evident. 

First, there are primarily three historical ideas that laid the framework for 

EIT; the Hanle effect [59], Fano interference [60], and Autler-Townes splitting 

[61]. The field of quantum coherence and interference dates back to 1921 with the 

Hanle effect and quantum beats. The Hanle effect occurs when a medium with a V 

shaped electronic structure [e.g. see Fig. 1.2(b)] absorbs linearly polarized light. 

When a magnetic field Zeeman shifts the excited states, the light emitted from the 

medium will have a precessing linear polarization [59]). Fano interference occurs 

when two ionization pathways (autoionization and photoionization) interfere to 

inhibit absorption and ionization. [60]. The difference between EIT and Fano 

interference is that, in Fano interference the medium starts in a single electronic 

level not a coherent superposition of atomic levels. 

Finally, Autler-Townes splitting is very closely related to EIT [61]. Autler- 

Townes splitting can be understood by considering any of the three level systems



shown in Fig. 1.2. When the atom is dressed by the coupling field, there are two 

dressed states |2 ′ 〉 and |3 ′ 〉 as shown in Fig. 1.4. These two dressed states are an 

Autler-Townes doublet with an energy difference of ∆U = |Ω c | = µ|E c |, where 

µ is the dipole moment and E c is the amplitude of the coupling electric field. In 

one paradigm for EIT, EIT is interpreted as the destructive interference between 

the Autler-Townes doublet. In three-level systems, like the one shown in Fig. 1.4, 

the distinction between EIT and Autler-Townes splitting is somewhat artificial 

because there is no way of isolating the two effects. 

In optical dense media where radiation trapping is significant it is impossible 

to use CPT to obtain EIT because CPT requires spontaneous emission [62–65]. 

Radiation trapping causes the spontaneously emitted photon to be recaptured and 

reradiated several times before leaving the region of EIT resulting in prohibitive 

decoherence and destroying EIT. In these cases stimulated rapid adiabatic passage 

(STIRAP) can be used to obtain EIT [62, 63]. In STIRAP the medium starts in 

a non coherent dark state (i.e. it starts in a single electronic state) and the fields 

are evolved adiabatically such that the system remains trapped in the dark state 

as it evolves to a coherent dark state [66, 67]. 

There is also a great deal of theoretical research studying EIT in cavities. 

There are two reasons why EIT in cavities is interesting. First, like EIT enhanced 

nonlinearities, cavities can help increase the strength of nonlinear optical inter-



actions [68–70]. 

Some researchers, mostly at the University of Arkansas, have 

performed experiments with EIT in cavities to study optical bistability [46]. 

The propagation of optical pulses in EIT media has been studied extensively, 

but it seems this field is still far from maturity. We have already mentioned that 

EIT can be used to obtain very slow group velocities. When the coupling field is 

cw and the probe field is pulsed, the probe pulse will propagate at a group velocity 

v g,p ≈ 

c 

1 + χ p /2 + ω p 

2 

∂χ p 

, 

∂ω p 

where ω p is the probe frequency, and χ p ≪ 1 is the probe susceptibility. However, 

for matched pulses § in an EIT media, all fields propagate with group velocities near 

c the speed of light in vacuum [71]. Also, when two pulses with sufficiently energy 

enter a coherently prepared EIT media, they will naturally evolve towards matched 

pulses[72]. There is a dark area theorem for EIT similar to the area theorem of 

self induced transparency that describes certian aspects of this evolution towards 

matched pulses [73]. Various solutions for temporal soliton have also been found 

for EIT media; simultons [74, 75], adiabatons [76–79], etc. [80–82]. Some of these 

solitons propagate at a slow group velocity, while others propagate with group 

velocity near c [81, 82]. For the EIT Kerr nonlinearity spatial Thirring solitons, 

have been predicted [83]. 

§ In matched pulses both the probe and coupling fields are pulsed with the same field amplitudes 

and phase modulation– i.e. Ω p (t) = κΩ c (t), where κ is a constant of proportionality.



F e = 1 

Ω (σ ) 

c 

+ 

Ω p 

(σ −) 

Applying 

B-field 

F g = 1 

m = -1 m =0 m = +1 

(a) 

(b) 

µ B B z 

Figure 1.5: Level diagram exhibiting the relation between nonlinear magneto-optic rotation and 

electromagnetically induced transparency. Right- and left-hand circular polarized light interact 

with an atom with a ground hyper-fine level F g = 1 and excited hyperfine level F e = 1. In the 

absence of a magnetic field (a) Raman resonance is established leading to coherent population 

trapping. When a magnetic field is applied (b) Zeeman splitting shifts the ground (excited) 

magnetic sublevels by mµ B,g B z (mµ B,e B z ), where mu B,i is the magnetic moment for level i 

and B z is the magnitude of the magnetic field which is directed along the direction of optical 

propagation. 

EIT-Kerr dynamics are also closely related to a considerable amount of work 

that has been performed under the broad umbrella of coherent transients in multilevel 

systems. Park et al. and Chen et al. have observed absorption transients in 

Λ-EIT media induced by rapid changes to the Raman (two-photon) detuning [51, 

84]. Similarly, Godone et al. have observed transients due to phase modulation 

of Raman beams in a Λ system [85]. Many others have also studied transients in 

three level EIT systems under a number of different contexts such as: coherent 

Raman beats [86], Zeeman splitting [87–89], EIT in semiconductors [16, 90] and 

coherent population trapping in closed loop systems [91, 92]. 

Nonlinear magneto-optical rotation (NMOR) shares many similarities with 

EIT. In fact there are some experiments that can either be explianed either as EIT



or as NMOR with different insights coming from the different perspectives. For 

example, consider the atomic level structure shown in Fig. 1.5(a). The probe and 

coupling fields are equal strength with the same frequency and orthogonal circular 

polarizations. 

Taken together, the probe and coupling field can be considered 

as one field with linear polarization, and the angle of the linear polarization is 

determined by the relative phase between the probe and coupling field. 

CPT 

causes the atom to become trapped in the dark state |−〉 = 1/ √ 2(|g, −1〉 + 

|g, +1〉) . Non-zero Raman detunings can be achieved using a magnetic field in 

the direction of optical propagation to Zeeman shift the magnetic sublevels [this 

is shown in Fig. 1.5(b)]. Away from Raman resonance the coupling and probe 

fields experience phase shifts with opposite signs rotating the linear polarization 

of the combined probe/coupling field. This rotation of the optical polarization is 

what is meant by NMOR. 

One of the slowest group velocity measured using EIT was in a NMOR experiment 

[21]. In this experiment, which used a room temperature rubidium vapor 

with paraffin coated walls, the atoms were allowed to establish equilibrium with 

a linearly polarized field. The polarization angle was then modulated; in the language 

of EIT one would say the probe field was phase modulated. The phase 

For this system the Clebsch-Gordon coefficients for the σ ± polarized light is ∓1/ √ 2. This 

difference in sign for the two circular polarizations accounts of the pulse sign in the dark state 

superposition.

1.5. APPLICATIONS OF THE EIT KERR NONLINEARITY 16 

modulation of the probe and coupling fields propagated through the medium at 

8 m/s.For a review of NMOR see Budker [93]. 

There are also several reviews of CPT and EIT [6, 94–96] and its subfields 

[66, 93, 97–100]. 

1.5 Applications of the EIT Kerr nonlinearity 

As we have already mentioned there are several applications that would benefit 

from larger Kerr nonlinearities. For example, there are classical applications 

such as all optical switching for fiber communication networks. 

In this work 

we have focused more on the quantum and few photon applications of the EIT 

Kerr effect. In particular we focus on the possibility of QND measurements of 

photon number with single photon resolution. The ability to detect and resolve 

individual optical number states would have several uses in quantum information 

processing (QIP) including; long-distance quantum teleportation [101], Bell-state 

measurements [102], quantum bit regeneration [103], optical Fock state synthesis 

[104], multiphoton correlation measurements [105], simultaneous closure of all of 

Bell’s inequalities loopholes [106, 107], and better-than-diffraction-limited photolithography 

[108]. We elaborate on a couple of these potential uses of few-photon 

Kerr nonlinearities.


1.5.1 Number resolving photon counters 

QND measurement of photon number may be the best way to improve upon the 

current state of photon counters/detectos and satisfy the demanding requirements 

of quantum computing and QIP. The current state of the art photon counters are 

Gieger-mode avalanche photodiodes with quantum efficiencies in the range of 0.2- 

0.7, and these detectors cannot distinguish between one photon and many photons. 

Also, most optical QIP experiments use weak coherent states and post-selection. 

Because weak coherent states have finite probabilities of having multiple photons, 

these experiments are operated at low photon flux (i.e. coherent states with much 

less than one photon on average) to minimize the number of multi-photon states 

post-selected as single photon states. By being able to distinguish between 1, 2 

and more photons, a QND measurement would enable QIP experiments to operate 

with higher photon flux. 

1.5.2 Quantum state preparation 

Kerr nonlinearities are helpful for quantum state preparation in two ways: as 

projective measurements and for entangling two previously uncorrelated wavepackets. 

QND measurements project the signal field into an eigen state of the 

measurement basis. For Kerr QND measurements the measurement basis is the 

number states. Thus, Kerr QND measurements provide a method for obtaining 

optical number states form any optical source (thermal or laser). Deterministic


optical quantum computing requires single photons on demand. Being able to 

preselect single photon states using a QND measurement would go a long was 

towards satisfying this demand. EIT Kerr nonlinearities can also entangle two 

previously uncorrelated wavepackets [57, 58]. This is accomplished using a QND 

measurement without a classical measurement of the probe/meter field. Finally, 

Kerr QND measurements could be used to create macroscopic quantum states 

(i.e. Schrödinger cat states) [96]. 

1.5.3 Quantum logic gates 

In their original paper Schmidt and Imamoğlu speculated the the EIT Kerr effect 

“should be beneficial for...quantum logic gate operation” [1]. This is an important 

application from the perspective of optical quantum computing because in 

many ways photons make ideal qubits. They experience little decoherence. It is 

straightforward to move photons between nodes of a quantum network or computer. 

Entangled photons are straightforward to obtain (i.e. via parametric down 

conversion). Finally, single qubit quantum logic operations are trivial (e.g. a 

Hadamard gate is simply a 50/50 beam splitter, and a phase shift is a wave 

plate). However, deterministic two-qubit quantum logic operations using photons 

have been elusive because photon-photon interactions are so weak (i.e. traditionally 

nonlinear optics has required intense optical fields). Linear optics quantum 

computing (LOQC) somewhat circumvents this issue using probabilistic two-qubit

1.6. OUTLINE OF DISSERTATION 19 

quantum gates, but LOQC has its own set of issues which are beyond the scope 

of this work [109, 110]. 

Several architectures based on optical Kerr nonlinearities have been proposed 

for quantum computing [111–115]. For example, a Kerr nonlinearity that is strong 

enough to induce a π phase shift with a single photon could be used as a controlled 

phase gate. Even a weak Kerr nonlinearity (i.e. much less π phase shift per 

photon) can be used to implement a CNOT gate [116]. 

Several groups have 

predicted high fidelity quantum logic gate operation using EIT Kerr nonlinearities 

[117–119]. Although, Shapiro found that a fully quantum theory of Kerr effect 

predicts poor fidelity for Kerr based quantum logic gates [120, 121], the fidelity 

of Kerr based quantum logic gates is still being debated. 

Even if high fidelity Kerr quantum logic gates are not possible, the EIT Kerr 

effect may still benefit quantum computing by enabling improved single photon 

sources and detectors. 

1.6 Outline of dissertation 

The most intriguing applications of the EIT Kerr nonlinearity are pulsed applications. 

Thus, a good understanding of the dynamics of the EIT Kerr nonlinearity 

is essential for successfully realizing these applications, and understanding what 

are the limitations of the EIT Kerr nonlinearity. 

Most papers discussing the EIT Kerr nonlinearity and its applications ignore


the rise-time of the EIT Kerr effect (i.e. treat it as though it is instantaneous). For 

example, Schmidt and Imamoğlu overlooked the rise-time of the Kerr nonlinearity 

when they predicted that the EIT Kerr effect “makes possible conditional phase 

shifts of the order of π with single photons” [1]. When the dynamics are considered, 

the largest phase shift per photon that can be achieved is about 1/100th of a 

radian even under the most ideal conditions. Overlooking the EIT Kerr rise-time 

is common even in papers explicitly discussing dynamical properties of pulsed 

EIT Kerr systems (see for example [56, 57, 122]). These “transient” analysies 

limit their discussion to group velocities and transmission bandwidths. Recently, 

there has been greater attention given to dynamics of the EIT Kerr nonlinearity 

[42, 120, 123]. 

One of the advantages of the EIT Kerr nonlinearity is that for cw applications, 

it can be arbitrarily large. 

The EIT Kerr effect can be made larger by either 

decreasing the linewidth of the EIT resonance or increasing the medium’s optical 

depth αL, both of which are theoretically boundless (α is the absorption coefficient 

and L is the thickness of the medium). However, for pulsed applications the correct 

figure of merit is the ratio between the size of the EIT Kerr effect and its rise-time. 

As we will show in this work, the EIT Kerr rise-time is directly proportional to 

the optical depth and inversely proportional to the EIT linewidth. Thus, the ratio 

between the size of the EIT Kerr effect and its rise-time is a constant with respect 

to both the EIT linewidth and optical depth.


The objective of this dissertation is to first explore the dynamics of the EIT 

Kerr effect, and consider how these dynamics relate to QND measurements of 

photon number based on the EIT Kerr effect. 

Chapter 2 is a theoretical discussion of EIT. We begin by considering the 

experimentally naive scenario of a three-level Λ EIT system with a single velocity 

class. We review two methods for calculating the steady state susceptibilities in 

EIT; the perturbative and Bloch-vector methods. The Bloch-vector method is 

then applied to the problem of transient solutions of EIT. Next, we generalize the 

three-level Bloch-vector solution to include Doppler broadening, buffer gas, and 

multiple magnetic sublevels in anticipation of our experiment. 

Chapter 3 is a theoretical discussion of the EIT Kerr nonlinearity and QND 

measurements using Kerr nonlinearities. We first discuss the cw aspects of the 

EIT Kerr nonlinearity before deriving expressions for its dynamical evolution. 

Next, we consider QND measurements of photon number using Kerr nonlinearities, 

and derive expressions for the signal to noise ratio for these QND measurements. 

This analysis of the EIT Kerr dynamics and QND measurements illustrates the 

limitations of the EIT Kerr effect due to its relatively slow rise-time. 

Chapter 4 describes our experimental observations of the EIT Kerr dynamics. 

The experiment is performed using rubidium 85 atoms with buffer gas in a room 

temperature vapor cell. The measurements of EIT Kerr dynamics and rise-time


show agreement between theory and experiment for variations in optical thickness, 

EIT linewidth, and size of the Stark shift. 

In Chapter 5 we consider a matched group velocity experiment designed to 

overcome some of the limitations imposed by the slow EIT Kerr rise-time. This is 

similar to the Lukin-Imamoğlu double Λ proposal shown in Fig. 1.3(d) [57]. We 

first derive a simple theory which accounts for signal and probe pulse propagation, 

EIT Kerr interaction, and EIT Kerr dynamics. Then, we present experimental 

observations of an EIT Kerr system with a slow group velocity for the signal field. 

Finally, we conclude by summarizing our results and their implications for 

potential EIT Kerr applications. We also discuss open questions related to this 

work and future directions.

23 

Chapter 2 

Theory of Electromagnetically 

induced transparency 

Electromagnetically induced transparency (EIT) is the result of quantum interference 

between multiple transitions to a single final electronic state. In order 

to obtain this interference the initial electronic states must be mutually coherent 

(i.e. the medium is coherently prepared). 

In this chapter we provide a detailed mathematical description of EIT in various 

systems and circumstances. The first three sections (i.e. Secs. 2.1-2.3) 

present steady state solutions for EIT assuming a single velocity class ∗ and the 

three-level system shown in Fig. 2.2. Section 2.1 provides a physical intuitive 

“back-of-the-envelope” discussion of EIT, but is of little utility for comparison 

with experiments. The perturbative solution in Sec. 2.2 also provides valuable 

insight, but is limited by the assumption of small probe fields (i.e. Ω p ≪ Ω c ). Fi- 

∗ Room temperature atomic vapors have multiple velocity classes due to Doppler broadening. 

A velocity class is defined as the collection of atoms whose velocity along the direction of laser 

propagation lie in the range v 1 < v ≤ v 2 , such that the doppler shift associated with frequency 

difference ∆v = v 2 − v 1 is equal to the atoms homogeneous linewidth γ (i.e. γ = ω 0 ∆v/c).

2.1. BACK OF THE ENVELOPE THEORY OF EIT 24 

nally, the Bloch-vector solution in Sec. 2.3 is too complicated to be very insightful, 

but it is accurate. 

Section 2.4 derives transient solutions of EIT using the Bloch-vector formalism. 

Finally, Secs. 2.5, 2.6, and 2.7 generalize the Bloch-vector solution to account 

for Doppler broadening, buffer gas, and the magnetic sublevel and hyperfine levels 

associated with real atomic systems. 

2.1 Back of the envelope theory of EIT 

A simple back-of-the-envelope calculation illustrates the dominant mechanisms of 

EIT and provides a valuable reference point before beginning a detailed theoretical 

treatment of EIT. Although there are other mechanisms, such as stimulated rapid 

adiabatic passage (STIRAP) [63], for obtaining the atomic coherence required for 

EIT, CPT is the most common mechanism for achieving EIT. To derive an approximate 

solution for CPT, it is sufficient to perform a detailed balancing analysis 

between pumping from the bright state to dark states (e.g. optical pumping) and 

pumping from dark to bright (e.g. decoherence). Figure 2.1 shows a three-level Λ 

EIT system in both the physical basis and the bright/dark basis. The bright and 

dark states are respectively defined as 

|+〉 ≡ Ω p|1〉 − Ω c |2〉 

Ω 

, and|−〉 ≡ Ω∗ c|1〉 + Ω ∗ p|2〉 

, 

Ω


γ/2 

3 

γ/2 

3 

γ 

γ /2 

/2 

Ω P 

Ω C 

Ω 

2 

Γ 

(a) 

1 

+ 

Γ 

(b) 

- 

Figure 2.1: 

To understand EIT in a three-level Λ system sometimes it is helpful to transform 

from the physical basis (a) to the bright/dark basis (b). Raman resonance is 

assumed, and γ (Γ) is the decay rate from the excited (ground) state. 

where Ω = √ |Ω p | 2 + |Ω c | 2 

is the “bright” Rabi frequency, and we recall that 

Ω p ≡ µ p E p / (Ω c ≡ µ c E c /) are the probe and coupling Rabi frequencies. The 

probe and coupling fields are cw monochromatic fields. 

There are two processes moving atoms from the bright to the dark state; 

optical pumping R and population exchange between ground states Γ. The optical 

pumping rate from the bright state to the dark state is 

R = 

γΩ 2 /4 

Ω 2 + ∆ 2 γ/γ ⊥ + γγ ⊥ 

, (2.1) 

where γ is the spontaneous decay rate out of state |3〉, γ ⊥ 

≡ γ/2 + γ ′ is the 

transverse decay rate (i.e. the decay rate for the coherence terms ρ 31 and ρ 32 ), 

and γ ′ is the decoherence due to dephasing (i.e. no change in population). Thus,


the total rate of transfer from bright to dark state is 

R |+〉→|−〉 = R + Γ/2, (2.2) 

There are also two processes transferring atoms from the dark to bright state; 

non-zero Raman detuning resulting in phase change for the coherence between 

ground states, and population exchange between ground states Γ. Thus, the rate 

of population transfer from the dark state to the bright state is 

R |−〉→|+〉 = Γ/2 + δ 2 R/R, (2.3) 

where δ R ≡ ω p − ω 2 − ω c + ω 1 is the Raman detuning. 

Using detailed balancing between the dark and bright states we see that the 

number density of atoms in the bright state is 

N + = 

NR |−〉→|+〉 

R |−〉→|+〉 + R |+〉→|−〉 

, (2.4) 

where the total number density of atoms is N = N + + N − , and the absorption of 

the optical fields is 

αL ∝ N + ∝ ΓR+δ2 R 

, (2.5) 

R 2 +ΓR+δR 

2 

where α is an absorption coefficient and L is the length of the medium.


Two important details are illustrated by Eq. 2.5. First, the medium only 

becomes “transparent” if the EIT condition R ≫ Γ is satisfied. Second, when 

the EIT condition satisfied and the probe and coupling fields are near singlephoton 

resonance, the line shape of the transparency resonance is approximately 

Lorentzian with a linewidth of 2R for full width at half maximum (FWHM). 

Finally, we note that the essential resource which enables EIT is ground states 

coherence ρ 21 being equal to the dark ground states coherence ρ (−) 

21 = −Ω c Ω ∗ p/Ω 2 . 

In practice, it is relatively trivial to achieve the desired ratio between ground state 

populations (i.e. ρ 22 /ρ 11 ≈ ρ (−) 

22 /ρ (−) 

11 = Ω 2 c/Ω 2 p, where ρ (−) 

22 = Ω 2 c/Ω 2 and ρ (−) 

11 = 

Ω 2 p/Ω 2 are the dark state populations for levels |2〉 and |1〉). For example, mutually 

incoherent † probe and coupling fields will achieve the desired ratio between ground 

state populations as long as the EIT condition is met and δ R ≪ γ, but there will 

not be EIT because ρ 21 = 0. Both coherent and incoherent optical pumping 

processes reach equilibrium with ρ 22 /ρ 11 = Ω 2 c/Ω 2 p, but only the coherent process 

(i.e. CPT) has an equilibrium of ρ 21 = ρ (−) 

21 . 

† Optical fields are incoherent when they have large uncorrelated phase fluctuations.

2.2. PERTURBATIVE SOLUTION FOR EIT 28 

2.2 Perturbative solution for EIT 

To make our discussion of EIT more rigorous we consider the master equation for 

the three-level system shown in Fig. 2.2, which is 

˙ρ = 1 [H, ρ] − D, (2.6) 

i 

where the coherent evolution is determined by the Hamiltonian 

⎛ 

H = 

⎜ 

⎝ 

δ R /2 0 − Ω∗ C 

2 

0 −δ R /2 − Ω∗ P 

2 

− Ω C 

2 

− Ω P 

2 

∆ 

⎞ 

. (2.7) 

⎟ 

⎠ 

The decay/repumping matrix is 

⎛ 

D = 

⎜ 

⎝ 

⎞ 

Γ(ρ 11 − 1/2) − γ ρ 2 33 Γ ⊥ ρ 12 γ ⊥ ρ 13 

Γ ⊥ ρ 21 Γ(ρ 22 − 1/2) − γ ρ 2 33 γ ⊥ ρ 23 

, (2.8) 

⎟ 

⎠ 

γ ⊥ ρ 31 γ ⊥ ρ 32 γρ 33 

where γ ⊥ 

= (γ + γ ′ + Γ)/2 is the transverse decay rate for the excited state 

coherences ρ 23 and ρ 13 . The rate of population exchange between ground states 

Γ is due to atoms diffusing between the optical region and a reservoir with ρ 11 = 

ρ 22 = 1/2 and ρ 21 = 0. The decay rate for the ground state coherence (GSC) ρ 21 

is given by Γ ⊥ = (Γ ′ + Γ), where Γ ′ is the dephasing rate for the GSC.


γ/2 

2 

−δ R /2 

Ω P 

∆ 

Γ 

3 

γ/2 

Ω C 

1 

δ R 

/2 

Figure 2.2: 

Diagram of a three-level Λ-type atom with a strong coupling field and a weak 

probe field. The detunings are defined as ∆ = (∆ P + ∆ C )/2, and δ R = ∆ P − ∆ C , 

where ∆ P = (ω 1 − ω 2 ) − ω P and ∆ C = (ω 1 − ω 3 ) − ω C . Also, the Rabi frequencies 

are defined as Ω i = µ i E i / where i = {P, C} (signifying probe and coupling fields 

respectively), the dipole moments µ i are assumed to be real, and E i is the the 

electric field. The spontaneous emission rate out of the excited state is given by 

γ and the ground state decoherence rate Γ is the same for both ground states. 

Written out explicitly the master equations are 

˙ρ 33 = −γρ 33 + Im {Ω ∗ 2ρ 32 + Ω ∗ 1ρ 31 } (2.9) 

˙ρ 22 = γ ( 

2 ρ 33 − Γ ρ 22 − 1 ) 

− Im {Ω ∗ 

2 

2ρ 32 } (2.10) 

˙ρ 11 = γ ( 

2 ρ 33 − Γ ρ 11 − 1 ) 

− Im {Ω ∗ 

2 

1ρ 31 } (2.11) 

˙ρ 32 = 

( 

i∆(Ω S ) − i δ ) 

R(Ω S ) 

− γ ⊥ ρ 32 (2.12) 

2 

+i Ω C 

2 (ρ 22 − ρ 33 ) + i Ω P 

2 ρ∗ 21


˙ρ 31 = 

( 

i∆(Ω S ) + i δ ) 

R(Ω S ) 

− γ ⊥ ρ 31 (2.13) 

2 

+i Ω P 

2 (ρ 11 − ρ 33 ) + i Ω C 

2 ρ 21 

and 

˙ρ 21 = (iδ R (Ω S ) − Γ ⊥ )ρ 21 − i Ω P 

2 ρ∗ 32 + i Ω∗ C 

2 ρ 31. (2.14) 

In its most general form, the steady state analytic solution of Eq. 

(2.6) is 

extremely complicated and provides little insight, which is why approximate solutions 

are typically used [6, 95, 96, 124–126]. One of the more common techniques 

uses perturbation theory and assumes a small probe field (i.e. Ω p ≪ Ω p ) to obtain 

a solution that is first order accurate in Ω P and accurate to all orders for all other 

parameters [124–126]). This perturbative solution is simple and insightful, but it 

breaks down for large single-photon detuning ∆ ≫ γ ⊥ and moderately large probe 

fields (i.e. Ω p ∼ Ω c /10). As we will show, these limitation make the perturbative 

solution poorly suited for calculations of EIT in Doppler broadened medium. 

Thus, for Doppler broadened media we use the Bloch-vector solution from Sec. 

2.3 instead of the perturbative solution. 

The steady state solution of the master equation is solved by setting all time 

derivatives in Eqs. (2.9)-(2.9) equal to zero and simultaneously solving these six 

equations constrained by the normalization equation ρ 11 + ρ 22 + ρ 33 = 1. We do


this by first rearranging the density matrix into a real vector 

⃗p = [ρ 11 , ρ 22 , ρ 33 , Re(ρ 21 ), Im(ρ 21 ), . . . 

Re(ρ 32 ), Im(ρ 32 ), Re(ρ 31 ), Im(ρ 31 )] T . 

The master equation can now be expressed as a simple matrix equation 

˙⃗p = (M + Ω P P )⃗p, (2.15) 

where M is a matrix containing terms independent of Ω p , and the matrix P has 

all the terms that are multiplied by Ω p . The the top three rows corresponding to 

the ˙ρ 11 , ˙ρ 22 or ˙ρ 33 equations are linearly dependent, so one of these rows must be 

replaced with the normalization condition ρ 11 + ρ 22 + ρ 33 = 1. The steady state 

solution is then 

⃗p (ss) = ( ˜M + Ω P ˜P ) −1 ⃗v, (2.16) 

where the tildes denote that one row matrix has been replaced by the normalization 

condition and ⃗v = [0, 0, 1, 0, 0, 0, 0, 0, 0] T arises from the 1 on the right hand 

side of the normalization condition. This exact steady state solution ⃗p (ss) is used 

for all numerical calculations discussed in this work, but it does not produce an 

insightful analytic result. 

The perturbative solution is found by defining the steady-state solution as a


power series in ɛ 

∞∑ 

⃗p (ss) = ɛ n ⃗p n , (2.17) 

n=0 

where ɛ is a dummy variable to indicate the order of the perturbation. Applying 

Eq. (2.17), the master equation becomes 

( ˜M 

∑ ∞ 

+ ɛΩ P ˜P ) ɛ n ⃗p n = ⃗v. (2.18) 

n=0 

Multiplying out, collecting terms of the same order in ɛ, and setting ɛ = 1, we 

find that the perturbation solutions are 

⃗p 0 = ˜M −1 ⃗v, 

(2.19a) 

and 

⃗p n = (−Ω P ) n ( 

˜M 

−1 ˜P 

) n 

⃗p0 . (2.19b) 

In most discussions of EIT the first order approximation ⃗p (a) = ⃗p 0 + ⃗p 1 is used. 

For the case of zero decoherence (Γ ⊥ = 0) the perturbative solution gives a 

probe susceptibility of 

χ (a) 

P (δ R) = Nµ2 p 

2ɛ 0 

ρ 12 

≈ α 0 

2δ R 

Ω 2 C /γ + 4∆ P δ R /γ − 2iδ R 

, (2.20)


Virtual level 

Primary level 

Ω C 

Ω P 

Ω P 

Undressed 

Picture 

Dressed State 

Picture 

Figure 2.3: Schematic of transformation to the dressed state picture in which the coupling field 

dresses the atom splitting in the excited energy level into to new energy levels; the primary 

energy level and a virtual virtual energy level. 

where α 0 = N|µ p | 2 /2ɛ 0 γ is the incoherent absorption coefficient, µ p is the dipole 

moment for the probe transition, N is the atomic number density, and we 

have assumed γ ′ = 0. Like Sec. 2.1, Eq. (2.20) also predicts an approximately 

Lorentzian transparency resonance when the EIT condition is satisfied and near 

single-photon resonance ∆ ≪ γ ⊥ . 

Near single-photon resonance the dominant feature of EIT is a transparency 

resonance in an absorbing background. However, far from single-photon resonance 

(i.e. ∆ ≫ γ ⊥ ) the dominant feature of EIT is an absorbing resonance in a mostly 

transparent background. The physics of this process is best understood using the 

dressed state picture shown in Fig. 2.3. When the atom is dressed by a large 

coupling field the excited energy level splits into two energy levels that we call the 

primary and virtual energy levels. Figure 2.4 shows the probe absorption due to 

transition to the primary and virtual energy levels. The primary energy level can 

be thought of as the original excited energy level ac-Stark shifted by Ω 2 C /4∆ C,


and the absorption due to the primary transition can be approximated by 

L 1 (δ R ) = 

α 0 Lγ 2 ˜ρ 2 gg 

(δ R + ∆ C + Ω 2 C /4∆ . (2.21) 

C) 2 + γ 2 ˜ρ 2 gg 

The virtual energy level is near Raman resonance δ R = 0, and the absorption due 

to the virtual transition is approximately given by 

L 2 (δ R ) = 

α 0 Lγ 2 ˜ρ 2 ee 

(δ R − Ω 2 C /4∆ , (2.22) 

C) 2 + γ 2 ˜ρ 2 ee 

where ˜ρ ee = Ω 2 C /4∆2 C and ˜ρ gg = 1− ˜ρ ee . EIT results from the interference between 

these two transitions. The total absorption is approximately 

( [ ( )] ) √L1 

L T otal = (δ R ) + exp i cot −1 δR − δ √L2 2 

vc 

(δ R ) , (2.23) 

δ vc /4 

where the phase term mimics the interference responsible for EIT, and δ vc = 

Ω 2 C /4∆ C is the Raman detuning at the center of the virtual transition. Figure 2.4 

(a) shows the agreement between Eq. (2.22) and the exact steady state solution 

of Eq. (2.6). 

Several of the features exhibited in Fig. 2.4 are important for systems with 

multiple velocity classes such as those discussed in Secs. 2.5 and 2.6. First, 

the Raman detuning corresponding to the center of the virtual transition (i.e. 

δ vc ≈ Ω 2 /4∆) moves closer to Raman resonance for larger single-photon detunings.


1 

0.8 

(a) 

L 2 Eq. (8) 

EIT exact 

Ω C 

=0 

Virtual 

Transition 

x10 

(b) 

2 

c 

Ω /4∆ 

c 

Probe Absorption 

(α L=1) 0 

0.6 

0.4 

Primary 

Transition 

Virtual 

Transition 

Absorption 

FWHM 

2 2 

γ Ω 

c 

/2∆ c 

0.2 

0 

-50 -40 -30 -20 -10 0 10 

Raman Detuning (GHz) 

-2 -1 0 1 2 

Raman Detuning (GHz) 

Figure 2.4: EIT in the far detuned case. Grey solid line shows the probe absorption for EIT 

when γ = 2π × 6 MHz and ∆ C = 5γ. The dashed grey curve shows the probe absorption in 

the absence of a coupling field, and the black solid line shows the probe absorption for a virtual 

transition as calculated in Eq. (2.22). 

For EIT in Doppler broadened systems there will typically be a few velocity classes 

near single-photon resonance, but most velocity classes will have large singlephoton 

detunings. The absorption due to the far detuned velocity classes occur at 

Raman detunings in the transparency window of the near resonant velocity classes 

(i.e. resonant velocity classes have a transparency window for Raman detunings 

|δ R | ≤ Ω 2 /γ while the far detuned velocity classes have peak absorption at Raman 

detunings δ R = δ vc ≈ Ω 2 /4∆ ≪ Ω 2 /γ.) Thus, the EIT linewidths in Doppler 

broadened media are much narrower than single velocity class EIT systems. 

Figure 2.5 shows how the superposition of absorption peaks from different 

velocity classes combine to create a narrower EIT resonance. When ∆ ≫ γ ⊥ , the 

absorption resonances become sufficiently narrow that they can be approximated


Probe Absorption (arb. units) 

5 

4 

3 

2 

1 

v=±v p 

v= - v p/6 

v= ± vp/3 

v= 0 

Doppler 

averaged 

EIT line 

Single 

velocity 

EIT line 

0 

-15 -10 -5 0 5 

Raman detuning (kHz) 

Figure 2.5: Probe absorption versus Raman detuning for a Doppler averaged EIT line shape 

and the probe absorption for several different velocity classes which contribute to the Doppler 

averaging. The parameters used to calculate these absorption curves are Γ = 0, γ =≈ 2π × 

6 MHz, v p = 50cγ/ω 0 = 222 m/sec. 

by Dirac delta functions–i.e. 

L 2 (δ R ) ≈ πα 0LΩ 2 γ ⊥ 

δ(δ 

4∆ 2 R − Ω 2 /4∆) 

≈ πα 0 Lγ ⊥ δ(∆ − Ω 2 /4δ R ), for ∆ ≫ γ ⊥ . (2.24) 

We will revisit this expression in Sec. 2.5 when we discuss Doppler broadened 

media. 

As mentioned previously the perturbative solution has limited utility (i.e. it 

is not valid when the probe field approaches the same order of magnitude as the 

coupling field and the single-photon detuning is large ∆ ≫ γ ⊥ ). For large single-


photon detunings and “large” probe fields the virtual transition is still valid if Eq. 

(2.22) is modified to obtain 

L 2 (δ R ) = 

≈ 

α 0 Lγ 2 ˜ρ 2 ee 

(δ R − FΩ 2 C /4∆ C) 2 + γ 2 ˜ρ 2 eeF , 2 

πα 0Lγ ⊥ 

δ(∆ − Ω 2 /4δ R ), for ∆ ≫ γ ⊥ , (2.25) 

F 

where F = √ 1 + |Ω P | 2 |Ω C | 2 ∆ 2 /γ 2 ⊥ Ω4 . Finally, in both Eqs. (2.22) and(2.25) 

we have assumed no decoherence. 

The effects of nonzero decoherence can be 

approximated by a second ad-hoc change such that 

L 2 (Γ ≠ 0) ≈ 2R2 L 2 (Γ = 0) 

( √ 2R + Γ) 2 , (2.26) 

where we recall that R = γΩ 2 /4(∆ 2 γ/γ ⊥ + γγ ⊥ ). 

An error bound for the perturbative solution ⃗p (a) can be calculated by considering 

the convergence rate of the perturbation power series from Eq. (2.17). From 

Eq. (2.19b) we have the inequality 

||⃗p n || ≤ B n ||⃗p 0 ||, for n > 0, (2.27) 

where B ≡ |Ω P | max |λ j | and max |λ j | is the maximum magnitude for the eigen-


values of matrix product ˜M −1 ˜P . Thus, the upper bound for the error is 

‖⃗p SS − ⃗p (a) ‖ 

‖⃗p SS ‖ 

≤ B 2 , if B < 1. (2.28) 

Through numerical experimentation we found that for a given probe detuning 

∆ p the maximum value for B is given by 

⎧ 

∣ 

⎪⎨ 

∣ Ω P ∣∣ 

∆P 

Ω C 

≪ γ ⊥ 

B ≈ 

∣ , (2.29) 

⎪⎩ 

4Ω 

∣ P Ω C ∆ P ∣∣ 

∆P ≫ γ 

Ω 2 C γ+8√ 2∆ 2 P Γ ⊥ 

where ∆ C = ∆ P − Ω 2 C /4∆ P for ∆ p > γ. There are a couple features of Eq. (2.28) 

worthy of mention. First, Ω p ≪ Ω c is typically given as the validity condition 

for the perturbative solution, which is correct near resonance. However, far from 

resonance B is large when Ω p ≪ Ω c and Γ ≈ 0. This fact makes the perturbative 

solution poorly suited to Doppler broadened media where the far detuned velocity 

classes play the dominant role in determining the EIT line shape and linewidth. 

In Doppler broadened media, the error bound is determined by the velocity 

class with the largest value of B (i.e. B max ≡ max(B)). Figure 2.6 shows how the 

ratio Ω p /B max varies as a function of the coupling amplitude for a medium with 

80 velocity classes. Two different decoherence rates are considered.

2.3. BLOCH-VECTOR SOLUTION FOR EIT 39 

0.03 

Large Probe 

0.02 

______ 

Ω P Ω C 

FWHM≈ 

3.5 γ 

=250 Hz Γ ⊥ 

ΩP 

max 

0.01 

FWHM 

≈Ω(Γ /γ) 1/2 

Γ ⊥ 

=0 

Weak Probe 

0 

0 0.5 1 1.5 2 

Ω C 

γ 

/ 

FWHM ≈ 

Ω 2 

____ 

N v 

γ 

Figure 2.6: Limiting value of the Ω P /B max versus coupling Rabi frequency for calculating 

accuracy limits using Eq. 2.29. The limits of the Doppler integration are ±40γc/ω ◦ , i.e. 

∆ = {−40γ, 40γ}. 

2.3 Bloch-vector solution for EIT 

The Bloch-vector method for calculating EIT susceptibilities has several advantages 

over the perturbative calculation. 

First, Bloch-vector solution makes no 

assumptions about the relative strengths of the probe and coupling field. Second, 

it becomes more accurate for large single-photon detunings. 

Third, it is 

straightforward to use the Bloch-vector formalism to derive transient solutions for 

EIT. 

The Bloch-vector method uses the fact that in most experimentally relevant 

EIT scenarios, the excited state coherences ρ 23 and ρ 13 can be adiabatically eliminated, 

and the master equation can be reduced to a Bloch-vector equation. This


approach to solving EIT problems was pioneered by Bennink [127]. The justification 

for adiabatically eliminating the excited state coherences is strengthened 

significantly by large single-photon detunings making it ideal for calculations in 

Doppler broadened media. 

The excited state coherences are adiabatically eliminated by setting the time 

derivatives in Eqs. (2.12) and (2.13) equal to zero and solving to obtain 

Ω P2 

(ρ 22 − ρ 33 ) + Ω C 

ρ 32 ≈ − 

2 

ρ ∗ 21 

(2.30) 

∆ + iΓ ⊥ 

and 

Ω C2 

(ρ 11 − ρ 33 ) + Ω P 

ρ 31 ≈ − 

2 

ρ 21 

. (2.31) 

∆ + iΓ ⊥ 

The density matrix can be rewritten as a Bloch-vector equation for the two ground 

states 

d 

{ 

dt ⃗ρ = (R + Γ) −⃗ρ + T ⃗ × ⃗ρ + (1 − 3ρ 33 ) F ⃗ } 

, (2.32) 

and a first-order differential equation for the excited state population 

d 

dt ρ 33 = −γρ 11 − (R + Γ)⃗ρ · ⃗F + (1 − 3ρ 33 )R. (2.33)


In the Eqs. (2.32) and (2.33) have used the following definitions 

⎛ 

⎞ 

⎛ 

⎞ 

⃗ρ = 

⎜ 

⎝ 

u 

v 

w 

≡ 

⎟ ⎜ 

⎠ ⎝ 

2 Re{e −iθ ρ 21 } 

2 Im{e −iθ ρ 21 } 

ρ 22 − ρ 11 

, (2.34) 

⎟ 

⎠ 

⃗F ≡ R (1 − 3ρ 33) 

R + Γ 

⎛ 

⎜ 

⎝ 

2e −iθ ρ (−) 

21 

0 

ρ (−) 

22 − ρ (−) 

11 

⎞ 

, (2.35) 

⎟ 

⎠ 

⃗T ≡ 

δ R 

R + Γŵ + ∆ 

F 

Γ (1 − 3ρ 33 ) ⃗ , (2.36) 

R ≡ Ω2 Γ ⊥ /4 

, (2.37) 

∆ 2 + Γ 2 ⊥ 

where ŵ = (0, 0, 1) T is the unit vector in the direction of w. The phase θ is defined 

as θ ≡ arg(ρ (−) 

21 ). 

It has been shown in Ref. [127] that the steady state solutions to Eqs. (2.32) 

and (2.33) are 

and 

⃗ρ ss = (1 − 3ρ 33 ) ⃗ F + ⃗ T × ⃗ F + ⃗ T ( ⃗ T · ⃗F ) 

1 + T 2 (2.38) 

ρ ss 

33 = 

R − (R + Γ) F ⃗ ∣ 

· ⃗ρ 

( 

γ + 3 R − (R + Γ) F ⃗ ∣ 

· ⃗ρ 

∣ 

ρ33 =0 

∣ 

ρ33 =0 

) (2.39) 

where T = | ⃗ T |.


In anticipation of the fact that ρ 33 

≈ 0 for the experimental conditions in 

chapters 4 and 5, we set ρ 33 = 0 throughout the remainder of this work. 

Finally, it is straightforward to relate the atomic parameters to the susceptibilities 

for the fields. We recall that the probe and coupling susceptibilities are 

χ P = N|µ|2 

ɛ 0 Ω p 

ρ 32 , (2.40) 

and 

χ C = N|µ|2 

ɛ 0 Ω c 

ρ 31 . (2.41) 

By assuming δ R ≪ γ ⊥ and ρ 22 /ρ 11 = |Ω c /Ω p | 2‡ , we can rewrite the expressions 

for the excited state coherences (i.e. Eqs. 2.30 and 2.31 in a more intuitive form; 

ρ 32 ≈ 

( 

ρ 22 Ω P 1 − ρ 21 /ρ (−) 

21 

2(∆ − δ R /2 + iΓ ⊥ ) 

) ∗ 

(2.42) 

and 

ρ 31 ≈ 

( 

) 

ρ 11 Ω C 1 − ρ 21 /ρ (−) 

21 

2(∆ − δ R /2 + iΓ ⊥ ) . (2.43) 

The crucial term in Eqs. 

2.42 and 2.42 is the ground state coherence (GSC) 

ρ 21 . In future sections it becomes evident that EIT, the EIT Kerr effect and its 

dynamics are all tied to ρ 21 and its evolution. The central role of the GSC is 

‡ The first assumption coupled with the EIT condition implies the second assumption. Recall 

that both coherent and incoherent optical pumping achieve the same ratio between ground state 

populations. Also, these assumptions are well justified for the experimental parameters used in 

this work.

2.4. TRANSIENT SOLUTION FOR EIT 43 

partially revealed by plugging Eq. (2.42) into Eq. (2.40) to obtain 

χ p (ρ 21 ) ≈ Nρ 22|µ| 2 

2ɛ ◦ 

( 

1 − 

) ∗ 

ρ 21 /ρ (−) 

21 

. (2.44) 

∆ − δ R /2 + iγ ⊥ 

2.4 Transient Solution for EIT 

The time dependent Bloch-vector equation in Eq. (2.32) can be decoupled into 

three independent ordinary differential equations: 

⎧ 

( d 

2 

dτ + 2 d ) ⎪⎨ 

2 dτ + 1 + T 2 

⎪⎩ 

u ′ − u ′ f 

v ′ − v ′ f 

⎫ 

⎪⎬ 

= 0 (2.45) 

⎪⎭ 

and 

( ) d (w 

dτ + 1 ) ′ − w f 

′ = 0, (2.46) 

where τ = (R + Γ)t and we have used the transformation of variables 

⎛ 

⎜ 

⎝ 

⎞ 

u ′ 

v ′ 

= 1 

⎟ T 

⎠ 

w ′ 

⎛ 

⎜ 

⎝ 

⎞ ⎛ 

T w 0 −T u 

0 T 0 

⎟ ⎜ 

⎠ ⎝ 

T u 0 T w 

u 

v 

w 

⎞ 

. (2.47) 

⎟ 

⎠ 

The initial conditions are given by ⃗ρ(t = 0) = ⃗ρ i , 

˙u ′ (0) = −T v ′ i − u ′ i + (1 + T 2 )u ′ f, (2.48)


and 

˙v ′ (0) = T u ′ i − v ′ i. (2.49) 

The subscript f denotes the final steady state values which are derived from the 

steady state solution Eq. (2.38) with ρ 33 = 0 (we are assume that for t ≥ 0 all 

parameters are constant.). 

The solutions to Eqs. (2.45) and (2.46) are 

u ′ (τ) − u ′ f = Ae −τ cos(T τ + φ), (2.50) 

v ′ (τ) − v ′ f = Ae −τ sin(T τ + φ), (2.51) 

and 

w ′ (τ) − w ′ f = ( w ′ i − w ′ f) 

e −τ , (2.52) 

where 

tan φ = v′ f − v′ i 

u ′ f − , and A = u′ i − u ′ f 

u′ i 

cos φ 

= v′ i − v ′ f 

sin φ . (2.53) 

When u, v, and w are plotted parametrically as a function of time, these equations 

describe an exponentially decaying spiral. 

For future discussions of the field susceptibilities it is convenient to transform 

from a Bloch-vector back to a density matrix. This is simplest if ∆ = 0, in which


case 

ρ 21 (t) = u+iv = ρ 21,f +(ρ 21 (0)−ρ 21,f ) exp [(iT − 1)(R + Γ)t] , for t ≥ 0; (2.54) 

where ρ 21,f = lim t→∞ ρ 21 (t) is the final steady state GSC. Also, for future reference 

we note that for ∆ = 0 the steady state GSC is given by 

ρ (ss) 

21 (δ R ) = 

ρ (−) 

21 R 

(R + Γ) − iδ R 

, (2.55) 

which is the equation for a circle when the real and imaginary parts of the GSC 

are plotted parametrically as a function of the Raman detuning. Finally, we note 

that thinking in terms of the GSC is an equivalent alternative to discussing EIT in 

terms of Lorentzian absorption lines with corresponding dispersion lines. Figure 

2.7 is a visual representation of the mapping between GSC and susceptibilities. We 

will change between these two representations depending on which representation 

yields greater insight. The dashed line in Fig. 2.7(a) is an example of a transient 

spiral calculated from Eq. (2.54) for the case when ρ 21 (t = 0) = ρ (ss) 

21 (δ R = 0) 

and ρ 21,f = ρ (ss) 

21 (δ R = 1.2R). The choice of other parameters can be found in the 

figure caption.

2.5. DOPPLER BROADENING 46 

0.5 

(a) 

Ground State 

Coherence 

21 

/ ρ 

(-)) 21 

(b) 

1 

Probe 

Susceptibility 

0.5 Re(χ ) ∝ -Im ( 

Im 

ρ 

( 21 (-) 

/ρ 21 

) 

0 

-0.5 

ρ (ss) (1.2R) 

21 

ρ (ss) (0 ) 

ρ 

(ss) (δ 

21 R 

) 

ρ 21 

(t) 

0 0.5 1 

Re 

ρ 

( 21 (-) 

/ρ 21 

) 

21 

ρ 

Im(χ ) ∝ Re( 1- 

P 

0.5 

0 

-4 -2 0 2 4 -0.5 

δ R /2R 

0 

P ) 

ρ21 

/ρ / 

21 

(-)) 

Figure 2.7: (a) Steady state (solid line) and transient (dashed line) ground state coherence are 

plotted in the complex plane, while (b) the probe susceptibility is plotted as a function of the 

Raman detuning δ R . All plots are theory with R ≈ 5Γ, and all variables except the Raman 

detuning are held constant. The transient ground state coherence curve (dashed) started with 

steady state for δ R = 0, then suddenly the Raman detuning changed to 2R. To show the 

mapping between ground state coherence and probe susceptibility four shapes corresponding to 

four Raman detunings have been plotted on both the steady-state ground state coherence and 

susceptibility curves. 

2.5 Doppler broadening 

Up to this point our model for EIT has only three energy levels, two dipole allowed 

transition, and one velocity class. The experimental systems discussed in Chapters 

4 and 5 are significantly more complicated than this, but the above theory is still 

applicable if we understand how it applies it to the real physical system. § In this 

section and the next two sections we generalize the theory to include Doppler 

broadening, velocity changing coherence preserving collisions (VCCPC), and the 

§ Many researchers work hard to make their physical system similar to the simplified theory. 

We have adopted the approach of embracing the complexity of our experiment and modifying 

our theory to accommodate our experiment.


complex structure of magnetic sublevels and hyperfine levels encountered in real 

atoms. 

In hot atomic vapors the velocity of the atoms parallel to the propagation of 

the optical fields shifts the atomic frequencies due to the Doppler effect. For a 

given velocity class of atoms (i.e. the set of atoms with velocity v in the range 

v 1 < v ≤ v 2 such that ∆vω 0 /c = 2γ ⊥ , where ∆v = v 2 − v 1 .) the detunings become 

δ R (v) = δ R (0) + vω 21 /c (2.56) 

∆(v) = ∆(0) + vω 0 /c, (2.57) 

where ω 0 = ω 3 − (ω 2 + ω 2 )/2, ω 21 = ω 2 − ω 1 , and ω n = E n / with E n being the 

energy of level n. We have assumed that the EIT fields are perfectly copropagating. 

For Doppler broadened media we must average quantities such as the probe 

susceptibility over all velocity classes. For example, 

χ p (δ R ) = 

= 

∫ ∞ 

−∞ 

∫ ∞ 

−∞ 

dvχ P [δ R (v), ∆(v)] exp(−v2 /v 2 p) 

v p 

√ π 

d∆χ P [δ R (∆), ∆] 2 ln 2 exp [−(∆ − ∆ 0) 2 4 ln 2/∆ 2 D ] 

∆ D 

√ π 

, (2.58) 

where δ R is shorthand for δ R (v = 0), ∆ 0 = ∆(v = 0), v p = √ 2k B T/m is the most 

probable velocity in the Maxwellian velocity distribution, m is the atomic mass, 

and ∆ D = 2 √ ln 2v p ω 0 /c is the Doppler broadened FWHM.


For simplicity, we will restrict our discussion to the case when ∆ 0 = 0. As 

we began discussing in Sec. 2.2 Doppler broadening results in a narrowing of the 

EIT resonance because absorption from the far detuned velocity classes fills in the 

transparency wings of the near resonant velocity classes making the transparency 

resonance narrow. 

In Doppler broadened media with greater than 20 velocity 

classes, the EIT line shape is almost entirely determined by the far detuned velocity 

classes. These far detuned velocity classes have narrow absorption spikes 

that can be approximated by Dirac delta functions [see Eqs. (2.24) and (2.25)]. 

In order to obtain the most accurate for EIT line shapes in Doppler broadened 

media we use the Bloch-vector solution. The steady state GSC is given by 

ρ 21 = F u 

2 

{ 

1 + 

[ ( 

¯δ 

i − 

1 + T 2 

¯δ + F )]} 

w∆ 

. (2.59) 

γ 

Applying this result to Eqs. (2.40) and (2.42) yields 

χ p ≈ − Nµ2 pρ 22 

2ɛ 0 

[ 

Γ⊥ + ¯δR ( 

1+T −i + ¯δ + 2 Fw ∆/γ ) ] ∗ 

, (2.60) 

(R + Γ ⊥ )(∆ − δ R /2 − iγ ⊥ ) 

where ¯δ R = δ R /(R + Γ ⊥ ), and we have used the identity F u = 2Rρ (−) 

23 /(R + Γ ⊥ ) 

in the last line. Under the assumptions of ω 21 = 0, Γ ⊥ = 0 and F w ≫ F u , we find


that the imaginary part of the susceptibility is 

Im(χ p ) ≈ 

32α 0 δ 2 R γ2 ⊥ 

( ) 

∆ 2 

+ 1 

γ⊥ 

2 

[ ( ) 

Ω 4 4δR (∆ 2 +γ⊥ 2 2 ( 

) 

Ω 2 γ ⊥ 

− ∆ γ ⊥ 

+ Fu 2 ∆2 + 1 

γ⊥ 

2 

)], (2.61) 

where we have defined α 0 = Nµ 2 /4ɛ 0 γ ⊥ and we have used the definition of the 

optical pumping rate R = Ω 2 γ ⊥ /4(∆ 2 + γ⊥ 2 ). In the limit of large single-photon 

detunings Eq. (2.61) describes a “virtual” absorption peak centered at Ω 2 /2∆ 

with a peak height of 

h = 2Nσ p /(1 + F 2 u∆ 2 /γ 2 ⊥) 

and the FWHM is 

w = Ω 2 γ ⊥ 

√1 + F 2 u∆ 2 /γ 2 ⊥ /4∆2 . 

This absorption peak has an approximately Lorentzian line shape, and its area is 

given by 

A(∆) = 

≈ 

∫ 2Ω 2 /∆ 

−2Ω 2 /∆ 

dδ R Im[χ p (δ R )] ≈ π 2 h w (2.62) 

πα 0 Ω 2 γ ⊥ 

4∆ 2√ 1 + F 2 u∆ 2 /γ 2 ⊥ 

.


Thus, the Doppler averaged absorption line is approximately 

Im(χ p ) 

≈ 

≈ 

∫ ∞ 

( 

A(∆) exp 

− ∆2 4 ln 2 

∆ 2 D 

) ( 

δ 

δ R − Ω2 

4∆ 

d∆ 

√ 

−∞ 

∆ D π/ ln 2/2 

√ ( ) 

2Nσ p γ ⊥ π ln 2 exp 

ln 2Ω4 

− 

4δR 2 ∆2 D 

√ . (2.63) 

∆ D 1 + F 

2 

u Ω 4 /16δR 2 γ2 ⊥ 

) 

This expression for the Doppler averaged line shape is valid for both the largeand 

weak-probe limits. In the weak-probe limit the absorption area in Eq. (2.62) 

is A(∆) ≈ πα 0 Ω 2 γ ⊥ /4∆ 2 and the line shape simplifies to 

Im(χ p ) ≈ 2Nσ pγ ⊥ 

√ 

π ln 2 

∆ D 

) 

ln 2Ω4 

exp 

(− , (2.64) 

4δR 2 ∆2 D 

and the EIT linewidth (FWHM) is Ω 2 /∆ D . In the weak-probe limit the effects 

of non-zero decoherence (i.e. Γ ⊥ ≠ 0) can be accounted with the substitution 

∆ D → ˜∆, where 

√ 

1 

˜∆ = 

1 

∆ 2 D 

+ Γ√ 2 

Ω 2 γ . (2.65) 

This substitution becomes necessary when not all velocity classes satisfy the EIT 

condition creating a coherence cutoff such that far detuned velocity classes with 

∆ > ˜∆ do not contribute to the line shape of EIT. 

In the large-probe limit the absorption area in Eq. 

(2.62) is A(∆) ≈


L / ) 

Absorption ( α max(α L) 

1 

0.8 

0.6 

0.4 

U-shape 

0.2 

weak probe 

V-shape 

large probe 

0 

1.5 1 0.5 0 0.5 1 1.5 

FWHM ) 

Raman Detuning ( δ R/ 

Lorentzian 

single velo. 

Figure 2.8: Absorption line shapes for the probe field in different regimes of EIT. The solidblack 

curve is a perfect Lorentzian corresponding to a single velocity class. The solid-grey and 

dashed-black curves show respectively the large-probe and weak-probe limits for the case of 

degenerate ground states (i.e. ω 21 = 0) and Doppler broadening with N v = 40. All curves 

assume single-photon resonance (i.e. ∆ 0 = 0) and Γ ⊥ = 0. 

πα 0 Ω 2 γ 2 ⊥ /4F u∆ 3 , and the line shape becomes 

Im(χ p ) ≈ 

2Nσ p γ ⊥ 

√ 

π ln 2 

∆ D 

√ 

1 + F 

2 

u Ω 4 /16δ 2 R γ2 ⊥ 

. (2.66) 

√ 

with a linewidth (FWHM) of Ω P Ω C /γ ⊥ 3. 

Figure 2.8 shows the probe absorption for EIT under three different sets of 

conditions: a single velocity class and multiple velocity classes in the weak- and 

large-probe limits. All three transparency lines are normalized to have the same 

FWHM in order to compare the line shapes. The Lorentzian shape for a single 

velocity class provides a reference for U and V shaped lines corresponding to the 

weak- and large-probe limits with multiple velocity classes.

2.6. COHERENCE PRESERVING COLLISIONS IN 

BUFFER GASSES AND WALL COATINGS 52 

2.6 Coherence preserving collisions in 

buffer gasses and wall coatings 

One of the primary obstacles in EIT related experiments is reducing the decoherence 

to tolerable levels. In hot atomic vapors this means reducing the transit-time 

decoherence. For example, room temperature rubidium atoms have a velocity of 

about v p − 250 m/s, resulting in a transit-time decoherence rate of 100 kHz for 

a beam diameter of 2.5 mm. Typically, either a buffer gas or a wall coating is 

used to keep atoms in the optical region without losing coherence. EIT atoms like 

rubidium can undergo thousands of velocity changing collisions with other buffer 

gas atoms or coated walls before they lose their GSC that is crucial for EIT. 

These velocity changing coherence preserving collisions (VCCPC) have the 

secondary effect of causing the GSC created in the near resonant velocity classes 

(i.e. the optical pumping rate R is largest near resonance) to diffuse to the far 

detuned velocity classes. If the collision rate R c is fast enough that on average 

each atom experiences all of the velocity classes in the coherence time τ c = 1/2Γ ⊥ 

(i.e. R c ≥ N v /τ c , where N v = ∆ D /2γ ⊥ is the number of velocity classes), then 

each atom will see the same average optical pumping rate 

R vccpc = 2√ ln 2 

∆ D 

√ π 

∫ ∞ 

−∞ 

d∆ R(∆) exp(−∆ 2 4 ln 2/∆ 2 D) 

≈ R(0)/N v (2.67)



regardless of its current velocity class. This leads to a uniform distribution of 

GSC, and several interesting effects for the EIT line shape. When the condition 

R c ≥ N v /τ c is satisfied we call the system a VCCPC system. 

To understand the implications of a uniform ground state coherence on the EIT 

line shape we use the Bloch-vector formalism of Sec. 2.3 with the assumptions 

Γ = 0 and Ω C ≥ 5Ω P and ∆(v = 0) = 0. Using Eq. (2.30) the GSC for all 

velocity classes can be expressed as 

ρ 21 = 

ρ (−) 

21 

1 − iδ R /R vccpc 

. (2.68) 

The probe susceptibility is found by plugging Eq. (2.68) into Eq. (2.44) to obtain 

( 

) ∗ 

χ p (∆) ≈ − Nµ2 p 

−iδ R 

, (2.69) 

2ɛ 0 (R vccpc − iδ R )(∆ − iγ ⊥ ) 

where we have assumed δ R ≪ γ ⊥ . Finally, integrating over all velocity classes the 

Doppler averaged probe susceptibility is 

δ R 

χ p ≈ −A 

, (2.70) 

R vccpc + iδ R



where A is a constant 

A = Nγ √ 

⊥µ 2 p ln 2/π 

2ɛ 0 ∆ D 

≈ 

∫ ∞ 

0 

d∆ 

( 

exp 

− ∆2 4 ln 2 

∆ 2 D 

∆ 2 + γ 2 ⊥ 

Nµ2 p 1 

. (2.71) 

4ɛ 0 γ ⊥ N v 

) 

In contrast to EIT in non-VCCPC Doppler broadened media, the farthest detuned 

velocity classes contribute least to the EIT line shape because for far detuned 

velocity classes (i.e. for |v| ≫ cγ ⊥ ω 0 ) the probe susceptibility has the symmetry 

χ p (v) ≈ −χ p (−v) such that χ p (v) ≈ +χ p (−v) ≈ 0. 

Equation (2.70) is simply a Lorentzian shaped transparency resonance with 

FWHM of 2R ave . It is as though the blurring of velocity classes has made Doppler 

broadened line into a single homogeneous line with linewidth 2N v γ ⊥ 

= ∆ D 

(FWHM). Empirical studies show that VCCPC systems are effectively approximated 

as a single homogeneous line for conditions deviating from the case we 

have considered, such as when the decoherence is non-zero (Γ ≠ 0), the system 

is away from resonance (∆(v = 0) ≠ 0), and the ground energy levels are not 

degenerate (ω 21 ≠ 0). 

Figure 2.9 shows how the linewidth and line shape change as a function of the 

collision rate. The black curve shows the FWHM of the transparency resonance 

and the grey curve shows the root mean square (rms) difference between the 

Approximating a Doppler broadened VCCPC system as a single homogeneous absorption 

line over looks the fact that Doppler-broadened lines is gaussian and not Lorentzian, but discrepancies 

resulting from this difference are typically small.



20 

0.1 

FWHM (kHz) 

16 

12 

Non-VCCPC 

8 

VCCPC 

4 

0 

0.1 1 10 10 10 10 

Velocity Changing Collision Rate (kHz) 

0.08 

0.06 

0.04 

2 3 4 0 0.02 

RMS difference from Lorentzian 

Figure 2.9: EIT linewidth (black) versus the rate of velocity-changing, coherence-preserving 

collisions (VCCPC). The grey curve shows the RMS difference between the EIT line shape and 

a Lorentzian line shape with the same amplitude and FWHM. 

numerical simulation of a system with velocity changing collisions and a Lorentzian 

transparency window of the same width and depth. The parameters chosen for 

this simulation are in the weak-probe limit. In the non-VCCPC region (i.e. the 

VCCPC condition R c ≥ N v /τ c is far from being satisfied), the system is a typical 

Doppler broadened system with linewidth Ω 2 / ˜∆. Once the VCCPC condition is 

satisfied there is no longer a coherence cutoff and the EIT linewidth decreases to 

Ω 2 /∆ D . Also, in the VCCPC region the line shape is essentially Lorentzian for 

collision rates satisfying the VCCPC condition.

2.7. MAGNETIC SUBLEVELS AND HYPER-FINE LEVELS 56 

2.7 Magnetic sublevels and hyper-fine levels 

In Chapters 4 and 5, the experiments use the D 1 line of rubidium which has many 

more electronic states than the three-level EIT systems considered up to this 

point. Both Rb 85 and Rb 87 isotopes are used in the experiments but only Rb 85 is 

considered here (Rb 87 is discussed briefly in the Appendix). The D 1 line of Rb 85 

has four hyperfine, 24 magnetic sublevels and 66 dipole allowed transitions making 

it substantially more complicated than the three-level EIT system we previoiusly 

considered. However, the essence of EIT in this system is the same as the threelevel 

Λ system (i.e. optical pumping and decoherence between bright and dark 

states). The theory developed for three-level systems is applicable to the D 1 line 

of rubidium if averaged quantities are used (e.g. the average optical pumping rate 

R ave and an effective homogeneous linewidth γ eff ; both are derived shortly). 

In anticipation of the experiments of Chapters 4 and 5, we consider orthogonal 

linear polarization for the probe and coupling fields ‖ . For the choice of quantization 

axis parallel to the coupling beam polarization, Fig. 2.10 shows the level 

diagram for Rb 85 with the coupling coefficients for the probe and coupling transitions. 

There are other dipole allowed transitions not shown in Fig. 2.10. The 

‖ There are several reason why the probe and coupling fields have orthogonal linear polarization, 

and all of them are important for achieving EIT with good contrast. First, choosing 

the the fields to have circular polarizations results in there being non-coherent dark state (i.e. 

dark states which are individual magnetic sublevels not coherent superpositions of sublevels for 

an appropriate choice of quantization axis). These non-coherent dark-states make the medium 

transparent regardless of the Raman detuning, and destroying the EIT contrast. Second, for 

parallel linear polarizations there are no dark states that are simultaneously dark for transitions 

to both excited hyper-fine levels. Finally, experimentally it is convenient to have different 

polarization for the two fields so that they can be separated using polarization.


probe and coupling fields’ interaction with these transitions creates an ac-Stark 

shifts the ground hyperfine levels and changes the hyperfine splitting. The energy 

shift to the hyperfine splitting is 

∆U = 

Ω2 p 

4∆ hf 

+ 

Ω2 c 

4∆ hf 

= 

Ω2 

4∆ hf 

, (2.72) 

such that the hyperfine splitting becomes ∆ hf (Ω) ≈ ∆ hf (0) + ∆U and for Rb 85 

we have ∆ hf (0) = 3.035 GHz. For now, we ignore these transitions, but we will 

consider them again in chapters 4 and 5. 

Although it is far from obvious by looking at the level diagram in Fig. 2.10, 

there is exactly one coherent superposition of ground states that is completely 

decoupled from both F ′ = 2 and F ′ = 3 excited hyperfine levels. This is the 

only truly dark state. 

There are also four other partially dark states that are 

only weakly coupled to the excited states, and seven other bright states that are 

strongly coupled to the excited states. Details about the completely dark state 

can be found in the Appendix. 

The coupling between the ground and excited states can be calculated by 

solving for the eigenvalues of the time independent Hamiltonian. The time independent 

Hamiltonian is obtained using a rotating reference frame with the rotating 

wave approximation, and by only considering those dipole allowed transitions 

explicitly shown in Fig. 2.10. The eigenvalues and eigenvectors are evaluated


hω P 

2 

27 

1 

3 

1 

3 

2 

27 

5 

8 

1 

8 

5 

27 

27 

3 

27 

27 

2 

−1 

−1 

2 

− 

− 

27 

3 

3 

27 

hω C 

m F =-3 m F =-2 m F =-1 m F =0 m F =1 m F =2 

m F =3 

F - 

(a) 

F’=3 

F’=2 

F=3 

F=2 

hω P 

(b) 

15 

10 

6 

1 

1 

27 

27 

27 

3 

27 

−1 

4 

−1 

1 

4 

1 

− 

0 

3 

27 

27 

27 

27 

3 

1 

27 

1 

3 

6 

27 

10 

27 

15 

27 

m F =-3 m F =-2 m F =-1 m F 

=0 

m F =1 m F =2 

m F =3 

F’=3 

F’=2 

hω C 

F=3 

F=2 

Figure 2.10: Transitions excited by the vertically polarized probe laser (grey lines) and 

horizontally polarized coupling laser (black lines). The quantization axis is chosen along 

the horizontal polarization. Also shown are the hyperfine matrix elements for each 

excited transition, which are related to the Clebsch-Gordon coefficients by the factor 

(−1) √ { } 

F ′ +J+1+I J J 

(2F ′ ′ 

1 

+ 1)(2J + 1) 

F ′ , where I = 3/2 is the nuclear spin, J = J 

F I 

′ = 

1/2 are the orbital plus spin angular momenta for the ground and excited state respectively, 

and we have use the Wigner 6-j symbol (see Ref. cite for details).


numerically for the case when δ R = 0 and ∆ ≫ Ω + , where 

Ω + = 2µ D1 

√E 2 P + E2 C /√ 3, (2.73) 

is the bright Rabi frequency for the D 1 line and for rubidium µ D1 = 2.54 × 

10 −29 Cm. The smallest eigenvalues are the Stark shifted energies for the ground 

states, and these energy shifts (i.e. ∆U i with i = 1, 2, . . . 12) can be calculated as 

a Rabi frequency for each ground state 

Ω i ≡ √ −4∆U i ∆, i = {1, 2, . . . , 12}. (2.74) 

The degree of coupling between ground and excited states is given by the ratio 

Ω i /Ω + , such that Ω i /Ω + = 1 is a bright state and Ω i /Ω + = 0 is a completely dark 

state. 

Figure 2.11 shows the degree of coupling for each of ground state energy eigen 

states plotted as a function of the ratio between probe and coupling amplitudes 

E P /E C . The experiments discussed in Chapters 4 and 5 lie in the region 0.2 ≥ 

Ω P /Ω C ≤ 0.6, and the decoherence rate Γ in Chapters 4 and 5 is large enough 

that the completely dark state is not appreciably darker than the partially dark 

states. 

The effective optical pumping rate can be approximated as the average of the 

optical pumping rates out of each bright state and into one of the five “dark”


Rabi Frequency (Ω /Ω ) 

i + 

Bright States 

1 

0.8 

0.6 

0.4 

Quasi-dark States 

0.2 

0 

Completely Dark State 

0 0.2 0.4 0.6 0.8 1 

E P 

/ E C 

Figure 2.11: Normalized Rabi frequencies Ω i /Ω + plotted versus the ratio E P /E C . There are 12 

Rabi frequencies; 7 bright states, 4 quasi dark states, and 1 completely dark state. 

states. The optical pumping rate out of bright state m is given by 

R m = 

5∑ ∑12 

P 

n=1 

l=1 

(|e〉 l 

∣ ∣∣|+〉m 

) 

R |e〉l →|−〉 n 

, (2.75) 

∣ ) ∣∣|+〉m 

where P 

(|e〉 l 

is the conditional probability that in steady state the atom 

will be in the excited state |e〉 l , given that it is in either an excited state or bright 

state |+〉 m . R |e〉l →|−〉 n 

is the spontaneous decay rate from excited state |e〉 l to dark 

state |−〉 n . The effective optical pumping rate is then 

R ave = 1 7∑ 

R m , (2.76) 

7 

m=1 

where we have assumed that all bright states are equally populated (in reality


0.42 

2 

B=R ave γ eff /Ω + 

0.40 

0.38 

5/12 

0.36 

0 0.2 0.4 0.6 0.8 1 

E P /EC 

Figure 2.12: Plot of B = R ave γ eff /Ω 2 + versus the ratio E P /E C . Also plot is the value 5/12=0.417, 

which is the ratio of the number of dark states to ground states. Over the range 0.2 ≥ E P /E C ≤ 

0.6 B is approximately 3/8. 

there are small variations in the relative populations of bright states but this is a 

small correction). 

Figure 2.12 shows the numerical calculation of R ave γ eff /Ω 2 + for the D 1 line 

of Rb 85 as a function of E P /E C . Naively one might expect R ave = 5γ eff /12Ω 2 + 

because there are five dark states out of a total of twelve ground states. However, 

calculations reveals that the optical pumping rate is slightly smaller than this 

because the branching ratios from those excited states coupled to the bright states 

are weighted towards spontaneous decay into the bright state. Evaluating Eq. 

(2.76) we find that over the range 0.2Ω C ≤ Ω p ≥ 0.6Ω C 

R ave ≈ 3Ω2 + 

8γ eff 

, (2.77)

2.8. MISCELLANEOUS EXPERIMENTAL CONSIDERATIONS 62 

where γ eff is the effective homogeneous linewidth that arises due to a combination 

of Doppler broadening and buffer gas effects. 

Finally, we need to calculate the effective homogeneous linewidth γ eff in order 

to apply the simple three-level EIT theory to rubidium. It can be calculated using 

the definition 

1 

γ eff 

≡ 

∫ 

γ ∞ 

√ ⊥ 

π/ ln 2∆D 

−∞ 

( 

exp 

− ∆2 4 ln 2 

∆ 2 D 

∆ 2 + γ 2 ⊥ 

) 

d∆, (2.78) 

where 2γ ⊥ is the homogeneous FWHM, and ∆ D is the inhomogeneous FWHM. 

For Rb 85 with a N 2 buffer gas the measured increase in homogeneous linewidth is 

16.3 MHz per Torr N 2 [128–130]. In the experiments, the rubidium vapor cells have 

20 Torr of N 2 gas resulting in a homogeneous linewidth of 2γ ⊥ ≈ 330 MHz. The 

inhomogeneous broadening is due to Doppler broadening and the Doppler broadened 

FWHM is ∆ D = 530 MHz for rubidium vapor at temperature of 350 K ∗∗ . 

Thus, the effective linewidth is γ eff ≈ 650 MHz. 

2.8 Miscellaneous experimental considerations 

There are various other considerations which affect real experimental systems. 

Some of these are four-wave mixing, various forms of decoherence, and spatial 

varying field amplitudes. We mention these briefly to make the reader aware of 

them, but will not go into detail. 

∗∗ For high enough buffer gas densities Doppler broadening can actually be reduced due to 

Dicke narrowing [131], but our experiments do not approach these densities.


5P 3/2 

F=3 

F=2 

Ω C 

↔ 

b 

b 

Ω 

P 

(a) 

∆ 5P 1/2 

Ω 4 

hf 

W 

↔ Ω C 

5S 1/2 

↔ 

Ω 

Ω C 

P 

b 

F=3 

F=2 

5P 1/2 

(b) 

∆ S 

↔ 

b 

Ω S 

Ω 4W 

5S 1/2 

Figure 2.13: Level diagrams for four-wave mixing created in our experimental EIT system. 

First, in the experiments there are two systems that exhibit four wave mixing, 

and these are diagramed in Fig. 2.13. In Fig. 2.13(a) only the probe and coupling 

beams are necessary to excite the four-wave mixing process and create a third field 

with Rabi frequency Ω 4W and frequency ω 4W = 2ω c − ω p . The four wave mixing 

process in Fig. 2.13(b) requires the probe and coupling fields with the signal field 

discussed in the next chapter, and it results in a fourth field with Rabi frequency 

Ω 4W and frequency ω 4W = ω p + ω s − ω c . In both of these cases the Maxwell wave 

equation can be written using the slowly varying envelope approximation (SVEA) 

as 

where in Fig. 

( ∂ 

c∂t + ∂ ) 

Ω 4W ∝ ρ 21γ ⊥ 

, (2.79) 

∂z ∆ i 

2.13(a), ∆ i is the hyperfine splitting of the ground states, and 

in Fig. 2.13(b), ∆ i is the signal detuning. In both cases the intensity of the


four-wave mixing product Ω 4W 

grows quadratically with optical depth. In Fig. 

2.13(a), the probe field also experiences some gain due to four-wave mixing, and 

in Fig. 2.13(b), the probe experiences absorption. The four-wave mixing gain is 

proportional to the magnitude of the GSC in contract to EIT which is proportional 

to Re(1−ρ 21 /ρ (−) 

21 ). Thus, four-wave mixing gain has a slightly wider FWHM than 

EIT and a non-Lorentzian line shape. Finally, as a function of Raman detuning 

There are several forms of decoherence that must be overcome if one is to obtain 

good EIT. Some of the sources of decoherence are limited dwell time in the beam †† , 

magnetic fields, spin exchange relaxation (i.e. Rb-Rb collision), buffer gas collisions 

(i.e. Rb-N 2 collisions), and imperfectly aligned probe and coupling beams ‡‡ . 

For the experiments discussed in this work, the dominant form of decoherence 

is the limited dwell time in the beams. The diffusion coefficient for rubidium in 

a nitrogen buffer gas at standard temperature and pressure is D 0 = 0.15 cm 2 /s 

[128, 130, 132]. It is possible to calculate that the Rb-N 2 collision cross-section is 

σ c ≈ 3 × 10 −15 cm 2 by using the approximate equation for the diffusion coefficient 

D ≈ 

π¯v 

8N N σ c 

, 

where ¯v = √ 8k B T/πm is the mean velocity of the atoms, N N 

is the number 

density of nitrogen. 

For 20 Torr of nitrogen, we see that the mean free path 

†† This is the same thing as transit-time decoherence. 

‡‡ If the probe and coupling beams are not perfectly copropagating parallel to one another, 

then they will see the same atom as having different velocities and different Doppler shifts.


length is l = 1/N N σ c ≈ 5 µm, and the collision rate is R c = ¯vl ≈ 10γ, where γ = 

5.75 MHz is the natural linewidth for the D 1 line of rubidium. The diffusion time 

for an atom to move from the center to out of the optical beam is approximately 

t d ≈ r 2 /l 2 R c , where r is the beam radius. For the above parameters and a 2 mm 

beam radius the average dwell time in the beam is about 0.5 ms corresponding to 

a decoherence rate of about 2 kHz. In Chapters 4 and 5, we see this decoherence 

rate corresponds well with the measured decoherence rates. The Rb-Rb collision 

cross-section is about 2×10 −14 cm 2 .which is larger than the N 2 -Rb collision crosssection. 

However, for the rubidium density discussed in this work is sufficiently 

small that the decoherence rate due to spin exchange collisions never exceeds the 

10’s of Hz. 

As a consequence of the decoherence encountered in experimental systems 

we are typically more concerned with the depth of the EIT ˜αL = αR/(R + Γ). 

Figure 2.14 shows experimental probe absorption data from Chapter 4 with our 

definitions for the linewidth FWHM, probe absorption αL = ρ 22 α 0 L, and EIT 

depth ˜αL, where α 0 = N|µ p | 2 /2ɛ 0 γ is the probe absorption coefficient if all the 

population were in the probe ground state. These definitions are used throughout 

the remainder of this work. 

The optical pumping rate has a spatial dependence due to spatial variations in 

the field intensities (i.e. the beams are approximately gaussian). Thus, quantities 

that depend on the optical pumping rate such as the rise-times and transparency


6 

Probe absorption (αL) 

4 

2 

0 

αL 

αL 

∼ 

FWHM 

Raman 

resonance 

-20 -10 0 10 20 30 

Raman Detuning (kHz) 

Figure 2.14: Visual representation of various terms used to describe EIT. 

FWHM must be averaged spatially. This is discussed in detail towards the end of 

chapter 4.

67 

Chapter 3 

Theory of the dynamics of the 

EIT Kerr nonlinearity 

The EIT Kerr nonlinearity arises when a signal field Ω s (see Fig. 3.1) ac-Stark 

shifts one or both of the EIT ground states perturbing the EIT Λ system away 

from Raman resonance. Perturbations of the Raman resonance change the index 

of refraction for both probe field Ω p 

and coupling field Ω c , resulting in phase 

modulation of the fields proportional to the intensity of the signal field. 

This type of Kerr nonlinearity has several advantages over more conventional 

Kerr systems. First, the cross-phase modulation can be large with minimal selfphase 

modulation, in contrast most other Kerr media which necessarily have large 

self-phase modulation. 

Second, the cross-phase modulation is one-directional. 

The signal intensity creates a phase modulation on the probe, but the probe does 

not phase modulate the signal. Third, for cw fields the size of the cross-phase 

modulation is tunable by changing the slope of the EIT dispersion. This slope 

can be increased by either making the EIT linewidth narrower or increasing the

68 

γ 

2 

3 

∆ 

γ 

2 

4 

∆ S 

γ 

Ω C 

Ω S 

2 

Ω P 

δ R 

2 

Γ 

δ R 

2 

1 

Figure 3.1: Simple EIT Kerr system in a four-level N-type system. The detunings are defined 

as ∆ P = (ω 3 − ω 2 ) − ω P , ∆ C = (ω 3 − ω 1 ) − ω C , ∆ S = (ω 4 − ω 2 ) − ω S , ∆ = (∆ P + ∆ C )/2, and 

δ R = ∆ C − ∆ P . Also, the Rabi frequencies are defined as Ω i = µ i E i / where i = {P, C, S}, µ i 

is the dipole moment, and E i is the the electric field. The spontaneous emission rate out of the 

excited state is given by γ and Γ is the decoherence rate for the ground states. 

optical depth of the EIT medium (e.g. by making the medium longer or more 

dense). Theoretically the EIT Kerr effect can be arbitrarily large for cw fields. 

Large optical nonlinearities are interesting because they open the door to quantum 

nonlinear optics (i.e. nonlinear optics with wavepackets at the few-photon 

level) and other low-light-level applications. Many of the most intriguing applications 

require a strong nonlinear response to pulses with small energies. This 

implies that the nonlinearities must be both fast and large. In these applications 

the correct figure of merit is not the size of the nonlinear susceptibility, but the 

ratio between the size and rise-time of the nonlinear susceptibility. 

The EIT Kerr effect is slow compared to parametric nonlinear processes. Also, 

the rise-time and size of the EIT Kerr susceptibility are directly proportional

3.1. THE EIT KERR NONLINEARITY 69 

making their ratio a constant. Thus, increasing the size of the cw EIT Kerr effect 

does not increase the figure of merit for pulsed EIT Kerr effects. 

This chapter is divided into two sections. First, in Sec. 3.1 we present the 

steady state and transient theory for the EIT Kerr nonlinearity. In Sec. 3.2, we 

discuss the application of Kerr nonlinearities to QND measurements and derive 

the signal to noise ratio for these measurements. In this chapter we only consider 

EIT Kerr QND measurements when the signal group velocity is approximately 

c the speed of light in vacuum. However, it is possible to generalize the theory 

to include arbitrary group velocities for the signal, which is done in Sec. 5.1 of 

Chapter 5. 

3.1 The EIT Kerr nonlinearity 

Figure 3.1 shows a typical system for refractive EIT Kerr nonlinearities. Various 

parameters such as the Raman detuning δ R , single-photon detuning ∆ and Rabi 

frequencies Ω i where i = {P, C, S} are defined in the figure caption. The EIT Kerr 

system consists of a Λ EIT system ∗ discussed in chapter 2 and a signal field Ω s 

which is detuned by ∆ s from the |1〉 → |4〉 transition. The signal field Stark shifts 

ground state |1〉 by Ω 2 S /4∆ S, which changes the Raman detuning and ground state 

to 

δ R (Ω S ) = δ R (0) + |Ω S | 2 /4∆ S . (3.1) 

∗ Figure 3.1 uses the same definitions for fields and detunings as Fig. 2.2 from chapter 2.


The decoherence rate of ground state is modified, 

Γ(Ω S ) = Γ(0) + |Ω S | 2 γ/4∆ 2 S, (3.2) 

due to weak optical pumping to state |4〉 followed by spontaneous decay back 

to the ground states. Typically, the signal detuning is chosen to be large (i.e. 

∆ s ≫ γ) such that the change in Raman detuning is significant while the change 

in decoherence is negligible and can be ignored. Throughout the remainder of this 

chapter we ignore the signal field contribution to the decoherence. 

The nonlinear susceptibility for the EIT Kerr effect is defined 

χ (3) 

p,ss ≡ ∂χ ∣ 

p ∣∣∣Es 

. (3.3) 

∂|E s | 2 =0 

Returning to the theory from chapter 2, we recall that on single-photon resonance 

∆ = 0 the probe susceptibility is 

χ p ≈ αc 

ω p 

δ R + iΓ 

, (3.4) 

R + Γ − iδ R 

where α = ρ 22 α 0 is the probe absorption coefficient and ρ 22 ≈ |Ω c | 2 /Ω 2 c is population 

in state |2〉. Combining Eqs. (3.1), (3.5), and (3.4) we see that the third-order


susceptibility “Kerr” susceptibility is 

χ (3) 

p,ss 

≈ 

|µ| 2 

2∆ s 2 ω p 

c 

v g,p 

, (3.5) 

where v g,p 

= 2R/˜α is the group velocity of the probe, and we have assumed 

δ R (0) = 0 † . 

As long as the Raman detuning stays in the linear regime (i.e. δ R (Ω s ) ≪ R) 

the cross phase modulation of the probe field will be 

φ = Lω p χ (3) 

p,ss|E s | 2 /2c, (3.6) 

where L is the length of the EIT medium, and we assume χ p ≪ 1 (this assumption 

is consistent with atomic vapors). However, this cross-phase modulation (XPM) 

can be written more generally as 

φ ≈ ω pL 

2c Re [χ p(Ω S ) − χ p (0)] . (3.7) 

Our primary focus is the linear regime for the EIT Kerr effect because of its 

applications to QND measurements, but there are also some interesting effects in 

the nonlinear regime. 

† The assumption of Raman resonance in the absence of a signal field is continued throughout 

this chapter.


3.1.1 EIT Kerr dynamics 

The EIT Kerr transients and rise times result from the slow evolution of the GSC 

ρ 21 when the signal field perturbs the Raman detuning. The dynamics of an EIT 

system were derived in Sec. 2.4. For the purpose of discussing the EIT Kerr 

dynamics we restrict our discussion to the case of zero single-photon detuning 

∆ = 0 and a single homogeneous line. We also assume γ ≫ 5Ω ≥ δ R . 

For optically thin media (i.e. ˜αL ≪ 1), the EIT Kerr dynamics can be determined 

by a straightforward application of the EIT transient theory of Sec. 2.4. 

To determine the rise and fall dynamics of the EIT Kerr effect we consider two 

cases. First, at time t = 0 the atoms are in steady state for Raman resonance. 

Then a signal field is suddenly turned on, Stark shifting the system away from 

Raman resonance. We call this the rising dynamics. The second case is simply 

the reverse. At time t = 0 the system is in steady state with the signal field on, 

and then the signal is suddenly turned off. This is the falling dynamics. In both 

cases the transients of the GSC are given by 

ρ 21 (t) = ρ 21,f + (ρ 21 (0) − ρ 21,f )e t(iδ R−R−Γ) , for t ≥ 0, (3.8) 

where the initial GSC is ρ 21 (0), the final steady state GSC is ρ 21,f , and δ R is the


0.5 

Im 

ρ 

(-) 

( 21 

/ρ 21 

) 

0 

(a) 

ρ (ss) (1.2R) 

21 

ρ 

(ss) (δ 

21 R 

) 

ρ 21 

(t) 

ρ (ss) (0 ) 

21 

0 0. 5 1 

Re 

ρ 

( 21 (-) 

/ρ 21 

) 

Porbe Susceptibility 

1 

0.8 

0.6 

0.4 

0.2 

(b) 

Im(χ p 

) 

Re(χ p 

) 

0 

0 2 4 6 8 10 

Time (t /R) 

Figure 3.2: The transients of the EIT Kerr nonlinearity plotted looking at (a) the ground state 

coherence (GSD) and (b) the probe susceptibility. In (a) the real part of GSC is plotted versus 

the imaginary part of the GSC parametrically as functions of time. The dashed red (green) line 

shows the rise (fall) dynamics, and the black solid line shows all possible steady state values. 

In (b) the rise dynamics for the real and imaginary parts of the probe susceptibility are plotted 

as a functions of time. The parameters used for these calculations are R ≈ 5Γ and for t ≤ 0 

δ R = 0, while for t > 0 δ R = 1.2R). 

Raman detuning for t > 0. We also recall that the steady state GSC is given by 

ρ (ss) 

21 (δ R ) = 

ρ (−) 

21 R 

(R + Γ) − iδ R 

. (3.9) 

Finally, we recall from chapter 2 that χ p ∝ i(1 − ρ 21 /ρ (−) 

21 ). 

Figure 3.2 shows two different ways of understanding the EIT Kerr dynamics. 

In Fig. 3.2(a) the rise of the EIT Kerr effect (red dashed line) is seen as the GSC 

spiraling from δ R = 0 to its new steady state δ R = 1.2R. The fall of the EIT 

Kerr effect (green dashed line) is a straight line back towards the dark state GSC. 

In Fig. 3.2(b) the rising transients for the probe susceptibility are plotted as a 

function of time. Both the real and imaginary parts of the susceptibility oscillate


Rise tiem (τR ) 

10 -1 

10 0 δ 

/ R 

τ Ref,rise 

τ Abs,rise 

10 -2 

10 -1 10 0 10 1 

Raman detuning ( ) R 

Figure 3.3: Solid lines show the 1/e rise time for absorption and refraction resulting from the 

EIT Kerr nonlinearity. Rise times are plotted logarithmically as a function of the magnitude 

of the Raman detuning δ R . All quantities are dimensionless and normalized by the optical 

pumping rate R, and Γ/R = .2. For small detunings the rise times asymptotically approach 

τ Abs = 2.15/(R + Γ) and τ Ref = 1/(R + Γ). For large two-photon detunings the rise times 

asymptotically approach τ Abs = 1.2/δ R and τ Ref = 0.63(R + Γ)/δR 2 . These asymptotes are 

shown as the dashed lines. 

as damped simple harmonic oscillators. For the probe susceptibility the falling 

transients, which are not shown, would be a decaying exponential. 

3.1.2 Rise-times in optically thin media 

From transient curves, like those in Fig. 3.2, it is possible to find the 1/e rise 

times for the refractive and absorptive parts of the EIT Kerr nonlinearity. Figure 

3.3 shows these rise times as a function of Raman detuning. Also shown as dashed 

lines are the asymptotic values for the rise times times. The fall times correspond 

to rise times with δ R ≪ R because regardless of the initial Raman detuning for


the fall dynamics, the final steady state is always in Raman resonance. As Fig. 

3.3 shows there are two regimes for which the asymptotic rise times are accurate: 

a nonlinear regime with δ R ≫ R and a linear regime with δ R ≪ R. 

For most applications of the refractive EIT Kerr nonlinearity, it is desirable to 

work in the linear regime near Raman resonance. This is the case we primarily 

focus on. Besides the obvious advantages of being linear, the linear regime also 

provides the maximum nonlinear susceptibility with the minimum absorption. In 

the linear regime, the asymptotic expressions for the 1/e rise and fall times are 

given by 

lim τ Ref, rise = lim τ Ref, fall = 1 

δ R →0 δR →0 R + Γ 

(3.10) 

and the absorptive rise and fall times are 

lim τ Abs, rise = 1 + c 1 

δ R →0 R + Γ 

(3.11) 

and 

lim τ Abs, fall = 1 

δ R →0 R + Γ , (3.12) 

where c 1 = ln{2 + ln[2 + ln(2 + . . .)]}, and the subscripts Ref and Abs refer rise 

times for the refractive and absorptive changes. 

These rise and fall times can be understood intuitively as the ratio ∆θ/ ˙θ, where 

∆θ is the total total phase change of the GSC and ˙θ is the rate at which the GSC


phase changes ‡ . The rate of GSC phase change is always ˙θ = δ R [this can be 

deduced from Eq. (3.8)]. The total phase change can be deduced by considering 

that in the linear regime Eq. (3.9) simplifies to 

( ) 

21 (z) ≈ ρ(−) 21 R 

R + Γ exp δ R 

i , (3.13) 

R + Γ 

ρ (ss) 

such that ∆θ = δ R /(R + Γ). Combining these two results we expect the rise time 

to be approximately 

τ ∼ ∆θ/ ˙θ = 1/(R + Γ). 

In the nonlinear regime the above analysis is more challenging an less insightful. 

In this regime the asymptotic rise times are 

lim τ Abs, rise = arccos (e−1 + (R + Γ)/δ R ) 

(3.14) 

δ R →∞ δ R 

and 

lim τ Ref, rise = (1 − e−1 )(R + Γ) 

. (3.15) 

δ R →∞ 

δ 2 R 

3.1.3 Rise times in optically thick media 

It is more challenging to calculate the EIT Kerr rise times in optically thick 

media because both the fields and atoms are simultaneously evolving. Adding to 

‡ For consistency throughout this work, we use θ for the phase of the GSC, and φ for the 

phase of the probe field


the complication is the fact that the evolution of the fields (atoms) is conditional 

on the state of the atoms (fields). This interplay between atoms and fields results 

in a longer rise time. We will show that the rise time is proportional to the optical 

thickness of the medium. 

In the linear regime, the rise time in optically thick media can be estimated 

as τ ≈ ∆θ/ ˙θ, similar to the optically thin case. To do this we must first calculate 

the steady state change GSC phase. 

We recall that the susceptibilities are 

χ p ≈ −i α ( 

) 

0c 

∗ 

ρ 22 1 − ρ 21 /ρ (−) 

21 

(3.16) 

ω p 

and 

χ c ≈ −i α 0c 

ω c 

ρ 11 

( 

1 − ρ 21 /ρ (−) 

21 

) 

. (3.17) 

In the SVEA, the Maxwell equations for the fields are 

( ∂ 

∂z + 1 ) 

( ) 

∂ 

Ω P = − α 0ρ 22 

1 − ρ∗ 21 

Ω 

c ∂t 

2 ρ (−)∗ P (3.18) 

21 

and 

( ∂ 

∂z + 1 ) 

( ) 

∂ 

Ω C = − α 0ρ 11 

1 − ρ 21 

Ω 

c ∂t 

2 ρ (−) C . (3.19) 

21 

Assuming steady state and negligible probe absorption, we can make the approx-


imation |Ω P (z)| ≈ |Ω P (0)|, and Eq. (3.18) can be integrated in space to obtain 

Ω P (z) ≈ Ω P (0)e − α 0 ρ(−) 22 

hz−Rz 

2 0 ρ∗ 21 (z′ )/ρ (−)∗ 

( 

≈ 

Ω P (0) exp 

− zα 0ρ (−) 

22 

2 

iδ R 

R + Γ 

21 (z ′ ) dz ′i 

) 

, (3.20) 

where we have used the fact that in steady state ρ 21 (z)/ρ (−) 

21 (z) ≈ exp(δ R /(R+Γ)). 

Similarly, 

( 

) 

zα 0 ρ (−) 

11 iδ R 

Ω C (z) ≈ Ω C (0) exp 

. (3.21) 

2 R + Γ 

Finally, the ground state coherence as a function of position is 

( 

ρ 21 (z) ≈ ρ (−) 

21 (0) exp i (1 + αz/2)δ ) 

R 

. (3.22) 

R + Γ 

From Eq. (3.22) we see that ∆θ(z) = (1 + α 0 z/2)δ R /(R + Γ) making the rise time 

approximately 

τ ≈ ∆θ/ ˙θ = 1 + α 0z/2 

R + Γ . 

In order to obtain a more accurate expression for the EIT Kerr rise time we 

numerically simulated the EIT Kerr effect in an optically thick system and calculated 

the rise time as a function of optical depth. The results of this simulation 

are shown in Fig. 3.4. By curve fitting the numerical derived rise times we found


8 

7 

6 

Absorp. 

rise-time 

Rise Time τ(R+Γ) 

5 

4 

3 

2 

1 

Absorp. 

fall-time 

Refractive 

rise & fall times 

0 

0 2 4 6 8 10 

Optical Depth (αL) 

Figure 3.4: Numerical simulations of the EIT Kerr 1/e rise time as a function of optical depth. 

Rise times were calculated using the fields. 

that the optically thick equivalents of Eqs. (3.10)-(3.12) are 

lim τ Ref, rise = lim τ Ref, fall = 1 + ˜α pz/4 

δ R →0 δR →0 R + Γ , (3.23) 

and the absorption rise and fall times are 

lim τ Abs, rise = 1 + (1 + ˜α pz/2)c 1 

δ R →0 R + Γ 

(3.24) 

and 

lim τ Abs, fall = 1 + 3c 1 ˜α p z/8 

. (3.25) 

δ R →0 R + Γ 

The EIT Kerr dynamics for optically thick media in the nonlinear regime

3.2. QND MEASUREMENTS & KERR NONLINEARITIES 80 

δ R ∼ R are not considered here. We also do not consider the effects of the signal 

group velocity, but they are discussed in Chapter 5. 

3.2 QND measurements & Kerr nonlinearities 

Quantum non-demolition (QND) measurement of photon number is only one of 

several application of the EIT Kerr nonlinearity. However in evaluating the effects 

of the EIT Kerr rise time on potential applications, it is insightful to focus on 

QND measurements because all potential EIT Kerr applications are either based 

on QND measurements or are based on similar operating principles. 

Some of the EIT Kerr applications based on QND measurements are creating 

optical number states, high efficiency photon counting detection, preselection for 

single photons for single-photon-on-demand sources (a prerequisite for deterministic 

QIP), and as a trigger for quantum bit regeneration [103, 104]. 

There are also several related applications of the EIT Kerr effect that do not 

directly involve QND measurements but operate on nearly identical principles. 

These applications include: quantum computing with optical nonlinearities, twoqubit 

quantum gates such as the controlled phase gate, quantum entanglement of 

two optical wavepackets, and quantum state preparation for macroscopic quantum 

states (i.e. schrodinger cat states). 

Each of these applications makes similar demands of the EIT Kerr nonlinearity 

to what would be required for a QND measurement capable of resolving a single


photon. QND measurements for large photon numbers have been observed previously 

[133, 134], but a QND measurement capable of resolving a small photon 

number has not been realized. Although we have not actually performed a QND 

measurement in this work, the measurements discussed here provide valuable insight 

into the limitations and potential of the EIT Kerr nonlinearity for QND 

measurements. 

3.2.1 QND measurement and SNR 

Before discussing EIT Kerr nonlinearities and QND measurements, we consider 

the simpler case of an idealized Kerr nonlinearity with instantaneous rise time. 

The primary figure of merit for the QND measurements is the signal to noise ratio 

(SNR) to be defined shortly. There are also several other criteria that must be 

satisfied in order for the measurement to be a QND measurement [135, 136]. We 

will show that these criteria can be expressed in terms of the SNR. 

Figure 3.5 shows an experimental arrangement for a photon number QND 

measurement using a Kerr medium. The signal and probe fields come together 

and interact in a Kerr medium, resulting in the phase modulation of the probe 

[i.e. 

E p (L, t) = E p (0, t) exp(iφ(t)), 

where φ(t) = ω p χ (3) 

p,ss|E s (L, t)| 2 /2c is the phase modulation]. The field amplitudes


Probe 

Beam 

Signal 

Pulse 

PBS 

Kerr 

Medium 

Local 

Oscillator 

PD 

PBS 

i sig(t) 

50/50 

BS 

PD 

PD 

i (t) 

diff 

Balanced 

Homodyne 

Detection 

Figure 3.5: Simplified diagram of a QND measurement of photon number using an optical Kerr 

nonlinearity and balanced homodyne detection. 

for the signal and probe fields are 

E l (z, t) = E l (z, t)e iω l(z/c−t) 

l = {s, p}, (3.26) 

where E l (z, t) are the slowly varying amplitudes of the fields. The probe field is 

a monochromatic cw field and the signal is a transform limited pulse that is zero 

for times t < 0 and z > 0. Also, we assume that the signal pulse has a temporal 

width τ sig that is long compared the length of the Kerr medium (i.e. τ sig ≫ L/c). 

The probe phase is measured via balanced homodyne detection, resulting in a 

difference current of 

i diff (t) = 2ηq eleA cɛ 0 |E p E LO | 

sin[φ(t)], (3.27) 

ω p 2 

where A is the area of the beams, E LO is the amplitude of the local oscillator,


η is the quantum efficiency of the photo diodes (PD), and q ele is the charge per 

electron. The signal of interest is the time integral of the difference current, and 

this charge can be expressed in terms of quantum mechanical operators as 

∫ t 

〈 ˆQ〉(t) = dt ′ i diff (t ′ ) (3.28) 

t−T int 

= ηq [( 

ele 

¯n LO + ¯n p + 2 √¯n LO¯n p 〈sin 

2 

ˆφ〉 

) ( 

− ¯n LO + ¯n p − 2 √¯n LO¯n p 〈sin ˆφ〉 

)] 

, 

≈ 2ηq ele 

√¯nLO¯n p C 1¯n s , 

where we have assumed the small angle approximation (sin φ ≈ φ), and the integration 

time is T int § . The other parameters in Eq. (3.28) are defined as 

¯n p ≡ 〈ˆn p 〉 = T int I p /Aω p and ¯n LO ≡ 〈ˆn LO 〉 = ¯n P /κ 2 (3.29) 

are the expectation value for the number of photon in the cw probe and local 

oscillator fields over a time interval T int . The local oscillator is much larger than 

the probe (i.e. κ 2 ≪ 1). C 1 ≡ φ/¯n s is the proportionality constant between the 

phase modulation φ averaged over the time interval [t − T int , t] and ¯n s ≡ 〈ˆn s 〉 

is the expectation value for the number of signal photons. C 1 ∝ χ (3) 

pss should be 

thought of as the “size” of the Kerr effect (i.e. in cw operation or if the Kerr 

effect is instantaneous then C 1 

is proportional to the nonlinear susceptibility). 

§ In order to achieve the maximum SNR the integration time which should be chosen such 

that T int ∼ (τ sig + τ Kerr ), where τ Kerr being the rise time of the Kerr nonlinearity.


The variance of the charge is given by 

√〈(∆ ˆQ) 2 〉 = q ele 

√ η 

[¯n p 

(1 + 1 κ 2 

) 

] 1/2 

+ 2¯n2 p 

C 2 1 〈(∆ˆn s ) 2 〉 , (3.30) 

κ 2 

such that the SNR is 

SNR = 

〈 

√ 

ˆQ〉 

√ 

≈ 2C 1¯n s η¯np , (3.31) 

〈(∆ ˆQ) 2 〉 

where it is assumed that the probe and local oscillator fields are both coherent 

states (i.e. they obey the statistics 〈(∆n i ) 2 〉 = ¯n i for i = p, LO, the signal field is 

a number state (i.e. 〈(∆n s ) 2 〉 = 0), and ∆X ≡ X − 〈X〉 . 

There are several quantifiable criteria that must be simultaneously satisfied 

in order for a measurement to be a QND measurement. Also, many of the EIT 

Kerr applications require a certain number resolution. Fortunately, it is relatively 

straightforward to express the resolution and QND criteria in terms of the SNR. 

The problem of resolving individual number states using a QND Kerr measurement 

was considered by Haus in Ref. [136]. It was found that given the signal 

field is in a particular number state n s , the probability for correctly identifying 

the photon number is given by 

P (n s |n s ) ≈ erf(SNR/2 √ 2n s ), (3.32) 

This SNR does not include technical noise.


where erf(x) = ∫ x 

0 dy2 exp(−y2 )/ √ π is the error function [136]. A SNR of 4 will 

provide 95% probability of correct detection, and a SNR of 10 gives a probability 

of a detection error is below 10 −6 . Because the resolution of a QND measurement 

is exponential in the SNR, QND measurements may be the ideal method to satisfy 

the demanding requirements for detection efficiency in linear optics quantum 

computing [109]. 

There are three criteria that must be satisfied for a measurement to be a QND 

measurement. 

1. A QND measurement should be non-demolition. This means that the signal 

should not be destroyed by its interaction with the probe. Tracing over the 

probe, the signal observable at the output should be highly correlated with 

the signal observable at the input. In our case, this signal observable is the 

number of signal photons. Expressed mathematically, the “nondemolition” 

condition is C s 

defined as 

= C(∆ˆn in 

s , ∆ˆn out ) ≈ 1, where the correlation function is 

s 

C(∆ ˆX, ∆Ŷ ) ≡ 〈∆ 

√ 

ˆX∆Ŷ 〉 . (3.33) 

〈∆ ˆX 2 〉〉〈∆Ŷ 2 〉〉 

2. The QND measurement should be a measurement (i.e. when we measure the 

probe observable, we should obtain information about signal observable). In 

our case, the probe observable is the charge ˆQ which is proportional to the 

probe phase. This “measurement” condition can be expressed mathemati-


cally using another correlation function (i.e. C m = C(∆ˆn in 

s , ∆ ˆQ) > 0). For 

a perfect QND measurement C m = 1. 

3. Finally, a QND measurement should be quantum (i.e. the probe and signal 

observables should be quantum correlated). In other words, the conditional 

variance between signal and probe observables should be below the standard 

quantum limit. This characteristic of QND measurements is often referred 

to as the quantum state preparation property, and it implies that if neither 

signal or probe observable is squeezed, then the probe and signal will be 

non-separable. This “quantum” condition can be expressed mathematically 

by the inequality 

V s|p = V (∆ˆn out 

s |∆ ˆQ) < 1 (3.34) 

( 

= 〈∆ˆn out2 

s 〉 1 − C 2 (∆ˆn out 

s , ∆ ˆQ) 

) 

/〈ˆn out 

s < 1〉, 

where V s|p is a conditional variance, and the last line is only true for observables 

with Gaussian statistics. Also, we follow the convention of Refs. 

[135, 137] by normalizing the conditional variance to the standard quantum 

limit, such that quantum correlations are implied by V s|p < 1. 

A measurement is considered to be QND if both C s + C m > 1 and V s|p < 1. 

With the exception of the first criterion, these QND criteria can be expressed 

in terms of the SNR. Satisfying the first criterion, non-demolition of the signal,


requires knowledge of the specifics of the Kerr measurement. For the EIT Kerr 

effect, the signal intensity after the medium is 

( 

I s (L) ≈ I s (0) exp − 

α 0Lγ 

∆ s (1 + κ 1 ) 

) 

, 

where κ 1 ≡ |Ω c /Ω p | 2 and ∆ s ≫ γ. Thus, the correlation function is C s ≈ 

exp[−α 0 Lγ/∆ s (1 + κ 1 )]. As long as α 0 Lγ ≪ ∆ s (1 + κ 1 ) the first condition C s ≈ 1 

is satisfied. This also enables the simplification 

C m = 1/ √ 1 + ¯n 2 s/SNR 2 〈(∆n s ) 2 〉, (3.35) 

and the conditional variance is 

V s|p = (1 − C m 2 ). (3.36) 

This implies that both the measurement condition C m ≈ 1 and the quantum state 

preparation condition are satisfied when SNR/¯n s ≫ 1/ √ 〈(∆n s ) 2 〉 ‖ . Assuming 

the signal, probe, and local oscillator fields are all coherent states, the conditions 

‖ All three of the QND measurement criteria seem to have been developed under the assumption 

that the input noise for both signal and probe observables is at the standard quantum 

limit-i.e. 〈(∆n s ) 2 〉 = 〈n s 〉 and 〈(Q) 2 〉 = 〈Q〉. All of the conditions lose their utility when one of 

the variances goes to zero. This is somewhat unfortunate because one of the primary conditions 

of a QND measurement is that repeated QND measurements obtain the same results. However, 

according to the three criteria given for QND measurements repeated non-ideal QND measurements 

become progressively more poor because the variance of the input signal is smaller with 

each repeated measurement.


for a good EIT Kerr QND measurement simplify to α 0 Lγ ≪ ∆ s (1 + κ 1 ) and 

SNR ≫ √¯n s . 

3.2.2 SNR for EIT Kerr Measurements 

In order for the results of Sec. 3.2.1 to be useful, we must apply these results to the 

EIT Kerr nonlinearity, which does not have an instantaneous rise time. In fact, 

the EIT Kerr rise time is typically rather slow (in the msec. to µsec. range for the 

experiments we discuss). Understanding the EIT Kerr rise time and dynamics are 

essential in optimizing the SNR and determining the limitations of the EIT Kerr 

effect. We consider only the linear regime δ R ≪ R, and in this section we focus 

primarily on optically-thin media ˜αL ∼ 1 although some of the results also apply 

to optically thick media. Most of the discussion of EIT Kerr QND measurements 

in optically thick media is reserved for chapter 5. Finally, in anticipation of the 

experimental systems in Chapters 4 and 5, we consider the specific case that the 

probe, signal, and coupling transitions are all on the D 1 line of an alkali metal 

vapor (i.e. |µ D1 | 2 = πγɛ 0 c 3 /ω0, 3 and ω 0 ≈ ω i with i = {c, p, s}) ∗∗ . 

For both optically thin and thick media the phase modulation of the probe 

∗∗ In the chapter 4 experiment the signal is on the D 2 line not the D 1 . The only significant 

change would be that the D 2 and D 1 dipole moments are related by |µ D2 | 2 = 2|µ D1 | 2 such that 

for chapter 4 C 1 in the last line of Eq. (3.41) should be a factor of two larger.


field can be written as 

φ(L, t) = ω P χ (3) ∫ t 

p,ssL 

c 2 ɛ 0 0 

dt ′ h(t ′ )I s (0, t − t ′ ), (3.37) 

where h(t) is the normalized impulse response function for the Kerr nonlinearity †† . 

For optically thin media the EIT Kerr impulse response function is 

⎧ 

⎪⎨ 

exp[−t/τ eit ] 

τ Kerr 

t ≥ 0 

h thin (t) ≈ 

⎪⎩ 0 otherwise 

, (3.38) 

where it is assumed L/v g,s ≪ τ eit and the signal’s propagation delay through the 

cell can be ignored [42]. The difference current then becomes 

i diff (t) ≈ 2ηq elen p ω P χ (3) 

p,ssL 

√ × . . . (3.39) 

T int κ1 c 2 ɛ 0 

⎧ 

( ) ∫ ⎪⎨ exp − 

t t 

τ Kerr 0 dt′ I s(t ′ ) 

τ Kerr 

τ Kerr ≫ τ sig 

. 

⎪⎩ I s (t) τ Kerr ≪ τ sig 

The maximal SNR is achieved when τ eit ≫ τ sig and T int = 5τ eit /4. 

Even though the time dependent current i diff is highly dependent on the EIT 

Kerr dynamics, the integrated charge 〈 ˆQ〉 is independent of the dynamics if the 

integration time is sufficiently long. 

In the limit of extremely long integration 

†† The assumption of an impulse response function implies that the Kerr response is a linear 

in the signal intensity i.e. the linear regime with δ R ≪ R.


times the charge becomes 

p,ssL 

lim 〈Q〉 = 2ηq elen p ω s χ (3) 

√ 

T int →∞ T int κ1 c 2 ɛ 0 

n s ω s 

A . (3.40) 

As long as the integration time is long enough to satisfy the inequality T int ≥ 

(τ sig + τ eit ), we can reasonably approximate 〈Q〉 ≈ lim Tint →∞〈Q〉. Throughout the 

remainder of this chapter, we assume T int = τ eit and τ sig ≪ τ eit . The fact that the 

signal 〈 ˆQ〉 is independent of the dynamics is important for ensuring the accuracy 

of the QND measurement. However, the SNR is not independent of the dynamics, 

which raises the question of how it can be maximized and what is its maximum 

value. 

Using Eq. (3.40) we find that for an EIT Kerr nonlinearity 

C 1 ≈ ω2 sLχ (3) 

p,ss 

2Ac 2 ɛ 0 T int 

, 

≈ 

πγ 

2∆ s A k 2 0 

˜αL 

4RT int 

, (3.41) 

where in the last line we have used the D 1 dipole moment µ D1 to simplify the 

expression. From the definition for ¯n p in Eq. (3.29) we have 

n p = (1 + ˜αL/4)Ak2 0 

π(1 + κ 1 ) 

T int 

τ eit 

, (3.42) 

where we have used the fact that τ eit = (1 + ˜αL/4)/(R + Γ). The SNR is largest


for optically thick media, in which case the SNR simplifies to 

SNR 

n s 

≈ 

√ √ √ πη 1 γ ˜αL τeit 

√ . (3.43) 

2 k 0 A ∆ s 1 + κ 1 T int 

There are three parameters that can be optimized to increase the SNR; optical 

depth ˜αL, beam area A, and signal detuning ∆ s . The beam area has a lower limit 

of A ≥ 2π/k0 2 due to diffraction. Also, the optical depth and signal detuning are 

not truly independent. We recall that the nondemolition criterion required that 

the inequality ∆ S /γ ≫ √ α 0 L/(1 + κ 1 ) be satisfied. 

Thus, the nondemolition 

criterion and measurement criterion have competing objectives. It is not possible 

to adjust the optical depth and signal detuning to increase the SNR without also 

increasing the absorption of the signal. 

Let us suppose that in a given experiment it is possible to focus down to the 

diffraction limit A = 2π/k 2 0 

and that 5% absorption of the signal field can be 

tolerated (i.e. αL/(1 + κ 1 ) ≤ 0.05∆ 2 s/γ 2 ). Also, assume that τ eit ≫ τ sig such 

that T int ≈ τ eit . Then, the SNR can be no larger than 0.16 √ η. The maximum 

phase shift per signal photon achievable under these circumstances is φ/¯n s = 

C 1 = γ/4∆ s ≪ √ 1 + κ 1 /4 √ α 0 L. Although the phase shift could theoretically 

be made large by decreasing the signal detuning, for practical reasons the signal 

detuning is typically ∆ s 

≥ 10γ, making the maximum phase shift per signal 

photon about two hundredth of a radian ‡‡ . From this discussion, it may seem 

‡‡ This result is similar to the result found by Harris and Hau [56] for a pulsed signal and


that the prospects are not good for single-photon resolution using EIT Kerr QND 

measurements. However, there are some work-arounds to this problem that are 

discussed in Chapters 5 and 6. 

For example, it is possible to decrease the integration time T int by slowing 

down the group velocity of the signal field. 

When the group velocities of the 

probe and signal are matched, the phase modulation on the probe will be much 

shorter temporally and a shorter integration time is required. The details of this 

process are discussed in Chapter 5. 

pulsed probe. In the case considered by Harris and Hau the signal and probe pulses had a 

limited interaction length due to temporal walk-off arising from the different group velocities for 

probe and signal fields. The probe has a very slow group velocity due to EIT, and the signal 

group velocity is approximately c. In our case the signal is pulsed but the probe is cw, and 

the limitation on the maximum phase shift is due to the dependence of the EIT Kerr rise time 

on optical thickness. There are greater similarities between the two cases than are immediately 

apparent, and these are discussed in chapter 5

93 

Chapter 4 

EIT Kerr dynamics Experiment 

Over the past ten years there has been considerable interest in the EIT Kerr 

nonlinearity and its potential applications. We have shown that the dynamics of 

the EIT Kerr effect play an important role in determining the limitation of the 

EIT Kerr effect for pulsed applications. This chapter discusses the first reported 

experiments measuring the dynamics of the EIT Kerr nonlinearity. 

The first section details the experimental parameters and apparatus. We then 

present steady state measurements of the EIT Kerr effect in Sec. 

4.2, before 

presenting the transient EIT Kerr measurements in Sec. 4.3. Finally, in Sec. 

4.4 we discuss the agreement between theory and experiment, and explain why 

certain small discrepancies exist. 

4.1 Experiment 

Using the Mach-Zehnder interferometer as shown in Fig. 4.1, we observed the 

transients of EIT on the D 1 line of rubidium 85. The transients are created

4.1. EXPERIMENT 94 

ECDL2 

AOM 

80 MHz 

Signal 

50/50 

BS 

ECDL1 

Homodyne 

Detection 

λ 

2 

PBS 

Pump 

Probe 

AOM 

1.5 GHz 

Computer 

DAQ 

50/50 BS 50/50 BS 

Spectral 

filter 

PBS 

Mach-Zehnder 

Interferometer 

Rb Cell 

PZT 

Magnetic Shielding 

PBS 

Figure 4.1: Schematic of the experimental apparatus. The probe beam is sent through a Mach- 

Zehnder interferometer to measure changes in the relative phase between the two arms of the 

interferometer. The coupling and signal beams interact with the probe in the rubidium cell before 

being separated from the probe via polarization and spectral filtering. A computer controlled 

data acquisition system records and averages the difference signal from balanced homodyne 

detection. 

by pulsing the signal beam that was about 2 GHz red detuned from the Rb 85 

D 2 line. Previous experiments have used frequency modulation spectroscopy to 

observe similar Stark shifts and refractive EIT Kerr nonlinearities [47], but such 

experiments are unable to observe the transient behavior because they require 

lock-in amplification. Thus, this is the first direct observation of the dynamics 

and rise times of the refractive EIT Kerr nonlinearity. 

Mutually coherent coupling and probe fields were obtained from the same external 

cavity diode laser (ECDL1) with wavelength λ = 795 nm. ECDL1 was


tuned to the coupling frequency. The probe frequency was obtained by double 

passing through a 1.5 GHz acousto-optic modulator (AOM) such that the frequency 

difference between probe and coupling was 3 GHz, which is the ground 

hyperfine splitting for Rb 85 . Deriving both EIT fields from a single laser ensured 

the excellent phase coherence necessary for good EIT. Additionally, we are able 

to control the frequency difference between probe and coupling precisely via the 

microwave frequency synthesizer driving the AOM. 

The signal was obtained from a second laser (ECDL2) with wavelength λ = 

780 nm, and pulses were created using a 80 MHz AOM with a rise time of 50 ns. 

Both ECDL lasers are New Focus Vortex lasers. The signal and coupling beams 

are combined on a 50/50 beam splitter (BS) and are both horizontally polarized in 

order to be transmitted through a polarizing beam splitter (PBS) into the probe 

arm of the Mach-Zehnder interferometer (MZI). 

The MZI provides measurements of relative phase changes between the two 

arms of the interferometer. The MZI has a strong reference/local-oscillator arm 

and a weaker probe arm ∗ . After the probe has propagated through the rubidium 

cell, both polarization and spectral filtering are used to isolate the probe field 

from the coupling, signal, and any four-wave mixing products. The mirror after 

∗ Both arms of the interferometer have the same power after the first 50/50 beam splitter 

(BS), but the probe arm becomes significantly weaker due to attenuation from the rubidium 

cell, spectral filter, and various other optics.


the rubidium cell is mounted on a piezo-electric actuator to control the relative 

phase between the MZI arms. 

At the output of the Mach-Zehnder interferometer the homodyne signal is 

measured using two reverse biased Hamamtsu S2830 photodiodes soldered for 

differential detection. After the photodiodes a 1 MHz low-noise transimpedance 

amplifier prepares the signal for computer controlled data acquisition (DAQ). CW 

measurements are recorded using the computer, and the fast transient data are 

recorded and averaged using a 1.5 GHz bandwidth oscilloscope. The homodyne 

signal is given by 

Q ∝ |E LO | |E p | sin [φ pzt + Re(χ p )k 0 L/2] , (4.1) 

where E LO is the amplitude of the field in the local-oscillator arm, E p is the 

probe field amplitude immediately before the last 50/50 BS, φ pzt 

is the piezo 

controlled phase difference between the MZI arms. The other parameters have 

been defined in early chapters. If amplitude measurements are desired instead of 

phase measurements, the local-oscillator beam and one of the photo diodes can 

be blocked. 

The apparatus for the rubidium cell consists of three layers of magnetic shielding 

with a long solenoid inside the innermost layer of magnetic shielding. 

In 

addition to containing 20 torr of nitrogen buffer gas, the rubidium cell contains


5P 3/2 

5P 1/2 λ = 780 nm 

ΩC↔ΩS 

ΩP 

λ = 795 nm 

F=3 

F=2 

↔↔ 

5S 1/2 

y-position (mm) 

2 

1 

0 

-1 

-2 

-2 -1 0 1 2 

x-position (mm) 

Figure 4.2: Level diagram of rubidium 85 D 1 and D 2 lines with the primary transitions excited 

by the probe, coupling, and signal fields. The inset shows the approximate beam sizes and 

shapes for the probe (w p,x = 1 mm and w p,y = 1.8 mm), coupling (w c = 2.1 mm), and signal 

(w s = 2.2 mm) fields. 

both rubidium isotopes in their natural abundance † . 

The cell is cylindrical in 

shape with a length of 30 mm and a diameter of 25.4 mm. The number density 

of rubidium atoms is controlled by heating the cell with electrical strip heaters 

positioned between the outer and middle layers of magnetic shielding. 

Figure 4.2 shows the hyperfine levels for the Rb 85 D 1 and D 2 lines with laser 

transitions that are driven in the experiment. The hyperfine splittings ‡ for the 

5P 1/2 (361 MHz) and 5P 3/2 (220 MHz) hyperfine levels are much smaller than 

the Doppler width (600 MHz), such that the individual hyperfine levels cannot 

be resolved. This is convenient because we can treat the excited states as a single 

† The natural abundance of rubidium isotopes are 27.8% Rb 87 and 72.2% Rb 85 

‡ The hyperfine splitting for Rb 85 and Rb 87 D 1 and D 2 lines can be found in Appendix 1.


hyper-fine level. Finally, the signal field Stark shifts both ground hyperfine levels 

resulting in a total change in Raman detuning of 

δ R (Ω S ) = δ R (0) + 

Ω 2 ∆ hf 

4∆ S (∆ S + ∆ hf ) , (4.2) 

where ∆ hf = 2π × 3.0357 GHz is the hyperfine splitting of the ground states. 

The inset in Fig. 4.2 shows the approximate 1/e 2 intensity beam shapes for the 

coupling, probe, and signal fields (in the experiment the beams were not perfectly 

centered). The coupling and signal beams were essentially identical with beam 

radius of w c ≈ w s ≈ 2.2 mm, while the probe beam was smaller and slightly 

elliptical w p,x ≈ 1.0 mm and w p,y ≈ 1.8 mm. 

Table 4.1: Several measured parameters for the EIT line shapes with no signal 

present. The beam powers were measured immediately before the PBS in front 

of the vapor cell. αL is the maximum absorption experienced far from Raman 

resonance. ˜αL is the difference between maximum absorption and the absorption 

minimum near Raman resonance, and EIT FWHM is self-explanatory. 

Temper- Power αL ˜αL EIT 

ature Coupl. Total EIT Depth FWHM) 

58 ◦ C 340(20) µW 5.2(3) 3.7(1) 11.5(7) kHz 

58 ◦ C 1.0(1) µW 5.3(2) 4.3(1) 31.9(1.2) kHz 

58 ◦ C 1.9(1) µW 5.5(2) 4.6(1) 58.1(1.8) kHz 

48 ◦ C 1.0(1) µW 2.0(1) 1.5(1) 31.9(7) kHz 

52 ◦ C 1.0(1) µW 3.3(1) 2.6(1) 33.2(7) kHz 

72 ◦ C 1.0(1) µW 20.4(9) 16.4(7) 41(3) kHz

4.2. CW MEASUREMENTS 99 

4.2 CW measurements 

To verify the dependence of the EIT Kerr dynamics on optical depth, EIT 

linewidth, and Raman detuning, we measured the transient and CW response 

of EIT to the signal field while varying the cell temperature (optical thickness 

αL), coupling intensity (EIT linewidth (R + Γ)), and signal intensity (Stark shift 

δ R ). The different parameters at which data was taken is summarized in Table 

4.1. Each row in Table 4.1 corresponds to a different set of data in which both 

cw and transient measurements were taken for several signal powers ranging from 

0.2 mW to 4.9 mW. In all measurements the probe power was 40 µW, and the 

signal detuning was ∆ S = 2π × 1.9 GHz. 

Figure 4.3 shows the cw probe absorption as a function of Raman detuning for 

several different signal powers, with a cell temperature of 58 ◦ C and a coupling 

power of 0.34 mW (first row in Table 4.1). Figures 4.6-4.11 also use this same 

data set. In addition to the shift in absorption minimum resulting from the Stark 

shift, the transparency also becomes broader and less deep with increasing signal 

power. These effects are mostly due to the additional decoherence created by the 

signal field (i.e. Γ(Ω S ) = Γ(0) + Ω 2 S γ/4∆2 S ). This additional decoherence is most 

noticeable when the optical pumping is of the same order of magnitude as the 

decoherence rate R ∼ Γ, which is the case for the data corresponding to row 1 

of Table 4.1. For all other sets of parameters in Table 4.1 the optical pumping R 

is a factor of 3-6 times larger than for row 1 and the broadening is significantly


5 

Probe absorption (αL) 

4 

3 

2 

P s = 4.5 mW 

P s = 2.5 mW 

P s = 1.5 mW 

P s = 0.4 mW 

P = 0 

s 

-20 -10 0 10 20 30 


Figure 4.3: Probe absorption versus Raman detuning for different cw signal powers. The signal 

detuning is ∆ S = 2π×1.9 GHz and the coupling (probe) power is 340 µW (40 µW). The fact 

that with no signal field Raman resonance is already shifted away from the expected zero is due 

mostly to the 20 Torr N 2 buffer gas shifting the hyperfine splitting (∆ω hf = 2π × 240 Hz per 

Torr N 2 ). The probe and coupling also cause a small (about 1 kHz) Stark shift at these powers.


Stark Shift (kHz) 

12 

10 

8 

6 

4 

2 

Strark Shift due to 

Signal Field Only 

theory 

αL=1.7, P C 

= 1 mW 

αL=2.6, P = 1 mW 

C 

αL=3.8, P C 

=340 µ W 

αL=4.3, P = 1 mW 

C 

αL=4.6, P =1.9 mW 

C 

αL=16.4, P C 

= 1 mW 

0 

0 1 2 3 4 5 6 

Signal Power (mW) 

Figure 4.4: Stark shift due to the signal field for several settings of coupling power and EIT 

optical depth. 

less noticeable. In discussing the measurement data we use row 1 of table 4.1 to 

show all of the steps in the experiment. The same experimental procedures and 

analysis were also carried out for the other rows of table 4.1 although they are 

not discussed in the same detail as row 1. 

From Fig. 4.3 it is possible to find the Stark shift as a function of signal power 

by determining the displacement of the absorption minimum for the probe. These 

measured signal Stark shifts are shown in Fig. 4.4 for all rows of Table 4.1. 

There is also a Stark shift due to the coupling field interacting with the probe 

transitions and vice-versa due to the probe field interacting with the coupling 

transitions. This Stark shift is plotted as a function of coupling power in Fig.


FWHM (kHz) 

Stark Shift (kHz) 

80 

EIT Linewidth 

60 (FWHM) 

40 

20 

0 

5 

4 

3 

2 

1 

0 

Stark shift due to Coupling 

and Probe fields only 

theory 

αL≈1.5 

αL≈2.6 

αL≈4.5 

αL≈16 

theory 

αL≈1.5 

αL≈2.6 

αL≈4.5 

αL≈16 

0 0.5 1 1.5 2 2.5 

Coupling Power (mW) 

Figure 4.5: Stark shift due to the coupling and probe fields as a function of coupling power. 

Also, shown is the FWHM of the EIT resonance versus coupling power. The signal was off for 

all of these measurements. 

4.5. Also, plotted in Fig. 4.5 is the transparency linewidth versus coupling power. 

The linewidth and Stark shift are also slightly dependent on the probe power, 

but probe power is sufficiently small that it can be neglected. Figures 4.4 and 4.5 

exhibit the expected linear dependence on intensity § . 

Using homodyne detection we measured both real and imaginary part of the 

probe susceptibility. Figure 4.6 shows cw homodyne data corresponding to the 

first row of table 1. The homodyne signal was measured by first adjusting the 

piezo voltage such that φ pzt = 0 and then stepping the the Raman detuning over 

§ Note that for large optical depth, ˜αL ≈ 16 the EIT linewidth was slightly wider than for 

smaller optical depth and all other parameters the same. This may be due to gain from four-wave 

mixing.


2 

Homodyne signal (arb. units) 

1 

0 

-1 

-2 

-3 

-30 -20 -10 0 10 20 30 40 50 


Figure 4.6: Raw CW homodyne data for coupling power of 340 µW. Two sets of data are shown; 

one for P s = 0 and the other for P s = 4.5 mW. The dotted lines show the homodyne envelope 

(i.e. the maximum and minimum values which the homodyne signal can obtain). 

the desired range of Raman detuning with a dwell time of 200 ms at each Raman 

detuning. 

In Fig. 4.6 the homodyne signal is plotted versus Raman detuning for the first 

row of Table 4.1 (black lines are in the absences of the signal field, and green lines 

correspond to a signal power of 4.5 mW). The dashed curves show the envelopes for 

the maximum and minimum homodyne signals. These envelopes are determined 

by measuring the homodyne signal while the phase φ pzt is scanned over several 

periods and fitting the measured homodyne current to a sinusoid. This is repeated 

at each Raman detuning. In Fig. 4.7 we have used the data from Fig. 4.6 to 

calculate the real and imaginary parts of the probe susceptibility. The homodyne


3 

2 

Probe susceptibility (αL/2) 

2 

1 

0 

3 

2 

1 

0 

-30 

Re(χ p 

) 

Im(χ p 

) 

δ = 8 kHz 

R 

δ = 0 

R 

-20 -10 0 10 20 30 40 50 


1 

0 

-1 

1 

0 

-1 

Probe susceptibility (∆φ in Rad.) 

Figure 4.7: Measured absorption αL and phase shift ∆φ derived from the homodyne data in 

Fig. 4.6. In addition to obtaining refraction information we are able to confirm the absorption 

measurements shown in Fig. 4.3. 

envelop gives the absorptive (imaginary) part of the probe susceptibility, and the 

refractive (real) part of the probe susceptibility is extracted using Eq. (4.1). 

Using homodyne detection to measure the absorption has a couple of advantages 

over measuring the transmitted intensity directly. 

First, it increases the 

dynamic range because we are measuring the field amplitude E P directly instead 

of measuring the probe intensity I P 

∝ |E P | 2 . Without this increased dynamic 

range it would not have been possible to measure EIT line shapes with optical 

depths of ˜α P = 16.4. Also, when we scan the piezo-controlled phase φ pv we obtain 

a sinusoidal signal that can be used for lock-in amplification to extract the signal 

out of the noise.

4.3. TRANSIENTS MEASUREMENTS 105 

Homodyne Signal (arb. units) 

1 

0 

-1 

Homodyne 

Envelope 

Signal 

0 100 200 500 600 700 800 

Time ( µ sec.) 

Figure 4.8: Homodyne measurements as a function of time (P s = 4.5 mW, P c = 340 µW). The 

signal was turned on at t = 0 and then turned off again at t = 500 µs. Similar to Fig. 4.6, the 

envelope shows the maximum and minimum values obtainable by the homodyne signal. 

4.3 Transients Measurements 

Homodyne detection was also used to measure the transient response of the EIT 

fields to the fast turn-on and turn-off of the signal field. In this case, the frequency 

difference between the probe and coupling field was optimized for peak probe 

transmission with the signal field off. The signal was then turned on abruptly and 

left on for approximately 1 ms, and then the signal is turned off abruptly. Figure 

4.8 shows the transient homodyne signal and homodyne envelope when the peak 

signal power was 4.5 mW and all other parameters are given by the first row in 

Table 4.1. The square dashed curve shows the shape of the signal pulse. From 

these measurements, the absorptive and refractive transients of the probe field can


Phase shift (rad.) 

Absorption ( αL/2) 

1 

0.8 

0.6 

0.4 

0.2 

0 

1.5 

1 

0.5 

0 

0 50 100 150 

Time ( µ sec.) 

Re(χ p ) and Im(χ p ) 

0 10 20 


30 

P s =4.5 mW 

P s =2.5 mW 

P s =1.5 mW 

P s =0.8 mW 

P s =0.4 mW 

P s =0.2 mW 

500 550 600 650 700 

Figure 4.9: Change in refraction and absorption as a function of time for different values of peak 

signal power. Both refraction and absorption are measured relative to their respective values 

when the signal is off. 

be extracted. Figure 4.9 shows the change in probe absorption and probe phase 

for different choices of peak signal power. These curves were extracted from the 

transient homodyne measurements identically to the process used to obtain Fig. 

4.7. The inset in Fig. 4.9 shows the cw absorption and dispersion curves from 

Fig. 4.7 in the absence of a signal . The vertical lines in the inset show the Stark 

shifts corresponding to several signal powers. 

This inset is intended as a tool 

to help visualize the change in the real and imaginary parts of the susceptibility 

corresponding to a given signal power and Stark shift. 

There are several characteristics of these transients worth noting. First, the


steady-state changes in absorption and refraction correspond well with the crossings 

in the inset where vertical Stark shift lines cross the susceptibility. For example, 

the change in refraction reaches a maximum near P s = 2.5 mW, and for 

signal powers greater than P s = 2.5 mW the steady-state change in refraction 

decreases. Also, for small signal powers (e.g. P S = 0.2 mW and P S = 0.4 mW) 

the steady state change in refraction is larger than the change in absorption. 

This corresponds with the fact that near Raman resonance the real part of the 

susceptibility is linear in Raman detuning while the imaginary part is quadratic. 

However, we will see that there is also a linear change in the absorption due to 

the signal field’s contribution to decoherence (i.e. Γ(Ω S ) = Γ(0) + 2Ω 2 S γ ⊥/∆ 2 S ). 

Finally, for small signal powers (i.e. in the linear regime with δ R ≪ R) the rise 

time is a constant, but in the nonlinear regime δ R ∼ R corresponding to large 

signal powers the rise times become slightly shorter and the oscillatory nature of 

the transients becomes more noticeable. 

Additional insight comes from considering the dynamics of the GSC. In Fig. 

4.10 we have used the inverse mapping of Eq. (2.44) to obtain the measured 

GSC transients. The real part of the GSC is plotted versus the imaginary part of 

the GSC parametrically as a function of time. The rise time dynamics look very 

similar to transient trajectory shown in the theory plot in Fig. 2.7. The fall time 

dynamics are essentially straight lines back towards the dark state GSC. In Sec. 

4.4 we return to this plot and compare it to the theory from chapter 3.


0.5 

0.4 

Dynamics of Ground State Coherence 

Rise Dynamics 

Im 

ρ 

( 21 (-) 

/ρ 21 

) 

0.3 

0.2 

0.1 

P s =4.5 mW 

0 

P s =2.5 mW 

P s =1.5 mW Fall Dynamics 

P s =0.4 mW 

-0.1 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 

ρ 21 

( ) 

Re (-) 

/ρ 21 

Figure 4.10: Trajectories of the ground state coherence GSC in the complex plane. GSC was 

derived using Eq. 4.1 and the refraction and absorption transients in Fig. 4.9. Steady state 

GSC values (black line) and transient data (non-black textured lines) are both shown. The 

curved/spiraling lines show the rise time dynamics (δ R ≠ 0), while the straighter lines show the 

fall time dynamics (δ R = 0).


200 

100 

Rise time ( µ sec.) 

50 

20 

10 

5 

2 

theory τ (Ref..) 

rise 

theory τ (Amp.) 

rise 

meas. τ (Ref.) 

rise 

meas. τ (Ref.) 

fall 

meas. τ (Amp.) 

rise 

meas. τ (Amp.) 

fall 

0.5 1 2 5 10 20 

Stark shift (kHz) 

Figure 4.11: Rise times of the refraction and absorption resulting from the EIT Kerr nonlinearity 

(P c = 340). Squares and triangles show the measured refractive rise and fall times. Diamonds 

and circles show the measured absorptive rise and fall times, and solid curves are theory for the 

rise times. The values ˜αL = 3.75 and R + Γ = 2π kHz are used to obtain the theory curve fit. 

Figure 4.11 shows measured 1/e rise and fall times for the transient measurements 

presented in Fig. 4.9 corresponding the the first row of Table 4.1. The 

solid lines are theoretical rise times from Fig. 3.3 modified to account for optical 

thickness [i.e. the refractive rise times are multiplied by (1 + ˜αL/4) and the 

absorptive rise times are multiplied by 1 + ˜αLc 1 /(2 + 2c 1 )]. The fall times are 

essentially independent of the size of the Stark shift as expected because δ R = 0 

always when the signal field is off. The theory curves agree very well with the 

measured rise times, except for the absorptive rise and fall times for small Stark 

shifts. For small Stark shifts the change in absorption is very small (see Fig. 4.9) 

making this discrepancy of little practical importance.


100 

(a) 

(b) 

Rise & fall times (µsec.) 

10 

2 

100 

10 

2 

αL ~ = 1.7 

R+Γ = 12 kHz 

~ αL = 4.3 

R+Γ = 12 kHz 

1 10 

(c) 

~ αL = 2.6 

R+Γ = 12 kHz 

~ αL = 4.6 

R+Γ = 20 kHz 

1 10 


(d) 

Figure 4.12: Rise and fall time measurements for the conditions (a) T = 48 ◦ C and P c = 1 mW, 

(b) T = 52 ◦ C and P c = 1 mW, (c) T = 58 ◦ C and P c = 1 mW, and (d) T = 58 ◦ C and 

P c = 1.9 mW. The definitions for measured data and curve fits are the same as in Fig. 4.11, 

and the parameters used in the curve-fits are given in each figure. 

Figure 4.12 shows the rise time plots for the data sets corresponding to the 

second through fifth rows of Table 4.1. For each data set in Fig. 4.12, we include 

the EIT half width at half maximum HWHM=R + Γ, and EIT depth which were 

used to calculate the theory curves. 

Finally, Fig. 4.13 shows the measured and theoretical rise times in the linear 

regime as a function of optical thickness. The measured refractive rise and fall 

times fit the theory curve well. For small optical depths (i.e. αL < 6) the 

absorptive rise time agrees with theory. However, the absorptive fall times differ 

from the theory, and both absorptive rise and fall times disagree with theory for 

large optical depths. It is not clear why these differences exist. However, these

4.4. DISCUSSION OF MEASUREMENTS 111 

100 

Rise Time (µs.) 

80 

60 

40 

20 

Ref. rise (meas.) 

Ref. rise (theory) 

Ref. fall (meas.) 

Ref. fall (theory) 

Abs. rise (meas.) 

Abs. rise (theory) 

Abs. fall (meas.) 

Abs. fall (theory) 

0 

0 2 4 6 8 10 12 14 16 

Optical Depth (αL) 

Figure 4.13: Rise and fall times for the refractive and absorptive parts of the EIT Kerr nonlinearity 

versus optical depth. The Stark shift is small (δ R ≪ (R + Γ)) for all measurements and 

theory. 

discrepancies only effect the absorptive rise time and they are not important for 

applications based on the refractive EIT Kerr nonlinearity. 

4.4 Discussion of Measurements 

The measured EIT Kerr transients show good agreement with the theory derived 

in chapters 2 and 3. However, there are a few small discrepancies between theory 

and experiment, which have rather simple explanations. They mostly arise from 

either approximations in the theory or the fact that in the experimental quantities 

are averaged over spatially varying intensities (i.e. the experiment uses gaussian 

beams rather the plane waves assumed in the theory). 

Most of the approximations made in the theory are well justified, but our


Im 

ρ 

( 21 (-) 

/ρ 21 

) 

Measured GSC Dynamics 

0.6 

0.4 

0.2 

0 

(a) 

0 0.2 0.4 0.6 0.8 

ρ 

( 21 

) 

Re (-) 

/ρ 21 

Im 

ρ 

( 21 (-) 

/ρ 21 

) 

Theoretical GSC Dynamics 

0.6 

0.4 

0.2 

0 

(b) 

0 0.2 0.4 0.6 0.8 

ρ 

( 21 

) 

Re (-) 

/ρ 21 

Figure 4.14: (a) measured transient (solid line with dots) and steady state (solid line) GSC and 

(b) transient (dashed line and steady state (solid line) GSC calculated using the theory from 

Chapter 3. Both plots show the normalized real part of the GSC plotted versus the imaginary 

part of the GSC. The GSC’s are plotted parametrically as a function of time (transient data), 

or as a function of Raman detuning (steady state data). The transient data show GSC starting 

out in steady state for zero Raman detuning, and evolving toward steady state when the Raman 

detuning is δ R ≈ 1.2(R + Γ). After which the Raman detuning is sharply switched back to zero, 

and the GSC again evolves toward steady state. 

choice of neglecting the increased decoherence created by the signal field [i.e. 

Γ(Ω S ) ≈ Γ(0) + Ω 2 Sγ/4∆ 2 S] (4.3) 

leads to some rather noticeable differences between theory and experiment . First, 

in Fig. 4.3 it is obvious that increased decoherence at large signal powers makes 

the transparency resonance broader and shallower. 

The effects of the signal dependent decoherence are also obvious in Fig. 4.10, 

Neglecting the signal dependent decoherence is a good approximation because it doesn’t 

change the predictions fundamentally and it makes the theory much simpler. However, this 

omission results in noticeable but small inaccuracy when the approximate theory is compared 

to the exact theory or experiment.


where this decoherence makes the steady state GSC non-circular. Figure 4.14 

compares the measured GSC (a) to the idealized theory (b). 

By ignoring the 

signal dependent decoherence, the theory predicts that the steady state GSC 

[Fig. 4.14(b) solid line] makes a circle. The measurements show a deformed 

circle that comes to a cusp near Raman resonance. An exact theory including the 

signal-dependent decoherence agrees with these measurements. 

The transients 

curves for theory and experiment are also slightly different due to the omission of 

signal dependent decoherence in the theory. In spite of these small differences the 

measured and theory curves look remarkably similar. 

The EIT Kerr theory assumes plane waves, but the experiment uses approximately 

Gaussian beams. Many of the EIT Kerr predictions for plane waves must 

be modified when we average over spatially varying field intensities. For example, 

when plane waves are considered the EIT rise time has a very simple relation to 

the EIT linewidth–i.e 

τ Ref = 1 + ˜αL/4 

HW HM , (4.4) 

where the EIT half width at half maximum is HWHM = R + Γ. However, when 

we average the rise dynamics become 

∫ ∫ 

Re[χ p (t)] ≈ 

{ [ 

dxdyRe[χ SS 

p (x, y)] 1 − exp − 1 + ˜αL/4 ]} 

, (4.5) 

R(x, y) + Γ


and EIT line shape becomes 

Im[χ p (δ R )] ≈ α p 

∫ ∫ 

δ 2 R 

dxdy 

δR 2 + R2 (x, y) . (4.6) 

These modification require that an extra multiplicative factor f is added to Eq. 

(4.4), such that 

τ Ref = f 1 + ˜αL/4 

HW HM . (4.7) 

Calculating the factor f is further complicated by the fact that spatial diffusion 

coarse grains the spatial averaging. The characteristic diffusion length is l D (x, y) = 

√¯vl/R(x, y), where ¯v is mean velocity, l is the mean free path length between 

Rb 87 –N 2 collisions. Most atoms in a characteristic volume ‖ encounter the same 

average field intensity during an optical pumping period regardless of variations 

in the field intensities such that 

∫ ∫ 

R(x, y) ≈ 

dx ′ dy ′ Ω2 (x ′ , y ′ ) 

γ eff 

exp [ (x − x ′ ) 2 + (y + y ′ ) 2] /2l D 2 (x, y). (4.8) 

A couple of simple observations allow us to calculate f for our experimental 

conditions. 

In the experiment we have described transit-time broadening the 

dominant source of decoherence Γ, which means that the diffusion length is related 

to the beam waist by R/Γ ∼ (w p /l D ) 2 (the probe beam waist w p is the smallest 

‖ The characteristic volume is defined simply as a cube with sides given by the characteristic 

diffusion length.


beam waist). Thus, when R ∼ Γ, the diffusion length is approximately equal to 

the beam diameter and f ≈ 1. By comparing the measured rise times with the 

measured linewidths, we are able to find f empirically. For coupling powers of 

P c = 0.34 mW, P c = 1 mW, and P c = 1.9 mW, we find that f ≈ 1, f ≈ 1.3, 

f ≈ 1.4 respectively. This agrees with the fact that as the optical pumping rate 

increases, the diffusion length decreases and the value of f should monotonically 

approaches its maximum value. The maximum value of f ≈ 1.5 can be calculated 

using spatial averaging while assuming an infinitesimal diffusion length ∗∗ . 

Finally, regardless of our efforts to decrease the decoherence Γ and increase the 

optical pumping rate R, the best transparency we could achieve is ˜αL ≈ 5/6αL. 

We believe that the maximum transparency was limited by the fact that 4 out of 

5 of the dark states are only partially dark states. For a ratio of Ω 2 P /Ω2 c = 0.2, the 

partially dark states have a transparency of about ˜αL ≈ 7/8αL. In the experiment 

there is about a 2 kHz decoherence rate due to limited dwell time for the atoms 

in the laser beams. 

∗∗ In order that calculate f = 1.5 we had to make a few educated assumptions about the 

overlap of the probe and coupling beams.

116 

Chapter 5 

EIT Kerr nonlinearity with slow 

signal group velocity 

In optically thick media (αL ≫ 1), the dynamics of the EIT Kerr effect depends 

on the relative group velocities of the probe and signal fields. In Chapter 3, 

when the EIT Kerr dynamics and the SNR for QND measurement were derived 

for optically thick media, it was assumed that the signal group velocity v g,s was 

close to c and that the EIT medium was short relative to the duration of the 

signal pulse (i.e. L ≪ v g,s τ sig ). In this chapter, we discuss how the group velocity 

of the signal affects the EIT Kerr dynamics and the SNR for QND measurements. 

The most important effect we discuss is that the maximum SNR and XPM are 

achieved when the signal group velocity is slowed till it equals the slow-light group 

velocity of the probe. 

Although matching the probe and signal group velocities is a necessary condition 

for achieving the maximum SNR and XPM, it is not a sufficient condition. 

To illustrate this point, consider the case when the probe and coupling fields are

5.1. SLOW LIGHT EIT KERR NONLINEARITY 117 

matched pulses. In this case, rather than slowing the signal’s group velocity, the 

probe’s group velocity is increased to approximately c; thus achieving matched 

group velocities. Although the group velocities are matched, the desired increase 

in phase shift and SNR is not achieved because the signal group velocity is not 

matched to the EIT Kerr dynamics. 

To achieve the effects we discuss in this 

chapter, the signal group velocity must be slowed to the slow-light group velocity 

of the probe. 

In Sec. 5.1, we generalize the EIT Kerr dynamics theory from Chapter 3 to 

account for the group velocities. In Secs. 5.2 and 5.3, we present the details of 

experimental observations for the EIT Kerr dynamics when the group velocity of 

the signal is slow. 

5.1 Slow Light EIT Kerr nonlinearity 

Our analysis of the EIT Kerr effect and dynamics in Chapter 3 led us to the 

conclusion that under the most ideal conditions, QND measurements using the 

EIT Kerr nonlinearity could only achieve a SNR of less than one per photon and a 

phase shift of a few hundredths of a radian per photon in the signal field. However, 

in Chapter 3 we did not consider the possibility of changing the group velocity for 

the signal. 

We recall from Chapter 3 that the QND signal is the integrated charge 

〈Q〉/¯n s = 2q ele η¯n p C 1 / √ κ 2 , (5.1)


where ¯n p ∝ T int is the expectation value for the number of probe photons in the 

integration time T int , C 1 = ¯φ/¯n s ∝ 1/T int is the XPM per signal photon and ¯φ is 

the average XPM over the integration time. If the integration time is sufficiently 

long, the signal will be independent of the EIT Kerr dynamics. 

However, the 

signal to noise ratio 

SNR ≡ 

〈Q〉 

√ 

〈(∆Q)2 〉 ≈ 2C 1¯n s 

√ η¯np ∝ 1/ √ T int , (5.2) 

is not independent of the dynamics. In Chapter 3, we assumed the impulse response 

function h(t) for the XPM had a width proportional to the refractive EIT 

Kerr rise-time τ eit = (1 + ˜αL/2)/(R + Γ), which is true when signal pulse propagates 

with group velocity v g,s ≈ c. Recall that the impulse response function was 

defined such that the time dependent phase modulation is 

φ(L, t) ∝ 

∫ t 

0 

dt ′ h(t ′ )I s (0, t − t ′ ), (5.3) 

where I s (z, t) is the signal intensity. However, when the probe and signal group 

velocities are similar, the impulse response function and the required integration 

time can change significantly in optically thick media. 

To calculate the XPM impulse response function, we consider the linearized


equation of motion for the GSC ρ 12 = ρ ∗ 21 

∂ 

( 

) 

∂t ρ 12 = Rρ (−) 

12 − (iδ R (Ω S ) + R + Γ)ρ 12 , (5.4) 

and the Maxwell wave equations for the fields in the SVEA are 

( ) 

∂ 

∂t + v ∂ 

g,s Ω S = 0, (5.5) 

∂z 

and 

( ∂ 

∂t + c ∂ ∂z 

) 

Ω P ≈ iω 0 

2 (χ pΩ P ) 

≈ −˜αc ( Ω P − ρ 12 Ω 2 /Ω ∗ C) 

. (5.6) 

In the last line of Eq. (5.6) we have used the fact that 

χ P ≈ i N|µ 32| 2 

ɛ 0 γ 

( 

1 − ρ12 Ω 2 /Ω ∗ CΩ P 

) 

. (5.7) 

Obtaining a general analytic solution for these equations is difficult, but they 

are straightforward to simulate numerically. Figure 5.1 shows several simulations 

of Eqs. (5.4)-(5.6) for different signal group velocities. The signal intensity (top) 

and probe phase (bottom) are plotted as a function of the time they exit the 

medium. The group velocity of the probe and all other parameters except v g,s


Normalized Signal 

Intensity (arb. units) 

1 

0.8 

0.6 

0.4 

0.2 

v g,s = 25v g,p 

v g,s = 2v g,p 

v g,s = v g,p 

v g,s = v /2 g,p 

Probe Phase (rad.) 

0 

-0.2 

-0.4 

-0.6 

-0.8 

0 L/v 

g,p 

Time (arbitrary units) 

2L/v g,p 

Figure 5.1: Numerical simulations of the EIT Kerr dynamics for various ratios of probe and 

signal group velocities. The signal intensity and probe phase are both plotted as a function of 

the time they exit the EIT Kerr medium. 

were held constant. The optical depth was ˜αL = 270, and the width of the signal 

pulse was F W HM = 10R. 

When the group velocities are matched, the phase pulse is approximately the 

same shape as the signal intensity, and the the largest peak phase modulation has 

is obtained. When the group velocities are not matched, the edges of the phase 

pulse correspond to times t 1 = t 0 + L/v g,p and t 2 = t 0 + L/v g,s where t 0 is the 

time that the signal peak enters the medium. Thus, the peak height of the phase 

pulse is roughly proportional to ∣ ∣ t1 − t 2 −1 ∣ 

= ∣vg,p − v g,s L ∣ ∣ ∣ , for ∣t1 − t 2 ≥ τsig . 

This result can be derived analytical by further simplifying Eqs. (5.4)-(5.6) under 

the assumptions of the linear regime (δ R ≪ 1), a weak probe field (Ω C ≫ Ω P ),


no decoherence (Γ = 0), and Raman resonance in the absence of the signal field 

[δ R (Ω s = 0) = 0]. Also, for simplicity we assume that at the medium entrance, 

the fields Ω P (0, t) and Ω C (0, t) are real positive constants. These assumptions allow 

us to write Ω P (z, t) ≈ exp(iφ(z, t))Ω P (0, t) and ρ 21 (z, t) ≈ exp(iθ(z, t))|ρ (−) 

21 |, 

where |φ(z, t)| ≪ 1 and |θ(z, t)| ≪ 1 such that Eqs. (5.4)-(5.6) simplify to 

∂ 

∂t θ ≈ R (φ − θ)) − δ R, (5.8) 

δ R (z, t) = Ω2 s(0, t − z/v g,s ) 

4∆ s 

(5.9) 

and 

( ∂ 

∂t + c ∂ ) 

φ ≈ − ˜αc (φ − θ) . (5.10) 

∂z 2 

These equations can be thought of as describing two separate processes; the 

generation of the probe phase and the propagation of the probe phase. By analyzing 

these two processes individually and then recombining them, it is straightforward 

to calculate an optically thick impulse response function h thick (t). 

First, let us consider the propagation of the probe phase modulation in the 

absence of the signal field. With δ R = 0 Eq. (5.8) can be integrated exactly to 

obtain 

( 

θ(z, t) ≈ R φ − 1 ∂φ 

R ∂t + 2 ) 

∂ 2 φ 

R 2 ∂t + . . . . (5.11) 

2


Plugging Eq. (5.11) into Eq. (5.10) and simplifying gives to first order 

( 

∂ 

n g,p 

∂t + c ∂ ) 

φ = 0, (5.12) 

∂z 

where the group index for the probe is n g,p = 1 + c˜α/2R and the probe group 

velocity is v g,p = c/n ∗ g,p . By leaving out higher order terms we are neglecting 

absorption and higher order dispersion terms. To understand the broadening of a 

phase pulse we consider that the index of refraction for a Lorentzian EIT resonance 

is 

√ 

1 + χp ≈ 1 + χ p /2 = 1 + αc ( 

− δ ) 

R 

2ω p R + δ3 R 

+ i αc δR 

2 

R 3 2ω p R + . . . , (5.13) 

2 

and the different orders of dispersion † are given by 

β j = ∂j 

ω n(ω). (5.14) 

∂ωj Agrawal [138] has shown that the pulse broadening due to dispersion alone is 

∗ It is interesting to note that nominally the group and phase velocities have reversed roles. 

The phase modulation propagates at the group velocity, while the probe energy propagates at 

the phase velocity. Typically we think of the energy propagating at the group velocity. This 

reversal of roles is in name only and the situation we describe is consistent with what is known 

of the group and phase velocities. However, this nominally role reversal illustrates the fact that 

one must be explicit and careful when discussing the velocity of light in EIT systems. 

† Technically these dispersion terms apply to amplitude pulses and not phase pulses. However, 

our assumption of 1 ≫ φ ensures that on average ∂ 2 φ/∂t j ≫ (∂φ/∂t) j for i ≥ 2, which means 

that to a good approximation these expressions also apply to the dispersion of a phase pulse.


given by 

[ ( 2 

∆t(z) 

∆t(0) = β2 z 

1 + 

+ 

2∆t (0)) 1 ( ) ] 2 1/2 

β3 z 

, (5.15) 

2 2 4∆t 3 (0) 

where the pulse width ∆t is the temporal variance of the pulse. Similarly, the 

broadening due to absorption alone is 

[ 

] 1/2 

∆t(z) 

∆t(0) ≈ αz 

1 + 

, (5.16) 

2∆t 2 (0)R 2 

where both Eqs. (5.15) and (5.16) assume Gaussian beams. In the limit of 

large optical depth the dispersive pulse broadening can be minimized by choosing 

∆t(0) = (3˜αL/4) 1/3 /R, which gives a dispersive broadened pulse width of 

∆t(z) = ∆t(0) √ 3/2. For this initial pulse width, absorptive broadening will be 

the dominant form of broadening for large optical thicknesses, and the final pulse 

width will be ∆t(L) ≈ √˜αL/R. This is a lower bound for the pulse width after 

propagating through a Lorentzian EIT medium regardless of the initial pulse 

shape if the phase modulation. 

When the Stark shift from the signal field is included in Eq. (5.8), it acts as 

a source term for the phase modulation φ. To understand the generation of the 

phase pulse, it is simplest to think about an optically thin medium αL ≈ 1. In 

an optically thin medium we can ignore the probe phase φ in Eq. (5.8), and the


medium phase θ is calculated by integrating Eq. (5.8) to obtain 

θ(z, t) ≈ 

∫ ∞ 

0 

δ R (z, t − τ) exp(−Rτ)dτ. (5.17) 

We are primarily interested in the case when v g,s ∼ v g,p such that v g,s ≪ c, and 

this allows us to ignore the time derivative in Eq. (5.10). Thus, Eq. (5.10) can 

be integrated in space to obtain 

∫ z 

φ 0 (z, t) ≈ ˜α θ(z ′ , t) exp [−˜α(z − z ′ )/2] dz ′ (5.18) 

2 0 

≈ ˜α ∫ z ∫ ∞ 

e−˜αz/2 dz ′ e˜αz′ /2 

dτe −Rτ δ R (z ′ , t − τ) 

2 

≈ 

∫ ∞ 

−∞ 

0 

0 

dτh(z, τ)δ R (t − τ, 0), 

where the impulse response function of the medium is 

⎧⎪ 

h(z, t) = 1 − exp [−˜αz 0/2(1 + v g,p /v g,s )] ⎨ exp[− ˜α v g,p 

2 v g,s 

(z 0 − z)] 

1 + v g,p /v g,s ⎪ ⎩ exp [ − ˜α (z − z 2 0) ] 

0 < z ≤ z 0 , L 

, 

z 0 < z ≤ L 

(5.19) 

and where z 0 = tv g,s . Equation (5.19) is approximately equal to Eq. (3.38) in the 

limit v g,s ≫ v g,p . 

To generalize our results to the case of optically thick media, we rewrite the


wave equation for the probe phase [i.e. Eq. (5.20)] to include a source term: 

( 

∂ 

n g,p 

∂t + c ∂ ) 

φ = P φ , (5.20) 

∂z 

where P φ (z, t) ∝ ∫ dz ′ h m (z ′ , t)δ R (z − z ′ , t) is the source term analogous to the 

polarization of the medium in a wave equation for the electric field. Also, 

h m (z, t) = 4˜α exp (−˜α|v g,st − z|/2) (5.21) 

is the optically thin impulse response function when the group velocities are 

matched (i.e. v g,s = v g,p ). Using Eqs. (5.9) and (5.21) and making the change 

of variables ξ = z and τ = t − z/v g,p allows us to integrate Eq. (5.20) exactly to 

obtain 

φ(L, t) = ω pχ (3) ∫ 

pssL 

c 2 ɛ 0 

dt ′ h thick (L, t ′ )I s (0, t − t ′ ) (5.22) 

where 

h thick (L, t) ≈ 

∫ L−t vg,p 

vg,s 

vg,p L−t v g,p 

dξ Rv g,p exp ( − ˜α 2 |ξ|) 

2(v g,p − v g,s )L . (5.23) 

It is interesting to note that in the limit 

lim h thick (L, t) = h m (L, t). (5.24) 

v g,s →v g,p 

The integration time must be as large as the final pulse width [i.e. 

T int ≈

5.2. EXPERIMENTAL APPARATUS 126 

(2 + √˜αL)/(R + Γ)], which results in a SNR of 

SNR 

n s 

≈ 

2γ √ 2˜αLη 

∆ S k 0 

√ 

A(1 + κ2 )) 

√ 

τEIT 

∆t(L) 

(5.25) 

≈ 

2γ √ 2˜αLη 

∆ S k 0 

√ 

A(1 + κ2 )) (˜αL)1/4 /2. 

5.2 Experimental apparatus 

To observe the EIT Kerr dynamics described in Sec. 5.1 and in Chapter 3, we used 

the experimental system shown in Fig. 5.2. This is essentially the same experimental 

apparatus used in Chapter 4, except the probe-coupling EIT is resonant 

with Rb 87 instead of Rb 85 and we have added a signal-coupling field to form an 

EIT pair with the signal field. The four laser frequencies used in the experiment 

are the probe, coupling, signal, and signal-coupling fields. Figure 5.3 shows these 

frequencies relative to the D 1 transitions of Rb 87 and Rb 85 . 

The probe and coupling beams form an EIT pair for Rb 87 , and they are close to 

Raman resonance for the Rb 87 (F = 2 ↔ F ′ = 2 ↔ F = 1) two-photon transition 

as shown in Fig. 5.3(a). In addition to Raman resonance condition, EIT also 

requires strong phase correlations between the probe and coupling fields, which 

we achieve by deriving both probe and coupling beam from the same laser. The 

laser is tuned to the coupling frequency, and part of the beam is picked off and sent 

through a 6.8 GHz electro-optic modulator (EOM) to create the probe frequency.


Homodyne 

Detection 

PD 

PD 

50/50 

BS 

Probe 

50/50 

BS 

Signal- 

Coupling 

Computer 

DAQ 

Spectral 

filter 

PBS 

Magnetic 

Shielding 

PBS 

Signal 

Coupling 

PZT 

APD 

Rb Cell 

Rb Cell 

Probe 

Signal- 

Coupling 

Coupling 

Signal 

Figure 5.2: Diagram of the experiment. At the first 50/50 beam splitter (BS) the probe field 

is divided into two arms of a Mach-Zehnder interferometer. In one arm of the interferometer, 

the probe interacts with the coupling, signal, and signal-coupling fields in a rubidium cell. 

These fields are then filtered using polarization (PBS), spatial (pick-off mirror), and spectral 

filters before the probe field is interfered on a 50/50 BS with the reference arm of the Mach- 

Zehnder interferometer. Balanced homodyne detection is recorded via a computer controlled 

data acquisition (DAQ) system to measure changes in the probe phase. (PD=photo diodes, 

APD= avalanche photo diode, BS=beam splitter, PBS=polarizing beam splitter, PZT= piezoelectric 

actuated mirror)


After the EOM, the probe field is obtained by frequency filtering ‡ to select only 

the lower side band of the frequency modulated field. We found that the best 

contrast for EIT of the probe is achieved when the probe beam diameter is small 

compared to the coupling beam diameter, and when coupling and probe beams 

are perfectly parallel to within experimental accuracy (i.e. slightly better than 

1 mrad). Also, EIT is best when the coupling and probe fields have orthogonal 

linear polarizations. The beam areas and their overlap are shown in Fig. 5.4. 

The signal and signal-coupling beams create EIT in the Rb 85 atoms, and are 

detuned from single-photon resonance as shown in Fig. 5.3(a). They are also 

derived from a single laser to ensure strong phase correlations. The signal-coupling 

frequency is obtained by upshifting the laser frequency by 3.2 GHz using a double 

passed 1.6 GHz acousto-optic modulator (AOM). The signal frequency is obtained 

by upshifting the laser frequency using a 200 MHz AOM. This AOM is also used 

to pulse the signal field with a minimum rise-time of 30 nsec. Both lasers used in 

the experiment were Newfocus Vortex lasers. Similar to the probe and coupling 

beams, the signal and signal-coupling also have orthogonal linear polarizations, 

and have parallel propagation directions to within experimental accuracy. Also, 

the signal beam was smaller than the signal-coupling to ensure good contrast for 

the signal EIT (see Fig. 5.4). 

The signal and probe converge in the cell at an angle of 7 mrad and overlap 

‡ The frequency filter is a temperature controlled solid etalon with finesse of 100 and free 

spectral range of 40 GHz.


F’=2 

F’=1 

(a) 

F’=2,3 

Rb 85 

Rb 87 Coupling 

Probe 

F=2 

Signal 

F=3 

Signal- 

Coupling 

F=1 

F=2 

Transmission 

1 

0.8 

0.6 

0.4 

0.2 

0 

Rb 87 

Rb 85 

Probe 

Coupl. 

Signal 

Signal- 

Coupl. 

F=2 

(b) 

F=3 

F=2 

-5 -4 -3 -2 -1 0 1 2 3 4 5 

Frequency - 377,107.46 (GHz) 

F=1 

Figure 5.3: (a) Level diagram and (b) laser frequencies relative to the Rb 85 and Rb 87 D 1 Doppler 

valleys.


1 

2 

Beam Circumfrance ( 1/e ) 

Sig.Coupl . 

Signal 

Probe 

Coupl. 

Position (mm) 

0 

-1 

-2 

-2 -1 0 1 2 

Position (mm) 

Figure 5.4: 1/e 2 intensity beam radius for the probe, coupling, signal, and signal-coupling 

beams. The probe and signal beams were smaller than the coupling and signal-coupling beams 

respectively in order to ensure good EIT for the entire probe and coupling beams. The probe 

beam is also smaller than the signal beam in order to obtain a uniform Stark shift. 

through the rubidium cell. The vapor cell contains both rubidium isotopes in their 

natural abundance (72% Rb 85 and 28% Rb 87 ). It also contains 20 Torr of nitrogen 

(N 2 ) buffer gas. The rubidium cell was a 30 mm long cylinder with a diameter 

of 25.4 mm, and was magnetically shielded. For the experiments discussed here, 

the optical depth on resonance with the Rb 87 (F = 2 → F ′ = 1) transition (i.e. 

the probe transition) is αL ≈ 7. For the Rb 85 (F = 3 → F ′ = {2, 3}) transition, 

(i.e. the signal transition) the on-resonance optical depth is αL ≈ 40. The large 

optical depth for Rb 85 transitions is why the signal and signal-coupling fields are 

not on resonance [see Fig. 5.3(b)].

5.3. SLOW SIGNAL EIT KERR MEASUREMENTS 131 

Probe Absorption (αL/2) 

3 

2 

1 

signal 

no sig. 

coupl. sig. 

coupl. no sig. 

0 

-150 -100 -50 0 50 100 150 

Probe Raman Detuning (kHz) 

Figure 5.5: Probe absorption as a function of Raman detuning. 

5.3 Slow signal EIT Kerr measurements 

Figure 5.5 shows the probe absorption as a function of Raman detuning for four 

different cases: 1) both signal and signal-coupling off, 2) signal on and signalcoupling 

off, 3) signal off and signal-coupling on, and 4) both signal and signalcoupling 

on. When the signal-coupling is off, the group velocity of the signal is 

approximately c, and when the signal-coupling is on, the signal group velocity 

is approximately c × 10 −4 . Ideally, slowing the signal group velocity would be 

the only effect of the signal-coupling field, but Fig. 5.5 shows that the signalcoupling 

field also has the side effects of making the probe EIT become shallower


Signal Absorption ( αl/2 ) 

and Refraction ( ω [n-1]/c ) 

s 

1 

0.5 

0 

αL/2 

ω (n-1)/c 

p 

-0.5 

-200 -100 0 100 200 

Signal Raman Detuning (kHz) 

Figure 5.6: Measured signal absorption αL as a function of Raman detuning. Also, plotted is 

the dispersion curve calculated as the Hilbert transform of the signal absorption. 

and Stark shifting the Raman resonance for the probe and coupling fields. The 

shape of the probe EIT resonance is qualitatively the same with and without the 

signal coupling field making qualitative comparisons possible. However, a direct 

quantitative comparison is difficult due to the signal-coupling’s effects on the line 

shape. 

In addition to the absorption curves shown in Fig. 5.5, we used homodyne 

detection to measure the probe phase as a function of Raman detuning (not 

shown). In the absences of the signal-coupling the dispersion was measured to 

have a slope of about 5×10 −2 rad/kHz and with signal-coupling on the slope was 

3×10 −2 rad/kHz. From these values we infer a probe group velocity of about 

4-6 km/s depending on whether the signal-coupling is on or off.


Figure 5.6 shows the signal absorption (blue) as a function of the Raman 

detuning for the signal and signal-coupling frequencies. The dispersion curve (red) 

was calculated by taking the Hilbert transform of the absorption curve. The EIT 

for the signal resulted in a signal group delay of about 1.5 µsec with about 80% 

transmission of the signal energy. For a 30 mm long vapor cell, this group delay 

implies a signal group velocity of 20 km/s. When we tried to obtain larger signal 

group delays by tuning the signal frequency closer to resonance EIT for both the 

signal and probe deteriorated quickly with only a marginal increase in the signal 

group delay. We believe this was due to radiation trapping in Rb 85 resulting from 

the combination of large optical density for Rb 85 and the larger beam diameters 

for the signal and signal-coupling beams. 

Thus, in the experiment, the signal 

group velocity was limited to the regime v g,s ∼ 5v g,p and the probe optical depth 

was limited to ˜αL ∼ 3 − 6. 

For our transient measurements of the EIT Kerr nonlinearity, we optimized the 

Raman detunings for both EIT pairs to obtain maximum transmission and then 

pulsed the signal field. Figure 5.7 shows the measured probe phase in the absence 

of a signal-coupling field [shown in Fig. 5.7(a)]. Also, shown in Fig. 5.7(b) is the 

signal intensity at the output of the rubidium cell. Time t = 0 is determined as 

the expectation time for the signal pulse to leave the vapor cell. For long signal 

pulses the probe phase has essentially the same pulse shape as the signal intensity– 

only slightly wider and delayed. For short signal pulses, the leading edge of the


Phase Shift 

(rad.) 

0.2 

0.1 

(a) 

Signal Intensity 

(arb. units) 

0 

5 

4 

3 

2 

1 

0 

-60 

-40 -20 0 20 40 60 80 100 

time (µsec.) 

(b) 

Figure 5.7: (a) measured EIT Kerr phase pulses and (b) measured signal intensity for different 

signal widths. 

phase pulse is approximately the integral of the signal intensity, and the falling 

edge of the phase pulse is a decaying exponential as predicted by Eq. 3.39. The 

probe phase data is also slightly rounded and delayed due to the transimpedance 

amplifier used after the homodyne detection. The transimpedance amplifier has 

a measured propagation delay of 2 µs and a rise-time of 5 µs. 

Deconvolving the effects of the transimpedance amplifier, we found that the 

1/e fall-time of the shortest phase pulses is about 13 µs, which is consistent with 

the observations from Chapter 4, where τ eit = f(1 + αL/4)/HW HM and where 

f ≈ 1.4, αL = 6, and R = 2π × 45 kHz . 

We repeated the measurements shown in Fig. 

5.7 with the signal-coupling


30 

w/o Sig.-Coupl. 

Phase Shift (mrad.) 

20 

10 

with Sig.-Coupl. 

0 

-10 0 10 20 30 40 50 60 70 

Time (µsec.) 

Figure 5.8: Measured (dots) and theory (solid) for the EIT Kerr phase pulses plotted versus 

time (with t = 0 defined as the expectation time for the signal pulse leaving the vapor cell). 

field on. The effects of the slow signal group velocity are most noticeable for short 

signal pulses. Figure 5.8 shows the probe phase as a function of time when the 

signal-coupling is either on or off (i.e. the signal group velocity is v g,s ≈ 2×10 4 m/s 

or v g,s ≈ c). The signal pulse width before the cell is τ sig = 7 µs, and t = 0 is the 

expectation time for the signal pulse leaving the cell. The dots are the measured 

probe phase and the solid lines are numerical simulations of Eqs. (5.4-5.6), where 

the simulations used the measured parameters from the experiment–e.g. optical 

depths, EIT Kerr rise-time, signal absorption etc. The good agreement between 

measurements and the numerical simulations give us confidence in the theory. 

There are two types of effects due to the signal-coupling: effects due to the 

slow signal group velocity, and effects due to changes in the probe EIT line shape.


Probe Phase (mrad.) 

20 

15 

10 

5 

v g,s = 0.5vg,p 

v g,s = v g,p 

v g,s = 5vg,p 

v = 20v 

g,s 

g,p 

0 

-20 -10 0 10 20 30 40 

Time (µsec.) 

Figure 5.9: Numerical simulations of EIT Kerr phase pulse for different signal group velocities. 

For these simulation ˜αL = 3, τ s ig=7 µs, τ eit = 1/R = 15 µs. 

The slower signal group velocity causes the peak phase to advance in time. Also, 

for slow enough signal group velocities, the rising slope of the probe phase will 

become shallower and the falling slope will become slightly steeper. The signalcoupling 

field also has the side effect of changing to the EIT line shape for the 

probe EIT. In Fig. 

5.8 these side effects make the phase pulse shorter with a 

slightly longer fall-time. 

Although experimental limitations prevented matching the signal and probe 

group velocities, using numerical simulations it is possible to explore the parameter 

space for matched group velocities. Figure 5.9 shows the probe phase for numerical 

simulations using the same parameters as the experiment when the signal-coupling


is on (i.e. τ eit =15 µs, αL = 3, τ sig =7 µs, v g,p = 6 km/s etc.) with different group 

velocities for the signal pulse. Phase pulses are plotted for four different signal 

group velocities. The v g,s = 20v g,p and v g,s = 5v g,p are qualitatively similar to 

the measured probe phase in Fig. 5.7. For slower group velocities with v g,s =< 

5v g,p , one can see the rising slope of the phase becomes shallower and the phase 

pulse becomes more symmetric about its peak. For this relatively small optical 

thickness, matching the group velocities does not increase the peak phase shift or 

decrease the optimal integration time as it would in an optically thicker sample. 

In order to achieve the benefits of slow group velocities, the EIT medium 

must be optically thick. For matched group velocities in optically thick media 

we expect an integration time of T match 

int 

≈ (2 + √˜αL)/R. While for fast signal 

group velocities (i.e. v g,s ≈ c), the integration time is T fast 

int 

≈ (1 + ˜αL/4)/R. For 

˜αL < 10 the integration time is shorter for a fast signal group velocity than for 

matched group velocities. 

Figure 5.10 shows the pulse-width of the phase plotted as a function of the 

ratio v g,s /v g,p , when the medium’s optical thickness is ˜αL = 3, ˜αL = 6, and 

˜αL = 12. For the optical depth used in the experiment (i.e. ˜αL = 3), there is 

no narrowing of the phase pulse FWMH when group velocities become matched § . 

However, for large optical thicknesses, the pulse-width decreases when the group 

velocities are matched. However, even for ˜αL = 12 the decrease in the pulse-width 

§ This is just a restatement of what is already obvious from Fig. 5.7

5.4. DISCUSSION 138 

35 

FWHM of Probe phase pulse 

30 

25 

20 

15 

αL = 3 

αL = 6 

αL = 12 

10 

0.5 1 2 5 10 20 

Normalized group velocity of signal (v /v g,p) 

g,s 

Figure 5.10: Phase pulse width (FWHM) plotted versus the ratio between the group velocities 

for the probe and signal fields. Numerical simulations were performed with three optical depths 

˜αL for the EIT. The signal pulse width was 7 µs and the optically thin EIT rise-time was 

τ eit = 1/R = 15 µs. 

is still less than a factor of two. Thus, the optical thickness of the medium must 

be very large before matched group velocities significantly improves the SNR for 

QND measurements. 

5.4 Discussion 

For QND measurement using the EIT Kerr nonlinearity, it is theoretically possible 

to improve the SNR by matching the group velocities for the probe and signal 

fields. However, the advantages offered by matched group velocities only become 

significant for very large optical thicknesses (i.e. αL > 30). 

We have observed some of the effects of nearly matched group velocities using

5.4. DISCUSSION 139 

EIT in two isotopes of rubidium. However, we have not been able to experimentally 

obtain the large optical depths and slow signal group velocities required to 

see decreased pulse widths for the probe phase modulation on the probe and large 

improvements in the SNR for QND measurements. 

It may be possible to improve on this experiment using a different distribution 

of rubidium isotopes and/or a longer rubidium cell in order to decrease the 

radiation trapping. Also, using cold atoms would eliminate the problems associated 

with overlapping Doppler valleys. However, using cold atoms would require 

trapping both isotopes of rubidium simultaneously. The benefits achieved by a 

better matched group velocity experiment probably would not justify the additional 

work.

140 

Chapter 6 

Conclusions and perspective 

As mentioned in the introduction, EIT enhanced Kerr nonlinearities may enable 

many exciting and challenging few-photon applications such as QND measurements 

with single photon resolution. However, some of the early optimism regarding 

the EIT Kerr effect should be tempered. Without resorting to optical 

cavities and/or slow group velocities for the signal field, the EIT Kerr effect can 

only achieve a maximum cross-phase modulation (XPM) of a few hundredths of a 

radian per photon under ideal conditions ∗ . Also under ideal conditions, the signal 

to noise ration (SNR) for QND detection of a single photon is less than one for 

the EIT Kerr effect † . 

The reasons for these limitations is the slow rise-time of the EIT Kerr effect. In 

∗ For the ideal conditions we assume a single velocity class of atoms with the atomic linewidth 

equal to the natural linewidth, the dipole moment for the signal field is for the D 2 line of an 

alkali atom. We also assume no decoherence, the fields are focused down to the diffraction limit, 

and the N system originally proposed by Schmidt and Imamoğlu [1]. 

† This result assumes a constraint of at least 90% transmission for the signal field

141 

the linear EIT Kerr regime (i.e. δ R ≪ R) the rise-time for XPM is proportional to 

the depth of the EIT ˜αL and the optical pumping time t R ≡ 1/R [i.e. τ ref = (1 + 

˜αL/4)/(R + Γ) ≈ ˜αL/4R, where it is assumed that the medium is optically thick 

(i.e. ˜αL ≫ 1) and the EIT condition R ≫ Γ is met]. For cw applications of the 

EIT Kerr effect it is theoretically possible to make the nonlinear susceptibility (i.e. 

χ (3) 

pss ∝ ˜αL/R) arbitrary large by either increasing the optical depth of the medium 

and/or decreasing the optical pumping rate. However for pulsed applications such 

as the few-photon applications that interest us, the figure of merit is the ratio 

between the nonlinear susceptibility and the rise-time, and this figure of merit 

cannot be increased by changing either the optical depth or the optical pumping 

rate. 

The EIT Kerr nonlinearity is still one of the best, if not the best, candidate for 

realizing few-photon applications, but researchers must be clever in their methods 

of utilizing the EIT Kerr effect. We have investigated using a slow group velocity 

for the signal field, and found that theoretically this method could be used to 

achieve arbitrarily large XPM and SNR for QND measurements. However, experimentally 

matching the signal and probe group velocities is challenging due to complications 

in creating EIT simultaneously for two nearly resonant atomic species. 

Also, matched group velocity experiments require very large optical depths before 

significant improvements in XPM and SNR can be observed. These large optical 

depths also pose significant experimental problems.

142 

Recently there have been a couple of proposals to create matched signal and 

probe group velocities using a single atomic species [58, 119]. These eliminate 

some of the problems we encountered in our two atomic species experiment, but 

they still require extremely large optical thicknesses to achieve their objectives. 

Also, they introduce new experimental problems by demanding precise control of 

magnetic fields. 

There are other proposals using EIT and slow light to achieve strong Kerr 

nonlinearities. For example, Friedler et al. have proposed using the long range 

dipole-dipole interactions between two slow-light pulses, where the slow light is 

due to EIT in a ladder system [83]. The large dipole moments required for this 

interaction result from the fact that the highest energy level in the ladder level 

system is a Rydberg state with large quantum number n. 

Some of the best results in XPM and QND measurements have used cavity 

QED [139, 140]. In fact, QND measurements of single photons have very recently 

been realized using microwave photons and super-conducting cavities [140]. However, 

in these experiments the photon can only be coupled into the cavity via the 

spontaneous emission from an excited atom travelling through the cavity. The 

photon remains trapped in the cavity until it is either absorbed by a probe atom 

traversing the cavity (more likely), or the photon leaks out of the cavity (less 

likely). This type of QND measurement is not useful for measuring the number 

of photons in a travelling wave.

143 

Figure 6.1: Conceptual diagram for an EIT Kerr experiment using cold atoms and an optical 

cavity. 

The best QND measurements of photon number for optical fields have also used 

optical cavities, but not in the strong-coupling regime of Cavity QED [133, 134]. 

In these experiments a conventional Kerr nonlinearity was used with an optical 

cavity to increase the intensity of the signal field interacting with the atoms ‡ . 

In EIT Kerr experiments an optical cavity could also be used to increase the 

XPM. This could be accomplished in an experiment like the one shown in Fig. 

6.1. In Fig. 6.1 two red detuned laser create a dipole trap for a cold atom cloud. 

The EIT fields (i.e. probe and coupling) enter from the top, and the signal field 

passes through a confocal Fabry-Perot etalon with finesse § Q. The atomic cloud 

is at the center of the optical cavity. The signal intensity inside the cavity is Q 

‡ This conventional Kerr nonlinearity is simply a three-level ladder system, and is exactly 

identical to the system Schmidt and Imamoğlu used to compare with the EIT Kerr nonlinearity 

and demonstrate 10 orders of magnitude increase in the size of the Kerr nonlinearity. 

§ The symbol Q is used to signify the quality factor of the cavity, which is the same thing a 

finesse.

144 

times larger than outside the cavity resulting in a XPM that is Q times larger. 

For QND measurements the results from chapter 3 for the rubidium D 1 line can 

be generalized to account for a cavity, such that Eq. (3.43) becomes 

SNR 

n s 

≈ 

√ √ 

πη 1 Qγ ˜αL 

√ , (6.1) 

2 k 0 A ∆ s 1 + κ 1 

where it is assumed that T int = τ eit . The nondemolition condition also must be 

modified such that ∆/γ ≫ √ Q˜αL/(1 + κ 1 ). Unlike the no-cavity scenario considered 

in chapter 3, the SNR can theoretically be increased indefinitely without 

violating the nondemolition condition by simultaneously increasing Q and ∆ while 

keeping the ratio ∆/ √ Q constant. 

Using realistic parameters for the experiment shown in Fig. 6.1 it should be 

possible to realize a QND measurement capable of resolving individual number 

states. Consider the case of a cold atomic cloud of Rb 87 , with EIT on the D 1 

line and the signal field on the D 2 line. Assuming the parameters are A = 8π 2 λ 2 0, 

κ 1 = 5, η = 0.7, Q = 10 4 , ˜αL = αL = 4 , and ∆ s ≈ 600γ, then the signal 

absorption is about 15% and the SNR for a single photon of about eight. Thus, it 

is realistically possible to achieve single photon QND measurements. To achieve 

this same result by matching group velocities would require an optical depth of 

˜αL ≈ 10 9 with perfect transparency for the probe field, which is not realistic. 

We are assuming that the atomic cloud has a length of 0.5 mm with a diameter of 0.1mm 

and about 10 5 atoms. The 5 to 1 aspect ratio of the atom cloud keeps the absorption of the 

probe field to a minimum while keeping αL large for the signal field.

145 

Finally, we return to the question of using the EIT Kerr nonlinearity as an 

optical two-qubit quantum logic gate. As we have already mentioned in chapter 1, 

there are several architectures for optical quantum computing that are based on 

Kerr nonlinearities and/or QND measurements [111–116, 141]. However, Shapiro 

has shown theoretically that optical gates based on Kerr nonlinearities have poor 

fidelity [120, 121]. This is in contrast to other researchers who have used a completely 

quantum analysis to predict good fidelity using the EIT Kerr effect in a 

quantum phase gate [118, 142]. 

It is interesting to contrast the conclusions of Refs. [120] and [118]. In Ref. 

[118] the conclusion is, that their “study shows that the implementation of efficient 

EIT-based nonlinear two-qubit gates for traveling single photons is possible” [118]. 

While Ref. [120], which is titled Single-photon Kerr nonlinearities do not help 

quantum computation, comes to the conclusion that “Unfortunately, the phase 

noise that is associated with the causal, noninstantaneous nature of the response 

function precludes high-fidelity operation” of EIT-based nonlinear two-qubit gates 

[120]. Both references use a completely quantum analysis, and both account for 

the dynamics of the Kerr nonlinearity, although their methods are significantly 

different. Ref. [120] derives general analytic expressions for Kerr nonlinearities 

in general, and Ref. 

[118] uses numerical simulations of the EIT Kerr effect 

specifically. 

The crux of the problem lies with understanding the dynamics of the Kerr

146 

nonlinearity and noise in the Kerr nonlinearity due to vacuum fluctuations. In 

this work we have discussed the dynamics of the EIT Kerr nonlinearity at length, 

but we have not discussed the vacuum fluctuations because our treatment is semiclassical. 

It is well known that the quantum vacuum AC Stark shifts the energy levels; 

this energy shift is typically referred to as the Lamb shift. Thus, the noise degrading 

the fidelity of quantum logic gates based on the EIT Kerr nonlinearity arises 

from fluctuations in the Raman detuning due to the Lamb shift. To our knowledge 

the affects of the Lamb shift noise on EIT have not been studied, but such 

a study should answer questions raised about the fidelity of EIT Kerr quantum 

gates and their utility for quantum computing and QIP. This noise may also be 

significant for other EIT Kerr applications.

147 

Bibliography 

[1] H. Schmidt and A. Imamoǧlu, “Giant Kerr nonlinearities obtained by electromagnetically 

induced transparency,” Opt. Lett. 21, 1936 (1996). 

[2] G. Alzetta, A. Gozzini, and G. Orriols, “An experimental method for the 

observations of R. F. transitions and laser beam tansitions in oreinted Na 

vapors,” Nuovo Cimento B 36, 5 (1976). 

[3] G. Alzetta, L. Moi, and G. Orriols, “Nonabsorption hyperfine resonances in 

a sodium vapor irradiated by a multimode dyelaser,” Nuovo Cimento B 52, 

5209 (1979). 

[4] H. R. Gray and C. R. Stround, “Coherent trapping of atomic populations,” 

Opt. Lett. 3, 218 (1978). 

[5] S. E. Harris, J. E. Field, and A. Imamoğlu, “Nonlinear optical processes 

using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107 

(1990). 

[6] E. Arimondo, in Progress in Optics, edited by E. Wolf (Elsevier Science, 

Amsterdam, 1996), Vol. 35, p. 275. 

[7] O. A. Kocharovskaya and Y. I. Khanin, “Coherent amplification of an ultrashortpulse 

in a three-level medium without a population inversion,” JETP 

Lett. 48, 630 (1988). 

[8] M. O. Scully, S.-Y. Zhu, and A. Gavrielides, “Degenerate quantum-beat 

laser: Lasing without inversion and inversion without lasing,” Phys. Rev. 

Lett. 62, 2813 (1989). 

[9] S. E. Harris, “Lasers without inversion: Interferences of lifetime broadened 

resonances,” Phys. Rev. Lett. 62, 1033 (1989). 

[10] E. S. Fry et al., “Atomic coherence effects within the sodium D 1 line: Lasing 

without inversion via population trapping,” Phys. Rev. Lett. 70, 3235 

(1993).

BIBLIOGRAPHY 148 

[11] Y. Zhu, “Lasing without inversion in a V-type system: Transient and steadystate 

analysis,” Phys. Rev. A 53, 2742 (1995). 

[12] P. Hemmer, S. Ezekiel, and J. C.C. Leiby, “Stabilization of a microwave 

oscillator using a resonance raman transition in a sodium beam,” Opt. Lett. 

8, 440 (1983). 

[13] M. Scully and M. Fleischhauer, “High-sensitivity magnetometer based on 

index-enhanced media,” Phys. Rev. Lett. 69, 1360 (1992). 

[14] M. Fleischhauer and M. Scully, “Quantum sensitivity limits of an optical 

magnetometer based on atomic phase coherence,” Phys. Rev. A 49, 1973 

(1994). 

[15] D. Budker, V. Yashchuk, and M. Zolotorev, “Nonlinear magneto-optic effects 

with ultranarrow widths,” Phys. Rev. Lett. 81, 5788 (1998). 

[16] M. Linberg and R. Binder, “Dark states in coherent semiconductor spectroscopy,” 

Phys. Rev. Lett. 75, 1403 (1995). 

[17] R.Wynands and A. Nagel, “Precision spectroscopy with coherent dark 

states,” Appl. Phys. B 68, 1 (1999). 

[18] F. Levi, A. Godone, J. Vanier, S. Micalizio, and G. Modugno, “Line-shape 

of dark line and maser emission profile in CPT,” Eur. Phys. J. D 12, 53 

(2000). 

[19] S. Brandt, A. Nagel, R. Wynands, and D. Meschede, “Buffer-gas-induced 

linewidth reduction of coherent dark resonances to below 50 Hz,” Phys. Rev. 

A 56, R1063 (1997). 

[20] M. Erhard and H. Helm, “Buffer-gas effects on dark resonances: Theory and 

experiment,” Phys. Rev. A 63, 043813 (2001). 

[21] D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, “Nonlinear 

magneto-optics and reduced group velocity of light in atomic vapor with slow 

ground state relaxation,” Phys. Rev. Lett. 83, 1767 (1999). 

[22] L. Hau, S. Harris, Z. Dutton, and C. Behroozi, “Light speed reduction to 

17 metres per second in an ultracold atomic gas,” Nature 397, 594 (1999). 

[23] M. Kash et al., “Ultraslow group velocity and enhanced nonlinear optical 

effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229 

(1999).


[24] C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent 

optical information storage in an atomic medium using halted light pulses,” 

Nature 409, 490 (2001). 

[25] D. Phillips, M. Fleischhauer, A. Mair, R. Walsworth, and M. D. Lukin, 

“Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783 (2001). 

[26] O. Kocharovskaya, Y. Rostovtsev, and M. O. Scully, “Stopping light via hot 

atoms,” Phys. Rev. Lett. 86, 628 (2001). 

[27] M. J. S. J. J. Longdell, E. Fravel and N. B. Manson, “Stopped light with 

storage times greater than one second using electromagnetically induced 

transparency in a solid,” Phys. Rev. Lett 95, 063601 (2004). 

[28] M. D. Eisaman, A. Andre, F. Massou, M. Fleischhauer, A. S. Zibrov, 

and M. D. Lukin, “Electromagnetically induced transparency with tunable 

single-photon pulses,” Nature 438, 837 (2005). 

[29] P. R. Hemmer, D. P. Katz, J. Donoghue, M. Cronin-Golomb, M. S. Shahriar, 

and P. Kumar, “Efficient low-intensity optical phase conjugation based on 

coherent population trapping in sodium,” Opt. Lett. 20, 982 (1995). 

[30] M. Jain, H. Xia, G. Y. Yin, A. J. Merriam, and S. E. Harris, “Efficient 

nonlinear frequency conversion with maximal atomic coherence,” Phys. Rev. 

Lett. 77, 4326 (1996). 

[31] Y.-Q. Li and M. Xiao, “Enhancement of nondegenerate four-wave mixing 

based on electromagnetically induced transparency in rubidium atoms,” 

Opt. Lett. 21, 1064 (1996). 

[32] M. D. Lukin, M. Fleischhauer, A. S. Zibrov, H. G. Robinson, and V. L. 

Velichansky, “Spectroscopy in dense coherent media: Line narrowing and 

interference effects,” Phys. Rev. Lett. 79, 2959 (1997). 

[33] G. Zhang, K. Hakuta, and B. Stoicheff, “Nonlinear optical generation using 

electromagnetically induced transparency in atomic hydrogen,” Phys. Rev. 

Lett. 71, 3099 (1993). 

[34] M. Jain, G. Yin, J. Field, and S. Harris, “Observation of electromagnetically 

induced phase matching,” Opt. Lett. 18, 998 (1993). 

[35] C. Dorman and J. Marangos, “Symmetric photon-photon coupling by atoms 

with zeeman-split sublevels,” Phys. Rev. A 58, 4121 (1998). 

[36] M. T. Johnsson and M. Fleischhauer, “Quantum theory of resonantly enhanced 

four-wave mixing: Mean-field and exact numerical solutions,” Phys. 

Rev. A 66, 043808 (2002).


[37] L. M. Duann, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum 

communication with atomic ensembles and linear optics,” Nature 414, 

413 (2001). 

[38] D. A. Braje, V. Balić, S. Goda, G. Yin, and S. E. Harris, “Generation of 

paired photons with controllable waveforms,” Phys. Rev. Lett. 93, 183601 

(2004). 

[39] V. Balić, D. A. Braje, P. Kolchin, G. Yin, and S. E. Harris, “Generation of 

paired photons with controllable waveforms,” Phys. Rev. Lett. 94, 183601 

(2005). 

[40] P. Kolchin, S. Du, C. Belthangady, G. Yin, and S. E. Harris, “Generation 

of narrow-bandwidth paired photons: Use of a single driving laser,” Phys. 

Rev. Lett. 97, 113602 (2006). 

[41] C. F. McCormick, V. Boyer, E. Arimondo, and P. D. Lett, “Strong relative 

intensity squeezing by four-wave mixing in rubidium vapor,” Opt. Lett. 32, 

178 (2007). 

[42] M. V. Pack, R. M. Camacho, and J. C. Howell, “Transients of the 


Theory,” Phys. Rev. A 74, 013812 (2006). 

[43] S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” 

Phys. Rev. Lett. 81, 3611 (1998). 

[44] N. Mulchan, D. G. Ducreay, R. Pina, M. Yan, and Y. Zhu, “Nonlinear 

excitation by quantum interference in a doppler-broadened rubidium atomic 

system,” J. Opt. Soc. Am. B 17, 820 (2000). 

[45] D. A. Braje, V. Balić, G. Y. Yin, and S. E. Harris, “Low-light-level nonlinear 

optics with slow light,” Phys. Rev. A 68, 041801(R) (2003). 

[46] H. Wang, D. Goorskey, and M. Xiao, “Controlling light by light with threelevel 

atoms inside an optical cavity,” Opt. Lett. 27, 1354 (2002). 

[47] H. Kang and Y. Zhu, “Observation of large Kerr nonlinearity at low light 

intensities,” Phys. Rev. Lett. 91, 093601 (2003). 

[48] H. Chang, Y. Du, J. Yao, C. Xie, and H. Wang, “Observation of crossphase 

shift in hot atoms with quantum coherence,” Europhys. Lett. 65, 485 

(2004). 

[49] M. Yan, E. G. Rickey, and Y. Zhu, “Observation of absoprtive photon 

switching by quantum interference,” Phys. Rev. A 64, 041801(R) (2001).


[50] Y.-F. Chen, G.-C. Pan, and I. A. Yu, “Transient behaviors of photon switching 

by quantum interference,” Phys. Rev. A 69, 063801 (2004). 

[51] S. J. Park, H. Cho, T. Y. Kwon, and H. S. Lee, “Transient cohernce oscillation 

induced by a detuned raman field in a rubidum Λ system,” Phys. Rev. 

A 69, 023806 (2004). 

[52] A. D. Greentree et al., “Intensity-dependent dispersion under conditions of 

electromagnetically induced transparency in coherently prepared multistate 

atoms,” Phys. Rev. A 67, 023818 (2003). 

[53] A. B. Matsko, I. Novikova, G. R. Welch, and M. S. Zubairy, “Enhancement 

of Kerr nonlinearity by multiphoton coherence,” Opt. Lett. 28, 96 (2003). 

[54] S. Rebic, D. Vitali, and C. Ottaviani, “Quantum theory of a polarization 

phase gate in an atomics tripod configuration,” Opt. and Spect. 99, 264 

(2005). 

[55] D. Petrosyan and Y. P. Malakyan, “Magneto-optical rotation and crossphase 

modulation via coherently driven four-level atoms in a tripod configuration,” 

Phys. Rev. A 70, 023822 (2004). 

[56] S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. 

Rev. Lett. 82, 4611 (1999). 

[57] M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement 

of ultraslow single photons,” Phys. Rev. Lett. 84, 1419 (2000). 

[58] D. Petrosyan and G. Kurizki, “Symmetric photon-photon coupling by atoms 

with zeeman-split sublevels,” Phys. Rev. A 65, 033833 (2002). 

[59] W. Hanle, “über magnetische beeinflussung der polarisation der resonanzfluoreszenz,” 

Z. Phys. 30, 93 (1924). 

[60] U. Fano, “Effects of configuration interaction on intensities and phase 

shifts,” Phys. Rev. 124, 1866 (1961). 

[61] S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. 

Rev. 100, 703 (1955). 

[62] M. Fleischhauer, “Optical pumping in dense atomic media: Limitations due 

to reabsorption and spontaneously emitted photons,” Europhys. Lett. 45, 

659 (1999). 

[63] M. Fleischhauer, “Electromagnetically induced transparency and coherentstate 

preparation in optically thick media,” Opt. Exp. 4, 107 (1999).


[64] A. B. Matsko, I. Novikova, M. O. Scully, and G. R. Welsh, “Radiation 

trapping in coherent media,” Phys. Rev. Lett. 87, 133601 (2001). 

[65] A. B. Matsko, I. Novikova, and G. R. Welsh, “Radiation trapping under 

conditions of electromagnetically induced transparency,” J. Mod. Opt. 49, 

367 . 

[66] K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer 

among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003 

(1998). 

[67] N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergman, “Laser-induced 

population transfer by adiabatic passage techniques,” Annu. Rev. Phys. 

Chem. 52, 763 (2001). 

[68] S. Rebić, S. M. Tan, A. S. Parkins, and D. F. Walls, “Large Kerr nonlinearity 

with a single atom,” J. Opt. B: Quant. Semiclass. Opt 1, 490 (1999). 

[69] T. Opatrny and D.-G. Welsch, “Coupled cavities for enhancing the crossphase-modulation 

in electromagnetically induced transparency,” Phys. Rev. 

A 75, 023805 (2001). 

[70] S. Rebić, An Investigation of Giant Kerr Nonlinearity, Ph.D. thesis, University 

of Auckland, 2002. 

[71] S. Harris, “Normal modes fro electromagnetically induced transparency,” 

Phys. Rev. Lett. 72, 52 (1994). 

[72] S. Harris and Z.-F. Lou, “Preparation energy for electromagnetically induced 

transparency,” Phys. Rev. A 52, R928 (1995). 

[73] J. H. Eberly and V. Kozlov, “Wave equation for dark coherence in three-level 

media,” Phys. Rev. Lett. 88, 243604 (2002). 

[74] M. Konopnicki and J. Eberly, “Simultaneous propagation of short differentwavelength 

optical pulses,” Phys. Rev. A 24, 2567 (1981). 

[75] M. Konopnicki, P. Drummond, and J. Eberly, “Theory of lossless propagation 

of simultaneous different-wavelength optical pulses,” Opt. Commun. 

36, 313 (1981). 

[76] F. Hioe and R. Grobe, “Matched optical solitary waves for three- and fivelevel 

systems,” Phys. Rev. Lett. 73, 2559 (1994). 

[77] R. Grobe, F. Hioe, and J. Eberly, “Formation of shape-preserving pulses 

in a nonlinear adiabatically integrable system,” Phys. Rev. Lett. 73, 3183 

(1995).


[78] E. Cerboneschi and E. Arimondo, “Transparency and dressing for optical 

pulse pairs through a double-Λ absorbing medium,” Phys. Rev. A 52, R1823 

(1995). 

[79] Kasapi, M. Jain, G. Yin, and S. Harris, “Electromagnetically induced transparency 

propagation dyanimcs,” Phys. Rev. Lett. 74, 2447 (1995). 

[80] M. Fleischhauer and A. S. Manka, “Propagation of laser pulses and coherent 

population transfer in dissipative three-level systems: An adiabatic dressedstate 

picture,” Phys. Rev. A 54, 794 (1996). 

[81] H. Eleuch and R. Bennaceur, “An optical soliton pair among absorbing 

three-level atoms,” J. Opt. A 5, 528 (2003). 

[82] A. V. Rybin, I. P. Vadeiko, and A. R. Bishop, “Theory of slow-light solitons,” 

Phys. Rev. E 72, 026613 (2005). 

[83] I. Friedler, G. Kurizki, O. Cohen, and M. Segev, “Spatial Thirring-type 

solitons via electromagnetically induced transparency,” Opt. Lett 30, 3374 

(2005). 

[84] Y.-F. Chen, G.-C. Pan, and I. A. Yu, “Frequency-modulation-induced transient 

oscillation in the spectra of electromagnetically induced transparency,” 

J. Opt. Soc. Am. B 21, 1647 (2004). 

[85] A. Godone, S. Micalizio, and F. Levi, “Rabi resonances in the l excitation 

scheme,” Phys. Rev. A 66, 063807 (2002). 

[86] K. Harada, K. Motomura, T. Koshimizu, H. Ueno, and M. Mitsunaga, “Coherent 

Raman beats from dark states,” J. Opt. Soc. Am. B 22, 1105 (2005). 

[87] A. Mair and M. D. Lukin, “Phase coherence and control of stored photonic 

information,” Phys. Rev. A 65, 031802(R) (2002). 

[88] P. Valente, H. Failache, and A. Lezama, “Comparative study of the transient 

evolution of Hanle electromagnetically induced transparency and absorption 

resonances,” Phys. Rev. A 65, 023814 (2001). 

[89] A. S. Zibrov and A. B. Matsko, “Optical ramsey fringes induced by zeeman 

coherence,” Phys. Rev. A 64, 013814 (2001). 

[90] Y. Wu and X. Yang, “Electromagnetically induced transparency in V-, Λ-, 

and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71, 

053806 (2005).


[91] E. A. Korsunsky, N. Leinfellner, A. Huss, S. Baluschev, and L. Windholz, 

“Phase-dependent electromagnetically induced transparency,” Phys. Rev. A 

59, 2302 (1999). 

[92] W. Maichen, F. Renzoni, I. Mazets, E. Korsunsky, and L. Windholz, “Transient 

coherent population trapping in a closed loop interaction scheme,” 

Phys. Rev. A 53, 3444 (1996). 

[93] D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuks, 

and A. Weis, “Resonant nonlinear magneto-optical effects in atoms,” Rev. 

Mod. Phys. 74, 1153 (2002). 

[94] S. E. Harris, “Electromagnetically induced transparency,” Physics Today 

50, 36 (1997). 

[95] J. P. Marangos, “Topical review electromagnetically induced transparency,” 

J. Mod. Opt. 45, 471 (1998). 

[96] M. Fleischhauer, A. Imamoǧlu, and J. P. Marangos, “Electromagnetically 

induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633 

(2005). 

[97] M. D. Lukin, P. Hemmer, and M. O. Scully, in Adv. At., Mol., Opt. Phys., 

edited by B. Bederson and H. Walther (Elsevier Inc., San Diego, 2000), 

Vol. 42, p. 347. 

[98] N. V. Vitanov, M. Fleischhauer, B. W. Shore, and K. Bergmann, in Adv. 

At., Mol., Opt. Phys., edited by B. Bederson and H. Walther (Elsevier Inc., 

San Diego, 2001), Vol. 46, p. 55. 

[99] A. B.Matsko, O. Kocharovskaya, Y. Rostovtsev, A. S. Z. G. R. Welch, and 

M. O. Scully, in Adv. At., Mol., Opt. Phys., edited by B. Bederson and H. 

Walther (Elsevier Inc., San Diego, 2001), Vol. 46, p. 191. 

[100] M. Lukin, “Trapping and manipulating photon states in atomic ensembles,” 

Rev. Mod. Phys. 75, 457 (2003). 

[101] J. Howell and J. Yeazell, “Nondestructive single-photon trigger,” Phys. Rev. 

A 62, 032311 (2000). 

[102] K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in 

experimental quantum communication,” Phys. Rev. Lett, 76, 4656 (1996). 

[103] I. L. Chuang and Y. Yamamoto, “Quantum bit regeneration,” Phys. Rev. 

Lett. 76, 4281 (1996).


[104] G. D’Ariano, L. Maccone, M. G. A. Paris, and M. F. Sacchi, “Optical fockstate 

synthesizer,” Phys. Rev. A 61, 053817 (2000). 

[105] J. C. Howell, A. Lamas-Linares, and D. Bouwmeester, “Experimental violation 

of a spin-1 Bell inequality using maximally entangled four-photon 

states,” Phys. Rev. Lett. 88, 030401 (2002). 

[106] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation 

of Bell’s inequality under strict einstein locality conditions,” Phys. 

Rev. Lett. 81, 5039 (1998). 

[107] M. Rowe et al., “Experimental violation of a Bell’s inequality with effiecient 

detection,” . 

[108] A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and 

J. P. Dowling, “Quantum interferometric optical lithography: Exploiting entanglement 

to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000). 

[109] E. Knill, R. Laflamme, and G. Milburn, “A scheme for efficient quantum 

computation with linear optics,” Nature 409, 46 (2001). 

[110] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. 

Milburn, “Linear optical quantum computing with photonic qubits,” Rev. 

Mod. Phys. 79, 135 (2007). 

[111] G. J. Milburn, “Quantum optical fredkin gate,” Phys. Rev. Lett. 62, 2124 

(1989). 

[112] I. L. Chuang and Y. Yamamoto, “Simple quantum computer,” Phys. Rev. 

A 52, 3489 (1995). 

[113] J. C. Howell and J. A. Yeazell, “Quantum computation through entangling 

single photons in multipath interferometers,” Phys. Rev. Lett. 85, 198 

(2000). 

[114] J. C. Howell and J. A. Yeazell, “Nondestructive single-photon trigger,” Phys. 

Rev. A 62, 032311 (2000). 

[115] G. M. DAriano, C. Macchiavello, and L. Maccone, “Quantum computations 

with polarized photons,” Fortschr. Phys. 48, 573 (2000). 

[116] K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled- 

NOT gate,” Phys. Rev. Lett. 93, 250502 (2004). 

[117] W. J. Munro, K. Nemoto, and T. P. Spiller, “Weak nonlinearities: A new 

route to optical quantum compution,” New J. Phys. 7, 137 (2005).


[118] C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Quantum phase-gate 

operation based on nonlinear optics: Full quantum analysis,” Phys. Rev. A 

73, 010301(R) (2006). 

[119] Z.-B. Wang, K.-P. Marzlin, and B. C. Sanders, “Large cross-phase modulation 

between slow copropagating weak pulses in 87Rb,” Phys. Rev. Lett. 

97, 063901 (2006). 

[120] J. H. Shapiro, “Single-photon kerr nonlinearities do not help quantum computation,” 

Phys. Rev. A 73, 062305 (2006). 

[121] J. H. Shapiro and M. Razavi, “Continuous-time cross-phase modulation and 

quantum computation,” New J. Phys. 9, 16 (2007). 

[122] H. Schmidt and A. Imamoglu, “High-speed properties of a phase-modulation 

scheme based on electromagnetically induced transparency,” Opt. Lett. 23, 

107 (1998). 

[123] J.-Q. Shen, Z.-C. Ruan, and S. He, “Influence of the singal light on the 

transient optical properties of a four-level EIT medium,” Phys. Lett. A 

330, 487 (2004). 

[124] S. Alam, Lasers without inversion and electromagnetically induced transparency 

(SPIE, Washington, 1999). 

[125] A. Javan, O. Kocharovshaya, H. Lee, and M. O. Scully, “Narrowing of 

EIT resonance in a doppler broadened medium,” Phys. Rev. Lett. 81, 3611 

(2001). 

[126] M. O. Scully and M. S. Zubairy, Qunatum Optics (Cambridge University 

Press, Cambridge, 1997). 

[127] R. S. Bennink, Frequency Conversion of Optical Signals Using Coherently 

Prepared Media, Ph.D. thesis, University of Rochester, Instute of Optics, 

2004. 

[128] J. Vanier and A. Audoin, The quantum physics of atomic frequency starndarads 

(Adam Hilger, Philadelphia, 1989). 

[129] M. D. Rotondaro and G. P. Perram, “Collisional broadening and shift of the 

rubidium D 1 and D 2 lines (5 2 s 1/2 → 5 2 p 1/2 , 5 2 p 3/2 ) by rare gases H 2 , D 2 , 

N 2 , CH 4 and CF 4 ,” J. Quant. Spectrosc. Radiat. Transfer 57, 497 (1997). 

[130] J. Vanier, J.-F. Simard, and J.-S. Boulanger, “Relaxation and frequency 

shifts in the ground state of Rb 85 ,” Phys. Rev. A 9, 1031 (1974).


[131] D. Budker and D. F. Kimball, Atomic physics: an exploration through problems 

and solutions (Oxford University Press, Oxford, 2004). 

[132] F. A. Franz and C. Volk, “Spin relaxation of rubidium atoms in sudden 

and quasimolecular collisions with light-noble-gas atoms,” Phys. Rev. A 14, 

1711 (1976). 

[133] P. Grangier, J.-F. Roch, and G. Roger, “Observation of backaction-evading 

measurement of an optical intensity in a three-level atomic nonlinear system,” 

Phys. Rev. Lett. 66, 1418 (1991). 

[134] J. P. Poizat and P. Grangier, “experimental realization of a quantum optical 

tap,” Phys. Rev. Lett. 70, 271 (1993). 

[135] J.-P. Poizat, J.-F. Roch, and P. Grangier, “Characterization of quantum 

nondemolition measurements in optics,” Ann. Phys. Fr. 19, 265 (1994). 

[136] H. A. Haus, Electromagnetic Noise and Quantum Optical Measurement (Spinger 

Verlag, Berlin, 2000). 

[137] H. A. Bachor, A Guide to Experiments in Qunatum Optics (Willey, Cambridge, 

1999). 

[138] G. P. Agarwal, Nonlinear Fiber Optics; Third Edition (Academic Press, San 

Diego, 2001). 

[139] Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, 

“Measurement of conditional phase shifts for quantum logic,” Phys. Rev. 

Lett. 75, 4710 (1995). 

[140] S. Gleyzes et al., “Quantum jumps of light recording the birth and death of 

a photon in a cavity,” Nature 446, 297 (2007). 

[141] K. Nemoto and W. J. Munro, “Universal quantum computation on the 

power of quantum non-demolition measurements,” Phys. Lett. A 334, 104 

(2005). 

[142] S. Rebic, C. Ottaviani, G. D. Giuseppe, D. Vitali, and P. Tombesi, “Assessment 

of a quantum phase-gate operation based on nonlinear optics,” Phys. 

Rev. A 74, 032301 (2006). 

[143] D. Steck, “Rubidium 87 D line data,” , unpublished, available on-line at 

http://steck.us/alkalidata, (unpublished). 

[144] D. M. Brink and G. R. Satchler, Angular Momentum (Oxford University 

Press, Oxford, 1962).

158 

Appendix A 

Appendix 1: EIT on the 

rubidium D1 line 

In a complicated level system like the D 1 lines of rubidium, EIT can be significantly 

more complicated than EIT in three-level systems. The purpose of this 

appendix is to supply some of the missing details for Chapter 3 by discussing 

EIT on the Rb 85 and Rb 87 D 1 lines. This appendix can also be looked at as a 

template for understanding EIT in other complicated level structures. We restrict 

our discussion to the case where the probe and coupling fields have orthogonal 

linear polarizations. 

A.1 Choice of Polarizations 

First, there are several reasons for our choice of orthogonal linear polarizations. 

The primary reason for our choice of polarization is that through experimentation 

with various polarizations we found that orthogonal linear polarizations for the 

probe and coupling fields provided the best contrast for EIT (by EIT contrast

A.1. CHOICE OF POLARIZATIONS 159 

we mean that the ratio between absorption on and off Raman resonance was the 

greatest). Following our empirical observations, we come up with Table A.1 to 

explain the reasons we should have anticipated good EIT contrast for orthogonal 

linear polarizations. 

Perhaps the most important criteria good EIT contrast is whether a particular 

choice of polarizations results in parasitic dark states (DS). By parasitic DS we 

mean a DS that is not a coherent superposition of both ground hyperfine levels. 

Such a DS would be dark regardless of Raman detuning between the probe a 

coupling fields resulting in poor EIT contrast. 

The second most important condition is the number of completely dark states. 

Although this criteria is somewhat invalidated by the existence of partial dark 

states, it still seems to be a good indicator of how good the transparency can be. 

For optically thick media four-wave mixing can become important. Unfortunately, 

orthogonal linear polarizations do create four wave mixing, and when the 

frequency filter was removed from our experiment we did detect a signal due to 

four-wave mixing. These four wave mixing product was an order of magnitude 

smaller than the probe, and were mostly ignored in the experiment. 

Finally, for the purposes of combining and separating the probe and coupling 

fields in the experiment, it is convenient that they have orthogonal polarizations. 

From Table A.1 it is evident that apart from four wave mixing, orthogonal

A.2. TRANSITION MATRIX ELEMENTS 160 

linear polarizations have all the desired features, whereas the other choices of 

polarization do not. 

A.2 Transition matrix elements 

For convenience and completeness we give the transition matrix elements here for 

the D 1 lines of Rb 85 . Much of this information can be found in numerous references 

[127, 128, 143]. Thus, in order to not duplicate information unnecessarily 

this section will be brief. If there is missing information, consult one of the many 

references on single electron atoms and rubidium. 

First, the Wigner-Eckart theorem states that radial and angular parts of the 

dipole moment expectation value can be factored such that 

〈n, F, m F |er q |n ′ , F ′ , m ′ F 〉 = 〈n, F ||er||n ′ , F ′ 〉〈F, m F |F ′ , 1, m ′ F , q〉, 

where 

⎧ 

〈n, F ||er||n ′ , F ′ 〉 = 〈J||er||J ′ 〉(−1) √ ⎪⎨ J J ′ 1 

F ′ +1+J+I 

(2F ′ + 1)(2J + 1) 

⎪⎩ F ′ F I 

⎫ 

⎪⎬ 

(A.1) , 

⎪⎭ 

and where we have used the Wigner 6-j notation (i.e. 

the curl bracket term). 

The angular momenta are defined as: F ≡ J + I and J ≡ L + S, where I is the 

angular momentum of the nucleus, S is the spin angular momentum of the electron

A.3. EIT IN RUBIDIUM 85 161 

and L is the orbital angular momentum of the electron. The prime denotes as 

excited state, and we have used the short hand of dropping the n and n ′ . The 

Clebsch-Gordon coefficients are given by 

⎜ 

⎝ 

〈F, m F |F ′ , 1, m ′ F , q〉 = (−1) F ′ −1+m F 

√ 

(2F + 1) 

⎛ 

⎞ 

F ′ 1 F 

⎟ 

⎠ , 

m ′ F q −m F 

(A.2) 

where we have used the Wigner 3-j notation. 

There are interesting properties 

and symmetries of these expression that are explained adequately elsewhere [127, 

128, 143, 144]. To add to the confussion, there are a couple of conventions for 

calculating the dipole matrix elements; we follow the more common convention 

(i.e. less Russian) of Brink and Satchler [144]. For Rb 85 the dipole matrix elements 

for the D 1 line are given in Table A.2 

A.3 EIT in rubidium 85 

By choosing the quantization axis parallel to the coupling polarization (i.e. ɛ c = 

z = ɛ 0 , where z is the unit vector for the quantization axis) we only need to 

consider the dipole allowed transitions shown in Fig. A.1. For concreteness we 

also choose the probe polarization in the y direction such that ɛ p = y = i(ɛ −1 + 

ɛ +1 )/ √ 2. 

When only one excited hyperfine level is considered at a time there are five dark 

states for the F ′ = 2 hyperfine level and five dark states for the F ′ = 3 hyperfine


(a) 

F’=3 

F’=2 

2 

1 

1 

2 

27 

3 

3 

27 

5 

8 

1 

8 

5 

27 

27 

3 

27 

27 

2 

−1 

−1 

2 

− 

− 

27 

3 

3 

27 

m =-3 F 

m F 

=-2 m F 

=-1 m F 

=0 m F 

=1 m F 

=2 m F 

=3 

F=3 

F=2 

(b) 

F’=3 

F’=2 

15 

27 

10 

27 

6 

27 

1 

3 

1 

27 

−1 

4 

−1 

1 

4 

1 

− 

0 

3 

27 

27 

27 

27 

3 

1 

27 

1 

3 

6 

27 

10 

27 

15 

27 

m F =-3 m F =-2 m F =-1 m F 

=0 

m F =1 m F =2 

m F =3 

F=3 

F=2 

Figure A.1: Dipole matrix elements for the transitions used in EIT with orthogonal linear 

polarizations.


level. For Doppler broadened atomic vapors in a buffer gas it is important that 

the dark state be simultaneously dark for both excited hyperfine levels because 

the Doppler broadened linewidth is greater than the hyperfine splitting. To find 

a completely dark state (i.e. a state that is simultaneously dark for both excited 

hyperfine levels) we must find a superposition of the five F ′ = 3 dark states that 

is also a superposition of the F ′ = 2 dark states. It is helpful to divide each set 

of dark states further into two sets; dark states involving only A) the | F =3 

m=−2〉, | 3 0〉, 

| 3 2〉, | 2 −1〉, and | 2 1〉 ground states, and B) dark states involving only the | F =3 

m=−3〉, | 3 −1〉, 

| 3 1〉, | 3 3〉, | 2 −2〉, | 2 0〉, and | 2 2〉 ground states. 

Transitions for dark states involving subset A of ground states are shown in 

Fig. A.2. The three dark states corresponding to subset A are 

|M〉 3A = 2 ( | −1〉 2 − | +1〉 ) 2 + a √ 5 ( | −2〉 3 + | 3 

(A.3) 

8 + 10α 2 

+2〉 ) 

|L〉 2A = 

|C〉 3A = | 3 0〉 

( 

1 

−1〉 + a √ 

6 

| 3 0〉 − 1 

√ 

1 + 11α2 /30 

| 2 

√ 

5 

| −2〉 

3 

) 

(A.4) 

(A.5) 

and 

( 

| +1〉 2 + a − √ 1 

6 

| 3 0〉 + 1 

|R〉 2A √ 

1 + 11α2 /30 

√ 

5 

| +2〉 

3 

) 

(A.6)


(a) 

− 

4 

27 

10 

27 

1 

3 

1 

3 

10 

27 

4 

27 

(b) 

5 

27 

2 

27 

1 

3 

−1 

3 

(c) 

1 

3 

1 

3 

5 

27 

− 

2 

27 

Figure A.2: Dipole matrix elements for dark states in Rb 85 D 1 line that involve the | F =3 

m=−2〉, | 3 0〉, 

| 3 2〉, | 2 −1〉, and | 2 1〉 ground states.


(a) 

(d) 

15 

27 

6 

27 

1 

27 

1 

3 

2 

27 

−1 

3 

1 

3 

1 

27 

6 

27 

15 

27 

− 

2 

27 

−1 

3 

(b) 

(e) 

15 

27 

−1 

3 

−1 

27 

8 

27 

1 

27 

− 

2 

27 

(c) 

(f) 

1 

27 

2 

27 

1 

27 

1 

3 

8 

27 

15 

27 

Figure A.3: Dipole matrix elements for dark states in Rb 85 D 1 line that involve the | F =3 

m=−3〉, 

| 3 −1〉, | 3 1〉, | 3 3〉, | 2 −2〉, | 2 0〉, and | 2 2〉) ground states. 

where a ≡ iΩ p /Ω c , and the subscript means hyperfine level F = 3 or F = 2 and 

the subset of ground states is subset A. There is no superposition of these dark 

states that is simultaneously dark for transitions to both excited hyperfine levels. 

There is a superposition of dark states associated with subset B of the ground 

states (i.e. magnetic sublevels | F =3 

m=−3〉, | 3 −1〉, | 3 1〉, | 3 3〉, | 2 −2〉, | 2 0〉, and | 2 2〉). The dark


states for subset B shown in Fig. A.3. These dark states are 

|M〉 2B = 1 √ 

3 

2 |2 0〉 + 

8 

( 

| 

2 

−2 〉 + | 2 2〉 ) 

(A.7) 

|L〉 2B = 

(√ ) 

8| 

2 

−2 〉 + a| −1〉 

3 / √ 8 + a 2 . (A.8) 

and 

|R〉 2B = 

(√ 

8| 

2 

2 〉 − a| 3 1〉) 

/ √ 8 + a 2 , (A.9) 

|M〉 3B = 

√ ( 

6 | 

2 

−2〉 + | 2 2〉 ) − | 2 0〉 + a √ 5 ( | −3〉 3 − | 3 3〉 ) 

√ 

13 + 10a 

2 

|L〉 3B = 

(√ 

6| 

2 

−2 〉 + a √ 5| −3〉 3 + a √ ) 

3| −1〉 

3 / √ 6 + 8a 2 (A.10) 

and 

(√ 

|R〉 3B = 6| 

2 

2 〉 − a √ 5| 3 3〉 − a √ ) 

3| 3 1〉 / √ 6 + 8a 2 . (A.11) 

The completely dark state is given by 

|DD〉 = 

√ 

2 

√ 

13 + 10a2 |M〉 3B + 3 √ 3 + 4a 2 (|L〉 3B + |R〉 3B ) 

2 √ 20 + 23a 2 . (A.12)

A.4. IMPORTANT PARAMETERS FOR RUBIDIUM 167 

4 

3 

2 

F=1 

5P 3 

2 

120.960(20) 

63.420(31) 

29.260(29) 

85 

Rb 

5P 1 

2 

F=3 

F=2 361.936(14) 

5S 1 

2 

F=3 

F=2 3035.732440(10) 

Figure A.4: Hyperfine splittings, in MHz, of the 5S 1/2 5P 1/2 and 5P 3/2 energy levels in rubidium 

85. 

A.4 important parameters for rubidium 

There are several parameters of rubidium that we have used in our calculations but 

are not part of the main text. We collect those parameters here for easy reference. 

The lifetimes for the D 1 (D 2 ) line is 27.70±0.04 ns (26.24±0.04 ns). These are 

the same for both isotopes of rubidium. From these lifetimes the dipole moments 

for the D 1 (D 2 ) transition is calculated to be 〈J||er||J ′ 〉=2.534(3)×10 −29 Cm 

(〈J||er||J ′ 〉=3.584(4)×10 −29 Cm). The mass and natural abundance of rubidium 

85 is 84.911800 amu and 72.17%. While the mass and natural abundance of 

rubidium 87 is 86.909184 amu and 27.84%. The hyperfine splittings for the Rb 85 

levels 5S 1/2 5P 1/2 5P 3/2 are shown in Fig. A.4. The hyperfine splittings for for 

the Rb 87 levels 5S 1/2 5P 1/2 5P 3/2 are shown in Fig. A.5.


3 

2 

1 

F=0 

5P 3 2 

266.650(9) 

156.947(7) 

72.218(4) 

87 

Rb 

5P 1 

2 

F=2 

F=1 816.656(30) 

5S 12 

F=2 

F=1 6834.682610904 29(9) 

Figure A.5: Hyperfine splittings, in MHz, of the 5S 1/2 5P 1/2 and 5P 3/2 energy levels in rubidium 

87. 

A.5 EIT in rubidium 87 

EIT in Rb 87 is slightly different from Rb 85 because in Rb 87 the hyperfine splitting 

for 5P 1/2 is 816 MHz which is larger than the Doppler broadening. Thus, the dark 

states do not have to be simultaneously dark for both excited hyperfine levels 

Tables A.4 and A.5. 

With the choice of orthogonal linear polarizations leads to the EIT transitions 

shown in Fig. A.6. Again we can divide the dark states into two subsets. Subset 

A has four dark states and involves magnetic sublevels | 1 −1〉, | 1 1〉, | 2 −2〉, | 2 0〉, and 

| 2 2〉. Subset B has two dark states includes magnetic sublevels | 1 0〉, | 2 −1〉, and | 2 1〉. 

Fig. 

A.7 shows the transitions involved in those dark states which require a 

superposition of multiple states (| 2 0〉 is also a dark state for the F ′ = 2 hyperfine 

level). There three dark states for the F ′ = 2 hyperfine level, and there are three 

dark states associated with the F ′ = 1 hyperfine level.


(a) 

F’=2 

F’=1 

1 

12 

1 

12 

1 

2 

−1 

12 

1 

3 

m F 

=-2 m F 

=-1 m F 

=0 m F 

=1 m F 

=2 

−1 

12 

1 

2 

F=2 

F=1 

(b) 

F’=2 

1 

2 

1 

2 

1 

12 

F’=1 

1 

− 

3 

−1 

12 

1 

12 

m F 

=-2 m F 

=-1 m F 

=0 m F 

=1 m F 

=2 

0 

1 

2 

1 

12 

1 

2 

1 

3 

F=2 

F=1 

Figure A.6: Dipole matrix elements for the transitions used in EIT with orthogonal linear 

polarizations in rubidium 87.


(a) 

(d) 

1 

1 

1 

1 

1 

− 

3 

2 

12 

12 

2 

1 

3 

−1 

12 

1 

2 

1 

2 

1 

12 

(b) 

(e) 

−1 

12 

1 

3 

1 

2 

1 

12 

−1 

12 

1 

2 

(c) 

1 

3 

1 

12 

Figure A.7: Dipole matrix elements for the dark states on Rb 87 D 1 . Not shown is the dark 

state | 2 0〉 because it is an incoherent dark state (i.e. it doesn’t require a superposition of ground 

states.)


Table A.1: Comparison of EIT for several choices of polarization 

Comparison of polarizations for EIT on the D1 line of Rubidium 85. 

Lin⊥Lin Lin‖Lin Cir⊥Cir Cir‖Cir 

# of Dark States (DS) 1 0 0 5 

Parasitic DS? No No Yes Yes 

4-Wave Mixing Yes Yes Yes No 

Probe⊥Coupling? Yes No Yes No 

Table A.2: Dipole matrix elements for Rb 85 D 1 line with F = 2 

Dipole matrix elements for rubidium D 1 transitions from the F = 2 ground hyperfine 

level. All coefficients must be multiplied by 〈J = 1/2||er||J ′ = 1/2〉/ √ 27. 

F=2 

m F 

√ 

= −2 m F 

√ 

= −1 m F 

√ 

= 0 m F 

√ 

= 1 m F = 2 

σ + 2 3 3 2 — 

F ′ = 2 π −2 −1 0 1 2 

σ − ) — − √ 2 − √ 3 − √ 3 − √ 2 

σ + −1 − √ 3 − √ 6 − √ 10 − √ √ √ √ √ √ 

15 

F ′ = 3 π 5 8 9 8 5 

σ 1 − √ 15 − √ 10 − √ 6 − √ 3 −1 


Clebsch-Gordon coefficients for rubidium D 1 transitions from the F = 3 ground 

hyperfine level. All coefficients are divided by 〈J = 1/2||er||J ′ = 1/2〉/ √ 27. 

F=3 

m 

√ 

= −3 m 

√ 

= −2 m 

√ 

= −1 m 

√ 

= 0 m = 1 m = 2 m = 3 

σ + 15 

√ 10 

√ 6 

√ 3 

√ 1 

√ — — 

F ′ = 2 π — 5 8 

√ 9 

√ 8 

√ 5 

√ — 

σ − 

√ 

— 

√ 

— 

√ 

1 

√ 3 

√ 6 

√ 10 15 

σ + 3 5 6 6 5 3 — 

F ′ = 3 π -3 -2 -1 0 1 2 3 

σ − — − √ 3 − √ 5 − √ 6 − √ 6 − √ 5 − √ 3



Dipole matrix elements for the Rb 87 5S 1/2 → 5P 1/2 transitions from the ground 

hyperfine level F = 1. To obtain the dipole moment each element must be multiplied 

by 〈J = 1/2||er||J ′ = 1/2〉/ √ 12. 

F=2 

m F = −1 m F = 0 m F = 1 

σ + -1 -1 — 

F ′ = 1 π 1 0 -1 

σ − ) — 1 1 

σ + −1 − √ 3 − √ √ √ 

6 

F ′ = 2 π 3 2 3 

σ 1 − √ 6 − √ 3 −1 


Dipole matrix elements for the Rb 87 5S 1/2 → 5P 1/2 transitions from the ground 

hyperfine level F = 2. To obtain the dipole moment each element must be multiplied 

by 〈J = 1/2||er||J ′ = 1/2〉/ √ 12. 

F=2 

m F 

√ 

= −2 m F 

√ 

= −1 m F = 0 m F = 1 m F = 2 

σ + 6 

√ 3 1 

√ — — 

F ′ = 1 π — 3 2 

√ 3 

√ — 

σ − 

√ 

— 

√ 

— 

√ 

1 

√ 3 6 

σ + 2 3 3 2 — 

F ′ = 2 π -2 -1 0 1 2 

σ − — − √ 2 − √ 3 − √ 3 − √ 2

Dynamics of Electromagnetically Induced Transparency Optical Kerr ...

Create successful ePaper yourself

Delete template?

Save as template?