Direct Frequency Comb Spectroscopy for Optical
Frequency Metrology and Coherent Interactions
byAdela Marian
B.A., University of Bucharest, 1998

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
2005

This thesis entitled:
Direct Frequency Comb Spectroscopy for Optical Frequency Metrology and Coherent
Interactions
written by Adela Marian
has been approved for the Department of Physics

Dr. Jun Ye

Dr. John L. HallDate

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

iii
Marian, Adela (Ph.D., Physics)
Direct Frequency Comb Spectroscopy for Optical Frequency Metrology and Coherent
Interactions
Thesis directed by Dr. Jun Ye

We take advantage of a phase-stable, wide-bandwidth femtosecond laser to bridge
the fields of high-resolution spectroscopy and ultrafast science. This approach, which wecall Direct Frequency Comb Spectroscopy (DFCS), involves using light from a comb of
appropriate structure to directly interrogate atomic levels and to study time-dependent
quantum coherence. In fact, DFCS may be effectively applied to determine absolute
frequencies for atomic transitions anywhere within the comb bandwidth, obviating theneed for broadly tunable and absolutely referenced continuous-wave (cw) lasers.
In this work, we apply DFCS to determine absolute atomic transition frequencies
for one- and two-photon processes in laser-cooled 87 Rb atoms. In addition, DFCS
enables studies of coherent pulse accumulation and multipulse interference, permitted
by the relatively long-lived excited states. These effects are well modeled by our density
matrix theory describing the interaction of the femtosecond comb with the cold atoms.
As in the case of precision spectroscopy performed with cw lasers, the use of
the femtosecond comb as a probe requires a careful understanding of all systematiceffects. We isolate and then mitigate the effects of the dominant sources of systematic
errors, which include the mechanical effect of the optical comb on the atomic motion,
Stark shifts by the probe laser, and Zeeman frequency shifts. The absolute frequency
measurement results are comparable to the highest resolution measurements made withcw lasers. In addition, by determining the previously unmeasured absolute frequencyof the 5S-7S two-photon transitions in 87 Rb, we show that prior knowledge of atomic
transition frequencies is not essential for DFCS.

Dedication

To my family.

Acknowledgements

I would like to offer my gratitude to the people who have made my time here at
JILA profitable and interesting. There are many people who have helped along the way
and I hope I have included all of them below.
Special thanks are due to my advisor Jun Ye who gave me the opportunity to workon a very new and challenging experiment. Under his guidance I have learned about
cold Rb, laser stabilization, mode-locked lasers and integrating cutting edge techniques
to probe new physics.
I would especially like to thank the people who worked closely with me on the Rb
project within the last two years: Matt Stowe, John Lawall and Daniel Felinto. This
thesis would not have been possible but for their help. I have gained a lot from ournumerous discussions, particularly during John’s three-month visit. In the early stagesof the experiment, I benefited from the involvement and helpful suggestions of Xinye
Xu, Tai Hyun Yoon and Long-Sheng Ma.
My experience here has been made more interesting and fun by the people who
were working on various other experiments in the lab and with me at times, in particular
Joe Berry, Kevin Holman, Seth Foreman, Jason Jones, Tara Fortier, Mark Notcutt
and David Jones. I also enjoyed interacting with and learning interesting things frommembers of the fs stabilization group, Kevin Moll, Darren Hudson and Mike Thorpe,
the cold strontium group, Tom Loftus, Andrew Ludlow, Marty Boyd and Tetsuya Ido,
and the cold molecule group, Eric Hudson and Heather Lewandowski.

viThanks are also due to the support staff at JILA, their help over the years wasinvaluable. Terry Brown and James Fung-A-Fat deserve double-thanks for the patience
they showed in guiding me through the art of electronics, while Alan Pattee and Hans
Green have always been very helpful in the machine shop.
I wish to thank all of my committee members, Chris Greene, Jan Hall, David
Jonas, Henry Kapteyn and Jun Ye, for their comments. I especially thank Jan for allof his suggestions and kind encouragement. I greatly appreciate Qudsia Quraishi’s help
in proofreading the final copy of this thesis, as well.
Much thanks to my dear friends Susan Duncan, Peter Schwindt, Anne Curtis,
Phil Kiefer and Malcolm Rickard for making my stay in Boulder both enjoyable andentertaining.
Finally, I wish to thank my family. My parents, my two sisters and Mike werealways there for me; I thank them for their constant support, love and patience.

vii

Contents

Chapter
1 Introduction

1

1.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

What about using the femtosecond comb directly for spectroscopy? . . .

4

1.3

Time- and frequency-domain description of mode-locked lasers . . . . .

5

1.4

Direct Frequency Comb Spectroscopy . . . . . . . . . . . . . . . . . . .

82 Density-matrix method for coherent accumulation
in a multi-level atom

12

2.1

13

Modeling coherent interactions . . . . . . . . . . . . . . . . . . . . . . .
2.1.1

The coherently excited density matrix ρc. . . . . . . . . . . . .

18

2.1.2

The incoherent feeding terms . . . . . . . . . . . . . . . . . . . .

20

2.2

Summary of the coherent interaction model . . . . . . . . . . . . . . . .

22

2.3

Coherent accumulation enables high resolution . . . . . . . . . . . . . .

23

3 Experimental Apparatus: the femtosecond Ti:S setup and the MOT
3.1

27

The femtosecond Ti:S laser and its stabilization setup . . . . . . . . . .

27

3.1.1

Stabilizing the two degrees of freedom of mode-locked lasers . . .

28

3.1.2

Stabilization of the laser repetition rate . . . . . . . . . . . . . .

29

3.1.3

Self-referencing . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

3.1.4

The ν-to-2 ν prism interferometer . . . . . . . . . . . . . . . . . .

32

viii
3.1.5
3.2

Stabilization of the laser offset frequency . . . . . . . . . . . . . .

34

The MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.2.1

Diode lasers: description and stabilization . . . . . . . . . . . . .

36

3.2.2

MOT details . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

4 Characterization of the systematic effects related to direct frequency comb spectroscopy

41

4.1

Doppler-free two-photon transitions with cw and pulsed sources . . . . .

41

4.2

Background for the technique of two-photon DFCS . . . . . . . . . . . .

43

4.3

Experimental method . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

4.4

Main systematic effects. . . . . . . . . . . . . . . . . . . . . . . . . . .

49

4.4.1

Stability of the reference for the pulse repetition rate . . . . . . .

51

4.4.2

Mechanical action of the probe . . . . . . . . . . . . . . . . . . .

53

4.4.3

AC Stark shift and power broadening . . . . . . . . . . . . . . .

62

4.4.4

Nulling stray magnetic fields . . . . . . . . . . . . . . . . . . . .

64Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

4.5

5 Two-photon absolute frequency measurements

66

5.1

Scanning fr gives a full spectrum . . . . . . . . . . . . . . . . . . . . . .

67

5.2

Absolute frequency measurements. . . . . . . . . . . . . . . . . . . . .

71

5.3

5S-7S absolute frequency measurements . . . . . . . . . . . . . . . . . .

72

6 One-photon absolute frequency measurements

76

6.1

Detection and timing scheme for the direct 5P measurements . . . . . .

76

6.2

5P3/2 frequency measurements . . . . . . . . . . . . . . . . . . . . . . .

80

6.3

5P1/2 frequency measurements . . . . . . . . . . . . . . . . . . . . . . .

83

7 Summary and outlook

87

ix
Bibliography

90

x

TablesTable
2.1

Relevant times for the interaction of the femtosecond laser with the
87 Rb atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

5.1

87 Rb level structure from direct frequency comb spectroscopy. . . . . . .

72

6.1

Example of a set of fr and f0 pairs used for the indirect scanning of theintermediate 5P3/2 F0 =3 state. . . . . . . . . . . . . . . . . . . . . . . .

82

xi

FiguresFigure1.1

Correspondence between the time and frequency domains for a Ti:S
mode-locked laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

7

Schematic of the fundamental premise of Direct Frequency Comb Spectroscopy (DFCS). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.1

Cascade configuration for a multi-level atom. . . . . . . . . . . . . . . .

13

2.2

Time evolution of the populations ρ22 and ρ33 for a three-level atominteracting with a train of hyperbolic-secant pulses. . . . . . . . . . . . .

2.3

17

Vector diagrams for the four terms present in the expression for the sample Dyson coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.4

Decay paths for the 5D levels of 87 Rb. . . . . . . . . . . . . . . . . . . .

21

2.5

Temporal dynamics of the 5D populations, illustrating the interplay of
the two timescales of the system, the excitation and the optical pumping. 24

2.6

Two aspects of coherent pulse accumulation: quadratic increase of the
5D population and linewidth decrease versus the accumulated pulses. . .

26

3.1

Optical layout of the KLM Ti:S laser used in the experiment. . . . . . .

29

3.2

Stabilization diagram used for phase-locking of the laser repetition rate
fr to a stable microwave reference. . . . . . . . . . . . . . . . . . . . . .

30

xii
3.3

Diagram of the ν-to-2 ν prism interferometer used for the detection of the
comb offset frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4

Stabilization diagram used for phase-locking of the laser offset frequency
f0 to a stable microwave reference. . . . . . . . . . . . . . . . . . . . . .

3.5

33

35

Saturated absorption spectra, showing the hyperfine structure of the
5S1/2 → 5P3/2 transitions for the two Rb isotopes. . . . . . . . . . . . .

38

4.1

Schematic of Doppler-free two-photon spectroscopy.

42

4.2

Schematic of the 87 Rb energy levels participating in the one- and two-. . . . . . . . . . .

photon transitions, including the wavelength and energy level information. 44
4.3

Resonantly enhanced two-photon transition with detunings from the intermediate and excited states. . . . . . . . . . . . . . . . . . . . . . . . .

45

4.4

Timing scheme used for the two-photon measurements. . . . . . . . . . .

47

4.5

Implementation of the setup for the two-photon studies: counterpropagating beams and subsequent PMT detection. . . . . . . . . . . . . . . .

4.6

Experimental setup for scanning f0 and typical closed two-photon lineshape obtained from such a frequency scan. . . . . . . . . . . . . . . . .

4.7

52

Schematic of the detunings from the intermediate and excited states relevant for the radiation pressure studies. . . . . . . . . . . . . . . . . . .

4.9

50

Comparison among two-photon lineshapes obtained with different rf references for the fr lock. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8

48

54

Momentum transferred by the optical frequency comb (in a single probe
beam configuration) to the 87 Rb atoms. . . . . . . . . . . . . . . . . . .

55

4.10 Momentum transferred by the fs probe to the atoms: comparison between
the experiment and a simplified theory.

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

57

4.11 Two-photon signal vs. radial coordinate r for a Gaussian probe beamprofile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

xiii
4.12 Momentum transferred by the fs probe to the atoms: comparison between
the experiment and the more complete theoretical model. . . . . . . . .

58

4.13 Momentum transferred by the fs probe to the atoms: line center shift
and broadening for the single-beam and the counterpropagating-beamconfigurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

4.14 Time evolution of the 5D fluorescence signal lineshape showing the mechanical action of the optical comb for two detunings from the 5P state.

60

4.15 Detailed comparison of the lineshape under the two detuning conditions
after an interaction time of 300 µs. . . . . . . . . . . . . . . . . . . . . .

61

4.16 Schematic of the detunings from the intermediate and excited states relevant for the Stark-shifted data.

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

62

4.17 Measurement of the line center frequency and linewidth for the closed
two-photon transition, showing AC Stark and radiation pressure shifts. .

63

4.18 Cancellation of the residual magnetic fields, using the two-photon fluorescence signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1

Experimental setup for scanning the laser repetition frequency fr andresulting two-photon spectrum. . . . . . . . . . . . . . . . . . . . . . . .

5.2

68

Experimental and simulated two-photon spectra obtained with a frequency scan of fr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3

65

70

Typical 5S1/2 F=2→ 7S1/2 F00 =2 two-photon Lorentzian lineshape obtained from an f0 scan and frequency shift of the transition line center
versus probing time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

6.1

Detection assembly for the direct one-photon measurements. . . . . . . .

77

6.2

Timing scheme for the direct one-photon measurements. . . . . . . . . .

78

6.3

Schematic of one- and two-photon DFCS, used for measuring singlephoton transition frequencies. . . . . . . . . . . . . . . . . . . . . . . . .

79

xiv
6.4

Lineshapes of the 5S1/2 F=2→ 5P3/2 F0 =3 transition, obtained from oneand two-photon DFCS. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.5

Theoretical plot of the time evolution of the ground state populations fortwo symmetric detunings from the 5P3/2 F0 =3 state. . . . . . . . . . . .

6.6

83

Theoretical plot of the time evolution of the ground state populations fortwo symmetric detunings from the 5P1/2 F0 =2 state. . . . . . . . . . . .

6.8

82

Lineshapes of the 5S1/2 F=2→ 5P1/2 F0 =2 transition, obtained from oneand two-photon DFCS. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.7

81

84

Raw and normalized data for the 5S1/2 F=2→ 5P1/2 F0 =2 transition,
obtained from two-photon DFCS. . . . . . . . . . . . . . . . . . . . . . .

85

Chapter 1

Introduction

1.1

Overview
The recent introduction [1,2] of phase-stabilized, wide-bandwidth frequency combs

based on mode-locked femtosecond (fs) lasers has provided optical frequency markers
that may be directly linked to optical or microwave standards. Many laboratories
have since constructed frequency combs for a variety of exciting applications. These include: measurements of absolute optical frequencies and precision laser spectroscopy [3],development of optical atomic clocks [4, 5], optical frequency synthesis [6] and broadband, phase-coherent spectral generation [7, 8], along with coherent synthesis of opticalpulses [9], phase-sensitive extreme nonlinear optics [10] and pulse timing stabilization.
Ultrashort pulses may also be applied to study coherent evolution in atomic andmolecular systems. In particular, coherent wave packet motion has been observed andeven actively controlled [11]. In addition, current work involves the use of fs lasers toachieve quantum coherent control in semiconductors [12] and in studying the temporal
dynamics of biological physics [13]. These studies may require pulses which are tailored
to probe or indeed selectively drive the dynamical process. This necessitates precise
control over the amplitude- and phase of the ultrashort pulses, otherwise known as
optical pulse synthesis [14].
On the road to tailoring optical pulses, one powerful demonstration was the phasecoherence established between two independent femtosecond lasers. Coherent optical2pulse synthesis from two fs lasers has been reported in 2001 [9], where light pulses weregenerated with durations shorter than those obtainable from the individual lasers. This
leads to an extended coherent bandwidth to be used in ultrafast experiments.
Before the advent of phase-stabilization, research showed [15] that the longitudinalmodes of a 73-fs Kerr-lens mode-locked Ti:S laser [16] are uniformly distributed infrequency space within an experimental resolution of better than 10−16 and that themode spacing differs from the pulse repetition frequency by less than 10−15 . Thus,
passively mode-locked Ti:S lasers can possess an inherent stability and that is one of
their most powerful features.
On an apparently independent path, precise spectroscopic studies of atoms andmolecules have always been performed with continuous wave (cw) lasers, refined over
the years to be very narrow spectrally, in order to enable probing of very narrow features [17]. Precision atomic and molecular spectroscopy, enabled by the progress in
cw laser stabilization, has been one of the most important fields of modern scientificresearch, providing the experimental underpinning of quantum mechanics and quantum
electrodynamics (e.g. detailed investigations of atomic and molecular structure, determination of fundamental constants, realization of time, frequency and length standards,
tests of special relativity, progress in optical communications etc.).
In the first experiments using mode-locked femtosecond lasers, they served only as
rulers for studying atomic systems and did not directly interrogate the atoms. Hänschand coworkers used the comb spectrum of a fs laser to span an optical interval of 50 nm,
and improved the measurement accuracy of the D1 resonance line in cesium by almost
three orders of magnitude [18], providing a new value for the fine structure constant. To
measure the D1 line, its frequency was compared against the fourth harmonic of a CH4 stabilized laser employed as the absolute frequency reference. The fs comb was used
as a frequency ruler to measure the resulting frequency difference of 18.39 THz (thisfrequency interval contained roughly 244 000 comb lines). To uniquely determine the3exact number of comb lines involved and therefore the unknown frequency, an opticalcavity was employed to select every 20th mode, thus increasing the pulse repetition rateby a factor of 20.
Diddams and colleagues at JILA later bridged a frequency gap of 104 THz [19],yielding an improved frequency measurement of the 1064 nm Nd:YAG laser by using
a 778-nm two-photon transition in 85 Rb as an absolute reference. In a following experiment, the output of the laser was spectrally broadened to an optical octave in a
microstructure fiber, permitting a direct measurement of an iodine-stabilized Nd:YAG,
and then measurements of ‘known’ frequencies within the laser spectrum, such as thewell established transitions at 633 nm and 778 nm [20].
Phase-stabilization of mode-locked lasers was enabled by the implementation ofself-referencing techniques [21] by means of highly nonlinear microstructure fibers [22].
A self-referencing technique, that will be discussed in Chapter 3, was first demonstrated
at JILA by Jones and coworkers [1]. They also made absolute frequency measurements
using a phase-stabilized frequency comb referenced to the primary Cs standard. The
stabilized frequency combs, providing a dense grid of reference frequencies that span
substantial parts of the visible to near-infrared, have rapidly become extremely powerful tools for optical frequency metrology, opening the door to a new exciting era ofremarkable progress. This is because atomic and molecular structural information can
now be probed over a broad spectral range, with vastly improved measurement precision
and accuracy enabled by this absolute frequency-based approach.
To date, however, these applications of fs lasers in frequency metrology used
the frequency comb only as a reference ruler, while still employing the traditional cw
laser-based spectroscopic approaches [23].

4

1.2

What about using the femtosecond comb directly for spectroscopy?
The main problem with using a pulsed laser for spectroscopy is the broad fre-quency bandwidth associated with a short pulse. The broad spectrum prevents highprecision measurements of state energies. This problem can be avoided by using a trainof phase-coherent pulses, which permits frequency resolution orders of magnitude better
than that associated with a single pulse.
T. Hänsch first introduced the idea of using coherent pulse trains from modelocked lasers for the measurement of optical frequency intervals and high-resolution
spectroscopy in 1976 [24]. A stable train of pulses forms a regular and discrete set ofmodes separated by the pulse repetition rate in the frequency domain. Hänsch showedthat a sequence of pulses results in signal enhancement over single-pulse excitation for
the atomic transition amplitude. In particular, he pointed out the resonant excitation
probability is proportional to the squared number of pulses arriving within the atomic
lifetime (for small incident light intensities).
The spectroscopic applications of this multi-pulse interference were experimentally demonstrated within the following two years. First, a train of phase-coherent pulses
produced externally to the laser, by multiple reflections of a single pulse injected into an
optical resonator, was utilized for Doppler-free two-photon excitation in Na [25]. Then,in 1978, the comb of a mode-locked picosecond dye laser [26] was used to measure the
Na 4d fine-structure splitting, i.e. frequency intervals of ∼ 1 GHz, via Doppler-free twophoton spectroscopy [27]. In this experiment, the axial mode separation was actively
controlled in the frequency domain (equivalent to stabilizing the repetition frequency ofa train of pulses). As the laser spectrum is scanned, resonant excitation occurs whenever
the sum of two frequencies is equal to the two-photon transition frequency. The combof lines was used as a ruler in frequency space to measure optical frequency differences.

5
The limitation of this technique was the small bandwidth of the available mode-locked
dye lasers, preventing measurements of large optical frequency differences.
The powerful combination of cw stabilization, passively mode-locked lasers andmicrostructure fibers, culminating with the phase-stabilization of femtosecond lasers,
has made it possible to meet some of the conditions necessary to probe atomic structure
directly with a frequency comb for precise frequency measurements: (i) short pulses,
i.e. wide bandwidth (ii) pulse-to-pulse coherence and (iii) absolute referencing of thefrequency comb.
Many new technologies have been enabled by the phase-stabilization of femtosecond lasers. In order to be able to phase-stabilize a femtosecond laser, one needs tounderstand its pulse characteristics. The connection between the short pulses emitted
from the laser and the powerful frequency comb generated is described in the following
section.

1.3

Time- and frequency-domain description of mode-locked lasers
The output of a mode-locked laser is a very regular train of ultrashort pulses, with

the interval between pulses τ usually ∼ 6 orders of magnitude higher than the pulse
duration. These short pulses are produced when a fixed phase relationship is establishedamong all the lasing longitudinal modes of the laser cavity (i.e. mode-locking), resulting
in their coherent addition and the formation of a light pulse. An isolated light pulse
can be represented as a sinusoidal optical carrier of angular frequency ωc , amplitudemodulated by an envelope function E(t), with the electric field thus given by E(t) =E(t) exp [ i (ωc t + φ0 )]. When traveling through a medium, the carrier propagates atthe phase velocity, while the envelope advances at the group velocity. In general, the

two velocities are different for a dispersive medium: vp = c /n and vg = c / n + ω ∂∂ ωn .
For instance, traversing a distance l in a dispersive material gives a phase shift of
ωc



l
l
vg − v p2= l ωcc ∂∂ ωn = −2π l ∂∂ nλ . This difference in the phase and group velocities

6
inside the laser cavity causes the phase between the optical carrier and the peak of theenvelope to evolve between successive pulses in the train. Let δφ denote the amount bywhich the carrier-envelope phase changes from pulse to pulse. After including this pulseto-pulse phase-shift, the electric field of a train of pulses for a fixed spatial coordinateisE(t) =X

En (t − nτ ) exp [ i (ωc t + φ0 − n ωc τ + n δφ)].

(1.1)

n

By taking its Fourier Transform [28] we obtain the following expression for the angularfrequencies
ωN =N 2π δφ
−
.
τ
τ

(1.2)

Thus, the spectrum of the mode-locked femtosecond laser consists of a wide comb
of discrete optical frequencies, offset from the exact harmonics of the repetition rateby a frequency proportional to the pulse-to-pulse carrier-envelope phase shift [Fig. 1.1].
The spectrum is described by the simple relation νN = N fr + f0 , where N is a large
1
integer on the order of 106 , fr = τ1 is the pulse repetition rate and f0 = − 2π
δφ fr is thecarrier-envelope offset frequency. Note that the relevant quantity here is δφ modulo 2π,i.e. f0 is bounded by fr .
The two laser parameters are dynamic quantities, sensitive to environmental perturbations affecting the laser cavity. Not surprisingly, there are no locking mechanisms
between the pulse envelope and the optical carrier inside an unstabilized laser, so theirrelative phase δφ experiences fluctuations between successive pulses. Consequently, forspectroscopic studies with a femtosecond laser, both comb parameters fr and f0 need
to be actively stabilized.
As a reminder, absolute optical frequency measurements are made with respect
to the ground state hyperfine splitting in 133 Cs at 9.192631770 GHz, which provides
the current definition for the second. This leap from the optical to the radio-frequency
domain has been a daunting task in frequency metrology for the last 30 years, before

7

δφ

E(t )

2δ φ

t
1 fr = τ

FT

I(ν )

fo
fr

ν
νN = N fr + fo
Figure 1.1: Correspondence between the time and frequency domains for a Ti:S modelocked laser. In the time domain, the carrier-envelope phase changes at a defined rate
between successive pulses in the train. In the frequency domain, the comb lines arespaced by the laser repetition rate fr and the comb is shifted from integer multiples of
fr by an offset frequency f0 . The Ti:S laser used in the experiment is centered at 778
nm and has a FWHM bandwidth of ∼ 55 nm.

8the introduction of the mode-locked approach. Because of the complexity involved,
only a few harmonic frequency chains were ever implemented, for very specific opticalfrequency measurements. A few examples are: the 88 THz transition in CH4 [29] (which
led to the speed of light measurement [30]), the I2 -stabilized He-Ne transition at 473
THz [31], the 455 THz intercombination transition in 40 Ca [32], the 5s 2 S1/2 → 4d 2 D5/2
transition at 445 THz in 88 Sr+ [33], the 2S-12D transitions in hydrogen and deuteriumat 799 THz [34]. Stabilized frequency combs allow one to phase-coherently link any
unknown optical frequency within the comb spectrum directly to a primary microwave
standard.

1.4

Direct Frequency Comb Spectroscopy
Following our own theoretical studies [35], we take advantage of the phase-stable,

wide-bandwidth femtosecond pulse train to bridge the fields of high-resolution spectroscopy and ultrafast science, in a spectroscopic study of laser-cooled 87 Rb atoms.
This approach, which we call Direct Frequency Comb Spectroscopy (DFCS), involves
using light from a comb of appropriate structure to directly interrogate atomic levels
and to study time-dependent quantum coherence.
By utilizing a stabilized fs comb, multiple atomic states may be simultaneously
and directly excited, by tuning the appropriate comb lines into resonance, and the subsequent dynamics may be probed. Furthermore, given that the comb has two independent
degrees of freedom, it is always possible to simultaneously satisfy two-photon as well as
one-photon resonance conditions.
In fact, DFCS may be applied to determine absolute frequencies for atomic transitions anywhere within the comb bandwidth. The entire transition spectrum can be
efficiently retrieved by a quick scan of the laser repetition frequency. This obviates theneed for a broadly tunable and absolutely referenced cw laser and is especially useful inthe study of multi-photon processes where several laser sources may be required. Thus,

9
optical frequency combs are a highly efficient tool for precise studies of atomic structure.
In this work, we apply DFCS to determine absolute atomic transition frequencies
for one- and two-photon processes, as illustrated in Fig. 1.2. DFCS enables studies
of coherent pulse accumulation and multipulse interference, permitted by the relatively
long-lived 5D and 7S states. The observed coherent pulse accumulation and interference
effects are well modeled by our density matrix theory describing the interaction of the
femtosecond comb with the atoms, as discussed in Chapter 2. We use the theory results
to construct transition spectra to compare against the experimental spectra.

3>
f r =100 MHz

5D or 7S
(240 ns or 88 ns)

2>

5P1/2 or 5P3/2(27 ns)

ν

1>

5S

87Rb

Figure 1.2: Schematic of the 87 Rb energy levels participating in the 5S-5D and 5S-7S
two-photon transitions and 5S-5P one-photon transitions. These are the relevant levels
probed using direct frequency comb spectroscopy (DFCS).

We present the details of the implementation of the femtosecond stabilization
scheme and the MOT setup in Chapter 3. These are integrated together for spectroscopic studies in Chapter 4.
As in the case of precision spectroscopy performed with cw lasers, the use of the
femtosecond comb as a precision probe requires a careful understanding of all systematiceffects. We find that the dominant effects are the mechanical effect of the optical comb

10
on the atomic motion, the light shift by the probe laser, and the Zeeman frequency
shifts. In Chapter 4, we address the mechanical action of the probe laser by using acounterpropagating beam configuration. We observe Stark shifts generated by detunings
from the intermediate states in the two-photon processes and we reduce their effect byalways probing on resonance with the intermediate state of interest.
We first study the 5S-5D two-photon transitions in 87 Rb, leading to high-resolution
spectroscopy for some of the transitions, as presented in Chapter 5. The measurement
results are comparable to the highest resolution measurements made with cw lasers.
By determining the previously unmeasured absolute frequency of the 5S-7S two-photontransitions in 87 Rb, we show that prior knowledge of atomic transition frequencies is
not essential for this technique to work, and indicate that it can be applied in a broad
context. When resonant enhancement is enabled by a comb component tuned near anintermediate 5P state, we observe two-photon transitions occurring between initial andfinal states that differ by one unit of the total angular momentum (∆F = ±1), which
are absent for far-detuned intermediate states. This capability of accessing adjacentexcited hyperfine levels from the same ground state allows for direct measurements ofhyperfine splittings.
We then use DFCS to measure one-photon transitions, and choose the 5S-5Ptransitions as an example, as detailed in Chapter 6. The measurement of 5P statesis also carried out indirectly via the 5S-5D two-photon transitions, by studying theirresonant enhancement when comb components are scanned through the intermediate
5P states. We compare the 5P measurements obtained via one-photon and two-photon
DFCS and clearly demonstrate the importance of population transfer in working withmultilevel systems probed by multiple comb components. Incoherent processes such as
optical pumping govern the population dynamics beyond the atomic decoherence time.
For the case of the indirect 5P frequency measurements via the 5S-5D two-photontransitions, the experimental data do not yield directly the lineshape. In this case, we

11
use the theory prediction to adjust the raw data and retrieve a Lorentzian lineshape, as
described in Chapter 6.
The optical coherence of a phase-stabilized pulse train provides a spectral resolving power approaching that of state-of-the-art cw lasers. At the same time, thenarrow linewidth of individual comb lines permits a precise and efficient determinationof the global energy-level structure, providing a direct connection among the optical,
terahertz, and radio-frequency domains.

Chapter 2

Density-matrix method for coherent accumulation
in a multi-level atom

We will now study coherent accumulation in a multi-level atom interacting witha train of ultrashort pulses. For the situation where the atomic relaxation times are
longer than the interpulse interval τ , the atomic coherence will survive between two
consecutive pulses, leading to coherent accumulation of excitation in the sample. Constructive/destructive interference among the coherences excited by the train of pulses
will determine the final atomic populations. We will focus on the accumulative effects
occurring in the sequential 5S-5P-5D two-photon transitions.
The interaction of the femtosecond comb with the atoms is modeled starting with
the Liouville equation for the density matrix of all the atomic states in the laser bandwidth accessible through two-photon absorption, with radiative relaxation included via
phenomenological decay terms. The density matrix equations are solved to a fourthorder perturbative expansion in the electric field. With the approximation of impulsive
optical excitation during the pulse, followed by free evolution and decay, an iterative numerical procedure is used to determine the state of the atomic system after an arbitrary
number of pulses.

13

3k>

5D

k = 1,82j >5P

j = 1,6

1i >

5Si = 1,2Figure 2.1: Cascade configuration for a multi-level atom.

2.1

Modeling coherent interactions
Consider an electric field E(t) interacting with a multi-level atom in which all

the levels are distributed in three different manifolds |1ii, |2ji and |3ki, in a cascadeconfiguration, as presented in Fig. 2.1. Here ‘1’ labels the ground 5S states, ‘2’ theintermediate 5P states and ‘3’ the excited 5D states, while i, j and k stand for thevarious fine and hyperfine levels of a specific state. We adopted this simplified notation
for the energy levels only in the theory chapter, because it will make the following
equations easier to follow.
b is given by the sum of the field-free atomic
The Hamiltonian of the system Hb 0 and the interaction potential between the atom and the electric field
Hamiltonian Hb int . Thus, H
b =H
b0 + Hb int , within the electric-dipole approximation H
b0 =
H

X

E1i |1iih1i| +ib int = −HX

E2j |2jih2j| +

j

X
i,j

µ1i,2j E(t)|1iih2j| +X

E3k |3kih3k|

and

k

X

µ2j,3k E(t)|2jih3k| + h.c.,

(2.1)

j,k

where E1i , E2j , E3k are the energies of levels |1ii, |2ji and |3ki, respectively, and µi,j
is the dipole moment of the i → j transition.

14
We are interested in the time evolution of the state of the system after a longsequence of pulses. This temporal evolution is described by the Bloch equations, derived
from the Liouville equations upon inclusion of the relaxation terms. The starting pointis therefore the Liouville equation for the density-matrix elements of the multi-levelatomic system, resulting in 9 families of equations for the ground-, intermediate- andexcited-state populations, the intra-ground-, intra-intermediate- and intra-excited-state
coherences, as well as the 1-2, 2-3 and 1-3 coherences. For clarity, I will write outsome representative Liouville equations, for the ground-state populations, ground-state
coherences and the 1-2 coherences, respectively:∂ρ1i,1i∂t∂ρ1i,1s∂t∂ρ1i,2j∂tib ρb]|1ii ground-state population= − h1i|[H,h̄ib ρb]|1si, s 6= i intra-ground-state coherence= − h1i|[H,h̄ib ρb]|2ji 1-2 coherence.
= − h1i|[H,
h̄

(2.2)

The 6 remaining families can be written in a similar manner.
The Bloch equations are obtained from the Liouville equations after including therelaxation rates Γi,j of the (i, j) density matrix components. Note that for our case,Γ1i,1i = 0, all the population decay rates for a given state are equal, i.e. Γ2j,2j = Γ5P ≡Γ2 and Γ3k,3k = Γ5D ≡ Γ3 , and Γi,j = 21 (Γi,i + Γj,j ). The subscripts indicating thestates in the Γ coefficients will all be kept in the following derivation, for symmetry of
the equations. As an example, I will write the Bloch equation for the 1-2 coherence:∂ρ1i,2jib int , ρb]|2ji − Γ1i,2j ρ1i,2j= −iω1i,2j ρ1i,2j − h1i|[H
∂t
h̄

(2.3)

b in Eq. 2.2, withThis was obtained by substituting the expression for the Hamiltonian H

ωi,j ≡ (Ei − Ej )/h̄. All the coherence equations look the same as Eq. 2.3, with only theappropriate change of indices.
Before writing expressions for the populations of levels |1i and |2i, we must also
consider the incoherent feeding of level |2i by level |3i and |1i by |2i present in a multi-

15level atomic system. Including these additional terms gives the populations as∂ρ1i,1i∂t∂ρ2j,2j∂t∂ρ3k,3k∂tXib int , ρb]|1ii − Γ1i,1i ρ1i,1i += − h1i|[Hγ1i,r ρr,rh̄rXib int , ρb]|2ji − Γ2j,2j ρ2j,2j += − h2j|[Hγ2j,r ρr,r
h̄
r

(2.4)

i
b int , ρb]|3ki − Γ3k,3k ρ3k,3k .
= − h3k|[Hh̄More details about deriving the incoherent feeding terms will be given in 2.1.2.
We can now integrate Eqs. 2.3 and 2.4, with the assumption of impulsive opticalexcitation during the interaction time: the pulse duration is ultrashort compared to thepulse repetition period or any relaxation time of the system, as seen from Table 2.1.
Pulse duration
Laser repetition period
5P lifetime
5D lifetime
1-2 coherence lifetime
1-3 coherence lifetime
2-3 coherence lifetime∼ 30 fs
10 ns
27 ns [36]240 ns [36]54 ns
480 ns48.5 ns

Table 2.1: Relevant times for the interaction of the femtosecond laser with the
87 Rb atoms.

b int changes on a very fast timescaleThis means that the interaction potential H

compared to the Γi,j relaxation rates, allowing us to neglect the temporal dependenceb int , and yielding:on Γi,j inside the integrals containing H"

ρ1i,1i (t) = e−Γ1i,1i ti
ρ01i,1i −h̄

−iω1i,2j t−Γ1i,2j t

ρ1i,2j (t) = e



Z t
0

b int , ρb]|1ii +dt0 h1i|[Hiρ01i,2j −
h̄

X

Z t

0 iω1i,2j

dt e
0

t0

#

dt0 eΓ1i,1i ρr,r (t0 )

γ1i,r

r

Z t

t0

0


b
h1i|[Hint , ρb]|2ji(2.5)

for the ground-state population and 1-2 coherence, respectively.
Numerical integration of Eq. 2.5 gives the time evolution of the system with arbitrary initial conditions, but is computationally challenging for the 87 Rb atom, where
there are two 5S initial levels, six 5P intermediate levels and eight 5D final levels,

16
leading to a total of 162 = 256 coupled Bloch equations. To avoid the long computational times, our collaborator Daniel Felinto [37,38] has developed an iterative numerical
procedure that determines the state of the atomic system after any number of pulses.
As previously mentioned, for laser repetition periods τ shorter than the relaxation timesof the system, as is the case of 87 Rb, the atomic sample will accumulate excitation in
the form of population and coherence.
In Daniel Felinto’s iterative solution, the state of the system before the (n + 1)th pulse in the train is obtained from the state prior to the n-th pulse. The relation
connecting ρn+1 to ρn is easily derived from Eq. 2.5 by setting t = τ . Again, making
the impulsive optical excitation assumption means setting t → ∞ in all the integrals
b int ; the lower limit can be made t → −∞, since E(t) = 0 for t < 0. For
containing H

the two sample elements of the density matrix appearing in Eq. 2.5 we obtain"−Γ1i,1i τ

ρn+1
1i,1i

= e

ρn+1
1i,2j

=

ρc1i,1i +

Z τ

X

γ1i,r
r
h
i
e−iω1i,2j τ −Γ1i,2j τ ρc1i,2j ,0 Γ1i,1i t0

dt e

#
0

ρr,r (t )

(2.6)

0

where
ρcr,s = ρnr,s −

i
h̄

Z ∞
−∞

0

b n , ρb]|si
dt0 eiωr,s t hr|[H
int

(2.7)

b n ), a function of the initialis the density matrix coherently excited by the n-th pulse (Hintstate prior to the n-th pulse ρn . Expressions similar to Eq. 2.6 are derived for the 7remaining families of equations.
As a demonstration, Fig. 2.2 (adapted from Felinto et al., reference [38]) shows
the temporal evolution of the ρ22 and ρ33 populations in a three-level atom interactingwith a train of hyperbolic-secant pulses. The total number of coupled Bloch equations
for this system is 32 = 9, so a direct numerical integration is not yet computationallyintensive. The solid lines are a result of the direct numerical integration (∼ 1.5 hours ofcomputational time), while the open circles arise from successive applications of Eq. 2.6
to the atom initially in the ground state (< 10 s). The transient behavior of the excited17
level populations is governed by their significantly different (factor of 10) lifetimes, as
demonstrated by the different timescales in the two frames of the figure.

Figure 2.2: (adapted from reference [38]) Time evolution of the populations ρ22 andρ33 for a three-level atom interacting with a train of hyperbolic-secant pulses. The twoframes are different timescales of the same evolution, where the solid lines are a resultof direct numerical integration of the Bloch equations and the open circles come fromsuccessive applications of Eq. 2.6.

After a large number of pulses, the system reaches a stationary state that repeats
every τ . The shape of the direct-integration curve accounts for the very fast change
in population during the pulse interaction time, followed by a slow decay in between
pulses. The numerical-iteration method agrees well with the direct integration, in thatit always matches the value at the minimum. This happens because, in the iteration

18
procedure, the initial state is always taken to be the state ρn just before the impulsive
excitation. The situation would be reversed and the two curves would match at the
maximum if the initial state were the state immediately after each individual pulse.
To summarize, in the simplified treatment of coherent accumulation for a twophoton process in a multi-level atom, there are two major steps: the first is obtaining thestate of the system after the interaction with a pulse in the train, for an arbitrary initialstate of the atom (Eq. 2.6). The second step is the successive application of Eq. 2.6,
yielding the atomic state after a sequence of pulses, or in other words, describing thetime evolution of the system.

2.1.1

The coherently excited density matrix ρcThere are two quantities left to calculate now, the coherently excited density
matrix in Eq. 2.7, associated with the impulsive excitation during the laser pulse, andthe feeding terms in Eq. 2.4, arising from incoherent decay.
We will first focus on ρc , realizing that it is a product of two probability amplitudes
in the interaction pictureρci,j = Ci × Cj∗ ,bI (t)|Ψ0 i,
Ci = hi|U(2.8)

with Ψ0 the initial state of the system, which is a superposition of all the atomic states,
and ÛI (t) the time propagation operator in the interaction picture, given by

Z t

 Z
Z 0
i
i 2 t 0 t 00 b I 0 b I 00
0 bI0b
UI (t) = 1 + −dt Hint (t ) + −dt dt Hint (t )Hint (t ) + . . .

h̄

h̄

0

0

(2.9)

0

b0 t/h̄ bb0 t/h̄b I (t), Hb I (t) = eiH
After inserting the relation for H
Hint (t) e−iH
, and specifying
int
int

the electric field E(t) = E(t)eiωc t+iφ0 + c.c. as defined in section 1.3, we obtain thefollowing general expressions for the three groups of probability amplitudes:
C3k (t) =X
s

0
A3k,3s C3s
+

X
j

0
A3k,2j e−iφ0 C2j
+

X
i

0
A3k,1i e−2iφ0 C1i(2.10)

19
C2j (t) =X

C1i (t) =X

0
A2j,3k eiφ0 C3k
+

X

2iφ0

X

0
A2k,2s C2s+sk

A1i,3k e

0
C3k
+

iφ0

A1i,2j e

j

k

X

0
A2j,1i e−iφ0 C1i

i
X
0
0
C2j +
A1i,1s C1i,
swhere Ai,j are the Dyson coefficients corresponding to the time evolution from state ito state j. This yields the final formula for any element (i, j) of the impulsively excited
density matrix ρcρci,j = e−iBi,j φ0X

Ai,r A∗j,s eiBr,s φ0 ρ0r,s ,(2.11)

r,swith Bi,j = 2

k (δ3k,i − δ3k,j ) +P

k (δ2k,i − δ2k,j ), where the δ s are Kronecker symbols.

P

We are interested in developing a theory in the weak-field regime: from the infinite
Dyson series only the terms up to the fourth order in the electric field will be kept. The
fourth order is the smallest order contributing to population in the most excited level
for two-photon transitions from the ground state. In this regime, making the rotatingwave approximation will result in expressions for the 9 groups of elements Ai,j present
in Eqs. 2.10 and 2.11. As an example, I will write out the expression for A3k,3s :iea0= δ3k,3s + −h̄A3k,3s



−

+

iea0
h̄

4 X

2 X

Z t

m3k,2r m2r,3s

r

×

X

m3k,2r m2r,3n m3n,2q m2q,3s
e

Z t00

Z t000

0

0

0

0

0

00

0

∗



00

000

0000

Z t Z t0

Z t00

Z t000

0

0

dt0

0
∗

dt00

0
000

dt000

0000

E (t )E(t )E (t )E(t ),0dt0000 eiδ3k,2r t

dt000

dt00

dt0

iea0
E (t )E (t )E(t )E(t ) + −h̄eiδ3n,2q t000 iδ2q,3s t0000 ∗

00

0

0

r,n,q

e

0

dt00 eiδ3k,2r t eiδ2r,3s t E ∗ (t0 )E(t00 )m3k,2r m2r,1n m1n,2q m2q,3s

r,n,q

× e

Z t0

Z t Z t0iδ2r,1n t00 iδ1n,2q t000 iδ2q,3s t0000 ∗

× e

dt0

4

0

00

dt0000 eiδ3k,2r t eiδ2r,3n t(2.12)

with δi,j ≡ ωi,j − ωc the detunings relative to the carrier frequency, mi,j = µi,j /ea
dimensionless dipole moments, e the electron charge and a the Bohr radius. The last
three terms in Eq. 2.12 are second-order and fourth-order contributions to the excitation,
related to the processes of absorption and stimulated emission on the lower and uppertransitions. They are schematically drawn in Fig. 2.3.
This messy expression can be greatly simplified if we consider the situation where

20

3>
2>
1>

Figure 2.3: Vector diagrams for the four terms present in Eqs. 2.12 and 2.13 for the
sample Dyson coefficient.

the pulse is ultrashort compared to the time variation determined by the atomic detunings, i.e. we neglect all the exponentials in Eq. 2.12, and if we limit ourselves to thereasonable case of a real envelope function. Under these approximations, the nested
integrals take a much simpler form, via successive integration by parts, with the result
A3k,3s = δ3k,3s −
+

θ4
4!

θ2 X
m3k,2r m2r,3s +2! r(2.13)!

X

m3k,2r m2r,1n m1n,2q m2q,3s +

r,n,q

X

m3k,2r m2r,3n m3n,2q m2q,3s .

r,n,qHere θ is the pulse area, defined as
θ=

2.1.2

ea
h̄

Z ∞

dt E(t)pulse area.

(2.14)−∞The incoherent feeding terms

There are two very different timescales over which the populations present in
the feeding-term integrals vary: first, there is a very fast change in population during
the pulse interaction time, followed by a slow, incoherent decay during the interpulseinterval. In this model, it is assumed for simplicity that level |2i is fed only by level |3i,
and |1i is only fed by |2i. Note that the population in the latter level varies in time
due to feeding from |3i. However, in reality, there are two decay channels for the 5D
population [36], as illustrated in Fig. 2.4: the first is the 5D-5P-5S radiative cascade21
and the second is the 5D-6P-5S cascade. In fact, we experimentally determine the 5D
population by detecting the 420 nm blue fluorescence emitted in the 6P-5S transition.
The 6P state is not included in the simulation.

5D38%
5.2 µm62%776 (762) nm

6P3/2 (1/2)
5P3/2 (1/2)

31% (27%)780 (795) nm

420 (422) nm

5S
Figure 2.4: Decay paths for the 5D levels of 87 Rb, with the corresponding transition
wavelengths and branching ratios specified. In the experiment, we detect the 6P-5S blue
photons at 420 nm, which represent roughly 10% of the 5D population. The 6P levels
also decay to the 6S and 4D levels, not pictured here.

For the situation where the pulse duration is ultrashort compared to the laser
repetition period τ and considering only the 5P decay channel for the 5D population,
we have:XZ τ

γ2j,3kk0

dt0 eΓ2j,2j t ρ3k,3k (t0 ) '0X

γ2j,3k ρc3k,3k

Z τXγ2j,3k ρc3k,3k

k

0

0

k

=

0

dt0 eΓ2j,2j t e−Γ3k,3k t

1 − e(Γ2j,2j −Γ3k,3k )τ
Γ3k,3k − Γ2j,2j(2.15)XZ τ

0dt0 eΓ1i,1i t ρ2j,2j (t0 ) 'γ1i,2j0j×X

γ2j,3k

k

=

X
j

Z τ

γ1i,2j ρc2j,2j

00 Γ2j,2j t00

dt e
0

0

0

dt0 eΓ1i,1i t e−Γ2j,2j t

γ1i,2j
0

j

Z t0

"

ρc2j,2j +X#00

ρ3k,3k (t )1 − e(Γ1i,1i −Γ2j,2j )τ X X γ1i,2j γ2j,3k+
Γ2j,2j − Γ1i,1i
Γ2j,2j − Γ3k,3k
j
k

22
"

×ρc3k,3k#

1 − e(Γ1i,1i −Γ2j,2j )τ
1 − e(Γ1i,1i −Γ3k,3k )τ
−.
Γ3k,3k − Γ1i,1i
Γ2j,2j − Γ1i,1i(2.16)

After realizing that Γ1i,1i = 0 and that all the population decay rates for a given state
are equal, i.e. Γ2j,2j = Γ2 , Γ3k,3k = Γ3 , after setting the feeding coefficients proportional
to the branching ratios of the upper levels down to the lower levels γ2j,3k = r2j,3k Γ3 ,2 −Γ3 )τ, D21 =
γ1i,2j = r1i,2j Γ2 and defining the following constants D32 = Γ3 1−exp(ΓΓ3 −Γ2Γ3
1 − exp(−Γ2 τ ) and D321 = ΓΓ22−Γ3h1−exp(−Γ3 τ )2τ )− 1−exp(−Γ
Γ3
Γ2

i

, the final expression forthe populations after the n-th pulse becomes:cρn+1
1i,1i = ρ1i,1i + D21

X

r1i,2j ρc2j,2j + D321

X

j

j

"

ρn+1
2j,2j

= e−Γ2 τ ρc2j,2j + D32

r1i,2jX

r2j,3k ρc3k,3k

k#X

r2j,3k ρc3k,3kk−Γ3 τ cρn+1
ρ3k,3k ,3k,3k = e(2.17)

where ρci,j are given by Eq. 2.11.

2.2

Summary of the coherent interaction model
The relevant laser parameters for describing the interaction of the femtosecond

comb with the atoms are the interpulse period τ , the carrier envelope phase evolution
between successive pulses δφ, the pulse duration and its associated area θ. The model
includes the fine and hyperfine structure of the 5S, 5P and 5D states, with energylevel information provided by the literature. The Zeeman substructure is averaged over,for linear polarization. In the simulation, calculated oscillator strengths and ClebschGordan coefficients are employed for all the involved transition pathways in an effort toaccurately predict the relative signal strengths. This general set of coupled Bloch-type
equations, evolving from one pulse to the next, leads to a global picture of coherent
population accumulation and incoherent optical pumping.
For times that are short compared to the atomic decoherence time, the femtosecond pulse train drives the atomic coherence in phase such that a multi-pulse excitation

23
is coherently built up for resonant atomic states. At longer times, however, the coherence in the optical field can no longer be transferred to the atom owing to the finite
atomic coherence time. Incoherent processes such as optical pumping then govern the
population transfer dynamics.
These two timescales of the system, excitation and optical pumping, can be clearlyseen in Fig. 2.5 (a) which shows the time evolution for two different two-photon resonances. The two curves have the same coherent excitation timescale of a few microseconds, set by the 5S-5D coherence lifetime of ∼ 500 ns. The slow decay rate is different
though for each transition. Figure 2.5 (b) illustrates the behavior of one of the transitions as a function of the detuning from resonance. On resonance, all the pulses inthe train contribute to the excitation, since the train of pulses is in phase with theatomic frequency. Away from resonance, the decrease in population due to destructive
interference is accompanied by oscillations in the excitation region, arising from thephase difference between the atomic frequency and the frequency of the closest comb
line. This oscillatory behavior at the frequency difference is more obvious in Fig. 2.5 (c),where the detuning from resonance is even larger.

2.3

Coherent accumulation enables high resolution
Temporal coherent control is best manifested in the 5D coherent population ac-

cumulation and transition linewidth evolution, which reach their asymptotic limits imposed by the atomic decoherence, through the coherent interaction with the train of
femtosecond pulses . The results of the model (Fig. 2.6), under the condition of a small
pulse area, illustrate the effect of pulse accumulation on signal strength and spectral
linewidth.
The upper graph [(Fig. 2.6 (a)] shows that the on-resonance 5D population increases as the square of the number of pulses until reaching approximately the decoherence time, ∼ 500 ns for the 5D states. This 5D signal scaling at short times is reinforced

24

0.4

decay

10 ρ33

0.3

excitation

0.2

5S1/2 F=2 to 5D5/2 F=4
5S1/2 F=2 to 5D3/2 F=1

0.1

(a)
0.0
0

5

10

0.4

25

30

5S 1/2 F=2 to 5D5/2 F=4 transition

0.3

10 ρ3315
20
Time (µs)

on resonance
0.7 γ 5D detuning

0.2

1.4 γ 5D detuning

0.1

(b)

0.0
0

5

10
Time (µs)

15

20

0.16
5S1/2 F=2 to 5D5/2 F=4 transition

50 ρ330.12
0.08

3.9 γ 5D detuning

0.04

(c)
0.00
0

2

4
6
Time (µs)

810Figure 2.5: Several aspects of population dynamics at short and long timescales:(a) Time evolution for two resonances, showing the short coherent excitation, followed by
the slow incoherent decay. (b) Temporal evolution for one of the previous transitions, fordifferent detunings from the atomic resonance, illustrating the destructive interference
that results in the out-of-resonance condition. (c) Oscillation resulting from the phasedifference between the atomic frequency and the train of pulses.

25in the figure by a quadratic fit. The proportionality of the transition probability to thesquared number of pulses N 2 is a more general result, valid in the weak excitation
regime, which was theoretically demonstrated for a two-level atom [39], in agreementwith earlier conclusions based on perturbation theory [40].
The rapid population increase is accompanied by the narrowing of the resolution
linewidth [(Fig. 2.6 (b)], with the obtained resolution limited only by the 5D naturallinewidth. After the very first pulse, the linewidth is the single-pulse bandwidth ofthe laser; as the pulses keep coming, the spectral components from successive pulses
interfere among themselves, resulting in the linewidth decrease. Thus, the atoms are
the spectrometer used to observe the frequency comb, and the atomic linewidth is the
bandwidth limit of this spectrometer.
The dependence of the 5D population and linewidth versus the number of pulses
is temporally analogous to the power density scaling and the spatial resolution versus
the number of slits in a multi-slit experiment [39, 40]. Again, this is a general result,valid for a broad range of pulse shapes, pulse areas or detunings. The two effects ofcoherent pulse accumulation on signal strength and spectral linewidth have both been
experimentally verified [41]. The quadratic increase of the excited state population
versus the accumulated number of pulses can be further enhanced with a larger fr ,where the shorter pulse interval allows for more pulses to be ‘added’ during the atomic
coherence lifetime.

The theoretical model we have developed is designed to predict the coherent
accumulation of population in the 5D states, followed by incoherent optical pumping.
Using this model we generate atomic transition spectra by tuning either fr or f0 . We
then compare them with the corresponding experimental spectra (covering the same
fr and f0 range), as presented in Chapter 5. In addition to comparing theory andexperiment, we also use the model prediction to adjust the experimental data. For the26case of the indirect 5P frequency measurements via the 5S-5D two-photon transitions,described in Chapter 6, we use the theoretical prediction to modify the raw data (after
the adjustment, the data fall on top of a Lorentzian).

D-State Population

1E-3

10

linewidth

1E-4

Quadratic Fit

0.15
0.12
0.09
0.06

20

(b)

40

1

Natural Linewidth(660 kHz)60

Number of Pulses

80

Linewidth (MHz)D-population

0.01

0

D-State Population

100

(a)

0.1

0.1
100

Total Pulses:
10
20
50
80
100

0.03
0.00
-2

-1
0
1
Frequency (MHz)2

Figure 2.6: (a) Left (right) axis shows calculated 5D population (linewidth) on resonancefor the closed two-photon transition versus total number of accumulated pulses. The
time between pulses is ∼ 10 ns, and the system reaches its asymptotic values of signal
amplitude and linewidth after ∼ 480 ns. The quadratic fit to the 5D population atshort times shows that the signal scales as the square of the number of pulses until
atomic decoherence limits the coherent pulse accumulation. (b) The corresponding 5D
resonance lineshape versus the number of pulses. During the first 10 pulses, the comb
structure is not sufficiently developed to offer appreciable signal or resolution.

Chapter 3

Experimental Apparatus: the femtosecond Ti:S setup and the MOTThis chapter discusses the two primary components of the experimental setup,the femtosecond laser and the MOT. I will first present the idea for the Ti:S laser
stabilization and then describe one of the most commonly used self-referencing schemes.
I will conclude by briefly discussing the stabilization of the cw lasers and the setup used
for cooling and trapping in the MOT. Chapter 4 will present the integration of the
femtosecond laser and the MOT.

3.1

The femtosecond Ti:S laser and its stabilization setup
Recent advances in controlling the pulse repetition period and the carrier-envelope

phase of mode-locked lasers have ushered in a series of novel applications using theirfrequency combs [42]. In the introduction I presented some of the fundamental aspectsof mode-locked lasers. In this section, I will review some of the basic concepts behind carrier-envelope phase stabilization and I will explain how the two dynamic laserparameters are optically detected and stabilized in the experiment. Stabilization of
the two degrees of freedom is a necessary step for using these lasers for spectroscopic
applications.

28
3.1.1

Stabilizing the two degrees of freedom of mode-locked lasers

For the experiment, we used a home-built titanium-doped sapphire (Ti:S) laser
pumped by a 5 W single-frequency, diode-pumped, frequency-doubled Nd:YVO4 laser(Coherent Verdi) operating at 532 nm. The Ti:S laser generates a 100-MHz pulse train
with pulse widths on the order of 15 fs and energies of about 5 nJ per pulse, using
Kerr lens mode-locking (KLM). A diagram of our KLM laser is shown in Fig. 3.1. The
configuration used is a standing-wave, folded geometry, with the cavity delimited by
two flat mirrors: the laser end mirror (EM) and the partially reflective output coupler(OC). The nonlinear gain element for the laser cavity is the Ti:S crystal. The passivemode-locking mechanism employed here is the nonlinear Kerr Effect, which gives anincrease in the index of refraction of the crystal as the optical intensity is increased (selfphase modulation, self-focusing, see for example [43]). The Ti:S acts like a nonlinear
lens, slightly focusing the intracavity beam, but it also spreads the pulses temporally
through normal dispersion (the long wavelengths travel faster than the short ones). As
a consequence, a sequence of two prisms is used to compensate for the group velocity
dispersion (GVD) in the crystal: the first prism spatially disperses the pulse, leading thelong wavelengths to traverse more glass in the second prism than the short wavelengths.
The beam is still horizontally dispersed after the second prism, though collimated. Uponreflection by the laser cavity EM, the pulse travels again through the pair of prisms,
thus canceling the spatial dispersion.

As mentioned before, fr determines the mode spacing and is inversely proporv

tional to the cavity length L, fr = Lg , while f0 sets the absolute comb position and isv

ωc
proportional to the intracavity dispersion, f0 = 2π
(1 − vgp ). They are both influenced

by the group velocity vg , but there is no apparent f0 dependence on the laser cavity
length. Ideally one would like to control the two parameters independently. One adjustable parameter can be the cavity length, for fr control; yet, an additional degree of

29
δL

β
EM

P1

P2

OC
PUMP
L CM1

Ti:S
crystalCM2

FM

Figure 3.1: Optical layout of the KLM Ti:S laser used in the experiment; the end mirror(EM) and the output coupler (OC) are the two flat mirrors bounding the cavity, used
for the f0 and fr stabilization, respectively. The two prisms P1 and P2 are used fordispersion compensation.

freedom is needed for changing vg . The one employed in my laser was first suggested
in 1999 by the Hänsch group [44] and uses small-angle rotations about the vertical axisof the laser EM. As previously stated, the different spectral components in the laser
beam are horizontally spread out at the EM, leading to a linear relationship between
the spatial coordinate and the wavelength. For small angle (β) changes, there is a linear
path length change, i.e. phase shift, with frequency. This is effectively a group delay,
linear with the angle. Under this assumption that the swivel mirror changes the groupdelay by a small amount ∼ β, it can be shown [28] that f0 is only controlled by β, while
fr depends on both β and L. Therefore, one can use the swivel angle β to control f0 (thiswill also change fr !), and the cavity length L to compensate for the subsequent change
in fr . Details of the experimental implementation of the vg , and hence f0 , control will
be given in 3.1.5.

3.1.2

Stabilization of the laser repetition rate

The laser repetition frequency fr is a radio frequency (rf) ∼ 100 MHz and ismeasured directly using a fast photodiode. To reduce the increase in phase noise thatis due to the large frequency multiplication factor up into the optical region (∼ 106 ), a30higher harmonic of the pulse repetition rate is used for stabilization. A portion of thepulse train is detected with a high-speed photodetector; the 10th harmonic of fr at 1
GHz is filtered and amplified (to 0 dBm, suitable for the rf port of a double-balanced
mixer) and then phase-locked to the 1 GHz provided by a stable microwave source.
The error signal obtained by comparing against this reference frequency is integrated,filtered and amplified in a JILA-built loop filter (LF) and then used for active feedback
to the laser, as shown in Fig. 3.2. Cavity length corrections are controlled with a small(∼ 5 mm) tube piezoelectric transducer (PZT), on which the output coupler of the laser
has been mounted. All the comb modes are moved together by this translating PZT
which varies the laser cavity length.

to f0 stabilizationand MOT

Ti:S Laser

1 GHz Wenzel

N fr
1GHz BPF rf amplifier
frequency counterto Output Coupler (translating PZT)DDSJILA LF
HV driver (1 kV)

Figure 3.2: Stabilization diagram used for phase-locking of the laser repetition rate fr ;BPF stands for band-pass filter, DDS for direct digital synthesizer and LF for loop filter.
More details are given in the text.

Several stable rf oscillators were tried as references for the repetition rate (I willaddress this issue further in Chapter 4) and in the end a custom-made Wenzel oscillator, based on a crystal oscillator that has a very good short-term frequency stability,
was settled upon. The short-term stability of the rf source is very important, given
the huge multiplication factor up into the optical domain. Unfortunately, the Wenzel
tuning range is only 3.2 kHz around 1 GHz, corresponding to a 320 Hz tunability of

31
the repetition rate very close to 100 MHz. One would like to have more freedom thanthat in choosing a suitable fr value, so an extra direct digital synthesizer (DDS) had tobe later added to the stabilization setup. Its widely tunable output (∼ MHz) is mixed
with the Wenzel and the repetition rate is forced to track the resulting frequency via
the phase lock described above.

3.1.3

Self-referencing

In 1999 Telle and coworkers suggested several methods allowing for direct measurement and stabilization of the frequency offset of the fs comb, without using anotheroptical standard [21]. These ‘self-referencing’ methods employ non-linear processes likesecond harmonic generation to compare the spectral extremes of the comb. In thatwork, it is shown that frequency-doubling of a number of modes from the infrared endof the spectrum and heterodyning them with the existing visible section of the spectrum
will result in a beat frequency, equal to the offset of the fs comb from integer multiplesof the repetition rate:
2 νN − ν2N = 2 (N fr + f0 ) − (2 N fr + f0 ) = f0 .

(3.1)

Thus, f0 can be derived directly from the beat note, with no knowledge of the laser
repetition rate. The requirement for this detection scheme to work is that the frequency
comb contain on the blue side the second harmonic of the red side, i.e. that the laser
optical bandwidth cover an optical octave. Along the same line of thought, the use ofmode-locked lasers for absolute metrology of optical frequencies, where the frequency of
every comb line is an integer multiple of the fixed comb spacing (f0 = 0), was suggested
in a NIST competence proposal from March 1999 [45].
Fortunately, the introduction in 2000 of special air-silica microstructure (MS) optical fibers [22], which broaden the output of the fs comb to span an entire optical octave
solved the bandwidth requirement. These photonic crystal fibers have a very small core

32
diameter (1.7 µm), providing a very tight spatial confinement that generates very highintensities (∼ TW/cm2 ). They have been engineered to exhibit zero GVD near 800 nm,
which minimizes the temporal spread of the pulses propagating in the fiber over significant distances (several cm). The main non-linear effect leading to spectral broadening
in the fiber is strong four-wave-mixing among the different frequency components in the
input pulse spectrum, which generates new spectral components, while maintaining theperiodicity of the incoming frequency comb. The spectral bandwidth at the output of
the MS fiber can extend well beyond one optical octave e.g. 400 to 1200 nm.
The first experimental implementation of the self-referencing technique using external broadening in a MS fiber, followed by doubling of the 1040 nm light and mixing
with the fundamental at 520 nm, was reported by Jones and co-workers in 2000 [1].
Ultra-broadband mode-locked lasers whose direct output spans on optical octave, allowing for direct self-referencing and stabilization of the pulse train, have also been
recently demonstrated [7].

3.1.4The ν-to-2 ν prism interferometerTo achieve spectral broadening, ∼ 120 mW of the fs laser output light (∼ 500mW) are coupled into a ∼ 7-cm piece of MS fiber. The output of the fiber is then sent
into an ν-to-2 ν interferometer for self-comparison.
The configuration chosen for the interferometer is the so-called ‘prism-pair’ geometry, presented in Fig. 3.3. The sequence of SF10 prisms (8-cm prism separation) spatially separates the green (e.g. 532 nm) and red (e.g. 1064 nm) components of thespectrum, which are then reflected by two mirrors located very close to each other. The
green-reflecting mirror is mounted on a translation stage, to easily adjust the temporal
overlap between the red and green pulses at the detector. When the interferometer iswell aligned and working, the distance between the end mirrors is ∼ 1 mm.
The beams retrace their path through the prism pair with a slight vertical offset

33
to MOT

Ti:S Laser
60 X

fr lock

60 X

MS fiberBBOν- to-2ν interferometer

532 nm filter(bandwidth 10 nm)

polarizer
f0 photodetector

Figure 3.3: Diagram of the prism interferometer used for the detection of the comb
offset frequency. The broad spectrum from the MS fiber is sent into a prism pair for
spatial dispersion. The light near 532 nm is reflected by one end mirror, while thelight near 1064 nm hits a second end mirror. Both beams are reflected back throughthe prism pair, recombined and then sent through a nonlinear BBO crystal tuned for
efficient second harmonic generation at 1064 nm. The output is filtered, and the f0 beatis detected on a PD.

in order to be picked off by a mirror positioned below the incoming beam. The pickoff mirror sends the recombined beams to a lens, which focuses them into a nonlinear
crystal (1-mm BBO, type I phase-matching), angle-tuned for efficient second harmonic
generation at 1064 nm.
Two types of green light are present at the output of the BBO crystal: thefrequency-doubled beam obtained from the infrared part of the spectrum and the fundamental green coming out of the MS fiber; the polarization of the frequency-doubled
green is rotated by 90 degrees with respect to the fundamental green. A 10-nm interference filter centered at 532 nm removes the extra, non-contributing spectral components,
while a polarizer ensures that the polarizations of the two green beams are aligned along
the same axis. Spatial overlap and mode-matching of the two beams are essential for agood signal-to-noise ratio (S/N) of the beat signal. A photodiode detects their difference
frequency, the f0 beat, along with the laser repetition rate. In reality, the detected beat34
arises from mixing of thousands of comb lines, not just the two frequencies mentioned,
giving a better S/N. In the experiment, the typical S/N of the offset beat is ∼ 45 dB in
a 100 kHz resolution bandwidth.
The prism pair geometry is advantageous to use compared to other configurations,e.g. a Mach-Zender interferometer, because most of its elements are common-mode (theonly differential path is the prism pair region), which makes it easier to align and tweak.

3.1.5

Stabilization of the laser offset frequency

The output of the ν-to-2 ν PD is sent through two 50 MHz low-pass filters (LPF),
to reduce all the unnecessary rf frequencies in the spectrum (otherwise, they would
saturate the subsequent amplifiers), then a small portion of the signal is sent via a
directional coupler to an rf spectrum analyzer for monitoring of the beat note. The
main signal is amplified to ∼ -5 dBm and then frequency-prescaled (divided by 64)
before its phase is compared to a stable rf oscillator such as a DDS. Unlike the fr lockwhich employs an analog phase detector, the stabilization circuit used for f0 employs adigital phase detector (DPD), which provides both a frequency and a phase lock. This
allows for a larger capture and locking range than the double-balanced mixer case andincreases the stability of the lock. Again, the error signal derived by comparing against
the reference frequency is integrated, filtered and amplified in a JILA LF (see Fig. 3.4)
and then sent to the laser for frequency correction. Note that the local oscillators used
for the fr and f0 stabilization are all referenced to the same local commercial cesium
clock.
As announced earlier in section 3.1.1, f0 control is carried out by fine-tuning thelaser end mirror angle using a twister PZT. The twister PZT used to tilt the end mirror
is a 0.5 inch diameter cylinder, about 0.5 inch long, that has been modified such that theouter electrical surface is split to allow for the two outer sides to have different voltages
with respect to the inner surface of the PZT. When applying opposite-sign voltages to35
to MOT

Ti:S Laser
fr lockDDSν-to-2ν
interferometerf0

DPD
50 MHz LPF

rf amplifier

spectrum analyzer
to End Mirror (tilting PZT)JILA LF

HV driver (±150 V)

Figure 3.4: Stabilization diagram used for phase-locking of the laser offset frequency f0 ;LPF stands for low-pass filter, DPD for direct phase detector, DDS for direct digitalsynthesizer and LF for loop filter. The details are given in the text.

the two sides, one contracts, while the other expands, with the net effect of an angle
change for the end mirror mounted on the PZT. As mentioned before, this is effectively
a group delay, which depends on the voltage sent to the PZT. Changing the voltage inthe 0 to ± 150 V range allows for scanning of the offset frequency beat over its full rangeof ∼ 50 MHz.
Another method of adjusting and stabilizing f0 is employing an acousto-optic
modulator (AOM) to control the pump laser power. Modulating the pump laser power
actually affects several laser parameters. Systematic studies of intensity-related dynamics in mode-locked Ti:S lasers have been recently undertaken [46, 47] to understand theeffects on fr and f0 .
To summarize, fr and f0 stabilization and referencing to an atomic clock resultin knowledge of the absolute frequencies of the ∼ 106 comb lines, with an accuracy only
limited by the frequency reference used. The stabilized comb can now be employed as aprecision frequency ruler for any optical frequency in the visible to near-infrared range.

36

3.2

The MOTThe magneto-optical trap (MOT) is currently the starting point for many ex-periments with laser-cooled atoms. Cold atoms are used for precision measurements,
atom interferometers, atomic clocks, to name just a few examples. Advances in laser
cooling of alkali atoms [48] have also paved the way to producing the first Bose-Einsteincondensates and degenerate Fermi gases [49–51]. In particular, vapor-cell traps [52] likethe one used in this experiment, are simple and versatile tools for many experimentswith cold atoms.
Our experiment was performed on a sample of laser-trapped and -cooled Rubidium 87 atoms (see Fig. 4.2 for a detailed 87 Rb level diagram, I=3/2). The 5S1/2 → 5P3/2
cooling transitions can be easily accessed with commercially available laser diodes centered at 780 nm, making 87 Rb a convenient choice for laser cooling and trapping. We
use two lasers to trap and cool the atoms. The cooling beams are ∼10 MHz red-detuned
from the 5S1/2 (F=2) → 5P3/2 (F0 =3) cycling transition. An additional laser, the socalled ‘repumper’, tuned to the 5S1/2 (F=1) → 5P3/2 (F0 =2) transition, is needed toreturn the atoms to the cooling cycle whenever they decay to the F=1 ground level,thus preventing them from falling dark due to optical pumping.

3.2.1

Diode lasers: description and stabilization

The laser diodes used are Sharp LT024MD (specified output power of 20 mW)
for the repump laser (RL) and Sanyo DL7140-201 (specified output power of 70 mW)
for the trap laser (TL). They are configured in an external cavity geometry, based on adesign described in reference [53, 54], with ∼ 3.5 cm extended cavity length. The lasers
employ a diffraction grating (1200 lines/mm) aligned in the Littrow configuration [55]to couple the first diffraction order back into the laser. One PZT controls the horizontal
tilt of the grating, permitting adjustments of the laser wavelength. A change in the PZT

37length alters the grating angle and the external cavity length at the same time, allowing
for continuous scans of typically ∼ 5 GHz. Additionally, the grating is mounted on afast disk PZT, which is servoed together with the laser diode current to stabilize thelaser frequency.
Both diode lasers are locked to the specified atomic transitions using standard
saturated absorption spectroscopy [53], which yields sub-Doppler hyperfine lines thatare used to frequency-stabilize the laser (see Fig. 3.5). The TL is locked to the red sideof the saturated absorption peak; a side lock is not ideal, since the locking set point
is inevitably dependent on power broadening of the transition lines. We use acoustooptic modulators (AOMs) to create beams at various detunings and also to control thetiming of the lasers in the experiment. The TL light is steered through two AOMs: thefirst AOM deflects a small amount of power (-1 order) into the saturated absorption
spectrometer, for frequency stabilization. The second AOM acts as a shutter for thelight, by directing the negative first order diffracted beam to the MOT cell. Thus, theMOT red detuning is set by the side lock along with the frequency difference of the two
AOMs.
The RL is locked to the peak of the crossover transition between the 5S1/2 (F=1)→
5P3/2 (F0 =1) and the 5S1/2 (F=1) → 5P3/2 (F0 =2) transitions, using modulation sidebands at 4.6 MHz, written directly on the diode-laser current (the laser output current
can be modulated rapidly by sending a voltage into the rf modulation input, this in turnwill modulate the frequency of the laser). The signal from the RL saturated absorption spectrometer is detected with a homemade resonant photodetector (PD) and thencompared in a double-balanced mixer with the local oscillator at 4.6 MHz, resulting
in dispersion curves for all the features in the saturated absorption spectrum [56, 57].
The value of these dispersion curves changes sign at the peak, allowing for a lock at thezero-crossing of the error signal. Since the RL is offset peak-locked, an AOM is needed
to shift the frequency into resonance with either the 5S1/2 (F=1) → 5P3/2 (F0 =1) or the38
87

Rb F=1 to F'=

RL
lock

1

2

0

85

Rb F=2 to F'=
2
1

85

Rb F=3 to F'=

2

87

3

3

4

Rb F=2 to F'=

TL
lock
3

2
1
2/3
1/2
1/3
700 MHz

Figure 3.5: Saturated absorption spectra, showing the hyperfine structure of the
5S1/2 → 5P3/2 transitions for the two Rb isotopes. The numbers labeling the hyperfine
peaks correspond to F0 states in the 5P3/2 manifold. Note that the spectra contain bothtrue Doppler-free resonances (numbered peaks) and crossover resonances. The crossover
resonances occur at frequencies (νa + νb )/2 for each pair of true resonances at frequency
νa and νb , as exemplified for the bottom spectrum.

39
5S1/2 (F=1) → 5P3/2 (F0 =2) transition. We opted for the latter and thus the positive
first order deflected beam is sent to the MOT vapor cell. In order to operate the trapat zero repump detuning, the driving frequency for the RL AOM is equal to exactly
half of the F0 =1– F0 =2 5P3/2 hyperfine interval (78.45 MHz). This AOM is a fast andconvenient switch used to turn off the RL beam whenever needed.

3.2.2

MOT details

The main TL beam is spatially filtered by a 70 µm pinhole, which reduces thehigh-frequency spatial modes, and collimated to roughly a 1-cm beam diameter witha beam-enlarging telescope. It is then divided into three separate, intensity-balanced
beams using λ/2 plates and polarizing beamsplitter cubes. The three beams are sent
to the experiment glass cell using 1-inch diameter optics, in a standard retro-reflected
MOT configuration [58]. The atoms are caught in the MOT using these three pairs oforthogonal, circularly-polarized beams in the typical σ + − σ − arrangement, and detuned∼ 10 MHz to the red of the atomic resonance. The RL beam is directed into the cellalong one of the TL beams. We monitor the beams in the cell and the position of thetrapped cloud with a low-cost CCD camera.
We load the MOT from the vapor produced by Rb dispensers (‘getters’) in acylindrical glass cell. The vapor cell is continuously pumped by an ion pump andtypically reaches a background pressure of 10−9 Torr. The source of Rb consists ofthree nichrome (Ni-Cr alloy) dispenser strips, connected in parallel and placed in asmall side arm of the cell. They contain a Rb salt and release atoms through a chemical
reaction when an electric current is passed through them and heats the nichrome. Only
one dispenser is used at a time, and the operating current is usually 3.5 A.
The quadrupole magnetic field necessary to operate a MOT is provided by a pairof anti-Helmholtz coils. The two coils are wound from ‘magnet wire’, i.e. Cu wire (∼ 1
mm diameter) coated with an insulating material, and are wrapped around the axis of

40
the cylindrical cell. Each coil is 7 cm in diameter and has 33 turns. The coils are spaced
5.5 cm apart and are supplied with ∼ 4.5 A, generating a magnetic field gradient of ∼10 G/cm.
Stray magnetic fields, due mainly to the Earth’s field, can shift the zero-point of
the MOT field. They are compensated by three pairs of orthogonal Helmholtz coils.
The glass cell is surrounded by these pairs of shim coils, each run by a different current
supply. In the early stages of the experiment, the beam alignment and ambient magnetic
field compensation were verified by monitoring the isotropy of the molasses expansion.
We were aiming for a slow and uniform expansion of the atomic cloud upon switching off
the quadrupole field. At a later stage, after obtaining the two-photon signal as describedin Chapter 4, we realized that we could use it to find the zero-point of the magnetic
field.
Resonance fluorescence from the atoms is detected with a photodiode (PD), leading to an estimate of the atom number based on the PD current, the PD responsitivity,
the solid angle of the light collected by the imaging lens, the energy of a photon andthe optical scattering rate. We typically trap 5 × 106 atoms in a 0.5 mm diameter MOT
with a 1/e lifetime between 3 and 5 s.
After collecting the atoms in the MOT, we transfer them to optical molasses [59]for further cooling using polarization gradients (PGC). To experimentally implement
PGC, the quadrupole field gradient is turned off, the TL is further red-detuned fromresonance (∼ 40 MHz) and the TL beam intensity is decreased. The resulting subDoppler cloud temperature established by PGC is ∼ 15 µK, providing an ideal initialcondition for spectroscopy.
I have presented here the basic operation details of the MOT. In Chapter 4 I will
describe how the trapped atoms are probed with the frequency comb of a femtosecond
laser.

Chapter 4

Characterization of the systematic effects related to direct frequency
comb spectroscopy

We would like to use a phase-stabilized femtosecond laser comb directly for highresolution spectroscopy and absolute optical frequency measurements of one- and twophoton transitions in laser-cooled 87 Rb atoms. We defer further discussion of the spectroscopy, however, until we have explored some of the important sources of possible
systematic error [60]. In section 4.4.2, I will discuss both experimentally and theoretically the mechanical action of the probe on the atoms. I will then present the AC
Stark frequency shifts induced by the femtosecond light in section 4.4.3. Finally, thelast systematic error we studied, arising from Zeeman shifts, will be presented in section 4.4.4. As briefly mentioned in Chapter 3, the short-term stability of the reference
used to stabilize the repetition rate also warrants attention. We will see that the main
contributions to the systematics are those arising from mechanical effects and the AC
Stark shifts.

4.1

Doppler-free two-photon transitions with cw and pulsed sources
Study of two-photon transitions to achieve high-resolution spectroscopy or to

maximize the excitation efficiency of a desired state has been an active field of researchin the past decades. Doppler-free multi-photon transitions have been theoretically studied by Cagnac et al. [61]. The first experimental demonstrations followed shortly, by42
observing two-photon absorption resonances in Na using tunable dye lasers [62–64]. The
effects of Doppler broadening in the vapor were eliminated by making the atoms absorb
one photon from each of the two equal-energy, counterpropagating cw beams (Fig. 4.1,
top panel).
In a closely related experiment, optical beams were arranged to both (i) propagate
in opposite directions as well as (ii) have unequal frequencies (Fig. 4.1, bottom panel).
This configuration was used to demonstrate resonantly enhanced two-photon absorptionin Na vapor [65]. The two-photon absorption cross-section was enhanced by over 7 ordersof magnitude, by letting one of the photons be nearly resonant with an intermediate
state. Doppler effects were greatly reduced by using counter-propagating beams, but
not totally eliminated. In a different experiment, the two-photon transition rate wasresonantly enhanced by using a fast atomic beam [66], where the high atomic velocities
allowed for Doppler tuning of an intermediate state into exact resonance in the rest
frame of the atom.

5D

DL

Equal Frequency

5P

cell

5S
5D

DL1

Two Color=> 107 enhancement

DL2

5P
5S

cell

Figure 4.1: Schematic of Doppler-free two-photon spectroscopy in a cell, with equalfrequency (top) and two-color (bottom) counterpropagating beams.

43
Doppler-free two-photon spectroscopy has also enabled the observation of optical Ramsey fringes. In the experiments of Salour [40, 67] the atoms have interactedwith two time-delayed phase-coherent laser pulses, while Baklanov et al. have suggested
two spatially separated light standing waves [68]. Note that Bergquist et al. have observed Ramsey fringes in the saturated-absorption signal from a fast beam of Ne* atoms
crossing three and four equally spaced spatially separated standing-wave beams [69].
With regard to employing pulsed sources, high-resolution two-photon spectroscopyusing picosecond pulsed light and hence, without any appreciable intermediate-state interaction or absolute frequency reference, has been previously demonstrated [27], with arecent extension to laser-cooled samples [70]. Cold atoms confined in a MOT provide analmost-Doppler-free environment for use in spectroscopic studies. For the case of twophoton transitions, where the absorption cross sections are relatively small and thus therequired intensities are rather large, the cold sample mitigates the Doppler broadening
arising from the strong focusing needed.
The advent of ultrafast lasers combined with pulse-shaping techniques has led to
new schemes [14, 71], where two-photon transitions which constitute the lowest-ordernonlinear interaction, are used as a benchmark system. In particular, Silberberg andcoworkers have shown that the two-photon absorption rate is maximized for transformlimited pulses only in the case of transitions without any resonant intermediate state.
For transitions involving a nearly resonant intermediate state, an enhancement by afactor of 7 of the two-photon absorption rate in Rb vapor has been obtained by shaping
the pulse [71]. Thus, the final population transfer was controlled by manipulating thecoherence properties of the optical field.

4.2

Background for the technique of two-photon DFCS
We directly interrogate a sample of laser-cooled 87 Rb atoms with a frequency

comb and induce two-photon transitions. This capability is enabled by the bandwidth

44

7S1/2

F’’=2
639.4 MHzF’’=1
18 224 929 MHz
F’’=1
F’’=2

741 nm

5D5/2

15.9 MHz22.9 MHzF’’=328.8 MHz

5D3/2

F’’=4

44.2 MHz27.9 MHz776 nm13.5 MHz

F’’=3
F’’=2

88 857 MHz

F’’=1
F’’=0

386 252 075 MHz

762 nm5P3/2

266.6 MHz
156.9 MHz72.2 MHz

780 nm

5P1/2

F’=3
F’=2
F’=1
F’=0
7 122 413 MHz
F’=2816.6 MHzF’=1795 nm377 104 390 MHz

5S1/2

F=2
6834.6 MHz

F=1Figure 4.2: Schematic of the 87 Rb energy levels participating in the one- and twophoton transitions we studied, including the wavelength and energy level information.
The levels highlighted in red correspond to the closed 2-3-4 transition pathway (section
4.4). We have investigated seven two-photon transitions, as detailed in section 5.2.

45
associated with the femtosecond pulse, which is sufficiently broad that many fine andhyperfine atomic states (see Fig. 4.2) can be excited by tuning the relevant comb components into resonance. The two independent parameters fr and f0 provide freedom inchoosing an appropriate frequency comb for control of resonant signal enhancement via
the intermediate 5P1/2 and 5P3/2 states.
Our experimental prototype system to study DFCS is therefore a set of twophoton transitions from the ground-state 5S1/2 to the excited 5D3/2 and 5D5/2 states,
enhanced by the 5P intermediate states as shown in Fig. 4.3. The dipole-allowed intermediate states, 5P3/2 and 5P1/2 , are located ∼ 2 and 17 nm (i.e. ∼ 1 and 8 THz) below
the energy degenerate virtual level for the two-photon transition, respectively (Fig. 4.2).
The lifetime of the 5P intermediate states is 27 ns, the lifetime of the 5D states is 241
ns and the lifetime of the 7S states is 88 ns [72].

phase-coherent
comb

δSD

5D

νPD = M fr + fo
fr
5P

…

f0

δSP

νSP = N fr + fo
5S

Figure 4.3: Resonantly enhanced two-photon transition with detunings δSP and δSDfor the pair of comb modes that makes the dominant contribution to the transition
probability.

As mentioned in Chapter 1, the optical frequency of a particular comb mode

46
can be expressed as νN = N fr + f0 , where fr is the pulse repetition rate, f0 is thecarrier envelope offset frequency, and N is an integer on the order of 106 . The sum
frequency of the light from two comb lines labeled by N and M is given by ν2γ =(N + M )fr + 2f0 . There are several hundred thousand comb pairs (N , M ) that yield
the same sum frequency and thus could contribute to the transition amplitude when ν2γ
is resonant with the two-photon transition (δSD ∼ 0 as shown in Fig. 4.3). However, inestimating their relative contributions, it is necessary to consider the intermediate 5P
states that provide resonant enhancement.
When one of the comb lines is tuned near resonance with one of the 5S-5P transitions (δSP ∼ 0 as shown in Fig. 4.3), the resonant enhancement causes the corresponding
pair to make the dominant contribution to the two-photon transition over all of the otherpairs. This dominant contribution is reinforced by destructive quantum interference between comb pairs symmetrically detuned on either side of the P state, resulting in a
∼ 180◦ phase difference [71]. For the non-resonant configuration of comb modes detuned ±kfr /2 (k ≥ 1, odd integer) away from the P state, with all pairs satisfying thetwo-photon resonance, the transition amplitudes associated with (+k) and (-k) modes
will again destructively interfere. However, there will be a net non-resonant contribution due mainly to the existence of multiple P states that break the symmetry of the
comb distribution. This destructive interference can be made constructive by flipping
the spectral phase about the P state [71].
The two-photon transition spectrum is obtained by scanning fr or f0 ; their precisely measured values, along with (N , M ), determine all relevant atomic energy levels
in absolute terms.

4.3

Experimental method
The experiment is performed with an optical frequency comb (emitted from a

20-fs, 100-MHz repetition rate, mode-locked Ti:sapphire laser) centered at 778 nm with

47

MOT
7.5 ms

PGC
2 ms

t

500 µson
Two-photon
timing diagramτ

fs light

Normalized Light Intensity
1.0
0.8
0.6
0.4

Shutter is
on at τ=6 µs

0.2
0.0
0

40

80

120

160

Time,τ (µs)
Figure 4.4: Timing scheme for the 100 Hz experiment cycle. The atoms are first loadedin the MOT for 7.5 ms and then, at the end of the loading time, the trapping quadrupole
magnetic field is extinguished (not shown). Additional cooling of the atoms is done withpolarization gradients (PGC) for 2 ms. All MOT-related fields are turned off at the endof the PGC stage. At this point, the femtosecond laser is switched on using the liquid
crystal shutter (30-µs response time). The 500-µs zoom window shows the variable
probe window (τ ) used for the two-photon measurements. Signals are averaged overhundreds of 10 ms experiment cycles.

48

Fs Shutter
Beam

TimingMOT
PMT

Computer

Photon Counter

Figure 4.5: Shown here is the optical configuration for the counterpropagating beams,
implemented in all our two-photon studies, including those described in Chapters 5 and6. The 420-nm blue fluorescence detected by the PMT is sent to a photon counter andthen read into the computer.

a full width at half maximum bandwidth of ∼ 55 nm (∼ 26 THz). There is ∼ 200 nWof power in each of the comb lines resonant with the 5S-5P and 5P-5D transitions. The
light is typically focused to a beam waist of 130 µm, giving an on-axis intensity of ∼ 0.8
mW/cm2 in each comb line. For time-resolved studies, the sample is exposed to lightby a liquid crystal with a 30-µs response time.
As described in Chapter 3, the atomic source is a cloud of 87 Rb trapped andcooled in a vapor-cell MOT. As shown in Fig. 4.4, the typical loading cycle used for the
MOT is 100 Hz and the sequence of the experiment is as follows: the atoms are loadedin the MOT for 7.5 ms, then the quadrupole magnetic field for the trap is switched off,
the atoms are further cooled with polarization gradients (PGC) for 2 ms, then all the
MOT beams are extinguished, the femtosecond comb beam is switched on for 500 µs
using the shutter, and finally, the 420 nm fluorescence is detected.
The cw MOT repumping laser controls the initial populations of the two groundstate hyperfine levels. During the actual probing period, the femtosecond comb itself
acts as a repumper, allowing one to maintain a stable population in the initial groundstate.

49
The excited 5D state population is determined from the 5D-6P-5S radiative cascade: the atoms relax to the 6P state and then decay down to the ground state, emitting photons at 420 nm. These blue photons are detected with a photomultiplier tube
(PMT) centered (Fig. 4.5) at 420 nm and are counted and subsequently time-binned
with a multi-channel photon counter, with a typical integration time of 10 µs. In general,signals are averaged over hundreds of 10-ms experiment cycles.
One of the main problems in acquiring reproducible data is that the femtosecond
laser is located on one optical table, and the Rb apparatus on a different optical table∼10 meters away. This creates serious beam-pointing instabilities. We have implemented a beam-steering servo based on a quadrant photodiode and a tip/tilt mirror toactively control the beam pointing.

4.4

Main systematic effects
As in the case of precision spectroscopy performed with cw lasers, the use of the

femtosecond comb as a precision probe requires a careful understanding of all systematiceffects. Indeed the dominant effects are the mechanical effect of the optical comb on theatomic motion, the light shift by the probe laser, and the Zeeman frequency shifts. The
short-term stability of the rf source used to reference fr is also important, as discussed
in the next section.
We have performed all studies of the systematics involved on one sample transition, namely the 5S1/2 (F=2)→ 5P3/2 (F0 =3)→ 5D5/2 (F00 =4) closed two-photon transition, which gives the best signal-to-noise ratio. A closed transition is defined to be atransition which decays back to the hyperfine level from which it was excited. I havehighlighted in Fig. 4.2, in red, the states involved in the only closed two-photon transition occurring in 87 Rb among the levels shown. This closed transition occurs when theatoms start in the F=2 state, are then excited to the F0 =3 state, and then to the F00 =4
state and finally decay back to the F=2 state (through either the 5P or the 6P channel).

50

f0 tuning andlocking electronics

f0 tuning of fs laser
to MOT

β
EM
OC

f0

2500

5D

Typical 2-3-4
two-photonlineshape

6P
5P
PMT

Signal (a.u.)

2000
1500
1000
500

5S

0
17

18

19

20

21

22

f0 (MHz)

Figure 4.6: The top panel shows the experimental setup for scanning f0 and the lower
panel presents a typical 5S1/2 (F=2)→ 5P3/2 (F0 =3)→ 5D5/2 (F00 =4) (‘2-3-4’) closed
two-photon lineshape obtained from such a frequency scan. The decay channel used for
detecting the population of the 5D excited state is also shown.

51
In contrast, note for example that the transitions F=2 to F0 =1 or 2 are not closed, since
they have a decay channel to the F=1 ground hyperfine level and thus, the atoms do
not all return to the initial F=2 state. In summary, the 2-3-4 is a closed transition
because for each of the two steps 5S-5P and 5P-5D, the electric dipole selection rule(∆F = 0, ±1) prevents the excited atoms from falling into the F=1 ground state.
As a convention, any two-photon process starting in F=2 that accesses the F0 =3
state will be referred to as having a closed first step. We will study all of the two-photontransitions that have a closed first step in Chapter 5.
For all of the systematic studies, we will scan the closed two-photon transition
repeatedly, taking note of how the line center and linewidth are affected by the changes
we implement. We will use an f0 scan, meaning fr is locked while f0 is swept (Fig. 4.6).
As a reminder, controlling f0 causes a rigid spectral shift of the optical comb, where thefrequency of all the lines is changed without modifying the comb spacing.

4.4.1

Stability of the reference for the pulse repetition rate

As detailed in Chapter 3, both the repetition rate and the carrier-envelope phaseof the fs laser are locked to radio-frequency oscillators with ultra-low phase noise.
At fast time scales, the femtosecond laser repetition frequency has less phasenoise, in other words it is more stable, than the best rf sources available. Therefore,
it may be that we are writing fast noise onto the laser if we lock it too tightly to an
external standard such as frequency synthesizer. Thus, after frequency multiplication
up into the optical domain, customary rf sources are not useful as a means to stabilize
the laser in the short term. As a reminder, we are using a custom-made crystal oscillator(Wenzel) that has very good short-term stability (fractional instability of 5 × 10−13 foran averaging time of 1 s). At long times though, the Wenzel drifts, which is a potential
inconvenience when doing a long data run. For all the important measurements wecount the repetition rate as we acquire data. The radio frequency counter and the local

52
oscillator used for the f0 stabilization are both referenced to the same local commercial
cesium clock. A slow servo control of the repetition rate is helpful in reducing the laser
frequency noise and drift resulting from lab vibrations and thermal drift of the laser
cavity.
We are able to lock the repetition rate to a number of different sources, including
two synthesizers referenced to a NIST hydrogen maser signal (fractional instability of
3 × 10−13 at 1 s), which were used as rf sources for the fr stabilization. The maser signal
was transferred to JILA through a single-mode fiber using amplitude modulation of acw laser [73]. Figure 4.7 has the resulting two-photon lineshape for the three standards
used. We observed that the two-photon linewidth for the case of the NIST synthesizers
was consistently broader than the Wenzel-referenced situation. We attribute this to theshort-term noise of the synthesizers being worse than the Wenzel, and possibly to fastphase noise in the optical fiber used for signal transfer.

Signal Amplitude (a.u.)

2500

W enzel Crystal O scillator
Maser-referenced HP8662A
Maser-referenced Agilent ESG

2000
1500
1000
500

16

17

18

19

20

21

22

C arrier Offset F req u en cy (MH z )Figure 4.7: Comparison among two-photon lineshapes obtained with different rf references for the fr lock, showing the importance of having a reference with good short-term
stability.

53
At a later time, we phase-locked one comb line near 1064 nm (282 THz) to a
Nd:YAG laser that was stabilized to molecular iodine. Although the short-term stabilityof the I2 -stabilized Nd:YAG is an order of magnitude better than that of the Wenzel
oscillator, we did not observe any improvement in the two-photon resonance linewidth.
We suspect that we are limited by the bandwidth of the PZT used for the fr lock.
This is a technical limitation, and in principle it can be overcome by improving theactuator bandwidth or by using pump-power control. Also, note that the I2 system has
a fast linewidth of ∼ 5 kHz (i.e. fractional instability of 1.8 × 10−11 ), which at longer
timescales falls down as the square root of the averaging time.

4.4.2

Mechanical action of the probe

I will now discuss an important systematic effect, arising from the mechanicalaction of the femtosecond probe on the atoms. Although the sub-Doppler temperatureestablished by polarization-gradient cooling provides an ideal initial condition for spectroscopy, it cannot be expected to survive the momentum transferred by the light fromthe comb. At best, the mean-square momentum will increase, leading to Doppler broadening. Worse, the momentum acquired may lead to systematic shifts in the resonanceline positions. Thus, we seek to understand momentum transfer from the comb and to
mitigate its effects insofar as possible.
We expect that the momentum transfer is dominated by the interaction of the
comb mode closest to resonance with the 5S-5P transition, given that the radiative
decay rate of the 5P-5S transition is an order of magnitude greater than that of 5D-5P,
and the population largely resides in the 5S state.
For ease of modeling, we use a single-beam (traveling wave) configuration andchoose a comb structure such that the closed 5S1/2 (F = 2) → 5P3/2 (F0 = 3) → 5D5/2(F00 = 4) transition is dominant (see Fig. 4.8). The temporal signal evolution is shownin Fig. 4.9 for the initial detunings of δSP = 0 (red curve), δSP = 1 MHz (green curve),

54

2δSP

5D

5P

δSP

5S

fr & fo fixed
Figure 4.8: Schematic of the detunings from the intermediate and excited states relevant
for the radiation pressure studies. fr and f0 are kept constant for these experiments,
and the temporal signal evolution is monitored for different blue detunings.

and δSP = 2 MHz (blue curve).
In the fully resonant case (red curve), the delayed signal peak at 30 µs reflects
the switching time of our liquid crystal shutter. The subsequent decay arises as theaccumulated momentum transfer gradually blue-shifts the atoms out of two-photon
resonance.
Although the 5S-5P transition controls the mechanical action, the two-photon
resonance condition plays a more decisive role in the observed signal decay due to the
10-times-narrower D-state linewidth. When δSP = 1 MHz (green curve), the atoms,
initially with δSD = 2 MHz, will only reach the peak of the two-photon resonanceafter they have reached the velocity at which the Doppler shift compensates the initial
detuning. Beyond this velocity, the signal contribution decreases. Similarly, whenδSP = 2 MHz (blue curve), the signal peaks still later.
The next section shows that all three experimental results agree well with ourtheory based on a simple two-level dissipative light force model. The peak position is

55
On resonanceLaser blue-detuned by 1 MHz
Laser blue-detuned by 2 MHz

Signal (a.u.)

800
600
400
200
0
0

50

100

150

200

250

300

Time (µs)

Figure 4.9: Momentum transferred by the optical frequency comb (in a single probe
beam configuration) to the cold 87 Rb atoms, observed via the time-dependent fluorescence signal from the 5D states. The optical comb has fixed values of fr and f0 to achieve
the desired detunings (with respect to atoms initially at rest) of δSP = 0, δSD = 0 (redcurve); δSP = 1 MHz, δSD = 2 MHz (green curve); and δSP = 2 MHz, δSD = 4 MHz(blue curve), respectively, for the dominant pair of comb modes.

determined by the scattering rate, photon recoil, and the initial detunings. The widthand shape of the peak are associated with the stochastic nature of radiation pressure,the intensity variation over the radial beam profile, and the finite laser linewidth.

4.4.2.1

Momentum transfer model

To understand the momentum transferred by the light from the frequency comb
to the atoms we used single-beam (traveling wave) excitation for the experiment.
In this case, the net force on the atoms is
F = h̄kΓs,
2 1 + s + [2 (δ + ωD )/Γ ]2(4.1)

where s is the saturation parameter i.e. the ratio of the light intensity I to the saturation
intensity Is . For the case of linear polarization in 87 Rb Is = 2.5 mW/cm2 . The detuningof the laser from resonance is δ = ωl − ωa , where ωl is the laser frequency and ωa is theatomic resonance frequency. The Doppler shift seen by the moving atoms is ωD = −k·v.

56
We can see from this equation that for a maximum scattering rate, i.e. at the peak of
the resonance, the Doppler-shifted laser frequency in the moving atoms’ reference frame
should match the atomic resonance frequency. In other words, for a given initial laser
detuning from resonance, the atoms will accelerate until they reach a threshold velocity,at which point the peak of the resonance is attained (the threshold velocity occurs when
the Doppler shift has compensated the chosen detuning).
For a constant intensity the above equation is easily separated and integrated,giving:Z v

4πv
dv 1 + s +
δ−
Γ
λ
v0




2 !

= vr

Γ
st
2

(4.2)

with vr = h̄km the recoil velocity of the atom. The right-hand side of this equation
becomes upon integration a cubic function of v, from which we can obtain an expression
for v(t) by comparing against the left-hand side. In reality, though, the intensity depends
on both the beam radial coordinate and time. The time dependence of the intensity
is determined by our liquid crystal shutter response and has been well characterized.
Taking it into account, along with the correct scattering rate, photon recoil and initial
detunings gives a relatively good agreement with the experimental data, as far as thepeak position is concerned, as presented in Fig. 4.10.
The width and shape of the peaks do not agree with the experiment yet. For a
more realistic model, we have to consider and include the intensity variation with thebeam radial coordinate; for a Gaussian spatial profile of the laser beam the intensity isgiven by"#r2
I(r) = I0 exp −2 2 ,
w0

(4.3)

where I0 is the on-axis intensity (∼ 0.8 mW/cm2 ) and w0 is the beam waist (∼ 130µm).
It is easy to show that ∼ 90% of the signal originates from within a radius of
100 µm. Actually, most of the signal comes from a radius of about 50 µm, as seen in
Fig. 4.11.

57
On resonanceLaser blue-detuned by 1 MHz
Laser blue-detuned by 2 MHz

Signal (a.u.)

800
600
400
200
0
0

50

100

150

200

250

300

Time (µs)

Figure 4.10: Momentum transferred by the frequency comb to the atoms. A comparison
between the experiment and the simplified theoretical model, which assumes a uniform
probing laser intensity.

0.12

Signal (a.u.)

0.10

r I2(r)

0.08
0.06
0.04
0.02
0.00
0

50

100

150

200

Radial coordinate r (µm)
Figure 4.11: Two-photon signal vs. radial coordinate r for a Gaussian probe beamprofile.

Including the r dependence, as well as a dependence on the laser linewidth (∼ 330
kHz) gives a reasonable agreement between data and theory (solid lines in Fig. 4.12),
as illustrated in Fig. 4.12. The dependence on the laser linewidth is introduced in the

58
simulation via a random change of detuning for each iteration, governed by a Gaussian
distribution centered at zero with a variance equal to the linewidth.
It is evident that the force exerted on an atom by the optical comb is well modeledby the radiation pressure due to a single comb mode tuned close to the 5S-5P resonance.
8000

Solid lines:
Data:

Amplitude (a.u.)

7000
6000

Theory0 MHz+ 1 MHz+ 2 MHz

5000
4000
3000
2000
1000
0
0

50

100

150

200

250

300

Time (µs)
Figure 4.12: Momentum transferred by the frequency comb to the atoms. A comparison between the experiment and the more complete theoretical model, which includes
integration over the radial probe beam profile and takes into account the laser linewidth.

As a first step in mitigating the effect of light-induced momentum transfer, we
use intensity-balanced counterpropagating beams generated by splitting the pulse witha beamsplitter. We have arranged the setup to establish this counterpropagating beamgeometry so as to have the focused beams spatially overlapped at the MOT. Once thesetup is well aligned, the beams reflected from the beamsplitter back toward the laser
cavity can disrupt the mode-locking process. Therefore, we employ an optical isolator
to prevent feedback to the laser. As mentioned earlier, the spatial positioning of theinterrogating beams is maintained with feedback control to an actuator on a mirror.
We have made no significant efforts to have the pulses overlap temporally inside

59

Signal Amplitude (a.u.)

300
250
200

τ = 0 µs
τ = 100 µs

Cold Rb

150
100

Cold Rb

50

16

18

20

22

Carrier Offset Frequency (MHz)Figure 4.13: Shown here is the narrowing of the lineshape due to the counterpropagatingbeam arrangement which counteracts the mechanical action of the probe. In addition,in the two-beam case, the line center shift is significantly reduced and the signal isenhanced by a factor of three compared to the single-beam case.

the atomic cloud (the pulse spatial extent is about 10 µm and the beam diameter at
the MOT is about 300 µm). Nonetheless, the pulses do interact with the same group of
atoms within the atomic coherence time, leading to signal enhancement by a factor of ∼3 compared to the single-beam case. We would expect a factor of 4, resulting from thesame quadratic signal scaling presented in section 2.3. We suspect that the difference
between our actual and expected enhancement is the result of both a slight intensity
imbalance between the beams and the presence of a small angle between the fs and the
coplanar (‘east-west’) MOT beam.
In addition to increasing the signal size when using a counterpropagating-beamconfiguration, the line center shift versus time is significantly reduced for this casecompared to the single-beam case, as clearly shown in Fig. 4.13.

60(a) δSP ≈ 0

Signal (a.u.)

7
6
5
4
3
2
1
0

5D

5P

300
250
200
150
100

μs

f - -1
0 f
0

0

,pe
ak (M 1

Hz) 2

50

Ti
m
e(

5S

)

-2

3

(b) δSP ≈ - 4 MHz

5

5D

Signal (a.u.)

4
3
2
1

5P

μs

)

-2

f0 - -1 0
f0
1
,pe (
ak M
Hz) 2

50

Ti
m
e(

5S

0

300
250
200
150
100
3

Figure 4.14: Time evolution of the 5D fluorescence signal lineshape showing the mechanical action of the optical comb for the case of two balanced counterpropagatingprobe beams. fr is fixed while f0 is scanned over the 5D resonance profile for two casesof P-state detuning, (a) δSP ≈ 0 and (b) δSP = −4 MHz. For (a) there is a clear asymmetry in the two-photon lineshape at longer probing times, as detailed in Fig. 4.15. In
contrast, for (b), the lineshape remains symmetric even after 300 µs of probing.

61
We continue to study the momentum transfer associated with the 5S1/2 (F = 2)
→ 5P3/2 (F0 = 3) → 5D5/2 (F00 = 4) transition, fixing fr and scanning f0 . Figure 4.14,(a) and (b), show time evolutions of the detected signal as f0 is swept, for two different
detuning conditions: δSP = 0 in Fig. 4.14(a) and δSP = −4 MHz in Fig. 4.14(b), when
f0 is tuned to the two-photon resonance peak. As f0 is scanned to recover the resonancelineshape, the value of δSP changes.
It is clear that the directed momentum transfer seen in Fig. 4.12 is greatly suppressed, and the heating is more evident. For the case of δSP ≈ 0, the lineshape profile
centered at 300 µs (Fig. 4.15, black circles) shows a marked asymmetry. This feature iseasily understood, because the comb lines are tuned blue relative to both the 5S-5P and
5S-5D resonances, although the detuning of the 5S-5P transition is less dramatic because its linewidth is an order of magnitude larger. Thus, Doppler heating accompanies
the probe of the blue side of the two-photon resonance.

10

Signal (a.u.)8

δSP ≈ 0
δSP ≈ - 4 MHz

6
4
2
0
-3

-2

-1

0

1

2

3

f 0 -f 0,peak (MHz)Figure 4.15: A detailed comparison of the lineshape under two detuning conditions after
an interaction time of 300 µs.

62δSD ≈ 05D

5P

δSP ≠ 05S
Figure 4.16: Schematic of the detunings from the intermediate and excited states relevant for the Stark-shifted data. fr and f0 are chosen such that the 5D state is always
on resonance, while the detuning from the intermediate 5P state of interest is varied.

For the case of δSP ≈ −4 MHz, the intermediate-state detuning is always red as
the two-photon resonance is probed. The red-detuned comb mode helps to maintain asymmetric absorption lineshape even after 300 µs, as confirmed by the corresponding
profile (Fig. 4.15, red squares). Thus, a judicious choice of comb structure can help tomitigate the heating of the sample caused by the probing beam. From Fig. 4.14, (a)and (b), we also observe quite different signal decay rates versus observation time.

4.4.3

AC Stark shift and power broadening

Another systematic source of error is the light-induced AC-Stark shift on various
atomic states to be measured. To assess this effect in the presence of many comb modes,
we again take advantage of the flexibility in control of fr and f0 to vary δSP while
keeping δSD nearly zero for the closed transition (see Fig. 4.16). In this near-resonancestepwise transition case, a non-vanishing value of δSP causes a shift in the measured twophoton transition frequency. Atomic energy shifts occurring in a resonantly enhanced
two-photon process were first observed by Liao and Bjorkholm in Na vapor [74].
To clearly distinguish the AC Stark shift from mechanical actions, we gradually

200

Ramp
Power

150
100

δSP

50

0 MHz
+4 MHz-4 MHz

0

(a)

-50
-100

Linewidth (FWHM, MHz)

Measured- Expected line center (kHz)

63

0

3.0

20

40
60
Time (µs)

100

δSP

Ramp
Power

2.5

80

0 MHz
+ 4 MHz- 4 MHz

2.0
1.5

(b)

1.0
0

40

80
Time (µs)

120Figure 4.17: Measurement of (a) the line center frequency and (b) linewidth for theclosed two-photon transition, revealing frequency shifts from both the AC Stark effectand mechanical action. Extrapolating to zero interrogation time and zero power yields
the corrected atomic transition frequencies (a) and the natural transition linewidths (b).

64
increased the power of the pulses as the laser shutter opened. For both δSP = +4
MHz (blue triangles) and δSP = −4 MHz (red squares) cases, the AC Stark shifts arepresent as soon as the laser is turned on and the transition frequency shift follows thetime evolution of the peak power of the pulse train [Fig. 4.17(a)]. When δSP = 0
MHz (black circles), the measured AC Stark shift is close to zero when the shutter justopens. The frequency shift measured at later times is attributed to the accumulated
photon momentum transfer, which is reduced in the detuned cases. The asymmetry infrequency shift between the red and blue detunings is caused by the presence of other5P hyperfine states that also perturb the 5S-5D transition.
Although the laser spectrum spans roughly 26 THz, the obtained spectroscopic
resolution approaches the atomic natural linewidth. This level of resolution is a resultof the use of ultracold atoms and careful control of the phase-stabilized comb parameters, stray magnetic fields, light-induced shifts, and photon-momentum transfer. We
typically measure two-photon linewidths on the order of 1 MHz [Fig. 4.17(b)], which
is roughly consistent with the convolution of the natural linewidth of 660 kHz and thelaser technical linewidth of 300 kHz. (Note that the laser linewidth is 600 kHz for a
two-photon transition.) The measured transition linewidth is slightly smaller for red
detuning (δSP = −4 MHz) than blue detuning (at the same δSP value) with the sameprobe power.

4.4.4

Nulling stray magnetic fields

A direct consequence of the previous two sections is that for all the spectroscopy
experiment, the line-center values are extrapolated to zero interrogation time to suppress
shifts associated with photon-momentum transfer and the transitions are probed on
optimal resonance (i.e., zero detuning for both the intermediate state and the final
state) to minimize the AC Stark shifts.
Additionally, stray magnetic fields are minimized by applying bias-coil compen-

65
sation in three orthogonal directions. To null the residual magnetic fields we make useof the two-photon signal itself: Zeeman-shifted spectra are obtained with right and left
circularly polarized femtosecond comb light along each direction and the zero-frequency
shift point is determined within ± 20 kHz, as presented in Fig. 4.18.

Zeeman Shift

σ+

1 MHz

σ-0.4

-0.2

0.0

0.2

0.4

Magnetic Field at the MOT (Gauss)
Figure 4.18: The two-photon fluorescence signal is used to cancel the residual magnetic fields: Zeeman-shifted spectra are obtained with right and left circularly polarized
femtosecond comb light and the zero-frequency shift point is determined.

4.5

Conclusion
Because the use of the femtosecond comb is novel as a direct spectroscopic probe,in this chapter we carefully characterized several prominent systematic effects. Indeed,
the dominant effects are the mechanical action of the optical comb on the atomic motion,
the light shift induced by the probe laser, and the Zeeman frequency shifts. Now that wehave isolated the biggest contributions to the systematic line center shift and linewidth
we can actually do precision spectroscopy! In fact, the bandwidth associated with the
femtosecond pulse is sufficiently broad so that many fine and hyperfine atomic states
can be excited by tuning the relevant comb components into resonance.

Chapter 5

Two-photon absolute frequency measurements

One desirable objective in atomic spectroscopy is the ability to obtain a global
energy level picture of the atomic transitions. In fact, the stabilized comb has tremendous possibilities for mapping numerous transitions in a straightforward way. In thischapter, I will show that it is possible to obtain all of the allowed single- and two-photontransitions within the laser bandwidth by a quick scan of fr (∼ 26 Hz). This scanning
method is efficient and eliminates the need for broadly tunable and absolutely referenced cw lasers. This is one of the main advantages of using a frequency comb directlyfor spectroscopy.
Previously, in Chapter 4, we scanned f0 and kept fr locked in order to characterize
the primary contributions to systematic effects on both the line center and the linewidth.
When these results are integrated into the the setup we have a system which can perform
precision mesurements. In this chapter, I focus on some representative 5S-5D transitions
for precision spectroscopy. Indeed, the measurement results approach the accuracy of
those obtained in the best cw measurements to date.
Also, if we consider that we have locked the Ti:S comb to a commercial cesium
clock and thus have generated an absolutely referenced set of optical frequencies, we
can use these frequencies for identification of previously unmeasured transitions. As
an example, we determine the previously unmeasured absolute frequency of the 5S-7S
two-photon resonances. We thus demonstrate that prior knowledge of atomic transition

67
frequencies is not essential for this technique to work, and indicate that it can be appliedin a broad context.

5.1

Scanning fr gives a full spectrum
In general, sweeping fr has the advantage of yielding all the transitions within

the laser bandwidth in only a ∼ 26 Hz scan. This is due to the fact that the ratio of onephoton optical transition frequencies (participating in stepwise two-photon transitions)to fr is ∼ 3.8 × 106 , so that for a change in fr of ∼ 26 Hz the resonant enhancement isrepeated by the next neighboring comb component.
However, the optical frequency for the two-photon transitions is ∼ 770 THz, an
fr harmonic on the order of 7.7 × 106 . Therefore, the two-photon resonance conditionis satisfied every time fr is changed by ∼ 13 Hz, corresponding to a change in the comb
frequency by fr /2 for the mode orders around 3.85 × 106 .
As an aside, I will discuss the two primary differences between scanning fr andscanning f0 : (i) simply stated, fr has the advantage of being multiplied by N , whereas
f0 does not and (ii) scanning fr can easily tune over 100 MHz in the optical region,
whereas scanning f0 in the current configuration produces a 80 MHz sweep at the
maximum. The trouble with scanning f0 is that the beat can reside anywhere between
0 MHz and fr /2 (there is of course a conjugate beat in the fr /2 to fr window) but,in order for us to lock f0 , we are restricted to the 7-47 MHz range. Thus, to obtainbroader tunability in the current experimental configuration, we scan f0 up to 47 MHz,
then change the sign within the servo to acquire lock on the conjugate beat and finally,
we proceed to scan this conjugate beat up to -47 MHz. This gives a total tuning of 80
MHz. (Note that we can adapt the current setup, for example by including acoustooptic modulators in the ν-to-2 ν interferometer, to obtain continuous tuning without
having to change the servo sign.)
Figure 5.1 presents the two-photon transition spectrum obtained by scanning68

fr tuning andlocking electronics

fr tuning of fs laser
to MOT

EM

δL
OC

fr

1100

5D5/2 F=4
(5P Resonant)Signal (Counts)

1000
900

5D5/2 F=4
(5P Non-Resonant)200

repeat

100
0

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

f r - 99 933 200.0 (Hz)Figure 5.1: The top panel shows the experimental setup for scanning the laser repetition
frequency fr and the lower one presents the two-photon spectrum obtained from sucha scan. All possible 5S-5D transitions are observed; this includes the non-resonant andresonantly enhanced transitions.

69
fr for a fixed value of f0 . It shows the 14 transitions (there are 39 possible pathways
for the case of linearly polarized light) that I identified for the 5S1/2 → 5D3/2 and5S1/2 → 5D5/2 two-photon resonances. The data clearly show that the larger, onephoton resonantly enhanced peaks repeat after a change of fr by ∼ 26 Hz, as expectedfrom the simple calculation above. As mentioned in section 4.2, for the resonantly
enhanced peaks, the pair of comb modes that is actually tuned onto the 5S-5P and
5P-5D resonances makes the dominant contribution to the two-photon transition rate.
In Fig. 5.2, we also present the theoretical spectrum corresponding to the set ofparameters used in the experiment. The two insets show signal magnitude on a logarithmic scale to enhance the visibility of the smaller peaks. The peak corresponding to
non-resonant excitation of the 5S1/2 (F = 2)→ 5P3/2 (F0 =3)→ 5D5/2 (F00 =4) transition,
made possible by the collective action of many comb modes, is larger than that theoretically predicted because the comb spectral phase is not flat and the comb spectrum is not
symmetric around the P state; thus, the destructive interference mentioned earlier insection 4.2 is reduced. Furthermore, as the detuning from the P state becomes greaterthan a few THz, the effect of phase mismatching between comb pairs over the spatial
dimension of the MOT can be non-negligible.
There are a couple of interesting spectral features that we notice when looking at
Fig. 5.2. The two resonance peaks observed in the experimental spectrum, which are notpresent in the theory model, are due to the 5S1/2 → 7S1/2 two-photon transitions. Theirobservation prompted us to consider both their theoretical modeling and experimental
investigation. However, it is more complicated to include these transitions in the model
than to include the 5S-5D two-photon transitions, primarily because their frequency has
never been measured before. In fact, this led us to actually experimentally measure theirabsolute frequencies, as presented in section 5.3. I had not initially intended to study
them, though this shows again one of the powerful features of our direct frequency comb
technique, the fact that we can observe all the transitions within the fs laser bandwidth.

2

4

6

8

12

14

16

18

5S → 7S

f r - 99 933 200.0 (Hz)10

Non-resonant S1/2 F=2 → D5/2 F=4

5S → 7S

20

22

24

26

1280 Pulses
2720 Pulses

first peakrepeated

1280 Pulses
2720 Pulses

Step-wise:
S1/2 F=2 → P3/2 F=3→ D5/2 F=4

28

10

8

6

4

2

0
0

2

4

6

8

10

Theory (a.u.)

Figure 5.2: Experimental (top) and simulated (bottom) two-photon spectra obtained with a frequency scan of fr . Shifting fr by 26 Hz
shifts the comb spectrum by 100 MHz, or fr , approximately repeating the original comb structure. The changes in relative peak sizes
from the spectrum obtained after 1280 pulses (in red) to that after 2720 pulses (in blue) illustrate population transfer dynamics by opticalpumping as well as heating effects. Two resonances in the data due to the 5S → 7S transitions are not included in our model.

0

Log Scale

Log Scale

Experiment (a.u.)70

71
For this fr scan, the initial ground-state population is in F=2. At longer times(blue curves), all the transitions starting on F=2 are decreasing in amplitude compared
to shorter time scales (red curves) due to ground-state population redistribution andheating. Most of the F=1 peaks remain unchanged or become larger. Our density
matrix simulation accounts for the time dependence of the shutter response and opticalpumping dynamics, but does not include any heating effects. The signal size shown in
Fig. 5.2 has been normalized against the square of the probe power. Not surprisingly,
both theory and experiment reveal that the most dominant transition pathway is theclosed 5S1/2 (F=2)→ 5P3/2 (F0 =3) → 5D5/2 (F00 =4) transition.
The relative size of the features in Fig. 5.2 after any fixed number of pulses
reflects intermediate-state resonant enhancement as well as population transfer. Thus,
the spectrum contains all of the fine and hyperfine structure pertinent to the 5D states.

5.2

Absolute frequency measurements
After accounting for the systematics discussed in Chapter 4, we have analyzed

spectra similar to the ones shown in Fig. 5.2 to construct a table of absolute transition
frequencies from 5S to 5D (see Table 5.1). We isolated the five two-photon transitions
which start in F=2, and go through 5P3/2 F0 =3, i.e. two-photon transitions which havethe 5S-5P step as a closed transition. This closed first step may help with the opticalpumping, giving better signal-to-noise ratios to determine their absolute frequency.
These representative transition frequencies are determined directly from the comb
structure and are given in the table, along with comparisons to available publishedvalues [75]. The measurement accuracy is currently a few kHz to a few tens of kHz forthe 5D states and on the order of 100 kHz for the 5P states, comparable to the highest
resolution measurements made with cw lasers. All the measurement errors reported
here are statistical errors (one standard deviation of the mean).

72
Measured transition
5S1/2 F=2→ 5D5/2 F00 =2
5S1/2 F=2→ 5D5/2 F00 =3
5S1/2 F=2→ 5D5/2 F00 =4
5S1/2 F=2→ 5D3/2 F00 =3
5S1/2 F=2→ 5D3/2 F00 =2
5S1/2 F=2→ 7S1/2 F00 =2
5S1/2 F=1→ 7S1/2 F00 =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 value (kHz)770 569 184 510.4 (16.0)
770 569 161 555.6 (16.0)
770 569 132 732.6 (16.0)
770 480 275 607.6 (10.0)
770 480 231 385.2 (10.0)
788 794 768 878.0 (40.0)
788 800 964 042.0 (40.0)

Table 5.1: 87 Rb level structure from direct frequency comb spectroscopy. All the values
are obtained by extrapolating the line center to zero probing time and power. The firstsix transitions presented here are resonantly enhanced by the 5P3/2 F0 =3 state and thus
have a closed first step. The final seventh transition passes through the 5P3/2 F0 =1
intermediate level.

We will now move on to discuss the measurement of the two extra resonancesthat appeared in the spectra shown in Fig. 5.2.

5.3

5S-7S absolute frequency measurements
After observing the two resonances, we reviewed the literature and discoveredthat in fact no one had measured these transitions with high precision. Without apriori information of the 7S energy levels, we have determined their absolute transition
frequencies [76]. We have used the transition wavelength reported in the online NIST
atomic database, which was of course covered by the fs laser spectrum, so we were able
to determine the resonance frequencies. A single optical comb thus provides atomic
structural information in the optical, terahertz, and radio-frequency spectral domains.
I will discuss in some detail now the 5S-7S two-photon frequency measurements.
They were performed in a similar manner to the 5S-5D measurements, with the important change that we used a Pockels cell with a 8-ns rise time instead of the liquid
crystal shutter (∼ 30 µs response time). As a consequence, we did not have to accountfor the shutter turn-on when obtaining the final frequency values, which made it easier to extrapolate to zero power. The experiment cycle and general method remained

73
the same, especially since the 7S levels also decay through the 6P levels, which meansthat we were able to use the same PMT. As seen from the general frequency diagram
in section 4.2, the wavelengths for the second step of the resonantly enhanced 5S-7Stransitions are 741 nm for transitions via 5P3/2 and 728 nm for transitions via 5P1/2 .
The energy degenerate two-photon transition is at 760 nm. This makes it more difficult
to cover them in our spectrum, optimized for 778 nm. This fact is especially true about
728 nm wavelength. Shifting the spectrum to smaller wavelengths is definitely doable,
the problem is having a sufficient signal-to-noise ratio for the f0 beat to properly maintain phase-lock of the Ti:S. The degradation in the signal-to-noise ratio of the f0 beatoccurs because the microstructure fiber is optimized for its zero GVD point near 800
nm. Thus, I usually center the fs spectrum at 770 nm for these measurements and I try
to use only transitions enhanced by a 5P3/2 intermediate level.
Shown in Fig. 5.3(a) is a typical 7S1/2 F00 =2 Lorentzian lineshape, generated bystepping the offset frequency f0 for a fixed value of the repetition rate fr and recording
the subsequent blue fluorescence corresponding to the 7S population. For each data
point, to obtain a better signal-to-noise ratio, the 80 ns-binned counts arising from thefluorescence are integrated over 2.4 µs. Alternatively, this lineshape is retrieved bysweeping fr with f0 fixed at some convenient value.
Once the lineshape has been acquired, what remains is to identify the correctmode number N associated with each transition. If the optical frequency is alreadyknown to within fr /2, this is a straightforward task. For the case of the 5S-7S twophoton transitions, where the resonance frequencies are not known a priori (νopt = (N1 +N2 )fr + 2f0 ), we scan the resonances for two different values of fr and unambiguously
deduce the sum of the two associated integers (N1 + N2 ) [77, 78]. In our case, the two
repetition rates used are separated by 600 kHz to eliminate possible uncertainties arisingfrom estimations of the f0 value corresponding to the peak of the resonance.
After identifying the comb numbers associated with the transition and reducing

74

Signal (a.u.)

1.0

Data
Fit

0.8

7S1/2
Lineshape

0.6
0.4

(a)

0.2
0.0
18

Measuredν0 (kHz)

1500

20
22
f0 (MHz)24

Line center shift
Fit

1000

500

(b)
ν0 = 788 794 768 921 kHz

0
0

20
40
60
80
Probing Time (µs)

Figure 5.3: (a) Typical 5S1/2 F=2→ 7S1/2 F00 =2 two-photon Lorentzian lineshape obtained from a scan of the offset frequency f0 for a fixed value of the repetition rate fr .
For each data point, the 80 ns-binned counts arising from the fluorescence are integrated
over 2.4 s. (b) Frequency shift of the transition line center vs. probing time resulting
from the momentum transferred by the femtosecond laser to the cold 87 Rb atoms. Extrapolation to zero interrogation time gives absolute atomic transition frequencies, aswell as roughly the natural linewidth, free of radiation pressure effects.

the systematic error arising from AC Stark shift (by probing on optimal resonance i.e.,zero detuning for both the intermediate state and the final state), the remaining error
from radiation pressure is suppressed by extrapolating to zero interrogation time, as

75
shown in Fig. 5.3(b). We determine for the 5S1/2 F=2 → 7S1/2 F00 =2 and the 5S1/2 F=1→ 7S1/2 F00 =1 two-photon transitions in 87 Rb the absolute optical frequencies of 788 794
768 921(44) kHz and 788 800 964 199 (122) kHz, respectively (see Table 5.1). The excitedstate hyperfine interval of 639 404 (130) kHz agrees very well with a previous differential
measurement performed with a picosecond pulsed laser [70]. The transition spectra
reported in [70], as well as a continuous-wave (cw) laser-based scan [79], indicatedthat the F=2 → F00 =2 and the F=1 → F00 =1, i.e. ∆F=0 transitions, were the only
allowed 5S-7S transitions in 87 Rb. However, we observe additional lines, as the resonantintermediate 5P state also enables the F=2 → F00 =1 and F=1 → F00 =2, i.e., ∆F=±1,
two-photon transitions. Similar ∆F=±1 S-S transitions have been previously observedin Na in a two-step excitation experiment employing two tunable cw dye lasers, whichenabled a direct measurement of the excited state hyperfine splitting [80].
We recently learned that conventional, cw laser-based measurements of the 5S-7Stransitions in Rb were reported by Chui et al. [81]. Our measurements agree with theirresults, within one standard deviation of the mean for the reported values, as can beseen from Table 5.1. Thus, we have demonstrated that by using DFCS, the absolutevalue of the 5S1/2 → 7S1/2 two-photon transitions in 87 Rb is conclusively determined,with no a priori knowledge about their optical frequency.
In the next chapter I will discuss single-photon frequency measurements that
we recently made. As described there, one indirect way to determine the absolute
frequencies of the 5S-5P transition is to scan the 5P state by using a set of fr and f0 pairsthat have a range of detunings from the 5P state and are all two-photon resonant. I willshow that detailed dynamics of population transfer driven by a sequence of pulses haveto be taken into account for the measurement of the 5P states via resonantly enhanced
two-photon transitions.

Chapter 6

One-photon absolute frequency measurements

Direct Frequency Comb Spectroscopy (DFCS) can be equally well applied tomeasure single-photon transitions. However, since we have the ability to dynamicallyaccess two-photon transitions (as shown in Chapter 4), we can measure the energyof a given state in two ways. We can either investigate an atomic state as part of aone-photon process or as a two-photon process. We have chosen the 5S-5P transitions
in 87 Rb to demonstrate this ability. The measurement of 5P states has been carriedout directly and also indirectly, via the 5S-5D two-photon transitions by studying theirresonant enhancement when comb components are scanned through the intermediate
5P states. We compare the 5P measurements obtained via one-photon and two-photon
DFCS and clearly demonstrate the importance of population transfer in working withmultilevel systems probed by multiple comb components.
Also, again with the technique of DFCS, we have the advantage of being able todetermine absolute atomic transition frequencies anywhere within the comb bandwidth,for one-photon processes.

6.1

Detection and timing scheme for the direct 5P measurements
For the one-photon studies we have investigated both the D1 and D2 transitions

in 87 Rb, namely transitions from the ground state to the 5P1/2 excited state at 795 nm
and to the 5P3/2 state at 780 nm (see general diagram, in section 4.2).

77
The three main timing sequences of the magneto-optical trap (MOT), the polarization gradient cooling (PGC), and the probing cycle are similar to those shown insection 4.3. However, here during the probing cycle, we implement a switching scheme
for the photomultiplier tube (PMT) and the fs laser.
The main challenge with the direct 5P measurements is to collect the fluorescenceemitted by the atoms, while avoiding gathering the light from the fs probe, since they
have nearly the same wavelength. In addition, because we use acousto-optic modulatorsfor fast switching of the MOT lasers, there is always some residual light that may affectthe measurements. Care was taken to ensure that these lasers did not influence thesignal.

pinhole to filterout fs beamz-axisaxis of fs beam
propagation

PMT

narrowband
spectral filter

Figure 6.1: Detection assembly for the direct one-photon measurements. Backgroundcounts are minimized by using spatial and spectral filters. As before, the PMT signal
is sent to a photon counter and the resulting counts are read into a computer.

We directly detect the fluorescence from the two 5P states with a near-infrared
PMT coupled with a 3-nm bandwidth interference filter centered at the appropriate
wavelength for the transition, as presented in the detection assembly in Fig. 6.1. We
use narrow band interference filters reduce the number of comb components that arereceived by the PMT and thus minimize detection noise. Background counts are further
minimized by spatial filtering with two pinholes, the first very small (1 mm), through78
which the MOT is imaged [Fig. 6.1]. Photon collection during the probe laser-on period
is disabled by switching off the PMT.
The loading cycle used for the MOT is 100 Hz, and the sequence of the experimentis similar to the two-photon experiments [Fig. 6.2]: the atoms are loaded in the MOT
for 7.8 ms, then the quadrupole magnetic field for the trap is switched off, the atoms arecooled with polarization gradients for 2 ms, then all the MOT beams are extinguishedand the femtosecond comb beam is switched on for 200 µs using the Pockels cell (8-nsrise time)
As illustrated in the one-photon timing diagram in Fig. 6.2, during the 200 µs
probe window, we have a sequence of short cycles with the probe laser on (200 ns)
followed by the PMT on (400 ns) to detect photons from the fast-decaying 5P states. A
2.6 µs interval (PMT switch-off time) is required before initiating the next laser cycle.
Both the two-photon and the single-photon signals are averaged over hundreds of 10 ms
experiment cycles.

MOT
7.5 ms

PGC
2 ms
200 µs...
200 ns, fs ont
One-photon
timing diagram

400 ns, PMT on

Figure 6.2: Timing scheme for the 100 Hz experiment cycle, where the 200-µs zoom
window shows the sequences for the direct one-photon measurements. We switch the
PMT off during the laser-on period, to minimize detection noise.

For the indirect 5P measurements, we use results from the theoretical model
describing the fs comb interaction with the atoms. As discussed in Chapter 2, the79theory accounts for detailed dynamics of population transfer among the atomic states
involved in transitions within the comb bandwidth. A numerical scheme is employed
to obtain the state of the atomic system after an arbitrary number of pulses [37, 60].
This model is applied to accurately predict the coherent population accumulation in therelatively long-lived 5D, followed by incoherent optical pumping. Especially important
for the indirect 5P measurements is the incoherent optical pumping to the ground state
hyperfine levels, which depends critically on the 5P state detunings and will be discussed
in detail below.
We will now compare one-photon and two-photon DFCS measurements for the
5P state energy levels. The one-photon DFCS employs radiative detection directly fromthe 5P states (Fig. 6.3, left panel), while the two-photon DFCS studies the 5P states
indirectly, via resonant enhancement of the 5S-5D two-photon transitions as a functionof the detuning from the intermediate 5P states (Fig. 6.3, right panel).

5D
6P

5P
PMT

5S

F=2
F=1

one-photon
DFCS

5P

PMT

5S

F=2
F=1

two-photon
DFCS

Figure 6.3: Schematic of one- and two-photon DFCS, used for measuring single-photontransition frequencies. The two-photon DFCS here differs from the two-photon processdescribed in Chapter 5 in that the 5D state is kept strictly in resonance. The 5Plineshape is obtained by scanning the detuning.

80

6.2

5P3/2 frequency measurements
First, we measure the 5S1/2 F=2 → 5P3/2 F0 =3 D2 transition with one-photon

DFCS, with the resulting transition line shown in Fig. 6.4(a) [76]. Frequency scans arecarried out similarly to those of the 5S-5D lines, that is, by stepping f0 continuously
while keeping fr fixed. The absolute optical frequency measured for this transition is384 228 115 271 (87) kHz, in agreement with a previous measurement done here at
JILA [82], which resulted in an absolute frequency value of 384 228 115 203 (7) kHz.
Once again, all the measurement errors reported in this chapter, and actually in thisthesis, are statistical errors (one standard deviation of the mean).
For the two-photon DFCS, we use a set of different pairs of fr and f0 specifically chosen to have varying detunings from the 5P state for each data point shown in
Fig. 6.4(b), while at the same time satisfying strictly the 5S-5D two-photon resonance(δSD = 0). In this case, we monitor once again the 420 nm fluorescence from the 5D
states. Such a set of fr and f0 is given as an example in Table 6.1. We immediately
see that fr must be known within 10 mHz. To obtain this level of accuracy, we always
reference the fr frequency counter to the cesium clock in our lab. As mentioned earlier,this is necessary because the fr excursions get multiplied by ∼ 8 × 106 . In contrast,
f0 must only be stable within 100 Hz.
The lineshape in Fig. 6.4(b) is retrieved by detecting the 420 nm signal as afunction of 5P state detuning and the optical frequency measured by this two-photon
DFCS is 384 228 115 309 (63) kHz, in agreement with the result obtained from onephoton DFCS within the standard deviation. Although the two methods for measuring
this 5P state, direct and indirect, agree well, the indirect one is more cumbersome fortwo main reasons. Firstly, fr must be changed by such a large amount as to require
the output coupler to be placed on a translation stage and thus, the laser box to beopened for each detuning value. Secondly, this required hand-tuning adds significant

81

Signal (a.u.)

1.0

Data
Fit

0.8
0.6

(a)

0.4

Single-photon P3/2

10

15

1.0

Data
Fit

0.8

Signal (a.u.)

20 25
f0 (MHz)35

P3/2
lineshape
via a
two-photon
scan

0.6
0.4

30

(b)

0.2
0.0
-10

0

10

5P State Detuning (MHz)Figure 6.4: (a) Lineshape of the 5S1/2 F=2→ 5P3/2 F0 =3 transition obtained from ascan of f0 for a fixed value of fr , by one-photon DFCS. (b) Same lineshape as in (a),retrieved using its resonant enhancement of the 5S1/2 F=2→ 5P3/2 F0 =3→ 5D5/2 F00 =4
closed two-photon transition, as a function of the detuning from the intermediate state.

time for retrieving a lineshape, so it is important that the data be taken under the same
conditions and thus, it all has to be acquired on the same day. The direct method,
though requiring an extra (switched) PMT, allows for a fast and smooth scan of the 5P
states.
It is also important to mention that for this scan we take advantage of the
5S1/2

F=2 → 5P3/2 F0 =3 → 5D5/2 F00 =4 being the only 5S-5D closed transition.

As shown in the theory plots in Fig. 6.5, this closed transition ensures that most of

82
5P 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

f0 (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

Fractional Ground State Population

Table 6.1: Example of a set of fr and f0 pairs used for the indirect scanning of theintermediate 5P3/2 F0 =3 state. All the detunings are given with respect to this P state.
The values for fr and f0 are chosen to ensure that the F00 =4 state is always resonant
for this indirect scan.

1.0

F=2, +4 MHz detuning
F=2, -4 MHz detuning

0.8
0.6
0.4

F=1, -4 MHz detuning
F=1, +4 MHz detuning

0.2
0.0
0

2

4

6

8

10

Time (µs)

Figure 6.5: Theoretical plot of the time evolution of the ground state populations fortwo (symmetric) detuning values in Fig. 6.4(b), showing that (i) most of the atoms
remain in the initial F=2 ground level for this closed two-photon transition and (ii) the
ground state populations are largely insensitive to the sign of the detuning.

the atoms initially starting out in the F=2 ground-state hyperfine level remain in thatlevel, while ∼20% of the atoms fall into the dark F=1 ground state due to opticalpumping and hence do not contribute to the signal. In addition, the probe laser power

83
is sufficiently reduced for the two-photon DFCS experiments to further decrease opticalpumping effects.

6.3

5P1/2 frequency measurements

1.0

Data
Fit

Signal (a.u)

0.9
0.8
0.7

(a)

0.6
Single-photon P1/2

0.5

Raw Number of Counts

10

500

15

20
f0 (MHz)

Raw P1/2
lineshape
via a
two-photon
scan

Data
Visual
Guide

400

25

300

(b)

200
100
-20

-10

0

10

20

5P State Detuning (MHz)Figure 6.6: (a) Lineshape of the 5S1/2 F=2 → 5P1/2 F0 =2 transition obtained from ascan of f0 for a fixed value of fr , by one-photon DFCS. (b) Raw counts for the samelineshape as in (a) by two-photon DFCS, along with a visual guide for the data.

Next, we employ DFCS to study another single-photon transition in the D1 manifold, 5S1/2 F=2 → 5P1/2 F0 =2, as shown in Fig. 6.6(a). Again, f0 is scanned while fr isstabilized to a convenient value. The absolute optical frequency for this transition isdetermined to be 377 105 206 563 (184) kHz, in agreement with a previous wavelength-

84
based measurement [83], which gave an absolute frequency value of 377 105 206 705(400) kHz for this transition.
For the corresponding two-photon DFCS experiment we map the 5S1/2 F=2→ 5P1/2 F0 =2 → 5D3/2 F00 =3 two-photon transition in the same manner employed forFig. 6.4(b). Figure 6.6(b) shows the raw data yielded by the (fr , f0 ) pair selections,along with a visual guide for the data. The apparent linewidth is significantly broader
than that associated with the 5P state.
Unlike the previous two-photon DFCS measurement reported in Fig. 6.4, thepairs of fr and f0 used to obtain each point in Fig. 6.6(b) lead to substantially different
detunings of the other 5P states and subsequently, varying optical pumping to the F=1
ground state. Indeed, the theory model applied to the actual experiment conditions
predicts significantly different ground state population transfer dynamics. As shown in
Fig. 6.7, the asymptotic values of the F=2 ground-state population are not the same

Fractional Ground State Population

for symmetric detunings from the intermediate state.

1.0

F=1, +4 MHz detuning
F=1, -4 MHz detuning

0.8
0.6
0.4
0.2
F=2, -4 MHz detuning
F=2, +4 MHz detuning

0.0
0

2

4
6
Time (µs)

810Figure 6.7: Theoretical plot of the time evolution of the ground state populations for two(symmetric) detuning values in 6.6(b), showing a significant difference in the populationtransfer between the ground state levels due to optical pumping caused by varying
detunings from the other 5P states.

85
Figure 6.8(b) presents the Lorentzian lineshape resulting from the normalizationof the raw data shown in Fig. 6.8(a) with respect to the theoretical value of (1 - ρF =1 ),where ρF =1 is the fractional ground state population in F=1, as shown in Fig. 6.7. After
we implement this normalization, the optical frequency for the transition measured by
the two-photon DFCS is 377 105 206 939 (179) kHz, which is within the error bars of

Raw Number of Counts

the corresponding one-photon DFCS result.

500

Raw P1/2
lineshape
via a
two-photon
scan

Data
Visual
Guide

400
300

(a)

200
100
-20

-10

0

10

20

Normalized Number of Counts

5P State Detuning (MHz)
6000

4000

2000

Data
Fit

Normalized
P1/2
lineshape
via a
two-photon
scan

(b)

0
-20
-10
0
10
20
5P State Detuning (MHz)Figure 6.8: Raw data (a) and normalized lineshape (b) obtained by using results fromthe theory simulation in Fig. 6.8 accounting for optical pumping.

We note that for all measurements reported in this chapter, the statistical errors(one standard deviation of the mean) associated with 5P3/2 are significantly smaller

86
than those associated with 5P1/2 . This is due to the stronger transition strength andless severe optical pumping effects for 5P3/2 F0 =3 (part of a closed transition), leading
to larger signal-to-noise ratios.
Direct frequency comb measurements of one-photon transitions in atoms otherthan 87 Rb have been recently reported. In Leo Hollberg’s group at NIST, they probed
Cs in an atomic beam with DFCS, but using a Ti:S laser with a 1 GHz repetition
rate [84]. Employing a 1-GHz fs laser has several advantages over a lower fr laser, as
stated in Chapter 7.
In conclusion, a phase-stabilized femtosecond comb has been used as an effective
tool to perform direct spectroscopy of one-photon transitions in cold 87 Rb atoms. We
have demonstrated that DFCS can be successfully applied to one-photon studies, bymeasuring 5S1/2 → 5P1/2,3/2 transitions both directly and indirectly, via their resonant
enhancement of the 5S-5D two-photon transitions. Additionally, we have shown theimportance of including the dynamic population changes arising from pulse-accumulated
population transfer in this indirect one-photon measurement.

Chapter 7

Summary and outlook

In this thesis, we have joined together the study of cold atoms, which are wellestablished as an ideal sample for spectroscopy with the field of ultrafast lasers, whichhave recently emerged as a powerful tool for frequency metrology.
We have used a wide-bandwidth, phase-stabilized femtosecond laser as a noveland efficient tool to perform Direct Frequency Comb Spectroscopy (DFCS) of onephoton and two-photon transitions in laser-cooled 87 Rb atoms. To begin with, we havestudied the dominant sources of systematic shifts and tried to reduce their effects asmuch as possible. We have then measured the absolute atomic frequency for several
5S1/2 → 5D3/2,5/2 two-photon transitions in 87 Rb. The two-photon spectroscopy is carried out efficiently despite the low laser power, because we take advantage of intermediate 5P states that are near-resonant with a comb line. We have demonstrated that theabsolute optical frequency of the previously unmeasured 5S1/2 → 7S1/2 resonances canbe determined unambiguously from the comb structure used in the experiment. Lastly,
we have shown that DFCS can be successfully applied to one-photon processes as well,by measuring 5S1/2 → 5P1/2,3/2 transitions both directly and indirectly, through theirresonant enhancement of the 5S-5D two-photon transitions. Detailed dynamics of population transfer driven by a sequence of pulses were uncovered and taken into account for
the indirect measurement of the 5P states. Although the current experiments involveone- and two-photon transitions, the advantages of DFCS should also apply to multi-

88photon excitations. In general, DFCS permits global, high-resolution spectroscopy of
all the atomic transitions within the comb bandwidth.
After we successfully implemented DFCS for the study of several one- and twophoton transitions, there is still room left for improvements. Firstly, the resolution of
DFCS can be enhanced by locking the femtosecond laser to a cavity, which has beenshown capable to reduce the linewidth to below 100 Hz [85], rather than the current 300
kHz. Secondly, a larger signal can be obtained by using a laser with a higher repetition
rate; for example, a 1-GHz laser with the same average power and spectral width could
increase the signal up to a 100-fold. This is a result of the quadratic increase of the excited state population versus the accumulated number of pulses, where the higher fr andhence the shorter pulse interval allow for more pulses to be ‘added’ during the atomic
coherence lifetime. Additionally, the power is distributed among fewer comb lines forthis fr ∼ 1 GHz case, and the decreased congestion of lines would reduce optical pumping effects. Achieving simultaneous one- and two-photon resonance conditions shouldstill be possible. Using a 1-GHz laser provides more latitude in identifying the mode
number of the comb component that is resonant with each optical transition. The larger
mode spacing makes this identification easier, because it only requires knowledge of the
optical frequencies with a precision of a few hundred MHz. One practical consequenceof these results is a method to control both degrees of freedom of the femtosecond comb
directly by an optical transition in cold atoms.
Another interesting application of the demonstrated pulse accumulation effectis using ultrafast pulse trains for laser cooling of atoms (e.g. hydrogen, deuterium,
antihydrogen) that require coherent ultraviolet light not easily accessible by conventional laser sources [86]. Toward this goal, an extension of frequency comb metrology to the deep-ultraviolet spectral region (where cw lasers are not readily available)
was recently achieved using an amplified train of phase-controlled pulses from a femtosecond laser [78]. The peak power of these pulses allowed for efficient harmonic

89
upconversion and thus, the 4th harmonic of the femtosecond laser was used for twophoton spectroscopy in krypton. Even more recently, phase-coherent frequency combs
in the vacuum-ultraviolet spectral region were demonstrated in our lab, without active
amplification of the standard femtosecond laser [87]. Its output was stabilized to acompletely passive optical cavity, with a gas jet at the intracavity focus, generating aphase-controlled frequency comb in the extreme ultraviolet region which preserves theoriginal repetition frequency. This technique, in combination with DFCS, opens thedoor to more accurate atomic optical clocks that operate on resonances with ultrahigh,vacuum-ultraviolet or extreme-ultraviolet, frequencies. These clocks would have a very
high stability, proportional to their transition frequency.
For general coherent control experiments, pulse accumulation (when enabled bylong coherence times) can complement spectral amplitude and phase manipulations,leading to improved efficiency in population control with the added spectral resolution
and longer experimental timescales due to maintaining phase coherence for many consecutive laser pulses. The precise and phase-coherent pulse accumulation may proveparticularly useful in efficiently populating atomic Rydberg states for quantum information processing. Multiple ultrafast lasers with optical spectra independently tailoredfor different spectroscopic features could be phase coherently stitched together [9,88] tofurther extend the utility of this approach.

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