Pushing the Limits for Directly Cooled MoleculesbyDavid L. Reens
S.B., Massachusetts Institute of Technology, 2011
M.S., University of Colorado, 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
2019This thesis entitled:
Pushing the Limits for Directly Cooled Molecules
written by David L. Reens
has been approved for the Department of Physics

Prof. Adj. Jun Ye

Prof. Eric Cornell

Date

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

iii
Reens, David L. (Ph.D., Molecular Physics)
Pushing the Limits for Directly Cooled Molecules
Thesis directed by Prof. Adj. Jun Ye

The subfield of cold and ultracold molecules is growing increasingly mature, with exciting
results from laser cooled species, from precision measurement experiments, and from atomic dipolar
competitors. These exciting developments have placed an appropriately stimulating pressure on
direct cooling experiments to hurry along and deliver on long expected promises of high number
densities and in-trap collisional e↵ects. Towards this end, this thesis reports on several fundamental
improvements in the key techniques employed. The first plugged magnetic traps for cold dipolar
molecules are presented. Deceleration is modified to enable unprecedented improvement in thetransverse trapping capability of the devices. Beam skimming is also made more e↵ective, trap
loading more efficient, and designs are initiated which will allow hold times of unprecedented
length.

iv

Acknowledgements

First and foremost I acknowledge my lovely wife. Abby, you have an incredible love andappreciation for the simple things in life- diligence, routine, and data collection! Without you as
a life partner, I am certain I would have traded in the PhD life for a flashy but fleeting trendy
job ages ago. I’m so glad we’ve stuck it out together. It’s been a worthwhile struggle, and we’recoming out stronger in so many ways. Kudos for finishing first.
I am also very grateful to my family for their support and encouragement of me as a scientist. I
owe much of my self-confidence as a researcher to my mother’s limitless confidence in my capabilities.
During our shared high school experience thanks to her role as math teacher, her joyful celebrationof my accomplishments shaped my goals and my future.

v

Contents

Chapter
1 Introduction

1

1.1

Atomic Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

Molecular Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3.1

Laser Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3.2

Direct Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

The OH Molecular Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4.1

The Rigid Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.4.2

The Vibrational Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.4

1.5

1.6

Angular Momentum Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5.1

Revisiting J=L+S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5.2

Coupling Many Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Stark and Zeeman E↵ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Creation
2.1

14Supersonic Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1

Rovibrational Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.2

Translational Disequilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.3

Experimental Transverse Temperature . . . . . . . . . . . . . . . . . . . . . . 20

vi
2.2

2.3

Pulsed Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1

Miniature Solenoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.2

Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.3

Lorentz Force Opening Mechanism . . . . . . . . . . . . . . . . . . . . . . . . 28

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Skimming

30

3.1

Aerodynamic Skimming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2

Cryogenic Skimming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3

Decelerator Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.1

Volcano Skimming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.2

Double Skimming. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4

Lower Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Hexapole Focusing

42

4.1

Phase Space Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2

History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3

Cryogenic Hexapole 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4

4.3.1

Hexapole Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3.2

Core Electrode Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3.3

Thermal Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3.4

Performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Cryogenic Hexapole 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4.1

Cryogenic Hexapole 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.5

Guided Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.6

Cryogenic Hexapole 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.7

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

vii
5 Optimized Deceleration

68

5.1

Conventional Decelerator Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2

Alternative Geometries and Configurations . . . . . . . . . . . . . . . . . . . . . . . 73
5.2.1

Flaws in Conventional Deceleration . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2.2

Alternative Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2.3

Alternative Field Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.4

The E↵ective Moving Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3

Transverse Trap Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4

Non-adiabatic Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.5

Decelerator Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.5.1

Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.5.2

Mechanical Considerations

5.5.3

High Voltage Arcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.5.4

High Voltage Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.5.5

Capacitance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.5.6

Di↵erential Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6 Molecular Trapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

105

6.1

History of OH traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2

Tricycle over Ring Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.3

6.2.1

Toroidal Destabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.2.2

Loading Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.2.3

Microwave Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.2.4

Limitations to Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.2.5

Ring and Tricycle Thermometry . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.2.6

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

Adiabatic Transitions during Loading . . . . . . . . . . . . . . . . . . . . . . . . . . 122

viii
6.3.1
6.4

Multi-pulse Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Pin Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.4.2

Loss Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.4.3

Experiments and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.4.4

Scaling Law Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.4.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.5

Trap Escape and Dynamical Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . 139

6.6

Next Generation Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.7

6.6.1

Pin Trap Farewell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.6.2

Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.6.3

Photon Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.6.4

High Voltage Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.6.5

Blackbody Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.6.6

Cloverleaf Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Trapping of Water Isotopologues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7 Detecting Collisions

152

7.1

Controlled Density Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7.2

Fitting Trap Decay Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.2.1

7.3

Electric Field Induced Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Comparison to Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.3.1

Simulating Electric Field Induced Loss . . . . . . . . . . . . . . . . . . . . . . 165

7.4

Anticipated Collision Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

7.5

Forward Minus Backward Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.6

Spatial Density Enhancements by Microwave Spectroscopy . . . . . . . . . . . . . . . 174

7.7

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

ix
Bibliography

177

x

TablesTable
1.1

Molecular Rotational Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

Molecular Vibrational Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1

Final Supersonic Expansion Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2

Final Supersonic Expansion Temperatures . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1

Phase Space Manipulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2

Guiding Coupling Efficiency Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.1

The Ye Group Molecule trapping endeavor. All but the last have been experimentallyrealized.

8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.2

Final Density Comparison in Ring and Tricycle . . . . . . . . . . . . . . . . . . . . . 121

6.3

E-field Enhanced Loss Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.4

Optimization Table for the Cryocycle Trap . . . . . . . . . . . . . . . . . . . . . . . 145

xiFiguresFigure1.1

Level Positions with 96 State Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1

Transverse Phase Space from a Supersonic Expansion . . . . . . . . . . . . . . . . . 19

2.2

Molecular Beam Imaging PSF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3

Inferring Transverse Temperature from Spot Size . . . . . . . . . . . . . . . . . . . . 22

2.4

Mounted Even-Lavie Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5

Even-Lavie Yield verse Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6

Lorentz Wire and Sealing Device of a Nijmegen-Style Valve . . . . . . . . . . . . . . 29

3.1

Cryogenic vs Conventional Skimmer Comparison . . . . . . . . . . . . . . . . . . . . 32

3.2

Volcano Skimmer Mechanical Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3

Double Skimmer Mechanical Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4

Skimmer Variations for Decelerator Coupling . . . . . . . . . . . . . . . . . . . . . . 35

3.5

Low Temperature Trends for Cryogenic Skimming . . . . . . . . . . . . . . . . . . . 36

3.6

5 K Flow Cryostat Skimmer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.7

Liquid Helium Cooled Skimmer Bouncing . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1

Hexapole Focusing Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2

Transverse Phase Space Acceptance Examples . . . . . . . . . . . . . . . . . . . . . . 46

4.3

No Hexapole E↵ect in Generation Two . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.4

Electric Fields in Hexapole 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

xii
4.5

Clamp Stress and Strain in Hexapole 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.6

Heat Conduction and Radiative Loads for Cryo-Hex 1 . . . . . . . . . . . . . . . . . 53

4.7

Assembled and Section Views of Hexapole 1 . . . . . . . . . . . . . . . . . . . . . . . 54

4.8

OH After the Hexapole

4.9

Hexapole Surface Arcing Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.10 OH post-Hexapole, Patch Charges Resolved . . . . . . . . . . . . . . . . . . . . . . . 58
4.11 Section View of Hexapole 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.12 Comparing Hexapoles 1 & 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.13 Section View of Hexapole 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.14 Clogging Behavior of Hexapole 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.15 Hexapole 2.5 Radiation Shielding Isolator . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1

Voltages, E-fields, and on-axis Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2

Phase Space for Di↵erent Final Velocities . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3

Phase Space for S = 1, S = 3, and S = 311 . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.4

A Magnetic Pin Stark Decelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.5

The Double Pin Decelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.6

Definition and Explanation of Alternate Field Distributions . . . . . . . . . . . . . . 81

5.7

S = 1 and F Mode Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.8

Traveling Potential Well Characteristics for di↵erent Modes . . . . . . . . . . . . . . 85

5.9

Longitudinal Phase Space for di↵erent Modes . . . . . . . . . . . . . . . . . . . . . . 86

5.10 Transverse Oscillation Diagram and Data . . . . . . . . . . . . . . . . . . . . . . . . 89
5.11 Transverse Guiding Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.12 Non-adiabatic Transitions O↵-Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.13 Non-adiabatic Losses during Switching . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.14 Fields Close to Zeros of SF Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.15 Decelerator Length Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

xiii
5.16 Decelerator Mounting Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.17 MACOR Surface Arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.18 Capacitance Matrix for Third Generation Decelerator . . . . . . . . . . . . . . . . . 102
5.19 Glass Ba✏e for Di↵erential Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.1

Ring and Tricycle Trap Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2

Ideal Trap Loading as a Quarter Rotation . . . . . . . . . . . . . . . . . . . . . . . . 110

6.3

On-Axis Loading for Ring and Tricycle . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.4

Microwave Coupling through Bias Tee . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.5

Spectra in Ring and Tricycle Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.6

Near Field Microwave Intensity in Ring and Tricycle Trap . . . . . . . . . . . . . . . 115

6.7

Spectroscopy from |f, 32 i to |e, 12 i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.8

Ring and Tricycle Collection Solid Angle . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.9

Spectroscopy with Di↵erent Pulse Numbers . . . . . . . . . . . . . . . . . . . . . . . 118

6.10 Nuclear Spin e↵ects in Microwave Spectroscopy . . . . . . . . . . . . . . . . . . . . . 121
6.11 Opposite Parities survive Deceleration . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.12 Opposite Parities survive Trap Loading . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.13 Trapped State and Spin-flip Partner . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.14 Trap loading for |f,3
2 i states.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.15 Multi-pulse Loading in the Ring trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.16 Spin Flip Loss Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.17 Pin Trap Geometry View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.18 Pin Trap Loss Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.19 Pin Trap Loss Removal with Pin Translation . . . . . . . . . . . . . . . . . . . . . . 135
6.20 Escape Dynamics in the Ring, Tricycle and Pin traps. . . . . . . . . . . . . . . . . . 141
6.21 Chaos Indicator Benchmarking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.22 Chaos Indicators in the Pin Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

xiv
6.23 Cryocycle Field Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.24 Stray Field Influences Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.25 Tricycle HV Tracking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.26 Permanent Magnetic Cloverleaf Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.1

Microwave Free Space Coupling Probe . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.2

Successful Coupling of Free-Space Microwave Cavity Modes . . . . . . . . . . . . . . 157

7.3

Decay Fitting for Collision Rate Extraction . . . . . . . . . . . . . . . . . . . . . . . 160

7.4

Electric Field Induced Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

7.5

Multiple Electric Field Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.6

Extracting Collision Rates from Decay Fits . . . . . . . . . . . . . . . . . . . . . . . 165

7.7

Successful Simulation of Electric Field Induced Loss . . . . . . . . . . . . . . . . . . 166

7.8

Simulating Escape Dynamics After E-field . . . . . . . . . . . . . . . . . . . . . . . . 167

7.9

Partially Trapped Substates in Electric and Magnetic Fields . . . . . . . . . . . . . . 168

7.10 Simulated Electric Field Loss with Secondary Substates . . . . . . . . . . . . . . . . 169
7.11 Mapping Molecule Number to Collisions . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.12 Forward Minus Backward Spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7.13 Simulating All Substates During Evaporation . . . . . . . . . . . . . . . . . . . . . . 174
7.14 Simulation of Forward Minus Backward Evaporations . . . . . . . . . . . . . . . . . 175

Chapter 1
Introduction

In the beginning, God created hydroxyl radicals, and it was good. God used a complex expansion and condensation process beginning with higher energy forms of matter such as quark gluon
plasma that thankfully are no longer relevant to anything, except certain other PhD theses. Lateron, God created atomic physicists, who neither know anything nor are aware of this. Fortunately,
through centuries of “serendipitous” discovery and other unscrupulous tactics, God managed toorchestrate that this same group of physicists would continue to make dramatic advances relevant
to life, the universe, and the obtaining of PhD’s, right up to the present day.

1.1

Atomic Physics
Atomic physics is a very exciting branch of physics, concerned with atoms or molecules asa whole and not their subatomic constituents, and especially the interaction of atoms with one
another and with light. Some of the core questions of the field have in some sense been solved forquite some time, such as the nature of an atom and all the finest details of the energy levels of
atoms and molecules. In other areas, the field has greatly matured only much more recently, suchas in the ability to cool an ensemble of atoms into a regime where not only their internal degrees
of freedom are quantized, but their external motional degrees of freedom as well, thereby enabling
the realization of Bose-Einstein Condensation [1, 2]. Another even more recent example of thedevelopment of the field can be found in the detection of gravitational waves [3], more specifically theinterferometer used for this purpose. This device relies on coherent and high power laser sources and2the finest understanding and management of the interaction between light and materials, especially
optical coatings, and even the quantum mechanical manipulation of quadratures to reduce noise
sources by spin-squeezing.
A significant contingent within atomic physics has in recent years moved in the directionof quantum information science, especially with regard to the trapped ion modality of quantum
computing, where two or several states within a trapped ion serve as a quantum bit of information [4,5]. While still only boasting rather modest near-term gains, such as the noisy, intermediate-scalequantum (NISQ) regime [6], this qubit modality at least provides some unique opportunities foratomic physicsts to compete with Josephson junctions and other modalities more the purview ofcondensed matter physics. Keeping a focus on scientific rather than technological aspects, quantum
information science a↵ords unique opportunities for application to scientific computing, and early
quantum computation demonstrations have focused on analog quantum simulations of real scientificphenomena [7, 8].
Lastly I’ll mention precision measurement, an amusing avenue of advancement whereby
atomic physicists leverage our measurement accuracy to learn something completely fundamental by understanding something simple to death and beyond. This branch of atomic physics has
been around since the beginning and is in some sense a natural evolution out of the spectroscopic
beginnings of the discipline [9]. Some of the highlights include the electron magnetic moment [10],providing a part per trillion comparison between standard model quantum field theory and experiment, and the measurement of time itself, which has seen incredible advances in the modern
era [11].

1.2

Molecules
Against this brief smattering of more modern directions of atomic physics, I am now pleased

to delve with more focus into the molecular subfield of atomic physics. Molecules boast a numberof exciting realized and hypothetical contributions, across all of the major focus areas of atomic
physics [12]. The recurring theme is that their rotational and vibrational degrees of freedom intro-

3
duce new features that are both challenging to control and very useful if controlled. For example,
in trapped ion quantum computing, the most realistic pathways to scalability demonstrated thus
far involve the performing of 2-qubit gates via the mechanical motion of the trapped ions. Thoughadvantageous as far as scalability is concerned since such manipulations can be achieved with conventional electronics, many challenges including heating issues, fidelities, and gate times exist [13].
Molecular trapped ions o↵er new opportunities for circumventing the need for motional manipulation, such as dipole-dipole couplings for gates [14].
In the study of quantum gases, manipulation of interactions between atoms have long enabledstudies of pairing and BEC-BCS crossover [15]. Nevertheless, availability of a quantum gas withconstituent particles themselves possessing anisotropic interactions has long attracted great interestin the field. Great strides have already been made in this direction, especially with magnetic
atoms with unusually strong magnetic dipole moments such as Erbium and Dysprosium. These are
already cutting into the territory once hoped for by molecular cooling enthusiasts with the startling
discoveries of self-bound quantum droplets [16] and liquid-He reminiscent roton modes [17]. Still, it
could be argued that these unexpected results connected to the dipolar character of the underlying
particle only confirm the validity of the quest for quantum degenerate systems of an ever widenedpool of candidate particles.
With regard to precision measurement, molecules o↵er a number of exciting possibilities,
perhaps most notably with respect to measurements of the electric dipole moment of the electron,
recently measured to zero within 10 28 e cm [18] and 10 29 e cm[19]. These results are fundamentally enabled by the internal electric fields present inside molecules. Additional possibilities include
extending such studies to polyatomics [20], studying time variations [21], or studying nuclear effects [22].

1.3

Molecular Cooling
Several strategies exist for getting molecules cold, which is the primary challenge associatedwith their further usefulness to atomic physics. Firstly there is the possibility of using already4
ultracold alkali molecules, associating them together loosely via scattering resonance techniques,but then utilizing precise spectroscopy of high-lying molecular states to coherently transfer themto ground states. A nice survey of these techniques is available here [23, Sec. 1.3], and exciting
recent work with polar potassium rubidium molecules produced in this manner is being performed
right here in the Ye group [24].

1.3.1

Laser Cooling

A di↵erent approach is to instead select a molecule with some favorability for traditional laser
cooling techniques. First proposed in titanium monoxide by Benjamin Stuhl during his dissertation work on this experiment [25], and later realized with yttrium monoxide [26], more advanced
magneto-optical trapping schemes exist which are amenable for use with molecules. Since then,numerous molecules have been successfully slowed and trapped, including strontium fluoride [27]and calcium fluoride [28], with the latter even loaded into optical tweezer arrays. Recently, it has
been exciting to observe the application of various advanced techniques to these molecules for rapid
further progress in cooling, such as lambda-enhanced gray molasses [29] and blue-detuned magneto
optical trapping [30]. These advances have been particularly satisfying for the community given
the long time spent waiting for these groups to master the molecule slowing process in order toload magneto optical traps, something which all groups involved spent something like four years
achieving. So far no group has brought these laser cool-able species below about 5 µK, although
the yttrium oxide has prospects for doing so via narrow-line cooling [31].
Another related but distinct technique that could in principle apply to a much broader classof molecules but still relies on lasers is sawtooth-wave adiabatic passage (SWAP) cooling [32].
Somewhat controversial in its reliance on stimulated emission and the opaque role of spontaneousemission in the process, SWAP cooling is nevertheless a valid technique which may be useful forsome species. For a time, the YO experiment considered its implementation, or a modification
using triangular waves, but this may have been superseded by opportunities with the techniques
described above.

5
1.3.2

Direct Cooling

Although sometimes used to refer to any technique other than the association of ultracold
molecules, I will use the term “Direct Cooling” to refer to those cooling techniques which do not rely
on lasers to transfer entropy, but instead make use of conventional and more broadly applicable
collision-based cooling mechanisms. Two primary directions in this regard are the supersonic
expansion [33] and the cryogenic bu↵er gas beam [34]. Although both rely on thermalization bycollisions between noble gas bu↵er atoms and the species of interest, their principles of operationand the characteristics of the molecules they generate are quite di↵erent. The supersonic expansion
technique is ideal for generating a high phase space density with a large lab frame velocity o↵set,
and the cooling happens all in one shot right from room temperature thanks to an expansive
flow process described further in Sec. 2.1. The cryogenic bu↵er gas beam generates moleculeswith greater breadth in velocity space but a much lower initial kinetic energy [34, Fig. 1]. These
properties make the cryogenic bu↵er gas beam ideal for experiments with good prospects for further
phase space compression by laser cooling, but with difficulty in performing slowing, exactly the casefor the laser cool-able diatomics. In contrast, the supersonic expansion is ideal for experiments withno clear path to higher phase space densities, but robust options for removing lab frame velocity
in a phase-stable manner, i.e. Stark or Zeeman deceleration. These techniques, though generalcompared to laser based methods, require a careful understanding of the molecule of interest, forwhich purpose we now transition into more directly practicable background information.

1.4

The OH Molecular Hamiltonian
All of our ability to successfully exploit and manipulate OH radicals is underpinned bya robust understanding of its internal structure undergirded by decades of careful spectroscopy.
Early spectroscopic results are still directly useful in determining where to set detection lasers [35],although many of these parameters are now more usefully catalogued in the NIST webbook [36]. As
in many areas of modern quantum physics, careful understanding begins with classical Hamiltonians

6followed by quantization and then a perturbative approach to better and better models of thesystem.
A well trodden pedagogical progression exists for the problem of the Hydrogen atom. One
begins with classical descriptions of electrons orbiting a hydrogen nucleus, moves to semiclassicalangular momentum quantization, and finally on to the exact Hydrogen quantum mechanical wavefunction solution [37, Sec. 4.2]. A similar progression exists for Molecules, and begins with thesimplified quantum mechanics of rigid rotors and springs. The former end up applying well tothe rotational degrees of freedom of the molecules, and the latter to their vibrational degrees of
freedom.

1.4.1

The Rigid Rotor

For the rigid rotor, we imagine some extended rigid body with moment of inertia Iz about
its center of mass in a particular direction z, which may be calculated in the usual way:Iz =Z

⇢r2 dAdz,

(1.1)

V

where the integral covers the whole volume of the body with density ⇢, and r gives the distancefrom the infinitesimal volume element to the axis parallel to z through the center of mass of the
body. Classically, lacking any environmental coupling, this body may indefinitely rotate about its
axis of rotation with any angular velocity !, and posses a rotational kinetic energy given by:1
U = Iz ! 2 ,
2

(1.2)

L = Iz !

(1.3)

and an angular momentum given by:

We can extend into the quantum domain following Bohr’s loose quantization rule and requiring theangular momentum to be quantized in units of ~:L = l~ = Iz !,

(1.4)

7
with l an integer, so that the allowed values of the angular velocity and energy become:! = l~/I,

U=

1 2 2
l ~
2I

(1.5)

with the natural consequence that for larger inertia I, the spectrum of allowed rotations becomesincreasingly continuous. This turns out to di↵er only slightly from the true situation, at least as it
pertains to the specific case of linear rotors and diatomic molecules, where we find:U=

1 2(l + l)~2 .
2I

(1.6)

It is instructive to follow this through more thoroughly [38, Sec. 2.8]. For the case of a linear
rotor, where the body’s mass is essentially confined to a single axis in space, a single momentum ofinertia I characterizes the system, and for the even simpler case where the mass is located at only
two points separated by R, we can evaluate Eq. 1.1 but with the following density distribution:⇢(r) = m1

✓

r

m2
R
m1 + m2

◆

+ m2

✓

m1
r+R
m1 + m2

◆

,

(1.7)

where m1 and m2 are the two masses separated by a distance R, and the values in the delta
functions create the situation where the radial coordinate r attains its zero value at the center ofmass of the distribution. Evaluating Eq. 1.1 then leads to the result:I=

m1 m22 + m2 m21 2
m1 m2
R =
R2 = µR2 ,2(m1 + m2 )m1 + m2(1.8)

where µ is the reduced mass of the system. Now the Hamiltonian for this system is simply therotational energy:
Ĥ =

1 2
r
2µ

(1.9)

which upon moving ahead with the Schrödinger Equation, gives the spherical harmonic eigenfunctions:
ĤYlm (✓, ) = El Ylm (✓, ),El = Bl(l + 1),

Here B is referred to as the rotational constant, and ✓,

B=

~2
.
2I

(1.10)

give physicist spherical coordinates relative

to an axis orthogonal to the axis of the linear rotor, i.e. an axis along which the system has angular

8
Table 1.1: Parameters leading to the value of the rotational constant B are shown, together with
its semi-classically inferred value. Spectroscopically determined values agree quite well. Radiiand spectroscopically determined rotational constants from [36], except RbCs from [39] and YbCs
from [40].
Moleculeµ (amu)R (Å)

Bsemi (GHz)

B (GHz)

H2
OH
LiH
YO
CaF
SrF
YbF
RbCs
YbCs

0.5
0.94
0.88
13.6
12.9
15.6
17.1
51.9
75.2

0.74
0.97
1.60
1.79
1.97
2.08
2.02
4.4
5.4

1850
573
227
11.6
10.1
7.52
7.24
0.503
0.230

1820
567
225
11.6
10.1
7.51
7.22
0.497
0.207

momentum I, not the axis of the rotor itself. It is interesting to ask what the wave-functions
associated with these eigenvalues are, in terms of their spatial probability distribution. In thecase of l = 0, quantum mechanically the molecule is in a spherically symmetric state with equal
probability of finding a nuclei anywhere on the surface of its own spherical shell, with a radiuscorresponding to the distance of that nuclei from the center of mass. This is a satisfyingly bizarre
quantum mechanical state of a↵airs, especially given that relative to the quantum mechanics of thelight electron, these coordinates describe the locations of heavy atomic nuclei.
It is interesting to compare the values of B calculated naı̈vely from Eqs. 1.10, 1.8 to thespectroscopically determined values, as shown in Tab. 1.1. Very good agreement is obtained, atleast when using the spectroscopically determined internuclear distance, which is therefore notquite fair as far as theory experiment predictions are concerned. Nonetheless, it is reasonable
to conclude that the rigid linear rotor model is a good description of the situation for diatomic
molecules. The bulk of the remaining deviations relate to centrifugal e↵ects, by which I broadly
refer to the interaction between the rotational and the vibrational components of the Hamiltonian,a nice segue into the following section.

9
1.4.2

The Vibrational Hamiltonian

In much the same way as the canonical simple harmonic oscillator problem in quantum
mechanics, at its core the vibrational Hamiltonian may be represented as a harmonic potential
describing the energetics of the internuclear distance of the molecule. This potential may be
specified beginning with the full Hamiltonian describing all of the electrons and nuclei in a molecule,and then separating the electronic and nuclear portions based on a hierarchy of timescales known
as the Born-Oppenheimer approximation [41, Sec. 8.1]. Essentially, the nuclear positions are taken
as parameters to the computation of the electronic wave functions, and so by parametrically solving
the electronic Hamiltonian for all nuclear distances and rotations, the energetic landscape governing
the dynamics of the nuclei emerges. It is therefore somewhat difficult to directly specify the formof the interatomic potential experienced by the nuclei, but the results from this procedure turn out
to be closely represented by the Morse potential:⇣
V (x) = D 1

e

x

⌘2

.

(1.11)

This potential has a minimum at x = 0, defined to be the equilibrium distance, often Re , of thevibrational oscillation. Below x =Re , this coordinate is unphysical, at which point the potential

is still finite, but usually quite large relative to the relevant energetics. It tends to D the dissociationenergy as x ! 1. For small x, expanding near zero gives a harmonic form as we would expect:
V (x) ⇡ D 1

(1

x+

2 2

x /2

...)

2

⇡ D 2 x2

D 3 x3(1.12)

This potential is therefore an anharmonic oscillator, with a cubic anharmonicity that enhances
restoring force in one direction but weakens it in the other. As far as energy level spacing isconcerned, the weakening direction has the dominating e↵ect, tending to reduce the energy levelspacing at higher energies. Another important property of the Morse oscillator is that only a finite
number of levels are supported, which is also the case for true molecular potentials, unlike electronicenergy levels, for which the so-called Rydberg series of their energy levels is countably infinite.
The Morse oscillator is a close enough match to true molecular potentials, that the results

10Table 1.2: Spectroscopically determined vibrational constants are shown for several molecules ofinterest [36].
Moleculeµ (amu)Re (Å)we (THz)we xe (GHz)

H2
FH
OH
N2
LiH
YO
CaF
SrF
YbF
RbCs

0.5
0.95
0.94
7
0.88
13.6
12.9
15.6
17.1
51.9

0.74
0.92
0.97
1.10
1.60
1.79
1.97
2.08
2.02
4.4

132
124
112
70.8
42.2
25.8
17.4
15.1
15.1
1.48

3640
2700
2550
427
696
87.0
82.2
66.0
66.0
-

of molecular spectroscopy are often expressed in terms of the coefficients that describe the Morseoscillator whose rovibrational energies best match the actual level spacings of the molecule. It isuseful therefore to write down the rovibrational level spacings predicted by the Morse oscillator:
E(⌫, J)/hc = we (⌫ + 1/2)

xe we (⌫ + 1/2)2(1.13)+Be J(J + 1)

De J 2 (J + 1)2(1.14)↵e (⌫ + 1/2)J(J + 1)

(1.15)

Here ⌫ gives the vibrational quantum number and J the rotational quantum number. Theseenergies come by using Eq. 1.11 in the rovibrational hamiltonian one can obtain by applying theBorn-Oppenheimer approximation [38, Eq. 2.157] to the full molecular Hamiltonian. The value ofeach of the parameters we , xe , Be , De , and ↵e may be written in terms of theand D of Eq. 1.11,

and the reduced mass µ and equilibrium nuclear distance Re of the molecule, see [38, Eq. 2.183].
The value of the vibrational parameters we and xe we for a few molecules of interest are shown in
Tab. 1.2.

11

1.5

Angular Momentum Coupling
By far the most significant challenge to understanding and working with molecules relates

to their complex angular momentum coupling. The various sources of angular momentum in themolecule are often described using a hierarchy of successively weaker couplings which specifies theHund’s case of the molecule, or rather of one particular state of the molecule.

1.5.1

Revisiting J=L+S

Before diving in, we begin with the simplest angular momentum coupling example, butwith an eye toward its generalization to the molecular situation. Electrons have orbital angularmomentum and spin in the hydrogen atom, but a state of definite orbit and definite spin is notnecessarily an eigenstate of the Hamiltonian, because of “coupling” that may exist between thesestates by the Hamiltonian governing the evolution of the system. Specifically, the orbit and thespin influence one another via the magnetic fields they generate, so that the Hamiltonian has aterm proportional to their dot product. This classic problem may be resolved by defining true
eigenstates of the Hamiltonian using a di↵erent quantum number which describes the total angularmomentum of the system. We say that L is “weakly” coupled to S in the sense that the level shift
caused by including or ignoring the coupling term is small compared to the e↵ects of the L and S
terms independently.

1.5.2

Coupling Many Spins

Every atomic nucleus or unpaired electron in the molecule may contribute angular momentum
to the coupling story, either spin or orbital. The latter is known as orbital angular momentum
when carried by electrons, and simply rotation when carried by nuclei. The coupling story is thenaddressed essentially by writing terms down in order of the magnitude of their leading coefficientin the full Hamiltonian. These terms can either be coupling terms or specific terms, and we arefree to treat them in order, defining as appropriate a combined quantum number to include a given

12

Figure 1.1: Inclusion of higher lying states of the OH Hamiltonian in [43] accounts for the mixed
Hund’s case and gives a significant 20% level correction to level crossings. A snippet from [44,Fig. 1] is shown in (a), and panel (b) has the results from [43].

coupling e↵ect before moving on to higher order, lower energy coupling e↵ects later on down theline. One way this is often indicated is to draw diagrams that indicate the magnitude of an e↵ect by
the length of a vector, and indicate the use of a coupled basis by the definition of a new vector whichis the sum of the other two. The first two vectors may be considered valid possible instantaneous
values of the angular momenta they represent were they measured, but only the final sum vector isactually conserved by the Hamiltonian. The ordering of these sources of angular and their couplingstrengths generally follow one of a few set patterns which are known as Hund’s cases. These are
well enumerated and diagrammed in [42, Sec. 2.2.1-4].

1.6

Stark and Zeeman E↵ects
These e↵ects may be well understood on their own, as described in [45, Sec. 2.3.1-2.1], but

combining them can be a very challenging thing to do properly. During my thesis, it was discoveredthat use of the Hund’s case (a) assumption, considered valid at least for the lowest rotational statesof OH [23, Sec. 2.2], led to significant errors in the location of avoided crossings between states.
Essentially, the magnetic moment of the molecule is not quite perfectly coupled to the internuclear
axis, so that rotational averaging of the molecule’s angular momentum from the molecule frame to13
the lab frame does not reduce the magnetic moment from 2 µB to 1.2 µB as one would calculate for
Hund’s Case (a), but instead only to 1.4 µB . One way to describe this is to simply include moreterms in the Hamiltonian used for computing the Stark and Zeeman e↵ect of the molecule. This wasdone in [43], where direct reference was made to one of the figures in our earlier work [44], see [43,Fig. 12], reproduced here as well 1.1. We had in fact discovered the correction much earlier in 2013,
but did not have a good way to publish our findings on their own. For future work, wherever itcounts, the full Hamiltonian has been implemented in Matlab and is available for future use in theexperiment.

Chapter 2
Creation

Like all things, Jun Ye’s hydroxyl radical experiment also begins with creation; specifically,
the creation of hydroxyl radicals via discharge during pulsed supersonic expansion. Such a supersonic expansion occurs when gas at pressure is allowed to escape into vacuum via a small orifice.
No better resource on the supersonic expansion or free jet seems to exist besides [33], although amore modern compendium of results does also exist [46]. I will begin with supersonic expansions
and the thermodynamics thereof, before moving to the implications for our work and some specificvalve technologies.

2.1

Supersonic Expansion
Suppose that an ideal gas is trapped in a certain volume V1 and at a certain pressure P1 ,when one of the walls enclosing the volume is suddenly removed, and the gas is allowed to expand
into a much larger volume V2 . To determine the long term properties of the system, one can use thefact that neither heat nor work were transferred out of the gas, so that its energy E must remain
fixed, and therefore also its temperature, following the well-known intrinsic formula relating thetemperature of an ideal gas to its internal energy, E = CV N T . Here CV denotes the specific heatof the gas at constant pressure, and N the number of particles present in the system. It would
follow that P2 = P1 · V1 /V2 , and that the entropy would be significantly increased. To determine
by how much, we can use the thermodynamic relation for the internal energy E of the system:dE = T dS

P dV,

(2.1)

15
which sinceE = 0 for this process since it transfers , leads us to:
1
S=
T

Z

1
P dV =
T

ZV2

V1

P1 V1
dV = N kB log
V

✓

◆
V2
,
V1

(2.2)

Since P (V ) = P1 V1 /V for fixed T and since P1 V1 = N kB T . This is also intuitively valid, since theentropy ought to correspond to the logarithm of accessible micro-states. With nothing else changing
about the momentum distribution before and after the event, only the system size contributes toan increase in accessible micro-states, by an amount given by the ratio of volumes before and after.
Although the foregoing example is ubiquitous in the problem sets assigned to physicists andmechanical engineers during thermodynamics classes, the corresponding short term behavior israrely if ever treated. At first glance, it may seem that in the short term, i.e. times t after barrier
removal smaller than that required for the reestablishment of a pressure equilibrium, the suddennessof the event and its obvious lack of reversibility would make it somewhat intractable to the methodsof statistical physics. To the contrary, much progress can be made once some additional details andassumptions are specified. Suppose that the initial volume is allowed to expand into a much larger
volume, but only through a rather small opening. Suppose further that the expansion process is ina flow equilibrium, so that gas is constantly supplied to the initial volume in order to maintain aconstant pressure, and gas is constantly pumped out of the back of the much larger volume. This
supposition includes what is known as the continuum hypothesis, which stipulates that the gas
may be treated as a fluid of smoothly varying density on all scales, and essentially requires a shortmean-free path and a suitably high local density [47, Sec. 1.2]. These assumptions create a very
similar environment to the one realized in many molecular beam apparatuses, known as a supersonic
expansion. Now consider a small sub-volume V1 within the high pressure volume, close to the small
opening. Since the speed of egress is fast compared to di↵usion timescales, we may assume that
this volume is isolated from the rest of the gas, and consider its fate as the molecules within follow
the flow lines of the fluid. As we follow the flow lines, this volume expands, performing work on thesurrounding environment, but not exchanging any heat with it. Paradoxically, the process is thenadiabatic, despite the significant and non-zero

S calculated above for the long term behavior. The

16
Table 2.1: Final supersonic expansion velocities for various gases according to Eq. 2.4, using T =300 K. For H2 I have used the ideal gas heat capacity since rotations are hardly populated at room
temperature. For UF6 , inclusion in this chart is largely fanciful in any case, but rovibrations areunlikely to be efficiently cooled so I have again used the ideal gas heat capacity.
Gas

Speed (m/s)

H2
3 He

2490
2030
1760
790
560
380
310
190

4 He

Ne
Ar
Kr
Xe
UF6

work done during expansion primarily serves to accelerate the volume just ahead along the flow,
and the magnitude of the acceleration is such that sound waves are significantly outpaced [33, 42],hence the “supersonic” in the expansion.
As the gas expands, it continues to work on the environment until it has exchanged all of its
internal energy, in principle reaching T = 0 and satisfying Bernoulli’s flow equation by converting
its initial enthalpy entirely to kinetic energy:
Hi = C P N k B T i =N
mvf2 .
2(2.3)

For a monoatomic ideal gas, CP = 5/2, and we have:vf =r

5kB Ti.
m(2.4)

The final velocities obtained from this equation for noble gas carriers are shown in Tab. 2.1. Inpractice, these speeds are not always obtained by the species of interest within a noble gas bu↵er,
often due to velocity slippage in cases of significant mass imbalance between the carrier and thespecies of interest. Another source of discrepancy comes from excess heating, either due to themechanism of operation of the valve [45, Sec. 3.1.3.1], or the mechanism of generating the species
of interest, see Sec. 2.2.2 below.

17
Table 2.2: Final supersonic expansion temperatures for noble gas carriers in an Even-Lavie valve
according to Ref. [49, Fig. 12], using T = 300 K.
Gas

Temp ( mK)

4 He

33
180
930
2400

Ne
Ar
Kr

As the gas rarifies, the continuum flow hypothesis breaks down, and the mean free path
between collisions of molecules becomes long enough that the assumption of an isolated sub-volumeof the flow becomes invalid. What this means is that below a certain pressure, the gas can no longer
perform work on itself during expansion, and a limiting nonzero expansion temperature is reached.
In practice, the remaining internal energy of the gas associated with this nonzero final temperature
is not a significant perturbation on the final velocity of the beam, but it does influence the phasespace density available for later use. The remaining internal energy of the gas manifests mostmeasurably as an extra spread on the final velocity obtained during the expansion. Also known as
the translational temperature, its direct calculation is possible but complex [48], but it is not overly
difficult to measure, at least approximately, using a pulsed source and measuring at a near anddistant point along the propagation of the pulsed expansion. When this is done with a miniaturizedsolenoidal Even-Lavie valve [49], the values shown in Tab. 2.2 are obtained for a selection of raregases. Various factors can influence this, and often clustering e↵ects can prevent utilization of thenarrowest possible distributions for a seeded beam, see discussion in [45, Page 47].

2.1.1

Rovibrational Temperatures

An additional e↵ect requiring careful accounting is the degree of thermalization of the particles in their frame of reference after the expansion. One piece of this is the degree of thermalizationof the rotational and vibrational degrees of freedom of molecules seeded in the beam with the translational degrees of freedom. Generally it is found that these degrees of freedom do not thermalize
as well as translation, with vibrations thermalizing much worse than rotations. Much attention has

18
been given to this subject, particularly given its very clear manifestation in an experimental observable, namely the relative population of the various rovibrational levels as probed by laser induced
fluorescence or other state selective means. It has even been found that the rotational levels fail tobe well described by any meaningful temperature, even towards the beginning of the expansion [50].
Studies specific to hydrides and even to the hydroxyl radical have been performed [51].

2.1.2Translational Disequilibrium

Compared to rovibrational levels where nearly complete information is available, it is rather
difficult to measure the translational temperature of the beam in the direction transverse (orthogonal) to the beam-line. Evidence exists for a disequilibrium between transverse and longitudinalframe of reference temperatures [52, 53, 33]. This is especially important for consideration of thee↵ective initial size of the beam as seen by devices downstream. In the continuum flow region ofa supersonic expansion, the fluid obeys a kind of hyperbolic flow pattern, which has the property
that after a few nozzle diameters flow-lines actually appear to originate from a single point [33]. As
the molecules leave the continuum regime and the cooling process halts at nonzero temperature,
this breaks down, so that individual molecules no longer appear to originate from a single point.
A useful way to understand these phenomena is to work in transverse phase space, where thevelocity of molecules in one of the directions transverse to the beam is plotted against their positionin that same direction. If the cooling were to continue indefinitely and all molecules appeared
to originate from a single point, the transverse phase space distribution function describing theensemble at a fixed large time t after passing through the orifice would look something like:d(x, v) = (vt22x)e (v /2 v ) ,where d is the phase space distribution function, and(2.5)

the Dirac delta function. The delta function

term enforces the criterion that molecules originate at x = 0, since with velocity v they may only
be found at x = vt. The additional gaussian term assumes some generic velocity spread with astandard deviation of

v . The nonzero final transverse temperature convolves extra spread in the

19

Figure 2.1: This scatter plot samples a distribution function designed to be reflective of that which
may be generated in an Even-Lavie valve, see Sec. 2.2.1. The quitting surface, where continuum
flow ceases, is assumed to lie 3 cm distant from the 500 µm nozzle, and a transverse temperatureof 200 mK in both transverse and longitudinal directions is used.

velocity direction over this narrow distribution, which is relevant for the use of hexapole focusers
as discussed in Chapter 4.
An example of transverse phase space generated by propagating molecules perfectly from asource point and then adding nonzero transverse temperature at a certain distance is shown in
Fig. 2.1. The distance is chosen to be representative of the transition between continuum andmolecular flow, which is often termed the quitting surface.

This distance is referred to as thequitting surface in the literature [33]. In Fig. 2.1, the molecules are back-propagated after the
extra temperature included at the quitting surface, so that the distribution shown can be thoughtof as an e↵ective source distribution in the same plane as the actual nozzle. The distribution is quite

20
tall vertically, which is a reflection of the fact that the beam expands on the surface of a sphere, sothat transverse velocities even approaching the forwards velocity magnitude of 810 m/s in this caseshould in principle be populated. In practice such large transverse velocities are unusable, sinceeven with an arbitrarily powerful focusing device molecules with very large transverse velocities
also have forwards velocities reduced relative to the better collimated portion of the beam. For
this reason, I use an unphysical directed-ness parameter in generating distributions such as theone shown in Fig. 2.1, which has a FWHM polar angle of ⇡/12, since the other molecules are not
relevant for subsequent focusing and deceleration. Actual directed-ness of supersonic expansions
tend to follow an angular distribution with polar angle ✓ according to cos3 (✓) [33], and some claim
increased directed-ness through the use of conical expansion regions [54, Fig. 11].

2.1.3

Experimental Transverse Temperature

It is possible to access some of this information experimentally by observing the angular
distribution of a beam, especially downstream of some additional collimating aperture. We canadopt similar terminology to that used in optics, although no corollary exists here to the possibilityof constructive and destructive interference. The point spread function gives the distribution ina detector plane after convolution with the distribution found in the aperture plane. Transverse
temperature controls the width of this point spread function, causing a sharp edged aperture tocast a nonetheless blurry shadow in the detector plane. See Fig. 2.2 for an illustration of this. One
way to measure this blurriness would be with an angle resolved, or adjustable position detector ofsome kind, but an easier alternative is to translate the distance between the source and aperture,
thereby increasing and decreasing the size of the blurry spot in the detector plane.
We detected this e↵ect during our work on skimmer cooling [55], where the skimmer, adevice designed to collimate a beam with minimum interference described in Chapter 3, serves as
the aperture. An important caveat is that any aperture has the potential to interfere with thebeam, so that the blurriness observed could be an artifact caused by the aperture, and not a true
representation of the original beam. Data from [55, Fig. 2c] are repeated here in Fig. 2.3 with

21

Figure 2.2: Transverse temperature lends a point spread function (PSF) to the image of aperturesin the detector plane of a molecular beam. These distribution functions reflect a 20 cm distance
between the source and the aperture, and a 3 mm aperture. This investigation was first performedin June 2018, and relevant files may be found in a code appendix I intend to add at some point.

an expanded discussion of these transverse e↵ects that is not available in [55].

For a suitably

small transverse temperature, we would expect that once the valve is close enough to the aperturethat the detector area is fully illuminated by the beam, signal no longer increases as a functionof reduced valve skimmer coordinate. This is at odds with the data shown in Fig. 2.3, where thesignal is seen to increase even below a 3 cm valve-skimmer distance. At this valve-skimmer distance,
considering the 3 mm skimmer opening and 40 cm detector distance, the zero temperature spot
size of the beam would be a 4 cm diameter circle, well overlapping our fast ion gauge detector(6 mm x 18 mm). One way to make sense of this discrepancy is to introduce a bimodal transverse
temperature distribution:
2

2

2

2

d(v) = ↵e v /2 c + e v /2 h ,

(2.6)

22

Figure 2.3: (a) Time of flight traces taken with varying distance between the valve and the aperture,with blue closest (1.8 cm) and red farthest (4.8 cm). The area under the curves are computed,
leading to the data-points in the next panel. (b) Fit attempts allowing a variable fraction of the
population in a second, warmer temperature distribution.

where d the distribution function consists of two normal distributions weighted by ↵ and , withwidthsc andh where the latter is hotter than the former and thus respecting

c <

h.

This

hypothesis has some scientific support and relates to the relatively hot collisions that can occur
for molecules which do happen to collide well after the quitting surface [52, 33]. Varying the fraction assumed cold and the hot temperature, various fits are obtained as described in the legend

23of Fig. 2.3b. The cold fraction is assumed to be at 200 mK, the measured longitudinal temperature, which well overlaps the detector at all valve-skimmer distances included here and essentiallycontributes only a baseline o↵set for the fitting. The warm fraction is assumed responsible for the
upturn at closest distances, leading to the various fractions and temperatures shown in the legend.
The temperatures required for obtaining any agreement with the experimentally observed upturn
are not surprising. In [52, Abstract, 3rd Paragraph], e↵ective transverse source size for the hot
distribution are claimed to be 4

8 times that of the cold distribution, implying a temperaturelarger by that factor squared. The fraction populating the hot distribution is much more surprising,with [52] reporting only 30

50%.

The primary alternative hypothesis is that the results obtained in Fig. 2.3 are not a purereflection of the transverse properties of the supersonic expansion, but rather are the result of somedegree of interference e↵ects from the aperture itself. The surprising thing here would be that theinterference e↵ects actually lead to an increase in signal, whereas the usual result of interference isonly the reduction of signal. It could be that this is a compound e↵ect of some kind, for example
that the close proximity of the valve may reduce interference with the aperture in a certain range.
This could be caused by a favorable match between the angle of the conical skimmer faces and theposition of the source, leading to a reduced cross section presented to the beam by the skimmer.
Another possibility is that ions generated in the discharge, see Sec. 2.2.2, have a cleaning e↵ectthat enhances the performance of the cryogenic skimmer. This is not out of the question, as in-situ
noble gas ion cleaning techniques are a legitimate surface preparation strategy [56].

2.2

Pulsed Valves
Pulsed valves solve what is otherwise a significant dilemma for a vacuum system, and enablesignificant improvements in brightness compared to their continuous analogs. The key challenge
for a pulsed valve is to rapidly transition from leak-tight to fully formed, minimally impeded flow.
Many strategies exist for achieving this, and many have been utilized right here in Jun’s OH
experiment. A current loop, solenoid, and drum-mode piezo valve have been used, well described24
in [45, Sec. 3.1.3.1-3]. The latter proved particularly well suited for stable long-term operation, andcontinued to serve our needs well into 2017. An impeccable description of this valve is available
here [57, Sec. 3.4.1.1], and of course its original publication [58]. In what follows, I focus on two
pulsed valves with which I have played a more direct role.

2.2.1

Miniature Solenoid

Towards the middle of my thesis work, a decision was made to lengthen the decelerator andswitch to a lighter neon carrier gas. The use of neon opens up many opportunities for possible
increases in source brightness, at the expense of increased beam forward velocity. Neon has alower viscosity, reduced propensity for clustering and colder final temperature relative to krypton,
the previously employed carrier gas. Additionally, velocity slippage, a phenomenon whereby the
species of interest reaches a di↵erent speed than the carrier, is reduced with OH due to the morecomparable mass. The reduced clustering issues also make it possible to consider running at
dramatically increased pressures, and also heating the valve to run at correspondingly enhanced
water pressures. The use of high pressures has an important e↵ect on the operating mechanism fora pulsed valve, since this can significantly increase the mechanical force which must be exerted inorder to open the valve. This seems particularly problematic for piezo valves, which seem alreadysomewhat limited in actuating distance and applied force. For this reason, together with the change
to a longer decelerator, we decided to purchase an Even-Lavie valve [49, 54], considered to be akind of gold standard in high density pulsed molecular beams, although with more challengers oflate a . Refer to Fig. 2.4 for a labeled view of the valve together with a custom designed andself-manufactured mounting structure.

The valve utilizes a highly miniaturized solenoid typedesign, which actuates a ferromagnetic plunger that actually maintains its magnetic susceptibility
down to cryogenic temperatures, allowing the valve to continue to function at those temperaturesand opening up new possibilities for Helium beams and other physical chemistry pursuits. It alsoaDavid Nesbitt claimed that this valve was comparable [59] to the Even-Lavie valve in email communication toHao, Jun, and I

25

Figure 2.4: The Even-Lavie valve, mounted in a thermally stabilized aluminum rig. (i) The stainlesssteel body of the valve itself protrudes from the front of the aluminum rig. (ii) This verticalaluminum cylinder is in fact a tube, containing within a water reservoir consisting of an NPT
fitting with glass fiber filter papers inside for localizing some liquid water vapor. (iii) A half-closed
cylindrical holder for the valve body sits down here, and connects to the valve via hose-clamps.
These hose clamps tighten onto copper sheets which help transfer heat to the upper aluminum
slices which surround the valve for optimum thermal equilibration. (iv) The valve mounts up to a
VCR connector, by means of a VCR plug with 1/4”-28 threaded hole feature. This VCR connector
formerly served as a gas line for the previous piezo valve. (v) A frighteningly frustrating eightpin mini-CF connector transfers wiring for two thermistors and two heaters. Teflon tubes prevent
unintentional grounding between pins. (vi) This Teflon block is original to the valve, and strainrelieves the feedthroughs for the wires which actuate the solenoid. (vii) This clamp tightens thealuminum cylinder onto the water reservoir within (the cylinder has a cut to allow flex). A small
vacuum compatible heater (Minco) is mounted underneath the clamp. (viii) The leads to an Omega
thermistor are connectorized to a teflon wire twisted pair and strain relieved via stainless steel wire
to the hose clamp. The thermistor is glued into the upper aluminum cover piece on the front, tothe left of this label and directly above the valve nozzle. An identical thermistor connects to theback of the water reservoir. (ix) High voltage RF feedthrough for the dielectric barrier discharge.
(x) Ultra-thin 1/16” Swagelok gas line for minimum dead volume at high pressure.

26

Figure 2.5: Free flight of OH radicals past a detection region in the source chamber as a functionof the temperature of the valve housing. An optimum is found close to 40 C, only ⇠ 20% above
the room temperature value.

boasts operation up to extreme pressures of 100 bar, potentially a game changer for high brightness
generation of OH radicals.
The purchase of the valve was not a pleasant experience, and involved over a year of waiting
and much haggling about paperwork and international agreements. The performance of the valve
was also somewhat lackluster, with OH brightness in the source chamber actually only half that
achieved with neon in our previous piezo valve. Tuning the temperature of the system did not yield
any significant change in the performance of the system either, see Fig. 2.5. Increased temperature
would have been necessary to capitalize on the possible gains of a higher pressure source region.
This is because OH radicals are formed from water as described below in Sec. 2.2.2, and the partialpressure of the water strongly depends on temperature close to room temperature (hence all theweather here on planet earth). Great care was taken to optimize all relevant parameters, including
discharge frequency, length, and delay; valve opening current and time; filament position andcurrent. Many plots similar to that in Fig. 2.5 may be found in the data folders and logbooks from

27
May-June of 2017. On the bright side, the valve was successful in delivering an impressively low
total gas load, and an impressive flux of neon atoms, potentially a crucial enabler of our skimmer
cooling results [55]. In addition, the low translational temperature of neon is essential for our as
yet unrealized dream of studying collisions between neon and OH in the midst of the beam [60,Chapter 7].

2.2.2

Discharge

For working with OH, or any other free radical, it is essential to integrate together withthe valve body a mechanism for generating the radicals during the supersonic expansion process.
Generally, their reactive lifetime is too short to enable any other solution, and therefore a delicate
balance arises between generating them far enough into the expansion process so that they do nothave time to react, but not so far that they don’t have time to be thermalized to the bu↵er gas
during the expansion. Various techniques exist, including both radio frequency and pulsed electricfields, and also UV photolysis of some precursor species. All three of these have been attemptedon this experiment, with the use of pulsed electric fields having been used over the majority of theexperimental history prior to the Even-Lavie valve, which was purchased together with a so-called
dielectric barrier discharge radio frequency source. One exception was an attempt made at using
photolysis of HNO3 , described in [45, Sec. 3.1.4.2], which produced only half the output of the piezo
valve, and led to some rather hazardous laboratory conditions b .
A surprising observation we made during the testing of the Even-Lavie valve was that production efficiency of OH radicals continued to improve well beyond the recommended operating
point of the device, yielding an additional factor of two by going beyond the maximum recommended RF discharge of 1.6 kV or so. This suggested to us that the RF discharging mechanism of
the Even-Lavie valve could be the primary limiting factor associated with its performance for ouroperation. This led us to be quite interested in the possibility of using a DC discharge with a highpressure neon source, something which has at this point not yet been tested.
b

Private Communication with Brian Sawyer during DAMOP 2019.

28
2.2.3

Lorentz Force Opening Mechanism

The Lorentz force exerted on a wire by a magnetic field containing the wire can also be
exploited for operating valves. A key advantage of this is the reduced obstruction of the required
hardware for gas flow. This style was designed in Nijmegen in the Netherlands [61], and a similar
design was undertaken in our shop, though not yet completed or tested at the time of this writing.
A primary motivation of the undertaking was the lead time of the Even-Lavie valve, but when thisat last arrived, the e↵ort was dropped. Based on the performance of the Even-Lavie valve however,the possibility of operating this valve remains interesting, and a potentially viable way of actuallyleveraging the possibilities in principle a↵orded by having switched to a supersonic expansion inneon.
The JILA valve design was performed by Kyle Thatcher, and contains many nice optimizations relative to the original [61], especially an externally tunable positioning of the sealing device,
which is shown in Fig. 2.6. The valve also features careful material engineering to enable operation
into the 100 bar regime, although the relation between opening current and stagnation pressure willneed to be explored. A traditional DC discharge stack sits in front, and is also capable of higher
voltages than usual, motivated by our experience with the Even-Lavie valve.

2.3

Conclusion
The major undertakings that have occurred during this and all theses of the OH experimentdemonstrate the challenge and difficulty associated with maximizing the performance of a beamsource. A new e↵ort associated with leveraging the seemingly promising characteristics of a neon
seeded beam has begun, with minimal results thus far. One thing worth noting is that switchingback to the older piezo valve should at least provide a decent two to four fold gain relative to theperformance of the Even-Lavie valve obtained thus far, although this valve may play much less wellwith the cryogenic techniques discussed in the next chapters because of its increased flux. Shouldan e↵ort be deemed worthwhile, the new JILA version of the Nijmegen valve could serve as an

29

Figure 2.6: This high spring coefficient aluminum alloy wire sits within a large magnetic field andpulls open a valve due to the Lorentz force resulting from a brief 1000 A current pulse. The design
was taken from [61] and manufactured by Kyle Thatcher in the JILA machine shop.

excellent starting point for exploring the frontiers of optimum OH production in the neon regime.

Chapter 3
Skimming

The creation was all too soon marred and cast into disarray. Molecules whose trajectories
led them into difficult circumstances, such as forceful impact with the chamber walls, did loudly
proclaim their misfortune by initiating damaging shockwaves upon the entire population. And so
it became necessary to separate these malcontents from their fellows, and to restrict from accessto the promised land all those who failed to follow the straight and narrow path that leadeth unto
salvation. The device chosen to perform this separation of straight line followers from their fellows,of the wheat from the cha↵, became known as a molecular beam skimmer.

3.1

Aerodynamic Skimming
Traditional skimming techniques focus on shaping a trumpet-like structure in just the rightway so as to minimize the chance of rejected molecules subsequently reentering the beam [62, 63].
To a good approximation, molecules encountering the surface may be assumed to leave with a cosine
distribution of trajectories. This cosine distribution also describes the the likelihood of a patch of
surface being struck by particles in a gas above that surface. For simple reasons of geometricalflux, a given patch of surface is more likely to encounter particles approaching along a normal to
the surface, than at shallower angles, and quantitatively, the distribution function is proportional
to the cosine of the angle between the normal and the actual line of incidence. It follows thatthe same should be true for molecules exiting a surface, at least in the thermodynamic limit fora gas interacting with a surface. For this reason it is advantageous to have the surface strongly

31
tipped away from the beam, and to have the surface razor thin around its opening, so that asmuch as possible the molecules which interact with the skimmer are directed strongly away fromthe beam-line. With this way of thinking, it might seem ideal to separate the beam with a very
long hypodermic tube, but interactions with the inner surface of the skimming structure are alsoa concern. Molecules impinge on the inner surfaces due to the inherent transverse temperatureand spreading of the unperturbed beam, and also due to molecules which interact unfavorablywith the tip of the skimmer, and for this region it is better to have the inside of the skimmer
open up and not be shaped like a tube [62, Sec. 2.4]. The precise details of the shape can beexactly computed in the case of a continuum flow of molecules [64], but since the advent of pulsed
supersonic expansions, these details are no longer relevant since skimmers are no longer positioned
in the regime of continuum flow of the valve, which only made sense with the much higher overall
fluxes and much lower instantaneous fluxes of continuous beam sources.

3.2

Cryogenic Skimming
Cryogenic skimming can make a significant improvement to the situation thanks to the highprobability of adsorption of molecules directly onto a cold surface, which much more thoroughly
addresses the problem of separating unwanted molecules without their having an opportunity tonegatively influence the others [65]. Even in the case of incomplete sticking, a partial accommodation of molecules to the low temperature surface reduces their velocity, making it less likely thatthey will travel back into the beam before it has passed. This work is thoroughly described in both
our PCCP publication [55] and in Hao’s thesis [60, Chap. 5]. I will focus here on the question of
how these gains may best be transferred to the loading and operation of a Stark decelerator. For
this purpose, the key result worth recapitulating from our skimmer cooling work concerns the absolute comparison between the state of the art in room temperature skimming and that in cryogenic
skimming, see Fig. 3.1. A gain of nearly a factor of thirty is found in the improvement that resultsfrom the use of the cryogenic technique. This gain may be understood to arise from a combinationof “geometric” and “cryogenic” gains, where the former refer to the reduction in valve-detector

32

Figure 3.1: (a) Experimental diagram for the skimmer comparison. The “X” marks the location of
LIF detection. (b) Spatial hydroxyl radical density is measured after cryogenic and conventionalskimmers with the valve-skimmer distance optimized for greatest density in both cases. A factorof thirty enhancement is observed.

distance a↵orded by the reduced valve-skimmer distance and correspondingly increased solid angle,
and the latter refers to the remaining gain directly associated with clogging mitigation. It should
be noted that this geometric gain is potentially suspect in the sense that at some point these gains
correspond to molecules which may contribute to the increased spatial density observed after theskimmer but are too hot transversely to be loaded into the decelerator anyway. To first order,
this is not the case for the molecules which contribute to the observation in Fig. 3.1, since witha minimal detector transverse size for optimum density measurement, and an assumed ⇠ 2 mmtransverse e↵ective initial source size, molecules hit the detector with a transverse velocity spreadof:
2 mm⇥ 810 m/s = 19 m/s.
8.4 cm(3.1)

By comparison, traditional deceleration accepts a spread closer to 10 m/s, but SF mode pushes
this out to 18 m/s, and V SF out to 24 m/s, see Fig. 4.2.

33

Figure 3.2: Mechanical drawing of the volcano skimmer, units are inches, 0.079” = 2 mm. With avalve mounted right at 2 3 cm from the skimmer tip, the optimum found with no decelerator, 9x
reduced flux from 50% hole diameter reduction and doubled distance of aperture to valve.

3.3

Decelerator Coupling
Our first attempt at coupling molecules into the decelerator did not show a significant increasein signal upon cooling down the skimmer, as had been observed with the skimmer by itself. This isa strong indication of clogging, since the cryogenic skimmers should feature severe clogging at room
temperature due to their rather blunt shape. The obvious hypothesis is that skimmer cooling was
successfully mitigating skimmer clogging, but the increased flux through the skimmer simply led todecelerator clogging instead. We first sought to address this through the use of modified skimmer
geometries, which I will describe here, before moving on to more advanced strategies involving
hexapolar focusing as described in the next chapter.

3.3.1

Volcano Skimming

One obvious way to reduce the flux of molecules encountering the room temperature surfacesof the electrodes present in the Stark decelerator is to require the molecules of interest to go througha longer, narrower hole than previously. To achieve this, we installed a modified skimmer which was

34

Figure 3.3: Mechanical drawing of the second skimmer attachment component of the double skimmer. This rather complex design was machined via CNC from a single block of 99.9995 (5N5)copper for high thermal conductivity. Dimensions are in inches, 0.03900 = 1 mm.

essentially just a cone with a hole drilled through, see the mechanical drawing shown in Fig. 3.2.
Using naı̈ve solid angle arguments, this ought to have reduced flux onto the decelerator rods bymore than an order of magnitude relative to the previous design. In practice, the volcano skimmer
also performed poorly, with no significant gain during cool-down. One likely explanation for this isthat the skimmer opening was too restrictive, and adsorption not complete enough to avoid clogging
in this geometry. To assess the relative contribution of volcano skimmer clogging and decelerator
pin clogging, we also performed tests with Argon as a carrier gas. This ought to have a significantly
improved adsorption probability relative to neon, and could have led to a performance enhancement
if clogging in the skimmer were truly to blame. A comparable ratio of hot and cold temperatureswith Argon was observed, suggesting that either both gases had rather poor first-bounce adsorption
probabilities or that clogging on the decelerator pins remained the primary issue.

35

Figure 3.4: Several Skimmers are attempted, and none are found to improve upon the best guided
molecule flux obtained via conventional skimming. The variation in arrival time of the measurements correspond to the di↵erent valve skimmer distances used in the various cases, and also to thevariation in the initial speed of carrier gases between the case where Argon is used and the others.
These data were collected on March 16 and April 26 of 2018.

3.3.2

Double Skimming

Suspecting clogging in the volcano structure, we designed an alternative with similar flux
reductions as far as the decelerator was concerned, but with much better egress options for limiting
any clogging within the structure itself, see the mechanical drawing shown in Fig. 3.3. The device
was designed to occupy a small space and integrate seamlessly with our existing mounting structurefor quick manufacturing turnaround. Essentially, a miniature skimmer was machined in one piecewith an outer ring for soldering directly on to the back of another skimmer. Any clogging associatedwith internal wall reflection in the volcano skimmer should be greatly reduced in the device, while
enacting even stricter flux limitations thanks to the further reduced 1 mm opening hole. In practice,
the device performed only slightly better than the volcano skimmer. Our key figure of merit for
performance is shown in Fig. 3.4. Here the detected OH molecules after guiding through thelength of the decelerator are shown comparing the conventional room temperature skimmer to thevolcano and double skimmers. In stark contrast to Fig. 3.1, here only a loss is found between theconventional and cryogenic devices.

36

Figure 3.5: The low temperature behavior for cryogenic skimming shows a distinct lack of saturation, and a broadening of signal variation. Binning data by phase of the pulse tube cooler
temperature variation reveals the strong role this plays. Results are somewhat encouraging ofprospects for a lower temperature measurement.

3.4

Lower Temperatures
Some of our results suggested a performance that was still strongly improving as a functionof further reduced temperature by the time our pulsed tube cryostat had reached its lowest temperature, see Fig. 3.5. This motivated us to pursue a liquid Helium cooled skimmer, so as to achievea further factor of two reduction in temperature, and potentially expose an even more dramatic
gain in molecule number relative to what had already been observed. These attempts did not leadto an improved performance, and were not reported in [55], but some lessons worth sharing werelearned.
One of our most surprising early observations was the wildly varying variance of the observed
signal as a function of skimmer temperature. In fact, at first the observation was even more bizarre
than what is shown in Fig. 3.5, where three distinct data branches were visible in the region between

37
35

20 K, as can still be seen in the experiment logbook. This turned out to be an artifact of

60 Hz noise on the thermometer spectrum, and is an interesting example of the ways that noise
sources can interfere with data interpretation especially on a parametric plot. In any case, with thisissue easily filtered away, it also became clear that the broad variation in signal variance, especiallybetween 20

8 K, was induced by the rather large 2

3 K amplitude temperature oscillations of

the second stage of our Cryomech PT807 pulsed tube cooler. Such fluctuations are not untypicalof a purely copper system, which has a negligible heat capacitance in these temperature ranges,
and therefore directly writes the fluctuating temperature of the cryogenic fluid onto the system
of interest. A similar observation was recently communicated [66], and resolved through the useof a helium heat transfer stage interrupting the direct copper heat link. We found a very strong
dependence of the observed flux through the skimmer on not only the instantaneous temperature of
the skimmer but whether it was in the midst of becoming colder or hotter. It is a bit of a mystery
why such a dependence would exist. I spent some time searching for any reason that a time-lag
might exist between the collection of OH signal and of cryostat temperature data, since this could
masquerade as such a dependence, but nothing turned up. Assuming the dependence is in fact
real, one reason could relate to details of the adsorption process and of the surface layer on the
cryogenic skimmer. For example, bringing the adsorbed neon back up from its lowest temperature
could function as an annealing step which changes the spring constant of the solid in a way that ismore favorable for further adsorption.
In any case, the observation suggests that lower temperatures could provide further benefit,
as mentioned above. I devised an unconventional flow type cryostat designed for bringing only
the skimmer and a small thimble worth of liquid Helium down to low temperature. The device ispictured in Fig. 3.6. A long 1/2” SS tube protrudes deep into the vacuum chamber, such that a
3/8” OD helium fill line is able to protrude all the way down without making significant contact
and while leaving room for cold Helium gas to escape back upwards, further cooling the fill line
during egress. In terms of manufacturing, the solder joint between copper and steel constitutes themost challenging step. This was performed using a more aggressive ZnCl based flux, and required a

38

Figure 3.6: A thimble sized flow cryostat at the bottom of a long, thin-walled Stainless Steel tube isseen soldered to a copper skimmer. (i) Copper skimmer, 3 mm orifice, 30 external included angle,
25 internal. (ii) Copper liquid Helium reservoir, soldered to a copper skimmer backplate and thence
to the skimmer. (iii) Thin-walled Stainless Steel tube, found in the shop. Not exorbitantly thin,something like 0.01”. (iv) Copper to SS solder joint. (v) Lakeshore DT-670 Silicon Temperature
Diode. (vi) Tungsten filament for seeding the discharge, taken from Hans’ Halogen bulb by removing
the glass on the diamond saw. (vii) Even-Lavie valve and aluminum mounting frame. (viii) Quadlead feedthrough for Temperature Diode. (ix) Un-anodized lens tube structure for mounting LIF
collection lens about 5 cm from the beam-line. (x) High voltage RF wire for driving the dielectricbarrier discharge during the supersonic expansion from the Even-Lavie valve. (xi) 1/16” OD gas
feed line for Even-Lavie valve just visible. (xii) Thermal mounting block containing water reservoir.

39
few times to get right. Another initial failure arose from the use of an UltaTorr fitting with gasket
directly contacting the OD of the 1/2” tube. This poor choice led to the freezing and failing of said
gasket due to the cold Helium gas flowing just within. A workaround was to use a 1” UltraTorr
fitting to bridge the vacuum barrier and provide some position tunability, and connect a 1” tube
to the thin walled 1/2” SS tube with the cold cryogen running through it via Swagelok, thereby
leaving plenty of room to apply a heater for the gasket in between. Further details are labeled inFig. 3.6.
Several of the first attempts at transferring Helium to this device were fraught with major
challenges, particularly with regard to temperature stabilization, which are worth discussing further here. In our experiments with the pulse tube cooled skimmer, we often detected a strong
sensitivity to whether or not the valve was operated during cool down. In particular, it was foundthat operating the valve was actually crucial to the resulting performance of the system. We wereusually unable to recover the performance of the system without first heating close to room temperature. This strongly suggests that the performance relates to the influence of water or otherhigh temperature freezing species on the cryo-adsorption process for neon onto the copper skimmer.
For example, suppose that a layer of solid ice is actually detrimental to neon adsorption, and theformation of such a layer is the key process leading to reduced performance under some operating
conditions. This could explain the observed requirement of operating the valve during cool down,
provided the regular bombardment of the skimmer surface with neon was sufficient to disturb theformation of such a layer. It could also explain the observed failure to restore performance by partial warming, since warming only above the neon freezing point could actually be a very efficient
way to help the water mixed in with the neon to settle down into a solid layer.
In any case, during our first attempts with the Helium cooled skimmer, we inadvertently
created a temperature bouncing phenomenon, see Fig. 3.7. The system temperature was found toerratically vary between 20 K and 5 K, as shown in panel (b) of Fig. 3.7. We hypothesize that thisresulted from the situation where the narrow size of the copper reservoir allowed a bubble of Heliumgas to sit between the base of the copper reservoir and the liquid Helium above. The bubble would

40

Figure 3.7: Neon flux and temperature are monitored during the cool down procedure for a liquid
helium cooled skimmer. (a) Neon signal, measured by fast ion gauge, creates a voltage dip on thecollector of the indicated magnitude over the measured timespan. (b) Temperature, measured via
DT-670 Lakeshore Silicon diode, exhibits an erratic quasi-periodic temporal variation, likely driven
by highly nonlinear characteristics of the liquid Helium flow including bubble formation and surface
tension inhibiting heat transfer. (c) Parametric plot showing strong hysteresis behavior in the sensethat the observed signal depends not only on the instantaneous temperature but its time history.

grow in size as the heat load on the skimmer continues to transfer to the Helium, until eventually
the bubble would interrupt the flow of liquid helium out of the transfer tube and into the thin-walled
stainless steel tube. At this point, the liquid Helium working its way up between the transfer line
and the thin-walled tube would quickly evaporate, allowing Helium gas to escape from the bubble
above the reservoir and allowing new liquid helium to drop suddenly down on to the reservoir once
more. By increasing the pressure of Helium gas applied to the Helium dewar during the transfer,we were able to overcome this temperature bouncing mode of operation, although the pressuresrequired (5 psi or so above room pressure) are very high as far as Helium dewars are concerned,
which usually have relief valves at 10 psi and very stern warnings about the dangers of overpressure.
The high pressures used for transfer also limit the temperature obtained at the skimmer, since theevaporation temperature of liquid Helium has a significant pressure dependence [67, Sec. A].

41

3.5

Conclusions
Although we have successfully demonstrated very significant gains associated with skimmer

cooling on its own, leveraging these gains for deceleration has proven elusive. A more reliable androbust method is desired for mitigating clogging not only in the skimmer but also in the ensuingcoupling process to the decelerator. While cooling of the decelerator itself certainly comes to mind
as a possibility, the best blend of manufacturing plausibility and prospects for clogging mitigation
were determined to be a cryogenic focusing hexapole, described further in the next chapter.

Chapter 4
Hexapole Focusing

And it came to pass that all those molecules who contrived to survive the great segregation
described in the previous chapter, were set about a grueling and strenuous exertion of the mental
faculties. Indeed, the focus requested of them was of such a severity as to literally bend theirwills and their trajectories. And in more modern terminology, the subject of this chapter is thefocusing of molecular beams for optimized coupling between their source and any ensuing scientificapparatus. The work described in this chapter is in progress, and has thus far only yielded a factorof two improvement in decelerator flux, when used together with the cryogenic skimming techniques
described in Chapter 3, and compared against conventional room temperature skimming with nofocusing. As a consequence of the ongoing nature of this work, I have deemed it particularly
worthwhile to include many details of potential relevance to those who may wish to carry this worka few steps farther.

4.1

Phase Space Matching
To understand the action and possible usefulness of a hexapole, it is essential to become

familiar with thinking and working in phase space. By phase space I simply mean the two dimensional space spanned by the position of molecules in the ensemble on one axis and their velocity
on the other. Technically speaking phase space ought to refer to the position-momentum space,but for our molecules the resulting units would be cumbersome. The usefulness of working inphase space stems from the straightforward correlation between the various transformations that43
may be applied to the ensemble and how they are manifested in phase space, as is summed up in
Tab. 4.1. Free propagation results in molecules with larger velocity coordinate deviating towards
larger spatial coordinate, and vice versa. I always work in the frame of reference of the ensemble, sothat the centroid of the initial ensemble is assigned the (0, 0) coordinate. The action of a hexapole
is to create a spring-like harmonic restoring force in the transverse directions, in which case it
may be proven that motion in phase space is circular, provided the velocity axis is scaled by the
angular frequency of the harmonic restoring force into spatial units: v‘ = v/!. Beginning with thespring-like restoring force:ẍ =kx/mOH =! 2 x,(4.1)

and obtaining the solution for the position and velocity under simple harmonic motion:x(t) = sin(!t + ),v(t) = ẋ(t) = ! cos(!t + ).

(4.2)

If we now scale the velocity units by !, we obtain:
✓

x(t)

◆2

+

✓

v(t)
!

◆2

=

sin2 (!t + ) + cos2 (!t + )

=

1.

(4.3)

By appropriately choosing the frequency of the hexapole and the length of time spent within, it ispossible to prepare the ensemble for a revival of the initial distribution after some additional freepropagation afterwards, see the third and fourth rows of Tab. 4.1.
Furthermore, it is possible to engineer the applications of hexapole rotation and free flights
to achieve compression along one axis, accompanied by density preserving dilation in the opposite
axis. These results parallel the magnification or demagnification of an image achieved by a lensbetween an image and object plane depending on whether the lens is positioned closer to the image
or the object plane. One key distinction however between the behavior of an optical lens and thatof a hexapolar lens is that the latter is almost never operated in the short-lens regime, which has
a significant impact on how one thinks about its behavior. Molecules are often rapidly transitinga hexapole in the longitudinal direction, which means that the device needs to be long in order toapply the desired rotation of the transverse phase space. By carefully following an ABCD matrix

44
Table 4.1: Phase space manipulations are described and visualized. Velocity is the vertical axis,
Position the horizontal. Gray shows the ensemble before the action, black afterwards. The first
four actions are applied subsequently, the last two are applied directly after initialization.
ActionE↵ect

Initialization

Some approximately GaussianDistribution

Free Propagation

Upper Region to Right, Lower
to Left

Hexapole Focusing

Clockwise Rotation

Additional Free Propagation

Spatial Density Revival

Free + Hex + Long Free

Velocity Compression

Free + Hex + Short Free

Spatial Compression

Visualization

approach, instead of a simple lens-maker equation for the locations of image and object planes, we
obtain [68, Eq. 9]:
L3 =L1 + 1 tan(L2 ),
L1  tan(L2 ) 1

(4.4)

45

Figure 4.1: Sample trajectories of molecules focused by a hexapole of length L2 after traveling L1
to arrive and an additional L3 afterwards.

where  = !/v is an inverse length unit directly related to the hexapole strength and inversely tothe beam speed, and the Li are as described in Fig. 4.1. In the limit of small L2 , this equation canbe reduced to something more familiar. First, using tan(x) ⇠ x,
L3 =L1 + L2
L1=
2
L1 L2 1
L 1 L 2 2

1

,

(4.5)

and now inverting,1= L 2 2
L3

1
,
L1

(4.6)

so that for fH = 1/L2 2 we recover the lens-maker formula:
1
1
1
+
=
.
L1 L3
fH

(4.7)

This focal length has the dependences we would expect, since it reduces (i.e. the lens strengthens)with larger L2 or with larger !. This recovery of the lens-make formula applies for spatial focusing,and if a similar focusing were desired in momentum space, this would correspond to collimation.
This compression of velocity and dilation of position or vice versa is often utilized for optimizing the transfer of molecules in an ensemble between a source and a device, or between devices,a procedure known as phase space matching. To flesh this out a bit further, all the devices worked
with in the laboratory can be characterized by a phase space acceptance, in principal a six dimensional region but in practice only ever considered in terms of two conjugate dimensions at a time.
The phase space acceptance typical of the Stark decelerator for example is indicated in Fig. 4.2, and46

Figure 4.2: Transverse phase space acceptance of a Stark decelerator operated in various modes.
Each mode has a di↵erent characteristic ratio of velocity width to spatial width. Some colors are
hard to perceive, but all have a width close to 2 mm and varying heights between 10 and 30 m/s.

discussed further in Chap. 5. Unfortunately the phase space generated by a supersonic expansion
is rather difficult to ascertain, as discussed previously in Sec. 2.1.2. One thing which can be said forcertain however is that the initial transverse velocity width generated by the supersonic expansion
is much larger relative to anything else in the experiment. This follows from simple intuition as thesupersonic expansion only generates a mildly directed beam, which therefore expands significantly
transversely away from the valve. Projected onto transverse phase space, such an ensemble shouldhave a variation of transverse velocity that is on the same order as the forward velocity of the beam,i.e. hundreds of meters per second. These velocities are much larger than the maximum stabletransverse velocities in decelerators, which are usually around ten meters per second. Nonetheless, the valve may still generate a distribution that is quite narrow in its initial transverse spatial
spread. If the initial distribution is narrower spatially than the 2 mm of the decelerator, there will
be opportunities for an improved loading of the decelerator with appropriate magnification of thedevice designed to compress velocity and dilate position as in row 5 of Tab. 4.1.

47

Figure 4.3: Three traces showing OH population after bunching with the hexapole set at di↵erent
voltages. A scant 17% e↵ect is observed.

4.2

History
In the first generation of the experiment, a 5 cm hexapole was employed and successfullyenhanced the molecule number just afterwards by as much as a factor of 16 [69, Fig. 6]. What isless clear however is whether this enhancement really exceeds what could have been had by simply
mounting the decelerator closer to the valve and skipping the hexapole entirely. By the time Hao
and I were running the second generation of the experiment [70], the hexapole was only providing
a 17% e↵ect, as shown in Fig. 4.3. I can find no earlier record of the performance of this hexapole,so it is difficult to say whether this observation of no e↵ect with the voltage on/o↵ stems from some
HV issue that developed later, from transition between Jordan and PZT valves, or an alignment
issue or some other e↵ect.
If it is indeed the case that the hexapole in the second generation decelerator never had asignificant e↵ect, there are a few possible explanations. One relates to the influence of skimmer
clogging, which can alter the trajectories of molecules so that they appear to have originated fromthe skimmer tip. The skimmer tip is too close to the hexapole relative to its focal length for it tohave much influence, hence the observation of little e↵ect. Another possibility is that the initial
distribution of molecules in transverse phase space is simply much larger than the phase space

48
acceptance of the decelerator, so that even with no hexapole and pure free propagation, the initial
distribution still thoroughly overlaps the acceptance of the decelerator. This explanation would
even jive with the observed influence of the hexapole in [69, Fig. 6], since that observation wasmade with a detector directly after the hexapole. This detector could have observed a legitimate
hexapole induced spatial density enhancement, but entirely composed of molecules with transverse
velocities above the threshold for decelerator acceptance.

4.3

Cryogenic Hexapole 1
Against this dubious backdrop, we decided in the summer of 2018 to begin work on a newhexapole for use with the third generation decelerator. Our motivations were quite di↵erent howeverfrom those that motivated the poorly performing hexapole of the second generation. Instead ofseeking to optimize coupling between the valve and decelerator, we were seeking to intentionallydistance the two for the sake of reduced clogging potential, without sacrificing the gains in phasespace density we had obtained by cryogenically cooling the skimmer. This was largely motivatedby observation of significant decelerator clogging, as described previously in Chap. 3. The newhexapole would this require cooling to cryogenic temperatures, so as to avoid the clogging observedin the decelerator. In a sense, one can think of the cryogenic hexapole as comparable to cryogenicallycooling the decelerator, only much more practically attainable.

4.3.1

Hexapole Fundamentals

A hexapole consists of six alternately charged poles, which generates close to the axis anelectric field with the functional form:
✓ 2◆
r
~
E(r)
= 3V0
,
r03

(4.8)

where V0 is the magnitude of the voltage to which each rod is charged, with alternating sign, andr0 is the smallest radius which reaches from the axis to the surface of a pin. For a molecule witha linear response to the electric field, this quadratic dependence of the electric field magnitude

49
results in a harmonic potential, with the exception of the relatively small region close to the axis
where the field is too small to polarize the molecules. This exception causes a perturbation to theideal focusing behavior, but is not very significant for OH [69, Fig. 2b], although more so for otherspecies [68, Fig. 1]. Specifically, with V0 = 13 kV and r0 = 5 mm, which lead to the typical fieldstrengths characteristic of our traps and decelerators, electric fields up to 100 kV/cm are generated,compared to 3 kV/cm for polarizing the radical.

4.3.2

Core Electrode Design

My first attempt at a cryogenic hexapole design leveraged some rather complex technologies.
Fearing the possibility of gas mediated discharge of the hexapole electrodes due to the high densities
of molecules shortly after the skimmer, I planned to have the electrodes encased in sapphire tubes,
so as to avoid the availability of conduction electrons at their surfaces. These sapphire tubes
would in turn sit within a copper block, for maximal thermal conductivity, mechanical alignment,
and to provide additional adsorbing surfaces for the Neon. The fields generated in the planned
configuration at its intended voltages are shown in Fig. 4.4. In panel (a), note how the fields inside
the sapphire are particularly small, a byproduct of its large dielectric constant close to 11. In panel(b), the electric fields along two axes are shown, one through the electrodes and the other between
them. A quadratic guiding curve is also shown, which corresponds to an angular trapping frequency
for OH radicals of ! = 20.6 kHz.
This all of course requires patterning of a honeycomb-like structure on the inside of a longcopper block, something which may be readily achieved using a technique known as wire electric
discharge machining, wherein a thin sacrificial wire at high voltage is used like a bandsaw blade, andis slowly moved through another metal. Electric discharges between the wire and the metal being
cut cause material to be removed, and the wire is constantly fed so as to avoid its disintegration.
This was achieved according to our specifications without any hitches by the CIRES shop, a shop
located at the time of this writing in the chemistry department at CU.

50

Figure 4.4: (a) Electric field magnitudes and geometric size of the first generation hexapole, composed of 1/16” steel electrodes mounted within 1/8” sapphire tubes mounted within a 1” copper
block. Note how the fields inside the sapphire are particularly small, a byproduct of its large dielectric constant close to 11. (b) Field Strength (also kV/cm) verse distance o↵ axis along various
axes as described in the main text.

4.3.3

Thermal Design

Some of the thermal considerations pertinent to the design are illustrated in Fig. 4.6. The final
mechanical design di↵ers slightly from what is shown in this figure, but the thermal considerations
remain the same. The large copper block within which the hexapole sits has a skimmer affixed to
the front and an aperture to the back, and a large clamp-like structure conveys sufficient thermal
conductivity without requiring Indium solder. Confirming this required careful consideration of theforce-dependent thermal contact between the clamp and the hexapole, and also between the braid
and the clamp. Thermal conductivity across gold-gold and copper-copper pressure interfaces havebeen measured [71, Fig. 2.7], and exceed grease joints above a certain pressure, see the discussionin [71, Sec. 2.6.4]. As can be observed below in Fig. 4.7, we settle for a gold-copper joint, which
naively would fall somewhere between the gold-gold and copper-copper performances. Of course

51
nothing is certain, see for example in [72, Fig. 5.8] how mixing copper particles into epoxy can
actually worsen its thermal conductivity in some temperature ranges. It is also important to figureout how tight the clamp should be made, so as to benefit from the pressure-dependent thermal
conductivity, without irreversibly deforming the copper parts. I studied this using finite element
analysis via Solidworks Premium Simulation package, see Fig. 4.5. One interesting point worth
mentioning is that the geometry of the device leads to a factor of 2⇡ hoop stress amplification, where
the force between the clamp and the cylinder is actually 2⇡ times bigger than the force applied
to the bolts. This can be derived by conceptually dividing the geometry in half through the bolts
and analyzing the static free body diagrams of each component, a standard exercise in statics. The
conclusion is that 1000 kg on each of two bolts would give 240 W/ K for gold-gold and 3 W/ K for
copper-copper, found by taking the 10 K 50 kg data in [72, Fig. 5.8] and multiplying up to 12000 kg
including the hoop stress amplification. Of course, this brings to mind the question of validity of
the linear scaling of conductivity all the way up to such large forces. In [73], extremely high loads
were investigated, and although their cryogenic hydraulic ram apparatus failed catastrophicallyduring the investigation, they were able to confirm that conductivity increased 50-fold for forces
increasing 20-fold up to 4700 lb, a super-linear scaling to very high forces.
The emissivities guessed for the surfaces shown in panel (b) of Fig. 4.6 turned out to be optimistic, perhaps because they were approximate room temperature emissivities and not cryogenic,or maybe due to adsorbed gas layers further increasing emissivities. A key challenge of the design
pertains to how voltage is to be fed into the electrodes located inside the sapphire tubes. A few
iterations were made on the design before this was achieved with acceptable surface path-length
reduction and suitable mechanical rigidity. The HV distribution rings visible in Fig. 4.6 ended uphaving too much flexibility on their own, so plastic Ultem cu↵s were glued directly onto the Sapphirerods, simultaneously increasing surface path length for arcs and adding the required rigidity.
An additional consideration for the hexapole is the matter of its alignment with the beam-line.
There are two primary strategies by which satisfactory alignment may be obtained:

52

Figure 4.5: (a) Stress resulting from clamp tightening. One of two pairs of bolts and nuts intersect
the shown plane near the bottom, and tighten on a rectangular block, the clamp, which thengrabs hold of a copper rod, the hexapole mounting rod. The hexapole is not visible within thisrod. Coloration indicates the stress induced by the tightening in the clamp, the rod, and the bolts.
Annealed copper yields at only 70 MPa or so, and this stress is exceeded in several places, indicatingthat 9500 N is on the upper end of what should be used on the bolts, and would accompany someyield on the edges of the clamp. (b) Strain in the vertical direction only, so as to view what happens
to the clamped hexapole mounting copper rod. Only 3 µm radial compression is observed, small
compared to the 3 mm diameter sapphire tubes later inserted into the rod.

(1) Contrive a way for alignment to be tweaked from outside of the vacuum chamber, so that
final alignment may be guided by actual system figures of merit such as molecule yield after
the hexapole.
(2) Provide in-vacuum fine-adjust knobs only, and engineer a protocol by which alignment
may be obtained before pump-down, and not subsequently lost during either pump-down
or cool-down.
Strategy (1) was employed for our cryogenic skimming work as described previously in Chap. 3,
but was deemed insufficient for this hexapole work, primarily due to unavailability of the requisite

53

Figure 4.6: (a) Conduction of thermal joints in W/ K. In the final design, two braids were used,
for a total conductivity closer likely above 1 W/ K. (b) Radiative heat loads on exposed surfaces
in mW.

degrees of freedom. Unlike the skimmer, which only really requires two translational degrees of
freedom in order to achieve beam alignment, the hexapole has four alignment parameters, andpotentially five depending on whether it is desirable to translate the device along the beam. There
seemed no straightforward way to achieve this across the vacuum envelope, especially given how
far above the hexapole the tuning bellows sits. Thus strategy (2) was pursued, and the problem ofcool-down induced misalignment addressed via a flexible heat linkagea .
The principal challenge of utilizing a flexible heat linkage is finding a di↵erent way to achievemechanical rigidity without violating thermal isolation. For this purpose, I selected Ultem (or PEI,
polyetherimide), a dimensionally stable engineering thermoplastic with poor thermal conduction
and good mechanical properties. Ultem rods 1/8” wide and 7/8” long can be seen in Fig. 4.7(i),which also illustrates other pertinent details of the mounting assembly. Many polymers featureadmirably low thermal conductivity, and the subject of polymer thermal conductivity is treatedmore fully in [72], Chap. 5. The same selection was once made for cryostats at the RHIC [74].
In order to make estimations of the thermal conductivity for long, thin cylinders made of unfilled
Ultem 1000 connected between room temperature and the cold apparatus, it is necessary to properly
treat the thermal gradient which will necessarily build up within the material. I address this firstaTechnology Applications Inc, braid P6-502. Tyler Link, at the time of this writing, is always ready to send longemails about his products, at any hour of the day. Contact him directly for a product manual.

54

Figure 4.7: (a) A view of Hexapole 1 after assembly. (i) 1/8” x 7/8” Ultem rods to insulate but
support the structure. (ii) Flexible heat linkage, 1.4 W. (iii) Springs enabling fine positioning, fourin total. (iv) Nichrome heater coil. (v) Blurry HV distribution ring barely visible. (vi) Invar plate
spacers behind steel 1/4-28 nuts. (vii) Custom Skimmer, 3 mm opening. (viii) Teflon kinematic
cylinder visible for translating but allowing twist. (ix) Fine adjust knob, 3/16-100, mounted in aphosphor bronze bushing, torr-sealed into a steel block. (b) A mechanical section view focusing
on the scientific portion of Hexapole 1. (i) A rear skimmer, one piece with an outer rim thatallows grease-only installation. (ii) Drill holes for slip-fitting thermally insulating Ultem mounts.
(iii) Recess drilled for increasing surface path between the open end of the sapphire rod and themounting block. (iv) Plastic caps glued onto the rods for further voltage safety. (v) One of six HVholes, 1/2”, used for feeding voltage to the electrodes. (vi) Plastic stubs for increasing surface arc
distance in the vicinity of the feedthroughs. (vii) HV Pin which pokes down through the stubs andcontacts electrodes. (viii) Distribution rings which HV pins thread into. (ix) Curved contact line
where plastic stubs were glued to sapphire tubes. (x) Recessed solder joint for cryogenic skimmer.

55
in a very general sense. Suppose a material has a thermal conductivity c(T ) which varies withtemperature T and is connected as a cylindrical rod between reservoirs at temperatures TL and
TH . The total heat transfer rate, Q̇, must be the same through every slice of the rod, neglecting
radiative heat loads, and will induce a temperature change through a thin slice dx of the rod of theamountdT =Q̇
,c(T ) · A/dx

(4.9)

where A is the cross sectional area of the rod. Rearranging and integrating:ZTHc(T )dT =

ZL

Q̇dx
,
A

(4.10)

0

TL

TH

c̄

= Q̇L/A,(4.11)

TL

where c̄ is the antiderivative of c. We may proceed using the data from [74] for slightly glass-filled
Ultem 2100, which shows 0.04 W/m K at 10 K, 0.4 W/m K at 300 K, and an approximately
straight line on a log-log plot in between. It follows that c(T ) may be approximated as:c(T ) = ↵T ,=0.4
log 0.04= 0.68,log 300
10↵=

0.4= 8.5 ⇥ 10 3 W/m/( K) +1 ,3000.68(4.12)

so that,
TH

c̄

=
TL

300 K
↵
T +1
= 73 W/m,+1
10 K(4.13)

and finally with A = 7.9 mm2 and L = 20 mm, we have rearranging Eq. 4.11:
Q̇ = 73 W/m ⇥ 7.9 ⇥ 10 6 m2 /(0.02 m) = 0.029 W.

(4.14)A table of thermal conductivity integrals with TL = 4 K and various TH is provided in [71,App. A2.1]. No value is included for Ultem, but the listed polymers have very similar values tothat found above in Eq. 4.13, for example Teflon is listed as 70 W/m. It is also worth pointing outthat the conductivity integral between room temperature and 4 K only di↵ers from that between
room temperature and 50 K by 10%. This is symptomatic of the asymmetric distribution ofthermal gradient, with most of it occurring in a very small section, and the majority of the plastic
closer to room temperature since this is where the thermal conductivity is larger.

56

Figure 4.8: Flight profiles of OH after the Hexapole. Voltage is seen to have an e↵ect. The e↵ect
is sometimes disordered, and shows strong hysteresis. Note how for the |ei state, signal decreases
monotonically with voltage, as one might expect, but for the |f i state, signal first decreases before
increasing. Upon returning to the 0 kV trace, the value is larger than both the original 0 kV and2 kV traces. These traces were recorded chronologically in the same order as the legend. The data
were collected on November 27, 2018, with a 16.5 cm valve position setting, which in this setup
corresponded to a 10 cm true valve skimmer distance, a 9.7 K hexapole temperature, and an ND1
filter on the fluorescence collection stack.

4.3.4

Performance

The hexapole did not even come close to performing as intended, particularly as far as highvoltage is concerned, with arcing behavior observed at only one third of the designed operating
voltage. One of the first datasets is shown in Fig. 4.8. A surprising e↵ect was observed, whereby
the signal was first decreased and then increased. At first it was suspected that this could relate
to the focusing behavior of the device in some way, since by analogy with lenses over-focusing canbe problematic. This does not make sense at such low voltages however, and does not explain theeven more surprising observation of hysteresis behavior, where the signal with the hexapole o↵ wasobserved to decrease dramatically after having operated the hexapole at voltage for a while.
Ultimately, these e↵ects were attributed to the buildup of very significant patch charges on
the surface of the sapphire, significant enough to simulate the hexapole having been charged toseveral kilovolts of voltage. This e↵ect was in fact anticipated during the design stage, and aworkaround developed- regular polarity reversal. In the design stage, it was expected that charges

57

Figure 4.9: This schematic diagram illustrates how surface charges donated from the conductive
grounded hexapole mounting block (blue) are able to rather efficiently coat the surface of thesapphire tubes insulating the positively charged hexapole electrodes (black).

might buildup due to ionization of carrier gas upon interaction with the electric field and thecryogenic sapphire surface, but in fact a more potent mechanism exists, see Fig. 4.9. Despite fillets
designed for minimizing surface currents, conduction electrons from within the grounded copper
block are likely able to migrate out onto the surface of the sapphire tubes. This also suggests thatcharging the hexapole with only negative voltage and ground could help reduce the e↵ect, since thecopper block only has negative charges to donate to the surface, not positive.
While true voltage alternation requires complex tri-polar switch output configuration, e↵ective alternation may be achieved with a unipolar setup by connecting each set of three hexapole
electrodes to the output of a switch which connects between ground and minus high voltage. The
switches can then take turns charging the hexapole, so that the field direction reverses each time,
rod polarity is always negative relative to the grounded copper block, and unipolar push-pull architecture is maintained. Operating in this manner did indeed resolve patch charge issues, and led
to the data shown in Fig. 4.10. One downside of this mode of operation is the reduced maximum
hexapole voltage which may be applied, assuming that availability of MOSFET switches limits the

58

Figure 4.10: Peak OH population, various valve-skimmer distances, several voltages. Much longer
averaging and runtimes without hysteresis and with sensible trace ordering. Vertical axis is arbitrary, the numbers may be derived from an area integrating script applied to raw data traces akinto those in Fig. 4.8.

maximum absolute value of voltage. There is also the important question of what influence thenon-bipolar hexapole voltages have on fringing field distributions. Generally speaking, the fringe
field magnitude should become much larger, since now on average the hexapole rods are charged
instead of neutral. To some extent, this may be a welcome e↵ect as far as maintaining polarizationis concerned, since without any electric field, the well focused |f, ± 32 i molecules would be liable tore-project into |f, ± 12 i after spending time in a region of little or no field.
As far as the thermal performance of the device, a thermal conduction of 1.4 W/ K wasachieved, based on the observed second stage cryostat temperature of 7.2 K, which suggests via
the capacity of the device that a 3 W heat load is incurred. This, together with the measured temperature of 9.4 K close to the hexapole suggests that 3 W leads to 2.2 K temperature rise, hence
the stated 1.4 W/ K. While the thermal conductivity is as intended, the heat load is much higher
than predicted. This prompted a thorough investigation that ultimately implicated blackbody radiation as the culprit. At one point we even disconnected the hexapole from its mounting structure
so as to take away any heat load due to the thermally insulating mechanical Ultem mounts. These
were completely exonerated, as the observed temperature after their removal did not detectably
improve.

59

4.4

Cryogenic Hexapole 2
Primarily motivated by the failure of the first Hexapole to hold the desired voltages, weundertook a new hexapole with a completely di↵erent high voltage strategy. Fearing significant
delays to our long term goals, I pushed for significant reuse of our earlier design, especially withregard to the mounting structure. The result is shown in Fig. 4.11. The mounting structure andthermal clamp are completely reused, but the copper hexapole mounting block is replaced witha thin copper sheath for transferring heat to the skimmer, inside of which sits an Ultem plastic
mounting tube, and inside of that a more traditional stainless steel hexapole. The steel is allowed
to reach some equilibrium temperature set by the balance between weak heat transfer through the
Ultem tube to the cryogenic copper transfer tube, and radiative coupling of the high emissivity
Ultem to the room temperature vacuum chamber. A room temperature skimmer is used as a backaperture, and HV distribution is simplified and achieved via steel setscrews that also fix the positionof the internal hexapole. These screws sit in a plane orthogonal to the section plane and are thus
not visible.
Cryogenic hexapole 2 was a very quick turnaround relative to hexapole 1, with first tests
occurring within one month of those for hexapole 1. The second hexapole performed significantly
better as far as voltage application, by more than a factor of two. However it seemed to be clogging
limited, showing an earlier fallo↵ with respect to reducing valve-skimmer distance, and a weakerpeak signal compared with hexapole 1. These results are summarized in Fig. 4.12. One obvious
explanation for this would be the increased temperature of the hexapole, a sacrifice made in lightof the importance of first verifying voltage capability before anything else. It is unsettling howeverthat the first hexapole appears to perform better even at larger distances where the clogging should
not yet dominate. This could be attributed to di↵erences in the alignment of the collection system
and laser, but it is difficult to say.

60

Figure 4.11: This section view shows Hexapole 2, and how it was designed to fit inside of thesame mounting structure used for Hexapole 1. (i) A copper transfer sheath transfers heat fromthe skimmer. (ii) An insulating Ultem tube extends the entire length of the hexapole, allowing a
stainless steel hexapole to be voltage isolated within. (iii) Ultem stalks are used like in Hexapole 1,
but this time with threaded PEEK 2-56 screws connecting them directly to the Ultem sheath. (iv)
One of four HV rings mount to every other hexapole rod in two places each. The rearmost rings
also connect to HV wires via setscrews mounted in the Ultem orthogonal to the section plane andthis invisible here. (v) One of six hexapole rods, about 50% longer than in Hexapole 1, chosen so
as to increase e↵ective hexapole position tuning flexibility by varying the turn-on and turn-o↵ ofthe device. (vi) Commercial rear skimmer, with nearby gas egress openings visible in the Ultem
tube.

4.4.1

Cryogenic Hexapole 2.5

Hoping to obtain the best of both worlds, an updated version of Hexapole 2 was createdcapable of reaching full cryogenic temperatures while leveraging the favorable voltage performanceof Hexapole 2. A section view showing this is available in Fig. 4.13. Relative to Hexapole 2,
Hexapole 2.5 features HV safety shoulders built together with the HV tube as a single piece, so61

Figure 4.12: Hexapoles 1 and 2 are compared regarding maximum signal with no applied voltage
as a function of valve-skimmer distance. The second hexapole turns over at larger distance andappears slightly lower in the large distance wing relative to the first. The original figure was
generated on January 7th, 2019.

that a cryogenic copper shield can slide over the entire Ultem HV tube instead of only coveringa small portion and serving as an interconnect to the cryogenic skimmer. To our chagrin, thedevice featured a significantly worsened high voltage performance relative to Hexapole 2, which
ultimately turned out to be a fault related to surface arcing on the Ultem tube itself, despite thisnot having limited behavior in Hexapole 2. This was confirmed by successively disassembling thedevice, pumping back down, and applying high voltage. Even with all of the shielding and eventhe mounting structure removed, leaving only the Ultem mount and the steel hexapole within, thearcing persisted.
The reason for the poor performance of the Ultem remains a mystery. Organic polymers
are not inherently bad, as evidenced by the ubiquitous use of polymethacrylate, or acrylic, in HV
situations [75]. In fact, organic polymers often outperform glasses and ceramics, possibly due totheir reduced dielectric constant, which controls field amplifications near voids and triple points [75,pp. 776]. Another possible explanation of the di↵erence in performance between Hexapoles 2
and 2.5 relates to their surface treatment, with the latter much more smooth than the former.
Evidence exists for a benefit to surface roughness [76, Sec. 8.3.11]. It is also possible that cryogenic
temperatures negatively influence performance, as observed for some polymers [77, Sec. 14.4].
As far as cryogenics are concerned, hexapole 2.5 performed better than any other, validating

62

Figure 4.13: This section view shows Hexapole 2.5. (i) The outer copper sheath extends past
the thermal block, all the way to the rear of the hexapole, and the Ultem tube inside is 1/16”
thick everywhere, rather than 1/8” for half of its length in Hexapole 2, see Fig. 4.11. (ii) The
rear skimmer is machined together with a cylindrical can designed to mate to the outer diameterof the outer copper sheath, like the rear aperture of Hexapole 1 except in that design the mate
was on the ID. Holes outside this aperture skimmer allow steady state gas egress. (iii) A large
box-like first-stage radiation shield is installed. (iv) An Ultem part directly mounts the radiation
shield to the second stage thermal clamp, but with extremely low thermal conduction thanks to
a 3D printed hole array. (v) A smaller shield boxes covers the front of the hexapole, except forthe skimmer cone. This screws into the larger shield box. (vi) Another shield box covers the rear,
except for an egress hole for the beam. These shield parts are all bent to shape and then annealed.
(vii) Macor bushings insulate 1/2” holes for HV feedthroughs. The other sits inside the smaller
rear shield box in the un-shown sector.

the blackbody shielding technique incorporated. Surprisingly, this did not lead to significant clogging mitigation relative to hexapole 2 at zero electric field where the high voltage performance is
not relevant. In fact, by varying temperatures, the magnitude of the cryogenic benefit was directly

63

Figure 4.14: Time of flight traces for di↵erent valve-skimmer distances and at two temperatures
elucidate the clogging performance of Hexapole 2.5.

investigated, and found to be far smaller than anticipated, see Fig. 4.14. One explanation for this
could be the negative influence of surfaces positioned orthogonally to the beam-line, such as thoseformed by the HV distribution rings in Hexapoles 2 and 2.5 but not on Hexapole 1. Another explanation could relate to the adsorptive properties of the materials involved in the two hexapoles. In
Hexapole 1, the surfaces presented to the molecules were lightly greased sapphire and semi-roughcopper from the wire-EDM process, while in Hexapole 2 and 2.5 the surfaces are un-greased highly
polished steel. It is conceivable that the former would perform better, in which case one simple
test would be to lightly grease the stainless steel.
Some of the technologies involved in the cryogenic performance of Hexapole 2.5 are worth
some further discussion. Firstly, the plastic component visible in Fig. 4.13(iv) demonstrates thepower of 3D printable thermoplastics for custom thermally insulating structures with good mechanical rigidity. Thermal simulations of this part indicate conductance in the vicinity of 10 K/mW,
which includes a factor of 2.5 achieved by the convenient removal of material in a hole pattern 4.15
This patterning not only reduces thermal conductivity, but also aids in the removal of 3D printingsca↵old material and reduces print time. It is also possible to reduce the thermal conductivity of
3D printed parts by reducing infill, so that voids are intentionally left inside of a structure. This
seems inadvisable given the likelihood of virtual leaks to the vacuum chamber, although this may

64

Figure 4.15: Thermal simulations of an Ultem isolator used as a mechanical interconnect between
Hexapole 2.5 and its radiation shields, with (b) and without (a) extra holes. In each case, a 1 mW
load is applied across the device, and the base is fixed at 0 K, so that the reported temperatureon the upper surface of the device rises with smaller conductance.

not be relevant for parts that will sit between two cryostat stages and therefore lie entirely below
the freezing point of all relevant gases after cool-down. Secondly, the radiation shields are manufactured using sheet-metal bending techniques. Sheet-metal bending techniques are not often utilized
in the laboratory, often reserved for high throughput manufacturing, but in Hexapole 2.5 they reallyenable geometries that would be vastly more challenging to achieve with traditional fastener-based
methodologies. Using appropriate design modules in Solidworks, the radiation shields are designed
to be folded into the desired shape after first being water-jetted out of copper sheet. At the timeof this writing, water-jetting is available in the physics shop with nearly next-day service and only
1-2 hour operator costs. Bending is hard to do accurately, but this is generally not required forradiation shielding, and if accuracy is required, commercial CNC bending is available with 0.005”
tolerances on hole positions for the final folded partb . The act of bending reduces thermal conductivity by work-hardening the joint, but where conductivity is critical, this can be addressed byannealing.
b

Rapid Sheet Metal o↵ers 10-day service and something like $500 per design, way less with quantity.

65
Table 4.2: The historical record of coupling molecules into the decelerator, with and without
hexapoles, is presented. Hexapole 1 is seen to provide the greatest peak signal by 48%, and thegreatest total flux by 75%.
Date

Situation

Velocity Spread(m/s)

Peak Signal(photons/shot)

Total Flux
(arb)

Aug. 2, 2018

No Hexapole,
8 K Skimmer

10.7

3.2

95

Aug. 21, 2018

No Hexapole,
Hot Skimmer

11.1

4.8

156

Dec. 04, 2018

Hexapole 1

14.1

7.1

275

Jan. 07, 2019

Hexapole 2

11.7

3.3

119

4.5

Guided Flux
One important way to gauge the performance of these hexapoles is to study the flux ofmolecules which they are able to couple into the decelerator when it is run in a guiding mode. We
can use this to compare the hexapoles together with earlier measurements without a hexapole to getat the key question of whether a cryogenic hexapole is capable of improving performance relative tothe non-cryogenic, hexapole-free, commercial skimmer approach. A selection of several measured
fluxes in guiding spanning the hexapoles discussed here are shown in Tab. 4.2. In all cases, theskimmer used features a 3 mm opening diameter, and the DC guiding voltage used is 6 kV, avalue which gives a transverse phase space acceptance comparable to that of F mode decelerator
operation, see Sec. 5.2.3. Overall, only Hexapole 1 is seen to outperform the commercial skimmer
option, and that by less than a factor of two. It is still an accomplishment to have exceeded theprevious best, but a far cry from the thirty-fold gain comparing a room temperature and cryogenic
skimmer without any coupling to the decelerator [55].

66

4.6

Cryogenic Hexapole 3
After an interlude for attempts at collisional studies in the decelerator and between parallel

plates, it was time to move on to other things, but the lessons learned on the first two and ahalf hexapoles could one day be brought to full fruition in a final iteration of the device. By
abandoning the emphasis in Hexapoles 2 and 2.5 of reusing the mounting structure developed for
Hexapole 1, it should be possible to further increase the surface path distance between electrodes,
by say an additional factor of 3-5 beyond the factor of 3 already implemented between Hexapoles
1 and 2. In addition, by carefully following the principles of triple point shielding laid out in [78,Sec. 4.3.3], extra safety should be reachable. In particular, true shielding of the triple points thatexist between the Ultem tube and the internal HV distribution mounts of Hexapoles 2 and 2.5 is
not really possible given the external copper tube and requirement of sliding installation inside.
In addition to making a further dramatic increase in surface arcing path-length and triple
point shielding, the next generation hexapole will need to address the observation of limited cryoadsorption performance. In addition to the reduction or removal of orthogonal surfaces and theinvestigation of surface materials and the performance of grease, another possibility would be to
reduce all surfaces, even parallel ones, presented to the beam-line. One way to move in this direction
would be to reduce the radius of the hexapole rods, which would introduce some dodecapolar
deviations from the ideal hexapole fields, likely a minor e↵ect on top of the clogging behavior.
Another rather drastic possibility would be to remove the front skimmer entirely. It is difficult
to predict what would result from this, but if the skimmer is still reducing the beam flux relative
to what it could be with no skimmer, a big win could be obtained. Simultaneously, without theskimmer the flux onto the hexapole rods would be significantly increased, exacerbating any clogging
issues associated with these surfaces.
An important factor to keep in mind is the e↵ective transverse size of the source coming fromthe valve. If the valve generates molecules with a broad enough spread in their initial transverse
coordinate relative to what the decelerator can accept, this can pose a limit on what can be obtained

67
by adding a hexapole. The reason for this is that with a suitably broad initial distribution, even an
extended time in free flight may be incurred without significant broadening of the portion of phasespace that overlaps with the acceptance of the decelerator. It is difficult to measure this initialtransverse spread of the molecules, but simulations of transverse oscillations in the decelerator
suggest the value may be somewhat large, see Sec. 5.3.

4.7

Conclusions
Several cryogenic hexapoles have been designed and tested. High Voltage issues have provenmore pervasive than initially expected. Clogging issues within the hexapole have proven difficult tofully understand. As far as the absolute comparison between what may be obtained with a cryogenic
skimmer and hexapole and what may be obtained with a traditional skimmer, thus far nearly afactor of two gain has been demonstrated, despite the high voltage and clogging underperformance.

Chapter 5
Optimized Deceleration

Over the past two decades, Stark deceleration has enabled groundbreaking collisional [79, 80,81] and spectroscopic [82, 83, 84, 85] studies of a variety of species [86]. Subsequent trap-loading
greatly enhances interrogation time for such studies [87] and opens the door for further coolingand manipulation [44, 88]. Stark deceleration preserves the high densities generated by supersonic
expansions, and these perform best with the lightest and fastest carrier gases. This therefore
promotes the continual hunt for more efficient and longer devices. In the following the conventional
decelerator geometry will be discussed, followed by an extensive discussion of alternative techniques
and improvements, which are shown both theoretically and experimentally to provide promising
improvements in molecule flux and decelerator efficiency. Finally some details of the manufactureof an actual device, the third generation of the OH experiment, are provided.

5.1

Conventional Decelerator Geometry
Much like related work in charged particle accelerators [89], a neutral particle decelerator isrequired to satisfy two key principles:(1) A particular candidate particle known as the synchronous molecule must be manipulated
as desired by the device, e.g. slowed from 800

50 m/s.

(2) Other particles of similar initial conditions to the synchronous molecule must also possess
similar final conditions.

69
This second principle is known as phase stability, and is crucial to the overall performance of thedevice as far as flux or brightness is concerned.
Neutral particle decelerators face an incredible setback relative to charged particle accelerators, in that it is vastly harder to apply forces on neutral particles. Specifically, the device I will soondescribe can accelerate hydroxyl radicals at ⇠ 200 km/s/s. If hydroxyl cations were instead placedin the largest electric fields generated by the device of about 100 kV/cm, they would experience anacceleration given by:
qe · Emax= 5.5 ⇥ 109 km/s/s.
mOH

(5.1)

This constitutes a factor of over ten million. But is this truly fundamental, or technically limited?
Applying forces on neutral particles requires the application of gradients in electric fieldmagnitude, and the magnitude of these gradients depends on the miniaturization of the geometry
the electrodes. It becomes challenging to develop a truly fair comparison, but underneath all of thedetails about electrodes and geometries, electric fields are always generated by charged particles,
regardless of whether they are conduction band electrodes in some material. The force on a neutral
particle by a charged one scales as r 4 , while that between charged ones scales as r 2 .
In any case, once the factor of ten million setback is taken into account, neutrals have thenice property that the forces which act on them depend only on the magnitude of the external
fields, and not its direction. This makes it possible to generate large magnitude electric fields withthe direction of the field orthogonal to the beam-line, which is precisely the trick utilized in thefirst Stark decelerators [90]. The design of this decelerator was replicated here in the Ye group,
first at a length suitable for expansion with Xenon, then later with Krypton, and most recently
with Neon. Pairs of cylindrical pins are arranged alongside the beam-line, so as to apply a large
but localized field magnitude across it for maximum field magnitude gradient. Successive pairs arerotated ninety degrees relative to one another, which is relevant to transverse confinement, andevery fourth pin is connected up to the same backbone electrode.
Diagrams such as that shown in Fig. 5.1 are useful in understanding the operation of the

70

Figure 5.1: Voltages, electric fields, and on-axis potential energy in a conventional pin decelerator [90]. Pin rotation indicated by representation as slight ellipses rather than circles. Electric fields
density corresponds to the potential experienced by molecules. The horizontal dashed line indicatesthe central axis of the device, and the potential energy of the target or synchronous molecule as it
transits this axis is shown below the pin pairs. A’ indicates the translation (and rotation) of thedrawn configuration of fields and voltages to the next pin pair(s). Vertical dashed lines indicate
change in potential energy of a molecule during a switching event between configurations.

device. Molecules approach a region of strong electric field, trading kinetic energy for internalpotential energy to compensate for their Stark shift, which describes the dependence of their internalenergy on external electric fields. In the case of molecules whose internal quantum state shifts
upwards in increasing external electric fields, i.e. with a positive Stark shift, entering a region ofstrong field results in a reduction of speed. At some point, the strong electric fields are turned o↵,and a di↵erent region of strong electric field is turned on further along the beam-line. If the strong
fields are turned o↵ before the synchronous molecule reaches the strongest fields, i.e. partway up
the hill, phase stability is obtained, because a molecule that is ahead of the synchronous one will
therefore exchange a greater quantity of kinetic energy for potential energy by climbing farther up
the hill before the turn-o↵ event. This molecule thus experiences a force restoring it to the locationof the synchronous molecule. Of course if a molecule is too far ahead, it may pass all the way
through the region of highest field prior to the turn-o↵ event. Such a molecule is no longer phase

71

Figure 5.2: Longitudinal phase space plots of a Stark decelerator with varying final velocity.
From left to right, vf = 33, 50, 100, 200, 500, 800 m/s. Note how the size of the populated area
reduces as the final speed is reduced. This is a direct reflection of how closely the synchronous
molecule approaches the charged pin-pair, which in turn controls how far ahead a non-synchronous
molecule may be while still experiencing a restoring force. Note also how the size reduces much
more significantly between 800 500 200 m/s then for any of the other speed reductions. This is
described further in the main text.

stable and will travel farther and farther from the synchronous molecule. Thus there is a tradeo↵
between the deceleration capability of the device and the volume of its phase stable region.
This can be seen more directly from simulations of its behavior, such as in Fig. 5.2. The areaof the region populated by molecules reduces as the deceleration is increased. However, the change
in area seems to plateau and then stop, somewhere between 200

500 m/s since all velocities below

200 m/s show nearly the same area. This can be attributed to the fact that all of these low speeds
require nearly the same energy to be removed per stage of the decelerator, since the total energy
removed can be calculated by the potential energy di↵erence before and after the decelerator:
1
PE = mOH vi2
2
Once vf << vi ,

vf2 .

(5.2)

PE no longer various strongly with vf :1
PE ⇠ mOH vi22

With no variation in

1

✓

vf
vi

◆2 !

.

(5.3)

PE, we expect no further change in the area of the regions populated inFig. 5.2.
An important corollary of this idea relates to the parameterization typically used for spec-

72
ifying decelerator operation. Since the devices are periodic, it is convenient to parametrize thelongitudinal coordinate with an angle, called the phase angle, and chosen to vary by 180 between
one pin-pair and the next, so that a single voltage configuration applied to the device repeats itself
after 360 . This parametrization also allows us to work independently from the choice of pin sizeand pin spacings for a given device. A decelerator is often operated so that a synchronous moleculewith a given speed would always experience a turn-o↵ event at the same phase angle(or +180 ),which then allows the operation of the device to be specified by only that angle. It follows from
Eq. 5.3 however that very slight changes in

PE can lead to dramatic changes in the final velocityof the synchronous molecule, which also applies tofar fromsince

PE / sin( ) and our device operates= ⇡/2. In earlier iterations of the code used for programming the switching sequence ofour decelerators, it was necessary to specify the phase angle down to the thousandth of a degree inorder to get the desired final speed within a few meters per second, which in turn required carefully
hand-exploring the function mapping phase angles to final speeds.
It is also worth discussing some other features of the longitudinal phase space plots shownin Fig. 5.2. Almost all final speeds show a bullseye type pattern, with a halo of outer particles
surrounding an inner core. This is discussed extensively in [91], and relates to the inefficiencies in thedevice that are the subject of the next section. However, there is additional structure evident within
the inner core, especially for the molecular packet with 800 m/s final velocity. These molecules have
not had their speed modified significantly by the simulated device, in which case the activity of thedevice is better described as “bunching” the molecules than “slowing” them. In the experiment,these bunched molecules would travel together with a large population of unaddressed moleculeswith initial longitudinal velocity and position unsynchronized with the timing of the device. These
unaddressed molecules are nonetheless transversely focused, and in some cases with better efficiency
than target molecules, and so constitute a rather large background signal relative to the moleculesof interest. The reason these unaddressed molecules are not evident in this simulation is that thesimulation only initializes molecules that are likely to be addressed, for reasons of runtime.
This in turn relates to the substructure evident in the inner core of the bunched molecules in

73
Fig. 5.2. This substructure is an artifact of incomplete phase space filling in the simulation, i.e. if
the right molecules had been included in the initialization, the inner core would be more uniformly
filled out and not feature any swirling. It is useful to keep an eye out for such artifacts as a wayof monitoring whether the initialization assumptions of a simulation is valid. This swirling is notnecessarily only an artifact however, and can in fact be relevant in the experiment as well. In [92],the rotating nature of the inner core in phase space is intentionally understood and exploited forminimizing the velocity spread of a beam exiting a decelerator.

5.2

Alternative Geometries and Configurations
Alongside the history of achievements enabled by Stark deceleration runs a parallel ongoingsaga surrounding their efficient operation. Many important steps have been made, not only inunderstanding the flaws of the conventional pulsed decelerator [91, 93], but also in addressing themthrough the use of overtones [94, 95], undertones [96], or even mixed phase angles [92, 97]. Even withthese advances, the outstanding inefficiencies of the pulsed decelerator, particularly with regard totransverse phase stability, have motivated alternative geometries such as interspersed quadrupole
focusing [93] and traveling wave deceleration [98, 99, 100]. Although traveling wave deceleration
takes a strong step in the right direction toward truly efficient operation, it comes with costs insystem complexity and high voltage engineering. These costs can be partially addressed by the useof combination pulsed and traveling wave devices [101], or even using traveling wave geometry withpulsed electronics [102, 103]. Others continue to pursue brand new geometries aiming to enhancetransverse acceptance without abandoning more reliable pulsed electronics [104].

5.2.1

Flaws in Conventional Deceleration

I begin here by describing more precisely these flaws in the conventional pulsed decelerator.
It is useful to distinguish between two primary categories of performance breakdown:(1) Breakdown associated with the discontinuous nature of the geometry.

74(2) Breakdown associated with the nature of the phase stable region.
The first breakdown mechanism follows directly from the geometry of the conventional pulsed decelerator. Focusing only occurs in a given transverse direction every other pin pair. So if the speed of
the beam is decreased enough, there is clearly a point at which molecules will be adversely a↵ected.
The same issue exists for the longitudinal direction. Switching events are required for molecules toexperience restoring force in the longitudinal direction. When molecules do not encounter pin pairs
with enough frequency, molecules again are adversely a↵ected. Discontinuity breakdowns are thus
further classified by their transverse and longitudinal manifestations– over-focusing and reflection
loss– discussed further in [93]. These breakdowns can in general be described by a lower bound on
the velocity of the beam:vz >> Lfosc ,(5.4)

where z indicates the propagation direction, L indicates the distance between successive pairs ofpins, and fosc is the oscillation frequency of the molecules about the synchronous molecule. Thislatter parameter is not strictly well defined across the entire ensemble, but accepts a range ofpossible values according to the anharmonicity of the restoring force experienced by molecules withdi↵erent deviations from the synchronous molecule. For typical operating conditions, f ⇠ 1 2 kHz.
This anharmonicity is at the core of the second kind of breakdown in performance of thepulsed decelerator, a breakdown which results in unwanted decreases in the flux of the device
even without any of the issues associated with achieving low final speeds. In fact, the e↵ect isalready clearly manifested for devices operated in bunching mode without any slowing, refer toFig. 5.3 for the following discussion. One way to understand these flaws is to think in terms of an
unwanted coupling between transverse and longitudinal modes [91]. In other words, molecules are
best focused transversely on their closest approach to the charged pin pair, and therefore molecules
which deviate significantly from the synchronous molecule in the longitudinal direction experience
better transverse focusing. It follows that the two directions of motion are strongly coupled to
one another, and any full description of their equations of motion will not be separable. It also

75

Figure 5.3: Longitudinal phase space plots after several types of bunching for several decelerator
lengths. Column-wise, operation is in S = 1, S = 3, and a hybrid of these that I term S = 311.
Indicated percentages are comparable by rows, but su↵er from a pernicious simulation artifact.
Some molecules are included which actually pass through decelerator pins. The e↵ect adversely
influences the comparative survival numbers, but not the observed projection of phase space onto
the longitudinal direction.

follows that molecules would survive better at greater distance from the synchronous molecule inlongitudinal phase space, which agrees nicely with the observed sparsity close to the center of theleft four plots in Fig. 5.3. The dead band and outer ring are best explained in terms of parametricamplification phenomena [91].
One way to address this is to allow the molecules to pass all the way through a charged pin
pair, and only switch the fields after they climb a second hill. This is known as overtone opera-

76tion [94], where the term stems from the fact that even without intending it, the standard operating
mode supports overtone operation for molecules that happen to begin at triple the intended speed.
Or one may intentionally reduce the switching frequency by a factor of three, typically indicatedby the overtone parameter S as S = 3 operation, in which case standard operation may be referred
to as S = 1. This gives rise to the much more homogeneous set of phase space plots in the center
column of Fig. 5.3. Despite improved homogeneity, the plots are thinner than those for S = 1 by apfactor of 3, which stems from the fact that molecules experience restoring force in the longitudinal
direction with one third frequency. They can therefore deviate from the synchronous molecule byp
only a third as much kinetic energy as before, or 1/ 3 less deviation in velocity. An even more
important drawback of S = 3 operation is that only a third as much energy may be removed perunit distance.
It is also possible to use a hybrid approach that we term S = 311. In this operating mode,
one allows the synchronous molecule to climb all the way over a hill only every third time. It
should be noted that S = 31 operation, where the synchronous molecule climbs all the way over ahill every other time, is not as useful because this molecule would only experience extra focusing
in one of the transverse directions. This mode has the advantage of sacrificing less deceleration
capability, since it is 60% as e↵ective as S = 311 since 3/5 of all possible hills are climbed, butalso addressing some of the coupling issues. As can be seen in the last column of Fig. 5.3, thesparsity closest to the synchronous molecule at the center of the phase space is addressed, althougha parametric amplification dead band still persists. Unfortunately this mode also features anincreased sensitivity to low speeds. Typically, S = 1 operation stops performing due to low speed
breakdown below 50 m/s, and S = 3 below 150 m/s. The cuto↵ has not been studied thoroughly
for S = 311, but a hybrid alternative exists, which is to use S = 311 for the bulk of the deceleration
except when the speed drops below 150 m/s, at which point one returns to S = 1 for a brief enoughtime that the transverse-longitudinal coupling does not lead to excessive loss. In practice, this has
yielded factor of 2

3 improvements in the number of molecules which may be slowed or trappedin our experiment. I spent some time seeking to determine the source of the discrepancy between

77
this improvement and the smaller one predicted by the simulations leading to Fig. 5.3, and only
years later caught a simulation bug relating to molecules with large transverse deviation ghosting
their way through decelerator pins without being removed.

5.2.2

Alternative Geometries

Perhaps the most important class of alternative geometries for deceleration are those that arecontinuous in nature. These include both Stark [98] and Zeeman [105] varieties, and in principle
provide a complete solution to breakdowns associated with the discontinuous nature of pulsed
geometries, although in practice they face various low-speed challenges of their own.
The Stark devices, also known as traveling wave decelerators, feature cylindrical symmetry
and use sinusoidal voltages to generate a macroscopic moving trap which can be translated or
decelerated, and even brought to rest, at least provided high voltage amplifiers with bandwidth
down to DC operation. The most common and as far as I can tell the only geometry used thus far,in three or four di↵erent groups, is that described in [98] with 0.8 mm diameter wires bent into
4 mm inner diameter rings, and connected up to eight di↵erent backbone electrodes. Their total
phase space acceptance, at least for lower decelerations, is an incredible improvement over S = 1
operation, skip ahead to Fig. 5.8 and compare S = 1 to T W in panel (b).
There also exist a host of proposals for alternative pulsed geometries, including this one withinterspersed quadrupole focusing stages [93], this one with charged wires for improved transverse
behavior [104], and even a magnetic device with interspersed hexapole stages for use with NH
molecules [106]. The first two may become less important thanks to the results discussed in thenext section. Perhaps the earliest attempted device appears to have used actual parallel plates andrelied on their fringing fields for focusing [107], as would have been necessary given the speed ofavailable switches at the time.
I would be remiss if I did not introduce a few new geometry proposals of my own, as have
my predecessors [45, Sec. 5.5], [23, Sec. 7.3]. The first, motivated by the trapping configuration
discussed further in Sec. 6.4, is to add magnetic focusing to each decelerator pin. The idea would

78
be to use dual domain magnetic pins to superimpose 2D magnetic quadrupole focusing fields on
top of the entire decelerator. This has the potential for a vey substantial benefit as far as transverse performance is concerned, see the vastly enhanced phase space volume shown in Fig. 5.4.
Unfortunately, this design would su↵er from very significant spin flip losses due to the presence oforthogonal magnetic and electric fields, see further discussion in Sec. 6.4.2. Molecules would likely
shu✏e back and forth between |f, 32 i and |f,3
2 i, with any favorable focusing e↵ect completely

washed out. A molecule besides OH with a J = 12 ground state (CH, CF, NO [108, 109]) could bemuch more likely to benefit from this scheme, since these molecules do not feature the electric fieldenhanced spin-flip behavior derived for J = 32 OH molecules in Sec. 6.4.4. I do not provide a formal
proof of this, but the path to the proof is that the spin-flip enhancement requires the Zeeman e↵ect
to have cubic order with orthogonal fields. This cubic order relates to the order of perturbation
theory required to get a coupling between the |f, 32 i and the |f,3
2 i states, since their m numberdi↵ers by 3. In a J = 12 molecule, the m number di↵ers by only one, so that the Zeeman e↵ectwould remain linear. I have also confirmed this by numerically diagonalizing sample Hamiltonians
with J = 12 .
And secondly, it is my distinct pleasure to introduce the double-pin decelerator, a brand new
design, specially featured thus far only here in this thesis. The double pin decelerator uses a pair ofpins everywhere that a normal decelerator would feature only a single pin. Instead of overlapping
each other, pairs of double pins point directly towards each other, with a well-controlled tip-to-tip
spacing. Manufacturing of the device is therefore somewhat more sensitive, although the regionsof the pins which need to be highly polished are reduced, and the tips of pins are by far the bestpolished portions when tumble polishing is performed. The key idea behind the use of pairs ofpins is to obtain a focusing e↵ect in both transverse directions each stage. It could also be thoughtof as a pulsed ring decelerator with each ring split in half to enable more significant deceleration
forces by building up field between the two halves. The idea is to make use of two di↵erent field
distributions obtained by charging the device up in di↵erent ways. One is like in conventional
deceleration, when a pair of double pins are oppositely charged; the other occurs when the pair are

79

Figure 5.4: By adding magnetic domains to the pins of a conventional Stark decelerator, dramatic
gains in transverse performance can be obtained. Here a factor of three is found, even with dramatically under-filled simulated phase space is evident by the swirling structure in the longitudinal
phase space.

charged to the same voltage, but preceding and following pairs of double pins are charged to an
opposite voltage. The field distribution resulting in the latter case would ideally resemble that ina ring decelerator. This concept is also very relevant for the conventional geometry, and discussed
at length in the next Section, Sec. 5.2.3.
The field distributions generated by the device are shown in Fig. 5.5. The transverse focusinge↵ect when pairs of pairs are charged to the same voltage is a big win compared with conventional
operation, and even a modest win over a conventional decelerator with alternative field distributions

80

Figure 5.5: (a) Geometry view showing arrangement of the double pins. (b-c) x y planar cuts
through a pair of double pins showing the magnitude of the electric field, color axis kV/cm, contours
every 10 kV/cm. Pins have 3 mm diameter, 1.5 mm adjacent spacing, 2 mm point to point spacing,and 2 mm stage to stage spacing. (b) The slowing hill with the pair oppositely charged. 40% lessheight than conventional deceleration and some defocusing. (c) Now with the pair charged to thesame voltage and un-shown preceding and following pairs of double pins charged to the opposite
voltage.

run in the most promising of the alternative modes described further in Sec. 5.2.3. However it
features a good bit of asymmetry, and has much better focusing for molecules that deviate along±~x±~y compared with ±~x or ±~y , see Fig. 5.5c. It also has 40% less slowing power than a conventional
decelerator, and some defocusing in one direction while slowing. These drawbacks could be partiallyaddressed by shaping the double pins more like half of a ring, but this would drastically reduce
manufacturability.

5.2.3

Alternative Field Distributions

As far as the conventional geometry is concerned, a much more e↵ective route to improved
performance is to mix alternate field distributions into the deceleration scheme that feature strong
restoring force in the transverse directions, refer to Fig. 5.6. In the conventional S = 1 operating
mode [86], molecules approach a charged pin-pair, climbing a hill in potential energy. The hill isabruptly switched o↵ partway up the hill, allowing molecules to have phase stability as discussed
above. But it follows that molecules spend a significant portion of their flight passing between
grounded pins. Conventionally, pins are always charged in bipolar pairs, in which case few field

81

Figure 5.6: Alternate field distributions which can be used alongside the conventional one forgreatly enhanced performance. Configurations B and C feature strong transverse focusing in theregions where molecules would normally pass between grounded pin pairs. On-axis energy diagrams
are shown for several modes of operation incorporating these alternate configurations, with primes
indicating translation to the next pin pair. In addition to S = 1 mode and its S = 3 overtone [94],a strong focusing (SF) and a very strong focusing (VSF) mode are introduced.

lines run toward the grounded pin-pairs, and those that do actually create a slight defocusinge↵ect. This defocusing is not obvious in Fig. 5.6 because those plots are generated by slicing so
as to include the decelerator axis but also make a 45 angle with all pin axes. This choice has thedistinct advantage of allowing the plane to intersect all pairs of pins, making it much more clear
where they are. However, the defocusing e↵ect is significantly stronger in the plane orthogonal to
the grounded pin pairs, and is thus obscured.
Useful alternate configurations can be created by applying voltage in a way that is not balanced between adjacent pin pairs. Once an imbalance exists, by charging up both pins in a pairto the same non-zero voltage, by only charging one pin in a pair, or even by unbalancing the de-82celerator power suppliesa , the field lines will run between pin-pairs. Near the grounded pin-pair,these field lines create a focusing 2D quadrupole structure, much like this one used intentionallyfor trapping and controlling spin-flip losses [88]. By implementing these configurations when thesynchronous molecule is flying between the grounded pin pair, but retaining the use of the conventional configuration for hill climbing, the longitudinal behavior of the device is una↵ected while
the transverse behavior is vastly improved. Regarding spin-flip losses, use of these configurations iscause for concern relative to the conventional S = 1 mode, since the molecules regularly pass close
to small regions of zero electric field. I have done some rudimentary modeling of this e↵ect, andfound it not to be a concern. This is discussed further in Sec. 5.4. Utilizing the field distributionthat results when both pins in a pair are brought to the same voltage gives rise to a new strongly
focusing operation mode (SF), and utilizing the distribution where the adjacent pin pair is alsobrought to the voltage opposite the first gives a very strongly focusing mode (VSF).
It is also possible to obtain some of these e↵ects without the need for rods being charged tothree di↵erent voltages, by making use of the field distribution generated by turning on only a single
pin, a somewhat focused mode (F). This is especially significant for being realizable immediately
on existing devices with no new electronics, and o↵ers 4-fold gains at trappable speeds, see theactual experimental data shown in Fig. 5.7. Operation in F mode requires alternating between thetwo choices of which pin in a pair should be charged and which should be grounded, in order tobalance out the asymmetry of the resulting field distributions. The field distributions and on axis
energy diagrams indicating when they are to be used are also shown in Fig. 5.7.
An important feature of the required time sequence of field distributions is that in switchingfrom A to E configuration, a single rod may be switched o↵ and the other left on, so that relative to
S = 1 operation, the total number of switching events for each rod is unchanged. This is importantbecause it means that the total energy dissipation in the high voltage system is unchanged, sothat no improvements to the thermal dissipation capability of existing high voltage systems area

It was once noted that unbalancing the power supplies led to improved performance on a conventional pulsed
Stark decelerator. S. Hoekstra, private communication.

83

Figure 5.7: (a) Experimental Data Comparing S = 1 and F mode. Fourfold or greater improvements
are found across many final velocities. F mode traces are shown with positive sign, and S = 1
with negative for visual comparison. Simulations are performed and reported as solid lines; good
agreement is obtained. (b) Diagrams like in Fig. 5.6, but for F mode. At bottom, the potential
energy of the alternate distributions (red) are still those for configuration B of Fig. 5.6 for reasonsof simplicity, but the labels are as appropriate for F mode.

necessary. This discussion ignores the fact that turning o↵ a single rod without the other also
turning o↵ actually requires more energy (⇠ 20%) than turning both o↵ at the same time, seeSec. 5.5.5. In practice however, extra capacitance in the decelerator will primarily influence thethermal load on the external limiting resistors, which are usually not chosen within 20% of failure,and can have a cooling fan added easily. For the high voltage FETs themselves, cooling power
can be a more sensitive parameter, but with the limiting resistors in place, most of their heat load
actually comes from discharging their internal parasitic drain source capacitance, which therefore
depends only on the number of switching events and not the capacitance of the load.

5.2.4

The E↵ective Moving Trap

In order to quantitatively analyze these and other operation modes, we can work in the noninertial moving frame of the molecules, where any feasible mode generates an e↵ective trapping
potential with deceleration included as a fictitious force. This approach is usually reserved for84
continuous deceleration schemes [98, 105], but is equally valid for pulsed schemes provided theirvelocity satisfies Eq. 5.4. The e↵ective trap for on-axis molecules in the longitudinal direction has
been discussed at length [110, 111], but computation of the full 3D e↵ective trap has not beenreported previously. The 3D e↵ective trap is evaluated numerically for various operation modes toobtain the equipotential surfaces in Fig. 5.8d. It is found that for S = 1, the e↵ective trap has holes.
Molecules moving away from the trap center along the x and y axes experience almost no restoring
force at all. This can be considered the underlying reason for the transverse-longitudinal coupling
problem discussed above. Such couplings are in some contexts useful for maintaining ergodicity ina trapping geometry [112], but with one dimension featuring a very low energy barrier, they leadto loss.
Motivated by the holes evident in S = 1 mode, a new figure of merit that may be used tocompare the performance of various modes of operation is introduced: the minimum depth oftheir e↵ective moving traps. Here minimum depth refers to the smallest energy above which amolecule with that energy can find a way out of the trap. Having a single value to characterizee↵ective traps allows doing so systematically across many modes and across many magnitudes of theapplied deceleration, see Fig. 5.8a. Remarkably, F mode o↵ers comparable trap depth improvement
to S = 3, but with no sacrifice in deceleration capability. The SF and VSF modes make still more
dramatic improvements, with the latter even rivaling traveling wave (TW) deceleration [98]. Notethat for SF and VSF, the alternate configurations are not utilized in a symmetric manner about
the grounded pin pair as for F. Instead, allowing them to be used asymmetrically opens up a newdegree of freedom, which is optimized so as to maximize the minimum depth.
It is useful to discuss this tuning a bit further. A very similar process was used in [96],where multiple switching events were employed during a single stage. The present case is similar inhaving two di↵erent switching events in a stage, one happening at the usual position, call it
another,

1 , and2 , happening earlier. In [96], extra switching events still only made use of configurations

A and A0 , but enabled greater transverse focusing power and also opened up other opportunities
for more specifically tailoring the e↵ective trapping potential. The same is true in the present case,

85

Figure 5.8: Characterizing the moving trap under di↵erent modes of operation. In addition to
the conventional S = 1 and S = 3 modes, newly defined strong focusing (SF), very strong focusing(VSF), and focusing (F) modes are shown. Traveling wave (TW) deceleration is also compared,
assuming 10 kV peak to peak, to our knowledge the largest voltage used to successfully decelerate
to rest with a TW device. In panel (a) the trap depth at the lowest point of escape is shown as afunction of deceleration for di↵erent operating modes. The stars on SF and VSF indicate 2 tuning,
see the text. In panel (b) the initial phase space volume remaining within these e↵ective traps aftera 3 ms hold time is shown, and in panel (c) a full decelerator simulation is performed as a functionof final velocities, with hold time fixed also at 3 ms and deceleration fixed as indicated by the graydashed lines in panel (b). For TW a full simulation is not performed, since low speed losses areless significant, but the dashed line shows the value corresponding to 67 km/s/s in panel (b). In
panel (d) equipotentials give a feeling for the e↵ective trap as a whole, which is helpful for visualizing
the derivation of the minimum trap depth in panel (a). The e↵ective traps shown correspond to= 45 , about 180 mK. The propagation axis is to the right, and transverse directions are up anddiagonally outwards.

as far as the increased opportunity for specific tailoring. Perhaps the simplest decision is to simply
set

2 =

1 , which has the extra benefit of making no change to the relationship between phaseangle and deceleration rate relative to S = 1 operation, since the molecule doesn’t gain or loose any
extra energy while in the alternate configuration. This decision gives good results across a wide
range of values of

1 , but not close to

1 = 90 , where

2 becomes nearly identical to

1 and the86

Figure 5.9: Longitudinal Phase Space Fillings are shown for several operation modes as labeled. All
modes are initialized with the same uniformly distributed phase space, and the surviving numberof molecules is indicated for each pair of panels. Note dramatic improvements in homogeneity andplanar density, without significant broadening to larger velocity classes except for VSF. Molecules
travel 333 stages, begin at 900 m/s, and slow at 200 km/s/s (67 km/s/s for S = 3).

alternate configuration is hardly employed at all. It is also a poor decision close toin this case

2 becomes nearly 180o↵ from

1 = 0 , since

1 and the normal configuration is hardly used. The

switch back and forth from the normal configuration is essential to obtaining restoring force in thelongitudinal direction, so this would also be a poor choice. Instead, by tuning the location of
optimize the minimum trap depth and not always choosing

2 =

2 to

1 , SF and VSF modes can be

more fully leveraged. This is evident in the phase space plots shown in Fig. 5.9. F mode can beseen to feature greater depth longitudinally than transversely based on the larger projected areaof the population, but for SF and VSF the projected area is comparable between the transverseand longitudinal planes. This is a direct reflection of the fact that

2 has been tuned for optimal

minimum trap depth, because what happens in this case is that longitudinal depth is exchanged
for transverse by increasing the time spent in the transversely focusing direction.
We can make further use of the e↵ective trap by directly employing it to simulate the fate ofparticles confined for 3 ms, the duration of a typical deceleration sequence (Fig. 5.8b). The results
show a very close qualitative match to the trends predicted by the minimum trap depth. The

87
most notable exception is found in S = 1 mode at low decelerations, where extremely deep holes
dominate the minimum trap depth, but the small e↵ective cross sectional area of the holes stillallows molecules to survive in greater number than at higher decelerations where the minimum trap
depth actually improves. As far as the comparison with TW is concerned, it is important to pointout that we use the rather small 2 mm pin-pair spacing and 2x2 mm2 opening area of our device,while TW devices use 4 mm diameter rings. If VSF mode were used with a 3x3 mm2 device [95]or a 4x4 mm2 [113], phase space volume would increase significantly, depending approximately on
the cube of pin-pair spacing, and thus outperforming TW. Deceleration does also reduce linearlywith pin-pair spacing, since this influences the number of pins that may be fit next to one anotherin a given longitudinal distance.
Of course the validity of using the e↵ective trap in this manner depends on the final speed
after the deceleration sequence. This e↵ect can be isolated by also performing a full Monte-Carlosimulation of the various deceleration modes, without use of the e↵ective trap approximation. By
varying only the final speed, and keeping deceleration and run-time exactly fixed by appropriately
varying initial speed and decelerator length, we obtain the results shown in Fig. 5.8c, an exact
isolation of the low speed e↵ects from other phase space e↵ects. The asymptotically flat profiles athigh enough speeds validate the e↵ective trap picture, as do the quantitative agreement between
the asymptotic values and the corresponding points at 200 km/s/s and 67 km/s/s in Fig. 5.8b.
The beginning of the low-speed breakdown depends on the intended use of the decelerator,and especially how far the molecules will be expected to travel unguided afterwards. In Fig. 5.8c,
the molecules still confined within a 3 mm diameter circle after 5 mm free flight after the end of
the sequence are shown. This is a conservative representation of what is required for trap-loading,but for collisional experiments a larger flight distance may be required. Note how F and SF cuto↵ at even lower speeds than S = 1, but VSF cuts higher. This can be attributed to the fact that
VSF actually features an increased transverse trap frequency relative to the others, while F and SF
improve over S = 1 mostly by plugging holes and not by increasing the trap’s depth or frequency.
This restricts the usefulness of VSF mode for OH or other strongly dipolar species at low speeds

88
to devices that combine with a TW device as in [101]. For less strongly dipolar species however,
VSF will more easily respect the low velocity transverse oscillation bound, and may not feature analtered low speed cuto↵ relative to SF or F.
Despite the lack of experimental validation for SF and VSF mode at the time of this writing,it is clear that alternate field distributions present an exciting and viable alternative to conventional
deceleration, obviating the need for more advanced geometries and o↵ering very significant gains
in flux in almost all cases. Brand new opportunities are opened up by the increased efficiency,such as more favorable outcomes for targeting heavier or less polar species, or even the decelerationof beams seeded in Helium. Alternate field distributions are sure to play a central role in Stark
deceleration moving forwards.

5.3

Transverse Trap Oscillations
A common technique for characterizing any trapping apparatus is to intentionally perturb

the population so as to directly observe oscillations caused by the trapping force. This is performed
for example in our OH magnetic trap [114], see Fig. 4, to ascertain the trapping frequency. The
same can be performed for the e↵ective moving trap formed by a given deceleration mode. We
first investigated this with the goal of better understanding the behavior of OH molecules in the
decelerator so as to extract the influence of collisions with Neon beams from single particle behavior. Experiments were performed with the goal of understanding and manipulating the trapping
behavior formed by the two dimensional potential created by the decelerator operated in a guiding
mode. The key idea is to initialize the trap with a broad position and low velocity distribution, sothat a breathing mode is excited, see Fig. 5.10a-e. Some time later the trap is briefly switched o↵.
Depending on whether this window corresponds to the outgoing or ingoing portion of the breathingmode, this brief lack of trapping will lead to more or less loss respectively, see Fig. 5.10f-h. Thistransverse trap may also be computed numerically, depending on the choice of field distribution,and gives rise to transverse potentials as shown in Fig. 5.11.
Breathing modes were detected as expected, and an e↵ort was undertaken to infer the initial

89

Figure 5.10: Transverse Oscillations in the Decelerator. (a) Schematic timing diagram for theexperiment. (b) Transverse temperature (velocity spread) as a function of propagation time. (c)Phase space views demonstrating the rotation that gives rise to velocity spread variations. (d)Schematic timing diagram for application of a detection o↵-window for time Tdet after a hold timeof tdet . (e) Resulting signal in simulation. (f-h) Experimental data for guiding with 6 kV (blue) or
2.2 kV (green) applied voltage. (f) Sine (solid) and damped sine (dashed) fits to the experimental
6 kV data (black dots). (g) Same for 2.2 kV. (h) Fourier transforms of the experimental data withdashed lines indicating the simulated breathing mode frequency.

Figure 5.11: Transverse Traps formed by Di↵erent Field Distributions. Field distributions are
generated in COMSOL, converted to internal potential energy, and then averaged longitudinally.

90
population distribution generated by the valve from the observed oscillations. This turned out toserve as a probe of skimmer interference, since it was found that a distant source of the expected
temperature could not reproduce the shallowness of the oscillation amplitude observed, although
there are of course other deviations between model and experiment which could lead to a reduction
in fringe contrast. It was also found that by intentionally turning the trap o↵ at the right moment
could serve as a manipulation of the transverse phase space distribution for the sake of reduced
transverse temperature.
It would be ideal to use this technique also as a means of studying the transverse or longitudinal frequencies in the traveling potential well in deceleration mode, but this has not yet been
undertaken. Greater signal to noise would be desirable, at the time of this writing unavailable due
to intentional space added between the source and the decelerator for studying collisions between
parallel plates.

5.4

Non-adiabatic Transitions
The Landau-Zener formula may be used to approximate the relevance of non-adiabatic tran-

sitions in a given situation:
P =e ⇡
Here

2/

(~ @E
@t ) .

(5.5)

gives the minimum energy gap between two approaching eigenstates, and @E/@t gives

their rate of approach. Working in frequency units, we can simplify this to say that non-adiabatic
transitions are negligible provided that:G >>p

E · ⌫,(5.6)

where G is the closest approach of the states in frequency units, E is the initial energy spacing
between the states in frequency units , and ⌫ is the frequency characterizing the variation speed of
the system. This is a convenient rule of thumb– just compare the minimum gap to the geometricmean of the inverse timescale and the typical energyscale.

91
For example, in the absence of any magnetic field, parity flips from |f i to |ei are highly
unlikely thanks to the lambda doublet spacing of 1.7 GHz. Compare this with the geometric meanof 100 GHz and 1 MHz where the latter is appropriate for the microsecond turn-o↵ of the fields
during switching events and the former is the energy scale set by the magnitude of electric fields
in the decelerator. The geometric mean is 300 MHz or so, below 1.7 GHz indicating that thesetransitions are not a concern.
On the other hand, transitions between |f, 32 i and |f, 12 i are a much greater concern, and havebeen observed during propagation of molecules in decelerators [115], and also during switching
events of decelerators [116]. The Stark e↵ect treats |f, ± 32 i the same, and also |f, ± 12 i, but thelatter have one third the response of the former, so transitions between them are essentially loss.
Moreover, these states are degenerate at zero field and don’t have any fixed barrier as do thestates of di↵erent parity. Of course, states being degenerate does not necessarily lead to loss if the
Hamiltonian driving their evolution does not have any amplitude connecting the states in question.
For states of di↵ering m quantum number, the Hamiltonian will only couple them when molecules
are experiencing a rotation of the electric field vector. In S = 1 operation, suitably fast rotations,
together with very small values of the electric field, cannot occur on-axis, but may occur for o↵-axis
molecules that experience weaker fields close to the grounded decelerator pins. I investigated theloss probability for o↵-axis molecules, for the first time as far as I can tell, and the results I obtained
are shown in Fig. 5.12. Only a small fraction of molecules are lost, perhaps something like 0.1%
over the length of the decelerator. Being more quantitative would require simulating a denser gridof trajectories close to the pins.
Although free propagation doesn’t lead to non-adiabatic transitions o↵ axis, there is still thequestion of non-adiabatic transitions during the switching of the electric fields. I also checked this,for the first time as best I can tell, for a 3D grid of possible positions relative to the synchronous
molecule, see Fig. 5.13. I find that non-adiabatic transitions are possible only if the molecule issignificantly behind the synchronous molecule, shielded by the grounded pins just prior to their
being rapidly switched on. These positions are not longitudinally phase stable anyway, unless the

92

Figure 5.12: (a) Minimum energy gap along straight line trajectories parallel to decelerator axis andintersecting the transverse plane as shown. (b) Resulting energy gap as a function of propagation
distance for a grid of trajectories. (c) Sample solution of the time-independent Schrödinger equation
for a trajectory that shows a 10% chance of hopping to the |f, 12 i state. Many sharp jumps are seen
between |f, ± 32 i, these are irrelevant and an artifact of an unphysically small bias magnetic field.
(d) Inferred hopping probability as a function of position in the transverse plane.

decelerator is being operated in an acceleration mode, in which case some locations o↵ axis doseem to feature a significant chance of non-adiabatic transitions. The real thing to do would beto repeat this study with alternate field distributions in mind. These would be likely to featurea significantly enhanced chance of free propagation transitions close to the electric quadrupole
created by the alternate configurations between the grounded pin pair. As far as the chance ofswitch-induced non-adiabatic transitions, those would seem to be reduced by the alternate field
distributions, since in S = 1 mode they were worst with the molecules beginning in a region ofsmall field, and the alternate distributions generally add electric field in the regions where it was

93

Figure 5.13: Loss per switching event in several x y planes of di↵erent z coordinate parametrizedby phase angle = 90 , 45 , 0 , 45 , and 90 , from left to right and top to bottom. Significant
loss only in the 90 plane, where phase stable molecules do not reside anyway.

94

Figure 5.14: Together these panels demonstrate that the electric field is zero not along a line but
only at a single point. (a) Electric field magnitude in the transverse plane bisecting a pair of pins.
A line of near-zero field is evident. (b) Electric field in the same plane, but only its projection on
the line of near-zero field, and zoomed in a few orders of magnitude in the grayscale axis. The
checker-like structure close to the central axis shows that there is not a line of zero field but actually
a 3D electric quadrupole structure, with very slight restoring force in the vertical direction. Thisis relevant to reducing non-adiabatic transitions through intentional misalignment.

previously small or absent. However significant the non-adiabatic transitions in the new operation
modes, one way to address them is to intentionally vary the length of decelerator pins, so thatone pin is longer than the other in a pair. What this does is push the electric field zero o↵ ofthe decelerator axis, creating some extra protective energy splitting even for molecules passing
dangerously close to the electric field zero. It is not obvious that there is a single point of zero
electric field, something I endeavor to demonstrate in Fig. 5.14. In panel (a), there appears to
be not a point but a line between the pins where the electric field is zero valued. However, if wezoom in on the energy axis (Fig. 5.14b) and consider the coefficient of the field which points along
this seeming linear electric field zero, we find that this component is in fact nonzero except in asingle point. Of course it comes very close to zero, but not much of an extra protective field is
required to reduce the e↵ect of the spin-flip loss significantly. Thanks to the variation in length ofour decelerator pins, a known side e↵ect of tapered pin mounting as described in Sec. 5.5.2, ourdecelerator may benefit from reduced sensitivity to these spin-flip issues.

95

5.5

Decelerator Manufacturing
Beginning in Fall of 2015, an e↵ort was undertaken to manufacture a new decelerator capableof slowing an OH beam generated with neon as a bu↵er gas. This e↵ort was begun in parallel withordering of a new Even-Lavie pulsed valve [54], which boasts significant gains especially for lighter
carrier gasesb . Initially argon was considered, but foolish zeal on the part of the author led toneon, although at least he was talked out of helium.

5.5.1

Modeling

Simulations were used to determine the optimal length for the device, or at least a lengththat would slow OH beginning at Neon speeds without comparable performance to the previous
generation, see Fig. 5.15a. It was determined that something closer to 350 stages should be ideal
for loading the magnetic pin trap at⇠ 55 , and so a 333 stage device was planned. Sensitivity ofdecelerator performance to pin pair spacing was also studied, and found not to have a significant
e↵ect, see Fig. 5.15b. Thinking in terms of the e↵ective trap, this reduction shrinks the spatial
extent of the longitudinal trap, but increases the velocity extent of the transverse traps, since
the transverse focusing becomes more frequent. The spacing was reduced from 5.461 mm in theprevious generation to an even 5 mm.

5.5.2

Mechanical Considerations

In order to address manufacturing costs and installation time associated with such a longdevice (up from 142 stages previously [70], and 69 before that [69]), we spent some time investigating the possibility of separate modules or of extensions mechanically affixed to the existing rods.
Ultimately it was decided that the challenges associated with getting such things right were notwarranted by our length, and that it would be best to tackle everything with one long set of rods,
mounted in a cage made of several stages, see Fig. 5.16. One significant change that was developedb
It is difficult to verify such a claim, which concerns the relative performance of di↵erent valves with di↵erent
carrier gases. The claim of better performance for an Even-Lavie valve only with lighter carrier gases comes fromverbal discussions with SYT van de Meerakker during his sabbatical at JILA in the winter of 2015-2016.

96

Figure 5.15: Simulation results for deceleration and loading verses decelerator design parameters.
(a) Loading efficiency is studied as a function of three initial speeds and various decelerator lengths.
A ramping of the phase angle is used in the final few stages for further optimized loading. Increased
length is observed not to facilitate improved trap population beyond some initial threshold value,
likely due to the inefficiencies of the deceleration modes used at the time. (b) The selection ofphase angle to achieve slowing to rest as a function of decelerator length is also shown. (c) The
interstage gap is also studied.

out of a partnership with a local manufacturing companyc was to use locking tapers on both thepins and their mounting holes to fix them in place. This halved manufacturing costs for the rods,
and removed the need for setscrews, which created a serious galling hazard on the previous device
and led to several near-disasters when re-polishing pins. A special tool was developed for pressing
the tapered pins into their holes, or removing them, without scratching any polished surfacesd .
We had also run into high voltage issues when using M oS2 powder dissolved in isopropyl alcohol
as a lubricating agent for said setscrews. A key downside of the tapers was that they ended up creating a large sensitivity of final installed pin length on pin polishing procedure, since an extra unit
thickness removed from the outside of the tapered pin causes the pin to sit 48 units deeper into itscdHirsh Precision Products, INC.
The design is publicly available on Instructables

97
hole (since the standard taper rate for locking tapered pins is a quarter inch per foot). Fortunately
the length of pins along their axis of symmetry has only a minimal e↵ect on the fields experienced
by molecules in a pin-style Stark decelerator. An additional challenge is that installation of many
tapered pins acts as a fairly e↵ective wedge for causing the main rod to bend ever so slightly. Afterinstallation, we worked very hard to adjust the mounting structure so as to force the decelerator
into proper alignment. Lacking all requisite degrees of freedom, we settled for a precision tuningof the pin pair spacing, in the hope that as long as that remained correct, the molecules would be
able to follow any slight bends in the device. Finding a way to work with a larger diameter rod
may have been preferable. The tapered pins may be a good choice for other designs, especially if
a vendor is used who can o↵er both surface quality and tolerancing at the same timee .
It is worth setting out a figure of merit to describe the extent to which such slight bends
would be acceptable. For velocity v, a bend radius r causes an acceleration a = v 2 /r. For a given
operation mode, we can characterize the transverse trap as a harmonic potential U = 12 m! 2 x2 . In
the case of a bent decelerator, this potential is a non-inertial frame, and the bending acceleration
can be included as a fictitious force which causes the molecules to shift o↵ of center in the trap.
The magnitude of this shift is given by the xs which satisfies:@Uv2= mOH ,
@x xs
r

(5.7)

from force balance. Solving:
xs =

v2
.
r! 2

(5.8)

For SF mode, ! ⇠ 1.0 kHz, for F mode, ! ⇠ 0.7 kHz. Assuming that xs ⇠ 0.2 mm is acceptable,
10% of the pin pair spacing, the allowed bend radius for 800 m/s molecules is 3.2, 6.5 km for SF,
F modes. Along the length of a 1.7 m device, such bends amount to o↵sets of 1, 0.5 mm. Basedon the observation that we are able to shine a laser beam all the way through the device, albeit
without the expected 2 mm square-like shape, bends in the device are unlikely to be significantlye

We worked with TriGON Precision for polish after having Hirsh Precision Products INC manufacture the pins.
They often expressed their preference to have manufactured the pins themselves in the first place, since we wereasking them to get a high quality polish without removing too much material.

98

Figure 5.16: Mounting structure of third generation Stark decelerator. Note large safety highvoltage surface distance along glass rods, blue. (a). Clamping pieces grab on to the actual decelerator rods. A mounting flat, a vestige of an earlier design, is visible on the decelerator rod. (b)Decelerator rods hidden, cage connection rods shown. (c) Full view of the cage structure.

larger than these numbers, but likely on this order.

5.5.3

High Voltage Arcing

With the new design, an e↵ort was made to address challenges that had developed with themounting of the previous generation, which had been facilitated by MACOR, a machinable glass
ceramic material. The MACOR was found to allow surface currents, which also were un-phased by
the surface path-length extension grooves that had been added to reduce their likelihood, as can beseen in Fig. 5.17. There was never a clear indication that these surface arcs were actually causing a
reduction in molecule yield, but nevertheless it seemed wise to make an attempt at improving the

99

Figure 5.17: MACOR surface arcs. At left, arcing on an insulator original to the manufacture of thesecond generation device [70]. At right, arcing on a redesigned version, undertaken concurrently
with a re-polishing of the last twenty decelerator pins performed in the fall of 2014. Here it isevident that the path-length extension grooves are not helpful.

situation. In reviewing literature [78, Sec. 4.3.3] and consulting other opinions, it was found that theprevious geometry did not respect recommended considerations regarding so-called “triple-points”
where vacuum, dielectric, and conductor meet. It is advised to have the triple-points recessed orotherwise manipulated so as to minimize the electric field at their locations [75]. This is achieved in
the new design, as well as increasing the safety distance and removing reliance on sharp insulator
features, by the geometry visible in Fig. 5.16 and heavily influenced by [91, Fig. 4]. Borosilicate
glass rods were selected for their vacuum and insulating properties, with alumina rods as a close
second.

5.5.4

High Voltage Electronics

In the design of the new decelerator, the capacitive load presented by the decelerator was
an important consideration. For a long enough device, it would in principle become necessary
to purchase a new set of switches capable of providing higher currents. Fortunately, prior to thedesign e↵ort which culminated in the third generation decelerator system, it was realized that the

100capacitance of the cables connecting the high voltage switches to the decelerator actually dominatedthe capacitance seen by the switches. It was thus possible to operate the original electronics with
the new decelerator at even lower load than previously, thanks to a systematic e↵ort of relocating
electronics so as to minimize all high voltage cabling, especially between the transistor switches andthe decelerator rods. Considerable influence in the design and execution of this e↵ort was takenfrom [117, Fig. 4.6].
An even more significant challenge constitutes the management of multiple output states
as required for the realization of the SF and VSF modes discussed above. Such an e↵ort was
successfully undertaken in the operation of certain traps discussed later in this thesis, but only atvery low repetition rate. In considering circuitry involving the use of multiple transistor switchesarranged in series or parallel it is essential that all parasitic elements of the devices are taken into
account, especially the parasitic capacitances in parallel with the drain and source terminals, which
can form the dominant current load as seen by other transistor switches in the circuit. At the timeof this writing, careful collaboration with engineers at Behlke GMBH resulted in the selection of adevicef that should be capable of operating SF mode, and VSF if a second such device is ordered.
The device has yet to arrive after requiring a rebuild following a miscommunication pertaining to
the specification of options for the switch.

5.5.5

Capacitance Matrix

It is also relevant to again consider the capacitive load associated with switching between
various distinct field distributions. In this case, it becomes necessary to treat the full system ofelectrodes not as a single capacitor but actually as a network of di↵erent capacitors most efficiently
captured in the form of a capacitance matrix Cij satisfying:Qi = Cij Vj ,fBehlke HTS-301-151-SiC, options HFB, ILC, ALL-OFF-BIPOLAR.

(5.9)

101
so that a vector of charges on each conductor can be obtained given a vector of voltages on each.
Surprisingly, in our system the capacitance between a pair of rods with parallel pins is actuallyslightly less than that between a pair of rods whose pins are orthogonal, see the matrices reported
in Fig. 5.18. The entries in these matrices are computed experimentally using a capacitance meter,
the SR720. It is not possible to measure Cij simply by connecting an LCR meter only to nodes iand j. What happens in this case is that the nodes other than i and j float to various unknown
voltages, and confound the measurement.
We opted instead to partition the nodes into two groups, short these groups to one another,and then measure the two point capacitance between those groups. Consider for example the casewhere rods 1-3 are shorted together, and 4 is grounded. When the capacitance meter applies atest voltage VT to rods 1-3 and measures the stored charge QT , this can be related to the Cij ’s asfollows:
QT =

3
X
i=1

Qi =

3 X
5
X
i=1 j=1

Cij Vj = VT

3 X
3
X

Cij .

(5.10)

i=1 j=1

That is to say, we measure the total sum of a sub-matrix of the full matrix Cij . It is clear thatsuch measurements may be added and subtracted so as to obtain each Cij independently.
Another interesting property of the capacitance matrices shown in Fig. 5.18 is their positive
diagonal and negative o↵-diagonal. This is as expected, as can be proven by considering the casewhere all Vi are zero except for one, Vj = V > 0. To achieve such a configuration, positive charge
must have flowed onto conductor j, which would then require negative charge to flow from groundonto conductors i 6= j. It follows that Cij < 0 whenever i 6= J.
An important corollary of the multi-capacitor system is that the transient dynamics of thesystem do not follow the usual RC behavior, as shown in panel (c) of Fig. 5.18. When one considers
the response of the system to a specific impulse vector of voltages, the system will respond on a few
di↵erent timescales according to the eigenvalues of the matrix. In panel (c), which is a first order
dynamical simulation applying just such an impulse using the experimentally measured capacitancematrix, the current through the resistor connecting to one of the rods is seen to first briefly increase

102

Figure 5.18: Capacitance matrices from LCR meter and simulation for third generation decelerator.
(a) Capacitance matrix by measurement and (b) by COMSOL simulation. (c) Example of the multitimescale dynamics that can occur when solving the transient response of a capacitive system tosudden changes in the voltages applied to the electrodes. Traces are simulated dynamics using theexperimentally determined capacitance matrix.

with time before then decreasing exponentially in a less surprising manner.

5.5.6

Di↵erential Pumping

One key downside of Stark decelerators compared to conventional alkali Zeeman slowers is thelack of feasible di↵erential pumping schemes. This challenge stems from the difficulty of mounting
anything close to the Stark decelerator without violating high voltage safety on some level. Electricfields also interact more strongly with most materials compared with electric fields, and so there is

103

Figure 5.19: Glass ba✏e for di↵erential pumping. Conductance of about 20 L/s for Neon. (a)Manufactured from a stack of water-jetted glass slabs, glued together with a low outgassing epoxy.
(b) Sample output from pipeConductance.m, a conductance calculation package developed forthis applicationg . One tenth of the chords used for calculation are shown. Calculation is wellconverged as far as chord inclusion is concerned. (c,d) Profiles used for specifying cross sections for
conductance calculation. Conductance is 24.2 L/s for (c), 12.7 L/s for (d).

no tube like structure that could be mounted inside the pairs of decelerator pin without shunting
away most of the field lines, not to mention surface arcing. Nonetheless, in the third generation of
the experiment a technique was developed for di↵erentially pumping the end region of the Stark
decelerator from the other vacuum regions, using a glass ba✏e structure which comes close to thedecelerator on all sides but without making any surface contact, see Fig. 5.19.
I devoted some e↵ort to determining the vacuum conductance of the unusual cross sectiong

See the MATLAB file exchange or my Github page.

104
one obtains by mounting something close to the outside of a Stark decelerator. While commercial
software exists for determining conductances and pump-down times for arbitrary geometries, these
options would require very high mesh densities and very long computation times given the small
scale and detail of the geometry. A better option is to work in two dimensions where general
expressions amenable to numerical integration exist [118, Eq. 14]. Of course the decelerator is notquite translationally invariant in the molecular direction, because the pins are discontinuous. A
worst case can be had by assuming no pins installed, and comparing this to the case of only two or
four rows of pins being installed, see panels (c) and (d) of Fig. 5.19. My conductance calculations
are nicely packaged and publicly availableg . Conductance would be 24.2 L/s without any pins,
12.7 L/s with two solid rows of pins, and 8.6 L/s with four solid pin rows. Given our 500 L/s
pumping speed in the chamber downstream of the ba✏e, the ba✏e therefore a↵ords something
like a twenty-fold di↵erential pressure reduction in the gas load coming through the ba✏e from thebeam source. This could be measured somewhat easily using a leak valve, but we have simply neverbothered. Excellent suppression of Neon is observed, with an RGA mounted in the trap chamber
not even detecting whether or not the pulsed valve is turned on, at least with a skimmer in placein the source chamber.

Chapter 6
Molecular Trapping

Some of the earliest successful trapping of neutral molecules began with CaH in John Doyle’s
group in a dilution fridge with 3 He bu↵er [119]. Stark decelerated and electrostatic trapped ammonia followed soon later [120]. Since then extensions to many species have occurred. In thischapter we focus on OH molecules, whose strong Stark shift to mass ratios make them favorable forattempts to attain high enough densities to observe collisions between members of the ensemble.

6.1

History of OH traps
The history of OH trapping in the Ye group has grown substantial enough to warrant atabular environment 6.1. Each attempt has brought new challenges and experiences, and steadyprogress has been made in a number of key areas. The Magnetoelectrostatic Trap (MET) installed
during Brian Sawyer’s time [70] was a heroic first attempt that included a few-turn magnetic coilrun at a startling 1400 A, and 2000 A briefly during loading. Later it was decided to trade the roleof the fields used for loading, and great gains in magnetic field strength and simplicity were attainedby switching to permanent magnets with the “Ring” trap [79]. A key improvement occurred when
it was discovered that patch charges on the Ultem mounts originally used for affixing the magnetic
trap to the Stark decelerator could have a significant impact on spectroscopic e↵orts [114, Fig. 6].
This was addressed by designing primarily stainless steel mounts, so that molecules only had line ofsight to grounded conductors, although still with insulators installed between the decelerator andtrap but relocated elsewhere.

106
Table 6.1: The Ye Group Molecule trapping endeavor. All but the last have been experimentallyrealized.
Name

Type

Depth (mK)

Uses

MET

Magnetic Quadrupole,
Electric Hexapole

250

First Demonstration [70]Ring

Magnetic Quadrupole

100

He, D2 , ND3
Collisions [87, 79]

Ring

Above, but new mounts

100

E-field Induced Collisions,
Evaporation [114, 44, 121]

Tricycle

Magnetic Quadrupole300

10x density, spin-flip loss

2D Magnetic and Electric
Quadrupoles

500

Solved spin-flip loss [88]Magnetic Quadrupole

200

Lifetime, Fluorescence
Enhancements

Pin
Cryocycle

6.2

Tricycle over Ring Trap
In pursuit of increases in density and molecule number, an iteration on the Ring trap was

performed, dubbed the “Tricycle” trap due to the three rectangles formed when examining planarcuts through the ring and rear magnets used to generate the trap, see Fig. 6.1. This was first
installed in 2014, and improved on the Ring trap in several key ways:(1) Replaced rear ring with its core, removing an outer toroidal trap and tripling depth.
(2) Significantly improved loading dynamics.
(3) Increased trap gradient, thanks to a ⇠ 40% reduction in size.

6.2.1

Toroidal Destabilization

This first achievement was one of the primary goals of the iteration, since the influence of thetoroidal minimum present in the Ring trap was difficult to ascertain. Molecules ought to have beenable to explore the toroidal region, but based on the observed distribution of molecules as a function

107

Figure 6.1: Cross Sections including the cylindrically symmetric axis for both the Ring and Tricycle
traps. Arrows indicate magnetization directions up to an overall sign. Magnetic trapping fields
are shown, demonstrating the destabilization of the toroidal minimum of the Ring trap. Contours
every 500 G. Steel mounting electrodes are not shown in either case, but are installed on the outer
diameter of the magnets and affixed to them with setscrews.

of magnetic field, they did not appear to be doing so for unknown reasons. At the time it was highly
desirable to be able to perform the same evaporation type experiments, but in a geometry withouta toroidal minimum. The removal of the toroidal minimum can be best understood by thinking
about the traps using the principle of superposition. The Ring trap may be thought of as the resultof superposing two di↵erent quadrupole traps, one formed by cylinders of diameter given by the IDof the rings, and the other formed by cylindrical magnets of diameter given by the OD of the rings.
The smaller quadrupole trap is nice and tight, but the magnets block the beam path. The larger
trap is larger and weaker, and its field lines move in the opposite direction, since its magnets areoppositely magnetized relative to the smaller quadrupole trap (so that they cancel each other out
in the centers of the rings, allowing molecules to pass through). The larger trap works against thesmaller, but is much weaker than it at least near the center of the geometry, so that the smaller
trap remains tight near the trap center. Further outside, where the fields generated by the smaller

108
quadrupole trap are less significant, the outer quadrupole dominates, creating the outer toroidal
minimum. If instead of overlapping a larger quadrupole trap with the smaller, we just overlap asingle disk magnet, then no significant outer trap is created, just as in the Tricycle trap.
In fact, in the limit of large OD of the single ring magnet of the Tricycle trap, the geometry
exactly approaches that of a pair of small disk magnets, but with the crucial modification of an
entry hole for the molecules to be delivered. This is because for a disk magnet, the on-axis magnetic
field is given by:
1
B(z) = Br
2z+tp
R2 + (z + t)2ztpR2 + (z

t)2

!

,

(6.1)

where Br is the remanent flux density of the permanent magnet, t is its half thickness, and z the
distance from the magnet center, and R gives the cylinder radius. In the limit of large R we obtain:
1
B(z) = Br
2✓z+t
R + (z + t)2 /2R)z t
R + (z t)2 /2R)

◆

,

(6.2)

and taking it to cubic order:
B(z) ==✓◆
1z + t (z + t)3 z t (z t)3
Br
+
2
R
2R3
R
2R3
✓
◆
1
2t 3z 2 t
t3
Br
+
.
2
R
R3
R3

(6.3)
(6.4)

So we see that as R grows, B(z) shrinks close to the magnet with 1/R, but the flatness of the field
increases, so that the term proportional to z 2 t/R3 which describes the second order fall o↵ of the
field away from the magnet reduces as 1/R3 . In other words, the magnetic field just outside of adisk magnet does indeed shrink as one increases its radius, although the volume occupied by thissmaller magnetic field increases. It is especially advantageous to have a small field from the large
superimposed disk of the tricycle trap, since this extra field acts to shift the trap center into thehole of the ring and out of view of the detection laser. For this purpose, the Tricycle trap features
a 19 mm outer diameter ring magnet, up from 12 mm in the Ring trap.

109
6.2.2

Loading Improvements

The Tricycle trap features some loading improvements over the Ring trap, although theprecise extent is difficult to pin down. The main reason for an expectation of improvement lies in
the phase space dynamical behavior of the two geometries. In analyzing any trap loading scenario,
it is important to pursue the analysis with both intuition and simulation. The latter on its own willlack the guidance necessary for truly identifying an optimal scenario, while the former is unlikely
to be able to fully disentangle interdependent factors influencing performance.
On the intuitive side, it is useful to apply the same reasoning as in the decelerator by focusing
on phase stability. Electric fields useful for loading are generated by charging up the surfaces of themagnets in the traps. These loading fields are mirror symmetric about the center of the Ring trap.
This means that if a loading sequence is designed so that the synchronous molecule ends up exactly
in the trap center, the synchronous molecule will be required to roll up to the top of a hill and thenstop there. Molecules with slightly more forward velocity than the synchronous molecule will end
up quite a ways down the other side of the hill by the time the loading fields are switched o↵, andvice versa. It follows that longitudinal phase stability during loading requires that the loading be
performed on a slope, not a peak.
It is possible to generalize these ideas further, while still remaining in the intuitive domain. If
it is true that the loading fields ought to feature a slope at the location of the trap center where thesynchronous molecule comes to rest, what is the ideal value for that slope? Also, is there a similar
ideal value for the slope of the loading fields in the region in front of and beyond the trap center?
We can answer these questions with a simplified thought experiment, refer to Fig. 6.2. Suppose
we have a population experiencing a harmonic trapping potential with a characteristic width z inreal space and v in velocity space, and centered at z0 = 0 and v0 > 0. One way to controllablytransfer this population to v0 = 0 without unwanted stretching of the population would be to load
it into a much larger harmonic potential with oscillation frequency ! centered at z = 0 and v = 0.
Because harmonic potentials always execute rotations in phase space with ellipticity controlled by

110

Figure 6.2: Phase space diagram depicting the action of the ideal phase space conservative loading
fields derived from a large external harmonic potential. The region of phase space acted on by theexternal potential is indicated as a light blue ellipse. The region of phase space populated initiallyis shown in red along the v axis. The region populated after rotation is shown along the z axis also
in red. Widths and origins are as indicated.

their oscillation frequency, the population would then be smoothly transferred over the course of aquarter oscillation to be centered at z = v0 /! and v = 0, with widths ! z and v/!. If ! is chosenequal to v/ z, i.e. to have the same strength as the trapping potential prior the initiation of theloading sequence, then the original widths in position and velocity are precisely maintained. On
the other hand, ! could be tuned so as to optimize coupling between the initial traveling trapping
potential and the trapping potential to be used after loading.
In addition to this harmonic loading potential, it would be ideal to also maintain a transverse
trap simultaneously. The ideal fields would have a magnitude with the following spatial dependence:
1|E(x, y, z)| = mOH !z2 z 2 + !r2 r2
2

(6.5)

where !z and !r are the longitudinal and transverse trap frequencies. Neglecting the nonlinearityof the Stark shift for OH molecules close to zero field, such a harmonic potential could be generated transversely with a hexapole, but to do it simultaneously in the transverse and longitudinal
directions would require an octopole moment, such as could be generated with three rings and two

111

Figure 6.3: Potential energy along axis for Ring and Tricycle traps. The trap center is located atz = 10 mm in both cases. Note the more favorable slope of the loading potential in the vicinityof the trap center for the tricycle trap. Field magnitudes arise from application of ±12.5 kV for
the Ring trap and also for the Tricycle. In practice almost half this voltage was applied on the
Tricycle for optimal operation, perhaps due to arcing e↵ects or non-adiabatic transitions duringloading discussed below in Sec. 6.3.

end-caps with the end-caps at +V , the outer rings grounded, and the inner one at

V /2 so as toapproximate spherical boundary conditions following the second Legendre polynomial [122]. This
would of course be very unlikely to be able to be crammed into the small space between the endof a decelerator and a trap, and unlikely to be able to be charged up to a high enough potential
energy for stopping an appreciable speed v. It is much more likely that an efficient solution would
be obtained by abandoning the harmonicity and instead focusing on the reduced criterion of loading fields that respect phase stability by having a slope at the location of what will be come thetrap center and which also provide some transverse confinement. The slope of the loading field at
the location of the trap center would ideally match the slope that the ideal harmonic trap withfrequency ! would have at that point, mOH !v0 .
In practice, Fig. 6.3 shows what we are able to obtain for loading fields comparing the Ring

112
and Tricycle traps. Note the role of the magnetic field, which is non-negligible. The tricycle trap
comes much closer to the harmonic ideal discussed above. For the ring trap, the extra e↵ect ofthe magnetic field actually depresses the potential energy at the trap center below a wider plateau,
making it formally impossible to bring the synchronous molecule to rest at the trap center. In
practice, the experimentally determined ideal application time of loading fields likely corresponds
to the synchronous molecule being brought close to rest but out in front of the trap center. This
unideal situation should result in a higher loaded temperature in the ring trap compared withthe tricycle trap, 87 mK compared with 61 mK in simulation. In practice however, the measured
spectra of molecules in the Ring trap fits better to a thermal distribution and to a lower temperaturecompared with the tricycle trap, see Fig. 6.5. Before discussing this further, we first revisit theprocess of spectroscopy in these traps.

6.2.3

Microwave Spectroscopy

A critical tool for understanding the behavior of populations in our magnetic traps is microwave spectroscopy performed on the parity changing transition between |f, 32 i and |e, 32 i states.
This transition is ideal for spectroscopy, because it has a very small but nonetheless nonzero differential Zeeman shift of 26.6 MHz/T, which allows molecules to be resolved according to theirmagnetic field, while also allowing the entire trap to be surveyed over a very narrow bandwidthof microwave frequencies. The narrowness of the band is crucial for avoiding the complexities associated with the delivery of microwaves to the trapping region. If it were necessary to instead
use microwaves to transfer molecules directly from a trapped to an un-trapped state by flipping
their magnetic quantum number, |f, 32 i to |f,3
2 i, scanning the trap would require scanning theapplied microwave frequency between 1.7 and 16 GHz. While this is no problem for the microwave
synthesizer itself, obtaining a suitably level passband in the components responsible for delivering
the microwaves to the molecules would be unfeasible. In particular, the microwaves are delivered tothe trap using a bias tee setup, nicely described in [114, Fig. 5], and the transfer function throughthe isolation capacitors is particularly troublesome, see Fig. 6.4. In contrast, with the entire pop-

113

Figure 6.4: Microwave transfer function from vacuum feedthrough to magnet surfaces. Large
variations are observed across the GHz scale required for performing spectroscopy along m-changing
transitions, but not on the MHz scale required for |f, 32 i to |e, 32 i spectroscopy.

ulation sitting below 0.5 T, only 13 MHz needs to be scanned out of 1.7 GHz, and no significant
attenuation variation is expected or measured.
In addition to the use of microwaves to transfer population, the spectroscopy relies on a few
further steps. First, molecules in the |e, 32 i state remain trapped, and must be removed. This isdone by first applying a bias electric field to allow molecules to escape via avoided crossings, likethose shown in Fig. 1.1. In [114], it was nicely demonstrated that this is e↵ective for removing
molecules in the |ei state. Second, the process is repeated many times, with varying wait times so
as to avoid any pathological resonance between the application of microwaves and the oscillationof molecules in the trap, which could result in certain privileged classes of molecule orbits always
avoiding the microwaves. Third, comparison is made between the total laser induced fluorescencewith and without the application of the spectroscopic sequence. When this is done, spectra suchas shown in Fig. 6.5 are obtained.

114

Figure 6.5: Spectra in the Ring and Tricycle traps after loading without forced evaporation. The
former was published in [44, Fig. 3a], though repeated many times by the author afterwards. The
latter was collected on Feb. 24, 2014.

6.2.4

Limitations to Spectroscopy

One key limitation to microwave spectroscopy via the procedure described is the spatial variation of sensitivity of the technique. The influence on microwave frequency dependent intensity has
already been claimed negligible, but microwave intensity at any given frequency varies significantly
across the cloud. This goes against the rule of thumb that electromagnetic intensity shouldn’t varysignificantly on length-scales below a wavelength, 18 cm in this case. This is because the microwaves
are in the near field regime with respect to the conductors that form the trapping potential. The
microwave fields may to a good approximation be taken to be dominantly electric and equal to
the DC electric field distribution generated during loading but with magnitude oscillating at the
microwave frequency. This means that their intensity should vary as shown in Fig. 6.6ab. In particular, the microwave intensity falls o↵ quickly as molecules approach the openings of the magnetsof the Ring trap. Furthermore, the efficiency of microwave transfer also features a polarization
dependence. Since the |f, 32 i to |e, 32 i transition has

m = 0, there is a dependence of the Rabi

frequency describing the intensity of the microwave drive on the cosine of the angle ✓ between theelectric and magnetic fields, see panel (c) of Fig. 6.6. This ends up favoring molecules on the axisof the trap.

115

Figure 6.6: The near field intensity close to the (a) Tricycle trap and (b) Ring trap. Colors andcontours are relative to the magnitude of the microwave drive, but may be thought of as ranging
from 0 to 100% as color ranges from black to red to yellow to white, with contours at 5% intervals.
Colors are comparable between the two traps, assuming the same microwave intensity on the surfaceof the conductors. These are also the DC field distributions for loading the traps, in which casecolor ranges from 0 to 200 kV/cm with contours every 10 kV/cm and 25 kV applied across themagnet surfaces. Rings are 3 mm apart for the Ring trap and in the Tricycle trap magnets are
2 mm apart. (c) The angle between electric and magnetic fields is shown, from red, 0, to blue, ⇡/2
radians. Magnetic field contours every 500 G are superposed.

In mid 2013, an e↵ort was made to perform spectroscopy along the |f, 32 i to |e, 12 i transition,with the goal of addressing the issue that |e, 32 i-state molecules with total energies below the lowest
crossing of the |e, 32 i and |f,3
2 i states at 410 G are unable to find their way out of the trap, seeFig. 6.7. The results were quite di↵erent from those performed along the |f, 32 i to |e, 32 i transition,
perhaps due to polarization di↵erences since this transition would feature dependence on sin(✓),
but more likely reflective of the microwave coupling bandwidth issues discussed above.
Another point is the spatial dependence of the fluorescence collection system. Fig. 6.8 shows
how the solid angle for photons to hit the lens varies with trap position, but beyond this there isof course the question of laser coverage. The laser is always optimized for signal, making it quite
likely to be well overlapped with the trap center, but for the larger magnetic field ranges, there areregions of the trap that are likely not well exposed to the laser, especially inside the rings of thetraps, since the laser impinges from the side.
Some of these spatial variations may be mitigated by an ergodic hypothesis. If the molecules
can be assumed to be exploring all regions of phase space, then variations in detection efficiency
may not significantly perturb the spectra or prevent it from representing the thermal state of the116

Figure 6.7: Depletion spectroscopy performed by microwave transfer from the |f, 32 i to |e, 12 i state.
The dip between 800 1000 G is likely an artifact of coupling efficiency challenges.

ensemble. Essentially, any blind spots in the spectroscopic procedure can be averaged away by
the motion of the molecules themselves, since a molecule which is flipped from |f, 32 i to |e, 32 i bymicrowave outside of the view of the laser may after just a partial trap oscillation find its way backinto view. Unfortunately, the ergodic hypothesis seems not well supported in this case, especially for
the Ring trap where cylindrical symmetry guarantees angular momentum as a conserved quantity,
unless strong elastic collisions capable of cross-dimensional thermalization were present. Since
ideally spectroscopy would be a tool to help confirm the thermalization status of the ensemble,
having this as an assumption becomes circular.
In any case, a final limitation of the spectroscopy worth discussing is that in principle it ought
to be performed in a regime where the distribution is not significantly altered by the spectroscopy.

117

Figure 6.8: Collection solid angle for the Tricycle (left) and Ring (right) traps. Colors do notquite match between the two panels but are labeled in steradians. For the tricycle trap, the lens islocated far below, while for the ring trap it is located far above. Electrodes are to the left and rightof the plots. In both cases, solid angle variations within 500 µm of the trap center are on the 10%
level only, although larger variations occur for molecules regularly reaching 3 5 kG, a deviationof over 1 mm from the trap center.

This is difficult to respect in practice, because the less significantly the population is perturbed,
the smaller the resulting signal and the longer the averaging time. In practice, the spectroscopies
performed in the Ring and Tricycle trap usually push the boundary with respect to the perturbationthreshold, with points collected at the peak of the distribution often featuring 30% population
removal. Pushing this boundary threatens to add an artificial smoothing on top of the true magnetic
field variation of the cloud, since even if one magnetic field were actually populated less fully thanits neighboring magnetic field regions, so many microwave pulses are applied that even molecules
which spend a small fraction of their time at that magnetic field end up having a decent likelihoodof being removed. We can observe this e↵ect to some extent by varying the number of microwave
pulses used in the spectroscopic sequence, see Fig. 6.9. For the most part, the techniques agree with
one another, but as far as the statistical likelihood of fitting to Maxwell-Boltzmann distributions,the single pulse spectra appears much more triangular, rising too far above the low and high fieldwings in the center.

118

Figure 6.9: Three spectra obtained with di↵ering numbers of microwave pulses. General agreement
is good, but the single pulse spectra deviates more noticeably. This is an indication of the potential
smoothing e↵ect of including many pulses as discussed in the text.

6.2.5

Ring and Tricycle Thermometry

The obtained spectra in the two traps were shown above in Fig. 6.5. Several importantfeatures are worth discussing here. First of all, the Ring trap fits nicely to a Maxwell-Boltzmann
distribution, which in a linear trapping geometry should scale with magnetic field in the followingway:p(B) / B 2 eµB B/kT .

(6.6)

The exponential term gives the expected Boltzmann suppression factor as a function of increasing
potential energy, while the B 2 term is proportional to the volume degeneracy factor. In otherwords, the volume corresponding to a magnetic field B within dB is an ellipsoidal shell in thelinear trap, whose surface area grows proportionally to B 2 since the trap is linear. Of course

119
there are already the limitations discussed above in Sec. 6.2.4, but in fitting Eq. 6.6 we must also
account for deviations from the linear quadrupole geometry. Close to 1100 G, molecules should be
exploring the region between the central and toroidal traps, a plateau which ought to contributea large volume degeneracy factor, although perhaps this should be suppressed due to polarization.
Close to the axis however, there is also a significant change in field gradient that occurs above andbelow ⇠ 1.5 kG. Contour lines are much more closely above than below, so that in principle thevolume degeneracy factor should be di↵erent for the two regions.
In the tricycle trap, the fit is more dubious. Points clearly fall below the fit at high magnetic
fields, and also seem to lie below at the lowest fields as well. On the other hand, given that thetricycle trap opens to the environment above 3 kG, it may be that falling below the MaxwellBoltzmann fit line is the more natural thing to occur in this situation. Alternatively, there isanother factor at play here worth mentioning- the influence of the nuclear spin of the molecule.
This comes not from the 16 O, which has a spin nucleus, but from the hydrogen. It does not create
any significant perturbation to the trapping potential experienced by the molecules at the energy
scales we are considering here, but because the di↵erential Zeeman shift between the |f, 32 i and|e, 32 i states is a hyperfine e↵ect, some relevant features emerge. Specifically, at low magnetic fields
we have to work in the coupled hyperfine basis F = I + J, and the |f, mJ = 32 i state actually
includes two substates, |F, mF i equal to |2, 2i and |2, 1i. The former experiences a fixed di↵erential
Zeeman shift relative to its opposite parity partner, but the latter experiences a varying di↵erential
Zeeman shift at low fields, see Fig. 6.10a. This has a significant e↵ect on how a Maxwell-Boltzmann
spectra ought to look for a population uniformly split between the two substates. Essentially, theshift grows significantly closest to zero field, so that as the microwave is scanned across what would
correspond to a 500 G for the |2, 2i state, only a third of that is scanned for the |2, 1i state. In
the Tricycle trap, including this e↵ect leads to an improvement in the fitting of the distribution
relative to excluding it as shown in Fig. 6.10c, while in the Ring trap the reverse is likely the case.
Uniform splitting is indeed what we would expect, even after carefully thinking throughadiabatic transitions during loading described soon in Sec. 6.3. None of the spin-flip and non-120
adiabatic e↵ects give significant bias to one nuclear spin orientation over the other; the nuclear spinorientation does often change after spin flips or non-adiabatic transitions, but with both populations
present before, both are present afterwards. This jives with thermodynamic intuition, since theensemble temperature of ⇠ 50 mK is far hotter than the energy splitting arising from the nuclear
spin of only 40 MHz or 2 mK. Equal population is also more or less consistent with the observationsin 6.24, where small stray electric fields perturb measured population distributions due to avoided
crossings which appear in two di↵erent locations corresponding to |2, 2i and |2, 1i.
6.2.6

Summary

The performance of spectroscopy deviates from the ideal in several key respects, which are
spatial variation of sensitivity due to polarization, fluorescence collection, and microwave intensity
variations; and oversampling beyond the small perturbation regime. The functional form of the
expected distribution deviates from the ideal given in Eq. 6.6 due to significant variations of thevolume element relative to a linear quadrupole, including the toroidal plateau, and the gradientenhancement above 1.5 kG. Nuclear spin e↵ects should influence this functional form still further.
In light of all this, the close fit of the microwave spectroscopy on the Ring trap at left in Fig. 6.5
to the functional form given in Eq. 6.6 is not a reliable indication of the degree of thermalization of
the ensemble. Similarly, the lack of a close fit to the microwave spectroscopy on the Tricycle trap
is not a reliable indicator of any lack of thermalization in that ensemble. This spectroscopy should
rather be used as a general indicator, assigning a representative temperature at the 10% level, but
not making any guarantee of the presence or absence of thermalization.
Thermalization aside, microwave spectroscopy may still be used to compare the spatial density achieved in the two traps. To do so, it is necessary to make some historical comparison ofrelative molecule numbers, and also to choose a size within which to count molecules for bothcases. Doing so leads to the numbers shown in Tab. 6.2.

Taking the measured photon number,

multiplying by the fraction below 1 kG, dividing by collection solid angle, and multiplying by gradient cubed gives the numbers shown in the density column of Tab. 6.2, for an overall fourfold

121

Figure 6.10: Inclusion of nuclear spin in di↵erential Zeeman shift calculations is shown to significantly influence microwave spectroscopy. (a) Microwave frequencies for targeting di↵erent states
are shown as a function of magnetic field for the four substates of what in the |J, mJ i basis are
the |f, 32 i and |f, 12 i states. (b) Distribution functions for the |F = 2, mF = 2i state and for the|F = 2, mF = 1i state versus microwave frequency. (c) The di↵erential Zeeman shift of the |2, 2i
state is used to convert between microwave frequency and magnetic field. Data collected on March
13, 2014 are shown. The green line shows what the Boltzmann distribution function accounting forthe presence of both states would look like, and is a fit to 37 mK. Blue only includes |2, 2i and fits
to 52 mK with a zero point o↵set.

Table 6.2: Historical Data are consulted in comparing the trap population between Ring andTricycle traps. In the tricycle trap, the number below 1 kG are counted, giving rise to the reported
fraction, while in the ring, the number below 1 kG are counted. This reflects the doubly strong
gradient in the Tricycle trap.
Trap
Tricycle
Ring

Num
78 ± 3110 ± 2Fraction
22 ± 3%28 ± 4%

⌦
0.38
0.34

Gradient
2
1

Density
360 ± 60
91 ± 13122

Figure 6.11: |ei parity molecules are observed after two di↵erent decelerations. (a)= 40deceleration. (b) = 50 deceleration. Insets zoom in on the decelerated packet. Note also therelative peak heights of the un-decelerated and decelerated molecules.

improvement in spatial density in the Tricycle trap.

6.3

Adiabatic Transitions during Loading
In early 2014, we were startled to discover that |ei state molecules were able to survive thedecelerator, and were appearing in significant numbers in our detection scheme at various final
speeds and even after loading the trap, see Fig. 6.11 This discovery led to a renaissance in our
understanding of transitions between states. Essentially what is happening is that molecules which
are in the |f,3
2 i state during the deceleration are able to adiabatically transition to the |ei state

due to the presence of the magnetic field in the detection region, despite the fact that the molecules
are not trapped and are only briefly flying through the trap. We realized that the magnitudeof electric field required to allow adiabatic transitions between the |f,3
2 i and |e,1
2 i states was

actually quite small, only a few V/cm. Since molecules pass through regions well in excess of 1200 G
where these states cross on their path into the trap region, all of the molecules in the |f,would deterministically make this transition assuming a rather small stray field.

3
2 i state

123
During the loading of the molecules, a similar fate may befall the |f,3
2 i molecules deliveredby the decelerator. When the loading fields are finally switched o↵, these molecules do not maintaintheir |f,3
2 i character, but rather follow a tangle of adiabatic crossings, and therefore retain theirenergetic hierarchy. That is to say, |f,3
2 i molecules remain in the second highest energy state,

which can be either |f, 12 i or |e, 32 i depending on the magnitude of the magnetic field at the molecule’s
location during the rapid turn-o↵ of the electric fields used for loading. In light of this, statements
such as the following, “perfect OH rotational, ⇤-doublet, and hyperfine state purity is achieved” [79,Page. 19061, Top Right] become somewhat ironic. An example of how significantly this turns out
to be false is shown in Fig. 6.12. Many molecules are remaining in |ei parity states, regardless ofwhether an extra bias field is left on or o↵ after loading. It is unclear whether this is a result ofmolecules regularly adiabatically transiting between |f, 12 i and |e, 32 i, or whether something di↵erent
is going on, such as a more complex orbital route including states of even lower energy. Either way,
one clear way to address the problem is to include regular switching of the electric field to address
any partially trapped orbitals including |ei parity segments. From Fig. 6.12, a single switching
event already significantly reduces the problem.
To complicate matters further, another kind of adiabatic transition, perhaps more appropriately termed a spin-flip, can also occur close to zero magnetic field in the presence of electric field.
At zero magnetic field, there is also a degeneracy between states, not of opposite parity but of thesame parity and di↵ering m number. Depending on the angle between the electric and magnetic
field a , these states are brought much closer together, facilitating spin-flips. This is describedfurther in the Appendices of both [121] and [88, App. A], and is also depicted visually in Fig. 6.13
The upshot for loading is that molecules which cross the plane including the magnetic zero almost
certainly flip between |f, ± 32 i given the very large magnitude of the loading fields.
It therefore follows that many of the molecules which end up in the |f, 32 i state after loadinga

In fact, all of the opening and closings of avoided crossings between states have dependencies not only on
the magnitude of the electric field but on this angle as well. Their exact functional form may be computed via
perturbation theory on the OH ground state Hamiltonian, which is actually analytically solvable since its eigenvalues
always come in pairs and therefore the system reduces to a quartic polynomial system, which has analytic expressions
for its roots. The perturbation has to be taken to various orders depending on how di↵erent the m number or theparity are of the two states that are interacting.

124

Figure 6.12: |ei parity molecules stably remain in the trap under either constant bias electric fieldor zero electric field, but not switching between the two.

Figure 6.13: The |f, 32 i and |f, 32 i state with varying B-field and three E-field conditions. The
states are squeezed together when the electric field is orthogonal to the magnetic, which in the Ring
and Tricycle happens along the plane parallel to the magnet surfaces through the trap center.

125
have in fact experienced the loading fields as |f,3
2 i molecules, and therefore with a di↵erent

on-axis potential than the two already shown in Fig. 6.3 which include or exclude the magnetic
field. This on-axis potential is added in red in Fig. 6.14. The magnetic field is seen to reduce themagnitude of the potential experienced by the |f,3
3
2 i states relative to |f, 2 i wherever the magnetic

field is non-zero. Although with near-unit flipping between the two across the trap center, the|f,3
3
2 i states actually experience a larger stopping potential than the |f, 2 i.

Because the |f,32i

molecules experience a reduced potential except past the trap center, they should arrive earlier,having traveled at a larger speed than their |f, 32 i counterparts. It should therefore be possible
to deterministically load |f, 32 i molecules just before the plane and |f,3
2 i molecules just after, sothat the latter actually convert to |f, 32 i during the loading. It would be interesting to demonstrate
this experimentally by sitting on the e-state LIF transition during the loading and seeing whether
tuning of the time-length of application of the stopping fields actually varies the initial loading of
the |ei state. It should be possible to observe variation, and it should be the case that a minimum
in loading of the |ei state would correspond to an optimally deterministic transitioning of |f,32i

molecules into |f, 32 i during the loading sequence.
6.3.1

Multi-pulse Loading

The field distributions used for loading the Ring trap feature an extra bump in between theend of the decelerator and the beginning of the trap, refer back to Fig. 6.3. This bump could in
principle be used to perform last minute slowing or bunching prior to trap loading, especially if
it were strengthened in magnitude by also turning on the last decelerator electrodes to the same
voltage. We attempted this in 2014, and obtained the disappointing results shown in Fig. 6.15,
among many other such datasets. This was also our first attempt at using Behlke switches in series,
at least for serious tri-polar output voltages and not only trap-floating voltages, and in the end led
to one of my first broken Behlke switches. At the time, we assumed HV issues, or perhaps that thebump was essential for transverse focusing, but the idea of loaded molecules actually existing as|f,

3
2 i during loading never occurred to us. With regards to |f,3
2 i molecules, the extra switching

126

Figure 6.14: The potential for molecules in the |f, 32 i state during trap loading. The trap centercan be seen close to 10 mm based on the single-point overlap of the three traces indicating an isolated
magnetic zero. Due to the spin-flip inducing behavior of superimposed electric and magnetic fields
close to magnetic zeros, molecules in the |f, 32 i state to the left of the zero transition uniformly
into |f, 32 i to the right of the zero, and vice versa. The kink seen close to 7 mm corresponds may
at first come as a surprise. This occurs in all three traces and is therefore an electric field e↵ect.
It turns out to be the zero of a weak electric quadrupole trap formed by the loading fields at thecenter of the frontal ring magnet of the Tricycle trap, as can be seen more clearly by studying
Fig. 6.6(a).

events and extra flight time with varying magnetic field and no electric field out in front of themagnets where the extra bump is created could lead to those molecules being transferred to various
other sub-levels. This would reduce the flux of |f,3
2 i molecules through the trap center to thenspin-flip into trapped |f, 32 i molecules.

6.4

Pin Trap
Originally motivated by an attempt to repeat the electric field inelastic collisions reported

in [121], but this time in the tricycle trap, it was discovered that electric field induced single
particle loss via spin-flips were in fact a dominant mechanism, overwhelming an e↵ect attributable
to inelastic collisions. In this section, we recapitulate the conversations also available in [88], which
describe this mechanism in the most coherent way we were able to contrive.

127

Figure 6.15: Breaking up the loading pulse into an extra slowing pulse followed by loading only
leads to a reduction in trapped molecules.

6.4.1

Introduction

A diverse array of direct cooling strategies have succeeded on various molecules [119, 90, 123,
105, 124, 125, 126, 127], including those now enabling molecular collision studies [79, 128, 129, 130].
Many of these molecules will require secondary strategies like evaporation or sympathetic cooling to
make further gains in phase space density [131, 44, 132]. They also may face a familiar challenge:
spin flip loss near the zero of a magnetic trap, but dramatically enhanced for doubly dipolar
molecules, relative to atoms, due to their internal spin dynamics in mixed electric and magnetic
fields.
Spin flips were directly observed for magnetically trapped atoms near 50 µK and overcome with a time-orbiting potential trap [133] or an optically plugged trap [2], enabling thefirst production of Bose-Einstein condensates. Non-laser-based molecular cooling experiments begin at modest temperatures and require trap strengths typically only attained with quadrupole
fields [119, 87, 134, 135, 136]. In the 2 T/cm magnetic quadrupole used in our previous studies
of hydroxyl radicals (OH) [44], spin-flips should not have had a significant influence until the µK

128
regime, but the application of electric field changes this. Electric fields applied to magneticallytrapped dipolar species o↵er interesting opportunities to study anisotropic collisions and quantum
chemistry [121]. They can also be useful for control over state purity [114]. But the electric field
can also dramatically enhance spin-flip losses, due to internal spin-dynamics that we corroborate
with direct experimental evidence for the first time in the present work. We achieve this with anovel trap geometry that also allows complete removal of the loss with minimal sacrifice of trap
strength.

6.4.2

Loss Mechanism

The internal spin-dynamics leading to spin-flip enhancement are subtle, having eluded two
previous investigations: In Ref. [137] the analogues of atomic spin-flip loss for molecules in mixed
fields were modeled. It was concluded that no significant loss enhancement due to electric fieldwould be evident. However, this conclusion holds only for the approximate Hamiltonian used inthat study, not more generally. In Ref. [138] it was correctly noted that Hund’s case (a) molecules
maintain a quantization axis in mixed fields. The states of the molecule were shown to align with~ ± = de↵ E~ ± µe↵ B~ b , µe↵ and de↵ theone of the two quantization axes set by the vector fields X
e↵ective dipole moments of the molecule in uncombined fields. The key idea is that Hund’s case (a)molecules have both dipole moments fixed to their internuclear axis, so that in the molecular frame,
the energy shifts from the two fields combine like vectors. It was also shown that the combined~ ± , depending
Stark-Zeeman energy shift of the molecule is proportional to the length of either X
on the choice of quantization axis, with proportionality given by an m quantum number. Thisbasis was dubbed “Hund’s Case X,” and it was asserted that the existence of the Hund’s Case X
quantization axis would prevent flips near the zero of a quadrupole trap. As we now describe, theloss is actually enhanced, but this Hund’s Case X basis actually proves very useful in explainingwhy.
b~ ± de↵ E.
~ We reverse this to provide a more physical connection to our experiment, where
The authors use µe↵ B
the electric field is fixed.

129

Figure 6.16: A uniform electric field, added to magnetically trapped molecules for dipolar studies
or other purposes, enhances spin-flip losses. Note in particular the increased size of the lowest
contour (red) in panel (f) relative to panel (e); this result can be understood by considering Zeeman shifts under various conditions as shown in panels (a-c) and described further now. Four
Zeeman split lines in OH’s X2 ⇧3/2 manifold are shown (a-c), with the doubly stretched state inblue and its spin-flip partner in dashed red. These states are shown with no electric field (a), with
~ = 150 V/cm and E~ kB~ (b), and with E~ ?B~ (c). Note the vastly reduced red-blue splitting in|E|the latter case. The ground state consists of two parities and four m states each, usually labeled|parity = f, e; m = ± 1/2, ± 3/2i. The negative parity, electrically strong field seeking manifold
sitsbelow (d). The application of electric field generally drives these parities further apart, but
within each parity manifold the exact modification depends on the relative field orientations (bc). In panels (e-f), energy splitting contours are shown every 40 MHz near the zero of a 2 T/cm
~ = 0 (e), and with uniform E = 150 V/cm
magnetic quadrupole trap for OH molecules [114] with E~ + sign(E~ · B)µ~ e↵ B,~ the proper
along the strong axis of the quadrupole (f). The vectors are de↵ E
quantization axis for well-trapped molecules as described in the text.

We begin with an intuitive picture. In order to remain well trapped in combined fields, amolecule must remain weak field seeking with respect to both fields, i.e. doubly stretched. This~ ± withmeans that the quantization axis to which it aligns should correspond to the vector field X
maximal length at the molecule’s location. In a geometry where the fields are continuously rotating,~ ± µe↵ B,~ depending on whether the fields are
the maximal length vector field can be either of de↵ E

130oriented closer to parallel or antiparallel. Consider a molecule in a magnetic quadrupole trap witha uniform electric field. This trap has two hemispheres, a parallel hemisphere where the fields~ ? B,~are closer to parallel, and vice versa. The hemispheres are separated by a plane where E
which intersects the trap center. Now suppose a molecule in the doubly stretched state begins in
the parallel hemisphere. To be doubly stretched, it must be aligned to the sum quantization axis
~ + = de↵ E~ + µe↵ B.
~ If its trajectory carries it near the trap center whereset by the vector field X
the magnetic field is small, the electric field does indeed maintain the quantization axis and themolecule’s alignment with it. However when the molecule enters the antiparallel hemisphere, themagnitude of this quantization axis now decreases with increasing magnetic field. This molecule
has therefore spin-flipped from the doubly stretched state to a magnetically strong field seeking~ ? B,~ leading to loss.
state. These states are degenerate in the plane where E
This intuition agrees with a more rigorous analysis of the energy splitting G between thetrapped state and its spin-flip partner. By diagonalizing the approximate eight state ground molecular Hamiltonian for OH, subtracting the relevant state energies and Taylor expanding, we find:3
G(B? , B|| , E) = 2B|| + 4.3·B?

2

E4

4
+ O(B||2 , B?
)

(6.7)

~ · ê?,|| , where ê is the unit vector in the labeled direction relative to the electric
Here B?,|| = µe↵ B~
field. E = |de↵ E|,

is the lambda doubling (see Fig. 6.16d). The relevant splitting is not quite zero

~ ?B
~ and B|| = 0 thanks towhere E, but nonetheless reaches a deep minimum; the remaining

Zeeman splitting is reduced from linear to cubic in magnetic field (Fig. 6.16). This Zeeman splitting
suppression is in fact a known phenomenon in the precision measurement community [139, 140],
and experimentalists have exploited it to suppress the influence of magnetic fields in electron EDM
measurements. However, in the case of applying mixed fields during trapping, this suppression is
not beneficial but rather detrimental.
To deduce the e↵ect of this loss plane on the ensemble, we consider molecular trajectories inlight of the Landau Zener formula:2

Phop = e ⇡ /2~Ġ ,

(6.8)

131
Table 6.3: Enhancements (⌘) and loss rates ( ) for OH with typical applied fields. Zero fieldvalues are equivalent to traditional spin-flip loss. Electric field is required during evaporation andspectroscopy to open avoided crossings [44, 114], or applied to polarize the molecules and study
collisions [121].

E (V/cm)
0
300
550
3000

55 mK
⌘
(s 1 )
1
0.02
5
0.1
17
0.3
1000
19

5 mK
⌘
(s 1 )
1
1.3
9
11
40
50
1600
2000

PurposeZero Field
Evaporation
Spectroscopy
Polarizing

which relates the probability of diabatically hopping between two states Phop to their energetic~ ?B~ plane,coupling  and their rate of approach Ġ = vz dG/dz. Here z and vz are normal to the E
and we neglect the components of Ġ due to the other coordinates since from Eqn. 6.7 it is clearthat G grows predominately in one direction. We can also set  to the minimum energy gap along
the trajectory, which is found in the plane. This facilitates direct numerical computation of theloss rate ( ) by integrating the molecule flux through the plane for a thermal distribution, weighted
by the hopping probability. The expression is more fully derived in Chapter 7. We perform these
integrations for OH over the velocity distribution in a 2 T/cm magnetic quadrupole [87] undervarious electric fields in Tab. 6.4.
We have also developed an algebraic scaling law, which yields the electric field induced loss
enhancement factor
3
⌘=
11

✓

d E
pe↵


◆8/3

,

(6.9)

see Sec. 6.4.4 for the full derivation. Here  represents a characteristic energy scale for spin-flips
that can be derived by setting Phop =1/e in Eqn. 6.8 and using a typical value of vz . This meanspthat for electric fields with de↵ E >  , the loss enhancement is almost cubic with electric field.
Crucially, it is notthat sets the relevant scale, as one might naively suppose given that this isp
the energy beyond which the Stark e↵ect is linear and the molecule is polarized. Instead it is  ,which is in general much smaller;  = 5 MHz for OH in our trap, while= 1.7 GHz.

Returning to the numerical approach, the direct integration of flux is a key improvement

132
relative to our previous work [121], where electric fields were applied to study collisions. The
mechanism of molecular spin-flip loss was identified, and an attempt was made to deconvolve itfrom the collisional e↵ect of the electric field. Revisiting this with the direct integration of flux,
we find a three-fold larger loss magnitude, enough to explain a significant portion of the e↵ect
previously attributed to collisions, see Chap. 7. In light of this, it becomes especially important toperform direct, un-convolved experimental verification of both the magnitude of the loss e↵ect andthe validity of our loss-flux calculations. We now present the new trap where this is achieved.

6.4.3

Experiments and Results

Our idea is to use a pair of 2D quadrupole traps, one magnetic and the other electric, withorthogonal centerlines (Fig. 6.17):~ = B 0 xŷ
B

B 0 yx̂

~ = E 0 y ŷE

E 0 z ẑ(6.10)

We achieve these fields in a geometry that matches our Stark decelerator [123]. This geometry~ ?B~ in both the x = 0 and y = 0 planes, and from Eqn. 6.7,
features large spin-flip loss, since E
3
G = B?

2 /E 4 will be generally very small due to the large E. However, by adding a small magnetic~ = Bcoil ẑ along the centerline of the magnetic quadrupole with an external bias coil, a drafield B~ ?B~ with minimal perturbation of the trapping
matic change can be made to the surfaces where E
potential.
~ ?B~ surface from a pair of planes into the hyperbolic sheet given by
Bcoil morphs the E~ ·B~ = 0). This means that E~ ?B~ is pushed away
x · y = z · Bcoil /B 0 (Substitute Eqn. 6.10 into E~ is smallest. In Fig. 6.18, the surfaces where E~ ?B~ for several Bcoil
from the z-axis where B
magnitudes are calculated and shown wherever G  . The loss regions ought to be tuned far
enough from the trap center that molecules cannot access them. This is indeed what we observe,note the striking di↵erence in trap lifetimes in Fig. 6.19a. With only 200 G bias field (the trap is
5 kG deep) the loss is suppressed below that due to background gas.
To further verify our understandings of the loss mechanism, we translated one of the magnetic

133

Figure 6.17: The last six pins of our Stark decelerator [87] form the trap (a), which is 0.45 K deep
with trap frequency ⌫ ⇡ 4 kHz (b). Along y the trap is bounded by the 2 mm pin spacing. The
yellow pins are positively charged and the central pin pair negatively, which forms a 2D electric
quadrupole trap with zero along the x-axis. This is shown for the x = 0 plane (c), with yellow
pins artificially projected for clarity since they don’t actually intersect the plane. The central pins
are magnetized, with two domains each. Blue indicates magnetization along +ŷ, red along ŷ.
These domains produce a magnetic quadrupole trap with zero along the z-axis, shown in the z = 0
plane (d).

pins along x̂. This pin translation disrupts the idealized 2D magnetic quadrupole by adding a small~ / B 0 z ẑ, which significantly alters the topology of the E~ ?B~ surface and the overall
trapping field B
loss rate in the trap. We also compute loss rates for all values of pin translation and Bcoil by numerically integrating the loss flux through these unusual loss surfaces via the Landau-Zener formula,just as for the simpler quadrupole geometry discussed previously. The calculated populations after
30 ms in the trap have a reasonable agreement with the measurements (Fig. 6.19b). The direct
integral calculation uses only the temperature of a purely thermal distribution as a free parameter,

134

Figure 6.18: Surfaces where spin-flips can occur are shown for three values of Bcoil in light gray,dark gray, and black. The magnetic pins are shown as in Fig. 6.17 for context. The purple starmarks the trap center, to which molecules are confined within a ⇠ 1 mm diameter.

and does not involve computation of any trajectories. The temperature fits to 170 ± 20 mK c . An
intuitive explanation for the intriguing double well structure in Fig. 6.16 is that Bcoil first translates
the magnetic zero along the z-axis, overlapping it with larger electric fields at first before moving
it out of the trap.
With strong experimental confirmation of the molecular spin-flip loss enhancement, we canmove on to generalize beyond OH. Hund’s case (a) states are most susceptible in the sense that
smaller electric fields are sufficient to cause a significant problem, but with enough electric field any
state exhibiting competition between electric and magnetic fields for alignment of the molecule oratom will be susceptible. One way to avoid competition is for the fields to couple to unrelated partsof the Hamiltonian, which happens to a limited extent for Hund’s case (b) states without electron
orbital angular momentum (⌃ states, ⇤ = 0) [138]. In these states, which include most laser-cooled
molecules thus far, the electric and magnetic fields couple to rotation and spin respectively, which
are only related by the spin-rotation coupling constant. This constant is usually in the tens of
MHz [132], so molecular spin-flip loss remains quite significant. The inclusion of hyperfine requirescCalculation performed in COMSOL: Source Code

135

Figure 6.19: Time traces (a) without bias field (black), with bias field (green dots), and with modulated density (green circles). One body fits (red) give loss rates of 200 s 1 without bias field and2 s 1 with full bias field at long times, in agreement with our background gas pressure. At the
fixed time 30 ms, population is shown as a function of both pin translation and bias field (b), forseveral values of pin translation, labeled relative to perfect alignment. Fits (red) are calculated by~ ? B.
~
integrating the molecule flux of a thermal ensemble through surfaces where E

a careful case-by-case investigation. For OH, it would initially seem to add an extra splittingthat could protect from spin-flips, but in fact the loss plane is only shifted slightly away from
~ ?B
~ and retains the same area. For Yttrium Monoxide [26], certain hyperfine states can avoidE136
spin-flip loss entirely when electric fields are applied. These states are characterized by significant
electron-spin-to-nuclear-spin dipolar coupling, which results in a protective gap regardless of fieldorientation.
It is also instructive to consider the related case of a pure electrostatic trap. Here there isalways some zero field parity splitting that prevents the orientation-reversing spin flips we havebeen discussing. However, this same splitting pushes all states with the same sign of m, the fieldalignment quantum number, very close to one another, leading to loss via Landau-Zener transitions
other than the m tom spin-flip [116]. Intriguingly, the addition of a homogeneous magnetic field

can actually suppress this loss [115].
The present trap, in addition to providing the desired experimental testing ground for molecular spin-flip loss, produces large 5 T/cm trap gradients useful for maintaining high densities tofacilitate collisional studies. This is in contrast with other strategies for plugging the hole of amagnetic trap which often lead to a reduction in trap gradient. With loss removed, we observe
a population trend whose initially fast decay rate decreases over time (Fig. 6.19a, green dots),suggesting a two-body collisional e↵ect. We test this by reducing the initial population fivefold
but without changing its spatial or velocity distribution, and then scale the resulting trend by five
(green circles). This technique is described further in Chapter 7. If collisions had contributed,
this new trend would show less decay, but we observe no significant change. This seeming lack ofcollisions is likely due to the much higher initial temperature of 170 mK, in contrast to the earlier
work at and below 50 mK [44]. An alternative hypothesis for the population trend is the existenceof chaotic trap orbits with long escape times [141]. The understanding of electric field enhanced
spin-flip loss brings an important consequence to the use of RF knife under electric field employedin the forced evaporation, especially at low temperatures. We present the e↵ect of evaporation
at intermediate temperatures (⇠ 30 mK) in Chapter 7. Moving forward, we aim to increase thedensity by means of several improvements [54, 65].

137
6.4.4

Scaling Law Derivation

Here we derive the loss enhancement scaling law presented in Eqn. 6.9, and repeated here:
⌘=

3
11

✓

d E
pe↵


◆8/3

.

(6.11)The key idea is to compare the surface areas of the loss regions with and without electric field.
There is no exact loss region where a molecule is guaranteed to spin flip, but rather its velocity anddirection contribute to the Landau-Zener probability (Eqn. 6.8). Nonetheless, for the purposes ofa scaling law, we can assume the average thermal velocity vT , and choose a probability thresholdof P > 1/e. These assumptions allow us to define the loss region as the contour surface of energy where=qp2~Ġ/⇡ = 4~vT B 0 /⇡.

(6.12)

Here Ġ is the rate of change in the energy gap between the trapped state and its spin flip partner,
and B 0 is the magnetic field gradient along the strong axis of the trap.
We assume that the electric field is applied parallel to the strong axis of the quadrupole trap,~ ? B,~ perpendicular to this axis. This matches thewhich makes the loss plane, as defined by E
geometry that has been realized in our experiment [121], and is the worst case, but by no morepthan a constant factor of 2 2 relative to other directions the electric field could have.
Before application of electric field, the  valued energy contour is the surface of an oblate
ellipsoid of long radius r0 = 2/µe↵ B 0 . Its area is then 2⇡↵ r02 , where ↵(e) = 1 + (1/ee)tanh 1 (e)generally for eccentricity e, and ↵ ⇠ 1.38 for the present 2:1 ellipsoid. When electric field is applied,the energy gap near the trap zero takes an unusual functional form. To derive it, we first assign
spatial coordinates r and z denoting directions within and normal to the loss plane, respectively.
Next we diagonalize the ground state hamiltonian of OH in mixed fields, see App. A of Ref. [114],or similarly for another species. Subtracting the energies of the trapped state and its spin-flip
partner, and then series expanding the result yields:
G = 2µe↵ B 0 |z| +(µe↵ B 0 r/2)3(de↵ E)42

f (de↵ E/ ),(6.13)

138
plus higher order terms in r and z. Here

= 625/144 = 4.3 and f is a rational expression thatpapproaches 1 for small arguments: f (x) = (1 + 1.28x2 )/ 1 + 1.44x2 . The key feature, as discussed
in Sec. 6.4.2, is the cubic dependence G exhibits on r which leads to much more severely oblate
contours.
Now we can use Eqn. 6.13 to compute the surface area of the G =  contour. We specialize, so that f (de↵ E/ ) ⇠ 1. The radial extent of the surface can beto the regime where de↵ E <solved by inverting  = G|z=0 :s
1 3 8(de↵ E)4rE =.
2µe↵ B 0(6.14)~ For large enough E, rE dominates over this axial
The axial extent remains z = /µe↵ B 0 for all E.
2 and the loss area enhancement becomes ⌘ = r 2 /(↵r 2 ).
extent, so that the area is e↵ectively 2⇡rE
0E

Putting everything together:
⌘

=

=
=

s

!2
1 3 8(de↵ E)42µe↵ B 0!2r
1
1 3 8(de↵ E)42↵ 2/3 2✓◆
3 de↵ E 8/3
p
.
11

1
↵

✓

2
µe↵ B 0◆2
(6.15)

Now we address the domain of validity of this result. When E is small, Eqn. 6.13 only has
a narrow range of validity, since the electric field only dominates in a very small region near thetrap center. Outside, G retains a nearly linear dependence on r. This means that Eqn. 6.14 only
holds for E above some threshold. For smaller E, rE will simply not be significantly perturbed
from its zero electric field value of r0 = 2/µe↵ B 0 . The implication for the enhancement factor in
Eqn. 6.15 is simply that it is only valid when it predicts an enhancement significantly greater thanpunity. In other words, Eqn. 6.15 holds when de↵ E > 1.6·  , but below this ⌘ gradually returns
to unity. Eventually when de↵ E >, the factor of f (de↵ E/ ) in Eqn. 6.13 is better approximatedby 1.1·de↵ E/ , which leads to the modification ⌘ = 0.26·(de↵ E)2 /4/3

2/3 . Thus for these larger

E-fields, the enhancement factor reduces in its dependence on electric field from order 8/3 to order
2. At this point, the loss is typically too large for trapping, see Tab. 6.4.

139
6.4.5

Conclusions

Molecule enhanced spin-flip loss arises in mixed electric and magnetic fields due to a competition between field quantization axes. We conclusively demonstrate and suppress this e↵ect using
our dual magnetic and electric quadrupole trap, which is also an ideal setting for further progressin collisional physics thanks to its large trap gradient. Our calculation of the magnitude of spin-flip~ ?B~ enables detailed predictions of how its location andloss via flux through surfaces where E
magnitude ought to scale with bias field and trap alignment, which we experimentally verify. Our
results correct existing predictions about molecular spin-flips in mixed fields and pave the way
toward further improvements in molecule trapping and cooling.

6.5

Trap Escape and Dynamical Phenomenon
Early during my thesis work, it was common to infer significant information pertaining to

the collisional behavior of ensembles based on the observed loss rates of molecules from traps. Onee↵ect which significantly influences the observed trends is the behavior of molecules in the |f, 12 i
and |e, 32 i states after loading. Molecules are loaded into these states from the |f,3
2 i state which

is decelerated, as described above in Sec. 6.3. An additional important e↵ect however is the escapeof molecules from the expected |f, 32 i state but with surprisingly long lifetimes. A common ruleof thumb floating around in the atomic physics community is that molecules or atoms which are
not well trapped ought to find their way out of the trap within a quarter trap oscillation. This
viewpoint neglects the potential complexities associated with the dynamics of molecular orbits.
In the e↵ective moving trap which characterizes deceleration for example, almost all molecules lie
above the true trap depth when operating in S = 1 mode, but many survive the ⇠ 5 oscillations they
undergo in the ⇠ 3 ms they spend in the decelerator, refer back to Chap. 5 for more information.
More generally, the stability of orbits in dynamical systems is an exciting question in classicalphysics, with applications to the planets of our solar system, chaotic systems, and so on [142]. One
intriguing study applies tools of dynamical system analysis to the trapping of atoms in a crossed

140
optical dipole trap [141]. Not only do they demonstrate chaotic behavior in this comparatively
simple trapping geometry, but as a corollary they find single particle trajectories with very longescape times over 104 times the oscillation period [141, Fig. 11]. I studied the escape dynamics
from the Ring, Tricycle, and Pin traps, generating the trajectory information shown in Fig. 6.20.
Energies of molecules are given in units of GHz, and there are 2.0 GHz/kG for OH molecules. It
is remarkable that molecules survive at these energies, which are therefore far above the minimum
trap depth of ⇠ 5 GHz in the Tricycle, similar in the Ring, and ⇠ 8 GHz in the Pin trap.
I therefore followed this up by studying chaos indicators for trajectories in the magnetic
pin trap. This can be done by studying the evolution of orbits with infinitesimally varying initialconditions, in which case a new higher order equation of motion may be found which describes theevolution of the variation itself. Specifically, if the evolution of the system is described by:@x= F (x, t),@t(6.16)

where x is a point in 6D phase space and F describes the “flow” of the system, then it is possible
to specify a modification to the initial position x and study:@x + x@t@x= F (x + x, t)@t

F (x, t),(6.17)

which after making various leading order subtractions and so on gives rise to:
@ x
@f
=
· x.
@t
@x

(6.18)

Various chaos indicators based on variational equations such as that in Eq. 6.18 may bestudied, such as the perhaps more well-known Maximum Lyapunov Exponent [143]. Using a higher
order variation called the OFLIT2 T [144], I determined the pin trap to be a highly chaotic system,as the panels shown in Fig. 6.22 indicate. A few stable orbits exist, but the majority are not. It isalso useful to verify that the software I developed for studying these indicators is in fact properly
functioning. In Fig. 6.21 I show how my results compare with those in [141]. The comparison
is nearly identical, which is especially remarkable given that I don’t make use of the analytic
expression for the crossed dipole trap [141, Eq. 6], since none is available for the magnetic pin trap.

141

Figure 6.20: Molecules with the indicated total energy are initialized and their escape times indicated in grayscale. Position within a panel corresponds to initial position of the molecule in thetransverse mid-plane of the trap.

142

Figure 6.21: The chaos indicator OFLI2T T nicely reproduced with a spline-based computational
tool, compare with [141, Fig. 5b]

Instead I use a spline to represent the potential. This is described further in my publicly available
code repository d .

6.6

Next Generation TrapsAt the time of this writing, an e↵ort is underway to develop a next generation magnetic trap

suitable for use with our 333 stage decelerator. Thanks to the development of our engineering andmanufacturing capabilities as far as cryogenics are concerned, I decided to push for a cryogenic
trap. This has the added benefit of addressing the vacuum and blackbody lifetime limits for OH
molecules and potentially enabling ten second study times. A number of features of this design are
worth discussing here.
dChaos Indicator Codebase

143

Figure 6.22: Chaos indicator OFLI2T T is evaluated for trajectories beginning in the pin trap. Several
islands of stability are engulfed within seas of chaos. Chaotic trajectories red, periodic blue, quasiperiodic green. The latter two vary on a color gradient. (a) 1.94 mm wide, all panels to the same
scale. Energies in this panel are 7.5 GHz above trap minimum. Trajectories begin in the x yplane with velocity in the z direction and normal to this plane. (b) 2.5 GHz, x y plane. (c)5 GHz, x y plane. (d) 5 GHz, y z plane.

6.6.1

Pin Trap Farewell

We decided to move back to the 3D magnetic quadrupole design, despite the success of themagnetic pin trap for addressing spin flip losses. Reasons for this include the following:(1) Difficulty in correctly selecting and orienting the magnetic pins, which required an invacuum translation stage and associated high voltage concerns. We also had to carefullyscan the field of the pins to identify a well-matched pair.
(2) Questions about the HV performance of the pins, which didn’t condition very well. No

144
major arcs during normal operation were occurring, but the loaded temperature was triplethat predicted by sims, a possible indication of arcing type e↵ects during loading.
(3) Maintaining the Bias coil, which required a high pressure water boosting system [145](4) Expectations about collisional behavior. The pin trap resolves spin-flip losses, but substates
nonetheless come quite close to one another over a broad spatial range, making inelastic
loss probable. At one point Goulven Quéméner actually calculated inelastic cross sections
for a map of fields and angles in the pin trap and confirmed this.
(5) Lifetime concerns for the magnetic pins themselves. We had a number of pins demagnetize
during conditioning, leading us to install an in-vacuum temperature probe which could bemade to touch the pins using a manipulator feedthrough or brought safely away from themduring the application of voltage.

6.6.2

Magnets

Neodymium magnets reversibly demagnetize to 70% of their room temperature strength
when brought to temperatures in the 10 K regime, but newer Praseodymium magnets actuallyenhance [146]. I was able to coordinate the manufacture of these for our applicatione .
An e↵ort was undertaken to more systematically study the variation of the performance ofthe system as far as various dimensions are concerned. For each parameter of interest, the fields
were exported to a MATLAB based deceleration, trap-loading, and trapping simulation to study
their e↵ectiveness. The results can be seen in Tab. 6.4. A tricycle style was selected, though with a
bit of a reduction in gradient. Radiusing the rear electrode was considered, as well as variations in
the thickness of extra surfaces designed to cover the imperfect surfaces of the magnets. In all cases,
deceleration and loading dynamics were included, with the exception of |f,3
2 i e↵ects. I have only

once made an attempt at including these e↵ects, for the sake of more precisely fitting the escapeeVacuumschmelze manufactures these and has a US distribution team based in Kentucky. Thanks to Ed Narevicius
for the referral. Grade 131 TP and 131 DTP.

145
Table 6.4: Various design parameters are studied in a loading and trapping simulation in order tomake the best selection. Loading is specified as a triplet of voltage labels, which apply to the lastpins, the front magnet, and the rear magnet respectively.
Name
Cryo1
Cryo2
Cryo3
Cryo4
Cryo5
Cryo6
Cryo7
Cryo8
Cryo9
Cryo10
Cryo11
Cryo12
Cryo13
Cryo14

Description
1 mm Front Magnet Thickness, +G- loading, 1 mm hole curvature
2 mm Front Magnet Thickness, +-+ loading, 1.25 mm hole curvature
Change Rear Magnet to NdFeB, +G- loading+-+ loading, 0.6 mm hole curvature
1.5 mm hole curvature, center hole from 2.5 to 2 mm
0.6 mm hole curvature, center hole 4 mm
Remove Rear CoverUse 28 kV loading fields
Rear Cover Protrusion
Back to Cryo6, but finer mesh
Try +G- again
Back to +-+. Rear radii to 7 mm from 10 mm
Front magnet OD to 16 mm from 25 mm
Rear radii back to 10 mm

N
3151
5007
4422
3961
4430
4655
4058
4341
2739
4720
4145
4612
4620
4710

T
30
45
41
40
40
41
42
43
35
41
41
41
41
42

Figure 6.23: Relevant information for the Cryocycle trap under manufacture at the time of thiswriting. (a) Magnetic field magnitude, 500 G per contour, trap depth of ⇠ 2200 G transversely.
(b) Electric field magnitude during loading. 10 kV/cm per contour. Curved rear surface acts toextend the region of steep trapping slope further towards the rear magnet relative to the Tricycle
trap, see Fig. 6.6a. (c) Ray-trace performed in COMSOL Multiphysics for determining collection
solid angle close to 4 sr for the Cryocycle.

dynamics of molecules in the tricycle trap with extra electric field turn-on events, described further
in Sec. 7.3.1. The final selected geometry is shown in Fig. 6.23, and is dubbed the Cryocycle.

146
6.6.3

Photon Collection

The cryocycle also features what will hopefully serve as an important improvement in fluorescence collection. Mounted together with the magnet is an elliptical reflector capable of directing
much of the fluorescence from the trapped molecules onto a PMT outside the vacuum. Collection
solid angles close to 4 sr should be possible, an order of magnitude improvement over the previous
traps, see Fig. 6.23c. Of course this assumes a highly reflective surface, only obtainable in the near
ultraviolet with aluminum [147], which is fortunately a good choice for thermal conductivity as
well. Aluminum may be a less ideal choice as far as high voltage is concerned however, since its
surface oxide layer should enable the formation of patch charges and stray electric fields to someextent. This could prove especially problematic in the event of any sensitive long-term studies,
since variations in the patch charges could lead to variations in the enhancement of spin-flip losses
near the trap center. One technique to address this is to detect the magnitude of the stray fieldby spectroscopy, see Fig. 6.24. Because of the small magnitude of the di↵erential Zeeman shift,slight perturbations of the electric fields on the magnetic field dependent line-shapes, especially inthe vicinity of avoided crossings, are clearly discernible. Performing a full spectroscopic sequence
would be slow, but similar observables could be designed to give sensitivity to stray electric fields,
for example a two-point spectroscopy comparing the frequency of the dip to that of the peak. Another possibility would be to look for population transfer at a more distant frequency that would
only address molecules close to avoided crossings.

6.6.4

High Voltage Considerations

In addition to the material and patch considerations mentioned above, I concentrated a significant e↵ort towards minimizing the exposure of the magnet surfaces to high electric fields. I was
motivated in this pursuit by observations of adverse e↵ects of the magnet surfaces on decelerator
electrodes, see Fig. 6.25. After the first data collection period with the tricycle trap in 2014, a vacuum break revealed significant arcing induced surface degradations. Pin pairs close to the end of

147

Figure 6.24: Stray fields are seen to influence spectroscopy. The black simulated line corresponds
to 10 V/cm. The two dips correspond to the inability of microwaves to address molecules at thosefrequencies due to avoided crossings perturbing the lines at those locations. Only one crossing issignificantly perturbed by small fields, that between |e, 32 i and |f, 12 i, but this crossing occurs in
two di↵erent locations for the two nuclear spin substates of |f, 32 i.

the decelerator where the trap was located showed degraded surface lines or tracks, with a ring-likestructure reminiscent of the cylindrically symmetric tricycle trap mounted in close proximity. Thissuggests that applying high voltage to the somewhat rough surfaces of the neodymium magnets
could have been causing arcs. It is also possible that merely the presence of the magnetic field exacerbates or facilitates arcing somehow. Although we never had direct evidence that the formationof these tracks corresponded to a decline in performance of the system, at the time this observation
contributed to a working hypothesis that the performance of the decelerator had degraded andneeded to be addressed. In the new cryogenic system, the magnets are mounted in such a waythat they are for the most part ensconced within the mounting structures, which can be better
polished than the magnets themselves, which are a brittle ceramic with no available high polishsurface preparation.

148

Figure 6.25: Decelerator pins after deinstallation of the Tricycle trap. Not only discoloration, butlinear scratch-like patterns or tracks, are evident. These tracks appear to form concentric rings
about the decelerator axis, suggesting that tricycle trap magnets were involved.

6.6.5

Blackbody Absorption

In order to benefit from a lifetime extension with cryogenic temperatures, it is necessary tocarefully control the radiation environment of the molecules. OH can absorb radiation primarily at
120 µm (2.5 THz) [148], where typical radiation shielding conductors have near perfect reflection.
In the case of an incomplete shield, as is required for our setup since the shield cannot approach toclose to the high voltage decelerator, this can mean that the radiation relevant for the molecules
is actually fully equilibrated to room temperature despite the cryogenic shield. Whatever thedominant absorbers are for the relevant wavelength will dominate the radiation environment, with
conductive surfaces playing little to no role. Many materials are surprisingly transparent, including
plastics such as Teflon which are actually used as lenses f . To address this, I opted for the additionof ceramic plates on the inside of our radiation shield to influence the radiation environment byabsorbing THz radiation otherwise reflected by the shields.
fTeflon Lenses

149

Figure 6.26: (a) Earlier Design from Matt Hummon, 32 one eighth arc segments. (b) Updated
design, 8 quarter arc segments, 2 rings, 1 disk. 4 mm gap between magnets. (c) Magnetic Trap.
Much longer than wide, unlike 3D quadrupole traps. Up to 1200 G deep or so, depending on tuningof extra disk magnet. (d) Loading fields, much less ideal than Tricycle trap, concave up across trap
center and steepest out front.

6.6.6

Cloverleaf Trapping

Of course it would be ideal to address the issue of electric field enhanced loss once and for allwith a lifted magnetic trap. This is possible with permanent magnets, and a streamlined design was
identified and simulated, see Fig. 6.26. Such a trap may one day be the right choice, but it would
seem unwise to tackle this simultaneously with the changes already being introduced. Moreover,without collisional e↵ects, it may not be worth the e↵ort. These e↵ects should be more clearly
evidenced in the Cryocycle, which should feature at least an order of magnitude improvement
over any cloverleaf style design in the densities which may be loaded. The reason for this is thatcloverleaf type geometries always require a delicate competition between the strengths of di↵erent
fields, and can never be generated with the same trap depths or gradients as the quadrupole trap.
The design shown in Fig. 6.26 loads only 20% of the Cryocycle, and is significantly less tight, likely
at least twofold reduction in each direction, for a forty-fold reduction in density relative to the
Cryocycle.
If at some point it becomes necessary to pursue the Cloverleaf trap in earnest, a few points
are worth mentioning. Firstly, always make sure to investigate cross sections through the trap inmany planes, since often the three principal orthogonal planes can miss the actual weak points ofthe trap. It should be sufficient to add in planes with ✓ = ⇡/4 and including the z axis, working

150
in cylindrical coordinates, for the Cloverleaf shown in Fig. 6.26b, but for that in Fig. 6.26a, I
believe the weak points may occur in the ✓ = ⇡/8 planes. Secondly, for any given geometry it isalways worth varying the magnitude of an extra bias magnetic field in the z direction before making
conclusions. This extra field can have a very significant e↵ect on the tightness and depth of thetrap, and in practice may even be tuned in the experiment with an external coil or a translatablepermanent magnet. I had some success with removing a small divot from the rear electrode so as
to reduce loss of molecules which can access that volume. It helps the loading significantly to add
conductor between the magnets and touching the front magnets closest to the decelerator, so as topush the loading hill further back and overlap the trap center better. One problem with this is thatthen the collection solid angle is negatively impacted. Mounting the magnets in aluminum framesthat are highly polished to better allow fluorescence to escape could help with this, as planned in
the Cryocycle. Finally, it will be important to study this trap outside of the the vacuum somehow,
especially in order to see how closely the magnets can be packed/glued together, what the resulting
field strengths are, and especially to make sure that the trap is truly plugged. This will require ahighly miniaturized magnetic field probe of some kind.

6.7

Trapping of Water Isotopologues
One potentially exciting capability of our new lengthier decelerator system and advanced

field distributions would be to apply it to a less Stark responsive molecule such as water [108,Sec. 2.4]. The precise feasibility depends on just how slow the beam is when produced in Xenon in
an Even-Lavie valve, which may be worth investigating in the near term. With earlier generationsof the experiment [69, Fig. 4], a wide range of initial speeds (323

465 m/s) were possible in a

Xenon expansion depending on the discharge timing. Although a discharge would not be used, thisindicates that at least with that valve a wide range of longitudinal Xenon speeds were present after
supersonic expansion. This is symptomatic of an inefficient supersonic expansion process, but goodnews in the sense that water molecules at the low velocity wings of the expansion will have a muchgreater decelerability. Some very relevant guiding work should prove a useful starting ground [149].

151
An electrostatic quadrupole trap would be most feasible, and this could be achieved in a very
similar system to the one currently planned for the Cryocycle, but with magnet-less electrodes. It
is best to have at least the rear electrode bend inwards towards the others, as in this electrostatic
trap [150]. The final pins of the decelerator can form one electrode of the quadrupole, which would
then require an additional ring and end-cap electrode. Detection could be performed with some
sort of REMPI scheme, but for initial tests of decelerability the RGA may suffice.

Chapter 7
Detecting Collisions

Collisions are an exciting area of study for low temperature molecular systems. Quantizationof intermolecular degrees of freedom leads to a host of interesting dynamics, which until relatively
recently could only be discerned via their contribution to measured integrated cross sections including averaging over many initial states and many partial waves of the interaction potential. However,with the ability to isolate single quantum states thanks to selective manipulation techniques, andwith the reduction of collision energies a↵orded by deceleration, new state-resolved e↵ects [151], andnew resolution on quantum mechanical interference type e↵ects are now achievable [128, 129]. It iseven possible to study e↵ects directly pertaining to the orientation or stereodynamics of molecularinteractions [152]. Simultaneously, studies of collisions in traps have evolved from total cross section
measurements [87, 124], to lower temperature determinations directly relevant to inelastic-elastic
ratios [125, 130, 153].
This last result [153] from the Narevicius group is particularly exciting, as it compares quite
directly with our own work, including cryogenic skimming techniques, decelerator optimization, andcryogenic trapping optimization, and is worth further discussion here. Realizing a similar observation would be an exciting outcome of the research in this thesis, though at the time of this writing
it remains unclear whether the optimizations we have pursued will achieve densities comparable
to [153]. In their Zeeman deceleration of oxygen, benefit is taken from oxygen’s much more directsourcing, and from the better integration of skimmer cooling with Zeeman deceleration thanks to

153
the possibility of affixing a long cryogenic Sapphire tube within the bore of the decelerator.acontrast, for OH our discharging yield from water may be only 1

In10%, and skimmer cooling has

only led to two-fold gains with the hexapole 4.5.
Another alternative strategy for phase space compression of molecules applicable to a widearray of species is found in the combination of centrifugal deceleration and optoelectric cooling [126].
The deceleration technique has also directly enabled collisional studies on its own [130], of CH3 F
and ND3 . The optoelectric sisyphus technique requires low spatial density flat-bottomed trap
geometries, but the velocity space compression is quite impressive, 420 µK is reached. Laser-based
techniques are also beginning to enable similar molecular collisional studies, notably the recently
observed sympathetic cooling of NaLi molecules by sodium [154], the first successful sympathetic
cooling of a molecule by an atom. Work on laser-cooled molecules has also recently progressed tothe collisional stage, with the Doyle group reporting on collisions between pairs of CaF molecules
loaded into optical tweezers [28].
A key challenge that has emerged during the course of my thesis work has been the verificationof collisional behavior of all kinds. For this purpose, several very general techniques exist, which
are now described in detail.

7.1

Controlled Density Reductions
Collisional e↵ects always influence a population in proportion to the likelihood of such col-lisions, which in turn scale with the square of the number density of the population. It therefore
follows that as one were to scale an experimental control parameter tuning the number density ofthe population, collisional e↵ects would grow quadratically in that parameter, while single-particlee↵ects would grow linearly. The challenge lies in whether it is really possible to achieve a tuningof this parameter which does not inadvertently lead to an unwanted change of a di↵erent nature.
Several di↵erent strategies have been pursued in this regard. The molecule source is perhaps thea

In private communication with Yair Segev, we discussed the application of skimmer cooling to deceleration, andthe use of such tubes in the Narevicius group.

154
most logically natural place to begin, but in practice all ways of tuning the initial molecule density
risk disturbing the experiment in unwanted ways. For example, tuning the current of the discharge
filament can easily reduce the measured number of molecules, and is simple to control and manipulate, but its most likely method of action is to make the discharging process more sporadic, inwhich case the measured reduction may actually be dominated by cases where no molecules are
generated at all. In this scenario, densities remain as always, but some fraction of the time thedischarge fails to ignite and no molecules are produced at all. This situation may masquerade as adensity tuning, but in fact does not at all investigate the physics of interest.
Other source parameters include the stagnation pressure in the pulsed valve; parameters
influencing the flux of the opening event such as coil current, coil voltage, pulse duration; or valve
temperature. In all of these cases, there is the risk that changes in the number of molecules
generated can change the efficiency of the supersonic expansion process, thereby changing themolecule distribution function across the population, and not only its number density. If we wereverifiably operating in a regime where the valve dramatically overfills the decelerator’s phase space
acceptance, these e↵ects could more plausibly be neglected, but this assertion is not justified,
especially when seeking small e↵ects and when using the relatively narrow initial distribution of
the Even-Lavie valve. To be more concrete, let us parametrize the most likely way that phasespace distributions resulting from di↵erent source parameter changes may vary from one another.
Assuming that in all cases we at least maximize the signal loaded into the decelerator as far as it
depends on timing parameters, we can approximate the phase space distribution downstream in the
decelerator where many rotations have cleaned out variations in the angular coordinate as follows:(r) = e

r2
2
2 r

,

(7.1)

where r is defined for a planar space-velocity slice of phase space as:
r2 = (zz0 )2 + (vzv0 )2 /! 2 ,z0 and v0 are the center coordinates of the phase space distribution, andof the distribution.

(7.2)
r parametrizes the width

155
Variations in thisthe gas, sincer can eventually contribute to subtle influences in the observed lifetime ofr in turn influences the extent to which the outer reaches of the trap are populated

after loading. Source variations which change the initial temperature of the supersonic expansion
are likely to increase the value ofr , leading to greater population in the wings, and faster decay

rates early on during loading, a classic indicator of collisional e↵ects, which also lead to faster decays
early on which later turn o↵ as the population reduces. The distribution proposed in Eq. 7.1 is infact a more benign representation of what is possible. Loading a narrow distribution o↵ of centercan easily lead the radially averaged phase space distribution to actually peak away from centerfor example.
Another possibility for achieving density variations without changing the distribution (r) of
molecules delivered eventually to the trap is to perturb some aspect of the deceleration process,but leave the source untouched. Any phase space manipulation based technique, such as operating
in di↵erent deceleration modes or introducing gaps in the coverage of the traveling potential, would
have the same possibility of disturbing the distribution ultimately loaded into the trap as justdescribed for the case of source variations. However, in the decelerator, the population is alreadystate selected, opening up the di↵erent possibility of directly reducing the density by spectroscopic
means. Were a second LIF laser available and a region of good optical access before the trapping
region, this would serve as a reliable means to achieve a 50% density reduction for example, although
at the expense of writing intensity noise and other shot-to-shot variation issues of the pulsed dyelaser onto the number density of the ensemble. Microwave transfer is a more compelling possibility,
especially since it a↵ords the possibility of directly transferring molecules to an un-trapped state.
Applying them during deceleration presents a bit of a challenge however, especially given the usualengineering constraints imposed by the high voltage system. In the past, we’ve addressed thisusing the biased tee strategy described further in [114], but hoping for a simpler workaround, we
attempted a microwave near-field probe coupler as shown in Fig. 7.1.
In the microwave engineering literature [155, Sec. 4.7], a probe is a near-field device usedeither for driving a waveguide or measuring its local field behavior. The chamber in which our

156

Figure 7.1: A microwave free space coupling probe used for addressing OH molecules even in the
midst of the Stark decelerator. The brass structure is in good mechanical and electrical contact
with the center pin of a coaxial vacuum feedthrough mounted in a 2.75” conflat vacuum connector.
This is achieved with the 0-80 setscrew just visible towards the base of the brass structure. The thinregion of the Brass structure has a 1/4” diameter, and the thick region is closer to 3/4” diameter.

decelerator sits has a 20 cm diameter, comparable to the microwave frequencies relevant for driving|f, 32 i to |e, 32 i transitions, where in zero field 1.7 GHZ corresponds to 18 cm wavelengths, so it isreasonable to expect that a simple current probe might successfully drive microwave modes of thevacuum chamber. This indeed turns out to be the case, and we successfully used this probe, notonly for microwave removal of population during deceleration, as shown in Fig. 7.2, but also to
perform the same in-trap spectroscopy described in the previous chapter.

157

Figure 7.2: Microwaves are successfully coupled into our vacuum chamber using the probe shownin Fig. 7.1, as evidenced by the response of hydroxyl radicals to increasing powers shown here.
These molecules were slowed to 43 m/s and then allowed to fly through the detection region withno trap loading. These data were collected on June 14th, 2016. Microwave powers are as measured
at the vacuum feedthrough. At a later point, we were able to apply even higher RF powers by
instead applying them in a pulsed manner so as to remain within the capabilities of our microwave
equipment. Fitted FWHM values in µs are reported in the legend and found not to vary withpower.

7.2

Fitting Trap Decay Curves
Another competing technique for observing the influence of collisions relates to the functional

form of the decay that is exhibited from a trapping geometry as a function of time. If we makethe naı̈ve assumption that collisions lead to loss uniformly over a population regardless of otherparameters, this can be described by the following di↵erential equation:
dN
=
dt

N2

↵N,

(7.3)

158
where the magnitude of the two-body process is parametrized by , and single-particle e↵ects suchas collisions with background gas molecules are included in the model and parametrized by ↵. Thiscan in turn be solved by the factorization method for rational polynomial functions:

Z

1
↵

✓

1
N

log N

Z

dN
N (↵ + N )
◆
dN
↵+ N

=

t + C,

(7.4)

=

t+C

(7.5)

↵t + C 0(7.6)

log (↵ + N ) =
N
↵+ N
1
↵/N +

= C 00 e ↵t

(7.7)

= C 00 e ↵t

(7.8)

↵/N +

= C 000 e↵t(7.9)

N (t) =↵C 000 e↵t

(7.10)

Now by requiring that N (0) = N0 , we have:N0 =↵(7.11)

C 000

↵
+ .
N0

(7.12)↵)e↵t(7.13)

N0 e ↵t
1 + N0 (1 e ↵t )/↵(7.14)C 000 =
Substituting and moving around:
N (t) =( N↵0 +=

This equation is rather ugly, but it can be seen to reduce to the more well known formulas describing
the decay of a population subject to either single or double particle e↵ects but not both. To see
this, we simply take the corresponding ↵ orparameters in Eq. 7.14 to zero:✓◆N0 e ↵tlim= N0 e ↵t!0 1 + N0 (1e ↵t )/↵✓◆
N0 e ↵t
N0
N0
lim
=
=
↵t
↵t
↵!0 1 + N0 (1e )/↵
1 + N0 (te |↵=0 )1 + N0 t
where the penultimate step for the two-body formula is achieved via L’Hôpital’s rule.

(7.15)(7.16)

159
With these functional forms in hand, it is now possible to use the fitting of measured trap
decays as an observable giving indication of the presence of collisional e↵ects. Historically, thisprocedure was used in our experiment in both the Ring and Tricycle traps, but never in the Pintrap due to thanks to the development of our capability of performing controlled density reductions.
When using decay fits, it is essential that these observations are made in the absence of any timevarying single-particle loss processes, such as the potentially slow escape dynamics of some classes ofmolecules from the trap geometry. In the Ring trap for example, we have experimental indications
that molecules do not explore the toroidal region of the trap, although they have access to it.
This confirms that the ensemble loaded in this trap is in a potentially dangerous regime for theapplication of loss functional forms, since there should certainly be a timescale for molecules to
explore the toroid which has not been easily satisfied in the first few milliseconds after trap loading.
I can take this a step further by approximating a trap with slow escape dynamics for some
populations as one characterized by two di↵erent single-particle decay times:
N (t) = A1 e ↵1 t + A2 e ↵2 t

(7.17)

Now consider a trap where half the population decays with a background pressure limited lifetime of
500 ms and the other half escapes the trap on a timescale that is slow relative to a 300 µs oscillation
time but faster than the background limit, say 50 ms. These numbers are somewhat representativeof historical values in our traps. In the absence of any collisional e↵ects, fitting this bi-exponential
decay with Eq. 7.14 is optimal with a zero-valued ↵, and with

= 9.9 ± 0.1/s/mol. Not only does

the hypothesis of decay according to Eq. 7.14 give a strong slant in favor of two-body e↵ects, itdoes so with high fidelity, fitting the data with an R2 = 0.99, see Fig. 7.3. I should mention thatin this fitting I fix the initial population to one, which reduces the ability of Eq. 7.14 to fit any
arbitrary shape.
Of course one contributing factor to the success of the two-body fit in matching the biexponential decay curve is the timescale probed in the experiment. One more e↵ective way for
distinguishing true two-body e↵ects could be to use a long-time measurement to fix the one-body

160

Figure 7.3: A biexponential decay curve is well fit by a two body, demonstrating the importanceof either removing single-particle e↵ects or ensuring that they occur on only a single timescale.

decay rate, since at long enough times the two-body e↵ects reduce in relevance while the one-bodyrates remain fixed. In practice, this approach can lead to more trouble, since at about 3 s [148],an additional one-body process kicks in, but as long as one-body e↵ects are well-known, pushing
to long timescales can improve the reliability of the decay-fitting method for detecting collisions.

7.2.1

Electric Field Induced Losses

One area where the fitting of decay curves played a very large role was in the study of what
was thought to be electric field induced inelastic loss. In the experiment [121], it was found thatapplication of electric field led to loss from the trap. It was further found that initiating the loss afterdi↵erent hold times in the trap was a way of varying the initial density so as to probe the two-bodyprocess in di↵erent parameter regimes. Some example fits from [121] are shown in Fig. 7.4a. At thetime, it was known that a single particle loss due to electric field existed, discussed in the appendix
of [121]. Its magnitude was approximated, and fixed in Eq. 7.14 when fitting decay curves. It was

161

Figure 7.4: Electric field induced decays with di↵erent turn-on times of the electric field are shown.
(a) Fits to decays initiated after application of a 3 kV/cm electric field after di↵erent wait times
yield two body parameters of (i) 40.4 ± 3.0, (ii) 41.2 ± 4.2, (iii) 48.4 ± 8.1, and (iv) 45.4 ± 5.5(photons/shot) 1 s 1 . Intrigued by the data shown in panel (a) from [121], we took data with anemphasis on simultaneous collection for reduced systematics at longer timescales for all datasets,
resulting in panel (b), collected in March of 2014 in the Tricycle trap with a similar electric fieldmagnitude.

argued that this strategy of fitting Eq. 7.14 while varying initial population would lend robustness
to the procedure despite the known difficulties of relying on Eq. 7.14 [121, Page. 1800, bottom
right].
During our reinvestigation, we pushed the decay measurements out to long times, and found162
the results shown in Fig. 7.4b. These results are a big challenge to the hypothesis of a fixed single
particle loss timescale, and strongly suggest a time-dependence of the single particle loss, since there
is no reason that either a one-body or two-body e↵ect, initiated after various wait times, should
lead to nearly overlapped traces in the long term. Note how near 180 ms in Fig. 7.4b, the 50, 100,
and 150 ms traces all approach the same level. Intrigued by this observation of what seemed likea fixed magnitude trap-loss induced by the turn-on of the electric field, we also experimented withthe application of multiple pulses of electric field, and obtained the results shown in Fig. 7.5. This
presents an additional challenge to the hypothesis of simple electric field dependent loss parameters,
which ought to respond only to the time integrated electric field application, and not whether it ispulsed or continuous. In contrast, in Fig. 7.5 we find that applying the electric field for the entiretyof the time sequence (red) causes far less loss than applying it for half of the time, but in a switchedmanner (yellow). This is strongly suggestive of a dynamical e↵ect, i.e. an e↵ect which concerns
the short term dynamics of molecule trajectories in their trapping potential, and not a collisionale↵ect which ought to respond generally speaking to thermodynamically and temporally averaged
quantities.
Specifically, one obvious dynamical e↵ect would be the behavior of the population in thepresence of a spin-flip loss region, discussed at length in Sec. 6.4.2. Molecules whose orbits regularly intersect the loss region are lost, but molecules may exist whose orbits never intersect the loss
region. Indeed, such a class of molecules is mathematically guaranteed, thanks to the cylindricalsymmetry of the ring and tricycle trap. This symmetry enforces the conservation of angular momentum about the axis of cylindrical symmetry under the evolution of the classical Hamiltonian
governing molecular trap orbits. Since the loss region occurs close to this axis, molecules withenough azimuthal angular momentum never intersect the loss region under classical Hamiltonian
trajectory evolution. With some collisional thermalization however, trajectories which intersect theloss region should gradually be repopulated, but at a rate that is slow relative to the initial escapedynamics. We can conclude that if thermalization is slow, electric field enhanced spin-flip loss cancause a decay rate that decreases over time. If thermalization is slow even compared to the total

163

Figure 7.5: Trap decay curves under various conditions related to the timing of application of electricfield. The conditions are (blue) no electric field applied, (red) constant electric field applied, (yellow)
5 ms pulses of electric field every 10 ms. Lines are five point running averages. Data collected in
March of 2014.

experiment runtime, spin-flip loss should lead to the removal of a fixed subset of the population overa certain dynamical timescale, regardless of any hold-time, exactly as observed in the long-term
behavior in Fig. 7.4b.

7.3

Comparison to Simulation
Comparison to simulations is a potentially favorable strategy for detecting the role of collisions

in an experiment. Fast mean-field based approaches to simulating collisional gases exist, with that
due to Bird [156] quite common in the literature. In the course of this thesis, significant simulation
endeavors were pursued, often exactly with this goal of elucidating the relative contribution of single
and double particle e↵ects. Simulation is particularly promising in light of the role of dynamicale↵ects described in the previous section, since any analytical forms are hopeless to shed light on
dynamics, while the simulation should serve well with a suitably meshed trapping potential.

164
Firstly, a note of caution is in order. Similar to the fitting of decay curves, comparison tosimulations can be plagued by the same sorts of fundamental flaws as are the comparison to decay
fits discussed in the last section. The simulation should certainly perform better than analytic decay
fits in that it does not need to make any homogeneous population or related assumptions, but at theexpense of abandoning these homogeneity assumptions the simulation becomes dependent on many
additional tuning parameters. Extracted collisional contributions can strongly depend on thesetuning parameters. As a thought experiment, suppose that an experimental parameter like the trap
strength, which may easily deviate between simulation and experiment due to manufacturing defects
or variability in the permanent magnets for example, is indeed di↵erent between the simulation andthe experiment. Suppose further that the simulation overestimates the trap depth by 10%. Thisparameter is important for the short-time dynamics of molecules from the trap, and an overestimate
would result in a reduced role of these dynamics, leading the simulation to underestimate the initial
decay rate due to dynamics. Consequently, it is likely that the best match between simulation andexperiment would occur for a simulation with a larger number of collisions than the experiment,
since these extra collisions would boost the initial decay rate to compensate for the trap depth
overestimate.
These issues could be behind the incredibly large collision rates predicted in the simulation
decay fitting shown in Fig. 7.6 for example. These are not unperturbed decays, but occur during
the application of various forced microwave evaporation sequences. It is seen that large collision
rates are rather successful at fitting the shapes observed in the trapping geometry. A key parameter
governing the comparison of simulation to experiment is the removal efficiency of the microwave
knife, which could have dependencies on polarization and position in the trap that are not treated,dependence on molecule velocity, etc. If this parameter is too small, the simulation will predictgreater numbers of molecules remaining, and the collision rate will be reduced to match this,since the collision rate also influences molecule number remaining by influencing how successful
the evaporation is. I believe the match of microwave knife between simulation and experiment is
responsible for the simulation landing at the large collision rate match of 130 s 1 mol 1 in Fig. 7.6.

165

Figure 7.6: The fitting of the decays of molecules from the trap with direct Monte-Carlo simulations
including collision rates are shown. Large collision rates of 130/s are found to fit best, primarily
due to the initially fast decay exhibited by the population. These data were collected on June 26th,
2013. The legend describes some details of the microwave knife,

In addition to parameter tuning, there is also the question of whether the simulation correctly
includes all relevant e↵ects, since simplifying assumptions are often essential for reasonable runtimes
and development times. Specifically, the influence of other substates of OH other than the doubly
stretched one are pervasive and pernicious. As discussed in Sec. 6.3, molecules in the |f,3
2 i state

end up in either |f, 12 i or |e, 23 i, and the dynamics of these molecules should also influence thedynamical escape processes involved during early trap decay.

7.3.1

Simulating Electric Field Induced Loss

One key success achieved during my thesis work was the use of simulations to demonstrate thedynamical e↵ects suggested by the experiments discussed above in Sec. 7.2.1. By carefully including
the electric field enhanced spin flip loss in simulations but with no appreciable collision rate, andthen applying the same fitting paradigm used in [121] to the decay traces obtained by simulation, avery close match is obtained to the experimental data, see Fig. 7.7. Each point corresponds to thefitting of a suite of decays occurring after di↵erent hold times at a specific electric field. We can

166

Figure 7.7: Two body fits from [121] to experimental data like that in Fig. 7.4 but at various electricfields. The blue data points and shaded region are repeated from Fig. 3 of [121], where the shading
indicates the variation that would be brought about by two-fold changes in the assumed one-bodyloss ↵ from spin-flip losses. The very close agreement highlights the ability of dynamical e↵ects tomasquerade as two-body decay curves.

also examine the match of individual experiments, instead of only looking at the full collection. An
example of the simulated trajectories arising from a specific electric field is also shown in fig. 7.8.
It is seen that the simulation overestimates the initial decay rate, but agrees fairly well with thetotal eventual loss.
At first it is perhaps tempting to attribute this discrepancy to the possible continued role ofcollisions. However, there is no reason that collisions should influence this short time discrepancy,
which is much more likely a higher order dynamical e↵ect, especially since the dynamical e↵ects
were just seen to so nicely explain the observation of fixed total electric field induced loss. And
indeed, just such a higher order dynamical e↵ect is available through the more careful inclusionof other substates of the OH ground state. Electric field induced spin-flip loss does not directly
remove molecules from the trap, but instead transfers them to lower lying states, some of which
remain fairly well trapped. This can be seen by viewing the energy of substates as a function of
position in the trap, animated in Fig. 7.9. Along the axis, all states are connected, but as theradial coordinate increases, mostly trapped states separate away from the others. Simulating this167

Figure 7.8: Experimental data on electric field-induced loss with an attempted overlap to spin-flip
loss simulations. The case of no electric field (black, solid, circles) is compared to electric fields of
3 kV/cm turned on after a wait time indicated in the legend.

has several associated challenges. Firstly, the dynamics associated with hopping between substates
must be carefully treated. This is made especially challenging by the fact that the location of
hopping points between the second trapped substate and lower states varies with the magnitude ofapplied electric field, unlike the hopping point between the primary trapped state and the second,which is always located at the trap center. Secondly, the detection probability of molecules inother states features strong position dependence, since these states have a positional dependence
to their parity character. This also increases the possibility of discrepancies between simulationand experiment associated with the power and frequency of the laser, since these will determine
how much |f i character is required for detection. Despite all of this, the key idea is that thetreatment of secondary and higher order partially trapped states should lead to a reduction in
the dynamical decay timescale after application of electric field, and this is indeed observed when
including the secondary trapped state, as shown in Fig. 7.10.

This figure also shows a few othertuning parameters that were considered in the simulation, such as laser beam diameter and initialmolecular ensemble center of mass velocity. I also spent considerable time on the question of theinitial population, at one point implementing a loading reflective of the |f, ± 32 i magnetic e↵ects,

168

Figure 7.9: Eight ground states of OH as a function of position in a magnetic quadrupole trapwith 3 kV/cm applied. In the top left, the energies of these eigenstates are plotted as a function of
their position along the cylindrical axis through the quadrupole trap. The doubly stretched state
appears as a dark blue V near the center, and the other states may be identified based on their
Zeeman e↵ect at larger distances. The red state is |e, 32 i, and has the same slope as the dark blue,
yellow is |f, 12 i, purple |e, 12 i, etc. In the other panels, position is plotted not along the axis, butparallel to it and o↵set by the distance labeled in the title of the panel, increasing in steps of 200 µs.
The avoided crossings open up even for very small deviations from the axis, demonstrating thatexcept for a few small openings, really the top four substates are predominantly trapped.

169

Figure 7.10: Decays after electric field turn-on are shown with and without the electric field andunder varying conditions. In the legend, final fractional di↵erences between the cases with andwithout electric fields are shown.

see Fig. 6.14 of Sec. 6.3. Comparing to Fig. 7.8, it is clear that a slower dynamical timescale has
been obtained, and we can conclude that the remaining discrepancy in that figure is capable ofbeing explained by the inclusion of higher order trapped states.
Simulation fitting is a powerful tool, but great care must be taken that single particle deviations between simulation and experiment are not falsely influencing the reported collision rate.
Such influences are not surprising, since often single particle deviations yield the same signature
as collisions. One way to address this is to explicitly study the influence of parameter variation on
the best fit collisional simulation parameter. Another is to use alternative means to constrain thecollision rate, such as density calibration [60, Chapter 4].

170

7.4

Anticipated Collision Rates
Given the challenges associated with each of the collision detection techniques discussed thus

far, it is especially useful to develop alternative means of anticipating reasonable collision rates in asystem of interest. One key way to do this is to estimate the molecule number via the experimental
detection scheme, use theory to approximate the collision cross section, and then rely on the wellknown and easily motivated expression to get an order of magnitude estimate for the collision
rate:

where n is a number density,= n vrel ,(7.18)a collision cross section,

the collision rate per molecule, and vrel

the average relative velocity.
As far as the molecule number, density calibration suggests 1.9 ⇥ 10 5 cm 3 molecule density
in free flight [60, Sec. 4.6]. Assuming that the laser beam fills a 2 mm3 volume or so, and detects
0.3 photons/shot during free flight, 1.5 photons/shot during trapping, we find:N=

2 · 10 3 cm3⇥1.9 · 10 5 cm 3⇥

1.5/0.3=

1900.

(7.19)

As far as the cross sections are concerned, we can take Goulven’s calculations [44, Inset of Fig. 1b]
at 50 mK giving an elastic cross section:= 2.5 ⇥ 10 12 cm2 .

(7.20)

Now at this stage it is a bit tricky to decide what exactly to put into Eq. 7.18, because
the density in the trap is not homogeneous, and in fact varies rather strongly given the tightly
confining linear quadrupole geometry. For this reason, I simulated this exact question by initializinga thermal distribution of molecules in a linear quadrupole trap and studying the collision frequency
per molecule, leading to the results shown in Fig. 7.11. These results are essentially a mapping thatgives the relationship between molecule number and collision rate for the geometry. The fitted value
scales directly with molecule number, so that in an ensemble with 106 molecules, each moleculewould have 68 collisions per second for example. This simulation was actually performed with a171

Figure 7.11: Collision rates are simulated as a function of molecule number. With 1000 molecules,
a rate of 0.068 s 1 mol 1 is found.

more ideal value of the collision cross section,= 2 · 10 11 cm2 , taken from the low temperature

limiting value computed by Goulven in [44, Fig. 1b]. It follows that for 50 mK temperatures, theexpected mapping from number to collision rate would be 8.5 s 1 mol 1 with 106 molecules in thetricycle trap.
With these numbers, the discussion surrounding the collision rates in Fig. 7.6 can be placedin much better context. We can anticipate that the molecule number required to achieve a collision
rate of 130 s 1 would be 1.5 · 107 . Our collection system ought to detect something like one photon
per three hundred molecules, or in this case 50, 000 photons, when in practice we detect only a few
photons per realization of the experiment.

7.5

Forward Minus Backward Evaporation
During our careful reinvestigation of the results reported in [44], we developed a rather

sensitive technique for probing the presence of slight but potentially nonzero collision rates in the

172

Figure 7.12: Microwave depletion spectra are shown with no evaporation, with evaporation, andwith a backward evaporation sequence.

experiment. The essence of our technique is to subtract the molecule number after an evaporation
sequence from the number we obtained by time-reversing the microwave frequency chirp, so thatthe population cut goes backwards from deep to shallow in the trap. This comparison subjects
all molecules to the same integrated microwave power, and thus the two conditions should beequivalent in a situation with only single particle e↵ects. With respect to collisional e↵ects, thetime-reversed case functions like a truncation, preventing molecules that would otherwise havecollisionally thermalized to lower temperatures from doing so, see [157] for a discussion of the usual
behavior of an evaporation. To whatever extent an evaporation is successful in facilitating beneficial
thermalizing collisions, the time-reversed condition should yield fewer molecules. We consistently
observed this in the Ring trap at the (6 ± 2)% level, pointing to an evaporative e↵ect despite thenegative influence of spin-flip losses.
It is also interesting to look at the spectra of molecules in the trap after such sequences.
This is shown in [88, App. B] only for a forward sequence, but I wish here to expand a bit on
the conversation o↵ered there, and to include a spectrum from the backward evaporation as well,see Fig. 7.12.

Both forward and backward evaporations show a single point well separated from173
the unevaporated spectra, and are remarkably similar overall. This similarity suggests that thetwo sequences are not significantly di↵erent from one another thermodynamically, and leaves one
wondering whether any systematic artifact in the forward minus backward comparison could becontributing to the measured di↵erence in molecule number.
I spent some time investigating this possibility, and eventually realized that any dynamicaltimescale for the escape of molecules removed by the microwave knife could indeed lead to a false
positive result of the forward minus backward evaporation. Since the backward evaporation removes
its molecules earlier than the forward, any extra dynamical time for their escape could result in
the forward evaporation artificially showing a larger remaining molecule number. The same kindsof multi-substate dynamics discussed above in Sec. 7.3.1 could also lead to an increased dynamicaltimescale relative to expectations. The naı̈ve expectation would be escape on the order of a few trap
oscillations, characterized by a single exponential decay, as discussed in [114] for |ei-state molecules.
From [114, Fig. 4], the electric field applied during evaporation of 300 V/cm should be sufficient
for removing |ei-state molecules in a few milliseconds. However, even in that work, signatures ofthe slow escape of some molecules are evident although not discussed, note for example the insetof [114, Fig. 3], where a single exponential is supposedly fit, but reaches a nonzero baseline value of0.1. If the dynamical timescale were slower for some molecules, say 10 ms or beyond, the forward
minus backward observable could skew positive by single particle means.
To simulate this possibility, I did the hard work of including all substates in a Monte-Carlosimulation, including their di↵erential Zeeman shifts so as to include the influence of the microwave
knife on other substates and other transitions besides the |f, 32 i to |e, 32 i transition directly targeted.
In Fig. 7.13, the di↵erence between forward and backward evaporation ramps for all substates asa function of time is reported in this simulation. Note how molecules persist in |f, 12 i at the final
step, and how there are actually fewer molecules in |f, 32 i with forward evaporation at the last step,at least in this realization of the Monte-Carlo simulation.
After setting up on the JILA computer cluster, I performed a suite of similar simulations
aimed at studying the forward minus backward collision observable, culminating in several panels

174

Figure 7.13: All eight substates of the OH ground state are included in a simulation of evaporation.

of data similar to that shown in Fig. 7.14.

As one would expect, the forward minus backward

observable really starts displaying a strong e↵ect for the largest molecule numbers and the longest
evaporation sequences. At lower molecule numbers, we see exactly the predicted e↵ect- the forward minus backward observable reaches a small but nonzero limiting value of approximately 5%di↵erence.

7.6

Spatial Density Enhancements by Microwave Spectroscopy
This last technique for detecting collisions plays a key role in the final claims of the legitimacyof the observation of some degree of thermalization presented in [88, App. B]. The idea is tocompare normalized microwave spectroscopies in order to determine whether there is any region
close to the center of the trap that ends up with a greater spatial density of molecules than it had

175

Figure 7.14: Contour plot indicating the percentage gain of forward evaporation compared withbackward as a function of evaporation length and molecule number. Slower evaporations with large
initial numbers show the most significant gains. Even very tiny numbers of molecules still show aforward minus backward di↵erence, indicating a small, single-particle contribution to this collisional
indicator.

originally. This is not a necessary condition for either collisions or evaporation, since phase spacedensity increase could happen primarily through the velocity dimensions of phase space. It is also
possible to enhance spatial density at the expense of velocity density, often referred to as adiabatic
compression, but this cannot happen by mistake and requires work to be performed on the ensemble
via the trapping potential. In this sense, the observation of a spatial density enhancement is anindisputable observable of elastic collisions in an ensemble.
A normalized spectroscopically determined density enhancement is indeed observed for theregion near 500 G in both forward and backward evaporation as evident in Fig. 7.12. Nevertheless,a few additional caveats should be kept in mind, as well as those in Sec. 6.2.4. First of all, it
would be ideal if a way could be found to eliminate some of the complexities of the normalization
procedure. To briefly recapitulate, depletion spectroscopy transfers only a fraction of molecules
to the lower parity state at a specific magnetic field value, so we integrate the total area enclosed

176
by the spectroscopy curve, which is scaled according to the observed total population by laser
induced fluorescence. This is necessary because depletion spectroscopy is performed with a trainof short microwave pulses lasting over a total time of about a quarter of a trap oscillation, so thatmolecules are not at all frozen in place. Relative to a very brief spectroscopy pulse that would
only deplete molecules in a given region at that particular instant, the use of a train of pulses overa longer period of time allows us to sample molecules more widely to boost the signal to noise
ratio of spectroscopy. The spectroscopy gives a value that is proportional to the true instantaneous
population in a specific magnetic field region, but with a scaling factor that allows the signal to beconstrained with the measured total number of molecules in the trap. Nevertheless, in any future
experiments, I would submit that claims of absolute density enhancement should be confirmed bymeans of a single transfer ARP spanning the low magnetic field region, and thus serving as a direct
witness on the spatial density in that region at that time, with no need to perform any careful
accounting of the total molecule distribution function.
Moreover, any future demonstration of spatial density enhancement would do well to include
additional observables to support the claim. One of these may be had through the use of resonantly
enhanced multi-photon ionization of the molecules, as pioneered for the use of measuring population
distributions recently in the Lewandowski lab [158]. The resonant process detects a small subregionof the trap, whose position may be tuned in order to access the population distribution function.

7.7

Conclusions
The detection of collisions is a challenging task, especially in the absence of a background-freeexit channel as for the collisions between OH molecules discussed in this chapter. Several methods
have been carefully investigated with regard to their potential for systematic e↵ects when employedin the detection of small e↵ects. With the possibility of enhanced molecule numbers in the near
future, these methods should prove very relevant for elucidating the full collisional story of the next
generation of trapped OH radicals.

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