Ultrafast Spectroscopy of Non-Markovian
Dynamics in a Dense Atomic Vapor
byVirginia O. Lorenz
B.A., University of Colorado, 2001
M.S., University of Colorado, 2005

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2007

This thesis entitled:
Ultrafast Spectroscopy of Non-Markovian Dynamics in a Dense Atomic Vapor
written by Virginia O. Lorenz
has been approved for the Department of Physics

Steven T. Cundiff

Chris H. Greene

Date

The final copy of this thesis has been examined by the signatories, and we findthat both the content and the form meet acceptable presentation standards ofscholarly work in the above mentioned discipline.

iii
Lorenz, Virginia O. (Ph.D., Physics)
Ultrafast Spectroscopy of Non-Markovian Dynamics in a Dense Atomic Vapor
Thesis directed by Prof. Steven T. Cundiff

This thesis presents results from ultrafast spectroscopy experiments and theoretical calculations of non-Markovian dynamics in a dense atomic vapor. For adense atomic vapor the excitation pulses are short compared to the duration ofcollisions and the time between collisions is long compared to the collision duration. Thus there are distinct timescales that correspond to regimes in which
the phase-coherence of the atomic superpositions is retained (called the nonMarkovian regime) and regimes in which the phase memory is lost (called theMarkovian regime). This is not the case for most condensed phase systems, inwhich the time between dephasing interactions, the interaction duration and thepulse width are of similar order. Using the current theory of stochastic fluctuations of the energy levels due to collisions, we simulate and fit the signaturesof phase memory from two-pulse transient four-wave mixing experiments, providing support for the theoretical model that is commonly used for more complex
condensed phase systems. Further three-pulse experiments in conjunction with atheoretical derivation of the correlation function of frequency fluctuations using
molecular dynamics simulations in an exciton picture provide insight into resonant
effects in dense atomic vapors. In addition, distinct signatures of local field effects are observed and compared with theory. The results presented in this thesis
provide support for theoretical models employed in more complex systems, contribute insight into the fundamental physics of dense atomic vapors, and exhibit
the benefits of ultrafast spectroscopic techniques through the clarity of signatures
representing system response.

Dedication

This thesis is dedicated to my parents.

v

Acknowledgements

I would like to thank Professor Steve Cundiff for being an excellent advisor,a sensitive mentor and an honest counselor. I deeply appreciate his uplifting words
and steadfast support, which opened up opportunities for scientific growth andachievement I never could have imagined.
I am very thankful to Professor Chris Greene and Professor Shaul Mukamel
for their inspiration, encouragement and generous support. I would also liketo thank my other committee members Professor Eric Cornell, Professor Ralph
Jimenez, Professor Allan Franklin and Professor John Bohn for their support andhelpful comments.
From the many JILA staff members who helped me I would like to especially thank Hans Green and Todd Asnicar, who brought essential expertise andoptimistic tenacity to numerous technical hurdles. I also thank Loree Kaleth for
her administrative assistance. I would like to thank the Cundiff group members,
past and present, as well as my graduate colleagues for their friendship and aid.
To my parents I owe a lifetime of thanks for their love, prayers and for
guiding me towards wisdom and knowledge. To them and my brother, thank you
for blessing my life. I give my utmost thanks to God.

Contents

Chapter
1 Introduction

1

1.1

Historical Background and Overview of Main Results . . . . . . .

2

1.2

Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2 Theoretical Background for Transient Four-Wave Mixing Experiments

9

2.1

Transient Four-Wave Mixing . . . . . . . . . . . . . . . . . . . . .

10

2.2

Density Matrix Formalism . . . . . . . . . . . . . . . . . . . . . .

13

2.3

Optical Bloch Equations for Homogeneously and Static Inhomogeneously Broadened Systems . . . . . . . . . . . . . . . . . . . . .

16

2.4

The Markov Approximation . . . . . . . . . . . . . . . . . . . . .

23

2.5

Non-Markovian Theory . . . . . . . . . . . . . . . . . . . . . . . .

27

3 Transient Four-Wave Mixing Experiments and Results

50

3.1

Experimental Components . . . . . . . . . . . . . . . . . . . . . .

50

3.2

The Two-Pulse Photon Echo Experiment and Results . . . . . . .

56

3.3

The Three-Pulse Photon Echo Experiment and Results . . . . . .

63

3.4

The Time-Resolved Three-Pulse Photon Echo Experiment and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

vii
4 Correlation Functions from Molecular Dynamics Simulations in an Exciton
Picture72

4.1

The Exciton Formalism . . . . . . . . . . . . . . . . . . . . . . . .

73

4.2

Molecular Dynamics Simulations . . . . . . . . . . . . . . . . . .

75

4.3

Resonant Exciton Couplings . . . . . . . . . . . . . . . . . . . . .

77

4.4

Diagonalizing the Exciton Hamiltonian . . . . . . . . . . . . . . .

79

4.5

The Correlation Functions and Peak Shifts . . . . . . . . . . . . .

81

5 The Local Field

87

5.1

Definition of the Local Field . . . . . . . . . . . . . . . . . . . . .

88

5.2

Frequency-Domain Manifestation of the Local Field . . . . . . . .

89

5.3

Time-Domain Manifestation of the Local Field . . . . . . . . . . .

91

5.4

Transient Four-Wave Mixing Experimental Results . . . . . . . .

93

5.5

Preliminary Calculations of Propagation . . . . . . . . . . . . . .

94

6 Conclusion

101

Bibliography

104

Appendix
A Optical Bloch Equations Derivation Details

110

B Non-Markovian Theory Derivation Details

112

B.1 Density Matrix Diagrams . . . . . . . . . . . . . . . . . . . . . . . 112
B.2 The Cumulant Expansion . . . . . . . . . . . . . . . . . . . . . . 114
B.3 Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 115
C Some Properties of Potassium

117

viii
D Experimental Details

120

D.1 The Pulse Picker . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
D.2 Experimental Components . . . . . . . . . . . . . . . . . . . . . . 123
E Novel High-Temperature Alkali Vapor Cell

127

E.1 Cell Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
E.2 Bonding Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 128
E.3 Loading Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 129
E.4 Previous Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
E.5 Transmission Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
F The Exciton Hamiltonian for a System of Two 2-Level Atoms

137

G The GROMACS Package

141

H Constants and Conversions

143

TablesTable
4.1

Simulation box side lengths for 10 and 20 atoms.

. . . . . . . . .

77

H.1 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Figures

Figure1.1

Schematic of fluctuation of energy levels during a collision . . . .

2

1.2

Schematic of fluctuation of a superposition during a collision . . .

4

2.1

Schematic of transient four-wave mixing . . . . . . . . . . . . . .

11

2.2

Calculated 2-pulse real-time TFWM signals for a homogeneously
broadened 2-level system . . . . . . . . . . . . . . . . . . . . . . .

2.3

19

Calculated 2-pulse real-time TFWM signals for a static inhomogeneously broadened 2-level system . . . . . . . . . . . . . . . . . .

20

2.4

Runners on a track: analogy to a photon echo . . . . . . . . . . .

21

2.5

Calculated 2-pulse real-time TFWM signals for a homogeneously
broadened and a static inhomogeneously broadened 2-level system
for two pulse delays . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6Calculated 2-pulse time-integrated TFWM signal for a homogeneously broadened 2-level system . . . . . . . . . . . . . . . . . .

2.7

23

Schematic of superposition phase jumps due to collisions in theMarkovian limit . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.9

23

Calculated 2-pulse time-integrated TFWM signal for a static inhomogeneously broadened 2-level system . . . . . . . . . . . . . . .

2.8

22

26

Double-sided Feynman diagrams for 3-pulse TFWM in a 3-Levelsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

xi
2.10 Schematic of stochastic correlation function . . . . . . . . . . . .

42

2.11 Calculated 2-pulse time-resolved TFWM signal for a non-Markovian
broadened 2-level system . . . . . . . . . . . . . . . . . . . . . . .

44

2.12 Calculated 2-pulse time-integrated TFWM signal for a non-Markovian
broadened 2-level system . . . . . . . . . . . . . . . . . . . . . . .

45

2.13 Calculated 2-pulse time-integrated TFWM signal for a non-Markovian
broadened 2-level system: varying ∆ . . . . . . . . . . . . . . . .

45

2.14 Calculated 2-pulse time-integrated TFWM signal for a non-Markovian
broadened 2-level system: varying Λ . . . . . . . . . . . . . . . . .

46

2.15 Calculated 2-pulse time-integrated TFWM signal for a non-Markovian
broadened 3-level system . . . . . . . . . . . . . . . . . . . . . . .

47

2.16 Calculated 3-pulse time-integrated TFWM signal for a non-Markovian
broadened 2-level system . . . . . . . . . . . . . . . . . . . . . . .

48

2.17 Calculated 3-pulse photon echo peak shift for a non-Markovian
broadened 2-level system . . . . . . . . . . . . . . . . . . . . . . .

49

3.1

The optically active energy levels of potassium . . . . . . . . . . .

51

3.2

Fluorescence intensity versus wavelength and temperature . . . .

53

3.3

Fluorescence intensity versus time . . . . . . . . . . . . . . . . . .

55

3.4

Theoretical linewidths for resonance and Doppler Broadening . . .

56

3.5

Schematic of the 2-pulse photon echo experimental setup . . . . .

56

3.6

Experimental 2-pulse photon echo signal . . . . . . . . . . . . . .

58

3.7

Experimental 2-pulse photon echo signals for various temperaturesand densities

3.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Typical experimental 2-pulse photon echo signals with fits basedon the stochastic theory . . . . . . . . . . . . . . . . . . . . . . .

61

xii
3.9

Stochastic theoretical fitting parameters versus temperature anddensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

3.10 Schematic of the 3-pulse photon echo experimental setup . . . . .

63

3.11 Schematic of the 3-pulse photon echo experimental geometry . . .

64

3.12 Time delays of the 3-pulse photon echo experiment . . . . . . . .

65

3.13 Dependence of the population decay on beam angle . . . . . . . .

65

3.14 Experimental 3-pulse photon echo signals . . . . . . . . . . . . . .

67

3.15 Schematic of 3-pulse photon echo regimes . . . . . . . . . . . . . .

67

3.16 Experimental 3-pulse photon echo peak shifts for a series of temperatures and densities . . . . . . . . . . . . . . . . . . . . . . . .

68

3.17 Experimental 3-pulse photon echo peak shift fitting parameters versus temperature and density . . . . . . . . . . . . . . . . . . . . .

69

3.18 Schematic of the 3-pulse time-resolved photon echo experimental
setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

3.19 Experimental time-resolved 3-pulse photon echo signals . . . . . .

71

4.1

Schematic of one- and two-exciton states for N atoms . . . . . . .

74

4.2

Potential energy surfaces used for the molecular dynamics simulations 76

4.3

Potential energy surfaces of the potassium dimer dissociating to
4s+4p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4

78

Potential energy surfaces of the potassium dimer dissociating to
4s+4s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

4.5

Potential energy surfaces used to determine the one-exciton couplings 79

4.6

Potential energy surface differences used to determine the oneexciton couplings . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7

80

Experimental 3-pulse photon echo peak shifts and theoretical correlation functions from molecular dynamics simulations . . . . . .

81

xiii
4.8

Experimental 3-pulse photon echo peak shifts and theoretical correlation functions from molecular dynamics simulations for a seriesof temperatures and densities . . . . . . . . . . . . . . . . . . . .

4.9

82

Correlation functions from molecular dynamics simulations, comparing short and long couplings . . . . . . . . . . . . . . . . . . .

83

4.10 Potential energy surfaces used to compare short and long couplings

83

4.11 Calculated peak shifts and their correlation functions . . . . . . .

85

5.1

Schematic of local field intensities interacting in a 2-pulse TFWM
experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2

91

Calculated 2-pulse TFWM signal for a 2-level system with andwithout local field . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

5.3

Experimental 2-pulse TFWM signal, including negative delay . . .

94

5.4

Structure considered for propagation calculations . . . . . . . . .

95

5.5

Calculated index of refraction of potassium vapor . . . . . . . . .

97

5.6

Incident pulse for propagation calculations versus frequency and time 97

5.7

Calculated reflected field intensity . . . . . . . . . . . . . . . . . .

98

5.8

Calculated transmitted field intensity versus distance . . . . . . .

98

5.9

Calculated transmitted field intensity for a particular distance . .

99

5.10 Calculated transmitted field intensity versus frequency . . . . . .

99

5.11 Calculated integrated transmitted intensity versus distance . . . . 100
B.1 Double-sided Feynman diagrams for 3-pulse TFWM in a 3-levelsystem for all time orderings . . . . . . . . . . . . . . . . . . . . . 112
D.1 Schematic of pulse picker . . . . . . . . . . . . . . . . . . . . . . . 120
D.2 Schematic of the Ti:sapph laser . . . . . . . . . . . . . . . . . . . 124
E.1 Schematic of the potassium cell components . . . . . . . . . . . . 128

xiv
E.2 Schematic of a loaded potassium cell . . . . . . . . . . . . . . . . 129
E.3 Photo of vacuum attachment for loading the cell . . . . . . . . . . 130
E.4 Photos of heating components of the potassium cell . . . . . . . . 131
E.5 Photo of vapor cell inside secondary glass cell . . . . . . . . . . . 131
E.6 Photos of pieces of cell lined up for assembly . . . . . . . . . . . . 132
E.7 Photo of an assembled vapor cell . . . . . . . . . . . . . . . . . . 132
E.8 Photo of a used cell . . . . . . . . . . . . . . . . . . . . . . . . . . 133
E.9 Schematic of thin cell manufacture . . . . . . . . . . . . . . . . . 135
E.10 Photo of thin cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
E.11 Transmitted TFWM signal showing dynamic beats . . . . . . . . 136

Chapter 1
Introduction

The research presented in this thesis falls under the general heading of“atomic, molecular and optical physics.” This field encompasses the study oflight-matter interaction on the atomic scale. The particular set of experiments
described in this thesis allows us to focus on how the electronic state of a dense
atomic vapor changes with time. In the experiments, an ultrashort pulse of light
impinges on a dense vapor of atoms at a resonant frequency, exciting electrons into
superpositions between a ground state and an excited state. Soon the electron
superpositions become “out of sync,” or dephase, due to changes in the atomic
energy levels from collisions. These changes are a form of quantum dissipation,
meaning they are not reversible. Probing with sequences of ultrashort laser pulses
results in a representation of the dephasing process even as it begins, a regime ofquantum dissipation often referred to as non-Markovian dynamics. This topic of
ultrafast spectroscopy of non-Markovian dynamics in a dense atomic vapor is theprimary focus of this thesis. A closely related phenomenon in which the emitted
light is redshifted due to the high density of atoms, known as the local field effect,
constitutes a secondary aspect of this thesis.

2
1.1

Historical Background and Overview of Main Results
This thesis research applies the relatively modern technique of ultrafast spec-

troscopy to the mature field of atomic vapors, resulting in an unusual perspective
on dense vapors as well as ultrafast techniques and the systems to which they arecommonly applied. In particular, the research of this thesis contributes to understanding the fundamental physics of atomic vapors, provides support to theoreticaltreatments commonly applied in condensed phase systems, and exhibits the benefits of ultrafast techniques through the clarity of signatures representing system
response. An overview of the research including some historical background is

Energygiven in this section.

h (w + dw (t ))hw 2w = w2 - w1
hw1
time

Figure 1.1: A schematic of the ground state and excited state energy levels of
an atom undergoing a collision. During the collision the transition frequency
fluctuates by a time-dependent amount δω(t).

This thesis research is related to the more general question of how to model
quantum dissipation [1]. Quantum dissipation generally refers to the irreversible
loss of order in quantum systems. In an atomic vapor, the cause of quantum
dissipation is atomic collisions. During a collision, the resonance frequency ofan atom fluctuates, as shown in Fig. 1.1. A collision can be thought of as thetransient formation of a dimer, with the amplitude of the resonance fluctuationin principle determined by the interatomic potential energy surfaces of the dimer.
If we consider the superposition between ground and excited state as a functionof time, shown in Fig. 1.2, the effect of the collision is to create a phase difference

3
between the superpositions of an atom that has undergone a collision and onethat has not undergone a collision. This loss of phase coherence is known asdephasing. If we overlay the superpositions of an unperturbed and perturbed
atom (see Fig. 1.2), we can define the correlation time tc as the time it takes
for a perturbed atom to become out of phase with an unperturbed atom. The
correlation time is a measure of the phase memory of the system, representing how
long an atom can retain approximately the same phase as an unperturbed atom.
In physics literature, dynamics which occur on timescales less than the correlation
time are termed non-Markovian dynamics, while those which occur in timescales
much greater than the correlation time are termed Markovian dynamics1 . For
non-Markovian dynamics the phase coherence is retained such that the evolution of
the superpositions of the atoms is reversible. Conversely, for Markovian dynamics
the phase coherence of the system is lost over the timescale of interest due to therandomization of the phases of the superpositions due to collisions, making theevolution irreversible. Stated from an experimental perspective, non-Markovian
dynamics arise in experiment whenever the finite duration of collisions must be
taken into account, while dynamics are Markovian whenever collisions can beconsidered to occur infinitely fast.
Historically, the effects of non-Markovian dynamics on frequency-domainlineshapes attracted interest about a century after the first studies of spectral
emission lines in the mid-1800’s. After decades of improving the technique ofwhite-light spectroscopy, in the 1920’s pressure-broadened lineshapes were resolved for a number of atomic vapors and theoretical treatments began to elucidate the underlying physics [2]. Initially, theories which approximated collisions
between atoms and perturbers as infinitely fast in what is known as the impact
1

The term “Markovian” is used because of its relationship to time evolution determinedsolely by the instantaneous state of the system. A more thorough definition is given in Ch. 2.

4Superposition

2p / w
unperturbed

perturbed
timetc

Figure 1.2: The superposition between a ground state and excited state as afunction of time for an uperturbed (top) and perturbed (bottom) atom. The
correlation time tc is the time it takes for a perturbed atom to become out ofphase with an unperturbed atom.

approximation were sufficient to describe the majority of experiments, with good
agreement for linewidths at low densities.
Studies took on new emphasis with the improved intensities and spectral
resolution provided by the advent of the laser in the 1960’s. Initially, theoreticaltreatments employing the impact approximation were still sufficient to explain
most experimental results [3, 4]. Starting in the 1970’s, predictions were madefor signatures of the finite duration of collisions to be manifested in frequencydomain lineshapes as a deviation from Lorentzian form, both near line center andat detunings from resonance on the order of the inverse duration of a collision[5]. These were quickly verified experimentally [6, 7]. A non-impact theory of thetechnique of frequency-domain four-wave mixing (FWM) was presented in 1986,
and FWM experiments revealed non-impact behavior in the wings of heliumbroadened sodium vapor lineshapes in 1989 [8, 9].
Because collisions occur on short timescales their frequency-domain manifestation is most dominant in the wings of the absorption lineshape [10]. Thus
frequency-domain experiments are limited in their ability to probe the fastest
dynamics by the inherently low signal-to-noise ratio and influence of other reso-

5
nances in the wings of absorption lineshapes. This is in contrast to time-domain
experiments in which the fastest dynamics dominate at short times and with highsignal-to-noise ratio. In addition, unlike lineshapes, which are integrated overmany collision durations, time-domain techniques can separate short timescales
from long ones, providing a different perspective from frequency-domain studies
of fast dynamics.
Time-domain techniques began with the first applications of pulsed lasers tospectroscopy in the 1970’s and the development of the theory of transient nonlinear optics [11]. However, applications of ultrafast spectroscopy were made largely
in condensed phase systems, with only a few ultrafast experiments in dense atomic
vapors, which mainly studied propagation effects in transmission [12]. Initially thetheory for ultrafast spectroscopy was based on the optical Bloch equations, which
relied on the impact approximation [13, 14]. Once sub-picosecond probes of relaxation phenomena began revealing ultrafast dynamics in the 1980’s theoreticaltreatments were proposed that modified the optical Bloch equations to include
the finite duration of interactions, with a unified approach to transient nonlinear
optical phenomena using Liouville operators published in 1983 [15, 16]. Ultrafast nonlinear spectroscopy is now commonly applied to such intricate systems
as biological light-harvesting complexes to result in an incredibly sophisticatedunderstanding of their structure and dynamics [17]. In molecular systems, studies
of non-Markovian interactions of atoms in molecules and between molecules andsolvents have provided detailed inter- and intramolecular information, such as how
bonds stretch and molecules conform [18, 19].
One of the main difficulties in ultrafast molecular studies is the dephasing
caused by interactions of the molecules with the solvent. These experiments areusually performed in liquid solutions, where the solvent-solute interactions often
occur at femtosecond time scales, causing ultrafast dephasing, which obscures the6non-Markovian dynamics at short times [20, 21, 22, 23]. In these experiments
it is also common to have dephasing times on the order of the pulse duration.
In contrast, in a dense atomic vapor the pulse duration, collision duration anddephasing time are all well-separated, making dense atomic vapors ideal systems
in which to apply ultrafast spectroscopy to the study of non-Markovian dynamics.
Considering the above opportunities for contribution, we study the phaserelaxation dynamics of a dense potassium vapor using ultrafast spectroscopy. The
experimental results exhibit signatures characteristic of both Markovian and nonMarkovian dynamics, which can be modeled using the non-Markovian theory commonly applied to more complex systems [15]. Studying such a model system lends
support to the use of the non-Markovian theory. Through further experiments inconjunction with detailed theoretical derivations we also gain insight into longrange interactions in resonance-broadened vapors. This work contributes to theunderstanding of excitonic interactions in dense media in general and a resonancebroadened vapor in particular. The research described in this thesis takes advantage of the simple electronic structure and tunable density of dense atomic vapors
to exhibit the benefits of ultrafast spectroscopic techniques through the clarity ofsignatures representing system response. Finally, we show that with time-domain
techniques we are able to probe other effects that have proven difficult to separate
from competing factors in the frequency domain, such as the local field effect.
1.2

Thesis Organization
This thesis is organized as follows. Chapter 2 includes an introduction to

the ultrafast technique of transient four-wave mixing, which is a general techniquethat encompasses the experiments of this thesis. The density matrix formalism
is described, leading to a derivation of the optical Bloch equations for homogeneously broadened and static inhomogeneously broadened systems. It is shown7
that the optical Bloch equations apply the impact approximation. Then the theory including non-Markovian dynamics is presented, which is used to model thetwo-pulse experimental results in the next chapter.
Chapter 3 contains the experimental setups and results. First the experimental components are described. Then the two-pulse TFWM experiment andresults are presented along with fits from the non-Markovian theory derived in
the previous chapter. The three-pulse and time-resolved experimental setups andresults are then presented. Although the data for low temperatures and densities supports the theoretical treatment used in the two-pulse results, the higher
temperature and density three-pulse data exhibits bi-exponential behavior that
requires a more detailed analysis, presented in the next chapter.
Chapter 4 describes a theoretical approach to modelling the three-pulse
results that derives the correlation function of frequency fluctuations in an exciton
picture based on molecular dynamics simulations. First an introduction is madeto the exciton formalism, then a description of the molecular dynamics simulation
is given, followed by the procedure of determining the correlation functions andthe calculated correlation functions and three-pulse photon echo peak shifts.
Chapter 5 describes the local field effect, with a derivation of the local field
effect in the frequency-domain and time-domain, followed by experimental results
and some preliminary calculations of propagation effects.
In the conclusion, Ch. 6, future ideas for buffer gas and transmission experiments are presented along with a summary of main results.
Appendices A-B.3 provide derivation details for Ch. 2. In particular, App. A
covers the details of the derivation of the optical Bloch equations for a two-levelsystem, App. B.1 describes the use of Feynman diagrams, App. B.2 provides
details of the cumulant expansion and App. B.3 gives further response function
derivation results.

8Appendices C-E supplement the experimental chapter. App. C lists some
attributes of potassium for reference. App. D describes the pulse picker mentioned
in the experimental chapter and other experimental components. App. E describes
the design and manufacture of the novel vapor cells used in the experiments.
App. F supplements Ch. 4 by providing the derivation of the exciton Hamiltonian for a 2-level system, while App. G gives detailed information about how to
run the molecular dynamics package GROMACS.
Finally, App. H lists some useful constants and conversions.

Chapter 2Theoretical Background for Transient Four-Wave Mixing Experiments

In order to provide a basis for interpreting the experimental results presented
in this thesis, in this chapter an overview of the signatures of non-Markovian dynamics in a dense atomic vapor is presented. We start with a description of
the transient four-wave mixing process and the fundamentals of nonlinear optics. The density matrix formalism is then presented to facilitate the theoreticalderivations of the material response through the optical Bloch equations. Firstthe cases of homogeneously broadened and static inhomogeneously broadened systems are considered and compared. Then it is shown that the typical optical Bloch
derivation of the homogeneously broadened system response makes the Markov
approximation for the dephasing process. Finally the theory that allows us to takenon-Markovian dynamics into account is presented and distinctions are drawn between the signatures of non-Markovian dynamics and those of homogeneously andstatic inhomogeneously broadened systems. The derivation of the non-Markovian
system response is carried out in detail for a three-level system and is the theoretical basis for much of the theoretical fitting made to the experimental data of
this thesis.

10
2.1

Transient Four-Wave Mixing
Transient four-wave mixing (TFWM) is a time-domain spectroscopic tech-nique used to study dynamics in a variety of systems. TFWM occurs whenthree input pulses of either the same or different frequencies successively excite
a medium. The radiative output, which is the fourth wave, depends on the time
sequence and separation of the input pulses. For the experiments presented inthis thesis the field frequencies are degenerate, and thus we will only consider thedegenerate case for the following derivations.
In the degenerate transient four-wave mixing process, three pulses withwavevectors k1 , k2 and k3 and degenerate frequencies ω1 = ω2 = ω3 are incident on a material sample. Figure 2.1 is a schematic of TFWM. The first pulse,with wavevector k1 , induces a polarization in the sample with a spatial phaseacross the vapor determined by its angle of incidence and wavelength. After somedelay τ , the second pulse, k2 , produces an excited state population whose amplitude is greatest in areas where the polarization left by the first pulse constructively
interferes with the second pulse and least where they destructively interfere. Thus
the excited state population is spatially modulated, forming an amplitude gratingwith periodicity k2 − k1 . After a second delay T the third pulse, k3 , scatters
off the grating (i.e. creates a polarization that radiates) into the new direction
k3 + k2 − k1 .
By varying the relative delay between pulse k1 and k2 , or pulse k2 and k3 ,the excited state dynamics of the system can be investigated. In particular, if
the time delay between pulse k1 and k2 , τ , is varied, the transient behavior of
the induced polarization can be investigated. If the time delay between pulse k2
and k3 , T , is varied, the transient behavior of the excited state population canbe investigated. Stated another way, varying the time delay between pulses al-11lows a characterization of the evolution of the intermediate density matrix states(described below). If the pulses are weak, the density matrix equations of motion can be expanded to third order in the field. As the fact that the signal
is radiated in a new direction attests, the signal is generated completely by thenonlinear polarization. Thus the detected signal energy is proportional to thesquare of the third-order nonlinear polarization. Varying the first delay τ probes
the off-diagonal density matrix elements ρnm representing the coherent superposition created between states n and m, while varying the second delay T probes
the density matrix elements ρnn representing population and, in the case of athree-level system, ρnm representing Raman coherence [24]. It is from this density
matrix expansion in orders of the field that the language used above, referring to
the sequence, polarization (first-order) – population (second-order) – polarization(third-order), stems.

xk4 = -k1+k2+k3
k1
k2
&

k3
I

sample
I1
I2
¿

I3
T

t

Figure 2.1: Schematic of transient four-wave mixing. The phase fronts of pulses
k1 and k2 are depicted with lines perpendicular to the wavevectors. The resulting
excited state population has an amplitude I which varies spatially across the
sample. The dark bars across the sample represent the maxima of the excitedstate population grating. At the bottom of the diagram are the pulses labeled bytheir intensities, shown as they are delayed in time.

It is clear from this description that to predict the response of the material

12
to the fields requires an understanding of how the polarization is coupled to thefields. The relationship between the material polarization and the incident fields is
described by the Maxwell equations. In particular, the nonlinearity of the material
is contained in the relation
D = E + 4πP,

(2.1)

where the polarization P (dipole moment per unit volume) depends nonlinearlyon the electric field E and contains all the information concerning the responseof the medium to the field in what is known as the constitutive equation. If thisequation were known, it would then be possible to find the solution of the Maxwell
equations with appropriate boundary conditions for any field and material system.
The constitutive equation is complicated, however, and therefore we must makeapproximations.
If the incident field is sufficiently weak1 , it is appropriate to express thepolarization P as a power series in the field strength E [25]. We do this specificallyto third-order with three different incident fields to represent degenerate TFWM:
P(k, ω) = P(1) (k, ω) + P(2) (k, ω) + P(3) (k, ω)withP(1) (k, ω) = χ(1) (ω) · E(k1 , ω),P(2) (k, ω) = χ(2) (ω) : E(k1 , ω)E(k2 , ω),(2.2)

P(3) (k, ω) = χ(3) (ω) : E(k1 , ω)E(k2 , ω)E(k3 , ω),
where χ(n) is the nth-order nonlinear susceptibility, a tensor of rank (n + 1). In
Eq. 2.3 the variable ω in the susceptibilities implies that each order of the polarization contains a sum over all the possible frequency combinations, including
1

A loose definition of “sufficiently weak” is that the incident field is much less than thecharacteristic atomic field strength Eat = e/a20 , where −e is the charge of the electron and a0 is
the Bohr radius of hydrogen.

13
negative frequencies, for that order. In addition, the dependence of the nthorder polarization P(n) on wavevector k means that each order of the polarization
contains a sum over all the possible combinations of wavevectors k1 . . . k3 . The
susceptibility (and therefore the polarization) is determined by the physical microscopic structure of the medium, and so it must be derived using quantum
mechanics. Thus if we wish to understand the response of a medium to third order in the field, as we do in TFWM, we must find a way of deriving the third-orderpolarization. Because in general a medium will be made up of many components
in different states, the density matrix formalism, which employs ensemble averages
over the states, proves useful for this task.
2.2

Density Matrix Formalism
The physical properties of a material system can in general be describedwith a wavefunction ψ(r, t). The wavefunction obeys the Schrödinger equationi~(∂ψ(r, t)/∂t) = Hψ(r, t), and therefore its time evolution and observable properties can be fully determined. For an ensemble system of quantum objects eachwith state i, it is convenient to use the density matrix formalism, which gives astatistical description of the system. The density matrix operator of the system isthen defined as the ensemble average of the outer product of each state vector of
the system with itself. The average is over all of the possible states of the system,
each state i having a probability weight wi :ρ≡Xi

wi |iihi| = |iihi|.

(2.3)

The corresponding density matrix elements ρnm give the probability of the system
to be in a certain physical state. This becomes clear if we use the eigenketsX|ni of the system as the base kets of all the states, |ii =ain |ni. Then the
diagonal elements ρnn = hn|ρ|ni = |ann2| give the probability that the system is in14energy eigenstate n. The off-diagonal elements ρnm ≡ hn|ρ|mi = an a∗m give thecoherence between the levels n and m, that is, ρnm is nonzero only if the system
is in a coherent superposition of the energy eigenstates n and m.
With the density matrix operator we can calculate the ensemble-averaged
value of any observable quantity, including the polarization P:
[P] ≡

X
i

wi hi|P|ii =

XX
n

m

hn|ρ|mihm|P|ni = T r(ρP).

(2.4)

To find how the expectation value of an observable quantity evolves with time inthe system, it is sufficient to determine how the density matrix evolves in time.
Because the wavefunction |ψi obeys the Schrödinger wave equation, we can alsofind the equation of motion for ρ. Thus
X
1 X
∂ρ
=
wj (|ψ̇j ihψj | + |ψj ihψ̇j |) =wj (H|ψj ihψj | + |ψj ihψj |H)
∂t
i~ j
j
=

1
1
(Hρ − ρH) = [H, ρ].
i~
i~

(2.5)

The Hamiltonian H typically consists of two terms,
H = H0 + Hint ,(2.6)

where H0 is the Hamiltonian of the unperturbed system and Hint represents theinteraction of the fields with the system. In the basis of the energy eigenfunctions,
H0 is diagonal and therefore [H0 , ρ]nm = (ǫn − ǫm )ρnm , where ǫj is the energy ofstate j. In the electric dipole approximation in which the spatial dimension of
the atom is considered much smaller than the incident wavelength, the electronicinteraction Hamiltonian is given by Hint = −µ·E(t), where µ = −er is the electric
dipole moment operator of the atom and E(t) is the incident field.
The equation of motion as it stands does not account for dissipative processes such as dephasing due to collisions and population decay. It is common
to add phenomenological terms to the equation of motion that represent decay

15
processes, using the assumption∂ρnm
eq= γnm (ρnm − ρeqnm ), where ρnm is the
∂t
relax



equilibrium value to which ρnm relaxes at rate γnm , with ρeq
nm = 0 for n 6= m, corresponding to the inability of incoherent thermal processes to produce coherent
superpositions of atomic states. Here γnm corresponds to the population relaxation rate T1−1 if the state is diagonal or the dephasing rate T2−1 if the state is off
diagonal. Although the population decay term can be justified by quantizing thefield [13], adding phenomenological dephasing terms is equivalent to making theMarkov approximation for the coherence decay process, as will be shown below.
Thus the final form of the density matrix equation of motion including
phenomenological decay terms isiρ̇nm = −iωnm ρnm − [−µ · E(t), ρ]nm − γnm (ρnm − ρeq
nm ),
~

(2.7)

where ωnm ≡ (En − Em )/~ is the transition frequency. Because of the similarityof this formalism with that used to describe a spin-one-half particle in a magnetic
field, these equations of motion are termed the optical Bloch equations2 .
To find the polarization P using the density matrix via Eq. 2.4, we mustfirst solve the density matrix equation of motion, Eq. 2.7. This is generally notpossible to do exactly, and thus we consider a perturbative approach.
For weak fields an expansion can be made in terms of the interaction Hamiltonian, replacing Hint by λHint , where λ ranges from zero to one to characterize
the strength of the perturbation. The solution of Eq. 2.7 is then in the form of apower series in λ:
(1)
2 (2)
3 (3)
ρnm = ρ(0)nm + λρnm + λ ρnm + λ ρnm .

(2.8)

For Eq. 2.8 to be a solution of Eq. 2.7 for all values of λ, the coefficients of each
2

The Bloch equations were initially developed by Bloch to simulate nuclear magnetic resonance experiments [14, 26]. They were later extended to provide a quantum-mechanical modelof the coherent response of a dilute atomic vapor to a laser field. In this context the equations
are called the optical Bloch equations [13, 14].

16
power of λ must satisfy Eq. 2.7 individually. The solutions starting with firstorder all take the same form; we obtain for the qth-order equation of motion
i
(q−1)
(q)
]nm .
ρ̇(q)
nm = −(iωnm + γnm )ρnm − [−µ · E(t), ρ
~

(2.9)

This system of equations can be integrated directly because each equation uses
previously solved orders and contains only linear terms of the current order. The
qth-order solution isiρ(q)nm (t) = −

~

Z t

−∞

′

dt′ e(iωnm +γnm )(t −t) [−µ · E(t′ ), ρ(q−1) ]nm .

(2.10)In the case of transient four-wave mixing, the third-order polarization is the quantity of interest. Following the above derivation, this means we must find the thirdorder solution to the density matrix equation of motion, ρ(3) , and take the traceof its product with the polarization operator P(ω) = N µ:[P(3) ]mn = N Tr(ρ(3) µ)mn .
2.3(2.11)

Optical Bloch Equations for Homogeneously and Static Inhomogeneously Broadened Systems
To make the physics of the polarization conceptually transparent let us firstconsider a two-level atomic system, with a ground state a and excited state b.
We will consider both a homogeneously broadened system as well as a static inhomogeneously broadened system in order to form a basis of comparison withthe theoretical approach to non-Markovian dephasing. Let there be only two
incident fields, such that the second pulse k2 creates both the second-order population and third-order polarization. The incident fields are of the form Ei (r, t) =1
E (t)ei(ki ·r−ωt) + c.c., i = 1, 2, with ω nearly resonant with the atomic transition
2 ifrequency ω0 = ωb − ωa . We specify that the third-order polarization of interest
is in the direction k4 = 2k2 − k1 .

17
The equations of motion ρba = ρ∗ab and ρD = ρaa − ρbb from Eq. 2.9 thenbecome in the rotating wave approximation (see App. A for details)


1
iµba
exp −
+ i∆ω t(2.12)
~
T2

 
Z t
1
′ (n−1)×E(r, t )ρDexp+ i∆ω t′ dt′
T
2
−∞


2iµ
t
ba
(n)
ρD (r, t) =(2.13)
exp −
~
T1
 ′
Z t
t
′ (n−1)∗′ (n−1)dt′ ,×{−E(r, t )ρ̂ab + E (r, t )ρ̂ba } exp
T1
−∞
(n)

ρ̂ba (r, t) =

(n)

(n)

where ∆ω = ω0 − ω and ρ̂ba = ρba exp(iωt).
Choosing the third-order coherence only in the direction k4 = 2k2 − k1
results in (details in App. A)(3)ρ̂ba (k4 ) =(2.14)


µba
~

3

Z t Z t′ Z t′′




1
exp −
+ i∆ω t + ik4 · r
dt′ dt′′ dt′′′
T2−∞ −∞ −∞
′′′



t
× E2 (r, t′ )E2 (r, t′′ )E∗1 (r, t′′′ ) exp γ(t′ − t′′ ) ++ i∆ω(t′ + t′′ − t′′′ )
T2
′′′


t
+ i∆ω(t′ − t′′ + t′′′ ) ,+E2 (r, t′ )E∗1 (r, t′′ )E2 (r, t′′′ ) exp γ(t′ − t′′ ) +T2−2iρ(0)where γ = T2−1 − T1−1 , µab = µba and the population was assumed to start in the
(0)

(0)

ground state, ρD = ρ(0) , ρba = 0.
We now consider delta-function field excitations separated by variable temporal delays. This assumption is valid when the pulse duration is much shorter
than the decay times T1 and T2 as well as the inverse detuning (∆ω)−1 . Then thefield envelopes become Ej (t) = δ(t − tj ), j = 1, 2, where tj is the pulse arrival
time; we also assume on-resonance excitation and set ω = ωba .
For the case of a homogeneously broadened system, the phenomenologicaldephasing constants already account for the homogeneous width. Thus the polar-

18
ization is P(3) (r, t) = P̂(3) (r, t) exp(−iωt) + c.c., with(3)

P̂(3) = N µab ρ̂ba (r, t, ω0 ),(2.15)

where N is the number density. Substitution of Eq. 2.14 with delta-function pulses
in the above equation yields the polarization in the homogeneous limit:
P̂(3) (r, t) = −iρ(0) CN µba θ(t − t2 ) exp[ik4 · r − T2−1 (t − t1 )](2.16)

where θ(t) is the Heaviside step function and C a constant.
We now consider the limit of the system being static inhomogeneously broadened [27]. In the case of an inhomogeneously broadened system, the product ofthe density matrix element and the inhomogeneous lineshape is integrated overfrequency; thus P̂(n) becomes
(n)

P̂

=N

Z ∞
0

(n)

µab ρ̂ba (r, t, ω0 )g(ω0 )dω0 ,(2.17)

R∞
where the lineshape function g(ω0 ) obeys the normalizing condition 0 g(ω0 )dω0 =1.

To model static inhomogeneous broadening we employ a Gaussian form
for the lineshape function, which is commonly used to represent the lineshapearising from a large number of frequencies (through application of the central
limit theorem):√g(ω0 ) = ( πδω)−1 exp[−(ω0 − ω)2 /∆ω 2 ].

(2.18)

This results in a polarization given byP̂(3) (r, t) =(2.19)
(t − t1 ) 1− (t − 2t2 + t1 )2 (∆ω)2 ,−iρ(0) DN µba θ(t − t2 ) exp ik4 · r −T2
4

where D is a constant. Note the homogeneous response can be retrieved by setting∆ω = 0.

19
The behavior of the real-time signal as a function of delay is shown for the
homogeneous case in Fig. 2.2 and the inhomogeneous case in Fig. 2.3. The realtime signal for the homogeneously broadened system is the free polarization decay,
while for the static inhomogeneously broadened system it is the photon echo. In
the latter case, the first pulse creates a coherent superposition between groundand excited state, which evolves according to each atom’s resonance frequency
until the second, conjugate pulse reverses the evolution to cause a rephasing,called the photon echo, the same delay later. The photon echo signal will have awidth in time which is the inverse of the inhomogeneous width of the frequency
distribution. Thus the pulses must be short, that is, wide in frequency such thatthey span the inhomogeneous width, for a photon echo to arise. An analogy is
described in Fig. 2.4. The temporal behavior for both the homogeneous and static
inhomogeneous limits are plotted in Fig. 2.5.

2-Level System

Increasing Delay

2-Pulse Real-Time TFWM Signal

Homogeneously Broadenedt2t, real time

Figure 2.2: Calculated 2-pulse real-time TFWM signals for a homogeneously
broadened system. The delay between pulses increases downward, with horizontal
ticks spaced by the delay step size.

Static Inhomogeneously
Broadened 2-Level System

Increasing Delay

2-Pulse Real-Time TFWM Signal

20

t

2

t, real time

Figure 2.3: Calculated 2-pulse real-time TFWM signals for a static inhomogeneously broadened system. The delay between pulses increases downward, withhorizontal ticks spaced by the delay step size.

The time-integrated signal is the integral over the square of the polarization:
Z ∞−∞|P(3) (r, t)|2 dt.

(2.20)In the homogeneous limit the time-integrated signal is2
J = Aθ(t − (t2 − t1 )) exp − (t2 − t1 )T2

(2.21)

where A is a proportionality constant independent of t2 − t1 .
The time-integrated signal for an inhomogeneously broadened system is




4
∆ω
J = Bθ(t − t2 − t1 ) exp − (t2 − t1 ) 1 + Φ √ (t2 − t1 )T2
π(2.22)

where B is a proportionality constant independent of t2 −t1 and Φ(x) ≡ √2π

R π −t2
e dt.
0

The time-integrated behavior for both types of broadening is shown in

Figs. 2.6 and 2.7. The decay of the TFWM signal for both the inhomogeneousand homogeneous cases is directly related to the dephasing of the coherent superposition between the ground and excited state. For the case of a homogeneously

21

Figure 2.4: Runners on a track as an analogy for the photon echo. The
runners represent oscillators, with their speeds the resonance frequencies, andthe gun shot a resonant pulse. The first pulse creates a coherent superposition between the ground and excited state. During the ensuing delay, theatomic superpositions dephase from one another due to their different frequencies. A second conjugate pulse causes the oscillators to reverse their evolution,
resulting in a rephasing of the coherence the same delay τ later, the photonecho. This interactive animation can be found on the Cundiff group website at
http://jilawww.colorado.edu/˜cundiffs/Research-SemiCoherence.html, and is anecho (pardon the pun) of the front cover artwork of the November 1953 edition of
Physics Today. [Created by J. Fall and V. Lorenz, JILA.]

broadened transition, the decay time of the TFWM signal is half the dephasing
time, T2 /2, while for a static inhomogeneously broadened transition it is T2 /4.
This difference of a factor of 2 between the homogeneous and inhomogeneous casecan be understood based on the real-time signals. In the homogeneous case thepeak of the real-time signal, the free polarization decay, occurs at t = t2 , while in
the inhomogeneous case the peak of the real-time signal, the photon echo, occurs
at t = t2 + (t2 − t1 ). When these signals are integrated over time, this differencein peak location results in the inhomogeneous response decaying twice as fast.

22
2-Pulse Real-Time TFWM Signals for(b)

Homogeneous

Response(c)

Inhomogeneous

Response(a)

Excitation Pulses

Homogeneously and Static Inhomogeneously Broadened 2-Level Systems
Zero delay

Finite delay

between pulses

between pulses

t, real time
t

1

& t

2

t, real time
t

1

t

2

Figure 2.5: Calculated 2-pulse real-time TFWM signals for (a) two-pulse excitation in the case of a (b) homogeneously broadened and (c) static inhomogeneously
broadened 2-level system. The two cases of zero delay between pulses and finite
delay are shown.

In both cases the decay rate is then doubled because we are detecting the fieldsquared, or the intensity. The time-integrated inhomogeneously broadened system
response exhibits a peak at approximately δω −1 . This behavior is a result of theincreasing area of the real-time signal at short times, shown in Fig. 2.7. There is
only “half” a photon echo at zero delay due to causality. With increasing delay
the “full” photon echo begins to develop, resulting in an initial increase of thetime-integrated signal.

2-Pulse Time-Integrated TFWM Signal

23

Homogeneously Broadened
2-Level System, delay between pulses
T /22

2-Pulse Time-Integrated TFWM Signal

Figure 2.6: Calculated 2-pulse time-integrated TFWM signal for a homogeneously
broadened system (natural log scale).

Inhomogeneously Broadened
2-Level System, delay between pulses
T /2
2

Figure 2.7: Calculated 2-pulse time-integrated TFWM signal for a static inhomogeneously broadened system (natural log scale).

2.4

The Markov Approximation
The above treatments of homogeneously and inhomogeneously broadened

systems contain some important assumptions about the microscopic timescales of24
correlations. In particular, the treatment of the homogeneously broadened system
above only accounts for exponential dephasing of the coherence, corresponding toa Lorentzian lineshape. In reality, no lineshape is perfectly Lorentzian. The finite
duration of collisions results in deviations from Lorentzian behavior both at line
center and in the wings of the absorption lineshapes [15]. This translates to nonexponential dephasing in time-domain experiments.
In this section the phenomenological approach used above is shown to beequivalent to making the Markov approximation for the dephasing process. In
physics literature, Markovian evolution refers to time evolution determined solely
by the instantaneous state of the system. Note this is not the same as the mathematical definition of a Markovian process, in which the time evolution is dependent
on the previous state (the precise definition is provided in the next section). In
this limit dephasing collisions are considered to occur infinitely fast, that is, thecorrelation time of the atomic superpositions is very short compared with thetime scales of the nuclear dynamics. If we consider the optically-induced coherent
superposition between ground and excited state as a function of time, the Markovian limit assumes collisions result in sudden phase jumps (see Fig. 2.8). In thissection the time-domain response will be derived for a two-level system using thedensity matrix formalism including finite frequency fluctuations in the Markov
approximation (also known as the impact approximation) [13]. It will be shownthat linear frequency fluctuations result in dephasing of the coherence, and that
the Markov approximation results in exponential dephasing. In the next section
this treatment will be extended to non-Markovian dephasing.
To calculate the optical response including the finite duration of collisions
the random frequency fluctuations at short times must be taken into account. To
do so we add a time-dependent fluctuation to the excited state frequency. We

25start with the equation of motion from Eq. 2.9:
i
(q−1)
(q)
]nm .
ρ̇(q)
nm = −(iωnm + γnm )ρnm − [−µ̂ · E(t), ρ̂~
We add a term to represent frequency fluctuations δω(t):
i
(q)
(q−1)
ρ̇(q)
]nm .
nm = −(iωnm + iδωnm (t) + γnm )ρnm − [−µ̂ · E(t), ρ̂
~(2.23)

Now solving this equation we findρ(q)nm (t)

i
= −
~

Z t

−∞

′

dt′ e(iωnm +γnm )(t −t) [−µ̂ · E(t′ ), ρ̂(q−1) ]nm(2.24)Z t
ih Z t′′′′′× exp iδωnm (t′ )dt′ .
δωnm (t )dt −00

We now apply the ensemble average, which only affects the random variations in δω(t):ρ(q)
nm (t)

i
= −
~

Z t

−∞

′

dt′ e(iωnm +γnm )(t −t) [−µ̂ · E(t′ ), ρ̂(q−1) ]nm(2.25)Z th Z t′i′′′′×hexp iδωnm (t )dt −δωnm (t′ )dt′ i.
00

To solve we approximate the ensemble average of the product of exponentials tobe the product of the ensemble averages and Taylor expand the exponentials. We
consider only one of the δω terms as the same treatment applies to both:
Z t

Z t

dt′ hδω(t′ )iδω(t )dt ]i = 1 + i
0Z 0t Z t
1dt′′ hδω(t′ )δω(t′′ )i + · · ·
dt′
−
2 0
0
Z t
2n Z t(−i)+
dt1 . . .
dt2n hδω(t1 ) . . . δω(t2n )i + · · · .
(2n)! 00hexp[−i′′(2.26)

Now we note that the fluctuations δω(t) have zero mean (the random fluctuations
are as often positive as negative), so terms odd in δω(t) are zero. Then if we
consider the first even term, the integrand, hδω(t′ )δω(t′′ )i, is the two time correlation function of frequency fluctuations and ultimately will be used to determine

26
the remaining terms. Thus it plays an important role in the theory and in theinterpretation of experiment. At this point we must decide what form it shalltake. To reproduce the results for a homogeneously broadened system above, we
now take the Markov approximation.
Detected
Superposition

Markovian Limit

Collision

Collision
time

Figure 2.8: Superpositions in the Markovian limit. Collisions result in phasejumps.

The Markov approximation assumes that variations in ω are rapid comparedto other changes such as population decay rates and dephasing rates such thatthe fluctuation at one instant in time can be considered completely uncorrelatedwith that of any different time. Thus we take the correlation function to be adelta function with an overall constant representative of the amplitude of thefluctuation:hδω(t)δω(t′ )i = 2γph δ(t − t′ ).
The constant γph can be set phenomenologically according to experimentally determined linewidths. The first even term of the expansion then is
1
2

Z t
0

′

dt

Z t
0

dt′′ hδω(t′ )δω(t′′ )i = −γph t.

(2.27)

We now assume that the fluctuations obey Gaussian statistics, that is, that they
have a normal distribution. This assumption is reasonable if the fluctuations are
made up of many small, random contributions with a finite mean and variance,
qualities which satisfy the requirements of the central limit theorem. Making this27
approximation the higher order terms are simply factors of the two-time correlation function, resulting in a factor of (2n)!/2n , equaling the number of permutations of 2n terms into n pairs. To restrict the permutations to distinguishable
products, we divide by n!, resulting in the 2nth term,(−i)2n
(2n)!

Z t

Z t

dt2n hδω(t1 ) . . . δω(t2n )i
Z t
Z t
(−1)n
n (2n)!
=
dt2n δ(t1 − t2 ) . . . δ(t2n−1 − t2n )dt1 . . .
(2γph ) n
(2n)!
2 n! 00(−γph t)n.
(2.28)=
n!
dt1 . . .

00Using this result we then find a simple form for the expansion:
hexp −i∞X(−γph t)nδω(t )dt i == exp(−γph t).
n!
0
n=0

Z t

′

′

(2.29)

Inserting this result back into the density matrix element for both fluctuation
terms we find the solution for the equation of motion:iρ(q)nm (t) = −

~

Z t

−∞

′

dt′ e(iωnm +γnm +2γph )(t −t) [−µ̂ · E(t′ ), ρ̂(q−1) ]nm .

(2.30)

Thus fluctuations with Markovian statistics result in exponential decay of thecoherence. This equation is just that formed using phenomenological dephasing
terms for a homogeneously broadened system, with the population and dephasing
rates separated into γnm − 2γph , γnm = 0, n 6= m. This approximation is not
valid when the experiment probes timescales in which fluctuations do not occur
instantaneously nor are obscured by fast dephasing dynamics. This brings us to
the theory used to treat nonlinear optical interactions in the non-Markovian limit.
2.5

Non-Markovian Theory
The Markov approximation only holds for the limit in which either theexperiment is insensitive to the retention of phase memory during collisions or

28
the fluctuations are fast compared with the dephasing dynamics. With ultrashort pulses we are capable of resolving the response of a system on the timescaleof collisions, and in a dense atomic vapor the dephasing time is longer than thecollision duration, meaning the signal will no longer exhibit purely exponential dephasing. Modeling non-Markovian dynamics theoretically ultimately comes down
to making a choice for the form of the fluctuations δω and thus the form of the
two-time correlation function of frequency fluctuations. To account for phasememory during collisions it is common to invoke the mathematical definition of a
Markov process. In particular, according to the mathematical definition a stochastic process x(t) is Markovian if its future probabilities P (x(tn )) for every n aredetermined by the most recent states:
P (x(tn ) ≤ xn |x(tn−1 ), . . . , x(t1 )) = P (x(tn ) ≤ xn |x(tn−1 )),(2.31)

with t1 < t2 ... < tn [28]. This equation can be read as “The probability of a systembeing less than or equal to a value xn at time tn given all previous states is equal
to the probability given only the most recent state, x(tn−1 ).” Applying this tothe frequency fluctuations allows the fluctuations to depend on their most recent
form, improving the treatment of the short-time limit in which non-Markovian
dynamics are important. Furthermore, in a theorem proven by Doob, any random
process x that is both Markovian and follows Gaussian statistics (also known asan Ornstein-Uhlenbeck process [29]) has a correlation function Cx (τ ) of the form
Cx (τ ) = σx2 e−τ /τr ,(2.32)

where τr is the relaxation time and σx the standard deviation of x [30]. Using
this correlation function is a common way of accounting for phase memory incondensed phase systems, often referenced simply as a stochastic model or the
Kubo model [15, 31, 32, 33]. Another way of achieving this result theoretically29
is to invoke the overdamped Brownian oscillator model in the high temperature
limit [16].
Note that in much of the literature as well as in this thesis the word “nonMarkovian” refers to the dephasing process, not the fluctuations themselves. Thisis an important distinction to make because as just shown the stochastic model
actually makes the Markov approximation for the dephasing mechanism. Thus theword “non-Markovian” is a descriptor for the dynamics of the coherence, not the
dynamics of collisions. The stochastic model does not explicitly account for the
dipole-dipole interaction between atoms and thus is limited in the physical insightit can provide. However, because of its frequent use in more complex systems
the well-defined signatures of non-Markovian dynamics in a dense atomic vapor
provide an important opportunity to test this theoretical description of dephasing[23].
Although it is possible to derive the dynamics including non-Markoviandephasing with steps similar to the previous section in which we insert the timedependent resonance frequency into the density matrix equations of motion, there
is a parallel approach that does not require us to keep track of the detailed form ofintermediate equations [15]. This approach derives the nonlinear response in Liouville space, which was originally developed by Zwanzig [34, 35, 36, 37]. A major
benefit of this formalism is that it does not require us to carry unused wavevector
combinations only to discard them later, as is done in the optical Bloch equations,in which all permutations of the wavevectors of the fields must be carried to thirdorder, at which point only terms in the direction of interest are chosen. By using
Liouville operators instead we are able to specify the perturbation sequences ofinterest and only derive response functions for those sequences. Here a summaryof the most important steps is provided for a three-level system in order to facilitate the discussion of the modeling of experimental results. The equations will

30
also be simplified to the solution for a two-level system for comparison with theabove results.
We begin with the Hamiltonian for a perturbed system from Eq. 2.6:
H = H0 + Hint ,(2.33)

where H0 is the Hamiltonian for the material system in the absence of the radiation
field and Hint represents the radiation-matter interaction and is given byHint (t) = E(r, t)V,

(2.34)

where V is the dipole operator of the atom. The evolution of the system isrepresented by the density matrix equation of motion, which takes the form of the
Liouville equationdρ= −i[H0 , ρ] − i[Hint , ρ],dt(2.35)

which becomes in Liouville-space notation,dρ= −iL0 ρ − iLint ρ.
dt(2.36)In Liouville space we employ Liouville (tetradic) operators, whose action on an
ordinary (dyadic) operator A is given by
L0 A ≡ [H0 , A],(2.37)

Lint A ≡ [Hint , A].

(2.38)

To simplify the coming expressions we now set ~ = 1 and define the Liouvilledipole operator
VA ≡ [V, A].

(2.39)We are interested in calculating the polarization P (r, t), which is given by the
expectation value of the dipole operator V :P (r, t) ≡ hhV |ρ(t)ii,(2.40)

31
where double-angle-bracket notation is used to denote an inner product of operators. For any two operators,
hhA|Bii ≡ Tr(A† B).

(2.41)We also define a Liouville-space matrix element by
hhA|L|Bii ≡ Tr(A† LB).

(2.42)The third-order polarization of interest is obtained by calculating ρ(t) perturbatively to third order in Lint :ρ(t) = U (t, −∞)ρ(−∞) ≡ ρ(0) + ρ(1) + ρ(2) · · · ,(2.43)

where U is the time-evolution operator, which when expanded takes the form
Z tdτ G(t − τ )Lint (τ )(2.44)U (t, −∞) = 1 − i
−∞
Z t
Z τ1
2
+(−i)
dτ1
dτ2 G(t − τ1 )Lint (τ1 )G(τ1 − τ2 )Lint (τ2 ) + · · · ,−∞−∞where the Green function G(τ ) is the formal solution of the Liouville equation in
the absence of the electromagnetic field:G(τ ) ≡ exp(−iL0 τ ).

(2.45)The third-order polarization, written in the time-domain, then is
P

(3)

3

(r, t) = (−i)

Z ∞
0

dt1

Z t1
0

dt2

Z t2

dt3

(2.46)

0×hhV |G(t3 )Lint (t1 − t3 )G(t2 − t3 )Lint (t1 − t2 )G(t1 − t2 )Lint (0)|ρ(−∞)ii.
Reading this equation from right to left reveals the evolution of the system: The
system starts at t = −∞ in state ρ(−∞). The interactions with the field are
represented by Lint (τ ) and occur at times 0 ≤ (t1 − t2 ) ≤ (t1 − t3 ) ≤ t1 . Eachinteraction is followed by an evolution in time according to G(τ ) for the intervals

32(t1 − t2 ), (t2 − t3 ) and t3 . At time t1 the polarization is calculated. This equation
for the polarization is the time-ordered expansion of the evolution operator. To
simplify the following steps we make a transformation of time variables and define
τ1 = t1 ,τ2 = t1 − t3 ,
τ3 = t1 − t2 ,(2.47)

These new time variables allow us to order the interactions 0 ≤ τ3 ≤ τ2 ≤ τ1 .
Equation 2.46 then becomes
Z τ2
Z τ1
Z ∞
(3)
3
dτ3
dτ2
P (r, t) = (−i)
dτ1(2.48)

00

0×hhV (τ1 )|G(τ1 − τ2 )Lint (τ2 )G(τ2 − τ3 )Lint (τ3 )G(τ3 )Lint (0)|ρ(−∞)ii.
By inserting the incident fields and the appropriate dipole operators we havean explicit expression for the polarization. We write the field in the formE(r, t) = [E1 exp(−iω1 t) + E2 exp(−iω2 t) + E3 exp(−iω3 t)] + c.c..

(2.49)

Here Ej is the amplitude of the jth field and ωj is its frequency. The total
polarization is the sum over all possible permutations of wavevector and frequency.
We choose to detect the polarization emitted in the direction k4 = −k1 + k2 + k3
with frequency ω4 = −ω1 + ω2 + ω3 . In this case we need only consider thepolarization
P

(3)

3

(k4 , t) = (−i)

Xm,n,q=1,2,3

Z ∞
0

dτ1

Z ∞
0

dτ2

Z ∞

dτ3

0



× exp −iωq (τ1 ) + iωn (τ1 − τ3 ) + iωm (τ1 − τ2 )×Em (τ3 )En (τ2 )Eq (τ1 )R(τ1 , τ2 , τ3 , τ4 ),(2.50)

where the summation is over all 3! = 6 permutations of the fields and we haveseparated the fields from the material response by defining the nonlinear response

33function, R(τ1 , τ2 , τ3 , τ4 ), which contains all relevant microscopic information:
R(τ1 , τ2 , τ3 , τ4 ) ≡(2.51)hhV (τ1 )|G(τ1 − τ2 )V(τ2 )G(τ2 − τ3 )V(τ3 )G(τ3 − τ4 )V(τ4 )|ρ(−∞)ii.
The above equation for the polarization can serve as a general formal expression for
any four-wave mixing process. Now if we make the assumption of delta-function
pulses Ei (τ ) = Ei δ(τ − τi ); i = 1, 2, 3; this expression simplifies toP (3) (k4 , t) =(2.52)

E1 E2 E3 exp[−iωq (τ1 ) + iωn (τ1 − τ3 ) + iωm (τ1 − τ2 )]R(τ1 , τ2 , τ3 , τ4 )This results in a real-time signal given by
S(k4 , t) = |E1 E2 E3 |2 |R(τ1 , τ2 , τ3 , τ4 )|2 .

(2.53)To this point the derivation has not required that we specify the level structure of the material system, but we have reached the stage at which we mustspecify the Hamiltonian of the system. We choose the most general level structurein which four different, arbitrary levels a, b, c, d are sequentially excited throughthe four-wave mixing process. This is the most general case because a third-orderexperiment can at most couple four levels; thus, four levels are sufficient. We now
specify the Hamiltonian H for this system interacting with a bath:
H = H0 + Hint + HB

(2.54)

with
H0 =


i 
|νi ǫν − γν hν|2
ν=a,b,c,dX(2.55)

and
HB =

X

ν=a,b,c,d|νiHBν (QB )hν|.

(2.56)

34
Here ǫν is the energy and γν the inverse lifetime of the level ν. HBν (QB ) is anadiabatic Hamiltonian describing the bath degrees of freedom QB interacting withthe system in the state ν. We assume the atoms are initially in the ground state,Pmaking |ρ(−∞)ii = a P (a)|aii, where P (a) is the thermal equilibrium groundstate population in the absence of a radiation field. To calculate the response
function we write the matrix elements of the Green function Eq. 2.45:1hhν ′ λ′ |G(t)|νλii = δνν ′ δλλ′ exp −iωνλ t − (γν − γλ )t ,2

(2.57)

with ν, λ = a, b, c, d and ωνλ ≡ ǫν − ǫλ . Now the bath Hamiltonian is combinedwith the Hamiltonian representing the interaction with the fields to simplify thefollowing expressions. This choice amounts to including frequency fluctuations δωby modifying the electronic dipole operator. Note modification could alternatively
be made to the Green function by adding a δω to H0 , where it can be seen more
directly to be a fluctuation of the resonance frequency [31]; the current approach
results in simpler expressions later on. The modified dipole operator then takes
the form
V (τ ) =Xν,λ=a,b,c,d;ν6=λµνλ exp+ −i

Z τ
0


dτ1 Uνλ (τ1 ) |νihλ|.

(2.58)

Here exp+ is the positive-time-ordered exponential and Uνλ ≡ Hν − Hλ . The
semiclassical approximation is made by replacing exp+ by an ordinary exponential
and treating Uνλ (τ ) as an ordinary function of time (not an operator) denotedδωνλ (τ ). Then
V (τ ) =Xν,λ=a,b,c,d;ν6=λ Z τdτ1 δωνλ (τ1 ) |νihλ|.
µνλ exp −i(2.59)0

Given the matrix elements of the Green function Eq. 2.57 and the form of
the dipole operator Eq. 2.59 we can now calculate the response functions. From
the definition of the Liouville dipole operator Eq. 2.39, we can write the expression

35
for the response function in terms of the dyadic dipole operator V :R(τ1 , τ2 , τ3 , τ4 ) =(2.60)hhV (τ1 )|G(τ1 − τ2 )[V (τ2 ), G(τ2 − τ3 )[V (τ3 ), G(τ3 )[V (τ4 ), |ρ(−∞)ii]]]
All of the response functions will take the same form, namely
R(τ1 , τ2 , τ3 , τ4 ) =Y
hhGiiF (τ1 , τ2 , τ3 , τ4 ),(2.61)

νλwhere the product is over the contributing matrix elements of the Green function
and F is the four-point correlation function of the dipole operator, defined for theexcitation of a four-level system as
F (τ1 , τ2 , τ3 , τ4 ) ≡ hVab (τ1 )Vbc (τ2 )Vcd (τ3 )Vda (τ4 )ρ(−∞)i.

(2.62)

Using the four-point correlation function allows us to use the same four dipole
operators for every response function. A minus sign accrues for every time adensity operator is acted on from the right, as evident from the definition [V, A] ≡V A − AV .
The real system in which we are interested for the experiments in this thesis
can be modeled as a three-level system with two excited states b and c radiatively
coupled to a ground state a but not radiatively coupled to one another. A given
four-point correlation function can only represent variations on four dipole operators, with each operator acting once. This means that to model a three-levelsystem we cannot use just one four-point correlation function. For example, it
might be thought that the appropriate four-point correlation function for a threelevel system with levels a, b, c is hVab Vba Vac Vca i. However, this function does notinclude two-level physics because each operator must act once, meaning sequences
involving only a and b or a and c are not covered. Another way of stating thisis that the formalism using four-point correlation functions assumes that the incident pulses are narrowband and only excite one transition at a time. In the36experiments of this thesis, the pulses are broadband and simultaneously excite
two excited states. This theoretical situation arises from our assumption of planewave fields as well as delta-function durations, a strange but useful fiction. Thus tomodel a three-level system we must use two 2-level four-point correlation functions
as well as the one given in the example. Thus we need the following correlation
functions:2

F (τ1 , τ2 , τ3 , τ4 ) ≡ hVam (τ1 )Vma (τ2 )Vam (τ3 )Vma (τ4 )ρ(−∞)i,(2.63)

3

F (τ1 , τ2 , τ3 , τ4 ) ≡ hVab (τ1 )Vba (τ2 )Vac (τ3 )Vca (τ4 )ρ(−∞)i,(2.64)

where the superscripts 2 and 3 represent 2- and 3-level physics, respectively, andm = b, c.
To choose the correct time sequences we must know which sequences of
dipole operators will contribute to a response in the chosen direction, k4 =−k1 + k2 + k3 . These sequences, called pathways, can be represented quite nicely
using diagrams, useful particularly for higher-order, multi-level response. Appendix B.1 gives an introduction to the use of these so-called double-sided Feynman diagrams. The pathways contributing to the material response are shown in
Fig. 2.9. The first (last) two diagrams correspond to two- (three-) level physics.
For two-level physics there will be two distinct pathways (for each of b and c)contributing to the response for the time ordering (−k1 , k2 , k3 ), corresponding todiagrams labeled 2 R2m and 2 R3m and the perturbation sequences (reading right toleft) (ma, mm, am, aa) and (ma, aa, am, aa), m = b, c. We can write the response
functions in terms of the four-point correlation function of the dipole operator 2 F
defined above. For three-level physics there also will be two pathways contributing
to the response for the time ordering (−k1 , k2 , k3 ), labeled 3 R2n and 3 R3n , corresponding to the perturbation sequences (reading right to left) (na, nm, am, aa),(na, aa, am, aa). We can write these response functions in terms of the four-point

37
correlation function of the dipole operator 3 F defined above. The R labels foreach response function have been written according to the pathways they represent. The first superscript represents 2- or 3-level physics. For 2-level physics, thesubscript represents an excited-state (2) or ground-state (3) population to second
order. For 3-level physics, the subscript represents a Raman coherence (2) orground state population (3) to second order. The second superscript is the final
excited state that forms a coherence with the ground state at third order. The
response functions then are(ma, mm, am, aa)

2R2m (τ1 , τ2 , τ3 ) = 2 F (0, τ2 , τ1 , τ3 )e−iωma (τ1 −τ2 −τ3 ) ,(ma, aa, am, aa)

2R3m (τ1 , τ2 , τ3 ) = 2 F (0, τ3 , τ1 , τ2 )e−iωma (τ1 −τ2 −τ3 ) ,(na, nm, am, aa)

3R2n (τ1 , τ2 , τ3 ) =3(na, aa, am, aa)3(2.65)F (0, τ2 , τ1 , τ3 )e−iωna (τ1 −τ2 )−iωnm (τ2 −τ3 )−iωma (−τ3 ) ,R3n (τ1 , τ2 , τ3 ) = 3 F (0, τ3 , τ1 , τ2 )e−iωna (τ1 −τ2 )−iωma (−τ3 ) ,where the exponentials are the matrix elements of the Green function, calculated
based on Eq. 2.57 for each duration and perturbation sequence; we assume thepopulation decay rates are slow compared to the dephasing rates and can beneglected. The total contribution from the Green functions for each perturbation
sequence ν1 λ1 , ν2 λ2 , ν3 λ3 can be calculated from the general function Gν3 λ3 (τ1 −τ2 )Gν2 λ2 (τ2 −τ3 )Gν1 λ1 (τ3 ), where Gνλ is the Green function matrix element for ν, λand the time arguments remain as written (no substitutions necessary). For thetime ordering (k2 , −k1 , k3 ) there will also be four response functions; a parallel
derivation for these is provided in App. B.2.
Using the four-point correlation function reduces the calculation of the response functions of all contributing pathways to finding the forms of (in our case)two four-point correlation functions, those for two-level and three-level physics. In
App. B.2 we derive the solution for the most general form of the correlation function 4 F , which assumes the excitation of four different arbitrary levels a, b, c, d (in

2 m
Re

m1

kc

mm

kb

3 n
Rr

m1

n 1

1 1

kc

1m
1 1

2 m
Rg

1m
1 1

ka

kb
ka

kc

1m
1 1

n 1

kc

nm

kb

38

3 n
Rg

1 1

kb

1m

ka

1 1

ka

Figure 2.9: Double-sided Feynman diagrams representing the relevant pathways
for 3-pulse TFWM in a 3-level system for signal in direction −k1 + k2 + k3 andtime ordering (−k1 , k2 , k3 ), with m, n = b, c; m 6= n. The diagrams for all timeorderings are shown in App. B.

that order) following the derivation of Mukamel [15]. The four-point correlation
function based on the definition in Eq. 2.59 for this case has the form
4Z τ2n hZ τ 1
F (τ1 , τ2 , τ3 , τ4 ) = µab µbc µcd µda hexp idτ δωab (τ ) +dτ δωbc (τ )
0
0
Z τ3
Z τ4
io
+
dτ δωcd (τ ) +dτ δωda (τ ) i(2.66)
00

In order to solve this expression without simply truncating its Taylor expansion
to second order we apply the cumulant expansion [38, 39, 40]. This is carriedout in detail in App. B.2. To overview the steps taken, first we make the ansatz
that F has an exponential form: F = exp(F). We then expand F in orders of
the fluctuation δω, and substitute this expansion back into the ansatz. Collecting
terms of like order of δω and comparing with the original Taylor expansion of
F allows us to solve for the orders of F in terms of the orders of F . We theninsert these solutions into the ansatz and truncate at F (2) . Thus instead of justneglecting higher order correlations starting with Eq. 2.66, this method allows
us to partially resum the perturbation series for F . The result for the 4-level39four-point correlation function is4

F (τ1 , τ2 , τ3 , τ4 ) = µab µbc µcd µda
n 1
× exp − gab (τ1 − τ2 ) + gab (τ1 − τ4 ) − gab (τ2 − τ4 )2

(2.67)−gac (τ1 − τ2 ) − gac (τ3 − τ4 ) + gac (τ1 − τ3 ) + gac (τ2 − τ4 )+gad (τ3 − τ4 ) + gad (τ1 − τ4 ) − gad (τ1 − τ3 )+gbc (τ1 − τ2 ) + gbc (τ2 − τ3 ) − gbc (τ1 − τ3 )+gbd (τ2 − τ3 ) − gbd (τ1 − τ4 ) + gbd (τ2 − τ4 )o+gcd (τ2 − τ3 ) + gcd (τ3 − τ4 ) − gcd (τ2 − τ4 ) ,where we have defined the line shape function
gαβ (τ ) ≡

Z τ
0

dτ1

Z τ1
0

dτ2 hδωαβ (τ2 )δωαβ (0)i.

Using this expression we can determine the 2-level physics correlation function by
substituting c → a:2

F (τ1 , τ2 , τ3 , τ4 ) = µ4am exp −gam (τ1 − τ2 ) + gam (τ1 − τ3 )(2.68)

−gam (τ2 − τ3 ) − gam (τ1 − τ4 ) + gam (τ2 − τ4 ) − gam (τ3 − τ4 ) ,where m represents the excited states b, c. For the four-point correlation function
representing three-level physics we make the additional substitution d → b to find3

n 1F (τ1 , τ2 , τ3 , τ4 ) = µ2am µ2an exp − 2gam (τ1 − τ2 ) + 2gan (τ3 − τ4 )2

(2.69)−gam (τ1 − τ3 ) + gam (τ1 − τ4 ) + gam (τ2 − τ3 ) − gam (τ2 − τ4 )−gan (τ1 − τ3 ) + gan (τ1 − τ4 ) + gan (τ2 − τ3 ) − gan (τ2 − τ4 )o+gmn (τ1 − τ3 ) − gmn (τ1 − τ4 ) − gmn (τ2 − τ3 ) + gmn (τ2 − τ4 ) .
Determining the response functions for a three-level system is now reduced
to inserting the time sequences of Eq. 2.65 into the above equations for each40
pathway. Doing so gives us the final results for the third-order response functions
for a three-level system in direction −k1 + k2 + k3 and time ordering (−k1 , k2 , k3 )(equations for the other time ordering are in App. B.3)3 :
22

3

R2m (τ1 , τ2 , τ3 ) = µ4am e−iωma (τ1 −τ2 −τ3 )n1−gam (−τ2 ) + gam (−τ1 ) − gam (τ2 − τ1 )
× exp
2
o
−gam (−τ3 ) + gam (τ2 − τ3 ) − gam (τ1 − τ3 ) ,R3m (τ1 , τ2 , τ3 ) = µ4am e−iωma (τ1 −τ2 −τ3 )n1−gam (−τ3 ) + gam (−τ1 ) − gam (τ3 − τ1 )
× exp
2
o
−gam (−τ2 ) + gam (τ3 − τ2 ) − gam (τ1 − τ2 ) ,R2n (τ1 , τ2 , τ3 ) = µ2am µ2an e−iωna (τ1 −τ2 )−iωnm (τ2 −τ3 )−iωma (−τ3 )n 1
× exp − 2gam (−τ2 ) + 2gan (τ1 − τ3 )2−gam (−τ1 ) + gam (−τ3 ) + gam (τ2 − τ1 ) − gam (τ2 − τ3 )−gan (−τ1 ) + gan (−τ3 ) + gan (τ2 − τ1 ) − gan (τ2 − τ3 )3

o+gnm (−τ1 ) − gnm (−τ3 ) − gnm (τ2 − τ1 ) + gnm (τ2 − τ3 ) ,R3m (τ1 , τ2 , τ3 ) = µ2am µ2an e−iωna (τ1 −τ2 )−iωma (−τ3 )n 1
× exp − 2gan (−τ3 ) + 2gam (τ1 − τ2 )2−gam (−τ1 ) + gam (−τ2 ) + gam (τ3 − τ1 ) − gam (τ3 − τ2 )−gan (−τ1 ) + gan (−τ2 ) + gan (τ3 − τ1 ) − gan (τ3 − τ2 )o+gnm (−τ1 ) − gnm (−τ2 ) − gnm (τ3 − τ1 ) + gnm (τ3 − τ2 ) ,with m, n = b, c; m 6= n. The sum of the above response functions gives thepolarization. Thus we have for the spectrally integrated polarization for time3

To convert the above equations to notation with times between pulses t1 , t2 and t3 (witht1 the time of pulse 1, t2 the delay between pulses 1 and 2 and t3 the delay between pulses 2
and 3), use the conversion equations τ1 = t1 + t2 + t3 , τ2 = t1 + t2 and τ3 = t1 .

41ordering (−k1 , k2 , k3 ):(3)
PSI (t) = 2 R22 (τ1 , τ2 , τ3 ) + 2 R23 (τ1 , τ2 , τ3 ) + 2 R32 (τ1 , τ2 , τ3 )(2.70)+2 R33 (τ1 , τ2 , τ3 ) + 3 R22 (τ1 , τ2 , τ3 ) + 3 R23 (τ1 , τ2 , τ3 )

+3 R32 (τ1 , τ2 , τ3 ) + 3 R33 (τ1 , τ2 , τ3 ) .

The TFWM signal can in turn be calculated by performing an integral over thesquare of the polarization,
S=

Z ∞

−∞

|P (3) (t)|2 dt.

(2.71)

Now that we have the response functions in terms of the lineshape functions
g the last step is to decide the form of the two-time correlation function of frequency fluctuations to find the form of g. To model a system with non-Markoviandephasing we use the stochastic model described at the beginning of this section.
In particular, we assume the modulation of the i to j transition δωij obeys thestochastic two-time correlation function of frequency fluctuationshδωij (t1 )δωij (t2 )i = ∆2ij exp(−Λij |t2 − t1 |)(2.72)

hδωij (t)i = 0,(2.73)

andwhere ∆ij corresponds to the root-mean-squared amplitude of the frequency fluctuations and Λ−1
ij is their time scale. The angle brackets denote an ensemble
average over the stochastic process. Figure 2.10 is a schematic of the stochastic
correlation function. The stochastic model results in a line shape function givenbygij (t) =

Z t
0

dτ1

Z τ1
0

dτ2 hδωij (τ1 − τ2 )δωij (0)i
2∆2ij = 2 exp(−Λij t) + Λij t − 1 .
Λij(2.74)

2 ¡¤jt0 j0

42

h±!(t )±!(0)i = ¢ e
±!(t)

¤¡1
¢

0

¢2

time

¤¡1

t0

Figure 2.10: Schematic of stochastic correlation function.

In order to compare these results with the two-pulse derivations above, we
reduce the equations to the two-pulse case by setting τ1 = τ2 . The response
functions for the two possible interaction pathways corresponding to 2- and 3level physics then are
2R2m = 2 R3m = µ4am e−iωma (τ1 −2τ2 ) exp 2gam (−τ2 ) + gam (−τ1 )− gam (τ2 − τ1 ) − gam (τ1 − τ2 ) ,n 1
3 n
R2 = 3 R3n = µ2am µ2an e−iωna (τ1 −τ2 )−iωma (−τ2 ) exp − 3gam (−τ2 ) + 2gan (τ1 − τ2 )2+ gam (−τ1 ) − gam (τ2 − τ1 ) − gan (−τ1 ) + gan (−τ2 ) + gan (τ2 − τ1 )o(2.75)+ gmn (−τ1 ) − gmn (−τ2 ) − gmn (τ2 − τ1 ) .

The sum of the response functions gives the polarization in Eq. (2.71) that isthen numerically integrated to obtain the TFWM signal. The results of thisnon-Markovian calculation for the case of a two-level system are plotted for thetime-resolved and time-integrated cases in Figs. 2.11 and 2.12.
The 2-pulse results including non-Markovian dephasing look very similar
to the static inhomogeneous broadening case derived earlier. Indeed, one way tothink about this result is in terms of the system being “inhomogeneously broadened” at short times and “homogeneously broadened” at long times. These two

43limits correspond to timescales in which fast modulations of the transition energy
dominate the response (homogeneous broadening) and short timescales in which
the static distribution of transition energies dominates (inhomogeneous broadening). Like the static inhomogeneously broadened case, the rise of the timeintegrated signal from zero delay is due to the integration up to the full area ofthe real-time signal, a photon echo [41]. This is similar to the inhomogeneously
broadened case derived earlier in that at zero time, there is only half of a photonecho because of causality; the photon echo cannot arrive until after the second
pulse, and if τ is zero, only half of the echo can form. Thus as τ is increased
from zero to positive delay, the signal rises because of the integration over thedeveloping photon echo.
The non-Markovian photon echo arises from varying atomic spacing at shortdelays, not Doppler broadening nor static inhomogeneous broadening, from which
the photon echo phenomenon was originally observed [42]. At short delay theatoms do not move and thus their resonance frequencies remain constant overthe delay. The atoms are at varying distances from one another, resulting ina distribution of transition frequencies due to the varying dipole-dipole interactions, making the vapor effectively inhomogeneously broadened. In this regime
the evolution of the coherent superposition between ground and excited state isreversible, and thus the real-time signal is a photon echo. For longer delays, collisions begin to take effect, randomizing the phases of the superpositions such thatthey cannot rephase as well to produce a photon echo. This intermediate regime
occurs when τ is approximately equal to the correlation time. This is in contrast
to the static inhomogeneous case, in which the peak of the signal is determined
only by the decay of the photon echo, not the loss of photon echo-like behavior.
The loss of photon echo-like behavior is clearly exhibited in the real-time signal
shown in Fig. 2.11. As the delay between pulses is increased, the photon echo

44signal decreases, resulting in peaks of the real-time signal at shorter and shorter
delays, until eventually the photon echo fails to arrive at all.
The parameters ∆ and Λ representing the amplitude and duration of fluctuations determine the timescale of the crossover from homogeneous to inhomogeneous broadening in the time-integrated signal. The decay constant at long times
is T2 = Λ/(2∆)2 . The effect of varying ∆ and Λ are shown in Figs. 2.13 and 2.14,
respectively. It can be seen that varying ∆ results primarily in an increase of thedephasing rate, which affects the location of the peak of the signal versus delay.
Varying Λ primarily affects the delay of the peak, faster fluctuations resulting inearlier crossovers from inhomogeneous to homogeneous broadening.

Non-Markovian Broadened

Increasing Delay

2-Pulse Real-Time TFWM Signal

2-Level Systemt

2t, real time

Figure 2.11: Calculated 2-pulse time-resolved TFWM signal for a non-Markovian
broadened 2-level system.

To see the effect on the TFWM signal of having two excited states insteadof just one, we calculate the 2-pulse TFWM signal for the case of a three-levelsystem using the response functions derived above. We choose the spacing of
the excited state energy levels to match that between the potassium D1 and D2

2-Pulse Time-Integrated TFWM Signal

45

Non-Markovian Broadened
2-Level System, delay between pulses

T /2
2

Figure 2.12: Calculated 2-pulse time-integrated TFWM signal for a nonMarkovian broadened 2-level system (natural log scale).

Non-Markovian Broadened

2-Pulse Time-Integrated TFWM Signal

2-Level System

=0.2

=0.3

=0.5

=1.0

0

1

2

3

4

5

6

7

8

9

, delay between pulses

Figure 2.13: Calculated 2-pulse time-resolved TFWM signal for a non-Markovian
broadened 2-level system (natural log scale). Varying the correlation function
parameter ∆, representing the root-mean-square amplitude of the fluctuations.

lines studied in the experiments. The resulting time-integrated 2-pulse TFWM

46
Non-Markovian Broadened
2-Pulse Time-Integrated TFWM Signal

2-Level System
-1

=0.25

-1

=0.5

-1

=1.0

-1

=2

0

1

2

3

4

5

, delay between pulses

Figure 2.14: Calculated 2-pulse time-integrated TFWM signal for a nonMarkovian broadened 2-level system (natural log scale). Varying the correlation
function parameter Λ−1 , representing the timescale of the fluctuations.

signal is shown in Fig. 2.15. The beating exhibited by the signal is a result of thequantum interference of the two excited state pathways, with the beat frequency
corresponding to the frequency difference between the levels. In addition, if thelevels have different dipole moments or are excited with different intensities theamplitude of the modulation in the beats will decrease.
Finally, we consider the case of 3-pulse TFWM. Although the particular sequence and arrangement of pulses will be described in detail in the experimental
chapter, we consider here how separating the second and third pulses gains added
insight into the physics of non-Markovian dynamics. In particular, it has beenshown that varying the delay τ between the first and second pulse for various values of the delay T between the second and third pulse allows the mapping out of
the correlation function of frequency fluctuations [43]. This may not be surprisingconsidering that the response functions are so intimately related to the correlation

2-Pulse Time-Integrated TFWM Signal

47

Non-Markovian Broadened
3-Level System, delay between pulses

Figure 2.15: Calculated 2-pulse time-integrated TFWM signal for a nonMarkovian broadened 3-level system (natural log scale).

function through the line shape functions. Physically this seems reasonable as wellconsidering that varying the second delay probes changes that occur while the system is in the intermediate density matrix state corresponding to population. Bychanging that time interval we have access to the timescales of the randomization
process that result in the loss of rephasing capability. That is, because the photonecho cannot arise until after the third pulse reverses the evolution of the coherent superposition created by the first pulse, varying the second-third pulse delay“exposes” the coherence to variable durations of randomizing collisions. It is justthese randomizing collisions that determine the form of the correlation function.
In terms of the actual signals, the effect of increasing the second delay is
to shift the peak of the TFWM signal versus the first delay τ towards zero. In
other words, increasing the second delay decreases the inhomogeneous system response that determines the peak location in the TFWM signal versus τ . The
time-integrated three-pulse TFWM signals for various second delays T are plotted in Fig. 2.16. By determining the peak delay of these signals and plotting this48
delay as a function of T , a form for the correlation function of frequency fluctuations can be found. This “photon echo peak shift” thus reveals the form of the
microscopic correlations involved in the dephasing process. As seen in Fig. 2.17,
for the stochastic model chosen in the above derivation we indeed find an exponential peak shift. The exact relationship between the peak shift and the correlation
function is just that exhibited by the response functions derived above. In somemolecular systems this relationship can be shown to take a more direct form, particularly for systems which exhibit both a fast and slow peak shift decay [43]. The
next two chapters on the experimental results and correlation function theory will
explore the relationship between the peak shift and the correlation function in
more detail for our system.

2-pulse Time-Integrated TFWM Signal

Non-Markovian Broadened
2-Level System

T = 0

T = 0.3

T = 0.8

T = 1.8
0.1

0.2

0.3

0.4

0.5

, delay between first and second pulse

Figure 2.16: Calculated 3-pulse time-integrated TFWM signal for a nonMarkovian broadened 2-level system (natural log scale).

49

Non-Markovian Broadened
2-Level System*, peak delay

3-Pulse Photon Echo Peak Shift

0

2

4

6

8

10

T, delay between second and third pulse

Figure 2.17: Calculated 3-pulse photon echo peak shift for a non-Markovian broadened 2-level system.

Chapter 3
Transient Four-Wave Mixing Experiments and Results

Transient four-wave mixing (TFWM) applied to the study of non-Markovian
dynamics is often called the photon echo technique, referring to the photon echo
as the signature of non-Markovian dephasing1 . In this chapter the experimental
setups and results will be presented for two- and three-pulse photon echo as well as
the time-resolved version of three-pulse photon echo. The results will be analyzed
in light of the theoretical results of the previous chapter. Through this analysis
the scope of the stochastic theory as well as the effects of resonance broadening
will become apparent.
3.1

Experimental Components
Potassium is the element of choice for these experiments because of its simple

electronic structure, with energy levels easily accessed by a Ti:sapphire laser (seeFig. 3.1 for a diagram of the optically active energy levels). Potassium is analkali metal containing 19 electrons, with one valence electron in the 4s shell.
The two lowest excited states of this valence electron correspond to the D1 andD2 lines at 769.896 nm (1.61 eV, 12, 985.17 cm−1 ) and 766.490 nm (1.62 eV,
13, 042.88 cm−1 ), respectively2 . These lines correspond to the 42 S1/2 → 42 P1/2
1

Two-pulse photon echo is also known as parametric scattering [27] or self-diffracted TFWM.
The phrase “D lines” comes from Frauenhofer’s study of the spectrum of the sun in theearly 1800’s, resulting in a paper in 1821. He labeled the strongest dark lines across the visible2

51
and 42 S1/2 → 42 P3/2 fine structure transitions that are split through the spinorbit interaction. Potassium was chosen over other candidate alkali metals to
simplify experiment and theory. In particular, it was chosen over sodium because
the potassium D lines are easily attainable by a Ti:sapphire laser, whereas the
D lines for sodium lie in the yellow part of the spectrum, which would require adye laser. Potassium was chosen over rubidium and cesium because the hyperfine
splitting of potassium is less than that of these other alkalis, with the potassium
hyperfine splitting negligible due to broadening at the experimental densities andtemperatures used [45]. Additional properties of potassium are given in App. C.
42P3/2
42P1/2D1

D2(769.896 nm) (766.490 nm)
42S1/2

Figure 3.1: Optically active energy levels.

In these experiments a Kerr-lens modelocked Ti:sapphire laser with an extended cavity was used because of its low repetition rate of ∼ 26 MHz to reduce
accumulation of population in the excited state. The laser pulses are dispersion
compensated by a prism pair just downstream of the laser output coupler, resulting in pulses ∼ 160 fs long by the time they reach the cell, approximately double
the Fourier-transform limited width of 70 fs. Simulations were run to determine
the effects of temporal chirp on TFWM and the signals were found to be unaffected by temporal chirp. The pulses are focused onto the interface between the
spectrum by the capital letters A in the red to H in the violet; the D line turned out to be adoublet, thus becoming D1 and D2 . It was not until twenty-seven years later that the connection
was made to the already-known spectral lines of sodium. His notation remains in use today for
sodium, and I assume that the similarity of the doublets in other alkali metals promotes their
being called the “D lines” also [44].

52
sapphire window and the potassium vapor by a 25 cm focal-length achromatic
doublet lens to minimize chromatic dispersion (a 15 cm singlet lens was used in
the two-pulse experiments, which are not as sensitive to dispersion). The broad(∼ 13 nm FWHM) laser spectrum excites both the potassium D1 and D2 lines,
which are well-separated, resulting in quantum beats at the difference frequency(as derived in the previous chapter). This is in contrast to molecular vibronic
spectra, which often contain multiple, overlapping transitions [23]. In addition
to the D lines the pulses off-resonantly excite other levels which were examined
via the fluorescence as a function of temperature and input laser power, shown in
Fig. 3.2. Intensity dependence was measured for the major transitions and bothlinear and square dependence was found, corresponding to the off-resonant singleand two-photon excitation of higher-lying states such as the atomic 6S and 4Dlines and a distinctive molecular band at 670 nm. These do not contribute to theresonant TFWM signal but do reveal the presence of potassium dimers.
The cell used to hold the potassium vapor is custom-made at JILA andconsists of a titanium body with a sapphire window for optical access. A detailed
description of the potassium cell, its historical development, and directions for its
manufacture can be found in App. E. To keep the potassium from condensing
onto the sapphire window, higher temperature is maintained at the window thanthe back of the cell. The densities of the vapor are determined using phenomenological formulae relating the vapor pressure of potassium to temperature, fromwhich the densities are then calculated based on the ideal gas equation3 [47].
Temperatures up to 800◦ C are achieved, corresponding to a number density of∼ 9 × 1018 cm−3 . For the data presented in this chapter the TFWM signals aredetected in reflection because the absorption length of the potassium vapor is on
3

Note the treatment of potassium as an ideal gas is less valid for higher densities and pressures at which phenomenological formulae have been shown to deviate [46]. Thus uncertainties
probably arise primarily from the assumption of ideal gas behavior.

53

o

200

350

400

450 500 550

600

T
e

m

p

e

ra

tu

re

(

C

770
750
700
650
600
550
500
450
400
350
300

)

Intensity

Vapor Fluorescence Intensity vs. Wavelength and Temperature

650 700

Wavelength (nm)Figure 3.2: Detected fluorescence from potassium vapor excited at 768 nm versus
wavelength for various temperatures.

the order of the wavelength of the incident light at these densities. Transmission
cells were also manufactured and data were taken in transmission as well; thetransmission cells and data are presented in App. E. This quality of short absorption length makes the vapor effectively thin for reflection experiments, which
minimizes reshaping of the excitation pulses and reabsorption of the signal. In addition, it has been shown that the reflected signal is equivalent to the transmitted
signal for an optically thin sample [48].
The nominal radiative lifetime of potassium is ∼ 26 ns, but this is known
to be extended due to the trapping of radiation (the emission and reabsorption ofreal photons) at high densities [49]. To determine the actual radiative lifetime we
employed a home-built pulse picker (see App. D for details) to divide down the54repetition rate and used a photo-multiplier tube to detect the vapor fluorescence
excited by laser pulses with center wavelength tuned to the D lines. The repetition rate was divided by 50, and data were collected using a fast oscilloscope fora series of temperatures ranging from 450◦ C to 600◦ C. The fluorescence decay
was most visible at 450◦ C, shown in Fig. 3.3(a), with a decay time of ∼ 0.16 µs.
The effects of radiation trapping decrease with temperature and density as shownin Fig. 3.3(b), which could be due to the decreasing optical depth in reflection.
Thus the extended cavity laser does not have a sufficiently low repetition rate
to allow all of the excited state population to relax back to the ground state between pulses. To determine the effects of radiation trapping on transient four-wave
mixing experiments, two-pulse TFWM data were taken with and without pulse
picking. It was found that the dephasing rate is slightly suppressed due to radiation trapping. The dephasing rate reaches a steady value versus repetition rate
starting around 1 MHz. Thus dividing down by 20 pulses seems to be sufficient to
reduce the influence of radiation trapping. The pulse picker was used in collecting
the two-pulse TFWM data presented here but was not used in the three-pulse
experiments due to the pulse picker’s heavy toll on pulse intensites and because
the signatures of interest in these experiments are not very sensitive to the change
in dephasing rate due to radiation trapping. Signal intensities were checked forcompliance with cubic dependence on intensity for all of the TFWM data to ensure the suitability of a perturbative treatment. The intensity was found to havea significant influence on the negative delay signal strength of 2-pulse TFWM
data, with a larger negative delay signal for lower intensities. This is importantfor studying the local field effect as described in Ch. 5 but does not affect thesignatures of interest in this chapter.
The major broadening mechanisms in a vapor of pure potassium atoms are
Doppler and resonance broadening. Doppler broadening arises from the blue- and450 C
repetition rate / 50oFluorescence,

1.1

1.2

time ( s)

3

1.

(b)

Decay Time ( s)(a)Intensity (natural log)

55
0.70
0.60
0.50
0.40
0.30

Radiative Decay Time

0.20
0.10
0.09
0.08
0.07
0.06
0.05

400 450 500 550o600

Temperature ( C)

Figure 3.3: (a) Oscilloscope trace of fluorescence intensity versus time. The vapor
is excited every 50th pulse, with remnants of the other pulses represented as thesmaller peaks. The exponential decay is the radiative decay of the excited state
population. (b) The measured decay times as a function of temperature.

red-shifted spectral contributions of atoms moving towards or away from the incident pulses. The varying atomic velocities result in contributions from a large variety of spectral components, resulting in a Gaussian (inhomogeneous) lineshape.
Resonance broadening is due to the dipole-dipole interaction between ground- andexcited-state atoms of the same species. If the vapor is resonance broadened, thedominant mechanism for dephasing is collisions. Because the dephasing collisions
arise from the behavior of like atoms, the vapor is homogeneously broadened inthis limit. In order to study the non-Markovian dephasing process we then workat densities at which resonance broadening dominates over Doppler broadening.
These densities correspond to temperatures above ∼ 350◦ C (see Fig. 3.4). Thus
the experimental results shown below were taken with the vapor above this temperature, which corresponds to densities > 8 × 1016 cm−3 .
We now introduce the two- and three-pulse photon echo experiments andresults.

56
Density (x10

FWHM (GHz)

2

4

1

7

-3

cm )

6

8

Doppler

4

Resonance

2

0
100

200

o

300

Temperature ( C)

Figure 3.4: Theoretical linewidths for resonance and Doppler broadening.
lens view
Ti:sapphire
26 MHz160 fs
768 nm

k-

delay

k+

detectors

k-= -ka+kb+kc

ka
kb

potassium
vapor
reflected signals
above input beams

k+= ka-kb+kc

Figure 3.5: The 2-pulse photon echo experimental setup.

3.2

The Two-Pulse Photon Echo Experiment and Results
As a description was given for three-pulse TFWM in the previous chapter,here we give only a brief description of the two-pulse transient four-wave mixing
process, which when applied to the study of non-Markovian dynamics is called thetwo-pulse photon echo technique (2PE). In the 2PE experiment, shown in Fig. 3.5,
two pulses with wavevectors ka and kb and center frequency tuned to resonanceare incident on the dense vapor. The first pulse, with wavevector ka , induces a
coherence between the ground and excited states of the atoms with a spatial phaseacross the vapor determined by its angle of incidence and wavelength. After somedelay τ , the second pulse, kb , produces an excited state population whose ampli-

57
tude is greatest in areas where the coherence left by the first pulse constructively
interferes with the second pulse and least where they destructively interfere. Thus
the excited state population is spatially modulated, forming an amplitude gratingwith periodicity kb − ka . At the same time it creates the grating, the second pulse
also scatters off the grating (i.e. creates a coherence that radiates) into the direction 2kb −ka . This background-free signal is time-integrated by a slow photodiodedetector. The scattered signal has an intensity proportional to the amplitude of
the population grating. By varying the time delay between pulses, τ , the transient behavior of the induced polarization can be investigated. Note that the useof subscripts a and b instead of numbers is to represent the fact that signals ineither direction can be detected, and thus there is a different ordering corresponding to positive and negative delay depending on which signal is detected. Thatis, for the signal in direction 2kb − ka “negative delay” corresponds to kb arriving
first, while for the signal in direction 2ka − kb negative delay corresponds to ka
arriving first. Thus we do not label the wavevectors with numbers in order toprevent confusion of the geometrical orientation with the pulse sequence.
Typical experimental results are shown in Fig. 3.6. The detected signal
intensity is plotted as a function of the delay between pulses. The signal exhibits
quantum beats due to that fact that two energy levels are excited, as was confirmed
theoretically in the previous chapter. The important aspect of this signal is thatit initially increases at short times, peaks and then exponentially decays at longtimes. This behavior corresponds to that predicted in the previous chapter fora non-Markovian broadened system. Thus the 2PE signal contains signatures ofboth non-Markovian and Markovian dynamics. Indeed, the dynamical signatures
are well-separated (see Fig. 3.6) as an initial rise of the signal caused by the
integration of the developing photon echo, a flat signal at short delays due tonon-Markovian scattering, and exponential dephasing due to pure homogeneous

58
broadening. The extra dip near zero delay is a result of the signal following the
integration of the pulses, which have a finite duration. The 2PE experiment was
performed for a series of densities and temperatures. The results are plottedin Fig. 3.7. As temperature and density increase, the range of non-Markovian
dynamics decreases. This agrees with the reasoning that the collision durationand time between collisions should decrease with increasing temperature and thusthat the range of non-Markovian dephasing should decrease with temperature.
This argument is made more quantitative below through the application of the
non-Markovian theory.

Non-Markovian scattering

Photon echo
integration 1

Pure homogeneous
broadening

TFWM Signal(arb. units)

Finite pulse width

0.1

-1 0

1

2

3

4

5

6

7

8

9 10

Time Delay (ps)Figure 3.6: Experimental two-pulse photon echo signal for potassium vapor at
500◦ C (number density ∼ 3.4 × 1017 cm−3 ) with dynamical signatures labeled.
Quantum beats are due to the excitation of both the potassium D resonancelines.

In this experiment the dephasing time, collision duration and pulse width
are all well-separated. Based on thermal velocities calculated assuming ideal gas
behavior and the range of the attractive part of a typical interatomic potential
energy surface, at 700◦ C the collision duration is approximately 0.5 ps. This collision duration is much less than the time between collisions of 2 ps (based on
the dephasing time) and much greater than the pulse duration of 180 fs. In ad-

59

750 C
N = 5.8x10 cm
o

T =

18

-3

700 C
N = 3.7x10 cmo

T =2PE Signal (arb. units)

18

650 C
N = 2.3x10 cm

-3

o

T =

18

-3

600 C
N = 1.3x10 cm
T =

o

18

-3

550 C
N = 6.9x10 7cm
T =

o

1

-3

500 C
N = 3.4x10 7cm
T =

o

1

0

1

2

3

4

5

6

7

8

9

-3

10

Time Delay (ps)Figure 3.7: Experimental two-pulse photon echo signals for various temperaturesand densities (log scale), with lines drawn to highlight the non-exponential behavior at short times. The non-Markovian behavior decreases with increasing
temperature and density.

dition, this estimated value for the collision duration agrees well with the delay
at which the signal peaks, which according to the theory corresponds to approximately the correlation time Λ−1 . In condensed phase systems the timescale offluctuations can be on the order of 200 fs [18], with dephasing due to solvent interaction also occuring at ultrafast timescales [20]. Thus the above experimental
results exhibit unusually well-separated contributions from homogeneous and inhomogeneous broadening as compared to molecules in the condensed phase. We
also note that we do not need to be concerned with contributions from other timeorderings that occur during pulse overlap. Indeed, with such a model system,

60
this is an opportunity to test the non-Markovian theory derived in the previous
chapter that is often used in more complex molecular systems. To do so we use
the response functions derived in the previous chapter for 2PE in a three-levelsystem. Least-square fits are made to the experimental data with the only fitting
variables being the parameters of the stochastic correlation function ∆ij and Λij
for the two transitions.
Theoretical results match quite well with the experimental results via leastsquare fits, as shown in Fig. 3.8. The discrepancy near zero delay is due to theassumption of delta-function pulses in time, which does not allow for finite pulse
width effects. The fit parameter Λij corresponds to the decay rate of the correlation function and determines the time scale of the deviation of the 2PE signal
from exponential behavior at short delays. To confirm that these signatures arise
from non-Markovian dephasing, fits were made to the data for various temperatures and as expected Λij increases with increasing temperature, as shown in
Fig. 3.9. This represents a decreasing collision duration. The magnitude of theexcited state frequency fluctuations is represented by the parameter ∆ij , which isalso plotted in Fig. 3.9. The ratio κ = Λij /∆ij is of order 1, which means thatthe broadening mechanism is intermediate and not in either the homogeneous orinhomogeneous limits.
The excellent fit between theory and experiment demonstrate that the stochastic model adequately describes the dynamics. The stochastic model does not take
into account memory in the bath, as occurs, for example, in semiconductors due
to electron-LO-phonon scattering [50]. Memory in the bath results in oscillatorydephasing, so the fact that we do not see oscillatory dephasing implies that amodel that includes such a microscopic description of the bath [51, 52] is notneeded4 . The stochastic model does not include resonance broadening effects;
4

For an atomic system “the bath” is all the other atoms.

61

500 C

ormalized TFWM Signal

o

o

N

600 C

700 C
o

0

5

10

Time Delay (ps)Figure 3.8: Typical experimental 2-pulse photon echo signals (solid grey lines) forpotassium vapor at 500◦ C (N ≈ 3.4 × 1017 cm−3 ), 600◦ C (N ≈ 1.3 × 1018 cm−3 ),and 700◦ C (N ≈ 3.7 × 1018 cm−3 ), with curves offset for clarity. The dashed lines
are least-square fits assuming a stochastic modulation of the electronic transition
frequency during collisions.

62

Density (x10ij

(1/ps)

ij(1/ps)

3.0

0.69

0.34
12

0

2.

13

0

18

-3

cm )

1.

30

2.2

6

600

650

3.71

5.76

700

750

(D )
1

(D )
2

1.

0.0
1.0

12
13

0.5
0.0

500

(D )
1

(D )
2

550

o

Temperature ( C)

Figure 3.9: Stochastic theoretical fitting parameters Λij , the timescale of theexcited state frequency fluctuations, and ∆ij , their magnitude, for the D1 (square)and D2 (triangle) line.

63
thus, differences will exist in comparison with a foreign-gas broadened vapor inwhich the interaction range and therefore the collision duration is relatively small.
The deviation of the inverse collision duration from square-root dependance on
temperature may indicate the importance of the resonant interactions. This canbe explored experimentally by introducing a buffer gas to the potassium vapor,which would decrease the interaction range and bring the system to the Markovian, homogeneously broadened limit (see Ch. 6 for a description). Performing
such an experiment would grant the ability to determine signatures unique to resonance broadening. In addition, further insight can be gained with a three-pulse
photon echo experiment, which as mentioned in the last chapter can enable themapping out of the correlation function of the system [43, 53, 54]. This latter
experiment is the subject of the next section.
3.3

The Three-Pulse Photon Echo Experiment and Results
lens viewk-Ti:sapphiredelay

26 MHz160 fs
768 nmdelayk+detectors

ka
kc
kb

k-= -ka+kb+kc
potassium
vapor

reflected signals
above input beams

k+= ka-kb+kc

Figure 3.10: The time-integrated 3-pulse photon echo experimental setup. The
signals are detected in reflection due to the small absorption length (∼ λ/2) of
the potassium vapor.

The experimental setup for three-pulse TFWM, known as three-pulse photon echo (3PE) in non-Markovian spectroscopy, is shown in Fig. 3.10. Figure 3.11

k-

64

k+
sapphire
windowlens

Figure 3.11: The three-pulse photon echo experimental geometry.

shows the pulse geometry. The 3PE technique extends the previous experimental
arrangement by adding a third pulse. The pulses once again are labeled alphabetically to imply that time ordering is dependent on the geometry of the detectedsignal. The first pulse excites a coherence between the ground and excited states.
During the ensuing delay τ , dephasing occurs due to collisions. Then the second
pulse arrives and converts the remaining coherence to excited-state or groundstate population. During the second delay T , loss of population and spectral
redistribution occurs due to natural decay and collisions, and the system’s ability
to rephase and form a photon echo decreases. The third and final pulse forms
a third-order coherence that radiates as the signal. The time-integrated 3PE is
measured by scanning the first delay τ for various fixed values of the second delay
T , with negative delay corresponding to the conjugate pulse coming second. A
schematic of the pulse ordering for different time delays is shown in Fig. 3.12. The
3PE signal is more sensitive to the beam geometry that the two-pulse case. In
particular, the beams must be parallel to observe the proper population decay as
a function of T , as shown in Fig. 3.13.
As exhibited by the data shown in Fig. 3.14, and considering only the k−signal for simplicity, for τ > 0 the overall signal versus τ (ignoring the quantum
beating due to simultaneous excitation of both excited states) rises, peaks, andthen exponentially decays. This is the same behavior as exhibited by the 2pulse photon echo signal. Across the peak, the real-time signal changes from a

65Time orderings for signal
k- = -ka+kb+kc, T > 0
positive delay:
Ia
Ib

Ic
t

T

t>0
zero delay:
Ia Ib

Ic
t

t=0 T
negative delay:
Ib
Ia
Ic

t

T

t<0

Figure 3.12: Time delays of the three-pulse photon echo experiment.

Effects of changing beam angle in

Parallel

3PE
Toed In

3PE Signal

Splayed Out

0

3

6

0

3

6

0

3

6

T, delay between second and third pulse (ps)Figure 3.13: Dependence of the population decay (taken by varying T for fixed τ )on beam angle. The beams must be aligned parallel before the focusing lens forthe population decay to be properly measured.

photon echo to free polarization decay, corresponding to the crossover from nonMarkovian to Markovian dynamics. As the second delay T is increased, the signals
reveal how the system’s ability to form a photon echo changes. For the case of
the first delay τ being less than the correlation time, we can divide the dynamics
into three regimes as a function of the second delay T (see Fig. 3.15). For T lessthan the correlation time, the non-Markovian dynamics at short times results in a

66
real-time signal that is a photon echo, but the intensity is slightly decreased due tothe collisions that occur during the second delay. The dotted line in the schematic
represents what the signal would be if T were zero. For T much greater than thecorrelation time, the dephasing during the second delay completely precludes a
photon echo from forming, as shown in the figure for the Markovian regime. In
the time-integrated signal, this results in a shift of the peak toward τ = 0. Thus
varying the second delay allows a probe of the loss of photon echo-like behavior
during the transition to homogeneous broadening. To determine the peak shift,data are taken as a function of τ for the second delay T at the beat maxima of the
population signal for best signal-to-noise. The scans of τ for each of these T arethen analyzed by fitting a Gaussian to each beat, determining the maxima delays
versus τ of the Gaussian fits, and fitting these delays in turn with a Gaussian
to find the overall peak of the 3PE signal. Three-pulse photon echo data weretaken for a series of temperatures and densities. The resulting 3PE peak shifts,
or 3PEPS, are shown for a series of temperatures in Fig. 3.16.
For some molecular systems the 3PEPS reveals the two-time correlation
function of frequency fluctuations directly [23]. However, this is only possible
due to a theoretical simplification arising from the presence of both fast and slow
modulation in the system. For a dense atomic vapor, such simplifications do notnecessarily occur, but a qualitative form for the correlation function can be foundfrom the 3PEPS. The experimental peak location versus τ is exponential at lower
temperatures and thus qualitatively supports the use of an exponential correlation
function in modeling two-pulse experiments, as was done for the 2PE signals above[55]. The peak shift is bi-exponential above 500◦ C. The fitting functions shownin the figure are of the form
∆21 exp(−Λ1 T ) + ∆22 exp(−Λ2 T ),

(3.1)

67

k-

3PE Signals (normalized)

k+

0

-2

2

-2

, delay btwn 1

st

& 2

nd

0

2

pulse (ps)Figure 3.14: Experimental 3-pulse photon echo signals at 600◦ C (1.3 × 1018 cm−3 )for T = 0.58 ps to 36 ps. The peak of the signal, marked by an arrow in the figure,shifts to τ = 0 ps due to the loss of rephasing capability.

T < tcT ~ tcT >> tc

non-Markovian regime

Intermediate regime

Markovian regime

Ia Ib

Ia Ib

Ia Ib Ic

t T t

time

t

Ic

T

t

time

t

Ic

T

time

Figure 3.15: Schematic of regimes for τ < tc (τ in the non-Markovian regime).
Real-time signals are shown in solid black for the cases of T in the non-Markovian,intermediate and Markovian regimes. Dotted lines represent the signal for perfectrephasing.

where the subscripts 1 and 2 refer to the fast and slow exponentials, respectively.

68

Experimental Peak Shift

00 C
550 C
600 C
650 C
700 C
750 C
5

*, peak delay (log ps)

1

o

o

o

o

o

o

0.1

0

10

T, delay between 2

nd

20
rd

and 3

pulse (ps)Figure 3.16: 3PEPS for a series of temperatures and densities with bi-exponential
fits (lines).

The amplitudes ∆2i and time constants 1/Λi for the exponentials are plotted inFig. 3.17. This bi-exponential behavior could represent the presence of two collision mechanisms, such as both short-range/binary/hard and long-range/manybody/soft collisions, the latter being due to resonance effects at these high densities [56]. Alternatively, it could be due to contributions from higher-order powersof the correlation function [57]. The current theory does not account for resonancebroadening, which dominates at these high densities [58]. To take resonance effects into account and gain a more complete description of the system response,
we performed molecular dynamics simulations in an exciton picture and calculated the correlation function of one-exciton frequency fluctuations. Using these
correlation functions we then calculated the 3PEPS signals [59]. This work ispresented in the next chapter.

69
18

-3

0.3 0.7 1.3 2.3

3 7

Density (x10 cm )

1.2
0.8
0.4
0.0
1.0
0.8
0.6
0.4
0.2

5

.

5

.8

00 550 600 650 700 750oTemperature ( C)

Figure 3.17: Experimental 3-pulse photon echo peak shift fitting parameters ∆and Λ for the fast (square) and slow (triangle) exponentials.

3.4

The Time-Resolved Three-Pulse Photon Echo Experiment and
Results

Ti:sapphire
26 MHz160 fs
768 nm

delay

delay

PMT

lens view

k-

BBO
delay

Ref

ka
kc
kb

k-= -ka+kb+kc
potassium
vapor

tracerFigure 3.18: The time-resolved 3-pulse photon echo experimental setup.

We also performed the time-resolved version of the 3PE experiment to gain
more insight into the correlation function experimentally. In the time-resolved
version of 3PE (time-gated 3PE), a reference pulse is correlated with the emitted
3PE signal at a particular τ and T in a BBO crystal for up-conversion, resulting
in a time-resolved scan of the nonlinear response. In the setup, shown in Fig. 3.18,
the beam from the laser is split into five arms, three corresponding to the three70
beams used for four-wave mixing, one used as a tracer for the TFWM signal fromthe vapor, and one used as a reference for upconversion of the TFWM signal.
Three delay stages are used to form the delays between pulses, with two stages
before the cell for the τ and T delays and one after the cell for signal upconversion
with the reference. Data are taken for various τ and T as a function of the delay
between the signal and the reference, t. Results are shown in Fig. 3.19.
The presence of the photon echo is clearly seen in the time-resolved signal(Fig. 3.19). As a function of τ we can see that the signal exhibits a free polarization
decay and a photon echo for small T . As τ is increased, the photon echo moves
to the corresponding delay. As T is increased this photon echo-like behavior goesaway. This is an unusually clear example of the utility of the three-pulse photonecho technique for probing non-Markovian dynamics.

71Time-resolved 3PE Signal (normalized)ccPhoton e ho de ays with

T = 0 ps=c

Photon e ho shifts with=T

T = 2.32 ps

T = 6.96 ps

0 ps

0.6 ps=1.76 ps=

2.92 ps

0

2

4

6

0

2

4

6

0

2

t, delay between signal & reference (ps)4

6

Figure 3.19: Experimental time-resolved 3-pulse photon echo signals at 600◦ C(1.3 × 1018 cm−3 ). Small arrows point to t = τ , the photon echo location for
perfect rephasing. Photon echo strength clearly decreases with increasing T .

Chapter 4
Correlation Functions from Molecular Dynamics Simulations in an
Exciton Picture

In this chapter the experimental three-pulse photon echo results are compared to calculated correlation functions of frequency fluctuations from molecular
dynamics simulations in the exciton formalism. Although fits using the stochastic model are excellent for the two-pulse photon echo experimental results, thebi-exponential behavior of the 3PEPS at high density cannot be explained by asingle exponential correlation function. Even if it could, the insight the stochastic theory provides into the dynamics of the system is limited by our assumed
correlation function. In addition, adding resonance effects is interesting from apurely theoretical point of view as to how it might change the 3PEPS as wellas potential experimental signals for varying buffer gas pressure and the spectra
obtained from 2D spectroscopy, in which crosspeaks reveal excitonic coupings (seeCh. 6 for a brief description of the latter). Understanding the influence of excitonic effects would provide insight into the interactions of a resonance-broadened
vapor in particular as well as other disordered excitonic systems such as molecularaggregates, semiconductors and biological systems. Potassium vapor is an ideal
system in which to study disordered excitons due to its electronic simplicity andits easily adjusted density and temperature.

73
4.1

The Exciton Formalism
The model used for the two-pulse experiment is the equivalent of a foreigngas model because we do not consider the interaction of atoms with atoms but
rather assume the interactions of a particular atom can be averaged into the fluctuations of “the bath,” which in the case of a vapor are “all the other atoms.”
This model holds for foreign-gas broadened vapors because dephasing collisions
are short-range and thus spatially extended couplings of energy levels do not arise.
For a resonance-broadened vapor, the range of interaction between atoms is significantly more long-range, meaning at the least binary interactions must be taken
into account. The exciton model facilitates binary as well as higher-order interactions [60]. The concept of excitons enables us to write a diagonal Hamiltonian fora system which fits neither the concept of a solid with bands nor that of a random
ensemble of separated atoms. This is done by working intermediately between
k-space and real space using the orthogonal states of excitons. An exciton can beconsidered a quasiparticle representing a system of coupled oscillators. Generally,
for a system of N atoms there will be N one-exciton (all electrons in the groundstate except one electron in the excited state) states and N (N − 1)/2 two-exciton(all electrons in the ground state except two) states, and so on. A schematic ofsuch a level scheme is shown in Fig. 4.1. An example derivation of the exciton
levels for a system of two 2-level atoms is given in App. F. The fluctuating exciton
Hamiltonian (including two-exciton states) is
m6=n

HS =

X
m

†
Ωm B̂m
B̂m +

X

†
†
(Jmn B̂mB̂n + Kmn B̂mB̂n† B̂m B̂n ),

(4.1)

m,n

where
†
= |m1ihm0|
B̂m

(4.2)

74

...

N 2-level atoms...

(N (N-1))/2
NFigure 4.1: Schematic of one- and two-exciton states for N atoms.

andB̂m = |m0ihm1|(4.3)

are the exciton creation and annihiliation operators, Jmn is the one-exciton coupling parameter between atoms m and n and Kmn that for the two-exciton. For
2 atoms (N = 2) we have the following relationship between the atom basis andexciton basis:Atom basis→

atoms m, states a|mai|mai

exciton basis:states em→|em em i→|00i

1 ground state:|10i|20i

N = 2 one-exciton states:|10i|21i→|01i|11i|20i→|10i

N (N − 1)/2 = 1 two-exciton states:|11i|21i→ |11i.
As a first step to improving the theoretical treatment we decided to consider
only the formation of one-exciton states. Within the one-exciton formalism there
are a number of ways we could imagine improving the theoretical treatment of

75
dense vapors. One approach is to attempt an analytical model in which N oneexciton states are assigned a distribution of values corresponding to the typicalenergies for a particular density. Unlike atoms in molecules that oscillate aboutsome mean distance from other atoms, atoms in a vapor vary drastically in distance from one another. Thus the experimental bandwidth of the incident pulses
would be the restraining factor on the range of energies such that only excitons
with energies within the laser bandwidth would contribute. The fluctuations of
the exciton energy levels then could be applied via a stochastic model for each
exciton energy level. This approach would represent binary interactions as well as
provide additional fitting parameters, but still assumes a form for the correlation
functions.
An alternative approach would be to use molecular dynamics (MD) simulations, in which we consider a cluster of atoms that undergo a dynamic trajectory.
This approach would not need the stochastic model as an assumption, but rather
would calculate a correlation function based on the configurations of the atoms
from the simulated trajectories using interatomic potential energy surfaces. Thisapproach and its results are described in this chapter. To calculate the correlation function from molecular dynamics (MD) simulations we took four main
steps: 1. simulate trajectories of an ensemble of atoms, 2. determine the resonant
exciton couplings Jm,n based on distances between atom pairs for each snapshotof the trajectory, 3. diagonalize the one-exciton Hamiltonian and 4. calculate thecorrelation functions and peak shifts. We begin with a description of the MD
simulation.
4.2

Molecular Dynamics Simulations
For the MD simulation, a 600 ps dynamic trajectory for a simulation unit

cell containing 10 to 40 potassium atoms in the gas phase was generated from

76
the ground state potential energy surface (PES) of K2 using the molecular dynamics package GROMACS [61] (see App. G for details about the package). The
ground state potential energy surface was approximated by a Buckingham potential function fit to the 11 Σ+
g ground state surface of K2 , which was calculated via
a one-electron non-empirical pseudopotential method by [62]. The Buckingham
potential function is represented as
V (r) = A exp(−Br) −

C
.
r6

(4.4)

The fit to the ground state is shown in Fig. 4.2. The ground state potentialfunction was truncated in the simulation at 5 nm, at which distance the potential
reaches a value 10−7 times less than the potential minimum.
11 + PES and Fit
Energy (kJ/mol)

g

0

PES

-5

0
0.0

Fit

0.4

0.8

1.2

R (nm)

Figure 4.2: The 11 Σ+
g ground state surface and Buckingham potential fit usedin the molecular dynamics simulation. [Surfaces provided in S. Magnier et al.,J. Chem. Phys. 121, 1771 (2004).]

Twenty runs were performed for each temperature, each run 6000 steps longwith a 0.002 ps time step, with the atom positions written to the output every
10 steps, resulting in a resolution of 0.02ps. Weak temperature coupling to abath at the experimental temperatures of the cell window was applied through a
Berendson algorithm provided by the GROMACS program. The Berendson algo-

77
Table 4.1: Simulation box side lengths for 10 and 20 atoms.
Temp (◦ C)
300
350
400
450
500
550
600
650
700
750

Density (m−3 ) Box (nm) 10 atoms Box (nm) 20 atoms
4.815 × 1021
127.585160.74822
1.8355 × 10
81.6736102.9025.660 × 1022
56.112770.697623
1.4775 × 10
40.7528
51.3453
3.375 × 1023
30.943938.986923
6.920 × 10
24.3574
30.688424
1.298 × 10
19.7504
24.88392.2625 × 1024
16.411120.676724
3.706 × 10
13.921917.5405
5.7610 × 1024
12.018115.1418

rithm slowly corrects for deviations of the system temperature T as determinedfrom atom velocities from the bath temperature T0 according to the equationdT /dt = (T0 − T )/τ , where τ was set to be 0.1 ps in the simulations. No pressure coupling was applied (i.e. the box size remained constant throughout thesimulation). The box size was scaled to include the same number of atoms foreach temperature (see Table 4.2 for a list of box side lengths for the temperatures considered). The temperatures used for this scaling were the experimental
temperatures at the vapor cell nut (where the potassium rests in liquid form anddetermines the vapor pressure). Atomic positions were set by the C commandrand() for each run, and initial velocities were randomly generated for each run
by the GROMACS program according to the Maxwell distribution for an ideal
gas.
4.3

Resonant Exciton Couplings
The resonant couplings arise from the attractive or repulsive nature of

atomic interaction, so interatomic potential energy surfaces represent the amplitude and range of the transition frequency fluctuations for the case of pairwise

78
interaction. Thus for each snapshot of the trajectory, the resonant exciton couplings Jm,n were determined for each pairwise atomic distance as the value of the
1 +difference between the 11 Σ+
u excited state potential energy surface and the 1 Σgground state surface of K2 [62]. Only the 11 Σ+u was used in the calculation; allother surfaces dissociating to 4s + 4p (as plotted in Fig. 4.3) were not taken into
account, nor were transitions from the other ground state (see both ground statesin Fig. 4.4). The surfaces used in this calculation are plotted in Fig. 4.5. The
Jm,n were input into the fluctuating Frenkel exciton Hamiltonian, defined by
m6=n

ĤS =

X

†
Ωm B̂mB̂m +X†
Jm,n B̂m
B̂n ,

(4.5)

m,n

m

where Ωm is the excitation energy of atom m, chosen to be Ωm = Ω = 13024.0162 cm−1[60]. Two-exciton states, or quartic couplings, were not taken into account.
21 S+ g
11 S+ u
4s+4p
30000

11 Pg
11 Pu
23 S+ u25000

13 S+ gE Hcm-1 L

13 Pu
13 Pg

20000

15000

10000
5

10

15
R Ha0 L

20

25

Figure 4.3: All of the potential energy surfaces dissociating to 4s+4p. [As provided
in S. Magnier et al., J. Chem. Phys. 121, 1771 (2004).]

79
4s+4s
12000

10000
11 S+ gE Hcm-1 L

8000
13 S+ u
6000

4000

2000

0
5

10

15
R Ha0 L

20

25

Figure 4.4: All of the ground state potential energy surfaces. [As provided in
S. Magnier et al., J. Chem. Phys. 121, 1771 (2004).]

1-1

PES (cm )

1

+

1

g

and 1

+
u

14000

States

1

+

1

+

1
1

7000

u

g

0
0.0

0.6

1.2

1.8

R (nm)

Figure 4.5: The PES whose difference was used to determine the resonant exciton
couplings.

4.4

Diagonalizing the Exciton Hamiltonian
By diagonalizing the exciton Hamiltonian we obtained the one-exciton statesused to calculate the frequency fluctuations δωi = Ω − HS [i], where i denotes theith diagonal one-exciton matrix element [60]. To attempt proper bookkeeping,80
One-exciton Coupling-1

Coupling (cm )

2000

0

-2000

-4000
0

2

4

6

8

R (nm)

Figure 4.6: The difference between the 11 Σ+
u excited state potential energy sur1 +face and the 1 Σg ground state surface, used to determine the resonant exciton
couplings.

the eigenvalues and associated eigenvectors for each snapshot were compared withthose of the previous snapshot. If any element of the eigenvectors was found tohave changed by an amount greater than ǫ = 0.3 cm−1 , the eigenvectors werereordered to minimize the number of elements that underwent such change. Although this removed most misorderings that masqueraded as avoided crossings,
there were some “real” avoided crossings within eigenvectors that could not bereconciled frame-to-frame. In experiment only fluctuations within the laser bandwidth (∼ 200 cm−1 ) contribute to the response; a survey of the range of exciton
energies revealed that most interactions stay within the laser bandwidth, representing the fact that the majority of collisions occur beyond the range of the most
attractive or repulsive parts of the potential energy surfaces.

81
4.5

The Correlation Functions and Peak Shifts
The correlation functions were calculated for each run using the formulaC(τj ) =

h

N
X
i=0

h

δω(ti − τj )δω(ti )i

N
X

,

(4.6)

δω(ti )δω(ti )ii=0

where the brackets denote an ensemble average over all the one-exciton states andN is the number of frames in the trajectory. The fluctuations δω are approximatedto be periodic with period equal to tN , and times only up to τN/2 are consideredin the final form of the correlation function. The resulting correlation functions,averaged over all runs for each of various temperatures and numbers of atoms, areplotted along with the experimental peak shift results in Figs. 4.7-4.8.

MD

Experimental Peak Shift

Correlation Function

o

500 C

1

o

o

550 C

300 C

o

o

600 C

350 C

o

o

650 C

400 C

o

o

700 C

450 C

o

o

750 C

500 C
o

550 C

0.1

o

600 C
o

650 C
o

700 C
o

750 C

0

T, delay btwn 2

10
nd

and 3

20
rd

pulse (ps)

0

10

20

time (ps)Figure 4.7: Experimental peak shifts (left) and theoretical correlation functions
from molecular dynamics simulations with 20 atoms, averaged over 20 runs (right).
The theoretical curves have been scaled to optimize their match with the experimental peak shifts.

In comparing the simulations with 10 atoms with the experimental results,
good agreement is apparent. However, when the number of atoms is increased to82
Correlation Function and Peak Shift

Correlation Function and Peak Shift500C Cfn, 10 atoms

1

1500C Cfn, 20 atoms

Correlation Function and Peak Shift

550C Cfn, 10 atoms

600

MD
MD
MD

(10 atoms)

550C Cfn, 20 atoms

600

400C Cfn, 20 atoms

450C Cfn, 20 atoms

600

500C Cfn, 40 atoms

550C Cfn, 40 atoms

600 exptl peak shift

500C Exptl Peak Shift

550C Exptl Peak Shift(20 atoms)(40atm,20rn)

0.1

0.1

0.1

0

10

T, delay between 2

nd

20

30
rd

and 3

pulse (ps)Correlation Function and Peak Shift

0

5

10

nd

20
rd

and 3

25

pulse (ps)Correlation Function and Peak Shift

650C Cfn, 10 atoms

700

650C Cfn, 20 atoms

700

650C Cfn, 40 atoms

700

650C Exptl Peak Shift
0.1

15

T, delay between 2

MD
MD
MD

0

10

nd

20

30
rd

and 3

pulse (ps)

10

20

nd

rd

and 3

pulse (ps)Correlation Function and Peak Shift
750C Cfn, 10 atoms(10 atoms)

750C Cfn, 20 atoms(20atm, 20rn)

750C Cfn, 40 atoms(40atm,20rn)

750C Exptl Peak Shift

700 exptl peak shift
0.1

T, delay between 20T, delay between 2

0.1

0

5

10

15

T, delay between 2

20

nd

25

30
rd

and 3

35

pulse (ps)09T, delay between 2

nd

18
rd

and 3

pulse (ps)Figure 4.8: Experimental peak shifts (triangles) and correlation functions frommolecular dynamics simulations with 10, 20 and 40 atoms (lines), scaled to optimize fit with experimental peak shifts.

20, the behavior is similar only when comparing with lower temperatures thanthose used in experiment. Simulations were performed for 40 atoms (as shown in
Fig. 4.8) and were found to be similar to the 20 atom case; thus, the correlation
functions can be considered to have converged with respect to the number ofatoms by 20 atoms. Simulations were also performed for 2 atoms, resulting incorrelation functions with much longer decay times, following the general trend
seen in going from 10 to 20. These correlation functions did not match as wellas the 10-atom case to the experimental peak shifts. The important features thesimulation and experimental results have in common are the single-exponential tobi-exponential transition across temperature as well as the similar ratio of slow tofast exponential amplitudes with temperature.
To determine the source of the two exponential time constants in the cor-83

Correlation Function

Calculated Correlation Functions
1

long
short

0.1

0.01

1E-3
1E-4
0

20

T, delay btwn 2

40
nd

& 3

rd

60
pulse (ps)Figure 4.9: Correlation function from molecular dynamics simulation for potassium vapor at 500◦ C (N ≈ 3.4 × 1017 ), with both short and long couplings.

-1
Potential Energy (cm )Potential Energy Surfaces

1000
0
- 000

1

long

-2000

short

-3000
-4000
-5000
0.5

1.0

1.5

R

2.0

2.5

3.0

(nm)

Figure 4.10: Potential energy surfaces used to compare short and long couplings.

relation function, we replaced the one-exciton potential energy surface with ashorter-range potential. This resulted in a single exponential decay, revealingthat the slow decay is due to long-range, or resonant, interactions (see Figs. 4.9
and 4.10). In addition we used this surface to check whether the change in goingfrom 10 to 20 atoms might be due to the box size being smaller than the rangeof the one-exciton PES. Using the short-range PES case, the box size was always

84much larger than the range of the PES. However, when the number of atoms
was decreased to 10, the correlation function exhibited a slower exponential decay
similar to that found when using the long one-exciton PES.
The correlation function temperature behavior does not agree as well withexperiment when the number of atoms is increased, which may imply that therange of the potential is less than that represented by the potential energy surface
used in the derivation. To check that the similar temperature behavior of theexperimental peak shifts and the calculated correlation functions is real, the peakshifts were calculated using the correlation function parameters derived above
within the theory of Ch. 2. There were thus four input parameters to the calculation corresponding to the ∆’s and Λ’s of the biexponential fits. Results areshown in Fig. 4.11. Only the correlation functions for 40 atoms were considered,
but the most important aspect of the results is their similar behavior with thecorrelation functions used to calculate them, showing that bi-exponential correlation functions result in bi-exponential peak shifts. Thus the fast decaying part ofthe experimental peak shifts does seem to arise from the correlation functions.
The main assumptions made in the theoretical calculations are 1. using only
ground-state atoms in the MD simulation and 2. the inclusion of only one of theeight dipole-allowed 4S+4P potential energy surfaces in calculating the exciton
Hamiltonian (this also means we ignore the fine structure and hyperfine splitting). The first assumption means we do not take into account the fact that theexperiment is sensitive only to the frequency fluctuations of excited atoms. The
way to account for excited atoms would be to take for each snapshot the overlap
integral between the exciton states for a system with all ground state atoms andthe exciton states for a system with all atoms in ground except one in the excitedstate. In this way the evolution of the system as detected by experiment would
be properly described. Concerning the second assumption, including all the po-

85

Calculated Peak Shiftswith their Correlation Functions
1
oCorrelation Fn 500 CoCorrelation Fn 600 CoCorrelation Fn 700 C

0.1oPeak Shift 500 CoPeak Shift 600 CoPeak Shift 600

C(single corrln fn)oPeak Shift 700 C

0.01

Peak shift fromsingle exponential
correlation function

1E-3

0

20

T, delay btwn 2

nd

rd

& 3

pulse (ps)Figure 4.11: Calculated peak shifts and their correlation functions.

86
tential energy surfaces would change the characteristic timescales and amplitudesof fluctuations in some way. To actually include all the surfaces properly seems to
be a challenging task that might not result in significant changes in behavior froma more qualitative approach, at least as a function of temperature and density.
To conclude, the above results show that peaks shifts based on correlation
functions calculated from MD simulations in an exciton picture exhibit similar
behavior as a function of temperature and density to experimental peak shifts.
Our theoretical approach to modeling uses classical trajectories to result in remarkably realistic simulations of quantum dissipation. In addition, the behaviorof the correlation functions versus temperature and density reveal the effects ofresonance broadening at high densities and temperatures. This unusually simple
electronic system has provided the opportunity to derive a correlation function
from first principles with relatively few simplifications compared to molecular systems. This work has implications for other disordered excitonic systems such as
molecular aggregates, semiconductors and biological systems.

Chapter 5
The Local Field

In 1880, Lorentz and Lorenz were both interested in the same problem: theconnection between the index of refraction of a transparent body, and its density[63, 64, 65]. They found that for high-density materials the microscopic field plays
a significant role in determining the index of refraction, which can be accounted
for using the concept of a local field. Doing so reveals that the local field causes
a redshift of the resonance lineshape called the Lorentz-Lorenz shift. It was not
until a century later that the Lorentz-Lorenz shift was detected in experiment,using resonant selective reflection in a dense potassium vapor [66]. The study
revealed a greater frequency shift than that predicted by the Lorentz and Lorenz,
which was attributed to collisional contributions. Later work disagreed with this
explanation of the extra shift [67, 68, 69]. In order to reduce the influence ofadditional frequency shifts we work in the time domain using transient four-wave
mixing (TFWM). The results of the TFWM experiment reveal a clear local field
contribution. In this chapter the influence of the local field will be derived forthe frequency domain and time domain, followed by experimental TFWM results
and a preliminary calculation of the propagated local field in a dense potassium
vapor.

88
5.1

Definition of the Local Field
When an electromagnetic field is incident on a material system, a typicalatom of the system experiences not only the externally applied field but also a
microscopic field due to the induced polarization of the surrounding atoms in thematerial. The effective, or local, field is the sum of the external and internal fields
incident on a typical atom in the ensemble, represented as Eloc = EM ax + Eint ,where EM ax is the macroscopic field determined by the Maxwell equations andEint is the internal field from the induced polarization of the surrounding atoms.
In order to count the field contributions of nearby atoms only once, we define
the internal field as the difference between the nearby field calculated through adetailed microscopic treatment and that calculated from the macroscopic polarization P . Thus Eint = Emic − EP , where Emic is the microscopic contribution of
nearby atoms and EP is the field from these same surrounding atoms calculated
from the macroscopic polarization P of the medium. In this way we replace themacroscopic field by the internal field that properly takes nearby contributions
into account. The range over which nearby atoms must be considered can be approximated as a sphere whose radius is large enough to encompass atoms whose
configuration and locations result in fields that cannot be averaged out, and large
enough such that the polarization resulting from the atomic dipoles can be considered evenly distributed throughout the volume. For a random configurationof atoms the field due to atoms inside the sphere, Emic , is zero [70], resulting in
the internal field Eint = −EP . The macroscopic field is the same as that from asphere with radius R containing a charge distribution with total dipole moment
3

P: EP = −LP [70], where L = 3ǫ10 (SI units), called the Lorentz conp = 4πR3stant. Thus in the end the total field amounts to the Maxwell field minus the89internal field calculated using the macroscopic polarization,
Eloc = EM ax − LP,(5.1)

hence the referencing of this treatment as a “mean-field” approach.
5.2

Frequency-Domain Manifestation of the Local Field
The local field is manifested in frequency-domain experiments as a redshiftof the resonance line center. This can be derived starting from the definition of
the polarization P(ω) = χ(ω)E(ω), with the local field E(ω) = EM ax (ω) + LP(ω):P(ω) = χ(ω)[EM ax (ω) + LP(ω)].

(5.2)

Solving for P, we find,P(ω) =

χ(ω)
EM ax (ω).
1 − Lχ(ω)(5.3)

Now substituting this into the equation for the total field E(ω) we findE(ω) = EM ax (ω) +Lχ(ω)
EM ax (ω)
EM ax (ω) =1 − Lχ(ω)
1 − Lχ(ω)(5.4)

Using the definition of χ for a two-level atomic system with resonance frequency
ω0 and population decay rate γ gives
11
ω02 − ω 2 − iωγ==
1 − Lχ(ω)(ω02 − LA) − ω 2 − iωγ
1 − ω2 −ωLA
2 −iωγ
0

LA
,
=1+ 2
(ω0 − LA) − ω 2 − iωγ(5.5)

where A represents the strength of material response and is proportional to thedipole moment squared. It can be seen from the above equation that accounting
for local field effects results in a redshift of the resonance frequency represented
by the LA term in the denominator. The local field thus results in a modified

90
definition of the dielectric constant ǫ of the material based on what is known as
the Lorentz-Lorenz law:
4π
ǫ−1
=
N α,
ǫ+2
3

(5.6)

where N is the density of the material and α is the atomic polarizability1 .
Maki et al. measured the reflectivity spectrum from the interface between
a dense potassium vapor and the sapphire window of an atomic vapor cell todetermine the resonant structure of the dielectric constant for both linear andnonlinear response. They worked at number densities high enough for local-field
effects to be important 2 [66]. In a method known as selective reflection [71],they measured the highly wavelength-dependent reflectivity near the D resonancelines. The ratio of reflected intensity to incident intensity was measured as afunction of the optical frequency. Their results reveal a shift greater than thatpredicted solely by the Lorentz-Lorenz shift, which they attribute to a collisional
shift contribution. To rule out the modifications that may arise from atom-window
interactions, they noted that the mean interatomic separation was much smaller
than the penetration depth of light into the medium, meaning the number of atoms
which interacted with the sapphire window was neglible compared to the numberof atoms contributing to the measured response. They support the assumption ofa collisional contribution by comparing the linear experimental results with nonlinear experimental results that contain local field contributions to higher order.
The collisional contribution is difficult to theoretically affirm, however, and later
papers have proposed other possible contributions beyond those due to collisions
between potassium atoms [67, 68, 69, 56].
1

This is also known as the Clausius-Mosotti relation for longer wavelengths.
Later papers showed that the Lorentz-Lorenz shift cannot be ignored at lower densities,
where inhomogeneous broadening dominates, either. [68, 67]
2

91
5.3

Time-Domain Manifestation of the Local Field
In 2-pulse TFWM the local field results in a signal for the “wrong delay.”

As was shown in the Ch. 3, the two-pulse TFWM signal from a 2-level system
not including local field effects is zero for negative delays, that is, for delays inwhich the conjugate pulse arrives last. Here we derive the nonlinear time-domain
response for a 2-level system including local field effects. We will see that the
local field allows a breaking of the “proper” time ordering, resulting in a signal
for negative delay [72].
Local Field Intensities for direction 2k2-k1
positive delay (t > 0) negative delay (t < 0)I 1loc I 2loc

t>0

time

I 2loc I 1loc

t<0

time

Figure 5.1: Schematic of local field intensities interacting in a 2-pulse TFWM
experiment for the signal in direction 2k2 − k1 . Including local field effects, forτ < 0 the tail of I2loc can act after I1loc to create a signal for negative delay.

To derive the time-domain response we use the local field defined above as
the incident field of the optical Bloch equations. Note that P depends on theoff-diagonal matrix elements ρnm , meaning there will arise a recursive relationshipin the optical Bloch equations that represents the polarization acting on itself.
This results in a negative delay signal due to the decaying tail of the induced polarization interacting with the next pulse such that signals for both time orderings
are allowed (see Fig. 5.1). Using the local field we have for the coherence at third92
order:−iLN µ2nm (2)(1)(2)[ρmm (t)ρ(1)mn (t) + ρmm (t)ρmn (t)~iLN µ2nm  (3)
(1)
(1)
(2)
−ρ(2)
(t)ρ
(t)
−
ρ
(t)ρ
(t)]
+
i(ω
−
ω
)
−
γ
+
ρmn (t)
nm
nm
mn
nn
mn
nn
~
iLN µnm (2)[ρmm (t) − ρ(2)(5.7)+
nn (t)]EM ax (r, t)
2~

ρ̇(3)
mn (t) =As seen in the above equation, the coherence at third order depends directly on thecoherence at first order, representing the recursive relationship of the polarization
acting on itself. This result is unlike that of the derivation which did not include
local field effects, in which the third-order coherence equation of motion Eq. 2.9
was only directly dependent on the current and penultimate order. The resultsof this derivation calculated for delta function pulses are shown in Fig. 5.2. A
negative delay signal appears when the local field is included. The decay rate
for negative delay is faster than that for positive delay by a factor of 2, whicharises from the convolution of the free polarization decay at first order with thatat third order. The ratio of the signal amplitude for positive and negative delay
is directly related to the Lorentz-Lorenz shift according to B/A = [∆ωL /γ]2 ,where A (B) is the amplitude of the signal at τ = 0 when extrapolated frompositive (negative) delay and ∆ωL = µ2 N L/~ is the Lorentz-Lorenz shift. Thuspthe shift ∆ωL = γ × B/A can be extracted directly from experimental TFWM

data. The original theory derived above predicts a value for the Lorentz constantof L = 1/3ǫ0 . It should be noted that in the derivation resulting in the above(1)∗

equation we have ignored a dependence on ρmn at second order; in addition, thetreatment may be improved through a more thorough inclusion of the recursive
relationship of the local field on the polarization for each order.

TFWM Signal

930Time delay t

Figure 5.2: Calculated 2-pulse TFWM signal for a 2-level system with (dashed)
and without (solid) local field.

5.4

Transient Four-Wave Mixing Experimental Results
A representative 2-pulse TFWM signal from the experiment described in

Ch. 3 is now shown in full including negative time delay in Fig. 5.3. Although theratio of the decay rate for positive and negative delay agree with theory, we find thestrength of the local field contribution L as determined by fitting the signal amplitudes for negative and positive delay B/A to be less than predicted by a factor of
20. This discrepancy could have a number of causes. Radiation trapping was one
candidate considered, but using the pulse picker to decrease the repetition rate
did not significantly modify the amplitude ratio. Excitation-induced effects (often
revealed by a decreasing dephasing rate with increasing excited state population)
was also considered [73], and indeed decreasing the incident intensity resulted in alarger local field effect, but still did not make a great enough difference to matchwith theory. It has been noted that at the depth at which these experiments
probe, the local field is not yet fully formed [56], which would support a decreased
shift. In addition, the opposite limit in which the fields penetrate further into the94vapor and are influenced by propagation effects is not included in the theoretical treatment. To test whether the discrepancy is due either to an incompletely
formed local field or propagation effects, we performed some preliminary calculations of pulse fields reflected from and transmitted through a dense atomic vapor,including local field effects. These calculations are briefly summarized below.

TFWM Signal

1

0.1

Ratio of slopes is right0.01

Signal for
negative delay

1E-3

-2

0

2

4

6

Time delay t (ps)Figure 5.3: Experimental 2-pulse TFWM signal for a 2-level system, with slopes
labeled.

5.5

Preliminary Calculations of Propagation
This section presents a calculation of pulse fields reflected from and trans-mitted through a dense atomic vapor, including local field effects. The structure
under consideration is shown in Fig. 5.4. This treatment is based on the formalismof Sipe [74] and to this point only includes the linear response.
+
We use the following equations for the reflected (Erefl (z, t)) and transmitted+(Etrans(z, t)) fields:

Z ∞

(5.8)

0



dω n1 − n2 (ω)
E− (ω)e−iω(t−n1 z/c)
2π n1 + n2 (ω)

Z ∞



dω
2n1
E− (ω)e−iω̃n2 (ω)z e−iωt
2π n1 + n2 (ω)

(5.9)

+
Eref
l (z, t) =and+
Etrans(z, t) =

0

95

E-

q

+

E refl

z>0

air

e1

vapor

e2

z=0
+

E transz<0Figure 5.4: Structure considered for the propagation calculations. In the calculations θ is set to zero (normal incidence).

where ω̃ = ω/c, and2

E− (ω) = e−(ω−ω0 ) /∆2

(5.10)

is the incident field for ω > 0 and ∆ << ω0 .
The index of refraction of the vapor, n2 (ω) (see Fig. 5.5), is defined to bepn2 (ω) = ǫ(ω), whereǫ(ω) = 1 + χ(ω) = 1 +Here ω̄21 =p

e2 f N/ǫ0 m
2
2
(ω̄21
+ γp12) − ω 2 − 2iωγp12(5.11)2
ω21− e2 f N L/ǫ0 m.

Note in these calculations ω is in units of rad/sec, ν in units of 1/s. Below
is a list of parameter values used in the calculations:• laser center frequency is on resonance: νlaser = ν0 = 3.9 × 1014 1/s• laser pulse FWHM δν = 1.02×1013 1/s, which corresponds to δλ = 20 nm(see Fig. 5.6)
• oscillator strength f = 1• index of refraction of “air” n1 = 1• local field parameter L = 1/3• density N = 3.7 × 1023 m−3 (corresponds to temperature T = 700◦ C)

96• dephasing rate γp12 = 1.16 × 1012 1/s.
Note the index of refraction of the vapor is determined with the parameters
f , L, N , and γp12 , but the resulting amplitude of the real and imaginary parts are
not consistent with experiment. In particular, the calculated index of refraction
is only a little above 1 at 700◦ C, compared to ∼ 6 in experiment. This meansthat in these results the fields are modified by the vapor resonance less strongly
than in experiment. The figures that follow are largely self-explanatory.
The results for the transmitted field shown in Figs. 5.8 and 5.9 exhibit
the well-known signature of propagation effects in the form of Bessel-function
behavior, sometimes referred to as dynamic beating [12]. Experimental results
measuring the linear response through upconverted reflected pulses show no signof dynamic beating, implying that propagation effects are not an issue in the reflected TFWM signals. This result in turn means the signatures of non-Markovian
dynamics are properly assigned and are not being confused with propagation effects. Experimental results in transmission using a 14 µm-thick transmission cellon the other hand exhibit pronounced dynamic beating, as shown in App. E.
These calculations of propagation effects can in turn be used to model the development of local field effects in the nonlinear response of dense vapors as a functionof penetration depth and density.

97

Calculated index of refraction of vapor

1.10

0.18

Index of Refraction

1.08

Im

Re

0.16

1.06

0.14

1.04

0.12

1.02

0.10

1.00

0.08

0.98

0.06

0.96

0.04

0.94

0.02

0.92
0.90

0.00
3.7

3.8

3.9

4.0

4.1

(10

3.7

14

3.8

3.9

4.0

4.1

-0.02

1/s)

Figure 5.5: Index of refraction of vapor, real (left) and imaginary (right) parts.

Intensity (arb. units)

Incident pulse intensity

0.8

0.8

0.4

0.4

0.0

0.0
3.6

3.8

4.0
14

(10

1/s)

4.2

-150
4.4 -100 -50

0

50

100

150

t (fs)

Figure 5.6: Incident pulse versus frequency (left) and time (right).

98Intensity

1(arb. units)

Reflected field intensity

0
1
2
3
4

5z (

m)

6
7
8
9

0

1

t (ps)

2

3+
2
Figure 5.7: Reflected field intensity |Erefl (z, t)| .

Intensity

1(arb. units)

Transmitted field intensity

0
-1
-2
-3
-4
-5

z ( m)

-6
-7
-8
-9

0

1

t (ps)

2

3+
Figure 5.8: Transmitted field intensity |Etrans(z, t)|2 .

99Intensity (arb. units)

Transmitted field for z = 1 m
0.01

1E-12

1E-22

0.0

0.5

1.0

1.5

2.0

2.5

3.0

t (ps)+
Figure 5.9: Transmitted field intensity |Etrans(z, t)|2 for z = 1 µm.

0.5Intensity

1.0(arb. units)

Transmitted field intensity

0
-1
-2
-3
-4
-5

z ( m)

-6
-7
-8
-9

3.7

3.8

3.9
14

(10

4.0

4.1

1/s)

+
Figure 5.10: Transmitted field intensity versus frequency |Etrans(z, ν)|2 .

100

Integrated intensity (arb. units)

Integrated transmitted intensity
120
Equation:y = y0 + A1*exp(-(x-x0)/t1)

100

y0 = 12.42904
x0 = -0.51268

80

A1 = 69.5306
t1 = 3.73168

60

40

20
-2

0

2

4

6

8

10

-z ( m)R∞ ++(z, t)|2 dt. FolFigure 5.11: Integrated transmitted intensity Etrans(z) = 0 |Etranslows Beer’s law (exponential, line in figure) beyond 2µm.

Chapter 6
Conclusion

In this thesis we have presented results from 2- and 3-pulse transient fourwave mixing experiments and theoretical calculations of the material response. In
Ch. 2 we derived the material response of a two- and three-level system to findthat Markovian dynamics results in exponential dephasing, while non-Markoviandephasing is distinctly non-exponential and can be modeled based on an assumed
two-time correlation function of frequency fluctuations. Chapter 3 on the photonecho experiments revealed distinct regimes of non-Markovian and Markovian dynamics in two- and three-pulse photon echo. The non-Markovian theory coveredin Ch. 2 results in excellent fits for the two-pulse data, but the three-pulse peakshift data for high temperatures and densities exhibits bi-exponential behavior
that cannot be modeled with a single exponential correlation function. In addition, time-resolved three-pulse data reveal clear photon echo-like behavior. To
improve our understanding of the three-pulse experimental results, in Ch. 4 we
presented a derivation of the correlation function of frequency fluctuations in an
exciton picture from molecular dynamics simulations. The resulting calculated
peak shifts exhibit bi-exponential behavior and reveal that the source of the slow
component is the long-range interaction due to resonance broadening. The lastchapter, Ch. 5, described the frequency- and time-domain signatures of the local field and presented experimental results with clear local field contributions.

102
A preliminary calculation of propagation effects was also presented that confirm
that the experimental signals in reflection are not altered by propagation effects.
In summary, transient four-wave mixing experiments in a dense potassium
vapor, which has the unusual quality of a long dephasing time compared to thecollision duration, reveal distinct signatures of quantum dissipation in the formof both non-Markovian and Markovian dynamics. These results serve as a basis
for understanding more complex systems such as molecular liquids due to theexcellent agreement between the two-pulse experiments and theory assuming thestochastic correlation function. In addition, calculated correlation functions revealthat resonance effects may be the cause of the bi-exponential behavior observed inthree-pulse photon echo peak shift experiments, giving insight into the dynamics
of resonance-broadened vapors and disordered excitonic systems in general. Finally, the utility of time-domain techniques in the study of dense atomic vapors
was supported throughout this research by the clear experimental signatures representing system response, both for non-Markovian dynamics and the local field
effect.
We end by considering other avenues for contribution that could be explored in the future. A systematic study of the TFWM signals as a functionof buffer gas density and temperature would yield insight into the difference between foreign-gas broadening and self-broadening. In conjunction with further
theoretical molecular dynamics simulations we could gain a fundamental pictureof the range of interactions in dense vapors. Once transmission cells (mentioned
in App. E) are manufactured with a thickness on the order of a wavelength, transmission experiments can explore the transition from purely local field effects to
propagation effects. In addition, with transmission cells dense vapors could thenbe studied with two-dimensional Fourier transform spectroscopy, which would enable the direct identification of excitonic effects, once again exhibiting signatures

103
with unusual clarity due to the simplicity of the atomic system.
Finally, we acknowledge the support of the NSF JILA Physics Frontier Center for the duration of the research presented in this thesis.

Bibliography[1] P. Hänggi and G.-L. Ingold, “Fundamental aspects of quantum Brownian
motion,” Chaos 15, (2005).
[2] H. Margenau and W. W. Watson, “Pressure Effects on Spectral Lines,” Rev.
Mod. Phys. 8, 2253 (1936).
[3] J. F. Ward, Correlation effects in the theory of combined doppler and pressure
broadening (Thesis, University of Colorado, Boulder, 1964).
[4] J. Cooper, “Broadening of Isolated Lines in the Impact Approximation Usinga Density Matrix Formulation,” Rev. Mod. Phys. 39, 167 (1967).
[5] J. H. V. Vleck and D. L. Huber, “Absorption, emission and linebreadths: A
semihistorical perspective,” Rev. Mod. Phys. 49, 939 (1977).
[6] K. Niemax, M. Movre and G. Pichler, “Near-wing asymmetries of the selfbroadened first Rb and Cs resonance lines,” J. Phys. B 12, 3503 (1979).
[7] R. E. Walkup, A. Speilfiedel and D. E. Pritchard, “Observation of NonLorentzian Spectral Line Shapes in Na-Noble-Gas Systems,” Phys. Rev. Lett.
45, 986 (1980).
[8] Y. Prior and A. Ben-Reuven, “Nonimpact theory of four-wave mixing andintracollisional dynamics,” Phys. Rev. A 33, 2362 (1986).
[9] N. Asida, M. Rosenbluh and S. Mukamel, “Nonimpact degenerate four-wave
mixing in Na perturbed by He,” Phys. Rev. A 40, 41234126 (1989).
[10] K. Niemax and G. Pichler, “New aspects in self-broadening of alkali resonancelines,” J. Phys. B 8, 179 (1975).
[11] A. Laubereau and W. Kaiser, “Vibrational dynamics of liquids and solids
investigated by picosecond light pulses,” Rev. Mod. Phys. 50, (1978).
[12] O. Kinrot and Y. Prior, “Four-wave mixing in optically dense media,” Phys.
Rev. A 50, R1999 (1994).

105[13] M. Sargent, III, M. O. Scully, W. E. Lamb and Jr., Laser Physics (Perseus,
Cambridge, Mass., 1974).
[14] F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460 (1946).
[15] S. Mukamel, “Nonimpact unified theory of four-wave mixing and two-photon
processes,” Phys. Rev. A 28, 3480 (1983).
[16] S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University
Press, New York, 1995).
[17] D. Abramavicius and S. Mukamel, “Quartic Interband Exciton Couplings in
Pump-Probe Spectroscopy of Light Harvesting Complexes,” J. Phys. Chem.
B 108, 10295 (2004).
[18] C. J. Fecko, J. D. Eaves, J. J. Loparo, A. Tokmakoff and P. L. Geissler,“Ultrafast Hydrogen-Bond Dynamics in the Infrared Spectroscopy of Water,”
Science 301, 1698 (2003).
[19] J. Stenger, D. Madsen, P. Hamm, E. T. J. Nibbering and T. Elsaesser, “A
Photon Echo Peak Shift Study of Liquid Water,” J. Phys. Chem. A 106,2341 (2002).
[20] G. R. Fleming and M. Cho, “Chromophore- solvent dynamics,” Annu. Rev.
Phys. Chem. 47, 109 (1996).
[21] R. Jimenez, G. R. Fleming, P. V. Kumar and M. Maroncelli, “Femtosecond
solvation dynamics of water,” Nature 369, 471 (1994).
[22] T. Joo, Y. Jia, J. Y. Yu, M. J. Lang, and G. R. Fleming, “Third-ordernonlinear time domain probes of solvation dynamics,” J. Chem. Phys. 104,
6089 (1996).
[23] E. T. J. Nibbering, D. A. Wiersma and K. Duppen, “Femtosecond NonMarkovian Optical Dynamics in Solution,” Phys. Rev. Lett. 66, 2464 (1991).
[24] J. G. Fugimoto and T. K. Yee, “Diagrammatic Density Matrix Theory ofTransient Four-Wave Mixing and the Measurement of Transient Phenomena,”
IEEE J. Quantum Electron. QE-22, 8 (1986).
[25] J. A. Armstrong, N. Bloembergen, J. Ducuing and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918
(1962).
[26] A. Abragam, Principles of Nuclear Magnetism (Oxford Univerity Press, Oxford, 1961).

106[27] T. Yajima and Y. Taira, “Spatial Optical Parametric Coupling of Picosecond
Light Pulses and Transverse Relaxation Effect in Resonant Media,” J. Phys.
Soc. Jpn. 47, 1620 (1979).
[28] A. Papoulis, Brownian Movement and Markoff Processes, 2nd ed. (McGrawHill, New York, 1984).
[29] G. E. Uhlenbeck and L. S. Ornstein, “On the Theory of the Brownian Motion,” Phys. Rev. 36, 823 (1930).
[30] J. L. Doob, “The Brownian Movement and Stochastic Equations,” Ann.
Math. 43, 351 (1942).
[31] S. Mukamel and R. F. Loring, “Nonlinear response function for time-domain
and frequency-domain four-wave mixing,” J. Opt. Soc. Am. B 3, 595 (1986).
[32] R. Kubo, in Fluctuation Relaxation and Resonance in Magnetic Systems,
edited by D. Ter-Haar (Oliver and Boyd, Edinburgh, 1962).
[33] R. Kubo, “A Stochastic Theory of Line Shape,” Adv. Chem. Phys. 15, 101(1969).
[34] R. Zwanzig, “Ensemble Method in the Theory of Irreversibility,” J. Chem.
Phys. 33, 1338 (1960).
[35] R. Zwanzig, “Memory Effects in Irreversible Thermodynamics,” Phys. Rev.
124, 983 (1961).
[36] R. Zwanzig, in Lectures in Theoretical Physics, edited by W. E. Brittin (Interscience, New York, 1961).
[37] U. Fano, “Pressure Broadening as a Prototype of Relaxation,” Phys. Rev.
131, 259268 (1963).
[38] W. Magnus, Commun. Pure Appl. Math. 7, (1954).
[39] R. M. Wilcox, “Exponential Operators and Parameter Differentiation in
Quantum Physics,” J. Math. Phys. 4, 962 (1967).
[40] U. Haeberlen, Advances in Magnetic Resonance (Academic Press, New York,
1976), suppl. I.
[41] K.-E. Süsse, W. Vogel and D.-G. Welsch, “Femtosecond photon echoes via
non-Markovian dephasing,” Chem. Phys. Lett. 162, 287 (1989).
[42] N. A. Kurnit, I. D. Abella and S. R. Hartmann, “Observation of a Photon
Echo,” Phys. Rev. Lett. 13, 567 (1964).

107[43] M. Cho, J.-Y. Yu, T. Joo, Y. Nagasawa, S. A. Passino and G. R. Fleming,“The Integrated Photon Echo and Solvation Dynamics,” J. Phys. Chem. 100,
11944 (1996).
[44] E. C. C. Baly, Spectroscopy (Longmans, Green and Co., New York, 1924).
[45] L. Tterlikkis, S. D. Mahanti, and T. P. Das, “Theoretical Analysis of theHyperfine Structure of Alkali Atoms,” Phys. Rev. 176, 10 (1968).
[46] O. D. Zakharova and A. M. Semenov, “The Pressures at Which the Properties
of Vapors of Alkali Metals Correspond to the Model of Ideal Reacting Gas,”
High Temperature 40, 150 (2002).
[47] A. N. Nesmeyanov, Vapor Pressure of the Chemical Elements (Elsevier, London, 1963).
[48] A. Honold, L. Schultheis, J. Kuhl and C. W. Tu, “Reflected degenerate fourwave mixing on GaAs single quantum wells,” App. Phys. Lett. 52, 2105
(1988).
[49] T. Holstein, “Imprisonment of Resonance Radiation in Gases II,” Phys. Rev.
83, 1159 (1951).
[50] W. A. Hügel, M. F. Heinrich, M. Wegener, Q. T. Vu, L. Bányai and H. Haug,“Photon Echoes from Semiconductor Band- to-Band Continuum Transitionsin the Regime of Coulomb Quantum Kinetics,” Phys. Rev. Lett. 83, 3313
(1999).
[51] M. Aihara, “Non-Markovian theory of nonlinear- optical phenomena associated with the extremely fast relaxation in condensed matter,” Phys. Rev. B
25, 53 (1982).
[52] J. P. Lavoine, A. J. Boeglin, P. Martin and A. A. Villaeys, “Non-Markovian
effects on quantum beats,” Phys. Rev. B 53, 11535 (1996).
[53] Y. J. Yan and S. Mukamel, “Photon echoes of polyatomic molecules in condensed phases,” J. Chem. Phys. 94, 179 (1991).
[54] P. Hamm, M. Lim and R. M. Hochstrasser, “Non-Markovian Dynamics of theVibrations of Ions in Water from Femtosecond Infrared Three-Pulse Photon
Echoes,” Phys. Rev. Lett. 81, 5326 (1998).
[55] V. O. Lorenz and S. T. Cundiff, “Non- Markovian Dynamics in a Dense
Potassium Vapor,” Phys. Rev. Lett. 95, 163601 (2005).
[56] J. A. Leegwater and S. Mukamel, “Self- broadening and exciton line shifts ingases: Beyond the local-field approximation,” Phys. Rev. A 49, 146 (1994).

108[57] D. A. Farrow, A. Yu and D. M. Jonas, “Spectral relaxation in pumpprobetransients,” J. Chem. Phys. 118, 9348 (2003).
[58] J. A. Leegwater and S. Mukamel, “Semiclassical Green function calculationof four wave mixing in polarizable clusters and liquids,” J. Chem. Phys. 101,
7388 (1994).
[59] S. Mukamel, “Multidimensional Femtosecond Correlation Spectroscopies ofElectronic and Vibrational Excitations,” Annu. Rev. Phys. Chem. 51, 691(2000).
[60] S. Mukamel and D. Abramavicius, “Many- Body Approaches for Simulating
Coherent Nonlinear Spectroscopies of Electronic and Vibrational Excitons,”
Chem. Rev. 104, 2073 (2004).
[61] D. van der Spoel, E. Lindahl, B. Hess, G. Groenhof, A. E. Mark and H. J. C.
Berendsen, “GROMACS: Fast, flexible, and free,” J. Comp. Chem. 26,
17011718 (2005).
[62] S. Magnier, M. Auber-Frécon and A. R. Allouche, “Theoretical determinationof highly excited states of K2 correlated adiabatically above K(4p) + K(4p),”
J. Chem. Phys. 121, 1771 (2004).
[63] H. A. Lorentz, “Ueber die Beziehung zwischen der Fortpflanzungsgeschwindigkeit des Lichtes und der Körperdichte,” Ann. Phys. Chem. 9,641 (1880).
[64] L. Lorenz, “Ueber die Refractionsconstante,” Ann. Phys. Chem. 11, 70(1880).
[65] H. A. Lorentz, The Theory of Electrons, 2nd ed. (Dover, New York, 1952).
[66] J. J. Maki, M. S. Malcuit, J. E. Sipe and R. W. Boyd, “Linear and nonlinear
optical measurements of the Lorentz local field,” Phys. Rev. Lett. 67, 972(1991).
[67] J. Guo, J. Cooper and A. Gallagher, “Selective reflection from a dense atomic
vapor,” Phys. Rev. A 53, 1130 (1996).
[68] J. Guo, A. Gallagher and J. Cooper, “Lorentz-Lorenz shift in an inhomogeneously broadened medium,” Opt. Comm. 131, 219 (1996).
[69] P. Wang, A. Gallagher and J. Cooper, “Selective reflection by Rb,” Phys.
Rev. A 56, 1598 (1997).
[70] J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975),
chap. 4.

109[71] J. P. Woerdman and M. F. H. Schuurmans, “Spectral narrowing of selective
reflection from sodium vapour,” Opt. Comm. 14, (1975).
[72] S. T. Cundiff, “Time domain observation of the Lorentz-local field,” Laser
Phys. 12, 1073 (2002).
[73] J. M. Shacklette and S. T. Cundiff, “Role of excitation-induced shift in the coherent optical response of semiconductors,” Phys. Rev. B 66, 045309 (2002).
[74] J. E. Sipe, “New Green function formalism for surface optics,” J. Opt. Soc.
Am. B 4, 481 (1987).
[75] S. Y. Yee and T. K. Gustafson, “Diagrammatic analysis of the density operator for nonlinear optical calculations: Pulsed and CW responses,” Phys.
Rev. A 18, 1597 (1978).
[76] R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, New York, 2003).
[77] N. B. Vargaftik, Handbook of Physical Properties of Liquids and Gases: puresubstances and mixtures, 2nd ed. (Hemisphere, Washington, 1975).
[78] D. S. Hughes and P. E. Lloyd, “Pressure Effects of Homogeneous K Vapor in
Absorption,” Phys. Rev. 52, 1215 (1937).

Appendix A
Optical Bloch Equations Derivation Details

This appendix contains details for the derivation of the optical Bloch equations for a two-level system.
Starting from Eq. 2.10, we have
Z
i t
′
(q)
ρnm (t) = −dt′ e(iωnm +γnm )(t −t) [−µ · E(t′ ), ρ(q−1) ]nm .
~ −∞

(A.1)

To third order we then have(3)ρba (t) = −2iρ(0)′′′µba 31exp[−(iωba + )]
~
T2
′′′

′

′′

Z t
′′′

′

dt

−∞

E(r, t )E(r, t )E(r, t ) exp[γ(t − t ) + t /T2 ]

Z t′

′′

dt

−∞

Z t′′

dt′′′

−∞

×{exp[iωba (t′ − t′′ + t′′′ )] + exp[−iωba (t′ + t′′ − t′′′ )]},

(A.2)

where γ = T12 − T11 , µab = µba and the population was assumed to start in the
(0)

(0)

ground state, ρD = ρ(0) , ρba = 0.
Now if we write the fields as Ei (r, t) = Ei (t) exp[−iωt + iki · r] + c.c. thenwe can separate the term in direction k4 = 2k2 − k1 and make the rotating wave
approximation in which we ignore terms with sums of frequencies ωba + ω and−ωba − ω. We have(3)ρba (t) = −2iρ(0)µba /~

3




1
exp −
+ iωba t + ik4 · r
T2

Z t Z t′ Z t′′
−∞

−∞


t′′′ 
E2 (t′ )E2 (t′′ )E1∗ (t′′′ ) exp[i∆ω(t′ + t′′ − t′′′ )]exp γ(t′ − t′′ ) +T2+E2 (t′ )E1∗ (t′′ )E2 (t′′′ ) exp[i∆ω(t′ − t′′ + t′′′ )] ,dt′ dt′′ dt′′′

−∞

(A.3)

111
where ∆ω = ωba − ω. This is equivalent to Eq. 2.14.
Now we let the field envelopes be delta-function pulses, defined by Ei (t) =δ(t − ti ):(3)ρba (t) = −2iρ(0)exp[ik4 · r −µba 3~(A.4)

1(t − t1 )] exp[−iωba t + i∆ω(2t2 − t1 )]θ(t − t2 )T2where we have used the definition
Z x−∞

δ(t − y)f (t)dt =

Z x−y
−∞

δ(t)f (t + y)dt = f (y)θ(x − y),(A.5)

where θ(t) is the Heaviside step function, θ(t) = 0 for t < 0, θ(t) = 1/2 for t = 0,
θ(t) = 1 for t > 0. Now if we consider pulses with center frequency on resonance,(3)ω = ωba , and use the definition P̂(3) = N µba ρ̂ba (r, t, ω0 ) we findP̂(3) (r, t) = −iρ(0) CN µba θ(t − t2 ) exp[ik4 · r − T2−1 (t − t1 )]which is Eq. 2.16.

(A.6)

Appendix B
Non-Markovian Theory Derivation Details

B.1

Density Matrix Diagrams
All of the relevant diagrams contributing to the response of a three-levelsystem in 3-pulse TFWM in direction −k1 + k2 + k3 are shown in Fig. B.1.
Time Ordering (-ka,kb,kc):
2 m
2 m
3 n
R2
R3
R2
m a kcma
n a
mmnm
kc a a kb
amkb a mkb a ma a kaa a kaa aTime Ordering (kb,-ka,kc):
2 m
2 m
R2
R3
m a kc kc m a
mm
a a
ka
m a kama
kb a akb a a

3 n
R3
n a

kc
kc
ka

a a

kc kc

3 n
R3
n a

R 2m

ma
ma

kb

a a

kb

am

3

ma

a a

ka

ka
kb

ka

a a
ma
a a

Figure B.1: Double-sided Feynman diagrams representing the relevant pathways
for 3-pulse TFWM in a 3-level system in direction −k1 + k2 + k3 for all timeorderings, with n, m = b, c; n 6= m. Orderings (−k1 , k3 , k2 ) and (k3 , −k1 , k2 )are the same as for (−k1 , k2 , k3 ) and (k2 , −k1 , k3 ), with 3 and 2 switched, whileorderings (k2 , k3 , −k1 ) and (k3 , k2 , −k1 ) have no diagrams.
Each diagram in Fig. B.1 is a unique pictorial representation of one of thethird-order density matrix terms [24]. These diagrams are called double-sided

113
Feynman diagrams because their representation of density matrix perturbations
is similar to that provided by Feynman diagrams used to determine perturbations
on wavefunctions [75]. The left-hand side of each diagram indicates the timeevolution of |ψi and the right-hand side indicates the time evolution of hψ|, withtime increasing vertically upward [76]. The incident fields are represented by wavylines labeled by the wavevector number of the field. Each interaction of the appliedfield with the material system is indicated by a vertex of a wavy line (the field)with a straight vertical line (the ket or bra component of the density matrix).
The letters before and after the vertex represent the density matrix state of thesystem before and after the interaction. From these diagrams we can write themathematical solution for ρ(3) by using the appropriate set of translation rules.
Double-sided Feynman diagrams give us the ability to form both a mathematical and physical description of a TFWM process. Reading off of the diagram
labeled 2 R3m for time ordering (−ka , kb , kc ) in Fig. B.1, we see that the system
begins in the ground state, which corresponds to the zeroth-order density matrix(0)element ρaa . The first pulse, −ka , interacts with the material system to induce a(1)

polarization represented mathematically by ρam . This intermediate density matrix state is off-diagonal and dephases in time T2 . This diagram corresponds tothe convolution integral over t1 . At a subsequent time t2 , the second field E∗2(2)

forms a ground state population represented by ρaa , correponding to the convolution integral over t2 . Finally, at a time t3 , the third pulse kc interacts withthe system to produce a polarization in the direction k4 = k1 − k2 + k3 rep(3)resented by ρma . This polarization, according to the definition of Eq. 2.11, is(3)

P(3) = N (ha|µ|miρma + c.c.). The other diagrams in Fig. B.1 can be read in asimilar manner. The bottom row of diagrams correspond to field interactions witha different time ordering.

114
B.2

The Cumulant Expansion
To show the steps taken in the cumulant expansion we derive as an example

the cumulant expansion of a one-point correlation function,
Z t
F (t) = µhexp[idt1 δω(t1 )]i.

(B.1)

0

The first step is to Taylor expand the expression for F to second order in thefluctuation (as in Eq. 2.26):
Z t
Z
Z t
n
1 t
dt1 hδω(t1 )i −F =µ 1+i
dt1
dt2 hδω(t1 )δω(t2 )i + · · ·
2 0
0
0
Z t
Z
o
(−i)2n t
dt2n hδω(t1 ) . . . δω(t2n )i + · · ·dt1 . . .
+(2n)! 00

(B.2)

Now we make the ansatz that the one-point correlation function has the formF ≡ exp[−F](B.3)

We expand F itself in powers of δω, with λ as the expansion parameter:F ≡ λF (1) + λ2 F (2) + · · ·(B.4)

If we now expand the ansatz in a Taylor series including this expansion of F andcollect terms of like order we find1
F = 1 − (λF (1) + λ2 F (2) + · · · ) − (λF (1) + λ2 F (2) + · · · )2 − . . .
2
1= 1 − λF (1) + λ2 F (2) − F (1)2 + · · ·
2

(B.5)

Comparing this equation with the expansion of F in Eq. B.2 and solving for F (1)and F (2) we find
(1)

Z t

dt′ hδω(t′ )i
Z0 t Z t
Z t
i2
1
1h
(2)
′′
′
′′
′
F =
dt hδω(t )δω(t )i − idtdt′ hδω(t′ )i220
Z t0 Z t0dt′′ hδω(t′ )δω(t′′ )i
=
dt′

F

=i

0

0

(B.6)

115
Thus our expression for F in the cumulant expansion truncated to second orderis
Z t Z t
h Z t
i
′
′
dt′
dt hδω(t )i +F (t) = µ exp idt′′ hδω(t′ )δω(t′′ )i
0

0

(B.7)

0

These steps are performed similarly for a four-point correlation function.
B.3

Response Functions
All of the relevant diagrams contributing to the response of a three-levelsystem in 3-pulse TFWM are shown in Fig. B.1. The response functions for timeordering (k2 , −k1 , k3 ) in terms of the four-point correlation functions defined in
Eqs. 2.63 and 2.64 and the lineshape function g of Eq. 2.74 are(ma, mm, ma, aa) 2 R2m (τ1 , τ2 , τ3 ) = 2 F (τ3 , τ2 , τ1 , 0)e−iωma (τ1 −τ2 +τ3 ) ,(ma, aa, ma, aa)

2(na, aa, ma, aa)

3R3m (τ1 , τ2 , τ3 ) = 2 F (τ1 , τ2 , τ3 , 0)e−iωma (τ1 −τ2 +τ3 ) ,(B.8)(ma, mn, ma, aa) 3 R2m (τ1 , τ2 , τ3 ) = 3 F (τ3 , τ2 , τ1 , 0)e−iωma (τ1 −τ2 +τ3 )−iωmn (τ2 −τ3 ) ,R3n (τ1 , τ2 , τ3 ) = 3 F (τ1 , τ2 , τ3 , 0)e−iωna (τ1 −τ2 )−iωma (τ3 )116
The response functions in terms of the lineshape function g are
22

3

R2m (τ1 , τ2 , τ3 ) = µ4am e−iωma (τ1 −τ2 +τ3 )n1
−gam (τ3 − τ2 ) + gam (τ3 − τ1 ) − gam (τ2 − τ1 )
× exp
2
o
−gam (τ3 ) + gam (τ2 ) − gam (τ1 ) ,R3m (τ1 , τ2 , τ3 ) = µ4am e−iωma (τ1 −τ2 +τ3 )n1
−gam (τ1 − τ2 ) + gam (τ1 − τ3 ) − gam (τ2 − τ3 )
× exp
2
o
−gam (τ1 ) + gam (τ2 ) − gam (τ3 ) ,R2n (τ1 , τ2 , τ3 ) = µ2am µ2an e−iωma (τ1 −τ2 +τ3 )−iωmn (τ2 −τ3 )n 1
× exp − 2gam (τ3 − τ2 ) + 2gan (τ1 )2−gam (τ3 − τ1 ) + gam (τ3 ) + gam (τ2 − τ1 ) − gam (τ2 )−gan (τ2 − τ1 ) + gan (τ3 ) + gan (τ2 − τ1 ) − gan (τ2 )3o+gmn (τ3 − τ1 ) − gmn (τ3 ) − gmn (τ2 − τ1 ) + gnm (τ2 ) ,R3m (τ1 , τ2 , τ3 ) = µ2am µ2an e−iωna (τ1 −τ2 )−iωma (τ3 )n 1
× exp − 2gam (τ1 − τ2 ) + 2gan (τ3 )2−gam (τ1 − τ3 ) + gam (τ1 ) + gam (τ2 − τ3 ) − gam (τ2 )−gan (τ1 − τ3 ) + gan (τ1 ) + gan (τ2 − τ3 ) − gan (τ2 )o+gnm (τ1 − τ3 ) − gnm (τ1 ) − gnm (τ2 − τ3 ) + gnm (τ2 ) ,with m, n = b, c; m 6= n.

Appendix C
Some Properties of PotassiumThis appendix lists some properties of potassium relevant to the experiments
and simulations. All frequencies are in inverse time (no 2π) unless otherwise
stated.
D1 lineD2 line769.89645 nm

766.48991 nm

389.393 THz

391.124 THz

1.6104 eV

1.61756 eV

12, 985 cm−1

13, 043 cm−1

2446.63 THz

2457.5 THz

Oscillator strength fik

0.339463548

0.6817342151

Stat. weight, lower lvl gi

2

2

Stat. weight, upper lvl gk

2

4

0.382

0.387

Physical Quantity
Lines
Resonance1

Angular Frequency ω0Strengths2

Transition prob. Aki

1
2

from A Physicist’s Desk Reference, Ed. H. L. Anderson, AIP Press, Springer (1989).
ibid.

118
Physical Quantity

D1 lineD2 lineLinewidths3
Natural0.0755279 × 10−3 nm 0.0758408 × 10−3 nm
38.2 MHz38.7 MHz

0.157982 µeV0.16005 µeVNatural lifetime τk

26.2 ns25.8 ns

Doppler (700 ◦ C)

1.37497 × 10−3 nm

1.36888 × 10−3 nm

0.695423 GHz

0.698513 GHz

2.87604 µeV

2.88882 µeV1.61786 nm

2.26709 nm

0.818269 THz

1.15685 THz

3.38409 meV

4.78435 meV

1.61927 nm

2.2685 nm

0.818984 THz

1.15757 THz

3.38704 meV

4.78732 meV

1/γtot1.22103 ps

0.86388 ps

Exptl Linewidths4 (700 ◦ C)

1.88622 nm

2.32422 nm

0.954 THz

1.186 THz

3.96294 meV

4.88317 meVCollisional shift coefficient β−0.05 m3 /ps−0.03 m3 /ps

Collisional shift (700 ◦ C)

0.0583112 nm

0.0346778 nm−29.4923 GHz−17.6954 GHz

Resonance (700 ◦ C)

Total (700 ◦ C)

Shifts5

3

ibid.
from Thesis of J. Shacklette, University of Colorado (2001).
5
from A Physicist’s Desk Reference, Ed. H. L. Anderson, AIP Press, Springer (1989).
4119
Physical Quantity

Lorentz-Lorenz Shift (700 ◦ C)

Shifted Resonance (700 ◦ C)

D1 lineD2 line−0.12197 meV−0.0731822 meV

0.08583 nm

0.170092 nm−43.4105 GHz−86.7943 GHz−0.179531 meV−0.358952 meV

770.041 nm

766.695 nm

389.32 THz

391.019 THz

1.6101 eV

1.61713 eV

Other Quantities
Level splitting

3.40654 nm
1.73059 THz
7.15717 meVShifted Level splitting (700 ◦ C)

3.34591 nm
1.69901 THz
7.02653 meVAtomic Mass
Thermodynamic properties6

6
7

from [77]at 760 mm Hg

39.0983 amu
melting 63.4◦ C

boiling 759.2◦ C7

Appendix D
Experimental Details

In this appendix the details of the pulse picker and other experimental components are given.
D.1

The Pulse Picker
The pulse picker is based on the use of acousto-optic modulators (AOMs). In

an AOM, acoustic waves create a strain-induced change in the index of refraction,
resulting in optical beam diffraction. By modulating the acoustic waves with an
RF signal it is possible to divide down the number of pulses in a pulse train
by refracting some pulses and transmitting others. The control is enabled by a
NIDAQ 6602 Counter/Timer on the computer. The pulse picker electronic design
is shown in Fig. D.1.

RF spectrum

bandpass
215 MHZ

mixer output
phase
shifter

from
laser

pulse
broadener
break-out box

timer/
counterpulse
shortener

R

L
I

Figure D.1: Schematic of pulse picker.

AOMs

121
Here we provide a brief overview of the components and appropriate inputs
at various stages. We start with the photodiode-detected pulses from the laser,which are divided into two arms, here referred to as “First” and “Second.”
First:(1) take to tunable bandpass filter (remember to impedance match at input)
and dial to ∼ 211 MHz RF frequency (use RF spectrum analyzer – remember to impedance match again). Note the filter is a linear device, so
it doesn’t matter which port is chosen for input/output.
(2) because the phase shifter cannot handle powers above 0 dBm, send the
211 MHz frequency to the phase shifter first(3) then amplify the 211 MHz carrier with one ZAD-1 amplifier to about 1 V
on the scope(4) send it to the R port on the mixer
Second:(1) take to pulse broadener and then to computer’s source pin (PFI39)(2) take from computer output and plug into input of pulse shortener; inside
are three chips:• a comparator (makes analog signal into digital) and• two inverters (shifts pulse and “AND”’s it with unshifted pulse to
create short pulse)• note the −5 V is used to remove any DC from the signal, and isconnected directly to the output; technically this is not the best way
to do it, but because the signal is so fast and short making a summer
with the AD8561 op amp would be more complicated122(3) take output from pulse shortener and send it to the mixer’s I port; notethe mixer is somewhat leaky, so make sure there is not too much of the
211 MHz leaking through (check by unplugging the modulating pulse andlooking at the L port on the scope); attenuate as needed(4) send about 1 V to the big amp from the mixer and then to the AOM; afinal output of about 5 V results in a good first-order diffraction; if more
power is needed, attenuate less at the big amp input
Typical values for input/output at various important junctions:• output from laser on scope: ∼ 140 mV direct from laser, ∼ 85 mV after
3-junction• output from bandpass filter: ∼ 10 mV on scope and into phase shifter• values going into / out of mixer:∗ R: 300 mV (after phase shifter and one amplifier ZFL-500LN)∗ I: 300 mV (attenuated by 10 dB after pulse shortener; is 1 V before
10 dB)∗ L: output ∼ 160 mV peak-to-peak (attenuated by 3dB before entering large amplifier; originally 225 mV peak-to-peak)• input to large amps: ∼ 160 mV peak-to-peak• output to AOMs: ∼ 9 V peak-to-peak (DC is ∼ 500 mV)• AOM power (% of input) with unmodulated input: input ∼ 136 mW,
output ∼ 13 mW; ∼ 9.5%.
• AOM power with modulated input: for 1/14, 210µW; for 1/21, 177µW

123
For second AOM, lower input/output of pulse shortener:• values going into / out of mixer:∗ R: 400 mV (after phase shifter and one amplifier ZFL-500LN)∗ I: 800 mV (attenuated by 10 dB after pulse shortener; is 1.6 V before
10 dB)∗ L: output ∼ 200 mV peak-to-peak (attenuated by 3dB before entering large amplifier; originally 300 mV peak-to-peak)• input to large amps: ∼ 200 mV peak-to-peak• output to AOMs: ∼ 11 V peak-to-peak (DC is ∼ 100 mV)• AOM power (% of input) with unmodulated input: input ∼ 125 mW,
output ∼ 20 mW; ∼ 16%.
• AOM power with modulated input: for 1/14, 445µW; for 1/21, 405µW
D.2

Experimental Components
In this section a detailed description of experimental components used forthe most general time-resolved three-pulse experimental setup is given.
The laser used for all of the experiments of this thesis is a home-built
extended-cavity Kerr-lens mode-locked Ti:sapph pumped by a green laser. The
pump laser is a Spectra Physics Millenia V diode-pumped CW laser (neodymium
yttrium vanadate, NdYVO4 , gain medium) centered at 532 nm with a maximum
power output of 5 W. The Millenia is usually set in power mode at 4 W for theexperiments to extend the lifetime of the diodes, which were replaced in April of2005. With this power the average modelocked Ti:Sapph output is approximately
380 mW. The Millenia laser is cooled by a Neslab RTE-III Chiller and controlled124
via the Millennia V control module. The T40 power supply is connected to a universal power supply to prevent damage from power outages, which had previously
led to diode failure.
The Ti:sapph laser has an extended cavity with curved telescope mirrors
from Newport (Part#10BV200UF.20, 25.4 mm dia, 2000 mm R, R > 99%, 0-20
degrees, 710-890 nm). Figure D.2 is a diagram of the components with approximate distances. The cavity mirors have a ∼ 2-inch focal length. The Ti:sapphirelaser rod is from Crystal Systems (3 mm φ × 10mm PL, specs α514 = 1.65). The
output coupler has 20% transmission (CVI PR2-750-80-1025).
Ti:sapphire Laser
18

19

Diode-pumped cw laser

1extended cavity

17

87 cm
1m

16
14
20

21
23

22
24

15

124 cm
36 cm

56 cm

10
30 cm
98 7

6

5

4 3 2

25 cm
11

12 13

Key:
1 periscope
7 cavity mirror
13 end mirror
19 beam sampler
2 adjustable mirror 8 iris
14 flat mirror
20 glass
3 iris9 beam stopper 15 2m ROC mirror 21 CCD camera
4 lens10 prism (SF10) 16 flat mirror
22 beam splitter
5 cavity mirror
11 prism (SF10) 17 2m ROC mirror 23 optical fiber
6 Ti:sapphire crystal 12 spatial filter 18 output coupler 24 photodiode detector

Figure D.2: Schematic of the Ti:sapph laser.

Immediately downstream of the laser two prisms (SF10) are used for dispersion compensation, with approximately 90 cm between prisms for the full setup.
In addition, a 1 m focal length lens (Thorlabs LA1464-B) is placed immediately
outside the laser to keep beam size small. The beam is then split up into fourarms, two of which are focused down by 10 cm lenses (Thorlabs LA1509-B) into
AOMs (Isomet 1250C) and recollimated. The reference arm includes two 10 cm

125
lenses and a thick piece of glass to compensate for the lack of an AOM. To keep
track of the spectral width without the need for an optical spectrum analyzer
a sample of one beam is used to obtain a real-time trace of the spectrum using a slit, diffraction grating (Optometrics U.S.A., 1200 g/mm, 7500 blaze) andCCD camera. The camera image is sent to a Tektronix TDS210 Digital Realtime Oscilloscope triggered against a video synchronization generator. In general
the mirrors used are Newport 10D20ER.2 silver-coated pyrex mirrors. Most of
the beam samplers are wedged at 30’ (CVI W1-IF-1012-UV-670-1064-45P) andthe beamsplitters are also mostly wedged and vary in percentage (typically CVI
FABS-800-45P-50-IF-1012-UV and CVI FABS-850-80-PW-45P). The stages usedin the experiments are all from Newport:• Newport MTM100CC.1, motor UE404CC, 25 cm stage• Newport IMS500CCHA, motor UE511S2, 600 cm stage with a 1” diam.
retroreflector (Newport UBBR1-2I)• Newport MFN25PP, motor UE16pp, 25 mm stepper stage
The longer stage can be used for population measurements. The stages are controlled with a Newport XPS-C4 controller.
The temperature of the cell is controlled with an Omega CN8200 temperature controller with multiple ramp/soak capability. Usually only one ramp/soak
sequence is used, typically taking 2 hours to reach 700◦ C from room temperature(∼ 0.1 deg/sec). A Marshall color CCD camera and a monitor are used to checkfor spatial overlap and as a visual check for changes in the cell. A 150 W illuminator from RAM Optical Instrumentation with a tube neck was also used to viewthe cell. The autocorrelation trace is taken in a BBO crystal, with the red light
blocked from entering the detectors by blue filters (e.g. Thorlabs FGB37). To obtain the upconverted signal a BBO crystal 0.5 mm thick was used, with a half-wave

126plate beforehand rotated such that optimal phase matching is achieved (i.e. thehorizontally-polarized beams are rotated to be vertically-polarized and are thenfocused on a plane parallel to the table onto the BBO). To reduce noise sometimes
the signals were focused with 5 cm lenses onto the detectors with 50µm pinholes.
The detectors used to detect the signals were built in JILA (model CS022A2-02).
They use a silicon photodiode, 1.5 mm diameter, 300 MHz, 6 pF; the signal gainis 1 MΩ. The detected signals are sent to two Stanford Research Systems 830
DSP lock-in amplifiers, connected via GPIB to a computer. The program used tocontrol these components is written in Visual Basic. The data were analyzed in
Origin or Mathematica.
Interference filters were used for spectrally-resolved excitation experiments(CVI F03-766.5-4-0.50 (3 nm BW) and CVI F1.5-769.9-4-0.50 (1.5 nm BW)).
In addition, a 300 mm achromatic doublet lens was obtained for three-pulse experiments (Thorlabs AC508-300-B) along with a regular lens for collimation intransmission experiments (Newport KPX205AR.16 PCX BK7). In the threepulse experiments of this thesis a 250 mm achromatic lens was used (Thorlabs
AC508-250-B) along with a regular lens for transmission experiments (Newport
KPX202AR.16 PCX BK7).

Appendix E
Novel High-Temperature Alkali Vapor Cell

Over the course of the work described in this thesis, novel high-temperature
vapor cells were constructed with ever-improving design. These improvements
are relevant for enabling the spectroscopy of alkali-metal vapors at high densities,
which has been hampered by the extremely corrosive nature of these vapors atelevated temperatures. Early work used magnesium oxide windows for studies ofpotassium vapor up to 500◦ C [78]. Alkali-resistant glasses (for example Gelaniteor 1720 glass) were developed for use at temperatures as high as 500◦ C. We haveconstructed novel potassium vapor cells for use in both reflection and transmission
experiments that survive temperatures ranging from 25◦ C up to 800◦ C (densities
up to 1019 cm−3 ). The high-vacuum sapphire-to-titanium seals are achieved via
diffusion bonding, without the need for gaskets or compression rings. The vacuum
seal maintains its integrity for an average of 90 hours above 500◦ C with about 10
heating / cooling cycles.
E.1

Cell Design
The materials of titanium and sapphire were chosen based on their resis-

tance to corrosion by potassium and their similar coefficients of thermal expansion.
Grade-2 commercially pure titanium from the McMaster-Carr Supply Co. is machined for the body. Sapphire windows custom-made by Meller Optics are used

128
for optical access. They have an anti-reflection coating on one side and are wedgedin order to spatially separate the front and back reflections. The specifications forthe sapphire windows are for c-axis orientation, 9.00 mm dia. × 0.50 mm thick,0.50 degree wedge, A/R coated for maximum transmission at 768 nm on one face.
See Figs. E.1 and E.2 for diagrams of the cell components. The screws are madeof Inconel. The face of the cell is polished for bonding to the sapphire window.
0.7“
Chamfered and smoothededges to prevent scratches
during polishing

Wrench flat
for tightening nut
Cell cap

Lapped and polishedfor sapphire window
bonding

1.7“Cell body
Titanium

Knife edge
for cap seal

Figure E.1: Potassium cell components.

E.2

Bonding Procedure
We bond the sapphire to the titanium through diffusion bonding (this canbe thought of as “welding” them together: apply pressure and heat to softening
point of titanium). First the cell body is baked without the window to make suredebris are not released during bonding. The sapphire window is cleaned withmethanol and set on top of the polished end of the cell, AR coating facing out.
The window is pressed against the cell with a custom-made holder that applies

Titanium cap

Inconelscrews129

Sapphire
window

K-type thermocouples
spot welded at these
two spots
Titanium nut,
washers and cap

Figure E.2: The loaded potassium cell.

constant pressure across the outer diameter and placed inside a vacuum tube witha cylindrical current-controlled heater surrounding the tube. The cell is heatedunder vacuum at 800◦ C for 2 hours to bond the window to the cell. The ramp
rate is set to reach 800◦ C from room temperature in about 4 hours.
E.3

Loading Procedure
To load the cell with potassium, we use a custom-made attachment on avacuum system, shown in Fig. E.3. The attachment is in the shape of a “T” andhas three openings, one to the pump, one for the empty cell and one for the nut,
washers and cap that seals the cell once it is filled with potassium. It also has acold trap to prevent particles from entering the pump.
Once the attachment is connected to the pump, we place the nut, washers
and cap together into a wrench socket connected to a steel rod that allows us
to tighten the nut with the washers and cap onto the cell inside the vacuum
attachment. This is inserted into an arm of the attachment. We then load thepotassium into the cell in a glove bag which has an opening connected to the130
Bonded cell with K

Nut with end cap
and washers
To pump

Cold trap to prevent
particles from entering pump

Figure E.3: Vacuum attachment for loading the cell.

other arm of the attachment. We fill the cell under an argon atmosphere toprevent the potassium from oxidizing and use a simple stainless steel scraper toplace an amount of potassium corresponding to about a 2 mm diameter sphere
into the back end of the cell. Once filled, we connect the cell, open side in, to its
end of the attachment. Then we evacuate the cell to about 7 × 10−7 Torr using
a turbo pump. At this point we screw on the nut to seal the cell while undervacuum until a torque meter reads about 60 inch-lbs and then close the valve tothe turbo pump, vent to atmosphere and take out the assembled cell.
Thermocouples from Omega Engineering are then spot-welded onto the titanium both at the front of the cell by the window and at the back of the cell on
the cap. We slip an oven custom-made for the cell onto the back end of the celland bind the leads of the oven and thermocouples together with a stainless steel
wire. The oven’s body is made out of Sauerisen. Six leads come out of the oven,which fit into the holes drilled parallel to the cell body. The oven is designed tokeep the temperature at the window a few degrees higher than the temperature
at the nut in order to keep the potassium from pooling at the window. Nickel

131

Figure E.4: Heating components attached to the potassium cell. Top: thermocouples are attached to the front and back of the cell, Middle: the oven leads fit
into holes drilled parallel to the cell, Bottom: the cell is wrapped in nickel foil forheat shielding.

foil is cut and crinkled to wrap around the cell for heat shielding. Figure E.4 is aseries of photos showing these steps.
Glass cell under rough vacuum
reduces turbulence around window
Heat shielding (Nickel foil)

Front view
Thermocouples
and heater wires

Figure E.5: Potassium vapor cell inside secondary glass cell to reduce turbulence
and for heat shielding.

The wrapped cell is then put inside a secondary glass cell to reduce tur-

132
bulence around the window and also for heat shielding (see Fig. E.5). The glass
cell is in the shape of a hollow cylinder with length ∼8.5” and diameter ∼1.25”.
One end of the glass cell is closed at an angle to prevent back-reflections and theother end is open and can be sealed with a metal plug that is removable. The cellis placed so that the sapphire window faces the closed end of the glass cell. The
wires from the oven and thermocouples are connected through the glass cell’s plug
for attachment to the temperature controller. The glass cell is pumped down tovacuum and the potassium cell is heated up to 300◦ C to allow the potassium topool at the back of the cell. Figure E.6 shows the various pieces of the cell lined
up for assembly and Fig. E.7 shows an assembled cell.
Sapphire
Inconel window
screwsPotassium Ti end cap

Ti nutTi washers
Ti capTi body

Sauerisen
heaterFigure E.6: Pieces of the vapor cell lined up for assembly.

Figure E.7: An assembled vapor cell.

If the process goes well, the cell can last up to 130 hours above 500◦ C.
The cells eventually fail at the titanium-sapphire interface. When the seal fails,
a yellow film forms on the outside of the cell. Figure E.8 shows a cell whosetitanium-sapphire seal failed. The lifetime (averaged over four cells) is about 90

133hours (ranging from 85 to 108 hours) with about 10 heating/cooling cycles.

Figure E.8: A used cell.

E.4

Previous Designs
Here we present a brief record of past designs and how they were improved.
The “first generation” of vapor cells had bodies made of poly-crystalline

alumina. Potassium was loaded in liquid form with a pipette into a molybdenum
cavity. The sapphire window was bonded to the polycrystalline alumina with ahigh-temperature frit. These cells were more difficult to load and did not last as
long as those of the current design.
The “second generation” of vapor cells had bodies made of titanium andsapphire windows like the current generation, but to assemble the cell, we set
the window on top of the titanium cell, AR coating facing out, and placed three
thin titanium washers on top of the window, followed by a cap and spring, andtightened it down with six screws. The cell was heated in this configuration forbonding. Using this method the rate of success was not as good as with thecurrent method in which in the bonding process we apply pressure more carefully

134with a custom-made holder. The current method also does not require the use of
the washers or pressure from the cap, making the process simpler. We also tried
depositing a titanium ring onto the sapphire before attaching the window to thecell; this method did not make for better results.
E.5

Transmission Cell
Potassium has an absorption length on the order of a wavelength at thedensities attained in these experiments. We made some progress towards a transmission cell thin enough for experiments at the lower densities. These cells would
enable transmission experiments such as 2D Fourier transform spectroscopy, which
would shed light on the coupling between levels and many-body effects. They
would also allow more precise studies of propagation and local field effects. Figure E.9 is a schematic of the manufacturing process and Fig. E.10 is a photo ofa thin cell. Initial success was fortuitous in that there was a leak in the holding
packet, then made of stainless steel, just small enough to retain potassium during
bonding and allow its release for experiments. A later attempt using silver as theholding packet successfully retained the potassium and released it as temperaturesapproached the melting point of silver, which is approximately 962◦ C, and seems
a more reliable route. The primary remaining technical problem is how to keep
the potassium from seeping into the thinnest parts of the cell, where it tends toremain due to capillary action. Figure E.11 shows transmitted TFWM data for a
14µm-thick cell, with dynamic beats due to propagation.

135

(a)

(b)

(c)

(d)

(e)

Figure E.9: Schematic of thin cell manufacture. (a) Two sapphire windows areused. (b) First a pocket is drilled into one of the sapphire windows. (c) Thena thin silver packet is filled with potassium and placed into the pocket. (d) A
6 µm-thin titanium washer is placed between the windows, which are then (e)sandwiched together and heated up to 800◦ C for bonding.

Figure E.10: Photo of thin cell.

136

Time-Resolved Signal
t = 0.12ps, T = 1.2ps

Time-Resolved Signal

0.1051026

RefA3_B

0.01

1E-3

1E-4

1E-5
-5 0

5 10 15 20 25 30 35 40 45 50 55 60 65
t, delay between signal and reference (ps)Figure E.11: Transmitted TFWM signal through a 14 µm-thick cell showing dynamic beats.

Appendix F
The Exciton Hamiltonian for a System of Two 2-Level Atoms

Here we derive the fluctuating exciton Hamiltonian for a system of two
2-level atoms. The fluctuating exciton Hamiltonian is
m6=n

HS =

X

†
Ωm B̂m
B̂m +

m

X

†
†
(Jmn B̂mB̂n + Kmn B̂mB̂n† B̂m B̂n ),

(F.1)

m,n

where
†
= |m1ihm0|
B̂m

(F.2)

B̂m = |m0ihm1|.

(F.3)

and

For 2 atoms (N = 2) we have:
Atom basis→

atoms m, states a|mai|mai

exciton basis:states em→|em em i→|00i

1 ground state:|10i|20i

N = 2 one-exciton states:|10i|21i→|01i|11i|20i→|10i

N (N − 1)/2 = 1 two-exciton states:|11i|21i

→

|11i.

138
We now write out the Hamiltonian explicitly:
HS |00i = 0
HS |01i = Ω2 |01i + J12 |10i

(F.4)

HS |10i = Ω1 |10i + J21 |01i
HS |11i = Ω1 |11i + K12 |11i + K21 |11i




0 0 0
0




 0 Ω J12

0


atom
HS
=



0
 0 J12 Ω
0 0 0 2Ω + 2K12

(F.5)

Now we diagonalize:
Solving to find the eigenvalues gives2(Ω − λ)2 − J12=02λ2 − 2Ωλ + Ω2 − J12=0q2) = Ω ± |J12 |→ λ = Ω ± 1/2 4Ω2 − 4(Ω2 − J12

The eigenvectors then can be found using  Ω J12 a = (Ω ± J12 )   .
b
J12 Ω

(F.6)

The equationΩa + J12 b = (Ω + J12 )aJ12 a + Ωb = (Ω + J12 )b(F.7)

gives b = a, and the equationΩa + J12 b = (Ω − J12 )aJ12 a + Ωb = (Ω − J12 )b

(F.8)

139
gives b = −a. Thus the eigenvectors are

 




 1

1 1  1 √  , √ 
2 −1 
 2 1


(F.9)

To find the polarization, we use the relation ĤSF = −P̂ · E(t), where P̂ =X
m†µm (B̂m+ B̂m ).

The eigenvectors written in the original space are then      




1 
1 1
10
 1   1 
|αi = √   = √  0 1     + 1 0    
2 1
2
1
1
1
0
1
= √ (|01i + |10i),2

and similarly,1|βi = √ (|10i − |01i)
2(F.10)

So now we can write the polarization:P̂ |00i = µ1 |10i + µ2 |01i
P̂ |01i = µ1 |11i + µ2 |00i
P̂ |10i = µ2 |11i + µ1 |00i
P̂ |11i = µ1 |01i + µ2 |10iP̂ |αi = √12 [µ1 |11i + µ2 |00i + µ2 |11i + µ1 |00i](F.11)

P̂ |βi = √12 [µ2 |11i + µ1 |00i − µ1 |11i − µ2 |00i]µ1 = µ2 = µ →√P̂ |αi = 2µ(|11i + |00i)P̂ |βi = 0




0µµ0




µ 0 0 µ
P atom = µ 0 0 µ
0µµ0(F.12)

140
The polarization in the exciton basis is (the columns and rows correspondto {|00i, |αi, |βi, |11i})√

2µ 0 00√√  2µ 0 0 2µ P exciton = .

0 0 0  0√0
2µ 0 0(F.13)

Appendix G
The GROMACS Package

This appendix gives directions on how to perform MD simulations using the
package GROMACS. To perform a molecular dynamics simulation (commands
are given in UNIX):• modify grompp.mdp for T, P: change dt, nsteps, ref t, ref p, gen temp
e.g. type emacs grompp.mdp &• modify topol.top for number of atoms• modify conf.gro for box size. To simplify this step, run input.c with gcc
input.c and then ./a.out, which will create a conf.gro file with random
atom positions• ensure there are not over ten or so trajectory files (traj.) in the output
folder, otherwise GROMACS won’t overwrite. That is, delete traj.trr,.xtc files.
• type grompp -v• type mdrun -v• type gmxdump -f > traj.dat to convert the output to data files

142• type scp traj*.dat username@jilau1:/. to secure shell copy to a moreaccessible location
To repeat this multiple times, run the program run.c with gcc run.c andthen ./a.out, which will ask you what temperature the run is for and will printout the input you would need for the number of runs you would like. Be sure totype gcc input.c after run.c and before inputing the lines from run.c.
To run interactively, type the command qsub -IV and be sure to exit when
done. If the program reads/writes quite a bit, cd to /data. To allow opening of
windows, type ssh -X jilau1.

Appendix H
Constants and Conversions

This appendix contains some handy constants and conversions related tothe calculations performed in this thesis.
• Wavelength (m) to frequency (Hz):λToν(λ) = λc
• Frequency (Hz) to wavelength (m):νToλ(ν) = λc
• Wavelength difference (m) to frequency difference (Hz):δλToδν(λ, δλ) = λc2 δλ• Frequency difference (Hz) to wavelength difference (m):δνToδλ(ν, δν) = νc2 δν• Frequency (Hz) to wavenumbers (cm−1 ):νToinvcm(ν) = 100ν c• Wavenumbers (cm−1 ) to frequency (Hz):invcmToν(invcm) = 100 · c · invcm• Temperature (K) to wavenumbers (cm−1 ):KToinvcm(K) = 100kBh c K

144• Wavenumbers (cm−1 ) to Temperature (K):incmToK(invcm) = 100 h kcB invcm• Joules (J) to wavenumbers (cm−1 ):Jtoinvcm(J) = 100Jh c• Electron volts (eV) to wavenumbers (cm−1 ):e·eVeVtoinvcm(eV) = 100
h c• Wavenumbers (cm−1 ) to electron volts (eV):invcmToeV(invcm) = 100 h c invcm/e• Hartree to wavenumbers (cm−1 ):h HartHartreeToInvcm(Hart) = E100
h c

• Wavenumbers (cm−1 ) to kcal/mole:
invcmTokCpM(invcm) = 0.0028591 · invcm• kcal/mole to wavenumbers (cm−1 ):kCpMToinvcm(kCpM) = kCpM/0.0028591

145

Table H.1: Constants
Physical Quantity

MKSA Units

CGS “Gaussian”

Speed of light (vacuum) c

2.99792458 × 108 (m/s)

2.99792458 × 1010 (cm/s)

Electron charge e

1.60217653 × 10−19 (C)4.803 × 10−10 (esu)Boltzmann constant kB

1.3806503 × 10−23 (J/K)

Electron mass m

9.1093826 × 10−31 (kg)Permittivity free spce ǫ09.1093826 × 10−28 (g)8.854187817 × 10−12 (F/m)

Planck’s constant h

6.6260693 × 10−34 (Js)Classical electron radius r

r = e2 /(4πǫ0 mc2 ) (m)Avogadro’s number NA

6.0221415 × 1023 (1/mol)r = e2cgs /(mcgs c2cgs ) (cm)