DYNAMIC PHASE AND POPULATION CONTROL OF STATE SELECTEDWAVE PACKETS IN Li2

byJoshua B. Ballard

University of Colorado
Boulder, Colorado
2003

Dynamic Phase and Population Control of State Selected Wave Packets in Li2

By
Joshua B. Ballard
A.B., Harvard University, 1995

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree of Doctor of Philosophy
Department of Chemistry and Biochemistry2003

This thesis for the Doctor of Philosophy degree by
Joshua B. Ballard
has been approved for the
Department of Chemistry and Biochemistryby_______________________________________________
Stephen R. Leone_______________________________________________David Jonas

Date_________________________________

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

Ballard, Joshua (Ph.D., Physical Chemistry)
Dynamic Phase and Population Control of State Selected Wave Packets in Li2
Thesis directed by Professor Stephen R. Leone

Pulse shaping of ultrafast pulses with a Liquid Crystal Spatial Light
Modulator (SLM) is used to control both transient and non-transient state-resolved
wave packet dynamics in Li2. In almost all of the experiments, a single launch state(generally A1Σu+ vA=11, JA=28) is prepared via excitation with a cw laser, from whicha pump pulse excites a superposition of states on an excited electronic potential
energy curve followed by a photoionizing ultrafast probe pulse. Using feedback andan Evolutionary Algorithm (EA), the weak field pump-probe photoionization signal
at a single time delay is optimized in Li2 for the state E 1Σg+ (vE=9, JE=27 & 29). First
order time dependent perturbation theory is used to explain the mechanism by which
the photoionization is maximized. Following this, the transient dynamics of
excitation of wave packets is studied in detail. A clear separation is made between
resonant and nonresonant effects. Both population and resultant phase in themolecule are transiently manipulated. By varying the polarization of the probe light,
population dynamics can be separated from interfering wave packet dynamics,
allowing precise determination of the instantaneous population and wave packet
dynamics. A pulse shaping scheme is described that implements a sign inversion for
one state of a two state superposition, and all sign inversion matrix elements arequantified. Elements of strong field coherent control are also explored in Li2. Fromthe launch state, the strong optical field couples the A and E electronic states,iii

inducing sequential ∆J=±1 transitions to populate states up to ∆J=±4. Taking
advantage of Rapid Adiabatic Passage, state selectivity is controlled by manipulating
chirp parameters on the excitation pulse, achieving selectivity of either Stokes or antiStokes quantum beats of nearly unity. Finally, wave packet dynamics on highly
excited electronic states is examined. Electronic wave packets consisting of beating
between bound states on the F1Σg+ and G1Πg electronic states are observed.

iv

“I thought of that while riding my bike.”-Albert Einstein on the theory of relativity

Acknowledgements
To fully describe the degree to which I feel indebted to my wife for her
support in this endeavor of studying graduate chemistry, a little background is inorder. By 1997, after my wife and I had lived in San Francisco for a few years, we
had a very comfortable existence. The internet and multimedia industries were
blooming in The City, and jobs were plentiful and high paying. Even though I wasenjoying a well paying position in high-tech, I had time to direct and produce my own
movies for personal expression while my wife Rebecca and I were living in the mostquaint neighborhood just two blocks from Golden Gate Park. From our front door,
we could walk to any of sixty restaurants within five minutes. Even in this very
urban environment, our house was surrounded by gardens, and we would wake to thesound of birds chirping without the usual din expected of the city. It was quite
peaceful spending the morning in our mature garden sipping coffee and collecting ourthoughts for the day.
On one particular trip home to visit the parents, I had a conversation with myfather-in-law Garth Gillan, who at the time was a Professor of philosophy at the local
university. I was telling him about how cool a life we were living, and he asked me if
I felt like I was getting anything out of it, or if I was just whoring myself out to thehighest bidder. Try to guess my answer…. Shortly after that, I quit my job to study
chemistry. Within three months, I was taking some undergraduate chemistry courses,
and we had moved to one of the roughest neighborhoods in San Francisco. It was
typical to have to wade through a group of thugs clad in ski-masks to get up the stoop
to our front door (actually, we’d just go to our back door). Now, instead of beingvi

surrounded by birds chirping and gardens, we were in the midst of what would beconsidered a war zone anywhere but in America. One night, as Rebecca and I wereon our way to BART, we passed by a bleeding stabbing victim. There were police
there, so we kept on walking. One block later, we passed a gunshot victim. Hey, I
wanted to go to graduate school. Rebecca knew that I needed to study to get tograduate school, so she endured this extreme change in situation—for me.
It is especially comforting to know that I have someone like Rebecca behindme. As might be expected, her support has not wavered since entering graduate
school.
One particularly large series of events in graduate school revolved aroundrelocating our lives and labs back to California. It would have been impossible todeal with if not for the collective impetus of everybody in the group. Laurie and Jodi
were especially nice to have around. We started graduate school together and spentcountless hours in the same room in lab. Seeing them in California is an important
constant, helping make the personal adjustments bearable.
In lab, it has been a pleasure to share so many conversations with Hans
Stauffer. He tends to get worked up when talking about science, and that can be very
motivating during the long haul of graduate school. On a personal level, it has beenvery nice to have Hans and the rest of his family around for so many important
events.
Through my five years of graduate school, there have been many others who
have helped me develop or endure. First of all, my advisor Steve has been quite asource of motivation, that is in addition to taking all of the responsibilities of the rolevii

of advisor that needs not be explained to anybody reading this text. It seems likealmost every conversation I had with him resulted in me wanting to do some of thebest science ever. There were several times I would go into a meeting with him
feeling like Dave Stoler [from Breaking Away] working in his father’s used car lot,
and leave feeling like Dave on the last lap of the Little 500.
I have to give a shout out to JILA, where it is very easy to do science. There
are so many people there whose main goal is to make the pursuit of knowledge
seamless. I particularly enjoyed participating in OSEP, where I got to follow the
progress of people in so many different groups. I can hardly imagine getting throughgrad school with nearly the perspective I have if it were not for everybody involved in
OSEP. It has truly been an enriching experience to be involved in such aninterdisciplinary group.
There are many others that deserve acknowledgement from me. We have tokeep perspective though: this isn’t the Oscars.

viii

Table of Contents
Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2

Experimetal Setup and Overview . . . . . . . . . . . . . . . . . . . . . 8

2.1

Experimental Apparatus and Techniques . . . . . . . . . . . . . . . . 8
2.1.1 Pulse Shaper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 Heat Pipe/Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2

Wavelength Subtraction Spectroscopy . . . . . . . . . . . . . . . . . . 17

Chapter 3

Evolutionary Algorithm . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 23

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 23

3.2

Algorithm Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3

Convergence Speed and Schema Size . . . . . . . . . . . . . . . . . . . 29

3.4

Pulse compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5

Optimizing Launched Wave Packet . . . . . . . . . . . . . . . . . . . . .40

3.6

Optimizing Transient Wave Packet . . . . . . . . . . . . . . . . . . . . . 50

3.7

Modeled Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.8

t=0.0 ps Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Chapter 4

Weak Field Population Control . . . . . . . . . . . . . . . . . . . . . . 66

ix

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Chapter 5

Dynamic Phase Control . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 89

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2

Theory . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.3

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.4

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Chapter 6

Strong Field Population Control:
Theory and Experiment. . . . . . . . . . . . . . . . . . . . . . . 110

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.2

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.2.1 Spatially Uniform Fields . . . . . . . . . . . . . . . . . . . . . . . 115
6.2.2 Alignment effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2.3 Spatially Inhomogenous Fields . . . . . . . . . . . . . . . . . . 130

6.3

Experimental Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

xChapter 7

Electronic Wave Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.2

Experimental and Theoretical Overview . . . . . . . . . . . . . . . . . 151

7.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.4

Discussion and Future Directions . . . . . . . . . . . . . . . . . . . . . . 164

7.5

Ultrafast transient behavior in Li2 with 680 nm excitation. . . . 167
7.5.1 Experiment and Results. . . . . . . . . . . . . . . . . . . . . . . . 169
7.5.2 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Chapter 8

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

xi

List of Figures

FigurePage

2.1

Typically excited potential energy curves of Li2 . . . . . . . . . . . . . . . 9

2.2

Schematic Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3

Cross correlation pulse profiles for pulse trains with various
interpulse delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4

Multi-color pulse train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5

Schematic spectra as used in Wavelength Subtraction
Spectroscopy (WSS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6

Comparison of excited state populations for a weak, phase shaped
input using narrow bandwidth excitation and WSS. . . . . . . . . . . . . . 21

3.1

Evolutionary algorithm flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2

Calculated pulse compression dynamics using an evolutionary
algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3

Optimized fitness (F) of a two part process showing double exponential
behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35

3.4

Optimized cross correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.5

WSS ({) spectrum with the probe set to t>5.0 ps . . . . . . . . . . . . . . 41

3.6

Two state wave packet signals for different optimization
conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.7Long time delay optimization convergence . . . . . . . . . . . . . . . . . . . 44

3.8

Multiple state wave packet optimization . . . . . . . . . . . . . . . . . . . . . 47

3.9

Short time delay optimization results . . . . . . . . . . . . . . . . . . . . . . . . 51

xii

3.10

Evolution of the fitness versus generation number for the 1.6 ps time delay
optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.11

Model optimization results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.12

Phase mask ( ) found by EA when optimizing photoionization at
t=0.0 ps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.13

Optimized photoionization with cross correlations . . . . . . . . . . . . . 63

4.1

Pump-probe traces at parallel (S) and magic angle (solid line) pump-probe
polarizations, along with the difference (U) between the traces for unshaped
pump pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2

Phase mask classes used in this chapter . . . . . . . . . . . . . . . . . . . . . . 76

4.3

Chirp-type pump spectrum transient . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.4

Phase-jump spectrum transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.5

Comparison of control capabilities using the phase-jump spectrum versus the
chirp-type spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.6

Comparison of model results with experiment . . . . . . . . . . . . . . . . . 86

5.1

States involved in excitation scheme . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2

Phase shaped population transfer traces . . . . . . . . . . . . . . . . . . . . . . 96

5.3

Resonant/nonresonant interaction scheme for inducing sign
inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.4

Phase evolution using the same phase mask for each resonance . . . 100

5.5

Sign inversion wave packet data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.6

Sign inversion wave packet data without time-shifting the resonant
contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.7

Wavelength Subtraction Spectroscopy traces for shaped andunshaped pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

xiii

6.1

Summary of resonant frequency positions . . . . . . . . . . . . . . . . . . . . 116

6.2

Dissipation of wavefunction over various J states for mJ=0. . . . . . . 119

6.3

Calculated WSS spectrum as a function of pulse normalized field strength fora two level system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.4

Evolution of J=28 population for various mJ states under excitation with alaser field with NFS=2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.5

Effects of alignment on total population in J=28 state. . . . . . . . . . . 127

6.6

Degeneracy dependent selectivity for high J vs low J. . . . . . . . . . . . 129

6.7

Rabi oscillation visibility for various probe sizes . . . . . . . . . . . . . . . 133

6.8

Calculated populations as a function of normalized field strength for
positively chirped pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.9

Fourier Transforms for quantum beats at two different peak fieldstrengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.10

Field strength dependence of quantum beat amplitudes . . . . . . . . . . 142

6.11

Phase and amplitude mask used to induce sequential Rapid Adiabatic
Passage events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.12

Implementation of Raman state selectivity based on sign of chirp. . 145

7.1

Potential energy curves in excitation region . . . . . . . . . . . . . . . . . . . 153

7.2

State locations accessible from the vA, JA=11, 28 launch state with a central
pump energy of 17,550 cm-1 to 18,100 cm-1. . . . . . . . . . . . . . . . . . . 155

7.3

Pump-probe signal for a pump energy of 17,850 cm-1 for a launch
state of vA, JA=11, 28. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.4

Fourier Transform quantum beat spectra for varying
pump energies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.5

Quantum beat energy as a function of pump pulse energy for 11,28 launch
state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
xiv

7.6

Quantum beat frequencies as a function of central pump energy from 11,16
launch state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.7

Pump-probe signal for Epu=14,700 cm-1 . . . . . . . . . . . . . . . . . . . . . . 169

7.8

Dependence of transient peak height on central
pump energy and JA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.9

Polarization dependence of peak photoionization for parallel,
perpendicular, and magic angle pumps . . . . . . . . . . . . . . . . . . . . . . . 172

7.10

Fluorescence spectrum with excitation by 606.955 nm cw laser . . . 176

xv

List of Tables

Page

Table
6.1

∆J=2 quantum beat frequencies for J=24 through 32. . . . . . . . . . . . 138

7.1

Expected quantum beat frequencies . . . . . . . . . . . . . . . . . . . . . . . . . 165

xvi

Chapter 1
Introduction

Ultrafast phenomena comprise a vast field of study in chemistry andphysics, with many types of applications, particularly the direct study of processes
happening on sub-picosecond timescales. The short timescales inherent toultrafast pulses correlate to a wide bandwidth in the frequency domain. During
excitation, especially to a first order approximation, several transitions can often
simultaneously become excited. In the case of dissipative or quasi-continuum
systems, such as a solid state semiconductor or a repulsive molecular electronic
potential, a predominant characteristic of the wide bandwidth excitation is a rapid
decay of signal[1-3]. In an atom or molecule in the gas phase, the energy levelstructure can contain narrow bandwidth resonances, implying the presence ofslowly decaying states. In the presence of resonances, the dynamics of ultrafast
excitation can drastically change from a monotonic exponential decay to series ofrecurrences, or temporal interferences[4-8]. Thus, if more than one state isexcited, a time dependent wave function, or wave packet, results. By defining awave packet as a superposition of multiple states, a continuum process—such as
molecular dissociation—represents the limit of wave packet behavior with an
infinite number of component states, but in this case a wave packet consists of alimited number of states[1, 2, 4, 9-11]. In the presence of a limited number oftransitions, an oscillatory behavior can show up in time as the time dependent
wave function returns to a configuration in state space that is favorable for1observation. This regular recurrence of the wave function to a favorable
configuration can be thought of as a re-phasing process. Once launched, thecomponent states of the wave packet accumulate variable amounts of phase as
determined by their energy and eventually they can all come into phase at somelater time. This re-phasing time depends on both the number of states involved
and their relative energy spacing, so it is not necessarily predictable.
The recurrence phenomenon of wave packets leads to the possibility of
directing the wave packet’s path as it travels through its configuration space andhas come to be realized in many forms. Known as coherent control, this study of
directing wave packet creation and propagation has seen attention in a wide rangeof systems from the relatively simple to the seemingly impossibly complex[9, 1226]. Although early hopes that coherent control would be viable as a synthetic
tool have yet to come to fruition, the process of performing coherent control canlead to a better understanding of many types of molecular and spectroscopic
processes[22]. Many examples of coherent control have been implemented in thepast, with one primary method of control revolving around manipulating thephase and amplitude of the component wave functions of a wave packet bymanipulating the excitation laser source[16, 21, 27-31]. In effect, a laser ismanipulated, and information from the laser is written onto the wave packet[32,33].
This thesis addresses several aspects of using ultrafast pulse shaping tocontrol both transient and stable wave packet dynamics. The dynamics to bestudied are state resolved, allowing very precise investigations into the spectral2

aspects of the dynamics of ultrafast excitation. Most of the studies involved inthis thesis address the creation and propagation of precisely prepared wave
packets. A wave packet is defined as a superposition of stationary state wave
functions that displays time dependent dynamics. A basic description of a wave
packet is included here, where the total wave function is a summation ofcomponent wave functions:Ψ (t ) ∝ ∑ ci (t )ψ i eiωi t ,(1.1a)iwhere n is a normalization constant, ci(t) is simply a weighting coefficient, and ψi
is a component stationary state wave function, and the exponential factor is a timedependent phase factor that depends on the wave function energy. Since anobservable consists not of a wave function but a modulus squared of the wave
function over some coordinate, the above equation needs to be rewritten to reflect
the qualitative act of observation:Ψ (t ) ∝ ∑∑ ci (t )ψ i + c j (t )ψ j + ci (t )c j (t ) cos(ωij t ) ,
2

2

i

2

(1.1b)jwhere ωij is the energy difference between states in a wave function, and theoverlap integral between the component wavefunctions along an observation
coordinate is assumed to be unity. In the above equation, the observable time
dynamics come from the cosine term representing an interference between states,
accounting for the “rephasing” processes mentioned above. In the weak field
limit the non-transient weighting coefficients are expressed asci ∝ ε (ωig )µig ,(1.1c)

3

where the excited state coefficient depends upon both the transition dipole
between the launch and excited state (µig) and the electric field strength at theresonant frequency between the launch and target state [ε(ωig)]. Note that in thepresence of an electric field, cn(t) can change but after an excitation pulse, thesefactors are effectively constant over timescales on the order of over a nanosecond.
Various permutations of Eq. (1) will be invoked several times in the text, andadditional details will be added in or expanded upon in the following chapters.
The equations above show that, qualitatively, we observe wave packets that getstronger as the excitation laser gets stronger and that contain an oscillatorycomponent in time. At the beginning of each chapter, a justification and a frameof reference will first be established, so that it will come as no surprise when aspecific detail is included.
To begin, a summary of important experimental considerations will beconsidered in chapter 2. The summary will include a short discussion of theexperimental setup, excitation scheme, and a somewhat detailed analysis of thelimitations of the pulse shaping apparatus. This pulse shaper is very similar to anumber of other pulse shapers and has been analyzed in the literature[30, 34], butsince its use plays such a large role in many of the experiments of later chapters, a
discussion is included. Another experimental technique, here referred to as
Wavelength Subtraction Spectroscopy (WSS), is examined. This techniqueconsists of looking at a signal as narrow bands within an ultrafast spectrum areattenuated. This is similar to observing an absorbance process, except

4interference effects between different wavelengths in the ultrafast pulse can be
examined with WSS.
Chapter 3 describes the use of an Evolutionary Algorithm (EA) to
optimize several types of processes in the weak field limit (i.e. peak power <1010
W/cm2 in these experiments in Li2). After a brief exposition of the routines usedby the EA, a sample pulse compression is performed. In this case, the EA
performs the same task as a pulse reconstruction algorithm in that it finds the
spectral phase of an input pulse. This is followed by an optimization of weak
field wave packet dynamics at times after the pump pulse is over, in effectaddressing issues related to resonant absorption processes within ultrafast pulses.
An optimization of transient weak field wave packets shows that during a pulse,
various nonresonant processes can be controlled, as verified by the good
agreement of a modeled optimization with experimental results.
Chapter 4 explores in more detail the limits of controlling transient
population transfer processes in the weak field. It is shown that the spectrum of
an ultrafast pulse can be divided into multiple spectral regions resulting in spectral
channels, or groups of wavelengths, that can be used for ultrafast pulse shaping.
Resonant and nonresonant effects are shown to be independently controllable,with the nonresonant effects providing the basis for controlling transient
dynamics. Even though this sounds similar to Wavelength Division Multiplexing,
it differs in that it really pushes the degree to which the control channels can beseparated, and it takes advantage of nonresonant effects in a way not previously
exploited.

5

Chapter 5 demonstrates a method for controlling the instantaneous phaseof a wave function, as observed by a change in wave packet phase. This is a caseof an application of the lessons learned in chapter 4. The observed dynamics areshown to transform a superposition of wave functions in a way that is consistent
with a quantum computational Z-gate.
One of the limitations of the wave function manipulations described in
Chapter 5 as applied to quantum computation is that the excited statesuperposition contains significant launch state character. One way around thiscoherence is to fully transfer the population out of the launch state. This isaddressed in chapter 6 as a goal of strong field coherent control (with peak laser
powers up to ~1012 W/cm2), where significant launch state population depletion ispossible. In this regime, Eqs. (1a-c) fail to describe the population dynamics, as it
is seen that both the ground and excited state populations oscillate as a function ofelectric field strength. In a two level system, this population oscillation is known
as a Rabi oscillation[5]. This chapter starts with a theoretical examination ofwave packet dynamics in the presence of both a uniform electric field and a
spatially Gaussian electric field. Finally, a chirped pulse Rapid Adiabatic Passage
scheme is proposed and implemented to transfer population through specificexcited states. Specifically, a Raman transition is excited such that the populationthat transfers through the E electronic states gets depleted to nearly zero alongwith the launch state. Stokes/anti-Stokes excitation is selected by simply
changing the sign of chirp.

6Finally, chapter 7 summarizes studies using a tunable ultrafast laser, withthe ultimate goal of identifying energetic regions favorable to studying electronic
wave packets. Chapter 7 begins with a study of wave packet dynamics between
two bound electronic states. In this low lying Rydberg state regime, it is shownthat quantum beats can be observed between individual states on the separate
curves. Additional work verifies that the observations are of primarily rotational
states, as progressions of rotational quantum beats for increasing vibrational
quanta are observed. Within this vibrational progression of rotational beats lie
additional coherences that are attributed to quantum beats between electronic
states. It is shown that the electronic beats are between states with ∆Λ=1,
suggesting a potential handle for further controlling the wave packet dynamics via
Stark shifting one state relative to the other. Chapter 7 concludes with a series ofexperiments exploring an ultrafast transient effect that is ultimately deemed to beatomic in nature. This final series of experiments is left with open questions to berevisited at a later date.

7

Chapter 2
Experimental Setup and Overview

2.1

Experimental Apparatus and techniques
In most of the experiments in this thesis, a three step excitation scheme isused to produce a time dependent photoionization signal, and hence observe timedependent wave packet dynamics as presented in the previous chapter. The
experimental setup uses a cw launch state preparation laser to excite the transition
for A 1Σu+←X 1Σg+, an ultrafast laser system to produce the pump and variably
delayed probe pulses, a pulse shaping apparatus, a heat pipe containing the Li2
sample, various detection apparatuses, and controlling software on a computer.
The laser system is pumped by an argon ion (Ar+) laser operating at ~27
W, which pumps a cw dye laser for launch state preparation, and various ultrafast
lasers, depending upon the experiment [see Fig. 2.1 for a sample excitation
scheme]. The experimental laser setup will be discussed in more detail in therespective section where data will be presented, since the setup changes
depending upon the experiment. Figure 2.2 summarizes the optical table layout,with notes where the laser setup is changed. In almost all cases, three lasersinteract to produce photoions, and pump-probe experiments are performed. It isduly noted where the excitation scheme deviates from a three laser interaction.

8

40

+ 2

+

1

+

X ( Σg )F ( Σg )

hν pr

1

20

+

E ( Σg )3-1

E (10 cm )

30

hν pu
10

1

+

A ( Σu )
hν cw

0

1

+

X ( Σg )
-10
2

4

6

R (Å)

8

10

Figure 2.1: Typically excited potential energy curves in Li2. In this scheme, thepump and probe pulses are of different colors. A cw laser excites from the X(1Σg+) electronic curve to a pure launch state on A (1Σu-). An 800 nm ultrafast
laser excites a superposition of states on the E (1Σg+) curve, which is subsequently
ionized by a time delayed 640 nm ultrafast probe pulse. At negative delays(dotted lines), the probe-pump pathway (for 640 nm probe pulse) through the F(1Σg+) state, as indicated by the dotted lines, has a much lower yield than thepositive time pathway due to unfavorable Frank-Condon overlap factors. Notethat in some experiments, as described in the appropriate chapters, the excitation
scheme will deviate from above.

9

CW
laser

Monochromator

Ar+
laser

Lock-In
Amplifier

Ultrafast
oscillator
AmplifierLock-In
Amplifier

OPA
delay line

PC

Pulse
shaperFigure 2.2: Schematic experimental setup. Sections in parentheses indicate
changeable portions of the setup. The output from the regenerative amplifier isdivided into up to two beams. One beam goes through the pulse shaper. This
beam can be either the pump or probe pulse, depending on the pulse timing, but isactively shaped only in experiments where it is acting as the pump pulse. The
second beam is either split unchanged or goes through the OPA to produce
tunable ultrafast output from 500-700 nm. The ultrafast and cw beams intersect
in the heat pipe which is equipped to detect ions and fluorescence. Focusing
optics can be placed into the beam lines to increase peak intensity.

102.1.1

Pulse shaper
Since many of the experiments described in this thesis depend exquisitelyon the manipulation of ultrafast dynamics by phase and amplitude shaping thespectrum of an ultrafast laser pulse, it is necessary to discuss the pulse shaping
setup in some detail. The pulse shaper consists of a zero dispersion, four focal
length imaging system, and uses a spatially addressable, double stack 128 pixel
SLM between crossed polarizers, in a setup similar to that described by Weiner etal.[30] and Wefers et al.[29, 34, 35] In the Fourier plane, the pulse is dispersed
linearly with respect to wavelength across the face of the SLM. Generally, thebandwidth of our pulses is dispersed across up to 40 pixels (FWHM) on the SLM.
With a pulse bandwidth of 9 nm, this results in approximately 4 cm-1 per pixel
resolution for frequency domain pulse shaping. By applying specific voltages tothe pixels, we can manipulate the phase and amplitude of each wavelength of thepulse.
With a single frequency imaged to a spot size of 170 µm, the broadband
pulses can be temporally shaped within a time envelope approximately 8 ps wide(referred to as the “pulse shaping window”), as derived previously [29] andobserved experimentally. This envelope, which will be an important experimental
constraint, is a result of both the sampling effects of the finitely sized pixels of the
SLM and the finite spot size for each frequency component. To shift a pulse intime, a linear phase is applied across the pulse according to ∆t=dϕ/dω. The
sampling aspect of the SLM gives an approximation to this formula to be11∆t=∆ϕ/∆ωpix where ∆ϕ is the phase shift per pixel, and ∆ωpix is the difference incentral frequencies imaged onto adjacent pixels. Because of wrap around ofphases, a phase shift of ∆ϕ can also be represented as ∆ϕ+2nπ, where n is an
integer. This produces replicas at ∆t=(∆ϕ+2nπ)/∆ωpix and attenuates the shifted
pulses with attenuation increasing along with ∆t. Another limitation is the finite
spot size: spatial overlap of more than one pixel by a monochromatic wavelength
causes diffraction of certain wavelengths out of the beam. This is because the ∆ϕ
between adjacent pixels introduces a certain amount of destructive interference.
We thus see some destructive interference in the direction of propagation, as lightthat is produced with the multiple phases is diffracted out of the beam to maintainmomentum matching conditions[29].
In many of the experiments here, we shape the pump pulse, and theelectric field can be written as
128 ∞E (t ) = ∑ ∫ dωi ' | ε i (ωi ' ) | Ti 2 exp[i (ωi ' t + ϕi )] ,
1

(2.1)

i =1 − ∞where the summation is over all 128 pixels across the SLM, and ε(ωi’) is theamplitude of the electric field at the central frequency ωi’ on each pixel. The
amplitude and phase controls over the pulse are included in the Ti½ and ϕi terms,
which represent the absolute value of the amplitude transmittance and applied
phase, respectively. Without modulation, Ti½=1, and ϕi=0. The εi(ωi’) term isapproximately Gaussian, to reflect the spectrum imaged onto a single pixel, and it
will be shown that this bandwidth is approximately 6 cm-1, as opposed to theseparation of central frequencies on adjacent pixels of 4 cm-1. Note that the12integrals of adjacent pixels will have significant overlap, resulting in attenuationof the field when phase shifts between adjacent pixels are applied. In typicalexperiments, excitation is in the weak field, but some experiments will examine
stronger field effects. In the experiments described in this thesis, peak electricfields range from 105-1011 W/cm2, depending upon the experiment.
As an experimental demonstration of the capabilities of the pulse shaping
apparatus, a series of phase locked pulses was generated. First, Fig. 2.3 shows theextent to which a phase locked pulse train can be created with our pulse shaping
setup. It is evident from the figure that longer time delays between pulses means
less intense pulses, as expected from the diffraction effect argument from above.
The longest time delay between pulses of about 8.5 ps demonstrates a limit ofpulseshaping. Notice that as the pulse separation increases, the pulse intensity
decreases; diffraction due to phase shifts of adjacent pixels decreases thetransmitted intensity of phase shaped pulses. In a phase locked pulse replica
scheme, a phase grating with an alternation between 0 and π phase between
adjacent pixels is applied to the SLM. This represents the maximum linear phasethat can be applied. Any more than a π phase shift between adjacent pixels,
because of wrap-around effect, will be equivalent to a smaller phase shift with the
opposite sign. In later chapters, several instances of pulse-shaping will be cited,
and the decrease in pulse intensity seen above will become important.
A second example of a pulse-shaping scheme can be useful for illustrating
the limitations of our pulse shaper. Figure 2.4 shows a two-color pulse train,

13

normalized intensity
0

-6

-4

-2

0

2

4

6

delay (ps)Figure 2.3: Cross correlation pulse profiles for pulse trains with various interpulse
delays. The pulse trains are generated by the application of a phase grating of
π*round[cos2(2πp)], where round indicates rounding to the nearest integer (0 or1), and p indicates a number of pixels from 2 to 16, for the longest to shortest
interpulse delay, respectively. The symbols represent scans for different delays
between phase locked pulses.

14

-1

E0~12575 cm-1normalized intensity

E0~12425 cm

0
-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

delay (ps)Figure 2.4: Multi-color pulse train. The pulse train was generated by applying
opposing linear phases to the high (E>Ei0) and low energy (E<Ei0) halves of the
input spectrum centered at Ei0.

15where two phase related pulses are generated that each correspond to a different
half of the input spectrum. The pulse train is generated by applying a linear phaseϕ(p)= ±mp where p is the pixel number and m is a constant. All wavelengths
with a shorter than average wavelength get a positive slope to the phase, and thosewith a longer than average wavelength get a negative slope. The net effect of thisis to create two pulses that are equally spaced around t=0 and have pulse durations
approximately twice as long as transform-limited pulses. This pulse train isdemonstrated here because of its potential importance in strong field control
schemes such as stimulated Raman adiabatic passage (STIRAP)[36].

2.1.2Heat pipe/detection
The lithium dimer sample is contained in a static heat pipe, which isoperated at 750o-800o C, with an argon buffer gas pressure of between 1.0 and 3.5
Torr, depending upon the experiment. The lithium atom pressure at thistemperature is approximately 1 Torr (130 Pa), which will give a Li2 pressure of
approximately 0.02 Torr (2.6 Pa). The lasers intersect between two parallel plateelectrodes that are separated by 1 cm with 10 V applied across them. The current
generated by the ionization of the molecules is measured by a lock-in amplifier,which is synchronized to the modulation of the cw laser at 750 Hz by amechanical chopper. This technique gives a maximum single scan signal-to-noise
ratio of up to 10:1.
Signals are generally normalized to the fluorescence from the launch state.
This is accomplished by observing fluorescence in a perpendicular geometry [see16Fig. 2.2]. All fluorescent light within the small solid angle bounded by a 2”
collection lens is imaged onto a fiber-optic bundle. This fiber bundle then couples
into a monochromator, and the light for a specific transition [e. g. A(11,28)→X(2, 29)] is detected with a photomultiplier tube driven at 950 V. The
output voltage is then filtered with a lock-in amplifier, and the photoionization isdivided by the fluorescence in real time. The net effect of this is that drift in the
power or wavelength of the cw laser becomes less important.

2.2

Wavelength Subtraction Spectroscopy
One of the main traits of ultrafast lasers that separates them from otherpulsed or continuous wave (cw) lasers is the wide spectral bandwidth inherent due
to the Heisenberg Uncertainty Principle. Since the ultrafast lasers occur on such ashort timescale, they are characterized by a wide spectral bandwidth. In studying
any sort of ultrafast transition, this wide bandwidth can affect the dynamics of thetransition. CW absorption spectroscopy can to a certain extent give complete
information describing a transition, but interactions between wavelengths can belost. As an alternative, Wavelength Subtraction Spectroscopy (WSS), where atransition is measured with a spectral notch in the excitation spectrum, can beused to look at the relationship of different frequencies in an excitation by an
ultrafast pulse. This becomes especially important in examining either strong
field or phase shaped pulses.
To study the effects of the wide bandwidth on a transition, there areseveral methods one can use. The first, and perhaps most obvious, is to perform a17

variant of tunable cw absorption spectroscopy, where a laser is tuned andabsorption is measured, giving an absorption linewidth, followed by performing
an Inverse Fourier Transform to infer the time dynamics. In this regime, thesignal will simply be a summation of the single frequency components in thesignal:S = ∫ dωA(ω )

2

(2.2)

where A(ω) is simply the signal amplitude—such as an excited state coefficient—expected for any single exciting wavelength; time dependence is implied. For thecase of a two state wave packet, it can easily be inferred that the two statesinteract if coherently excited, so that S=|A(ω1)+A(ω2)|2, but as an excitation gets
more complex, making inferences such as this get more and more complex, as
well.
In contrast, WSS can give direct information about the nature of an
excitation. As implemented in our experiments, one pixel at a time is blocked on
the SLM, as shown in Fig. 2.5. For each gathered data point, the signal ischaracterized by the following:
SWSS (ω ') ∝ ∫ dωA(ω ) − ∫ dω ' A(ω ')
ω

2

(2.3)

ω'

where the integration over ω is the base spectrum, the integration over ω’ is thesubtracted part of the spectrum, and again A(ω) is a frequency dependent
amplitude with the time dependence implied. The crucial difference between thisspectrum and what one would expect from a cw type experiment is that there are
now cross terms between the frequency components ω’ and ω:
18

1.0

Spectral Intensity

0.8
0.6
0.4
0.2
0.0

52

56

60

64

68

72

76

80

Pixel number
Figure 2.5: Schematic spectra as used in Wavelength Subtraction Spectroscopy(WSS). A signal is acquired for each spectrum as one single pixel at a time isblocked. In the experiment, the light on each pixel is completely attenuated, butwhen the SLM mask is converted to wavelength units, a single frequency is notcompletely attenuated due to focusing effects. Three wavelength spectra areshown in the figure as pixels 63, 65, and 67 are sequentially blocked.

19

SWSS (ω ') ∝∫ dω A(ω ) + ∫ dω ' A(ω ') − ∫ ∫ dωdω '[A * (ω )A(ω ') + A * (ω ')A(ω )]
2

∞

.

2

∞'

(2.3b)∞ ∞'In a case where there is only one resonance present, and the field is weak, thecross terms are significant only when ω and ω’ are resonant frequencies[7]. In thetwo resonance case mentioned above, the only cross term is where ω and ω’represent the two resonant frequencies, and it only modulates the depth of thesignals. In more complex cases, the cross terms can become numerous,
drastically changing depth and width of the observed signal. New information,
including the amplitude of an excitation relative to all other excitation pathways,
can now be extracted from a simple spectrum. Since, in pulse shaping schemes,interferences between spectral pathways are often invoked to manipulate asystem, this technique gives a measure of the interference (see chapter 5).
To illustrate, Fig. 2.6 shows a comparison of the two cases for examining
an excitation at a specific time during an excitation of a two level system. In bothcases, the same phase mask is applied to the SLM. In one case, it is assumed that
the SLM transmits the light from only one pixel at a time, and in the second case,
it is assumed that the SLM attenuates the light on one pixel at a time. A
perturbative calculation is then performed to find an excited state population at atransient time (details of the calculation procedures are presented in chapter 6).
The main result to be considered here is that the two types of spectroscopies
present qualitatively different results under certain conditions. WSS is basically atype of optical heterodyne detection. In WSS, phase information is gleaned fromthe signal since the data is gathered with a certain coherent background, as

20

1.25

|ce(t)|

2

1.00
0.75
0.50
0.25
0.00
56

60

64

68

72

76

80

Pixel number
Figure 2.6: Comparison of excited state populations for a weak, phase shaped
input using narrow bandwidth excitation and WSS. The dots (z) represent thecalculated excited state population at t=0.0 ps with an excitation laser bandwidth
equal to the spectrum imaged onto a single pixel of an SLM, scanned pixel by
pixel. The squares ( ) represent the calculated excited state population at t=0.0
ps using the WSS method. In both cases, an identical phase mask is used.

21

represented by all other frequencies in the full pulse. In following chapters, the
WSS spectroscopic technique will be used in both the limited-cross-term regime
and the highly coupled regime.

22Chapter 3
Evolutionary Algorithm

3.1

Introduction
Ultrafast laser sources are promising new tools for research in the area of

coherent phase control[15, 25]. The wide bandwidth inherent to these lasers is
conducive to detailed manipulation of the amplitudes and phases of the fields thatinteract with the system of interest. Several pulse shaping techniques employ
frequency domain manipulations to generate a desired field[26, 30, 37]. Thesemethods provide a large number of amplitude and phase parameters by dividing
the frequency bandwidth into numerous segments and adjusting the phase andamplitude of each frequency component of the pulse. Directing a system
response from such a complex parameter space can be tedious, even if a detailed
mechanistic knowledge of the system is known[13, 38]. Learning algorithms havethus far proved to be an advantageous way to control a physical process togenerate desired photo-induced products, and these algorithms can be
implemented with little a priori knowledge of the system of interest.
There are several recent applications of learning algorithms (such as
evolutionary or genetic algorithms or simulated annealing) to find optimal
solutions to various physical problems[20, 25, 26, 39-41]. Three main aspects ofthese algorithms have been studied in the context of chemical physics: (1) theability to control molecular processes[20], (2) the methods for reaching the23

optimized solution[25, 39], and (3) what can be learned about a physical processfrom the form of the optimized driving laser field[39]. The benefits of using thesealgorithms to glean physical information from an unknown system have beendifficult to realize, and thus far they have been limited to evaluating just a few
parameters such as degree of chirp or pulse train timing[20, 39]. Additionally,
few examples exist that address concepts such as the competition between
optimization pathways or motion through solution spaces. The effect ofmanipulating physically irrelevant parameters has been addressed only on alimited scale. This chapter will take these considerations a step further byapplying a simple evolutionary algorithm to a system with variable complexity,
illustrating multiple mechanisms at play simultaneously and evaluating noncontributing parameters.
This chapter will primarily examine optimization conditions for the
preparation step of a wave packet in lithium dimer followed by ionization of the
wave packet. The Li2 E 1Σg+ state has been studied both spectroscopically andthrough wave packet dynamics.[7] Here we use phase and amplitude
manipulation of an ultrafast preparation pulse to control a wave packet on the E
state with learning algorithm optimizations of the ionization probability by a timedelayed probe pulse. Wave packets composed of either two or ten rovibrational
states are investigated. The physical processes for one optimization condition
involve using only a limited number of wavelengths, and in the second involve
optimization of the entire spectrum of the preparation pulses. Actual wave packet
traces will be compared to frequency domain solutions to help identify

24optimization mechanisms and to show which mechanisms become optimized first.
It will be shown that non-resonant wavelengths play a role in photoionizationeven in the weak field limit, but only at pump-probe time delays when the
preparation pulse may temporally overlap the probe pulse.
Sections 2 and 3 of this chapter will provide a characterization of the
optimization routine used in the experiments in this and subsequent chapters. A
simple pulse compression calculation will be examined in order to performed toperform an analysis of what can be learned about the physical system from the
dynamics of the EA operations. Section 4 will show an experimental
implementation of the pulse compression calculation of section 2. Section 5 willshow experimental results of an optimization of wave packet dynamics under the
restriction that transient phase and amplitude effects can not play a role in the
optimization. It will be shown in this section that only resonant wavelengths playa role in the dynamics of long time wave packets in the weak field regime.
Sections6-8 will show a similar optimization as in Section 4, except transientphase and amplitude effects will both be optimized. Included in this section is amodeled optimization using first order time dependent perturbation theory.
Finally, an optimization of the photoionization at 0.0 ps will be analyzed. The
solution of this final optimization will form the basis for the work of upcoming
chapters.
3.2

Algorithm characterization
The experiments described in this chapter use a multiple memberedevolutionary algorithm (EA), similar to several that have been used before[3, 23,25

25, 26, 42, 43]. The EA begins with a population of 25 random SLM phasepatterns (128 phases each), called individuals. The EA assigns each individual afitness value proportional to the total ionization signal at a specific pump-probe
time delay, then reproduces the fittest individuals and operates on the new
population. The process is repeated with each new population until a stopping
condition is reached [usually simply a pre-determined number of generations]. In
general, any number of operators, or population manipulators, can be used toachieve optimal configurations[43], among which can be both time and frequency
domain mutation, crossover, smoothing, averaging, and many others specific to aphysical system of interest[25, 40, 41]. In the experiments described here, only
mutation is used significantly.
The algorithm used here consists of an elitist reproduction method,
followed by a simple mutation operator, as shown in Fig. 3.1. All specificparameters below were found to both quickly and reliably optimize a pulse
compression [see section 3.3], so these parameters are used in all optimizations.
The reproduction method used here chooses the top 24% of the population,
ranked according to their fitness, and propagates them to the next generation. The
proportion of the subsequent generation that descends from a particular individualis further weighted according to the rank, R (R=1, 2, 3,…), of the parent, giving
the number of children descended from a single individual as approximately 8(R0.6). Individual pixels are chosen at random from among the entire new

population regardless of individual and are mutated. Approximately one in26

Reproduce(select best individuals)

Evaluate

F

Photoion signal
∆t
0.2

0.1

0.0

Mutatephases

-0.1

-0.2

-2

0

2

t

F ≈ photoionization at ∆tallindividuals
evaluated

ϕ

1.5
1.0
0.5
0.0
5

10

1.5

ϕ

15

20

25

30

λ (pixel number)

1.0(random pixels of
random

0.5
0.0
5

10

15

20

25

30

λ (pixel number)Phase pattern
for single individual

ϕ

2.0
1.5
1.0
0.5
0.0
5

10

15

20

25

30

Mutated phase patternFigure 3.1: Evolutionary Algorithm Flowchart. First, a population of individuals
consisting of phase masks is created and evaluated. The best individuals are
reproduced then mutated. This procedure is repeated until a suitably optimized
condition is met.

27

twenty pixels gets mutated each generation. Here, mutation means that a random
phase is added to the phase already on a pixel. The random additional phase isgenerated according to a normal distribution about zero with a distribution widththat goes as the generation number to the -0.75 power, or δϕ=A(G-0.75), where δϕis the standard deviation of the mutation amplitude for a single generation, A is aconstant phase (pre-determined to be about 30o), and G is the generation number.
This rate of mutation proceeds the same throughout all the optimization
experiments so as to act consistently on different systems. The complexity of the
EA is kept to a minimum for ease in interpretation of the optimization dynamics.
Additionally, physical intuition concerning the well characterized Li2 system isusually left out of this EA to better simulate the experimental conditions for
optimizing unknown processes. The lack of physical intuition written into the EA
has the tradeoff of longer convergence times, but the simplicity of the processfacilitates interpretation of solutions during optimization.
The fitness function is simply the ratio of an individual’s total ionization
signal to a reference: F(i)=S(τ, i)/S(τ, 0,), although later experiments not included
here use more complex fitness functions[6]. The pump-probe delay is set to asingle delay (τ), and the total chopper-modulated ionization signal (S) for eachindividual (i) is monitored. To correct for experimental instabilities, each fitness
value is normalized to a signal generated by an unshaped pump pulse [S(τ, 0)],probed at the same delay. The simplicity of the fitness function can give insight
into the physical processes being optimized. As individual pixels that haverelatively large effects on the total ionization signal are manipulated, there is a28

correspondingly large change in the fitness function upon optimization, so thesepixels are the first to be optimized.
The actual signal is the average current generated by the fixed delay pumpand probe pulses. Typically, each individual is evaluated for 0.5 seconds,
resulting in a maximum coherent oscillation signal-to-noise ratio of 10:1.
Occasionally, individuals are evaluated for up to 4 seconds each. Whencomputation and system control times are taken into account, a single generation
with 0.5 second averaging takes 40 seconds. When the EA has nearly converged,
the 10:1 signal-to-noise ratio masks the differences in the individuals, so that iswhen a longer averaging time might be used. There are advantages to averaging
for shorter periods since low frequency drift in the cw preparation laser has been
observed to change the signal on a timescale of approximately a half hour. This
will have little effect on the relative fitnesses of individuals within a single
generation, but it may affect the average fitness of one generation with respect toanother.
3.3

Convergence speed and schema size
One of the advantages of having a fitness value that is linearly related tothe physical observable is that when coupled with a simple optimization routine,
predictable optimization dynamics can be observed. It is intuitive that
optimization of a complex parameter space would take longer to converge thanthe optimization of a very simple parameter space. It has been empiricallyobserved that the simple EA used in the experiments described here does indeed
take a shorter time to converge for problems of reduced dimensionality.

29

average fitness(a)O ptim ized Spectral
Intensity (w = 25)
1

030 40 50 60 70 80 90
pixel

0

1/e rise time

40

20

40

60
generation

80

100

20
30
40
spectral fwhm (pixels)

50

(b)

30
20
10
10

Figure 3.2: Calculated pulse compression dynamics using an evolutionary
algorithm, according to Eq. (3.1a) in text. Panel (a): average fitness (F) versus
generation number (z) and single exponential rise fit for the inset pulse with 20
pixel FWHM Gaussian spectral profile. The fit has a 28 generation 1/e rise time.
Panel (b): relation between model spectral FWHM and optimization rise time.
This shows that the more pixels required for the optimization, the longer it willtake to reach the optimum.

30

Evidence for the processes occurring in the physical system can beinferred from the dynamics of an optimization by this simple EA. To illustrate,first we examine the effects of changing schema—i.e. groups of pixels that arecoupled and physically relevant, or active—on the time required to optimize acertain fitness value. Different processes can be seen by examining the widths of
independent schema that are being optimized. For example, optimizing a nontransient incoherent ionization signal for a single quantum state on the E1Σgsurface requires an optimization of approximately two pixels. The fact that there
are two relevant pixels can be attributed to an effect of the single frequency spot
size on the SLM being approximately Gaussian and 1.7 pixels wide.
The dynamics of this EA have been studied using a simple model of ourcurrent system, as illustrated in Fig. 3.2. In the model, the same EA is used asunder experimental conditions, with the determination of the fitness value being
the only difference. A few different processes are modeled to generate a fitness
value (F), one mimicking a pulse compression, S(t)pc , and one mimicking a twostate wavepacket optimization, S(t)wp, whose form is based on perturbation
theory:S(t)pc = |Σjajexp(i(ωjt+ϕj))|2(3.1a)

S(t)wp = |Σk(ck exp(iϕk))|2+|Σl(cl exp(iϕl))|2+|Σk(ckexp(iϕk))||Σl(clexp(iϕl))|cos[(ω0c-ω0(c-10))t+ϕ0c-ϕ0(c-10)](3.1b)

aj = exp[-(2/log(2))(ωj-ω0a)2/w2](3.1c)

ck = exp[-(2/log(2))(ωk-ω0c)2/n2](3.1d)

cl = exp[-(2/log(2))(ωl-ω0(c-10))2/n2](3.1e)

31

where ωj (ωk) is the central frequency on pixel j (k), ϕj (ϕk, ϕl) is the phaseapplied to light on pixel j (k,l), ω0a is the central frequency of the pulse, w is thepulse FWHM in pixels, ω0c and ω0(c-10) are the frequencies resonant withtransitions to the wavepacket states, ϕ0c and ϕ0(c-10) are the phases applied to thosefrequencies, and n is the resonant frequency imaged spot size (in pixels). In Eq.
(3.1b), the summation factors with the square of the same index account for theincoherent ionization signal, analogous to the ai term of Eq. (3.1c), and the cosine
term accounts for the coherence between wave packet states. They include
variable resonant linewidths and the fact that phase jumps between adjacent pixels
on the SLM result in some light being diffracted out of the beam. Theseequations have been normalized so that at their maxima, S(t)pc and S(t)wp haveequal amplitude. For simplicity, all models have a pump-probe delay set to t = 0,
and the wavepacket state spacing is assumed to be 10.0 pixels. The integrals over
the range ω=- ∞ : ∞ from Eq. (2.1) have also been left out of the above equations,
resulting in a sampling of the wavelengths on each pixel, giving the summation
expressions.
Using Eq. (3.1a) as the model system response [i.e. F = S(t=0)pc], we
investigated the effect of the schema size (w from equation 3.1c) on theconvergence to the optimum. As shown in Fig. 3.2a, the convergence of a single
optimization to the final solution fits very well to a single exponential decay. As
w is varied, the decay time does increase as seen in Fig. 3.2b, although thecorrelation between schema size and decay time is not analytical. One conclusionthat can be drawn from this data is that “inactive” pixels have minimal effect on32

the EA optimization dynamics. The constant rate of selection of pixels to be
mutated assures that this is the case. In every physical case, each active pixel has
the same probability of being mutated each generation, regardless of how manyother pixels are involved in the EA. These experiments were conducted on
various spectral profiles (i.e. Lorentzian and square spectra) and qualitatively
similar results were found, with only slight deviations from exponential decay
seen in the evolution of the fitness functions, and increasing optimization times
for increasing spectral widths.
In this case of pulse compression, every pixel that is involved in the
optimization depends on every other one, so there is a high dependence of eachpixel’s phase on all of the other pixels’ phases. This is in contrast to a situation
where there are groups of optimized pixels that do not interact, or that interact
minimally. One intuitive case of this would be to optimize average laser power,
which is simply the sum of the intensities of all laser frequency components.
Expressed algebraically (from eq. 3.1a), |Σiaiexp(i(ωit+ϕi))|2 ≠ Σiai2, whichindicates that the maximum intensity expression contains cross terms between
pixels [i.e. |aiak|cos(ϕi-ϕk)], and the average intensity does not. The presence ofthe cross terms defines the pulse compression as a single process that tries tomake ϕi=ϕk for all i and k, whereas optimization of average power is more of aseries of “independent” parallel processes, each process being the optimization ofa single spectral amplitude component. The effect seen would be that the
optimization of peak intensity will occur more slowly than for average power.

33

A multiple mechanism optimization was modeled by including both Eqs.
(3.1a) and (3.1b) in the optimization [i.e. F = S(t=0)pc + S(t=∞)wp]. Although thisparticular case is a highly simplified example of the true S(t)wp, it is still
instructive for showing the dynamics of multiple embedded processes. As thewidth (n in eq. 3.1d) of the wavepacket absorption schema increases for thismultiple mechanism process, we see that there is a transition in the dynamics ofthe optimization from double exponential to single exponential behavior. The
solid line in Fig. 3.3 shows the evolution of the average fitness function for asimultaneous optimization of a pulse compression with a 30 pixel FWHM schema
size and a wave packet with two schema of 1.5 pixels each. In this curve, theaverage fitness F avg(g) increases quite quickly in the first few generations, thenfollows a different exponential decay to the optimum, for a form F avg(g)=a0a1exp[-g/g1]-a2exp[-g/g2]. The two lifetimes (g1 & g2) in the double exponential
correspond to schema sizes expected from Fig. 3.2b. This effect will be mostpronounced when the interacting schema are of vastly differing sizes. As theprocesses become more comparable in schema size and interaction, theseparability of the two components in the optimization, and hence the fitness, is
not apparent. Another instructive trait of the EA is that in this case of two largely
different schema sizes, the leading coefficients (a1 & a2) correspond to the relative
influence of the two processes at play. Since the offset at g=0 is not necessarily 0in experimental cases, information can only be gathered from the ratio of the twocoefficients. In the calculation of Fig. 3.3, the two leading coefficients are bothvery nearly 1.0, as is expected from this analysis.

34

2
F

τ=65 gen1
τ=8 gen00

50

100
Generation #

150

200

Figure 3.3: Optimized fitness (F) of a two part process showing double
exponential behavior. The optimization (solid line) consists of one small schema
size process embedded in another larger schema process. The average fitness as afunction of generation number was fit with a double exponential (dotted line).

35

3.4

Pulse compression
As a simple test case, a pulse compression of the pump pulse was carriedout, similar to work previously done in other groups[3, 28, 44]. The motivation
for performing this experiment is twofold. First, the intuitive dynamics of a pulse
compression makes the mechanism of optimization by the EA easy to grasp—weare minimizing an intrinsic phase profile. This makes it easy to verify that the EA
is actually performing as expected. A second motivation is that the optimized
pulse compression phase mask contains information about the phase applied bynonidealities in the pulse shaping optical setup. The EA optimization acts as a
type of analytical instrument for understanding our input pulse’s phase function.
The experimental procedure is as follows. First, the pulse shaping setup isslightly misaligned so that it induces a small amount of dispersion into the pumppulse. This slightly modified output pulse and the reference pulse are thenfocused onto a GaAsP photodiode (bandgap~14,000 cm-1) and the peak in the
two-photon absorption signal as a function of interpulse delay is found. Next, theEA is allowed to run as described in the previous section with 32 effective pixels(each group of 4 pixels is treated as a single unit), with the pump-probe delay set
to the peak in the signal as found with the unoptimized pump pulse. The total raw
current from the photodiode is the fitness. Since the raw current is the crosscorrelation signal, the fitness F at τ=0 is proportional to the following, assuming
no saturation of the signal:
S (τ ) = ∫ E pu (t ) E pr (t − τ ) dt
2

2

(3.3)

t

36

where pu and pr correspond to the optimized pulse and the reference pulse,
respectively [these are the same as pump and probe in the rest of the text].
Assuming that the reference pulse is either transform limited or contains only
linear chirp (quadratic phase versus wavelength), the greatest signal willcorrespond to the optimized pulse being transform limited. The EA searches for asolution that compensates for any intrinsic phase on the pulse to be optimized.
After performing the optimization as described above for 50 generations,the cross correlation time was indeed reduced slightly from the unshaped case, as
shown in Fig. 3.4. The cross correlation of the unshaped pulses with the reference
pulse was approximately 331 fs FWHM, and the optimized cross correlation
showed a FWHM of 316 fs. This change is indeed small, indicating that the pulse
shaper imparts a small degree of chirp in these experiments. The overall
amplitude of the cross correlation is not quite as high as in the unshaped case,which is attributed a limitation of the EA. The reduced timescale of the optimized
pulses shows that the overall signal response deviates slightly from Eq. (3.3). In
contrast to most experiments, here we did not adjust the alignment of the pulse
shaper to minimize the intrinsic dispersion it adds, so we can assume that theadded phase in the experiments described here represents a maximum expected
added phase.
The optimized phase pattern contains information that can be helpful for
analyzing the pulse shaper output. Assuming the reference pulses are transform
limited, the optimized phase pattern essentially extracts identical information(within a factor of -1) as a frequency resolved optical gating (FROG)—a phase37

o p t p hases
sp ectral intensity

2

optim ized
(fwhm =316 fs)
zero phase(fwhm =331 fs)

ϕ (rad/π )photoion signal (arb. units)

O ptim ized X -correlation

1

0

20

40

60

80

100 120

Pixel #

-1

0

1

delay (p s)Figure 3.4: Optimized cross correlation. The dotted line is the cross correlation
for an unshaped pulse, and the solid line is the cross correlation for the shaped
pulse. Notice the small decrease in pulse length in the optimized case. The inset
shows the optimized phase profile (+) and fit (z) along with the pulse spectrum.

38and amplitude spectrum of the pulses to be studied[45]. Since a cross-correlation,
instead of auto-correlation, technique is used, there is no time ambiguity
characteristic of several FROG techniques[45]. At t=0.0 ps, the output from thepulse shaper can be rewritten as the following based on Eq. (3.1a):|E(t=0)|2 = |Σjajexp(iϕ'j)|2

,

(3.4)

where ϕ'j refers to the intrinsic phase applied by the pulse shaper apart from that
intentionally added by the SLM. To maximize the output, the negative of the
intrinsic phase must be applied, so that the depleting phase contributions areminimized, or ϕintrinsic(ω)-ϕoptimized(ω)=C, where C is a constant. The optimized
phase mask in the inset of Fig. 3.4 compensates for the intrinsic phase imparted
by the pulse shaping assembly. A fit to the data between pixels 25 and 95 (where
there is input intensity) shows an intrinsic group velocity dispersion (GVD) fromthe pulse shaper of Dν of -577 fs2 ± 100 fs2, which was calculated based on a fit to
the data of d2β/dω2=0.0033*(∆pixel)2, where Dν=-2πd2β/dω2[46]; the third orderdispersion is calculated to be approximately -102 fs3 based upon the samepolynomial fit. The lack of observed high order dispersion distortions such as
pulse replicas in the cross correlation verifies a small maximum error (~20 fs2)
due to reference pulse distortions; the large GVD error is approximate and resultsfrom the assumption in the fitting procedure that all pulse distortions arecharacterized by second and third order dispersion. A second order only fit gives

39Dν=-347 fs2. This data fits well with the observation that the cross correlation
FWHM was decreased by 15 fs. Using the EA as an analytical tool has proven to
be useful for later interpretations, setting a benchmark for how much chirp to
expect on our input pulses.
3.5

Optimizing launched wave packet
To analyze the EA even further, we optimize the relative phases of the two

eigenfunctions in the two state wave packet. By setting the pump-probe delay toa time outside of the temporal pulse shaping region, we can optimize the wave
packet signal without concern for the temporal shape of the ultrafast pulses. The
advantage of this is that the system responds only to two resonant frequencies.
This permits quantification of the relative accuracy of the EA, and the results are
straightforward to interpret. Additionally, an optimization of the ten state wave
packet, also at long time delay, will be briefly discussed. This multiple discrete
state wave packet will show a more obvious global maximum of thephotoionization signal than the phase shifting of a single beat, while still requiring
manipulation of only resonant wavelengths.
First, we analyze the two state wave packet, which has a beating period
corresponding to a 42 cm-1 (or 11 pixel) energy separation between the
eigenstates. In Fig. 3.5 is shown the WSS spectrum for the two state wave packet
at a long time delay. Also shown is the normalized transmittance recorded with amonochromator while blocking one pixel at each resonant transition wavelength.
From the excellent agreement, the effective full width at half maximum (FWHM)
attenuation width of each resonantly excited state is approximately 1.7 pixels,

40

16
1.2

total ion signal (arb units)

14
0.8
0.6
12

0.4
0.2

10

relative spectral intensity (arb units)

1.0

0.0

50

60

70

pixel number
Figure 3.5: WSS ({) Spectrum with the probe set to t>5.0 ps. Spectral
attenuation (U) of the pump pulse light [1-I(unattenuated)+I(attenuated)]
recorded with a monochromator while blocking pixels 56 and 67 on the SLM.
The linewidth of approximately 1.7 pixels is attributed to non-ideal focusing
conditions in the pulse shaper.

41

which is almost entirely an effect of the single frequency spot size on the SLM
and not an effect of the absorption lineshape of Li2. It is confirmed that there are
two discrete resonances within the bandwidth of the pump pulse.
Figure 3.6 shows the coherent ionization signals of the two state wave
packet following various EA optimizations. In Fig. 3.6(a), the pump-probe delay(dashed line at 5.4 ps) is set so that the wave packet should be in phase with thesignal generated by unshaped pulses. This optimization was run for 30
generations, with all 128 pixels involved in the optimization. A full signal versus
pump-probe delay was then acquired using the optimized phase pattern. Note
how the optimized wave packet signal builds up over a long period. Additionally,
the optimized wave packet shows no obvious multiphoton burst as seen at 0.0 ps
pump-probe time delay in the unshaped wave packet signal. The traces in Fig.
3.6(b) are optimized for 20 generations under conditions chosen to be out of phasewith the reference signal, verifying that an arbitrary wave packet phase can beachieved. In the top trace, only those pixels near the resonances were allowed tobe manipulated by the EA, while the rest were kept at a constant phase of π. In
the middle trace, the same pixels were manipulated by the EA, but the light on theother pixels was attenuated. The amplitude of the coherent oscillations in all threeof these optimized cases is similar in amplitude to the unshaped wave packet
signal, regardless of whether or not the non resonant frequencies are manipulated
during the optimization.
To analyze further the ability of the EA to find the expected result, shownin Fig. 3.7 are several optimizations that were run to find the repeatability of the42

aphotoion signal

zero phase wavepacket
optimized wavepacket

-2

0

2

4

6

delay (ps)bphotoion signal

optimized 8 phases, inactive set to pi
optimized 8 phases, inactive attenuatedzero phase wave packet

-2

0

2

delay (ps)4

6

Figure 3.6: Two state wave packet signals for different optimization conditions.
In both panels, the vertical dotted line marks the pump-probe time delay at which
the signal optimization is sought. Panel (a): Wave packet signal for optimized
phases ({) for a pump-probe delay of 5.4 ps, using all 128 pixels in the
optimization, and an unshaped wave packet (solid line). The optimized wave
packet has been offset for visualization purposes. Note the multiphoton signal
burst at zero time delay in the zero phase (unmodified pulse) wave packet signal,
which is absent when the phases are manipulated. This is due to the overlap of
the pump and probe pulses at t=0. Panel (b): wave packet signal optimized at 4.2
ps when only two regions of four pixels each centered at each resonance (i.e.,
λresonant ±2 pixels) are used, with non-resonant wavelengths either held at aconstant phase of π ({), or attenuated (V). Wave packet signals are offset
vertically for clarity.

43

optimized phases at 5 generations, all pixels active
phases offset so average of pixel 52, 53 = 4.47radoptimized phase (rad)

6

4

2

0
45

50

55

60

65

70

pixel
1.05

average fitness

1.00

0.95

exponential decay fit to data with
τ =1.4 generations

0.90

0.85

2

4

6

8

10

12

14

16

18

20

generation

Figure.3.7. Long time delay optimization convergence. Panel (a): optimized
phases after five generations for five different optimizations with the pump-probe
delay set to 5.1 ps. Phase profiles have been offset so that the average phase of
pixels 52 and 53 is the same for all optimizations. Resonant wavelengths aremarked with vertical lines. Note the convergence of optimized phases at pixels 63
and 64. Panel (b): evolution of the average fitness (z) by generation number for
one of the long time delay optimizations. The fit line is an exponential with atime constant of 1.4 generations.

44

solutions. Figure 3.7a shows the optimized phase difference of the two resonant
wave packet states for the best individual after five generations of each
optimization, allowing all 128 pixels to be manipulated. The average phasedifference for the best individual from each of the five separate optimizations is
1.0π ±0.12π radians, where the pump-probe time delay was set to be π radians outof phase with an unshaped wave packet. This shows that even after just a few
generations, the EA converges to a consistent phase difference between thetransition frequencies. Another characteristic of the optimized phases in Fig. 3.7ais that at least two pixels are optimized around each transition frequency. This
reflects the width of a monochromatic spot on the SLM, which is approximately
1.7 pixels.
Figure 3.7b shows the evolution of the average fitness of each generationfor one of the optimizations above, extended to 20 generations. From this data,
there is little change in the average fitness after five generations. This indicatesthat the optimization has largely converged, with a fit to an exponential
characterized by a lifetime of 1.4 generations.
In the two state optimizations above, the complexity of the EA is varied byallowing either 128 or 8 pixels to be involved in the optimization of the two state
wave packet. The success of the optimization is similar for both cases. This
supports the notion that in the weak field limit the EA only sees the pixels
relevant for the transition wavelengths. However, the speed of the optimization
can be affected by how many pixels are involved in a particular optimization.

45

This dependence of the optimization time on the number of active pixels
can be seen experimentally in Li2. For the long time delay optimization of thetwo state wave packet, which has about 4 relevant pixels, the average fitness ischaracterized by a rise time of 1.4 generations. A similar optimization performedon the ten state wave packet showed a rise time of approximately 13 generations.
In this case, the 128 pixel resolution was divided into ten regions (blocks) witheach block centered on a transition for each of the ten states in the wave packet,
giving ten effective pixels.
To better illustrate that the EA can in fact find a global maximum, thephotoionization signal of a ten state wave packet was optimized for 30
generations at a time delay of 6.9 ps, again manipulating all 128 pixels. The EA
was able to optimize the photoionization signal, as shown in Fig. 3.8. Notice how
the EA has essentially time-shifted the unshaped wave packet dynamics by 6.9 ps.
A similar time shift can be accomplished with a knowledge of the state energies
and their positions on the SLM, but the result has been obtained here with none ofthat knowledge applied.
The two state wave packet photoionization signal, which consists of bothincoherent and coherent components, was optimized at pump-probe time delays
outside of the temporal pulse shaping window. Assuming that little of theresonant wavelength light gets directed out of the beam from abrupt phase shifts,
the incoherent signal from the first term in Eq. (1.1) will remain a constant. This
leaves the coherent oscillation as the predominant component of thephotoionization signal that can be optimized.

46

1719

1717

1619

1617

1519

1517

1419

1417

1319

1317

intensity

1

0
20

40

60

80

100

pixel num ber
0.6

optimized wavepacket
zero phase pulse(shifted to 6.9 ps)

photoion signal (arb. units)

0.5
0.4
0.3

unshaped τ =0 contribution

0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
4

6

8

10

12

14

16

18

20

delay (ps)Figure 3.8. Multiple state wave packet optimization. Panel (a): wave packet
resonances (vertical lines) superimposed on the pump pulse spectrum. Panel (b):the unshaped wave packet (V) shifted by 6.9 ps to display its similarity to thewave packet optimized at 6.9 ps ({). Note the multiphoton burst at zero delay inthe unshaped wave packet, similar to that in Fig. 3.6. The optimized wave packet
has been offset for better visualization.

47

The main process by which the coherent oscillation gets optimized is theestablishment of the correct relative phases of the wave packet states to produce
the maximum of a recurrence at the desired time delay. The presence of thesingle, phase shifted cosine wave in the coherent signal indicates that the EA is
setting the relative phases of the two participating eigenstates to be a specificvalue, with no requirement as to the absolute phase of either state relative to therest of the pulse. The optimized phases for multiple runs by the EA, as shown in
Fig. 3.7, all show a convergence of the relative phase of the resonant wavelengths.
The actual phases of pixels 52 and 53, which have been offset in the figure, were
distributed fairly evenly between 0 and 2π radians, indicating that the phaserelative to any non-resonant wavelength is irrelevant.
These experiments were set up to optimize the photoionization signal after
the pump pulse light is no longer present. As can be seen from the onsets of theoscillatory signals in Fig. 3.7, the timescales for the wave packet coefficient
amplitude buildup under three different optimization conditions is not defined by
the EA, with the E←A population transfer happening variably over a period of ~6
ps (i.e., -2 to 4 ps). The buildup of the coefficients depends in a complex way on
both resonant and non-resonant wavelengths, but the similarity of the final wave
packet amplitudes under several conditions where the non-resonant wavelenths
were either restricted (or not) shows that at long time delays non-resonant
frequencies in the pump pulse play little to no role in the optimization.
Consequently, how the wave packet amplitude builds up does not matter as longas it reaches its maximum value by the optimized pump-probe time delay, which48

is the same as its value predicted by the t=∞ approximation implied in Eq. (1.1).
Numerous trials show that the wave packet amplitude at long times cannot be
distinguished between the unshaped and optimized cases, even though manydifferent coefficient buildup timescales have been observed in using various
optimized and randomized phase pulses.
Since the EA uses a constant mutation rate, the mechanism for
optimization of the simple two state system is computationally equivalent to thatof a more complex system; during any one generation, the EA is only aware ofthat generation, so historical information such as the number of relevant pixels
has no effect on which pixels are mutated. Using this fact, we can observe just a
few pixels of the 128 pixel optimization and treat the system as if it contained
only four parameters, which in this case are the pixels on which the resonant
wavelengths have been imaged. With this simplified physical case, we candevelop a measure of the precision of the EA in general. Figure 3.7a shows thephases from the best individual for each of five different optimizations after fivegenerations. The standard deviation of the relative phases between the pixels of
the resonant transitions is ± 0.12π radians. We expect the precision of the bestindividual of each generation to have a lower bound defined by the averagefitness of all individuals in each generation of a single optimization, which
experimentally shows a 1.4 generation rise time. Assuming the modulated signal
is simply the phase of a cosine wave, the root mean square deviation from the true
optimum is ∆θ = acos (1-exp[-gen/τ]) where gen is the generation number, and τ
is the optimization time constant. If we take τ to be 1.4 generations, then at five49

generations, the lower bound to the precision of the optimizations should be±0.09π radians. This slight discrepancy with observation is most likely a result ofa slight miscalibration of the applied phases on the SLM in the experiments,
adding error to the readout of the applied phases.

3.6

Optimizing transient wave packet
In this section, we consider a more complex process that has multiple

pathways and requires the optimization of the phases of many more frequencies
than the previous experiments. This is accomplished by setting the pump-probe
time delay to be within the temporal pulse shaping window, as was described in
section 2.1. We consider a two state wave packet, and the EA again uses the full
128 pixel resolution, with elitist reproduction and mutation operators only.
We set the pump-probe time delay to the peak of the second recurrence(1.6 ps) of the unshaped coherent oscillation signal. This time delay was chosen
for optimization so that the maximum possible transient component of thephotoionization signal would be equal to the t=∞ oscillation amplitude. By
optimizing the ionization signal at the second recurrence, as produced byunmodified laser pulses, the results from the previous section indicate that thephases of the wavelengths at the two resonant transitions should be the same, i.e.,∆ϕij=0.
An EA optimization at this short time delay was carried out for 30
generations, and the results are shown in Fig. 3.9. In Fig. 3.9(a), from the50

p h o to io n s ig n a l (a rb u n its )1ao p tim iz e d w a v e p a c k e t
unshaped w ave packet

32
2

p u m p /p ro b e
c r o s s c o r r e la t io n

0
-2

0

2

4

6

d e la y (p s )1.0bϕ (rad/2π )

0.8

0.6

0.4

0.2

0.0
50

60

70

pixel num ber

80

90

Figure 3.9. Short time delay optimization results. Panel (a): optimized wave
packet signal (V) for a pump-probe delay of 1.6 ps. The solid line shows thewave packet signal produced by unshaped pulses. The shaded region shows thepump-probe cross correlation on a photodiode. Peak 1 has been optimized andclearly exceeds the maximum multiphoton burst (peak 3) of the unshaped trace.
Peak 2 was not monitored in the fitness function. Panel (b): optimized phases fora pump-probe delay of 1.6 ps, after 30 generations. The resonant transitions aremarked with solid vertical lines, and the spectral intensity is denoted with thedotted line. Note the regions with a linear slope of the phase (most of region
shown), the marked deviation from linear slope of the phase around the resonanttransitions, and the unequal phase of the resonant transitions.

51

complete pump-probe signal versus time delay, we see that the EA clearly
optimizes the ionization, but at a slightly longer time delay than was desired.
Because of the effects of time-dependent wave packet coefficient phase andamplitude maximization, the signal at 1.6 ps is actually larger than the t=∞ twostate recurrence alone. For comparison, a wave packet generated using unshaped
pulses is included as a reference. As can be seen, the ionization maximum at 1.6
ps far exceeds that of the unshaped pulse at the same time delay, but at timedelays longer than 4 ps the amplitudes are the same in both cases. The wave
packet signal also shows no discernible phase shifting relative to itself over thecourse of its buildup. There is a constant 0.4 radian phase shift of the optimized
wave packet recurrences relative to the unshaped wave packet from -2 ps to 7 ps.
This phase shift is an artifact of competing processes that apply a phase shift tothe overall signal in early generations, and it is observed that the wave packet
phase does converge on the desired phase as the EA progresses.
A cross correlation of the shaped pulse with an unshaped pulse obtainedon a GaAsxP1-x photodiode, also shown in Fig. 3.9(a), indicates that the peak lightintensity at the 1.6 ps optimized time is much less than what occurs in theunshaped, near-transform limited case (not shown). At first glance, this may
appear to be a highly suboptimal solution compared to an intuitive solution of atemporally shifted transform limited pulse. However, the photoionization signal
at 1.6 ps significantly exceeds the signal at t=0 for the unshaped pulse, in spite of
the very low optimized light intensity.

52

Examination of the phase pattern for the optimization shows some of theprocesses that are occurring. The phase pattern in Fig. 3.9(b) shows a roughly
linearly phase (modulo 2π) as a function of wavelength across much of the pulse,which is characteristic for a pulse that is being shifted in time. Indeed, theshifting of light intensity to longer times is what is observed in the crosscorrelation. Looking at the phases near the wave packet resonances, however, thelinear phase is far from preserved. Finally, the relative phases at the wavelengths
resonant with the wave packet transitions show a slight phase shift (0.4 radians)
which agrees with that seen in wave packet signal.
Figure 3.10 shows an evaluation of the solutions found by the EA atvarious stages of an optimization. In Fig. 3.10(a) the average fitness versus
generation number is shown; note that the optimization takes much longer toconverge than in the long time delay optimization cases presented earlier. In Fig.
3.10(b), it is shown that the manipulation of the long time wave packet at 5.6 ps isachieved on a shorter timescale than the optimization of the photoionization at the
short pump-probe delay time of 1.6 ps.
At short pump-probe delays (<4 ps) the pulse shaping apparatus has theability to produce a pump pulse electric field that overlaps in time with the probe
pulse. The simultaneous presence of electric field from both beams can createtransient coefficient amplitudes that depend on frequencies that are not resonantwith the wave packet preparation frequencies [see Eq. (1.1)]. The EA finds aphase pattern that creates a maximum in the photoionization signal at 1.6 ps, and

53

1 .2

a

fitness (au)

1 .1

1 .0

0 .9

0 .8
5

10

15

20

25

30

generationb

6

ionization signal

5

4

3

total ionization at
optimization time: τ =16.8 (+/-0.7) gen
long time wave packet
fitness (x4): τ =8.3 (+/-0.9) gen

2

1
0

10

20

30

40

50

60

generation

Figure 3.10. Evolution of the fitness versus generation number for the 1.6 ps timedelay optimization. Panel (a): average fitness (z) as monitored during the
optimization. The solid line is a double exponential fit with time constants of 5
and 35 generations. Panel (b): total coherent ionization signal at 1.6 ps ({) and
5.6 ps fitness value ( ) assigned after completion of the optimization using
phases from the individuals with the highest fitness values of selected generations.
The long time fitness value was 4 times the ratio of the total ionization signal
divided by the reference (i.e. unoptimized) ionization signal at 5.6 ps (note that
this value was not monitored during the optimization). The fits show a 17
generation rise time at 1.6 ps pump-probe delay and an 8 generation rise time at
5.6 ps pump-probe delay.

54

it also generates substantial electric field at 1.6 ps, possibly indicating thepresence of multiphoton burst effects.
The EA finds the best maximum in the ionization signal when the pumpprobe delay is set within the temporal pulse shaping window. As seen in Fig.
3.9(a), the amplitude of the oscillating photoionization signal around 1.6 ps [peak(1)] is larger than the long time oscillation amplitude; additionally, it exceeds thepeak signal in the unshaped case [peak (3)]. At the optimized time delay, thecross correlation data shows a peak in the pump pulse intensity, which suggeststhat the EA does optimize somewhat a time-ambiguous (i.e. pump-probe + probepump), or multiphoton burst, component of the signal. However, the large
amplitude of the peak [peak (2)] preceding that at the optimized time delay does
not coincide with significant pump pulse intensity. This peak cannot then be
attributed to a multiphoton burst. This observation is consistent among repeated
optimizations, with little obvious correlation of the wave packet oscillations andthe light intensity (except at the optimized time), verifying that the wave packet
amplitude coefficients of Eq. (1.1) are transiently pumped while maintaining theircorrect relative phase relationship.
Notice in Fig. 3.9b that the relative phase of the resonant frequencies is 0.4
radians, which is consistent with the observed phase shift of the optimized
experimental wave packet oscillations. Even though first order perturbation
theory predicts that wave packet coefficients can change phase during theirbuildup [see chapter 5], it is not apparent in this data. The phases of each55

coefficient may be changing during the pump pulse, but it appears that thetransient relative phase (∆ϕij) remains relatively constant at time delays after -2
ps, as judged by the positions of the optimized wave packet recurrences relative tothe unshaped wave packet recurrences. The constant non-zero phase shift is due
to competition in early generations of the optimization of the wave packet
recurrences, wave packet amplitude coefficients, and a multiphoton burst effect.
Although there is no obvious physical requirement that the coefficientphase evolution during the pump pulse must be the same for both wave packet
states, the effect of the EA is such that both states have the same transient phaserelative to long time [i.e., ϕi(t=1.6ps)-ϕj(t=1.6ps) = ϕi(t=∞)-ϕj(t=∞)]. As the EA
begins to optimize the photoionization signal at 1.6 ps, one dominant early effectto be optimized is the relative phase of the wave packet states. The reason is
twofold: the phase shift of the wave packet can cause a large change in the fitness,and very few pixels are involved in determining the phase of the wave packet.
Out of random phase configurations, the correct relative phases of the few
resonant pixels will be more likely to occur than the correct phases of all theresonant and non-resonant phases. Figure 3.10 shows the evolution of the fitnessof an optimization at a time delay of 1.6 ps. It is shown in panel 3.10(b) thatwhile the average fitness rises with an approximate lifetime of 16 generations, theoptimization of the long time delay component of the signal (representative ofresonant wavelengths) proceeds about twice as fast as the optimization of the peakionization signal. As the EA proceeds through many generations, the optimal t=∞phase persists, for it represents a “ridge” in the total photoionization solution

56

space, leading to a final solution that is a combination of the correct resonant
frequency phase shift, transient wave packet amplitude, and possibly multiphoton
burst effects.

3.7

Modeled optimization
A calculated model of this system was studied in a situation free fromnoise and where the optimization could be extended up to 200 generations inorder both to compare the form of the perturbation theory solution of Eq. (3.1)
and to investigate the competition between coefficient buildup and multiphoton
effects. The EA was chosen to modulate 128 phases, similar to how it would
normally modulate phases on the SLM. The fitness function given to thecomputer included both a wave packet oscillation term and an undefined
multiphoton burst term, respectively:f(i,t) = |a1(t)||a2(t)|cos[ϕ1(t)-ϕ2(t)] + 1.25|Σjbjexp[i(ωjt+ϕj)]|2,(3.5)

where the a1 and a2 are derived using time dependent perturbation theory as in Eq.
(3.1), ϕ1 and ϕ2 are from Eq. (3.1), and bj is the amplitude of light with central
frequency ωj and phase ϕj imaged onto pixel j. The multiphoton burst term isnormalized so that |Σjbj|2 = |a1(∞)||a2(∞)|. The spectral profile (bj) is Gaussian andhas a FWHM of 35 pixels (156 cm-1), centered at 12500 cm-1. States 1 and 2 are
set to have energies of 12480 cm-1 and 15020 cm-1, respectively.
After 200 generations, the calculated optimal solution has a linear phaseapplied across almost the entire pulse, with a deviation from linear phase applied
near the resonant wavelengths (see Fig. 3.11), and no phase shift between wave
57

a

c

3 .0

1.2

spectral amplitude for each pixel
optimized phases after 200 generations

n o rm a liz e d w a v e p a c k e t
c o e ffic ie n ts |a i |, |a j |sim u la te d w a v e p a c k e t

2 .5

norm alized signal

1.0

ϕ (rad/2π )

0.8

0.6

0.4

2 .0

1 .5

1 .0

0 .5

0.2

c a lc u la te d p u m p p u lse
in te n sity (x 4 )

0 .0
-2

b

40

50

60

70

0

80

1

2

3

4

5

d e la y (p s)

d

0.0

1.4

-1

pixel

normalized fitness

*

1.3
1.2

d|a 1 |/dt

1.1

pulse intensity*

Fitness has been offset (.75 units) to reflect
experim ental background ionization effects.

1.0

0

0.9

cross correlation com ponentof fitness (x4) + offset (.75 units)

0.8
0

20

40

60

80

100 120 140 160 180 200

generation number

-2

-1

0

1

2

3

4

5

delay (ps)Figure 3.11. Model optimization results. Panel (a): optimized phases ( )as modeled by Eq. (4) after 200 generations. Resonant transitions are marked bysolid vertical lines, and the spectral intensity is indicated by the dotted line.
Notice the linear phase applied across most of the pulse, except in regions near
the resonant transitions. Panel (b): progression of the average fitness ({) during
the optimization (scaled to reflect experimental background effects in Li2) and
four times the average multiphoton component of the fitness (solid line), offset by
0.75. The average long time fitness was not a component of the observed fitness
and is simply a measure of the long time wave packet amplitude and phase. Panel(c): wave packet coefficients |a1| and |a2| (dotted line), four times the optimized
electric field intensity (shaded region), and a simulated wave packet (solid line).
The coefficients have been normalized to their values at t=∞, and the pulse
intensity has been normalized to reflect its correct contribution to the calculated
fitness at the optimized time delay. Both coefficients follow the same buildup
path, obscuring one in the figure. Notice the very low light intensity at the
optimized time. Panel (d): optimized pulse intensity at 200 generations (solidline), with the derivative of the absolute value of the wave packet coefficient(dotted line). Notice that immediately after the optimized time the wave packet
coefficients are driven down to their final values.
58

packet states: ϕ1(1.8 ps)-ϕ2(1.8 ps) = ϕ1(∞)-ϕ2(∞). This behavior closely mimics
the result found in the experimental optimization in Li2. One important result
obtained in this modeled optimization is that the laser pulse intensity at 1.8 ps isvirtually zero, as can be seen in Fig. 3.11c. Instead, its peak intensity is found at
1.73 ps. Note that even though there cannot be a multiphoton burst contribution
to the photoionization at 1.8 ps delay, the signal exceeds by 50% what would be
predicted assuming a simple temporal pulse shift coupled with the normal wave
packet oscillation amplitude at long time delays. The difference is the timedependent wave packet amplitude coefficients.
In this model, the wave packet amplitude coefficient buildup is in directcompetition with the multiphoton effect, with the coefficient buildup dominating
the calculated photoionization signal. Figure 3.11b shows that even though themultiphoton term plays a role early in the optimization with a maximum
contribution at generation 16, it is later overwhelmed by the wave packet
oscillation term. Experimentally, a -70 fs time shift of the peak intensity from theoptimized time is difficult to resolve, especially given that our experimental crosscorrelations are convolutions of the shaped pulse with an unshaped pulse with awidth of up to 180 fs. Also, the experimental optimization was concluded at 30
generations, at which point it is reasonable to expect that the multiphoton effectmay not have been fully suppressed yet. Regardless, this calculation does showthat the optimal pulse shape has low intensity at the optimized pump-probe delay.
The average fitness for the modeled optimization also shares several
common traits with the experimental optimization of Li2 photoionization,

59

including an initial quick rise attributable to a wave packet phase shift followed
by a slower further rise to the optimum. To illustrate, when the modeled fitness isfitted with a double exponential decay function, two lifetimes of 7.2 and 60
generations can be seen. This is similar to the experimental average fitness in Li2,
which also shows a quick initial rise followed by a slower decay, with a double
exponential rise fit with lifetimes of 5 and 35 generations (Fig. 3.11). The
evolution of the model and experiment show qualitatively similar behavior,suggesting a similar mechanism of optimization.
Given that this model calculation qualitatively reflects the Li2 system,
hypotheses regarding the excited state coefficient evolution can be put forward.
In the calculation, before the optimized time delay, most of the pulse intensity
goes into building up the amplitude of the coefficients, as shown in Fig. 3.11d.
Immediately after the optimized time, there is a π phase shift of the carrier
frequency of the pulse, with all subsequent light intensity driving the coefficient
amplitude down. This is made possible by correctly timing the buildup
attributable to non-resonant frequencies. In other words, the phases of nonresonant oscillations in |apu,i| are optimized so that, collectively, they come into
phase at 1.6 ps and quickly dephase just after that.

60

3.8

t=0.0 ps Optimization
The previous sections showed that indeed it is possible to optimize thephase of a coherent oscillation as well as transient population transfer typeprocesses. Even in the case of the model optimization, whose solution shows
little noise, the nature of the optimized phase mask seems somewhat hidden by
the linear part of the applied phase mask. To try to reduce this difficulty, an
optimization was performed to get the greatest photoionization signal at t=0.0 ps.
As in the previous sections, the launch state of (vA, JA)=(11,28) was used so that awave packet consisting primarily of (vE, JE)=(9, 27) & (9,29) gets optimized.
Figure 3.12 shows the optimal phase mask found by the EA while
monitoring the photoionization at t=0.0 ps. For the positively and negatively
detuned frequencies, there is a clear phase shift of approximately 1.2π radiansaround each resonance. This phase shift is a result of the EA finding an optimal
population transfer condition. Additionally, right at each resonance, there isapproximately the same phase. Since the size of the resonant spot on the SLM isabout two pixels, the exact phase of the resonant frequency can not be known
exactly in this case, but it is approximately π for each resonance. Additionally,
there is a significant jump in the optimized phase around pixel 75. This is
attributed to a competition between optimizing the phase of the quantum beatwhile simultaneously optimizing the transient wave function coefficients. This
will be studied in more detail in the next chapter.
Figure 3.13 shows the pump-probe trace corresponding to the optimalphase mask compared to a pump-probe trace with an unshaped pump pulse. Also

61

2.0

1.0

9,29

9, 27

ϕ (rad/π )

1.5

0.5

0.0

60

70

pixel

80

90

100

Figure 3.12: Phase mask ( ) found by EA when optimizing photoionization at
t=0.0 ps. Resonances are marked (°), and the laser spectrum is noted by thesolid curve. Note the phase shifts of the frequencies to the left and right of theresonant frequencies is approximately 1.2π radians.

62

photoionization signal (arb units)

2.0

zero phase
EA solution

1.5

1.0

0.5

0.0
-3

-2

-1

0

t (ps)

1

2

3

4

Figure 3.13: Optimized photoionization with cross correlations. The top traces
show the optimized photoionization ({) pump-probe trace along with anunshaped pump-probe trace (solid line). Below are cross correlations with theunshaped probe pulse for the optimized pump pulse (z), and the unshaped pumppulse (dotted).

63

shown in the lower part of the figure are cross correlations with the probe pulse of
the optimized and unshaped pump pulses. The photoionization shows a large
peak at the optimized time, as should be expected from earlier work. The peak at
t=0.0 ps barely exceeds that found in the transform limited case, but there aresome interesting traits to the optimized dynamics. First, even though thephotoionization does show a maximum similar to the transform limited case, thebackground offset and the coherent oscillation amplitude are both smaller for the
optimized case. Since the optimization considers the photoionization at only one
time, the physical requirements for optimization only “see” an optimal transientstate. In this case, the transient coefficient is optimized at the cost of the final
state amplitude.
The cross correlation measurements (Fig. 3.13, heavy line) also show aninteresting feature. The optimized cross correlation shows that the amplitude at
t=0.0 ps is only approximately one quarter that in the transform limited case.
Even so, the photoionization remains strong. In the optimized case, it could beargued that the transient effects exceed the transform limited case even with thislow peak intensity. This observation hints at the power of using transient
processes to control dynamics, as will be explored in the next chapters.
Unfortunately, at t=0.0 ps, the actual mechanism for the optimized
photoionization is unclear, since at that time, the shaped pulse is simultaneously apump pulse as well as a probe pulse. A mechanism for simultaneous optimizationof the probe pulse can be imagined, introducing some ambiguity into thephotoionization mechanism. At the very least, we should expect minimization of64

diffraction of the light at those frequencies of light that are resonant with theprimary autoionizing Rydberg states in the probe section of the pulse[47]. Also,
we should expect that the phase difference between the resonant probe regions forstates (9, 27) and (9, 29), respectively, should be approximately zero[48]. These
considerations complicate the optimization, so a means to eliminate the pumpprobe ambiguity at t=0.0 ps has been used, and it will be examined in thefollowing chapter.

65

Chapter 4
Weak Field Population Control

4.1

Introduction
Coherent control remains a primary goal of ultrafast laser dynamics

studies [1,2]. Research so far addresses several key categories, such as the use ofresonant frequencies to control quantum wave packet interferences [2-5] andnovel pulse shapes to control multi-order processes [2,6-10]. Optimization ofcoherent control has been achieved using various types of learning algorithms[8,11-13] [see previous chapter], but the nature of the mechanisms for the optimal
results is not always readily apparent. This chapter will identify a class of pulse
shapes that enhance the population transfer from a single state to multiple excitedstates, providing a basis for understanding the nature of optimal pulse shapes forboth population transfer and quantum interferences in the superpositions. Pulse
shapes that have been shown to enhance transient excited state population in a
two level system will be applied to multiple excited states. All experiments will
be analyzed using first order perturbation theory.
Much work has been devoted to the field of population transfer dynamics
in the continuous wave regime [14], but only recently have well-establishedpopulation transfer theoretical tools been applied to the wide bandwidth inherent
to ultrafast spectroscopy [13,15-18]. For example, it has been shown that chirped
pulses can be used to enhance population transfer dynamics in a multiphoton
process [15,19] or even transiently in a single photon process [13,17]. It has been

66

shown that optimal enhancement of multiphoton dynamics can be achieved not by
chirped pulses, but by applying constant phases to blocks of nonresonant
frequencies relative to resonant frequencies [13,16,18], based on the t=0.0 ps
optimization described in the previous chapter. By using a frequency domain
analysis, the optimal pulse shape is shown to depend primarily on frequencies thatare near resonant, as will be discussed in detail in a later section.
This chapter considers the simultaneous control of both time dependent
population transfer dynamics and superposition state quantum interferences using
analytically optimal pulse shapes, in comparison to simple chirped pulse shapes
and near transform-limited pulses. Of interest are the effects of resonant andnonresonant frequencies involved in the excitation of superposition states. By
independently controlling not only quantum interferences between wave packet
states, but also the population transfer dynamics to each state, the coherently
controlled ionization yield at specific time delays is more than double that
produced by a transform limited pulse in a two-state superposition pump-probe
experiment. A two-color ultrafast laser experimental system, with pulse shaping,is used to isolate the coherent population transfer dynamics of a two-levelsuperposition in Li2 molecules. The criteria for independent control of thecoherent population transfer dynamics and quantum interferences are established.
Extensions to more than two states are also discussed.
Simultaneous coherent control of two well-defined excited states unifies
concepts concerning the dynamics between interfering superposition states andpopulation transfer dynamics to individual states. As long as the interfering states67

are sufficiently separated in energy from each other, the population transfer
dynamics to each state can largely be independently controlled. Additionally, therelative phases of the multiple states are controlled during the process of
population transfer. In the following sections, a theoretical framework for theseexperiments is first presented, followed by a summary of experimental
considerations. Finally, experimental results are presented and discussed in thenext section, along with a quantification of the limitations of enhancing
photoionization using various pulse shapes. The degree to which the populationtransfer for two states can be independently controlled is quantified in that final
section.

4.2

TheoryIn the weak field limit, first-order time-dependent perturbation theory canbe used to describe a single photon absorption. The excited state amplitude
coefficient for a single state is described in the time domain [20]:cn (t ) =µegt

∫ ε (t ) exp(iωeg t ' )dt ' ,
iη −∞

(4.1)

where µeg is the dipole moment matrix element between the excited and groundstates, ε(t) is the electric field as a function of time, and ωeg = (Ee−Eg)/ η is thetransition frequency. Using similar logic to earlier work [16,18,21], this excitedstate amplitude can be approximated for positive time t by the frequency domain
expression68cn ( t ) =∞µeg  ∗iε (ω ) exp[i ( ∆ )t ] dω  ,℘∫
ε~ (ωeg ) −
iη 
∆
2π −∞


(4.2)

where ε~ ∗ (ω eg ) is the amplitude of the electric field at the transition frequency,
ε (ω ) is the amplitude of light at frequency ω, ∆ = ω − ω eg is the detuning, and ℘is the Cauchy principal value. The first term in Eq. (2) represents the resonant
contribution to the excited state amplitude, and the second term represents
nonresonant contributions. As t approaches infinity, the second term averages tozero, and Eq. (2) reduces to the resonant component; however the second term
does not necessarily average to zero for small t (i.e., within the pulse width). In
the absence of any phase manipulation (i.e., transform limited pulses), thenonresonant term will be greatly diminished for all t, since the ∆ term for ω>ωeg
is π out of phase relative to ω<ωeg [18]. Additionally, the i/2π leading factor of
the nonresonant term places the nonresonant contribution π/2 out of phase withthe resonant term. This phase relationship suggests that to bring all frequency
components into phase at small t, an additional +π/2 and −π/2 phase must be
added to the nonresonant frequencies above and below ωeg, respectively.
Additionally, the 1/∆ dependence of the nonresonant term shows that the mostinfluential frequencies will be those with the smallest detuning. These results
may then be generalized to multiple excited states.
The 1/∆ dependence of the nonresonant wavelengths lends itself to thepossibility that frequencies with a large ∆ can be manipulated with minimal
influence on the excited state coefficient dynamics. This will allow for the
manipulation of multiple states, provided that the energy separation between
69

states, ∆ω, is great enough that all frequencies near one state have little effect on
any other states. For a two-state superposition, a lower limit of the degree towhich the population transfer to state i can be independently controlled, IPTi, isquantified as±∞2

Ι PTi = 1 − 2n ∫ d∆ ' ε ( ∆ i ' ) ∆ i ' ,∆ω / 2(4.3)

where ∆ω is the difference in energy between the superposition states(∆ω=ωi−ωj), the second limit of integration (±∞) has the same sign as ∆ω, andε ( ∆ ' ) is the amplitude of the electric field at a specific detuning. The factor of 2

accounts for destructive interference, and n normalizes the expression so that thesecond term reaches a maximum of 1. Physically speaking, IPTi is a measure of
the cumulative intensity of all nonresonant contributions to a particular population
at a specific time relative to the ideal case (i.e., single control channel). In thecase where two states have degenerate energies (i.e., ∆ω=0) with a uniform
energy spectrum, ΙPT=0, the states can not be independently controlled. On theother hand, in the limit of ∆ω=±∞, the second term goes to zero, and the states arecompletely independent of each other. This analysis assumes a regime where thespectrum is divided into an arbitrary number of “control channels,” with a
specified bandwidth devoted to the control of each state. The spectrum is divided
into two or more regions, and the integrated value of ε (∆ i ' ) ∆ i ' inside the control
channel for state i is compared to ε (∆ i ' ) ∆ i ' outside of its control channel.
Increasing the number of states involved in a control scheme will likely decrease
the spectral width of the control channel for any state, decreasing the degree to70which that state can be independently controlled. In the discussion below, we use“control channel” to refer to large blocks of wavelengths surrounding eachresonant transition and “resonant control channel” or “resonance” to refer to verysmall bandwidth regions centered on each resonant transition.
Lithium dimer (Li2) is used as a model system to investigate the extent towhich nonresonant frequencies can be used to control multiple excited statecoefficients with a single pulse. The relevant potential energy curves in Li2 areshown in Fig. 2.1 [22-24]. From a single launch state, two rovibrational states(vE=9, JE=27 & 29) are accessible within the bandwidth of the pump laser,
creating a time-dependent wave packet. A probe pulse ionizes the coherent
superposition at various pump-probe time delays to obtain the signal. For parallelpump and probe polarizations (referred to as “parallel probe”), the presence ofmultiple rotational states in the wave packet produces a coherent superposition
state with oscillations at a frequency equal to the energy difference between thestates [25] [see Fig. 4.1]:
S ( t ) ≅ pr1

2

c1 ( t )

2

+ pr2

2

c2 ( t )2[]+ 2 pr1 pr2 c1 ( t ) c 2 ( t ) cos ∆ωt + ∆ϕ (t ) (4.4a)

where pr1 and pr2 are constants related to the probe step (not discussed here); theenergy separation, ∆ω, is 42 cm-1 (1.5 Thz); and ∆ϕ is the relative phase between
the wave packet states at time t. In contrast, for a probe polarization oriented at
the magic angle (~55ο) with respect to the pump polarization (“magic-angle
probe”), the coherent oscillation is completely suppressed, so
S(t) = |pr1|2|c1(t)|2 + |pr2|2|c2(t)|2.

(4.4b)

711.2

photoionization

1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-2

0

2

t (ps)

4

6

Figure 4.1: Pump-probe traces at parallel (S) and magic angle (solid line) pumpprobe polarizations, along with the difference (U) between the traces forunshaped pump pulses. These traces correspond to the total, incoherent, andcoherence portions of the signal, respectively, with the raw signals normalized toan average value of 1.

72

This suppression is possible because the coherent oscillation is rotational innature.[6,24] Hence, by manipulating the probe polarization, we can selectwhether the signal reflects a full wave packet trace or simply the ground toexcited state population transfer dynamics. Furthermore, by subtracting themagic-angle probe signal from the parallel probe signal, we are left with just thecoherent oscillation pieceS(t) = 2|pr1pr2||c1(t)c2(t)|cos(∆ωt + ∆ϕ).

(4.4c)The probe step consists of an ultrafast pulse of a different color than thepump pulse [26]. This aspect of the signal was not considered above forsimplicity. The color of the probe pulse is chosen so that the probability for aprobe-pump (i.e., t<0) ionization pathway is below the noise level, allowing timedependent ionization to occur only at positive time delays. There is a constantbackground component to the photoionization signal at all time delays, and thishas also been subtracted out of the signals. Hence, the observed photoionization
signal only shows the time dependent coefficient buildup behavior and thequantum interference of the superposition state as described in Eq. (4.4).
4.3

Experiment
The lithium sample is contained in a heat pipe at 1050 K and isphotoionized by a three step excitation process. An overview of the laser system
is given below, but details can be found in previous work [13,26], and in chapter
2. In this experiment, a narrow bandwidth continuous wave (cw) laser, anultrafast amplifier system, and an optical parametric amplifier (OPA) are used,
and all but the OPA are pumped by a 27 W argon ion laser. The frequency of the73

cw laser is tuned to a specific A1Σu+←X1Σg+ resonance (606.954 nm) of Li2,
producing a pure launch state (vA=11, JA=28) on the first excited electronic state.
The ultrafast regenerative amplifier is seeded by a 76 MHz oscillator to produce
180 fs [full width half maximum (FWHM)] ultrafast pulses at a 200 kHz
repetition rate, near the 800 nm central wavelength with 8 nm (FWHM)
bandwidth. These ultrafast pulses are split into two beams, with 25% of the lightbecoming the pump pulse and 75% of the light frequency doubled to pump the
OPA. The resulting 15 mW OPA output is a train of nearly transform limited
pulses with a 200 fs FWHM pulsewidth and a central wavelength of 640 nm. The
OPA output is variably time-delayed relative to the pump pulse via a precisiondelay stage. This OPA wavelength is used since in the pump-probe regime, ityields a strong photoionization while producing very little photoionization in theprobe-pump regime [see Fig. 2.1].
To shape the pump pulse, it is sent through a dispersion free pulse shaper[27,28]. In the pulse shaper’s Fourier plane, a liquid crystal spatial light
modulator (SLM) is used to shape the spatially dispersed pump light byindependently attenuating and/or applying phase to 128 individual frequency
components (SLM pixels) of the pump light. The central frequencies imaged onto
the SLM pixels are separated by approximately 4 cm-1 with a single frequency
spot size of approximately 1.7 pixels. Normally, the 8 nm FWHM pulses areimaged onto the SLM to achieve a bandwidth of 40 pixels FWHM. Pixellationand finite spot size effects limit the temporal pulse shaping to be within a windowof about ± 4 ps around time zero.

74

The cw laser, pump, and probe pulses intersect in the interaction region in
the center of the heat pipe. The ions are produced between two parallel plates
separated by 1 cm with a 10 V potential applied across the interaction region. The
cw light is optically chopped, and the resulting current is detected with a lock-in
amplifier that is synchronized to the cw modulation.
To study the effects of nonresonant frequencies, two classes of pulse
shapes have been used to enhance transient photoionization effects, shown in Fig.
4.2. The first class, which has previously been studied in the context of a single
resonance [17] (“chirp-type spectrum”), has a quadratic phase applied across thefrequency spectrum around each resonance with the form ϕ(ω)=c∆2 + ϕk, where∆=ωi-ω is the detuning from the nearest resonance, and ϕk is a constant phase [seeFig. 4.2a]. The constant phase, ϕk, is applied to the entirety of one of the control
channels to control the phase of the wave packet interferences. Here, the timedelay is adjusted by varying the magnitude, c, of the quadratic phase. An optimalphase mask for population transfer comprises the second class, called the “phasejump spectrum”; this phase mask involves the application of π/2 and -π/2 radians
above and below resonance, as established in several recent works [see Fig. 4.2b][13,18]. The resonant portion of the control channel is defined here as a block of
two pixels (~8 cm-1 FWHM) centered on a specific transition, chosen to avoid
attenuating the long time delay signal by diffraction [13]. So, in effect,“resonance” actually refers to the smallest practical bandwidth achievable by ourpulse shaper around each transition. To vary the timing of the optimal
photoionization, a linear phase versus wavelength is added to the nonresonant

75

2.0

a
(9, 29)phase (rad/π )1.5(9, 27)

1.0
0.5
0.0

phase (rad/π )

2.0
1.6

50

60
pixel number

70

50

60

70

b

1.2
0.8
0.4
0.0

pixel number

Figure 4.2. Phase mask classes used in this chapter. In both panels, quantum state
resonances (vE=9, JE=27 & 29) are marked with vertical lines. Relative phaseoffset of 0.0 and π radians between control channels are represented by crosses(X) and squares ( ), respectively. Panel a: Chirp-type spectra. The degree ofquadratic phase, c, is defined by ϕ(pixel) = c x (∆pixel)2 where ∆pixel is theseparation in pixels from the nearest resonance. Panel b: Phase-jump spectra. A
phase of +π/2 and -π/2 is added to the frequencies above and below resonance,
respectively. A linear phase of ϕ(pixel) = c x ∆pixel is added to the phase masks to
induce a time shift in the population transfer (not shown). The phase on pixel 56
is always the same as that on pixel 55, and the phase on pixel 66 is always thesame as that on pixel 65.

76

portion of the phase-jump phase profile. As in the case of the chirp-type
spectrum, the relative phase of the wave packet states using the phase-jump
spectrum is manipulated by adding a constant phase to only one of the control
channels.

4.4

Results and Discussion
This section will first analyze the ability to control quantum beats while

applying a previously studied chirp-type phase mask around each state, but
extended to a quantum superposition. Next it will be shown that the phase jump
spectrum gives a much greater enhancement in the transient photoionization. It
will further be shown that the timing of the peak photoionization for the phasejump spectrum can be much more precisely controlled than the chirp-type
spectrum. Finally, the nonresonant contributions to the photoionization will becompared to a simple model to quantify the degree to which the quantum states inthe superposition can be independently controlled.
To demonstrate the separability of the population transfer dynamics fromthe wave packet oscillations, we first examine the previously studied case of the
chirp-type spectrum, with results for the two states at the magic angle [Eq. 4.4b]
and parallel probe minus the magic angle [Eq. (4.4c)], shown in Fig. 4.3[49]. The
population transfer dynamics in the upper part of the figure show several traits ofthe buildup produced by a strictly chirped pulse. After time zero, there is a peak inthe excited state population followed by a short period of ringing. After just a few
picoseconds, this ringing decays, leaving a constant value for the

77

78

1.4

photoionization

1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2

-2

0

t (ps)

2

4

6

Figure 4.3. Chirp-type pump spectrum transients. The upper solid line and circles (c)represent the magic-angle probe transient for π and 0 radians relative phase between
control channels, respectively, and a chirp factor of 0.2. The lower solid line and circles(c) represent just the coherent oscillations produced with the same phase masks as
above.

79

excited state population. Notice that these traits are present for both zero and π relative
phase between the control channels, and in both cases there is a peak in the excited state
population around a planned delay time of 1.2-1.3 ps, depending on the degree of chirp,
in this case 0.2 chirp factor. The coherent oscillations, which are simply the difference
between the parallel and magic-angle probes, are also predictably controlled. Clearly, a πphase shift of one coherent oscillation relative to the other can be induced, which issimply a result of adding π phase to only one of the resonant control channels.
Additionally, around 1.2 ps, the amplitude of the coherent oscillations passes through amaximum, as expected from Eq. (4.4a). The combination of the population transfer
dynamics and the wave packet interferences produces a clear peak in the photoionization
signal at 1.2 ps despite the slightly decreased population transfer using zero phase offset
between control channels.
The analysis of the chirp-type spectrum above establishes that both the populationtransfer dynamics and wave packet interferences can be independently controlled. Thissection now explores more optimal phase masks to maximize the photoionization due tothe population transfer dynamics in conjunction with the interference of two states at
specific short pump-probe time delays.
Several magic angle and parallel pump-probe data are summarized in Fig. 4.4. A
pump-probe transient obtained with the phase-jump spectrum of Fig. 4.4 clearly shows anenhancement in the excited state population, in addition to the proper phasing of thecoherent oscillation at t=0, shown in Fig. 4.4a. In these signals, the background signal
has been subtracted out, and the remaining80

photoionization

2.0
1.6
1.2
0.8
0.4
0.0

a

photoionization

-2
2.0
1.6
1.2
0.8
0.4
0.0

0

photoionization

2

4

2

4

2

4

b

-2
1.6

t (ps)0t (ps)

c

1.2
0.8
0.4
0.0
-2

0

t (ps)

Figure 4.4. Phase-jump pump spectrum transients. Panel a: Magic angle probe traces
produced using zero phase offset (z) and π phase offset (solid line) between the control
channels, and parallel probe photoionization signal (c) for zero control channel phaseoffset. A trace produced by an unshaped pump pulse with magic angle probe (+) is
included for reference. Panel b: Population transfer traces adding varying degrees oflinear phase to nonresonant frequencies with π phase offset between wave packet
resonances. The nonresonant contributions were timeshifted to have peak photoionization
occur at 0.0, 0.25, 0.5, 1.0, and 2.0 ps (c, X, V, +, and U, respectively). The solid line
represents an attempt to shift the nonresonant contribution to a time beyond thecapabilities of the experimental apparatus, resulting in a purely resonant excitation effect.
Panel c: Nonresonant effects on the population transfer at various time delays. Traces
were generated by subtracting the purely resonant contribution of the population transfer
trace from the total population transfer signal (same symbols as in panel b).
photoionization signal has been normalized to the signal at t=∞, which has been shown todepend only on resonant wavelengths and hence is constant in these studies[25]. Two81population transfer traces (magic-angle probe) are shown, along with a total wave packet
signal (parallel probe) that is designed to have a maximum in the quantum beat at thesame delay (0.0 ps) as the maximum in the population transfer. The population transfer
traces use phase masks with 0 and π relative phase between two control channels, and thefull wave packet signal is set to have 0 relative phase between the two control channels[the X’s in Fig. 4.2b]. Both population transfer traces show a peak at the same pumpprobe time delay, with the case of π relative phase between control channels [the’s inFig. 4.2b] having the larger peak amplitude.
Of all relative phases between control channels of the phase-jump class, the caseof zero relative phase is the least optimal for population transfer. This is so because all of
the wavelengths between the two wave packet states in one control channel maximally
destructively interfere with the buildup of the other control channel. For example, thelight on pixels 57-60 maximally and destructively interferes with the buildup of state(9,29), where the control channel includes pixels 61-128. Even so, the wave packet
interference produces a global maximum in the photoionization at 0.0 ps pump-probe
delay. A net gain in the photoionization relative to a non-interfering condition canalways be achieved as long as the wave packet interference amplitude exceeds one minus
the degree of independent control, or when |pr1pr2||c1(t)c2(t)|>1-IPTi from Eqs. (4.3) and(4.4). Note that for a multiple state superposition, 1-IPTi increases, but simultaneously
multiple coherent oscillations can constructively interfere, counteracting the decrease indegree of independent control inherent to the smaller control channel sizes of a multiple
state system.

82

In Fig. 4.4b several population transfer traces are shown for the phase jump
spectra with varying amounts of linear phase added just to the nonresonant frequencies;
these traces are generated using only π relative phase between the wave packet
resonances. The peak height and time for each trace is noted and will be discussed later.
Evident in this figure is that for each trace there is a slow, monotonic buildup (>4 ps) ofpopulation attributable to resonant (and very near resonant) frequencies coupled with amuch shorter timescale population transfer attributable to nonresonant frequencies. By
subtracting the resonant portion of the traces from the full traces, purely nonresonant
effects can be analyzed [see Fig. 4.4c]. The time-shifted nonresonant contributions to the
population transfer show peak widths between 750 fs and 790 fs FWHM. This data will
be used later to quantify the degree to which the superposition states can be
independently controlled.
The phase-jump spectrum permits a higher degree of coherent control than the
chirp-type spectrum by obtaining a greater enhancement in the population transfer and by
allowing more precise timing of the peak of the photoionization signal [see Fig. 4.5].
Figure 4.5a shows the peak excited state populations as a function of time for the twopulse shape classes. At short times, the phase-jump spectrum doubles the excited state
population, whereas the chirp-type spectrum shows a maximum of 30 percent
improvement of the photoionization signal. The

83

peak photoionization

2.0

a

1.8
1.6
1.4
1.2
1.0

5

b

t (ps)

2

peak time (ps)peak time (ps)

0

4
3
2

4

2.4
2.0
1.6
1.2

1

0.8

0

0.4
0

1
2
3
subtracted
phase (rad/pixel)c

0.0 0.1 0.2 0.3 0.4 0.5
chirp factor

Fig. 4.5. Comparison of control capabilities using the phase-jump spectrum versus the
chirp-type spectrum. In all panels, peak photoionization was monitored for various
degrees of chirp type pulse shapes (triangles) and time shifted phase jump pulse shapes(circles). Filled symbols denote 0 relative phase between control channels, and open
symbols represent π relative phase. Panel a: Peak photoionization comparison as afunction of designed peak time delay. Data is collected by analyzing peak heights asshown in Fig. 4.4b, for example. The peak photoionization for the unshaped spectrum(i.e., degree of chirp = 0) is represented by the chirp type spectrum data point at 0.4 ps
and is normalized to 1. Panel b: Peak times as a function of degree of linear phase added
to the standard phase-jump spectrum. Panel c: Peak times as a function of the degree ofquadratic phase applied around each resonance for the chirp-type spectrum.

84

phase-jump spectrum shows greater population transfer enhancement than the chirp-type
spectrum because the phase-jump shape assures that all non-resonant contributions comeinto phase simultaneously. This is in contrast to the chirp-type situation, in which only
certain nonresonant contributions are in phase at any one time. For the phase jump
spectrum, the timing of the peak of the photoionization signal can be controlled precisely,
as shown in Fig. 4.5b. Note that in this regime the time delays of the peak
photoionization signals are chosen by time-shifting the nonresonant contributions to the
population transfer by simply adding a linear phase to the basic phase-jump pattern,
based on the relation ∆t = dφ dω . Relative to the phase-jump spectrum, the timing of thepeak population transfer in the chirp type spectrum is not controlled as successfully.
Notice in Fig. 4.5c that the peak timing follows a roughly linear relationship with thedegree of chirp, but the deviation from a linear relationship is substantial. This canpartially be attributed to wrap-around effects (i.e. the phase can only be defined modulo2π) and “overlap” between the separate shaping channels of the phase mask. For
example, the truly quadratic phase for the control channel around state (vE, JE)=(9,27) isdefined to stop at pixel number 60, but with chirp factor c=0.2 and π relative phasebetween wave packet states [see Fig. 4.2b], pixel 61 fits well onto the quadratic
progression. This in essence extends the size of the control channels. While this will betrue for some chirp factors, it will not be true for others, accounting for the scatter in Fig.
4.5c.
The mechanism for the increase in population transfer for the phase jump
spectrum can be attributed to off resonant Rabi oscillations. In the weak field limit, the



[ (

) ] 
2

1/ 2

Rabi frequency Ω( ∆ ) = ∆2 + µeg ε ∆ + ω eg / η  reduces to Ω(∆)=∆, since85[µ ε(∆ + ω )] << ∆ for values of ∆ with non-negligible electric field amplitudes [14].
2

eg

2

eg

Additionally, given that the amplitude of any nonresonant oscillation is proportional toε(∆+ωeg)/∆ [29], the total of all nonresonant contributions to the excited state populations
∞

[(

) ]

can then be formulated as c(t ) ∝ ∫ d∆ ε ∆ + ω eg ∆ sin 2 (∆t 2 + φ (∆ )) . Notice the similarity−∞

2of this representation to Eq. (4.2). To bring all of these sine waves into phase at t=0, aphase mask with +π/2 and -π/2 radians applied to the positive and negative detuned
wavelengths, respectively, must be implemented. At the resonant wavelength (i.e., where∆=0), the optimal phase will be intermediate between the positive and negative detuning,or zero degrees. Thus we have the same optimal phase pattern as predicted by Eq. (4.2).
In contrast, for the chirped phase pattern, not all nonresonant Rabi oscillations come into
phase simultaneously. Rather, there is a series of partial recurrences in time; this isobserved as a ringing in the signal.
To quantify the degree to which the two states in the experiments described here
can be independently manipulated, a model based upon Eqs. (4.1) and (4.2) is
implemented. This model accounts for the nonresonant and resonant contributions to thesignal and is summarized in Fig. 4.6. Equation (2) is used to account for the nonresonant
contributions to the population transfer:cn (t ) =

µeg
h

∑
p

ε (ω p ) exp[i( ∆ p )t − iϕ p ]
∆p

,

(4.5)

86

a

1.2

1.2

ϕ (rad/π )

0.4

0.4

0.0

0.0
-0.4

-0.4

-0.8

-0.8
-1.2
80

-1.2
50

pixel number

70

1

2

b

2.0

photoionization

60

normalized amplitude

0.8

0.8

1.5
1.0
0.5
0.0
-0.5
-2

-1

0

3

4

t(ps)

Fig. 4.6: Comparison of model results with experiment. Panel a: phase ( ) andamplitude (•) [ε(ωp)/∆p from Eq. (4.5)] of the oscillations created by each pixel used togenerate the nonresonant contribution to the model population transfer dynamics.
Resonances are marked with vertical lines, and the electric field spectrum is given by thesolid line. A second state (dashed line) is included for reference. Panel b: Comparison ofmodel and experiment. The experimental nonresonant ( ) and full population transfer
traces ({) are compared to the model nonresonant (dotted line) and full populationtransfer (solid line) traces. The model resonant contribution is given by the dot-dashed
line.

87

where the summation is over all pixels (p) of the SLM, with each pixel representing lightof a specific frequency ωp and phase ϕp. In essence, each pixel creates an oscillation withfrequency ∆p and amplitude ε(ωp)/∆p. Transform limited pulses with Gaussian spectral
widths equal to experimental observations are assumed, giving ε(ωp), and the quantum
states are assumed to be symmetrically located under the spectral envelope. In Fig. 4.6a,the values of ε(ωp)/∆ and ϕp are plotted. The amplitude of the light at the two pixels
around each resonance is assumed to be zero here, but it is accounted for in the resonant
portion of the model. The signal is then computed as a convolution of the excited state
population, |cn(t)|2 with a 180 fs FWHM Gaussian probe pulse. The model populationtransfer dynamics fit very well with the experimental data, as shown in Fig. 4.6b. Using
this model to quantify the influence of nonresonant frequencies, Eq. (4.2) shows that eachstate can be controlled up to 90% independently (i.e. IPTi=0.90).
The resonant contribution to the signal is modeled using Eq. (4.1). The electricfield is assumed to be transform limited with a bandwidth for excitation equal toapproximately 10 cm-1 FWHM, which corresponds to the light imaged onto two pixels.
Each quantum state is assumed to be located at the central frequency of its respective
narrow bandwidth region to best mimic experiment and to minimize computational
artifacts related to the singularity at ∆=0. This component of the modeled signal shows aslow, monotonic increase in the excited state population, just as one would expect from aseveral picosecond transform limited pulse. When the resonant and nonresonant
components of the model are added together, they closely follow the experimental signal,as shown in Fig. 4.6b. A similar analysis was performed on the results using phase jump
spectra with 0 radians relative phase, with similar outcomes, further supporting the88validity of this model. In future work, the effects of changing the energetic spacing
between states will be used to experimentally verify the bandwidth considerations
presented earlier.
We have shown the ability to independently implement the coherent control ofboth the population transfer coefficients and quantum interferences to two states in asuperposition. Additionally, we have shown that the optimal pulse shape for transient
population transfer in the weak field regime is characterized by nonresonant wavelengths
that are shifted by π/2 and -π/2 radians relative to the resonant wavelengths. Using asimple model based upon the assumption that an ultrafast transition can be described by acollection of driven oscillators, we have quantified the degree to which the two states inthe superposition can be independently controlled. This work should be instrumental for
designing coherent control pulse schema and for understanding solutions found byvarious learning algorithms.

89

Chapter 5
Dynamic Phase Control

5.1

Introduction
The study of coherent control to manipulate molecular systems has

generated numerous tools that can be useful for directing a system under optical,
especially ultrafast, excitation[12, 13, 22, 26, 32, 50-57]. The tools of coherent
control may be applicable to quantum computing, where superpositions ofquantum states, or qubits, are used to perform computational operations[58].
Crucial to being able to perform quantum operations is the ability to manipulate
individual states within qubits. Basic computations have been performed on spin
systems using Nuclear Magnetic Resonance with considerable success[59-62].
Optically manipulated systems using trapped atoms and ions also show promisefor quantum computation by using narrow bandwidth lasers to resonantly control
particular transitions between states[63, 64]. Wide bandwidth, ultrafast pulses
have been proposed as a means to implement quantum computational operations
using the relatively dense states inherent to single quantum dots and evenmolecules[32, 54-57]. Given the relatively short decoherence times (<100 ns) ofthese more complex systems, the use of ultrafast laser sources is valuable toperform sufficient numbers of operations for meaningful computations. This
chapter explores a mechanism for using ultrafast pulse shaping techniques tomanipulate individual states in a molecular superposition, specifically multiple

1rotational states in an excited electronic state of the lithium dimer. As an example
of using nonresonant frequencies to control the amplitude of molecular rotational
states, a transient Z-gate is implemented. Using an ultrafast pulse with sufficient
bandwidth to encompass resonant transitions to two excited states, one state of thetwo state superposition will have its sign unchanged while the other will undergo
a sign inversion. The operation is transient in nature, so it can not be used for
actual computation, but the results illustrate new possibilities of using opticalpulse shaping techniques for individual state manipulation.
In the previous chapter, it has been shown that resonant frequencies can beused to control the phases of states involved in a wave packet[7, 48], and it has
also been shown that nonresonant frequencies can be used to transiently
manipulate the absolute amplitude of resonant transitions, even in the weak field[33, 65]. One additional aspect of transient, ultrafast control of wave packet
phases has not been demonstrated: i. e. the transient phase of a wave packet canbe controlled by using nonresonant frequencies. As in any first order process,though, the final wave packet phase and amplitude at long times are determinedby the resonant frequencies, as is expected[12]. This will be demonstrated byobserving the coherence only portions of the signal as written out in Eq. (4.4c)
We show in this chapter that the wave packet phase can be induced toundergo a π phase shift at time zero relative to long times. Assuming that one
contributing state in the superposition has a constant phase at all times, the signal
shows a change in sign because of a transient sign inversion of the second state.
This dynamics is accomplished by summing resonant and nonresonant2

contributions with the same sign for one state while summing resonant andnonresonant contributions with opposite sign for the other. The amplitudes of therelative contributions of each effect are manipulated so that the nonresonant
contribution dominates at short times for the opposite sign but disappears at longtimes. This process of inverting the sign of one state of a superposition whileleaving the second unaltered is characteristic of a quantum computational Z-gate,
which in this case is constructed in a transient fashion.
This example of controlling the dynamic phase of a wave function in the
weak field regime provides a rather clear illustration of the limitations andpossibilities of precise control of molecular systems. As long as the intensity ofthe electric field is weak enough to disregard intensity dependent Rabi oscillation
effects, pure phase shaping of the ultrafast pulses fully controls the dynamics ofthe effects observed. In Sec. 5.2, a theoretical framework for these experiments ispresented using first-order perturbation theory. In Sec. 5.3 is presented acategorization of the resonant and nonresonant effects that are used as tools inthese experiments. Finally, experimental results are presented, discussed, andquantified in Sec. 5.4.
5.2

TheoryThis chapter develops a method to induce a transient sign change in one state of atwo state superposition, or wave packet, using language consistent with theformalisms of quantum computing. The wave packet consists of two states thatare excited coherently, so that:Ψ ' (t ) = c0 (t ) 0 + c1 (t ) 1 + c 2 (t ) 2 ,(5.1a)

3

where c0(t) is the launch state wavefunction amplitude, c1(t) and c2(t) are thecomplex excited state wavefunction amplitudes to be described below, and |0>,
|1> and |2> are the time dependent rovibrational wavefunctions [i. e.
1(R, θ , t ) = 1(R, θ ) e iω1t ] in the superposition [see Fig. 5.1]. The detection
scheme is only sensitive to |1> and |2>, so the effective wave packet can bereformulated as
Ψ (t ) = c1 (t ) 1 + c 2 (t ) 2 .

(5.1b)As described previously[33, 48], the photoionization signal using parallel, linear
pump and probe polarizations can be written as
S ( t ) ≅ pr1

2

c1 ( t )

2

+ pr2

2

c2 (t )

2[]

+ 2 pr1 pr2 c1 ( t ) c 2 ( t ) cos ∆ωt + ∆ϕ (t ) , (5.2)

where the only change from Eq. (4.4a) is the explicit time dependence of ∆ϕ(t).
Since the multiphoton ionization pathway from the launch state (pr0) is weak,pr0<<pr1, pr2, so all launch state terms are left out of the signal, and the timedependent signal only shows the dynamics of the excited states.
In these experiments, the probe step consists of an ultrafast pulse of adifferent color (645 nm) than the pump pulse (801 nm) [33] so as to avoid any
ambiguities in the contributions to the signal when the probe comes before thepump. With the 645 nm probe pulse, the probability for a probe-pump (i. e., t<0)
ionization pathway is below the noise level, allowing time dependent ionization to
occur only at positive time delays. This allows the transient populations andcoherent oscillations to be measured reliably around t=0.
In Eq. (5.2), the cn(t) factors have the same functional form as Eq. (4.2),
with nonresonant contributions to the coefficient being ±π/2 out of phase with4

probe

ELi

2

|2>
|1>
pump
|0>
cw

Figure 5.1. States involved in excitation scheme. A cw laser excites from the X
electronic state to a pure launch state (state |0>) on the A electronic state. A wide
bandwidth 801 nm ultrafast pump pulse excites a superposition of states (states|1> and |2>) on the E electronic state, which is subsequently ionized by a timedelayed 645 nm ultrafast probe pulse.

5respect to the resonance and with a 1/∆ dependence. Notice that these effects
result in a complex amplitude cn(t), which under the conditions imposed in theseexperiments will remain primarily real. The manner in which the various
contributions interact is explored in the next section where it is demonstrated how
to change the wavefunction sign.
In this chapter, a transformation of the qubit in Eq. (5.1b) ismathematically equivalent to a quantum computational Z-gate. Using a matrix
formulation, the action of the ultrafast pulse on the prepared superpositiontransforms the system as a single qubit Z-gate: c1 ( f ) 1 0   c1 (i )
c ( f ) ≈ 0 − 1 c (i ) .
 2 
 2  (5.3)

First, a superposition is prepared such that at t=0.0 ps, the two excited states haveamplitudes of opposite sign so that c1≈1, and c2≈-1. The electric field then drives
the system so that the sign of c2 changes, leaving c2=1. Although thistransformation can be analyzed in terms of a series of three correlated pulses (i.e.
pulses with a defined relationship between the carrier frequency phases), it will be
explained in terms of the spectral contributions to the signals, as given in Eq.
(4.2).

5.3

Experiment

In the experiments described in this chapter, the experimental setup is the same as
chapter 4. What are varied are the pulse shapes that are investigated as well as theanalysis of the data. As a result, this section will focus on the basics of different

6

types of excitation mechanisms and pulse shapes, as if they were simply
experimental tools.
To investigate phase mask effects, the available spectrum is defined asconsisting of two spectral channels separated at the energetic midpoint between
the two resonant transitions. A global phase offset can be added to the channel
for state |2> in order to control the long time wave packet phase. Within eachchannel, i. e. for each state, resonant and nonresonant contributions to the excitedstate wave function coefficient are independently controlled. See Fig. 5.2 for one
phase mask that is used to add the resonant and nonresonant contributions of theexcited state coefficients along with the corresponding observed time domain
evolution. In Fig. 5.2a, it is shown that a small bandwidth region around eachresonance (~8 cm-1 each) is assigned a phase halfway between the positive andnegative detuned nonresonant frequencies. The effect of these narrow bandwidth
regions is a monotonic increase in the excited state coefficient, as shown by thetriangles in Fig. 5.2b , taken with magic angle probe so there are no wave packet
recurrences. This will be referred to as the resonant contribution. The applied πphase shift between positive and negative detuned frequencies around eachresonance compensates for the intrinsic spectroscopic π phase shift of thenonresonant contributions above and below the resonances. This allows thenonresonant contributions to cooperate on short timescales, as seen by the circles
in Fig. 5.2b. The solid line in Fig. 5.2b gives |c1(t)|2 + |c2(t)|2, which has beennormalized. Assuming that the crosstalk between control channels for the twocoefficients is insignificant [see previous chapter][33], which is achieved when7

ϕ (deg)

2.0π

a

1.5π
1.0π

|1>

|2>

65

70
75
pixel number

0

2

0.5π
0.0π
2.0

60

b

80

1.5
1.0
0.5
0.0
-2

t (ps)

4

Figure 5.2. Phase shaped population transfer traces. Panel a: Phase mask used tomaximize population transfer at t=0.0 ps. The applied phase mask is given by( ), and resonances are marked with (X). The applied phase masks near eachresonance (i. e .in each channel) are identical, assuring that both states follow thesame excitation pathway. Panel b: Magic angle traces (to suppress the rotational
quantum beats) using the applied phase mask of panel a. The solid line represents
the total population transfer signal, the (U) shows the resonant only contribution,and the ({) gives the nonresonant contribution, which is just the difference
between the two traces.
8the two resonances are spaced far apart energetically, |c1(t)| ≈ |c2(t)|. By taking
the square root of the signal and renormalizing, a measure of each coefficient isobtained. This signal shows that at the times of interest in this paper, t=0 andt=∞, |c1(t)| = 1.4 and 1.0, respectively. This information will be used later toquantify the manipulations of coefficient c2(t).
Figure 5.3 is a schematic diagram of the signed amplitude [i.e., cn(t), not|cn(t)|2] of the excited state coefficients. The traces in Fig. 3a-b are calculated
from parameters of the square root of the cooperative signal as described above.
Given that these contributions add to yield the amplitude of cn(t), the resonant andnonresonant effects within each channel can be subtracted as well as added. Thisis done by multiplying either the resonant or nonresonant contribution by -1.
Fortuitously, in the complex plane, multiplication by -1 is identical to the additionof π phase, which is accomplished by the application of π phase to the appropriate
pump wavelengths. As can be seen in the modeled coefficients in Fig. 5.3b,
simply subtracting the nonresonant contribution from the resonant gives a valueof -0.7 for c2(0). In these experiments, state |1> is manipulated to be a sum of theresonant and nonresonant contributions with the same sign, while state |2>
consists of a sum of contributions with opposing signs. Since the coefficient for|2> reflects a difference, the absolute amplitude of this negative coefficient can beincreased in one of two ways: by increasing the negative part or by decreasing thepositive part. Here, the positive contribution due to the resonant frequencies has
been time shifted by 1.0 ps by applying a simple linear phase mask to the resonant
regions [solid line in Fig. 5.3a] with the nonresonant contribution unaltered. To

9coefficient amplitude

1

a

|1>

resonant0

nonresonant|2>-1
1 b |1>

photoionization

0

|2>

-1
2

c

1
0

-2

0

2

t (ps)

4

Figure 5.3. Resonant/nonresonant interaction scheme for inducing sign inversion.
Panel a: Calculated resonant and nonresonant contributions to the populationtransfer. The solid line is the resonant contribution time shifted by 1.0 ps. The(U) and (V) reflect the nonresonant contribution for states |1> and |2>,
respectively, when inducing a sign inversion. Panel b: Total excited statecoefficient evolution using contributions from panel a. The (z) show the time
evolution for state |1>, and ({) shows the time evolution for state |2>. Panel c:Experimental photoionization traces for a single state with a phase mask that adds
resonant and nonresonant contributions (), and a phase mask that subtracts theresonant and nonresonant contributions ({). Note that the photoionization
correlates with the square of the amplitude coefficients.
10

demonstrate the effects experimentally, each type of phase mask was applied to apulse exciting the vE, JE=7, 39 state, and the results are shown in Fig. 5.3c. The
light near state vE, JE=7, 41, spaced over 62 cm-1 away, was attenuated to makesure that it was not excited. Excitation with the additive phase mask shows the
expected transient peak at t=0.0 ps. Excitation with the subtractive phase maskshows the photoionization goes nearly to zero at 0.42 ps. It is at this time that thesign of the coefficient inverts and the phase quickly changes. These traces show
the total photoionization, so they actually reflect |c(t)|2, but it is quite clear fromthe data that the subtractive phase mask changes the coefficient sign of thisisolated state.

5.4

Results and Discussion
To serve as a reference, a case where the dynamics of the populationtransfer is controlled identically for both states involved in the wave packet isshown in Fig. 5.4 {see [33]}. Here, the resonant and nonresonant contributions tothe signal are added together with the same sign for both states |1> and |2>. The
upper part of Fig. 5.4a shows that there is a peak in the excited state coefficientsat t=0.0 ps. The difference between the full signal and the excited statecoefficient signal yields purely the coherence signal, which is shown as the circles
in the lower part of Fig. 5.4a. This is obtained from parallel and magic angle
pump-probe polarization signals, respectively. Notice that the coherence signal
only appears after the excited state coefficients become nonzero (i.e. after t=-1.0ps), and that it has an overall constant temporal phase of π radians.

11

a

2.0

signal (arb units)

1.5
1.0
0.5
0.0
-0.5
-1.0
-2

2

4

0

2

4

b

1.5π

phase (rad)

0

1.0π
0.5π
0.0π
-2

t (ps)

Figure 5.4. Phase evolution using the same phase mask for each resonance. Panel
a: Pump-probe signals using phase mask of Fig. 5.2. Total signal ( ) andpopulation transfer (solid line) traces are shown on the top. Also shown is thecoherence-only signal ({) as well as a single cosine fit (dotted line). The
coherence-only signal and fit have been vertically offset for visibility. Panel b:
Instantaneous wave packet phase. The maxima and minima of the coherence-only
signal are compared to the fit to generate the instantaneous phase. The wave
packet phase remains relatively constant from the onset of the signal until longtimes.

12

To show that the phase of the wave packet oscillation remains constantduring the population transfer process, the signal up to 6.0 ps pump-probe delay
was compared to a cosine wave that is generated to fit the coherence signal
between t=6 ps and t=30 ps. The signal is fit only at times after 6 ps to avoid
potential uncertainties introduced by a changing wave packet phase during the
population transfer process. The maxima and minima of the coherence signal arecompared to the fit, and the time offset of each peak relative to the fit is used todetermine the instantaneous phase, plotted in Fig. 5.4b. As is evident in Fig. 5.4b,
the phase shift of the experimental trace is centered around 1.16π ± 0.25π(maximum range) radians over all time delays up to 6 ps. This comes as no
surprise, since no effects have been introduced to change the instantaneous wave
packet phase.
The case of subtracting the nonresonant contributions from the resonant
contribution is the main point of this paper and is shown in Fig. 5.5. The phasemask that is used to induce a sign change in state |2> is shown in Fig. 5.5a. Notethat the phase applied to the nonresonant contributions around |1> is not changed
from the additive case, as shown in the previous section. In contrast, the phaseapplied to the nonresonant contributions around |2> is inverted; the nonresonant
contributions have an additional π phase shift relative to the resonant ones, whichis equivalent to a multiplication by -1, as explained previously. The
corresponding signal traces from Fig. 5.5b show the expected population peakaround t=0.0 ps, just as when the contributions were added together. Note that
this trace must remain positive, since the observed signal is a sum of squares for13phase (rad)

1.0π

0.5π

|2>

|1>

0.0π
60

80

Pixel number

2.0

signal (arb. units)

70

b

1.5
1.0
0.5
0.0
-0.5
-1.0
-2

2

4

0

2

4

c

1.5π

phase (rad)

0

1.0π
0.5π
0.0π

-2

Figure 5.5. Sign inversion wave packet data. State |1> is programmed to behave
the same as in Fig. 5.4, but state |2> is programmed to change sign. Panel a:
Phase mask applied to induce sign inversion. The applied phase is given as ( ),and the resonances are given by (X). Notice that the relative phase between
resonances is 0.0 radians. Panel b: Pump-probe signals using the phase mask ofpanel a. Population transfer (solid line) trace is shown in addition to thecoherence-only signal ({) as well as a single cosine fit (dotted line). The
coherence-only signal and fit have been vertically offset for visibility. Panel c:
Instantaneous wave packet phase from onset of excited state population. The
wave packet phase undergoes a π shift from t=0.0 ps to t>1.0 ps.

14

both states |1> and |2>. A critical difference in this case occurs in the coherence
only portion of the signal, which contains information about the wavefunction
sign. In Fig. 5.5b it is apparent that around t=0.0 ps, the quantum beat portion of
the signal exhibits a phase of ~π even though the relative phase between theresonant frequencies is 0.0 radians. The instantaneous phase graph of Fig. 5.5c
displays a π phase shift in the region around t=0.0 ps relative to t>2.0 ps. Thisshows that there is indeed a temporally dynamic phase of one state in the wave
packet relative to the other. In fact, it shows that the sign of one state has changed
temporarily relative to the other.
A full discussion of the effects seen here can not be made without a brief
mention of the nonidealities at play in this pulse shaping scheme. As was
mentioned earlier, the resonant contribution to the signal was shifted to a morepositive time to minimize the interactions between it and the nonresonant
contributions. An example when the nonresonant contribution to the signal is notshifted in time is shown in Fig. 5.6. It is quite clear in this figure that the phasedeviates substantially from either 0 or π phase. The dynamics at play here arehighly sensitive to the positions of the resonances relative to the cutoffs between
those spectral regions treated as “resonant” and “nonresonant.” Within the“resonant” portion of the bandwidth, there are variable amounts of positive andnegative detuned wavelengths that have no compensating phase added. The net
effect of this is that the instantaneous phase of the nonresonant contribution is notnecessarily zero or π. When this nonideality interacts with the transiently
prepared state, there can be a large shift in instantaneous phase, as demonstrated

15

photoionizationaphase (rad)

b

0

2

0

2

4

0.0π

-0.5π

-1.0π

t (ps)

4

Figure 5.6: Sign inversion wave packet data without time-shifting the resonant
contribution. All symbols are the same as Fig. 5.5. Notice the deviation in
instantaneous phase away from 0 or π phase at short times. Also notice that theoscillation amplitude fails to approach zero between 0.0 and 2.0 ps.

16

in Fig. 5.6b. For this reason, the instantaneous phase is most predictable as theresonant and nonresonant contributions are separated in time.
Around t=0.0 ps, when the sign of the wave function for state |2> isinverted from that expected for the purely resonant phases, there is a certain
degree of destructive interference in the contributions to the signal. By
performing Wavelength Subtraction Spectroscopy (WSS) with the shaped pulses,
a measure of the relative contributions of the various spectral contributions can be
made. The traces shown in Fig. 5.7 illustrate destructive interference at playaround t=0.0 ps for state |2>. The data were acquired by setting the pump-probe
delay to a specific time delay, either 6.0 ps or 0.0 ps, with either unshaped or
shaped pulses, respectively, and attenuating the light imaged onto one pixel at atime. The circles in Fig. 5.7 show the WSS spectrum for unshaped pulses at a
pump-probe time delay of 6.0 ps. The trace shows a narrow bandwidth ofcontributing pixels, as one would expect since resonant frequencies determine theultimate behavior of each state at long times in the weak field limit. The
approximately two-pixel bandwidth is primarily an effect of spot size on thepulseshaper and not intrinsic transition bandwidth.
The squares show the spectral contributions to the signal at t=0.0 ps forboth states |1> and |2> using the phase mask of Fig. 5.5a. The overall lineshapefor state |1> only partially reflects the 1/∆ behavior expected from Eq. (4.2), but it
unambiguously shows an increase in the contributing bandwidth. The increase ineffective bandwidth is attributed to the application of ± π/2 phase to positively
and negatively detuned frequencies. The spectral asymmetry is a result of a17

signal contribution (arb. units)

0.0

-0.5

|2>

|1>
-1.0

58 60 62 64 66 68 70 72 74 76 78 80 82

Pixel number

Figure 5.7: Wavelength Subtraction Spectroscopy traces for shaped and unshaped
pulses. Here, the signal reflects the change in photoionization relative tobackground. The ({) is the WSS photoionization spectrum for an unshaped pulse
at a fixed pump-probe delay of greater than 6.0 ps. The observed linewidths
represent the resolution of the pulse shaper. The () show the WSS
photoionization spectrum for the phase mask of Fig. 5.5a at a fixed pump-probe
delay of 0.0 ps. The () spectrum around pixel 65 shows the effect of adding
resonant and nonresonant contributions, and that around 75 represents subtracting
resonant from nonresonant. Note the increased linewidth of the shaped pulses,
and the increase in signal while blocking pixel 74.

18

combination of diffraction effects in the SLM and an asymmetric laser pulse
spectrum. The WSS spectrum at t=0.0 ps for state |2> similarly shows that there
is an increase in the contributing bandwidth relative to the long time delay. Moreinterestingly, the results show that blocking the resonant wavelength actually
increases the signal at t=0.0 ps, indicating a destructive interference. Even
though the lineshape remains ambiguous, the increase in signal by blocking theresonant frequency on pixel 74 relative to the baseline is unambiguous. Since thewave packet signal at long times consists of recurrence oscillations with a phaseoffset of approximately zero, the resonant contribution to the signal can beconcluded to have a positive sign relative to state |1>. This suggests that the total
wave function coefficient for state |2> around t=0.0 is negative, in agreement withthe wave packet phase data.
The results can be quantified in terms of the Z-gate representation as in
Eq. (5.3), with the initial state taken to be the superposition at t=0.0 ps, and thefinal state taken to be the superposition at t=∞:res1 (∞ )< .1 × res1 (∞ ) 

(
)
c
f
 1 
 c1 (i )
res1 (0 ) + nonres1 (0)=c ( f )
,
res 2 (∞ ) c 2 (i )
 2   < .1 × res (∞ )1
res 2 (0 ) − nonres 2 (0 )

(5.4a)

where resi(t) and nonresi(t) are the resonant and nonresonant contributions to statei at time t. The off diagonal elements are a result of cross-talk between control
channels which have previously been quantified with respect to the resonant
contribution to the coefficients[33]. These off diagonal elements are essentially ameasure of the degree to which the dynamics of each state can be independently

19controlled. The diagonal elements of Eq. (5.4a) can be taken directly from theexperimental data. The first diagonal element is determined to be 0.71, based
upon the data from Fig. 5.2. The second diagonal element can be determinedfrom the coherent oscillation amplitude shown in Fig. 5.5. Around t=0.0 ps, thepeak-to-peak amplitude of the coherent oscillation is approximately 1.1 times
greater than that at long time delays. Given that the coherent oscillation
amplitude is proportional to |c1||c2|, and that |c1|≈1.4, then |c2(t=0)| ≈ 0.79|c2(t=∞)|.
This gives a value of -1.3 for the second diagonal element of the transient Z-gate
matrix, to yield an experimental transformation of< .1   c1 (i )
 c1 ( f ) 0.7 ± 0.2.
c ( f ) =  < .1
− 1.3 ± 0.2 c2 (i )
 2  (5.4b)

Ideally, as long as a phase of ± π/2 is applied to the various nonresonant
contributions to the wave packet, there will only be an addition or subtraction of
those components relative to resonance, so the matrix components in this case canbe assumed to have very little imaginary component. In other words, allcontributions to the signal have been manipulated to have phases of either zero orπ, which lie along the real axis of the complex plane. Experimentally, we observe
the phase to be centered around zero or π, but limitations in the instantaneous
phase detection introduce substantial error, see Fig. 5.5.
The instantaneous wavefunction phase could be better controlled anddetermined if the energy spacing between states were greater. The ability tocontrol one state versus another increases as the state spacing increases as a resultof the intrinsic 1/∆ Rabi oscillation amplitude of Eq. (4.2)[33]. In the current

20

case, state |1> is used for a type of heterodyne detection of the phase of state |2>,
where the first state is used to serve as a reference for the second. Theinterference between |1> and |2> permits effective viewing of the phase of state|2>, but only at the maxima and minima of the interference signal, giving a data
point every 2/ω12 ps, where the state spacing ω12 is given in units of THz. With
the current state spacing of 42 cm-1, one data point is retrieved every 0.3 ps, which
is nearly half of the timescale over which the phase of state |2> changes from π to
0 radians. Increasing the state spacing, or increasing the coherent oscillation
frequency, will provide a more accurate probe for how the phase change occurs.
Additionally, a more rapid coherent oscillation will provide a more accurate
amplitude of the coherent oscillation.
In this chapter, we outlined a system for temporarily switching the relative
sign of one state in a superposition while keeping the sign of a second state
constant. We showed that this can be done by simply using weak field, phaseshaped ultrafast pulses. Additionally, we showed, using established spectral
techniques, that the resonant and nonresonant contributions to a specific transition
can be programmed to add together or to work in opposition on short timescales.
This work expands upon established methods for controlling transient processes
in molecular systems and presents a case where both the phase and amplitude ofan excited state depend upon more than resonant frequencies.

21

Chapter 6
Strong Field Population Control: Theory and Experiment
6.1

Introduction
Ultrafast lasers have come to be used all over the world to study chemicaland physical dynamics on the shortest of timescales. With the short timescales
inherent to ultrafast lasers often comes high peak laser power. This cansometimes be problematic, since a system response to a strong field, such as that
produced for a short time by an ultrafast laser, can cause interesting, orconfounding, effects. Among these effects are multiphoton absorption[6, 20, 23],transition saturation effects[66], ultrashort X-ray generation[26, 67], or evennuclear fusion[68, 69]. Before the advent of ultrafast lasers, studies into strong
field dynamics with nanosecond lasers have illuminated several of the sameprocesses now also observed by ultrafast lasers[5], and there have even been
examples of coherent control using cw lasers[36].
The study of wave packet dynamics and coherent control has a fairly
broad base with respect to weak field processes where the outcomes areconveniently described using perturbation theory[9, 12, 17, 18, 70, 71]. There is awell established formalism for describing the response of molecules in thestronger field regime using the dressed state approach[36]. Somewhere in
between these two regimes lies what would be considered stronger than
perturbative, but which is not conveniently described using the dressed state
approach. This chapter will investigate this middle ground, first theoretically andthen experimentally.

22

One of the major goals of the analyses and experiments described here isto verify the complete depopulation of a single state in a molecule. This has beenshown in many types of regimes, with and without pulse shaping[5, 36, 72].
Besides dynamic processes such as population inversion in lasers, stable
population inversion can be seen in isolated systems with the application of asingle pulse, producing Rabi oscillations in an excited state population[5].
Besides simple atomic systems, this effect has been observed or predicted incomplex systems such as semiconductor quantum dots or molecules[52, 73].
With a little bit higher fields and even rudimentary pulse shaping, full populationtransfer to an excited state via Rapid Adiabatic Passage or to a third state withstimulated Raman via adiabatic passage (STIRAP) has been demonstrated on
numerous systems[36, 66, 72, 74-79]. Often, in these cases, the systems arecharacterized by a single resonance or a series of resonances that are well
separated in energy.
One way the theory and experiments in this chapter will expand upon thisprevious groundwork is that all transitions are embedded within the spectrum of asingle broadband laser pulse, introducing a level of richness only recently
addressed[23, 75, 80, 81]. Another is that these experiments are rotationallyresolved, allowing a precise level of bookkeeping. Since the theory andexperiments described here focus on second and third order coherences (i. e.
involving two or three photons, respectively), some similar types of coherences
can be created using various four wave mixing techniques[39, 82]. Two traits
separate these experiments, however. First, these experiments will show evidence

23for a depopulation of states, and second, these methods can be extended to study
an arbitrary order of coherence without any intrinsic changes in methodology.
The successful development of this higher order pumping could be ameans for easily selecting rotational quanta by coherent control. It is also a goodmodel system for studying moderately dispersed processes such as populationtransfer among several vibrational states in a real molecule[24, 83, 84]. In fact,
the system described here simply represents the dynamics at play for a single
rotational state in a typical ground state ensemble—this study provides a basis forunderstanding the limitations of population transfer within an ensemble ofrotational states. The techniques described in this chapter create what isessentially a rotational analog to vibrational ladder climbing[72, 78, 84-87]. In
essence, the lessons learned here can eventually be used to generate highly
excited rotational states, but with state specificity. One other important
application of the lessons learned here is the potential for manipulating quantum
information systems. In some material systems (such as trapped ions), series ofunshaped population inverting pulses are used to switch qubit states, but analternate method of population switching based upon ladder climbing withchirped pulses is presented here[63].
This chapter will only focus on the requirements that need to be met tomeasure state-resolved population transfer. We study an optically coupled system
consisting of a progression of J values for a single vibrational level on each of twoelectronic states, so that the total wavefunction consists of a superposition of
optically excited J states on the A1Σu+ and E1Σg+ electronic potential energy

24curves. Experimentally, to measure populations, quantum beating between states
will act as the primary sensor. One difficulty with excitation by higher fields isthat the strong excitations on a theoretically isolated system may induce too manytransitions. To see state resolved population transfer within the coupled system
requires that the finite starting population not spread out among many different
rotational states in the coupled system via optical pumping. This process of
spreading out of wave function via optically induced ∆J=±1 transitions will bereferred to as wavefunction dissipation. This wavefunction dissipation reduces
the amplitude of a single observed quantum beat between two states since eachindividual state will have a reduced population, even if no population is lost out of
the target coupled system. A second requirement for observation is that
population not get optically excited out of the coupled system; this will bereferred to as wavefunction depletion since the population in the coupled systemdecreases. In a molecule exposed to strong field laser pulses, processes such as
multiphoton ionization and excitation to continuum states can play a significant
role in population losses. Besides dissipation and depletion effects, certain
physical phenomena such as the spatial beam profile and molecular alignment
make observing oscillations in population as a function of electric field strength
difficult[88-90], since these effects introduce inhomogeneities in the overall
transition strength.
This chapter will begin with a theoretical treatment of the backgroundissues required to consider strong field dynamics. The first section focuses on
calculations related to the effect of field strength on the populations in the coupled25

system. The calculations will start with a simple model, with the model
complexity increasing step by step to account for experimental nonidealities. It
will be shown that as the field strength increases, the degree of dissipation of the
population over various J states increases to create a superposition of many J
states at surprisingly low field strengths. Next, the influence of a spatiallynonuniform electric field, as expected by a Gaussian beam, will be addressed, as
well as the effect expected from using a specific probe beam profile. The
reduction in the visibility of molecular Rabi oscillations because of alignment
effects will also be discussed. Following this theoretical section, the field strength
dependence of rotational quantum beats will be analyzed, and an example of wave
packet dissipation will be shown. Depletion mechanisms are proposed, and ameans to overcome this via spectral filtering is tested. The possibilities ofcontrolling time and field strength dependent dynamics will be considered in boththe theoretical and experimental sections with a simple case of implementing a
Rapid Adiabatic Transfer Passage (RAP) using a chirped pulse[91], with thesecondary goal of finding a population transfer scheme that shows robustness
with respect to electric field strength.

6.2

TheoryThis section will focus on the interactions between an electric field andtwo optically coupled progressions of J states. The goal of this theoretical section
is to define and then control wavefunction dissipation. First, the theoreticalformalism will be established, followed by an analysis of the response to a strong26

ultrafast laser pulse of a system with numerous (~15) transitions embedded within
the laser bandwidth. Here, the term strong refers to a breakdown of the normal
weak field assumption that the population in the launch state remains near unity.
Instead, the observed population of the launch state goes through a series of
decaying oscillations as the total wavefunction dissipates to larger and larger
numbers of J states on the A and E potentials. This will be followed by ananalysis of expected trends in electric field strength dependence assuming
selected probe pulse characteristics and alignment issues.

6.2.1

Spatially uniform fields

The dynamics used to model the system under the assumption of a
spatially uniform laser intensity are summarized below. First, it is assumed that apure launch state is used (vA,JA=11,28), as in all previous chapters. Disregarding
depletion of the population out of the A1Σu+(vA=11) and E1Σg+(vE=9) vibronic
states, the time dependent wave function has the following form:∞Ψ (t ) = ∑ c j (t )ν j J j
j =− J0∞∑ c j (t )j =− J02.

(6.1a)

where each wavefunction ν j J j represents one of the rovibrational
wavefunctions shown in Fig. 6.1, and cj(t) represents the amplitude of eachwavefunction. It is instructive to note that all odd J states are on the E electronic
potential, and all even J states are on the A electronic potential. All J quantum
numbers are offset relative to the launch state of J=28, so the summations go from-28 to ∞. To reduce calculation times, the summations are over J=22-34, and P

27

a E1Σ+gstates

3-1

Energy (x10 cm )

29.8
29.6

-4

29.4
17.4

-3

-2

-1

1

17.2
17.0
1.0

1 +

A Σ u states
20

22

b
-7

ϕ (rad/2π )

4

3
2

-5

24

26

-3 -12

4

6

28

30

32

J state
-6

-4

-2

1

3

34

5

36

7

0.5

0.0

56

60

64

68

72

76

pixel number
Figure 6.1: Summary of resonant frequency positions. Panel a: Energy levels
considered in this theoretical treatment. Resonant transitions are notated by theirgreatest proximity from the launch state of J=28. For example, a transition
between J=31 & 32 is noted 4 (32-28=4) regardless of whether it is a “pump” or“dump” process. Panel b: Resonance locations as imaged onto the SLM. The
dotted line represents the Gaussian spectral envelope.

28

and R branch pump (and dump) transitions are notated relative to their greatest
proximity to the J=28 state in J space. In all of the calculations presented here,
the wavefunction fails to dissipate to the J limit, justifying the J space truncation.
Each of the coefficients cj(t) are calculated as∞

c j (t ) ∝ i ∫ dtε (t )∑ ck (t )σ jk e−∞iω jk t(6.1b)kwhere the summation is over all states k, ε(t) is the electric field, σjk is thetransition moment, and ωjk is the resonant frequency between any two of thecoupled states. With a typical minimum pulse length of ~150 fs compared to theoptical cycle of ~2.87 fs, the slowly varying envelope approximation is applied,giving a temporally varying electric field as

ε (t ) ≈ N FS E (t ) cos[ω (t )t + φ (t )] .

(6.1c)

where |E(t)| is the pulse envelope, and NFS is a field strength normalization
parameter, to be described below. The time dependence of the frequency and thephase has been made explicit to account for pulse shaping situations. Since only
P and R branch transitions are strongly allowed between the A and E states, theoscillator strength varies only slightly with J (<5% for mJ=0)[92], so if we assume
that all states are in mJ=0, σjk =δ(j, k=j±1). Following the first few calculations,the mJ dependence of the transition dipole will be made explicit. With the JA=28
launch state, this selection rule establishes that all even J states are on the A state,
and all odd J are on the E state. The exploitation of the ∆J=±1 selection rulesimplifies the above scheme to 14 coupled differential equations. For a two levelsystem, this dynamical model reduces to the classic Rabi formula:

29

[

]

ce (t ) = 1 − cg (t ) = ∫ d∆ 1 − (σ egε (∆ ) Ω eg ) sin 2 (Ω eg t )
2

2

[

2

]

Ω eg = ∆2 + σ eg ε (∆ ) ,2(6.2a)(6.2b)

where ∆=ωeg-ω is the detuning away from resonance, and σeg is the transition
moment from the ground to excited state[93].
In the dynamics modeled here, we express the electric field as outlined in
the literature[34], and summarized in Eq. (3.1a), with a central wavelength of
801.7 nm and a Gaussian bandwidth of 9 nm, imaged onto the SLM over 35
pixels with a single frequency spot size of 1.7 pixels, with all extents expressed as
FWHM (see Fig. 6.1). In many of the calculations and experiments in thischapter, the system dynamics as a function of the normalized field strength, NFS
from Eq. (6.1c), are explored. The excitation pulses are normalized so that for a
two level system [i.e. σ28,29=1, all other σjk=0, see Eq. (6.2)] with an unshaped
pulse, the time integrated product of ε(t)σ28,29t=π holds for NFS=1. The unit NFS isused instead of pulse area to avoid confusion with the beam size (see [5]). Given
that several of the calculations and experiments involve phase and amplitude
shaped pulses, NFS for a shaped pulse is expressed in terms of the unshaped parent
pulse. This allows the comparison of shaped and unshaped pulses.
In contrast to the two state calculation, when all of the strongly allowedtransition moments (σJ,J±1) are set to 1, the wavefunction will dissipate to anincreasing number of rotational states as NFS increases. Figure 6.2 shows theresults of this dynamical model as applied to this system for an unshaped input
pulse. In this calculation, probe beam and alignment effects have not yet been

30

0.4

a

|cn(t)|

2

0.3
0.2
0.1
0.0

0

0.3

|cn(t)|

2

0.4

1

b

NFS 2

3

NFS=3

0.2
0.1
0.0
-200

-100

t (fs)0100

200

Figure 6.2: Dissipation of wavefunction over various J states for mJ=0. An
unshaped spectrally Gaussian pulse is used, and the populations are normalized.
In both panels, the launch state (z), ∆J=±1 (U,V), ∆J=±2 (ª,±), and ∆J=±3(solid lines) are included, along with the average of the shown E state populations( S). Panel a: The t=∞ population distribution as a function of field strengthnormalization parameter NFS. Panel b: The time evolution of the J-state
distribution as a function of time for NFS=3. Notice in both panels how states at+J and –J are approximately the same.

31

implemented. The population transfer in this unshaped case shows similar
dynamics as a function of both time and NFS, demonstrating oscillating
populations. First, the launch and ∆J= ± 1 states show an immediate onset ofoscillatory behavior, with a phase shift between the two, and the oscillations forthe ∆J=±2 and ±3 states show a delayed onset of oscillations. This approximatelymeans that starting with ∆J=0, population goes to ∆J=±1, then to ∆J=0, ±2, thento ∆J=±1, ±3, etc., all via ∆J=±1 transitions. The J states more remote from thelaunch state (not shown) show a similar delayed onset of population followed by
the expected oscillatory behavior. Since all even states are on the A state, andodd states on E, this will produce oscillations in the population between electronic
states, with almost complete inversion at high field strengths as all of the even(and odd, respectively) oscillations come into phase at the same field strengths.
One interesting observation here is that the oscillation frequency for electronic
inversion is twice that for an isolated two state system. Of course, this is easily
explainable because in the full model, each state gets depopulated via twochannels, namely the P and R branch transitions.
The observed signal is slightly more complicated than a simple
observation of populations, since the observable is actually a quantum
interference, or coherence, between states that share a common final state[33, 48].
In the experimental system, photoionization occurs via a weak excitation to anarrow band of incoherently autoionizing Rydberg levels, converging on the v=0vibrational state of the ion[47]. Although multiphoton ionization out of the E
state has been observed, the primary coherence producing mechanism involves a32

single photon. To observe a wave packet, this means that the observed wave
packets will have a ∆J=2 component (i.e. JE=27 interfering with JE=29), as thesestates share the common final state of the average J between the interfering state
via the ∆J=±1 single photon ionizing probe, which is accessible by half of the
population from each of the interfering states. Similar arguments suggest that the
primary wave packets seen on the A state will have a ∆J=2, even though they arephotoionized with at least two photons of the probe pulse which has a selection
rule of ∆J=0, ±2[6, 94]. As applied to Fig. 6.2, these beats will follow pairs of
populations that satisfy the ∆J interference requirements for a common final state.
In an analysis of beat amplitudes as a function of pulse amplitude, the oscillatory
behavior will be seen, with beats on the A and E states having a phase shift intheir oscillations, like individual populations on the A and E potentials. At lower
pulse amplitudes, this behavior will be more complicated, but the qualitative trend
remains.
As will be discussed later, the ability to observe wave packet dynamics in
the high field requires little dissipation of population within the coupled system ofinterest. One potential method for minimizing this dissipation is to attenuate
wavelengths of light that do not participate in the dynamics that one wants toinvestigate. For this reason, it is useful to quantify the importance of various
spectral contributions to the signal. To perform this, WSS spectra were calculated
for a variety of pulse amplitudes in a two level system, shown in Fig. 6.3. The
bottom of the figure represents the excited state population as it would appear in
our current setup with the sequential attenuation of light on single pixels. As

33

6

N

FS

4

2

56

58

60

62

64

66

68

70

P ix el n u m b e r

Figure 6.3: Calculated WSS spectrum as a function of normalized field strength
for a two level system. Here, NFS is equivalent to the total pulse area multipliedby π. The figure shows the WSS spectrum based on the experimental traits of thepulse shaping setup, with a single resonance centered on pixel 63. White
indicates excited state population. The population inversion seen while blocking
pixel 63 indicates a nonresonant Rabi oscillation. Note how, at the lowest
powers, a notch in the excited state population appears when blocking the pixel
63.

34

highly nonresonant frequencies are attenuated, we see no significant change in the
excited state population: Rabi oscillations in the excited state population are
observed. As the resonant frequency is blocked (pixel 63), the population
inversion occurs at NFS= 4 instead of at NFS=1, indicating that the Rabi
oscillations have a frequency of approximately 8 instead of 2. This is primarily aresult of some resonant frequencies being imaged onto pixels adjacent to that
where the spectrum is centered. To try to get a more refined sense of what ishappening near resonance, sub-pixel resolution on the SLM was assumed. This
calculation confirms that at relatively low powers, the population transfer
dynamics can be considered almost as resonant processes. Unless the electricfield amplitude exceeds 4π, the effective bandwidth for each transition remains
below 4 pixels. This suggests the possibility of attenuating frequencies more than
4 pixels from any resonance without concern for significantly changing theenergy dependent dynamics. Of course, as should be expected from previous
chapters and other published work[6, 8], the highly nonresonant wavelengths
become more important if the pulses are phase shaped, as will be addressed infuture work not included here.

6.2.2

Alignment effects

An additional mechanism for limiting the visibility of Rabi-type
oscillations is attributed to alignment effects, or mJ effects. In all of thecalculations in this and the previous section, it was assumed that mJ=0, which is
not necessarily the case. The presence of multiple mJ states in the initial35

wavefunction will result in two different types of effects: dispersion of Rabi
oscillation frequencies and a tendency to drive population toward high J. The
dispersion of Rabi oscillation frequencies stems from the polarization of theexcitation laser. Defining the laboratory coordinate relative to the excitation laser
polarization, the transition strength for a parallel transition, such as between the A
and E states, decreases as |mJ| increases[92]. In order to study this, the above
model was modified to fully account for the presence of multiple mJ states:∞Ψ (t ) = ∑j∑ c j (t )ν j J j mJ

j = − J 0 mJ = − j∞∑ c j (t )

2

(6.3)

j =− J 0

where all terms are the same as Eq. (6.1) except the mJ quantum numbers are
explicit. The coupled equation scheme of Eq. (6.1c) does not change except thatσjk is futher modified by Clebsch-Gordon coefficients coupling the various
rotational states. It should be noted that intrinsic to the transition strengths is therequirement that |mJ|≤J, so a transition from J,mJ=29,29 to J,mJ=28,29 has atransition strength of zero, since the 28,29 state does not exist. Since the time or
NFS dependence of population evolution show similar dynamics for an unshaped
Gaussian pulse (see Fig. 6.2), the time evolution of the population dynamics will
be shown here and inferred to reflect the NFS evolution.
If multiple mJ states are populated, there will be multiple mJ dependent
Rabi oscillation frequencies, as seen in Fig. 6.4. Shown is the evolution of thelaunch state population for three selected mJ from a uniform distribution of mJ
states under an excitation by a field with NFS equal to 2 with the transition dipole
moment and electric field polarizations parallel to mJ=0. At short times, all mJ

36

1.0

J=28 population

0.8

0.6

0.4

0.2

0.0
-400

-200

0

200

400

t (fs)
Figure 6.4: Evolution of J=28 population for various mJ states under excitation
with a laser field with NFS=2. mJ quantum numbers of 0, 14, and 28 arerepresented as the solid line, (+) and ({), respectively.

37

states are equally populated, but at longer times the mJ distribution clearly
deviates from uniform. To describe the nature of the evolution of the mJ
distribution, a measure of alignment (Al) for a specific J follows:
AlJ (t ) = ∑ mJ cJ , m J (t )2mJJ ∑ cJ , m J (t )

2

(6.4)

mJ

where the cJ,mJ(t) coefficients describe J, mJ states, and the J in the denominator is
included since the approximate angle of a single mJ state depends on J. Assumingthat the total observed population is an incoherent sum of mJ states, the presenceof this distribution will reduce the depth of modulation of any observed Rabi-type
oscillations and cause the molecules to come into and out of alignment as afunction of time or NFS[89, 90].
Figure 6.5 summarizes the population evolution as a function of time for
J=28 and NFS=2. In Fig. 6.5a, the limiting case of a uniform mJ distribution caseshows a significant decrease (~20%) in the observed depth of modulation of theoscillations in population. In experiment, the cw and probe lasers are mostsensitive to small |mJ| for the same reasons cited above, in effect sampling small|mJ|, so the calculation was also performed assuming an additional alignment
inducing laser. The time dynamics of the J=28 population initially prepared by aweak field cw launch laser shows a depth of modulation of ~95% of the mJ=0
case, also shown in Fig. 6.5a. A second weighting laser (such as an ultrafast
probe) will further sample small |mJ|, but subsequent calculations take aconservative approach and assume a single mJ sampling laser. Pulse shaping
strategies can be put into effect to account for this effective spread in mJ
dependent transition strengths, and they will be analyzed in section 6.3.2.

38

alignment

J=28 population

1.0
0.8

0.2

a

0.6
0.4
0.0

0

0.2
0.0
-300
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
-300

-200

-100

-200

-100

t (fs)

200

0

100

200

300

0

100

200

300

b

t (fs)

Figure 6.5: Effects of alignment on total population in J=28 state. Panel a: total
J=28 population as a function of time for NFS=2. Circles represent the population
with a uniform distribution of mJ starting states. Plusses represent the J=28
population assuming the presence of a mechanism that samples the mJ space—such as a single weak probe pulse. The solid line shows the evolution of the mJ=0
state. All populations have been normalized to a starting population of 1. The
inset expands the peak around t=50 fs. Panel b: alignment as a function of time
for J=28 state. Circles and plusses indicate the same conditions as in panel a. In
the sampled case, the alignment more closely approaches zero, which is explainedby preferentially sampling low |mJ| states. The peaks around -20 fs represent
times when population has been transferred out of the low |mJ| states, leaving
primarily high |mJ|.

39

In accounting for the total mJ distribution, an interesting alignment trait
comes out of the calculation. Consider the J=28 launch state, for example. Earlyin the evolution of its population, the first |mJ| states to be taken out of its total
population will be parallel small, leaving large |mJ|, as is shown in Fig. 6.5b. For
NFS=2, the alignment, as defined by Eq. (6.4), goes through a maximum at t=-20
fs, in effect creating an incoherent rotation, or wobble, as the incoherent sum ofmJ states oscillates between a parallel and perpendicular net alignment. Even in
the presence of a low |mJ| weighting laser that preferentially aligns the population
parallel to the laser polarization, this wobble still exists at short times.
Alignment effects can also influence the preferred direction (+J instead of–J) of a transition with linear laser polarization, since we expect a selection rule of∆mJ=0. Given this selection rule and the basic degeneracy limitation that |mJ|≤J,
choosing a launch state of J=28 imposes the condition that |mJ|≤28 for J≥28 and|mJ|≤J for J<28, referred to here as population “degeneracy filtering”. Figure 6.6
shows that high J states are favored in this rotational system due to thesedegeneracy effects using spatially uniform pulses. The time dynamics for the∆J=±3 (i.e. J=25&31) states under excitation with an NFS=2 pulse is shown in Fig.
6.6a, with the J=25:J=31 population ratio equaling 0.81. This ratio is very close
to the ratio of the J state degeneracies between the states, but it should be notedthat the population ratio does not rigorously have to match the ratio of J statedegeneracies. For example, after excitation using NFS=2 the +∆J:-∆J population
ratios for the ∆J= ±1, ±2, ±3, ±4, ±5, ±7 states are 0.97, 0.95, 0.81, 0.90, 0.98,

40

0.20

a

Population

0.15

J=31

0.10
J=25

0.05

Σ P(J<28)/Σ P(J>28)

0.00
1.00

0

100

200

300

t (fs)

0.96

b

0.92
0.88
0.84
0.80

0

1

2

NFS

3

4

5

Figure 6.6: Degeneracy dependent selectivity for high J vs low J. Panel a:
population evolution for J=25 ({) and J=31 (+) for a pulse amplitude with NFS=2
and an initially uniform mJ. The final J=25:J=31 population ratio is about 0.8.
Panel b: Ratio of the sum of all J state populations below the launch state versus
those above the launch state as a function of pulse amplitude NFS.

41

0.98, and 0.94, respectively. This deviation is primarily a result of the fact that
this degeneracy filtering is occurring where |mJ| is large, and hence has arelatively weak transition probability. As the optical transitions become stronger,
the population ratio for those states with low J versus high J will decrease[Σ(J<28):Σ(J>28)], as shown in Fig. 6.6b. Plotted is the ratio of the sums of
populations for J<28 versus J>28, where it is shown that increasing NFS increases
the selectivity to higher J. In the figure, the low to high J ratio for low NFS
approaches .965 and reaches 0.84 at NFS=5. At lower NFS, the ratio is fairly
insensitive to changes in the field amplitude since the primary states to be
involved in a transition will have small |mJ|. The ratio of low to high J
approaches the average ratio of degeneracies between the low and high J states,
0.75, only at the highest powers. This can only occur in a system that can besaturated, such as this case of a 15 J state model, illustrating a limitation of thismodel at the highest powers. In the experimental system, since J can approachinfinity, saturation will not occur, so the ratio of populations will remain above
the ratio of J state degeneracy. Additionally, our experimental laser fields only
approach the equivalent of NFS=~2, reducing the importance of this effect.

6.2.3

Spatially Inhomogenous Fields

In the above analysis, a single value of NFS represents the strength of theelectric field. In experimental situations, a single value often fails to describe theelectric field[88]. In a typical case, the laser beam used to excite any transition is
not uniform, rather spatially Gaussian. In our gas phase sample, the nonuniform

42

beam excites molecules in different regions of space. The molecules respond as if
the excitation comes from a number of independent lasers with varying powers
since the molecules are basically uncorrelated with respect to our method ofsignal detection. For this reason, the spatial extent of our laser beam will be madeexplicit in further calculations. Since our signal is a result of a pump-probe
process, the excitation produced by the pump laser gets spatially sampled by the
non-uniform probe pulse, so that pulse will be included in the calculations.
The spatial cross section of a Gaussian beam is described by thefollowing:[ε (R) = N FS ε 0 exp − (2 / ln 2)(R − R0 ) FWHM 2
2

]

(6.5)

where NFS |ε0| is the normalized electric field amplitude at the center of the beam,
the spatial extent R is relative to the center of the beam, and FWHM is the full
width at half maximum of the beam radius. This effective field amplitude thengets plugged into Eq. (6.1c) and Eq. (6.2) giving a spatially dependent Rabi
frequency. Since the signal has spatial dependence, each of these frequencies getsweighted by a radial factor R. Now the time dependent wave function of Eq.
(6.1a) contains an R dependence, and the overall signal is related to it:
S (t ) ∝ ∫ RdR Pr (R, t )Ψ (R, t )2(6.6a)R∞

Ψ (R, t ) = ∑ c j (R, t ) v j J j
j =− J 0∞∑ c (R, t ) ,j =− J 02

j(6.6b)

Above, Pr(R t) simply refers to a spatial sampling condition fulfilled by thespatially inhomogenous probe pulse, and all summations are relative the launch
state J0=28. In this particular subsection, alignment effects will be ignored to43

more accurately illustrate spatially specific effects. Additionally, the detectedsignal is an incoherent integral over R since the ions are detected via an
incoherent autoionization of weakly excited Rydberg states[47], so there will be
no evidence for spatial coherences in R when taking the modulus squared of the
wave function. The R coordinate will only act as a weighting factor.
In experiment, the probe pulse is approximately spatially Gaussian of thesame size as the pump, but this assumption carries with it a few caveats. The
calculated dependence of the Rabi oscillation visibility versus the size of the beamin the probe step for a two level system is given in Fig. 6.7. In these calculations,the diameter of the probe pulse changes relative to the pump pulse, but the
integrated flux remains constant, as if the probe is focused relative to the pump.
As the probe size decreases, the observed excited state population behavior
evolves from a monotonically increasing type dynamic to a more oscillatory typedynamic. This simply results from sampling a smaller range of field amplitudesof the pump pulse as the probe size decreases. The limit of excitation with auniform field would look similar to the smallest probe size, except that probe size
would not affect depth of modulation of the Rabi oscillations. On the other hand,using a detection scheme that does not sample a small volume in the pump beamwould resemble the large probe limit, with virtually no depth of modulation of the
NFS dependent Rabi oscillations. An example of such a detection scheme would
be in an experiment studying laser induced fluorescence with a single photon
excitation.

44

w=.25detected population

w=.5
w=1.0

w=2.0

0

1

2

NFS

3

4

5

Figure 6.7: Rabi oscillation visibility for various probe sizes. The probe is
assumed in all cases to have a constant fluescence, but changing relative focusing
conditions. The detected signal is further assumed to be linear in probe power.
The full width at half maximum of the probe is given by w in units of the pumpwidth.

45

To demonstrate the possibilities for using pulse shapes to control the
dynamics of population transfer in this particular system, we examine chirped
pulse excitation as a function of NFS. The form of the basic wavefunction now
contains aspects of Eq. (6.6b) and Eq. (6.3):∞

Ψ (R, t ) = ∑∑ c (R, t )ν J m

j =− J 0 mJ =− j

j

j

j

2

∞

j

J

∑ c (R, t ) ,j =− J 0

j(6.7)

where both alignment and beam inhomogeneity effects have been taken into
account. In these calculations, an alignment inducing (or mJ sampling) cw laser isassumed. For a probe, a beam with the same spatial extent as the pump isassumed, and a single photon is used to photoionize all E (odd J) states, and two
photons are used to photoionize all A (even J) states. In experiment, the A state is
photoionized via a two-photon process at approximately 75% of the efficiency of
the E state since its signal reflects a smaller sampling region of the excitation
region [see Fig. 6.10]. To account for this, the spatially integrated observed
population for the A state is set to be 70% of an equally populated E state.
Figure 6.8 shows several states’ population dependence on NFS using an
excitation pulse with a chirp of 4500 fs2. This chirp was chosen for its energetic
robustness relative to other chirps not shown here. This figure deviates
substantially from the example of excitation with transform limited pulses. First,
recurrences in observed populations no longer exist except at fairly high NFS;
instead the population transfers to the ∆J=2 state for a large variety of NFS[86]. In
fact, at specific NFS values, there is essentially full population transfer between the
launch and final states, but it is not completely visible in the presence of anonuniform excitation pulse. The populations of the ∆J= ±1 states show signature46

J0

relative signal∆ J=+2∆ J=+1∆ J=-1
0

1

2

3

4

5

NFS

Figure 6.8: Calculated populations as a function of normalized field strength for
positively chirped pulse. J0 indicates the launch state of J=28. Note that all evenJ states lie on the A electronic state and odd lie on the E electronic state. The
chirp is set to 4500 fss. The ∆J=-2 state is essentially zero at all NFS. The E (odd)and A (even) states are probed by a beam with the same diameter of pump with 1
and 2 photons, respectively.

47

dynamics of going through a maximum at fairly low NFS then slowly decaying tonear zero, with a small recurrence at very high NFS, and the ∆J=-2 state gets very
little population transferred to it at all times shown here. Again, the only reason
the ∆J=±1 populations do not go to zero is that the electric field is not uniform. In
the case of a uniform electric field, the amplitude of all states except ∆J=2 will goto zero over a wide range of energies. The mechanism behind this energeticallyrobust process is that there are two sequential Rapid Adiabatic Passage events thatare separated in time because of the overall chirp of the excitation pulse[52, 66,
71, 72, 78, 86]. This effect has been shown in theory and in atomic excitation
where only a single pathway is possible[95], and theoretically with respect tovibrational excitation[81]. Inverting the sign of the chirp will reverse the timeordering of the wavelengths in the pulse, subsequently inverting the Rapid
Adiabatic Passage events, giving final state selectivity (not shown).

6.3

Experimental Observations

The goal of this section is the show dissipation of the wave function froma single rotational pair into a much more complex set of states, and then to control
the dissipation. Next, it is shown that different quantum beats have different
electric field amplitude dependence, reflecting the order of the beats. Finally, asequential Rapid Adiabatic Transfer dynamical scheme is implemented.
Selectivity between final state populations is shown to depend on the sign of the
chirp.

48

To accomplish the experiments described here, the only change to theexperimental setup relative to experiments described in previous chapters is thatthe pump and probe beams pass through a 25 cm focal length lens just at their
input to the heat pipe. This gives a spot size of approximately 100µm FWHM.
For the pump beam, the average power in the heat pipe is approximately 200 mW.
Given a pulse duration of approximately 200 fs, and with a repetition rate of 200
kHz, the peak power density is approximately ~1011 W/cm2, several orders ofmagnitude greater than the weak field experiments described in earlier chapters.
As it has been shown in the previous calculations, many of the effects that we
expect to observe depend primarily on resonant wavelengths. This allows us toattenuate much of the nonresonant pump light, resulting in a peak power closer to1x1010 W/cm2. Still, this intensity will adequately induce Rabi oscillations as
described above.
As the power of the pump pulse increases, higher order states become
populated at the expense of the population in the launch state. One way ofobserving this is to identify quantum beats at new frequencies, as summarized inTable 6.1, and shown in Fig. 6.9. In Fig. 6.9, the quantum beat spectrum for two
different pump energies using the JA=28 launch state clearly shows the populationof a new set of states when the peak pump power increases. The high field traceshows not only the quantum beat between the JE=27&29 states, but also a set ofbeats between JA=28&30 and JE=29&31. The JE=29&31 beat require at least athird order process to populate the JE=31 state, so its appearance at higher fields

49Beat frequency(cm-1 . ±0.7 cm-1)

Electronic
stateJstatesBeat
Order*

39.8

E

29-31

4

42.5

E

27-29

2

43.8

A

30-32

6

44.8

E

25-27

4

46.9

A

28-30

2

50.0

A

26-28

2

53.1

A

24-26

6

Table 6.1: ∆J=2 Quantum beat frequencies for JA=24through 32.
*Beat order is defined as the sum of the |∆J| of the states involved in the quantum
beat relative to the launch state. From Eq. (6.7), this means that the beat order isdefined relative to NFS instead of pulse intensity.

50

Amplitude

JE=27-29

JE=29-31

30

35

40

JA=28-30

45

-1

50

55

60

Frequency (cm )Figure 6.9: Fourier Tranforms for quantum beats at two different peak fieldstrengths. The ({) represent the FT for a peak pump power of ~108 W/cm2(unfocused beam), and the (ª) represent the FT for a peak pump power of ~1011
W/cm2. Notice the appearance of new beats in the stronger field case. In the low
power case, the A state beats were not observed due to insufficient probe power todrive multiphoton ionization. At the high power case here, the low J beats(JE=25-27 and JA=26-28) are minimal due to spectrally weak probe light at thefrequencies that photoionize these states, i.e. the optimal pulse for E ←A
population transfer is slightly suboptimal for low J photoionization.

51

should come as no surprise. An additional quantum beat at 44.8 cm-1 between
JE=25&27 on the E state is sometimes present, but is not observed here for two
reasons: the probe spectrum favors the higher J quantum beats, and the beatfrequency slightly overlaps the JA=28&30 frequency. Also, the lack of theJA=26&28 beat at 50 cm-1 suggests that photoionization yield for low J is small.
This occurs here because the wavelength to which the laser is tuned to optimallypump the states of interest is suboptimal for multiphoton ionization of the lower J
part of the rotational manifold of states.
Quantum beats that are composed of states excited via high order process
will show energy dependent dynamics that vary wildly from linear[66]. The
quantum beats as we observe them consist of an interference between two states.
As a reminder, the coherence term from Eq. 1.1 has a cosine term with a leading
factor that contains the wave function coefficients for each interfering state:
Coherence ∝ |pr1pr2||c1||c2|cos(ω12t+ϕ)(6.7)

where all terms have been previously defined. Of particular interest is therelationship between the coefficients cn and the electric field amplitude. In the
weak field, first, second, third, etc. order processes are related to the electric field
to the first, second, third, etc. power. The quantum beat for a pair of first orderstates (i.e. JE=27-29) goes as the electric field squared, so it is considered a second
order beat. The primary type of quantum beat with the launch state on the A state
will also go as the electric field squared, since it involves a product of a second
order and a zero order state, giving a net second order process. As the transition
strength moves out of the weak field, these simple assumptions become more52

difficult to apply. Figure 6.10 shows the quantum beat amplitudes for three typesof beats as a function of pulse amplitude.
To gather this data, first the spectrum was manipulated so that the only
light to pass through the SLM was a narrow band of ±8 cm-1 around eachresonance between the JA=28 and JE=27 and 29 states, enough bandwidth toencompass all first, second, and third order transitions [see Fig. 6.1b]. Next, thebandwidth allowed to pass was attenuated, and a pump-probe trace was taken foreach pump intensity. Next, we performed a Fourier Transform of the timedependent signal. The 42 cm-1 beat (JE=27&29) shows an initial rise followed bya longer decay with a recurrence at the highest NFS. The 47 cm-1 (JA=28&30) beatshows similar behavior, except it shows no recurrence at the highest energies. If
both of these beats are second order processes, they should follow the same
trajectories vs. NFS at the lowest powers. A third beat at 39 cm-1 (JE=29&31) isshown in the figure. This beat clearly differs from the first two, verifying that thisis indeed a fourth order beat as defined in Table 6.1. It shows a threshold at
which the quantum beat appears, much like the threshold behavior of calculated
third order populations in Fig. 6.2. At the point where the quantum beat appears,
significant population has departed from the state of JA=30 to create the thirdorder (JE=31) state.
The fact that the fourth order quantum beat starts to approach the same
amplitude as the second order beats suggests that it is being produced in parts of
the laser beam where the pulse area approaches 2π (see Fig. 6.7). Thisobservation suggests that pulse shaping schemes may be applied to the system in

53

6
-1

FFT amplitude

42 cm

4

-1

47 cm

-1

39 cm

2

0
0.0

0.2

0.4

0.6

0.8

1.0

NFSx|E|max
Figure 6.10: Field strength dependence of quantum beat amplitudes. A spectrum
containing only first order resonances (ω±1±8cm-1) and nearby wavelengths wascreated by attenuating all light except for one window four pixels wide aroundeach first order resonance. The relative electric field amplitude dependence of thequantum beats was then recorded. The 42 cm-1, 47 cm-1, and 39 cm-1 beatscorrespond to the coherences between JE=27-29, JA=28-30, and JE=29-31,
respectively.

54

order to control the population transfer to various states. To test the degree towhich the populations can be transferred through the first order states, asequential Rapid Adiabatic Passage pulse shape was tested[95]. To perform thistest, the input pulse was attenuated outside a band that contained all of theresonant transitions [see Fig. 6.11]. The light that was able to pass was chirped by
the application of a quadratic phase in the frequency domain. The net chirp
amounted to ±4500 fs2, as in the calculation of Fig. 6.11.
The result of the application of this pulse is illustrated in Fig. 6.12. Figure
6.12a shows a Fourier Transform of the recurrence signal for a pulse with zero
phase as well as for positively and negatively chirped pulses; all traces have thesame intensity spectrum. The unshaped case shows second and fourth order
beating. In contrast, the positively chirped case shows a very large quantum beatcorresponding to a second order beat (JA=28&30). Also observed is a sixth order
beat (JA=30&32) at 43.75 cm-1. The second order JE=27&29 beat is almost
completely attenuated, indicating that the second (and fourth) order states are
populated at the expense of the first order states. Here, the beat at 46.9 cm-1 is
identified as an A state beat since there is a large beat involving the ∆J=32 state,
which is very far removed in J space from the J=25 state involved in the E state

55

2π
1.0

ϕ (rad)Transmittance

0.8
0.6

1π

0.4
0.2
0.0

40

50

60

0π

Pixel number
Figure 6.11: Phase and amplitude mask used to induce sequential rapid adiabatic
passage events. A window of 84 cm-1, centered to encompass all resonances up to
6th order, allowed light to pass through the pulse shaper ({). Within that window,
a chirp of 4500 fs2 was applied (). First order resonances are marked (°).

56

Amplitude (arb units)

E27,29A30,32

A28,30E25,27
E29,31

A26,28
A24,26

40
2

45

-1

50

55

Frequency (cm )
-1

-1

ratio

47cm :42cm-1-150cm :42cm

1

0
-4500

0

2

4500

chirp (fs )Figure 6.12: Implementation of Raman state selectivity based on the sign of chirp.
Panel a: Fourier Transform of quantum beats for non-phase-shaped pulse (),positively chirped pulse ({), and negatively chirped pulse (+). Quantum beats
are labeled by their electronic state, as well as the J state quantum numbers. Panelb: Ratio of A state beats and the E state beat at 42 cm-1. In both chirped cases, theratio of A state beats is measured to be greater than 20, and is limited by thesignal to noise ratio.

57

beat also at 44.8 cm-1. Similar results, except with the opposite direction of the J
transfer, is seen for negatively chirped pulses. For the negatively chirped case,
the E state beat is reduced significantly, but the overall amplitude of the A state
quantum beats is also small since the probe wavelength for small J is suboptimal,
as in Fig. 6.9.
Fig. 6.12b contains a comparison of the amplitudes of various beats as thesign of the chirp is changed. It is quite clear that changing the sign of the chirp
effectively selects the pathway being excited by the sequential Rapid Adiabatic
Passage mechanism proposed above, with a selectivity between the Stokes andanti-Stokes quantum beats approaching unity. Also remarkably striking is the factthat the observed ratio of the A state quantum beats relative to the JE=(27&29)
beat approaches a factor of 2, even in the presence of the nonuniform laser beam.
Based on the calculations of Fig. 6.8, this selectivity can only be reached at fieldstrengths above 2π, and even at the maximum the ratio in the calculations only
reaches 5. This number should be even lower in the experimental situation, since
the A state beats are detected via a two-photon ionization, as opposed to the firstorder photoionization of the first order beats.
In addition to the ladder climbing technique described above, anothertechnique has been developed by other groups that shows a high degree ofrobustness with respect to power: STIRAP[79]. In the STIRAP Λ excitation
scheme—where there are two ground states and one excited state (e.g. only JA=28& 30, JE=29 in the above model)—the three level system is “dressed” by two
resonant excitation sources corresponding to the pump and dump frequencies. In

58the case where the dump frequency slightly precedes the pump frequency, robust
excitation of the Stokes state is seen. Unfortunately, in this experimental system,
the dump then pump pulse order that corresponds to the excitation of the ∆J=2
state is nearly identical to the pump then dump excitation to ∆J=-2. This pathway
ambiguity prevents the implementation of STIRAP in the previously describedrotational system.
In these experiments, the degree of population transfer is not the observed
quantity. Instead, the observed characteristics are coherences between states, as
in Eq. (6.7); if all population were transferred to a single state, then the coherence
would be zero. Still, it can be tentatively concluded that most of the population istransferred to the ∆J=±2 states as a result of the large amplitude of the second andsixth order beats. Also, the good agreement with the theoretical mechanism
supports the conclusion that most population finds its way to the target states.
Further experiments that are sensitive to individual populations, such as
fluorescence detection, will verify these final state populations. One expected
loss in quality of data by using fluorescence detection is the probe pulse beam
sampling effect, as described in Eq. (6.7) will be lost. This can be overcome bycarefully creating a spatially uniform electric excitation field, or by using avariant of Stimulated Emission Pumping to detect final state populations.
These experiments can also be improved by more carefully selecting thechemical system being studied to minimize loss mechanisms such as pump pulse
multiphoton ionization. An attempt to overcome this in our current experiment byattenuating nonresonant wavelengths in the pump has shown some promise, but59more study is necessary to fully characterize the spectral extent to all multiphoton
ionization pathways. We have shown that quantum beat amplitudes can beincreased by up to 70% by attenuating nonresonant frequencies. A potential
solution could be to perform these experiments on a system with a much higher
ionization potential, or even to lower the energy of the launch state so that anadditional photon of light is needed to reach the ionization continuum.

60Chapter 7
Electronic wave packets

7.1

Introduction
Wave packet dynamics have been studied extensively on single electronic

states (see previous chapters) and on multiple highly excited molecular Rydberg
states[96]. Interactions between electronic states have also been observed in thecontext of observing unimolecular reactions such as dissociation[2, 4]. Yet theinteractions between simultaneously propagating wave packets on intermediate
energy electronic surfaces has received little attention[97-101]. Vast amounts ofresearch have been undertaken to study interactions of electronic wave packets inatoms, but little has been addressed yet in molecules[102]. Calculations haveaddressed simple molecular systems such as H2[97-99], and some experiments
have been performed on the hyperfine splitting in molecular NO[101]. In all ofthese cases, the approximation can be made that the differences between
interacting electronic states are small; all electronic states are approximately
identical except for a total energy offset. Recently, experiments have been
performed that control the angular momentum of electronic wave packets inatoms[103]. The primary reasons for this lack of study of electronic wave packets
in molecules include the lack of efficient detection schemes or overall signal
strength and lack of knowledge of the states[96]. In the case of Li2, both of these
conditions can be overcome. Much study has gone into the optimization of state
resolved photoionization of Li2, and significant excited state spectroscopy

61

has been performed, making the understanding of wave packets on highly excitedelectronic states possible[47, 104-110]. The experiments described in this chapter
will examine an energetic region of Li2 that is at the upper end of those states thathave proven theoretically tractable using standard spectroscopic experiments
coupled with ab initio theory, while at the same time is at the lower end of thosestates whose description using “top down” techniques such as multichannel
quantum defect theory is valid.
Using ultrafast pump-probe spectroscopy, we observe quantum beating
between rovibrational states on the G1Πg+ and F1Σg+ electronic curves of Li2.
Using a narrow bandwidth cw laser, a launch state is prepared on the A1Σu+electronic potential energy curve (vA=11, JA=28). From this launch state, a wave
packet is prepared on these low lying Rydberg levels using the output from the
OPA (λ0 = 550 - 590 nm) and is subsequently probed by a second near IR ultrafast
pulse (λ0 = 800 nm). As the OPA wavelength is scanned, quantum beats between
P, Q, and R branch rotational states for a progression of vibrational states on the
G curve is observed. At a pump wavelength of 560 nm (Ge=35,000 cm-1,
IP=41490 cm-1), in addition to rotational beating on the G curve (vG=10-14,
JG=27-29), electronic beating between the G (vG=13, JG=27-29) and F (vF=26,
JF=27) states is observed. As a result of the relatively low lying electronic states
involved (n~4), there is a clear breakdown of the simplifying ∆v=0 propensity
rule for quantum beating in high lying molecular Rydberg wave packet work[96,102], indicating a complex vibrational structure.

62

The fact that the interfering states occupy different electronic states withdifferent angular momenta suggests a potential handle for controlling dynamics.
It has been shown in atoms that states with higher angular momentum are moresensitive to external fields than those with less angular momentum[111]. Throughthis mechanism, electric fields can be used to Stark shift one electronic state
relative to the other[112]. This process has been exploited in designing molecularwave packets based upon the splitting of an electronic fine structure within aseries of states with identical angular momentum[113].
This first part of this chapter will be divided into several sections,
including a summary of the experimental techniques and a general overview ofthe energetic region being studied followed by presentation of the experimental
results. Quantum beating between rotational states for a progression ofvibrational levels on the G 1Πg potential energy curve is shown as well as beating
between states on the G and F1Σg+ potential energy curve. An analysis anddiscussion of the experimental results including prospects for improvements in theexperimental scheme will be posited. This chapter will then conclude with asummary of some experiments performed to understand a spurious transient
signal observed with excitation by 670-690 nm light.

7.2

Experimental and Theoretical Overview
The experiments here focus primarily on the observation of wave packet

dynamics in two excited electronic states that are accessible from launch state
A1Σu+, vA,JA=11,28 (Etot=17,151 cm-1). After preparation of this launch state, as

63

described in chapter 2, output from the OPA pumps accessible excited states, then~800 nm ultrafast pulses photoionize those states pumped by the OPA output.
The central wavelength of the OPA output is scanned over a range from 17,000 to
18,000 cm-1, with a bandwidth of ~150 cm-1 FWHM. The pump-probe delay isscanned over a range from about 0.0 ps to greater than 30 ps. In the weak field,
parallel polarization, non-transient limit, the coherent signal can then be expressed
as a variation of Eq. (1.1):
S (t ) ∝ ∑∑ | a fm |2 | ami |2 + ∑∑ ∑ a fm ami a fn ani cos(ωnm + φnm , f ), (7.1a)
f

m

f

m n>m

where i, f, and m (n) refer to the launch, final, and intermediate states,
respectively. All other terms have previously been defined, but an additional
detail pertaining to the an coefficients will be helpful:a21 ∝ ε (ω21 ) ν 2 J 2 µˆ elec (R )ν 1 J1 ,(7.1b)

where 2 and 1 refer to the final and initial states of a particular single photontransition. Note that the transition dipole moment operator µ̂elec (R ) is now
dependent on both the specific pair of states in a transition and the vibrational
coordinate R[105]. The implications of this are twofold: the transition dipole
moment operator may lead to an element that can be controlled, and the FCF is
more complex than a simple vibrational overlap[105].
Several bound electronic states are energetically accessible from thelaunch state with a pump pulse energy of 17,800 cm-1[104, 107-110]. See Fig. 7.1
for the potential energy surfaces in the region examined here. Five singlet
electronic states are bound with a total energy around 35,900 cm-1 total energy.

64

40
1 +

38

6 Σg

36

5 Σg

3

-1

E (10 cm )

1 +

34

1

+

G Πg

1 +

F Σg

32

30

2

3

4

R (Å)

5

6

Figure 7.1: Potential energy curves in excitation region. Also shown is the launch
state wavefunction superimposed on the relevant energy regions, as bounded by
the horizontal dotted lines. The F, G, and 5 & 6 curves are taken from references[104], [109], and [108], respectively. At the relevant energies here, the low lying
E1Σg+ state is bound by less than 500 cm-1, so it is not included in the figure.

65

displaying the discrete wave packet dynamics that will be examined later. The
The E 1Σg+ state is bound, but is very near its dissociation limit (35,700 cm-1) inthe experiments examined here, and its near continuum behavior precludes it from51Σg+ and 61Σg+ states are included in the figure, but with the v=11 launch state,
the low lying vibrational levels that are accessible have small vibrational overlap
with this launch state wavefunction. The inner and outer turning points of the51Σg+ and 61Σg+ states fall within the spatial extent of the launch state wave
function that is completely oscillatory. In contrast, the inner walls of the F and G
potentials nearly touch, and overlap well with the inner part of the launch state
wave function. In fact, calculations predict three orders of magnitude smaller
vibrational overlap between the launch state and accessible states on the 5 and 6
states compared to accessible levels on the F and G states. This analysis is only afirst approximation, neglecting overlap with the vibrational wavefunction of the
final, autoionizing Rydberg state. Also, small deviations in the functional form of
the transition dipole operator with respect to R can introduce uncertainties to thesimple calculation referred to above, but for the large differences in vibrational
overlap such as this, the above conclusions remain qualitatively valid.
Working from the assumption that only the F and G states play a role inthe dynamics seen here, a portion of the spectrum of states that can be excited via
strongly allowed transitions from the launch state is shown in Fig. 7.2. The F andG states are based upon fits to experimental data and are calculated according to
the following expression[109, 110]:
E tot = Gv + Bv [J (J + 1) − 1] − D[J (J + 1) − 1] + q[J (J + 1)]
2

(7.2)
66

1

G Πg

17.6

17.8

14,27
14,28
14,29
27,29

27,27

26,29

13,27
13,28
13,29
26,27

25,29

25,27

24,29

12,27
12,28
12,29

1

F Σg

18.0
3

-1

Etot-Elaunch (10 cm )Figure 7.2: State locations accessible from the vA, JA=11, 28 launch state with acentral pump energy of 17,550 cm-1 to 18, 100 cm-1. P and R branch transitions
are included for the F1Σg+ state, and P, Q, and R branches are included for the
G1Πg+ state. States are labeled as (v, J).

67

where Gv is the energy for vibrational level v, Bv is the rotational constant forvibrational level v, D is the centrifugal distortion constant, and q is the Λ doubling
parameter. The centrifugal distortion constants are calculated based upon the
Dunham fit parameters for the rotational constant Y01 and vibrational frequency
Y10 so that D=4Y013/Y102[114]—approximately 4.31x10-6 cm-1 and 7.84x10-6 cm-1
for the F and G states, respectively. The Λ doubling parameter only applies tovibrational levels on the G state, splitting each rotational level into two, hence theterm “doubling” parameter. For even JG, the higher lying state has (+) parity, andthe lower lying state has (-) parity, and for odd JG, this is reversed. To maintainthe +↔- parity selection rule, this means that the P and R transitions excite thehigher level, and the Q branch the lower level, of each JG doublet[112]. Thismeans that in Eq. (7.2), the Λ doubling parameter only applies to the P and R
branches of the G state. All constants for the F state are taken from refs. [107,115], except for the centrifugal distortion constant which is taken from ref. [116].
V, B, and q for vG=11 and 12 are taken from ref. [109], and the rest of theconstants for the G state are taken from ref. [110]. The state energies for the Fstate around vF, JF=26, 27&29 have not been experimentally determinedelsewhere, so the fit values have been given with an error of 0.25 cm-1 assumed.
In the G state, similar errors should be assumed.
7.3

Results
As stated previously, the observed wave packets show very slowly

decaying behavior, as shown in Fig. 7.3. Figure 7.3a shows the time domain
pump-probe signal using a pump pulse with a central energy of 17,850 cm-1, and a68photoionization

a

1.32
1.30
1.28
0

amplitude (arb. units)

4.0
3.5

10

20

t (ps)

30

b

3.0
2.5
2.0
1.5
1.0
0.5
0.0

0

10

20

30

-1

beat frequency (cm )40

50

Figure 7.3: Pump-probe signal for a pump energy of 17,850 cm-1 for a launch
state of vA, JA=11,28. Panel a shows the photoionization signal as a function of
pump-probe delay. Panel b shows the Fourier Transform of the signal from panel
a. Beats at energies of 4, 21, 41, and 45 cm-1 are clearly present. Possible peaks
are also located at 8 cm-1 and 25 cm-1.

69

bandwidth of 150 cm-1 FWHM. Notice that there is a very pronounced low
frequency oscillation present along with several higher frequency oscillations. To
analyze the nature of the beats, an FT is performed on the signal after t=0.0 ps.
The Fourier Transform of this signal shows four pronounced peaks in the beat
spectrum. Beats are observed at approximately 4, 21, 41, and 45 cm-1. Each ofthese peaks is very narrow, in fact their width is defined by the pump-probe scan
length. This narrow linewidth suggests that the interfering wavefunctions are
relatively stable and do not quickly decay to secondary states.
To identify the quantum beats in Fig. 7.3, the pump pulse energy wasvaried in an attempt to view any spectral trends. Figure 7.4 shows the FT spectraof the quantum beats observed for a variety of central pump energies ranging
from 17,100 cm-1 to 18,200 cm-1. In the figure, each spectrum has been offset on
the vertical axis to show the relative pump pulse energy. The spectral amplitudes
have been normalized so that the beats around 45 cm-1 have similar amplitudes.
In the case where no beats were observable above the noise level, the spectra werenormalized so that the strongest peak in the spectrum coincides with theamplitude of the observable beats around 45 cm-1 [see spectra at 17,500 cm-1 and
17,600 cm-1]. Due to large instabilities in the experimental apparatus, theamplitudes of the beats are generally unreliable, so an analysis of the spectral
amplitudes is generally intractable. However, the frequencies of the quantum
beats can be reliably observed since, in the weak field limit, the frequencies are
independent of the laser intensity or overlap and only need to be sufficiently
above the noise to detect [see Eq. (7.1a)]. Each of the peaks that have amplitude

70

18.2

3

-1

pump E0 (x10 cm )

18.0
17.8
17.6
17.4
17.2

0

10

20

30

40

50

-1

beat frequency (cm )Figure 7.4: Fourier Transform quantum beat spectra for varying pump energies.
All traces have been rescaled to show peaks at around 45 cm-1, if present. Whereno significant peaks are visible (solid lines), the spectra have been amplified toshow the noise baseline. All spectra have been offset to reflect their respective
pump pulse energy. Where peaks are clearly above the noise level, the frequency
is recorded.

71

more than twice the noise level were fit with a Gaussian lineshape, and the central
frequency is obtained.
A summary of the observed beats as a function of central pump energy isplotted in Fig. 7.5. The error bars for the beat frequencies reflect the widths of thepeaks as recorded in Fig. 7.4, except the lowest frequencies which have a higher
error due to laser drift as a function of delay stage position. Also included in the
figure are the predicted beat frequencies assuming rotational quantum beating in
the G1Πg+ state, and where they are expected to be at their respective maxima.
Several trends in the quantum beat frequencies should be noted. First, the beataround 45 cm-1 shows a trend toward lower frequencies as the pump energy isincreased. The beat around 21 cm-1 shows similar behavior, although the degreeof change falls within the error bars of the quantum beats. Around pump energiesof 17,700 cm-1 to 18,100 cm-1, additional beats around 2, 4, 25, and 41 cm-1 areobserved. These beats are clearly not a part of the overall progression, lending
credence to the possibility that the beats are something other than simple
rotational beats in the G state. Also included in the figure is a schematic spectrumwith a central wavelength of 17,880 cm-1 and a FWHM bandwidth of 200 cm-1.
This illustrates that quantum beats can be present even if the central pump pulse
energy is shifted from the location of the peak beat amplitude. Given that thequantum beats have energies around 45 cm-1, it is concluded that the predominantstate at play in this case is the ∆J=2 rotational beat in the G 1Πg+ state. This fits
well with previous cw type spectroscopy since the ∆J=2 rotational spacing in the
F 1Σg+ and 5 1Σg+ states are 34 cm-1 and 52 cm-1, respectively. Additionally, the

72

-1

beat frequency (cm )

50

40

20

10

0
17.2

17.4

17.6
3

-1

17.8

18.0

E0 (10 cm )Figure 7.5: Quantum beat energy as a function of central pump pulse energy for
11,28 launch state. A sample spectrum with central energy of 17,880 cm-1 isincluded as a measure of pump pulse bandwidth. Closed circles are experimental
quantum beat frequencies for different pump energies. Open circles are predictedrotational quantum beats in the G state, and stars are predicted beat frequencies
between the F and G states around 17,850 cm-1 and 18,000 cm-1 as tabulated inTable 7.1.

73

progression of rotational beats around 21 cm-1 fit well with a ∆J=1 beat
(JG=27&28) that is only possible in the G state due to the ∆Λ=1 electronic
component of the transition from A 1Σu+ state. It is interesting to note that the
JG=28&29 beat around 23 cm-1 is predominantly absent from these spectra. Since
this beat is observed with a rather weak amplitude in several of the traces, thepredominant absence of these higher frequency ∆J=1 beats is attributed to asystematically weak transition dipole moment to the Q-branch JG=28 state as well
as a preferential excitation to JG=27 over JG=29. Even though the observed
quantum beat amplitudes involving the Q-branch states are less than predicted by
Hönl-London factors, the line positions coincide with the expected positions[92].
To further verify that the main progressions seen are indeed rotationalbeats in the G state, a similar series of experiments was carried out for a different
launch state. To make sure that any effects related to FCFs are minimized, thesame vibrational launch state is chosen but the J state is changed, so that vA,JA=9,
16. Assuming the main progression is due to pairs of rotational states(JG=15&17), the progression should be reproduced. Due to different rotational
constants for the F and G potentials, a change in J would result in a shift of thestates on one potential relative to the other. Figure 7.6 shows that the vibrational
progression of rotational beats is reproduced, with no other beats present, failingto disprove the “extra” beats of Fig. 7.5 are beats between electronic states.
Again in this lower J case, the rotational quantum beats involving the Q branch
are weaker than those between the P and R branch states and only show the lower
frequency beat, except for one low amplitude 14 cm-1 beat at E0=17,530 cm-1.

74

30

26

-1

beat freq (cm )

28

14
12
10

17.2

17.4

17.6

17.8

-1

Pump E0 (cm )Figure 7.6: Quantum beat frequencies as a function of central pump energy from
11,16 launch state. Closed circles are experiment, and open circles are predictedrotational beats for different vibrational levels of the G state. The beat at
E0=17,530 cm-1 has large error bars because its presence was determined in thepresence of significant noise.

75

The absence of additional beats in this progression is attributed to a shift in the Fstate energy levels due to different rotational constants for the F versus the Gstates. This further verifies that the beats are in fact rotational and not vibrationalin nature. If the additional beats seen for the 11, 28 launch state were confined to
the G state, they would be observed from this second launch state, but they are
not.

7.4

Discussion and Future Directions
Previously completed high resolution spectroscopy suggests the wave

packet excited with 17,850 cm-1 contains contributions from both the G and F
states, although the reported state to state energy fits deviate up to 0.3 cm-1,
depending upon the source[107, 109, 110, 115, 116]. Since the G←A transition
consists of ∆Λ=1, single photon selection rules allow ∆J=0, ±1 transitions, hence
the presence of the P, Q, and R branch states. The F←A transition corresponds to∆Λ=0, so parity selection rules allow ∆J=±1, so only the P and R branch states of
the F state are excited. Again, here only the line positions are shown; information
regarding transition strength is not included. To identify quantum beats, energy
differences must be taken into account, and these values have been summarized inTable 7.1.
Upon examining the expected quantum beat frequencies, the participatingstates can be inferred. Around 17,850 cm-1, the P-R beat in the G state [i.e.
G(13,27)-G(13,29)] is seen as the 45.0 cm-1 beat, as was established before. The
P-Q beats are near their expected values at slightly less than half the P-R
76

Epu=17,850cm-1

State (v, J)

State (v, J)
G(13,27)
F(26,27)

EtotElaunch/
17816.0
17821.3

G(13,28)

17836.6

F(26,29)17854.9

F(26,27)

G(13,28)

F(26,29)

G(13,29)

17821.3(5.3)

17836.6
20.6(15.3)

17854.9
38.9(33.6)

17859.7
43.7(38.4)

18.3

23.1
4.8

Epu=18,000cm-1

State (v, J)F(27,29)

G(14,27)

G(14,28)

G(14,29)

State (v, J)

Etot Elaunch/

17993.017996.3

18016.5

18039.4

F(27,27)

17959.9(33.1)(36.4)(56.7)(79.5)

F(27,29)

17993.0

3.3(23.5)(46.4)

G(14,27)

17996.3(20.3)

43.1

G(14,28)

18016.5(22.8)

Table 7.1: Expected quantum beat frequencies. For pump pulses centered around
17,850 cm-1 and 18,000 cm-1, optically allowed rovibrational states on the F and G
electronic curves are listed, along with their total energies. The quantum beatfrequency is the difference in energies. Unobserved quantum beats are inparentheses.

77frequency due to the Λ doubled spectrum of the G state. Interestingly, when they
are observed, the Q-R beats are very weak and have a frequency over 1.0 cm-1
away from the predicted frequency. Almost all of the beats involving the F(26,27) state are observed, with the beats at 4.2 cm-1 and 40.9 cm-1 observed. The
absence of beats involving state F(26, 29) indicate that the photoionization
pathway for that state is weak. Calculations show the vibrational overlap between
the F(26, 29) and the ion ground state X+(25, 28) is very near zero. At theenergies observed here, ionization proceeds through the X+(24, 28) state which
has nonzero vibrational overlap with F(26, 29), but it should be remembered that
there is a small difference in the meanings of the vibrational contribution to atransition dipole and the vibrational overlap between states: µ̂elec (R ) . Vibrational
overlap calculations assume that µ̂elec (R ) is constant over all R, but even at thesmall R where there is appreciable vibrational wavefunction in the states observed
here, there will be small variations in the transition dipole operator. This meansthat the vibrational overlap calculations provide qualitative information, but fail toquantitatively characterize the transition strength.
Future experiments can be undertaken that can improve or exploit theeffects seen here. In these experiments, quantum beat frequencies are observed as
the pump wavelength is tuned. Since the probe light is the 800 nm fundamental
from the regenerative amplifier, the final energy of the system is tuned along with
the intermediate energy. In order to reduce the number of degrees of freedom in
the analysis of the data, it would be advantageous to keep the final energy fixed.
This could be overcome somewhat by simultaneously tuning the regenerative78

amplifier output along with the OPA wavelength. This will technique will belimited since the optimization of the OPA output depends a great deal on the input
pulse characteristics. An alternate means to keep the final energy fixed would be
to include the addition of either a second OPA or the use of a second order of
output from the OPA in these experiments. Since the signals are still rather weakat this point, the addition of a second OPA is possibly not feasible due to losses in
the production of OPA light.
In this series of experiments, we have shown progressions of P-R, Q-R,
and Q-P rotational quantum beats on various vibrational levels of the G1Πg+ stateof Li2. In addition to these single electronic surface quantum beats, we haveobserved and identified various quantum beats between states on the G state andthe F1Σg+ state. This observation of low lying Rydberg wave packets begins tobridge the gap between relatively low energy, hence isolated, wave packets on
single electronic curves and high lying, nearly ionic Rydberg levels.
Additionally, this work sets a benchmark for the types of signals to be expected as
the total energy of a system increases towards the ionization limit.

7.5

Ultrafast transient behavior with
680 nm excitation

In the previous sections of this chapter, a type of electronic interaction was
examined, where bound wave functions on separate electronic states beat against
each other. In an effort to identify as many regions with observable wave packet
dynamics such as in the previous section, we scanned other wavelength ranges.

79

In the process, another interesting spectroscopic feature was found: we observe avery large, transient, photoionization (PI) signal for a pump energy around 14,700
cm-1 while exciting the vA, JA=11,28 launch state. In fact, we see up to 100 times
the total PI signal at short times relative to longer times, as shown in Fig. 7.7.
Since this effect is so strong, it is studied to try to identify potentially useful traits.
Much of the data support the conclusion that the fast decaying signal is an atomic
ionization, motivating us to preempt further study of this effect with several of thestudies presented in chapters 5 and 6.
The following section will begin with an examination of the spectral
extent of the large transient peak. To verify that the transient effect is not a resultof an isolated predissociative rovibrational state, launch states with JA=16, 28, and
40 are compared. Next, the dependence of the transient signal on the pumppolarization is presented. Finally, the results are discussed, and a very simple
atomic mechanism will be proposed to explain the ultrafast effect. A significant
unanswered question regarding the observed decay time is also presented.
7.5.1

Experiment and Results

The experimental setup for this section is nearly identical to that in theprevious section, except that the wavelength range scanned by the OPA is slightly
longer. In the experiments described here, a central wavelength range ofapproximately 650-700 nm (14000-15400 cm-1) will be examined. Generally, thepump bandwidth is about 200 cm-1 FWHM. Since the wavelengths studied here
are at the edge of the spectral output from the OPA, the bandwidth can be greatly

80expanded due to nonidealities in the OPA alignment, further limiting thereliability of the spectral studies in this section. In all discussions, the central

120

photoionization

100
80
60
40
20
0
-8 -6 -4 -2 0

2 4 6

8 10 12 14 16 18 20 22 24

t (ps)
Figure 7.7: Pump-probe signal for Epu=14,700 cm-1. Probe polarization isparallel, and the photoionization (PI) has been normalized so that PI=0 for t<-4.0
ps, and PI=1 for t>20 ps.

81

energy of the approximately Gaussian spectral envelope will be cited as Epu.
Typical OPA output power is approximately 60 mW, but the power at the heatpipe is considerably less (~20 mW) due to losses along the optical path.
To test whether the transient peak was unique to a specific cw frequency,
launch states with vA=11 and JA=16, 28, and 40 are excited with cw light at 603,
606.955, and 613. nm, respectively. Figure 7.8 shows the transient peak
amplitudes for Epu for JA=16, 28, and 40. From all launch states, pump-probetransients show a similar transient behavior with a quickly decaying peakfollowed by a stable incoherent photoionization signal. Obviously, from thisfigure, there is no reliable JA dependence to the transient signal. The spectral
profile of the JA=28 data shows a small (~100 cm-1) shift relative to JA=16 and 40,
but there is still significant overlap of the profiles.
Finally, the pump polarization was rotated relative to the cw and probe
pulses from parallel to the magic angle and perpendicular. Figure 7.9 shows asemilog plot of the pump-probe traces at each of these three critical polarizations
for Epu=14700 cm-1, giving unexpected results. In the Fig. 7.9a, the traces havebeen normalized so that the signal for t<4.0 ps is set to zero, and the signal fort>10 ps is set to 1. Clearly, the transient signal goes down relative to the longtime signal. The traces are characterized by an initial exponential rise time wheret1/2=0.16 ps, as should be expected from the timescale of the pump pulse. Aftert=0.0 ps, all of the traces exhibit exponential decay behavior with halflives ofbetween 0.43 and 0.59 ps. The parallel and perpendicular signals both show
slower (t1/2~8 ps) decays, but with much lower amplitude, making quantitative

82

1000

photoionization

750

500

250

0
14200

14400

14600

14800

15000

15200

-1

Epu (cm )Figure 7.8: Dependence of transient peak height on central pump energy and JA.
The peak photoionization is been plotted as a function of central pump energy for
JA=16 (), JA=28 (c,z), and JA=40 ( ). In all cases vA=11. The solid circle
dataset was taken before a lab relocation, and the open symbols were taken after;
this is given to show the repeatability of the data. The error bars given for one
data set are similar for the others.

83aphotoionization

100o

90 , t1/2=.59 pso

55 , t1/2=.43 pso

0 , t1/2=.55 ps

10
1

0.1

-5

0

5

peak photoionization

1.0

10
t (ps)

15

20

25

b

0.8
0.6
0.4
0.2
0.0

0

30

60

90

probe polarization (deg)Figure 7.9: Polarization dependence of peak photoionization for parallel,
perpendicular, and magic angle pumps. Panel a: Pump-probe traces for eachprobe polarization. The solid lines are single exponential decay fits, withhalflives ranging from 0.43 to 0.59 ps. Panel b: Peak photoionization (relative tot>20 ps) as a function of pump polarization.

84

conclusions difficult. Figure 7.9b shows the peak amplitude as a function ofprobe polarization. It is clear that there is a dramatic decrease in the transient
signal as a function of degrees off parallel.

7.5.2

Discussion

The data suggest that the main transient peak is an atomic, not molecular,
effect. The predominant transient time dependent effect appears to come from an
excitation to the quickly decaying 2p→2s atomic transition, resulting in a 1+3
photoionization pathway. To ionize, a total energy of 43,300 cm-1 is needed.
This means that the 2p electron needs an additional 28,400 cm-1, so a three photon
resonance enhanced photoionization pathway with the 800 nm probe is suggested.
The resonant enhancement may be possible since the energy between the 2p and3p electronic levels is only 13,000 cm-1, very near the probe energy, and to reachthe ion an additional 15,900 cm-1 are needed. Three primary pieces of evidence
support this atomic photoionization pathway conclusion: the polarization
dependence of the signal, the fluorescence linewidth for the 2p→2s transition, andthe similarities of the transient signal with and without the cw laser
present. The observed lifetime of the transient peak is three orders of magnitude
shorter than the literature value for spontaneous decay for the suggested
transition; this large deviation from literature values requires further study before
making a definitive conclusion for the transient effect seen.
That the main transient effect is atomic in nature is supported by thedependence of the photoionization on the polarization of the pump and probe

85

pulses. As it was shown in Fig. 7.9, the transient signal is nearly completely
depleted by a simple rotation of the pump polarization relative to the probe. In a
molecular system, this should not necessarily be the case. Assuming that there isa non-uniform mJ distribution in the state prepared by the probe (i.e. the state isaligned), the perpendicular signal should not go to zero. By setting the probe
polarization to perpendicular relative to the probe, or rotating the frame ofreference by 90o, the effect (disregarding coherences) is approximately a swap of
the effective mJ assignments from high to low and vice versa. We can quantify
the alignment dependent contribution to the signal, assuming a uniform mJ
distribution in the launch state, as the following:
S align =

2

J

∑ M

pu = − Jprfµˆ pr M pri M puf µˆ pu M pui

,

(7.3)

where M is the aligned spherical harmonic wavefunction with a specific J, i and frefer to the initial and final states of a transition, and pu and pr refer to the pumpand probe steps. Here, it is assumed that all transitions follow the selection rule∆mJ=0. In the parallel case, the mJ quantum numbers pu=pr. For theperpendicular case, the quantum numbers can be approximated by |pr|≈ J-|pu|, sothat in effect, the mJpu=0→mJpr=±(J-1) and mJpu=±(J-1)→mJpr=±1. The effect of
this transformation is that for J=27, the signal decreases by at most 13%. In Li2,
only the electronic ground state of the ion, which is known to have sigma
symmetry, is reachable with a single photon from each of the pump and probe
pulses. This requires that both pump and probe dipole moment operators lie along
the same axis relative to the molecular axis, further justifying the analysis above.

86

This is all in contrast to the atomic case, where it is well known that a 1+3 photon
absorption by an electron shows cos2(θ) dependence where θ is the angle between
photon polarizations. This is the first bit of evidence that the transient effect seenin Fig. 7.7 is an atomic effect.
In the dispersed fluorescence spectrum under excitation by the 606.955
nm cw laser, many doublet lines appear that correspond to P and R branch decays
from the launch state, as well as regularly spaced collisionally induced satellite
lines. An additional line appears that is not part of these progressions at 670.5
nm. Figure 7.10 shows a portion of the dispersed fluorescence spectrum with adoublet at 14,770 and 14,830 cm-1 along with an additional prominent line at
14,910 cm-1. The line at 14,910 cm-1 corresponds to the exact expected
wavelength of the 2p→2s atomic transition, and in fact appears in the dispersed
fluorescence spectra for many different excitation wavelengths. Higher resolution
spectra even show the spin orbit splitting of less than 0.3 cm-1 for the 2p state.
The linewidth is clearly wider than the molecular lines (the molecular linewidth
here is primarily determined by the instrument response). The observation of abroadened line in the dispersed fluorescence spectrum only suggests a short
lifetime. Since the linewidth is so much greater than expected from tabulated
lifetimes, a significant problem exists with concluding a mechanism based uponthis evidence.
One final piece of evidence supports the argument that the observed effectis a result of an atomic, not molecular process. The pump-probe traces observed
above were gathered using a three beam interaction, but they can also be seen87

fluorescence

2p→ 2s

14700

14800

14900

15000

-1

photon energy (cm )Figure 7.10: Fluorescence spectrum with excitation by 606.955 nm cw laser. The
narrow peaks are molecular fluorescent lines with widths limited by the
monochromator. The marked line corresponds to the 2p→2s line of atomic
lithium.

88without any cw laser present. While optically chopping the pump pulse, it is seenthat the timescales and spectral profiles of the transient peaks match remarkably
well with those in the presence of the cw laser. This evidence supports themechanism where the OPA excites the 2p←2s transition from the atomic groundstate followed by multiphoton ionization by the 800 nm pulse.
The proposed mechanism for the transient effect is a simple excitation of afast decaying atomic state is supported by four primary pieces of evidence: thespectral profile of the ultrafast transient peak height coincides with the 2p→2stransition in atomic lithium, the dramatic polarization dependence of the peakheight, the presence of a wide bandwidth fluorescent line, and the presence of theeffect without the cw laser. The fact that the effect is seen to depend on the cw
laser (when optically chopping the cw laser) indicates that the cw laser somehow
participates in the dynamical effect, but a mechanism is not proposed. Due to theevidence supporting the ultrafast effect being primarily atomic, further study on
this system was preempted in favor of studies that are more directly molecular innature [see chapters 1-6].
This chapter has summarized a series of experiments that address atransient signal in our pump-probe signals. Several series of experiments were
performed in an attempt to infer the nature of the transient effect, but with little
success since all of the molecular explanations are characterized by significant
flaws. This argument was supported by several pieces of evidence including a
dramatic pump-probe polarization dependence, the presence of an atomic

89

fluorescence line in an ostensibly molecular fluorescence spectrum, and thepresence of the effect both with and without the cw laser.

90

Chapter 8
Conclusion

This thesis demonstrates several aspects of coherent control of the welldefined lithium dimer using ultrafast lasers and pulse shaping. Primaryexperimental results include a demonstration of feedback mediated control ofphotoionization of Li2 using an evolutionary algorithm. Lessons learned by using
the evolutionary algorithm were used to control both the transient amplitudes ofcontributing wavefunctions in a two-state wave packet. The effects of resonant
and nonresonant wavelengths have also been elucidated. Further study into
strong-field dynamics were also performed, showing that as the field strength
increases, the complexity of the produced wave packets increases as the driving
electric field increases. Finally, the possibilities for creating electronic wave
packets in lithium dimer were studied, with some electronic wave packets
observed.

91

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