Production of Ultracold, Absolute Vibrational Ground State ...

Production of Ultracold, Absolute Vibrational Ground State NaCs Molecules

by

Patrick J. Zabawa

Submitted in Partial Fulfillment of the

Requirements for the Degree

Doctor of Philosophy

Supervised by

Professor Nicholas Bigelow

Department of Physics and Astronomy

Arts, Sciences and Engineering

School of Arts and Sciences

University of Rochester

Rochester, New York

2012

Dedicated to my father, L. George Zabawa (1948-2010), for instilling in me a

passion for the sciences.

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Biographical Sketch

The author was born in Norman, OK. He attended the University of Ok-

lahoma and graduated with a Bachelor of Science in Physics and Math with

Distinction. He began doctoral studies in Physics at the University of Rochester

in 2005, and received a Master of Arts degree from the University of Rochester

in 2007. He pursued his research in ultracold polar molecules under the direction

of Professor Nicholas Bigelow.

The following publications were a result of work conducted during doctoral

study:

1. J. Kleinert and C. Haimberger and P. Zabawa and N. P. Bigelow, “Manufac-

turing a thin wire electrostatic trap for ultracold polar molecules,” Review

of Scientific Instruments 78, 113108 (2007).

2. J. Kleinert and C. Haimberger and P. Zabawa and N. P. Bigelow, “Trapping

of Ultracold Polar Molecules with a Thin-Wire Electrostatic Trap,” Physical

Review Letters 99, 143002 (2007).

3. C. Haimberger, J. Kleinert, P. Zabawa, A. Wakim and N. P. Bigelow, “For-

mation of ultracold, highly polar X1Σ+ NaCs molecules,” New Journal of

Physics 11, 055042 (2009).

4. P. Zabawa, A. Wakim, A. Neukirch, C. Haimberger, N. P. Bigelow, A. V.

Stolyarov, E. A. Pazyuk, M. Tamanis, and R. Ferber, “Near-dissociation

photoassociative production of deeply bound NaCs molecules,” Physical

Review A 82, 040501 (2010).

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5. A. Grochola, P. Kowalczyk, W. Jastrzebski, W. J. Szczepkowski, P. Zabawa,

A. Wakim, and N. P. Bigelow, “The spin-forbidden c3Σ+(Ω = 1)← X1Σ+

transition in NaCs. Investigation of the Ω = 1 state in hot and cold envi-

ronment,” Physical Review A 84, 012507 (2011).

6. A. Wakim, P. Zabawa, and N. P. Bigelow, “Photoassociation studies of ul-

tracold NaCs from the Cs 62P3/2 asymptote”, Physical Chemistry Chemical

Physics 13, 18887 (2011).

7. P. Zabawa, A. Wakim, M. Haruza, and N. P. Bigelow, “Formation of ultra-

coldX1Σ+(v′′ = 0) NaCs molecules via coupled photoassociation channels,”

Phys. Rev. A 84, 061401 (2011)

8. A. Wakim, P. Zabawa, M. Haruza, and N. P. Bigelow, “Luminorefrigeration:

vibrational cooling of NaCs,” Optics Express, Vol. 20 Issue 14, pp.16083-

16091 (2012)

9. P. Zabawa, A. Wakim, M. Haruza, and N. P. Bigelow, “Investigation of

molecular states of NaCs dissociating to the Cs 6P asymptote with PA

spectroscopy,” paper in preparation

v

Acknowledgments

There are many people who have made this work possible.

Prof. James P. Shaffer, my undergraduate advisor at the University of

Oklahoma, gave me my first opportunity to work in an experimental research

laboratory, and (subtly and not so subtly) directed me toward the University of

Rochester for graduate school. A testament to the small world of experimental

ultracold physics, I completed my PhD research using the same basic experimen-

tal apparatus as Prof. Shaffer. Thanks to my following in his footsteps, I now

have an additional verb in my vocabulary: ”to Shafferize”, defined as tightening

an item (such as a bolt or screw) to the point that future generations of graduate

students must use unorthodox methods to remove said item.

I would like to thank my advisor, Prof. Nick Bigelow, for giving me a

chance to work in his laboratory and for the financial, scientific, and emotional

support that he provided throughout my graduate career.

When I first arrived in the summer of 2005, former graduate student

Michael Holmes took time to teach me about electronics and soldering, with

which I had previously had very little experience. His kindness and patience is

much appreciated. Later, after I decided to work in the molecule lab, I got to

know Christopher Haimberger and Jan Kleinert (then graduate students). From

these two, I learned the skills and began to appreciate the fortitude necessary to

successfully operate the lab. I thank them for their patience and good humor,

and for laying much of the foundation for the work contained in this thesis.

After the graduation of Jan Kleinert in 2008, I was left with the daunting

vi

task of running a highly complex lab largely by myself. This state of affairs con-

tinued for several months until Amy Wakim, a fellow graduate student of Prof.

Bigelow who was at the time running an even more complicated experiment by

herself, agreed to join me in the molecule lab next door. Our productivity sky-

rocketed soon after thanks to our complementary experience, methods, attitudes,

and musical tastes. My experience at the University of Rochester has been far,

far richer for having had Amy as a colleague and a friend.

Marek Haruza, the newest member of the molecule lab, has already con-

tributed much to our research. I thank him for providing his keen intellect as a

sounding board for new ideas, and for his editorial assistance.

I would like to thank all of the former and current staff at the University

for everything that they do, including (but certainly not limited to): Barbara

Warren, Laura Blumkin, Sondra Anderson, Janet Fogg-Twichell, Shirley Brignall,

Connie Jones, Susan Brightman, Connie Hendrix, Michie Brown, and Ali DeLeon.

I am profoundly grateful to my parents, Pamela and George Zabawa, for

encouraging my early interest in the sciences. I have many fond memories of

science fair projects and engineering fairs. Without these experiences, I may

never have turned toward one of the most challenging and rewarding careers

imaginable. I also thank Christopher Zabawa and Julia Narramore, my siblings,

for their support.

Melanie Carter, my spouse and best friend, has been ever understanding

and encouraging throughout our stay in Rochester. Along the way, she has shared

countless adventures with me. It is a great comfort and an amazing thing to know

that after a late night in the lab, or a conference in some distant city, I will be

going home to be with the joy of my life.

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Abstract

This dissertation describes a progression of experiments that are based on

the association of ultracold (∼250 µK) Na and Cs atoms with laser light. One of

the primary goals of the experiment is to form molecules in the absolute vibra-

tional ground state. The work begins with our attempts to label, with certainty,

spectral lines obtained from tuning either the photoassociation (PA formation)

and Resonance Enhanced Multi-Photon Ionization (REMPI detection) lasers. To

this end, we develop a technique that has heretofore never been used in the ultra-

cold molecule community: pulsed depletion spectroscopy (PDS). Traditionally,

depletion spectroscopy involves the use of narrow-linewidth CW lasers. However,

the narrow linewidth and limited tuning ranges of diodes used for CW depletion

spectroscopy mean that this technique is only helpful if the expected transitions

are known to some degree in advance, and even then is primarily useful for de-

termining closely-spaced rotational ground state populations. In contrast, the

broad linewidth and flexible tuning range of a pulsed dye laser makes it suit-

able for the detection of vibrational progressions, allowing fast determination of

ground state populations even without a priori knowledge of the transitions in-

volved. We also use this technique in our investigation of excited state potential

energy curves (PECs). We also investigate a range of PA resonances detuned

from the Cs D1 and D2 lines. We find and label PA structure associated with at

least 6, and possibly all 8 electronic states corresponding to both of the Cs 6P

fine structure asymptotes. From the PA and depletion spectra, we obtain infor-

mation on the PA scattering process and the excited electronic states. Among

the PA spectra, we find several channels which directly form vibrational ground

viii

state molecules in the singlet electronic state. Finally, we manipulate the in-

ternal states of molecules created with PA using laser light. We use broadband

laser sources to pump higher-lying singlet vibrational levels into the vibrational

ground state. We also find a set of nearly-closed transitions which allow rotational

pumping into the absolute rovibrational ground state.

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Table of Contents

Foreword xx

Chapter 1. Introduction 1

1.1 A brief history of photoassociation . . . . . . . . . . . . . . . . . . 1

1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Heteronuclear, diatomic molecules . . . . . . . . . . . . . . . . . . 4

1.3.1 The Born-Oppenheimer approximation . . . . . . . . . . . . 5

1.3.2 Hund’s cases . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.3 Term symbols . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.4 Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.5 Basis states and interactions . . . . . . . . . . . . . . . . . 10

1.3.6 Hyperfine structure . . . . . . . . . . . . . . . . . . . . . . 15

1.3.7 Electric dipole moment . . . . . . . . . . . . . . . . . . . . 16

1.4 Molecule formation and detection . . . . . . . . . . . . . . . . . . 17

1.4.1 Magneto-Optical Trap . . . . . . . . . . . . . . . . . . . . . 17

1.4.2 Interatomic forces and cold collisions . . . . . . . . . . . . . 19

1.4.3 Photoassociation . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4.4 Resonance Enhanced Multi-Photon Ionization . . . . . . . . 23

Chapter 2. Experimental Design 25

2.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1.1 Vacuum chamber and atomic sources . . . . . . . . . . . . . 25

2.1.2 Laser sources . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.3 Ion detection . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.1.4 Timing and data collection . . . . . . . . . . . . . . . . . . 32

2.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.1 Photoassociation and REMPI spectroscopy . . . . . . . . . 33

2.2.2 Continuous wave (CW) and pulsed depletion spectroscopy . 33

2.2.3 Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2.4 Electrostatic trapping . . . . . . . . . . . . . . . . . . . . . 36

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Chapter 3. REMPI and PDS: vibrational spectra and analysis 38

3.1 REMPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.1.1 Initial analyses of NaCs REMPI spectra . . . . . . . . . . . 38

3.1.2 Extending the REMPI spectrum . . . . . . . . . . . . . . . 39

3.2 PDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.1 Implementation of PDS . . . . . . . . . . . . . . . . . . . . 46

3.2.2 Labeling deeply bound X1Σ+ molecules with PDS . . . . . 47

3.2.3 Investigation of the B1Π and c3Σ+Ω=1 electronic state with

PDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2.4 Labeling a3Σ+ molecules with PDS . . . . . . . . . . . . . . 52

Chapter 4. PA spectra and analysis: Ω > 0 states correspondingto the Cs 62P3/2 asymptote 56

4.1 Initial analyses of NaCs PA spectra . . . . . . . . . . . . . . . . . 56

4.2 A new investigation . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.1 The c3Σ+Ω=1 electronic state . . . . . . . . . . . . . . . . . . 58

4.2.2 The B1Π electronic state . . . . . . . . . . . . . . . . . . . 65

4.2.3 Extension and reanalysis of the b3ΠΩ=2 PA spectra . . . . . 68

Chapter 5. PA spectra and analysis: the Ω = 0 and (2)Ω = 1 elec-tronic states 73

5.1 The Ω = 0+ states . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.1.1 Characteristics of the Ω = 0+ A1Σ+−b3Π complex electronicstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.1.2 The A1Σ+ state . . . . . . . . . . . . . . . . . . . . . . . . 74

5.1.3 Dispersion coefficients for the A1Σ+ state . . . . . . . . . . 78

5.1.4 The b3ΠΩ=0+ state . . . . . . . . . . . . . . . . . . . . . . . 80

5.1.5 Ω = 0+ hyperfine structure . . . . . . . . . . . . . . . . . . 82

5.2 The c3ΣΩ=0− state . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.3 The b3ΠΩ=0− and b3ΠΩ=1 electronic states . . . . . . . . . . . . . . 86

Chapter 6. Excited state coupling and shape resonances 88

6.1 Scattering waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.1.1 Partial wave analysis using the Ω = 0 rotational spectra . . 88

6.1.2 Shape resonances . . . . . . . . . . . . . . . . . . . . . . . . 90

6.2 Excited state coupling . . . . . . . . . . . . . . . . . . . . . . . . 95

6.3 Observation of coupling in the NaCs PA spectra . . . . . . . . . . 97

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Chapter 7. Production and detection of X1Σ+(v′′=0)NaCs moleculesvia PA 102

7.1 Absolute vibrational ground state formation and detection channels 102

7.1.1 The B1Π electronic state . . . . . . . . . . . . . . . . . . . 102

7.1.2 Finding candidate X1Σ+(v′′=0) REMPI detection lines . . 103

7.1.3 Unambiguous labeling of X1Σ+(v′′ = 0) REMPI detectionlines with PDS . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.2 Investigation of the X1Σ+(v′′=0) rotational populations . . . . . . 106

Chapter 8. Optical pumping of ultracold NaCs molecules 109

8.1 Vibrational cooling of molecules with broadband light . . . . . . . 109

8.2 Narrow line optical pumping of NaCs . . . . . . . . . . . . . . . . 116

Chapter 9. Conclusions 121

Bibliography 125

Appendix A. 137

A.1 X1Σ+ REMPI lines . . . . . . . . . . . . . . . . . . . . . . . . . . 137

A.2 a3Σ+ REMPI lines . . . . . . . . . . . . . . . . . . . . . . . . . . 142

A.3 c3Σ+Ω=1 PDS lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

A.4 A1Σ+ PA lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

A.5 b3ΠΩ=0+ PA lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

A.6 B1Π PA lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

A.7 b3ΠΩ=2 PA lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

A.8 c3Σ+Ω=1 PA lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

A.9 c3Σ+Ω=0− PA lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

A.10 b3ΠΩ=2 RKR PEC . . . . . . . . . . . . . . . . . . . . . . . . . . 156

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List of Tables

1.1 Dipole selection rules for angular momenta (AM) and other prop-erties of heteronuclear molecules. [1] . . . . . . . . . . . . . . . . 8

4.1 Comparison between observed and calculated rotational constantsand binding energies for J ′ = 1 levels in the B1Π state. All values(except v) are given in cm−1. Calculations are performed usingthe experimental PEC in [2] and LEVEL [3]. The uncertaintiesin the observed energies are ∼ ±1 GHz, while the standard errorsobtained for the rotational constants are < 0.0006 cm−1. . . . . . 66

5.1 Dispersion coefficients and parameters obtained from the improvedLeRoy-Bernstein fit of near-dissociation A1Σ+ vibrational levels.These are compared to the ab initio A1Σ+ dispersion coefficients.All parameters except vD are given in atomic units. . . . . . . . . 79

5.2 Comparison between observed and calculated rotational constantsand binding energies for J ′ = 0 levels of the c3ΣΩ=0− and c3Σ+

Ω=1

states. All values (except v) are given in cm−1. We calculatelevels and rotational constants using Level. The uncertainties inthe observed energies are ∼±1 GHz, while the standard errorsobtained for Bv in the experimental rotational fits are < 0.00005cm−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.1 Summary of partial waves responsible for the Ω = 0+ rotationallines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.2 Shape resonances for the two NaCs ground states as calculated byGonzalez Ferez and Koch [4]. . . . . . . . . . . . . . . . . . . . . 95

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List of Figures

1.1 The empirical PECs of the lowest-lying electronic states of NaCs[2, 5–7]. The A1Σ+− b3Π complex PECs are diabatic, and thus donot all approach the fine-structure asymptotes. The b3ΠΩ=2 PECis the adjusted version that is discussed in 4.2.4. . . . . . . . . . . 12

1.2 An avoided crossing between the A1Σ+–b3ΠΩ=0+ electronic statesof NaCs. The diabatic (dashed) lines are taken from [6], and theadiabatic PECs are approximated by diagonalizing the interactionmatrix. In this case, the experimentally determined R-dependentS-O function is used. . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 A plot of the X1Σ+ PEC with the dominant interactions labeledin each region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4 Diagram showing how PA is used to create ground state molecules. 23

1.5 Schematic of single color, 2-photon REMPI. . . . . . . . . . . . . 23

2.1 A photograph of the MOT chamber. The red arrows indicate theMOT beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Schematic of the REMPI spectroscopy experiment. E indicatesthe electric field. In this experiment, the REMPI laser is beingscanned while the PA laser is fixed. Each set of data from themulti-channel scaler is typically composed of 10-200 ionizer shotsat the same laser frequency. The total number of NaCs ions iscounted from each set, and makes up a single point in the scanshown on the computer. When performing PA spectroscopy, theREMPI wavelength is fixed and each point in the scan representsa single frequency of the PA laser as it is scanned. . . . . . . . . . 34

3.1 Combined REMPI scans taken while setting the PA to the 32 GHzline. These scans require three different laser dyes: Pyrromethene567, Coumarin 540, and Coumarin 522. Note that the backgroundlevel is low, perhaps 5-10 ions on this scale, and so all featuresabove this level indicate molecular structure. . . . . . . . . . . . . 40

3.2 Single-color RE2PI scan with DCM laser dye taken with the PAfixed to the 32 GHz line. . . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Two-color RE2PI (532 nm + red) scan taken using Pyridine I dye.The PA is fixed to the 32 GHz line. Essentially no NaCs signalwas detected without the addition of the green photon. . . . . . . 43

3.4 Single-color RE3PI scan taken with LDS 821 dye while setting thePA to the 32 GHz line. . . . . . . . . . . . . . . . . . . . . . . . . 44

3.5 Single-color RE3PI scan taken with LDS 867 dye while setting thePA to the 32 GHz line. . . . . . . . . . . . . . . . . . . . . . . . . 45

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3.6 PDS scans taken with the REMPI laser set to various X1Σ+ de-tection lines: a) v′′ = 4, b) v′′ = 5, c) v′′ = 6, d) v′′ = 19. ThePA is set to the 32 GHz line. Solid bars indicate both the positionand relative transition moments of calculated transitions from [8],arbitrarily scaled for visibility. . . . . . . . . . . . . . . . . . . . . 48

3.7 Franck-Condon map of the B1Π ← X1Σ+ and c3Σ+Ω=1 ← X1Σ+

transitions, calculated with LEVEL [3] using empirical PECs [2,5, 7]. Note the overlap of near-dissociation vibrational levels witha wide range of X1Σ+ vibrational levels for both states. . . . . . . 49

3.8 PDS scan taken while setting the PA to the 32 GHz line and theREMPI laser to a X1Σ+(v′′ = 5) detection line. Vertical dashedlines indicate B1Π(J ′ = 1)← X1Σ+(v′′ = 5, J ′′ = 0) transitions,as calculated from [2]. . . . . . . . . . . . . . . . . . . . . . . . . 50

3.9 The unpublished (dashed) and published (solid) experimental PECs,compared to the ab initio calculation (dotted). . . . . . . . . . . . 51

3.10 PDS scan taken with the PI set to 598.32 nm and the PA lockedto 1009 GHz detuned from the Cs 62P3/2 asymptote. Many of thedips correspond to transitions from a3Σ+(v′′ = 17) to B1Π (dashedvertical lines) or c3Σ+

Ω=1 (dotted vertical lines) vibrational levels. . 53

3.11 PDS scan taken with the PI set to 598.32 nm and the PA lockedto 1009 GHz detuned from the Cs 62P3/2 asymptote. Many of thedips correspond to transitions from a3Σ+(v′′ = 17) to B1Π (dashedvertical lines) or c3Σ+

Ω=1 (dotted vertical lines) vibrational levels. . 54

4.1 PDS scan taken while setting the PA to the 32 GHz line and theREMPI laser to a X1Σ+(v′′ = 5) detection line. Vertical dashedlines indicate B1Π(J ′ = 1)← X1Σ+(v′′ = 5, J ′′ = 0) transitions,as calculated from [2]. Vertical dotted lines indicate c3Σ+

Ω=1(J′ =

1)← X1Σ+(v′′ = 5, J ′′ = 0) as determined from our PA spectraand the Docenko et al. ground state PEC [5]. . . . . . . . . . . . 59

4.2 Experimentally determined IPA c3Σ+Ω=1 (dashed) and B1Π (dot-

ted) PECs from [2] and [7] around the region of a crossing. . . . . 60

4.3 Empirical and ab initio long range PECs for the (4)Ω = 1 (upperset of curves) and (3)Ω = 1 (lower set of curves) electronic states.For the empirical long-range PECs, we refer to the B1Π from Ref.[2] as (3)Ω = 1 and the c3Σ+

Ω=1 from [7] as (4)Ω = 1. Solid linesindicate the experimentally determined PECs, dashed lines areestimated from Marinescu and Sadeghpour [9], and dotted linesare from Bussery et al. [10]. Here, the energy origin is taken to bethe Cs 62P3/2 atomic asymptote. . . . . . . . . . . . . . . . . . . . 61

4.4 PA scan of the a) 32 GHz PA line and b) 958 GHz PA line. Therotational levels are labeled for the 958 GHz scan. . . . . . . . . . 62

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4.5 Depletion scans (5-point moving averaged) taken while locked tothe rotational lines in Fig. 4.4 b): a) J ′ = 1, b) J ′ = 2, c)J ′ = 3, d) J ′ = 4, e) J ′ = 5. The REMPI laser is set to detectX1Σ+(v′′ = 4) molecules. The vertical dashed lines indicate thatcalculated wavelengths of the B1Π(v′ = 48, 49)←X1Σ+(v′′ = 4) Rtransitions with the ground state rotational levels labeled. . . . . 63

4.6 CW depletion scans (5-point moving averaged) taken while lockedto the rotational lines in Fig. 4.4 b): a) J ′ = 1, b) J ′ = 2.The REMPI laser is set to detect X1Σ+(v′′ = 4) molecules. Thepositions of the vertical dashed lines are computed from a fit tothe rotational spectrum, and we have labeled the ground staterotational level for each. The energy of the A1Σ+(v′ = 29, J ′ =2)←X1Σ+(v′′ = 4, J ′′ = 1) transition calculated by Ref. [8] is11747.154 cm−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.7 PA scan across the B1Π(v′ = 13) line with rotational labels. Thevertical bars indicate the positions of the calculated energy levelsfrom [2] shifted by 6.7 GHz to the red such that the J ′ = 1 levelsmatch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.8 PA scan across the b3ΠΩ=2(v′ = 85) line with rotational labels. . . 68

4.9 The diabatic (red dashed) and RKR (blue solid) b3ΠΩ=2 PECs.We focus on the significant deviation along the outer wall, whichat its maximum is approximately 16 cm−1. The inset shows thefull depth of the two PECs, which are difficult to discern from oneanother on that scale. . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.10 Comparisons between observed b3ΠΩ=2(J′ = 2) energy levels and

rotational constants extracted from PA spectra (Xs) and calcula-tions from the RKR PEC using LEVEL (open squares). . . . . . . 70

4.11 Comparison of the long range behavior of RKR (crosses), de-perturbed (dots), Marinescu et al. (dot-dashed), Bussery et al.(dashed) b3ΠΩ=2 PECs. The solid curve is the result of a fit toseveral of the outer RKR points. . . . . . . . . . . . . . . . . . . . 71

5.1 J ′ = 0 energy levels and rotational constants for the A1Σ+ (X) andb3ΠΩ=0+ (+) electronic states. Solid lines indicate values calculatedfrom ab initio PECs using LEVEL [3]. . . . . . . . . . . . . . . . 75

5.2 An example PA scan of an A1Σ+ vibrational level with rotationalstate (J) labels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.3 A fit of the NDE equation (solid line) to observed (circles) and cal-culated (squares) A1Σ+ vibrational levels [8]. The inset highlightsperturbations in the vibrational progression. . . . . . . . . . . . . 77

5.4 An example PA scan of an b3ΠΩ=0+ vibrational level with rota-tional state (J) labels. The 1′ label indicates hyperfine structureof the J ′ = 1 state in the hypefine ghost channel. . . . . . . . . . 81

xvi

5.5 Hyperfine structure in A1Σ+(v′ = 117, J ′ = 0− 2), observed withvarious MOT and dark-SPOT configurations. The PA scans ina) and b) cover the FNa = 1, 2;FCs = 3 hyperfine entrance chan-nels. Scans c) and d) cover the FNa = 1, 2;FCs = 4 entrancechannels. The x-axis origin for scans c) and d) are shifted by 9.19GHz relative to a) and b) so that the various hyperfine peaks arealigned between the two Cs hyperfine channels. Primes indicatethose transitions from the FNa = 2 entrance channel, no primesindicate the FNa = 1 channel. Dotted, dot-dashed, and dashedlines indicate hyperfine structure associated with J ′ = 0, 1 and 2respectively, and are included to highlight differences in observedstructure throughout the scans. . . . . . . . . . . . . . . . . . . . 83

5.6 An example PA scan of an c3ΣΩ=0− vibrational level with rota-tional state (J) labels. . . . . . . . . . . . . . . . . . . . . . . . . 85

6.1 An example PA scan of an A1Σ+ vibrational level with rotationalstate (J) labels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2 An example PA scan of an c3ΣΩ=0− vibrational level with rota-tional state (J) labels. . . . . . . . . . . . . . . . . . . . . . . . . 91

6.3 Centrifugal barriers for the a3Σ+ scattering channel. . . . . . . . . 92

6.4 Calculated s-wave (lower amplitude) and f -wave (higher ampli-tude) scattering wavefunctions for the a3Σ+ electronic state atE/kB=249.810841 µK. . . . . . . . . . . . . . . . . . . . . . . . . 93

6.5 A example of the tunneling effect of an f -wave shape resonance.The amplitude ratio is the ratio between the first anti-node alongthe inner wall to the long range scattering amplitude. . . . . . . . 94

6.6 Potential energy curves and approximate wavefunctions for inho-mogeneously coupled states producing X1Σ+(v′′ = 0) molecules.Ground state potential curves are from [5] and the adiabatic 0+

states are approximated perturbatively from the diabatic PECs in[11] and long range dispersion coefficients in [9]. The experimentalB1Π state is from [2]. Wavefunctions were calculated using Level8.0 [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.7 PA scan across B1Π(v′ = 13) (no primes) and a likely b3ΠΩ=1

(primes) state with rotational labels. This scan was taken withthe REMPI laser fixed to an X1Σ+(v′′ = 0) detection line (seeChapter 7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.8 Photoassociation scan depicting the heterogeneous mixing betweenB1Π(v′ = 17) and A1Σ+(v′ = 125) rotational levels. Primed num-bers indicate B1Π(v′ = 17) labels, non-primed numbers indicateA1Σ+(v′ = 125) rotational labels. The full scan is taken using anX1Σ+(v′′=0) detection line (see Chapter 7); the REMPI frequencyused for the inset scan is sensitive to a3Σ+ and X1Σ+ ground statemolecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.9 PA scan across the coupled c3Σ+Ω=1(v

′ = 65) (no primes) andb3ΠΩ=2(v

′ = 100) (primes) lines with rotational labels. . . . . . . . 100

xvii

7.1 Franck-Condon factors for B1Π←X1Σ+(v′′=0) transitions. . . . . 103

7.2 REMPI scans detecting the absolute vibrational ground state. a)RE3PI scan with several labeled ground state vibrational levels. b)RE2PI scan where D1Π-f 3∆Ω=1 ←X1Σ+(v′′ = 0) transitions areindicated and compared toD1Π Dunham expansion from [12]. Theobservation of two perturbed electronic states gives the appearanceof doubling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.3 PDS scan taken while setting the PA to the B1Π(v′ = 13) and theREMPI laser to the IR X1Σ+(v′′=0) detection line. Bars indicatetransitions to A1Σ+ − b3Π levels, as labeled. . . . . . . . . . . . . 105

7.4 PDS and REMPI scans (not calibrated) detecting X1Σ+(v′′ =0).The REMPI scan is taken while locked to the B1Π(v′ = 13) PAline, and the depletion scan is taken while REMPI detects theinfrared X1Σ+(v′′=0) detection line. . . . . . . . . . . . . . . . . 106

7.5 CW depletion scans showing rotational population of photoasso-ciated X1Σ+(v′′ = 0) molecules. A1Σ+(v′ = 29) ←X1Σ+(v′′ = 0)rotational lines are labeled with calculations from [8]. The cal-culated line positions are shifted overall by 4.45 GHz to matchthe observed spectra. We set the PA frequency to the mixedA1Σ+(v′ = 125)-B1Π(v′ = 17, J ′ = 2, 1) levels in a) and b), andthe B1Π(v′ = 13, J ′ = 2, 1) levels in c) and d). . . . . . . . . . . . 107

8.1 Spectrum of the diode lasers used in the broadband OP experi-ment. In a), the spectra for the 4 independent 980nm laser diodesare combined. In b), the spectrum for the 1206 nm laser diodeis shown. The vertical line in a) denotes the b3ΠΩ=0+(v

′ = 0) →X1Σ+(v′′ = 0) transition . . . . . . . . . . . . . . . . . . . . . . . 111

8.2 RE3PI scans taken with: a) no OP, b) 1206 nm OP light, c) 980nm OP light, and d) 1206 and 980 nm OP light. . . . . . . . . . . 112

8.3 RE3PI scans taken without (blue) and with (red) OP: a) 1206 nmOP light, b) 980 nm OP light, and c) 1206 and 980 nm OP light.The system was better optimized for the scans in c), hence thelarger signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8.4 CW depletion scans a) with and b) without broadband OP, withrotational state labels. In a), the RE3PI laser is set to detectX1Σ+(v′′=4), while in b) it is set to detect X1Σ+(v′′=0). Diago-nal hatching indicates the range of the noise. . . . . . . . . . . . . 115

8.5 Franck-Condon factors for transitions between the lowest X1Σ+

and b3ΠΩ=0+ vibrational levels. . . . . . . . . . . . . . . . . . . . 116

8.6 Narrow-line vibrational OP scans. For the upper scan, the RE3PIlaser is set to detect X1Σ+(v′′ = 0), and for the lower scan toX1Σ+(v′′ = 1). A 2-point moving average is used to smooth thescans. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

8.7 Diagram of the combined broadband and narrow-line OP experi-ment. Green arrows indicate spontaneous emission. . . . . . . . . 118

xviii

8.8 Rotational transfer fromX1Σ+(J ′′ = 2) to J ′′ = 0 via b3ΠΩ=0+(v′ =

0, J ′ = 1). We include both the raw data points (×) and a 4-pointmoving average (solid line). . . . . . . . . . . . . . . . . . . . . . 119

xix

List of Acronyms

B-O Born-Oppenheimer

CEM Channel Electron Multiplier

CW Continuous Wave

DAVLL Dichroic Atomic Vapour Laser Locking

DFB Distributed FeedBack

DS Depletion Spectroscopy

ECDL Extended Cavity Diode Lasers

F-C Franck-Condon

FM Frequency Modulation

GPIB General Purpose Interface Bus

MOT Magneto-Optical Trap

OAM Orbital Angular Momentum

OODR Optical-Optical Double Resonance

OP Optical Pumping

PA PhotoAssociation

PDS Pulsed Depletion Spectroscopy

PEC Potential Energy Curve

PZT Piezo-Electric Transducer

RE2PI Resonance Enhanced 2-Photon Ionization

RE3PI Resonance Enhanced 3-Photon Ionization

REMPI Resonance Enhanced Multi-Photon Ionization

SPOT Spontaneous Force Optical Trap

TOF Time-of-Flight

TPA TaPered Amplifier

TWIST Thin-WIre electroStatic Trap

xx

Contributors and Funding Sources

This work was supervised by a dissertation committee consisting of Pro-

fessors Nicholas P. Bigelow (advisor), Antonio Badolato and Stephen Teitel of

the Department of Physics and Astronomy and Professors James M. Farrar and

Patrick L. Holland of the Department of Chemistry. All of the data presented

in this thesis were taken and analyzed in a collaboration between the student

and Amy Wakim of the Department of Physics and Astronomy, unless otherwise

stated. In particular, the implementation of pulsed depletion spectroscopy to

label ground state detection lines, and the analysis of the c3Σ+Ω=1 and b

3ΠΩ=2 PA

structure were very much a collaborative efforts. The acquisition of experimental

data and its subsequent analysis concerning the B1Π and Ω = 0 photoassocia-

tion spectral lines, and the production of vibrational ground state molecules via

PA as described in Chapters 4-7, was performed primarily by the student. The

simulation, design, and implementation of the broadband optical pumping ex-

periments described in Chapter 8 was primarily the work of Amy Wakim. Much

of the narrow-line optical pumping data, described in Chapter 8, was taken by

Marek Haruza. Melanie Carter provided Figs. 1.4, 1.5, and 2.2. We also collab-

orated with external groups in two of the publications listed in the Biographical

Sketch: the University of Latvia, Moscow State University, and Warsaw Univer-

sity. The University of Latvia and Moscow State groups performed calculations

based on NaCs A1Σ+−b3Π complex data, which allowed us to assign some of our

pulsed depletion spectra. We provided spectral data to the University of Warsaw

group, which they then used to complement their own data in order to calculate

an accurate c3Σ+Ω=1 PEC. The NSF and ARO provided support for the research

xxi

contained in this thesis.

1

Chapter 1

Introduction

1.1 A brief history of photoassociation

Thorsheim et al. presented the theory for the photoassociation (PA) of

ultracold Na atoms [13] soon after the first neutral atom optical molasses experi-

ments [14]. This proposal was experimentally realized in the 1990s for Na2, Rb2,

K2, Li2, and Cs2 [15–20]. With this work, PA spectroscopy became an important

technique in the study of scattering properties and previously inaccessible excited

states in bi-alkali molecules.

After the successes in producing ultracold homonuclear molecules, inter-

est grew in the study of heteronuclear dimers [21]. Unlike the homonuclear

species, heteronuclear molecules have permanent dipole moments if prepared

in the right quantum states [22]. The dipolar interaction between molecules

is strong, anisotropic (as with all dipoles), and tunable, making such molecules

suitable for a number of applications including quantum information schemes,

dipolar quantum degenerate gases, ultracold chemistry, and the investigation of

unique phases of many-body systems [23–37]. Most of these applications require

the molecules to be prepared in the rovibrational ground state in order to avoid

rovibrational quenching collisions. 2004 saw the first production of ultracold,

polar NaCs and RbCs molecules via photoassociation [38, 39], but not in the

absolute ground state. Efficient production of rovibrational ground state polar

molecules would not be achieved until the late 2000s.

2

In 2008, magnetoassociated 40K87Rb (fermionic) molecules were coherently

transferred via Stimulated Raman Adiabatic Passage (STIRAP) into the absolute

rovibrational ground state [40]. This experiment allowed the production of high-

density, nearly quantum degenerate samples of polar molecules. We note that this

approach, while highly successful, is technically challenging. Nevertheless, there

is currently much ongoing work along similar lines for a multitude of heteronuclear

species.

In the late 2000s and early 2010s, direct PA pathways to the production

of absolute vibrational ground state molecules were discovered in LiCs, NaCs,

39K85Rb and RbCs [41–44]. While not as efficient as the STIRAP technique, this

method has the advantage of simplicity (it requires only a single laser beam), and

it allows for continuous production and accumulation. Also, vibrational cooling

with broadband light can significantly enhance the ground state population while

purifying the sample [45, 46]. In this thesis, we describe the experiments that al-

lowed us to create a sample of ultracold, vibrational ground state NaCs molecules

via PA and vibrational cooling.

1.2 Motivation

The motivations for the current drive to produce, trap and manipulate

ultracold molecules are manifold and interdisciplinary. In this section we will

review some advantages to working with molecules at microkelvin temperatures,

as well as some possible applications.

As will be seen throughout the rest of this thesis, one of the most obvious

applications for molecules produced at ultracold temperatures is spectroscopy.

Of course, there are a number of techniques that are used to extract Doppler-

free spectra from thermal vapours. These generally involve the use of pump and

3

probe beams which must interact with the same velocity class for an effect to be

observed (e.g. saturated absorption, electromagnetically induced transparency,

laser-induced fluorescence, etc.). A major limitation to using hot samples for

spectroscopy is a lack of control over the ground state rovibrational distribution,

determined by Maxwell-Boltzmann statistics. Any change to the internal energies

will quickly be reverted due to collisions between molecules or with the wall. An

ultracold, dilute sample, on the other hand, consists of a ‘frozen’, athermal rovi-

brational distribution. Also, formation channels can be selected which produce

levels that are normally inaccessible in a thermal sample (e.g., near-dissociation

levels). Ultracold samples, which provide inherently Doppler-free spectra, are

thus an attractive complement to thermal samples in the investigation of molec-

ular systems.

Another important application for ultracold molecular samples involves

scattering. Specifically, certain energy-dependent scattering resonances may only

be accessed or resolved for samples with very low kinetic energies. Such reso-

nances can have a significant impact on elastic and inelastic collision rates [47].

Understanding chemical processes at low temperatures is not only necessary for

attempts to control chemical reactions in the laboratory, it also helps us un-

derstand low-temperature reactions which occur in interstellar space. Although

even the coldest current interstellar temperatures are orders of magnitude higher

than those achieved experimentally [48], ultracold chemical reactions can be used

to investigate dynamics that will likely occur in a future cosmic age due to the

expansion of the universe [47].

There is also interest in using a lattice of ultracold, polar molecules as a

quantum simulator [49–51]. While many systems in condensed matter physics

are difficult to describe using numerical simulations, polar molecules can be used

4

to model their behavior. Indeed, it has been shown that lattice-spin systems can

be modeled using polar molecules with a single valence electron trapped in an

optical lattice [50]. In that work, examples are given for systems that produce

topologically ordered states of matter which support anyonic excitations [52], as

well as those which are required for topological quantum computing [53].

Aside from the strong, tunable, and anisotropic interaction between the

molecules, several properties of these molecules make them attractive as simula-

tors and other applications. First, because the rotational energy level structure

is anharmonic, dressing the states with microwave radiation does not lead to

unwanted multi-photon excitations. Also, the first excited rotational state is

long-lived and experiences essentially the same trapping potential as the ground

state [50].

Recent experiments are pushing toward creating samples of dipolar molecules

close to the quantum degenerate regime [28–30]. Dipolar molecules are expected

to exhibit exotic behavior upon Bose-Einstein condensation. In particular, the

stability of the BEC depends on the strength of the dipolar interaction, which is

tunable [47]. Chromium, which has a large magnetic dipole moment has already

been condensed, and was shown to collapse anisotropically in a “Bose-nova” [54].

The dipole interaction is also expected to bring about novel phenomena in stable

BECs, both rotating and non-rotating [47].

1.3 Heteronuclear, diatomic molecules

In this section, we present a brief and primarily qualitative introduction

to the theory and nomenclature of heteronuclear dimers. For a more complete

description of the physics of diatomic molecules, see [55] and [1], in which much

of the information in this chapter can be found.

5

1.3.1 The Born-Oppenheimer approximation

The Born-Oppenheimer approximation is an assumption that the nuclear

and electronic variables in the molecular Hamiltonian are separable. This ap-

proximation is valid due to the high mass ratio of the slow moving nuclei to the

electron cloud and allows simplified treatment of otherwise incredibly complex

equations. In this picture, the potential energy for the atoms can be found for

a particular electronic configuration by fixing the internuclear separation R. A

set of electronic energy eigenvalues along the internuclear axis forms a molecular

potential energy curve (PEC). Because the Hamiltonian is rotationally invariant

about the center of mass, the vibrational (v) modes of the nuclear system are

calculated for a given rotational mode (ℓ) by solving the one-dimensional radial

Schrodinger equation

− ~2

2µ

d2Ψv,ℓ(R)

dR2+

[V (R) +

~2

2µ

ℓ(ℓ+ 1)

R2

]Ψv,ℓ(R) = Ev,ℓΨv,ℓ(R) ,

where µ is the reduced mass and V (R) is the PEC. Since the PEC does not depend

on angular coordinates, the rotational eigenfunctions are spherical harmonics.

The computer program LEVEL 8.0, by Robert J. Le Roy, numerically solves the

radial Schrodinger and outputs rovibrational eigenvalues for a given input PEC.

We use LEVEL extensively throughout this work.

Atoms can have electronic angular momenta which influence the nuclear

motion. To fully describe rotational energy spectra, the details of how the various

momenta couple must be considered. The Hund’s cases describe the limiting

behavior for several different types of coupling [1]. We will only refer to Hund’s

cases (a), (b), and (c), all of which assume weak interaction between molecular

rotation and electronic angular momenta.

6

1.3.2 Hund’s cases

The strong, axially symmetric electric field of the nuclei interacts strongly

with the electronic orbital angular momentum (OAM) L, which results in the

quantization of the OAM projection about the internuclear axis, Λ. Values for Λ

(0,1,2...) are labeled with capital Greek letters (Σ, Π, ∆, Φ...), rather than the

Roman letters used to label L in the atomic case (S, P , D, F ...). States with the

same magnitude of Λ which point in opposite directions are degenerate in energy

and not generally distinguished. The electronic spin S can also be coupled to the

internuclear axis through interaction with the magnetic moment of the OAM, in

which case its projection Σ (= S, S − 1, ...,−S) is a good quantum number. In

Hund’s case (a), the S-O interaction is strong enough to couple the spin to the

internuclear axis, but not strong enough that Λ and Σ lose their meaning via

coupling to neighboring electronic states. In this case, the resultant total angular

momentum including nuclear rotation, J , is quantized, and the energy spectrum

is similar to that of a symmetric top, Ev,J ≈ Ev,Ω + Bv(J(J + 1) − Ω2), where

J = Ω, Ω+1, Ω+2..., Ev,Ω is the rotationless energy, Bv is the rotational constant

for the vibrational state, and Ω = |Λ + Σ|. If Ω = 0, the energy spectrum is

similar to that of a rigid rotor.

In Hund’s case (b), interaction between the spin and the internuclear axis

is missing (primarily when Λ = 0), but coupling between the spin and rotation

results in multiplet lines for each rotational state. Because the spin-rotation

interaction is weak, the multiplet Σ rotational energy spectrum is essentially

that of a rigid rotor with small splittings.

In Hund’s case (c), the S-O interaction is large such that Λ and Σ are

mixed, and only total electronic angular momentum projected on the internuclear

axis, Ω, is a good quantum number. The rotational energy spectrum, however, is

7

the same as in case (a). Case (c) coupling is typically valid at long range, where

the S-O interaction becomes comparable to the difference in energy between the

electronic states.

1.3.3 Term symbols

Molecular electronic states are represented by term symbols which de-

scribe the angular momenta and symmetry of the state. The term symbols for

both Hund’s cases (a) and (b) are 2S+1Λ±, where ± gives the symmetry of the

state under coordinate inversion, and 2S + 1 is the spin multiplicity of the state.

The symmetry is only labeled in the non-degenerate case Λ = 0; if Λ = 0,

both even and odd parity states (corresponding to ±Λ) are present but nearly-

degenerate for molecules with low rotational quanta. Electronic states with the

same multiplicity are differentiated by letters in order of increasing energy. In

heteronuclear molecules with two valence electrons, for example, singlet states

are ordered X1Σ+, A1Σ+, B1Π, etc., and triplet states are ordered a3Σ+, b3Π,

c3Σ+, etc. In Hund’s case (c), the electronic states are identified by the term

symbol Ω±, where ± is only given if Ω = 0. States with the same angular mo-

mentum and symmetry are differentiated by numbers. For example (1)0+, (2)0+,

(3)0+, etc. describe successive states with no angular momentum and even par-

ity. Often times, however, we will label a component of a multiplet state with

the Hund’s case (a) labels with the value of Ω as a subscript. These components

are typically not ’pure’ Hund’s case (a) states, but it is helpful to identify the

primary contributor to the state in this basis.

8

Property Selection Rules

Total AM ∆J = 0, ±1, J = 0 ; J ′ = 0

Electronic AM ∆Λ = 0, ±1 and ∆S = 0; ∆Ω = 0,±1Parity +⇔ −, + < +, −< −

Vibrational Franck-Condon Factors: |⟨v′| v⟩|2

Table 1.1: Dipole selection rules for angular momenta (AM) and other propertiesof heteronuclear molecules. [1]

1.3.4 Selection rules

Dipole selection rules for transitions in heteronuclear molecules are sum-

marized in Table 1.1. As in the case for atomic transitions, dipole selection rules

are determined by calculating the off-diagonal matrix elements of the dipole op-

erator d. The strongest single-photon transitions have non-zero dipole matrix

elements, and are called dipole allowed transitions. Weaker, higher-order transi-

tions (e.g. magnetic dipole, electric quadrupole, etc.) are possible, but these are

beyond the scope of this introduction.

The non-zero matrix elements of the dipole operator can be determined

by using symmetry and conservation arguments. In order to conserve angular

momentum in the process of radiating or absorbing a photon, the total angu-

lar momentum J for the molecule can only change by 0,±1 (with the exception

that J = 0 ; J ′ = 0 as this transition cannot conserve angular momentum),

where J includes electronic and rotational angular momenta, but excludes nu-

clear spin. Though technically all of the momenta included in J are coupled,

selection rules for the individual components of the total angular momentum are

often helpful. As in the atomic case, the photon does not couple to electronic

spin (absent strong spin-orbit coupling), so ∆S = 0. In general, the electronic

orbital angular momentum and its projection Λ and the projection of the total

9

electronic angular momentum Ω can change by 0,±1. However, these selection

rules are constrained by parity and total angular momentum conservation, and

thus must be applied carefully for specific cases. Because the dipole operator is

antisymmetric, non-zero matrix elements only occur when the initial and final

molecular wavefunctions have opposite parity.

There are no strict selection rules for vibrational transitions. The vibra-

tional wavefunctions are always symmetric because, by definition, they do not

depend on angular coordinates and thus do not contribute to parity or angular

momentum considerations. However, the vibrational wavefunctions of the initial

and final states modulate the overall dipole matrix element. This modulation is

described by the Franck-Condon principle. Semi-classically, this principle states

that the internuclear separation and kinetic energy of vibration cannot change

drastically as a consequence of the transition. More formally, we can approximate

the dipole matrix element M:

M = ⟨ψ′e−r|⟨v′| d |v⟩|ψe−r⟩ ≈ ⟨ψ′

e−r| de |ψe−r⟩ ⟨v′| v⟩ ,

where |ψe−r⟩ and |v⟩ represent the electronic-rotational and vibrational compo-

nents of the total wavefunction, respectively. Only the electronic part of the

dipole operator is retained in the second step, as any purely nuclear contribu-

tion to the operator will be lost due to the orthogonality of the initial and final

electronic wavefunctions. The approximation lies in treating the dipole moment

function, ⟨ψ′e−r| de |ψe−r⟩, as independent of internuclear coordinate. For most

cases, the dipole moment function varies only smoothly with R, and so the vi-

brational overlap integral,

⟨v′| v⟩ =∫ ∞

0

v′∗(R)v(R) dR,

10

accurately describes the vibrational contribution to the dipole matrix element.

Because the intensity of a transition depends on the square of the transition

dipole matrix element, the Franck-Condon Factor is defined as the square of the

overlap integral.

1.3.5 Basis states and interactions

All electronic structure calculations by necessity are approximations and

thus neglect terms in the molecular Hamiltonian. As discussed in Section 1.1.1,

the use of molecular PECs assumes separability, which is in some cases a poor

approximation. However, these PECs can be used as basis states in more rigorous

calculations. There are multiple approximations that maintain the separability of

the electronic, vibrational and rotational wavefunctions, but the validity of a par-

ticular interaction picture can vary considerably, even between electronic states

of the same molecule. Thus, while the choice of basis is technically arbitrary,

some choices are more sensible than others.

A typical approach to improving the accuracy of molecular structure calcu-

lations is to start with a basis set and treat the neglected terms in the Hamiltonian

as perturbations. Molecular perturbations can be separated into two categories:

homogeneous, where the interacting states have the same angular momentum Ω,

and heterogeneous, where the coupling occurs between states with ∆Ω = ±1

[55]. Homogeneous perturbations arise from a number of different interactions

(spin-orbit, electrostatic, spin-spin, etc) that depend on the choice of approxi-

mate Hamiltonian. Heterogeneous perturbations arise from electronic-rotational

interactions. There are significant differences between these two types of per-

turbations. Homogeneous perturbations can be very strong and do not depend

on the quantum number J , and thus tend to shift all of the rotational structure

11

of the vibrational states equally (also called vibrational perturbations). Con-

versely, electronic-rotational coupling is weak and J-dependent, and so is often

only observed between a few rotational lines in a series (called rotational pertur-

bations). Also, vibrational perturbations are independent of J , while rotational

perturbations are approximately proportional to J .

While choice of basis set is arbitrary in the context of performing elec-

tronic state calculations, it is often helpful to discuss one or two specific inter-

action pictures to provide physical insight into experimental data. In this work,

we refer to two different interaction pictures and their associated PECs: non-

relativistic (no S-O interaction) [11] and relativistic (including S-O interaction)

[56]. As discussed in 1.1.3, The non-relativistic PECs are appropriately labeled

with Hund’s case (a) term symbols, as these contain the valid Λ and S quantum

numbers. S-O coupling renders these quantum numbers less valid, so Hund’s case

(c) term symbols label the relativistic PECs. The non-relativistic PECs are most

valid for predicting rovibrational energy levels when there is weak interaction and

large separation between interacting states, such as the X1Σ+ state in NaCs, or

deeply bound vibrational levels in certain excited states. Transitions between

non-relativistic electronic states follow the strict spin selection rule ∆S = 0.

The relativistic PECs, on the other hand, are more valid everywhere than non-

relativistic PECs, particularly at long range where the electronic states are closely

spaced. Because spin is not a good quantum number in this case, the only strict

selection rule for transitions between relativistic PECs is ∆Ω = 0,±1. However,

while the relativistic PECs may be more realistic than the non-relativistic PECs,

they do not, by themselves, contain sufficient information to calculate transition

dipole moments. To do this, the relativistic electronic states must be expressed

as (R-dependent) mixtures of nonrelativistic states [6]. Transitions can then be

12

4 6 8 10 12

-5000

0

5000

10000

15000

Internuclear separation HÞL

En

erg

yHc

m-

1L

X1S+

a3S+

A1S+

c3S+

B1P

b3P

Na 3S + Cs 6S

Na 3S + Cs 6P

Figure 1.1: The empirical PECs of the lowest-lying electronic states of NaCs [2, 5–7]. The A1Σ+ − b3Π complex PECs are diabatic, and thus do not all approachthe fine-structure asymptotes. The b3ΠΩ=2 PEC is the adjusted version that isdiscussed in 4.2.4.

interpreted as taking place only between components of these mixtures that are

spin allowed. For example, a Ω = 1→ Ω = 0 transition is nominally allowed, but

if the Ω = 0 state is mostly spin singlet in character, while the Ω = 1 is mostly

spin triplet in character, the transition is unlikely to occur.

Most ab initio PECs are not useful for making precise assignments of ex-

perimental spectra, but can be used as starting points in calculations of empirical

PECs. Empirical PECs have been found for numerous electronic states of the

13

NaCs molecule, using a variety of experimental and computational techniques

[2, 5–7, 12, 57, 58]. The lowest lying of these empirical PECs are given in Fig.

1.1 Molecular PECs calculated using the inverted perturbation (IPA) or Rydberg-

Klein-Rees (RKR) methods simply recreate the best PEC which reproduces the

observed (and labeled) rovibrational spectra. These are best suited to electronic

states which are relatively unperturbed. In cases where these approaches are

insufficient due to large perturbations, a deperturbation analysis may be used.

With this technique, the goal is to obtain the set of non-relativistic PECs and

interaction terms in a coupled Hamiltonian model which reproduce the data. We

emphasize that this technique does not yield PECs which can be used to di-

rectly compute eigenvalues with programs like LEVEL; the fully coupled model

must be utilized to interpolate/extrapolate the results for comparison with other

experiments.

An important phenomenon occurs when non-relativistic PECs of the same

symmetry (∆Ω = 0) cross. Once the S-O interaction is added, the PECs will

avoid the crossing. The paths of the two PECs will switch compared to the non-

interacting case at the crossing. This effect may be approximated [59] by using

the non-relativistic PECs and the interaction terms to construct a matrix:

V =

V1 + ξ11 ξ12 . . . ξ1iξ21 V2 + ξ22 . . . ξ2i...

.... . .

ξi1 ξi2 Vi + ξii

,

where ξmn includes the total interaction between each pair of electronic states,

as well as the on-diagonal shifts for m = n, and Vi is the PEC corresponding

to the electronic basis function |ψi⟩. Diagonalization yields a set of mixed PECs

in which interacting states do not cross. The amount of mixing between states

is determined by evaluating the R-dependent eigenfunctions of the matrix. An

14

3 3.5 4 4.5 5 5.5 6

5000

5500

6000

6500


En

erg

yHc

m-

1L

Figure 1.2: An avoided crossing between the A1Σ+–b3ΠΩ=0+ electronic states ofNaCs. The diabatic (dashed) lines are taken from [6], and the adiabatic PECsare approximated by diagonalizing the interaction matrix. In this case, the ex-perimentally determined R-dependent S-O function is used.

example of an avoided crossing for NaCs is given in Fig. 1.2.

The validity of crossing and anti-crossing potential curves depends on the

vibrational frequency of the molecule and the strength of coupling between the

states. A high vibrational frequency associated with levels above the crossing

(i.e., small spacing between adjacent vibrational levels) and/or weak coupling

may cause the electronic state to remain constant across the interaction region

[1]. In this scenario, the electronic states may be represented more accurately

as diabatic. The adiabatic representation is better suited to cases of strong

coupling and/or low vibrational frequency. Simply stated, if many molecular

vibrations must occur before the electronic state changes (possibly due to weak

coupling), then it is best to approximate the states as crossing, and vice versa

15

for non-crossing states. Both the diabatic and adiabatic pictures are unrealistic

right at the (avoided) crossing, as the B-O approximation breaks down when the

electronic states are nearly degenerate. From an experimental standpoint, the

choice between adiabatic or diabatic representations is made based on practical

considerations: whichever approach best fits the data. In real systems, interacting

states may have properties of both crossing and anti-crossing potential curves.

When interacting electronic states have different dissociation asymptotes,

predissociation may occur. Predissociation is a non-radiative transfer from a

bound molecule to a free atom pair via change in electronic state [1]. Quali-

tatively, such a state change is made possible because the near-degeneracy in

energy leads to non-trivial Franck-Condon overlap between the free and bound

vibrational wavefunctions. Predissociating states have a finite lifetime as a bound

molecule, causing line broadening. For predissociation to occur during spectro-

scopic experiments, these lifetimes must be on the order of or shorter than other

processes, such as spontaneous emission (typically > 10 ns).

Heterogeneous coupling terms in V are typically ignored while calculating

PECs; however, they are included in more rigorous calculations, such as those in

[6], and give rise to perturbations when two different rovibrational levels with the

same J are nearly degenerate. The resulting rotational levels, while still having

a definite value for J , will no longer have a definite value for Ω. This means that

the selection rules for the electronic transitions are relaxed (e.g. nominal Ω = 2

states can decay to Ω = 0 if mixed with an Ω = 1 state).

1.3.6 Hyperfine structure

The nuclei have electric and magnetic moments which interact with var-

ious fields produced within the molecule. The most prominent coupling occurs

16

between the magnetic dipoles of the nucleus and electrons. The electrons have

both spin and orbital angular momentum which can result in a magenetic dipole

moment, but the strongest electronic-nuclear spin coupling is the Fermi contact

interaction, caused by non-zero electron spin density at the nuclei. Other sources

of hyperfine structure include nuclear spin-nuclear spin, nuclear spin-molecular

rotation, and the electric quadrupole moment of the nucleus with the electric field

within the molecule. The magnitude of all these interactions vary considerably,

and most depend on the rotational, electronic, and vibrational configuration of

the molecule. In our work, we only resolve hyperfine structure caused by the

magnetic dipole-dipole interaction.

1.3.7 Electric dipole moment

One of the properties of the NaCs molecule that makes it attractive for the

applications discussed previously is its strong electric dipole moment, calculated

to be ∼4.6 Debye for the vibrational ground state [22]. This dipole moment

arises from the difference in electronegativity between the Na and Cs atoms. The

larger Cs nucleus attracts the shared electrons slightly more than Na, resulting

in a charge asymmetry, and thus a permanent electric dipole moment pointing

along the internuclear axis. However, this dipole moment will average out due

to the rotational symmetry of the molecule unless oriented by an external field.

Fortunately, the typical field required to accomplish this is reasonable (∼104

V/cm).

Polar molecules interact with external electric fields via the Stark effect.

There is no linear Stark shift for molecules with Λ = 0 ground states because

each rotational state has definite parity, while the dipole-field interaction Hamil-

tonian has odd parity [1]. However, a quadratic effect occurs via mixing between

17

neighboring rotational states of opposite parity. We exploit this effect to electro-

statically trap neutral, ground state NaCs molecules [60].

1.4 Molecule formation and detection

In the following section, we introduce the concepts essential to creating and

detecting ultracold NaCs molecules via PA. First, we prepare ultracold atoms in a

Magneto-Optical Trap (MOT). Photoassociation occurs when a pair of atoms in

the trap absorb a photon during a collision. These PA molecules can then radia-

tively decay into the ground state. A pulsed laser then ionizes certain molecules

in the sample which can be detected with a channel electron multiplier (CEM).

1.4.1 Magneto-Optical Trap

The MOT has been described in detail elsewhere [61–63]. Briefly, single

frequency laser beams are aligned along the x, y, and z axis, with mirrors po-

sitioned to retro-reflect the beams back along each axis. The laser is slightly

detuned from an atomic resonance such that each beam is blue-shifted back into

resonance in the reference frame of atoms moving counter to its propagation via

the Doppler shift. Independent of direction, atoms moving over a certain range of

velocities in the overlap region will encounter a resonant beam and cycle between

the ground and excited states. Stimulated emission can be ignored for sufficiently

low laser intensities, and thus most of the emission is spontaneous. Each time

a photon is absorbed the atom receives a small momentum kick ~k in the di-

rection opposite its motion, and a subsequent kick from spontaneous emission in

a random direction. Over many cycles the randomly oriented momentum kicks

from spontaneous emission will average out, resulting in a net force opposite the

motion. Energy is conserved as the spontaneously emitted photons will be more

18

energetic than the absorbed photons due to differences in their respective Doppler

shifts. This results in a velocity-dependent optical force, also known as an optical

molasses.

In a MOT, the optical molasses beams are circularly polarized beams

and a magnetic quadrupole field is added to create spatially-dependent force.

The magnetic field Zeeman shifts the levels of an atom moving away from the

trap minimum. On one side of the field minimum, the Zeeman sublevels are

shifted such that the ∆MF = +1 transitions driven by the σ+ beam are closer

to resonance than the ∆MF = −1 transitions driven by the σ− beam, pushing

the atom back to the center. The roles are reversed on the other side of the

magnetic field minimum, with the σ− beam providing the restoring force. The

use of σ± polarized light also ensures that the atoms are optically pumped by

the appropriate beam into closed cycling transitions that maximize the scattering

rate.

The alkali atoms have two hyperfine ground states, only one of which

is addressed by the trapping light. After a sufficient number of cycles, some

of the atoms in the cycling transition may decay into the dark state, at which

point they are lost to the MOT. To prevent this, a separate beam is used to

repump the atoms back into the cycling transition. We usually operate in the

dark-SPOT (spontaneous-force optical trap) configuration [64], where the center

of the repumper beam is masked. The atoms in the center of the MOT are then

optically pumped to the dark state, but repumped on the edges before they are

lost from the trap. This technique increases the ground state fraction of the

MOT, and reduces trap loss from light-assisted collisions. This also prevents

the spontaneous formation of autoionizing molecules in our dual species Na+Cs

MOT [65].

19

4 6 8 10 12 14 16 18 20

-6000

-4000

-2000

0

2000

4000


Energ

yHc

m-

1L

Nuclear repulsion

Electron exchange

Dispersion

Figure 1.3: A plot of the X1Σ+ PEC with the dominant interactions labeled ineach region.

1.4.2 Interatomic forces and cold collisions

Atomic collisions are central to any technique which creates molecules

from ultracold atoms. The interaction between two different alkali atoms is pri-

marily described by three types of interatomic interactions: the van der Waals,

electron exchange interactions, and nuclear repulsion. The ground state PEC of

NaCs is given in Fig. 1.3 with the dominant interatomic interaction labeled for

each regime. The attractive van der Waals (or dispersive) forces arises from the

instantaneous Coulombic interactions between the two atoms averaged over the

electronic motion [1]. These are often weak compared to the other two interac-

tions, but dominate at large internuclear distances. The potential energy curve

associated with dispersion is expressed as an expansion in 1/R with coefficients

Cm determining the interaction strength. The theoretical framework for finding

the coefficients of expansion in bialkali molecules can be found in Refs. [9, 66].

We note that the leading order of the expansion is not necessarily the same for

different pairs of atomic states. In this thesis, we primarily focus on heteronuclear

20

pairs in which one of the atoms is in the S state and the other is in the S or

P state. In these cases, the leading order is always 1/R6 due to symmetry con-

siderations. The electron exchange interaction is a quantum effect arising from

Coulomb forces. Because the valence electrons are identical fermions, the total

wavefunction including spin must be antisymmetric. The total orbital wavefunc-

tion can be symmetric or antisymmetric under exchange, meaning the spin must

take on the opposite for each case. This pair of states is degenerate in energy at

long range, where the atoms do not interact. As the two atoms approach, the

Coulomb interaction between the electrons and nuclei lifts the degeneracy. This

interaction can be very large for medium-range internuclear separations, result-

ing in deep PECs (thousands of cm−1). For small internuclear separation the

energy of the PEC increases steeply. This repulsion increases as the wavefunc-

tions of the filled-shell electrons of each nucleus begin to overlap one another. The

PEC represents the sum of all of these interactions as a function of internuclear

separation.

Because collisions in a MOT occur at ultracold temperatures, where the

de Broglie wavelength of the colliding atoms is on the order of the extent of the

molecular potential well, we utilize quantum mechanical scattering theory. The

interaction between the two atoms depends only on the internuclear separation,

so it is useful to write the Schrodinger equation in spherical coordinates. This

is the same situation as given in 1.1.1, except now the solutions are not bound.

The typical approach begins by assuming a solution that consists of an incoming

plane wave plus an outgoing (scattered) spherical wavefunction. The incoming

plane wave is the solution for the Schrodinger equation with no interaction, and

can be expanded as a sum of incoming and outgoing spherical eigenfunctions (this

is the partial wave expansion). Each spherical eigenfunction has angular momen-

21

tum ℓ, and the corresponding incoming and outgoing waves are phase shifted

relative to one another by ℓπ due to the centrifugal term in (1.3.1). Adding in

the attractive potential between the atoms results in complicated radial functions

at short range; however, the solution must still approach that of a free particle

at long range. The only long-range effect of the scattering potential then is to

cause another ℓ-dependent phase shift in the outgoing waves. Qualitatively, as

the atoms approach the attractive potential their kinetic energy increases, thus

altering the de Broglie wavelength of the system. It is this alteration which

contributes the phase shift in the outgoing waves. For sufficiently low scatter-

ing energies and high angular momentum ℓ, the atoms must tunnel through the

centrifugal barrier, significantly reducing the wavefunction amplitude that pen-

etrates to short-range. Thus, partial waves with large angular momentum ℓ do

not interact with the scattering potential, are not phase-shifted with respect to

the outgoing components of the plane wave, and thus do not contribute to the

scattered part of the wavefunction. ℓ = 0 collisions have no such barrier.

1.4.3 Photoassociation

PA is a process in which two atoms absorb a photon during a collision,

forming a bound, excited state molecule [67]. As discussed earlier, the application

of this technique to ultracold atoms was proposed and realized shortly after early

atomic cooling and trapping experiments, allowing high-resolution spectroscopic

investigation of long-range states which were previously inaccessible. These states

are of interest because they can be used to measure the strength of the van der

Waals interaction, and because they exhibit different properties when compared

to deeply bound molecules; the atomic wavefunctions overlap very little, and

so the system resembles more closely a pair of weakly interacting atoms than

22

a molecule. PA is well-suited for investigation of these states as free-bound F-

C overlap is highest when the excited state has more wavefunction amplitude

at long-range, where the interaction is weak and the scattering wavefunction

amplitude is relatively large.

PA spectra also reveal information about collisions. Dipole selection rules

require that a reaction with angular momentum ℓ is excited to a state with

J = ℓ, ℓ± 1. Additionally, the parity of the ground state collision is simply (-1)ℓ

[68], while the parity of the excited state is typically known. Further, the PA

transition must conserve total momentum, such that J ′ may not differ from ℓ by

more than l + s, where l and s are the excited state electronic orbital and spin

angular momentum, respectively. Thus, analysis of the excited state rotational

spectra gives information about the partial waves involved in the collision. For

the most part, PA of ultracold atoms involves only the lowest partial waves, as

the collision must have sufficient energy to overcome the centrifugal barrier. An

exception occurs when the presence of quasibound states in the scattering channel

allows the wavefunction to penetrate the barrier. These ’shape resonances’ are

often observed with PA spectroscopy [41, 42, 69–72].

Another important application of PA, as discussed earlier, is the formation

of ultracold, ground state molecules. Assuming no predissociation or secondary

excitation, a newly formed PA molecule will spontaneously emit a photon and,

if the transition is favorable, decay into a stable, bound molecular state (see

Fig 1.4). The excited PA molecule can decay into a number of ground state

vibrational levels. The probability of populating any particular ground state

vibrational level for a given transition is determined by the transition strengths,

which are heavily influenced by F-C factors.

23

Figure 1.4: Diagram showing how PA is used to create ground state molecules.

Figure 1.5: Schematic of single color, 2-photon REMPI.

1.4.4 Resonance Enhanced Multi-Photon Ionization

Resonance Enhanced Multi-Photon Ionization (REMPI) is a well-established

spectroscopic technique [73] and is widely used in PA experiments. In REMPI,

ions are detected while a tunable, pulsed laser is aligned to the sample. The

intensity is such that a molecule will not be ionized unless the laser is resonant

with one of the bound-bound molecular transitions (see Fig. 1.5). As the laser is

24

scanned, peaks corresponding to molecular transitions appear in the ion signal.

This technique allows one to choose the ground state that is being detected, and

is thus is often referred to as state-sensitive or state-selective detection [39].

25

Chapter 2

Experimental Design

In this chapter we, discuss apparatus and experimental design. The dis-

cussion on the experimental apparatus will be relatively brief and general as

details can be found in the theses of Jan Kleinert and Chris Haimberger [74, 75].

The production and detection of ultracold NaCs molecules requires an ultra-high

vacuum (UHV) chamber, atomic sources, a variety of laser sources, an ion de-

tection system and timing circuitry. We perform several experiments with this

apparatus, including PA and REMPI spectroscopy, pulsed and continuous-wave

depletion spectroscopy, optical pumping and electrostatic trapping.

2.1 Apparatus

2.1.1 Vacuum chamber and atomic sources

To achieve high MOT and molecule densities, collisions with background

gases must be reduced or eliminated. Thus, atomic and molecular samples are

kept under UHV pressures, roughly defined as < 10−9 torr. Achieving such

low pressures requires great care: the chamber must be constructed of clean

materials that do not outgas (e.g. stainless steel, oxygen-free copper, glass), and

these components must be ’baked out’ after exposure to atmosphere to eliminate

contaminants. Our vacuum system consists of two mechanical roughing pumps

and three turbo-molecular pumps. The mechanical roughing pumps alone achieve

a vacuum of ∼10 millitorr in the chamber. Turbo-molecular pumps, or turbos,

consist of a series of fan-like blades spinning at tens of thousands of RPM. The

26

blades kick atoms or molecules through a series of stages of increasing pressure

towards the outlet. Turbos operate most efficiently in a pressure regime in which

collisions between the gas molecules are negligible, necessitating the mechanical

pumps. In our system, the final chamber pressure is ∼ 5× 10−10 torr.

We have two separate atomic sources for Na and Cs. The melting point

for Cs is only 28C, and it has a relatively high vapor pressure (∼ 10−6) at this

temperature [76]. This means that our Cs MOT can be loaded from an atomic

vapor source attached directly to the chamber. Na has a higher melting point of

98C with a lower associated vapor pressure of ∼ 10−7 [77], making direct loading

much less efficient. Instead, we employ a Zeeman slower, which longitudinally

slows a collimated atomic beam of Na atoms before it reaches the MOT. The

design, construction and operation of this Zeeman slower has been detailed in

Jan Kleinert’s thesis [74].

Our chamber and vacuum system (much of which can be seen in Fig. 2.1)

is constructed as follows. The main chamber is stainless steel and has two 8 in.

and 16 2.75 in. ports. Attached to one of the 8 in. ports is a reducer to a 6

in., 4-way cross. A Pfeifer TPU-110 turbo is attached to this cross. An Alcatel

5080 turbo is in series with the primary pump to increase the overall compression

ratio and reduce pressure in the chamber. An Alcatel ZM2008AC rotary vane

pump provides the rough pressure for the main chamber vacuum line. One of

the 2.75 in. ports opens to the Zeeman slower, which consists of a 0.8 m long

stainless steel tube with a series of exterior magnetic coils providing the Zeeman

field. At the far end of the tube is a tee with one port leading to the Na atomic

source and the other leading to a Pfeifer TPU 050 turb. This pump is backed

by a Pfeiffer Trivac D16A rotary vane pump. This secondary vacuum line serves

two purposes: to lower the differential pressure across the Zeeman slower tube,

27

Figure 2.1: A photograph of the MOT chamber. The red arrows indicate theMOT beams.

and to minimize accumulation of sodium on the tubing leading from the source

to the Zeeman slower. The chamber pressure is measured using a Varian UHV-24

cold-cathode ion gauge and Varian 845 Vacuum Ionization Gauge controller. The

ion gauge is attached to a 2.75 in. tee that is in turn attached to one of the main

chamber ports. Another of the 2.75 in. ports is taken by a stab-in lamp used for

bake-outs, while another is used for ion detection. The rest are glass viewports

for optical access, or simply blanked off if not in use.

2.1.2 Laser sources

We utilize a wide variety of laser sources in the experiment: diode-pumped

Nd:YVO4, ring Ti:Sapphire, ring dye, Q-switched Nd:YAG, pulsed dye, dis-

tributed feedback diode, in-Littrow ECDLs and free running diode lasers. We

28

will briefly describe each laser system and how it is used in the overall setup.

The diode-pumped Nd:YVO4 Coherent Verdi V10 systems provide up to

10.5 watts of stable, narrow-line (< 5 MHz) laser light at 532 nm. Each system

consists of a laser head and power supply. The power supply contains control

electronics and high-power 808 nm diode bars. An umbilical of bundled electrical

cable and optical fiber connects the power supply to the head. The head contains

the Nd:YVO4 gain medium, cavity, and frequency doubling crystal. The entire

system requires little maintenance and is essentially turnkey. One of the V10

lasers pumps the ring Ti:Sapphire laser, while the other pumps a ring dye laser.

The 899-21 Coherent Ti:Sapphire laser provides up to 2 watts of narrow-

line (< 1 MHz), infrared light for PA. The 899 cavity is in a ring configuration

(as is the Verdi), with four mirrors causing the light to travel in a figure eight.

This results in a traveling wave with no nodes or discrete modes, allowing for

more efficient gain and better tunability when compared to a linear cavity with a

standing wave. The Ti:Sapphire crystal has an extremely broad emission profile,

which allows the laser to operate between 600 and 1100 nm with the appropriate

optics and pump power. Intracavity line-narrowing elements include a birefrin-

gent (Lyot) filter, and thin and thick etalons. The birefringent filter and thin

etalon also provide coarse tuning, while a scanning Brewster plate allows fine

tuning. External control of the frequency is available through the laser control

box. Frequency stability is achieved by locking to a thermally stabilized Fabry-

Perot cavity. The cavity is not completely stable over longer time scales, however,

and can drift tens of MHz over the course of an hour. In situations where long-

term frequency stability is required, the laser is locked relative to the signal of

another stabilized laser using a scanning Fabry-Perot cavity. The combination of

narrow linewidth, high power, and broad tunability makes the 899 Ti:Sapphire

29

laser ideal as a PA laser.

We measure the wavelength of the 899 using a Burleigh WA-1000 wave-

length meter, which consists of a scanning Michelson interferometer and uses a

stabilized He-Ne laser for internal calibration. While running the experiment,

we pick off a small fraction of the beam to free-space couple into the wavemeter.

Because the wavemeter readout is sensitive to the input angle of the beam, we

calibrate all of the scans to the nearest Cs atomic transition. With this tech-

nique, the typical absolute uncertainty for our PA scan frequency is < 1 GHz

(the specified absolute accuracy is ±0.3 GHz. If needed, we can also use the ex-

ternal Fabry-Perot cavity to calibrate the relative frequency of the scan, typically

achieving ∼10 MHz accuracy.

The 699-21 Coherent dye laser is very similar to the 899 model, but in

this case the gain medium is dye in solution. An optical quality ribbon of dye is

passed in free-space through the cavity from a nozzle at ∼ 80 PSI. The dye is then

collected in a catcher tube and recirculated. We use this 699 with Rhodamine 6G

dye dissolved in ethylene glycol for the Na MOT and Zeeman slower beams. The

laser is externally stabilized using a frequency-modulated saturated absorption

lock (details of this locking system can be found in Jan Kleinert’s thesis [74]).

Use of a dye laser is necessary because no laser diodes have been developed that

operate at 589 nm. Recently, diode-pumped solid state lasers with sufficient

power have been developed for Na, but these are still prohibitively priced.

We operate two Q-switched Nd:YAG lasers: a Continuum Minilite II and

a Spectra-Physics INDI. Both lasers produce < 10 ns pulses at 532 nm and a

repetition rate of 10 Hz. To produce the pulses, a high energy Xe flashlamp

pumps an Nd:YAG rod. As the flashlamp pumps the gain medium, the Q of

the cavity is kept low using a Pockels cell and polarizer. Once the gain medium

30

is saturated, a pulse to the Pockels cell switches the cavity to high Q, and the

energy built up in the rod is released via stimulated emission as a high-intensity

pulse. We use these lasers to pump pulsed dye lasers.

The pulsed dye lasers are both Lambda-Physik FL3002 models, and pro-

duce < 10 ns duration pulses with ∼0.5 cm−1 linewidth. The lasers are broadly

tunable over visible and near-infrared wavelengths with the appropriate dyes.

Oscillator and amplifier dye cells are transversely pumped by a pump laser. Flu-

orescence from the oscillator cell seeds a cavity which includes a grating for line-

narrowing. The light from the cavity is then amplified twice, once as it passes back

through the oscillator cell and again as it passes through the amplifier cell. The

wavelength can be tuned externally through a GPIB (General Purpose Interface

Bus) connection. We use these lasers to perform REMPI and pulsed depletion

spectroscopy. The wavelength of the REMPI laser is set by the computer via

the GPIB interface. However, this wavelength must be calibrated externally. We

calibrate scans by comparing observed atomic REMPI line positions to the tran-

sition energies published in the literature [78, 79]. This process gives us a typical

uncertainty of < 1 cm−1.

Diode lasers are ubiquitous in the ultracold molecule lab. We use them

in a number of different configurations and for a multitude of purposes. The

Cs MOT laser system is composed of three diodes and a tapered amplifier chip

(TPA). Two of the diodes lasers are in a Littrow extended cavity configuration

(ECDL), with a grating mounted on a PZT (piezo-electric transducer) providing

optical feedback to the diode for frequency stabilization. Electronic feedback to

the PZT is provided with a signal from a Doppler-free DAVLL setup, which is a

combination of saturated absorption and Dichroic Atomic Vapour Laser Locking

(DAVLL) [80]. In DAVLL, an atomic reference cell is wrapped in an electromag-

31

netic coil, splitting the Zeeman sublevels [81]. The σ+ and σ− components of

the linearly polarized reference beam are then absorbed at slightly different fre-

quencies. The output is split into the circular components, and measured on two

different photodiodes. One of the signals is then subtracted from the other, creat-

ing an approximately linear signal that passes through zero at the line center. In

Doppler-free DAVLL, we retro-reflect the beam that is weakened by absorption

back through the cell before sending it to the detectors. This creates an effective

pair of counter-propagating pump and probe beams. The pump and probe beams

only interact with atoms in the same velocity class when they are at precisely

the frequency of the atomic transition (or half-way between atomic transitions).

When this happens, the probe beam is essentially not absorbed because the pump

beam is saturating the atoms of the same velocity class, producing Doppler-free

spectra.

We also utilize laser diodes for PA and depletion spectroscopy. Home-built

Littrow lasers with off-the-shelf diodes rarely have the mode-hop free tunability

necessary for performing such experiments. Distributed feedback (DFB) diode

lasers have a diffraction grating etched on the device that provides wavelength

selection. These diodes are tunable over several nanometers, with large mode-

hop free scanning regions (∼ 20 GHz) and linewidths of several MHz. We utilize

an 852 nm DFB (sometimes amplified with a TPA) for both PA and depletion

experiments. We also borrowed a Toptica DL-Pro diode laser system to perform

rotational and vibrational state transfer experiments. This particular DL-Pro

model covered the 900-1000 nm range with mode-hop free scanning regions of

∼10 GHz.

Finally, we employ free-running, multi-mode diodes for broadband optical

pumping experiments [46]. These diodes have several nm wide spectral widths

32

and operate without any optical feedback. We use four diodes covering the 979-

990 nm range, and another covering 1204-1208 nm. All of the diodes output ∼2

W of optical power.

2.1.3 Ion detection

Ions are detected with a Sjuts KBL 408 channel electron multiplier (CEM).

The CEM consists primarily of a continuous dynode structure, essentially a glass

tube. The entrance of the dynode is held at a large negative voltage, collecting

positive ions created from REMPI. The ions impact the dynode wall with a

large amount of kinetic energy and eject electrons. These secondary electrons

then impact the opposite side of the dynode, and the process continues. This

electron cascade is directed down the CEM to the anode end, resulting in a

measurable electronic pulse. These pulses are counted with respect to the timing

of the REMPI laser pulse using an SR430 multi-channel scaler. The positive ions

created by the REMPI pulse are immediately funneled down to the detector by

the combined electric field of a ring electrode and the CEM. The time it takes

each atomic or molecular ion (all with the same charge +e) to reach the CEM

depends only on its mass. This time-of-flight (TOF) mass spectrometer allows

us to distinguish between the different species of ions created in the experiment.

2.1.4 Timing and data collection

The maximum repetition rate of the experiment is 10 Hz, set by the

Nd:YAG pulse frequency. We trigger the flashlamp at 10 Hz with an Array

Analysis MFI-1000 timing computer. At this rate, the pulse resolution is 100

microseconds, sufficient for all current experiments. This computer also supplies

pulses to acousto-optic modulators controlling the MOT light and the CEM volt-

33

age control circuit. Ion detection data are collected from the scaler on a computer

running Labview 6 using a GPIB interface. This computer also scans the tunable

laser systems.

2.2 Experiments

2.2.1 Photoassociation and REMPI spectroscopy

The PA and REMPI techniques, as described in the previous chapter,

are central to all of the work described herein. Both PA (molecule formation)

and REMPI (molecule detection) may also be utilized as spectroscopic tools. By

fixing the REMPI wavelength to a known detection line and scanning the PA

frequency, we investigate the available free-bound transitions. Conversely, fixing

the PA to a known free-bound transition and scanning the REMPI laser reveals

the bound-bound structure of the ground state molecules.

The PA and REMPI spectra are considerably different from one another.

The pulsed dye REMPI laser can cover tens of nanometers in a single scan, but

has a broad linewidth. The Ti:Sapphire PA laser scans only tens of GHz at a

time, but with substantially better resolution. Thus, PA spectroscopy reveals the

rotational and hyperfine structure of the excited state, but scanning the entire

accessible free-bound spectrum requires a significant time investment. REMPI

spectroscopy, on the other hand, is ideal for detecting a large number of bound-

bound transitions quickly, but does not reveal any information on rotational or

hyperfine structure.

2.2.2 Continuous wave (CW) and pulsed depletion spectroscopy

While the REMPI technique allows vibrational state-sensitive detection

of ground state molecules, REMPI spectra alone are not always sufficient for

34

Figure 2.2: Schematic of the REMPI spectroscopy experiment. E indicates theelectric field. In this experiment, the REMPI laser is being scanned while the PAlaser is fixed. Each set of data from the multi-channel scaler is typically composedof 10-200 ionizer shots at the same laser frequency. The total number of NaCsions is counted from each set, and makes up a single point in the scan shownon the computer. When performing PA spectroscopy, the REMPI wavelength isfixed and each point in the scan represents a single frequency of the PA laser asit is scanned.

35

unambiguous state labeling. This is because PA populates a large number of

ground state vibrational levels, and because each ground state vibrational level

is associated with many transitions to excited states. Even when the positions

of the excited states are known from other experiments, the uncertainty of the

REMPI wavelength and high line density prevents accurate assignment of the

spectra. Depletion spectroscopy provides a way to unambiguously label these

REMPI lines.

Depletion spectroscopy has been used in a number of ultracold molecule

experiments [41, 42, 82–84]. In this technique, a pulsed or CW laser illuminates

the sample prior to ionization. The depletion laser is scanned while the REMPI

laser is set to detect a single ground state vibrational level. Molecules in the

sample that are driven to an excited state by the depletion beam decay to a broad

range of ground state vibrational levels, depleting the ground state. The molecule

may also be ionized or dissociated. Either way, a dip in the ion signal occurs

when the depletion laser drives molecules out of the state that the REMPI laser

is set to detect. In this way, we record only the transitions from a single ground

state vibrational level. A pulsed depletion laser can scan across widely spaced

vibrational levels very quickly, making it well-suited for assigning vibrational

structure. Narrow-line CW depletion lasers are necessary for resolving rotational

structure, but are better suited to situations in which the transition wavelengths

are already approximately known.

2.2.3 Optical pumping

Once the molecular states are assigned using the combination of PA,

REMPI and depletion and spectroscopy, we can perform experiments to con-

trol the molecular states present in the sample using optical pumping (OP). OP,

36

or Luminorefrigeration [85], refers to the general method of using light to manip-

ulate the internal states of a sample of atoms or molecules. The OP light is tuned

to a frequency (or frequencies) that couples to certain transitions in the atom or

molecule in such a way that the system is driven into a single, predetermined

state.

We perform both broadband and narrow-line optical pumping experiments

on ground state sample of NaCs molecules. In broadband OP, a number of high

intensity, multi-mode diode lasers interact with all but the lowest ground state

vibrational levels (v = 0, 1, 2). This eventually drives the molecules into these

lowest ‘dark’ states, vibrationally purifying the sample. This technique was pre-

viously used to optically pump homonuclear molecules into the absolute ground

vibrational state [45]. Rotational pumping of molecular ions has recently been

achieved in MgH+ and HD+ [86, 87]. However, narrow-line OP is not generally

used to optically pump neutral molecules, with laser cooling experiments on SrF

being the major exception [88]. For narrow-line OP to be be practical, the num-

ber of transitions in the system must be relatively small, otherwise many different

lasers are needed. In general, molecules have many rovibrational levels which are

accessible to the excited state via spontaneous emission, making such experiments

difficult. In NaCs, like SrF, the lowest several ground and excited state vibra-

tional levels form a relatively closed system, making narrow-line OP feasible. By

exploiting selection rules and F-C factors, we can transfer significant fractions of

population from one rovibrational level to another using a single-frequency laser.

2.2.4 Electrostatic trapping

Molecules formed by PA are not confined by the magnetic field or trapping

light and thus drift away from the MOT after several milliseconds. To hold the

37

molecules longer, we constructed a Thin-Wire Electrostatic Trap (TWIST). The

design and construction of the trap is presented elsewhere [60, 74, 89]. Briefly, the

trap consists of two ring electrodes 8 mm in diameter and spaced 2 mm apart.

The rings are made of 75 µm tungsten wire. The MOT sits at the center of

the trap; the wires only slightly perturb the MOT beams. Both rings are held

at +1 kV, creating an electrostatic quadrupole field. This holds the weak-field

seeking (J < 0) molecules at the field minimum. Two more rings, situated 3 mm

apart from the trapping rings on each side, are held at a slightly positive voltage.

To detect the molecules in the trap, the center rings are switched to 0 V, the

REMPI pulse ionizes the sample, and the combined electric field of the two outer

rings, the ring electrode, and the CEM help push the molecules through the space

between the grounded inner rings toward the detector. All of the spectroscopy

presented in this thesis was performed without the use of the TWIST.

38

Chapter 3

REMPI and PDS: vibrational spectra and

analysis

In this chapter, we describe how REMPI and PDS are used to unambigu-

ously label vibrational levels of the X1Σ+ and a3Σ+ states of NaCs populated

via PA. We also discuss how this technique provides information on the B1Π and

c3Σ+Ω=1 excited states.

3.1 REMPI

3.1.1 Initial analyses of NaCs REMPI spectra

Haimberger, et al. [38, 90] attempted to label the range of vibrational

states created by certain NaCs PA lines. A lower bound on the ground state

binding energy was given based on the REMPI photon energy and the dissociation

energy of the first ionic state. It was argued that the detection of bound molecular

ions could only occur if the energy of two REMPI photons was insufficient to

dissociate an ion from the ground state, giving a lower bound of ∼ 2000 cm−1.

However, this explanation excludes the possibility that the electron carries away

the excess energy kinetically, or that the molecule absorbs another photon that

places it in a higher-lying, but still bound, ionic state [91]. Further, an attempt

was made to label a series of evenly spaced peaks in the REMPI spectrum. These

were attributed to an electronic state with a series of vibrational levels that fit

the observed spacing and predicted binding energy. This assignment gave a more

restricted range of the ground state binding energy of between 2500 and 3200

39

cm−1, or v = 18− 23 in the X1Σ+ state.

The possibility that the sample consisted of polar (e.g., deeply bound)

molecules led to the construction of the TWIST [60, 89]. Only deeply bound

X1Σ+ molecules have a sufficiently large dipole moment for trapping, so the

success of the trap confirmed that PA was populating these states. However, not

all of the observed PA lines created trappable samples. In fact, these untrappable

states included vibrational levels that we had previously labeled as X1Σ+(v =

18 − 23). This apparent contradiction motivated us to develop a more precise

technique for labeling ground states.

3.1.2 Extending the REMPI spectrum

Given that our previous REMPI assignments were incorrect, we attempt to

gain more information by extending the REMPI spectrum. In particular, we look

at portions of the spectrum that we can compare to calculations from empirical

PECs. We also look to see if there is a blue cut-off for molecular ion formation.

For all of the scans in this section, we fix the PA laser to a line detuned 32 GHz

from the Cs 62P3/2 asymptote (which will henceforth simply be referred to as the

32 GHz line). We choose the 32 GHz line because it is the most efficient PA line

that produces trappable molecules.

We first extend the REMPI scans to the blue in order to determine whether

or not there is an experimentally accessible wavelength at which molecules are no

longer detected. If molecular ions are not formed with two-photon REMPI when

the energy is sufficient to dissociate the ion, then we should detect no molecular

ions for 2hν > Ebinding + ECs+ , where ν is the RE2PI frequency, Ebinding is the

binding energy of the ground state molecule, and ECs+ is the ionization energy of

the Cs atom. The ionization energy of Cs is used here because Cs+ + Na 32S1/2

40

17600 17700 17800 17900 18000 18100 182000

50

100

150

200

250

18300 18400 18500 18600 18700 18800 189000

50

100

150

19000 19100 19200 19300 19400 19500 196000

50

100

150

NaC

s+ sig

nal (

arb.

uni

ts)

REMPI wavelength (cm-1)

Figure 3.1: Combined REMPI scans taken while setting the PA to the 32 GHzline. These scans require three different laser dyes: Pyrromethene 567, Coumarin540, and Coumarin 522. Note that the background level is low, perhaps 5-10 ionson this scale, and so all features above this level indicate molecular structure.

is energetically the lowest-lying ionic pair of the system. The largest photon

energies that should still produce molecular ions from the absolute rovibrational

ground state would then be 18180 cm−1. However, we detect molecular ions with

energies up to 19650 cm−1, limited only by the range of our laser dyes (see Fig.

3.1). Clearly, the excess energy does not go into dissociation of the molecule,

meaning the earlier analysis placing a lower bound on the detected ground state

binding energy was flawed.

41

NaC

s+si

gnalHa

rb.u

nitsL

15 400 15 500 15 600 15 700 15 8000

50

100

150

200

15 900 16 000 16 100 16 200 16 3000

50

100

150

200

REMPI laser wavelength Hcm-1L

Figure 3.2: Single-color RE2PI scan with DCM laser dye taken with the PA fixedto the 32 GHz line.

Assignment of the structure in Fig. 3.1 is complicated by a lack of ex-

perimentally determined excited state PECs for this spectral region. Dunham

coefficients for the D1Π state are published in Ref. [12], but these only accurate

for the lowest 4 vibrational energy levels because of mixing with f3∆Ω=1 lev-

els. Due to the uncertainty in the predicted line positions and the high density of

transitions in the data we can only conclude that the presence of RE2PI structure

in this region is consistent with D1Π←X1Σ+(v′′ = 4 − 15) transitions without

more information.

We then extend the REMPI scans further to the red/infrared, where there

are fewer molecular transitions and more experimentally determined PECs. From

42

2009 to early 2010, the time period in which all of the scans presented in this

section were taken, two experimental papers on the lower-lying excited states

of NaCs were available. These included the B1Π state and the heavily mixed

A1Σ+ − b3Π complex [6, 92]. This early B1Π PEC, however, is only accurate

(within 1 cm−1) for v ≤ 8. The A1Σ+ − b3Π complex calculations, on the other

hand, are accurate over more than 70% of the depth of the wells.

In the search for B1Π←X1Σ+ transitions, we perform single-color RE2PI

scans from 613-651 nm with DCM laser dye. However, assignment of the resulting

spectrum is complicated by several issues. First, the spectrum is very dense, as

seen in Fig. 3.2, and likely representative of a number of excited electronic states.

Second, it is possible that some of the B1Π-c3Σ+Ω=1←X1Σ+ vibrational levels

predissociate via mixing with the b3ΠΩ=1 electronic state. Finally, the precise

ionization threshold for NaCs is unknown1, such that we cannot be sure that two

photons between 613 and 650 nm provide sufficient energy for ionization for the

lowest vibrational levels of the X1Σ+ state. Several portions of the scan were

retaken with two-color ionization by adding the 532 nm pump light, but these

scans did not appear to reveal any new lines. Without more information, we are

unable to definitively attribute any of the observed structure in this wavelength

range.

The wavelengths for A1Σ+ − b3Π←X1Σ+(v′′ ≤ 30) transitions with non-

negligible transition moments are> 710 nm, precluding single-color RE2PI. Many

of the transitions involving non-predissociating B1Π levels are also in this wave-

length range. We then proceed further to the red using 2-color RE2PI. In these

experiments, we use two temporally separated pulses. This is done to control

1There are several predicted values for the ionization threshold of NaCs ranging from 26809to 29712 cm−1 [91, 93–95].

43

14100 14200 14300 14400 14500 146000

50

100

150

200

NaC

s+ sig

nal (

arb.

uni

ts)


Figure 3.3: Two-color RE2PI (532 nm + red) scan taken using Pyridine I dye.The PA is fixed to the 32 GHz line. Essentially no NaCs signal was detectedwithout the addition of the green photon.

which pulse is absorbed first by the molecules. The first pulse is from the dye

laser, with the second pulse coming from the 532 nm pump. With Pyridine I dye,

we scan the 681− 706 nm range (see Fig. 3.3). We note that all of the structure

in this scan requires the addition of the 532 nm pulse. Even assuming the lowest

published ionization threshold for NaCs, this suggests that the observed ground

states must be in the range of X1Σ+(v′′ = 0− 40). However, our attempts to as-

sign the structure through comparison with LEVEL calculations of B1Π←X1Σ+

transitions are inconclusive. While there are some calculated transition energies

that match line positions in the experimental spectrum, there are no clear in-

dications2 of complete or nearly complete vibrational progressions (i.e., series of

transitions from the same ground state to the excited state manifold).

We extend our 2-color scans to the IR with LDS 821, between 792 and

2With the more recent investigation of the c3Σ+Ω=1 PEC, we find likely vibrational pro-

gressions for c3Σ+Ω=1←X1Σ+(v′′ = 4 − 6). These have not been confirmed with depletion

spectroscopy.

44

NaC

s+si

gnalHa

rb.u

nitsL

11 950 12 000 12 050 12 100 12 1500

50

100

150

200

250

12 200 12 250 12 300 12 3500

50

100

150

12 400 12 450 12 500 12 550 12 6000

20

40

60

80

100

120

140


Figure 3.4: Single-color RE3PI scan taken with LDS 821 dye while setting thePA to the 32 GHz line.

837 nm, in an attempt to find A1Σ+ − b3Π←X1Σ+ transitions. Unexpectedly,

we discover that scans taken with the 532 nm pulse do not significantly differ

from the single-color spectrum shown in Fig. 3.4. These lines must then be

due to efficient single-color, RE3PI. We also use LDS 867 dye to cover the 845-

878 nm range of the RE3PI spectrum, as seen in Fig. 3.5. RE3PI transitions

are most efficient when there is an ‘accidental’ resonant secondary transition.

Thus, with the laser intensities used in the experiment, we expect that many

45

NaC

s+si

gnalHa

rb.u

nitsL

11 400 11 450 11 500 11 550 11 6000

100

200

300

400

500

600

11 650 11 700 11 750 11 8000

100

200

300

400

500

600

700


Figure 3.5: Single-color RE3PI scan taken with LDS 867 dye while setting thePA to the 32 GHz line.

of the A1Σ+ − b3Π←X1Σ+ transitions will be missing from the spectrum. The

energy of the second transition is also not necessarily going to be at the same

wavelength as the first, in which case the RE3PI line may be shifted from the

expected A1Σ+− b3Π←X1Σ+ transition. These uncertainties make it difficult to

use the calculations from Ref. [6] to make definitive spectral assignments.

3.2 PDS

From the previous section, it is clear that the labeling of NaCs REMPI

spectra with any certainty requires a different approach, one that can make use

of the limited number of experimentally derived PECs. We therefore implement

the PDS technique as described in 2.2.2. In the following section, we describe

experiments using this technique to labelX1Σ+ and a3Σ+ REMPI detection lines,

46

as well as investigate the B1Π and c3Σ+Ω=1 electronic states.

3.2.1 Implementation of PDS

The delay between the pulsed depletion and REMPI laser pulses must be

at least as long as the lifetime of the excited states to avoid effective two-color

REMPI. Ideally, the delay should also be long enough to avoid detection of any

ions created by the depletion pulse, as these can can interfere with the TOF

spectrum. The latter timescale is on the order of 10 µs, which would require a

delay line of several km if using the same pump for both dye lasers. Fortunately,

we have two pump lasers and so can easily set an arbitrarily long delay (typically

∼100 µs). To implement this experiment, however, we first had to purchase,

repair, and modify a used3 Lambda-Physik FL3002 to serve as the depletion

laser. The laser came with pump mirrors coated for an excimer laser, so these

had to be replaced with optics compatible with our 532/355 nm pump lasers.

Also, we constructed a pump mirror mount to replace one that was missing.

The two lasers are overlapped using a polarizing beam splitter (PBS) be-

fore entering the chamber. We can check to see that the two beams are interacting

with the molecular sample by checking the REMPI signal for both. Once the two

pulsed lasers are aligned, we verify that PDS is working by scanning the deple-

tion laser across the REMPI line. In this way we can make adjustments to the

intensity and spatial mode of each laser to maximize the depletion signal.

3The word “used” here is very much a euphemism; the laser was covered in dirt and dye,missing optics, and looked as though it had been left out in the rain and perhaps kicked a fewtimes for good measure.

47

3.2.2 Labeling deeply bound X1Σ+ molecules with PDS

We proceed to take several depletion scans for REMPI lines in the 800-

830 nm range. We choose this range because the density of REMPI lines here

is relatively low, and because we have accurate calculations for the A1Σ+ − b3Π

complex levels from A. Stolyarov [8]. Our first successful PDS scans are shown in

Fig. 3.6. We note that these scans can be shifted to match one another, with the

offsets indicating a ground state vibrational spacing of ∼ 95 cm−1, consistent with

X1Σ+(v = 4− 6). These initial assignments are confirmed by the calculations.

We use this same technique to label many other X1Σ+ detection lines

corresponding to a broad range of X1Σ+ vibrational levels produced by the

32 GHz line in the infrared portion of the spectrum. In total, we find v′′ =

4, 5, 6, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 34. Other PA lines create very simi-

lar ground state distributions. For example, the line at 958 GHz detuning pro-

duces v′′ = 10, 24, 31 in addition to those levels listed for the 32 GHz line. A

complete table of labeled X1Σ+ REMPI lines is given in Appendix A.1.

The production of deeply bound vibrational levels (v′′ < 10) via PA reso-

nances to weakly bound excited states has (to our knowledge) been observed in

only two heteronuclear species: NaCs and LiCs [84, 96]. For LiCs, the relatively

shallow B1Π electronic state provides these resonances. This would not be pos-

sible with the A1Σ+ − b3Π PECs, which are substantially deeper than the B1Π

state. The near-dissociation A1Σ+−b3Π wavefunctions are highly oscillatory and

have smaller classical inner turning points compared to the deeply bound X1Σ+

levels, making the F-C overlap between them negligible. The situation is similar

for NaCs: the only excited states with near-dissociation vibrational levels that

have non-negligible F-C overlap with X1Σ+(v′′ = 4 − 6) are B1Π and c3Σ+Ω=1.

This F-C overlap can be seen in Fig. 3.7. Note that both of these electronic states

48

800 805 810 815 820 825 8300

100

200

300

400

a)

800 805 810 815 820 825 8300

100

200

300

400

500

800 805 810 815 820 825 8300

100

200

300

400

800 805 810 815 820 825 8300

100

200

300

400

500

600d)

NaC

s+ sig

nal (

arb.

uni

ts)

Depletion laser wavelength (nm)

b)

c)

Figure 3.6: PDS scans taken with the REMPI laser set to various X1Σ+ detectionlines: a) v′′ = 4, b) v′′ = 5, c) v′′ = 6, d) v′′ = 19. The PA is set to the 32 GHzline. Solid bars indicate both the position and relative transition moments ofcalculated transitions from [8], arbitrarily scaled for visibility.

49

Figure 3.7: Franck-Condon map of the B1Π ← X1Σ+ and c3Σ+Ω=1 ← X1Σ+

transitions, calculated with LEVEL [3] using empirical PECs [2, 5, 7]. Notethe overlap of near-dissociation vibrational levels with a wide range of X1Σ+

vibrational levels for both states.

50

15450 15500 15550 15600 15650200

300

400

500

600

700

15700 15750 15800 15850 15900200

300

400

500

600

700

15950 16000 16050 16100 16150200

300

400

500

600

700

NaC

s+ sig

nal (

arb.

uni

ts)


Figure 3.8: PDS scan taken while setting the PA to the 32 GHz line and theREMPI laser to a X1Σ+(v′′ = 5) detection line. Vertical dashed lines indicateB1Π(J ′ = 1)← X1Σ+(v′′ = 5, J ′′ = 0) transitions, as calculated from [2].

can decay to the singlet ground state due to S-O mixing. Other heteronuclear

species have B1Π-c3Σ+Ω=1 states with similar properties [11]; however, these levels

may predissociate in other systems [39].

51

5 10 15 20

9500

10000

10500

11000

11500

12000

Ene

rgy

(cm

-1)

Internuclear separation (Å)

Figure 3.9: The unpublished (dashed) and published (solid) experimental PECs,compared to the ab initio calculation (dotted).

3.2.3 Investigation of the B1Π and c3Σ+Ω=1 electronic state with PDS

With the assignment of ground state detection lines, it becomes possible

to investigate the excited electronic states with PDS. In the scan shown in Fig.

3.8, we set the REMPI detection wavelength to ionize X1Σ+(v′′ = 5) and scan

the depletion laser across the Cs 62P3/2 dissociation asymptote. We note that the

line density increases as the REMPI wavelength approaches the asymptote, then

becomes a continuum as the molecule dissociates. Observation of the dissociation

limit provides independent confirmation of the X1Σ+(v′′ = 5) assignment. For

redder wavelengths, the spectrum becomes more sparse and we see two clear

vibrational progressions. As previously mentioned, there are only two electronic

states with weakly bound vibrational levels that have good F-C factors with

X1Σ+(v′′ = 5): B1Π and c3Σ+Ω=1.

In Fig. 3.8, we compare our depletion scan to B1Π←X1Σ+ LEVEL calcu-

52

lations using the two published experimental B1Π PECs and the well-established

X1Σ+ PEC [5]. The spacings and positions of the vibrational progression cal-

culated from the Zaharova et al. PEC [92] do not agree with our scan of the

higher-lying B1Π levels. In contrast, levels calculated using the B1Π PEC from

Grochola et al. [2] match the experimental line positions well, allowing us to

make precise assignments.

To further investigate the B1Π-c3Σ+Ω=1 complex, we extend our depletion

scans by utilizing different dyes and various X1Σ+ detection lines (these lines

can be found in Appendix A.3). We then use Robert LeRoy’s RKR1 2.0 program

[97] to estimate the c3Σ+Ω=1 PEC. Lacking a complete set of rotational constants,

however, we use the ab initio value [11] for the potential minimum and allow

RKR1 to estimate the inner wall of the PEC. This allows us to create a PEC

that reproduces our vibrational spectra. Simultaneously with our work, Grochola

et al. calculated an IPA c3Σ+Ω=1 PEC using data acquired while investigating the

B1Π state [2]. Transitions calculated from this initial PEC match a majority

of our observed transitions, but deviate significantly for v′ > 48. Additionally,

this PEC does not include two vibrational levels observed in our spectra that are

lower than the potential minimum. In collaboration with our group, Grochola et

al. calculate a new PEC that matches all of the data from both groups [7]. This

PEC is shown in Fig. 3.9 along with the previous versions.

3.2.4 Labeling a3Σ+ molecules with PDS

We revisit the REMPI spectra analyzed in Haimberger et al. [90]. These

REMPI spectra are associated with PA states that are labeled as b3ΠΩ=2 levels.

We note that these PA lines are not the same as those which we found to produce

deeply bound X1Σ+ molecules in 3.2.2. Indeed, a comparison of the REMPI

53

11 350 11 400 11 450 11 500 11 5500

200

400

600

800

11 600 11 650 11 700 11 7500

200

400

600

800

Depletion laser wavelength Hcm-1L

NaC

s+si

gnalHa

rb.u

nitsL

Figure 3.10: PDS scan taken with the PI set to 598.32 nm and the PA locked to1009 GHz detuned from the Cs 62P3/2 asymptote. Many of the dips correspond totransitions from a3Σ+(v′′ = 17) to B1Π (dashed vertical lines) or c3Σ+

Ω=1 (dottedvertical lines) vibrational levels.

spectra shown in Fig. 3.11 suggests that the b3ΠΩ=2 PA lines do not produce any

deeply bound molecules. This is expected for a purely triplet character excited

state, which should decay exclusively to the a3Σ+ ground state. To confirm this,

we perform PDS with LDS 867 in the depletion laser. By scanning near the Cs

62P3/2 asymptote, we should observe transitions between weakly bound ground

and excited states, allowing us to make assignments using the B1Π and c3Σ+Ω=1

PECs.

54

NaC

s+si

gnalHa

rb.u

nitsL

aL

11 550 11 600 11 650

200

300

400

500

600

bL

11 550 11 600 11 6500

100

200

300

400

500

600


Figure 3.11: PDS scan taken with the PI set to 598.32 nm and the PA locked to1009 GHz detuned from the Cs 62P3/2 asymptote. Many of the dips correspond totransitions from a3Σ+(v′′ = 17) to B1Π (dashed vertical lines) or c3Σ+

Ω=1 (dottedvertical lines) vibrational levels.

We set the PA frequency to the 1009 GHz and fix the REMPI frequency

to several peaks that were used in the Haimberger et al. analysis in order to

perform PDS. An example PDS scan is given in Fig. 3.10. We label the ground

state by comparing the dip positions to calculated B1Π-c3Σ+← a3Σ+ transition

energies. Using several different b3ΠΩ=2 PA, we label a range of a3Σ+ detection

lines: v′′ = 10, 12, 14, 15, 17 [98]. A complete list of labeled a3Σ+ REMPI lines is

given in Appendix A.2. We note that each high-lying b3ΠΩ=2 state is expected to

55

produce a narrow range of ground state vibrational levels from F-C calculations.

Our data appear to be consistent with these calculations (see the discussion in

Ref. [98]).

56

Chapter 4

PA spectra and analysis: Ω > 0 states

corresponding to the Cs 62P3/2 asymptote

In this chapter, we discuss our investigation of Ω = 1 and Ω = 2 PA spectra

that dissociate to the Cs 62P3/2 atomic asymptote (∼ 11732.3079 cm−1). These

include the first PA resonances observed in NaCs [38] as discussed in Chapter 3.

Through analysis of the spectra, we obtain information on the B1Π, c3Σ+Ω=1, and

b3ΠΩ=2 excited electronic states and the PA formation process.

4.1 Initial analyses of NaCs PA spectra

Our first attempt to assign the NaCs PA spectra is described in Refs.

[75, 90]. First, all of the rotational lines in this portion of the spectrum exhibit

dense hyperfine structure1, meaning that they belong to electronic states with

Ω > 0. The rotational structure is analyzed by performing least-squares fits to

the data using the equation for the symmetric top,

E(N) = E0 −Bv((N + Ω)(N + Ω+ 1)− Ω2), (4.1)

where the ‘rotationless’ energy E0, rotational constant Bv and electronic angular

momentum projection Ω are left as parameters. For the Ω > 0 rotational lines,

1If the atomic samples are not completely polarized, molecules may be photoassociated fromdifferent hyperfine ground states, producing hyperfine ”ghosts” in the spectrum [99]. Theseoccur with predictable splittings, and can typically be distinguished from the excited statehyperfine structure.

57

we take the center of mass of the hyperfine structure to be the line position. In

this way, we can sometimes find the real integer value for Ω, though local pertur-

bations or even small errors in the extracted line positions can make the results

ambiguous. Many of the extracted Ω values are for these spectra are between

2 and 3, indicating a likely Ω = 2 state. We also compare the hyperfine struc-

ture and number of rotational lines for each vibrational state, and measure the

vibrational splittings to look for likely progressions. Using this information and

comparing the rotational constants and vibrational spacings to LEVEL calcula-

tions using an ab initio PEC [11], we identify this progression as coming from

the b3ΠΩ=2 electronic state. The remaining lines are not yet assigned due to the

sparsity of the data.

As described in Section 3.1, the REMPI lines in Ref. [90] are incorrectly

attributed to deeply bound X1Σ+ vibrational levels. These data were taken with

the PA fixed to one of the labeled b3ΠΩ=2 resonances. There are no neighboring

singlet Ω = 2 electronic states with which the triplet b3ΠΩ=2 state can mix,

meaning the excited PA molecules have essentially pure spin character2. Thus,

b3ΠΩ=2→X1Σ+ transitions are dipole forbidden, and it was proposed that the

ground state molecules were formed after a secondary excitation from the b3ΠΩ=2

state to a higher-lying Ω = 1 state with mixed spin character. The molecule

could then decay to the X1Σ+ ground state. With the implementation of PDS

we now know that this explanation is unnecessary as none3 of the pure b3ΠΩ=2

levels decay to the X1Σ+ ground state.

2We find a b3ΠΩ=2 vibrational state that exhibits singlet spin character due to heterogeneouscoupling with a c3Σ+

Ω=1 state, but this phenomenon is rare.3There is b3ΠΩ=2 state that is heterogeneously mixed with a c3Σ+

Ω=1 level which decays tothe X1Σ+ ground state. This phenomenon is discussed further in Section 6.3

58

4.2 A new investigation

Several upgrades to the experiment allow us to further investigate the PA

spectra associated with Cs 62P3/2 asymptote. First, the tuning range, output

power, and stability of the Ti:Sapphire laser is improved by replacing the argon-

ion pump laser with a Coherent V-10 Verdi. Also, as discussed in Chapter 3, PDS

allows assignment of the ground state vibrational distribution created via PA.

These experimental upgrades, as well as the availability of empirically determined

PECs, allow us to build upon the previous work outlined in Section 3.1.

We first reinvestigate all of the PA structure discussed in [75], as well as

some previously discovered lines which were omitted due to poor signal/noise

ratio. All of these levels have binding energies of less than 6000 GHz. We rescan

many of the lines using known a3Σ+ or X1Σ+ REMPI detection lines to ensure

that the assignments are correct and to improve the quality of the data. We then

extend the PA scans in the hunt for deeper b3ΠΩ=2 and B1Π levels.

4.2.1 The c3Σ+Ω=1 electronic state

We first consider a series of PA resonances which produce X1Σ+ state

molecules. These PA lines form a clear and complete vibrational progression

beginning at -4 GHz relative to the Cs 62P3/2 asymptote and ending at -958 GHz

(see Appendix A.8 for a complete list). Performing a least squares fit for each

vibrational line to Eq. 4.1 yields values for Ω which are for the most part between

1 and 2, confirming that these are Ω = 1 levels.

In order to obtain rotational constants, we again use 4.1 but this time fix

Ω to 1. There is not normally a large difference between the rotational constants

59

16120 16130 16140 16150

200

400

600

800

NaC

s+ sig

nal (

arb.

uni

ts)


Figure 4.1: PDS scan taken while setting the PA to the 32 GHz line and theREMPI laser to a X1Σ+(v′′ = 5) detection line. Vertical dashed lines indicateB1Π(J ′ = 1)← X1Σ+(v′′ = 5, J ′′ = 0) transitions, as calculated from [2]. Verticaldotted lines indicate c3Σ+

Ω=1(J′ = 1)← X1Σ+(v′′ = 5, J ′′ = 0) as determined from

our PA spectra and the Docenko et al. ground state PEC [5].

obtained in either fashion, but using the actual integer value for Ω is preferable4.

The results of these fits are given in table.

There are two excited states which dissociate to the Cs 62P3/2 asymptote:

B1Π and c3Σ+Ω=1. To determine which of these is responsible for the Ω = 1 PA

progression, we compare the B1Π-c3Σ+Ω=1←X1Σ+(v = 5) depletion spectra, the

positions of the Ω=1 PA lines, and the calculated line positions of the Grochola

et al. B1Π state (see Fig. 4.1) [2]. As none of the PA lines in this progression

correspond to the predicted B1Π levels, we conclude that these are c3Σ+Ω=1 states.

As noted in Chapter 3, Grochola et al. incorporate the PA data into their

calculation of the first experimentally derived c3Σ+Ω=1 PEC for NaCs. Unlike

the ab initio PECs, this new version has an unexpected feature: it crosses the

4Although rotation N and angular momentum Ω are not always integers, J is (ignoringhyperfine interactions). For the purposes of fitting, though, we always assume N is an integer,so it would not be physical to allow Ω to be non-integer.

60

10 1511500

11550

11600

11650

11700E

nerg

y (c

m-1)


Figure 4.2: Experimentally determined IPA c3Σ+Ω=1 (dashed) and B1Π (dotted)

PECs from [2] and [7] around the region of a crossing.

B1Π state at ∼9.4 A as seen in Fig. 4.2. The diabatic picture is of course an

approximation here, but one that recreates the data well.

The incorporation of our PA spectra into the c3Σ+Ω=1 PEC calculation also

allows Grochola et al. to calculate dispersion coefficients. The details of their

fitting procedure are given in Ref. [7]. We compare the empirical and ab initio

long-range PECs in Fig. 4.3. Marinescu and Sadeghpour do not include disper-

sion coefficients for the relativistic electronic configurations, so we approximate

these PECs perturbatively [59] assuming the asymptotic value for the Cs S-O

interaction. The apparent match between the empirical and ab initio PECs in

Fig. 4.3 belies a complication: the spin character of the states do not correlate

unless the crossing at ∼9.4 A is assumed to be avoided. The higher-lying ab initio

electronic states are mostly B1Π in character, while the lower-lying are primarily

c3Σ+Ω=1 in character.

61

15 20 25 30

-7

-6

-5

-4

-3

-2

-1

0

Internuclear SeparationHÅL

Ene

rgyH

cm-

1 L

Figure 4.3: Empirical and ab initio long range PECs for the (4)Ω = 1 (upper setof curves) and (3)Ω = 1 (lower set of curves) electronic states. For the empiricallong-range PECs, we refer to the B1Π from Ref. [2] as (3)Ω = 1 and the c3Σ+

Ω=1

from [7] as (4)Ω = 1. Solid lines indicate the experimentally determined PECs,dashed lines are estimated from Marinescu and Sadeghpour [9], and dotted linesare from Bussery et al. [10]. Here, the energy origin is taken to be the Cs 62P3/2

atomic asymptote.

We have no doubt that the crossing and mixing between the c3Σ+Ω=1 and

B1Π electronic states influence the long-range PECs and the PA process. For in-

stance, there would be no c3Σ+Ω=1→X1Σ+ transitions in the absence of coupling

with the B1Π state. We also note the sharp5 cut-off in the c3Σ+Ω=1 PA spectra

at 958 GHz detuning (well above the crossing), as well as the lack of B1Π lines

alongside the c3Σ+Ω=1 spectra. It seems unlikely that poor free-bound F-C overlap

is alone responsible for so many missing lines; we later find B1Π PA structure

at much greater binding energies and with shorter-range wavefunctions. Instead,

we suspect that predissociation along the Cs 62P1/2 asymptote causes the PA

molecules to break apart before they can decay to the ground electronic state.

5We call this cut-off sharp because the reddest PA line in the progression is also one of thestrongest.

62

NaC

s+si

gnalHa

rb.u

nitsL

aL

-34 -33 -32 -31 -30 -29 -280

200

400

600

800

bL

-960 -955 -950 -945

200

400

600

800

PA laser frequency - 351730.9 HGHzL

1 2 3 4 5

Figure 4.4: PA scan of the a) 32 GHz PA line and b) 958 GHz PA line. Therotational levels are labeled for the 958 GHz scan.

Transitions to predissociating states appear line-broadened due to the shortened

lifetime of the excited states, but our PDS resolution does not allow us to observe

this phenomenon. For comparison, a 500 ps predissociation lifetime would give a

natural linewidth of 1 GHz, much smaller than the pulsed laser linewidth of ∼15

GHz. CW depletion or trap loss spectroscopy (where dips in the fluorescence sig-

nal indicate trap loss from molecule formation/dissociation), on the other hand,

might allow us to characterize the predissociation lifetimes of these states. Wave-

functions obtained from a full deperturbation analysis of the interacting system

would also help us to better understand the PA spectrum in this region.

We examine the ground state rotational structure populated by the c3Σ+Ω=1

PA resonances using pulsed and CW depletion spectroscopy. As seen in Fig. 4.4

63

NaC

s+si

gnalHa

rb.u

nitsL

aL

16 224 16 226 16 228 16 230200

300

400

500

600

5 4 3 2 10 5 4 3 2 10

bL

16 224 16 226 16 228 16 230

400

500

600

700

5 4 3 2 10 5 4 3 2 10

cL

16 224 16 226 16 228 16 230

200

300

400

500

600

5 4 3 2 10 5 4 3 2 10

dL

16 224 16 226 16 228 16 230400

500

600

700

800

5 4 3 2 10 5 4 3 2 10

eL

16 224 16 226 16 228 16 230200

250

300

350

400

450


5 4 3 2 10 5 4 3 2 10

Figure 4.5: Depletion scans (5-point moving averaged) taken while locked to therotational lines in Fig. 4.4 b): a) J ′ = 1, b) J ′ = 2, c) J ′ = 3, d) J ′ = 4, e) J ′ = 5.The REMPI laser is set to detect X1Σ+(v′′ = 4) molecules. The vertical dashedlines indicate that calculated wavelengths of the B1Π(v′ = 48, 49)←X1Σ+(v′′ =4) R transitions with the ground state rotational levels labeled.

64

a), the rotational splittings of the most weakly bound states are similar to the

extent of the hyperfine structure, making precise assignments difficult. Fortu-

nately, the rotational structure of the more deeply bound PA states, as seen in

Fig. 4.4 b), are clearly defined and labeled. We see five rotational states, along

with a weak hyperfine ghost of the J ′ = 1 level. Initially, we utilize PDS to

investigate the ground state rotational populations created by these resonances

[98]. PDS does not typically resolve individual rotational transitions due to the

large bandwidth of the pulsed laser compared to the rotational splittings. To

work around this problem, we exploit the large difference in rotational constants

between the deeply bound X1Σ+ and weakly bound B1Π levels. We lock to each

of the rotational lines of the vibrational state in Fig. 4.4 b) and scan the de-

pletion laser. In Fig. 4.5, these scans are compared to calculations using the

empirical B1Π PEC. From these data we confirm our assignment of the highest

three rotational PA states: J ′ = 3, 4, 5. We note that all three of these excited

states have even parity, as they all decay only to odd parity ground states. This

suggests that the excited states are all populated by odd-parity components of

the photoassociative collisions. Because angular momentum conservation con-

strains the difference between J ′ and ℓ to 2 units of angular momentum (1 from

electron spin and 1 from the photon), ℓ = 3 (f -wave scattering) must be entirely

responsible for J ′ = 4 and 5.

While the PDS technique is successful in resolving J ′′ ≥ 3, it is not enough

to determine the lowest rotational state populations. In Fig. 4.6, we use CW

depletion spectroscopy to label the ground states populated by the J ′ = 1, 2 PA

lines labeled in Fig. 4.4 b). Unlike J ′ = 3, 4, 5, these PA states decay to both

even and odd parity ground states. This indicates that both even and odd partial

waves contribute to the lower rotational levels. Because the higher rotational lines

65

NaC

s+si

gnalHa

rb.u

nitsL

aL

11746.95 11747 11747.05 11747.1 11747.15 11747.2

600

700

800

900

1 0 1 23

bL

11746.95 11747 11747.05 11747.1 11747.15 11747.2

400

600

800


1 0 1 23

Figure 4.6: CW depletion scans (5-point moving averaged) taken while lockedto the rotational lines in Fig. 4.4 b): a) J ′ = 1, b) J ′ = 2. The REMPI laseris set to detect X1Σ+(v′′ = 4) molecules. The positions of the vertical dashedlines are computed from a fit to the rotational spectrum, and we have labeledthe ground state rotational level for each. The energy of the A1Σ+(v′ = 29, J ′ =2)←X1Σ+(v′′ = 4, J ′′ = 1) transition calculated by Ref. [8] is 11747.154 cm−1.

show no indication of ℓ = 2 partial wave contributions, ℓ = 0 (s-wave scattering)

must be the even-parity wave. A more complete analysis of the partial wave

contributions to the PA spectra is given in Section 6.1.1.

4.2.2 The B1Π electronic state

The B1Π electronic state is of particular interest for producing X1Σ+(v′′=

0) molecules. Many of its vibrational levels have good Franck-Condon overlap

with X1Σ+(v′′ = 0) (see Fig. 3.7), and it is mixed with the c3Σ+Ω=1 electronic

state, meaning that its wavefunctions have some triplet character and can thus

be excited from the more efficient a3Σ+ scattering channel. Fortunately, the

experimental B1Π PEC constructed by Grochola et al. and verified by PDS

66

v Eexp Ecalc Bexpv Bcalc

v

3 1133.70 1133.78 0.0418 0.0420

12 737.11 737.06 0.0347 0.0342

13 699.50 699.29 0.0338 0.0334

16 592.76 592.37 0.0316 0.0312

17 559.29 558.83 0.0284 0.0306

39 92.17 93.09 0.0181 0.0158

Table 4.1: Comparison between observed and calculated rotational constantsand binding energies for J ′ = 1 levels in the B1Π state. All values (except v)are given in cm−1. Calculations are performed using the experimental PEC in [2]and LEVEL [3]. The uncertainties in the observed energies are ∼ ±1 GHz, whilethe standard errors obtained for the rotational constants are < 0.0006 cm−1.

allows us to narrow our search for these PA resonances.

As mentioned in Section 4.2.1, no B1Π PA resonances are found in the

same range as the c3Σ+Ω=1 progression. However, we find that one of the weak

resonances in Haimberger’s thesis [75] at 2754 GHz detuning is actually B1Π(v′ =

39). Our scans reveal a full rotational progression (v′ = 1 − 5) which matches

the characteristics of the deeper B1Π levels (see Fig. 4.7 for an example). This

resonance is a bit of an oddity as it is within 2 cm−1 of the predicted positions of

both the c3Σ+(v′ = 55) and b3ΠΩ=2(v′ = 90) levels. However, these nearby levels

have not yet been detected in the PA spectra. Further study will be required to

understand what role coupling plays in the observation of this lone B1Π PA line.

We find no other B1Π PA resonances blue-detuned from the Cs 62P1/2

asymptote, although we have not yet executed an exhaustive search. Neverthe-

less, most are clearly missing, indicating that these levels have some property in

common that makes them poor PA channels. As discussed in Section 4.2.1, we

suspect that predissociation prevents these states from decaying to the ground

state once formed.

67

-4360 -4355 -4350 -4345 -4340 -43350

100

200

300

400


NaC

s+si

gnalHa

rb.u

nitsL 1 2 3 4 5

Figure 4.7: PA scan across the B1Π(v′ = 13) line with rotational labels. Thevertical bars indicate the positions of the calculated energy levels from [2] shiftedby 6.7 GHz to the red such that the J ′ = 1 levels match.

We continue our search red-detuned from the Cs 62P1/2 asymptote and find

PA resonances corresponding to B1Π(v′ = 3, 12, 13, 16, 17). An example B1Π PA

scan is given in Fig. 4.7, and a complete list of line positions is given in Appendix

A.6. All of the observed vibrational levels exhibit similar hyperfine structure and

rotational intensity patterns, and their binding energies and rotational constants

match well with the predicted values as seen in Table 4.1. Despite the fact that

B1Π levels bound by more than the Cs fine-structure splitting cannot predisso-

ciate, we still observe only a fraction of them in the PA spectrum. We attribute

these missing lines to either weak c3Σ+Ω=1-B

1Π mixing, inferior free-bound F-C

factors, or some combination of these. More PA beam power might bring out

some of the deeper lines (v′ < 10); we had < 200 mW over much of this spectral

region in our recent investigation.

68

-6420 -6415 -6410 -6405 -6400

100

200

300

400

500

600

700


NaC

s+si

gnalHa

rb.u

nitsL 2 3 4 5

Figure 4.8: PA scan across the b3ΠΩ=2(v′ = 85) line with rotational labels.

4.2.3 Extension and reanalysis of the b3ΠΩ=2 PA spectra

As discussed in Section 3.2.4, b3ΠΩ=2 PA lines produce a3Σ+ molecules.

The b3ΠΩ=2 PA lines are also easily distinguished from other states because they

typically have four rotational lines J ′ = 2− 5, and all exhibit a similar pattern of

hyperfine structure (see the example in Fig. 4.8). We first rescan some of the PA

resonances presented in Ref. [75] in order to verify electronic state assignments.

We then extend our PA scans to the red. Rather than performing continuous

scans, we extrapolate the behavior of the observed vibrational progression with

a polynomial fit to predict the position of next level. In this manner, we find

b3ΠΩ=2 vibrational levels with binding energies from 11 to 1134 cm−1. There may

be PA resonances at redder detunings, but the Ti:Sapphire power is limited at

these wavelengths.

In previous work, we compared the b3ΠΩ=2 PA spectra to positions calcu-

lated with the ab initio PEC and found close agreement [90]. Our extension of

the spectra, however, shows that this agreement does not exist for deeper levels.

We also compare our spectra to calculations using the b3ΠΩ=2 state obtained from

69

10 2011400

11500

11600

11700

11800

5 10 15 20 25

6000

8000

10000

12000

14000

Ene

rgy

(cm

-1)


~16 cm-1

Ene

rgy

(cm

-1)


Figure 4.9: The diabatic (red dashed) and RKR (blue solid) b3ΠΩ=2 PECs. Wefocus on the significant deviation along the outer wall, which at its maximum isapproximately 16 cm−1. The inset shows the full depth of the two PECs, whichare difficult to discern from one another on that scale.

the A1Σ+ − b3Π complex deperturbation analysis [6]. We note that the diabatic

approximation is appropriate in this case because there are no neighboring Ω = 2

states to perturb it. The b3ΠΩ=2(v′ = 73 − 75) levels calculated from this PEC

are within 3 cm−1 of three of observed binding energies. This agreement is unex-

pected because the range of energies used in the deperturbation analysis covered

only up to v′ = 36. However, the divergence between the calculated and observed

vibrational levels becomes much larger for higher vibrational quantum numbers,

upwards to 16 cm−1 for v = 90 and 92. To label vibrational quantum numbers in

our PA spectra, we assume that the b3ΠΩ=2(v′ = 73− 75) levels are correct and

fit the NDE equation to the full b3ΠΩ=2 PA progression, altering the numbering

until the residuals of the fit are minimized.

The levels obtained via PA spectroscopy cover a significant portion of the

70

11100 11200 11300 11400 11500 11600 117000.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

Bv (

cm-1)

Vibrational Energy (cm-1)

Figure 4.10: Comparisons between observed b3ΠΩ=2(J′ = 2) energy levels and

rotational constants extracted from PA spectra (Xs) and calculations from theRKR PEC using LEVEL (open squares).

b3ΠΩ=2 potential well and complement the range of states that was investigated

in the deperturbation analysis. Thus, we can use our data to correct the upper

portion of the diabatic PEC. To this end, we utilize the Rydberg-Klein-Rhys

method. We calculate the energies and rotational constants for v′ = 0 − 63

of the diabatic b3ΠΩ=2 state, leaving a gap of 9 vibrational levels between the

two data sets. This gap allows for a smooth interpolation. We shift all of the

J ′ = 2 vibrational energy levels by −Bv(J(J + 1) − 2) to obtain approximate

‘rotationless’ numbers as defined in Zaharova et al. [6] and Kato [100]. We then

perform another least-squares fit using the NDE equation, using more coefficients

(including terms in the denominator) than when numbering the vibrational levels

to account for the inclusion of the deeper levels. We then take the coefficients and

appropriate physical constants and feed these into Le Roy’s RKR1 program [97].

71

12 14 16 18 20 22 24 2611 726

11 727

11 728

11 729

11 730

11 731

11 732

Internuclear Separation HÞL

Ene

rgyHc

m-

1L

Figure 4.11: Comparison of the long range behavior of RKR (crosses), deper-turbed (dots), Marinescu et al. (dot-dashed), Bussery et al. (dashed) b3ΠΩ=2

PECs. The solid curve is the result of a fit to several of the outer RKR points.

We allow the program to automatically smooth over irregularities in the inner

wall. The resulting RKR PEC is shown in Fig. 4.9 and compared to the diabatic

PEC (see Appendix A.10 for the RKR points). As expected, the two PECs are

essentially the same from the bottom of the well to ∼10600 cm−1, where the PA

data is incorporated.

To check the quality of the RKR PEC, we calculate its energy levels and

rotational constants using LEVEL. The input PEC fed to LEVEL consists of the

bare RKR points plus the centrifugal term [J(J + 1) − 2]~2/2µr2. We then use

the automated cubic spline interpolation included in the LEVEL package. The

J ′ = 2 energy levels and rotational constants used to create the RKR input and

the subsequent LEVEL output are compared in Fig. 4.10. The standard error

for the J ′ = 2 PA data is 0.015 cm−1, which is better than our experimental

72

uncertainty of ±1 GHz. Comparison of the rotational constants give a standard

error of 0.0007 cm−1.

Unfortunately, we do not observe enough vibrational levels with suffi-

ciently small binding energies to calculate a reliable C6 coefficient. Nevertheless,

we compare the extrapolated6 long-range behavior calculated by the RKR1 pro-

gram to the ab inito potential curves in Fig. 4.11. The ab initio PECs appear to

be reasonably consistent with our calculation, particularly the curve from Mari-

nescu and Sadeghpour [9]. We note that the long-range behavior of our RKR

PEC is closer to both ab initio versions than is the original diabatic potential.

6While such an extrapolation may be worthwhile in estimating the long-range behavior ofthe PEC, any attempt to use this to find dispersion coefficients would be pointless; such fitsare highly sensitive to error.

73

Chapter 5

PA spectra and analysis: the Ω = 0 and (2)Ω = 1

electronic states

There have been no previous investigations of the Ω = 0 PA spectra for

NaCs. These electronic states, along with the largely absent b3ΠΩ=1 state, will

be the primary focus of this chapter.

5.1 The Ω = 0+ states

In contrast to the Cs 62P3/2 asymptote, the dominant PA structure de-

tuned from the Cs 62P1/2 asymptote consists of states with Ω = 0+. We assign

these states by examining the rotational structure and vibrational energy spac-

ings. We also use PDS to assign some of the ground state vibrational levels they

populate.

5.1.1 Characteristics of the Ω = 0+ A1Σ+ − b3Π complex electronicstates

The deeply bound portion of the A1Σ+ − b3Π complex was studied in

detail by Zaharova et al. [6]. The Ω = 0+ components of the A1Σ+ and b3Π

electronic states cross near the bottom of their respective potential wells and their

eigenstates are heavily mixed due to strong homogeneous coupling. This coupling

is obvious from our investigation of the PA spectra: all of the Ω = 0+ vibrational

levels have mixed singlet-triplet spin character and predissociate above the Cs

62P1/2 asymptote.

74

No PECs can be calculated that accurately recreate the Ω = 0+ levels of

the A1Σ+ − b3Π system due to the strong coupling, but the nomenclature is still

useful for labeling. Calculations of the Ω = 0+ A1Σ+ − b3Π levels from Andrei

Stolyarov include percentages indicating the level of mixing for each wavefunction

[8] (e.g., 25% A1Σ+/75% b3ΠΩ=0+). We use these numbers to assign the vibra-

tional levels to either the A1Σ+ or b3ΠΩ=0+ electronic states. Although we lack

calculations for the states observed in the PA spectra, we assign those levels that

dissociate to the Cs 62P1/2 asymptote as A1Σ+ and those that dissociate to the

62P3/2 limit as b3ΠΩ=0+ . As the (2)Ω = 0+ state is mostly A1Σ+ in character and

the (3)Ω = 0+ state is mostly b3ΠΩ=0+ in character, this approach is consistent

with the assignment of the calculated values.

5.1.2 The A1Σ+ state

We label 17 vibrational levels of the A1Σ+ state with binding energies

ranging from 0.1 to 204.5 cm−1. To assign these levels, we first note that the

rotational lines associated with this progression do not exhibit as much resolvable

hyperfine structure as the Ω > 0 states [98]. Indeed, most of the lines consist of

single peaks in our PA scans (excepting very near-dissociation levels), indicating

that these are likely Ω = 0 levels.

We then proceed to analyze all of the rotational structure which appears

to be Ω = 0+, an example of which is given in Fig. 6.1. We see that there

are four rotational lines, with one missing between the third and fourth J (see

Chapter 6). We fit Eq. 4.2 to each observed rotational progression, first leaving

Ω as a parameter. Unlike the Ω > 0 states, the extracted value for these fits is

consistently |Ω| < 0.2. We then perform the fit with the rigid rotor equation,

E(J) = E0 − Bv(J(J + 1)), to extract rotational constants as shown in Fig.

75

-220 -200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0

0.004

0.008

0.012

0.016

0.020

0.024

0.028

0.032

(2) =0+ ab initio

(3) =0+ ab initio

11078

112

115114

8081

83

82

117

Bv (

cm-1)

J = 0 position relative to Cs 62P1/2(F=4) (cm-1)

119-127

Figure 5.1: J ′ = 0 energy levels and rotational constants for the A1Σ+ (X) andb3ΠΩ=0+ (+) electronic states. Solid lines indicate values calculated from ab initioPECs using LEVEL [3].

5.1. We can see from this figure that one of the progressions clearly dissociates

to the Cs 62P1/2 asymptote, and is similar to the predicted behavior for the ab

initio (2)Ω = 0+ state. The Hund’s case (c) (2) and (3)Ω = 0+ PECs shown in

Fig. 5.1 are derived from nonrelativistic ab initio PECs [11] that are smoothly

connected to the dispersive potentials from Ref. [9]. The spin-orbit PECs are then

approximated using the perturbative approach given in [59]. We note, however,

that the predicted behavior of the b3Π0− state near the Cs 62P1/2 asymptote is

similar to that of the A1Σ+ state, so more information is required to definitively

label the progression.

We probe the ground states to which these PA states decay in order to

confirm that this progression corresponds to the A1Σ+ electronic state. The

76

-322 -320 -318 -316 -3140

100

200

300

400

500

600 421

NaC

s+ sig

nal (

arb.

uni

ts)

PA frequency - 335121.7 (GHz)

0

Figure 5.2: An example PA scan of an A1Σ+ vibrational level with rotationalstate (J) labels.

Ω = 0− states should have essentially pure triplet multiplicity, in which case the

populated ground states would be exclusively a3Σ+ in character. We determine

that this is not the case using PDS: the PA lines in this progression produce

X1Σ+(v = 29, 31) levels. These observations are consistent with A1Σ+ − b3Π to

ground state transition moment calculations [8], confirming the A1Σ+ electronic

state assignment.

To label vibrational quantum numbers of the A1Σ+ levels, we interpolate

between the highest-lying line positions of the A1Σ+−b3Π calculated by Stolyarov

[8] that are mostly A1Σ+ in character and our PA data. As shown in Fig. 5.3,

we fit these to an NDE (Near-Dissociation Expansion) equation,

77

90 100 110 120 130

-1000

-800

-600

-400

-200

0

-100

-50

0

Bin

ding

ene

rgy

(cm

-1)

Vibrational quantum number v

perturbations

Figure 5.3: A fit of the NDE equation (solid line) to observed (circles) and cal-culated (squares) A1Σ+ vibrational levels [8]. The inset highlights perturbationsin the vibrational progression.

D − E ≈

X0,n

[(µ)n(Cn)2]1

n−2

× (vD − v)2nn−2 ×

[1 +

L∑i=t

pi(vD − v)i],

where the constant X0,n is found in [101], µ is the reduced mass, Cn is

the coefficient of the leading n order of the asymptotic behavior, D is the dis-

sociation limit, and vD is the non-integer vibrational level at dissociation. The

values for C6, vD, and D are given in Section 5.3, and we set i = 1 and L = 5.

In choosing the range and number of coefficients (i, L) we seek to avoid over-

parametrization while including terms of higher order to follow the (dissociative

and non-dissociative) behavior over the relatively large range of states. We then

perform fits while varying the overall vibrational numbering of the PA states and

78

vD until the residuals are minimized. Because of the gap between the PA data

and the calculated positions, the quality of the fit is not significantly decreased

by shifting the PA numbering by ±1, setting the certainty of the absolute la-

beling. We note that, as expected, some of the highly perturbed levels deviate

considerably from the fit.

Even when using the dark-SPOT configuration, several hyperfine “ghosts”

appear in the spectrum. These are easily accounted for as they occur to the red

of the primary lines at precisely the atomic hyperfine ground state splittings. In

the case of the Na hyperfine ghosts, particularly J ′ = 1, we observe multiple

lines to the blue of the primary ghost. We attribute these lines to excited state

hyperfine structure, which will be discussed further in Section 1.5. Sometimes

the J ′ = 1 Na hyperfine ghost lines appear very close to the J ′ = 0 rotational

line, complicating assignment. To distinguish between the two, we utilize the Na

MOT configuration to significantly reduce the J ′ = 0 signal while maintaining or

enhancing the lines associated with the hyperfine ghost.

5.1.3 Dispersion coefficients for the A1Σ+ state

The exchange interaction for the A1Σ+ state has the longest range of the

6P states of Cs, meaning that any calculation of dispersion coefficients must

only include very long range states, and/or take into account the non-asymptotic

behavior. Indeed, we calculate the modified Le Roy Radius [102] for A1Σ+ to

be ∼16 A, giving an estimate of the minimum classical outer turning point for

purely asymptotic states. We can then find the binding energy corresponding to

the LeRoy radius using the ab initio long range PECs from Refs. [9, 10]. These

two dispersive PECs give binding energies of 3.7 and 4.5 cm−1 at the modified

LeRoy radius, respectively. We may include some more deeply bound states into

79

Source C6 C8 γ vD

This work 10375 841872 0.06037 132.089

Marinescu et al. [9] 11393 1477812 — —

Bussery et al. [10] 14165 1492595 — —

Table 5.1: Dispersion coefficients and parameters obtained from the improvedLeRoy-Bernstein fit of near-dissociation A1Σ+ vibrational levels. These are com-pared to the ab initio A1Σ+ dispersion coefficients. All parameters except vD aregiven in atomic units.

the fit by using the improved LeRoy-Bernstein expansion formula given in [103].

The improved equation includes higher order correction terms that account for

both asymptotic and non-asymptotic behavior, as well as the next highest order

coefficient in the long-range multipolar expansion (in this case C8). To avoid

perturbed levels, however, we can only use the 6 bound states observed closest

to dissociation v′ = 125− 130.

We then fit Formula 26 from [103] to the data:

D − E ≈(vD − vH−1

n

) 11−β

[1− 1

1− β1

vD − v

(γ

(vD − vH−1

n

) 11−β

+(−Cn)

δ−1/2

n

Cm

Cn

δ − 1/2

1− δ

(vD − vH−1

n

) 1−δ1−β

B (δ, 1/2)

)],

where

H−1n =

√2µ

π

(−Cn)1/n

~(n− 2)

Γ(n+22n

)

Γ(n+1n),

D is the dissociation limit, vd is the non-integer vibrational level at dissociation,

Cn and Cm are the first two dispersion coefficients in the expansion, µ is the

reduced mass of the system, B is the Euler beta function and δ and β are con-

stants dependent on n and m. In our case, n = 6, m = 8, δ = 1/3 and β = 2/3.

We choose the dissociation energy to be 11178.4577 cm−1, which corresponds to

80

the Cs 62P1/2(F′ = 4) ← 62S1/2(F

′′ = 3) transition. This choice is a practical

one; we are unable to distinguish between molecular hyperfine states which corre-

spond to a particular atomic hyperfine state, so we choose the excited hyperfine

level which gives the best fit. We then leave vD, C6, C8, and γ as parameters,

where vD is initialized to the vD acquired with the traditional Leroy-Bernstein

fit, C6 and C8 to the calculated values given in Ref. [10], and γ to 0. Using the

nonlinear Levenberg-Marquardt fitting procedure in Mathematica, we obtain the

values which are given in Table 5.1. We note that the value obtained for the

C8 coefficient, like γ, is not likely to be physically meaningful. It is included to

improve the overall accuracy of the fit and the extracted C6 coefficient, but is

too sensitive to error of any kind to be trustworthy. The standard error for the

C6 parameter computed from the fit is ∼4000 Hartree a60. The estimated values

for the parameters are compared to the calculated values from [9] and [10]. As

the former reference excludes S-O coupling, we fit the analytic form of the long-

range potential (C6/R6 + C8/R

8) to our Hund’s case (c) approximation in order

to extract estimates for the C6 and C8 values that are given in Table 5.1. The C6

coefficient approximated from Ref. [9] is closest to our own value, easily within

the standard error.

5.1.4 The b3ΠΩ=0+ state

We observe 5 vibrational levels from the b3ΠΩ=0+ electronic state with

binding energies ranging from 13.7 to 163.7 cm−1. Levels blue-detuned from the

Cs 62P1/2 asymptote predissociate, and thus do not appear in our PA spectrum.

The b3ΠΩ=0+ vibrational states are similar to the A1Σ+ states in appear-

ance. As seen in Fig. 5.4, the same rotational states appear with similar relative

amplitudes. The parities of the rotational states are also the same for both elec-

81

-4910 -4905 -4900 -4895

50

100

150

200

250

300

350

400421

NaC

s+ sig

nal (

arb.

uni

ts)

PA laser frequency - 335121.7 (GHz)

0

1'

Figure 5.4: An example PA scan of an b3ΠΩ=0+ vibrational level with rotationalstate (J) labels. The 1′ label indicates hyperfine structure of the J ′ = 1 state inthe hypefine ghost channel.

tronic states and they decay to a similar range of ground state vibrational levels.

To identify the b3ΠΩ=0+ levels, we must consider the rotational constants and

vibrational spacing.

We calculate the rotational constants and label vibrational levels for the

b3ΠΩ=0+ by fitting the rigid rotor equation to the data. As seen in Fig. 5.1, the

b3ΠΩ=0+ levels all have larger rotational constants than the neighboring A1Σ+

states, although the two most heavily mixed states exhibit considerably smaller

rotational constants than would be expected in the absence of resonant coupling.

To label the vibrational quantum numbers, we use the same procedure as for

the A1Σ+ state, but using the calculated b3ΠΩ=0+ energy levels. As before, the

certainty of the assignment is ±1.

82

5.1.5 Ω = 0+ hyperfine structure

In the Hund’s (a) and (c) cases for magnetic hyperfine interactions, the

Ω = 0+ states exhibit no structure [104]. These limiting cases, however, are not

complete descriptions, and the molecular states must be described as a mixture

using one of these basis sets. The b3ΠΩ=0+ state, for instance, mixes slightly with

the b3ΠΩ=1 state. This can also be thought of as decoupling of the electronic

spin from the internuclear axis [1]. This weak spin decoupling can lead to a non-

zero Fermi contact interaction between the electronic and nuclear spins. A1Σ+

levels can also have hyperfine structure due to spin-orbit mixing with b3ΠΩ=0+

levels. In work by Ashman et al. [58], optical-optical double resonance (OODR)

spectroscopy of the (5)3ΠΩ=0+ state of NaCs did not reveal observable hyperfine

structure at the resolution of the experiment. In their analysis, it was proposed

that the lack of hyperfine structure was due to the applicability of the Hund’s

case (a)/(c) coupling models to the system. This suggests a difference between

coupling regimes. Our study involves vibrational levels that are relatively close to

the dissociation limit, while Ashman et. al investigated deeply bound (5)3ΠΩ=0+

states.

Fig. 5.5 shows the excited state hyperfine structure exhibited byA1Σ+(v′ =

117). These scans are calibrated by comparing the frequency of the Ti:Sapphire

laser with the Cs trapper beam on a scanning Fabry-Perot interferometer, which

is itself calibrated using both Cs hyperfine structure and the spacing between

hyperfine ghosts. This brings the certainty of our relative frequency of the scan

to ∼10 MHz, though it does not improve our absolute frequency calibration. As

noted earlier, hyperfine multiplets are only observed in the entrance channels

where Na is in the F = 2 state, and only for the J ′ = 0, 1, and 2 rotational levels.

We investigate all four possible combinations of MOT and dark-SPOT configu-

83

-1835 -1834 -1833 -1832 -1831 -1830

0

200

400

600

800

-1835 -1834 -1833 -1832 -1831 -1830

0

200

400

600

800

1000

-1844 -1843 -1842 -1841 -1840 -1839

0

100

200

300

400

-1844 -1843 -1842 -1841 -1840 -1839

0

200

400

600

800

1.77 GHz

J = 2

Cs MOTNa MOT

1.77 GHz

1.77 GHz

J = 2

J = 1

J = 1

J = 0J = 0

NaC

s io

n si

gnal

(arb

. uni

ts)

PA detuning from Cs 62P3/2 asymptote (GHz)

Cs MOTNa Dark-SPOT

Cs Dark-SPOTNa Dark-SPOT

Cs Dark-SPOTNa MOT

d)

c)

b)

a)

Figure 5.5: Hyperfine structure in A1Σ+(v′ = 117, J ′ = 0 − 2), observed withvarious MOT and dark-SPOT configurations. The PA scans in a) and b) coverthe FNa = 1, 2;FCs = 3 hyperfine entrance channels. Scans c) and d) cover theFNa = 1, 2;FCs = 4 entrance channels. The x-axis origin for scans c) and d) areshifted by 9.19 GHz relative to a) and b) so that the various hyperfine peaks arealigned between the two Cs hyperfine channels. Primes indicate those transitionsfrom the FNa = 2 entrance channel, no primes indicate the FNa = 1 channel.Dotted, dot-dashed, and dashed lines indicate hyperfine structure associated withJ ′ = 0, 1 and 2 respectively, and are included to highlight differences in observedstructure throughout the scans.

84

rations. These combinations enhance certain entrance channels while minimizing

others, facilitating assignment of the ground state hyperfine scattering channels.

Each hyperfine entrance channel can be represented as a mixture of molec-

ular electronic states, but we do not perform that analysis here. We note, how-

ever, that the choice of entrance channel likely restricts the observed excited

states via weakened selection rules. Because of the sparsity of the data and un-

certainty in the various sources of electronic state coupling, we do not attempt

to label the excited state hyperfine structure.

The strength of the hyperfine peaks seems to correlate with the partial

wave of the entrance channel. The strongest appears for s-wave scattering, less

so for p-wave scattering, and is non-existent for f -wave scattering, despite the

fact that these are associated with the strongest PA lines. The absence of hy-

perfine structure for f -wave scattering may be due to the sensitivity of the shape

resonance to the height of the centrifugal barrier, which is shifted for different

hyperfine entrance channels. A detailed discussion of the scattering channels and

shape resonance can be found in Chapter 6.

5.2 The c3ΣΩ=0− state

We observe only three vibrational levels of the c3ΣΩ=0− electronic state,

with binding energies of 271, 568, and 602 cm−1. This electronic state dissociates

to the Cs 62P3/2 asymptote, but we only find one of its levels to the blue of Cs

62P1/2. We have not conducted a thorough search for more c3ΣΩ=0− levels in the

PA spectrum, but have not found any within 30 cm−1 of the Cs 62P1/2 asymptote.

The lack of higher-lying states may be due to predissociation caused by coupling

with the b3ΠΩ=0− state.

85

-8120 -8115 -8110 -8105 -8100 -809550

100

150

200

250

300

350

400

NaC

s+ sig

nal (

arb.

uni

ts)


1 2 3 5

Figure 5.6: An example PA scan of an c3ΣΩ=0− vibrational level with rotationalstate (J) labels.

To determine that these are c3ΣΩ=0− levels, we analyze the rotational

structure and vibrational splittings. As seen in Fig. 6.2, the rotational states

consist of narrow peaks, indicating that the electronic angular momentum Ω of

the state is 0. However, the structure fits best when the first rotational state in the

progression has J ′ = 1, unlike the Ω = 0+ levels. The difference in the observed

rotational levels suggests that the electronic state has opposite parity, Ω = 0− (see

Chapter 6). There are two Ω = 0− electronic states which dissociate to the Cs 6P

asymptotes: c3ΣΩ=0− and b3ΠΩ=0− . One of the observed Ω = 0− vibrational levels

is to the blue of the Cs 62P1/2 asymptote, while the two levels red-detuned from

the 62P1/2 asymptote have large rotational constants and vibrational spacings

which are consistent with the c3ΣΩ=0− state. Thus, we confirm the c3ΣΩ=0−

assignment.

We compare binding energies and rotational constants for the observed

86

vΩ=0− Ecalc Eexp Bcalcv Bexp

v vΩ=1 Ecalc Bcalcv

32 610.47 601.87 0.0294 0.0281 33 617.07 0.0317

33 576.87 568.33 0.0289 0.0277 34 582.70 0.0312

44 280.18 271.09 0.0223 0.0215 45 275.77 0.0245

Table 5.2: Comparison between observed and calculated rotational constantsand binding energies for J ′ = 0 levels of the c3ΣΩ=0− and c3Σ+

Ω=1 states. Allvalues (except v) are given in cm−1. We calculate levels and rotational constantsusing Level. The uncertainties in the observed energies are ∼±1 GHz, while thestandard errors obtained for Bv in the experimental rotational fits are < 0.00005cm−1.

levels to values calculated from the c3ΣΩ=0− and c3Σ+Ω=1 PECs with LEVEL. As

we did for the (2) and (3) Ω = 0+ states, we estimate the Hund’s case (c) c3ΣΩ=0−

PEC from the nonrelativistic ab initio PECs in Refs. [11] and [9]. The experi-

mental c3Σ+Ω=1 PEC is taken from Ref. [7]. The c3Σ+

Ω=1 state interacts strongly

with B1Π, as well as b3ΠΩ=1, resulting in a large difference in the positions of the

two spin components for the same vibrational level.

5.3 The b3ΠΩ=0− and b3ΠΩ=1 electronic states

There are a number of features in the PA spectra which do not correspond

to the states discussed thus far. These features are difficult to assign with cer-

tainty because they are weaker when compared to the other states. It is likely that

at least some of this structure corresponds to b3ΠΩ=1 or b3ΠΩ=0− states, the only

two electronic states in the 6P manifold that we have not already investigated.

One of the possible b3ΠΩ=1 levels is in close proximity with B1Π(v′ = 13).

Homogeneous coupling between these states is evidenced by the fact that the

b3ΠΩ=1 level decays to the X1Σ+ ground state. There seems to be a large amount

of hyperfine structure associated with the b3ΠΩ=1 rotational levels, though much

of it is barely discernible from the background. We take the positions of the

87

tallest hyperfine clusters for each observed rotational level, J ′ = 1, 2, 4, 5, and

find a rotational constant of ∼0.027 cm−1. Considering that this is a perturbed

level, the observed rotational constant is in fairly good agreement with the value of

∼.020 cm−1 calculated from our approximated Hund’s case (c) potential extracted

from the ab initio PECs [9, 11].

Another apparent b3ΠΩ=1 state occurs in close proximity to b3ΠΩ=0+(v′ =

82). Again, we utilize a series of prominent hyperfine clusters to extract a ro-

tational constant. The extracted rotational constant of ∼0.011 cm−1 is in good

agreement with the value of ∼0.013 cm−1 calculated from the ab initio PECs.

We note that unlike the level mixed with B1Π(v′ = 13), this level only appears

when using the excited Na hyperfine channel. This is likely due to a difference

in the wavefunctions for these channels, which was also observed in Cs2 [105].

We observe some unidentified lines near the b3ΠΩ=0+(v′ = 82) and c3ΣΩ=0−(v

′ =

32) vibrational levels that could be associated with the b3ΠΩ=0− electronic state.

Unfortunately, analysis of this structure is complicated by weak signals or hyper-

fine structure from neighboring states.

88

Chapter 6

Excited state coupling and shape resonances

Many of the levels we find detuned from the Cs 62P1/2 asymptote are

deeply bound, and thus their wavefunctions have relatively short spatial extent.

Scattering waves, on the other hand, typically have most of their wavefunction

amplitude at long range. This apparent conundrum is resolved by considering

shape resonances in the scattering channel and coupling in the excited states.

6.1 Scattering waves

PA spectroscopy provides a tool for the investigation of the NaCs scatter-

ing channels. In this section, we exploit the symmetry of the Ω = 0 vibrational

levels to determine these channels and then consider the nature of an apparent

shape resonance.

6.1.1 Partial wave analysis using the Ω = 0 rotational spectra

An example of a PA scan of an A1Σ+ level is shown in Fig. 6.1. Through-

out the A1Σ+− b3Π PA spectra, we observe a mostly consistent intensity pattern

in the rotational structure. However, due to uncertainties in absolute detection

rates and saturation effects, we attempt only a qualitative assessment of the PA

line strengths and their likely collision channels. The strongest lines are typically

the J ′ = 2 and 4 lines, followed by the J ′ = 1 line, and the weaker J ′ = 0 line.

The J ′ = 3 line is consistently missing. Because the Ω = 0+ rotational states have

89

-322 -320 -318 -316 -3140

100

200

300

400

500

600 421

NaC

s+ sig

nal (

arb.

uni

ts)

PA frequency - 335121.7 (GHz)

0

Figure 6.1: An example PA scan of an A1Σ+ vibrational level with rotationalstate (J) labels.

definite (−1)J parity, they will only be populated by collisions with opposite par-

ity such that total angular momentum is conserved. The scattering wave parity is

simply (−1)ℓ [67]. Thus, J ′ = 0 is only populated by p-wave scattering, J ′ = 1 by

s-wave and d -wave scattering, J ′ = 2 by p-wave and f -wave scattering, J ′ = 3 by

d -wave and g-wave scattering, and J ′ = 4 by f -wave and h-wave scattering. We

see that the dominant PA scattering channel is then f -wave scattering. d -wave

scattering appears to be absent, so J ′ = 1 is exclusively populated by s-wave

scattering, the next strongest production channel. p-wave scattering is weaker,

but produces detectable amounts of J ′ = 0 molecules in many of the observed

levels. These observations are summarized in Table 6.1.

We can verify the preceding analysis by considering the c3ΣΩ=0− rota-

tional structure. In Fig. 6.2, we see J ′ = 1, 2, 3 and 5, with 1,3 and 5 being

90

ℓ ℓ Parity J′ Observed?

0 + 1 ×1 − 0,2 ×2 + 1,3

3 − 2,4 ×4 + 3,5

5 − 4,6

Table 6.1: Summary of partial waves responsible for the Ω = 0+ rotational lines

the most prominent rotational lines. This is consistent with our Ω = 0+ partial

wave analysis because the c3ΣΩ=0− rotational states have opposite parity (−1)J+1.

Again, f -wave scattering appears to be the dominant entrance channel, populat-

ing J ′ = 1, 3 and 5 while s-wave scattering produces a weak J ′ = 2 line (the

ℓ = 0 to J ′ = 0 transition is forbidden due to angular momentum conservation).

p-wave scattering is also likely contributes, but forms J ′ = 1 and 3 which are

already dominated by f -wave scattering.

6.1.2 Shape resonances

Our analyses find that f -wave contributions tend to be strong throughout

the PA spectra, regardless of electronic state. However, as seen in Fig. 6.3, there

is an effective centrifugal barrier for any partial wave higher than ℓ = 0. The

barrier heights for ℓ = 1− 4 in the a3Σ+ scattering channel are 0.3, 1.6, 4.6, and

9.9 mK, respectively. Classically, ultracold Na and Cs atoms (T < 500 µK) have

insufficient energy to overcome barriers with ℓ > 1, and we would thus not expect

these collisions to penetrate to short range and allow PA. Quantum mechanical

tunneling allows some wavefunction amplitude to penetrate, but this explanation

alone is insufficient to explain why, in most cases, the f -wave scattering appears

to be a more efficient channel for PA than s-wave scattering.

91

-8120 -8115 -8110 -8105 -8100 -809550

100

150

200

250

300

350

400

NaC

s+ sig

nal (

arb.

uni

ts)


1 2 3 5

Figure 6.2: An example PA scan of an c3ΣΩ=0− vibrational level with rotationalstate (J) labels.

In order to explain the strong f -wave contribution to the spectra, we must

consider quasibound vibrational levels. Technically, any vibrational wavefunction

calculated for energies above the atomic asymptote are scattering waves. How-

ever, if we consider only the effective potential on the inside of a centrifugal

barrier, there may exist a “bound” vibrational level above the dissociation limit.

When taking into account the complete potential, such a level satisfies weaker

boundary conditions than a truly bound state. Molecules driven to quasibound

levels often have shorter lifetimes because they will dissociate via tunneling before

spontaneous emission can occur, causing these states to be energy-broadened.

In the context of scattering, quasibound levels can give rise to shape res-

onances [69]. Shape resonances occur when the collision energy is at or near the

position of the quasibound state, and can significantly enhance the wavefunction

92

0 50 100 150 200 250 300-10

-5

0

5

10

Internuclear Separation HÞL

Ene

rgy

k BHm

KL

Figure 6.3: Centrifugal barriers for the a3Σ+ scattering channel.

amplitude inside the centrifugal barrier. This can be understood (roughly) as an

interferometric effect: scattering events with energies that give rise to wavefunc-

tions not supported by the inner part of the well are rejected, while those that

satisfy the boundary conditions are accepted. With a sufficiently long-lived qua-

sibound state, the short-range wavefunction amplitude due to a shape resonance

can greatly exceed that of the s-wave wavefunction.

We use the a3Σ+ ground state PEC from Ref. [5] to perform calculations

showing the consequences of a shape resonance in NaCs. Notably, LEVEL finds

a quasibound state at ℓ = 3 for the a3Σ+ PEC. However, LEVEL also predicts

the position of this state to be 0.0023 cm−1 above the asymptote, which at ∼3

µK is more energy than is available in the atomic collisions. We attribute this

discrepancy to a slight inaccuracy of the long-range portion of the Docenko et al.

PEC.

For the purpose of illustration, we adjust the long range (R > 10.2 A) part

of the a3Σ+ PEC until the quasibound level approximately matches our tempera-

93

5 10 15 20 25 30-1.0

-0.5

0.0

0.5

1.0

Internuclear SeparationHÅL

Rel

ativ

ew

avef

unct

ion

ampl

itude

Figure 6.4: Calculated s-wave (lower amplitude) and f -wave (higher amplitude)scattering wavefunctions for the a3Σ+ electronic state at E/kB=249.810841 µK.

ture of T = E/k = 250 µK. This is achieved by adjusting the C6 coefficient higher

by 2.95%1. We then use this PEC in the radial, time-independent Schrodinger

equation to compute the wavefunction for the f -wave shape resonance. To this

end, we write a Mathematica program which finds the shape resonance and cal-

culates its wavefunction. We already know the approximate energy of the shape

resonance from performing a LEVEL calculation. LEVEL, however, does not

propagate the quasibound wavefunction far enough beyond the classical outer

turning point to see the asymptotic behavior. For illustrative purposes, we also

want to find the wavefunctions for energies around the resonance. We then use

the finite difference form of the Schrodinger equation to propagate the wavefunc-

tion from a pair of initial values. We initialize these values of the grid to small

amplitude inside of the classical inner turning point (to account for the decay of

1The shift causes a slight mismatch between the intermediate and long range parts of thePEC. We do not correct for the resulting slope discontinuity as this is simply an illustrativecalculation. Also, we do not account for the small splittings between spin components.

94

180 200 220 240 260 280 300

0

1

2

3

4

Ek HΜKL

Am

plitu

dera

tio

Figure 6.5: A example of the tunneling effect of an f -wave shape resonance. Theamplitude ratio is the ratio between the first anti-node along the inner wall tothe long range scattering amplitude.

the wavefunction beyond the inner wall) and then propagate the wavefunction

for a number of energies near the center of the quasibound level. In Fig. 6.4, we

show the results of these wavefunction calculations. The wavefunctions are scaled

to one another such that their long-range maximum amplitudes match. This is a

best case scenario, of course, as it assumes that collisions occur precisely at the

shape resonance. Nevertheless, Fig. 6.4 illustrates the significant impact that a

shape resonance can have on the scattering wavefunction.

As a final illustrative point, we calculate wavefunctions for energies at

and around the shape resonance. This is done by initializing the wavefunctions

to small values inside of the inner wall and propagating them outward. These

wavefunctions are likely not as accurate as those produced using the shooting

method, but still show the behavior of the resonance. In Fig. 6.5, we have taken

the maximum of the first anti-node on the inside of the well and divided it by

the maximum long-range amplitude. This plot illustrates the effective ‘width’ of

95

Ground state ℓ Energy (µK) Lifetime (ns)

X1Σ+ 1 609.7 4.38

X1Σ+ 5 13566.8 5.26

X1Σ+ 9 34015.2 NA

a3Σ+ 3 3125.6 10.5

a3Σ+ 7 13259.0 NA

Table 6.2: Shape resonances for the two NaCs ground states as calculated byGonzalez Ferez and Koch [4].

the resonance.

Using the PECs in [5], Gonzalez-Ferez and Koch also calculate shape

resonances for NaCs [4]. These values are summarized in Table 6.2. As with our

calculation using LEVEL, their f -wave resonance appears to be too far from the

asymptote. Other resonances are found for both of the ground states (as with

our calculations), but none of these appear to play a significant role in our PA

spectra.

We note that even without the f -wave shape resonance, the a3Σ+ state

provides the best channel for PA. Its potential well is shallow (217.17 cm−1) com-

pared to the X1Σ+ state (4954.24 cm−1), resulting in a shorter s-wave scattering

length [5].

6.2 Excited state coupling

An understanding of coupling in the excited state manifold of electronic

states is vital in the analysis of the PA spectra. The subject of basis states and

interactions was treated more generally in Section 1.1.5. In this section, we focus

on the effects that these interactions can have on PA. We note that the treatment

of coupling between electronic states depends on the basis set used. Because the

X1Σ+ and a3Σ+ ground states can be approximated as having pure singlet and

96

triplet spin multiplicities, respectively, it is useful to talk about the excited states

in terms of mixing between Hund’s case (a) states.

To first order, we can approximate homogeneous S-O mixing as hap-

pening between electronic states rather than between individual rovibrational

eigenstates. We do this when we calculate relativistic PECs from ab initio non-

relativistic PECs. A result of these calculations is that the diagonalized set of

PECs are mixtures of the non-relativistic basis states, with the amount of mixing

varying with internuclear separation. Quantification (i.e., looking at the eigen-

vectors corresponding to each PEC) of this mixing is useful in determining the

spin character of excited states, and therefore what transitions to expect in our

spectra.

A simple PEC, however, cannot represent an interaction between neigh-

boring vibrational levels if the coupling between them is anomalously strong.

These instances are referred to as perturbations. Heavily perturbed levels are of-

ten few in number, and are thus simply neglected in the construction of empirical

PECs. However, sometimes perturbations exist throughout an electronic system,

in which case a more complicated description is required (e.g., the A1Σ+ − b3Π

complex). Because perturbations require that the pair of interacting states be

nearly resonant in energy (the strength of the interaction determining how reso-

nant), the interaction is also referred to as resonant coupling. Such coupling can

alter the spin character, wavefunctions, and eigenenergies of the pair, and can

occur due to homogeneous or heterogeneous interactions.

In PA, resonant coupling between states can result in enhanced molecule

formation rates [106]. A long-ranged vibrational level, which often has better

Franck-Condon overlap with a scattering wavefunction, can couple into a short-

ranged vibrational level, which often has better Franck-Condon overlap with

97

5 10 15 20

-5000

0

5000

10000

15000

Na + Cs*

B 1

0+

0+

a3

Ene

rgy

(cm

-1)

Internuclear coordinate (Å)

X 1

Na + Cs

PASpontaneous emission to (v )

Figure 6.6: Potential energy curves and approximate wavefunctions for inho-mogeneously coupled states producing X1Σ+(v′′ = 0) molecules. Ground statepotential curves are from [5] and the adiabatic 0+ states are approximated per-turbatively from the diabatic PECs in [11] and long range dispersion coefficientsin [9]. The experimental B1Π state is from [2]. Wavefunctions were calculatedusing Level 8.0 [3].

deeply bound ground states (see Fig. 6.6). Greater spin mixing between res-

onantly coupled states can also enhance molecule formation rates by improving

free-bound and bound-bound transition moments. For instance, if the scatter-

ing channel is triplet and the desired ground state is singlet, a mixed-character

excited state is ideal.

6.3 Observation of coupling in the NaCs PA spectra

The effects of excited state coupling can be seen throughout most of the

Cs 6P PA spectra. The a3Σ+ source of the f -wave shape resonance requires that

any of the excited states populated through this channel have some triplet spin

98

-4360 -4355 -4350 -4345 -4340 -43350

50

100

150

200

250

300

350


NaC

s+si

gnalHa

rb.u

nitsL 1 2 3 4 5

1¢ 2¢ 3¢ 4¢ 5¢

Figure 6.7: PA scan across B1Π(v′ = 13) (no primes) and a likely b3ΠΩ=1 (primes)state with rotational labels. This scan was taken with the REMPI laser fixed toan X1Σ+(v′′=0) detection line (see Chapter 7).

character. Due to S-O coupling, all electronic states in the Cs 6P complex have

some triplet character. The b3ΠΩ=2 electronic state, for instance, has essentially

pure triplet spin character, and we see these levels regularly over a large range of

binding energies. B1Π levels occur less frequently in the PA spectrum, but the

ones that do appear are populated primarily by the f -wave shape resonance as

indicated by the observed rotational levels J ′ = 1 − 5. It is difficult to say with

certainty whether any particular B1Π level is missing due to poor free-bound

Franck-Condon overlap, to predissociation, to a lack of triplet spin character, or

to some combination of these without performing a full deperturbation analysis

of the Ω= 1 complex of states. However, we know that mixing with the triplet

character Ω=1 states must be sufficient to allow coupling with the triplet ground

state scattering channel in those cases that we observe. We also see the effects of

A1Σ+-b3ΠΩ=0+ mixing: the higher-lying b3ΠΩ=0+ states predissociate, and con-

tributions from the triplet f -wave scattering channel is apparent in the A1Σ+ PA

spectra.

99

-160 -155 -150 -145 -140 -135 -1300

100

200

300

400

500


NaC

s+si

gnalHa

rb.u

nitsL

1¢ 2¢ 3¢ 4¢ 5¢

01 2 4

Figure 6.8: Photoassociation scan depicting the heterogeneous mixing betweenB1Π(v′ = 17) and A1Σ+(v′ = 125) rotational levels. Primed numbers indicateB1Π(v′ = 17) labels, non-primed numbers indicate A1Σ+(v′ = 125) rotationallabels. The full scan is taken using an X1Σ+(v′′=0) detection line (see Chapter7); the REMPI frequency used for the inset scan is sensitive to a3Σ+ and X1Σ+

ground state molecules.

We also see evidence for homogeneous resonant coupling between the

Ω = 0+ states of the A1Σ+ − b3Π complex. The coupling is most apparent

in the b3ΠΩ=2(v′ = 82, 83) vibrational levels, as can be seen by their markedly

decreased rotational constants. Resonant coupling with the long-ranged A1Σ+

states also likely explains why we see such strong PA lines for the deeply bound

b3ΠΩ=0+ vibrational levels, including rotational lines that are populated with the

less efficient s-wave and p-wave scattering channels.

We also find a case of resonant coupling between a b3ΠΩ=1 level and

B1Π(v′ = 13), as seen in Fig. 6.7. The b3ΠΩ=1 state is almost completely

triplet in character, and its near-dissociation levels have vanishingly small Franck-

Condon overlap with deeply bound X1Σ+ levels. The lack of dips corresponding

to b3ΠΩ=1 levels in the PDS scan shown in Fig. 3.8 confirms that these transi-

tions simply do not occur. Thus, extension of the wavefunction and mixing of

100

-336 -334 -332 -330 -328 -326 -324

100

200

300

400

500

600


NaC

s+si

gnalHa

rb.u

nitsL

1 2 3 4 5

2¢ 3¢ 4¢

Figure 6.9: PA scan across the coupled c3Σ+Ω=1(v

′ = 65) (no primes) andb3ΠΩ=2(v

′ = 100) (primes) lines with rotational labels.

the spin-multiplicities are necessary to allow the transitions observed in Fig. 6.7.

The coupling between these two levels may also improve the free-bound transition

strength to the B1Π level via spin mixing.

We find two confirmed cases of heterogeneous resonant coupling in the PA

spectra. In Fig. 6.9, we show a PA scan of heterogeneously mixed c3Σ+Ω=1 and

b3ΠΩ=2 vibrational levels. The mixing is strong enough to allow otherwise forbid-

den transitions between the b3ΠΩ=2 and X1Σ+ electronic states. This mixing also

alters the b3ΠΩ=2 wavefunction, which would otherwise have poor Franck-Condon

overlap with deeply bound X1Σ+ states. In Fig. 6.8, we show an example of het-

erogeneous mixing between B1Π(v′ = 17) and A1Σ+(v′ = 125). In this figure,

we compare scans taken with different REMPI wavelengths: the frequency is set

to detect only deeply bound X1Σ+ molecules in the main scan, while the inset

is a scan taken while detecting both triplet and singlet ground state molecules.

B1Π(v′ = 17) has significantly better Franck-Condon overlap with deeply bound

X1Σ+ levels, while the A1Σ+ level has a better free-bound transition strength.

The strongly mixed A1Σ+(v′ = 125, J ′ = 2) level is therefore the most efficient

101

line in the scan for producing deeply bound X1Σ+ molecules.

The other case of heterogeneous resonant coupling observed in the PA

spectra occurs between the b3ΠΩ=2(v′ = 100) and c3Σ+

Ω=1(v′ = 65) vibrational

states (see Fig. 6.9). b3ΠΩ=2(v′ = 100) produces a substantial number of

deeply boundX1Σ+ molecules, transitions which are otherwise strongly forbidden

(∆Ω = 2).

102

Chapter 7

Production and detection of X1Σ+(v′′=0) NaCs

molecules via PA

One of the primary goals of our research is to create a sample of ultra-

cold molecules in the absolute rovibrational ground state. Using the techniques

developed in previous chapters, we find and confirm PA channels which produce

X1Σ+(v′′=0) molecules. We also investigate the lowest ground state rotational

levels populated in the vibrational ground state and discuss the scattering waves

involved in their formation.

7.1 Absolute vibrational ground state formation and de-tection channels

7.1.1 The B1Π electronic state

The B1Π electronic state provides the most promising channels for pro-

ducing X1Σ+(v′′ = 0) molecules via PA. As seen in Fig. 7.1, many of the B1Π

vibrational levels have good Franck-Condon overlap with the absolute vibrational

ground state. This is partly due to the fact that the B1Π PEC is shallow com-

pared to other states in the Cs 6P complex.

We discuss the six B1Π vibrational levels observed in the PA spectrum

in Chapter 4. From Fig. 7.1, we see that the deeper levels of the B1Π state,

most of which are detuned from the Cs 62P1/2 asymptote, should have the best

F-C overlap with the vibrational ground state. However, to confirm that these

channels produce absolute ground state molecules, we must find a X1Σ+(v′′=0)

103

0 10 20 30 40 50 600.00

0.01

0.02

0.03

0.04

0.05

0.06

B1P vibrational level

Fran

ck-

Con

don

fact

or

Figure 7.1: Franck-Condon factors for B1Π←X1Σ+(v′′=0) transitions.

REMPI detection line.

7.1.2 Finding candidate X1Σ+(v′′=0) REMPI detection lines

To determine candidate X1Σ+(v′′=0) detection lines, we first take RE3PI

scans with LDS 867 dye. This frequency range has some of the strongest tran-

sition dipole moments for excitation from X1Σ+(v′′ = 0) to the A1Σ+ − b3Π

complex according to calculations from Stolyarov [8]. Unfortunately, our inves-

tigations with PDS turn up no X1Σ+(v′′ =0) detection lines in this part of the

spectrum. Because ionization with LDS 867 requires 3 photons, our inability

to detect the vibrational ground state may indicate that there are no secondary

transitions available. We also attempt to ionize X1Σ+(v′′ = 0) molecules using

B1Π levels, again without success. These transitions may or may not require 3

photons, depending on the NaCs ionization limit. Even with a sufficiently low

ionization threshold, some of these levels may predissociate before they have a

chance to absorb the second ionizing photon.

104

18200 18300 18400 185000

10

20

30

12300 12350 12400 124500

10

20

30 0 1 46

Exp. v = 0:


NaC

s+ sig

nal (

ions

/ pu

lse)

Ref. 21 v = 0:b)

v =

a)

Figure 7.2: REMPI scans detecting the absolute vibrational ground state. a)RE3PI scan with several labeled ground state vibrational levels. b) RE2PI scanwhere D1Π-f 3∆Ω=1 ←X1Σ+(v′′ = 0) transitions are indicated and compared toD1Π Dunham expansion from [12]. The observation of two perturbed electronicstates gives the appearance of doubling.

We then investigate a different portion of the RE3PI spectrum with LDS

821 dye. The A1Σ+−b3Π←X1Σ+(v′′=0) transitions are weaker in this frequency

range than with LDS 867, but still sufficient for detection. In Fig. 7.2 a), we

show the PI spectrum recorded while locked to the B1Π(v′ = 13, J ′ = 2) PA

line. We immediately note two lines in the REMPI scan that do not correspond

to previously detected X1Σ+ vibrational levels, but do correspond to transitions

from X1Σ+(v′′ = 0, 1).

In Fig. 7.2 b), we use the Coumarin 540 dye to investigate the RE2PI

spectrum using D1Π as the intermediate state. We find good agreement between

line positions recorded in this scan and those predicted by the Dunham expansion

of D1Π in Ref. [12]. This expansion is only accurate for v′ = 0− 2; higher lying

vibrational levels are perturbed by the crossing f3∆Ω=1 state [11].

105

11450 11500 11550 11600 11650 117000

2

4

6

8

10

12

14

16

18

212019181716

16151413

NaC

s+ sig

nal (

RE

MP

I ion

s/pu

lse)

A1 +

Depletion laser wavelength (cm-1)

b3

Figure 7.3: PDS scan taken while setting the PA to the B1Π(v′ = 13) and theREMPI laser to the IR X1Σ+(v′′=0) detection line. Bars indicate transitions toA1Σ+ − b3Π levels, as labeled.

7.1.3 Unambiguous labeling of X1Σ+(v′′ = 0) REMPI detection lineswith PDS

We now have multiple candidate X1Σ+(v′′=0) REMPI detection lines to

investigate with PDS. In Fig. 7.3, we fix the RE3PI wavelength to 12295.97 cm−1

and scan the depletion laser across several A1Σ+−b3Π←X1Σ+(v′′=0) transitions

with LDS 867 dye. To within our experimental uncertainty, these line positions

match perfectly with the calculated values. We find similar agreement for the

X1Σ+(v′′=1) RE3PI line.

To label the RE2PI detection lines, we detect with the RE3PI v′′ = 0 line

and perform PDS over the same range as in Fig. 7.2 a). Matches between PDS

dips and RE2PI peaks indicate v′′ = 0 detection lines (see Fig. 7.4). All of the

labeled REMPI detection lines are given in Appendix A.1.

106

18 300 18 350 18 400 18 450 18 500 18 550 18 6000

100

200

300

400

500

600

700

REMPIDepletion laser wavelength Hcm-1L

NaC

s+si

gnalHa

rb.u

nitsL

Figure 7.4: PDS and REMPI scans (not calibrated) detecting X1Σ+(v′′=0). TheREMPI scan is taken while locked to the B1Π(v′ = 13) PA line, and the depletionscan is taken while REMPI detects the infrared X1Σ+(v′′=0) detection line.

7.2 Investigation of the X1Σ+(v′′ = 0) rotational popula-tions

To experimentally determine the rotational population of our v′′ = 0 sam-

ple, we utilize CW depletion spectroscopy by driving a (2)Ω = 0+←X1Σ+(v′′=0)

transition and compare our results with calculations [8]. Four depletion scans con-

ducted while photoassociating different B1Π states are shown in Fig. 7.5. We

note that significant power broadening (∼200 MHz) of these transitions was nec-

essary to achieve an adequate signal-to-noise ratio, precluding the resolution of

hyperfine structure. In Fig. 7.5 a) and c), we excite to resonantly coupled states

labeled as A1Σ+(v′ = 125, J ′ = 1, 2) levels because these are stronger than the

corresponding B1Π rotational levels. In the depletion scans shown in Fig. 7.5

a) and b), we see that both J ′ = 2 levels decay to the X1Σ+(v′′ = 0, J ′′ = 1, 3)

states, confirming that both of the excited states have even parity. In Fig. 7.5

c), the odd parity A1Σ+(J ′ = 1) state should decay to both J ′′ = 0 and 2, but

we only detect J ′′ = 2. This failure to produce absolute rovibrational ground

107

0.0 0.5 1.0 1.5 2.0 2.5

8

10

12

0.0 0.5 1.0 1.5 2.0 2.5

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5

4

6

8

10

0.0 0.5 1.0 1.5 2.0 2.5

12

13

14

(v' = 13, J ) (v' = 13, J )

( ) (v' = 17, J )

b) d)

c)

NaC

s+ si

gnal

(ion

s / p

ulse

)

( ) (v' = 17, J )

Depletion laser frequency - 352249.956 (GHz)

a)

Figure 7.5: CW depletion scans showing rotational population of photoassociatedX1Σ+(v′′ = 0) molecules. A1Σ+(v′ = 29) ←X1Σ+(v′′ = 0) rotational lines arelabeled with calculations from [8]. The calculated line positions are shifted overallby 4.45 GHz to match the observed spectra. We set the PA frequency to themixed A1Σ+(v′ = 125)-B1Π(v′ = 17, J ′ = 2, 1) levels in a) and b), and theB1Π(v′ = 13, J ′ = 2, 1) levels in c) and d).

state molecules may simply be caused by a slightly weaker transition strength;

the overall signal is weak to begin with.

In Fig. 7.5 d), we observe decay which appears to exclusively popu-

late J ′′ = 1, indicating that the B1Π(v′ = 13, J ′ = 1) state has even par-

ity. Population of odd-parity ground state rotational levels indicates excitation

from even partial waves. Therefore, we infer that the f -wave shape resonance

is the dominant formation channel for this state. This feature is noteworthy

because it provides a channel for production of a rotationally pure sample of

X1Σ+(v′′ = 0, J ′′ = 1) molecules. Combining information from the ionization

108

rate and CW depletion scan, we estimate the formation rate of v′′ = 0, J ′′ = 1

molecules to be between ∼104 and 105 molecules/second.

109

Chapter 8

Optical pumping of ultracold NaCs molecules

As a method for producing absolute ground state molecules, PA is lim-

ited. Even when a path to X1Σ+(v′′ = 0) is found, many of the molecules end

up in a range of excited rovibrational levels. In this chapter, we investigate tech-

niques which optically transfer these excited rovibrational levels into the absolute

rovibrational ground state.

8.1 Vibrational cooling of molecules with broadband light

The use of broadband light to pump ultracold molecules into their vibra-

tional ground state was first achieved by Matthieu et al. [45]. In this experiment,

Cs2 was formed in a range of X1Σ+g levels via PA and were cycled through B1Πu

vibrational levels using shaped femtosecond laser pulses. The spectrum of the

pulse was shaped in such a way that it did not excite the absolute vibrational

ground state, effectively making this the only dark state. The laser drove many

transitions in each molecule, with a significant fraction piling up in the dark

ground state. A drawback to this technique is that there is no control over the

rotational transitions; the rotational quantum number essentially underwent a

random walk weighted by the Honl-London factors. This work was followed up

with a simpler and cheaper approach using a commercially available free-running

laser diode [107]. In this experiment, the diode was run below lasing threshold

to provide broadband light, then spectrally filtered with a grating and mask.

110

To apply the vibrational cooling technique to NaCs, we must select suit-

able excited states through which to pump. We must consider transition strengths,

diode availability, and loss channels such as predissociation and decay to dark

ground states. Using these criteria, we find that the Ω = 0+ A1Σ+−b3Π complex

states are the most attractive for pumping. To avoid any pumping out of the

absolute vibrational ground state, the broadband light spectrum must be to the

red of ∼978.6 nm. Fortunately, high power 980 nm diodes are relatively cheap

due to their use as pumps for Erbium-doped fiber lasers and in biomedical ap-

plications. A1Σ+-b3ΠΩ=0+↔X1Σ+ transitions are also some of the strongest for

the molecule. The only drawback related to using this state is the possibility

of spontaneous emission to the a3Σ+ ground state. However, given a reasonably

broad optical pumping spectrum even the a3Σ+ state is not dark, meaning that

these states can be pumped back into the X1Σ+ state through mixed character

A1Σ+ − b3Π complex levels.

In order to pump a maximal number of excited X1Σ+ vibrational levels,

we need a broad OP light spectrum. To achieve this, we employ 4 2.5 W, free-

running multimode laser diodes from Intense Ltd., temperature tuned to cover a

range of ∼12 nm. This spectrum is shown in Fig. 8.1 a). Clearly, the laser diode

spectrum does not have perfect coverage; there are many deep intensity minima

between the noise-broadened laser diode modes. Nevertheless, we proceed to

overlap the beams with a series of beam-splitter cubes and waveplates. Using

this simple polarization system, we cannot overlap all 4 beams with full intensity.

We thus input the two pairs of beams on a 50/50 beamsplitter cube such that all

of the beams are overlapped, but the total intensity is split between the two exit

ports. To use the full intensity, we bring the output of both exit ports into the

chamber. If we need to free up optical access, we can block one of these beams

111

Figure 8.1: Spectrum of the diode lasers used in the broadband OP experiment.In a), the spectra for the 4 independent 980nm laser diodes are combined. Inb), the spectrum for the 1206 nm laser diode is shown. The vertical line in a)denotes the b3ΠΩ=0+(v

′ = 0)→ X1Σ+(v′′ = 0) transition

without sacrificing spectral content.

Our first attempt to optically pump molecules into the vibrational ground

state from levels populated by the 32 GHz PA line was successful using the

980 nm OP light. Fig. 8.2 a) depicts a RE3PI scan in the absence of OP. In

112

NaC

s+si

gnalHa

rb.u

nitsL

aL

12 300 12 320 12 340 12 3600

200

400

600

800

1000

1200

4569

21

bL

12 300 12 320 12 340 12 3600

200

400

600

800

1000

1200

345689

142021

cL

12 300 12 320 12 340 12 3600

200

400

600

800

1000

1200

012

dL

12 300 12 320 12 340 12 3600

200

400

600

800

1000

1200

Rempi wavelength Hcm-1L

012

Figure 8.2: RE3PI scans taken with: a) no OP, b) 1206 nm OP light, c) 980 nmOP light, and d) 1206 and 980 nm OP light.

113

NaC

s+si

gnalHa

rb.u

nitsL

aL

11 550 11 600 11 6500

100

200

300

400

500

5

8

9

15

19

21

23

25

27

34

bL

11 550 11 600 11 6500

100

200

300

400

500

5

8

9

15

19

21

23

25

27

34

cL

11 550 11 600 11 6500

200

400

600

800

1000

Rempi wavelength Hcm-1L

589

15192123252734

Figure 8.3: RE3PI scans taken without (blue) and with (red) OP: a) 1206 nmOP light, b) 980 nm OP light, and c) 1206 and 980 nm OP light. The systemwas better optimized for the scans in c), hence the larger signal.

114

Fig. 8.2 c), the OP light continuously interacts with the sample. Clearly, most

of the observed population ends up in the lowest vibrational levels, X1Σ+(v′′ =

0, 1, 2). Accumulation of molecules in v′′ = 1 and 2 is expected from the transition

moment calculations. With the wavelength range of the OP diodes, v′′ = 1 and

2 can only be excited to the lowest levels of the A1Σ+ − b3Π complex. These

transitions are relatively weak because the excited states have significant triplet

spin character, and therefore do not contribute much to the broadband pumping.

However, these transitions are useful for narrow-line OP (see Section 8.2).

The calculated transition moments provided by A. Stolyarov [8] show that

higher-lying vibrational levels of X1Σ+ can be pumped more efficiently with

deeper infrared light, particularly in the 1200-1300 nm range. To this end, we

install a 1206 nm, 3.5 W Thorlabs multimode laser diode. The spectrum for this

laser is given in Fig. 8.1 b). The addition of this beam increases our optical

pumping rate into X1Σ+(v′′=0) by ∼70%, as seen by comparing Fig. 8.2 c) and

d). We note that the 1206 nm diode alone does not efficiently populateX1Σ+(v′′=

0) as seen in Fig. 8.2 b). This is simply because, based on transition moment

calculations, the light does not not interact with X1Σ+(v′′ < 25). According to

simulations developed in Mathematica by Amy Wakim [46], the 1206 nm laser

primarily improves the OP efficiency by pumping higher lying vibrational levels

into states that are accessible to the 980 nm light.

In Fig. 8.3, we take LDS 867 RE3PI scans with and without OP light. Al-

though there are no X1Σ+(v′′=0) detection lines in this portion of the spectrum,

these scans allow us to see the transfer of population in a larger set of vibrational

levels. In Fig. 8.3 a), the 1206 nm light increases the population in nearly all of

the observed states. In contrast, the scans in Fig. 8.3 b) confirm that nearly all

of the same levels are depleted by the 980 nm light. In Fig. 8.3 c), some of the

115

NaC

s+si

gnalHa

rb.u

nitsL

aL

447 448 449 450 451

600

700

800

900

1000

X1S+Hv²=

4LJ²= 0 1 234

bL

526 527 528 529 5301200

1400

1600

1800

2000

2200

Depletion laser frequency - 351722 HGHzL

X1S+Hv²=

0L

J²= 21304

Figure 8.4: CW depletion scans a) with and b) without broadband OP, withrotational state labels. In a), the RE3PI laser is set to detect X1Σ+(v′′ = 4),while in b) it is set to detect X1Σ+(v′′ = 0). Diagonal hatching indicates therange of the noise.

lines are depleted more than others, indicating that the relative optical pumping

rates of the two diode systems vary by vibrational level. Nevertheless, we see

depletion of essentially all of the lines when both diodes are on, showing that

there are very few truly dark states in the range of detected vibrational levels.

Unfortunately, we have not yet labeled REMPI lines for higher-lying vibrational

levels, and therefore cannot experimentally confirm the depletion of these states.

We also directly measure the rotational states populated via broadband

OP. In Fig. 8.4 a), we see the rotational distribution created by the 958 GHz PA

line: J ′′ = 0, 1 and 2. While we use the 32 GHz PA line for OP in Fig. 8.4 b),

the initial rotational distribution is the same as for the 958 GHz line (the latter

116

Figure 8.5: Franck-Condon factors for transitions between the lowest X1Σ+ andb3ΠΩ=0+ vibrational levels.

is shown in a) simply because the data has a superior signal to noise ratio). The

final rotational state for a molecule after OP depends on how many transitions it

undergoes before it is pumped to the dark state. Each total transition strength

depends on Honl-London factors [1], so the problem can be modeled effectively by

a weighted, J-dependent random walk. A. Wakim developed a simple simulation

in which each molecule undergoes a randomly selected but constrained number

of transitions that we can compare to the experimental results to determine the

likely number of transitions. Our data is consistent with the simulation if the

molecules undergo 1-6 transitions (assuming uniform probability). According to

our vibrational pumping model, such a small number of transitions is sufficient

to pump only ∼10% of the total population into the ground state. Nevertheless,

the signal is sufficient to nearly saturate our detection system, suggesting that

our relatively low pumping efficiency is overcome by a high overall PA molecule

production rate.

8.2 Narrow line optical pumping of NaCs

Once the vibrational degree of freedom of the NaCs molecule has been

cooled by the broadband OP, we seek to rotationally cool the sample. An ap-

117

303382.4 303382.6 303382.8 303383 303383.2 303383.4

200

400

600

800

Vibrational transfer laser frequency HGHzL

NaC

s+Ha

rb.u

nitsL

Via B3PW=0+Hv

¢=0, J¢=0L

detecting v²=0 detecting v²=1

Figure 8.6: Narrow-line vibrational OP scans. For the upper scan, the RE3PIlaser is set to detect X1Σ+(v′′ = 0), and for the lower scan to X1Σ+(v′′ = 1). A2-point moving average is used to smooth the scans.

proach suggested in [108] involves using a beam that would pump any excited

rotational levels of the absolute vibrational ground state, leaving X1Σ+(v′′ =

0, J ′′=0) as the only dark state. A drawback to using this approach is that there

is a chance the excited state or states will decay to a different vibrational level.

Given that our broadband cooling does not drive very many cycles on average,

this could greatly reduce our ground state production efficiency. Fortunately, the

lowest vibrational levels in the Ω = 0+ A1Σ+ − b3Π complex states provide a

system of nearly closed vibrational transitions.

F-C factors are given in Fig. 8.5 for the lowest vibrational levels of the

A1Σ+− b3Π and X1Σ+ electronic states. F-C factors for emission to levels above

X1Σ+(v′′ = 3) are negligible. These b3ΠΩ=0+ vibrational levels are unique be-

cause they are deeper than the A1Σ+ electronic state, so their wavefunctions

do not extend out far from coupling with this state, and thus have vanishing

118

Figure 8.7: Diagram of the combined broadband and narrow-line OP experiment.Green arrows indicate spontaneous emission.

Franck-Condon overlap with the a3Σ+ electronic state and most of the X1Σ+ vi-

brational level. Fortunately, these levels still have enough singlet spin character

to couple to X1Σ+ levels, if only weakly. The result is a system of transitions

that are relatively closed, making them excellent candidates for narrow-line OP.

In particular, the X1Σ+(v′′ = 0) ←b3ΠΩ=0+(v′ = 0) is almost completely closed,

making it promising for rotational OP. These system are also good candidates for

optically pumping the v′′ = 1 and 2 molecules back into the vibrational ground

state.

To implement rotational OP using the X1Σ+(v′′ = 0) ←b3ΠΩ=0+(v′ =

0) manifold, several lasers are necessary to drive the lowest several ∆J = −1

transitions. Considering the rotational distribution in Fig. 8.4 b), driving of the

J ′ = 1← J ′′ = 2 and J ′ = 3← J ′′ = 4 transitions should be sufficient to transfer

a majority of the X1Σ+(v′′=0) molecules into the rovibrational ground state.

In our preliminary investigation of these narrow-line OP manifolds, we use

119

´

´

´´

´

´

´

´

´

´

´

´

´

´

´

´

´

´

´

´

´´

´ ´ ´

´

´

´ ´

´

´´

´

´

306324 306324.2 306324.4 306324.6

600

700

800

Rotational transfer laser frequency HGHzL

NaC

s+Ha

rb.u

nitsL

v²=0; J²=2 to J²=0 transfer

Figure 8.8: Rotational transfer from X1Σ+(J ′′ = 2) to J ′′ = 0 via b3ΠΩ=0+(v′ =

0, J ′ = 1). We include both the raw data points (×) and a 4-point moving average(solid line).

a DL pro tunable diode laser borrowed from TOPTICA Photonics AG. Using the

B1Π(v′ = 13, J ′ = 1) PA line, we create rotationally pure samples of X1Σ+(v′′ =

0 − 2) molecules. We then fix the REMPI laser to detect these states and scan

the diode laser across several vibrational pumping transitions. Because these are

vibrational transfer scans, we can monitor either the depleted or the enhanced

vibrational populations. An example of one such vibrational transfer line is shown

in Fig. 8.6. We see that the depleted and enhanced signals match, confirming

that there is essentially no loss to other vibrational levels.

We also use the rotationally pure sample to test transfer between ro-

tational states in the same vibrational level. Because our REMPI laser does

not distinguish between rotational levels, we must use the CW depletion beam

to monitor rotational pumping. We accomplish this by setting the CW deple-

tion beam to a frequency that depletes the destination state. We then scan the

pumping beam across a transition, and look for further depletion as molecules

120

are transferred into the depleted state.

Once our technique for monitoring rotational OP is successfully proven,

we attempt to transfer vibrationally cooled molecules into the rotational ground

state. A diagram of this experiment is given in Fig. 8.7. We note that all of

the beams are interacting with the molecular sample simultaneously. This means

that a molecule that is depleted from the X1Σ+(v′′ = 0, J ′′ = 0) state can be

returned via the broadband OP lasers. This experiment works, however, because

the broadband OP process is much slower and less efficient than CW depletion.

We also know that the rotational optical pumping beam is not simply depleting

the sample by itself; we only see a depletion dip when the CW depletion and

rotational optical pumping beams are interacting with the sample at the same

time. This scan is shown in Fig. 8.8. Taking into account background ions, we

estimate that ∼35% of the v′′ = 0 population is in J ′′ = 0 after OP.

121

Chapter 9

Conclusions

In this thesis, we described the experiments and analysis leading towards

continuous production of X1Σ+(v′′ = 0, J ′′ = 0) NaCs molecules. In earlier work,

we had very little understanding of the PA process itself or the ground states

detected via REMPI. The design and implementation of different spectroscopic

methods, along with the work of other molecular spectroscopy groups, allowed

us to label most of the observed PA structure and many REMPI detection lines.

With these methods, we also greatly expanded the number of known NaCs PA

resonances, particularly those detuned from the Cs 62P1/2 asymptote. Through

our analysis of the spectroscopic data, we contributed to the construction of

high quality empirical PECs. Some of the observed PA resonances were found

to produce absolute vibrational ground state molecules, though only as low as

J ′ = 1 because of the partial waves involved in these collisions.

Understanding of the PA and REMPI structure allowed us to implement

broadband vibrational cooling, the first such demonstration for polar molecules

[46]. We also demonstrated our ability to transfer population between the lowest-

lying rovibrational levels using narrow-line optical pumping. It is important to

note that our methods for producing rovibrational ground state molecules are

relatively simple and inexpensive compared to techniques which employ STIRAP.

Now that we are able to produce ultracold, rovibrational ground state molecules

at high rates, we can turn our attention to trapping and further cooling.

122

There are multiple approaches to trapping and accumulation of molecules.

In previous work, we constructed a thin-wire electrostatic trap (TWIST) [60].

This trap functioned, but was not mechanically stable, and has since been re-

moved. We also could not characterize the distribution of rovibrational states we

were creating via PA. Now we can use PA channels which are known to produce

the lowest electrostatically trappable state, X1Σ+(v′′ = 0, J ′′ = 1). Using an

improved version of the TWIST, we can accumulate these molecules. Optical

traps are able to confine the high-field seeking states (J ′′ = 0), as well as atoms,

making this another attractive option. Either way, a large sample of trapped

molecules would allow for a number of interesting experiments.

First, a large sample of X1Σ+(v′′ = 0, J ′′ = 1) molecules could be directly

imaged [109]. In general, molecular transitions are not cycling transitions, so this

technique is not as efficient as for the atomic case. However, the b3ΠΩ=0+(v′ =

0, J ′′ = 0) ↔ X1Σ+(v′′ = 0, J ′′ = 1) transition is nearly closed, with 97.63% of

excited molecules returning to the ground state via spontaneous emission. With

this branching ratio, on average a single molecule would absorb and spontaneously

emit ∼40 excitations before being lost to a dark vibrational level. We note that

the transition is weak relative to atomic lines used in absorption imaging; the

excited state lifetime for b3ΠΩ=0+(v′ = 0, J ′′ = 0) is 2.5 µs. For a beam at the

saturation intensity, a population of 105 molecules held in a trap with 1 mm

diameter, and an exposure time of 90 µs, one would expect ∼5% absorption of

the beam in the interaction region. This amount of absorption is easily detectable

but the absorption beam intensity is fairly low (.1775 W/m2), and any imaging

system would have to be carefully constructed. Nevertheless, direct imaging of

the cloud would allow us to determine the actual molecule number, trap size and

sample temperature (via thermal expansion). Imaging would also be useful in

123

matching the spatial modes of different traps for loading purposes (for instance,

the MOTs and the TWIST).

A molecule trap also allows the investigation of molecular collisions, and

the extraction of scattering rate constants [110]. Such collisions are of interest

to achieve state purification, as only molecules in their absolute rovibrational

ground state are immune to rotational or vibrational state quenching collisions

with ground state atoms. Any inelastic collision will impart sufficient energy

to kick the excited state molecule out of the trapping region. Elastic collisions

between absolute ground state atoms and molecules are also of interest as atoms

cooled to sub-Doppler temperatures could be used to sympathetically cool a

trapped sample of molecules, assuming there are no chemical reactions between

the species [111]. With sufficient densities, rovibrational quenching molecule-

molecule collisions could also be observed. While ground state NaCs is not a

reactive species, collisions between molecules in rovibrationally excited states

can result in chemical reactions [112].

Many of the interesting experiments and applications involving polar molecules

(see Chapter 1) require the implementation of an optical lattice [113]. Further

cooling would be necessary to load our NaCs molecules into an optical lat-

tice, which typically have trap depths of a few µK. As discussed previously, a

trapped and sub-Doppler cooled sample of (non-reactive) atoms could be used

to sympathetically cool the molecules to these temperatures. Another option

would be to form the molecules from colder atoms, though this would require

a significant upgrade to the lab. A third possibility is to use the near-cycling

X1Σ+(v′′ = 0, J ′′ = 1) ↔b3ΠΩ=0+(v′ = 0, J ′ = 0) transition to create an op-

tical molasses for the molecules. This approach is related to narrow-line laser

cooling schemes for atoms with complex electronic structure [114]. Narrow-line

124

cooling has the inherent benefit of having a low Doppler limit, which for NaCs

would be ∼1µK. However, the ideal values for the bandwidth and detuning of

the laser be very small, on the order of 10 and 100 KHz respectively. These

values are technically feasible, but because this is a molecule, the laser cannot be

locked atomically. Instead, a stabilized cavity must be used. Also, the hyperfine

structure of the ground and excited states in this cycling transition has not been

investigated. If the hyperfine structure is on the order of the detuning then this

method would be complicated by the fact that the cooling laser would address the

different hyperfine states at different detunings. Nevertheless, it should techni-

cally be feasible to obtain a sample of rovibrational ground state NaCs molecules

on the order of 1 µK.

BIBLIOGRAPHY 125

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137

Appendix A

A.1 X1Σ+ REMPI lines

The following is a list of X1Σ+ REMPI detection lines. Note that IR

wavelengths are RE3PI and visible wavelengths are RE2PI. When possible, we

label the excited state. † indicates that the line was labeled without the use of

PDS, and is thus the assignment is not as certain. In the case that there is no

vibrational label, the electronic state assignment is based on our consideration

of ab initio PECs and selection rules. The N column places the level (when

applicable) in the context of the entire A1Σ+ − b3Π complex system, starting

with N = 1.

# v′′REMPI λ−1

(cm −1)Excited state v′ N

1 0 12295.97 A1Σ+ 32 99

2 0 18233.24 D1Π 0

3 0 18298.36 D1Π 1

4 0 18361.81 D1Π 2

5 0 18382.36 f3∆Ω=1 0

6 0 18423.14 D1Π 3

7 0 18437.35 f3∆Ω=1 1

8 0 18480.54 f3∆Ω=1 2

9 0 18493.95 D1Π 4

10 0 18533.50 f3∆Ω=1 3

11 0 18552.24 D1Π 5

12 1 12328.70 A1Σ+ 34 106

13 1 12366.62 A1Σ+ 35 108

14 1 18199.58 D1Π 1

15 1 18264.03 D1Π 2

16 1 18283.92 f3∆Ω=1 0

17 1 18325.12 D1Π 3

18 1 18339.18 f3∆Ω=1 1

138



19 1 18382.36 f3∆Ω=1 2

20 1 18396.07 D1Π 4

21 1 18454.61 D1Π 5

22 2 12330.30 A1Σ+ 36 112

23 2 12368.46 b3ΠΩ=0+ 26 113

24 2 12371.03 b3ΠΩ=0+ 26 113

25 3 16573.84 C1Σ+/d3ΠΩ=0+

26 4 11471.42 A1Σ+ 24 76

27 4 11471.47 A1Σ+ 24 76

28† 4 12036.28 A1Σ+ 34 106

29† 4 12076.01 A1Σ+ 35 108

30† 4 12195.05 A1Σ+ 37 114

31† 4 12294.96 A1Σ+ 39 120

32† 4 12400.66 A1Σ+ 41 126

33 4 12441.73 b3ΠΩ=0+ 29 127

34 4 12656.23 A1Σ+ 46 140

35 5 11679.41 b3ΠΩ=0+ 21 91

36 5 11706.53 A1Σ+ 30 94

37† 5 12042.22 A1Σ+ 36 112

38 5 12173.45 b3ΠΩ=0+ 27 118

39† 5 12259.88 b3ΠΩ=0+ 28 123

40 5 12464.17 A1Σ+ 44 135

41 5 12704.01 A1Σ+ 48/48 147/149

42 5 16907.03 C1Σ+/d3ΠΩ=0+

43 5 17996.39 f3∆Ω=1 2

44 5 18009.96 D1Π 4

45† 6 11985.76 b3ΠΩ=0+ 26 113

46† 6 12211.50 A1Σ+ 41 126

47† 6 12250.42 b3ΠΩ=0+ 29 127

48† 6 12262.77 A1Σ+ 42 129

49† 6 12312.93 A1Σ+ 43 131

50 6 12336.85 b3ΠΩ=0+ 30 132

139



51† 6 12369.06 A1Σ+ 44 135

52† 6 12411.02 b3ΠΩ=0+ 31 136

53† 6 12470.25 A1Σ+ 46 140

54† 6 12565.13 b3ΠΩ=0+ 33 145

55 8 11591.61 A1Σ+ 33 103

56 8 11761.03 A1Σ+ 36 112

57† 8 11977.31 b3ΠΩ=0+ 28 123

58† 8 12022.28 A1Σ+ 41 126

59† 8 12500.06 A1Σ+ 50 152

60 9 11631.82 b3ΠΩ=0+ 25 109

61 9 11667.11 A1Σ+ 36 112

62† 9 11981.83 A1Σ+ 42 129

63† 9 12033.54 A1Σ+ 43 131

64 9 12054.57 b3ΠΩ=0+ 30 132

65† 9 12308.64 A1Σ+ 48 147

66 9 12505.81 A1Σ+ 52 158

67 10 11445.22 b3ΠΩ=0+ 24 104

68 10 11633.76 A1Σ+ 37 114

69 11 11525.23 b3ΠΩ=0+ 26 113

70† 11 12062.84 A1Σ+ 47 143

71 11 12125.43 A1Σ+ 48 147

72† 11 12163.87 A1Σ+ 49 149

73† 11 12281.14 A1Σ+ 51 155

74⋆ 11 15553.74 B1Π 45

75 13 11435.03 b3ΠΩ=0+ 27 118

76 13 11514.19 A1Σ+ 40 122

77 13 11694.23 b3ΠΩ=0+ 30 132

78 13 11723.20 A1Σ+ 44 135

79 13 11787.73 A1Σ+ 45 138

80† 13 12198.97 A1Σ+ 53 161

81† 13 12297.97 A1Σ+ 55 167

82† 13 12317.72 b3ΠΩ=0+ 38 168

140



83 15 11428.44 b3ΠΩ=0+ 29 127

84 15 11439.43 A1Σ+ 42 129

85 15 11544.87 A1Σ+ 44 135

86 15 11547.46 A1Σ+ 44 135

87 15 11550.27 A1Σ+ 44 135

88 15 11605.30 A1Σ+ 45 138

89† 15 12272.11 A1Σ+ 58 176

90 17 11472.83 A1Σ+ 46 140

91‡ 17 11638.75 A1Σ+ 49 149

92 17 12196.78 A1Σ+ 60 181

93 19 11552.51 b3ΠΩ=0+ 35 154

94 19 11577.78 A1Σ+ 51 155

95 19 11579.92 A1Σ+ 51 155

96 19 11618.91 A1Σ+ 52 158

97 19 11640.17 b3ΠΩ=0+ 36 159

98† 19 11947.97 b3ΠΩ=0+ 40 177

99 19 12101.36 b3ΠΩ=0+ 42 186

100† 19 12326.51 b3ΠΩ=0+ 45 200

101† 19 12393.88 b3ΠΩ=0+ 46 204

102 21 11542.71 b3ΠΩ=0+ 37 163

103 21 11560.66 A1Σ+ 54 164

104† 21 12010.44 b3ΠΩ=0+ 43 191

105 23 11215.76 b3ΠΩ=0+ 35 154

106 23 11492.49 A1Σ+ 56 170

107 23 11614.87 b3ΠΩ=0+ 40 177

108 23 11690.10 A1Σ+/b3ΠΩ=0+ 60/41 181/182

109 23 11690.25 A1Σ+/b3ΠΩ=0+ 60/41 181/182

110 24 11686.26 b3ΠΩ=0+ 42 186

111 25 11530.38 b3ΠΩ=0+ 41 182

112 27 11523.75 b3ΠΩ=0+ 43 191

113 29 11552.22 A1Σ+ 67 202

114 31 11596.20 A1Σ+ 71 214

141



115 34 11659.51 A1Σ+ 77 233

142

A.2 a3Σ+ REMPI lines

The following is a list of a3Σ+ REMPI detection lines. Note that IR

wavelengths are RE3PI and visible wavelengths are RE2PI. When possible, we

label the excited state. In the case that there is no vibrational label, the electronic

state assignment is based on our consideration of ab initio PECs and selection

rules.

# v′′ REMPI λ−1

(cm −1)Excited state v′

1 8 12292.62 d3Π/C1Σ+/e3Σ+

2 10 12330.77 d3Π/C1Σ+/e3Σ+

3 10 16717.71 f3Π/G1Σ+

4 10 16802.03 f3Π/G1Σ+

5 12 11570.54 b3ΠΩ=2 85

6 12 11680.61 c3Σ+Ω=1/B

1Π/b3ΠΩ=2 55/39/90

7 12 16699.27 f3Π/G1Σ+

8 14 11546.36 c3Σ+Ω=1 48

9 14 11636.55 c3Σ+Ω=1 53

10 14 16684.01 f3Π/G1Σ+

11 15 16677.98 f3Π/G1Σ+

12 17 11511.28 c3Σ+Ω=1/b

3ΠΩ=2 47/84

13 17 11531.80 c3Σ+Ω=1 48

14 17 16668.36 f3Π/G1Σ+

15 17 16702.11 f3Π/G1Σ+

16 17 16715.95 f3Π/G1Σ+

143

A.3 c3Σ+Ω=1 PDS lines

The following is a list of observed c3Σ+Ω=1←X1Σ+ PDS transitions that

were used in the construction of the c3Σ+Ω=1 PEC [7]. We also compare the

observed transition energies to those calculated from this and the Docenko et al.

[5] using LEVEL and assuming J ′ = 1← J ′′ = 0 rotational transitions.

# v′′ v′ Depletion laserλ−1 (cm −1)

Calc λ−1 (cm−1) ∆exp−calc

1 4 1 14198.29 14197.94 0.35

2 4 8 14550.53 14550.88 -0.35

3 4 9 14598.54 14598.72 -0.18

4 5 2 14154.68 14155.25 -0.57

5 5 3 14206.77 14207.00 -0.23

6 5 4 14257.20 14258.04 -0.84

7 5 5 14307.58 14308.35 -0.77

8 5 6 14357.50 14357.97 -0.47

9 5 7 14406.32 14406.94 -0.62

10 5 10 14550.11 14550.51 -0.40

11 5 11 14597.26 14597.42 -0.16

12 5 30 15426.15 15425.81 0.34

13 5 33 15535.91 15535.59 0.32

14 5 34 15570.26 15569.96 0.30

15 5 35 15603.55 15603.24 0.31

16 5 36 15635.75 15635.38 0.37

17 5 37 15666.12 15666.36 -0.24

18 5 38 15695.88 15696.20 -0.32

19 5 39 15725.25 15724.95 0.30

20 5 40 15752.50 15752.64 -0.14

21 5 41 15778.84 15779.28 -0.44

22 5 42 15804.78 15804.94 -0.16

23 5 43 15829.30 15829.74 -0.44

24 5 45 15876.29 15876.89 -0.60

25 5 46 15898.76 15899.08 -0.32

26 5 47 15920.27 15920.36 -0.09

27 5 48 15941.34 15940.88 0.46

144



28 5 49 15960.93 15960.78 0.15

29 5 50 15979.31 15979.93 -0.62

30 5 51 15998.72 15998.06 0.66

31 5 52 16015.63 16015.11 0.52

32 5 53 16031.29 16031.15 0.14

33 5 54 16045.96 16046.06 -0.10

34 5 55 16060.64 16059.84 0.80

35 5 56 16073.29 16072.54 0.75

36 5 57 16084.15 16084.27 -0.12

37 5 58 16095.28 16095.03 0.25

38 5 59 16104.62 16104.73 -0.11

39 5 60 16113.44 16113.29 0.15

40 5 61 16120.45 16120.70 -0.25

41 5 62 16126.95 16127.15 -0.20

42 5 63 16132.68 16132.79 -0.11

43 5 64 16137.36 16137.60 -0.24

44 5 65 16141.53 16141.57 -0.04

45 9 0 13673.90 13672.40 1.50

46 9 1 13727.02 13726.88 0.14

47 9 2 13779.23 13779.75 -0.52

48 9 14 14360.18 14360.54 -0.36

49 9 15 14405.69 14406.21 -0.52

50 9 16 14451.08 14451.66 -0.58

51 9 17 14496.12 14496.89 -0.77

52 9 18 14541.01 14541.89 -0.88

53 9 20 14630.79 14631.11 -0.32

54 9 21 14675.02 14675.25 -0.23

55 9 11 14221.51 14221.92 -0.41

56 13 22 14354.00 14354.64 -0.64

57 13 25 14483.31 14483.38 -0.07

58 15 19 14043.56 14044.37 -0.81

59 15 23 14219.49 14220.10 -0.61

145



60 15 24 14262.49 14263.02 -0.53

61 15 25 14304.72 14305.47 -0.75

62 15 26 14346.58 14347.41 -0.83

63 15 27 14388.28 14388.76 -0.48

64 15 28 14428.97 14429.40 -0.43

65 15 29 14468.85 14469.21 -0.36

66 15 30 14507.47 14508.04 -0.57

67 15 31 14545.45 14545.78 -0.33

68 15 33 14617.75 14617.82 -0.07

69 19 30 14160.29 14161.08 -0.79

70 19 32 14235.08 14235.40 -0.32

71 19 33 14271.04 14270.86 0.18

72 19 34 14305.13 14305.23 -0.10

73 19 35 14338.36 14338.51 -0.15

74 19 36 14370.29 14370.65 -0.36

75 19 37 14400.71 14401.63 -0.92

76 19 38 14431.06 14431.47 -0.41

77 19 39 14459.85 14460.22 -0.37

78 19 40 14487.50 14487.91 -0.41

79 19 41 14513.58 14514.55 -0.97

80 19 42 14539.11 14540.21 -1.10

81 19 43 14564.10 14565.01 -0.91

82 19 44 14587.89 14589.02 -1.13

83 19 45 14611.12 14612.16 -1.04

84 19 46 14633.36 14634.35 -0.99

85 19 47 14655.24 14655.63 -0.39

86 19 48 14675.88 14676.15 -0.27

87 19 49 14695.73 14696.05 -0.32

88 21 47 14486.04 14486.69 -0.65

89 21 48 14506.84 14507.21 -0.37

90 21 49 14526.65 14527.11 -0.46

91 21 50 14547.15 14546.26 0.89

146



92 21 51 14564.31 14564.38 -0.07

93 21 52 14581.30 14581.44 -0.14

94 21 53 14597.05 14597.48 -0.43

95 21 54 14611.77 14612.39 -0.62

96 21 55 14625.66 14626.16 -0.50

97 21 56 14638.93 14638.87 0.06

98 21 57 14650.51 14650.60 -0.09

99 21 59 14670.50 14671.05 -0.55

100 21 60 14679.11 14679.61 -0.50

101 21 61 14686.66 14687.03 -0.37

102 21 62 14692.92 14693.47 -0.55

103 21 63 14698.75 14699.12 -0.37

104 21 64 14703.72 14703.92 -0.20

105 21 65 14707.83 14707.90 -0.07

147

A.4 A1Σ+ PA lines

The following is a list of the observed A1Σ+ PA lines. The Line column

indicates the method used to obtain the line center. MAX = peak maximum,

FIT = Gaussian fit, COM = center of mass of the hyperfine peaks.

# v′ J′ PA λ−1

(cm−1)Line

1 110 0 10973.967 MAX

2 110 1 10974.016 MAX

3 110 2 10974.116 MAX

4 110 4 10974.467 MAX

5 112 1 11024.138 MAX

6 112 2 11024.435 MAX

7 114 0 11066.651 MAX

8 114 1 11066.690 MAX

9 114 2 11066.765 MAX

10 114 4 11067.032 MAX

11 115 1 11087.086 MAX

12 115 2 11087.162 MAX

13 115 4 11087.428 MAX

14 117 0 11117.282 MAX

15 117 1 11117.315 MAX

16 117 2 11117.381 MAX

17 117 4 11117.616 MAX

18 119 0 11141.508 MAX

19 119 1 11141.542 MAX

20 119 2 11141.615 MAX

21 119 4 11141.854 MAX

22 120 1 11148.781 MAX

23 120 2 11148.831 MAX

24 120 4 11149.002 MAX

25 121 1 11155.603 MAX

26 121 2 11155.649 MAX

27 122 0 11161.000 MAX

28 122 1 11161.024 MAX

# v′ J′ PA λ−1

(cm−1)Line

29 122 2 11161.073 MAX

30 122 4 11161.241 MAX

31 123 0 11167.725 FIT

32 123 1 11167.747 FIT

33 123 2 11167.793 FIT

34 123 4 11167.956 FIT

35 124 0 11170.739 FIT

36 124 1 11170.757 FIT

37 124 2 11170.790 FIT

38 124 4 11170.910 COM

39 125 0 11173.273 FIT

40 125 1 11173.285 FIT

41 125 2 11173.314 FIT

42 125 4 11173.403 FIT

43 126 0 11175.184 COM

44 126 1 11175.195 COM

45 126 2 11175.215 COM

46 126 4 11175.290 COM

47 127 0 11176.556 COM

48 127 1 11176.567 COM

49 127 2 11176.582 COM

50 127 4 11176.642 COM

51 128 A 11177.488 COM

52 129 A 11178.043 COM

53 130 A 11178.333 COM

148

A.5 b3ΠΩ=0+ PA lines

The following is a list of the observed b3ΠΩ=0+ PA lines. The Line column


FIT = Gaussian fit, COM = center of mass of the hyperfine peaks.

# v′ J′ PA λ−1

(cm−1)Line

1 78 0 11014.719 MAX

2 78 1 11014.774 MAX

3 78 2 11014.885 MAX

4 78 4 11015.274 MAX

5 80 1 11078.641 MAX

6 80 2 11078.741 MAX

7 80 4 11079.090 MAX

8 81 0 11109.494 MAX

9 81 1 11109.540 MAX

10 81 2 11109.633 MAX

11 81 4 11109.961 MAX

12 82 0 11136.295 MAX

13 82 1 11136.331 MAX

14 82 2 11136.407 MAX

15 82 4 11136.679 MAX

16 83 0 11164.731 MAX

17 83 1 11164.761 COM

18 83 2 11164.822 COM

19 83 4 11165.017 COM

149

A.6 B1Π PA lines

The following is a list of the observed B1Π PA lines. The Line column


FIT = Gaussian fit, COM = center of mass of the hyperfine peaks. We compare

the line positions with those calculated from the Grochola et al. PEC [2].

# v′ J′ PA λ−1 (cm−1) Calc λ−1 (cm−1) ∆exp−calc Line

1 3 2 10598.951 10598.867 0.085 MAX

2 3 3 10599.205 10599.119 0.086 MAX

3 3 4 10599.538 10599.455 0.083 MAX

4 3 5 10599.957 10599.875 0.082 MAX

5 12 1 10995.235 10995.418 -0.184 COM

6 12 2 10995.368 10995.555 -0.187 COM

7 12 3 10995.576 10995.760 -0.184 COM

8 12 4 10995.857 10996.033 -0.176 MAX

9 12 5 10996.202 10996.375 -0.172 MAX

10 13 1 11032.981 11033.185 -0.204 COM

11 13 2 11033.114 11033.319 -0.204 COM

12 13 3 11033.319 11033.519 -0.200 COM

13 13 4 11033.587 11033.786 -0.199 COM

14 13 5 11033.927 11034.120 -0.194 MAX

15 16 2 11139.851 11140.233 -0.381 COM

16 16 3 11140.043 11140.420 -0.377 COM

17 16 4 11140.292 11140.670 -0.378 COM

18 16 5 11140.610 11140.983 -0.373 COM

19 17 1 11173.195 11173.653 -0.458 COM

20 17 2 11173.304 11173.775 -0.471 MAX

21 17 3 11173.488 11173.958 -0.470 COM

22 17 4 11173.703 11174.203 -0.500 COM

23 17 5 11173.920 11174.508 -0.589 COM

24 39 1 11640.311 11639.393 0.917 COM

25 39 2 11640.382 11639.456 0.925 COM

26 39 3 11640.489 11639.551 0.938 COM

27 39 4 11640.633 11639.677 0.956 COM

28 39 5 11640.773 11639.835 0.938 MAX

150

A.7 b3ΠΩ=2 PA lines

The following is a list of the observed b3ΠΩ=2 PA lines. All line positions

were determined by taking the center of mass of the hyperfine structure. We

compare the line positions with those calculated using the new b3ΠΩ=2 PEC.

# v′ J′ PA λ−1 (cm−1) Calc λ−1 (cm−1) ∆exp−calc

1 73 2 11085.227 11085.210 0.017

2 73 3 11085.471 11085.426 0.044

3 73 4 11085.764 11085.714 0.050

4 73 5 11086.125 11086.074 0.051

5 74 2 11130.145 11130.132 0.013

6 74 3 11130.362 11130.344 0.018

7 74 4 11130.652 11130.626 0.026

8 75 2 11173.592 11173.582 0.009

9 75 3 11173.782 11173.790 -0.008

10 75 4 11174.055 11174.066 -0.011

11 79 3 11332.101 11332.057 0.044

12 79 4 11332.355 11332.308 0.047

13 80 2 11367.326 11367.319 0.007

14 80 3 11367.509 11367.502 0.008

15 80 4 11367.756 11367.745 0.011

16 80 5 11368.063 11368.050 0.013

17 81 3 11401.218 11401.194 0.023

18 81 4 11401.455 11401.431 0.024

19 81 5 11401.760 11401.727 0.034

20 82 2 11433.013 11432.981 0.032

21 82 3 11433.181 11433.153 0.029

22 82 4 11433.421 11433.382 0.039

23 85 2 11518.396 11518.397 -0.001

24 85 3 11518.553 11518.551 0.001

25 85 4 11518.760 11518.757 0.002

26 85 5 11519.022 11519.015 0.007

151


27 86 2 11543.329 11543.307 0.022

28 86 3 11543.485 11543.455 0.030

29 86 4 11543.689 11543.653 0.036

30 86 5 11543.932 11543.900 0.032

31 87 2 11566.438 11566.429 0.009

32 87 3 11566.551 11566.571 -0.020

33 87 4 11566.741 11566.761 -0.019

34 87 5 11567.002 11566.997 0.004

35 88 2 11587.793 11587.776 0.016

36 88 3 11587.943 11587.912 0.031

37 88 4 11588.126 11588.093 0.033

38 90 2 11625.225 11625.215 0.010

39 90 3 11625.352 11625.338 0.014

40 90 4 11625.515 11625.501 0.014

41 90 5 11625.719 11625.705 0.014

42 92 2 11655.866 11655.856 0.010

43 92 3 11655.983 11655.965 0.018

44 92 4 11656.120 11656.110 0.010

45 92 5 11656.303 11656.292 0.012

46 93 2 11668.753 11668.752 0.001

47 93 3 11668.867 11668.854 0.013

48 93 4 11668.990 11668.990 0.000

49 93 5 11669.153 11669.160 -0.008

50 94 2 11680.180 11680.170 0.010

51 94 3 11680.273 11680.266 0.007

52 94 4 11680.407 11680.393 0.013

53 94 5 11680.563 11680.552 0.011

54 96 2 11698.849 11698.862 -0.012

55 96 3 11698.946 11698.943 0.003

56 96 4 11699.050 11699.052 -0.002

57 98 2 11712.215 11712.205 0.011

58 98 3 11712.289 11712.273 0.016

59 98 4 11712.379 11712.363 0.016

152


60 98 5 11712.489 11712.476 0.013

61 100 2 11721.470 11721.452 0.018

62 100 3 11721.534 11721.507 0.026

63 100 4 11721.614 11721.581 0.033

64 100 5 11721.703 11721.673 0.031

153

A.8 c3Σ+Ω=1 PA lines

The following is a list of the observed c3Σ+Ω=1 PA lines. All line positions

were determined by taking the center of mass of the hyperfine structure. We

compare the line positions with those calculated using the Grochola et al. PEC

[7].


1 61 1 11700.513 11700.535 -0.021

2 61 2 11700.570 11700.584 -0.015

3 61 3 11700.651 11700.658 -0.007

4 61 4 11700.761 11700.757 0.004

5 61 5 11700.896 11700.880 0.016

6 62 1 11707.037 11706.982 0.056

7 62 2 11707.089 11707.028 0.061

8 62 3 11707.165 11707.097 0.069

9 62 4 11707.267 11707.188 0.079

10 63 1 11712.669 11712.623 0.046

11 63 2 11712.713 11712.665 0.048

12 63 3 11712.787 11712.729 0.058

13 63 4 11712.878 11712.814 0.065

14 63 5 11712.993 11712.920 0.073

15 64 1 11717.380 11717.429 -0.049

16 64 2 11717.424 11717.468 -0.044

17 64 3 11717.486 11717.525 -0.040

18 64 4 11717.567 11717.602 -0.035

19 65 1 11721.391 11721.402 -0.011

20 65 2 11721.443 11721.437 0.006

21 65 3 11721.508 11721.489 0.019

22 65 4 11721.585 11721.558 0.027

23 65 5 11721.682 11721.645 0.037

24 66 1 11724.556 11724.633 -0.077

25 66 2 11724.598 11724.664 -0.066

26 66 3 11724.649 11724.711 -0.062

27 66 4 11724.710 11724.772 -0.062

154


28 66 5 11724.800 11724.849 -0.049

29 67 1 11727.108 11727.185 -0.077

30 67 2 11727.135 11727.213 -0.077

31 67 3 11727.183 11727.253 -0.070

32 67 4 11727.237 11727.307 -0.070

33 67 5 11727.309 11727.375 -0.066

34 68 1 11729.164 11729.140 0.024

35 68 3 11729.234 11729.198 0.036

36 68 4 11729.290 11729.244 0.045

37 69 1 11730.520 11730.538 -0.017

38 69 2 11730.542 11730.557 -0.014

39 69 3 11730.581 11730.585 -0.005

40 69 4 11730.629 11730.623 0.006

41 69 5 11730.685 11730.671 0.014

42 70 A 11731.449 11731.477 -0.028

43 71 A 11732.072 11732.086 -0.014

44 72 A 11732.339 11732.398 -0.060

155

A.9 c3Σ+Ω=0− PA lines

The following is a list of the observed c3Σ+Ω=0− PA lines. All line positions

were determined by taking the peak maximum.

# v′ J′ PA λ−1 (cm−1)

1 32 1 11130.6673

2 32 2 11130.78105

3 32 3 11130.94916

4 32 5 11131.45438

5 33 1 11164.20897

6 33 2 11164.31948

7 33 3 11164.48753

8 33 5 11164.98531

9 44 1 11461.43743

10 44 2 11461.52279

11 44 3 11461.65095

12 44 5 11462.03868

156

A.10 b3ΠΩ=2 RKR PEC

R (A) E (cm−1) R (A) E (cm−1)

2.5731 13928.70265 2.703637095 11294.40927

2.6081 13120.68044 2.705988643 11255.59954

2.6431 12391.53955 2.708453613 11215.20444

2.678172937 11732.2776 2.711032008 11173.26211

2.678176725 11732.21 2.713723993 11129.80939

2.678189201 11731.98735 2.716529655 11084.88564

2.67821725 11731.48683 2.719449061 11038.5317

2.678267825 11730.58445 2.722482325 10990.78881

2.678346397 11729.18281 2.725629644 10941.69798

2.678457274 11727.20542 2.72889133 10891.2995

2.678604547 11724.57994 2.732267833 10839.63266

2.678792712 11721.22712 2.735759751 10786.7356

2.679026921 11717.05645 2.739367847 10732.64512

2.679312964 11711.96663 2.743093055 10677.3967

2.679657072 11705.84926 2.746936493 10621.0244

2.680065869 11698.58993 2.750899466 10563.56092

2.68054918 11690.01861 2.754983476 10505.03754

2.681119845 11679.91374 2.759190229 10445.48422

2.681766412 11668.48529 2.763521642 10384.92961

2.68249732 11655.59216 2.767979853 10323.40107

2.683320835 11641.09855 2.772567225 10260.92475

2.684241044 11624.94454 2.777286364 10197.52564

2.685261134 11607.08807 2.782140122 10133.22759

2.686383865 11587.49659 2.787131614 10068.05337

2.687611649 11566.14572 2.792264231 10002.02475

2.688946576 11543.01879 2.797541653 9935.162484

2.690390418 11518.10674 2.802967869 9867.486435

2.691944629 11491.40813 2.808547193 9799.015572

2.693610313 11462.92947 2.814284286 9729.768041

2.695388184 11432.68592 2.820184179 9659.761213

2.697278536 11400.7016 2.826252297 9589.01173

2.699281789 11367.00011 2.83249469 9517.535553

2.701400576 11331.56991 2.838916806 9445.348006

157

R (A) E (cm−1) R (A) E (cm−1)

2.845523472 9372.463818 3.217016663 6645.606511

2.852318919 9298.897161 3.237141854 6554.603695

2.859306836 9224.661677 3.258407757 6463.156223

2.86649045 9149.770514 3.280978557 6371.266745

2.873872643 9074.236341 3.305071626 6278.937991

2.881456095 8998.071371 3.330981556 6186.172805

2.889243448 8921.287369 3.359118045 6092.974197

2.897237497 8843.895667 3.390069383 5999.34538

2.905441389 8765.907164 3.42471789 5905.289824

2.913858846 8687.332333 3.46447515 5810.811299

2.922494381 8608.181219 3.511847369 5715.913927

2.931353522 8528.463441 3.572209653 5620.602235

2.940443034 8448.188191 3.602821039 5582.362648

2.949771124 8367.364232 3.620466932 5563.218398

2.959347629 8285.9999 3.640392988 5544.057902

2.969184192 8204.103107 3.663666363 5524.881204

2.979294393 8121.68134 3.692615198 5505.688352

2.989693874 8038.741669 3.735237089 5486.47939

3.000400409 7955.290754 3.79448421 5476.868883

3.01143396 7871.334852 3.854909217 5486.47939

3.022816697 7786.879829 3.899965706 5505.688352

3.034572994 7701.931172 3.931448401 5524.881204

3.046729419 7616.494008 3.957348216 5544.057902

3.059314721 7530.573113 3.979986972 5563.218398

3.072359848 7444.17294 4.000425847 5582.362648

3.085898014 7357.297634 4.03684176 5620.602235

3.099964862 7269.951057 4.112761561 5715.913927

3.114598775 7182.136814 4.176780892 5810.811299

3.129841393 7093.858274 4.233886802 5905.289824

3.145738445 7005.118606 4.286310445 5999.34538

3.162340992 6915.920799 4.335276589 6092.974197

3.179707261 6826.267702 4.381548592 6186.172805

3.197905288 6736.162053 4.425646885 6278.937991

158

R (A) E (cm−1) R (A) E (cm−1)

4.467950724 6371.266745 5.585630142 9149.770514

4.508750609 6463.156223 5.619110754 9224.661677

4.548277351 6554.603695 5.652870676 9298.897161

4.586719189 6645.606511 5.68693625 9372.463818

4.624232421 6736.162053 5.721334017 9445.348006

4.660948357 6826.267702 5.756090819 9517.535553

4.696978118 6915.920799 5.791233946 9589.01173

4.732416133 7005.118606 5.826791325 9659.761213

4.767342824 7093.858274 5.862791761 9729.768041

4.801826774 7182.136814 5.899266407 9799.015572

4.835926547 7269.951057 5.936248792 9867.486435

4.869692257 7357.297634 5.973775071 9935.162484

4.903166953 7444.17294 6.011884296 10002.02475

4.936387849 7530.573113 6.050618715 10068.05337

4.969387424 7616.494008 6.090024106 10133.22759

5.002194403 7701.931172 6.130150144 10197.52564

5.034834621 7786.879829 6.171050813 10260.92475

5.067331784 7871.334852 6.212784883 10323.40107

5.099708122 7955.290754 6.255416433 10384.92961

5.131984944 8038.741669 6.299015462 10445.48422

5.164183096 8121.68134 6.343658583 10505.03754

5.196323329 8204.103107 6.389429807 10563.56092

5.228426583 8285.9999 6.436421444 10621.0244

5.260514192 8367.364232 6.48473512 10677.3967

5.292608024 8448.188191 6.534482929 10732.64512

5.324730552 8528.463441 6.585788725 10786.7356

5.356904884 8608.181219 6.638789566 10839.63266

5.389154744 8687.332333 6.693637303 10891.2995

5.42150442 8765.907164 6.75050032 10941.69798

5.453978697 8843.895667 6.809565376 10990.78881

5.486602771 8921.287369 6.871039472 11038.5317

5.519402163 8998.071371 6.935151531 11084.88564

5.552402639 9074.236341 7.002153444 11129.80939

159

R (A) E (cm−1)

7.072319477 11173.26211

7.145943132 11215.20444

7.223340494 11255.59954

7.304949408 11294.40927

7.391708049 11331.56991

7.484381918 11367.00011

7.582494752 11400.7016

7.686567373 11432.68592

7.797625197 11462.92947

7.916592565 11491.40813

8.044416917 11518.10674

8.182167327 11543.01879

8.331089454 11566.14572

8.492656025 11587.49659

8.668626635 11607.08807

8.861121759 11624.94454

9.072709438 11641.09855

9.306450193 11655.59216

9.565238603 11668.48529

9.84557624 11679.91374

10.15859102 11690.01861

10.53424279 11698.58993

10.95170256 11705.84926

11.42203644 11711.96663

11.95879941 11717.05645

12.578364 11721.22712

13.30496752 11724.57994

14.17851763 11727.20542

15.26995466 11729.18281

16.71475837 11730.58445

18.79034897 11731.48683

22.10298032 11731.98735

Production of Ultracold, Absolute Vibrational Ground State ...

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