Measurement of Cold Molecular Collisions between
co-trapped OH and Rb
byJohn M. Gray
B.S., Georgia Institute of Technology, 2014

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2022

Committee Members:
Heather Lewandowski, ChairJohn Bohn
Eric Cornell
Margaret Murnane
Robert Parson

ii
Gray, John M. (Ph.D., Physics)
Measurement of Cold Molecular Collisions between co-trapped OH and Rb
Thesis directed by Prof. Heather Lewandowski

Cold molecules are an important tool in the study of precision measurement, chemical physics,
and quantum collisions. Polar radical molecules, like the hydroxyl radical OH, are particularlyinteresting due to their importance as short lived intermediates in extreme chemical environments,
such as the interstellar medium, the atmosphere, and combustion. In this thesis, we describe
the OHRbIT Experiment (OH and Rb Interacting in Traps). With this experiment, we seek
to measure the cross sections of collisions between cold OH molecules and ultracold Rb atoms
in a co-trapped environment. Rubidium atoms are trapped in a quadrapole magnetic field at atemperature of T ∼ 500µK. The atoms are transported through the vacuum chamber to overlap
with an electrostatic quadrapole trap that is used to trap OH molecules. The OH molecules are
slowed using a Stark Decelerator and trapped in the electric quadrapole trap with a temperature of∼ 50mK. The cross sections of the interaction are extracted from measurements of the evolution of
the number and energy distribution of the trapped OH molecules. We describe the 1+1’ resonantlyenhanced multiphoton ionization (REMPI) detection scheme that allows us to measure the energy
distribution of the OH molecules. We also discuss the production of vacuum ultraviolet (VUV)
light at 118nm that is used in the REMPI detection.

Dedication

To my parents.

iv

Acknowledgements

First, I would like to thank my advisor, Heather Lewandowski, for the advise, expertise,and support that she has provided throughout my PhD. I would also like to thank the rest of the
Lewandowski group. Travis Briles and Yomay Shyur were both great mentors who helped me learn
a lot about experimental physics. Jason Bossert and Tommy Garcia both worked directly with meon the OH and Rb apparatuses, and were both great scientists and a pleasure to work with. The
post-docs on the ion experiment, Philip Schmidt and Catherine Catani, were always there to help.
And to the rest of the Lewandowski Group who has helped me during my time here; Maya, Cam,Greg, Kyle, Olivia, Chase, Ben, Andres, and Ian.

v

Contents

Chapter
1 Introduction

1

1.1

Why Cold Molecules? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Cold Molecular Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

Cold Molecule Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.4

The OHRbIT Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 The OH Molecule: Structure and Production

6

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.2

Molecular Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.2.1

OH Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.3

Hund’s Coupling Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.4

OH Rotational Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4.1

2.5

2.6

Lambda Doubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

OH in External Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5.1

OH in Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5.2

OH in Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5.3

OH in Combined Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

OH Creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

vi
3 Stark Deceleration

26

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2

Stark Deceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3

Operation of a Stark Decelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4

Alternate Deceleration Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.5

Deceleration Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Detection of OH

40

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2

LIF of OH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3

LIF Detection Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4

REMPI of OH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.5

Future Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5 Produciton of 118 nm light

52

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.2

Theory of Non-Resonant Third Harmonic Generation . . . . . . . . . . . . . . . . . . 53

5.3

Experimental Setup for Third-Harmonic Generation . . . . . . . . . . . . . . . . . . 57

5.4

Phase Matching and Conversion Efficiency Limitations . . . . . . . . . . . . . . . . . 60

5.5

Absorption Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.6

Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.7

Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6 Electrostatic Trapping of OH

75

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.2

REMPI of OH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.3

Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.4

Molecular Dynamics Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

vii
6.5

6.6

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.5.1

Trap loading dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.5.2

Ballistic Expansion of OH cloud . . . . . . . . . . . . . . . . . . . . . . . . . 88

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7 Measuring Cold Collisions Between OH and Rb

91

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7.2

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7.3

Experimental Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.4

Collision Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.5

Data Taking Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.6

Co-Trap Alignment and Characterization . . . . . . . . . . . . . . . . . . . . . . . . 114

8 Conclusion

120

8.1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.2

Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Bibliography

124

viii

FiguresFigure2.1

OH Electronic Energy Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92.2

Hunds Case (a) Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3

Hunds Case (b) Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4

OH Rotational Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5

Stark Shifts of OH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6

Zeeman Shifts of OH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.7

Supersonic Expansion Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.8

Velocity vs mass of carrier gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.9

OH Beam Creation Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1

Single Pin Pair Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2

Picture of Decelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3

Stark Potential Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4

Potential Energy Path of Deceleration . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.5

Separatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.6

OH Electronic Energy Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.7

Deceleration to various velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.8

Comparison between S=1 and Focusing deceleration schemes for slowing to 33.7 m/s. 39

4.1

PMT Quantum Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

ix
4.2

OH REMPI Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3

OH REMPI Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.4

VUV Ionization spectrum from R21 branch . . . . . . . . . . . . . . . . . . . . . . . 49

4.5

VUV Ionization spectrum from Q1 branch . . . . . . . . . . . . . . . . . . . . . . . . 50

5.1

Diagram showing the parameters of the gas cell for generating 118 nm light (blue).
The 355 nm beam (red) is focused into the gas cell (gray box) by a lens with focal
length f . The confocal parameter b and beam waist w0 are noted. . . . . . . . . . . 55

5.2

Schematic of the experimental setup. A lens focuses the 355 nm light into the gas
cell where 118 nm light is created. A window placed above the focus of the 355
nm beam allows perpendicular optical access to the tripling region. The MgF2 lensfocuses the 118 nm light and defocuses the 355 nm in the detection region. 118
nm flux is detected via ionization of deuterated ammonia (ND3 ). A low flow rateof ND3 fills the detection chamber to a partial pressure of 9 × 10−8 Torr. ND+
3ions are accelerated through a time-of-flight mass spectrometer and detected by amicro-channel plate (MCP).

5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Phase-matching curves for different pressures of xenon (labeled above). 118nm flux
is measured while Ar is added to a fixed pressure of Xe and plotted as a function of
total pressure.

5.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Plot of optimum Ar:Xe ratio vs. Xe pressure. The solid line is the output of a fitof Eq. 5.8 to the data with b = 2.3 ± 0.3 cm as the only fit parameter. The dashedline represents the asymptotic limit of 11.48 predicted from the model.

5.5. . . . . . . 62

Plot of 118 nm flux and the fluorescence signal vs total pressure for PXe = 50 Torr.
The fluorescence signal is correlated with the presence of 118 nm light. The inset
show a picture of the fluorescence streak through a conflat window. . . . . . . . . . . 63

x
5.6

Comparison of the 118 nm signal and fluorescence streak signal at different xenonpressures for a phase-matched mixture with Ar (Ar:Xe Ratio = 11.3) and Kr (Kr:Xe
Ratio = 4.6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.7

Spectra of the fluorescence streak created in a phase-matched (a) Xe:Ar mixture(Ar:Xe Ratio = 11.3) and (b) Xe:Kr mixture (Kr:Xe Ratio = 4.6). Lines corresponding to Ar I, Kr I, and Xe I are assigned according to criteria described in the
text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.8

Comparison of simulated and experimental results for Xe/Ar. Experimental 118
nm flux and fluorescence signal are plotted as green circles and squares respectively.
Simulated 118 nm flux and fluorescence signal are plotted as red dashes and dots
respectively. The simulated fluorescence curve is independently scaled by eye.

5.9

. . . 71

Comparison of simulated and experimental results for Xe/Kr. Experimental 118
nm flux and fluorescence signal are plotted as green circles and squares respectively.
Simulated 118 nm flux and fluorescence signal are plotted as red dashes and dots
respectively. The simulated fluorescence curve is independently scaled by eye.

6.1

. . . 73

Energy level diagram of OH showing the relevant levels for the 1 + 1’ REMPI
detection scheme. The lowest two levels are the Λ-doublet states of X 2 Π3/2 . The
first photon is resonant with the A2 Σ+ , ν = 1 ← X 2 Π3/2 , ν = 0 transition and isdenoted by R21 (1.5). The second photon transfers the molecule into the A3 Π, 3d
Rydberg state, from which it autoionizes. J is the total angular momentum, and pis the parity. Vertical spacing between energy levels is not to scale. . . . . . . . . . . 78

6.2

Experimental setup showing the last two rod pairs of the Stark decelerator, four
trap electrodes, detection lasers, and MCP detector. The red rectangular prism
represents the 281 nm laser beam, while the blue cone represents the 118 nm light.
The molecular beam propagates along the z-axis. (Not to scale) . . . . . . . . . . . . 80

xi
6.3

Diagram of trap loading process for both an optimized and a non-ideal sequence. The
location of the final decelerator electrode and the four trapping electrodes are shownat the top. The longitudinal potential energy curves seen by the OH molecules
are shown in the panels for three different configurations of voltage applied. The“Trap on” configuration has the following voltages applied in order from left to right(decelerator, left trap endcap, left trap ring, right trap ring, right trap endcap) = (0,-10, +10, +10, -10) kV. “Trap Half on” has the following voltage configuration (0,0, +10, 0, -10) kV, and “Trap off” has all electrodes grounded. The blue (top) curve
represents the potentials seen by the central molecule in the optimized sequence andthe orange (bottom) curve represents the non-ideal sequence. The switches between
configurations are shown by vertical lines. . . . . . . . . . . . . . . . . . . . . . . . . 81

6.4

Measured central column density during the trap loading process with corresponding
molecular dynamics simulations. The measured central density is shown with error
bars connected by lines to guide the eye and the simulations results are shown withdashed lines. (a) Results from the optimized loading scheme. The final trap configuration is turned on at 4.34 ms. (b) Results from the non-ideal loading scheme. The
final trap configuration is turned on at 4.30 ms. The plots at the right correspondto the simulated longitudinal phase-space distribution after 15 ms of trapping. . . . 84

6.5

Width of the trapped OH cloud as a function of time for the (a) optimized loadingand (b) non-ideal loading scheme. The squares represent the experimental measurements, while the solid lines represent simulation results. For both cases, thesimulations reproduce the modulation in the width of the cloud after it is trapped.
The optimized loading scheme results in fewer oscillations in the cloud width. The
statistical uncertainty is on the order of the point size or smaller. . . . . . . . . . . . 86

xii
6.6

Measured density profiles of the OH molecules in the trap using the non-ideal loading
sequence. These data were recorded directly after the trap was loaded (4.30 ms)
and at 450 µs later (4.75 ms). The lines are Pearson IV distribution fits to theexperimental data. The widths of the experimental density profiles are determinedby extracting the FWHM of a Pearson IV distribution fit.

6.7

. . . . . . . . . . . . . . 87

Width of the cloud as it expands after the trap is turned off. The molecules weretrapped using the optimized loading scheme and held for 15 ms before release inboth the experiment and simulation. The initial size of the cloud in the trap differs
between the experiment and simulations. This is likely due to the lack of knowledgeof the initial distribution of molecules leaving the decelerator.

. . . . . . . . . . . . 89

7.1

OH-Rb Interaction Potential Energy Surfaces . . . . . . . . . . . . . . . . . . . . . . 93

7.2

Quantum States in the OH-Rb interaction . . . . . . . . . . . . . . . . . . . . . . . . 95

7.3

Theoretical Cross sEctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.4

Zeeman Shifts of OH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.5

Temperature vs. Number of Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.6

OHRbIT Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.7

Collisions Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.8

Collisions Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.9

OH Width Over Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.10 Rb atoms z-axis alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.11 118nm Laser z-axis alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.12 Overlap between Rb atoms and OH molecules along the x-axis . . . . . . . . . . . . 118
7.13 Rb atoms x-axis alignment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.14 Rb atoms y-axis alignment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Chapter 1

Introduction

1.1

Why Cold Molecules?
Cold and ultracold molecules have been an area of intense research interest in the past few

decades. As an extension of the study of cold atoms, cold molecules offer many of the samechallenges, tools, and goals. However, owing to the complex quantum energy level structure due
to rotational and vibrational degrees of freedom, cold molecules offer a rich environment for the
study of quantum systems and their applications. The fundamental goal of studying cold molecules
is the understanding and manipulation of individual quantum states to probe complex physics
in a controlled manner. Cold molecules have been used in precision measurements for tests offundamental physics and to probe the variation of fundamental constants, such as the fine structure
constant[1], the electron electric dipole moment[2, 3, 4], and CP-violation[5]. Cold molecules arealso useful for studying the physics of chemical reactions, and allow for control of the quantum
state of the reactants, collision energy, and even the orientation of reactants.[6, 7]
The study of cold molecules is also important for studying the natural chemical processes
found in many extreme environments. The isolated environment in which cold molecules are often
produced is an ideal environment for studying radicals and ions which can be highly chemicallyreactive. A plethora of chemical species have been found to exist in the interstellar medium, which
has a temperature of ∼ 4K.[8] Laboratory experiments with cold molecules can be used to study
the spectroscopic and chemical properties of these species to better understand the chemistry in
the interstellar medium, predict molecules that may be found, and identify molecules in data from2
interstellar spectra. Studying cold molecules in the lab can also help understand the chemistry of
combustion and the atmosphere by isolating short-lived intermediate radicals and ions that play animportant role[9, 10]. The OH radical, which is the subject of this thesis, plays an important role inthe chemistry of the atmosphere,[11] combustion,[12, 13] and even the interstellar medium.[14, 15]Laboratory experiments involving spectroscopy of cold OH radicals have been used alongside datafrom interstellar OH megamasers to constrain the time variation of fundamental constants.[1]1.2

Cold Molecular Collisions
The study of collisions involving cold neutral molecules is also an area of particular researchinterest. In the cold-collision regime, the cross sections and products of interactions become largely
dependant on the details of the interaction potential. Resonances have been observed due to thequantization of internal states of the constituents and the partial waves of the interaction energy.
[16, 17, 18]The rapid advancement of experimental techniques has begun to catch up with theoreticaltechniques for approximating interaction potentials. A major contributor to the complexity of cold
molecular collisions is the large anisotropy that can dominate long-range interactions. Experimentsthat push the energy of collisions to colder regimes where theoretical models may differ can help
refine these models. [19].
In order to study an interaction, an experimenter needs a method to bring the species together
and a method to detect the products of the interaction. Experiments that study interactions
between cold molecules usually bring collision partners together in three main ways. First is tomerge or cross two molecular beams. Merged beams can allow for high resolution tunability of
the collision energy by changing the beam crossing angle or by directly controlling the velocityof the species with a decelerator.[20, 21, 22] Higher resolution can also be gained by exploiting
the correlation between velocity at different positions in a moving beam. Second is to have amolecular beam impinging on a trapped sample. For both types beam experiments, the interaction
time is limited by the length of the interaction region and the velocity of the beams. Beam-trap
experiments do, however, allow many pulses of the beam to impinge on the trapped sample.

3
The third method is to trap both collision partners in the same region of space. This method
can increase the sensitivity to more rare collision events by increasing the interaction time. The
co-trapped environment can also allow inelastic and elastic collisions to be distinguished more
directly. Elastic collisions will lead to a thermalization of the collision partners, and this can beseen by probing the energy distribuition. Inelastic collisions, on the other hand, will cause a change
in the quantum state. If the molecule is transferred to a non-trapped state, its loss from the trap
can be detected. Of particular interest to this experiment is the case of a trapped molecule colliding
with a trapped atom. This type of experiment has recently been achieved for Li+O2 [23], N+NH[24],Rb+ND3 [25, 26], and Rb+CaF[27]. In this experiment, we study collisions of cotrapped Rb+OH.

1.3

Cold Molecule Production
The production of cold molecules can be accomplished by a wide range of techniques. These

techniques are typically categorized into direct and indirect methods. Direct methods take a preexisting molecule and cool it down from some high initial temperature. The most common techniqueis known as supersonic expansion. Using a supersonic expansion, any molecule that can be vaporized can be entrained in a beam of rare gas can be cooled to the Kelvin regime. Other direct coolingmethods involve removing heat energy from the molecule through interactions with light, fields,
or atoms. Recent advancements in this field have involved direct laser cooling of molecules, evenallowing molecules to be trapped in a magneto-optical trap (MOT). [28, 29, 30] Direct laser coolingof molecules, however, requires particular criteria of the energy level structure. The branchingratio for spontaneous emission into the ground cooling state must be large enough and the numberof transitions with a non-trivial branching ratio must be low enough to minimize the number of
repumping laser wavelengths needed. These criteria are not met in most molecules. The processof directly cooling one species with another is known as sympathetic cooling. Sympathetic coolinghas been accomplished by using ions to cool ions[31], ultra cold atoms to cool atoms,[32] and ultra
cold atoms to cool ions.[33] The current frontier in sympathetic cooling is the prospect of using
ultra cold atoms to cool cold molecules, moving these molecular species from the millikelvin to4
microkelvin regime.
Indirect cooling methods bring together ultracold atoms to form cold or ultracold molecules.
The molecules are created from ultracold atoms by either photo-association [34] or magnetoassociation[35]. Molecules formed by indirect methods can be have high densities and ultracold
temperatures (100’s of nK). However, the number of species that can be created via these methods is severely limited. Currently, the molecules are limited to diatomic molecules that can beformed from atoms that can be easily laser cooled and trapped, such as alkali and alkaline-earth
elements.[36, 37, 38, 39] Furthermore, the accessible quantum states are limited by the potential
energy surfaces, which must allow free to bound transitions for association and bound to boundlevels for stimulated emission in an energy regime that is accessible by high precision lasers. Indirect
cold molecule production techniques have led to many high impact studies of ultracold physics andchemistry.[40, 41, 42, 43] However, in order to produce cold molecules that are traditionally chemically important, direct cooling is necessary and most molecules are limited to the 1K temperature
regime from supersonic expansion.

1.4

The OHRbIT Experiment
In this thesis, we describe the OHRbIT Experiment (OH and Rb Interacting in Traps). With

this experiment, we seek to measure the cross sections of collisions between cold OH molecules andultracold Rb atoms in a co-trapped environment. Rubidium atoms are trapped in a quadrapole
magnetic field at a temperature of T ∼ 500µK. The magnetic coils that produce the trapping fields
sit outside the vacuum chamber on movable tracks. The atoms are transported through the vacuum
chamber to overlap with an electrostatic quadrapole trap that is used to trap OH molecules. Afterthe atoms are in place, the cold OH molecules are created in a supersonic expansion. The OH
molecules are slowed using a Stark Decelerator and trapped in the electric quadrapole trap with atemperature of ∼ 50mK. The cross sections of the interaction are extracted from measurements ofthe evolution of the number and energy distribution of the trapped OH molecules.
The contents of this thesis are organized as follows. Chapter 2 describes the quantum me-5chanical properties of the OH radical and how it behaves in external fields. We also detail theprocess we use to produce a molecular beam of cold OH radicals. Chapter 3 describes how we
exploit the behaviour of OH in an electric field to slow our beam of cold OH radicals in a process
known as Stark Deceleration. In Chapter 4, we discuss how OH can be detected in a cold molecularbeam experiment. We describe a 1+1’ REMPI detection scheme for OH radicals and the advantages of using REMPI over LIF. In Chapter 5, we describe the production of the 118 nm VUV lightrequired for the REMPI scheme. In Chapter 6, we describe how we trap OH in an electrostatic
trap. We also discuss how the new REMPI scheme can be used to measure the density distributionof the trapped OH to elucidate the dynamics of the cloud. In Chapter 6, we discuss the combination of our OH trapping experiment with a Rb trapping experiment to study collisions between Rb
and OH. We discuss the theoretic predictions and the expectations of the experiment. We present
measurements that demonstrate collisions between OH and Rb and estimate cross sections fromthese measurements. We also discuss future work to extract the inelastic and elastic cross sections
using Monte Carlo simulations.

Chapter 2

The OH Molecule: Structure and Production

2.1

Introduction
In this thesis, the main molecule of interest is the neutral OH radical. In order to understandthe methods used to manipulate and detect OH, some understanding of the molecular structureis necessary. In this chapter, I will discuss the molecular structure of OH and how the relevantmolecular states react to electric and magnetic fields. I will also describe the process of making ourmolecular beam of nearly state-pure OH radicals.

2.2

Molecular Structure
The description of molecular energy level structure is much more complicated than atomic

structure. The structure of atoms is well described by the behavior of the electrons orbiting anapproximately fixed nucleus. Whereas for molecules, the relative motion of different nuclei needs
to be accounted for as extra degrees of freedom for energy to be stored in. The rich spectroscopic
structure of molecules , while complicated to describe, is one of the factors that make the study ofmolecular spectroscopy interesting and useful.
In order to simplify the description of molecular wavefunctions, we can apply the BornOppenheimer approximation. Since the mass of the nuclei is many orders of magnitude larger thanthe mass of the electrons, the motion of nuclei is much slower compared to the orbital motion theelectrons. The Born-Oppenheimer approximation assumes that the motion of electrons happens
instantaneously relative to nuclear motion. The Hamiltonian describing the molecule, Hmol , can

7
be separated into the parts that describe the electron orbit of the nuclei, Helec , and the part that
describes the relative motion of the nuclei, Hnuc . The nuclear motion can further be separated into
rotation about the center of mass and vibrations that change the equilibrium bond lengths
Hmol = Helec + Hvib + Hrot ,(2.1)

The total molecular wave function, Ψmol , is then separable into the product of the eigenfunctionsof the separate electronic, vibrational, and rotational hamiltonians|Ψmol ⟩ = |Ψelec ⟩ |Ψvib ⟩ |Ψrot ⟩ .

(2.2)

The energy of the molecule is then the sum of the separate electronic, vibraitional, and rotationalenergies
Emol = Eelec + Evib + Erot ,(2.3)

The energy associated with each of these separate degrees of freedom also occurs on different
energy scales. The energy difference between different electronic levels is typically on the orderof eV’s, or 10,000’s of cm−1 . Exciting a molecule to the next electronic level typically requires alaser with wavelengths in the visible or UV part of the spectrum. Vibrational energy is the next
largest scale, with vibrational energy levels typically being separated on the order of 1000 cm−1 .
Excitation between vibrational levels is typically achieved with an infrared laser. Rotational energy
level splittings are the lowest energy, being typically on the order of 10’s-100’s of cm−1 . The energy
scale of rotations cooresponds to the microwave region of the spectrum.

2.2.1

OH Electronic Structure

In molecular orbital theory, molecular orbitals are described as linear combinations of atomic
orbitals. The ground electronic states of the atomic constituents of OH are• H: (1s)2S• O: (1s)2 (2s)2 (2p)43P

8These combine to form the molecular orbitals• OH: (1sσ)2 (2sσ)2 (2pσ)2 (2pπ)3This combination of orbitals represents an open shell radical with an unpaired electron in
the one of the (2pπ) orbitals. This combination of orbitals gives four possible term symbols:
2 Σ− ,4 Σ− ,4 Π,2 Π. The potential energy curves for OH are shown in Fig.2.1. The only bound statecorresponding to the this configuration is the 2 Π which is the electronic ground state.
This electronic state can be further divided by fine structure splitting. The value of the
projection of the orbital angular momentum onto the internuclear axis, Λ is given by the letter of
the term symbol, Π → |Λ| = 1. The superscript of the term symbol is 2|Σ| + 1 = 1 → |Σ| = 1/2.
The total angular momentum can take on the values J = |Λ| ± |Σ| = 1/2 and 3/2.
There are a total of four bound states, with the first excited state A2 Σ+ resulting fromthe promotion of a (2pσ) electron to occupy the vacant (2pπ) orbital. This excited state lies
approximately 4 eV above the ground state and is accessible by excitation by UV lasers. Resonant
excitation to the A2 Σ+ state is typically used in detection of OH.

2.3

Hund’s Coupling Cases
In atoms, the electronic spin angular momentum S couples with the electronic orbital angularmomemtum L to provide an energy splitting between elecronic energy levels with the same L but
different S. This spin orbit coupling arises from the considering the frame of reference of theelectron, which sees a nucleus orbiting it: a moving charge which creates a magnetic field thatinteracts with electrons magnetic dipole moment. Likewise, in molecules the relative motion ofelectrons and nuclei leads to coupling between different spins and angular momenta. The various
angular momentum vectors that lead to coupling interactions are:[45]• L, Orbital electronic angular momentum• S, Electronic spin angular momentum

9

Figure 2.1: Energy curves showing the possible states of the OH molecule. From[44] Boundand unbound molecular states are derived from combinations of atomic states. The energy of amolecular state at large internuclear distance approaches the combined atomic energy.

10• J, Total angular momentum• N, The total angular momentum excluding electron spin, N = J -S
• R, Rotational angular momentum, R = J - L -S
Clearly, the existence of so many competing couplings would be overwhelming to attempt afull description. However, by considering the relative strengths of the interactions, one can constructa simplified picture that can roughly diagonalize the spin orbit Hamiltonian with nicely definedquantum numbers. Five such cases have been described and are known as Hund’s cases. Labeled(a) through (e), Hund’s cases (a) and (b) are the most common and relevant to the OH molecule.
The first and most common Hund’s case, case (a) depicted in Fig. 2.2, is defined by two
dominant interactions. First, the orbital angular momentum L is strongly coupled to the internuclear axis by electrostatic forces. Second, a strong spin-orbit interaction couples the electronic spin
angular momentum S to L, effectively coupling S to the internuclear axis. The projection of S and
L onto the internuclear axis, Σ and Λ respectively, sum to give Ω = Σ + Λ which is a good quantum
number. The total angular momentum is then defined by J = Ω + R. The ground electronic stateof OH (2pπ) is best described by Hund’s case (a), and the good quantum numbers are Σ, Λ, Ω,
and J.
In Hund’s case (b), the spin orbit coupling is much weaker. Depicted in Fig. 2.3 L is coupled
to the internuclear axis, but S is not. The good quantum numbers in this case are Λ, N, and J.
The first excited state of OH, A2 Σ+ , is best described by Hund’s case (b).

2.4

OH Rotational Structure
The rotation of diatomic molecules is often treated with the rigid rotor model, where the two

atoms with masses m1 and m2 are assumed to have a fixed distance, r. From classical mechanics,
the energy associated with rotations in this case is1
Er = Iω 2
2

(2.4)

11

Figure 2.2: Diagram of vector coupling for Hund’s Case (a). From [46]

12Figure 2.3: Diagram of vector coupling for Hund’s Case (b). From [46]

13
where ω is the frequency of rotation. I is the rotational moment of inertia given by
I = m1 r12 + m2 r22 = µr2(2.5)

where r1 and r2 are the distances from the center of mass to the respective nuclei and µ is the
reduced mass
µ=

m1 m2
m1 + m2(2.6)

The angular momentum of the system, given by L = Iω, can be substituted into Equation (2.4) to
give
Er =

L2
2I

(2.7)

The quantum Hamiltonian is defined by substituting the rotational angular momentum operator, R, for the classical angular momentum
Hr =

ℏ2 2
R = BR2
2I(2.8)

where B is the rotational constant. In a Hund’s case (a) molecule, like the ground state 2 Π statesof OH, R = J − L − S and the Hamiltonian becomes
Hr = B(J2 + L2 + S2 − 2J · L − 2J · S + 2L · S)

(2.9)

The rotational energy is given by
Hr = B(J(J + 1) + S(S + 1) + 2ΛΣ + Λ2 − 2Ω2 )(2.10)The rotational constant B will be different for each electronic and vibrational energy level,
and for the ground state of OH B = 18.534cm−1 . The rotational energy levels for the OH groundstate are plotted in Figure (2.4). Each new rotational level increases the value of J by 1 while theother values are constant. Levels can be split between the 2 Π3/2 and 2 Π1/2 manifolds.

2.4.1

Lambda Doubling

Another important factor in the structure of diatomic molecules is parity. The parity, ϵ, ofa state determines whether the sign of the overall wavefunction changes sign upon inversion, or a

14

Figure 2.4: Rotational energy levels of the ground state of OH. From [46]15reflection of all particles at the origin. If the wavefuction is unchanged, the parity is ϵ = +1. If
the sign of the wavefunction flips, the parity is ϵ = −1. The parity of a rotational state alternates
between + and - for even and odd levels of J respectively. For electronic states with Λ ̸= 0, likethe ground 2 Π state of OH, there are two possible orientations of Λ. Parity states are made fromlinear combinations of the two orientations, and each rotational state, J, has a positive and negative
parity level. The degeneracy between the two parity states is lifted by a weak spin-rotation coupling,giving rise to a Lambda Doublet splitting. In the 2 Π3/2 state of OH, the ground rotational state
has a lambda doublet splitting of ∆ = 0.056cm−1 . In Figure 2.4, the lambda doublet splitting is
exagerated, but demonstates the important characteristics. The splitting grows with increasing J,
and the parity of the upper and lower states flips based on the parity of the rotational state.

2.5

OH in External Fields

2.5.1

OH in Electric Fields

When a molecule with an electric dipole moment encounters an electric field, it will experiencea Stark shift. The Hamiltonian for the Stark effect is given by
HS = −de · E,

(2.11)

where de is the molecule’s electric dipole moment and E is the magnetic field vector. For a diatomic
molecule electric dipole moment points along the internuclear axis. As the molecule rotates, thecomponents of the dipole moment perpendicular to the rotation axis average out to zero. The time
averaged component that survives is the projection of de onto J,

dK< dJ >= |de |cos(z, J) = p
J(J + 1)

(2.12)

where K is the projection of J onto the internuclear axis, which for a Hund’s case (a) moleculeK = Ω. The Hamiltonian then becomes
HZ = −dJ J · E,

(2.13)

16
We can evaluate the Hamiltonian in the |JMJ Ω⟩ basis by taking the magnetic field to bealong the z-axis

EZ = ⟨JMJ Ω|HS |JMJ Ω⟩ = −dE

MJ Ω= −def f E
J(J + 1)(2.14)

where MJ is the projection of J along the z-axis.
The above treatment is a first-order approximation that ignores the effects of parity. In theparity basis, Equation (2.14) represents the off diagonal elements, and the diagonal elements givethe zero-field lambda doublet splitting.
 −∆/2 −def f E 
H=−def f E∆/2
Solving for the eigenvalues of this matrix gives the Stark energy splittings∆∆
MJ Ω 2∆ES =∓ ( )2 + (dE)
2
2
J(J + 1)(2.15)(2.16)

where the ∓ symbol is used to denote that the sign is opposite of the parity state (the energy ofpositive parity states decreases with electric field and negative parity states increase). Stark shifts
are plotted in Figure 2.5. For small electric fields, the first order Stark shift is comparable to theLambda doublet splitting and the over all shift is quadratic in the electric field. For large electricfields, the effect is linear.

2.5.2

OH in Magnetic Fields

In the OHRBIT experiment, the OH molecules will also experience magnetic fields on theorder of 100’s of gauss. So, we must also understand how magnetic fields affect the internal stateof OH. The interaction of a molecule’s magnetic dipole moment with an external magnetic field isknow as the Zeeman effect. The Hamiltonian for the Zeeman efect is given by
HZ = −µm · B,

(2.17)

17

Figure 2.5: Stark Shifts in OH for (a) small and (b) large electric fields.

18
where µm is the molecule’s magnetic moment and B is the magnetic field vector. The magnetic
moment of a moleculeµm = −µb (L + ge S),(2.18)

includes contributions from both the orbital angular momentum, L, and the spin angular momentum, S, where µb is the Bohr magneton and ge ≈ 2 is the electron g-factor. For a Hund’s case (a)molecule, L and S are coupled to the internuclear axis with projections Λ and Σ. The magnetic
moment points along the molecular axis with magnitudeµΩ = µB (Λ + ge Σ)(2.19)

While the molecule rotates, the molecular axis precesses around J. The time averaged magnetic
moment along J due to this precession is given by< µJ >= µΩ cos(z, J) = µB (Λ + ge Σ) pΩJ(J + 1)

(2.20)

Equation (2.17) then becomes
HZ = −µJ J · B,

(2.21)

We can evaluate the Hamiltonian in the Hunds case (a) basis |JMJ Ωϵ⟩, in which the Hamiltonian is diagonal. By taking the magnetic field to be along the z-axis

EZ = ⟨JMJ Ωϵ|HZ |JMJ Ωϵ⟩ = µb B(Λ + ge Σ)ΩMJ.
J(J + 1)(2.22)

where MJ is the projection of J along the z-axis.
In this experiment we are mainly concerned with the X 2 Π3/2 ground state of OH, in whichcase Equation (2.22) reduces to

4
EZ = µB MJ B,
5(2.23)

Equation (2.23) is plotted in Figure 2.6 for each MJ level in each parity state. Crossings between
Zeeman levels with different parities occur at 496 G, 744 G, and 1488 G.

19Figure 2.6: Zeeman Shifts in OH. Black curves are negative parity, Red curves are positive parity.

20
2.5.3

OH in Combined Fields

In the presence of both electric and magnetic fields, the Hamiltonian is the sum of the Stark
and Zeeman
HX = −µm · B − de · E.

(2.24)

While the electric and magnetic dipole moments are alligned to the internuclear axis and will always
point in the same (or opposite) direction in the molecular frame, the electric and magnetic fields
are free to point whereever the experiment demands. We can let the magnetic field B lie along thez-axis and define the angle β to be the angle between B and E. The Hamiltonian can for the 2 Π3/2
state of OH can be written [47] ∆ 6
− 2 − 5 µB B

0

0

0


0
−∆
− 25 µB B
0
0
2

∆
2

0
0
− 2 + 5 µB B
0

∆
6

0
0
0
−
+
µ B
2
5 B

√
 3 dEcosβ − 3 dEsinβ
0
0
5
 5√
 − 3 dEsinβ 1 dEcosβ − 2 dEsinβ
0
 5
5
5
√

0
− 2 dEsinβ − 1 dEcosβ − 3 dEsinβ
5

0

0

√5
− 53 dEsinβ5− 53 dEcosβ√3dEcosβ − 53 dEsinβ5√− 53 dEsinβ 15 dEcosβ0− 52 dEsinβ

0

0



− 25 dEsinβ

0

0

0
0













√
− 15 dEcosβ − 53 dEsinβ√− 53 dEsinβ − 35 dEcosβ00∆− 56 µB B
2

0

0

∆
− 25 µB B
2

0

0

0

∆
+ 25 µB B
2

0

0

∆
+ 65 µB B
200(2.25)

where the first four indices are the positive parity states with MJ = −3/2, −1/2, +1/2, +3/2 andthe last four are the negative parity states with MJ = −3/2, −1/2, +1/2, +3/2. Note that the
diagonal contains the lambda doublet splitting and Zeeman shift, because they do not mix parity
states. The off diagonal elements contain the Stark shift, which do mix parity states.

2.6

OH Creation
When studying the physics of atoms and molecules, it is important to have a sample of cold

gas-phase particles in order to minimize the potential interactions between particles. This presentsa challenge, since particles tend to exit the gas phase when they are cooled to the sub Kelvintemperature regime that affords the unique quantum control that makes studying cold molecules
so interesting. The solution that experimentalists have been using for decades is to entrain themolecules they want to study in a carrier gas that can collide with and cool the molecule while
limiting collisions between the molecules that would cause them to cluster into liquids. The method

21
employed in this experiment is to entrain our molecules in a supersonic expansion of a rare gas.
Illustrated in Figure 2.7, a supersonic expansion occurs when a gas is allowed to pass through anapperature between a region of high pressure and a vacuum. Collisions will tend to convert heat
energy into forward velocity, making a beam that has a large velocity in the forward direction but
a narrow spread of velocities cooresponding to a temperature on the order of 1K. The rare gas
can also quench the internal molecular degrees of freedom, which leaves the molecules near therovibrational ground state. The expanding gas will often be passed through another aperature inorder to separate a useable portion of the beam and reduce the background gas farther down thebeamline.

Figure 2.7: Illustration of a supersonic expansion.

The velocity of the beam can be calculated by considering the change in enthalpy across theaperature. As the gas expands into the vacuum, the pressure differential accelerates particles whileconverting enthalpy into forward velocity. Using the enthalpy per unit mass the equation describing
this enthalpy change is[48]1h0 = h + v 2
2(2.26)

where h0 is the initial enthalpy of the gas reservoir, h is the final enthalpy after expansion, and v

22is the velocity. This can be solved to give the velocity[48]s2kB γv(T ) =
(T0 − T )m γ−1

(2.27)

where m is the mass of the carrier gas and γ is the ratio of the heat capacities at constant pressureCPand volume, γ = C. When using a monoatomic gas, such as a rare gas, as the carrier gas γ = 53 .
V

The approximation is made that the intial temperature is much larger than the final temperature,or TT0 = ∞, which leads to a velocitysv∞ =2kB T0 γm γ−1(2.28)The velocity can be tuned by changing the temperature of the gas reservoir, T0 . The velocity canalso be changed by changing to a carrier gas with a different mass. Most often, a rare gas is usedas the carrier gas. Figure 2.8 plots v∞ versus carrier gas mass, noting the velocity for the different
rare gasses at T0 = 300K.
The molecular beam apparatus used in this experiment is displayed in Figure 2.9. We use
Krypton as our carrier gas at a pressure of 2200 Torr. The Krypton is bubbled through a reservoirof distilled water, to seed the gas with 1% H2 O. The flow of the gas into the vacuum chamberis controlled by a piezo electric transducer (PZT) valve.[50] The PZT valve opens when a voltage( 300-500V) is applied. We pulse the opening voltage for 100 µs, creating a pulsed supersonic
expansion of H2 O seeded in Kr.
OH radicals are created in a discharge across the expanding gas at the exit aperature. Collisions between high energy electrons and H2 O molecules cause H2 O to fragment. For collisionenergies on the order of 100 V, the process that dominates is
e− + H2 O → OH(X) + H + e−

(2.29)

where the OH(X) denotes that OH is left in the ground electronic state. A voltage of 900V is applied
across the discharge electrodes for a duration of 20 µs. The relatively short pulsed discharge allows
the OH to be created with a tunable mean velocity. Since the time of flight from the valve exit to the23

Figure 2.8: Velocity vs mass of carrier gas.

24

Figure 2.9: Diagram of the apparatus used for producing OH molecular beam. From [[49]]25
discharge region is dependant on the velocity of the molecules, the delay between valve opening anddischarge can be varied to select a different portion of the Kr/H2 O gas pulse to create the OH from.
The short discharge also produces a smaller velocity distribution. Under our typical experimental
conditions, the OH beam has a velocity of 420 m/s and velocity spread of 50 m/s. The beam iscollimated by a skimmer as it passes into the portion of the vacuum chamber containing our Stark
Decelerator.

Chapter 3

Stark Deceleration

3.1

Introduction
As described above, a supersonic expansion can produce a relatively dense sample of cold

molecules. However, these cold molecules are produced with a large forward velocity, which reduces
the amount of time an experimentalist has to interact with the molecules. A technique known as
Stark deceleration has been demonstrated to slow fast moving polar molecules to a tunable final
velocity. Stark deceleration uses spatially inhomogeneous time varying electric fields to decelerate
neutral polar molecules. By changing the timing of the electric fields, the final velocity can bechanged to fit the desired experiment. In this chapter, we describe the Stark Deceleration process
and how we slow our OH molecules down to a velocity where they can be trapped at rest.

3.2

Stark Deceleration
To decelerate something, it must experience a force. The force to decelerate a polar moleculecan be produced by exploiting the Stark potential experienced by polar molecules in an electricfield. The force experienced by a particle in some potential field is given by the gradient of that
potential

F = −∇UStark .

(3.1)

27
In the approximation of high electric fields where the Stark potential is linear, the force becomes
F = −∇(de · E) = de ∇|E|.

(3.2)

We can see from Eq. 3.2 that the force a polar molecule feels from an electric field comes fromthe gradient of the electric field. Whether this force produces an acceleration or deceleration isdependent on the internal state of the molecule. In the ground state of OH, if the sign of the
product ΩMJ is negative, the state is weak-field-seeking.
In order to do this in the lab, we can use electrodes to produce a large electric field in one
region, like shown in Figure 3.1. When a molecule in a weak-field-seeking state approaches theregion between the electrodes, it is moving from a region of low electric field to a region of highelectric field. This produces a potential hill with a gradient that opposes the motion of the molecule.
As it climbs the potential hill, the molecule is decelerated while trading kinetic energy for Stark
potential energy. If the molecule were allowed to pass through the electrodes to the other side
and over this peak in the potential energy, it would begin to accelerate back to its initial velocity.
However, if the voltages on the two electrodes were quickly turned off while the molecules were still
experiencing the Stark potential, that potential energy will go away without doing any work to themolecule.
The amount of energy that can be taken out by a single pair of electrodes is somewhat
limited, but this can be overcome by combining many stages one after another. In this way, themolecules are constantly climbing a potential hill and losing kinetic energy, then losing the potential
energy when the hill is turned off. The electric dipole moment of typical polar diatomic and simple
polyatomic molecules is around 1-2 debeye, with the electric dipole moment of OH being 1.67 D.
Electric fields that can be reasonably produced in a vacuum chamber between two stainless steel
electrodes is 50-150 kV/cm. This means that approximately 2K can be removed per stage, and150 stages are needed decelerate down to rest.
The Stark decelerator used in this experiment consists of 149 pairs of pin electrodes. A
picture of the decelerator can be found in Figure 3.2.

28

Figure 3.1: Single Pin Pair Diagram. With the electrodes held at ±HV, the molecule experiencesa potential energy hill due to the Stark effect.

29

3.3

Operation of a Stark Decelerator
Four rods run down the length of the decelerator, in which the electrode pins are inserted.

The rods are numbered 1 through 4. Pins connected to rods 1 and 4 oppose each other and makea pin pair. Pins connected to rods 2 and 3 also oppose each other and make a pin pair. The pinsfrom rods 1 and 4 are oriented at a 90 degree angle from the pins from rods 2 and 3. The electricfields tend to keep the beam from expanding in the direction between the pins but not along thepins. So the 90 degree angle alternates between which of the transverse directions the molecules
are being confined.
The voltages on the rods are switched between ground and ±12kV by individual high voltage
switches with a rise/fall time on the order of 100ns. Rods 1 and 3 will hold -12kV, while rods 2
and 4 hold +12kV. When operating the decelerator the voltages on the rods are switched between
two configurations• Rods 1 and 4 at ±12kV , Rods 2 and 3 grounded• Rods 1 and 4 grounded, Rods 2 and 3 at ±12kV
To understand how the decelerator is operated in order to control the final velocity ofthe molecules, we define the synchronous molecule. The synchronous molecule is a hypothetical molecule that travels directly along the axis of the decelerator around which the timing of thedecelerator is designed. The Stark potentials of an OH molecule traveling directly along the axisof the decelerator can be found in Figure 3.3. The potential is usually parameterized by W (ϕ) asa function of the phase angle
ϕ = 180◦

z
L

(3.3)

where z is the position of the molecule along the decelerator axis and L is the distance between
similar pin pairs.
Figure 3.4 shows the path of the synchronous molecule along the potentials during deceleration. The zero of ϕ is defined to be the position where the potentials for both configurations are

30

Figure 3.2: A picture of the Stark Decelerator.

31

Figure 3.3: Stark Potential for the two voltage configurations. Because the rods electricallyconnect every 4th pin, a periodic potential is created that spans the entire decelerator.

32

Figure 3.4: Demonstration of Deceleration. Path of the synchronous molecule along the potential
energy curves are is the black line. When the sychronous molecule reaches the desired phase angle,
the potential configuration is switched. The molecules lose an amount of energy ∆K, which is thedifference between the potential energy curves at that phase angle.

33
equal, which is also halfway between the pin pairs. For a phase angle of ϕ = 0◦ , the energy lostwhen the potentials switch is ∆K = 0. The maximum amount of energy is lost when switched at
ϕ = 90◦ , since this phase angle lines up the maximum of the initial potential with the minimumof the switched potential. The potential after switching between configurations is just the original
potential shifted by π, and the energy lost is given by∆K(ϕ) = W (ϕ) − W (ϕ + π).

(3.4)

If the phase angle of switching is kept constant, then the velocity after a number of stages, n, isgiven by1
1
mvn (ϕ)2 = mvi2 + n∆K(ϕ)
22r2n∆K(ϕ)vn (ϕ) = vi2 −
m

(3.5)
(3.6)

The phase space acceptance of the decelerator is the region of the 6-D phase space that will beremain in the vicinity of the synchronous molecule and be successfully decelerated. A description of
the behaviour of the molecules in the transverse dimensions can be found in [51]. In the longitudinal
direction, the phase space acceptance can be derived from the potentials above. The energy lostper stage, ∆K(ϕ), varies from 0 to Wmax between ϕ = 0◦ and 90◦ and to first order is given by∆K(ϕ) ≈ −Wmax sin(ϕ)(3.7)

The average force on a molecule per stage is given byF̄ =∆K−Wmax sin(ϕ)
=
L
L

(3.8)

A molecule that differs in phase angle by an amount ∆ϕ will experience a force relative to thesynchronous molecule ofF̄rel = F̄0 − F̄ =−Wmax[sin(ϕ0 ) − sin(ϕ0 + ∆ϕ)]
L

(3.9)

34
This force is negative for phase angles larger than ϕ0 , and positive for phase angles smaller than ϕ0 ,representing a restorative force that pushes the molecule back towards the synchronous molecule.
Using Newton’s second law, we can derive an equation of motion

mL d2 ϕ Wmax−[sin(ϕ0 ) − sin(ϕ0 + ∆ϕ)] = 0
π dt2
L(3.10)

By numerically integrating Equation 3.10, we can find the boundary of phase space for whichorbits around the synchronous molecule are stable, known as the separatrix. The separatrices fora few different phase angles are plotted in Figure 3.5. Molecules in the phase space outside of

Figure 3.5: Separatrices for OH at various phase angles. From [?].

the separatrix are accelerated away from the synchronous molecule, while the molecules within
the separatrix are decelerated with the synchronous molecule, forming a packet of decelerated
molecules. The phase space volume shrinks as ϕ0 increases, meaning the number of molecules thatcan be decelerated decreases as the target velocity decreases.

35

3.4

Alternate Deceleration Schemes
One of the main problems with the traditional pulsed pin decelerator is loss due to transverse

motion of the molecules.[52, 51] Coupling between the longitudinal and transverse motion of themolecules leads to regions of the phase-space acceptance that are unstable. Alternative decelerationgeometries and schemes have been designed to eliminate or overcome this problem. For pulsed pin
decelerators, changing the sequence that voltages are applied can help mitigate the transverse
loss.[53, 54, 55] Travelling wave ring decelerators can eliminate the transverse loss, since they
produce a three dimensional trapping potential that moves along with the molecules.[56] A ring
decelerator can also be operated in a pulsed mode, which still exploits the cylindrical symmetry ofthe decelerator to greatly reduce potential transverse loss.[57]Recently, it has been shown that mixing in alternate voltage configurations can help provide a
restoring force in the transverse direction.[58] In the traditional deceleration mode, after the voltage
is switched between configurations, the molecules spend a significant amount of time between thetwo grounded electrodes. However, the field lines from the charged electrodes create a defocusing
potential along the grounded electrodes. By applying an alternative voltage configuration between
slowing stages, the transverse potentials can be changed to be focusing. Dave Reens describes in
his PhD Thesis different configurations that can be achieved with varying degrees of experimental
complexity.[58] These configurations are dubbed Very Strong Focusing (VSF), Strong Focusing(SF), and Focusing, in order of decreasing complexity and improvement. The voltage configurations
and switching schemes are summarized in Figure 3.6.
For each of these schemes, there is an intermediate focusing stage where the molecules are
better confined in the direction along the pin pair. A single stage of the deceleration sequence
consists of the molecules traveling from a phase angle of ϕ = ϕ0 − 360 to ϕ = ϕ0 . The focusing
potentials are symmetric about ϕ = −180◦ , and ∆K is symmetric about ϕ = 0. So, in each of
the focusing schemes, the molecules travel from a phase angle of ϕ = ϕ0 − 360 to ϕ = −ϕ0 on
the focusing potential without a change in energy. The molecules then travel along the slowing

36

Figure 3.6: Different voltage configurations for Focusing modes. From [58].

37
potential from ϕ = −ϕ0 to ϕ = ϕ0 . There is an energy drop at both potential switches, which totals
to ∆K(ϕ0 ).
The main difference in experimental complexity is the number of high voltages switches
needed to achieve the voltage configurations. In the traditional (s=1) switching scheme, a single
high voltage switch is needed to switch each rod between ground and either negative or positivehigh voltage. This requires only four high voltage switches. In the VSF configuration, two of therods now need to be able to switch between ground and both positive and negative high voltage.
This requires a total of six switches. It also requires two pairs of switches to be operated in parallel,
which is prone to breaking the switches. In the SF configuration, only one of the rods needs tobe able to switch between ground and both positive and negative high voltage. This configuration
requires a total of five switches, with a single pair being run in parallel. The F configuration
requires no new switches and no switches to be run in parallel. The only changes that need to bemade from the s=1 configuration are in the times when electrodes turn on and off. Reens saw afourfold improvement in the number of decelerated molecules across a wide range of final velocities,
including slower velocities that can be loaded into a trap.[58] For these reasons, we adopted the
Focusing slowing scheme in 2019.

3.5

Deceleration Data
Figure 3.7 shows OH molecule signal for various final velocities decelerated from vi = 420m/s

using the s=1 slowing scheme. The plot for each velocity has three distinct peaks, corresponding todifferent potential wells of the decelerator. The synchronous molecule is in the middle of the threepeaks. Our goal in decelerating OH is to load the molecules into an electrostatic trap (discussed
below). The trapping scheme is designed to accept molecules at ∼ 33.7m/s. Figure 3.8 showsslowing to 33.7m/s using S=1 slowing and the Focusing scheme. The focusing scheme increases thetotal number of molecules in the decelerated packet by a factor of ∼3.

38

Figure 3.7: Deceleration of OH to different final velocities from an initial velocity of vi = 420m/s.
For vf = 420m/s the decelerator is operated at a phase angle of ϕ0 = 0.

39

Figure 3.8: Comparison between S=1 and Focusing deceleration schemes for slowing to 33.7 m/s.
The S=1 data is plotted upside down. The peak that arrives later in time around 4.2s contains
the sychronous molecule and would be loaded into our trap. Integrating this later peak shows anapproximately threefold increase in signal.

Chapter 4

Detection of OH

4.1

Introduction
State selective detection of molecules is an important tool for performing science with cold

molecules. Detection requires a way to interact with the molecule, and a way to determine if theinteraction has happened. For a detection scheme to be state selective, either the interaction or
the product of the interaction must be state selective in some way. Often, the resonant absorptionof a photon is chosen as the state selective interaction. From this excitation, three main methods
are often employed to measure the interaction. First, the fraction of light that was absorbed canbe measured. This requires a combination of molecule density and absorption cross section thatis large enough to absorb enough light to have a good signal-to-noise ratio. However, this is notpossible in most cold molecule experiments. Second, the excited molecule will fluoresce, producing
photons that could be measured. This process is referred to as laser induced fluorescence (LIF).
Third, the resonantly excited molecule can be ionized by another photon, and the amount of ionscan be measured. This process is known as resonantly enhanced multiphoton ionization (REMPI).
In this chapter, I will discuss how OH is traditionally detected via LIF. I will also describe our
UV-VUV REMPI detection scheme for OH. The advantages and disadvantages of LIF vs REMPI
are discussed.

41

4.2

LIF of OH
The most common technique for detecting OH is via laser induced flouresence on the A← X

transition. The strongest absorption line for this electronic transition is to the ν ′ = 0 vibrational
state using a UV laser wavelength of 308 nm. However, a large Frank-Condon factor leads thestrongest emission to be along the the ∆ν = 0 (ν ′ = 0 −→ ν = 0′′ ) transition. Thus, most of
the fluorescing light is the same wavelength as the excitation laser. While LIF detection is often
accomplished in atoms by collecting fluorescence at the same wavelength as the excitation, this is
not as easy in the detection of a cold molecule like OH. The UV photon in a cold molecule detection
scheme will usually be produced by a pulsed laser with a pulse width too fast and repetition rate
too slow to produce multiple cycles of excitation and emission. The large amount of laser lightbeing pumped into the vacuum chamber, compared to the relatively small amount of light emitted
from the molecules, would limit the signal to noise ratio of light emitted by LIF vs backgroundlight reflected from elsewhere. This would tend to limit the sensitivity of detection. Therefore, LIF
detection of OH is typically acheived by exciting to the A2 Σ+ (ν ′ = 1) manifold using 281 nm light.
The light emitted is mostly along the ∆ν = 0 (ν ′ = 1 −→ ν = 1′′ ) transition at 313 nm, which canbe spectrally filtered from the laser light.

4.3

LIF Detection Efficiency
One of the main disadvantages of LIF is a low detection efficiency that limits the signal-to-noise ratio. When the molecules fluoresce, the photons are emitted in all directions with equal
probability. Only those photons that can be collected into a detection device will be counted
towards the detection of the molecules. The fraction of emitted photons that can be collected tothe detection device will be limited by the geometry of the system. The fraction of photons that canbe detected is fundamentally limited by the solid angle around the molecules from which photons
can be efficiently collected. Since the molecules being detected are in a vacuum chamber andoften surrounded by electrodes or other shiny metal objects, the options for collecting a significant

42
fraction of the solid angle are limited. Even in the best case scenarios, where a lens can be placed
inside of vacuum very close to the fluorescing region, and a parabolic mirror is used to reflect
photons emitted away from the detector back towards the detector, collecting more than 5-10%
is extremely difficult and often experimentally unfeasible. In our experiment, the trap electrodes
and chamber design present a fundamental limit to the solid angle that can be collected. The gap
between the trap electrodes in our experiment is 3 mm. There is no room to fit a lens inside thevacuum chamber, and the closest a lens can be placed is 37 mm away from the center of the trap.
For a 25mm diameter lens, the fraction of the solid angle of emitted photons is limited to 4 ∗ 10−3 .
Another problem that limits the signal-to-noise of an LIF detection scheme is the filteringthat is needed to distinguish between fluoresced signal photons and background noise photons. The
typical instrument used to detect fluoresced photons is a photomultiplier tube (PMT). Photons
impinging on the detector of a PMT create photoelectrons which are amplified through a systemof cascading electrodes that produce measurable current. PMTs can be extremely sensitive andallow for detection of single photons; however, that sensitivity to signal also makes them sensitive
to noise sources. Photomultiplier tubes can be designed for detecting light in the UV, but the
spectral response still presents problems for our LIF scheme.
The PMT used earlier in my thesis work is a 9125B from ET Enterprises, the quantum
efficiency of which can be found in Figure 4.1. The quantum efficiency for a single 313nm photon
to create an electron cascade is around 30%, limiting the sensitivity to signal. While the quantum
efficiency falls off quickly below 300 nm, there are 8.5x1015 photons per 6mJ pulse of our 281nm
laser. Even the small fraction of the light that is reflected off windows, the chamber, and trap
electrodes would present a significant amount of background photons. Furthermore, the quantum
efficiency extends into the visible light, where orders of magnitude more photons are present fromroom lights. In order to block out these undesired wavelengths, spectral filters can be added to theoptical system collecting the photons. Since the 281 nm light at shorter wavelength and the room
light at longer wavelength both need to be filtered while allowing the 313 nm to be transmitted,
a combination of high pass, low pass, and band pass filters are required. An analysis of the filters

43

Figure 4.1: Quantum efficiency vs wavelength of the PMT (labeled B) used in previous experiments. The quantum efficiency at the flourescnence wavelength 313 nm is 26% and at the laser
wavelength 281 nm is 5%.

44
used in our prior experiments can be found in the thesis of Travis Briles [?] (Chapter 6.5). Based on
the filters available at the time, a stack of filters was designed that sufficiently blocked out possible
noise wavelengths, but the trasmission at 313 nm was only 35.5%.
From these different loss sources, we can estimate an overall detection efficiency for LIF. For
every molecule that absorbs a photon at 281 nm, the probability of detecting a photon at 313 nm
using the PMT isη = α ∗ β ∗ χ ∗ γ = 2.5 ∗ 10−4

(4.1)

where α is the fractional solid angle, β is the transmission of 313 nm through the filters, χ is thequantum efficiency of the PMT at 313 nm, and γ is the branching ratio for the excited OH toemit along the ∆ν = 0 transition. This means approximately 4000 molecules need to be excited
to detect a single photon. Since so many excited molecules are needed to detect a single photon,
the entire cloud of molecules must be illuminated in order to produce a detectable signal. While
this can be useful for determining information about the total number of molecules in the cloud, it
does not allow the density profile of the cloud to be probed.

4.4

REMPI of OH
Resonantly enhanced multi-photon ionization (REMPI) is the photoionization of a moleculein a multi step process where the first step is a resonant excitation. The molecule in the excitedstate is subsequently ionized by the absorption of another photon. The resonant excitation step
can be due to the absorption of a single photon or multiple photons. The ionization step can be
due to a photon of the same color in the same laser pulse as the excitation photons, or it can befrom photons of a different color. The convention for describing a REMPI scheme is A+B, where
A is the number of photons required for excitation and B is the number of photons required for thesubsequent ionization. If the ionization step utilizes a different color than the excitation step, thiswill be denoted by B’ (B prime). For example, a 2+1’ REMPI scheme would have a two photon
absorption at one wavelength and a single photon ionization from the excited state at a different

45
wavelength.
The ionization potential of OH (13.017 eV) would correspond to a photon wavelength of
95.2 nm, which is in the vacuum ultraviolet (VUV) part of the spectrum. All of the excited states,
except for the A2 Σ+ , also lie high enough above the ground state to require a VUV energies toaccess. A few REMPI techniques have been developed that utilize a single UV color for a 2+1 or
3+1 scheme through Rydberg states of the OH molecule, summarized in Figure 4.2. While ionized

Figure 4.2:
[59]

Energy diagram showing different REMPI schemes observed for ionizing OH. From

OH was able to be detected with these schemes, the efficiency is limited. The excited Rydberg
states have lifetimes on the order of 100 ps resulting from potential energy surfaces that crossrepulsive states, which lead to dissociation before the molecule can be ionized.
A REMPI scheme that proceeds through the A2 Σ+ state used in LIF would seem to be agood candidate, but the ionization step is a bit more complicated. The electron configuration of
the ground state of the OH + ion and the A2 Σ+ state are given by• OH + X 3 Σ− : (1sσ)2 (2sσ)2 (2pσ)2 (2pπ)246• OHA2 Σ+ : (1sσ)2 (2sσ)2 (2pσ)1 (2pπ)4Ionizing the neutral A2 Σ+ state through the ground ionic state would require two electrons tochange state and is forbidden. However, there are Rydberg states of the neutral OH molecule athigher energy than the ionization potential that subsequently autoionize. This was used by Beames
et al to demonstrate a 1+1’ REMPI scheme, which we use in this experiment. In this study, OH wasproduced in a supersonic expansion and detected by 1+1’ REMPI. The first photon is a UV photonwith a tunable wavelength near 281 nm. The second photon is a VUV photon of fixed wavelength at
118 nm, the 9th harmonic of a Nd:YAG laser. The UV wavelength was varied while measuring theionization signal. The spectroscopic peaks were assigned to transitions between different rotational
levels of the ground and excited states. For our experiment, we are only concerned with detectionof OH in the J=3/2, ϵ=-1 state. The peak ionization signal from this state was detected at awavelength corresponding to the R21 rotational branch.
Figure 4.3 shows the experimental setup for our OH detection. Light at 281 nm is produced
using a pulsed dye laser (R6G), with the fundamental being frequency doubled by a KDP crystal.
Light at 118 nm is produced by creating the 9th harmonic from an Nd:YAG laser. The YAG laser
is internally frequency tripled to 355nm, then the 355nm light is tripled in a gas cell of Xenon and
Argon. The process of producing this 118nm light is detailed later in this chapter. The ions areaccelerated through a time of flight mass spectrometer (TOFMS) and detected by a microchannel
plate (MCP). Ions are extracted from the ionization region to the MCP with near unit transfer
efficiency. A grid on the front of the MCP needed for the final acceleration of the ions into the
MCP plates reduces the detection efficiency to 90%. Since the detection efficiency is so much higher
than for LIF, instead of trying to detect the entire cloud, we can reduce our detection volume byfocusing the ionization beam. This will allow us to gain information about the position profile ofthe molecules in the trap. This information can elucidate dynamics of the molecules in the trap[60] and is important for understanding the results of trapped collision experiments [25, 61]
The 118nm light is focused through the trap with a beam waist of 30 µm. The last mirror

47

Figure 4.3: Experimental set up for REMPI detection of OH.

48
before the 355nm light enters the gas cell contains a computer-controlled closed-loop picomotor(New Focus Model 8310). The motor is used to scan the ionization position across the OH in
the trap to measure the density as a function of position along the y-axis. The 281 nm light iscylindrically focused through the trap, with the focused direction being along the molecular beamaxis. This produces a high intensity to saturate the excitation step, while also keeping the beam
spread out along the y-axis in order to make sure the molecules are still being excited when the
118 nm light is scanned along that dimension. The two laser pulses are temporally separated by
20 ns, with the 281 nm light arriving first.

4.5

Future Improvements
It is possible that more OH ionization signal could be achieved by tuning the wavelengthof the VUV. In [59], the authors resonantly excite OH to different rotational levels of the of theA2 Σ+ (ν = 1) state then measure the ionized OH signal while varying the VUV wavelength. This
produces a rotationally resolved spectrum of transitions to the auto-ionizing Rydberg state. Figure
4.4 shows ionization from the J = 5/2 level of the excited state vs VUV wavelength, with thedashed purple line corresponding to the wavelength of the 9th harmonic of a YAG laser (discussed
later in this chapter, this is how our VUV is produced). The strong ionization signal originatingfrom the J = 5/2 state seems to be due to a coincidence of this 9th harmonic with one of therotational levels of the Rydberg state. This wavelength sits on the side of the peak assigned to the
P Q branch.

Moving to the peak of this branch, or to the peak assigned to the R2 branch could

yeild more ionization signal in our experiment.
It also possible that varying the UV light to a different wavelength corresponding to excitation
to a different rotational level in A2 Σ+ (ν = 1). There are 3 allowed transitions from our trapped
OH state: R21 , Q1 , and P21 . Ionization signal after excitation along the Q1 branch can be found in
the middle plot of Figure 2 in [59] (reproduced here in Figure 4.5) while corresponding plot for theP21 branch is in Figure 1 of [59]. The OH ionization signals are all reported as relative signal andmay not be directly comparable. However, the signal to noise ratio for the Q1 branch plot appears

49

Figure 4.4: VUV Ionization spectrum of OH, From [59]. The vertical dashed line cooresponds tothe fixed 118 nm light from the 9th harmonic of a Nd:YAG. The middle plot shows the ionization
spectrum from the excited rotational state used in this work.

50

Figure 4.5: VUV Ionization spectrum of OH, From [59]. The vertical dashed line cooresponds tothe fixed 118 nm light from the 9th harmonic of a Nd:YAG. The middle plot shows the ionization
spectrum from the J’=1.5 state, which is accessible from the J”=1.5 state along the Q1 rotational
branch.

51
to be nearly an order of magnitude better than the other two branches. The enhanced ionization
signal is likely due to a the Q1 branch being the most efficient branch in terms of excitation. This
may indicate that tuning the UV light to the Q1 branch and the VUV to the peak of the R Q branchwould yield even the most efficient ionization of decelerated and trapped OH.

Chapter 5

Produciton of 118 nm light

In this chapter, we present a reproduction of our paper ”Characterization of a vacuum ultraviolet light source at 118 nm” by Gray et al. published in JCP in 2021 [62] The abstract andacknowledgements have been removed, and the bibliographical references and formatting have beenchanged to match the format of this thesis.

5.1

Introduction
Vacuum ultraviolet (VUV: λ ≈ 100-200 nm) light sources [63, 64, 65] have important applica-

tions in a wide variety of experiments in atomic physics and physical chemistry, including molecular
spectroscopy for the understanding of chemical bonds, [66, 67] high resolution studies for metrology,[68, 69, 70, 71, 72] and as a universal ionization source in mass spectrometry[73, 74]. High fluxes
of VUV light are available from synchrotrons, but the complexity, cost, and beam-time constraintsof these sources makes a more accessible tabletop VUV source desirable. With the exception of
excimer lasers,[75] which operate at only a few set wavelengths, tabletop sources of VUV light require use of nonlinear optical techniques. Nonlinear optical frequency mixing, [76, 77, 78] of whichharmonic generation is a special case, is a powerful technique for producing coherent radiation at
wavelengths where small commercial laser sources do not typically operate.[79]One VUV source is made by frequency tripling the third harmonic output of a Nd:YAG
laser in a xenon-argon gas mixture to create 118 nm light. The 118 nm photon can be used insingle or multi-photon ionization applications. Specifically, chemical dynamics studies use 118 nm

53light in ion-pair photodissociation spectroscopy,[80] ion pair production,[81] anion photoelectron
spectroscopy,[82] and one-photon ionization processes. [83, 84, 85, 86] Additionally, 118 nm photons
have been used to perform multi-photon ionization of cold radical molecules.[87, 88, 60]
Since the conversion efficiency of 355 nm light to 118 nm light is low (∼ 10−5 )[89] and may
constrain the detection sensitivity of an experiment, it is important to understand the limits of thetripling technique in order to experimentally optimize 118 nm light production. The basic theoryof 118 nm generation in a xenon-argon gas cell [90, 91, 92] and experimental setup[93] have beendetailed in previous papers. However, the experimentally observed 118 nm photon production has
been observed to significantly diverge from theory at high gas pressures and has not been modeledso far.
Here, we describe the basic theory and experimental setup for generating 118 nm light andcharacterize the limitation of the VUV production at high xenon pressures. We present evidence
that absorption at 118 nm by xenon limits 118 nm production. We develop a model that accounts
for this absorption, whose predictions are in good agreement with our experimental results.

5.2

Theory of Non-Resonant Third Harmonic Generation
Anisotropic crystals with large second-order nonlinear optical susceptibilities (χ(2) ) and awide optical transparency range can be efficient at producing visible, near-infrared, and soft UVlight. Since these crystals start to absorb light below 200 nm, generation of VUV light via nonlinear
optical frequency mixing generally requires a gaseous nonlinear medium, which is transparent atthese wavelengths. However, the structural symmetry of isotropic media such as gases means that
only odd-order nonlinear susceptibilities are non-vanishing,[94] leaving the third-order susceptibility,χ(3) , as the first nonlinear term. [76]Third-harmonic light produced by the large χ(3) in these systems can extend into the VUVand extreme ultraviolet regions of the spectrum. For this process to be efficient, phase matchingis important, as the fundamental light and third-harmonic light must maintain a fixed phase relationship in order to coherently build the tripled light. In the gaseous system discussed here, phase54
matching is accomplished by mixing a negatively dispersive gas that acts as the nonlinear medium
with positively dispersive gas that can be used to tune the phase matching. In the remainder of thissection, we discuss the basic model of third-harmonic generation (THG) and conditions to optimize
phase matching.
The basic setup for THG is depicted in Fig. 5.1. The fundamental light at 355 nm is focused
into a gas cell of length, L, with a beam waist of w0 and a confocal parameter of b =

2πw02
λ .

The

third-harmonic light is shown in blue. The total power of the third-harmonic light produced by afocused Gaussian beam with incident fundamental power Pω is given by [91, 92]

P3ω =8.215 × 10−2 (3) 2 3
χef f Pω |F1 (b∆k)|2 .
(3λ)4

(5.1)
(3)

Here, all powers are in watts, λ is the third-harmonic wavelength in nm, and χef f is the macroscopic
nonlinear coefficient. F1 (b∆k) is a dimensionless function that describes phase matching, and ∆kis the wave-vector mismatch between the fundamental and third-harmonic light.
Experimentally, there are four parameters that can be varied to optimize the third-harmonic(3)

power (Eq. 5.1): (1) the macroscopic nonlinear coefficient χef f , (2) the confocoal parameter, and(3) maximizing the power in the fundamental (Pω ). However, independent maximization of ∆k(3)

in a single component gas is impossible because both χef f and F1 (b∆k) are dependent on N , the(3)atomic number density of the gas. χef f is given by(3)

χef f = N × χ(3)
a ,

(5.2)

(3)

where N is in units of atoms/cm3 and χa is the third-order nonlinear coefficient per atom in esu.
Generally F1 (b∆k) must be evaluated numerically. However, in the tight-focusing limit, where thenonlinear medium is much longer than the confocal parameter of the focused fundamental beam(b ≪ L), F1 (b∆k) can be solved analytically to give [92, 90]|F1 (b∆k)|2 =π 2 (b∆k)2 eb∆k∆k < 0,0∆k ≥ 0,

(5.3)

55

Figure 5.1: Diagram showing the parameters of the gas cell for generating 118 nm light (blue).
The 355 nm beam (red) is focused into the gas cell (gray box) by a lens with focal length f . The
confocal parameter b and beam waist w0 are noted.

56
which has a maximum at ∆kmax = − 2b .
∆k is related to N through the index of refraction, and is given by

∆k =

2π
[n1 − n3 ] .
λ

(5.4)

In this notation, n1 and n3 are the index of refraction at the third-harmonic wavelength λ and thefundamental wavelength 3λ, respectively. For a gas, the effective macroscopic index of refraction,n, is given byn(λ) = 1 + N χ(1)a (λ),(5.5)(1)

where χa is the atomic linear susceptibility at wavelength λ. Therefore, ∆k can be expressed as∆k = CN ,
h

(1)

(1)

(5.6)

i

where C = 2π
λ χa (λ) − χa (3λ) is a constant describing the microscopic dispersive properties of
the gaseous medium. C can be thought of as the effective wave-vector mismatch per atom. Phasematching is adjusted by tuning ∆k, and thus F1 (b∆k), via the pressure of the gas.
As can be seen from Eqn. 5.3, ∆k must be negative to generate third-harmonic light. Thisoccurs only in media that exhibits negative dispersion (i.e., n3 > n1 ). Xenon was chosen as thenonlinear medium for THG of 118 nm because it is the only rare gas that exhibits negative dispersion(3)

at this wavelength. With only a negatively dispersive gas, ∆k and χef f are both dependent on N .
(3)

If a positively dispersive gas with negligible nonlinearity is added, ∆k and χef f can be decoupled.
Argon is typically used as the positively dispersive gas for THG of 118 nm light.[93] For a mixtureof the two gases, the total wave-vector mismatch at the nominal third-harmonic wavelength, λ0 , isgiven by summing over the partial pressures of each gas,∆k(λ0 ) = NXe CXe (λ0 ) + NAr CAr (λ0 ),= NXe [CXe (λ0 ) + RCAr (λ0 )] ,

(5.7)

57
(3)

NAr
where R = N
is the pressure ratio of the two gases. With the addition of argon, χef f can beXeadjusted using NXe , while ∆k can be maximized using NAr .
Since F1 (b∆k) is maximized when ∆k(λ0 )max = − 2b , phase matching is optimized when thepressure ratio is

Ropt =−2
1
CXe (λ0 )−.
b NXe CAr (λ0 ) CAr (λ0 )

(5.8)

(3)

For large xenon pressures, which are desirable for increasing χeff and thus P3ω , the optimum ratio
Xeapproaches the constant value of C
CAr and is independent of the value of confocal parameter, b.

Using the values CXe (118.2 nm) = −6.12 × 10−17 cm2 and CAr (118.2 nm) = +5.33 × 10−18 cm2 from
Mahon et al., [95] this ratio is 11.48.

5.3

Experimental Setup for Third-Harmonic Generation
The experimental apparatus used to generate VUV light is shown in Fig. 5.2. Light at 355

nm from a Q-switched Nd:YAG laser (Continuum: Inlite III-10, internally tripled) is focused into a
mixture of xenon and argon contained in a stainless steel cell using a 50 cm focal length lens. The
gas cell is made of standard 304-SS conflat vacuum parts with an overall length of ∼ 40 cm. A UV
fused silica window seals the input end of the cell, where the 355 nm light enters. The output end,
between the gas cell and the UHV detection chamber, is sealed using a 25 cm focal length MgF2lens, which also serves to focus the VUV light into the detection region.
The focal length of the lens and size of the cell must be chosen to avoid burning optics andmaximize gas conductance. The cell is three standard conflat crosses connected together and has
a length of ∼ 40 cm. The focal length of the 50 cm lens, external to the cell, was chosen to beapproximately the length of the gas cell. The focus is designed to be in the center of the gas cell, sothat the 355 nm intensity at the input window and output lens are low enough to prevent damaging
the optics. The position of the focus also allows for perpendicular optical access through a conflat
window into the focal region. Although LiF windows have a higher transmission at 118 nm, a58

Figure 5.2: Schematic of the experimental setup. A lens focuses the 355 nm light into the gas
cell where 118 nm light is created. A window placed above the focus of the 355 nm beam allows
perpendicular optical access to the tripling region. The MgF2 lens focuses the 118 nm light anddefocuses the 355 nm in the detection region. 118 nm flux is detected via ionization of deuterated
ammonia (ND3 ). A low flow rate of ND3 fills the detection chamber to a partial pressure of 9×10−8
Torr. ND+
3 ions are accelerated through a time-of-flight mass spectrometer and detected by a microchannel plate (MCP).

59
MgF2 window was chosen for the output lens due to its hardness and because it is not hygroscopic,
allowing the window to handle pressure differentials in excess of 1 atm. The large positive dispersion
in MgF2 causes the focal length of the lens at 118 nm to be about 58% shorter than that at 355
nm. This property is exploited to minimize the intensity of 355 nm in the detection region by
defocusing the 355 nm light, which helps to prevent multi-photon processes. Additionally, mostof the defocused 355 nm light is blocked from even reaching the detection region by a macor iris
placed after the MgF2 lens.
The gas cell is constructed from 2-3/4 inch conflat pieces to provide adequate gas conductance.
Attempts at making portions of the gas cell out of 1/4 inch ID tubing led to trapped pockets ofunmixed xenon, which slowly diffused out over several days. The release of trapped xenon led todrifts in the gas ratio, changing the phase matching and leading to drifts in the amount of 118 nm
light produced.
Xenon and argon are introduced into the gas cell through a leak valve and needle valve. The
needle valve is important for separating the gas mixing cell from the low conductance elements
such as the pressure gauges and gas lines connecting the leak valve. Leaving open the needle valve
between the gas mixing cell and leak valve increases the gas mixing time by a factor of three.
Gas mixing in the cell can be aided by a turbine, which causes gas to flow in a loop of 2-3/4
inch conflat pieces (Fig. 5.2). Gas mixing using the turbine did not reach equilibrium noticeably
faster than without the turbine, but the 118 nm flux varied more smoothly while mixing. Without
the turbine, adding argon to the system would cause large spikes and dips in the measured 118
flux, as the introduction of argon rapidly changed the Xe/Ar ratio in the tripling region. Unaided
mixing in the cell took only 15 minutes with no signs of continued mixing or signal loss observed
over the course of several weeks.
The 118 nm light is detected indirectly by using it to ionize deuterated ammonia (ND3 ) andsubsequently detecting ND+
3 . 118 nm light is focused by the MgF2 lens between two electrodes in
the UHV chamber. Ionized ND3 is accelerated through a time-of-flight mass spectrometer (TOFMS) and is detected by a micro-channel plate (MCP) (Fig. 5.2). The number of ND+
3 ions produced

60
is proportional to the number of 118 nm photons that are focused into the detection region, since
the ionization is a one-photon process and no saturation effects are observed.

5.4

Phase Matching and Conversion Efficiency Limitations
To experimentally determine the phase matched ratio of partial pressures of Xe/Ar, we add a

set amount of xenon to the mixing cell and then add increments of argon in order to map out eachphase-matching curve. Fig. 5.3 shows phase-matching curves for several xenon pressures. Each
phase-matching curve was fit to Eq. 5.3 to extract the location of the peak 118 nm flux and the
optimum Xe/Ar ratio for a given xenon pressure. The optimum Xe/Ar ratios are plotted in Fig.
5.4 along with a fit to Eq. 5.8, which gives b = 2.3 ± 0.3 cm. As shown in Eq. 5.8, the optimum
Xe/Ar ratio depends on the confocal parameter and xenon pressure, but for high xenon pressures
the ratio approaches the asymptotic limit of 11.48 (horizontal dashed line).
Although the optimum pressure ratios are consistent with the model, the conversion efficiencyof 355 nm light to 118 nm does not agree with model predictions. Eq. 5.1 predicts that for a phasematched mixture, the 118 nm light produced should scale as the square of the xenon pressure.
However, mixtures with more than 24 Torr of xenon (290 Torr total pressure) yielded decreasing
118 nm light with increasing Xe pressure. At xenon pressures >70 Torr, almost no 118 nm light was
detected. This same turnover in 118 nm light production versus total pressure has been observedin Shi et al.. [85] In other studies, optimal conversion efficiencies have been observed at lower
total pressures of 20 Torr,[84] 50 Torr,[82] and 200 Torr. [86] The prevalence of optimal conversion
efficiencies at low total pressures and the large discrepancy between the predicted and measured
yield of 118 nm light at high pressures indicates that some process, which is not included in themodels, is significantly limiting the production of 118 nm light.
The deviation from theory for the expected 118 nm flux of a phase-matched mixture of Xe/Ar
at higher total pressures is accompanied by the appearance of a fluorescence streak inside the gas
cell (inset Fig. 5.5). It appears near the focus of the 355 nm light and along the propagation axis.
The fluorescence streak does not appear when 355 nm light is focused in to the cell filled with only

61

Figure 5.3: Phase-matching curves for different pressures of xenon (labeled above). 118nm flux ismeasured while Ar is added to a fixed pressure of Xe and plotted as a function of total pressure.

62

Figure 5.4: Plot of optimum Ar:Xe ratio vs. Xe pressure. The solid line is the output of a fit of
Eq. 5.8 to the data with b = 2.3 ± 0.3 cm as the only fit parameter. The dashed line represents
the asymptotic limit of 11.48 predicted from the model.

63

Figure 5.5: Plot of 118 nm flux and the fluorescence signal vs total pressure for PXe = 50 Torr.
The fluorescence signal is correlated with the presence of 118 nm light. The inset show a pictureof the fluorescence streak through a conflat window.

64
argon or only xenon. Instead it appears only when xenon and argon are both present and near aphase-matched ratio. The fluorescence is collected through a vaccuum window using a lens outside
the gas cell to focus the light onto a photodiode. The fluorescence intensity follows the shape of thegenerated VUV light flux throughout a phase-matching curve as shown in Fig. 5.5, indicating that
118 nm photons play a role in the fluorescence. Fig. 5.6 shows the 118 nm flux and fluorescencesignal as a function of the xenon pressure for a phase-matched mixture. The fluorescence appears to
begin at some threshold xenon pressure and increases roughly linearly in intensity with increasing
xenon pressure.
118 nm light was also created using krypton as the phase-matching gas. Using the same
apparatus and measurements described above, xenon and krypton were phase matched with an
optimum Kr/Xe ratio of 4.6, which agreed with the krypton dispersion calculated in Mahon et al.
[95] This mixture also produced a maximum of 118 nm light followed by a decrease with increasinggas pressure. The fluorescence streak was also present only with a phase-matched mixture of thetwo gasses. (Fig. 5.6). Spectra of the fluorescence were obtained for the Xe/Ar (Fig.5.7 (a)) and
Xe/Kr (Fig.5.7 (b)) systems using an Ocean Optics USB2000 spectrometer. Spectral lines wereassigned using data from the NIST Atomic Spectra Database. [96] We mark the location of all lines
in the NIST database that are above a certain threshold relative intensity [96] in order to identify
observed lines and note which lines are missing from the spectra. For Xe and Kr, a threshold of
500 was used, and for Ar a threshold of 20000 relative intensity was used. The Xe/Ar spectrum(Fig. 5.7 (a)) shows predominantly Xe I spectral lines and weak Ar I spectral lines. The lines are
similar in relative signal to those observed from a xenon arc lamp, indicating that the fluorescenceis likely from electron-ion recombination in xenon. The spectrum for the Xe/Kr system (Fig.5.7(b)) also shows predominantly Xe I spectral lines, with Kr I spectral lines appearing relatively
weakly. Xenon is ionized preferentially despite being present in a lower concentration than thephase-matching gasses (Ar or Kr). This would seem to indicate that ionization is not due to justthe high electric field of the 355 nm light, but a resonant process involving xenon. The energetics
suggest a two-photon process is causing the ionization of xenon. The energy of a single 118 nm

65

Figure 5.6: Comparison of the 118 nm signal and fluorescence streak signal at different xenonpressures for a phase-matched mixture with Ar (Ar:Xe Ratio = 11.3) and Kr (Kr:Xe Ratio = 4.6).

66

Figure 5.7: Spectra of the fluorescence streak created in a phase-matched (a) Xe:Ar mixture (Ar:Xe
Ratio = 11.3) and (b) Xe:Kr mixture (Kr:Xe Ratio = 4.6). Lines corresponding to Ar I, Kr I, and
Xe I are assigned according to criteria described in the text.

67
photon (10.5eV) is below the ionization potential of xenon (12.1 eV). However, the combined energyof one 118 nm photon and one 355 nm photon (14 eV) is enough to ionize xenon, while being below
the ionization potential of argon (15.8 eV). The ionization potential of krypton (14 eV) is low
enough to possibly be ionized in this system, however strong ionization of kryton is not evident.
Two-photon ionization of xenon, where one of the photons is 118 nm and the other is a 355 nm
photon, is likely the mechanism producing xenon ions. The intensity of the 355 nm light is muchlarger than that of the third harmonic, so we can assume the absorption of the 118 nm light is
the rate limiting step. We model this process as two-photon, two-color ionization model where theabsorption of the 118 nm photon is far from resonance and is affected by the pressure in the gas cell
due to pressure broadening. This pressure broadened absorption is the proposed mechanism forloss of 118 nm light in the detection region for high cell pressures. In Xenon the two lines closest 
to resonance with the 118 nm light are the Xe (2 P3/2 )5d 23 line at 119.2 nm and the Xe (2 P3/2 )7s
 line at 117 nm. We also must consider the Kr (2 P1/2 )5d 12 resonance line at 116.5 nm, which
can lead to additional absorption in the case of phase matching with Kr. Argon does not have anytransitions near enough to 118 nm to cause significant loss.

5.5

Absorption Cross Section
We consider far off-resonance absorption in the presences of pressure broadening. The atomic

absorption cross section for light at frequency ω due to an absorption line at ω0 is given byσ(ω) =(γ/2)2f e2,
mϵ0 cγ (ω − ω0 )2 + (γ/2)2(5.9)

where f is the oscillator strength, e is the electron charge, m is the mass of the electron, ϵ0 isthe permittivity of free space, c is the speed of light, and γ is the linewidth. Far from resonance,(ω − ω0 )2 >> (γ/2)2 , and Eqn. (5.9) can be approximated asσ(ω) =

γf e2,
4mϵ0 c (ω − ω0 )2(5.10)

noting that the cross section is proportional to the linewidth. For gasses at high pressures, collisions

68
between atoms contribute to the Lorentzian linewidth of an absorption in a process known aspressure broadening. Neutral atoms undergoing collisions induce a dipole that perturbs the energy
levels and broadens the linewidth of transitions. Collisions between atoms of the same species
experience resonant collisions that enhance this broadening. Same-species broadening is typically∼ 1 GHz×2π per Torr of gas vs. ∼ 10 MHz×2π per Torr of gas for collisions between different
species.[97, 98] In our two-species gas, although the phase matching gas is present in much higher
quantities than xenon, resonant Xe-Xe collisional broadening will dominate the pressure broadening.
The pressure broadened linewidth is given byγ = γn + γc ,(5.11)

where γn is the natural linewidth and γc is the collisional linewidth, which is proportional to thedensity, γc = βN , where β is the collisional broadening coefficient and N is the density of the gas.
For the lines considered in this paper γn ∼ 100 MHz×2π, while γc ∼ 1 GHz×2π per Torr of gas.[99]So, for pressures used in this experiment γn << γc , and γn can be ignored such that γ ∼= γc . This
allows us to write our pressure broadened cross section asσ(ω) =

f e2
βN,
4mϵ0 c (ω − ω0 )2= αN ,(5.12)

where α is defined as our pressure broadening constant.

5.6

Simulation Results
In order to compare our pressure broadened absorption model to the experimental data, we

simulate the simultaneous tripling and absorption processes in our gas system. The electric fieldof an electromagnetic wave with frequency, ω, and wavenumber, k, propagating along the z-axis isgiven byE(z, t) = Aei(kz−ωt) + c.c.,(5.13)

69
where A is the electric field amplitude. The intensity of the electric field is related to the electricfield amplitude by I = 2nϵ0 c|A|2 . The change in the intensity of the 118 nm beam as it propagates
through a medium can be simulated by calculating the change in the electric field amplitude. The
differential equation describing the propagation of the 118 nm beam is given by

d|A118 |11= χ(3) NXe k118 |A355 |3 − σNXe |A118 |,dz
2
2(5.14)

where χ(3) is the third order susceptibility, NXe is the density of xenon, k118 is the wavenumber ofthe 118 nm light, A355 is the field amplitude of the 355 nm beam, σ is the absorption cross section,
and A118 is the field amplitude of the 118 nm beam. The first term describes the third harmonic
generation of 118 nm light assuming perfect phase matching and the second term describes the lossof 118 nm light due to absorption. Using Eqn (5.12), this becomesd|A118 |112= χ(3) NXe k118 |A355 |3 − αNXe
|A118 |.
dz
2
2

(5.15)The 355 nm power is kept constant, assuming the third harmonic generation leads to negligible
loss. To account for the fact that the 355 nm beam is focusing through the gas cell, the electricfield amplitude along the z axis is scaled by the radius of the beam
|A355 (z)| = |A355 (0)|where ω0 is the beam waist, ω(z) = ω0

ω0
,
ω(z)

(5.16)p
1 + (λz/πω02 )2 is the beam radius as a function of z, andA355 (0) is the field amplitude at the focus calculated from experimental parameters.
Although the 118 nm beam also has a transverse spread that changes through the focus, the118 nm amplitude is not scaled by the beam radius. This assumes that the effects of the transverse
spread are negligible on relative measurements across our parameter space. The majority of thethird harmonic generation takes place within the Rayleigh range, where the beam radius varies from√
ω0 to 2ω0 . We assume this variation is negligible and the 118 nm light is created with roughly
the same beam radius. Linear absorption does not affect the width of a Gaussian beam, thus the70transverse expansion of the beam is the same for all parameters, and ignoring the expansion does
not effect relative measurements.
The differential equation is numerically integrated over the 40 cm length of the gas cell, withthe focus at the center. The 118 nm field amplitude at the end is used to calculate a beam flux,
since our experimental results are a measure of the flux of the 118 nm photons. Additionally, we
assume that the amount of light from the fluorescence streak is proportional to the loss of 118 nm
light in a region around the focus of the 355 nm beam that approximates our fluorescence collection
region.
Using our simulations, we want to model the experiment shown in Fig (5.6), then compare
the simulated and experimental results to extract a value for α. The two free parameters in thissimulation are the pressure-broadening constant, α, and an overall vertical scaling of the magnitudeof the signal. For a single value of α, the simulation is run for a range of xenon pressures to
produce a curve of 118 nm flux vs pressure similar to Fig (5.6). The simulated curves are fit to
the experimental data by calculating a sum of squared residuals between the experimental andsimulated results for a given value of α and vertical scaling factor.
The fitted simulation results for the Xe/Ar case are shown in Fig 5.8. The model accurately
reproduces the shape of the experimental 118 nm flux. A value of αXe = 0.84(1) × 10−37 cm5 is
extracted from the fit, where the error bar represents only the statistical uncertainty in the fit.
We can compare this value of αXe to one calculated using an oscillator strength f = 0.379 from
Chan et al. [100] and a collisional broadening coefficient β = 5.6GHz/T orr extracted from theexperimental data in Ferrel et al. [99]. The two lines in xenon nearest to the 118 nm light are the 
Xe (2 P3/2 )5d 32 line at 119.2 nm and the Xe (2 P3/2 )7s line at 117 nm. The contribution due to the
117 nm line is negligible since the oscillator strength is a factor of 4 smaller and the collisional line
width is a factor of 5 smaller than the 119.2 nm line. The expected value of αXe is calculated to be
1.18 × 10−37 cm5 , in reasonable agreement with our experimentally determined fit value considering
the simple model used.
In the case of phase matching with krypton, the Kr (2 P1/2 )5d

1
2

resonance line at 116.5 nm

71Figure 5.8: Comparison of simulated and experimental results for Xe/Ar. Experimental 118 nm
flux and fluorescence signal are plotted as green circles and squares respectively. Simulated 118
nm flux and fluorescence signal are plotted as red dashes and dots respectively. The simulated
fluorescence curve is independently scaled by eye.

72
also leads to absorption of 118 nm light. This can be accounted for with an additional loss term of

d|A118 |dz

12= − αKr NKr|A118 |,
2
Kr

(5.17)

where αKr is the pressure-broadening coefficient due to krypton and NKr is the density of krypton.
Since the density of xenon and krypton are related by the phase matching ratio, Eqn. (5.15) canbe rewritten as

1
1
d|A118 |2= χ(3) NXe k118 |A355 |3 − αXe/Kr NXe|A118 |.
dz
2
2

(5.18)

where αXe/Kr = αXe + αKr R2 . The results of this model are fit to the experimental Kr results in
Fig 5.9, with the model reproducing the shape of the experimental curve. From the fit, a value of
αKr = 4.6(2)×10−39 cm5 , can be compared to an expected value of αKr = 7.1×10−39 cm5 calculatedusing an oscillator strength f = 0.193 from Chan et al. [100] and a collisional broadening coefficientβ = 3.0GHz/T orr extracted from the experimental data in Ferrel et al. [99].
For both Xe/Ar and Xe/Kr, the fluorescence results from the model do not match the experimental results well. However, there is some qualitative agreement. The fluorescence appears toturn on near the pressure where the 118 nm flux starts to turn over. The experimental fluorescenceappears to increase linearly while the simulated results appear to increase non-linearly. This may be
attributed to uncertainties in the fluorescence measurement. The fluorescence streak is several cm
long, and extends beyond the window through which we have optical access to collect fluorescence.
We collect only a small portion of the fluorescence streak around the focus.

5.7

Discussion and Conclusions
Third harmonic generation of 118 nm light in a Xe/Ar gas cell is a widely used table top

source of VUV radiation. The yield of 118 nm light has been observed to diverge from simple
theoretical predictions. The observation of ionized xenon at the focus of the 355 nm beam in thepresence of 118 nm light suggests two photon ionization, led by absorption of 118 nm light, is the73

Figure 5.9: Comparison of simulated and experimental results for Xe/Kr. Experimental 118 nm
flux and fluorescence signal are plotted as green circles and squares respectively. Simulated 118
nm flux and fluorescence signal are plotted as red dashes and dots respectively. The simulated
fluorescence curve is independently scaled by eye.

74
mechanism limiting the amount of 118 nm light that exits the gas cell. To confirm this mechanism,
we developed a model based on pressure-broadened absorption that can be modeled using a simple
1-D simulation. The simulated results are fit to experimental data to extract a pressure-broadening
constant, which agrees with values calculated from theory.
A 118 nm light source could be optimized by considering design options that limit the amountof absorption. Minimizing the amount of xenon the 118 nm light must pass through will increase
the yield. This can be accomplished by shortening the gas cell and tightening the focus, but thedamage threshold of optical components must be considered. Another option might be using a
jet of gas rather than a gas cell. While third harmonic generation has been studied in a pulsed
supersonic jet of pure xenon, [101, 102] jets of phase matched mixtures have not yet been studied.

Chapter 6

Electrostatic Trapping of OH

In this chapter, we present a reproduction of our paper ”Measurements of trap dynamics ofcold OH molecules using resonance-enhanced multiphoton ionization” by Gray et al. published in
PRA in 2017 [60] The abstract and acknowledgements have been removed, and the bibliographicalreferences and formatting have been changed to match the format of this thesis.

6.1

Introduction
Confining cold molecules in electric, magnetic, or optical traps allows for precision studies ofmolecules and their interactions due to long interrogation times. Neutral molecule trap lifetimes
can range from seconds to minutes [103, 104, 105], depending on the residual background gas
pressure and blackbody optical pumping rates [106]. These long interaction times are required to
study molecular interactions in cold systems, as the density of trapped samples is typically very low(103 − 108 cm−3 ) [104, 103, 107, 108]. The long lifetimes increase the probability that interactions
can take place before the molecules are ejected from the trap by collisions with background gas orby photon scattering [106]. There have been many recent experiments that have explored various
electrostatic [109, 110, 111, 112] and magnetic [113, 114, 115] trapping geometries, and others
that demonstrated additional in-trap cooling [116, 117, 118, 119, 120, 121]. Importantly, several
experiments have taken advantage of trapped molecules to study the properties of the molecules
themselves, including vibrational relaxation of OH [122] and NH [123]. Additonally, studies ofmolecular interactions have benifited from trapped samples including cold atom-molecule collisions

76[25, 124, 125], molecule-molecule collisions [126], and collisionally assisted evaporative cooling [127].
(Here, we restrict our discussions to cold molecules not created by assembly of ultracold atoms.)
One of the most challenging aspects of these experiments is detecting the trapped molecules.
The two main methods are laser induced florescence (LIF), where emitted photons are detectedusing a photo-multipler tube, and resonant ionization, where the resulting ions are detected using
a microchannel plate (MCP) detector. Resonant ionization typically has a higher efficiency, as ionscan be detected with nearly unit efficiency, while photon collection in LIF is limited by the solid
angle of the detection optics, which is only a few percent at best. LIF detection not only produces
a lower signal, but can also have a larger noise contribution, as the molecules are trapped close toreflective surfaces, which increases the scattered light impinging on the detector. For ion detection,
the ions are usually sent through a time-of-flight mass spectrometer (TOFMS), nearly eliminating
the contribution to the noise from other molecular species. Finally, because of the low signal-to-noise(S/N) ratio, experiments that use LIF must illuminate the entire cloud of molecules to increase thenumber of photons emitted by the molecules. These measurements produce information about only
the total trapped number or average density and not about the spatial distribution of the moleulesin the trap. Resonance enhanced multiphoton ionization (REMPI), on the other hand, has a highenough S/N ratio that the ionization laser can be focused to illuminate a small portion of the trappedsample. Then, by making measurements of the molecular density at different laser positions, we
can profile the density distribution in the trap. The density distribution in the trap can be used todetermine the energy distribution or temperature, which are important parameters to measure formany cold molecule experiments. The importance of measuring the density distribution, and notjust the number in the trap, was shown in the study of collisions between electrostatically trappedammonia molecules and magnetically trapped rubidium atoms [25]. Since the density, and thus
the collision rate, varies spatially in the trap, extracting collision cross-sections requires a detailed
understanding of the molecular distribution. Therefore, studies of trapped molecules benefit greatly
from an efficient and spatially selective resonant ionization scheme.
Convenient REMPI schemes have been used to measure the density distribution and under-

77
stand collisional dynamics with trapped ammonia [25]. However, until recently, the OH radical,
which is a fundamental molecule in atmospheric [11], interstellar medium [14, 15], and combustion
chemistry [12, 13], did not have an efficient REMPI scheme identified and all trapped studies used
LIF [104, 126, 127, 122, 128, 129]. In 2011, the Lester group demonstrated a new UV + VUVREMPI scheme to state-selectively ionize ground-state OH radicals using 118 nm light as the VUVphoton [130, 131, 59]. They used the REMPI scheme to examine the OH products from unimolecular dissociation of Criegee intermediates [9, 10]. Here, we present the first measurements of trapped
OH molecules using this ionization detection scheme. This method allows us to measure not only
the relative number of molecules in the trap, but also the temporal evolution of the spatial density
distribution. We use this new capability to measure trap loading dynamics to optimize the loading
process and determine the energy distribution in the tap. Proper trap loading and characterizationof the energy distribution are important for future atom-molecule collision experiments.

6.2

REMPI of OH
The 1+1’ REMPI scheme used to detect trapped OH molecules uses two photons; the first

photon causes a bound-to-bound transition around 281 nm and the second photon (118 nm) excites
the molecule to an autoionizing Rydberg state whose lifetime is on the order of a picosecond [59].
The relevant energy structure of OH is shown in Fig. 6.1. The trapped molecules are in the groundrotational (J = 3/2) and vibrational (ν = 0) state of the X 2 Π3/2 electronic state. The first photon(UV) is tuned to the A2 Σ+ , ν = 1 ← X 2 Π3/2 , ν = 0 transition. There are several possible rotatonal
states to tranfer the molecule into with the UV photon. The stongest A ← X transision is Q1 (1.5)
shown in Fig. 6.1. Beames et. al observed that in a supersonically expanded jet of OH, the highest
ionization signal was achieved when the UV photon was resonant with the Q21 (1.5) transition [131].
However, this transition originates from the lower Λ-doublet level and only the upper Λ-doublet
level is trappable with the static electric fields used in our experiments. The strongest transition
that originates from upper Λ-doublet level is the R21 (1.5) transition, which is used in the work
presented here. The second step in the process transfers the molecule to an autoionizing Rydberg

78(A 3Π, 3d)

118 nm

7
2
5
2

-

5
2
3
2

+
+

3
2
1
2
1
2

A 2Σ+ (ν = 1)+J

281 nmQ1(1.5)

3

2

+

-

Q21(1.5)

R21(1.5)

X 2Π3/2 (ν = 0)

J p
Figure 6.1: Energy level diagram of OH showing the relevant levels for the 1 + 1’ REMPI detection
scheme. The lowest two levels are the Λ-doublet states of X 2 Π3/2 . The first photon is resonantwith the A2 Σ+ , ν = 1 ← X 2 Π3/2 , ν = 0 transition and is denoted by R21 (1.5). The second photontransfers the molecule into the A3 Π, 3d Rydberg state, from which it autoionizes. J is the total
angular momentum, and p is the parity. Vertical spacing between energy levels is not to scale.

79
state using a 118 nm photon.

6.3

Experimental Setup
The experimental setup is shown in Fig. 6.2. OH radicals are produced at the exit of apiezoelectric transducer (PZT) valve by electric discharge of 1% water seeded in krypton. The
molecules are cooled both internally and translationally in the supersonic expansion, producing
a beam of OH radicals moving with a mean velocity of 415 m/s and mostly in their rovibronic
ground state. The molecular beam is slowed to 34 m/s by a pulsed Stark decelerator, which uses
time varying inhomogeneous electric fields to decelerate polar molecules [132]. An electrostatic trap
located at the end of the decelerator is used to reduce the mean velocity of the molecules to nearlyzero by switching the potentials on the four trap electrodes on and off at different times during theloading process, (Fig. 6.3). The final potentials on the trap electrodes form a 3D trap for OH witha trap depth of 700 µK in the z direction and 250 µK in the x and y directions (Fig. 6.2).
Molecules in the trap are detected using the 1+1’ REMPI scheme. The two detection laser
beams enter the center of the trap perpendicular to the molecular beam axis through a 3 mm gap
between the two central trap electrodes. A frequency-doubled pulsed dye laser, which produces
light at 281 nm, excites the OH molecules on the R21 (1.5) transition as shown in Fig. 6.1. The
281 nm beam has a beam waist of 2 mm in the y direction and is focused by a 50 mm focal length
cylindrical lens to have a beam waist of around 0.5 mm in the z direction at the trap center. The
pulse energy of this laser is 4.7 mJ per pulse, which is enough to saturate the transition.
To create the 118 nm light, the third harmonic of a pulsed Nd:YAG laser at 355 nm is
frequency tripled in a gas cell containing xenon and argon [133, 134, 135]. The 355 nm light isfocused with a 500 mm focal length lens into the gas cell containing 24 torr of xenon mixed withargon at an Ar:Xe ratio of 10.3:1. The xenon provides the non-linear medium, while the argon
is used to satisfy the phase-matching condition. We estimate that we create 1 nJ/pulse of 118
nm light. A 250 mm focal length MgF2 lens, which serves as the vacuum window and separates
the gas cell from the ultrahigh vacuum trapping chamber, focuses the 118 nm light to a diameter

80

Figure 6.2: Experimental setup showing the last two rod pairs of the Stark decelerator, four trap
electrodes, detection lasers, and MCP detector. The red rectangular prism represents the 281 nm
laser beam, while the blue cone represents the 118 nm light. The molecular beam propagates along
the z-axis. (Not to scale)

81

Final
Decelerator Rods

Trap Geometry
Trap Electrodes

0.5 Trap On

Longitudinal Potential Energy (cm-1)

0.0
Optimized
Non-Ideal

Trap Off0

0.5 Trap On

0.0
Trap Off0

0.5 Trap Half On

0.0
0.5 Trap On

0.0
-15

-10

-5

0

Distance (mm)510Figure 6.3: Diagram of trap loading process for both an optimized and a non-ideal sequence. The
location of the final decelerator electrode and the four trapping electrodes are shown at the top.
The longitudinal potential energy curves seen by the OH molecules are shown in the panels forthree different configurations of voltage applied. The “Trap on” configuration has the following
voltages applied in order from left to right (decelerator, left trap endcap, left trap ring, right trap
ring, right trap endcap) = (0, -10, +10, +10, -10) kV. “Trap Half on” has the following voltage
configuration (0, 0, +10, 0, -10) kV, and “Trap off” has all electrodes grounded. The blue (top)curve represents the potentials seen by the central molecule in the optimized sequence and theorange (bottom) curve represents the non-ideal sequence. The switches between configurations areshown by vertical lines.

82
estimated to be 30 µm at the center of the trap. The chromatic dispersion of the MgF2 lens is used
to collimate the residual 355 nm light, the majority of which is blocked by an alumina aperture
before the trap. The position of the focus of the 118 nm laser can be scanned both vertically andhorizontally using a computer-controlled closed-loop picomotor (New Focus Model 8310).
The central molecular column density and density distribution along the y-axis is measured
during the trap loading process, after the trap has been loaded, and after the trap has been turned
off to allow the cloud of molecules to expand. During ionization, the potentials on the four trap
electrodes are rapidly switched off during the laser pulses, and then changed to (3, 3, 2, 0) kV
to create a uniform electric field and accelerate the ions through a TOFMS to an MCP detector.
The signal from the MCP corresponding to mass/charge = 17 is recorded via a multichannel
scaler (Stanford Research Systems SR430). For all experimental data presented below, each point
represents 1000 individual laser shots (100 seconds of integration) with error bars representing thestatistical uncertainty in the measurement.

6.4

Molecular Dynamics Simulations
To help interpret the experimental results, three-dimensional molecular dynamics simulations

are performed in two stages. The first stage simulates the trajectory of the molecules from the nozzleof the PZT valve to the end of the Stark decelerator. The molecules experience time-varying forces
due to the interaction of the molecules’ electric dipole moment and the time-varying electric fieldcreated by the high-voltage electrodes. More information on this simulation can be found in Refs.
[25, 61]. Molecules are then propagated through the trapping region using a second simulation
using the phase-space distribution of the molecules from the decelerator simulation as the input.
The vector force fields for each configuration of the potentials on the electrodes are generated via
a commercial finite element solver, COMSOL, and linearly interpolated to the location of eachmolecule at every time step. A second-order, symplectic, velocity-Verlet integrator is used topropagate the molecules forward in time. A integration time-step of 2 µs is used to minimize
computation time without sacrificing required accuracy. To mimic experimental measurement

83
conditions, every 50µs, we create a 2D histogram of the number of molecules in the y-z plane,
where each bin of the histogram is 50 µm wide in the y- and z-dimensions and is 6 mm long in thex-dimension. This simulated measurement scheme allows us to record data with high spatial andtemporal resolution to compare with experimental data. All trapping simulations were run withan initial number of 3.5×106 molecules.

6.5

Results and Discussion

6.5.1

Trap loading dynamics

Efficiently loading cold molecules from a decelerator into a trap is critical to be able toperform subsequent experiments, as the number density exiting a decelerator is already low. We
use information about the time evolution of the central molecule column density and the in-trap
density distribution to optimize loading into a quadrupole electrostatic trap. Previously, only the
total number was used in this optimization procedure, which could cause situations where thecentral column density of the sample was not at a maximum, leading to poor initial conditions forcollision studies.
Figure 6.4 shows the column density, which was produced by integrating through the xdimension, at the center of the trap during the loading process for two sample loading procedures(See Fig. 6.3). The optimized scheme (Fig. 6.4(a)) was created by varying trapping stage times tomaximize the column density at the center of the trap at long times in the trap (15 ms). The nonideal scheme (Fig. 6.4(b)) was made by using a similar optimization procedure, but timings werevaried to maximize the column density about 1 mm away from the trap center in the longitudinal
direction. This produced an example of a scheme that traps a large number of molecules, butdoes not produce a high density in the trap center. The non-ideal trapping scheme also induces
both longitudinal and transverse oscillations as the trap turns on with the center-of-mass of themolecular cloud displaced from the center of the trap (See Supplemental Material [136] for phasespace movies of both trapping schemes). Typical LIF measurements have used an excitation laser

84

1 0

E x p e r im e n t
S im u la tio n

O H C o lu m n D e n s ity ( a r b . u n its )

(a )

2 0
1 0

5

-1 0
-2 0

0
2 0

(b )

1 0

5

z V e lo c ity ( m /s )

0

0
-1 0
-2 0

0
3 .5

4 .0

4 .5

5 .0

5 .5

T im e ( m s )

6 .0

6 .5

-2

-1

0

1

2

z P o s itio n ( m m )Figure 6.4: Measured central column density during the trap loading process with corresponding
molecular dynamics simulations. The measured central density is shown with error bars connectedby lines to guide the eye and the simulations results are shown with dashed lines. (a) Results fromthe optimized loading scheme. The final trap configuration is turned on at 4.34 ms. (b) Resultsfrom the non-ideal loading scheme. The final trap configuration is turned on at 4.30 ms. The
plots at the right correspond to the simulated longitudinal phase-space distribution after 15 ms oftrapping.

85
with a beam waist on the order of millimeters. So, the two schemes would produce similar LIF
signals as they have similar numbers of molecules in the same volume even though the local density
within the trap is different.
The results of molecular dynamics simulations are also shown for these loading schemes,
and show good agreement of the time evolution of the central column density. Small mismatches
between the results of the simulations and the experimental measurements are likely due to thesmall inaccuracies in the initial molecular distribution used for the trapping simulations. The
simulated final phase-space distribution in the trap for the two loading schemes is shown at theright of Fig. 6.4. For the optimized loading scheme, the most dense portion of the phase-space
distribution resides at low energies, whereas the molecules are at higher energies in the non-ideal
loading scheme. The non-ideal loading scheme produces a distribution that would be unsuitable forcollision experiments due to its lower number of molecules in the center of the trap and its higher
most probable velocity.
In addition to measuring the peak density as a function of time for the loading, we can alsomeasure the dynamics of the density distribution along the y-axis as shown in Fig. 6.5. Here,the widths are determined by measuring the ions created as the 118 nm laser is scanned along they-axis of the trap. Since the trapping potentials are not well approximated by either a harmonicor linear potential, an analytic model for the distribution can not be derived from our trapping
potentials, so a Pearson IV function was chosen to describe the distribution [137]. The widthsof the experimental and simulated distributions were found by fitting to a Pearson IV function
and calculating the full width at half maximum (FWHM) from the fit parameters (Fig. 6.6). For
the optimized loading scheme, there are few oscillations in the transverse cloud width or the peakdensity. However, even after a ms in the non-ideal case, there are observable oscillations in boththe cloud width and peak density, indicating the cloud was loaded with excess energy.

86

2 .0

(a )

E x p e r im e n t
S im u la tio n

1 .5

F W H My a x is ( m m )

1 .0
0 .5
0 .0
2 .0

(b )
1 .5
1 .0
0 .5
0 .0
4 .3

4 .4

4 .5

4 .6

4 .7

4 .8

4 .9

5 .0

T im e ( m s )Figure 6.5: Width of the trapped OH cloud as a function of time for the (a) optimized loadingand (b) non-ideal loading scheme. The squares represent the experimental measurements, while thesolid lines represent simulation results. For both cases, the simulations reproduce the modulation in
the width of the cloud after it is trapped. The optimized loading scheme results in fewer oscillationsin the cloud width. The statistical uncertainty is on the order of the point size or smaller.

87

4

O H C o lu m n D e n s ity ( a r b . u n its )4 .3 04 .3 04 .7 5
4 .7 53m s D a ta
m s F itm s D a ta
m s F it

2

1

0
-1 .0

-0 .5

0 .0

0 .5

1 .0

y P o s itio n ( m m )Figure 6.6: Measured density profiles of the OH molecules in the trap using the non-ideal loading
sequence. These data were recorded directly after the trap was loaded (4.30 ms) and at 450 µs later(4.75 ms). The lines are Pearson IV distribution fits to the experimental data. The widths of theexperimental density profiles are determined by extracting the FWHM of a Pearson IV distribution
fit.

88
6.5.2

Ballistic Expansion of OH cloudWith the ability to measure the density profile, the energy distribution can be determined
experimentally from the expansion of the molecules after the trapping potentials have been turned
off. Although this method of determining the temperature is standard practice in cold atom
experiments, it is not often used in cold molecule experiments. Previous experiments have measured
the ballistic expansion of Rydberg molecules from a moving trap; however, their method is uniqueto molecules that can be easily field ionized [138, 139]. Using REMPI to ionize the molecules
provides a more general method to measure energy distributions in cold molecule experiments.
a Figure 6.7 shows the change in the width of the cloud after the trap turns off for the
optimized loading scheme after 15 ms of trapping. We fit the width of the cloud as a function ofptime to the following model σy (t) = (σy (t = 0))2 + (Vrms t)2 , where σy (t = 0) and Vrms (rootmean-square velocity) are fitting parameters, and t is the time after the trap is turned off. Fitting
σy (t) to the experimental and simulated results gives transverse Vrms of 5.5 ± 0.2 m/s and 6.2 ±0.2 m/s respectively. This model assumes the cloud is in thermal equilibrium and is contained ina harmonic trapping potential. Although neither of these assumptions are satisfied by our system,
we use this model to determine a reasonable value for the Vrms . Additionally, one can calculate
the rms velocity using the initial size in the trap and the linear potential energy gradient, while
employing the virial theorem. Using this, the rms velocity for the experimental measurements (5.92± 0.07 m/s) and results of simulations (5.13 ± 0.02 m/s). These measured velocities correspond toa temperature of 50 mK. All the uncertainties stated are statistical. Because the system does notsatisfy the conditions of thermal equilibrium or harmonic trapping, the systematic uncertainties
likely dominate over the statistical ones. However, all measurements are self-consistent, leading toreasonable confidence in the results.

89

1 .2

y a x is ( m m )0 .8

F W H M

1 .0

0 .4

0 .6
S im u la tio n
E x p e r im e n t

0 .2
0 .0
0

5 0

1 0 0

F r e e E x p a n s i o n T i m e ( µs )1 5 0

Figure 6.7: Width of the cloud as it expands after the trap is turned off. The molecules weretrapped using the optimized loading scheme and held for 15 ms before release in both the experiment and simulation. The initial size of the cloud in the trap differs between the experiment andsimulations. This is likely due to the lack of knowledge of the initial distribution of molecules
leaving the decelerator.

90

6.6

Conclusion
Cold, trapped samples of OH molecules could be a useful starting point to explore molecularinteractions and cold chemistry. Being able to characterize the trapped samples beyond just total
trapped number will be important for these studies. Using a recently developed 1 + 1’ REMPI
scheme, we have been able to measure the non-uniform density distribution in a trapped sample
of OH. This detailed information has allowed us to optimize the loading of OH into an electrostatic trap. Additionally, the new measurement technique provides information about the trappedmolecules’ energy distribution from the in-trap cloud size and from the rate of expansion after thetrapping fields have been turned off. Previously, these dynamics of trapped OH molecules havebeen inferred from simulations, but never before observed in experiments.
Future work utilizing this detection method includes studying collisions between OH andultracold Rb atoms. Measuring the density distribution of trapped OH molecules will allow forexperimental determination of elastic and inelastic scattering cross-sections in the cold moleculeultracold atom system. These measured cross-sections will help to improve theoretical models of
atom-molecule collisions in the 100 mK temperature regime.

Chapter 7

Measuring Cold Collisions Between OH and Rb

7.1

Introduction
One of the most fundamental rules of thermodynamics is that heat flows from hot to cold. Inorder to cool something, you must bring it in contact with with something that is colder. In orderto further cool cold molecules with temperatures on the order of tens of millikelvin, we need tobring the molecules in contact with something even colder. Since large density samples of ultracold
atoms can be easily made in the laboratory, ultracold atoms are often used as the thermal reservoir
in which the heat of another species can be dumped. The process of using one cold gas sample tocool another species is known as sympathetic cooling. Sympathetic cooling has been accomplishedby using ions to cool ions[31], ultra cold atoms to cool atoms[32], and ultra cold atoms to coolions[33]. The current frontier in sympathetic cooling is the prospect of using ultra cold atoms tofurther cool cold molecules, moving these molecular species from the millikelvin to microkelvinregime.
Sympathetic cooling of trapped molecules requires a pair of collisions partners with an elastic
cross section that is much larger than the inelastic cross section. Elastic collisions will lead to atransfer of kinetic energy from the hotter species to the cooler species, which drives the cooling.
Inelastic collisions, however, are collisions that cause a change in the internal quantum state ofone of the species. The large anisotropy of atom-molecule collisions can lead to a large inelastic
collision cross section, which presents a unique problem for sympathetic cooling of molecules. [140]In order to use static fields to trap a molecule, the molecule must be in a weak-field-seeking

92
state. However, since energy splittings in a field come from energy levels repelling each other,a weak-field-seeking state cannot be the absolute ground state of the molecule. For a trappedmolecule, there will always be an energy level below the trapped state for the molecule to fall into
due to inelastic collisions. Thus, inelastic collisions will lead to the molecule changing state into an
untrapped state and being lost from the system. Since it can take 10-100’s elastic collisions to cool
the molecules but only a single inelastic collision to knock it out of the trap, the elastic scatteringcross section must be much larger than the inelastic to have efficient cooling.

7.2

TheoryThe large anisotropy of atom-molecule interactions leads to large inelastic scattering crosssections since angular momentum quantum numbers are no longer conserved.[140] The combinationof the 2 Π OH molecule and the 2 S Rb atom ground states leads to the 1 A′ , 1 A”, 3 A′ , 3 A” potential
energy surfaces.[141] There is also an ionic state with the same 1 A′ symmetry as one of the covalentstates, which leads to an avoided crossing that could have a large effect on the scattering dynamics.
Figure 7.1 shows a couple of these potential energy surfaces, demonstrating the large anisotropy of
the interaction.
Lara et al. [141] generated and used the above potential energy surfaces to calculate theinelastic and elastic cross sections of the Rb-OH interaction. They use a basis set that considers
the hyperfine state of the Rb atom,|Rb⟩ = |fa mfa ⟩(7.1)

where f is the total spin angular momentum f = s + i, with i being the nuclear spin angularmomentum. The subscript (a) is used to denote the quantum numbers of the atomic species, vs a
subscript of (d) being used when describing the diatom species, the diatomic OH molecule. The
basis for the OH molecule is93

Figure 7.1: Potential Energy Surfaces of the OH-Rb interaction. From [140].

94|OH⟩ = sd λ̄ω̄ϵfd mfd(7.2)

where f = j + i, and linear combinations of the spin-orbit states are used to produce statesof good parity1λ̄ω̄ϵ = √ (|λω⟩ + ϵ |−λ − ω⟩).
2

(7.3)

With the inclusion of the partial wave quantum numbers |LML ⟩ to account for the relative orientation of the collision pair in the lab frame, the total basis is given bysd λ̄ω̄ϵfd mfd |fa mfa ⟩ |LML ⟩ .

(7.4)

The incident channels that are considered are shown in Figure 7.2, with channel C1 being
the most relevant to our experiment. C1 includes both Rb and OH in their fully stretched states,|OH⟩ = |ϵ = f, fd = 2, mfd = +2⟩ and |Rb⟩ = |fa = 2, mfa = +2⟩. In our experiment, Rb is trappedin the |2, +2⟩ state. The hyperfine states of OH that are decelerated and trapped are all of thosethat are fully stretched by electric fields, or with Ωmj = −9/4. So the states of OH that are present
in the trap are |OH⟩ = |ϵ = f, fd = 2, mfd = ±2, ±1⟩. [142]The elastic and inelastic cross sections for channel C1 can be found on the top of Figure 7.3.
The cross sections are generally in agreement with two different semiclassical models that describe
the behaviour for a higher and lower energy regime. In the lower energy regime, below 10−4 K, thebehaviour of the cross sections follows the prediction of the Wigner threshold law. [140] The Wigner
threshold law states that as the energy of a collision goes to zero, the cross sections related to anelastic or isoenergetic process will approach a constant value. For an exoergic process, the cross√section will vary as 1/ E. For our system, the OH and the Rb being in the fully stretched statesleads to inelastic collisions being an exoergic process. The constants that elastic cross sections
approach and the constant of proportionality for exoergic processes are strongly dependent on thepotential energy surfaces. The cross sections that Lara et al. derive from their potential energy

95

Figure 7.2: Quantum States in the OH-Rb interaction. The states relevant to the experiment are
represented by Channel C1. From [140].

96

Figure 7.3: Theoretical Cross Sections. From [140]97
surfaces paint a grim picture for the prospect of sympathetic cooling. The inelastic cross section isalready a factor of 2 above the elastic cross section at 1mK, and they quickly diverge to the pointthat inelastic scattering is already an order of magnitude more likely at 10 µK.
For higher energies where many partial waves can contribute, above 10−2 K, the inelastic crosssection is predicted by the Langevin capture model. [143] In this model, the centrifugal barrier
defined by the long range potential
V (R) =

−C6 L(L + 1)
+
R6
2µR2

(7.5)

gives a maximum partial wave L(E) for a given energy E that will exceed the centrifugal barrier.
The Langevin model assumes that any collision that overcomes this centrifugal barrier leads toan inelastic collision with unit probability. When the atom and molecule get near enough to eachother, the internal state of both is scrambled and can change to any energetically allowable state.
This produces a Langevin cross sectionσLangevin = 3π

C6
4E

1/3
.

(7.6)

where C6 = 325Eh a60 is the isotropic vanderwalls coefficient of the PES.[141] For our trapped OH
with a temperature of 50mK, we are within the regime where the Langevin model seems to bevalid, and the expected cross sections is σLangevin ≈ 2 ∗ 10−13 cm2 or 2000Å2 . Efficient cooling toany point below the starting temperature of the OH does not seem feasible. While the elastic crosssection does appear to be larger than the inelastic cross section in the tens of millikelvin region,
the elastic and inelastic cross sections are on the same order of magnitude and fast loss from thetrap would likely obscure any observable effect of cooling.
There are some areas where the theory used to calculate these cross sections deviates fromthe conditions used in this experiment. The calculations described above are field free and do notaccount for the effects an electric field would have. The collision partners in this experiment could
experience fields on the order of tens of kV/cm due to the electrostatic trap used to confine themolecules. Electric fields have been shown to strongly affect the inelastic cross sections of cold

98
atom-molecule collisions. The effect is attributed to a competition between the field, which tends
to quantize the molecule at long distances along a lab fixed axis, and the potential anisotropy,
which tends to quantize the molecule at short distances along the intermolecular axis. At someintermediate distance during a collision, the two effects balance which leads to an indeterminate
quantization axis. This would tend to increase the inelastic cross section and decrease the elastic
cross section.
Another difference between the conditions of this experiment and the theoretical calculations is the hyperfine levels included. The calculations above use OH in the state |OH⟩ =|ϵ = f, fd = 2, mfd = +2⟩ as the incident channel, and consider any change of mfd state to induce aloss from the trap. The states of OH that will be decelerated and trapped in our experiment are thestates |OH⟩ = |ϵ = f, fd = 2, mfd = ±2, ±1⟩. [142] The mfd = −2 and ±1 states would need to beconsidered as incident channels to produce a more detailed picture of the theoretical expectationsof this experiment. These cross sections would be further complicated by the presence of fields.
While states with the same |mf | are degenerate in electric fields, the magnetic fields would breakthis degeneracy. [142]
Furthermore, collisions that change the state of the OH molecule from one trapped state toanother trapped state would not lead to trap loss and could appear in the experiment as elastic
scattering rather than inelastic scattering. However, this effect may be negligible based on thetheory presented by [141]. The states of OH that are trapped have the same product mj Ω = −9/4
and the quantum numbers can be broken down as follows• mf = +2 −→ Ω = −3/2, mj = +3/2, mi = +1/2
• mf = +1 −→ Ω = −3/2, mj = +3/2, mi = −1/2
• mf = −1 −→ Ω = +3/2, mj = −3/2, mi = +1/2
• mf = −2 −→ Ω = +3/2, mj = −3/2, mi = −1/2
The nuclear spin projection mi are untouched by potential energy couplings and will not be changed

99
in a collision,[141] which rules out many of the possible transitions between two trapped states. The
transitions where mi is conserved involve a very large change of ∆mj = ±3, which would further
lessen this effect. The cross sections calculated by Lara et al. are seen to decrease the more that
mf is changed, with the cross section of a ∆mj = −3 transition being an order of magnitude lessthan for ∆mj = 0. So this possible effect would likely not significantly increase the experimentallymeasured cross section nor significantly decrease the inelastic cross section.

7.3

Experimental Expectations
Due to the complex dynamics of the co-trapped environment, extracting cross sections fromexperimental measurements cannot be accomplished with analytical models, but instead requires
comparison to simulated results. However, we can construct an approximate analytical model toestimate what we expect to see from our experiment. To determine if the cross sections calculatedby Lara et al. are experimentally feasible to measure, we can estimate the rate of collisions andwhat effect that will have on our measured signal. Since the theoretical predictions indicate efficientcooling is unlikely, we start by considering loss from the trap due to inelastic collisions assuming
elastic collisions have a negligible effect. We assume that the colder rubidium atoms are approximately stationary and the contribution to the collision energy is negligible. We approximate all of
the molecules to have a constant velocity equal to the average velocity of the cloud. The densityof Rb atoms that the OH molecules move through is approximated as the measured peak density.
The inelastic collision rate per OH molecule is given by
Γi = σi vOH ρRb .

(7.7)

where vOH is the average velocity of the OH molecules, which we measured in Chapter 6 to be
5m/s, and ρRb is the peak density of Rb, which we measure to be 1010 atoms/cm3 . This evaluates toeach molecule experiencing an inelastic collision at a rate of approximately 1Hz, which corresponds
to a 1/e lifetime
τi =

1
≈ 1s
Γi

(7.8)

100
induced by inelastic collisions causing loss from the trap. In this measurement of the OH molecule
lifetime, inelastic collisions with Rb are not the only process causing loss from the trap. There
are also collisions with background gas and blackbody radiation pumping [144], which lead toa background lifetime, which we measure to be 2.49 ± 0.15s. Lifetimes add in parallel and themeasured lifetime will be given by the expression

1
1
1
= +
τm
τi τ0

(7.9)

where τm is the lifetime measured in the presence of rubidium and τ0 is the lifetime measured in
the absence of rubidium.
We can use Equation 7.9 to estimate the experimental sensitivity to the inelastic collisioncross section. Figure 7.4 plots the estimated lifetimes that would be expected to be measured for
different inelastic collision lifetimes with τ0 = 2.49s. In order to distinguish a statistically significant
difference between the measured lifetime and the background lifetime, the measured lifetime would
need to differ by 2στ¯0 from τ0 , where στ¯0 = 0.15s is the standard error of the mean of τ0 . The shaded
red area in Figure 7.4 represents this area that excluded by the sensitivity of our experiments The
plot of Equation 7.9 crosses this level at τi ≈ 18s, which corresponds to an inelastic cross section ofσi ≈ 111Å2 . So, the experiment should be sensitive enough to measure inelastic cross sections on
the order of 10−14 cm−2 and set an upper limit on the cross section if a significant effect were notseen. On the lower end of τi and higher end of σi , we are only limited by how small of a lifetime wecan measure. This will be determined experimentally by the uncertainty in the interaction time,
which is limited by eddy currents induced in the stainless steel vacuum chamber when the magnetic
coils turn off. These eddy currents persist for ∼ 1 − 10ms. To measure a lifetime, we need several
time points, so we could measure an inelastic lifetime down to ∼10ms and set an lower limit on the
inelastic cross section on the order of 10−10 cm−2 .
Estimating the effect of elastic collisions on the signal that we see will be a bit more complicated. Elastic collisions will tend to shrink the width of the OH cloud, which also increases the101

Figure 7.4: Measured lifetime vs collision induced lifetime. As the inelastic cross section decreases(increasing τi ) the measured lifetime τm approaches the background lifetime τ0 . The shaded red
area represents the values of τm excluded by our estimated sensitivity.

102
density in the center of the trap. The distribution of the molecules in our trap seems to fit best
to a Pearson IV distribution, since the trapping potentials are not well approximated by a linear
or harmonic potential and the molecules are not in thermal equilibrium.[60] However, to simplify
the math for our estimations, we will model the density as a 3D gaussian. The density of the OH
molecules in the trap is then given by


 2

N (t)
x
y2
z2
1
n(x, y, z, t) =
exp −
+
+
2 σx2 (t) σy2 (t) σz2 (t)(2π)3/2 σx (t)σy (t)σz (t)(7.10)

where N(t) is the total number as a function of time, and σx , σy , and σz are the gaussian widths ofthe cloud in the x, y, and z directions. The total number over time is simply an exponential decay
with a lifetime of τm from Equation 7.9

N (t) = N0 e−t/τm .

(7.11)

In our quadrapole electric field trap with cylindrical symmetry, the widths in all three directions
can be related to a single width by1
σx = σy = σz = σ0 .
2

(7.12)


2N0 e−t/τm
1 x2 + y 2 + 4z 2n(x, y, z, t) =exp −2
σ02 (t)(2π)3/2 σ03 (t)(7.13)and Equation 7.10 becomes

When detecting the OH molecules, the ionization laser integrates through the y-axis to give acolumn density. Assuming that the transverse extent of the laser is much smaller than the cloud,the column density is given bync (x, z, t) =
2N0 e−t/τm
1 x2 + 4z 2exp−2(2π)σ02 (t)σ02 (t)(7.14)

103
By measuring this column density as a function of x by scanning the ionization laser with thepico-motor, we can get a measure of the width of the cloud by fitting to a gaussian. By measuring
the change in the width over time we can get an estimate of the elastic cross section.
To estimate the effect of the elastic cross section on the width of the cloud, we need to relate
the width of the cloud to the elastic collision rate. From the Virial theorem, we say that the
average potential energy is half of the thermal energy. The potentials are approximately linear, so
we can relate the average potential energy to the average position to get the width as a function oftemperature.

1
kB T = ⟨Epot ⟩ = d∇E⟨x⟩
2kB T
σ(T ) = ⟨x⟩ =2d∇E(7.15)(7.16)

where d is the dipole moment of OH and ∇E is the gradient of the electric field.
The temperature loss due to the average kinetic energy lost by the molecules per elastic
collision with the atoms is given by [145]∆Tm =−(Tm − Ta )
κ

(7.17)

where Tm is the temperature of the molecules, Ta is the temperature of the atoms, and κ isthe ratio of the sum of the masses to the reduced massκ=

(mOH + mRb )2
2mOH mRb(7.18)For the OH-Rb system, κ = 3.66. When the temperature of the molecules is much larger
than the atoms, 1/κ = 0.27 is the fraction of the temperature lost from the molecules undergoing
one elastic collision on average. Since the width of the cloud is proportional to the temperature,
the width will decrease by the same fraction as the temperature. The temperature as a function of
the average number of collisions is given by

104

Tm (N ) = Ta + (T0 − Ta )e−N/κ

(7.19)

Equation 7.19 is plotted in Figure 7.5 for a Rb temperature of Ta = 500µK and OH temperature T0 = 50mK. It takes 2.5 collisions to drop the temperature by a factor of 2 and 8.5
collisions to drop the temeprature by an order of magnitude. In our experiment, we do not expect
the molecule cloud to survive long enough to see many elastic collisions. In the region where theaverage number of collisions is low, Ta << T0 and Equation 7.19 can be approximated byTm (N ) = T0 e−N/κ

(7.20)The temperature could be expressed as a function of time by noting that the number of collisions
is the amount of time elapsed multiplied by the elastic collision rate
Tm (t) = T0 e−Γe t/κ = T0 e−σe vOH ρRb t/κ

(7.21)

This model predicts the temperature and width of the cloud decay exponentially over time with alifetime of1= σe vOH ρRb /κτσ(7.22)

This model neglects the effect of the cooling on the velocity of the OH molecules, which will also
be a function of the temperature
1
1
2
kB T = ⟨Ekin ⟩ = mvOH
2
2r
kB T
vOH (T ) =m(7.23)(7.24)As the cloud cools the average velocity decreases, and the elastic collision rate will slow. Since theinelastic collision rate depends in the same way on the velocity of the OH molecules, the inelastic
collision rate will also be slowed by elastic collisions. The velocity scales as the square root of thetemperature, so the velocity will drop at a slower rate than the temperature, limiting the effect thecooling has on the collision rates.

105

Figure 7.5: OH molecule cloud temperature vs average number of elastic collisions per molecule for
an initial OH temperature of 50 mK and initial Rb temperature of 500µK. The red stars represent
the values for integer number of collisions to help guide the eye.

106
The main measurement in our experiment is the column density at the center of the trap,
which in our model is given by Equation 7.14 when taking x and z to be 0nc (t) =2N0 e−t/τm(2π)σ02 (t)(7.25)

Substituting in our expression for the width we can see that the column density as a function oftime is proportional to the product of three different exponentials

nc (t) ∝ e−t/τ0 e−t/τi e2t/τσ(7.26)

an exponential decay due to the background lifetime, an exponential decay due to inelastic
collisions, and an exponential growth due to the shrinking of the width. The measured lifetime of
the column density will be a combination of these exponential factors

1
1
1
2
1
1
=
+ −
=
+
τm
τ0 τi τστ0 τcol(7.27)

where τcol is the modified lifetime induced by collisions, which can be expressed in terms of theelastic and inelastic collision cross sections
1
2
2= Γi − Γe = vOH ρRb (σi − σe )
τcol
κ
κ

(7.28)

This expression implies that the elastic collisions can have a large effect on the measured lifetime of
the column density if the elastic cross section is on the order of the inelastic cross section. Equation
7.28 equals 0 when the elastic cross section is greater than the inelastic cross section by a factor of
κ/2 = 1.88, which is not unlikely based on the field free theoretical predictions around 50mK.

7.4

Collision Measurement
The apparatus used to measure cross sections is illustrated in Figure 7.6. Rubidium atoms

are trapped and cooled in a magneto-optical trap (MOT). Once the MOT is loaded to the desired
level, the atoms are optically pumped and the anti-Helmholtz coils that produce the quadrapole

107

Figure 7.6: OHRbIT Apparatus. The Rb atoms are trapped in a separate glass chamber, separated
by a gate valve. The coils are mounted on movable track that transports the coils outside of vacuum
while holding the atoms inside the vacuum chamber. The path of the atoms inside of vacuum is
an L shape, represented by the dashed red line. The atoms are loaded through an aperture in theback of the electrostatic trap.

108
magnetic trapping field are ramped to 500A. This traps a total of N ∼ 109 atoms with a peakdensity of n ∼ 1010 atoms/cm3 at a temperature of T ∼ 500µK. The magnetic coils sit outside thevacuum chamber on a series of two movable tracks. The first track moves the atoms along the atom
trapping chamber through an all-metal gate valve and into the science chamber to align with theelectrostatic molecule trap in y direction. The second track moves the atoms through the back of
the molecule trap to align with the center of the trap. After the atoms are in place, the molecules
are fired from the valve, decelerated, and trapped as described in Chapter 6. The molecules aredetected via REMPI as described in Chapter 4.
The ionization signal of the OH is plotted in Figure 7.7 as a function of interaction time withand without atoms. The signal with atoms present decreases much quicker than without atoms.
This is clear evidence that we are observing inelastic collisions between OH and Rb. When fitted
to a simple exponential decay, the lifetime without atoms is τ0 = 2.49(15)s and the lifetime withatoms is τm = 0.94(7)s. The lifetime of the column density due to collision is then τcol = 1.51(18)s,
or 1/τcol = 0.66(9)Hz.
From this collision induced lifetime, we can estimate the elastic and inelastic cross sections.
Equation 7.28 can be rearranged to give
σi =

1
τcol

vOH ρRb+2
σeκ

(7.29)

which is a relation between the cross sections that depends on the measured collision induced
lifetime. The values of σi and σe will lie along a line with slope 2/κ. The values in Equation 7.29
that vary with experimental conditions and results only change the intercept of the line. Equation
7.29 is plotted as the black line in Figure 7.8. The horizontal dashed red line represents the inelastic
cross section predicted by the Langevin model. The vertical dashed red line represents the elastic
cross section predicted by Lara et al. at 50mK.
The use of the average velocity for vOH and the peak density for ρRb are both overestimatesof the conditions the OH molecules most of the time in the trap. A particle in a trap spends
most of its time at a slower velocity near the edges of the trap. To produce a likely underestimate

109

Figure 7.7: OH Signal vs Interaction Time. The blue points are without Rb atoms present andthe red data is with Rb atoms present. Fit to an exponential decay, the lifetime without Rb atoms
is τ0 = 2.49(15)s and the lifetime with atoms present is τm = 0.94(7)s.

110

Figure 7.8: Relation between Inelastic and Elastic Cross Sections. The Inelastic and elastic crosssections lie on a line, with the intercept being determined by the measured OH decay curve. The
solid black and dotted blue lines represent under and overestimates of the intercept described in
the text. The horizontal and vertical dashed red lines represent the theoretical predictions of theinelastic and elastic cross sections respectively.

111
of these parameters, we take the product vOH ρRb to be an order of magnitude lower than our
initial estimate. This shifts the intercept to produce the dotted blue line in Figure 7.8. The true
cross sections likely lie in the region between the solid black and dotted blue lines. To constrain
the inelastic cross section, we could use the change in the width of the OH cloud to estimate theelastic cross section. Preliminary data (Figure 7.9 shows no significant change in the width over aninteraction time of 1s, so the elastic cross section is on the lower side of this graph and the inelastic
cross section is likely on the order of 10−13 cm−2 . The theoretical predictions in Lara et al. areclaimed to be within a factor of two, and the estimates based on our experimental data seem toagree with those prediction. In order to extract values and quantitative uncertainties for the crosssections from our data, we will need to use Monte Carlo simulations of the co-trap environment.
This process is described below in the Future Work section of this chapter.

7.5

Data Taking Procedure
In order to isolate the effects of the atoms on the molecules, a procedure was designed toensure the only change between shots with atoms and without atoms was the presence of the atoms.
For the shots without atoms, the only difference in procedure is that the laser that repumps atoms
in the MOT is shuttered, preventing atoms from being trapped. The atom trapping apparatus stillgoes through the procedure of waiting the ∼15 seconds it takes the MOT to load, ramping up themagnetic fields, and moving the magnetic coils to overlap with the molecule trap. The componentsof the molecule trapping apparatus that may pick up noise still sees all of the possible noise sourcesfrom the atom trapping apparatus, such as vibrations caused by the movement of the track andelectrical pick up noise from the large changing magnetic fields from the trapping coils.
The time points were chosen to have a variable interval meant to capture the fast decay
expected for the signal with atoms and the slow decay expected for signal without atoms. From
preliminary data, we determined an interval of 100ms at short times would be appropriate tomeasure the fast decay. The lifetime without atoms is ∼ 2.5s, and time points were chosen to be
far enough out in time to fit the lifetime appropriately. The data is interleaved, alternating between

112

Figure 7.9: Preliminary scans of the OH cloud density profile at interaction times of 0s and 1s. The
red squares are the initial distribution at interaction time of 0s. The blue circles are the distribution
after interacting with Rb atoms in co-trap for 1s. The similarly colored lines are a Gaussian fit to
the data. The Gaussian width for 1s, σ(1s) = 1.02(6)mm, does not differ significantly from thewidth at 0s, σ(0s) = 1.00(6)mm

113
a shot with atoms and a shot without atoms. A pair of shots with and without atoms takes ∼ 50s.
We take 60 shots for each time point, for a total integration time of ∼ 16 hours.
Due to the long integration time, the procedure was designed in order to minimize the possible
effect of systematic error sources that may change over time, such as laser powers and wavelengths.
A vector of the time points is created in random order with no repeats. A pair of interleaved shots
is taken at each time point before creating a new randomized vector to repeat each time point.
In this way, the number of shots taken for each time point is the same on the time scale thatexperimental drifts may happen. Each single shot of the experiment is recorded with a time stamp.
Some experimental conditions, such as the status of the laser locks, are also recorded with a timestamp which allows data to be rejected if there are known problems that might affect the qualityof the data. A histogram of the single shots at each time point can be compared to a Poisson
distribution to check if the data appears to follow counting statistics. The data can also be checked
for an abundance of 0 count shots, which may indicate an intermittent problem destroying thesignal.
Data taking is interrupted at regular intervals, corresponding to a set amount of shots at eachtime point. During these interludes, some benchmark signals are measured in order to check thatthe experiment is operating in a consistent manner. This is important to ensure that all of the data
is directly comparable across the long integration time. First, the molecule signal at zero interaction
time is checked to ensure there has not been a change in the initial number of molecules in thetrap or the REMPI detection efficiency. The most common problem that may affect the measured
molecule signal is drifts in the wavelength of the dye laser that produces the 281nm light. The Xe:Ar
gas mixture for producing the 118nm light can also require refreshing. Next, the number of atoms
is checked for interaction times of 0s and 1s. The atom number is determined by transporting theatoms to overlap with the electrostatic trap for a set amount of time before transporting the atoms
back to the glass cell where the atoms are measured using absorption imaging. The atom number
is measured at 0s interaction time to check the initial number of atoms that are trapped. Deviationfrom previously measured atom numbers are most likely due to wavelength and alignment drift of

114
the optical pumping laser. The atom number is then measured at 1s interaction time as a checkon the alignment of the magnetic trap to the electrostatic trap. If the atom number at 1s deviatesfrom what was previously measured, the likely cause is an error in the positioning of the track
along the z-axis. While this track mostly operates within the expected encoder error on the orderof 10µm, we have occasionally observed the track to spontaneously generate an error of 1mm andrequire realignment.

7.6

Co-Trap Alignment and Characterization
In order to measure collisions effectively, and accurately simulate the measurements madeby the apparatus, we need an accurate co-alignment of our electrostatic trap, magnetic trap, andionization laser. The electrostatic trap is stationary, so we need to move the magnetic trap and
118 nm beam to overlap with the center of the electrostatic trap. Figure 7.6 shows the geometryof the combined trap apparatus. The electrostatic trap creates a quadropole electric field withapproximately cylindrical symmetry. The strong axis of the electrostatic trap is aligned to thez-direction, while the radial direction of the trap is in the x-y plane. The anti-Helmholtz coils of
the magnetic trap are not aligned along the same axis as the electrostatic trap. The magnetic
quadropole field has its strong axis aligned to the x-direction and the radial symmetry is in the y-z
plane.
In the co-trap environment, the Rb atoms and OH molecules both will experience Zeeman
and Stark shifts due to the magnetic and electric fields present. The OH molecule trapped in the
fully stretched Stark state includes mf levels that are both weak and strong field seeking. However,the Zeeman shifts the molecules will see are small enough relative to the Stark shifts to be negligible,and do not affect the trap depth or the dynamics of the molecules in the trap. The Rb atoms,
however, are polarizable and experience a Stark shift that significantly affects the dynamics of theatoms in the co-trap. The Hamiltonian describing the dc Stark effect on Rb is1
1
3J 2 − J(J + 1)
HS = − α0 Ez2 − α2 Ez2 z
2
22(2J − 1)(7.30)

115
where α0 and α2 are the scalar and tensor polarizabiliteis of Rb.[146] When J=1/2 the second term
vanishes. For the S1/2 state of Rb 87, the scalar polarizability is α0 = h ∗ 0.0794Hz/(V /cm)2 . This
Stark potential decreases with increasing electric field magnitude and is quadratic. Near the centerof the co-trap the linear Zeeman potential dominates and the Rb atoms remain trapped. However,there is a point around 1-2 mm from the trap center where the Stark shift overcomes the Zeeman
potential, leading to holes in the trap that cause loss from of Rb atoms form the co-trap. The
number of Rb atoms follows a double exponential decay of in the presence of the electric fields.
The initial fast decay is due to the loss of atoms by the reduction of the trap depth. While this loss
does limit the number of atoms available for collisions, the effect can be exploited to help align thecenters of the traps. If the centers of the traps are offset in some direction, the co-trap potential
will be asymmetric with one side being weaker than a properly aligned trap. Misalignment leads
to a faster trap loss, and the lifetime of the atoms is maximized when the co-trap is well aligned.
One might think that alignment along the z-axis could be achieved by scanning the ionizaiton
laser over the trapped molecular cloud and assigning the center to the point where OH+ ion signal
is maximized. However, attempting this alignment would introduce a few problems. First, thecounterpropogating 281 nm laser beam, which provides the first photon in the REMPI detection
scheme, is focused through the trap with an estimated beam waist of 500µm. Thus, the 281 nm
beam would need to be realigned by hand for each new alignment of the 118 nm beam in orderto deconvolve the beam width from the measurement of the cloud. This realignment would betedious to repeatedly perform. It could also potentially introduce systematic errors with constantly
having to re-optimize the detection efficiency at each new position. Second, the time of flight of
the OH+ ions would change with ionization laser position. To reduce the effect of noise peaks,
only peaks that have an arrival time within a small time window are assigned as OH+ and counted
as part of the ion signal. This time window would need to be identified and shifted for each newlaser alignment. With a single laser ionization scheme, these potential detection problems would
be alleviated.
Instead, we first align the atoms along the z-axis to the electric field then align the ionization

116
laser to the atoms. To align the atoms along the z-axis we exploit the fact that the lifetime of theatoms is reduced when the electrostatic trap and magnetic trap are not aligned. The fast loss of
atoms from the co-trap in the first 100’s of us is the main cause of reduction in atom number.
The alignment procedure involves (1) moving the atoms into the electrostatic trap (2) pulsing theelectric fields on for 1s (3) moving the atoms back to take an absorption image (4) fitting theabsorption image to determine the remaining amount of atoms. The surviving atom number isplotted in Figure 7.10 This procedure is repeated while varying the final positions of the magnetic
trap. Loss of atoms due to the electric fields is minimized when the center of the magnetic trapaligns with the center of the electric fields. Since the z-axis is the strong axis of the electrostatic
trap and one of the weak axis of the magnetic trap, the z-axis is the most sensitive to this alignment
with a FWHM of 200um. This suggests our determination of the center of the trap can be on theorder of 10’s of um from a Gaussian fit, when our 118nm laser beam waist is 50um. We should be
able to align the center of the atom distribution to the center of the electric trap well enough to use
the atoms as a tool to align the ionization laser. Using the picomotor to scan the ionization laser
while measuring the Rb+ ion signal, we find the center of the distribution by fitting to a Gaussian.
The Rb Ionization signal is plotted in Figure 7.11
Alignment of the 118 nm laser in the x-direction (lab vertical) is easily achieved using apicomotor scan of the molecules in the trap, shown in Figure 7.12 This scan is fit to a Gaussian
whose center is at the trap center, which is determined with an error of 10s of um. The laser isthen set to this center and its position acts as a reference for aligning the atoms to the verticalcenter of the trap. While monitoring the Rb+ ion signal, we vary the difference between currents
applied to the magnetic trapping coils, shown in Figure 7.13. This has the effect of scanning theatoms across the laser beam, and the center of a Gaussian fit of the Rb+ ion signal corresponds tothe alignment of the magnetic trap to the electric trap.
Alignment of the atoms in the y-direction is limited to measuring atom loss due to E-fields atvarious alignments. The y-axis is a weak axis for both traps, and the atom loss is not as sensitive
to misalignment as the axes. We move the atoms into the electrostatic trap and hold for 8 seconds

117

Figure 7.10: Alignment of the Rb atoms along the z-axis. The Rb atoms are brought to theelectrostatic trap and the trapping fields are turned on. After 1s the electrostatic trap is turned off
and the Rb atoms are transported back to the glass cell where the survival number is determinedby absorption imaging.

Figure 7.11: Alignment of the 118nm laser along the z-axis. After alignment of the Rb atoms to theelectrostatic trapping fields along the z-axis, the 118nm laser position is scanned along the z-axis
to find the center of the Rb atom cloud.

118

Figure 7.12: Overlap between Rb atoms and OH molecules along the x-axis. A Gaussian fit to the
OH molecules is used to determine the correct alignment of the ionization laser along the x-axis.

Figure 7.13: Alignment of the Rb atoms along the x-axis. With the ionization laser kept at
the center of the electrostatic trap, the Rb atoms are displaced along the x-axis by changing thedifference between the trapping coil currents.

119
to enhance the sensitivity. The signal is shown in Figure 7.14 This signal still has a FWHM on theorder of mm, but a Gaussian fit gives us the center with an error of 10s of um.

Figure 7.14: Alignment of the Rb atoms along the y-axis. The Rb atoms are brought to theelectrostatic trap and the trapping fields are turned on. After 8s the electrostatic trap is turned off
and the Rb atoms are transported back to the glass cell where the survival number is determinedby absorption imaging.

Chapter 8

Conclusion

8.1

Summary
In this thesis, we have demonstrated a measurement of collisions between cold OH molecules

and ultracold Rb atoms in a co-trapped environment. Rb atoms are trapped in a quadrapole
magnetic field at a temperature of T ∼ 500µK. The atoms are transported through the vacuum
chamber to overlap with an electrostatic quadrapole trap that is used to trap OH molecules. The
OH molecules are slowed using a Stark Decelerator and trapped in the electric quadrapole trapwith a temperature of ∼ 50mK. Collisions are inferred from changes in the distribution of the
OH molecules in the presence of Rb atoms. Initial estimates of the cross sections from this data
can place the inelastic cross section on the order of O(10−13 )cm2 . This is in good agreement withthe full theoretical calculations by Lara et al. and the Langevin model. [141] This supports theidea that the internal state of the OH molecule is ”scrambled” anytime the molecule is near an Rb
atom. Preliminary measurements show no change in the OH energy distribution over a 1 second
interaction time, indicating the elastic cross section is much lower than the inelastic. In order tomore precisely extract values for the cross sections from our measurements, the experimental dataneeds to be compared to simulated data. This process is described in the Future Work section
below.
A new 1+1’ REMPI scheme was developed for detecting decelerated and trapped OH molecule.
The advantages of this REMPI detection were discussed, which give it a much higher signal-tonoise ratio than the typical LIF detection. This high signal-to-noise ratio allows the ionization

121
laser to be focused to a much smaller detection volume, which gives the ability to measure thedensity distribution of OH molecules in the trap by scanning the ionization laser across the cloud.
The usefulness of measuring the density distribution of the OH molecules was demonstrated bymeasuring the temperature of the trapped OH, as well as examining the dynamics of trapping.
The 1+1’ REMPI scheme requires light at 118nm, which is in the vacuum ultraviolet regionof the electromagnetic spectrum. We described the theory and experimental implementation of
producing 118nm light by frequency tripling 355nm light in a phasematched mixture of Xe and Ar.
The power of 118nm light that can be produced was shown to be limited by pressure broadened
absorption of 118nm light by Xe.

8.2

Future work
In order to extract the elastic and inelastic cross sections from the experimental data, we needto simulate all of the complex dynamics of the molecules, atoms, and collisions that are occurring
simultaneously. This can be achieved using trajectory simulations that propagate the atoms andmolecules on the trapping potentials and incorporate conditions that change the populations due
to collisions. This will be done in a multi-stage process that uses a purely atomic simulation to
generate an analytical model of the Rb density that can be used to generate a probabilistic model ofatom-molecule collisions based solely on the molecular motion. The atomic simulation is separated
from the molecular simulation for a few reasons. First, the number of Rb atoms present in the cotrap outnumbers the number of molecules by four orders of magnitude. The number of simulated
OH particles would be severely limited by the memory and speed requirements of simulating a
proportional number of Rb atoms. Second, particle collision models are inherently slow due to theneed to search for pairs of particles that are close enough to trigger a collision event. This is further
compounded by the need to include Rb-Rb collisions that are important to the dynamics of thetrapped Rb atoms. Third, the velocity of the Rb atoms contributes negligibly to the probability orenergy of collision. Finally, the effect of OH-Rb collisions on the Rb distribution are negligible dueto the significantly larger number of Rb atoms present. An average atom-molecule collision rate

122
for the OH molecules of 1Hz would mean an average atom-molecule collision rate of 10−4 Hz for
the Rb atoms. Therefore, the Rb trajectories can be simulated independent of the OH molecules,
and the Rb density can be treated as a mean field that the OH propagates through.
The Rb trap simulations will be modeled by generating an initial distribution of particles
expected from the purely magnetic trap, then propagating those particles along potentials basedon the Stark and Zeeman shifts in the co-trap. The potentials are calculated from electric andmagnetic fields generated by COMSOL, a commercially available finite element modeling software.
The initial distribution is generated from random initial velocities and positions weighted by aBoltzmann distribution. The distribution of the Rb as a function of time is given by 2NRb (t)1
xy2
z2n(x, y, z, t) =
exp −
+
+
2 σx2 (t) σy2 (t) σz2 (t)(2π)3/2 σx (t)σy (t)σz (t)(8.1)

where NRb is the total number of Rb atoms, and σx (t), σy (t), and σz (t) are the Gaussian widths ofthe cloud in the x, y, and z directions. The number is determined from absorption imaging of thecloud after transporting back to the MOT cell. The number is fit to a double exponential decay
with the form
τ +τ−t τ1 τ 2

NRb (t) = (N0 − N1 )e

1 2+ N1 e−t/τ2(8.2)

The widths can be modeled to evolve in time as a double exponential with the form
σi (t) = λ1 e−t/β1 + λ2 e−t/β2 + λ0

(8.3)

The width can be measured in the x dimension by scanning the ionization laser across the cloudusing the picomotor. However, the other dimensions cannot be measured easily in our experiment.
The measurements in the x dimension are used to fit the simulations, which give the fit parameters inother two dimensions. Once these fit parameters are generated, the density of Rb can be calculated
at any position and time in the trap.
The collisions simulations will propagate OH trajectories along the trapping potentials. The
initial phase space distribution of OH molecules comes from Monte Carlo simulation of molecules
propagating through the decelerator and trap. As the particles are propagated in the trap, the123instantaneous molecular velocity and Rb density are determined for each particle during each time
step. The probability of either type of collision collision happening during a given time step is
Pi = Γi · δt = ρRb vOH σi δt(8.4)

Pe = Γe · δt = ρRb vOH σe δt(8.5)

The time step needs to be chosen such that the probabilities are much less than 1. For eachmolecule, a random number is generated between 0 and 1 for each time step. If the number is lessthan Pi , an inelastic collision event has occurred and the molecule is removed from the simulation.
If the number is between Pi and Pi + Pe , an elastic collision event has occurred, and the kinetic
energy of the molecule is reduced according to Equation 7.17 with a random recoil angle. If thenumber is greater than Pi + Pe , no collision has occurred and the molecule is unchanged.
To extract the cross sections from the experimental measurements, the simulation will be
repeated with many different pairs of values for σi and σe . Each simulation produces a simulated
decay curve which can be compared to the experimental decay curve to determine a quality of fit
via a reduced-chi-squared fit. From the landscape of fitness values, the likely range of cross sections
can be determined within a confidence interval.

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