Direct Frequency Comb Spectroscopy andHigh-Resolution Coherent Control
byMatthew C. Stowe
M.S., Rensselaer Polytechnic Institute, 2001
B.S., Rensselaer Polytechnic Institute, 1999

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulﬁllmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2008Stowe, Matthew C. (Ph.D., Physics)
Direct Frequency Comb Spectroscopy and High-Resolution Coherent Control
Thesis directed by Professor Adjoint Jun Ye

We present the ﬁrst experiments demonstrating absolute frequency measurements of one- and two-photon transitions using direct frequency comb spectroscopy (DFCS). In particular we phase stabilized the inter-pulse period andoptical phases of the pulses emitted from a mode-locked Ti:Sapphire laser, creating a broad-bandwidth optical frequency comb. By referencing the optical comb
directly to the cesium microwave frequency standard, we were able to measureabsolute transition frequencies over greater than a 50 nm bandwidth, utilizing thephase coherence between wavelengths spanning from 741 nm to 795 nm.
As an initial demonstration of DFCS we studied transitions from the 5S to
5P, 5D, and 7S states in 87 Rb. To reduce Doppler broadening the atoms were laser
cooled in a magneto-optical trap. We present an overview of several systematic
error sources that perturb the natural transition frequencies, magnitudes, andlinewidths. These include radiation pressure from the probe beam, AC-Stark
shifts, Zeeman shifts, power-broadening, and incoherent optical pumping. After
careful study and suppression of these systematic error sources, we measured
transition linewidths as narrow as ∼1.1 MHz FWHM and ≥10 kHz linecenter
uncertainties. Our measurements of the 5S to 7S two-photon transition frequency
demonstrated the ability to determine the comb mode order numbers when theinitial transition frequency is not known to better than the comb mode frequency
spacing.
By modifying the spectral phase of the pulses we demonstrated high-resolution
coherent control. Our ﬁrst coherent control experiment utilized a grating based

iv
pulse stretcher/compressor to apply a large, ±250,000 fs2 , chirp to the pulses. We
measured the two-photon transition rate as a function of linear frequency chirp.
The results illustrate the diﬀerences between similar classic coherent experiments
done with a single femtosecond pulse and ours conducted with multiple pulses.
Furthermore, we show that it is possible to reduce the two-photon transition rateby tuning the comb such that the two-photon amplitudes from all comb modepairs destructively interfere.
One of the unique features of DFCS is the large bandwidth over which
atomic coherence may be established. We tuned the comb frequencies to not
only be two-photon resonant, but also resonant with two diﬀerent intermediate
states separated by 7 THz. In this experiment we demonstrate the phase sensitive
excitation of a closed-loop four-level system in a diamond conﬁguration. Using aspatial light modulator based pulse shaper, we adjusted the relative phase of thetwo diﬀerent two-photon transition pathways. We measured a sinusoidally varying
two-photon transition rate as a function of the pulse shaper phase, with a fringe
visibility of up to 69%. As a ﬁnal example of high-resolution coherent control,
we adjusted the spectral phase of the pulses to force constructive interference
between the two-photon amplitudes that arise from the many thousands of modepairs detuned from an intermediate state. This resulted in an increase of thetwo-photon transition rate by approximately 250%.

Dedication

To my dear wife, Lin.

Acknowledgements

I have taken a rather long and varied road from ﬁrst earning my Bachelors
degree in Physics at Rensselaer to ﬁnally my PhD at JILA. Along the way I haveworked with many diﬀerent people. Certainty my experience working with thesepeople has contributed to my knowledge, and especially shaped my approach tophysics.
In particular I’d like to thank Ralph Nightengale for his patient mentoring
during my time at General Dynamics. I can only hope his phenomenal ability to
break a problem down neatly and logically has rubbed oﬀ on me. Working at the
Jet Propulsion Laboratory was an especially enjoyable and educational experience. In particular working in the group of Jon Dowling was always intellectuallystimulating. I’d especially like to thank Dmitry Strekalov for teaching me about
quantum optics, and Bill Klipstein for allowing me the opportunity to work on
laser frequency stabilization. It was working on frequency stabilization of a YAG
laser to Iodine with Bill that prompted my interest in joining Juns group.
Here at JILA I had the pleasure of working on a very interesting atomic
physics project. Thanks to my advisor Jun Ye for providing me this opportunityand guidance. He has established a very nice laboratory environment with great
people, resources, and projects. There are probably few laboratories in the world
with such varied expertise. Between all the projects in Juns group, it really ispossible to learn about the A, M, and O in AMO physics. I also enjoyed the manyvii
lunch time talks, not just for the pizza, but to learn about a variety of physics
experiments.
Everyone in the lab has an excellent sense of humor, which helped make ita great place to work. I’m indebted to Mike Thorpe, Thomas Schibli, and Seth
Foreman for listening to me talk about my project ad nauseum, and returning thefavor by telling me about theirs. I’d also like to thank Andrew Ludlow, Marty
Boyd, Gretchen Cambell, Tom Loftus, Eric Hudson, Dillan Yost, Brian Sawyer,
Darren Hudson, Kevin Holman, Lisheng Chen, and Thomas Zanon. I’ve hadinteresting and entertaining conversations with all of these people and learned
something new from each person.
There have been several people involved with the direct frequency comb
spectroscopy during my stay at JILA. First, I would like to thank Adela Marian
who began the project. I’d like to thank Avi Pe’er, not only for assistance constructing a pulse shaper, but for the many helpful conversations. Thanks also to
Flavio Cruz, Milan Maric, John Lawall, Ennio Arimondo, and Daniel Felinto. Allof whom contributed to the success of this project during their visits to JILA.
Finally, inﬁnite thanks to my family for their support through my ups anddowns on this long road. If I think back to the things that led me to pursue science
they were all provided by my family. Thanks to my brother Mike for buying a
VIC-20 and getting me interested in computers. Thanks to my parents Tom andSandi, for the many opportunities you’ve provided, love, understanding, patience,and support. Special thanks to my wife, Lin, for the sacriﬁces you made along theway for us both. You have always been there for me with your love and support.
Fortunately, I love you all enough to not make you read my thesis!

Contents

Chapter
1 Introduction

1

2 Experimental Apparatus

9

2.1

2.2

Ti:Sapphire Laser . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.1.1

Optical Frequency Comb . . . . . . . . . . . . . . . . . . .

10

2.1.2

Repetition Rate Stabilization . . . . . . . . . . . . . . . .

12

2.1.3

Oﬀset Frequency Stabilization . . . . . . . . . . . . . . . .

15

Magneto-Optical Trap . . . . . . . . . . . . . . . . . . . . . . . .

18

3 Theory

24

3.1

Second-Order Perturbation Theory . . . . . . . . . . . . . . . . .

25

3.2

Density Matrix Treatment . . . . . . . . . . . . . . . . . . . . . .

32

4 Direct Frequency Comb Spectroscopy

39

4.1

Coherent Accumulation Experiment . . . . . . . . . . . . . . . . .

43

4.2

Systematic Eﬀects . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

4.2.1

Laser Linewidth . . . . . . . . . . . . . . . . . . . . . . . .

56

4.2.2

Radiation Pressure . . . . . . . . . . . . . . . . . . . . . .

57

4.2.3

AC Stark Shift and Power Broadening . . . . . . . . . . .

66

4.2.4

Zeeman Shift . . . . . . . . . . . . . . . . . . . . . . . . .

75

ix
4.2.5
4.3

4.4

Incoherent Optical Pumping . . . . . . . . . . . . . . . . .

78

Two-Photon Spectroscopy Results . . . . . . . . . . . . . . . . . .

80

4.3.1

Repetition Rate Scan . . . . . . . . . . . . . . . . . . . . .

82

4.3.2

Absolute Frequency Measurements . . . . . . . . . . . . .

83

4.3.3

Two-Photon Selection Rules . . . . . . . . . . . . . . . . .

90Single Photon DFCS . . . . . . . . . . . . . . . . . . . . . . . . .

92

4.4.1

95

5P Measurements via Two-Photon Transitions . . . . . . .

5 High-Resolution Coherent Control: Chirp

102

5.1

Chirped Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.2

Pulse Stretcher Compressor . . . . . . . . . . . . . . . . . . . . . 103

5.3

Dressed Atom Picture . . . . . . . . . . . . . . . . . . . . . . . . 110

5.4

Chirped Pulse Experiment . . . . . . . . . . . . . . . . . . . . . . 113

6 High-Resolution Coherent Control: Discrete Phase
6.1

126

Spatial Light Modulator Pulse Shaping . . . . . . . . . . . . . . . 126
6.1.1

Alignment Procedure . . . . . . . . . . . . . . . . . . . . . 127

6.1.2

Calibration Procedure . . . . . . . . . . . . . . . . . . . . 130

6.2

Four-Level Interference Experiment . . . . . . . . . . . . . . . . . 140

6.3

Four-level Enhancement Experiment . . . . . . . . . . . . . . . . 161

7 Future Directions

169

Bibliography

172

x

TablesTable
4.1

Angular part of the reduced dipole matrix elements for the 5S1/2 F=2
to 5P3/2 F=3 to 5D5/2 F=4 transition, q=0 . . . . . . . . . . . . .

4.2

Measured absolute two-photon transition frequencies and comparison to reported values from literature

4.3

71

. . . . . . . . . . . . . . .

89

Angular part of the reduced dipole moment matrix element for eachpossible q=0 transition from 5S1/2 F=1 to 7S1/2 =2 . . . . . . . . .

91

4.4

Example set of fr and fo pairs for indirect 5P measurements . . .

96

6.1

Angular part of the reduced dipole matrix elements for the four
possible transitions from 5S1/2 F=2 to 5D3/2 F=1 . . . . . . . . . . 164

Figures

Figure1.1

Multi-pulse Ramsey Spectroscopy . . . . . . . . . . . . . . . . . .

1.2

Multi-mode excitation of a three-level ladder (a) and lambda (b)4type system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1

Comb degrees of freedom . . . . . . . . . . . . . . . . . . . . . . .

13

2.2

Saturated absorption lines for diode laser stabilization . . . . . . .

21

3.1

Three-level system excited by two modes . . . . . . . . . . . . . .

27

4.1

Full hyperﬁne energy level structure of 87 Rb (5S,5P,5D,7S) . . . .

41

4.2

Comb excitation of a model three-level system . . . . . . . . . . .

42

4.3

Timing diagram for the pulse picking experiment . . . . . . . . .

46

4.4

Measured 5D population scaling versus pulse number . . . . . . .

49

4.5

Quadratic dependence of 5D population; signature of coherent accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6

51

High-resolution via coherent pulse accumulation; linewidth measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

4.7

Measured optical comb mode linewidth . . . . . . . . . . . . . . .

58

4.8

Energy level diagram for radiation pressure experiment . . . . . .

60

4.9

Experimental and theoretical results for radiation pressure experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

xii
4.10 Reduced radiation pressure using counter-propagating probe beams 67
4.11 Three-level system excited by two or four modes, reduction of Stark
shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

4.12 Stark shift experiment results from two-photon spectroscopy . . .

70

4.13 Theoretical prediction for Stark shift with comb excitation . . . .

72

4.14 Measured two-photon transition linewidths for power-broadening
experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4.15 Theoretical predictions for power-broadening experiment . . . . .

76

4.16 Zeeman shift example using σ + and σ − two-photon excitation . .

79

4.17 Eﬀect of incoherent optical pumping on measured two-photon transition spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.18 Complete scan over all 5S to 5D hyperﬁne transitions using fr . .

84

4.19 Theoretical prediction for ground state populations; for 5P3/2 F=3
measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

4.20 Theoretical prediction for ground state populations; for 5P3/2 F=3
measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

4.21 Improved 5P1/2 F=2 lineshape using normalization for incoherent
optical pumping from theory . . . . . . . . . . . . . . . . . . . . . 101
5.1

Grating pair diagram for the derivation of the second-order dispersion105

5.2

Double pass grating pulse stretcher / compressor

5.3

Dressed states relevant for chirped pulse excitation experiment . . 112

5.4

Energy level diagram of resonant states for chirped pulse experiment115

5.5

Steady state measured signal and calculated excited state popula-. . . . . . . . . 107

tion versus chirp and the single pulse theoretical response . . . . . 120
5.6

Measured signal and calculated excited state population versus
pulse number for large positive and negative chirps . . . . . . . . 122

xiii
5.7

Phase control of destructive interference experiment results andtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.1

Spatial light modulator in a folded 2f-2f geometry pulse shaper . . 128

6.2

Measured transmission and extracted phase versus SLM voltage
count using the repump laser at 780.2 nm . . . . . . . . . . . . . 135

6.3

Map of two-photon photocurrent versus second and third orderdispersion applied to the SLM . . . . . . . . . . . . . . . . . . . . 137

6.4

Eﬀect of polarization rotation of SLM on pulse spectrum . . . . . 139

6.5

Relevant 87 Rb level subsystem for the diamond coherence experiment141

6.6

Pulse spectrum and SLM phase mask used in the four-level interference experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.7Time dependence of four-level interference results . . . . . . . . . 147

6.8

Example of SLM phase vs. voltage calibration eﬀect on fringe period150

6.9

Final results for four-level interference experiment . . . . . . . . . 152

6.10 Table of relevant dipole moments and two-photon amplitudes forfour-level interference experiment . . . . . . . . . . . . . . . . . . 157
6.11 Multi-mode standing wave eﬀect for the diamond coherence experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.12 Pulse spectrum and SLM phase mask used for the four-level optimization experiment . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.13 Four-level enhancement experiment results . . . . . . . . . . . . . 167
6.14 Four-level enhancement theoretical predictions . . . . . . . . . . . 167

Chapter 1
Introduction

In the last decade it became possible to stabilize the optical phases of thepulses from a mode-locked laser [1]. This technological achievement, together
with the stabilization of the pulse repetition rate, led to the optical frequency
comb. An optical frequency comb is a set of equally spaced discrete frequency
modes generally spanning 1 to 100 THz with an inter-mode spacing of 100 MHz
to 1 GHz. Perhaps the most revolutionary aspect of this new technology is thatit provides a phase coherent link between microwave and optical frequencies [2].
Whereas previous frequency chains used for the same purpose required immense
resources [3], the frequency comb is a rather simple device providing a vastly moreelegant solution.
The impact of the optical frequency comb in the last decade on the ﬁeld ofhigh-precision spectroscopy has been signiﬁcant [4, 5]. It has enabled the absolute frequency measurement of atomic [6] and molecular transitions [7]. In thesemeasurements the optical comb served as a frequency ruler with which the absolute frequency of the cw-laser used for spectroscopy was determined. Similarly
the comb can be used to make an optical clock. By locking a comb mode to a
cw-laser, which in turn is locked to an optical transition, the repetition frequency
may be counted electronically to serve as a clock. This has been demonstrated
on a single trapped Hg+ ion [8] and in molecular Iodine [9] as couple of examples.

2
Recently a fractional frequency uncertainty of 1 part in 1016 was measured using aneutral Strontium lattice clock [10]. In this measurement the optical comb served
a second purpose, it allowed for a phase coherent comparison in the optical domain
between the Sr transition frequency and a remote Ca clock optical transition.
We present some of the ﬁrst experiments demonstrating direct frequency
comb spectroscopy (DFCS). In the previously mentioned applications of opticalfrequency combs the atoms or molecules were excited by a cw-laser, the comb was
only used to measure the absolute frequency of the cw-laser. Direct frequency
comb spectroscopy uses the comb itself to excite the atomic transitions. One of
the major advantages of using the comb directly to excite the atoms is the broadbandwidth of the comb spectrum. For example, in our experiments we measureone- and two-photon transitions using wavelengths from 795 nm to 741 nm fromour single Ti:Sapphire laser.
Figure (1.1) shows the time- and frequency-domain pictures of direct frequency comb spectroscopy. The time-domain picture is very similar to Ramsey
spectroscopy extended to multiple phase coherent pulses. For a two-level atom
initially in the ground state the ﬁrst pulse excites some population, which also
creates an atomic coherence. Between pulses the atomic coherence acquires phaseat natural frequency. Depending on the inter-pulse delay and optical phase thesecond pulse will either excite more population or de-excite population. In Fig.
(1.1) the left panels correspond to a comb tuned on-resonance with the two-leveltransition, and in the right panels the comb is oﬀ-resonance. The middle panels
show the excited state populations after each pulse. After the ﬁrst pulse there isan equal amount of population excited regardless of the comb detuning. This isbecause the resonance condition is established by the interference between two or
more pulses. When the comb is on-resonance the population grows quadraticallyversus the number of excited pulses. Strictly speaking, the quadratic scaling is3
only for small excited population and without any dephasing processes. When the
comb is tuned oﬀ-resonance the population excited by the ﬁrst pulse is de-excitedback to the ground state. Notice in the bottom panels the spectrum after two
pulses is modulated sinusoidally, with peaks separated by the inter-pulse period.
After more than two pulses the peaks become sinc2 functions with a width inversely proportional to the total duration of the pulse train. In the limit of an
inﬁnite number of pulses the peaks are given by delta functions, this is the originof term frequency comb. The general idea behind using multiple pulses for highresolution spectroscopy is that the resolution is no longer limited by the spectral
width of a single pulse, but rather the total coherent excitation time. In principle
the total coherent excitation time is only limited by the natural lifetime of theatomic transition under study.
This idea of using multiple phase-coherent pulses for high-resolution spectroscopy was ﬁrst proposed in 1976 by Hänsch [11]. In his proposal the atomic
sample was to be placed inside an optical resonator in which the short pulses passthrough the atoms multiple times. By tuning the resonator length the inter-pulse
delay could be adjusted for resonance with a speciﬁc atomic transition. Shortly
after this proposal two experimental demonstrations were conducted. In 1977
Salour and Cohen-Tannoudji conducted Doppler-free two-photon spectroscopy of
the 3S to 4D transition in sodium using two phase coherent 5 ns pulses [12, 13].
Peaks in the 4D excitation were observed as a function of the inter-pulse delay,
which was adjusted by an optical delay line. At the same time Teets, Eckstein, andHänsch used multiple phase coherent pulses to measure the 3S to 5S two-photontransition [14]. In their experiment the atoms were excited by multiple passes ofa single 6 ns pulses bouncing between the end mirrors of a variable length FabryPerot cavity. This principle was then applied in 1978 by Eckstein, Ferguson, andHänsch to measure the 4d ﬁne-structure splitting in sodium to be 1028 MHz [15].

4Light Off Resonance

Time

Time

EnergyExcited State
Population

Light
Electric Field

Light On ResonanceegFigure 1.1: Time and frequency domain pictures of excitation of a two-level system
by a series of pulses. The top two panels show the time domain picture of thepulses, the only diﬀerence between the left and right side is the optical phase of
the pulses. In the case of the left panels, there is a comb mode on resonance.
For the right panels there is no comb mode resonant. The middle panels show
the excitation in terms of population for the two-level atom. Notice that theﬁrst pulse has the same eﬀect for both on and oﬀ-resonance. In the case of theon-resonance pulse train the second pulse adds to the population. For the oﬀresonance case the second pulse de-excites the population. In the bottom panels
we show the positions of the spectral peaks after 1,2,3 pulses in relation to theatomic transition frequency.

5Perhaps the most closely related experiment to our own was conducted in 1996
by Snadden et. al. [16]. They used a cavity stabilized mode-locked Ti:Sapphirelaser to conduct two-photon spectroscopy of Rb atoms in a magneto-optical trap.
In particular they measured the 5S to 5D and 7S transitions and improved theestimate of the 7S hyperﬁne constants for both 86 Rb and 85 Rb. One very important diﬀerence between all the above experiments and our own is that they only
conduct relative measurements. In other words only frequency splittings between
closely spaced levels are determined. This is due to the fact that they only stabilize the frequency diﬀerence between comb modes, not the absolute frequency ofthe modes. We will discuss this important point further in section (2.1.1).
Figure (1.2) shows the two possible conﬁgurations for two-photon transitions. So far all of the experimental demonstrations discussed excited a laddertype conﬁguration, Fig. (1.2)(a). The total two-photon transition frequency isgiven by the sum of two comb mode frequencies. So to determine the absolutetransition frequency, the absolute frequency of each mode must be known. Dueto the equal spacing of comb modes if any pair of comb modes is two photon
resonant, then all modes are part of a two-photon resonant pair. In this ﬁgure
we give an example where the two-photon transition is twice the frequency of the
N th comb modes. Therefore, the N − 1 and N + 1 modes also form a two-photon
resonant pair. The second conﬁguration is shown in (b), this is known as a Raman
transition. In this case a photon from one mode is absorbed and a second photon
is stimulated to emit into a second mode. The two-photon resonance condition is
only a function of the diﬀerence in frequency between two modes. So the absolute frequency of this type of transition may be determined from only the spacing
between comb modes.
In 1981 Mlynek et. al. conducted high-resolution spectroscopy of a Raman
transition in sodium using a mode-locked laser [17]. They achieved very narrow ∼1

6

(a)

(b)
2

3
QN

QN1

QN+1

QN2

Q2J
QN1-1
QN2-1

2
QN
1

QN-1
1

Q2J

3

Figure 1.2: (a) Ladder type two-photon transitions in which the total transition
frequency is the sum of two comb mode frequencies. (b) Raman type two-photontransition in which the total transition frequency is the diﬀerence between two
comb mode frequencies. Due to the equal spacing of comb modes if any twomodes are two-photon resonant then all modes in the spectrum are part of a
two-photon resonant pair.

7
kHz FWHM two-photon transition linewidths, much narrower than the bandwidthof the laser used for excitation. This was possible due to the fact the two-photontransition frequency in a Raman transition is only a function of the inter-mode
frequency spacing. A particularly interesting use of multiple pulses to excite aRaman transition was conducted in 1991 by Weiner et. al. [18]. They generated
a phase coherent pulse train with a repetition rate of the order 1 THz. This wasdone by passing a single femtosecond duration pulse through a pulse shaper witha sinusoidally varying phase mask. One property of this type of phase mask isthat the single femtosecond pulse becomes a short burst of weaker pulses withan inter-pulse spacing given by the period of the sinusoidal phase mask. They
conducted selective excitation of Raman transitions in a molecule by adjusting
the inter-pulse period.
The direct frequency comb spectroscopy that we present in this thesis represents a distinct evolution of these previous experiments. In all of the previously
mentioned experiments the absolute frequency of the comb modes was not known
and subject to drift. This placed a limit on the types of measurements that werepossible to either Raman transitions or only measuring the diﬀerence between two
ladder type transitions. We not only stabilize the inter-mode spacing of the comb
but also the absolute frequency of each mode to a cesium reference. This allowed
us to demonstrate high-resolution spectroscopy of ladder type two-photon transitions and measure the absolute transitions frequencies. In chapter (4) we present
an explanation of several sources of systematic error and the DFCS results forboth one- and -two-photon transitions.
Not only can the frequencies of the comb modes be adjusted, but the phases
as well. Recall that for a two-photon transition all modes in the comb may
contribute to the excitation. By adjusting the phases of the comb modes, it ispossible to control the two-photon excitation rate using the interference between

8multiple two-photon transition pathways. Coherent control is the ability to control
the excitation of a particular quantum state by changing the phases of multiple,
interfering, transition pathways. In chapter (5) we demonstrate coherent control ofthe excitation to a particular hyperﬁne level by applying linear frequency chirp toour pulses. This is an example of quantum interference between many thousandsof two-photon transition amplitudes arising from diﬀerent optical pathways. Dueto the multi-level structure of Rb, it is also possible to have quantum interference
between transitions via diﬀerent intermediate states. In chapter (6) we tuned
the comb to be resonant with a particular 5D hyperﬁne state, via two diﬀerent
intermediate states. This experiment utilized four resonant comb modes in adiamond type excitation conﬁguration. By adjusting the phase of the comb modes
near-resonant with one of the intermediate states we controlled the two-photontransition rate. Much like an optical interferometer we recovered a sinusoidally
varying excited state population with a fringe visibility of 69%.

Chapter 2
Experimental Apparatus

In this chapter we brieﬂy present the basic apparatus used in our experiments. The two main components are the Ti:Sapphire laser with its stabilization
subsystem, and the rubidium MOT. There are many resources for more in-depth
explanations of both the laser system, frequency comb stabilization, and construction of a MOT. For our coherent control experiments we provide more detailed
explanations of the diﬀerent pulse shaping apparatus in sections (5.2,6.1).
2.1

Ti:Sapphire Laser
The foundation for all of the experiments presented in this thesis is the

optical frequency comb [19]. In this section the design and frequency stabilizationof our Ti:Sapphire laser will be explained. Our optical frequency comb is generated
from a mode-locked Ti:Sapphire laser. The Ti:Sapphire crystal is pumped with
5.5 Watts of 532 nm light from a Coherent Verdi laser. It uses passive modelocking based on the nonlinear refractive index of the Ti:Sapphire crystal itself.
This design is referred to as Kerr lens mode-locking (KLM) because of the role ofthe Kerr nonlinearity.
Mode-locking is when many thousands of the longitudinal modes of thelaser are not only lasing but are in phase at some spatial location so a pulse isformed. The large gain bandwidth of the Ti:Sapphire crystal and intra-cavity

10dispersion compensation allowed us to operate with a FWHM spectrum from 30
nm to 65 nm, depending on the experiment. With such a large bandwidth of
longitudinal modes oscillating in-phase the peak ﬁeld intensity in the laser can bevery large, particularly in the crystal. Due to the Kerr eﬀect the radial intensity
proﬁle causes a radial index gradient; much like a gradient index lens this causes
focusing. However, for the low peak ﬁeld of cw-lasing there is no Kerr lens eﬀect.
By oﬀsetting the cavity focusing mirrors to account for the extra Kerr lensing thelaser preferentially lases with high peak ﬁelds. Typically we operate the laser suchthat the power output is at least 20% higher mode-locked versus cw.
In our cavity design we use a prism pair to compensate for the dispersionof the Ti:Sapphire crystal. Although it is possible to operate in a soliton mode,
we typically do not have the signature sech2 spectrum of a soliton. Instead thespectrum usually has a tail to the IR, see for example Fig. (6.4). The typical
output power is about 450 mW mode-locked at a repetition rate of ∼100 MHz
and center wavelength of 778 nm. About 30% of the output power is used forthe interferometer to stabilize the oﬀset frequency, see subsection (2.1.3). Before
detailing the laser stabilization we explain the properties and relevant parametersof an optical frequency comb.
2.1.1

Optical Frequency Comb

An optical frequency comb is simply an electric ﬁeld composed of manyequally spaced discrete frequencies (comb modes), typically in the visible to infrared regions of the spectrum. The beauty of the optical frequency comb is theease with which the comb mode frequencies can be referenced to a microwave reference such as a cesium. This allows typically >100,000 separate comb modes eachwith an absolute optical frequency referenced to a convenient frequency standard.
The frequency spectrum of an ideal (noise and drift free) mode-locked laser

11
is an optical frequency comb. In practice, there are noise sources that aﬀect thetime between pulses and optical phase of the pulses on short and long time scales.
So the trick to producing a useful frequency comb is to reduce or counteract thenoise sources. In this section, we focus mainly on counteracting the slow noise
sources, especially slow drift, with a bandwidth of up to ∼5 kHz. Other techniques
for reducing the linewidth of the comb by locking to a stable optical laser or cavity
can be found in reference [20].
Each comb mode frequency is uniquely speciﬁed by an integer mode ordernumber and the two comb degrees of freedom [21]. Strictly speaking a comb has
many more degrees of freedom when one considers the phase of each mode. In ourcoherent control experiments, we extend the usual picture of the comb to include
the phases of each mode. However, if the spectral phase of each pulse in the pulse
train remains the same, the comb mode frequencies are unaﬀected.
Figure (2.1) shows the electric ﬁeld for a few ultrashort pulses in time domain(a) and the frequency spectrum corresponding to an inﬁnite train of such pulses(b). Each pulse in time domain has the same envelope, shown as the dashedline, under which is the carrier electric ﬁeld. Notice that the peak of the electricﬁeld for the ﬁrst pulse is aligned with the peak of the envelope. However, in thesecond pulse the carrier electric ﬁeld is slightly shifted from the envelope peak by
an amount ∆φceo . The carrier-envelope oﬀset phase is deﬁned for each pulse as
φceo, the change of this phase from pulse to pulse is ∆φceo.
The second important parameter is the inter-pulse period labeled τrep , its
inverse is the repetition frequency frep . In frequency domain the comb mode
frequencies are deﬁned by frep and fceo, the two comb degrees of freedom. The
spacing between modes is frep and the oﬀset of any mode frequency from an integer
multiple of frep is fceo . In terms of the rate of change of the carrier-envelope phasefceo =frep ∆φceo/(2π). Note that for the remainder of this thesis the repetition

12
frequency is abbreviated to fr , and fceo to simply fo . With this deﬁnition ofcomb degrees of freedom the frequency of a comb mode with order number N is
νN = N × fr + fo .
We will return to the origin of the carrier-envelope oﬀset phase in the context
of locking fo . However, given the time domain picture of the pulses one canalready understand the eﬀect ∆φceo has on the frequency spectrum. Consider theinterference between the ﬁrst two pulses. If we take time equal to zero at thepeak of the ﬁrst pulse, it has a Fourier transform given by the spectral envelopewith a constant zero phase. The second pulse is identical to the ﬁrst except it is
temporally delayed by (1/fr ) and has a phase oﬀset of ∆φceo. Its Fourier transformhas the same amplitude spectrum with a spectral phase given by,φ(ω) = ω ×

1− ∆φceo.
fr(2.1)

We chose a negative phase oﬀset to be consistent with our deﬁnition of fo , it does
not change the frequency spectrum. The interference between these two pulses
will have maxima at every frequency for which φ(ω) = 2πN. Solving the above
equation, we get νN =Nfr + ∆φceofr /(2π) as expected. Note that the frequencyof any comb mode can be deﬁned with either positive or negative fo , we use bothsigns in our deﬁnition of comb frequencies so the magnitude is always less thanfr /2.
2.1.2

Repetition Rate Stabilization

The repetition rate of the laser is servoed by adjusting the cavity length
directly. The output coupler is mounted on a fast ring PZT such that the beam
passes through the center. Considering our laser center wavelength is about 778
nm and fr is 100 MHz the comb order numbers N are of the order 4×106 . So
it is necessary to stabilize fr very well because any phase noise in the repetition

13

Figure 2.1: (a) Time domain picture of a pulse train with the relevant degrees of
freedom labeled. ∆φceo is the phase oﬀset of the electric ﬁeld from the peak of theenvelope, notice the ﬁrst pulse has no oﬀset phase. The time between pulses isτrep and is the inverse of the pulse repetition frequency fr . (b) Frequency domain
spectrum (amplitude only) of the comb corresponding to an inﬁnite number ofpulses. fr sets the frequency spacing between modes and fceo is the oﬀset frequency
of every mode from an integer multiple of fr . fceo is given by ∆φceo × fr /(2π) andthe N th comb mode has an optical frequency given by N × fr + fceo .

14frequency is multiplied to the optical domain by N. To achieve better phasesensitivity, we lock to the 10th harmonic of fr rather than the fundamental.
A small fraction of the output power is reﬂected oﬀ of a Brewster window toa fast >1 GHz photodiode. The majority of this signal is then bandpass ﬁlteredand ampliﬁed before it is mixed with the local oscillator. Some of the power is sent
to a cesium referenced counter that counts the 100 MHz repetition rate for our
adjustments of the comb mode absolute frequencies. Our local oscillator signal
for locking fr is primarily from the 1 GHz output from a Wenzel crystal oscillator.
To allow for easy tuning of fr the 1 GHz Wenzel signal is mixed with the signal
from a direct-digital synthesizer (DDS). We only operated the frequency of the
DDS at about 10 MHz or less. So for example, if we desired a fr of 99 MHz, we
locked the laser to the 990 MHz sideband produced by mixing the Wenzel with
the DDS at 10 MHz. Although our typical fr locking was done no more than
±0.5 MHz from 100 MHz. The error signal produced from mixing the measured
fr harmonic and the local oscillator was then ampliﬁed and ﬁltered using a JILA
loop-ﬁlter. The output of which was further ampliﬁed with a 1 kV PZT driver toservo the high-voltage PZT and thus change the cavity length.
Clearly this is just an overview of basic fr locking. Reducing the laser
linewidth was not of great importance for our particular spectroscopy applications.
We have locked the laser directly to a Iodine stabilized NPRO:YAG laser with ashort time linewidth of ∼5 kHz at 1 ms. However, this did not improve ourmeasured two-photon transition linewidths. Primarily because the transition is
power-broadened to just over the spectral resolution limit of our laser linewidth,see section (4.2) for a details. Tremendously better comb linewidths (<1 Hz) may
be obtained by locking the laser directly in the optical regime to a cavity stabilized
diode laser [22].

15
2.1.3

Oﬀset Frequency Stabilization

We used a technique called self-referencing to measure the fo of our frequency comb [23]. This technique gets its name from the fact the fo signal is generated from the comb itself, rather than from a heterodyne beat with a separate
laser. A typical implementation of self-referencing requires an octave spanning
pulse spectrum. Our implementation uses an f-2f interferometer. The principle
is simple, if the pulse spectrum has two colors that are spaced in frequency by
one octave the long wavelength color may be doubled and mixed with the shortwavelength color, producing a fo beat note. More speciﬁcally, suppose we have
two frequencies in the broadened pulse spectrum,νN = N × fr + fo
ν2N = 2N × fr + fo .

(2.2)

If we double the frequency at νN , the heterodyne beat frequency between the
doubled light and ν2N is,
2νN − ν2N = 2(N × fr + fo ) − (2N × fr + fo )= fo .

(2.3)

So the trick is to generate a phase coherent pulse spectrum that spans one octave.
It was the development of dispersion modiﬁed micro-structured ﬁber [24,25] that enabled the ﬁrst experimental demonstrations of self-referencing [1, 26].
Micro-structure ﬁber, also called photonic crystal ﬁber, exhibits two importantfeatures for eﬃciently producing an octave spanning bandwidth. Micro-structure
ﬁber gets its name from the regular pattern of air holes that extend the length of
the ﬁber and surrounding the core, forming a photonic band gap to conﬁne thelight. By tuning the waveguide dispersion of the ﬁber the net dispersion can bemade zero near the desired input pulse wavelength. Typically such a ﬁber has a16core diameter of about 1.7 µm for a zero dispersion point at 800 nm. Conﬁning
the pulse to such a small area can generate extremely large peak intensities, which
due to the zero dispersion property may persist through the ﬁber with minimal
pulse stretching. This combination of high intensity over a long interaction length
allows for eﬃcient four-wave mixing. In practice we generate the necessary octaveof spectrum using 7-10 cm of ﬁber with about 150 mW of power.
An alternative approach to generating an octave spanning spectrum is fromthe Ti:Sapphire laser itself. With a careful, and perhaps lucky, set of laser optics
and high intra-cavity power people have demonstrated fo detection using thespectrum directly from the Ti:Sapphire laser [22, 27]. Another technique only
requires 2/3 of an octave spectrum. The fo beat is formed by doubling the highfrequency portion of the comb and mixing it with the tripled light from the low
frequency portion [28].
The design of our f-2f interferometer is based on a prism pair to separate thetwo wavelengths we use for our beat note, 532 nm and 1064 nm. After the outputof the ﬁber the continuum is spectrally dispersed using two prisms. There are
two retro-reﬂection mirrors in the dispersed beams such that the two wavelengths
are reﬂected from diﬀerent mirror. One of the mirrors is on a translation stage
to compensate for the diﬀerent group delays of 532 nm and 1064 nm, so thatthey overlap temporally. After a second pass through the prism pair the two
wavelengths are combined, ideally into one spatial mode. Both wavelengths arefocused through a 1 mm BBO crystal to double the 1064 nm portion of thespectrum to 532 nm. To ﬁlter out all but the desired wavelength the beam issent through a 10 nm FWHM optical interference ﬁlter centered at 532 nm. Dueto the fact the doubled 1064 nm and original 532 nm are orthogonal polarization
after the BBO crystal we project the two polarizations modes onto a common axis
using a polarizer. The resulting heterodyne beat frequency is then measured on

17a photodiode.
The output of the photodiode is low-pass ﬁltered at 50 MHz to attenuate
the signiﬁcant amount of RF power at fr , which would saturate our ampliﬁers.
The ﬁltered signal is then ampliﬁed to about -5 to 0 dBm. A directional coupler
is used to divert a small amount of the fo beat to an RF spectrum analyzer tomonitor the beat note. Usually we operate with no less than 35 dB signal to noise
in a 100 kHz resolution bandwidth, the best we’ve obtained is about 55 dB above
the noise ﬂoor.
Most of the fo signal is then sent to a digital phase detector and divided
down by 64. We divided down the signal from a typical fo of 18 MHz to about∼280 kHz. This was done to make phase locking easier, phase excursions fromnoise are reduced by 64 so it’s easier to retain locking, although less sensitive.
The phase stability is not as essential for fo as it is for fr , simply because fo is
only multiplied by unity in the comb mode frequency equation.
To understand how to servo the laser and thus control fo we must ﬁrst
understand the origin of the phase, φce . Recall that φce is deﬁned between thepulse envelope, which travels at the group velocity vg , and the underlying electric
ﬁeld, which travels at the phase velocity vp . In the laser cavity, the diﬀerence of vg
and vp results in a generally nonzero ∆φce given by ∆φce = lc ωc ( v1g − v1p ), wherelc is the cavity round trip length and ωc is the carrier frequency.
With this knowledge in mind it is clear that tuning the dispersion of thecavity will also change fo . This was accomplished by tilting the end mirror of
our cavity, at which point the spectrum is spatially dispersed via the intra-cavity
prisms. The resulting wavelength dependent extra path length changes the groupvelocity. Experimentally this was done by mounting the end mirror after the
intracavity prism pair on a split PZT that was driven diﬀerentially such thatthe mirror tilts in response to the error signal [19]. The error signal generated

18
by mixing the measured fo /64 with a local oscillator provided by a DDS was
ampliﬁed and ﬁltered like the fr lock. To drive the split PZT this error signal
from the loop-ﬁlter was again ampliﬁed to ±0-150 V with a PZT voltage driver.
There are two outputs from the PZT voltage driver of equal and opposite voltages.
So that one half of the split PZT contracts when the other expands, causing theend mirror to tilt. Alternatively, one can change the dispersion of the laser cavity
via control of the pump power [29]. This has the advantage of a much larger servo
bandwidth of ∼100 kHz.
2.2

Magneto-Optical Trap
All of our experiments were conducted on a laser cooled and trapped sampleof 87 Rb using a magneto-optical trap (MOT). Many experiments using cold atoms
are conducted with a MOT, in particular using rubidium atoms [30]. There has
been a great deal of literature published on the experimental techniques and theoretical treatments of laser cooling. A basic laboratory guide to making a MOT
can be found in reference [31]. Informative references on the classical and quantum theory of Doppler cooling may be found in references [32, 33]. The scalinglaws applicable to a MOT have been studied and presented in reference [34] andsimilarly for the dynamics of the trapped atom cloud [35].
In our experiments we used a Rb vapor cell MOT [36], as opposed to a
trap loaded using an atomic beam. Experimentally this is simpler than using an
atomic beam, although typically less atoms are trapped. The atoms are sourced
from Rb getters, Ni-Cr alloy metal containing a salt of Rb atoms [37]. Inside theglass chamber there are three getters, each can be supplied current separately, sothat when one is depleted another may be used. Typically we operate a single
getter at a current of 3.5 Amps, however, in one experiment this was lowered to2.75 Amps to reduce the amount of hot background atoms, see section (6.2).

19
We used grating stabilized laser diodes for the trap and repump light, withwavelengths of about ∼780 nm. Both lasers were built by Dr. Adela Marian, whostarted the DFCS experiments. I will not detail the design of the lasers in thissection, refer to her thesis [37] for some more detail and reference [38]. Instead I
will provide more of a hands on explanation of the diode laser system.
Both the lasers have two feedback mechanisms. The grating is mounted ona fast disk PZT, and the diode laser current may be servoed as well. For coarse
adjustments of the laser wavelength the baseplates are cooled via Peltier TECs
below room temperature, however the current control is unipolar so they may
not be heated. Another coarse wavelength adjustment is the grating horizontal
control, which is usually adjusted by hand, but there is a PZT as well for ﬁnetuning. Finally for optimizing the grating feedback there is the vertical grating
adjustment, also controllable via PZT.
A typical alignment procedure is as follows. Initial coarse adjustments to
set the diode laser operating at the desired wavelength is done using a wavemeterand the horizontal adjustment of the grating feedback. If in the tuning of thegrating the laser mode hops over the desired operating wavelength the baseplate
temperature should be adjusted, tuning the cavity length. Once this is done itshould be possible to tune the laser wavelength near the desired wavelength using
only the grating horizontal adjustment. The laser current may need to be adjusted
somewhat coarsely ±10 mA to obtain the exact wavelength. The usual current
settings are ∼75 mA for the trap laser and -65 mA for the repump. At this point
the grating feedback should be optimized using the vertical grating adjustment.
After careful adjustment of the vertical tilt the laser threshold current should be∼24 mA, indicated by a clear jump in laser output power. With a voltage sweep
driving the fast PZT the saturated absorption spectrum is monitored on a scope.
Using both the fast PZT voltage oﬀset and laser current we search for the sub-

20
Doppler features. Adjustment of the laser current is usually necessary so that thefull set of hyperﬁne lines can be swept over without encountering a mode hop.
The trap and repump laser frequencies were stabilized using saturated absorption spectroscopy [39]. Figure (2.2) provides an example of the measured
sub-Doppler transmission lines using saturated absorption spectroscopy. With alaser wavelength of about 780 nm, we measured four sets of lines from 5S1/2 to
5P3/2 . In our spectrometer vapor cells there are two isotopes, 87 Rb and 85 Rb. For
each isotope there are two sets of lines separated by the ground state hyperﬁnesplitting, about 6.8 GHz for 87 Rb. We chose to lock our lasers to 87 Rb, and thus
optically cool only that isotope. Each line is labeled by its hyperﬁne quantum
number in Fig. (2.2).
Notice that there are several extra unlabeled transitions, these are referred toas cross-over transitions and they occur at the average frequency of two hyperﬁnetransitions. This is due to the Doppler width of the atoms in the vapor cell.
Basically each cross-over line corresponds to saturating a velocity group of atoms
with a Doppler shift equal to half the spacing between two hyperﬁne levels. For
example, we lock the repump laser to the F=2-F=3 cross-over transition. The
hyperﬁne spacing between F=2 and F=3 is 156.9 MHz, so the repump laser output
is 78.45 MHz detuned from either hyperﬁne level for atoms at rest in the lab frame.
We used a few diﬀerent types of saturated absorption spectrometers over thecourse of our experiments. The repump laser was always frequency modulated atabout 4.6 MHz via direct current modulation. So in this case both the pump andprobe beams for the repump laser spectrometer are frequency modulated. For thetrap laser we either used oﬀ-set locking or frequency modulated the pump beam.
A side-lock was achieved simply by providing a stable voltage oﬀset to the F=3transition shown in Fig. (2.2). By adjusting the magnitude of the voltage oﬀset
the detuning of the trap laser from the resonance can be tuned. the drawback

21

Figure 2.2: Examples of the four sets of lines measured as transmission peaks using
saturated absorption spectroscopy. In our experiments we always lock to the 87 Rb
isotope lines. The spectra correspond to the 5S1/2 to 5P3/2 hyperﬁne transitions,
there are two sets for each isotope. Unlabeled peaks are from crossover transitions,as explained in the text. The blue bars in the bottom panel indicate the cross-overtransitions and the corresponding pair of hyperﬁne transitions.

22of this type of locking is that for a given voltage oﬀset changes in signal size canalter the detuning. For a more stable detuning we used the frequency modulation
technique, however, in this case the modulation was done using an EOM [40, 41].
In the case of FM spectroscopy the measured signal is demodulated, with theappropriate local oscillator phase, the resulting lineshape is the ﬁrst derivativeof those shown in Fig. (2.2). So there is a zero crossing at the peak of thetransmission through the cell, this provides a more stable reference.
To act as a shutter and shift the repump laser to the resonant F=2 transition,
an AOM operating at 78.45 MHz was used in the beam path of the repump laser.
We used the -1 order to shift to F=2 resonance. Similarly for the trap laser weused two AOMs. The ﬁrst AOM is operated with low RF power at a frequency
of about 88 MHz and the +1 order beam is sent to the spectrometer. When thelaser is subsequently locked to the atoms its frequency is pushed below that of
the F=3 transition by 88 MHz. The light transmitted through the ﬁrst AOM isthen passed through a second, operated with the maximum +1 order diﬀraction
eﬃciency. By setting the frequency of this AOM to 80 MHz the resulting +1 orderbeam is detuned 8 MHz below the F=2 to F=3 cooling transition.
In the context of each experiment we explain the speciﬁc MOT operation
timings. However, in general we cooled the atoms for only about 7 ms every 10 ms
using standard Doppler cooling. In our spectroscopy experiments a subsequent
polarization-gradient cooling (PGC) period is used to cool the atoms further to
sub-Doppler temperatures [42, 43]. With careful nulling of the magnetic ﬁeld at
the atom sample temperatures of 20 µK may be reached using PGC of 87 Rb. The
procedure for nulling the ﬁeld is explained in section (4.2.4). During the PGC
cycle the trapping magnetic ﬁeld was turned oﬀ and the trap laser was attenuatedand far-detuned (∼40 MHz) below the resonant F=2 to F=3 transition. There isno central trapping force in PGC, however, there is a viscous force [44]. So the23
expansion of the atom cloud is slowed using the PGC cycle.
The description provided in this section is only meant to be an overview. In
actual practice the characteristics of the cooling and trapping were not routinely
measured, such as MOT diameters and number. Instead the alignment of thetrapping beams, magnitude of the trap magnetic ﬁeld, and trap laser detuning
were optimized by monitoring our two-photon signal. Although for reference someof the initial MOT characterization was done with a getter current of 3.5 A, atrap laser detuning of 10 MHz, and a magnetic ﬁeld current of 4.5 A. In this casethe measured atom number was ∼5×106 in a diameter of ∼0.5 mm [37].

Chapter 3
TheoryIn this chapter we present two of the primary approaches used to solve forthe interaction of an optical comb with a multi-level atom. The diﬀerent models
are best suited for understanding diﬀerent aspects of our experiments. In the limit
of weak excitation, we used a model based on second-order perturbation theory
to calculate the two-photon transition amplitude. At higher ﬁelds we used a nonperturbative approach based on the numerical solution of the Liouville equation
for the density matrix that includes optical pumping due to spontaneous decay.
When the primary interest is in the population distribution of all the hyperﬁnestates in the 5S, 5P, and 5D manifolds we used a fourth-order perturbation model
based on the Liouville equation. The following two sections provide a detailed
derivation of the two main models used and the particular situations for which
they are most suited.
There has also been a signiﬁcant amount of literature published studying
diﬀerent aspects of pulsed ﬁeld interactions with atoms. Most of the work done
consisted of theoretical studies of the interaction of a two-level atom with highintensity ultra-short pulses. In the context of an ultrashort pulse interacting witha two-level atom, eﬀects beyond the common rotating-wave approximation havebeen investigated [45, 46, 47]. Deviations from the classical area theorem havebeen predicted for intense ultrashort optical pulses [48]. Using the exact solution

25
to the Liouville equation, including the counter-rotating terms, we have checkedthat our experimental parameters are not in the range of those studied in the cited
works. Analytic expressions have been obtained for the interaction of a two-level
atom with a train of phase coherent pulses as well [49, 50]. One of the reasons
we present the models in this chapter is because the exact analytic solution for a
simpliﬁed system of three-levels and two-modes is complicated. However, we dorecommend the work of Brewer and Hahn [51] for an approximate solution andVitanov for a complete solution [52], and also Carroll and Hioe [53].
3.1

Second-Order Perturbation Theory
In this section, we provide a derivation of the equation for the excited state

amplitude in a three-level system under excitation by two-modes. The equation
is derived using second-order perturbation theory with phenomenological decay
included via a modiﬁed Hamiltonian. The focus is only on the derivation of
the relevant equation and the approximations made in this section. We use thisequation in subsequent chapters were the primary interest is in the phase of thetwo-photon amplitudes corresponding to many diﬀerent transition pathways. Oneof the limitations of this approach is that it does not predict the Stark shift ofthe excited state or power-broadening eﬀects. Furthermore, the equation is only
applicable for the steady excited state population. We will address the modelingof these non-perturbative and transient eﬀects in the context of the Liouvilleequation for the density matrix of the system in section (3.2).
The derivation we present follows most closely that found in the text byRodney Loudon, The quantum theory of light [54]. There are two key diﬀerences,
our derivation includes two time varying electric ﬁelds, and phenomenologicalrelaxation. Our goal is to derive the value of c3 (t), the excited state amplitude,
valid to second order in the interaction Hamiltonian for the system shown in Fig.

26(3.1).

27|3>

B A A B

µ23
Z3
|2>

A B A B

µ12

Z2

|1>
Figure 3.1: Three-level cascade system with two dipole allowed transitions excited by two modes labeled A and B. For completeness we show all four possible
conﬁgurations of excitation, each gives rise to a two-photon amplitude. There is
only one conﬁguration that is resonant with the intermediate and ﬁnal states, theamplitude due to this conﬁguration is dominant.

28
We start with the Hamiltonian for the system modiﬁed to include relaxation,Ĥ(t) = Ĥo + ĤI (t),

(3.1)

with,
Ĥo = (ω2 − iγ2 /2)|22| + (ω3 − iγ3 /2)|33|ĤI (t) = −µ12 Ẽ(t)|21| − µ23 Ẽ(t)|32| + H.C.,(3.2)

where we have deﬁned,
1Ẽ(t) = (EA e−iωA t + EB e−iωB t ).
2

(3.3)

The relaxation is given in terms of the scattering rates for each excited state suchthat γi =2π∆νi where ∆νi is the natural linewidth of state i in Hertz. Of course
electric ﬁelds are real quantities, so the relevant experimental ﬁeld is given by,
E(t) = Ẽ(t) + Ẽ ∗ (t), where the magnitude of the ﬁelds are EA and EB .
The amplitude c3 (t) is given exactly by the expression,
3|e−iĤ(t)/|1,(3.4)

for an atom initially in the ground state. We wish to ﬁnd the leading ordercontribution to this expectation value relevant for three-level system. Within the
dipole approximation the ground state is only coupled to the excited state by a
two-photon interaction. Although we typically excite the 5S to 5D transition in87

Rb we are ignoring the weak quadrupole transition. So the leading order term

for two-photon absorption will be second order in the interaction Hamiltonian,and thus Ẽ(t).
At this point we introduce a diﬀerential equation that is commonly used toﬁnd an arbitrary order solution in perturbation theory. It can be easily veriﬁedby carrying through the diﬀerentiation that,eiĤo t/ĤI e−iĤt/ = id  iĤo t/ −iĤt/
.
e
e
dt

(3.5)

29
To put the previous equation into a useful form we integrate both sides and aftersome rearranging arrive at,
i t iĤo t1 /
−iĤt/
−iĤo t/iĤo to / −iĤto /
−iĤ(t1 )t1 /eĤI (t1 )e
=e
e
−
e
dt1 .
e
 to

(3.6)

If we make the interaction Hamiltonian zero at a time inﬁnitely in the past
and assume all population is initially in the ground state, we can set to =−∞ andsimplify the above equation to,
i t iĤo t1 /
−iĤt/
−iĤo t/−iĤ(t1 )t1 /1−ĤI (t1 )e
=e
e
dt1 .
e
 −∞

(3.7)

The LHS of the above equation is the correct expression to solve for c3 (t), however,the RHS is a function of the total Hamiltonian Ĥ. At this point a common trickin perturbation theory is used to arrive at an equation valid for any order ofthe interaction Hamiltonian. Equation (3.7) can be substituted into itself for
e−iĤ(t1 )t1 /, with the variable of integration changed in the substitution to t2 . The
portion of the resulting equation second order in ĤI becomes, t1−e−iĤo (t)/ t iĤo t1 /
−iĤt/
−iĤo t1 /ĤI (t1 )e=e
dt1
eiĤo t2 /ĤI (t2 )e−iĤo t2 /dt2 .
e
2
−∞
−∞
(3.8)
Strictly speaking, after only one substitution the last term in the above equation
should be, e−iĤt2 /. However, notice that upon one more substitution we would
only retain the ﬁrst term in Eq. (3.7) for an expression second order in ĤI . So it
is valid to simply substitute e−iĤo t2 / for −iĤt2 /. The resulting equation is exactly
what is needed to solve for c3 (t) at second order in ĤI . In the derivation of thisequation we dropped several terms that were either not a function of ĤI or only
ﬁrst order in this operator. It is not an approximation to drop those terms, theexpectation value for c3 given by any operator lower than second order in ĤI iszero.
Now that we have the appropriate operator equation, the expectation value
can be calculated. This is done by inserting the unity operator as a sum over all30
states in between the two ĤI operators. In other words the equation for c3 (t) isnow,
−e−iĤo (t)/ t iĤo t1 /ĤI (t1 )e−iĤo t1 /dt1
c3 (t) =
3|
e
2

−∞
i
 t1
×
eiĤo t2 /|ii|ĤI (t2 )e−iĤo t2 /dt2 |1.


(3.9)

−∞

For the purposes of this derivation we are including only one intermediate state,
so the summation over |i can be dropped. It is of course rather straight-forward
to see that the case for multiple intermediate states is just given by the coherent
sum of the amplitudes calculated using each state separately. After evaluating
the operators given by Eq. (3.2) we arrive at the c-number equation,
 t1−µ12 µ23 −(iω3 +γ3 /2)t te
dt1
dt2 e(iω3 +γ3 /2)t1 e−(iω2 +γ2 /2)t1
c3 (t) =2−∞−∞×e(iω2 +γ2 /2)t2 Ẽ(t1 )Ẽ(t2 ).
Performing the integral over t2 the c-number equation becomes,−µ12 µ23 −(iω3 +γ3 /2)t t
c3 (t) =e
dt1 e(iω3 +γ3 /2)t1 e−(iω2 +γ2 /2)t1 Ẽ(t1 )
2
2
−∞


t1 (i(ω2 −ωA )+γ2 /2)
EB et1 (i(ω2 −ωB )+γ2 /2)
EA e
+
.
×
γ2 /2 + i(ω2 − ωA )γ2 /2 + i(ω2 − ωB )(3.10)(3.11)

We can start to interpret this result by noticing the two resonant denominators. These two denominators correspond to a photon from either mode A or
B exciting the electron from the ground to intermediate state. Considering these
two resonant denominators are to be multiplied by Ẽ(t1 ) we can see how the four
possible conﬁgurations shown in Fig. (3.1) arise. The main reason for mentioning
this is so that we may simplify the expression from this point, the full expression
is complicated and not necessary.
In all of our experiments we are only interested in the near-resonant orresonant two-photon amplitudes. The two conﬁgurations that correspond to twophotons absorbed from the same mode are in general not two-photon resonant

31
and can be safely neglected. This leaves only two conﬁgurations, absorption of aphoton from mode A followed by B, and vice-versa. With this simpliﬁcation the
two-photon amplitude becomes,ei(ω3 −ωA −ωB )µ12 µ23 EA EB e−iω3 t
c3 (t) =42(ω3 − ωA − ωB ) − iγ3 /2


1
1
×
+
.
(ω2 − ωA ) − iγ2 /2 (ω2 − ωB ) − iγ2 /2(3.12)

The above form of the equation is used in the subsequent chapters. Sometimes we approximate this equation further when we are considering only theordered absorption from two modes, meaning we retain only one of the intermediate resonance denominators. The condition of two-photon resonance is met whenωA + ωB = ω3 . In this case notice that the amplitude evolves phase at its natural
frequency of ω3 , often this phase evolution is not of importance and therefore
neglected.
This treatment based on second-order perturbation theory proved to bequite useful in understanding our experimental results. One of the great advantages of this method is that a total two-photon amplitude may be easily constructed by coherently summing the c3 calculated for each mode combination andintermediate state [55]. Furthermore, it is the only model we use explicitly expressed in frequency domain, so it allows easy interpretation. For example, it iseasy to see the phase dependence of the two-photon amplitude as a function ofintermediate state detuning.

32
3.2

Density Matrix Treatment
A state vector in Hilbert space is useful for describing a quantum system ina coherent superposition of eigenstates. However, it is not suﬃcient to describe
arbitrary quantum states of an ensemble, which in general may be in a mixed
state.
In our experiments we must treat the population relaxation due to spontaneous emission. Suppose at some initial time an ensemble of atoms is preparedsuch that every atom is in an excited electronic state. After some period of time,
half of the atoms will have undergone spontaneous emission and the electron in
those atoms will relax to the ground state. In this example, half of the atoms
are excited and half are in the ground state. The probability of ﬁnding an atom
in a certain state is entirely classical, as opposed to a quantum superposition ofstates. If one considers the subsequent interaction of this system with a resonant
pulse of area π/2 the diﬀerence becomes more clear. More generally the state ofan ensemble can be have quantum superpositions and classical distributions, thisis referred to as a mixed state.
To account for incoherent population distribution due to spontaneous emission, we model the state of our system using a density matrix [56]. In terms of theamplitudes of the familiar state vector based approach, the elements of the density
matrix are deﬁned as, ρij =ci c∗j . So all density matrix elements are deﬁned as
ensemble averages. Matrix elements in which i = j are typically called population
terms, more appropriately they correspond to the probability of the system being
in state i. For a closed system, like ours, the sum of the population terms is unity.
The oﬀ-diagonal terms in which i = j are referred to as the coherences.
Coherences are in general complex numbers that range in magnitude from 0 to
0.5. In our previous example of spontaneous emission causing half of the atoms to

33
be in the ground state the coherence term would be zero. The maximal coherencecase of 0.5 would correspond to all the atoms in the ensemble being in an equalsuperposition of the ground and excited states with the same phase.
The time evolution of the density matrix is governed by the Liouville equation. A convenient form of this equation including relaxation can be found in
the text by R. W. Boyd, Nonlinear Optics [57]. In the remainder of this section,a brief overview of the density matrix model is introduced highlighting points
relevant to our implementation.
We start with the following form of the Liouville equation,
ρ̇ij =−i 
1
[Hik ρkj − ρik Hkj ] −[Γik ρkj − ρik Γkj ] + F eedingT erms (3.13)

2
k

k

where the sum is taken over all states and the relaxation operator is deﬁned
as Γij = γij δij . The feeding terms account for incoherent population relaxationfrom higher energy states to lower energy states. The only loss of population
we expect in our experiment is due to ionization, which over the duration of ourmeasurements is negligibly small [58]. Therefore with the inclusion to the feedingterms the total population remains unity.
The full set of Liouville equations for a four-level system in a diamondconﬁguration are calculated using Eq. (3.13) and listed below. We present the
four-level equations because they include the eﬀect of branching ratios and caneasily be simpliﬁed to a two- or three-level system.
Populations:

34

iE(t)
[µ12 (ρ21 − ρ12 ) + µ13 (ρ31 − ρ13 )] + (γ2 ρ22 + γ3 ρ33 )iE(t)[µ12 (ρ12 − ρ21 ) + µ24 (ρ42 − ρ24 )] − γ2 ρ22 + γ42 ρ44
=

iE(t)
=
[µ12 (ρ13 − ρ31 ) + µ34 (ρ43 − ρ34 )] − γ3 ρ33 + γ43 ρ44iE(t)[µ24 (ρ24 − ρ42 ) + µ34 (ρ34 − ρ43 )] − γ4 ρ44=ρ̇11 =(3.14)

ρ̇22(3.15)

ρ̇33
ρ̇44(3.16)(3.17)

Coherences:ρ̇12 =

ρ̇13 =

ρ̇14 =

ρ̇23 =

ρ̇24 =

ρ̇34 =−i[ρ12 (E1 − E2 ) + µ12 E(t)(ρ11 − ρ22 )
γ2−µ13 E(t)ρ32 + µ42 E(t)ρ14 ] − ρ122−i[ρ13 (E1 − E3 ) + µ13 E(t)(ρ11 − ρ33 )
γ3−µ12 E(t)ρ23 + µ43 E(t)ρ14 ] − ρ132−i[ρ14 (E1 − E4 ) + E(t)(µ24 ρ12 − µ12 ρ24
γ4+µ34 ρ13 − µ13 ρ34 )] − ρ142−i[ρ23 (E2 − E3 ) + E(t)(µ13 ρ21 + µ43 ρ24
γ2 + γ3ρ23−µ12 ρ13 − µ24 ρ43 )] −2−i
[ρ24 (E2 − E4 ) + E(t)µ24 (ρ22 − ρ44 ) + E(t)(µ24 ρ23
γ2 + γ4ρ24−µ12 ρ14 )] −2−i
[ρ34 (E3 − E4 ) + E(t)µ34 (ρ33 − ρ44 ) + E(t)(µ24 ρ32

γ3 + γ4ρ34−µ31 ρ14 )] −2

(3.18)(3.19)(3.20)(3.21)(3.22)(3.23)In the above equations, the energy of state i is Ei and the electric ﬁeld E(t)is the real electric ﬁeld. The equations for the populations contain the previously
mentioned feeding terms. For example, the equation for ρ11 has been modiﬁedwith +(γ2ρ22 + γ3 ρ33 ), so all of the population relaxation from states 2 and 3

35branches to the ground state. A slightly diﬀerent case is for ρ22 , this has been
modiﬁed with +γ42 ρ44 . Where γ42 is the decay rate of state 4 multiplied by thebranching ratio to state 2. In this case, the population in state 4 relaxes to thetwo intermediate states, so the proper branching ratio must be included. One
thing to note is that the coherences are complex quantities that obey the relation,
ρ12 =ρ∗21 .
We use a numerical approach to solving the set of coupled ordinary differential equations. Generally a fourth-order Runge-Kutta (RK4) algorithm is
suﬃcient. However, in the current form of the coherences they evolve phase at
approximately their natural frequencies, i.e. very fast. To signiﬁcantly reduce
the numerical error it is necessary to make a change of variables. We use thesubstitution, ρij =ρ̃ij eiωji t , where ωji=ωj − ωi . Under this change of variables thenatural phase evolution of the coherences is explicitly in the equations of motion.
Even after the change of variables for the coherences the time step setting
for the RK4 algorithm must be less than an optical period. This is due to the E(t)term. In some of our applications the ﬁeld in time domain is generated directly
by Fourier transforming the measured pulse spectrum with some arbitrary phasemask, so it is a numerically generated time series. If this is not the case and we usean analytic expression for the ﬁeld, we can invoke the rotating-wave approximation(RWA). This is done by expressing the ﬁeld as, E(t) = Eo A(t) 12 (eiωo t + e−iωo t )where A(t) is the slowly varying envelope and ωo is the carrier frequency of thepulse. After this substitution, terms that evolve at ∼ 2 × ωo may be dropped fromthe equations of motion. Having performed a comparison between the solutions
with and without the RWA, for our typical experimental conditions, the diﬀerenceis entirely negligible. After invoking the RWA the time step need only be shortenough in duration to sample the pulse envelope. Note that this explanation is
only valid for transform limited pulses, we did not use the RWA when considering

36
chirped pulses.
So far we have only described the approach used to numerically solve theequations of motion during excitation by a single pulse. To extend this model for
excitation by a coherent pulse train we break up the solution into a numerical partand an analytic part. Clearly this is necessary considering the time scales of the
problem. The numerical solution evolves the initial density matrix from time zero
to some time after one pulse, where the pulse peak ﬁeld occurs at the middle ofthis time window. The coherences and populations may be solved for analyticallybetween pulses because the driving ﬁeld is zero. The analytic evolution of thecoherences is simply free-evolution of phase at the natural frequency and decay
due to spontaneous emission.
Due to the large time between pulses compared to the pulse duration, mostof the incoherent population redistribution occurs between pulses. During thisﬁeld-free time, the equations for the populations with E(t)=0 can be analyticallysolved. The resulting equations are,γ2 γ42 ρ44 (0) 1 −γ2 t1[ (e− 1) − (e−γ4 t − 1)]
γ2 − γ4 γ2γ4γ3 γ43 ρ44 (0) 1 −γ3 t1[ (e− 1) − (e−γ4 t − 1)]+(1 − e−γ3 t )ρ33 (0) +γ3 − γ4 γ3γ4ρ11 (t) = (1 − e−γ2 t )ρ22 (0) ++ρ11 (0)γ42ρ44 (0)[e−γ4 t − e−γ2 t ]γ2 − γ4γ43−γ3 t
ρ33 (0) +ρ44 (0)[e−γ4 t − e−γ3 t ]ρ33 (t) = e
γ3 − γ4ρ22 (t) = e−γ2 t ρ22 (0) +ρ44 (t) = ρ44 (0)e−γ4 t .

(3.24)(3.25)(3.26)(3.27)

Where the time, t, is the inter-pulse period and all initial populations are ρii (0).
The initial populations are given by the density matrix after the previous pulse.
This two step process of solving for the density matrix is then simply repeated forsome desired number of pulses.

37
For a complex level structure the direct numerical solution for the density
matrix becomes increasingly diﬃcult. The number of equations that must be
solved grows as the square of the number of states. So for example in 87 Rb
there are 16 hyperﬁne levels to consider in the full 5S to 5P to 5D interaction,
this would correspond to solving 162 =256 coupled ordinary diﬀerential equations.
One alternative is referred to as the quantum Monte Carlo wavefunction approach[59]. In this type of treatment the number of equations grows linearly with thenumber of states. However, it is a stochastic model in which spontaneous emission
events occur at random times. So to obtain high quality results representing an
ensemble of atoms the model must be solved many times.
To model the full set of 16 hyperﬁne levels we used fourth-order perturbation
theory based on the Liouville equation. I will not present the details of this model
but rather some of the key approximations made in its derivation. The model wasderived and implemented in C-code by one of our very talented collaborators, Dr.
Daniel Felinto [60]. In this model the excitation of the full set of hyperﬁne levels is
calculated for an impulsive pulse train to fourth-order in perturbation theory. The
impulsive approximation is used when the spectral width of the exciting pulse issigniﬁcantly larger than the frequency spread of the hyperﬁne levels [50]. Clearlya delta function (impulsive) electric ﬁeld has an inﬁnitely broad spectrum andno carrier-frequency need be deﬁned. The only input to this model to describe
the ﬁeld is an area parameter, which is deﬁned as the pulse area with a dipole
moment given by, µ = eao , where e is the electron charge and ao is the Bohrradius. It should also be noted that the dipole moments used in the model are
given by the root-sum-of-the-squares of the individual dipole moments for eachmagnetic sub-level transition. These are calculated directly in the software, soeither polarization q=(1,0) may be used. However, note that this approximation
forces all dipole moments to be positive, see section (6.2) for an example of when

38
this approximation breaks.
One of the useful features of this model is that it does remarkably well at
predicting the population distribution due to incoherent optical pumping. In fact
we used this model to normalize one of our measured lineshapes in section (4.4)
to account for optical pumping eﬀects. An introduction to this model outlining
the general approach may be found in the thesis of Dr. Adela Marian [37].

Chapter 4
Direct Frequency Comb Spectroscopy

In this chapter we present our experiments demonstrating single and twophoton direct frequency comb spectroscopy (DFCS). We have conducted all of
our experiments using 87 Rb atoms trapped and cooled prior to measurement ina MOT. This choice of atom is not only convenient from an experimental pointof view for laser cooling and trapping, but also the energy level structure of the
5S, 5P, and 5D states is well known from previously published experiments. Furthermore, this is the ﬁrst experiment to use an optical frequency comb to directly
conduct high-resolution absolute frequency measurements, so we may easily compare our results to those obtained with traditional cw-laser spectroscopy.
The full set of hyperﬁne levels that may be excited by our Ti:Sapphire laser
spectrum is shown in Figure (4.1). We indicate the 5S1/2 F=2 to 5P3/2 F=3 to
5D5/2 F=4 transition by the red lines. This is a particularly important two-photontransition, it will be the basis for our experimental demonstration of coherent
accumulation in section (4.1) and our studies of systematic errors in section (4.2).
Not only does this transition have the largest dipole moments, but it is also
closed, meaning all excited population returns to the 5S1/2 F=2 ground state. The
broadband nature of DFCS can be understood directly from this ﬁgure, notice
the range of wavelengths span ∼54 nm, we in fact measure transitions over all of
the indicated wavelengths. One important wavelength not indicated in this ﬁgure

40
corresponds to half of the 5S to 5D two-photon transition frequency at 778 nm, we
generally center our spectrum at this wavelength. The 5P3/2 intermediate states
are at ∼1 THz below the half two-photon frequency and the 5P1/2 states at ∼8
THz below. The lifetime of population in a 5P states is taken to be 27 ns, 241 ns
for the 5D states [61], and 88 ns for the 7S states [62]. So the natural linewidthsof the 5P states are 6 MHz, ten times broader than the 5D state linewidth of 660
kHz. Keep these numbers in mind as we will manipulate the comb frequencies
and spectral phases around these real intermediate states.
We use the femtosecond pulses emitted by the mode-locked Ti:Sapphire laser
to directly excite the one or two-photon transition we wish to study. Generally
the laser is operated with a ∼55 nm FHWM spectrum centered at 778 nm that
we approximate to be Gaussian for modeling purposes. At our repetition rate offr ∼100 MHz, this yields about 200 nW of power per comb mode near the center
wavelengths used to excite the 5S to 5D transition. To probe the atoms the light
is focused with a 1 m focal length lens to a beam waist radius of ∼160 µm, so theon-axis per mode intensity is ∼0.8 mW/cm2 . These numbers are only meant toprovide a rough idea of the intensities we used, over the course of our experiments
this has changed depending on the particular application. Figure (4.2) illustrates
the basic idea of resonantly enhanced two-photon DFCS. For all of our two-photon
spectroscopy experiments we set the fr and fo of the comb such that only twomodes from the comb laser spectrum were resonant or near resonant with threestates. We will discuss the eﬀect of the many thousands of other comb mode pairsthat are two-photon resonant but detuned from an intermediate state in Chapters(5) and (6). A comb mode with mode order number N1 has an optical frequency
given by νN 1 = N1 × fr + fo and is detuned from the closest intermediate stateby δSP . Similarly a second comb mode N2 completes the two-photon transition.

41

Figure 4.1: Energy level structure of all the relevant 87 Rb states for our experiments. The three red levels indicate the only closed transition.

Qoptical

42

5D

GSD

6P
N2

QN2=N2fr+fo
420 nm

N1

5P

fr

GSP

fo

0

QN1=N1fr+fo

5S

Figure 4.2: Diagram of the optical frequency comb and a near resonant threelevel sub-system of 87 Rb. All of the DFCS experiments we present utilize only
two resonant comb modes as shown in this ﬁgure. The notation and equations
for the comb mode frequencies νN 1 and νN 2 , and detunings δSP and δSD , areused throughout this chapter. For two-photon spectroscopy we detect the 420 nm
spontaneously emitted photons indicated by the dashed blue line from 6P to 5S.

43
The total two-photon optical frequency is given by,
ν2γ = (N1 + N2 ) × fr + 2 × fo .

(4.1)

The two-photon transition lineshape and frequency is measured by shifting either
fr or fo , and thus changing the excited state detuning δSD . A convenient detection
technique for our 5D and 7S two-photon spectroscopy is via the 5D to 6P to 5S
cascade ﬂuorescence at 420 nm. We count photons using a PMT sensitive to this420 nm light to infer the 5D or 7S populations, strictly speaking we only need toknow our signal is directly proportional to the excited population, the absolutevalue is not essential. In section (4.4) we also measure some 5S to 5P transition
frequencies using single photon DFCS, we measured the 5P ﬂuorescence directlyin those experiments. The details of our data acquisition process vary dependingon the speciﬁc measurement, and therefore are discussed in the context of eachexperiment.
4.1

Coherent Accumulation Experiment
The ﬁrst experiment we discuss provides a clear demonstration of the princi-ple of coherent accumulation to achieve high spectral resolution. Chronologicallythis experiment was actually conducted after the spectroscopy experiments presented later in this chapter. However, it is useful to present this work ﬁrst because
it sets the conceptual foundation for direct frequency comb spectroscopy. In particular, we show the quadratic scaling of the excited population versus pulse number at short times, a characteristic of fully coherent atom-light interaction. We
also measure the two-photon transition linewidth versus the comb oﬀset frequency
fo and total number of accumulated pulses to demonstrate the high-resolution we
obtain with direct frequency comb spectroscopy.
We excite the 5S1/2 F=2 to 5P3/2 F=3 to 5D5/2 F=4 resonant two-photon

44transition versus a variable number of femtosecond pulses and fo settings. This
excitation is achieved by locking the Ti:Sapphire with fr =100.41356730 MHz and
fo =18.14 MHz. For this set of comb mode frequencies, mode number 3,826,456
is resonant from 5S1/2 F=2 to 5P3/2 F=3, and mode number 3,847,566 is resonant
from 5P3/2 F=3 to 5D5/2 F=4 within ∼ ±20 kHz. All other allowed two-photontransitions within the ∼30 nm FWHM power spectrum of the laser are detuned
at least 8 MHz, and at least 13 MHz for single photon transitions. Given the ∼6
MHz 5S to 5P transition linewidth and 660 kHz 5S to 5D linewidth the mentioned
comb modes may be considered exactly resonant with the respective transitions.
This particular transition is used quite a bit in our experiments, for example
all the characterization of our systematic errors is conducted on this transition.
Due to the single photon transition hyperﬁne selection rule, ∆F =0,±1, this transition may only be excited via the 5P3/2 F=3 intermediate state. More importantly
5D5/2 F=4 may only decay back to 5P3/2 F=3 or 6P3/2 F=3, both of which may only
decay to 5S1/2 F=2. We refer to this as a closed transition because all the excited
population relaxes back to the same ground state. All other one and two-photontransitions from this ground state may relax to the 5S1/2 F=1 state, thus reducing
the signal. This process of incoherent pumping to the oﬀ-resonant ground state
will be discussed further in the context of our spectroscopy experiments.
A second advantage of this two-photon transition is that it has the largest
dipole moments. We probe with linearly polarized Ti:Sapphire light in a nominallyzero magnetic ﬁeld, so we can assume to excite only ∆mF =0 transitions. This typeof ∆mF =0 selection rule is often referred to as a transition with q=0. As opposedto transitions with ∆mF =±1, which correspond to q=±1. The process of zeroing
the magnetic ﬁeld is explained in subsection 4.2.4. There are in fact a total of 5
separate transitions from 5S1/2 F=2 to 5D5/2 F=4 corresponding to mF =±2,±1,0.
As mentioned in section 3 we do not model all of these states in our density

45
matrix, but rather treat all the magnetic sub-levels as a single hyperﬁne level.
Each transition in this model then has an eﬀective dipole moment given by the
root-sum-of-the-squares of each individual transitions dipole moment. Implied inthis approximation is that the magnetic sub-levels are uniformly populated, andthe signs of the dipole moments are irrelevant, an approximation that will be liftedin section 6.2.
The timing diagram for this experiment is shown in Fig. (4.3). Similar to
the rest of our experiments we form a MOT for 7 ms every 10 ms. This means thetrap and repump lasers are turned on at time zero and the current to the antiHelmholtz coils is turned on for the trapping magnetic ﬁeld. At 7 ms the trap
and repump lasers are turned oﬀ in ∼3 µs via AOMs and the current to the antiHelmholtz coils is turned oﬀ. For the next 2 ms the atoms are freely expanding as
the magnetic ﬁeld turns oﬀ, refer to section ?? for a description of the magnetic
ﬁeld systematics. During part of the remaining 1 ms the data acquisition occurs.
This 1 ms window is split up into many identical probing windows each of total
duration 5.09891 µs, which is equal to ∼ 512 × f1r . At the beginning of each ∼5 µs
sub-window the Pockels cell, used to turn on/oﬀ the Ti:Sapphire laser light to theatoms was switched on. A variable duration TTL signal generated by a SRS-535
digital delay generator was used to trigger the Pockels cell high voltage driver, andthus transmit a variable number of femtosecond pulses to the atoms, denoted as
M-pulses in Fig. (4.3). To ensure that the number of transmitted pulses remains
constant the SRS-535 was triggered synchronously with every 512th femtosecond
pulse. This was done by digitally dividing down the fr signal directly from theTi:Sapphire laser so the Pockels cell trigger was always phase locked to the 512th
sub-harmonic of fr . We only conducted this experiment with total transmitted
pulse numbers between 1 and 150, so the longest the Ti:Sapphire was ever on
during the 5 µs sub-window is 1.5 µs. That means there is at least 3.5 µs for the

46

3 ms

10 ms
MOT

t
5 µs

Fields Off…
Excite w/ M fs
Pulses1 ms

Count Photons

Figure 4.3: Timing diagram for the coherent accumulation experiment. The 1
ms duration measurement window is divided into separate excitation periods andphoton counting periods. During the ﬁrst part of the 5 µs window the atoms are
excited by M femtosecond pulses transmitted via the Pockels cell. Immediately
following this period of excitation the laser is switched oﬀ and photons are counted.
This 5µs excitation and probing window is then repeated for 1 ms before another
MOT is formed.

47
atoms to fully relax between sub-windows, many times the 5D and 6P excitedstate lifetimes.
The signal for all of our two-photon excitation experiments is obtained bycounting 420 nm photons from the 6P to 5S spontaneous emission with PMT
as shown in Fig. (4.2). In this experiment we used a multichannel SRS-430
photon counter to count and temporally bin our signal. It was operated with
2048 sequential bins each of duration 40 ns, with the ﬁrst bin triggered 500 ns
before the ﬁrst Pockels cell trigger. So we measured the signal during the ﬁrst
81.92 µs of the 1 ms window when the MOT optical and magnetic ﬁelds were oﬀ.
Each 5.09891 µs sub-window then corresponds to about 127.5 bins. With an errorof about 20 ns, or one-half a bin duration, we can then assign a time stamp toeach bin modulo 5.09891 µs. Although it is possible to plot the data as a functionof time, we are only interesting in the total integrated ﬂuorescence after excitation
by M pulses.
To transmit a variable number of femtosecond pulses we used a four-crystal
KD∗ P Pockels cell model 25D manufactured by Conoptics Inc as a high speed
shutter. It is designed to operate with a low half-wave voltage, which allows highrepetition rate switching. The advertised rise and fall times are ∼ 7.5 ns, which
is just slightly less than our 10 ns inter-pulse period. Our model Pockels cell has
one high quality polarizing beam splitter cube on the output, so it is necessary toensure high quality linear polarization of the input light. There are two voltage
settings on the high-voltage drivers itself, the bias voltage and maximum switchedvoltage. Our high-voltage drive only supplies 175 Volts peak, slightly less thanthe half-wave voltage. So depending on the bias voltage setting it can be operated
to either optimally attenuate power, or transmit the most power. We chose tooperate with the bias voltage set to minimize the transmitted power when thecell is switched oﬀ. With the switched voltage set to maximum, the transmitted

48
power is approximately 90% of the input power and the steady state on/oﬀ power
ratio is 100:1, using the entire 30 nm FWHM spectrum.
The pulse picking performance was measured with a fast-photodiode on asuitably fast oscilloscope. While observing the photodiode voltage on the scope
the trigger delay and duration to the Pockels cell was adjusted for the best single
pulse transmission performance. We found that the best performance we could
achieve was to actually set the trigger duration to 53 ns with a 3 ns delay, thedelay is dependent on the cable lengths, so this is only relevant to our setup.
With this 53 ns duration trigger one pulse was transmitted with the two adjacentpulses down by a factor of 6.5 in power, and all other pulses down by a factorof 100. It is not clear why the single pulse trigger duration is so much morethan 10 ns, however, increasing the duration by N×10 ns transmits N additional
pulses. Although the dispersion of the Pockels cell, including the output PBS, isapproximately 6400 fs2 we do not compensate for this chirp. The eﬀect of thisamount of chirp on the excited 5D5/2 F=4 population is small, refer to section (5.4)
for more details.
The ﬁrst result we present is the measured signal scaled to the theoretical
5D5/2 F=4 population versus pulse number. The data shown in Fig. (4.4) shows
the scaled signal as squares for pulse numbers from 1 to 140. This data was takenwith the probe laser split into two equal power beams using a non-polarizing
beam splitter. Just prior to the beamsplitter the Ti:Sapphire light passed thougha 1 m focal length lens. These two probe beams were then counter-propagatedthrough the Rb atoms with careful attention to overlapping the beams spatially,
although not necessarily temporally overlapping in the atom cloud. Figure (4.4)
shows two theoretical predictions corresponding to the experimental peak electricﬁeld strength of Eo ∼107 V/m (solid line), and asymptotically low power Eo ¡¡107
V/m (dashed line). Where Eo corresponds to the peak electric ﬁeld strength

49

Figure 4.4: Measured signal (squares), scaled to theoretical 5D population (solidline), versus pulse number. Asymptotically low power signal (not to scale) versus
pulses is shown as the dashed line.

50
assuming a transform limited Gaussian pulse. From this data we can already see
the 5D population reaches steady state at the experimental electric ﬁeld strengthof 107 V/m sooner than for asymptotically low power. After approximately 80
accumulated pulses the mean 5D population is constant, this saturation is thetime domain signature of power broadening. Figure (4.5) shows the ﬁrst 15 pulsesof this data set to illustrate the scaling law versus pulse number at times shortcompared to the 5S-5D coherence time of ∼480 ns. In this ﬁgure the solid line is
not theory but a ﬁt to a function of the form c1 M2 , where M is the pulse number,the purpose of this ﬁt is to illustrate the quadratic scaling law.
For a two-level system driven on resonance the population exhibits the familiar process of Rabi ﬂopping in the absence of any decoherence. We deﬁne theaccumulated pulse area from time t=0 to t=t’ to be,µA(t ) =



 t
0

Eo (t)dt(4.2)

where µ is the relevant dipole moment and Eo (t) ﬁeld envelope magnitude. With

this deﬁnition of the pulse area the excited population varies as sin( A(t2 ) )2 , forexample, a pulse area of A(t ) = π inverts the population. If we extend this deﬁnition of pulse area to include multiple phase coherent pulses, the total accumulated
pulse area after M pulses is simply M×A(tpulse ), where A(tpulse ) is the total area ofa single pulse. So for small total accumulated pulse area, M×A(tpulse ) << π, thedependence of the excited population on pulse number goes as M2 . This analogy
holds true for two-photon excitation as well in the case of intermediate and ﬁnal
state resonance. The population in the 5D5/2 F=4 state is proportional to M2 ,however, the exact dependence is more complicated than in the two-level case so
we don’t have an analytic expression. We have checked this scaling law in thecase of weak excitation of 5D5/2 F=4 driven by a series of phase coherent pulses
using our three-level density matrix model. So by considering the familiar case

51

Figure 4.5: Measured signal (squares) versus number of accumulated pulses for
short times, the data is ﬁt to a function of the form C1 M2 where M is the pulse
number.

52
of a coherently driven two-level atom we can understand the quadratic scaling of
the excited population for accumulated pulse numbers < 15 in Fig. (4.5). One interesting conclusion from this analysis and experimental result is for a given laser
power a higher repetition rate comb excites more 5D population. In frequency
domain this can be understood by considering a higher repetition rate comb has
more power per comb mode. Our results in section (5.4) demonstrate that thisquadratic scaling law holds true for chirped pulse excitation as well.
The second part of this experiment is to demonstrate high-resolution spectroscopy. In general there are two ways to change the detuning from two-photon
resonance, either by shifting fr or fo . Figure (4.6) shows the signal scaled to thetheoretically calculated 5D5/2 F=4 population versus two-photon detuning for 4
sets of accumulated pulse number. In this result we scanned fo , due to the fact it
is a two photon transition, the detuning is given by
δ2γ = 2 × (18.14 − fo )

[MHz]

(4.3)

where fo =18.14 MHz is the resonance condition. The measured lineshapes after
10, 15, 20 total accumulated pulses is shown with the corresponding 5D5/2 F=4
population on the left y-axis, similarly the population after 80 pulses is on theright axis.
This result simply provides a clear picture of how the measured lineshape
narrows as more pulses are used for excitation, or in other words how we obtainhigh-resolution from coherent pulse accumulation. There are a few things thatin general can limit the linewidth of the measured transition in our experiments.
For the results presented in Fig. (4.6) the linewidth is limited by probing time,or the number of pulses, for less than ∼80 pulses. After ∼80 pulses have beenused to excite the transition the linewidth no longer continues to decrease due topower-broadening, recall that the population after 80 pulses was in steady state.

53

Figure 4.6: Measured signal (squares), scaled to theoretical 5D population (solidline), versus fo and for several values of accumulated pulse number.

54
Traditionally most high resolution spectroscopy has been done with cw-lasers.
A line-broadening phenomena common to cw-laser spectroscopy is transit time
limiting. This occurs when the duration of excitation of the atom is limited,
usually by motion through the probe beam, to be less than either the electronicor laser dephasing times. In this sense excitation by only 10, 15, 20 pulses isanalogous to transit time broadening. We chose to operate with the most laser
power possible for this experiment to increase the signal, particularly for themeasurements taken with very few pulses. In principle the resolution obtainablewith direct frequency comb spectroscopy is only limited by the natural linewidthof the excited state. Experimentally the narrowest linewidth we have achieved
was ∼1.1 MHz FWHM, most likely limited by power-broadening.
4.2

Systematic Eﬀects
In this section we discuss several of the most important sources of errorthat may either broaden or shift the lineshapes we measure using DFCS. In general we are not interested in determining the natural linewidths of the transitions
we measure so the main reason to reduce line-broadening is to improve our linecenter estimates. In particular, we will discuss the eﬀect of radiation pressure
on the atoms due to the probe laser, Zeeman shifting of the ground and exitedstates, probe laser linewidth, and Stark shifts. Power-broadening of the excitedstate linewidth was already discussed brieﬂy in section (4.1). After careful minimization of the systematic eﬀects presented in this section we generally measured
power-broadened linewidths ranging from 1.1 MHz to 2.5 MHz in our spectroscopy
results.
Before addressing the previously mentioned systematics, it is worth estimating the eﬀect of the atomic cloud temperature on the linewidths. As discussed
previously we conduct all of our experiments on a sample of 87 Rb laser cooled in55
a MOT. All of the one and two-photon spectroscopy results we present were conducted after an additional ∼ ms polarization gradient cooling (PGC) stage while
the trapping magnetic ﬁeld was turning oﬀ. Although we did not measure theatomic cloud temperature directly, we can estimate the range of Doppler widths
we expect for temperatures from 20 µK for excellent PGC, to 100 µK without
PGC. In the range of these temperatures, we expect the Doppler widths for single
photon transitions from 5S to 5P to be ∼130 to 300 kHz, much less than the 6
MHz transition linewidth. The Doppler width for the two-photon transitions isapproximated to be twice that of the single photon case, ∼260 to 600 kHz. A
reasonable approximation considering the resonant wavelengths are typically very
similar. So we can expect that the 660 kHz natural linewidth of the 5S to 5D transition will be slightly Doppler broadened, however even at 100 µK the Doppler
width is still less than the typical power-broadened linewidths we measure.
A common misconception people have about the eﬀect of the intermediate
state on the two-photon transition is that it may broaden the measured linewidth.
In our experiments we generally set the comb frequencies such that a mode issingle photon resonant with an intermediate state, and another resonant fromthat state to the excited state. As mentioned previously, we measure the twophoton transition linewidth by scanning either fr or fo , which shifts not only the
excited state detuning but the intermediate state detuning as well. So if we scanthe excited state detuning by ±4 MHz using fo , we recover most of the ∼2 MHz
FWHM lineshape, and the intermediate state detuning only changes by ±2 MHz.
Considering the intermediate state natural linewidth of 6 MHz the detuning is notvery signiﬁcant. The eﬀect of changing the intermediate state detuning as we scan
the two-photon lineshape does not broaden the line, but it does in fact change
the lineshape slightly. For example, using Eq. (3.12) to estimate this eﬀect, thedeviation from a Lorentzian lineshape between ±2 MHz detuning is at most 5%,

56
qualitatively the eﬀect is to make the tails of the lineshape approach zero fasterthan that of a Lorentzian lineshape.
4.2.1

Laser Linewidth

To conduct high-resolution DFCS it is important to ensure the linewidthof each comb mode is suﬃciently narrow. Recall that the order numbers of theresonant modes, corresponding to ∼ 780 nm, are of order 4×106 . So any noise in
fr is multiplied up in the optical domain by this factor. We typically locked the
comb fr by phase stabilizing the 10th harmonic of fr to the Wenzel low phase noise
crystal oscillator, as described in section (2.1). With the comb fr locked to theWenzel oscillator and fo locked via self-referencing we can measure the linewidthof an individual comb mode by forming a beat note with a stable cw-laser.
We have done this measurement using two diﬀerent cw-lasers, an Iodine
stabilized NPRO-YAG laser, which typically has a narrow 5 kHz linewidth at
1 ms [9]. And also a cavity stabilized diode laser with a signiﬁcantly narrower
linewidth [63]. In both cases the linewidth of the cw-lasers is much less thanthat of the comb over our measurement time. Figure (4.7) shows a typical beatsignal between the comb and the cavity stabilized diode laser at 698 nm. Using a
Lorentzian ﬁt to the data we extract a 638 kHz FWHM comb linewidth measured
at 698 nm. Similarly when we measured the beat between the comb and the
NPRO-YAG at 1064 nm, we recovered a 330 kHz FWHM linewidth. In both ofthese measurements the required wavelength to form a beat with the cw-lasers
was generated from supercontinuum generation in a highly nonlinear ﬁber; we areassuming negligible extra phase noise in this process.
It has been shown that the spacing between comb modes is extremely uniform [64]. In other words the frequency noise of diﬀerent comb modes is highly
correlated and is generally due to noise in fo and fr . Neglecting the smaller fo

57
noise we can attribute the measured linewidths primarily to ﬂuctuations of fr andestimate the comb linewidth at 780 nm. Under these approximations the comb
linewidth at 780 nm is between 570 kHz, using the 698 nm measurement, and450 kHz using the 1064 nm measurement. One thing to note is that these measurements were conducted about two years apart so the diﬀerence in estimatedlinewidths may be attributed to diﬀerences in the fr lock. Although the linewidthof our comb modes is relatively wide, it does not present a problem due to the fact
we conduct most of our experiments with suﬃcient intensity to power broaden the
two-photon transition linewidth to about ∼1.5-2.2 MHz.
4.2.2

Radiation Pressure

One of the most important systematic eﬀects in our experiments is due toradiation pressure exerted on the atoms by the probe beam itself. The radiationpressure we consider in this section is due to the momentum imparted on an atomfrom incoherent photon scattering out of the probe laser mode. Coherent photon
scattering, responsible for the dipole force, gives rise to a much weaker force that
we ignore in this section. When an atom is excited it receives a momentum kick inthe direction of the probe beam propagation, however, the spontaneously emitted
photon imparts a momentum kick of the same magnitude but with a random
direction. The net eﬀect on an ensemble of atoms probed from one direction is
primarily to increase the mean velocity in the direction of the probe. Due to therandom directions of emissions and number of photons scattered after some periodof time, the width of the velocity distribution of the atom ensemble also increases.
These two eﬀects lead to a Doppler shift of the measured center frequency of atransition and an increased linewidth due to Doppler broadening.
In the case of our two-photon transition measurements with a resonant intermediate state this eﬀect is quite signiﬁcant. Recall that the linewidths for the

58

Figure 4.7: Comb linewidth measured by the beat frequency linewidth between acomb mode and a cavity stabilized diode laser, solid line is a Lorentzian ﬁt with
638 kHz FWHM.

59
5P states is 6 MHz, ten times larger than for the 5D states. Furthermore, thetwo-photon excitation probability ρ5D arises at second order in perturbation theory and is in general much less than the one-photon excitation probability ρ5P .
The total incoherent scattering rate from a particular state is,γi = 2π∆νi ρii(4.4)

where ∆νi is the linewidth of state i and ρii is the population. So the incoherent
scattering rate from the resonant 5P state is much larger than the 5D, typically bya factor of at least 100. In other words for every 420 nm photon emitted from the
5D-6P-5S radiative cascade there are on average at least 100 photons scatteredfrom the resonant intermediate 5P state. On average for every photon scatteredfrom 5P at 780 nm the atom obtains a recoil velocity kick of,
vr =

k
m

(4.5)

where m is the 87 Rb mass, and k is the probe laser wavevector. Substituting for thelaser wavevector and atomic mass we ﬁnd the recoil velocity is about 5.88 mm/s,
which causes a Doppler shift at 780 nm of 7.54 kHz. Clearly the accumulated
Doppler shift from many photon recoils can amount to a signiﬁcant shift of themeasured linecenter frequencies.
To better understand and develop a model of the radiation pressure eﬀects
we studied the 5S1/2 F=2 to 5P3/2 F=3 to 5D5/2 F=4 transition using excitationfrom only one direction. As mentioned previously the large dipole moments andclosed nature of this transition makes it the best choice for studying systematic
eﬀects. The experiment is simply to measure the signal at 420 nm versus excitation
time under three diﬀerent initial detuning conditions. Figure (4.8) shows therelevant three-level system and the two comb modes nearest to resonance. The
detuning of this comb mode pair from 5P3/2 F=3 is denoted as δSP . For this60

5D5/2F=42GSP
776 nm

5P3/2F=3

GSP
780 nm

5S1/2F=2
Figure 4.8: Energy level diagram of the relevant three-level system for studying
the radiation pressure. All detunings are relative to atoms stationary in the labframe, radiation pressure will Doppler shift the light and therefore change thedetunings.

61experiment we initially set fr and fo such that δSP =0 and the comb modes areresonant. By shifting fo by either 1 MHz or 2 MHz we make the detuning from
5D5/2 F=4 either 2 MHz or 4 MHz. When we conducted this experiment we did
not have the Pockels cell as a shutter. Instead the Ti:Sapphire was switchedon or oﬀ using a liquid crystal shutter with a much slower, ∼30 µs, response
time. Our measurement results are shown in Fig. (4.9) as symbols, together
with the theoretical predictions of our model (solid lines). The three detuningcases correspond to the initial detunings from 5P3/2 F=3, or more speciﬁcally to
atoms at rest in the lab frame. Note that a positive detuning corresponds to blue
detuning from intermediate resonance, so the atoms will eventually be Doppler
shifted into resonance.
Considering we only measure the 420 nm ﬂuorescence from the two-photontransition, it can be thought of as a probe for the mean velocity of the atoms.
When the Doppler shift is equal to ∼2δSP the 5D5/2 F=4 state is resonant andstarts to ﬂuoresce. With this in mind the characteristic shapes of the resultsin Fig. (4.9) may be interpreted. For an initial detuning δSP =0 the LC shutter
slowly turns on for the ﬁrst ∼40 µs, when the shutter is completely on the signal isa maximum. The signal then rapidly decays over ∼150 µs as the mean velocity of
the atom ensemble Doppler shifts atoms oﬀ of two-photon resonance. Similarly forthe blue detuned cases, initially the signal is essentially zero due to the 5D5/2 F=4
detuning, however at later times the atoms are Doppler shifted onto two-photon
resonance. The two initially detuned cases show signiﬁcantly broader peaks thanthe δSP =0 case due to the fact the atoms have undergone more heating.
To go beyond simply a qualitative understanding of the radiation pressure
we developed a more comprehensive model. We have already discussed two of the
key pieces to our model, the incoherent scattering rate Eq.(4.4), and the single
photon recoil velocity Eq.(4.5). Due to the fact the radiation pressure is almost

62
entirely from 5P3/2 F=3 incoherent scattering, and the time scale for accumulatinga signiﬁcant Doppler shift is much larger than the 5P lifetime, we only need toconsider the steady state value of the 5P3/2 F=3 population. Solving for the density
matrix of a two-level system using the Optical Bloch Equations in steady state[30], we calculate the scattering rate to be,

=
γ5P

γ
so
2 1 + so + ( 2(δ+ωD ) )2

(4.6)

γ

where so is the on-resonance saturation parameter, δ is the angular frequency
detuning of the laser, ωD is the Doppler shift, and γ=2π∆ν5P . The on-resonancesaturation parameter can be written in terms of the saturation intensity, Is , asso = IIs where Is =2.5 mW/cm2 for linearly polarized light. The detuning δ is thedetuning for an atom at rest in the lab frame. It is deﬁned as, δ=ωl -ωa , where
the subscripts refer to the laser and atom angular frequencies. The Doppler shift
is given by the familiar formula, ωD =-k · v, where k is the probe beam wavevector
at 780 nm.
Ultimately we are interested in deriving an equation for the velocity of an
atom versus time given some initial conditions. Recall that the change in velocityof an atom after one incoherent photon scattering event is mk . Equation (4.6)
gives us the rate at which the photons are scattered. So the diﬀerential equation
for the rate change of velocity as a function of time is simply,
kγ
so
dv(t)
=
dt
2m 1 + so + ( 2(δ−kv(t)) )2

(4.7)

γ

where we have simpliﬁed the vectors to scalars by only considering the probe beam
propagation direction. Given an initial velocity along the probe beam direction
this equation can be solved to calculate the detuning from 5D5/2 F=4 versus time.
Finally, the equation for the two-photon transition amplitude from second-order
perturbation theory, Eq. (3.12), may be used to calculate the expected signal
versus time.

63
It is possible to solve Eq.(4.7) analytically for the velocity versus time assuming the initial velocity of the atom is zero. This method correctly predicts
the positions of the peaks in Fig. (4.9), however, it does not account for several
eﬀects that broaden the peaks observed in our results. One of the most important simpliﬁcations made is assuming a constant on probe beam intensity. In our
experiment we expect to have a radial intensity dependence due to the Gaussianmode of the beam, in the focal plane the intensity varies as,
I(r) = Io exp[−2

r2
]
wo2

(4.8)

where Io corresponds to the on-axis intensity (∼0.8 mW/cm2 ) and the beam waist
radius wo is ∼ 130 µm. If we assume a constant radial density of atoms and a
two-photon transition rate that varies as I2 , it is possible to calculate the signal
strength versus radial displacement. The number of atoms in a thin ring at ﬁxed
radius is proportional to r, so the radial dependence of the signal is proportional
to rI(r)2 .
To incorporate the radial intensity variation into so and several other stochastic processes we turn to a numerical simulation based on Eq. (4.7). Although we
also modeled the longitudinal dependence of the signal, due to absorption andbeam focusing, it is suﬃcient to consider only the radial degree of freedom to generate the results in Fig. (4.9). We divided the radial direction into 20 segmentsfrom r=0 to r=3wo , and each segment had 20 atoms. Each atom was assigned
an initial velocity, assuming a Maxwell-Boltzmann distribution at T=20 µK. The
atom velocity, and two-photon signal weighted by r∆r, was solved for versus time
to 300 µs in 100 steps. To incorporate the stochastic nature of the number of photons scattered per unit time, ∆t, each atom obtained a random velocity kick ineach time segment. The magnitude of the random velocity kick is 2×vr multiplied√
by a random number from a Normal distribution of standard deviation N , where64
N is the mean number of photons scattered given by γ5P∆t, see Eq.(4.6). Sum-

ming over the predicted two-photon signal from each atom at each radial segmentprovides quite a good estimate of the total signal versus time. However, one ﬁnal
stochastic process that was included is the variation of the probe laser frequency
given by the comb linewidth. We assumed the laser noise to be a slow processcompared to 300 µs, meaning for each of the above 300 µs simulations the laser
was at a ﬁxed frequency in the lab frame. The ﬁnal theory result shown in Fig.
(4.9) was generated by repeating the above simulation 50 times, and randomly
selecting a laser frequency consistent with the Lorentzian comb mode linewidth.
Clearly the agreement between this model and the measured signal versus time inquite good.
For all of our spectroscopy experiments we would like to reduce the net
momentum transfer to the atoms as much as possible. We do this by balancing
the radiation pressure on the atoms along the probe beam direction with anothercounter-propagating probe beam at the same intensity. Before the Ti:Sapphire
beam enters the MOT chamber it is focused with a 1 m lens then split into two
equal intensity beams with a non-polarizing 50:50 beam splitter. By far the mosttedious alignment issue with this experiment is overlapping these beams at theatoms to optimally balance the radiation pressure. One of the reasons this istedious is that to have suﬃcient signal to noise in our two-photon signal requires∼1 s of real time data acquisition, or 100 MOT cycles. First, the signal fromone beam is maximized by ﬁnding the center of the atom cloud, adjustments ofthe focusing lens may also be necessary for this step initially. Then the spatial
overlap of the two beams in the center of the atom cloud is found by maximizing
the two-beam signal. To obtain our best results we would then further optimize
by scanning a two-photon transition line and adjusting the beam overlap so as tomeasure the narrowest linewidth. Figure (4.10) shows the measured two-photon

65µFigure 4.9: Measured signal and theoretical prediction versus probing time forthree detuning conﬁgurations. Detunings are from the intermediate 5P3/2 F=3
state, the 5D5/2 F=4 detuning is twice as large.

66
transition lines (symbols) together with a Lorentzian ﬁt (solid lines) under counterpropagating and single beam excitation. We set the two channel photon counterto record the signal from 0-20 µs and also 100-120 µs to observe the linecenter
and width after some probing time. Clearly the use of well overlapping counterpropagating beams signiﬁcantly reduces the linecenter shift versus time seen in
the single direction excitation. Despite the fact the net momentum transfer tothe atoms is nearly zero, there is still heating of the atoms, which appears as anincrease of the linewidth versus time in the counter-propagating case.
4.2.3

AC Stark Shift and Power Broadening

Ideally we would like to measure the frequency and linewidth of a particular
two-photon transition without any perturbations. In this section we discuss two of
the most important fundamental perturbations from the probe light itself, the ACStark shift and power-broadening. Unlike the case of an ideal two-level system, fora two-photon transition in a three-level atom there is almost always a Stark shift ofthe excited state. In the case of 87 Rb this is due to the fact that the intermediate
state is shifted ∼1 THz from exactly half of the two-photon transition frequency.
So if we were to try and resonantly excite a 5D state with two-modes, of equal
or dissimilar frequencies, there is always a mode detuned from intermediate state
resonance, resulting in an AC-Stark shift. Figure (4.11)(a) shows a diagram of
the relevant three-level system with a pair of one and two-photon resonant modes.
Only one conﬁguration of the modes is resonant with the 5P intermediate state,
the other conﬁguration is detuned and causes a Stark shift. In this case the second
conﬁguration, while two-photon resonant, is detuned from 5P3/2 F=3 by 2 THz.
The near intermediate resonance Stark shift has been studied using two-mode
excitation in Alkali vapors by Liao and Bjorkholm [65].
To study the Stark shift and power-broadening we measured the 5S1/2 F=2

67

250

Counter-propagating Probe

0 µs – 20 µs

Rb

Signal Amplitude (a.u.)

100 µs – 120 µs
200

150

Single Direction Probe
100

Rb
50

16

18

20

22

fo Frequency (MHz)Figure 4.10: Measured signal (symbols) and Lorentzian lineshape ﬁts (solid lines)
under counter-propagating and single direction excitation. The color of the symbols and lines correspond to diﬀerent measurement delays after the probe beamis turned on; red for zero delay and blue for 100 µs delay. The fo for two-photon
resonance of zero velocity atoms is 18.14 MHz.

68(a) 2-Modes
5D
B

(b)

A

1 THz

5P3/2
A

4-Modes

2 THz
2 THz

B

5S
Figure 4.11: (a) Three-level system under two mode excitation, due to the 1 THz
oﬀset of the 5P3/2 states from half the two-photon frequency there are two diﬀerentmode pair conﬁgurations. (b) Same three-level system under four mode excitation
illustrating the possibility to approximately cancel the Stark shift arising in (a).

69
to 5D5/2 F=4 transition frequency and linewidth with three diﬀerent intermediateresonance conditions. We have already discussed the fr and fo frequencies for
excitation with one and two-photon resonance. In addition to the on intermediate resonance set of comb frequencies we also used two other pairs of fr and fo
corresponding to ±4 MHz 5P3/2 F=3 detuning. For all three cases of varying intermediate state detuning the 5D5/2 F=4 state is two-photon resonant. Note thatfor these experiments we used counter-propagating probe beams in an attempt
to balance the radiation pressure, however, there is some residual imbalance. We
also used the LC shutter in this experiment to turn on/oﬀ the Ti:Sapphire, so forthe ﬁrst 40 µs of data the probe power is increasing to its maximum value.
Figure (4.12) shows the measured linecenter shift versus probing time for thethree cases of intermediate state detuning. The 420 nm ﬂuorescence was counted
using the SR-430 multichannel photon counter to obtain time resolution. For eachcase of intermediate state detuning, fo was scanned to measure the two-photontransition lineshape. Using this record of signal versus both fo and time we ﬁt eachline with a Lorentzian function and extracted the linecenter and FWHM linewidth.
Clearly after 40 µs of probing time the ±4 MHz detuned cases exhibit much lessof a lineshift versus time. This decrease is due to the reduced 5P populationand thus radiation pressure compared to the on-intermediate-resonance case. To
obtain the best possible estimate of the unperturbed linecenter frequency for our
spectroscopy results we extrapolated to zero probing time.
We model the eﬀect of the Stark shift and power-broadening due to the
comb excitation using the three-level density matrix theory discussed in section(??). The driving ﬁeld was a Gaussian envelope electric ﬁeld with a 35 fs pulse
duration. Although we have developed a version of this model that uses excitationby counter-propagating pulses, we only used excitation from one direction for thisnumerical study. The reason being that properly treating the counter-propagating

70

Figure 4.12: Measured linecenter shifts from the literature value of the S1/2 F=2
to 5D5/2 F=4 two-photon transition frequency versus time and intermediate state
detuning. During the ﬁrst ∼40 µs the LC shutter is opening and therefore thepower is increasing.

71
pulse excitation requires an added spatial integration. Computationally this is
too lengthy considering the fact we wish to vary fo and the peak ﬁeld strength
independently. Another shortcoming of this three-level model is that the magnetic
sub-levels are not included. Table (4.1) contains the angular part of the reduced
dipole moments for transitions relevant with linear polarized light.
Intermediate Stateµgi

5P3/2 F=3 mF =2

1
3

5P3/2 F=3 mF =1
5P3/2 F=3 mF =0µ
ifµgiµif√2√30.0976

3
√1
5

2
√2
35

2

5

3
35
7

0.13800.1512

Table 4.1: The angular part of the reduced dipole matrix elements for the
5S1/2 F=2 to 5P3/2 F=3 to 5D5/2 F=4 transition with linear polarized light (q=0).

Due to the large variation of the dipole moment products between diﬀerent
mF levels, optical pumping could potentially change the two-photon transition
rate. Instead of running the model for each mF level and incoherently summing
the populations we used the average dipole moments. Analytic expressions for
the Stark shift and power-broadening have been derived for a three-level system
driven by two-modes by Brewer and Hahn [51]. However, to arrive at an analytic
expression several simpliﬁcations were made, such as equal Rabi frequencies foreach transition step.
Figure (4.13) shows the predicted linecenter shift versus peak ﬁeld strengthfor each 5P detuning case. The linecenter shifts at 107 V/m seen for ±4 MHz
detuning agree well with the ± ∼150 kHz shift measured in the experiment. For
the case of zero intermediate state detuning the Stark shift is signiﬁcantly reduced.
We attribute this to the fact all comb modes are equally spaced, so the shift fromthe pair detuned N×fr below 5P3/2 F=3 is canceled by the shift from the pairtuned symmetrically above 5P3/2 F=3.
At large electric ﬁeld strengths even the on-intermediate-resonance case

72

δ
δ
δ

Figure 4.13: Theoretical predictions for the AC-Stark shift of the 5S1/2 F=2 to
5D5/2 F=4 two-photon transition frequency versus peak ﬁeld strength and intermediate state detuning. We used a 35 fs FWHM Gaussian driving ﬁeld centered
at 778 nm to drive our three-level density matrix model.

73
starts to exhibit some Stark shift, this appears to be due to the ﬁnite spectral
width and center wavelength. For example, the Stark shift using a peak ﬁeld
strength of 3×107 V/m is (-148 kHz, -98 kHz, -36 kHz, 30 kHz, 89 kHz) for center
spectrum wavelengths (776 nm, 778 nm, 780nm, 782 nm, 784 nm). Referring
back to Fig. (4.11)(b), in the case of four mode excitation it is possible to cancel
some of the Stark shift. We discussed the 5P resonant two mode case previously,in that case the Stark shift arises from a mode pair detuned by 2 THz from theintermediate state. If we introduce two more modes of equal strength detuned 2
THz below intermediate resonance, canceling that particular source of Stark shift.
In this four-mode example, the Stark shift arises from a mode pair detuned 4 THz
above the intermediate resonance, and gives rise to less of a shift. The Stark shift
cancellation due to comb excitation may be thought of as an extension of this fourmode example, although with >100,000 modes in the ∼55 nm FWHM Gaussian
spectrum. This explanation is only meant to provide a conceptual picture. Further study is required to understand the extent to which the Stark shift diﬀers
between two-mode and comb-based two-photon excitation.
We also measured the power-broadening versus time and intermediate state
detuning in the same experiment. Figure (4.14) shows the measured FWHM
linewidths from the Lorentzian ﬁts of each line versus time. Similar to our studyof Stark shifts the on-intermediate-resonance case exhibits an increasing linewidthversus time even after the LC shutter is fully opened. This is due to the heating
caused by the counter-propagating probe lasers discussed in subsection (4.2.2).
At the expense of an increased Stark shift we can retain narrower linewidths bydetuning from intermediate resonance, seen in the ±4 MHz detuned results. We
observe what appears to be a signature of Doppler cooling, or more accurately
suppressed heating, in the case of -4 MHz (red) detuning from the intermediate
state. This is evident in the reduced linewidth measured with red detuning ver-

74

Figure 4.14: Measured linewidths of the S1/2 F=2 to 5D5/2 F=4 two-photon transition versus time and intermediate state detuning. During the ﬁrst ∼40 µs the
LC shutter is opening and therefore the power is increasing.

75
sus equal blue detuning. Of course regardless of intermediate state detuning all
linewidths tend to increase versus time so there is still heating of the atomic ensemble due to the 5S-5P incoherent photon scattering. At very short times, before
the radiation pressure has caused signiﬁcant heating, the linewidths corresponding
to ±4 MHz intermediate state detunings are narrower than the on-resonance case.
This is indicative of reduced power broadening, and is due to a reduced resonant
enhancement of the two-photon transition rate.
The theoretical prediction for power-broadening was obtained in the samemanner as the previously discussed Stark shifts. Figure (4.15) shows the results ofour theoretical model for a range of peak electric ﬁelds. For the on-intermediateresonance case the power-broadened FWHM linewidth at 107 V/m is 1.1 MHz.
This serves to provide an estimate of the eﬀect of power-broadening only, theexperimental results for this case exhibit a large line-broadening due to heating
as well. Unfortunately this set of experiments was among the ﬁrst conducted and
we did not have the Pockels cell yet. Perhaps better agreement would be obtained
between theory and experiment if the peak ﬁeld strength was not a functionof time, due to the slow turn on of the LC shutter. Then power-broadening
could be more clearly separated from radiation pressure eﬀects which take time toaccumulate. Note that the coherent accumulation experiment presented in section(4.1) was conducted with a Pockels cell, in that case we observed good agreement
between theoretically predicted and experimentally measured power-broadened
linewidths. But the atoms were excited for at most ∼13 µs, corresponding to the
80 pulse excitation case.

4.2.4

Zeeman Shift

There are two primary sources of magnetic ﬁeld that may Zeeman shift theatomic energy levels. The anti-Helmholtz magnetic ﬁeld used to trap the atoms

76

δ
δ
δ

Figure 4.15: Theoretical predictions for power-broadened linewidth of the
5S1/2 F=2 to 5D5/2 F=4 two-photon transition versus peak ﬁeld strength and intermediate state detuning. We used a 35 fs FWHM Gaussian driving ﬁeld centered
at 778 nm to drive our three-level density matrix model.

77
in the MOT and the ambient magnetic ﬁeld, for example from the Earth. Ideally
we would like to conduct spectroscopy in a magnetic ﬁeld free environment so
we use three sets of large Helmholtz coils to compensate for the ambient ﬁeld.
These Helmholtz coils are mounted around the MOT and oriented to provide
three relatively homogeneous orthogonal static magnetic ﬁelds.
To adjust the three pairs of Helmholtz coil magnetic ﬁelds we probe the
5S1/2 F=2 to 5P3/2 F=3 to 5D5/2 F=4 two-photon transition with left and rightcircular polarized light. This particular two-photon transition is well suited forthis study because the population will tend to be pumped to the stretched states,
mF =±F, under σ + or σ − excitation.
As we discussed previously, the trapping magnetic ﬁeld is turned oﬀ before
we probe the atoms, however, this process takes some time. We determined areasonable delay between the time we turned oﬀ the trapping magnetic ﬁeld andprobe the atoms. This was done by turning on the Ti:Sapphire and measuring
the linecenter shift at various delays after the trapping magnetic ﬁeld was turnedoﬀ. At times less than ∼0.5 ms we observed large linecenter shifts, from 4 MHz
at zero delay to 100 kHz at 0.5 ms delay. To be certain the Zeeman shift from thetrapping magnetic ﬁeld was negligible we measured all of our data with at least a
2 ms delay.
The Helmholtz coils used to compensate for the ambient ﬁeld are oriented
such that it is relatively easy to probe the atoms with a beam parallel to thegenerated magnetic ﬁeld. For example, there is a pair of Helmholtz coils oriented
to create a vertical magnetic ﬁeld, to adjust this ﬁeld we probe the atoms with
the Ti:Sapphire beam vertically. For each of the three directions we record thetwo-photon transition lineshape with left and right circular polarized light versus
the current applied to the Helmholtz coil. Experimentally great care was taken to
ensure the quality of the polarization, meaning we reduced the ellipticity as best

78
we could.
Figure (4.16) shows the resulting linecenter shift from some arbitrary value
versus the applied magnetic ﬁeld. The magnetic ﬁeld at the atoms is weak so
we are in the anomalous Zeeman regime, meaning the linecenter shift is a linear
function of magnetic ﬁeld. The optimal ﬁeld value is given by the crossing of two
linear ﬁts to the data, one for left circular the other for right circular. This process
is repeated at least once for all three directions, in practice we did each direction
twice to ensure the best results.
4.2.5

Incoherent Optical Pumping

Incoherent optical pumping is a particularly important, and complicated,
process in DFCS. Recall that the only closed two-photon transition is 5S1/2 F=2
to 5P3/2 F=3 to 5D5/2 F=4. Population excited to either the intermediate or ﬁnal
state in this transition will eventually spontaneously decay back to the 5S1/2 F=2
ground state. We are simplifying the optical pumping process somewhat hereby ignoring the magnetic sub-levels, the populations of which are determinedby incoherent optical pumping as well. With the exception of this one closed
transition, the signal size of all the other transitions is inﬂuenced by incoherent
optical pumping.
In general all of the two-photon resonant transitions are resonant from only
one of the two 5S1/2 hyperﬁne levels. So the signal strength depends on theamount of population in the resonant ground state. Because of the signiﬁcantly
faster scattering rate of the 5P states compared to the 5D states, most of the
optical pumping occurs due to decay from the 5P states. For our typical probe
intensity the two ground states reach a steady state population distribution in atime of order ∼1-10 µs. So any initial state preparation we may do with the trapor repump laser before probing is quickly undone.

79

Zeeman Shift

V+

1 MHz

V-0.4

-0.2
0.0
0.2
Magnetic Field at the MOT (Gauss)

0.4

Figure 4.16: Example linecenter crossing for σ + and σ − excitation of the 5S1/2 F=2
to 5D5/2 F=4 two-photon transition.The x-axis scaling is calculated from the linecenter shifts and atomic properties.

80
In our early experiments we noticed many lines were missing from our measurements. Suspecting incoherent optical pumping to the oﬀ-resonant groundstate we employed a repuming scheme interleaved with our data collection. Instead of recording data continuously we gated the photon counter to only record
data for 2.5 µs every 5.0 µs, over a total time of 0.5 ms for every MOT cycle.
After each 2.5 µs counting window we turned on the repump laser for 2.5 µs
to pump population from 5S1/2 F=1 back to 5S1/2 F=2. Figure (4.17) shows thelines we measured by scanning fo with and without the interleaved repumping
scheme. Clearly this greatly increases our signal for all but the one closed transition. The reason that the incoherent optical pumping process is complicatedto predict is that it is not only due to the decay from the resonant intermediate
state. The steady state population distribution is also a function of the rate at
which the comb itself repumps population. This point will be made quite evident
in subsection (4.4) where our optical pumping model plays a key role.
4.3

Two-Photon Spectroscopy Results
With a clear understanding of the systematic eﬀects we now proceed to use

the comb for direct frequency comb spectroscopy. There are two unique advantages of using the comb directly for spectroscopy. The ﬁrst and perhaps more important is that we may determine the absolute frequencies of the transitions withreference to the cesium standard. Initially we studied two-photon transitions to aset of hyperﬁne 5D5/2 and 5D3/2 hyperﬁne levels. As this was the ﬁrst experimental demonstration of DFCS these transitions served as excellent proof-of-principle
measurements because all of them had been measured previously by conventional
cw-laser spectroscopy. In fact the uncertainty in the previous measurements ofthese transitions was 10-16 kHz, much less than our laser fr . This allowed us todetermine the comb mode order numbers without additional measurements. We

81

1200

With Burst Repumping
Without Burst Repumping
1000

Amplitude (a. u.)

Closed Transition
800

600

Recovered Lines

400

200

0
-15 -10 -5

05

10 15 20 25 30 35 40 45 50 55 60

Relative Frequency (MHz)Figure 4.17: Measurement of several two-photon transitions illustrating the eﬀect
of incoherent optical pumping. Black symbols are measurements made with interleaved repumping, blue symbols without repumping. The only closed transition,
5S1/2 F=2 to 5P3/2 F=3 to 5D5/2 F=4 is unaﬀected by the repumping.

82
also conducted measurements of the 5S to 7S two-photon transitions, the frequencies of which were not known to better than 100 MHz. To resolve the absolutetransition frequencies for 7S we had to determine the mode order numbers using
multiple measurements. A second advantage of DFCS is the broad spectrum of
the femtosecond pulses. We show that simply scanning fr over ∼26 Hz excited
all allowed two-photon transitions within the comb spectrum.
4.3.1

Repetition Rate Scan

After reducing the systematic errors such as Zeeman shifts and radiationpressure by studying the strong transition to 5D5/2 F=4 we scanned fr of thelaser to observe the full set of two-photon transitions. The two-photon transition
frequency to all of the 5D states is ∼770.5 THz, as a very rough estimate. The
resonance criteria for two-photon transitions is given by,
ν2γ = fr × (N1 + N2 ) + 2 × fo

(4.9)

where N1 and N2 are the integer mode order numbers. Of course this equation has
many solutions within the spectrum of the comb. However, the sum N1 +N2 is thesame for all mode pair combinations with a given fr and fo . Considering we use
fr ∼100 MHz, the sum of mode order numbers N1 + N2 ∼7.7×106 . Similarly forthe intermediate state transitions the mode order number is ∼3.8×106 . Becauseof the fact fr /3.8 × 106 ∼26 Hz, changing fr by 26 Hz shifts the comb such thatthe one-photon transition is resonant with the next comb mode, N1 − 1. In thecase of the two-photon transitions the change of fr needs to only be 13 Hz before
the two-photon resonance condition is again satisﬁed. Therefore when we scan
the comb fr we expect the two-photon transitions to repeat every 13 Hz and thesingle-photon transitions to repeat every 26 Hz. Of course this is not true if fr ischanged so much that the resonant mode order numbers change appreciably.

83
In Fig. (4.18) we present the measured signal versus fr over a range of 28
Hz with a ﬁxed fo =-19.0 MHz. This result provides a very clear demonstrationof the versatility of DFCS for measuring many diﬀerent transition frequencies.
Between the two ground hyperﬁne levels and eight 5D hyperﬁne levels there are atotal of 14 allowed two-photon transitions. Considering we know the frequenciesof these transitions from previous measurements it is possible to identify each of
the 14 transitions in this result. There are also two extra small peaks in this data
set that do not correspond to any 5S to 5D transition, we attribute these to 5S
to 7S transitions and will discuss these in further detail in subsection (4.3.2).
The largest peak corresponds to when the comb is resonant from 5S1/2 F=2
to 5D5/2 F=4 with perfect intermediate state resonance with 5P3/2 F=3. For thisparticular experiment we used a value of fo and scanned fr over a range such that
this three-level resonance would occur. There is a second clearly identiﬁable peakcorresponding to a transition to the same ﬁnal state, however in this case thetwo-photon resonant comb pair is detuned from the intermediate state. Notice
there are two peaks that appear virtually identical in signal size and width at∼0.8 Hz and ∼26.8 Hz. These peaks are due to both the two-photon and single
photon resonance conditions repeating after a change of fr by 26 Hz.
4.3.2

Absolute Frequency Measurements

To conduct high-resolution absolute frequency measurements of several twophoton transitions we need to focus on individual lineshapes. We have selected 5
two-photon transitions from 5S to 5D to measure as an initial test of DFCS. After
we successfully measured these transitions we shifted the center wavelength andexpanded the spectral width of our Ti:Sapphire laser to be able to measure the
5S to 7S transitions. We will discuss how to determine the relevant mode ordernumbers in the context of our 7S measurements.

84

Figure 4.18: Full two-photon transition spectrum recovered by scanning fr by 26
Hz. The strongest peak is for intermediate and ﬁnal state resonant excitation of
the 5S1/2 F=2 to 5D5/2 F=4 transition. The two nearly identical peaks at ∼0.8 Hz
and ∼26.8 Hz correspond to the same excited hyperﬁne level and intermediate
state detuning.

85
As was discussed previously we know the frequencies of all of the 5P and 5D
hyperﬁne levels from previously published literature measurements. So insteadof scanning fr over all of the transitions we calculate speciﬁc values of fr andfo such that the comb is not only two-photon resonant but also resonant with anintermediate state. This allows us to reduce the Stark shift discussed in subsection(4.2.3). With a ﬁxed value of fr we measure the two-photon lineshape by scanning
fo around the resonant frequency.
To estimate the linecenter frequency corresponding to a relatively unperturbed atom we record the signal versus time and estimate the linecenter using
a ﬁt to the data. Speciﬁcally we counted photons using the SR430 multichannel
photon counter with a per bin duration of 160 ns. We then post-process the data
into new bins yielding a moving average, this was done by adding up 40 bins of
160 ns duration each and shifting by 5 bins. So the end result is a set of bins
each with a 6.4 µs in 800 ns intervals. These times were chosen so that there isenough signal to noise in each 6.4 µs bin to reasonably ﬁt the data to a Lorentzianlineshape and extract the linecenter.
Due to the fact the power transmitted through the LC shutter is a functionof time we can not simply extrapolate the linecenter data to zero time using alinear ﬁt function. Instead we ﬁt the linecenter data minus the expected linecenter
value with a function of the form,∆ν  (t) = c1 + c2 × P (t) + c3 × (P (t) × t)

(4.10)

where ∆ν  (t) is the linecenter shift from the expected value versus time, P (t) is
the power versus time measured after the LC shutter, and t is the time after datacollection initiates. The coeﬃcient c1 is the estimate of the unperturbed linecenter
frequency shift from some reference frequency. The last two coeﬃcients are used
to empirically model the Stark shift and radiation pressure Doppler shift of the

86linecenter frequency. We expect the Stark shift to be solely a function of power
for a ﬁxed probe beam mode and intermediate state detuning, so it is modeled as
c2 × P (t). The Doppler shift due to radiation pressure on the other hand is an
accumulated eﬀect, at short times we expect this to vary linearly with power andtime, so it is modeled as c3 × (P (t) × t). This type of ﬁtting function assumes
that the atom is not Doppler shifted from intermediate state resonance, as we
discussed in subsection (4.2.2), the actual shift versus time is nonlinear. Initially,
when the LC shutter is just beginning to transmit the probe light the signal tonoise ratio is not suﬃcient to reasonably ﬁt the lineshape data. For this reason
we only ﬁt the linecenter data with Eq.(4.3.2) from about 20 µs to 100 µs. Thisdata analysis technique proved quite successful, the one-sigma uncertainty in ourestimates of c1 ranged from ∼11 kHz to ∼50 kHz.
Conducting direct frequency comb spectroscopy of the 5S to 7S transition
required several additional considerations beyond our 5D measurements. The
ﬁrst of which is that the optimum Ti:Sapphire center wavelength is diﬀerent. Like
our previous measurements, we wish to conduct two-photon spectroscopy with aresonant intermediate state. The resonant wavelengths are shown in Fig. (4.1),
it can be seen that to excite 7S via a 5P3/2 state requires 780 nm and 741 nm
light. To provide more power at the 741 nm wavelength the center wavelength of
the Ti:Sapphire was shifted to 770 nm. Unfortunately further blue shifting of thespectrum deteriorates the fo beat signal too much.
A second important diﬀerence in our 7S measurements is that the twophoton transition frequency was not initially known to better than ∼ fr /2, therefore we must determine the resonant mode order numbers. Recall the equation
for the two-photon transition frequency is,ν2γ = fr × (N1 + N2 ) + 2 × fo .

(4.11)

87
We know the 5S to 5P3/2 transition frequencies very well and can therefore assign
a mode order number to N1 . The question is how do we determine N2 ? If
the measured two-photon transition linecenter frequency had zero uncertainty
we could simply measure the line with a slightly shifted fr and the peak would
appear at a shifted fo . Clearly the frequency shift of ν2γ is a function of N2 so thiswould allow the determination of N2 . The problem is that there is uncertainty
in our linecenter frequency measurements. So diﬀerentiating between say, N2 andN2 +1 requires a change of fr by at least the uncertainty in our estimate of fo
corresponding to the linecenter.
From previous measurements of the 5S to 7S transition we know that thefrequency is of order ∼788.8 THz. This corresponds to an estimated N1 + N2 of
7.888×106 , so the two-photon resonance condition repeats every ∆fr ∼12.6775 Hz
for an fr ∼100 MHz. In our experiment we measured the two-photon transition
linecenter at two values of fr separated by 600 kHz. Considering this change in
fr is much greater than 12.6775 Hz, the resonant mode order numbers changed
between the two measurements. In principle one could measure the change inmode order numbers between the two measurements by observing how many times
the peak repeats as fr is scanned over 600 kHz. This is of course very tedious
experimentally. Fortunately for the 5S to 7S measurements we can determine thechange of mode order number N2 based on previously published cw-laser based
measurements.
Suppose we deﬁne the integer M=N1 + N2 where we know N1 , and N2 is
to be determined. We denote our uncertainty in the linecenter frequency as δfo .
The two equations we have to solve for the change in M are,ν2γ = fr × (M) + 2 × (fo + δfo )ν2γ = (fr + ∆fr ) × (M + ∆M) + 2 × fo(4.12)

88
where fo =fo + δfo + ∆fo + δfo . We have introduced ∆fo to denote the change in
fo between the two measurements and δfo is the error in our linecenter frequency
estimate. Note that the change in fr is given by ∆fr and can be measured at the
sub-mHz level. The above set of equations can be solved for ∆M, the change intotal mode order number,∆M = −

M∆fr + 2∆fo + δfo.
fr + ∆fr(4.13)

Considering the denominator is a large number of order 108 and ∆M must be
an integer we can interpret this equation. We know 2∆fo well, by deﬁnition theerror is given by δfo , so we can calculate the ∆M from this contribution. Our
uncertainty in the second linecenter estimate, δfo , is 120 kHz or less, so when
divided by (fr + ∆fr ), is much less than an integer. The remaining term is,
−

M∆fr
.
fr + ∆fr(4.14)

Because of the fact ∆fr << fr any uncertainty in our estimate of M is suppressed.
Using numbers from our experiment, fr ∼100 MHz and ∆fr ∼600 kHz, it would
take an uncertainty in M of the order ∼160 to cause a shift of ∆M by 1. Given
our laser repetition rate of 100 MHz this would correspond to a 16 GHz initial uncertainty in the two-photon transition frequency. Fortunately this is not the casefor the 5S to 7S transitions which have been roughly measured [66]. For further
reading on how to estimate an absolute transition frequency using a frequency
comb refer to references [67, 68].
Now that we have estimated ∆M it can be used to solve Eqs. (4.12) unambiguously for M and thus the two-photon transition frequency ν2γ . Of course
there is still the uncertainty δfo in our lineshape ﬁt which can not be avoided,
so our knowledge of ν2γ is limited to δfo . Experimentally the exact value of fr
is not known perfectly, however, we generally measure it on a Cesium referenced

89
counter. If we count long enough to reduce the uncertainty in fr to the 1 mHz
level the resulting error in the two-photon linecenter is of order ∼7.7 kHz, lessthan our lineshape ﬁt uncertainty.
The resulting absolute frequency measurements for several 5D and 7S hyperﬁne levels are shown in Table (4.2). We report the linecenter frequencies wedetermined from the previously discussed ﬁt to the linecenter versus probing time,along with the one-sigma uncertainty in our estimate. The quoted literature values were all measured using traditional cw-laser spectroscopy. Absolute frequency
measurements of the 5S to 5D transitions are reported in reference [69]. In thecase of the 5S to 7S transitions an optical frequency comb was used to measure
the absolute frequencies of the cw-lasers used to excite the transitions [70]. Our
study agrees well with this cw-laser based study which was conducted and published about the same time we made our measurements. This provides strong
validation of our technique for determining the mode order numbers and thus absolute transition frequencies. Note that all of the reported two-photon transition
frequencies, with the exception of 7S1/2 F=1, are via the 5P3/2 F=3 intermediate
state. Therefore the 5S to 5P transition is closed and we obtained a better signal
to noise than for the 7S1/2 F=1 transition, which was via 5P3/2 F=1.
Transition
5S1/2 F=2→5D5/2F=2
5S1/2 F=2→5D5/2F=3
5S1/2 F=2→5D5/2F=4
5S1/2 F=2→5D3/2F=3
5S1/2 F=2→5D3/2F=2
5S1/2 F=2→7S1/2 F=2
5S1/2 F=1→7S1/2 F=1

Measured Frequency (kHz)770 569 184 527.9 (49.3)
770 569 161 560.5 (11.1)
770 569 132 748.8 (16.8)
770 480 275 633.7 (12.7)
770 480 231 393.9 (38.1)
788 794 768 921.4 (44.5)
788 800 964 199.3 (121.9)

Literature Frequency (kHz)770 569 184 510.4 (16)770 569 161 555.6 (16)770 569 132 732.6 (16)770 480 275 607.6 (10)770 480 231 385.2 (10)788 794 768 878.0 (40)788 800 964 042.0 (40)

Table 4.2: Measured absolute two-photon transition frequencies and comparison
to reported values from literature. Note that all of our reported values and ﬁterrors are from the extrapolation to ∼0 probing time and power.

90
4.3.3

Two-Photon Selection Rules

Our measurements of the two-photon transitions from 5S to 7S are particularly well suited for a discussion of two-photon transition selection rules. Previous
picosecond [16] and cw-laser based measurements [66] of this two-photon transition did not observe any transition corresponding to 5S1/2 to 7S1/2 in which
the hyperﬁne quantum number changed by 1. In other words the only change
in total hyperﬁne quantum number observed was ∆F =0. However, we observed
weak ∆F =±1 transitions, and other authors have as well with S-S transitions in
Sodium [71]. The explanation of this two-photon selection rule in the literatureis rather incomplete. Essentially it amounts to a two-photon transition selection
rule that is dependent on the intermediate state detuning, however, no explicit
dependence is given. The only observed ∆F=±1 transitions occurred when the
two-photon transition was resonantly enhanced via an intermediate state.
A good question is, how does this eﬀective two-photon transition selection
rule vary as a function of intermediate state detuning? The answer to this question provides a good example of how to treat two-photon transitions via multiple
intermediate states when comb mode pairs are detuned from any intermediateresonance, an important aspect of our experiment in chapter (6).
For the purpose of this explanation we will discuss only excitation by linear
polarized light, so we may use q=0 selection rules. It is important to ﬁrst understand any two-photon transition between 5S and 7S is via an intermediate state,in this case the 5P states need only be considered. In other words, regardlessof the intermediate state detuning, this two-photon transition is the sum of several two-photon transition amplitudes via the diﬀerent intermediate states. Using
standard single photon selections rules, in particular ∆F=±1,0, we can determine
the possible intermediate states for a transition from 5S1/2 F=1 to 7S1/2 =2 as an

91
example. Table (4.3) shows the angular part of the reduced dipole moment matrix
elements for the ﬁrst and second step of the two-photon transition as a functionof mF and intermediate state.
Intermediate State
5P3/2 F=2
5P3/2 F=1
5P1/2 F=2
5P1/2 F=1

mF =-1( 2√1 3 , −1)
6√

( 65 , 2√115 )√ , −1 )
( 2−1
6
3
1
√
( −1
,
)
6
2 3

mF =0
( 13 , 0)(0 , 3√1 5 )
( −1
, 0)
3
(0 , 13 )mF =1( 2√1 3 , 16 )√( −6 5 , 2√115 )√ , 1)( 2−1
6
3( 16 , 2√1 3 )Table 4.3: Angular part of the reduced dipole moment matrix element for eachpossible q=0 transition from 5S1/2 F=1 to 7S1/2 =2.

The ﬁrst term in each parenthesis corresponds to the 5S to 5P part of thetransition and the second term to the 5P to 7S step. For any given mF andintermediate state, the corresponding two-photon transition amplitude goes as
the product of dipole moments of each single photon step. The total two-photontransition amplitude can be calculated using these dipole moments and Eq. (3.12)
based on second-order perturbation theory. Clearly for the mF =0 transitions, via
any intermediate state, this two-photon transition is not allowed. Suppose weonly use two-modes to excite this transition, then the two-photon amplitudes
due to each intermediate state would be a function of the mode detuning. So
for large intermediate state detunings, compared to the 5P3/2 F=2 to 5P3/2 F=1
hyperﬁne splitting of ∼160 MHz, the two-photon transition amplitudes would
be weighted equally for both 5P3/2 F=2 and 5P3/2 F=1 transition pathways. The
important thing to notice is that for linear polarized light the product of dipole
moments for the F=2 and F=1 intermediate states (for either 5P1/2 or 5P3/2 )are equal and opposite. So the total two-photon transition amplitude is zero due
to complete destructive interference. This approach to calculating the total twophoton transition amplitude from 5S1/2 F=1 to 7S1/2 =2 can be used for arbitrary

92
intermediate state detunings. And of course, if the detuning in comparable to thehyperﬁne splitting the two-photon amplitude from one intermediate state will bestronger than the other, leading to incomplete destructive interference. Note thatthe same cancellation occurs for the other possible ∆F =0 transition, 5S1/2 F=2
to 7S1/2 =1.
Although we don’t present the dipole moments relevant for circular polarized
light, q=±1 transitions, a similar cancellation occurs. However, in it is not between hyperﬁne levels of a particular ﬁne-structure intermediate state, but rather
between diﬀerent ﬁne-structure states. So the cancellation only occurs when theintermediate state detuning is much larger than the 5P ﬁne-structure splitting of
7 THz.
A ﬁnal interesting observation from this analysis is that a single femtosecond
pulse, with a spectrum that does not cover the 5P1/2 to 7S transition, should not
excite any population from 5S1/2 F=1 to 7S1/2 =2. This is somewhat at odds
with the idea that the two-photon selection rule is solely a function of detuning.
In this case the spectrum is of course equally resonant with both intermediate
states, however, the two-photon transition amplitudes via 5P3/2 F=2 and 5P3/2 F=1
should be equal and opposite and still cancel out. Further discussion on twophoton transition selection rules may be found in [72].
4.4Single Photon DFCS
We have also performed direct frequency comb spectroscopy of single photontransitions in 87 Rb. There are some advantages and disadvantages of single photon
DFCS relative to our previous two-photon measurements. A signiﬁcant advantage
is that the single photon transition rate is much higher than the two-photontransition rate. Recall in the context of the radiation pressure, subsection (4.2.2),
we estimated that the 5S-5P incoherent scattering rate is at least 100 times that93of the 5S to 5D. However, the ﬂuorescence that we measure as our signal is ateither 795 nm for the 5P1/2 states or 780 nm for the 5P3/2 states. Both of thesewavelengths are of course in the bandwidth of the Ti:Sapphire laser, after allthat is how we excite the transitions. So we had to alter our data acquisition
technique and apparatus to avoid measuring scattered Ti:Sapphire light. The last
aspect of the 5P measurements we will discuss is how to use the intermediate
state enhancement of a two-photon transition to determine a 5P lineshape andlinecenter frequency indirectly.
Similar to our two-photon spectroscopy measurements we still used a 100
Hz MOT cycle. For the ﬁrst 7.5 ms the trap and repump lasers and magnetic ﬁeld
were on to load and cool atoms in the MOT. For the next 2 ms while the trap
magnetic ﬁeld was turned oﬀ the atoms were further cooled using polarization
gradient cooling (PGC), meaning the trap laser was attenuated and detuned fromthe cooling transition. This period of PGC cooling is primarily intended to holdthe atoms in optical molasses, the temperature gives rise to negligible Doppler
broadening relative the the 5P 6 MHz linewidth. Finally, the atoms are probed
and the 780 nm or 795 nm ﬂuorescence counted on a PMT for 0.5 ms, after
allowing the the trap and repump lasers to fully extinguish.
We had to use a somewhat more complicated probing cycle for the measurements at 780 nm and 795 nm to avoid saturating the PMT with scattered
Ti:Sapphire light. The imaging apparatus before the PMT was modiﬁed so thatan image of the atom cloud was formed on a ∼1 mm aperture. After the aperture
the light was passed through a 3 nm bandwidth optical interference ﬁlter centeredat the desired detection wavelength, and ﬁnally focused onto the PMT. This combination of spectral ﬁltering and imaging helped reduce the amount of scattered
Ti:Sapphire light incident on the PMT.
Unfortunately with the design of our MOT chamber there is a lot of scattered

94
Ti:Sapphire light, so we also have to time gate the data acquisition to avoid
saturating the PMT. This was accomplished by using a fast high voltage switchingcircuit to drive the PMT. The total 0.5 ms data collection window for each MOT
cycle was divided into sub-windows of total duration 3.2 µs. Using the Pockels cellto switch the Ti:Sapphire laser we only transmitted the probe light and excitedthe atoms during the ﬁrst 200 ns of this sub-window. During the next 400 ns
the PMT was switched on and we counted photons as the excited 5P population
decayed. Due to the relatively long turn-oﬀ time of the PMT the next 2.6 µs
was simply a delay before turning on the Ti:Sapphire again. This data acquisition
cycle was repeated for the ﬁrst 200 µs of the 0.5 ms window before the MOT ﬁelds
were turned back on. A typical signal was the result of repeating this process for
hundreds of MOT cycles, or several seconds per point.
Using this direct detection technique we measured the transition frequencies
from 5S1/2 F=2 to 5P3/2 F=3 and 5P1/2 F=2. One simpliﬁcation over the 5S to
5D transition frequency measurements is that we no longer used the slow turnon LC shutter but rather the Pockels cell to switch the Ti:Sapphire light. So
the probe power was constant versus time for all of these measurements. Again
using the multichannel SR-430 photon counter we measured the signal versus
fo as we scanned over the transition lineshape. From this data the linecenter
was estimated from a Lorentzian ﬁt versus time and the unperturbed linecenter
frequency determined by linear extrapolation to ∼0 probing time. We measured
an absolute transition frequency for the 5S1/2 F=2 to 5P3/2 F=3 transition of 384
228 115 271 (87) kHz. This agrees well with a previously published cw-laser
based measurement of 384 228 115 203 (7) kHz in reference [73]. Similarly forthe 5S1/2 F=2 to 5P1/2 F=2 transition we measured an absolute frequency of 377
105 206 563 (184) kHz. An improvement from a cw-laser based measurement
reporting a value of 377 105 206 705 (400) kHz in reference [74].

95
4.4.1

5P Measurements via Two-Photon Transitions

In this subsection we present an alternative technique for measuring the 5Plineshapes and center frequencies. Instead of measuring the 5P-5S ﬂuorescence
directly we infer the intermediate state lineshape based on changes in the 420
nm ﬂuorescence from the 5D states. Recall that in Eq. (3.12) the two-photontransition amplitude is enhanced as a function of the intermediate state detuning.
Therefore if we measure the two-photon transition rate for various intermediate
state detunings while maintaining two-photon resonance we can infer information
about the intermediate state.
One of the complications of this technique is determining a set of fr andfo values corresponding to exact two-photon resonance but with varying intermediate detunings. The second complication was alluded to in subsection (4.2.5)
concerning incoherent optical pumping to the oﬀ-resonant ground state. Unlike
the 5P3/2 F=3 state, the 5P1/2 F=2 state population may decay to the oﬀ-resonant
5S1/2 F=1 ground state.
Using our knowledge of the 5P and 5D energy levels we created a listof fr and fo pairs corresponding to diﬀerent intermediate state detunings from
5P3/2 F=3, but all two-photon resonant with 5D5/2 F=4, see Table (4.4). We also
selected only those combinations of fr and fo such that all other intermediate andﬁnal states were detuned at least ∼2.5 linewidths.
The above set of comb fr and fo was used to measure the 5P3/2 F=3 lineshape
via the two-photon transition 5S1/2 F=2 to 5D5/2 F=4. Another set of fr and fo
frequencies were used to measure the 5P1/2 F=2 lineshape via the two-photontransition 5S1/2 F=2 to 5D3/2 F=3.
Initially one may expect that fr and fo may be simply scanned together ina smooth fashion to achieve the desired two-photon resonance and intermediateDetuning (MHz)
-13
-10
-9
-4
-3
-2
+1
+3
+4
+5
+9
+12

fr (MHz)99.89260605
99.92066423
99.93474869
99.92482030
99.93890785
99.91518350
99.94325356
99.86747878
99.88155032
99.89562280
99.84803930
99.84303590

fo (MHz)-16.4737-22.4503-16.2599-9.9189-11.1280-21.3653-24.2433-22.2673-24.0872-14.1944-11.4370-24.3973

96

Table 4.4: Set of fr and fo pairs with varying detunings from the intermediate
5P3/2 F=3 state and two-photon resonant with 5D5/2 F=4.

state detuning. While this is true in a three-level system, it becomes complicatedby the many other possible transitions the comb may excite in an actual Rb atom.
More speciﬁcally, take the 5P3/2 F=3 lineshape measurement as an example. This
intermediate state is oﬀset from half the two-photon transition frequency by 1
THz, so considering we use fr ∼100 MHz, the diﬀerence in resonant comb modeorder numbers is ∼104 . So a change of fr by ∆fr only results in a diﬀerentialchange of frequency between the two resonant modes of ∼104 × ∆fr . Supposethat we wish to have a 10 MHz intermediate state detuning, this would requirechanging fr by ∼1 kHz. This change is large enough that the detunings from themany other intermediate and ﬁnal states will in general change, and may become
resonant with other comb modes. It is for this reason extra care must be taken in
selecting the values in Table (4.4).
In Chapter (3) we brieﬂy discussed one of our theoretical models based on
fourth-order perturbation theory using the Liouville equation. Due to the factthat this model includes all of the 5S, 5P, and 5D states it is ideal for calculating the eﬀects of incoherent optical pumping. In fact, as we will show below,

97

Figure 4.19: Theoretically predicted population distribution between the two hyperﬁne ground states for ±4 MHz detuning from 5P3/2 F=3. Note that the intermediate state detuning was set using values of fr and fo to ensure two-photon
resonance with 5D5/2 F=4 and reduce excitation to all other transitions.

98
it is essential for our indirect 5P1/2 F=2 measurement. Figure (4.19) shows thetheoretically predicted population distribution between the two hyperﬁne groundstates for ±4 MHz detuning from the 5P3/2 F=3 state. We expect the majorityof the population to be in the F=2 ground state initially. From this theoreticalprediction we see that there is some optical pumping, due to other intermediate
states. However, the amount of population that remains in the F=2 state is essentially independent of the 5P3/2 F=3 detuning. Using the set of fr and fo pairs
shown in Table (4.4) we measured the 5P3/2 F=3 lineshape via its enhancementof the two-photon transition. With this indirect measurement technique we estimated the 5S1/2 F=2 to 5P3/2 F=3 transition frequency to be 384 228 115 309 (63)kHz, within ∼40 kHz of the frequency measured with the direct single photon
technique at 780 nm.
Figure (4.20) shows the same simulation for the resonance with the 5P1/2 F=2intermediate state. Recall that population in this state may decay to the oﬀresonant ground state, 5S1/2 F=1. Clearly in this case optical pumping is an
important eﬀect, not only does most of the ground state population end up in
F=1, but it depends on the intermediate state detuning. Note that in both cases
the intermediate state detuning was achieved using our predetermine fr and fo
values, like in Table (4.4). The steady state value of the ground state population
distribution is not only determined by the optical pumping rate from F=2 to F=1,
but also the rate for the reverse process, which in general occurs at a diﬀerent
rate. One of the reasons the incoherent optical pumping is diﬀerent between the±4 MHz detuned cases is the optical repumping due to the comb itself from F=1
to F=2. Of course if there was no optical repumping all the population would
eventually end up in the F=1 ground state.
Unfortunately the measured 5P1/2 F=2 appears noisy due to the apparently
random ground state population distribution for diﬀerent fr and fo pairs. To

99

Figure 4.20: Theoretically predicted population distribution between the two hyperﬁne ground states for ±4 MHz detuning from 5P1/2 F=2. Note that the intermediate state detuning was set using values of fr and fo to ensure two-photon
resonance with 5D3/2 F=2 and reduce excitation to all other transitions.

100
improve the quality of our signal, meaning make the lineshape closer to Lorentzian,
we normalized all of our measured 5P1/2 F=2 data. This was done by dividing thesignal measured at each intermediate state detuning by the theoretically predictedvalue of (1-ρF =1 ), where ρF =1 is the population in the oﬀ-resonant ground state
5S1/2 F=1. Note that this is not the same as normalizing by ρF =2 because some
population is excited. The raw lineshape measured prior to normalization is shownin Fig. (4.21)(a). After applying our normalization scheme the improvement in theLorentzian ﬁt is quite dramatic, see Fig.(4.21)(b). Using the normalized linshape
we estimate the 5S1/2 F=2 to 5P1/2 F=2 transition frequency to be 377 105 206 939(179) MHz, close to the single photon direct measurement. Although this provides
an alternative means to measure an intermediate state it is not an improvement
over the direct measurement technique applied in the previous section. However,it does provide a remarkable validation of the theoretical model used to calculate
the optical pumping eﬀects.

101

Figure 4.21: (a) Raw signal versus intermediate state detuning measured using
the 420 nm ﬂuorescence enhancement from the 5D3/2 F=2 state. (b) Improvedlineshape recovered by normalizing the raw data by the theoretically predictedvalues of (1-ρF =1) to compensate for the variable optical pumping to the F=1
ground state.

Chapter 5
High-Resolution Coherent Control: Chirp

5.1

Chirped Pulses
A simple non-transform limited pulse shape to consider is a pulse in time-

domain with a linear frequency sweep. This means that the instantaneous frequency of the pulse is either increasing or decreasing versus time at a constantrate. Although the time domain deﬁnition is intuitively clear, the pulse shaping
and atomic physics we wish to discuss are best considered in frequency domain.
We use a frequency domain deﬁnition of chirp given be the Taylor expansion of the phase of a Gaussian ﬁeld envelope pulse about some center frequency.
With our convention, changing the chirp of the pulse only changes the quadraticcomponent of the phase in frequency domain, the power spectrum remains unaltered but the temporal duration is increased. The quadratic spectral phase term
is given by,
φ =

1 ∂2φ
|ω=ωo
2 ∂ω 2(5.1)

where ωo is the center angular frequency of the pulse spectrum. This term issometimes also called the group velocity dispersion, usually in the context ofmaterial dispersion, because it represents a change of group velocity as a functionof frequency. With the given deﬁnition of chirp, lower frequencies precede higher
frequencies for positive chirp, and vice-versa for negative chirp.

103
A Gaussian pulse with arbitrary chirp has an electric ﬁeld given by,E(ω) =

√

2

2



2

πEo To e−(ω−ωo ) To /4 eiφ (ω−ωo )(5.2)

where 2To is pulse duration deﬁned by the 1/e values and Eo is the peak ﬁeld
magnitude of the transform limited pulse. In time domain this ﬁeld becomes,

To
2
2
cos((ωo + αt)t + φceo)e−t /T(5.3)
E(t) = EoTwithT  = To

1 + (4φ2 /To4 )

(5.4)

2φ
.
To4 + 4φ2

(5.5)

and
α=

It can be seen from Eq.(5.3) that the instantaneous frequency changes at a constant rate given by α, and for either sign of α the chirped pulse duration, T , isgreater than the transform limited pulse duration To . This expression for the electric ﬁeld in time domain is used when solving numerically for the density matrixof the atomic system.
5.2

Pulse Stretcher Compressor
To generate pulses with large, tunable linear frequency sweep we use a grat-

ing based pulse stretcher and compressor [75]. Typically grating based pulse
shapers are used to compress pulses after long travel through dispersive ﬁber [76]or for reducing the peak intensity of a pulse in a chirped-pulse-ampliﬁer system[77]. In prism and grating based pulse shapers, the vast majority of the dispersion arises due to the angular dispersion, as opposed to the material itself. It
was shown in [78] that angular dispersion normally gives rise to negative groupvelocity dispersion, but by using a telescope in the pulse shaper it is possible
to have both negative and positive dispersion. The large positive and negative

104
dispersion possible with a telescopic grating-based pulse shaper is ideal for our
experiments, which required of order ±100,000 fs2 chirp.
It is worth going through a brief derivation of the second-order dispersion
of a grating pair stretcher/compressor, just to highlight one of the subtleties. As
mentioned previously, the second-order dispersion, responsible for the chirp of apulse, is deﬁned as the second frequency derivative of the optical phase. Therefore
we must derive an expression for the phase as a function of frequency. Figure (5.1)
shows a diagram of a basic two grating pulse stretcher/compressor with two input
rays of the center wavelength λo used to approximate a ﬁnite width plane wave.
There are three lengths deﬁned: Lperp is the distance between gratings along thegrating perpendicular direction, Lλ is the distance traveled by a ray of wavelength
λ, and y is the extent of the beam along the second grating.
The total optical phase arises from two parts, the path length deﬁned as
ABCD and an extra phase due to phase matching. The grating equation for ﬁrst
order diﬀraction is,sin(θi ) + sin(θr ) =

λ
d

(5.6)

where the incident θi and reﬂected θr angles are deﬁned in Fig. (5.1) and d is thegroove spacing. Using the grating equation it is not hard to determine that the
wavelength dependent path length ABCD, now deﬁned as l, isl = Lperp sec(θr )(1 + cos(θi − θr )).

(5.7)

The second contribution is due to phase matching, and is somewhat lessobvious, for this explanation we must consider two parallel input rays of the same
frequency. There are two dashed blue lines perpendicular to the optical rays inFig.(5.1) that denote the optical phase fronts, or in other words planes of constantphase. Notice that one of the rays must travel further than the other, denoted as
the dashed red segments of the ray path. For the phasefronts to be at constant

105

D

C
y
LO

Lperp

Oo

B

A
Tr

Oo

Ti

X=0

x

Figure 5.1: Grating pair diagram for the derivation of the second-order dispersion.

106
phase there must be an extra phase included due to the grating. This extra phasecan be determined by calculating the extra path length, given by the two dashed
red segments. With some geometry this extra length can be shown to be,
lextra = y(sin(θr ) + sin(θi )).

(5.8)

Notice that the previous equation can we rewritten using the grating equation as,yφextra = 2π ,
d

(5.9)

with φextra ≡ 2π lextra. This can be interpreted as an extra phase shift of 2π perλgroove covered on the second grating. Note that due to the fact the beam has notbeen spectrally dispersed before the ﬁrst grating, we need to only consider thiseﬀect after reﬂection from the second grating. Combining the results for l andlextra and noting that y = Lperp tan(θr ) the total phase as a function of frequency
can be written as,φ(ω) =tan(θr )ωLperpsec(θr )(1 + cos(θi − θr ))+ 2πLperp.
cd(5.10)

After taking two frequency derivatives, substituting for Lperp , the second-orderdispersion is found to be,
4π 2 c Lλo.
φ = − 2 3d ωo cos(θr )2(5.11)

From this formula it can be seen that the chirp is a linear function of displacement L
 λ . Further calculations for grating pair pulse eﬀects can be found in
the literature for beams of ﬁnite size [79], Gaussian beams [80], and distortions
due to misalignment [81].
We used a grating with 1400 lines per mm in a folded 2f geometry, seeFig.(5.2). Although the telescope can be used with a magniﬁcation of greater thanunity to increase the dispersion of the shaper, we used a magniﬁcation of unity forthe purpose of allowing both signs of chirp. The apparatus was constructed with

107

f+ x
O1

f
O1

Oo
Oo

O1

Figure 5.2: Double pass grating pulse stretcher / compressor. This ﬁgure shows
the design used in the our experiments with a telescope of unity magniﬁcation.

108
both the lens and the retro-reﬂection mirror mounted on one rigid platform that
could be translated along the optical axis via a long travel micrometer. Usinga long focal length lens allows large excursions of the lens+mirror pair from thegrating, and thus a large range of ±∆x. In this geometry the grating separation
Lλ used in Eq. (5.10) must be substituted with ∆x [78]. Figure (5.2) illustrates
the oﬀset of wavelengths that occurs after one pass through a grating pair. The
input ray contains all wavelengths, but the rays corresponding to wavelengths λoand λ1 traverse diﬀerent paths due to the angular dispersion of the gratings andexit with a spatial oﬀset. Spatial chirp, or wavelength dependent propagation
directions, is an issue for our experiments. We focused the Ti:Sapphire beam
into the atom cloud, if the modes corresponding to diﬀerent wavelengths do not
overlap well in the focal plane there will be reduced interference in the atomic
excitation. Fortunately, the spatial chirp incurred by one pass through the shaper
can be canceled out by a second pass. In our case the retro-reﬂection mirror shownin Fig.(5.2) is tilted slightly such that a pick-oﬀ mirror can be used outside theshaper to reﬂect the beam back for a second pass, note that this also doubles thetotal dispersion of the pulse stretcher.
For alignment purposes a 3 nm bandwidth optical interference ﬁlter wasused at 778 nm, the center wavelength of the pulse, before the beam entered
the pulse stretcher. The grating was rotated such that the specular reﬂection
directly retro-reﬂected onto itself, this allows for a reference position to determine
the angle of incidence. From this position the grating was then rotated to set theangle of incidence for optimum diﬀraction eﬃciency, approximately 90%. To alignthe grating grooves to the beam polarization the full spectrum of the pulse wasused. This is done by ensuring the dispersed spectrum remains horizontal to thetable, the grating mount allows for the tilt of the grooves to be adjusted.
The lens plus retro-reﬂection mirror, both mounted on a rigid platform on

109
a translation stage, was set such that the center wavelength of 778 nm traveled
directly along the optical axis of the lens. The lens and retro-reﬂection mirror
were both mounted on tip-tilt stages such that the 778 nm impinges normally
onto the lens and is reﬂected slightly down from the mirror, allowing the beam
to be picked oﬀ after one or two passes. A second short travel translation stage
was used to mount the retro-reﬂection mirror, thus allowing ﬁne tuning of thelens to mirror distance to be one focal length. As mentioned previously the beam
was sent twice through the pulse stretcher by reﬂection from a pick-oﬀ mirror, toreduce spectral walk-oﬀ.
Any residual spatial chirp of the pulse must be reduced. This was accomplished by diverting the full spectrum beam onto a card about 8 meters after twopasses through the pulse stretcher. Then the retro-reﬂection mirror position was
adjusted via micrometer until the Ti:Sapphire mode dimmed uniformly as diﬀerent spectral components were blocked in the fourier plane of the lens. When there
is signiﬁcant spatial chirp a dark line can be observed in the mode, the positionof which depends on the color blocked.
Finally the remaining calibration was to determine the necessary grating
oﬀset, ∆x, to compensate for the dispersion of the optics such that the pulse wastransform limited at the atomic sample. For this we optimized the two-photon
photocurrent on a GaAsP photodiode. The beam was diverted from the MOT
chamber and focussed onto the photodiode. As the distance of the grating fromthe lens+mirror system was scanned via micrometer a clear peak was observedin the photocurrent corresponding to an approximately transform limited pulse.
Note that for this experiment the variations in excited population we observe areon the scale of many thousands of fs2 chirp, so a rough calibration is suﬃcient.

110
5.3

Dressed Atom Picture
The dressed atom picture is a useful tool for visualizing and estimating someof the basic physics involved with chirped pulse excitation of a three-level atom. In
the case of large positive chirp the excitation of rubidium by one or more chirped
pulses there is a characteristic oscillation of the excited population versus chirp.
This characteristic oscillation was the main eﬀect observed in the original single
pulse experiments and also appears in our comb experiments. The ﬁrst period of
this oscillation can clearly be seen in our results presented in Fig. (5.5) of section(5.4). Theoretical calculations indicate that it continues beyond the range of chirp
used in our experiment, which was limited by our pulse stretcher/compressor.
An intuitive derivation of the rough dependence of the oscillation period, in
terms of chirp, can be arrived at from a dressed atom picture. Figure (5.3) shows
the dressed energy levels of a three-level atom in a cascade conﬁguration. We areinterested in the interference between the two paths indicated by the red dashedarrows in the ﬁgure. Both paths start from the ground state before the pulse andend in the excited state after the pulse, however, one path passes through thetriangle deﬁned by ABC and the other goes directly from A to C. The path via
ABC is said to be diabatic because the state vector passes between two dressed
eigenstates, this will be referred to as path i. The other path is adiabatic because
the state vector remains in the same dressed state, this will be referred to as
path ii. In the actual experiment both paths are traversed simultaneously forthe range of chirps and intensity used in our experiment. Clearly the energy ofthe state-vector changes as a function of laser frequency and thus time. We areinterested in the diﬀerence in phase acquired, calculated from the energy of the
wavefunction, between the two paths.
The dashed lines in Fig. (5.3) represent the dressed levels in the limit of very

111
weak instantaneous intensity, such that the Rabi frequency between adjacent levels
is low and thus the coupling. In the limit of large chirps, this condition of weakcoupling is satisﬁed and the dressed states, indicated by the solid lines, approachthe energies given by the dashed lines. In this limit the diﬀerence between thetwo paths is just the triangle ABC. The total phases acquired along the two paths
is given by,
 t
φi =
φii =

0
 t
0

ωi (t)dt(5.12)

ωii (t)dt

where t is the time it takes to sweep the frequency between points A and C, andωi,ii (t) is frequency of the wavefunction along path i or ii. Note, ωii (t) is constantand equal to ω2 and that,
ωi (t) = ω2 + αt,(5.13)



between times t=0 to t= t2 . The phase diﬀerence can be simpliﬁed to t /2tdt.
∆φ = 2α(5.14)0

As noted previously the rate change of frequency of a Gaussian envelope pulse forlarge chirps is given by α. The diﬀerence in frequency between points A and C isgiven by δ, the diﬀerence in frequency between the ﬁrst and second transitions.
So the time t , is just αδ . In the limit of large chirps, Eq. (5.5) for the frequency
sweep rate gives the relation α = 2φ1 . Using these substitutions and solving for∆φ = 2π gives the amount of chirp φ corresponding to one oscillation period,
φ =

4π
.
δ2

(5.15)

So from this rather simple interfering paths argument assisted by the dressed atom
picture we arrive at the correct dependence of the characteristic oscillation periodon δ.

EigenEnergy (a.u.)112|1+³Z>

A

B
G

C

|2>

|3-³Z>Z21

Z32

Photon Energy (a.u.)
Figure 5.3: Diagram representative of the dressed atom states and the two interfering paths from the ground to excited state.

113
5.4

Chirped Pulse Experiment
One of the most common and simplest applications of coherent control iscalled ladder climbing. Ladder climbing refers to the subset of coherent control
techniques relying upon a linearly chirped broadband pulse to excite some target
state via one or more intermediate states. In general this type of multi-photontransition can be in either a cascade (ladder) type or Raman type conﬁguration,
the theory being nearly identical for the two cases. Ladder climbing has found application in molecular systems [82] for dramatically increasing dissociation. Somework has been conducted demonstrating a frequency selectivity much narrower
than the bandwidth of the pulse(s) used in the experiment [83, 84]. If the chirp is
suﬃciently large, and the intensity high, the state vector follows some eigenstate
adiabatically, this regime is referred to as adiabatic following [85]. Eﬃcient population transfer in a three-level system can be achieved using two time delayedtransform limited pulses, referred to as stimulated Raman adiabatic passage (STIRAP) [86], or a single chirped broadband pulse [87]. At pulse intensities such thatadiabatic following is not present the interaction of a single broadband chirped
pulse with a three-level system has been studied in rubidium [58], and Sodium[88, 89].
In this section we describe our ﬁrst experiment demonstrating high-resolution
coherent control. This experiment extends the previous work done on chirped
pulse excitation of a three-level atom to the multi-pulse regime by using a femtosecond comb. All previous experiments studying the interaction of a shaped
femtosecond pulse with an atomic or molecular system have been limited to only
a few pulses with temporal separations of much less than a nanosecond. For example, pump probe experiments generally use an optical delay line to generate
the time between pulses, signiﬁcantly limiting the delay. Due to the relatively

114
long coherence time of the phase stabilized Ti:Sapphire laser we are not limited toshort inter-pulse delays. As demonstrated in section (??), as long as the interaction between the femtosecond pulse train and the atom electronic states remains
coherent the resolution obtained increases versus pulse number. We use this fact
to study the eﬀects of a series of linearly chirped pulses with a three-level subsystem of rubidium levels at high-resolution. In particular we excite only the
5S1/2 F=2 to 5P3/2 F=3 to 5D5/2 F=4 cascade transition, see Fig. (5.4), all otherallowed transitions are detuned by at least a few linewidths.
Figure (5.4) shows the resonant two comb mode wavelengths at approximately 776 nm and 780 nm and the frequency oﬀset of the intermediate levelfrom exactly half the two-photon transition frequency, we refer to this as δ. The
comb was locked with an fr of 100.41356730 MHz and fo of 18.14 MHz, which
results in the required resonance condition of one comb mode directly resonantwith the 5S1/2 F=2 to 5P3/2 F=3 transition and another 5P3/2 F=3 to 5D5/2 F=4.
As mentioned in the context of spectroscopy given the two degrees of freedom forthe comb frequency spectrum it is possible to satisfy the above resonance condition exactly. Unlike our previous spectroscopy experiments, the entire spectrumof the femtosecond pulse plays a role in this experiment.
Next let us consider the role of the hundreds of thousands of remaining comb
modes detuned from an intermediate state. The two-photon transition frequency
from 5S1/2 F=2 to 5D5/2 F=4 is ν2γ = 770.569132733400 THz. Due to the fact it
is a two-photon transition the sum of comb mode frequencies must add to thisfrequency such that,ν2γ = fr (N1 + N2 ) + 2fo(5.16)

where N1 and N2 are two integer mode order number of order 4 million. Notethat adding a constant quadratic spectral phase to each pulse does not change the115

5D5/2 F=4Amp

776 nmG

778 nm

5P3/2 F=3)780 nmO

778 nm

5S1/2 F=2
Figure 5.4: Energy level diagram of resonant states for chirped pulse experiment.

116
frequency spectrum of the comb, only a change in spectral phase as a functionof time would modify the spectral content. From this equation it can be seenthat if any particular pair of modes add to two-photon resonance then due to theequal spacing of comb modes all modes in the spectrum are part of a resonant
pair. In this set of chirped excitation experiments the spectrum was approximated as Gaussian with a FWHM of 30 nm centered at 778 nm. This results inroughly a 15 THz FWHM spectral width supporting approximately 150,000 comb
modes. If we take each separate possible two-photon transition amplitude to befrom an ordered pair of modes there are ∼300,000 two-photon resonant transition
amplitudes within the FWHM of the comb spectrum.
Each two-photon resonant transition amplitude is proportional to,
CN1 ,N2 ∝|EN1 ||EN2 |ei(φN1 +φN2 )×i(ωgf − 2π(N1 + N2 )fr − 4πfo ) + π∆νf
1i(ωgi − 2π(N1 )fr − 2πfo ) + π∆νi(5.17)

where |EN(1,2 ) | and φN1,2 are the magnitude and phase of the N1 and N2 modes,ωgf (ωgi ) are the transition frequencies from ground to ﬁnal(intermediate) states,and ∆νi(f ) are the intermediate(ﬁnal) state linewidths. Note that this simpliﬁed
form of the equation is for the N1 mode from the ground state, i.e. it constitutes
the ﬁrst step in a speciﬁc ordering of absorption. The other possible ordering has
N2 exciting the 5S to 5P transition. The full formula is not necessary to explain
the relevant physics for this section.
Inspection of Eq. (5.17) reveals that the phase of the excited state amplitude is a function of the phases of the two modes and also the detuning fromintermediate state resonance. In particular we notice that for a transform limited
pulse, if the frequency of the N1 mode is much less(greater) than ωgi the phase of
the amplitude is -90o(+90o ). The nearest mode pairs oﬀ-resonant from the intermediate state are detuned by ±fr ∼100 MHz, for a transform limited pulse these117
give rise to transition amplitudes of equal magnitude but ∼180o out of phase andtherefore destructively interfere. All other modes are detuned further than 100
MHz from the intermediate state and therefore give rise to two-photon amplitudes
with a phase of ±90o as well. Altering the spectral phase from that of a transform
limited pulse (constant phase) by introducing chirp, changes the phases of thetwo-photon amplitudes corresponding to diﬀerent wavelengths.
Due to the fact the spectrum is centered at 778 nm, close to the 5S1/2 F=2
to 5P3/2 F=3 transition wavelength of 780 nm, it is a good approximation to saythat for a transform limited pulse all mode pairs detuned from the intermediateresonance completely destructively interfere. Of course there is a second reason
why the mode pairs detuned from an intermediate resonance can be ignored to aﬁrst approximation. The second term in Eq. (5.17) predicts the amount of resonant enhancement of the two-photon transition as a function of intermediate state
detuning. So any mode pair detuned from the intermediate state has a greatly
diminished amplitude compared to the intermediate state resonant transition amplitude.
In this chirped excitation experiment we are interested in the eﬀect of thehundreds of thousands of mode pairs detuned from intermediate resonance. The
experimental apparatus used in the section is very similar to that of the spectroscopy experiments. In particular the Pockels cell is used as a pulse picker inthe same manner as section (4.1). As mentioned previously the Ti:Sapphire isoperated with a spectral width of approximately 30 nm FWHM, at a center wavelength of 778 nm, resulting a transform limited pulse duration of about 80 fs. The
average power at the MOT was measured to be approximately 55 mW. This beam
was focused into the MOT, from one direction only, to a beam waist estimated to
be about 160 µm. The peak transform limited electric ﬁeld resulting from thesebeam parameters is approximately 1.3×107 V/m.

118
The MOT was operated in 10 ms cycles like our previous experiments. It
was loaded for the ﬁrst 7 ms, then the trapping and repump lasers and magnetic
ﬁeld were turned oﬀ for the remaining 3 ms. In the last 200 µs the atoms wereprobed with the Ti:Sapphire laser. Due to the fact the atoms were probed fromone direction only there is a signiﬁcant momentum transfer to the atoms. We only
probed the atoms from one direction because we want all the atoms to be excitedby the same exact pulse shape. We will discuss the eﬀect of counter-propagating
pulses in the context of coherent control in section (6.1). For this reason the data
was collected during only the ﬁrst 80 µs of the 200 µs window. This 80 µs was
divided into two windows of equal 40 µs duration. During the ﬁrst 40 µs the atoms
were excited by the Ti:Sapphire laser that passes through the Pockels cell and thepulse stretcher, this is the chirped beam. During the second 40 µs window the
Pockels cell is in the oﬀ position, during this time the Ti:Sapphire laser is diverted
out the side exit of the cell and bypasses the pulse stretcher. This diverted beam
is chirped from the Pockels cell and general optics, but remains at constant chirp
throughout the experiment. We used this constant chirp beam as a reference toreduce the eﬀect of MOT atom number ﬂuctuations during the data collection
process. Using a SR-400 two-channel photon counter we were able to normalize
the counts taken on channel A during the ﬁrst 40 µs to those taken on channel B
during the second 40 µs. Each data point was taken at a set displacement of thelens and retro-reﬂection mirror, ∆x in Fig. (5.2), and is the sum of 20,000 MOT
cycles, 200 s of operation time, or 0.8 s of atom interrogation time.
The main result of this experiment is presented in Fig. (5.5). We scanned
the displacement, ∆x, of the pulse stretcher from 22 mm to -33 mm. Using Eq.
(5.10) for the dispersion of a single pass through the stretcher and multiplying by

119
two for double pass we get the dispersion per unit ∆x,φ (∆x) = −6.774 × 106 ∆x.

[f s2 ]

(5.18)

This range of displacements gives -150,000 to 225,000 fs2 chirp, enough to observe the characteristic oscillation. Figure (5.5) shows a theoretical calculation(red lines) for both the single pulse response (not to scale) and the asymptotic
steady state 5D5/2 F=4 population versus chirp. The measured signal is shown as
open squares and has been scaled to the theoretically predicted population by one
global scale factor that takes into account things such as photon collection eﬃciency and atom number. Although the dressed state approach and second-order
perturbation theory yield useful information about the physics, the theory model
used here is based on solving for the density matrix of the three-level system. In
particular we solve for the density matrix of the system numerically as outlined
in section (3.2) of chapter (3) for a series of linearly chirped femtosecond pulses.
There are two main features of the data in Fig. (5.5) that separate our
comb experiment from previous single pulse experiments. The ﬁrst of which is
the large diﬀerence in the excited population between one pulse and 100 pulses
for negative chirps. Intuitively the very low population predicted for a single
negatively chirped pulse can be understood from the order of frequencies that
arrive at the atom.
For a rubidium atom the 5S to 5P wavelength is less than that of the 5P
to 5D wavelength. Initially the electron is in the 5S1/2 F=2 ground state, with noprobability to be in the 5P states. So the natural time ordering of wavelengths
to excite an electron from 5S to 5P then 5D, is longer to shorter (positive chirp).
In fact that is exactly what previous experiments in Na and Rb found for single
pulse excitation [88, 58]. If the intermediate level lies above the two-photon halffrequency then the ordering would need to be reversed.

120

Figure 5.5: Steady state measured signal and calculated excited state population
versus chirp and the single pulse theoretical response.

121
Coherent excitation by multiple pulses, like in our experiment, results in asigniﬁcant 5D population at negative chirps. Initially we did not expect this result.
However, after some thought this is a nice example of how coherent interaction
with a chirped comb diﬀers from a single pulse. This eﬀect is due to the 5S1/2 F=25P3/2 F=3 electronic coherence that persists between pulses, with a coherence timeof about 50 ns. This coherence essentially ties together the electric ﬁeld fromseveral pulses. So despite the fact individually the pulses are chirped from short
to long wavelengths, between the end of the ﬁrst pulse and the beginning of thesecond the ﬁeld jumps from long to short wavelengths. This inter-pulse frequency
jump is similar to a positively chirped ﬁeld, and is therefore capable of exciting
the 5S to 5P to 5D transition in the proper time ordering. To demonstrate thiseﬀect we measured the 5D population versus the number of femtosecond pulses
at ± 67,000 fs2 chirp. Our results are shown in Fig. (5.6) starting from a single
pulse excitation by 1 to 9 pulses. The theoretical predictions are the lines throughthe data points, which are shown as symbols. At both large positive and negative
chirps the population grows as the square of the number of pulses, a signature ofcoherent accumulation. However, the scale factor for the negative chirp results isless population transfer than that of the positive. Unfortunately the single pulse
signal is dominated by noise so the single pulse versus two pulse response does
not show the dramatic increase expected from theory.
The second feature of signiﬁcance is indicative of the high-resolution obtained in our experiment. Recall that the oscillation of the excited population is. Where δ is the detuning of the intermepredicted to occur with a period of 2π
δ2diate state from half of the two-photon transition frequency. In previous single
pulse experiments there are two main sets of intermediate states due to the ﬁne
structure splitting, for example the 5P states in Rb. Excitation of Rb by a single
pulse will result in population transfer to both the 5D3/2 and 5D5/2 excited states.

122

Figure 5.6: Measured signal and calculated excited state population versus pulse
number for large positive and negative chirps.

123
The 5D3/2 excited state manifold is accessible via both the 5P1/2 and 5P3/2 states.
These two sets of intermediate states have very diﬀerent detunings from half thetwo-photon transition frequency. The 5P3/2 states have detunings, δ, in the rangeof 1 THz and the 5P1/2 states have much larger 7 THz detunings. Therefore itis expected that there should be two diﬀerent periods for the oscillation of 5D
population versus chirp, these diﬀerent periods were observed in the Sodium single pulse experiment [88]. The high-resolution of the comb allows us to tune to
speciﬁc transition pathways. In this particular experiment we chose the 5S1/2 F=2
to 5P3/2 F=3 to 5D5/2 F=4 transition. Nearly all of the measured signal can thenbe attributed to population solely in the 5D5/2 F=4 state, which is only accessible from the 5P3/2 F=3 intermediate state. We measured an oscillation period ofφ2π =14.5×104 fs2 , very close to the theoretical value of φ2π =14.3×104 fs2 . Notethat if transitions via the 5P1/2 states contributed to the signal, there would also
be a much shorter oscillation period of φ2π =2.4×103 superposed on the result.
To further demonstrate the capability of using a comb for high-resolution
coherent control we conducted a ﬁnal chirped pulse experiment in which we signiﬁcantly reduced the nonlinear absorption. As explained previously the phasedependence as a function of intermediate state detuning predicted by Eq. (5.17)
results in destructive interference. Speciﬁcally, recall that for a transform limited
pulse the two-photon amplitude from a comb pair detuned below an intermediate
state is canceled by an equal and opposite amplitude from a pair detuned symmetrically above the intermediate state. This prediction is especially valid forour choice of two-photon transition, 5S1/2 F=2 to 5P3/2 F=3 to 5D5/2 F=4. As was
discussed in section (4.1) this transition has a closed excitation pathway and istherefore a reasonable approximation to a three-level system, if we can ignore themF sublevels. So for a transform limited pulse all of the excitation is essentially
due to only those two modes that are exactly resonant with a transition step. In

124this experiment we demonstrate phase sensitive cancellation of the excited state
population due to this interference eﬀect.
By shifting the oﬀset frequency fo of the comb by −fr /2 to -31.86 MHz, twophoton resonance is maintained, but there is no mode resonant from 5S1/2 F=2 to
5P3/2 F=3. All mode pairs are symmetrically detuned around the intermediateresonance by fr × (1/2 + N), where N is an integer from zero. Figure (5.7) shows
the resulting 5D population as a function of chirp between the range of about 75,000 to 150,000 fs2 . Like our previous chirped excitation results the theoretically
predicted value is shown as a solid red line and the measured data points are
symbols. The ﬁgure inset is a diagram depicting the detuning of comb mode pairsthat are two-photon resonant but symmetrically detuned around 5P3/2 F=3. There
are two reasons why the signal is reduced in this experiment from our previous
chirped excitation results shown in Fig. (5.5). First and foremost, is that theresonant enhancement of the two-photon transition is signiﬁcantly reduced. For
example, the ratio of the resonant enhancement from a mode pair directly on
resonance to that of a mode pair detuned by 100 MHz is about 1000. The second
reason for a reduction in the signal is the phase sensitive cancellation we areinterested in demonstrating. From Fig. (5.7) it can be seen that by changing
from zero to about 25,000 fs2 the signal is increased ﬁve fold, a clear signature of
the phase sensitive destructive interference.

125

Figure 5.7: Steady state measured signal and calculated excited state population versus chirp for a comb tuned symmetrically about the intermediate state
resonance.

Chapter 6
High-Resolution Coherent Control: Discrete Phase

6.1

Spatial Light Modulator Pulse Shaping
In this section the design and operation of the spatial light modulator (SLM)

based pulse shaper is described. There are several crucial calibrations that must
be conducted on both the SLM itself and the pulse shaper as a whole. It should be
noted that these calibrations greatly inﬂuenced the results of our coherent control
experiment. Although mundane, and perhaps obvious to some, the alignment andcalibration procedures presented in this section constitute a great deal of the thenecessary considerations for obtaining high quality coherent control results.
The design of our SLM based pulse shaper is shown in Fig. (6.1). It is basedon a folded 2f-2f conﬁguration with the SLM liquid crystal element at the Fourier
plane. There are four optical elements shown on each side of the pulse shaper
in Fig. (6.1). There is a ﬂat mirror used to divert the beam to the grating, thismirror is on a removable mount such that the beam can pass directly though twoirises for alignment. A 1200 grooves per mm grating is used to diﬀract the lightto a pickoﬀ mirror. The pickoﬀ mirror is used to fold the optical path, this saves
space and allows the grating to be adjusted for its optimum diﬀraction eﬃciency.
Light is reﬂected from the pickoﬀ mirror to the spherical mirror with a focal lengthof 25 cm, used to focus into the SLM. There is a f/2 length distance between thegrating and pickoﬀ mirror, f/2 distance from the pickoﬀ mirror to spherical mirror,

127
and one focal length from the spherical mirror to the center of the SLM element.
Before the pulse shaper we used a telescope to expand the Ti:Sapphire beam toapproximately 3.5 mm diameter, resulting in a beam radius in the Fourier planeof about 30 µm. Ideally the spacing and angles of the elements on either sideof the SLM are equal. Note that the grating, spherical mirror, and SLM are all
mounted on micrometer translation stages for ﬁne tuning.
6.1.1

Alignment Procedure

Proper alignment of the pulse shaper optics is essential for ensuring that thebeam exiting the shaper has as little spatial chirp as possible. Although it is inprinciple simple to set the distances and angles of the optics on either side of the
SLM to be equal, in practice this proves to be more diﬃcult. Even with carefulmeasurements of all the distances and beam heights, after the initial alignment, I
found the beam exiting the pulse shaper to generally have a signiﬁcant amount of
spatial chirp and astigmatism. So it is useful, to explain the alignment procedure
we used.
There are two alignment irises mounted on the same optical breadboard as
the pulse shaper at equal heights. The ﬁrst step is to ensure the input beam isaligned to these irises without passing through the pulse shaper. This is done byremoving the two ﬂat input/output mirrors. Next replace the ﬂat mirrors, theﬁrst mirror is angled to direct the beam to the grating with a shallow downward
slope, such that the beam is diﬀracted at the same angle to the pickoﬀ mirror,
but not clipped by the mirror.
For this part of the alignment it is best to aperture the input beam to a small
diameter and ﬁlter the center wavelength with an optical interference ﬁlter. The
grating rotation angle should be adjusted such that the diﬀracted beam is almost
reﬂected back on itself, but tilted slightly down such that the full, unapertured

128

f/2
f/2
f/2

Figure 6.1: Spatial light modulator in a folded 2f-2f geometry pulse shaper. The
input beam is expanded prior to entering the pulse shaper to ∼3.5 mm diameter, however it is shown as a thin line. The broad line represents the spectrally
dispersed beam after the grating. The dispersed beam is picked-oﬀ with a mirror
and diverted to the spherical mirror to focus into the SLM elements. Note thatthe pick-oﬀ mirrors are below the beam sent to the SLM.

129
mode, is incident on the center of the pickoﬀ mirror. If this is done properly
the full mode will not be clipped by the pickoﬀ mirror, but all vertical angles
are kept to a minimum. The pickoﬀ mirror is aligned to reﬂect the beam 90o in
the horizontal plane and vertically upward such that the beam hits the sphericalmirror at the the same height as the input beam at the ﬁrst iris. The sphericalmirror should be angled slightly upwards such that the beam after reﬂection isparallel with the table.
At this point the SLM position can be roughly adjusted so that liquid crystal
elements are in the focus of the beam and roughly centered such that pixel number∼160 corresponds to the center wavelength. It is normal for the full spectrum at
the SLM to be rotated about the propagation direction such that each wavelength
is at a slightly diﬀerent height, this is due to the diﬀerent heights at which thewavelengths are reﬂected from the pickoﬀ mirror. One half of the shaper should
be roughly aligned after this procedure and the output half can be aligned in thesame fashion to be a mirror image.
After the previous rough alignment is completed the beam should exit thepulse shaper at a ﬁxed height through the alignment iris. Assuming all the distances were initially set properly the interference ﬁltered beam should be collimated, if it is not the second spherical mirror should be adjusted via micrometer.
The ﬁnal step in the pulse shaper alignment is to reduce the spatial chirp ofthe output mode. This is best accomplished by walking the vertical and horizontal
angles of the second grating and pickoﬀ mirrors with the full spectrum passing
through the shaper. To assist in this step and ﬁne tune the spatial chirp it is very
useful to observe the mode via a CCD camera in the focus of a lens. We simply
picked oﬀ the beam after the lens used to focus into the MOT and diverted it toa camera. The idea is to walk the vertical and horizontal angles of the second
pickoﬀ mirror and grating, it may be necessary to adjust the last ﬂat mirror to130
ensure the beam exits the center of the iris. With this adjustment combination it
should be possible to obtain a Gaussian mode on the CCD. Prior to alignment it
generally appears as a larger rotated ellipsoid due to astigmatism.
Once a small round spot is obtained, check for spatial chirp by blocking
narrow spectral regions directly in front of the SLM. For example, as a screwdriveris passed through the spectrum generally a tilted dark line will appear on thecamera, this is due to spatial chirp. Using the same degrees of freedom it should
be possible to make the entire mode on the camera dim uniformly as diﬀerent
spectral regions are blocked, meaning all wavelengths focus to approximately thesame spot. Although we never tried this idea, it might be best to use a frame
grabber with the camera and check the overlap of the modes corresponding to
diﬀerent wavelengths. This could be done using the SLM directly between crossed
polarizers to select diﬀerent spectral regions.
6.1.2

Calibration Procedure

We used a Jenoptik SLM-S320 SLM with a near-IR anti-reﬂection coating in
our pulse shaper. This SLM consists of 320 separately addressable liquid crystal
elements. Each element is a a rectangle of 97 µm width and 13 mm height. There isa gap between each element of 3 µm width, in this region there is always a constantphase retardation. Both the input and exit faces of each liquid crystal element
are coated with thin optically transparent electrodes. When there is no voltage
applied to the electrodes the liquid crystals are all aligned vertically, meaning
the long crystal axis is vertical. Under an applied voltage these crystals rotate,
the higher the applied electric ﬁeld the more they align to the DC electric ﬁeld
direction. This rotation causes a change in birefringence of the crystal, and thus
a change in the phase retardation a vertically polarized optical ﬁeld experiences.
There are two voltage operating ranges for the SLM voltage, low is from 0 to 5

131volts and high is 0 to 8 volts. Each voltage range is divided into 4096 steps called
the voltage count, so 0 to 4095 voltage count covers the entire range. Within justthe low range the phase retardation may be varied approximately 12 radians overthe entire spectral width of our pulses, however, 2π radians is of course suﬃcient.
So we set the SLM, via software, to the low range for our experiments to give ahigher phase step resolution.
In the alignment section the SLM position was roughly adjusted such thatthe middle of the pixels corresponds to the center of the Ti:Sapphire spectrum.
Next we use the repump laser to align the position of pixel 160 (middle pixel)
with respect to the input beam spectrum. The repump laser is conveniently
locked to the 87 Rb 5S1/2 F=1 to 5P3/2 F=2 transition at approximately 780.25 nm,
so it serves as a frequency reference. For this calibration the repump laser wasdiverted into the pulse shaper colinearly with the Ti:Sapphire. It is not necessary
to mode match the two lasers, the repump laser need only be collimated and havea diameter large enough that the focal spot is ideally less than a pixel width.
Although normally the SLM is operated as a phase mask with no wavelength
dependent attenuation, it is possible to use the voltage controlled birefringenceto operate as amplitude mask. This is done simply by placing crossed polarizers
around the SLM element, one at +45o with respect to vertical and the otherat -45o. Note that the polarizers should be placed directly before and after the
SLM to avoid any polarization rotation due to reﬂections from other optics. The
transmitted repump laser power is monitored on a power meter at any point after
the second polarizer, and the Ti:Sapphire is blocked for this calibration.
Using a simple LabVIEW program written for this purpose the phase of avariable width region of the SLM pixels can be controlled. It is generally best
to determine the voltage counts corresponding to maximum transmission andattenuation by setting all pixels to a uniform voltage, this will determine the132best case scenario for operating the entire SLM as an on/oﬀ switch. Once thesevoltages are known a narrow region of the mask around pixel 160, for example
pixels 145-175, should be set to attenuate while all other pixels set to transmit.
By repeatedly adjusting the SLM position via the micrometer and reducing thewidth of the attenuating region the majority of the repump power can be blocked
by only pixel number 160. This procedure allows one to determine that pixel
number 160 corresponds to a wavelength of 780.25 nm.
The position of the SLM with respect to the focal plane of the sphericalmirror is not yet calibrated. There are two ways to do this, either by adjusting
the SLM distance from the spherical mirror by hand or adjusting the mirror via
micrometer. Generally it was suﬃcient to adjust the SLM position by hand, thisalso avoids misalignment of the pulse shaper when adjusting the mirror position.
When the SLM is positioned at the focus of the spherical mirror the attenuation
will be maximum, because more of the focused repump laser mode ﬁts on pixel
number 160. It may be necessary to do one more iteration of the previous calibration to ensure that pixel 160 still corresponds to 780.25 nm after the SLM has
been moved by hand.
One of the most important calibrations is to determine the phase retardation
versus voltage count at the operating wavelength. The manufacturer provides thisdata measured at a wavelength of 633 nm. I feel it is necessary to point out for
future users of this SLM that the operating manual states the phase retardation
at any wavelength is calculable given the measurement data at 633 nm. The
argument proceeds as follows. Clearly for a given index of refraction diﬀerent
wavelengths have varying retardations given by,∆φ(V, λ) =2πd∆n(V, λ)λ(6.1)

where V is the voltage applied to the pixel that wavelength λ passes through,133
and d is the thickness of the liquid crystal element, 10 µm in our case. The ITO
electrodes, glass plates, and anti-reﬂection coating of course cause an extra phaseretardation, however, it is constant versus voltage. The manufacturer claims thebirefringence of the crystal has approximately constant wavelength dependence,
so the change in refractive index may be rewritten as,∆n(V, λ) = ∆n(λ)f (V )(6.2)

where the function f (V ) contains all the voltage dependence. At a voltage of 0
all the liquid crystals are oriented vertically and the resulting refractive index isa maximum, at this voltage f (V ) is taken to be unity. From measurements of theindex of refraction versus wavelength an empirical formula for the index dispersion
is found to be,∆n(λ) =∆no λ
λ2 − λ2o(6.3)

with ∆no =0.2002 and λo =327.44 nm. It is then possible to calculate the retardance at any voltage and wavelength using,∆φ(V, λ) = Γ(V )mλm ∆n(λ)λ ∆(λm )(6.4)

where Γ(V ) is the given measurement of phase versus voltage at 633 nm, λm isthe calibration wavelength of 633 nm, and λ is the wavelength of interest. We
found that using this method to calibrate the SLM phase versus voltage at ourwavelength of 780 nm is too inaccurate.
Determining the phase retardation at 780 nm directly via measurement
proved to be a much more accurate calibration technique. This was done byoperating the SLM as an attenuator between crossed polarizers. Unlike the previous calibration of the pixel versus wavelength, for this measurement high qualitypolarizers must be used and oriented at exactly ±45o from vertical. We conducted
this calibration two ways. The ﬁrst was to use the repump laser again and mea-

134
sure the transmitted power via a single photodiode, giving a calibration at one
wavelength.
To check for variations over the bandwidth of our Ti:Sapphire spectrum asecond technique was used in which the attenuation versus voltage was measured
via an optical spectrum analyzer. Software was written to measure the spectrum
power versus voltage and extract the full phase retardation across the pulse bandwidth. In principle we could have used this data directly to calibrate the SLM,
however, we found using Eq.(6.4) in conjunction with the repump laser measurement was suﬃcient over the 55 nm bandwidth of our spectrum. The measured
transmission curve, normalized to unity, and the extracted phase dependence versus voltage count using the repump laser technique is shown in Fig. (6.2). Notethat this was measured with SLM operating in low voltage mode. A speciﬁc example illustrating the necessity of this calibration is shown in Fig. (6.8) of section(6.2) using the results of the coherent control experiment directly.
We have discussed some of the calibration necessary to change the spectralphase of the Ti:Sapphire pulses with the SLM based pulse shaper. However, for
our coherent control experiment we wish to know the actual spectral phase of thepulses at the Rb atoms. Dispersion due to optical components such as lenses, thePockels cell, and the pulse shaper must be compensated for by the SLM phasemask to ensure the pulses at the MOT have a ﬂat spectral phase. The amountof dispersion possible with the SLM phase mask is much less than the grating
pulse stretcher used in section (5.2). It is limited by the use of a pixelated phasemask to approximate a smooth dispersion function. For example, if the phaseversus wavelength varies by 2π radians between adjacent pixels the approximation
completely breaks down.
The two-photon photocurrent was measured on a GaAsP photodiode versus
the second, third, and fourth order dispersion applied to the SLM. More speciﬁ-

135

Figure 6.2: Measured transmission and extracted phase versus SLM voltage countusing the repump laser at 780.2 nm.

136
cally, to search for a global maximum in the photocurrent and thus the shortest
pulse, a two dimensional map versus both the second and third order dispersion
was measured. Figure (6.3) is an image constructed from the two-dimensionalscan, the lighter areas correspond to higher photocurrent. One particularly important feature is that there are local maxima, so it is necessary to survey a
suﬃciently large region to ﬁnd a global maximum with high conﬁdence. In ourcase we varied the second order dispersion from about 1000 fs2 to -8000 fs2 andthe third order from 100,000 fs3 to -250,000 fs3 . The global maximum was foundand set on the SLM to be -4350 fs2 and -74,000 fs3 . We also checked for fourthorder dispersion and found an optimum value of -37,500 fs3 with the second andthird orders ﬁxed. Considering the negligible amount of fourth order dispersion
we can conclude that this method is suﬃcient to ﬁnd the shortest pulse. Notethat this calibration neglects any spectrally sharp dispersion that may occur, forexample due to the rubidium atoms themselves.
Once the spectral phase of the pulses is adjusted to be nearly ﬂat at the
MOT the spectral amplitude must be calibrated as well. One of the calibration
techniques is to use the SLM as a variable attenuator by placing the liquid crystal
element between crossed polarizers and varying the birefringence. Nominally the
input polarization is vertical, this is set by the PBS at the exit of the Pockels cell.
However, due to the fact some of the grating and mirror reﬂections in the pulse
shaper occur with both s- and p-polarizations there is some ellipticity induced in
the polarization, due to slightly diﬀerent phase shifts between the two polarization
projections upon reﬂection. This is particularly true for the dielectric pickoﬀ
mirror. The slightly elliptic polarization of the Ti:Sapphire light will become
more elliptic after passing through the liquid crystal of the SLM and in general
any subsequent optics before the MOT.
For this experiment we used a PBS cube as a beam splitter in conjunction

137

Figure 6.3: Map of two-photon photocurrent versus second and third order dispersion applied to the SLM.

138
with a zero order half-wave plate before the MOT two create two equally balanced
probe beams. Recall that to obtain a ﬂat spectral phase at the MOT requires
dispersion compensation, which is done via a spectral phase mask on the SLM.
Figure (6.4)(a) shows the spectrum measured on an OSA placed in one output
port of the PBS before the MOT with just dispersion compensation applied to
the SLM and one of the phase masks used in section(6.2). The details of thephase mask are not important at this point, only the fact that the spectrum
changes when the phase mask applied to the SLM is changed. This is of course
a problem for conducting a coherent control experiment in which we desire tochange only the spectral phase of the pulses. Placing a polarizer before the halfwave plate remedies this problem to a large extent, see Fig. (6.4)(b). There issome residual reshaping of the spectrum that occurs due to the pulse shaper, forexample the plateau at 770 nm, however we were unable to determine the cause.
One remaining unavoidable source of spectral amplitude modulation as a functionof the SLM phase mask is due to diﬀraction. Diﬀraction occurs for wavelengths
that have a focal spot covering two adjacent pixels with diﬀerent phases. For
example, if half the mode undergoes 0 phase shift and half π radians phase shift
the wavefront will be distorted from that of a Gaussian mode and therefore diﬀract
diﬀerently.

139

Figure 6.4: (a) Power spectrum of Ti:Sapphire measured on one exit port of
the PBS before the MOT versus applied SLM phase mask (no polarizer between
SLM and PBS) (b) Power spectrum measured the same as in (a) with a polarizer
between the SLM and PBS before the MOT.

140
6.2

Four-Level Interference Experiment
In our ﬁrst coherent control experiment we demonstrated the eﬀects of fre-quency chirp on a three-level system. The quantum interference that occurred
was due only to the diﬀerent optical modes of the comb. In this experiment we
use the high-resolution of the comb to excite two diﬀerent two-photon transitions
to the same ﬁnal state with diﬀerent resonant intermediate states. This type oflevel structure is referred to as a closed-loop diamond conﬁguration, see Fig. (6.5).
There have been several theoretical studies of close-loop four-level systems predicting the optical phase sensitive response of the excited population in double-Λ[90, 91] and diamond conﬁgurations [92, 93, 94]. Applications of close-loop fourlevel systems have been proposed to enhance and control four-wave mixing [95, 96],
ampliﬁcation without inversion [97], and lasing without inversion [98] for cw-VUV
lasers. This type of level sub-structure can also be of importance to studies ofmolecules, for example the sign of the product of dipole moments is of importance for two-dimensional spectroscopy [99], for which there are molecular states
excited in a closed-loop conﬁguration. Controlling the interference of two twophoton transitions can also be used for changing molecular dissociation pathways[100].
There have been a limited number of experiments done using cw-lasers to
excite a closed-loop four-level system. For example, to control population trapping[101], and to phase correlate independent lasers [102]. However, these previous
experiments were severely limited by the optical bandwidth of their lasers. The
bandwidth was limited by the phase modulation frequency of a cw-laser via anelectro-optic modulator. In our comb experiment we demonstrate phase controlof the two-photon absorption with a visibility of 69% using resonant wavelengths
spanning 32 nm, far surpasing the bandwidth of previous experiments.

141

5D3/2F=1762 nm

776 nm

5P3/2F=2
5P1/2F=2
5P1/2F=1794 nm

780 nm

5S1/2F={2,1}
Figure 6.5: Relevant 87 Rb level subsystem for the diamond coherence experiment.
The dotted lines indicate an extra oﬀ-resonant transition via the 5P1/2 F=1 state.

142
It is possible to ﬁnd two combinations of fr and fo , both resonant with afour-level diamond conﬁguration subset of 87 Rb levels. In particular, by tuning fr =100.59660605 MHz and fo =+16.94 MHz the four resonant states are:5S1/2 F=2, 5P1/2 F=2, 5P3/2 F=2, and 5D3/2 F=1. This choice of levels is well suitedfor our coherent control experiment because the diﬀerent transition wavelengths
are easily resolved with our SLM pulse shaper. Furthermore these two resonant
two-photon transitions can be nearly equal in amplitude for a suﬃciently broad
pulse spectrum. Figure (6.5) shows a simpliﬁed energy level diagram of the resonant four levels with the corresponding transition wavelengths indicated by solid
red lines. This ﬁgure also shows the intermediate state 5P1/2 F=1; the nearest
comb mode is detuned about two linewidths from this state, however, the importance of this transition path will be demonstrated in our subsequent calculations.
A slightly shifted value of fr =100.59660525 MHz with the same fo createsa diﬀerent four-level diamond consisting of the same intermediate and excitedstates but from the 5S1/2 F=1 ground state. All 7S and 5D states with the exception of 5D3/2 F=1 are detuned at least 6 MHz for either choice of fr . For bothfour-level subsystems the corresponding resonant wavelengths are approximately:
794 nm, 762 nm, 780 nm, and 776 nm. The ﬁrst two wavelengths correspondto the transition via 5P1/2 F=2 and the second two via 5P3/2 F=2. So the largest
separation between resonant wavelengths is 32 nm, the diﬀerence between the
wavelengths resonant with the 5P1/2 F=2 transition. We would like the electricﬁelds strength of the resonant modes to be approximately equal so for this experiment the Ti:Sapphire was operated with a spectrum centered at 778 nm witha spectral width of about 63 nm FWHM, the largest we could achieve. Recall
that the pulse shaper reshapes the spectrum somewhat, as seen in Fig.(6.4). So in
our calculations we used electric ﬁeld values derived from this reshaped spectrum
measured directly before the MOT.

143
The data collection procedure and general experiment setup is very similar
to our previous experiments. Rubidium atoms are cooled and trapped in a MOT
for 6.5 ms. The trap and repump lasers, and magnetic ﬁeld, are then switched oﬀ.
To allow suﬃcient time for the magnetic ﬁeld to turn oﬀ there is a 3 ms delay.
After 3 ms the Pockels cell is switched on and transmits the Ti:Sapphire laser
pulses through the pulse shaper and to the Rb atoms for 0.5 ms. This sequence
results in a total MOT operation plus probing cycle time of 10 ms, and is repeatedcontinuously so that we retrap and probe the atoms 100 times per second. During
the 0.5 ms probing time the cascade ﬂuorescence at 420 nm is measured via a
PMT and counted using a SR430 multi-channel photon counter. The SR430 multichannel photon counter was operated with 1024 separate channels (or bins), eachwith a temporal width of 640 ns and triggered simultaneously with the Pockels
cell. All data points presented in this section were accumulated over 4000 MOT
cycles, or 40 s per point in terms of experiment operation time.
The Ti:Sapphire probe beam was split into two paths using a PBS cube
and a zero-order half wave plate just before the MOT. This allowed us to eithertransmit all the power into one probe beam, or balance the power between two
counter-propagating probe beams. A second zero-order half wave plate was placed
in one of the two paths to rotate the polarization 90o, so that the atoms are excitedby two horizontally polarized, equal intensity, counter-propagating probe beams.
For this experiment one of the two probe beam paths passes through an adjustable
optical delay line, this was set such that the optical path from the PBS to thecenter of the MOT is equal with an estimated uncertainty of about ±5 mm.
We demonstrate coherent control of the excited state, 5D3/2 F=1, population
by adjusting the phase of one resonant path of the diamond with respect to theother. This demonstration was accomplished using the phase mask shown inFig.(6.6) to apply a variable phase, denoted as ∆Φ, to the spectral region from144
772 nm to 784 nm. The phase shifted spectral region covers the comb modepair resonant with the 5S1/2 F=2 to 5P1/2 F=2 to 5D3/2 F=1 transition. It also
covers the spectral region up to 4 nm shorter than the resonant wavelength 776
nm and 4 nm longer than 780 nm. By choosing a large region over which toadjust the phase we can assume all comb mode pairs with an intermediate state
detuning of up to ∼4 nm acquire the same phase, ∆Φ. This is important for thisexperiment because we would like to neglect the comb mode pairs detuned froman intermediate resonance, which can only be done if the net amplitude from thesepairs cancels to zero. Note that although it is not shown in Fig.(6.6) the SLM
is always operated with the necessary dispersion to compensate for the optics inthe beam path from our previous calibration. So the actual spectral phase of thepulses at the Rb atoms is ideally just due to the phase mask shown in Fig.(6.6).
It is useful to cover some of the initial results of this experiment to demonstrate how we arrived at our ﬁnal data. For this experiment, in particular theroad to the ﬁnal results is most useful to an experimentalist. One of the ﬁrst
aspects of the data to optimize is the fringe visibility as we varied the phase step∆Φ. Initially we used a two-channel photon counter and integrated the counts
for relatively long durations, between 100 µs and 500 µs per MOT cycle. During
this time we also only probed the atoms from one direction, so that all atoms
are excited by the same exact pulse shape, something we will discuss later. The
important point is that when probing from one direction the atoms are pushed oﬀ
of resonance, as we expected from our experience with radiation pressure. However, we noticed that the fringe visibility varied dramatically as a function of our
photon counting window width. The reason for this proved to be a surprise.
Figure (6.7) shows the signal versus phase step ∆Φ versus time, acquired
using the SR430 multi-channel photon counter. At short times, less than 100 µs,
we measure a clear fringe with a peak near ∆Φ=0 radians. At slightly longer

145

Figure 6.6: Spectrum of a pulse (directly from Ti:Sapphire) and the phase mask
used in the four-level coherent control experiment. Red arrows indicate the positions of the resonant transition wavelengths with respect to the phase step of ∆Φradians.

146
times the signal rapidly drops to near zero and remains there for about 150 µs,
as expected. The surprising feature is that at about 300 µs we again measurea fringe with a smaller signal but clearly shifted in phase, by almost ∆Φ=90o .
Clearly integrating the signal versus time will result in a reduced fringe visibility
for times longer than about 300 µs, because it is in fact two separate fringes
that are completely out of phase. Keeping in mind that we are accelerating theatoms in one direction as we scatter photons the comb fr in the frame of theatoms appears on average to be reducing versus time, or more accurately versus
velocity. This is how we discovered the second fr resonance condition mentioned
previously, which happens to correspond to excitation from the 5S1/2 F=1 groundstate. However, this does not explain the obvious phase shift between the two
fringes, we defer that discussion until we present the ﬁnal fringe results and amore complete analysis of the transitions.
Still with the intention of increasing the fringe visibility we considered theeﬀect of the hot Rb vapor in the MOT chamber. Although the comb is tuned tobe resonant with a speciﬁc subset of four Rb levels, this is only true for the atoms
that are cooled in the MOT to have negligible velocity in the lab frame. Due tothe fact that the comb mode spacing of 100 MHz is much less than the Doppler
width of the hot Rb atoms, all allowed transitions within the bandwidth of the
comb will in general be excited in these hot atoms. For example, the strongest
two-photon transition in terms of dipole moments is the 5S1/2 F=2 to 5P3/2 F=3
to 5D5/2 F=4 transition. In terms of the product of dipole moments, meaning
the dipole moment from 5S to 5P multiplied by the dipole from 5P to 5D, thistransition is about 9 times as strong as either transition path in the intended
four-level system.
The transitions in the hot atoms will in general yield a constant backgroundsignal that does not vary as a function of the applied ∆Φ. For example, none of theSignal (a.u.)

147

0.5
0
1.0

300
400

e (P
s)

Ph

Tim

500
600

a se

200

Ste
p

100

(S)

0.0

1.5

Figure 6.7: Measured single beam signal versus SLM phase step versus excitation
time. The initial fringe is due to transitions from 5S1/2 F=2. At longer times
the atoms are Doppler shifted onto resonance with the same transition but fromthe 5S1/2 F=1 ground state. Notice the clear phase shift between the two fringes,
clearly averaging the signal too long can reduce the fringe contrast.

148
transitions to 5D5/2 states can be accessed via the 5P1/2 intermediates states, andtherefore will not be aﬀected by shifting the phase of the 5P3/2 transition paths.
It was easily veriﬁed that indeed the hot atoms were contributing a signiﬁcantbackground signal to our initial results by simply measuring the signal without
ever forming a MOT.
We tried to reduce the signal due to these hot atoms by two methods.
First, we modiﬁed the optics before the PMT to form an image of the cooled
atomic cloud onto an aperture that was adjustable via micrometers, thus blocking
light originating from outside the cooled atom cloud. This was only moderately
successful due to the fact that in the 3 ms after cooling the atoms, but before
probing them, the hot atoms have ample time to redistribute throughout thechamber. So when we actually probe there are hot atoms distributed throughout
the cold atomic cloud. This forced us to operate the Rb getter at a signiﬁcantly
lower current, thus lowering the density of hot Rb vapor. At a getter current of
about 2.5 Amps, really only relevant for our speciﬁc setup, we were able to reduce
the signal from the hot atoms. At this getter current the signal from the hot
atoms was not the dominant contribution to the fringe minimum, and there was
suﬃcient signal from the reduced number of trapped atoms.
Now that we have an understanding of how to increase the fringe visibility
we turn our attention to the fringe period. Due to the fact the phase mask covers
both resonant wavelengths of the transition through 5P3/2 F=2 the fringe period
in terms of ∆Φ is expected to be π radians. Ruling out any physics of the lightmatter interaction that could give rise to a period other than π, we attribute any
measured deviation from this periodicity to miscalibration of the SLM. Figure(6.8) shows the fringe measured with two diﬀerent SLM phase versus voltage
calibrations. The details of this calibration procedure were discussed in section(6.1.2). Initially we used the manufacturers SLM calibration done at 633 nm and149
used Eq. (6.4) to determine the phase versus voltage at our desired wavelengths
centered around 778 nm. With this calibration technique the measured fringe
period was approximately 1.15×π radians. Using direct calibration of the SLM
at 780 nm with the repump laser we repeated the measurement and recovered amuch improved fringe period of (1 ± 0.01)×π radians.
We now proceed to discuss our ﬁnal data sets demonstrating the ability tocontrol the excited 5D3/2 F=1 population via our SLM phase mask. Figure (6.9)
shows four fringes taken under diﬀerent experimental conditions to demonstratesome of the relevant physics. The fringes in the top two panels labeled as (a)and (b) were measured using traveling wave excitation, meaning the atoms wereprobed from only one direction with the Ti:Sapphire. The bottom two panels
labeled as (c) and (d) present the results under what we refer to multi-mode
standing wave excitation, meaning if we probe the atoms with counter-propagating
beams.
As mentioned previously, there are two diﬀerent resonance conditions obtained using slightly shifted values of fr . The left panels were taken with fr
corresponding to excitation from the 5S1/2 F=2 ground state and the right panels using the fr for excitation from the 5S1/2 F=1 ground state. In the case of
traveling wave excitation we have discussed the reduced fringe visibility versus
probing time, due to the radiation pressure Doppler shifting the comb to anotherresonance. For this reason we only show the signal accumulated during the ﬁrst
20.48 µs of probing time, short enough to avoid signiﬁcant Doppler shifting.
After aligning the counter-propagating probe beams very well, we found thatthe atoms remain on resonance for about 250 µs, much longer than the ∼40 µs
for the traveling wave data. So in the case of excitation via counter-propagating
beams the atoms may be probed longer with essentially no reduction in fringe
visibility. Despite this fact, the data presented in the bottom two panels for

150

Figure 6.8: Measured fringe versus phase step, corresponding to ∆Φ in Fig. (6.6),
using SLM phase vs. voltage calibrations at 780 nm and 633 nm. The period of thefringe using the 633 nm calibration is too long, 1.15π radians. Using calibration
at our operating wavelength of 780 nm produced a much better 1.0±0.01π radians
period.

151
counter-propagating excitation correspond to only the ﬁrst 20.48 µs of excitation
for consistency. To avoid changing anything about the experiment except theexcitation scheme the traveling wave data was measured simply by blocking thecounter-propagating beam. So the Ti:Sapphire probe power used in the travelingwave case is half of the 75 mW used in the counter-propagating case. Both probe
beams were focused into the MOT by the same 750 mm focal length lens mounted
before the PBS cube used to split the probe beams. This results in about a 180 µm
diameter focal spot at the center of the cold atom cloud. In general we measure
the MOT atom number stability with a photodiode setup to image the MOT, it
is not necessary to calibrate for the absolute number of atoms. This photodiodemeasures the scattered trap and repump laser light during the last 1 ms of MOT
formation, and is a useful tool for normalizing any slow drift in atom number.
Each fringe in Fig. (6.9) is shown with a ﬁtting function of the form c1 +c2 cos(Φ + c3 )2 , with the free variable Φ, which is the phase step applied via our
SLM mask. After determining that the observed period was very nearly π, we
dropped any dependence in the ﬁt function on the period. The parameter c1
represents the fringe minimum, c2 is simply the magnitude of variation due to theinterference, and c3 is the fringe oﬀset phase from zero. Combining the c1 and c2
2parameters we can write the fringe visibility, deﬁned max−min, as 2c1c+c. In each
max+min2

panel of Fig. (6.9) the extracted fringe oﬀset and visibility is shown. Our best
measured fringe visibility is 69%. This is a signiﬁcant improvement from our intial
results which were as low as 25%, clearly the previously mentioned techniques for
improving fringe contrast were successful.
We have used two theoretical approaches to understand our results andcompare them with the predicted fringe visibility and phase oﬀset. The ﬁrst andsigniﬁcantly more complicated approach is to directly solve the Liouville equation
for the density matrix of the system under shaped pulse excitation. To this end

152

Figure 6.9: Measured signal fringes versus SLM phase for four excitation conﬁgurations. (a) Traveling wave excitation from F=2, (b) traveling wave excitationfrom F=1, (c) multimode standing wave excitation from F=2, and (d) multimodestanding wave excitation from F=1.

153
we initially developed a four-level model which incorporated only the four resonantly excited states. The driving ﬁeld in this model was constructed by taking
the Fourier transform of the measured spectrum, shown in Fig. (6.4), with aspectral phase given by the applied SLM mask mask. As a practical side notethe sharp steps in the pixelated phase mask were smoothed, this is a necessary
step to ensure the time domain ﬁeld falls oﬀ to zero in a reasonable amount oftime for our simulation. The results of this simulation were only partially successful. In particular the predicted fringe visibility was nearly perfect, approximately
99%, and the diﬀerence in phase oﬀset between excitation from 5S1/2 F=2 versus
5S1/2 F=1 was 90o . We will come back to the source of this phase oﬀset in thecontext of second-order perturbation theory.
Based on our prior observation of the fringe visibility being reduced whenaccumulating data from both ground states, due to the Doppler shift of the comb,
it stands to reason that both ground states should be included in the model.
To try and account for the reduced fringe visibility we observed compared totheory we expanded our density matrix simulation to include ﬁve-levels. The ﬁfth
level being the second ground state. First it must be pointed out that when the
comb is tuned to be resonant from the 5S1/2 F=2 ground state, transitions fromthe other ground state to the same intermediate states are only 6 MHz detuned,slightly oﬀ-resonance. So a non-perturbative model which takes into accountpower broadening may predict some eﬀects only in the presence of both groundstates. In fact this ﬁve-level model does predict a reduction in fringe visibility
as a function of ﬁeld strength, however, this occurs at slightly higher ﬁelds thanused in our experiment. To check this we measured the traveling wave fringes
with reduced powers, but the fringe visibility was not improved. At this point
we determined that perhaps other intermediate states needed to be included in
the model and turned to using a treatment based on second-order perturbation

154
theory.
The simplicity of using second-order perturbation theory makes it an ideal
method to model many diﬀerent simultaneous two-photon transitions. In our
experiment this is particularly true due to the fact that we are far from saturating
the 5D3/2 F=1 population. Furthermore the majority of the incoherent opticalpumping that occurs is due to relaxation of the population in the intermediate
states, and is therefore not phase sensitive. As a reminder the resonant two-photon
amplitude for absorption from two modes with one intermediate state is given by,
cgf ∝|En || Em | ei(φn +φm ) µgi µif
×i(ωgf − (m + n)2πfr − 4πfo ) + π∆νf
1+i(ωgi − 2π(nfr + fo )) + π∆νi1,i(ωgi − 2π(mfr + fo )) + π∆νi

(6.5)

where |EN( 1,2) | and φN1,2 are the magnitude and phase of the N1 and N2 modes,ωgf (ωgi ) are the transition frequencies from ground to ﬁnal(intermediate) states,and ∆νi(f ) are the intermediate(ﬁnal) state linewidths in Hertz. Equation (6.5)
is valid only for mode pairs that are two-photon resonant. The total two-photon
amplitude is then reasonably approximated by the summation of two-photon amplitudes for each transition path, for example in the four-level diamond there are
only two main transition paths corresponding to each intermediate state. Notethat we refer to these as the main transitions, it is also possible to have much
weaker transitions, for example absorption from 5P3/2 F=2 to 5D3/2 F=1 followed
by stimulated emission to 5P1/2 F=2.
One of the beneﬁts of using a comb for coherent control is that there aremany modes; in this experiment it allowed us to use four phase coherent modes
to excite the four-level diamond system. The remaining hundreds of thousandsof modes can lead to complications and must be considered. In particular we155show the 5P1/2 F=1 level in Fig. (6.5) and two modes resonant from either groundstate to the 5D3/2 F=1 excited state via this intermediate state. In our initial
analysis of the physics the two-photon amplitude due to this transition pathway
was disregarded because the closest modes to this state are detuned by nearly two
linewidths, about 12 MHz. Due to the fact this transition path has larger dipole
moments than the other two resonant paths it may not be neglected.
The information necessary for estimating a fringe phase oﬀset and visibility,
assuming no residual pulse chirp, is tabulated in table (6.10). The top section
is for transitions from the 5S1/2 F=2 ground state and the bottom for transitions
from the 5S1/2 F=1. It is important to notice that the angular part of the dipole
moment matrix elements for the 5S to 5P transitions are a function of the particular ground state. Corresponding to each single photon transition step the angularpart of the dipole moment matrix element is given by LmF F ||r̂||L mF F  , these
are labeled as µgi and µif in Table (6.10), for the 5S to 5P, and 5P to 5D transitions respectively. The next column in the table denoted ∆(MHz) is the detuningof the nearest comb mode for that particular transition from the intermediate
state. Using the dipole moments, intermediate state detuning, and measured
pulse spectrum, we calculated the corresponding two-photon amplitude. The lasttwo columns of the table are the magnitude (only relative values) and phase of theamplitude, assuming excitation by a transform limited pulse. Each intermediate
state is labeled by its mF quantum number, since we are exciting with linearlypolarized light with nominally zero magnetic ﬁeld we use q=0 selection rules.
Noting that transitions in which ∆F =0 there can be no mF =0 transition for q=0
we only need to include mF =0 states for excitation from 5S1/2 F=1. We do nottabulate the mF =-1 transitions because, although each step may have diﬀerent
dipole moment signs, the two-photon amplitude is the same as for mF =1 transitions. Concerning estimates of the fringe properties for transitions from 5S1/2 F=1,

156
in which mF =0,±1, we assume an equal population distribution between all mF
ground states.
Using the two-photon amplitudes in Table (6.10) we can estimate the total
two-photon amplitude due to transitions through the 5P1/2 states. For example,
in the case of transitions from the 5S1/2 F=2 ground state, the total two-photon
amplitude due to transitions through 5P1/2 states is given by,
c5P1/2 = 3.4ei(−7.6π/180) + 4.3ei(75.4π/180) .

(6.6)

The phase of this amplitude is 39.8o, the phase of the transition through 5P3/2 F=2
is -3.8o, so even for a transform limited pulse we expect the fringe peak to beooshifted by ∆Φ = 39.8 2+3.8 , or 21.8o. We attribute the extra phase shift beyond
this value, as seen in Fig. (6.9)(a), to residual pulse chirp. Within the scope of
this theoretical model the visibility is given by,
2|c5P1/2 c5P3/2 ||c5P1/2 |2 + |c5P3/2 |2(6.7)

Again using the values from Table (6.10) the predicted visibility is 82%, clearlymuch larger than the observed 69%. We do not know what sets the limit on
the measured fringe visibility, however we have determined that is not a trivial
background source such as stray light or hot atoms, due to the fact that thebackground level is present only with the Ti:Sapphire probing cold atoms. We havealso used our theory model based on four-order perturbation and all the Rb 5S, 5P,
and 5D states to estimate the population exited to all other 5D levels. However,this yields only a very small background contribution at our experimental ﬁeld
strength.
We can use the same estimation procedure for transitions starting fromthe 5S1/2 F=1 ground state. In this case the total two-photon amplitude for thediﬀerent mF states do not interfere and therefore are only added incoherently.

157

µ

µ

∆

θ

Figure 6.10: Relevant reduced dipole moments, detunings, and two-photon amplitudes for the 5-level system from both hyperﬁne ground states.

158
One unknown quantity is the population distribution between the diﬀerent mF
states. If we neglect population in the mF =0 states we calculate a fringe visibilityof 99% and a phase oﬀset of ∆Φ=78.2o . The majority of this phase shift isdue to sign of the angular part of the dipole moment matrix elements. Morespeciﬁcally the dipole moments for the resonant transitions from 5S1/2 F=2 are all
negative. However, in the case of excitation from 5S1/2 F=1 one of the four dipole
moments is positive. This is an example of the importance in considering not only
the magnitude but also the sign of the dipole moments in closed-loop excitation
schemes.
As mentioned previously there is an overall phase shift regardless of groundstate due to the chirp. However, we can make a comparison with experimentin terms of the diﬀerence in phase oﬀsets between excitation from 5S1/2 F=2 and5S1/2 F=1. In this case the agreement between our model and experimental results
is quite good. The theory prediction is 56o and the measured diﬀerence in the caseof traveling wave excitation is 54o . If we include the mF =0 transitions and assume
equal population distribution the fringe is shifted by about 8o and the visibility is
reduced to 92%. So the analysis based on second-order perturbation theory gives
quite a good understanding of the relative phase shifts between fringes.
So far we have only discussed the fringes measured under traveling wave excitation. Panels (c) and (d) in Fig. (6.9) show the fringe measured using counterpropagating beams. The fringe visibility in both data sets is clearly reduced fromthe case of traveling wave excitation. Panel (c) corresponds to excitation from the
5S1/2 F=2 ground state and shows a fringe with a visibility of 26%, compared to
68% for travling wave excitation. Similarly, panel (d) corresponds to excitationfrom the 5S1/2 F=1 ground state and shows a visibility of only 23%. Generally weonly use counter-propagating probe beams to balance the radiation pressure on
the atoms and reduce the net Doppler shift, which usually results in an increased

159
signal to noise ratio.
Unlike our previous spectroscopy experiments, we must now consider theeﬀect of standing wave electric ﬁelds within the atom cloud. Recall that we aligned
the path lengths of the two probe beams to be equal such that the pulses overlap
temporally and spatially in the center of the atom cloud. Figure (6.11) illustrates
the standing waves created by the two resonant frequencies for transitions via
5P3/2 , and similarly for 5P1/2 , as a function of position from the center of the cloud.
At the point in space where the pulses overlap temporally the average intensityof each comb mode is a maximum. In either direction along the probe beams theaverage intensity of a particular comb mode will vary spatially, just like a single
frequency standing wave. The important point is that the resonant wavelengths
used in our experiment have signiﬁcantly diﬀerent standing wave periods. So theelectric ﬁeld driving an atom is a function of the atom position with respect to thepoint in space where the pulses overlap temporally. For example, there are spatial
regions of the atom cloud in which there is no average electric ﬁeld at 780 nm.
In this four-level experiment we wish to observe the interference produced by all
four resonant comb modes, so if the average electric ﬁeld corresponding to any of
the four modes is zero the interference will not occur. However, it is still possible
to excite population to 5D3/2 F=1 if the electric ﬁelds for a pair of two-photon
resonant modes is non-zero. In spatial regions where this occurs, the 5D3/2 F=1
population, and thus the ﬂuorescence at 420 nm, will not be phase sensitive.
To expand our theory model to include the multi-mode standing wave eﬀect
we simply average predicted the signal over the spatial dimension of the atom
cloud. We did this over a distance of 0.5 mm corresponding to the atom cloud diameter, and assumed a uniform atom number spatially. The electric ﬁeld strengths
used in Eq. (6.5) were modulated spatially with a standing wave period appropriate to the particular comb mode. Again using the information from Table (6.10)

160

1.0

Product of Field Envelopes

P1/2 Path
P3/2 Path
Total

0.8

0.6

0.4

0.2

0.0
0

20

40

60

80

100

120

140

Distance (Pm)
Figure 6.11: The black line is the average product of electric ﬁelds two-photon resonant via the 5P1/2 intermediate state, corresponding to wavelengths 795 nm and
762 nm. Similarly, the red line is for transitions via the 5P3/2 intermediate stateand corresponds to wavelengths 780 nm and 776 nm. Notice that the two standing waves generated (black and red) have very diﬀerent periods, due to the larger
diﬀerence in wavelengths for the transition via 5P1/2 . The blue line is the productof the red and black lines, this represents the spatial regions, and extent to which,all four resonant wavelengths are present. Clearly there is a spatial dependence
not only on the expected 5D population (only two wavelengths required), but also
on the phase dependence of this population (all four wavelengths required).

161
we calculated fringe visibilities of 36% for excitation from the F=2 ground stateand 44% for the F=1 ground state.
This concludes the discussion of the four-level interference experiment. We
have demonstrated coherent control of the 5D3/2 F=1 population by tuning theinterference between two diﬀerent two-photon transition pathways. This interference is nominally due to only those comb mode pairs that are resonant or
near-resonant with an intermediate state. The relative phase shift of the interference fringes can be explained within second-order perturbation theory using threeintermediate states and six comb modes. This general scheme of enhancing andcontrolling a two-photon transition rate via multiple resonant or near-resonant
pathways can in principle be extended to incorporate many more intermediate
states. For example, this forms the basis of a proposal from our group to enhancea Raman type two-photon transition between highly excited and deeply boundmolecular vibrational states [103]. The proposal relies on forcing constructive interference between two-photon amplitudes, due to diﬀerent intermediate states,
to a speciﬁc target state.
6.3

Four-level Enhancement Experiment
In the previous four-level interference experiment, the two-photon transition

rate was solely due to the comb mode pairs near an intermediate state resonance.
The goal of this experiment is to enhance the total two-photon transition rateby incorporating the many thousands of comb mode pairs detuned from an intermediate resonance. Recall that for a transform limited pulse the two-photon
amplitudes from these modes pairs detuned from an intermediate state tend tocancel out, see Eq. (5.17) and the following explanation. This cancellation isdue to a relative phase shift of 180o in the two-photon amplitudes for mode pairs
detuned above versus below an intermediate state. To compensate for this 180o

162phase shift we use the SLM phase mask shown in Fig. (6.12).
This experiment follows closely the experiment in reference [104] in which it
was shown that the two-photon transition rate could be increased by pulse shaping
if a resonant intermediate state is present. In that experiment a single femtosecond
pulse from a Ti:Sapphire laser was used to excite all of the allowed 5S to 5P3/2 to
5D transitions in a Rb vapor cell. Due to the fact that the spectral resolution inthat experiment is limited by the single pulse bandwidth any intermediate state
covered by the spectrum is considered resonant. They show a seven fold increase in
the measured ﬂuorescence from the 5D states by applying a π2 phase step between
the resonant wavelengths of 780.2 nm and 776.0 nm. Clearly a signal increase bya factor of 7 is signiﬁcant and could potentially be used to improve the signal tonoise ratio in DFCS experiments. Especially because this outweighs the loss insignal incurred due to the power attenuation in the pulse shaper.
In our experiment, we use pulse shaping in a similar manner to increase thetwo-photon transition rate of the four-level system shown in Fig. (6.5), particularly from the 5S1/2 F=2 ground state. The fact we use a comb to resonantly excite
only the 5D3/2 F=1 hyperﬁne level changes the physics signiﬁcantly from the single
pulse case. This is due to selection rules and dipole moments, analogous to our
discussion of two-photon selection rules in chapter (4.3), subsection (4.3.3). In the
single pulse excitation of Rb the dominant transition is to the 5D5/2 F=4 hyperﬁnelevel due to the fact the dipole moments are larger than any other two-photontransition. Compared to either of the resonant transitions in our experiment, theproduct of the angular part of the dipole moments for the ﬁrst and second steps
to 5D5/2 F=4, is a factor of 10 larger. Therefore the signal due to a single pulse is
primarily from the 5D5/2 F=4 population. Furthermore, the only transition pathto 5D5/2 F=4 is via 5S1/2 F=2 to 5P3/2 F=3, due to the ∆F=0,±1 selection rule.
So in the application of Eq.(6.5) to calculate the total two-photon amplitude only163

πFigure 6.12: Spectrum of a single pulse (directly from Ti:Sapphire) and the phasemask used to enhance the two-photon transition rate (optimal 4-level mask). Red
arrows roughly indicate the positions of the resonant transition wavelengths withrespect to the phase steps of π radians.

164
one intermediate state is necessary regardless of detuning, when only 5D5/2 F=4
is resonantly excited. In other words a three-level model with the corresponding
dipole moments for the transition to 5D5/2 F=4 correctly predicts a two-photontransition rate enhancement by a factor of 7 for the single pulse experiment. This
is not the case for all two-photon transitions as we will show below.
In our comb experiment the number of possible intermediate states is four.
Table (6.1) shows the angular part of the relevant reduced dipole matrix elements denoted µgi and µif for the ground to intermediate and intermediate to
ﬁnal state transitions. Equation (6.5) can be used to sum over all possible twophoton transition pathways using the dipole moments and states in this table. In
the calculation of the two-photon amplitude the correct (measured) electric ﬁeld
strength and phase of each mode must be used.
Intermediate Stateµgi

5P3/2 F=2
5P3/2 F=1
5P1/2 F=2

1
6
−1
√
2 15
1
6

5P1/2 F=1

−1
√
2 3

µif
√2

−

15

5
1√
15 2
−1
√
2√30
6

5
2

µgi µif-0.0122-0.0061-0.0152-0.0761

Table 6.1: Angular part of the reduced dipole matrix elements for the four possible
transitions from 5S1/2 F=2 to 5D3/2 F=1, the ﬁnal column is the product of thegiven dipole moments.

In our previous experiment with the SLM pulse shaper the resolution was notimportant because we used a broad phase mask covering the resonant wavelengths
for the transition via 5P3/2 F=2, so knowing the exact wavelengths of each phasestep was not essential. However, in this experiment we would like the π phasesteps to occur as close as possible to 5P3/2 F=2 and 5P1/2 F=2. Our pulse shaper
has a per pixel resolution of approximately 150 GHz at 780 nm, meaning eachpixel corresponds to about 150 GHz of spectral width. This corresponds to the165
resolution at which we can set the phase steps in Fig. (6.12) via software.
For ﬁner resolution we mounted the entire SLM on a micrometer driven
translation stage. By translating the SLM through the spectrally dispersed beam
we are able to ﬁne tune the frequencies at which the π phase steps occur, however
not independently. The focal spot diameter with our given 3.5 mm input beam
diameter and mirror focal length of 22 cm is about 60 µm. This is about half
the pixel width so roughly speaking we may ﬁne tune the frequency of one phasestep with about a 75 GHz resolution. Given the ﬁxed per pixel resolution of thepulse shaper once the frequency of one phase step is ﬁne tuned the frequency ofthe second step is ﬁxed for a set phase mask. Unfortunately this means that if we
optimize the position of the phase step, around 5P1/2 for example, the position of
the step near 5P3/2 is also ﬁxed.
Using the phase mask shown in Fig. (6.12) the phase of the mode pairsthat are detuned below either the 5P3/2 or 5P1/2 sets of states gain an extra πphase shift. We used the repump laser method described in section (6.1.2) tocalibrate the wavelength corresponding to the center of pixel number 160. This
results in the phase step near 5P3/2 occurring at λ=775.984 nm and the resonant
wavelength eﬀected is λres,1=776.157 nm. Similarly for the phase step near 5P1/2
we have λ=761.706 nm and the resonant wavelength eﬀected is λres,2=762.103
nm. The diﬀerence in frequency oﬀsets between the phase steps and resonant
wavelengths is about 111 GHz, so this places a limit on how well we can optimize
around the two ﬁne-structure sets of states simultaneously.
The experiment was run similarly to our resonant four-level coherent control experiment. We used the same MOT timings, 6.5 ms load time, 3 ms formagnetic and MOT optical ﬁelds to be turned oﬀ, then probed for 0.5 ms. The
full Ti:Sapphire power of 70 mW was focused into the atoms with a 750 mm focal
length lens and used to probe the atoms from one direction only. Because we166
are only interested in the enhancement of the two-photon transition rate due tothe phase mask, we used a diﬀerential measurement technique. The atoms wereprobed for 1000 MOT cycles with the phase mask in Fig. (6.12) applied, thenfor 1000 MOT cycles with only dispersion compensation on the phase mask. Using the two-channel photon counter we measured probing times of 0-25 µs and
0-500 µs. Figure (6.13) shows the ratio of the signals with and without the phasemask, versus an arbitrary frequency oﬀset. Each frequency oﬀset corresponds toa diﬀerent micrometer setting, or equivalently position of the SLM within thedispersed spectrum. Using the fact that each pixel center is separated by 100 µm
and corresponds to a 150 GHz diﬀerent frequency we were able to calibrate thefrequency versus position.
We achieved a maximum enhancement ratio of 2.56 using the short probingtime data from 0 to 25 µs. Using a simple theoretical model which summed all
the possible two-photon amplitudes for each comb pair and intermediate state weestimated the enhancement versus frequency oﬀset. The theoretical enhancement
is shown in Fig. (6.14), it has a peak value of 2.85 using the phase mask from Fig.
(6.12) and the measured power spectrum from Fig. (6.4) with no diﬀraction loss.
There are two peaks in the theory results corresponding to enhancement of the
5P1/2 transitions (larger peak), and oﬀset by 120 GHz, the transitions throughthe 5P3/2 states are enhanced. We tried including diﬀraction loss to account forthe dip below unity enhancement seen in our measurements at about 300 GHz,
however this only reduced the peak enhancement ratio and did not change thecharacteristic shape signiﬁcantly. As seen in our results for longer probing times
from 0 to 500 µs the dip below unity is no longer present. The only diﬀerence that
we can attribute this to is the radiation pressure Doppler shifting the atoms onto
resonance from the 5S1/2 F=1 state. Why this would change the characteristic
shape of the enhancement versus frequency though is unknown.

167

µ
µ

Figure 6.13: Measured signal enhancement ratio versus SLM position for shortand long integration times. The x-axis is the frequency oﬀset of the SLM pixels
from some reference value.

∆νFigure 6.14: Theoretically predicted enhancement ratio assuming no diﬀraction
loss using second-order perturbation theory.

168
In this experiment we have demonstrated the ability to increase the twophoton transition rate from 5S1/2 F=2 to 5D3/2 F=1. This was done by using aphase mask to force constructive interference between the two-photon amplitudes
from comb modes pairs detuned from an intermediate state. Unlike the single
pulse excitation we have the ability to select a speciﬁc excited hyperﬁne level,
which in turn determines the possible intermediate states due to selection rules.
In our experiment we must consider four possible intermediate states.
This type of coherent control can be extended much like the previous resonant experiment in section (6.2) to optimize the total two-photon transition ratewith many intermediate states, for example in a molecule. One of the limitationsof our experiment is the SLM pulse shaping resolution. To truly optimize theSLM phase mask for a speciﬁc transition would require the ability to control thephase of each individual comb mode. The ideal phase mask would compensate forthe phase shift due to intermediate state detuning of each two-photon amplitude,
which varies from 0o for a comb pair on intermediate resonance, to ±90o for pairs
detuned ±fr and further. Due to the limited per pixel resolution of our SLM
pulse shaper, we are not able to make the phase of the amplitudes from modepairs detuned from intermediate resonance the same as those that are resonant.
Therefore with the phase mask used in this experiment the two-photon amplitude
via only intermediate resonant transitions is 90o out of phase with the two-photon
amplitudes from mode pairs detuned from intermediate resonance. For this reason
future experiments of this type would beneﬁt from using a pulse shaper with permode resolution, a feat that is already possible with high-repetition rate combs[105].

Chapter 7
Future Directions

In this thesis we have demonstrated the ability to use an optical frequency
comb to conduct absolute frequency measurements. All of the transitions were
excited directly by the frequency comb. Although the range of wavelengths usedin our experiments was quite wide, ∼50 nm, it is possible to vastly extend the
bandwidth of the comb.
Due to the extremely high peak intensities possible with a femtosecond
laser it is well suited for non-linear frequency conversion. For example, the supercontinuum generation in a micro-structured nonlinear ﬁber is a result of four-wave
mixing. The super-continuum spectrum is far wider than the original Ti:Sapphire
spectrum, recall we used two wavelengths separated by one octave for our fo measurement. This is just one simple example of how the frequency comb spectrum
may be extended for spectroscopic use at wavelengths that are diﬃcult to reachwith cw-lasers.
There has already been some experiments conducted using the principle of
DFCS at very short wavelengths. Using traditional non-linear frequency conversion techniques the group of Eikema has conducted spectroscopy of two-photontransitions in krypton and xenon with a burst of up to six phase-coherent pulses.
The ﬁrst demonstration of this technique used two-photon absorption at 212 nm in
krypton by a pair of phase coherent pulses [68]. The most recent example demon-

170
strates multi-pulse DFCS of the 5p6 1 So → 5p5 (2 P3/2 )5d[ 12 ]1 transition in xenon
at 125 nm, see reference [106]. The required VUV wavelengths were generated byﬁrst doubling a series of ampliﬁed Ti:Sapphire pulses in BBO, and subsequently
tripling them in either oxygen or krypton. Using this technique the authors wereable to scan fr and recover a transition linewidth as narrow as 7.5 MHz FWHM
after excitation by six pulses. One of the drawbacks of this technique is that only
a limited number of pulses were generated, limiting the comb mode linewidths.
In an eﬀort to extend the number of phase coherent pulses generated atvery short wavelengths researchers in the group of Ye have turned to using highharmonic generation. Instead of amplifying the phase-coherent pulse they arecoupled into a broad-band optical cavity to increase the peak intensity. Within
the cavity there are two spherical mirrors to focus the pulses into a xenon gas jet.
In the intense peak electric ﬁeld of the femtosecond pulses the xenon atoms undergo high-harmonic generation. Using this technique it was possible to generate
wavelengths as short as 114 nm, and could in principle be extended to shorter
wavelengths by increasing the peak intensity [107], also see reference [108]. We
have published a theoretical study on the use of short wavelength optical combs
for high-resolution two-photon helium spectroscopy [109]. An interesting proposalthat is closely related to DFCS, is to Doppler cool atoms at traditionally hard toreach wavelengths using a femtosecond frequency comb [110].
In the context of coherent control we have mentioned the possibility of extending our technique to molecules. The idea is to combine pulse shaping withthe comb, to enhance and control a Raman transition between a highly excited
vibrational state and a deeply bound vibrational state. Instead of forcing the constructive interference between only a few intermediate states like in our research,
we proposed to use many intermediate vibrational states [103]. On a more generalnote, the combination of very-broad-bandwidth frequency combs with the possi-

171bility of line-by-line pulse shaping allows for very complicated coherent control
experiments. I’m sure many imaginative experiments will utilize this technology
in the future.

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