The Laser Cooling and Magneto-Optical Trapping of the
YO MoleculebyMark Yeo
B.S., University of Michigan, 2006

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
2015This thesis entitled:
The Laser Cooling and Magneto-Optical Trapping of the YO Molecule
written by Mark Yeo
has been approved for the Department of Physics

Prof. Jun Ye

Prof. Deborah S. Jin

Date

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

iii
Yeo, Mark (Ph.D., Physics)
The Laser Cooling and Magneto-Optical Trapping of the YO Molecule
Thesis directed by Prof. Jun Ye

Laser cooling and magneto-optical trapping of neutral atoms has revolutionized thefield of atomic physics by providing an elegant and efficient method to produce cold dense
samples of ultracold atoms. Molecules, with their strong anisotropic dipolar interaction
promises to unlock even richer phenomenon. However, due to their additional vibrationaland rotational degrees of freedom, laser cooling techniques have only been extended to asmall set of diatomic molecules.
In this thesis, we demonstrate the first magneto-optical trapping of a diatomic moleculeusing a quasi-cycling transition and an oscillating quadrupole magnetic field. The transverse
temperature of a cryogenically produced YO beam was reduced from 25 mK to 10 mK via
doppler cooling and further reduced to 2 mK with the addition of magneto-optical trapping
forces.
The optical cycling in YO is complicated by the presence of an intermediate electronic
state, as decays through this state lead to optical pumping into dark rotational states. Thus,
we also demonstrate the mixing of rotational states in the ground electronic state using
microwave radiation. This technique greatly enhances optical cycling, leading to a factor of 4
increase in the YO beam fluorescence and is used in conjunction with a frequency modulated
and chirped continuous wave laser to longitudinally slow the YO beam. We generate YO
molecules below 10 m/s that are directly loadable into a three-dimensional magneto-optical
trap. This mixing technique provides an alternative to maintaining rotational closure andshould extend laser cooling to a larger set of molecules.

Dedication

To Teow.

Acknowledgements

There are many people who made this work possible. First and foremost is Jun Ye
for taking me on in his group and providing such a great science environment and providing
support at every level.
I also had great team working on the YO experiment. Matt Hummon was the senior
post-doc and has overseen the experiment from the empty optics table phase to what it
is today. Yong Xia was instrumental in the initial building of the experiment. Alejandra
Collopy, was instrumental in building up the 3D MOT apparatus. Bo Yan is a magician withthe lasers. Also I must thank visiting students Max Kuhnert from TU Wien and Maurice
Petzold from the University of Hanover. Also I would like to welcome Yewei Wu who has justjoined the crew in recent months and with Alejandra, will take this experiment to dizzying
heights.
This experiment was done in collaboration with John Doyle’s group at Harvard University. Their group shared their expertise which let us build our own advanced cryogenic
buffer gas beam source. Thanks to Boerge Hemmerling, Chae Eunmi, Garrett Drayna,
Aakash Ravi, David Patterson, Hsin-I Lu and John Doyle for all their help and for letting
me visit their lab..
In my time at JILA, I also spent time on the OH stark deceleration experiment and I
must thank the great people I worked with on this project, Brian Sawyer, Benjamin Stuhl,
Benjamin Lev, Dajun Wang, Hao Wu, David Reens and Timothy Langen.
The Ye group has been a great font of knowledge and great friendships, whether it issharing equipment or more importantly sharing knowledge, I have never wanted in either

vi
aspect from this great group. In particular I must thank Florian Adler and Kevin Cossell for
their help in setting up the frequency comb. Thanks to everyone who provided great advice
and letting me borrow stuff at various points.
I must also thank the great support staff here in JILA. Thanks to the great machinists,
Hans Green, Todd Asnicar, Blaine Horner, Kim Hagen, David Alchenberger and Kels Detra.
Thanks to the electronics shop staff, Terry Brown, Carl Sauer, Chris Ho, James Fung-A-Fat,
Mike Whitmore and Paul Beckingham. Also, I must say thanks to JILA computing for
maintaining our computer systems, especially J.R. Raith for fixing countless computers andbeing a great concert buddy.
I also want to thank Chris Monroe and the group at the University of Michigan (nowat JQI), for giving me my first experience in cutting edge AMO research.
Outside of JILA, I made many friends but in particular I must shout out to the mostaptly named band, Cryogen (please see Chapter 4) for the time I played in the band and forletting me be a member of the extended family. It was a great experience recording CDs andplaying shows all over Colorado. Thanks go to Brian Sawyer (again) for recruiting me into
the band and for inspiring my love for the more extreme side of metal. Ryan “Darkness”
Conner, vocalist and recording extraordinaire. for making my drums sound like trapped
thunder and all of my other bandmates, Mike Papajohn, Greg Burgess, Chris Gugglioti and
Aaron Lande, my predecessor. The band is no more but I hope our legacy lives on.
Back in Singapore, my parents, Khee Leng and Margaret, raised me in an environmentthat always emphasized the importance of learning and gave me all the opportunities topursue my dreams. Thanks to my brother Shaun for all the good times growing up wrestlingand bowling and is himself graduating in a month.
Here in Colorado, my wife, Teow Lim Goh has been my rock from the very beginning,
thank you for all the support and understanding. Also thanks to my two personal assistantcats, Gingertoes and DaFuzz.

Contents

Chapter
1 Introduction1

1.1

Why polar molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Molecule production techniques . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

Laser cooling of molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.4

Why YO? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 YO molecular structure
2.1

6

Angular Momentum Coupling . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1.1

Case (a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.1.2

Case (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2Electronic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.3

Hyperfine and Zeeman structure of the ground state . . . . . . . . . . . . . .

12

2.4

Hyperfine and Zeeman Structure of the A2 Π1/2 state. . . . . . . . . . . . . .

18

2.5

Vibrational structure of YO . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.6

Rotational Closure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.7

Calculating branching ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.7.1

Converting the X 2 Σ+ state to the Hund’s case (a) basis. . . . . . . .

27

2.7.2

Calculating dipole moments. . . . . . . . . . . . . . . . . . . . . . . .

30

2.7.3

X 2 Σ+ , N = 1 ← A2 Π1/2 , J = 1/2, E1 transition. . . . . . . . . . . . .

32

viii
2.7.4

X 2 Σ, N = 0, 2 ←A02 ∆3/2 , J = 1/2, E1 transition. . . . . . . . . . . . .

34

2.7.5

X 2 Σ, N = 0, 2 ← A02 Π1 /2, J 0 = 1/2, M1 transition. . . . . . . . . . .

37

3 Laser setup.

41

3.1

An introduction to frequency combs . . . . . . . . . . . . . . . . . . . . . . .

41

3.2

Erbium Doped Fiber Frequency Comb . . . . . . . . . . . . . . . . . . . . .

43

3.2.1

Octave spanning supercontinuum . . . . . . . . . . . . . . . . . . . .

48

3.2.2

Measuring fCO with an f-2f interferometer . . . . . . . . . . . . . . .

52

Light for the main cooling transition at 614 nm . . . . . . . . . . . . . . . .

54

3.3.1

Dye Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

3.3.2

Raman Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.3.3

Stabilization of 614 nm lasers. . . . . . . . . . . . . . . . . . . . . . .

57

3.4

Vibrational repumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

3.5

Frequency to Voltage Convertor . . . . . . . . . . . . . . . . . . . . . . . . .

63

3.6

Frequency scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

3.7

Putting them together. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

3.3

4 Generating a Cold YO Beam

69

4.1

The Cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

4.2

The Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

4.2.1

Gas flow regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

4.2.2

Slower Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

4.2.3

In-cell buffer gas density . . . . . . . . . . . . . . . . . . . . . . . . .

77

Charcoal Cryopumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

4.3

5 Magneto-Optical Trapping of YO
5.1

81

Introduction to Laser Cooling . . . . . . . . . . . . . . . . . . . . . . . . . .

81

5.1.1

83

2 Level Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

5.2

5.1.2

MOT Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

5.1.3

Doppler Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . .

87

A 2D MOT for molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

5.2.1

Recap of YO structure . . . . . . . . . . . . . . . . . . . . . . . . . .

88

5.2.2

Beam Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

5.2.3

Polarization chopping and Doppler cooling . . . . . . . . . . . . . . .

92

5.2.4

Doppler cooling of YO . . . . . . . . . . . . . . . . . . . . . . . . . .

92

5.2.5

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

98

5.2.6

A YO MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

6 Microwave remixing and longitudinal slowing.
6.1

6.2

105

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.1.1

Capture Velocity Estimation . . . . . . . . . . . . . . . . . . . . . . . 105

6.1.2

MOT Loading in a cell . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Microwave mixing and slowing . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.2.2

Optical cycling enhancement via microwave mixing. . . . . . . . . . . 119

6.2.3

Radiation pressure slowing . . . . . . . . . . . . . . . . . . . . . . . . 122

7 Prospects for a 3D YO MOT.

125

7.1

3D MOT Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.2

A narrow line YO MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.2.1

The A02 ∆3/2 state’s linewidth . . . . . . . . . . . . . . . . . . . . . . 129

7.3

Narrow-line cooling on the X 2 Σ+ to A02 ∆3/2 transition . . . . . . . . . . . . . 132

7.4

Narrow-line MOT simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

xBibliography

137

Appendices150

1 He-YO collisions simulations

151

1.1

Simulation Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

2 Beam Source Machine Drawings

157

3 Electronic Schematics

159

TablesTable
2.1

Rotational and hyperfine constants for X 2 Σ+ state. . . . . . . . . . . . . . .

12

2.2

Equlibrium rovibrational molecular parameter . . . . . . . . . . . . . . . . .

21

2.3

Calculated Franck Condon Factors. . . . . . . . . . . . . . . . . . . . . . . .

22

2.4

X 2 Σ+ ← A2 Π1/2 branching ratios for E1 transitions. . . . . . . . . . . . . . .

33

2.5

X 2 Σ+ ← A2 Π1/2 branching ratios for E1 transitions summed over MF andaveraged over MF0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

2.6

X 2 Σ+ , N = 0 ← A02 ∆3/2 branching ratios for E1 transitions. . . . . . . . . .

35

2.7

X 2 Σ+ , N = 2 ← A02 ∆3/2 branching ratios for E1 transitions. . . . . . . . . .

36

2.8

X 2 Σ+ ← A02 ∆3/2 branching ratios for E1 transitions summed over MF andaveraged over MF0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

X 2 Σ+ , N = 0 ← A2 Π1/2 branching ratios for M1 transitions. . . . . . . . . .

38

2.10 X 2 Σ+ , N = 2 ← A2 Π1/2 branching ratios for M1 transitions. . . . . . . . . .

39

2.9

2.11 X 2 Σ+ ← A2 Π1/2 branching ratios for M1 transitions summed over MF andaveraged over MF0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1

Forward beam velocities in effusive and supersonic regime compared to measured velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1

40

76

Narrow line transition frequencies. . . . . . . . . . . . . . . . . . . . . . . . . 130

xii
7.2

Comparison of characteristic laser cooling temperature between the main optical cycling and the narrow line transition. . . . . . . . . . . . . . . . . . . . 131

7.3

Vibrational branching ratios form A02 ∆3/2 , v 0 = 0. . . . . . . . . . . . . . . . 131

A1

2nd stage leak area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Figures

Figure2.1

Hund’s Case (a) coupling scheme. . . . . . . . . . . . . . . . . . . . . . . . .

10

2.2

Hund’s Case (b) coupling scheme. . . . . . . . . . . . . . . . . . . . . . . . .

11

2.3

X 2 Σ+ Hyperfine structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.4

X 2 Σ+ Zeeman Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.5

Vibrational Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.6

Vibrational wave functions. . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.7

Rotational Closure Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3.1

Illustration of a mode-locked laser. . . . . . . . . . . . . . . . . . . . . . . .

42

3.2

Er3+ energy diagram.

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

44

3.3

Frequency comb oscillator schematic and spectrum . . . . . . . . . . . . . .

46

3.4

frep stabilization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.5

Super-continuum generation schematic. . . . . . . . . . . . . . . . . . . . . .

49

3.6

Super-continuum spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

3.7

fco detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.8

fco stabilization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

3.9

Photograph of dye laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.10 Heterodyne beat generation. . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

3.11 Dye to Raman Lock. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

xiv
3.12 Littrow configuration ECDL mount and in-loop linewidth. . . . . . . . . . .

62

3.13 Laser-comb heterodyne beat generation. . . . . . . . . . . . . . . . . . . . .

64

3.14 Feed-forward schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

3.15 Longitudinal slowing laser setup. . . . . . . . . . . . . . . . . . . . . . . . .

68

4.1

Schematic of YO cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

4.2

2 stage cell realization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4.3

YO forward velocity distribution. . . . . . . . . . . . . . . . . . . . . . . . .

80

5.1

Optical Molasses force on 2-level system. . . . . . . . . . . . . . . . . . . . .

86

5.2

Type I vs Type II MOT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

5.3

2D MOT laser cooling structure. . . . . . . . . . . . . . . . . . . . . . . . . .

89

5.4

2D MOT apparatus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

5.5

Final temperature vs polarization chopping rate. . . . . . . . . . . . . . . . .

95

5.6

Optical Molasses force for YO system. . . . . . . . . . . . . . . . . . . . . .

96

5.7

Doppler cooling of YO beam in 2D..

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

97

5.8

Polarization chopping schematic. . . . . . . . . . . . . . . . . . . . . . . . .

98

5.9

Photo and circuit diagram for resonant Baseball coils. . . . . . . . . . . . . . 102

5.10 1-D MOT and Doppler Cooling. . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.11 1D and 2D MOT comparisons. . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.1

MOT capture simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2

Fluorescence signal for three bugger gas density regimes. . . . . . . . . . . . 110

6.3

Trap lifetime as a function of He collision rate. . . . . . . . . . . . . . . . . . 111

6.4

YO Level Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.5

Slowing Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.6

Loss rate measurement, microwave enhancement . . . . . . . . . . . . . . . . 121

6.7

Slowing results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

xv
7.1

3D MOT apparatus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.2

MOT Imaging setup and image of YO molecules. . . . . . . . . . . . . . . . 128

7.3

Narrow line MOT scheme based on the X 2 Σ+ → A02 ∆3/2 transition. . . . . . 135

7.4

Narrow line MOT simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . 136

A1

MOT forces simulation without He collisions.

A2

Diffusion simulation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

A3

Rotational state thermalization simulation. . . . . . . . . . . . . . . . . . . . 156

A4

2 stage cell machine drawing. . . . . . . . . . . . . . . . . . . . . . . . . . . 157

A5

YJ132 high speed frequency to voltage convertor. . . . . . . . . . . . . . . . 160. . . . . . . . . . . . . . . . . 153

Chapter 1
Introduction

Laser cooling and trapping is one of the great cornerstones of modern atomic physics.
These techniques have facilitated the efficient creation of cold and dense samples of atoms [1]and have been extended to about twenty atomic species. This abundance of species makes
ultracold atomic systems a powerful tool for studying various phenomena, from quantumdegenerate gases [2, 3], physics beyond the Standard Model [4], and strongly correlated
systems [5], to applications in quantum information [6, 7] and simulation [8, 9], quantum
sensing [10], and ultraprecise optical clocks [11, 12].
Polar molecules have additional features over atoms, most notably the permanent bodyfixed electric dipole moment. This leads to a range of tantalizing applications that have motivated the demonstration of several cold and ultracold molecular production techniques.
1.1

Why polar molecules
Polar molecules possess a strong anisotropic interaction potential due to their body

fixed electric-dipole moment. This leads to the possibility of controlling collision dynamics [13, 14] and cold and ultracold chemical reaction rates [15–17]. Electric field controllable
cross sections have been shown in collisions between magnetically trapped OH with 5 K ND3
molecules [18]. Here, a 40% increase in collision cross section was observed upon the application of an electric field large enough to polarize both species. In addition, the inelastic collision cross section for magnetically trapped OH was shown to exhibit a quadratic dependence

2
on their effective dipole moment [19]. Also, experiments in the ultracold KRb have shown
electric field controllable chemical reaction rates in the single partial wave regime [20, 21].
Cold molecules have found great utility in precision measurements. Due to the enormous internal electric fields in heavy polar molecules on the order of 100 GV/cm, hugeenhancements in constraining the electron electric dipole moment can be gained [22, 23].
This has been exploited to improve measurements of the electron electric dipole moment in
ThO [24] and YbF [25] and have placed further constraints on physics beyond the Standard
Model. In addition, the rich internal structure of molecules promises improvements in measurements of physical constants such as the proton to electron mass ratio and time variationsof the fine-structure constant [26–28].
Due to the dipolar interaction, polar molecules have been identified as potential qubits [29–31] and promise to give rise to novel many-body quantum systems and to further advancecondensed matter simulations. [32, 33]
1.2

Molecule production techniques
The techniques used to generate cold and ultracold molecules are generally separated

into “indirect” and “direct” methods [34]. Indirect methods have come to mean the assemblyof pre-cooled atoms into molecules using the methods of photoassociation [35,36] or magnetoassociation via a Feshbach resonance [37]. To date the 40 K89 Rb samples formed by magnetoassociation and stimulated Raman adiabatic passage (STIRAP) [37,38] is the only such polar
molecular with sub µK temperatures. However, there has been much effort to extend thesetechniques to other molecules including RbCs [39, 40], NaK [41], LiCs [42], LiYb [43, 44] andNaCs [45]. A clear limitation to this class of methods is that the constituent atoms arelimited to readily laser cooled ones, thus limiting the chemical diversity of the molecules.
The so called direct methods take preexisting molecules and cool them down. Suchmolecules can be initially produced via photodissociation, electric arc discharge, laser ablation or even straight out of a gas bottle. The first step involves forming a molecular beam for3which two methods have gained prominence. The first is the supersonic expansion [46, 47],
where the internal degrees of freedom of the molecule are cooled at the expense of a highforward velocity (several 100 m/s). The second is buffer gas cooling [48], which was refinedin the Doyle group in Harvard [49]. Here, the molecule species is thermalized via a helium
buffer gas to a cold cell at a temperature on the order of 1 K. Beams with velocities of50-200 m/s, depending on source configuration [49], are routinely produced.
One of the most prominent techniques to further slow these beams is Stark deceleration [46, 50]. Weak electric field seeking molecules ride up a potential hill generated froma pair of electrodes. Fields are switched off before the molecules cross the other side of
the hill, which removes kinetic energy. In addition to crossed beam experiments [51], thistechnique can decelerate molecules all the way to rest and can therefore be used to loadconservative traps [50]. Zeeman deceleration using switched magnetic fields have also been
demonstrated [52, 53]. Such systems have been used to load conservative traps, and havebeen used to probe both atom-molecule and atom-atom collisions [18, 50, 54–56] as well as
measuring diatomic molecular properties [57, 58].
These deceleration and trapping techniques are conservative implying that phase spacedensity cannot be increased. However, recent work, using the rf evaporation of OH [59, 60]
has shown evidence of increasing the phase space density by several orders of magnitude. In
addition, the polyatomic CH3 F molecule was cooled down to 29 mK using a Sisyphus-type
cooling scheme on a vibrational transition [61, 62].
1.3

Laser cooling of molecules
Extending laser cooling to polar molecules was known to be very difficult due to theplethora of dark rovibrational branching paths that would stop the required optical cycling [63]. While there was some initial optimism in the idea of cavity-mediated laser cooling [64], where the Purcell effect [65] would enhance the coherent Rayleigh scattering for
only the cooling transitions, it turned out that collective effects were also required to ensure

4
optical cycling [66], this however requires a prohibitively high initial phase space density toinitiate and therefore still remains a dream.
However, it turns out that there is a way to free-space laser cool polar molecules. Di
Rosa was able to identify several laser cooling candidates,for example CaH, based on theirvery diagonal vibrational branching [67]. A theoretical proposal in our group using a socalled Type II MOT transition [68] further showed the way for the closure of rotational
branching. This technique uses a transition whose total angular momentum J in the groundstate is larger than in the excited state, i.e. J 00 → J 0 = J 00 − 1, which is the opposite oftypical atomic laser cooling transitions [1]. This eliminates rotational branching because theelectric dipole selection rule, ∆J = ±1 will only allow decays back to the original rotational
state. This idea inspired the DeMille group at Yale to perform the first demonstration of
optical cycling [69] and doppler cooling [70] of a polar molecule, SrF. The first magnetooptical trap was demonstrated on the YO molecule in our group [71] on a beam of YO. The
full realization of a full three dimensional MOT for SrF was realized very recently in [72]and a further improvement on the SrF MOT, based on a MOT beam polarization scheme
derived from a detailed calculation by Mike Tarbutt [73] was realized [74].
In addition, the advent of optical cycling in polar molecules allows for radiation pressure
slowing in molecules which was demonstrated in Hind’s group at Imperial college whichslowed the longidtudinal velocity of CaF by 30 m/s [75]. For SrF, broadband laser radiation
was used to slow molecules to be loaded directly into a 3D MOT [72].
1.4

Why YO?
The Yttrium Monoxide (YO) molecule was chosen for several reasons. The X 2 Σ+ →A2 Π1/2 transition is strong, with a linewidth of Γ = 2π × 4.8 × 106 MHz. YO also has very
diagonal vibrational branching, such that addressing the 3 lowest vibrational levels allows
for more than 106 photon scatters, which is sufficient for demonstrating MOT action. The
electronic structure is also favorable for shutting down rotational branching as described5
above.
In addition, as YO has been detected in main sequence stars including the sun [76,77], much of its molecular structure is well characterized [77–88]. YO has an intermediate
electronic state between the two states used for optical cycling. In this thesis we found amethod to circumvent the losses due to branching through this state [89] but this intermediate
state may also be used to create a narrow line MOT, which could lead to temperatures 100
times smaller than previously observed [90] in laser cooled molecules.
The remainder of the thesis is structured as follows: In Chapter 2 we will describe thesalient molecular structure for the laser cooling of YO. Chapter 3 describes the various laser
systems used to address the complicated YO structure, and the stabilization scheme to a self
referenced frequency comb. Chapter 4 describes the 2 stage cryogenic buffer gas YO beamsource that was developed in collaboration with the Doyle group at Harvard University. In
Chapter 5 we present the first magneto-optical trap (MOT) for a molecule [71]. Here, the YO
beam is doppler cooled and confined transversely by MOT forces. Chapter 6 demonstrates
the microwave mixing of rotational states in the electronic ground state. This techniqueaddresses losses to dark rotational states due to decays through the intermediate A02 ∆3/2
state and allows us to radiation pressure slow YO to 10 m/s, which is loadable into a 3D
MOT [89]. The microwave mixing presents an alternative method to close off rotational
branching and opens the door to the laser cooling and trapping of a larger set of molecules.
Chapter 7 discusses the next technical steps to demonstrating a 3D MOT in YO as as well
as discussing the prospects for the narrow-line MOT.

Chapter 2
YO molecular structure

In this chapter, we will present some general concepts in diatomic molecular structureand specify this for the YO molecule. We begin with the relevant Hund’s angular momentum
coupling cases for the three electronic states for YO that are engaged in the optical cycling.
We will then derive the hyperfine structure in the ground state of YO as well as its Zeeman
structure. Then we will calculate the vibrational structure and show how this is favourable forlaser cooling. With these pieces in place, we describe the method by which rotational closure
is maintained during the optical cycling. Finally we present branching ratio calculations forvarious transitions that which will be used in later chapters.
2.1

Angular Momentum Coupling
Diatomic molecules with their rotational degrees of freedom have far more complicatedangular momentum structure than atomic species. In addition, the internuclear axis breaks
the spherical symmetry and thus the good quantum numbers are different from that of
atoms. To help understand this mess of angular momenta coupling, Friedrich Hund cameup with idealized angular coupling schemes [91] now known as the Hund’s coupling cases.
We will present the two most common coupling cases, which also describe the states of the
YO molecule that are encountered during optical cycling.

7
2.1.1

Case (a)In Hund’s case (a), the electron orbital angular momentum L is strongly coupled tothe internuclear axis as shown in Fig. 2.1. The electron spin S is in turn strongly coupled
to L via spin-orbit coupling. Their projection onto the internuclear axis are denoted as Λand Σ and the sum of these two components is denoted as Ω. The rotation of the nuclei isdenoted by the vector R and couples to Ω to form the total angular momentum J.
As L may precess about the internuclear axis in a clockwise or anti-clockwise sense,
the projection can take on values of ±Λ. This leads to a two-fold degeneracy for a given
value of |Ω| that is known as Λ-type doubling and this degeneracy is lifted due to molecularrotation. The good quantum numbers for a case (a) state can be summed in ket notation as|η, Λ; S, Σ; J, Ω, MJ i

(2.1)

where η denotes the electronic, vibrational or any other quantum number required to define
the molecular state, MJ is the projection of J on some space fixed axis and is used when
considering the effects of external fields on the molecule.
The rotational energy is given by
Hrot = BR2 = B(J − L − S)2(2.2)

where B is the rotational constant.
We note that case (a) is a good approximation for states where the spin-orbit couplingis much larger than the rotational energy, i.e. γSO Λ  BJ where γSO is the spin-orbit
coupling constant. The relative rotational energies to first order have the form
Erot = BJ(J + 1)

(2.3)

where B can expressed as
B=~2
2
2µRnuc(2.4)

8where Rnuc is the internuclear separation and µ is the reduced mass given by
µ=

m1 m2
m1 + m2(2.5)

and m1 and m2 are the masses of the two nuclei. For a given electronic state, the lowest
rotational level has J = Ω.
States that are well approximated in this coupling scheme are labelled as 2S+1 Λ|Ω| wherethe value of Λ is denoted by the Greek letters Σ, Π, ∆, Φ... for |Λ| = 0, 1, 2, 3.... Note that
there is an unfortunate naming degeneracy between the electron spin projection quantum
number and |Λ| = 0 but the context should always make it clear which is which.

2.1.2

Case (b)Hund’s case (b) occurs most often when Λ = 0. In such a case, S is not coupled tothe internuclear axis as spin-orbit coupling is zero. In some light molecules, for Λ 6= 0 the
spin-orbit coupling is sufficiently weak that S is not coupled to the internuclear axis andsuch cases are also appropriately described by this coupling scheme. The angular momenta
couple as follows, Λ couples to R to form N. For Λ = 0 (which is indeed the case for groundstate of YO), N and R are equivalent and as such the former can be correctly referred as therotational angular momentum. N then couples to S to form J, the total angular momentum.
Basis functions are written as|η, Λ; N, S; J, MJ i

(2.6)

and the rotational Hamiltonian is given by
Hrot = BR2 = B(N − L)2 .

(2.7)

Relative rotational energies are given by
Erot = BN (N + 1)

(2.8)

9
States with Λ = 0 are denoted by the spectroscopic symbol 2S+1 Σ± where ± denotes theparity of rotational states with even N with the parity alternating with each successive
rotational level.
Additional Hund’s cases, i.e. (c)-(e) coupling cases are presented in section of Brown
and Carrington [92], and in [91] but are outside the scope of this work.
2.2

Electronic Structure
The YO molecule has one abundant naturally occurring isotopologue, 89 Y 16 O and has

three electronic states of interest, the ground X 2 Σ+ state, the excited A2 Π1/2 state and themetastable intermediate state A02 ∆3/2 . For the ground state, Λ = 0 hence it is well describedwith the Hund’s case (b) coupling scheme. Rotational states with even (odd) N have even(odd) parity.
The A2 Π1/2 (|Λ| = 1|) and A02 ∆3/2 (|Λ| = 2|) are good Hund’s case (a) states. The
quasi-cycling laser transition occurs between the lowest vibrational levels of the X 2 Σ+ andA2 Π1/2 states where the transition energy is 488.492 THz or 614 nm. The radiative lifetimeof the A2 Π1/2 state is γ −1 = 33 ns [93], which allows for fast optical cycling. As we shallsee in chapter 6, on out of every ≈3000 optical cycles, the molecule falls into the A02 ∆3/2
intermediate state and removes itself from the optical cycling, which is a most undesirable
outcome. In chapter 6, we will describe how to avoid this loss via microwave rotation state
remixing.
While the ground state is a Hund’s case (b), it is a pathological case because its
hyperfine splitting is much larger than the spin rotation coupling. We will treat this in thefollowing section.

10

J
Λ

Σ

R

S

L
Ω

Figure 2.1: Angular momentum coupling diagram for Hund’s case (a)

11

S
J

N
Λ

R

L
Figure 2.2: Angular momentum coupling diagram for Hund’s case (b)

12
2.3

Hyperfine and Zeeman structure of the ground state
The Frosch-Foley effective Hamiltonian [94] for the YO ground state is given by

Hds = BN2 − DN4 + γSO S · N + bI · S + cIz Sz + CI I · N(2.9)

where B is the rotational constant, D is the centrifugal distortion constant, γSO is the
spin-rotation coupling constant, and CI the nuclear spin rotation term. The constants b
and c are related to the electron-nuclear dipole and Fermi contact interactions and will be
discussed in further detail shortly. The numerical values of these constants are tabulated inTable 2.1
Parameter

v=0

v=1

v=2

v=3

v=4

B [GHz]11.6336(2)11.5818(3)11.5302(4)11.4782(5)11.4258(5)D [kHz]9.581(15)9.533(54)9.69(13)9.71(15)

9.67(15)

γ0 [MHz]−9.2254(1)−7.8465(1)−6.3256(1)−4.9905(3)−3.514b0 [MHz]−762.976(2)−763.054(2)−763.104(2)−763.118(4)c [MHz]−28.236(5)−28.349(2)−28.461(5)−28.586(8)CI [MHz]

2.57(3) × 10−3

2.30(3) × 10−3

1.83(7) × 10−3

γ1 [MHz]1.7738(5) × 10−4

1.8288(4) × 10−4

1.8858(7) × 10−4

b1 [MHz]−3.09(5) × 10−5

−2.76(4) ∗ 10−5

−2.34(5) × 10−5γ2 [MHz]5.96(6) × 10−10

6.07(6) × 10−106.49(8) × 10−10

1.946(5) ∗ 10−4

Table 2.1: Rotational [81]and hyperfine constants [95] for the X 2 Σ+ state of YO.

The rotational energy levels are given by
Erot = BN (N + 1) − D(N (N + 1))2 .

(2.10)

Note that this includes the small centrifugal distortion. The spin-rotation (γSO ) and the

13
I · S term b do also have small rotational dependence given by
γSO = γ0 + γ1 N (N + 1) + γ2 N 2 (N + 1)2
b = b0 + b1 N (N + 1)

(2.11)(2.12)(2.13)

To interpret the meaning of the coupling terms b and c, Brown and Carrington [92]gives the following expressions 8π
1hη, Λ| δ(r) |η, Λi − hη, Λ| 3(cos2 θ − 1) |η, Λib = gS µB gN µN (µ0 /4π)323c = gS µB gN µN (µ0 /4π) hη, Λ| 3(cos2 θ − 1) |η, Λi .
2

(2.14)

here gs = 2.0023 is the absolute value of the electron g-factor, µB is the Bohr magneton,gN and µN are the nuclear g-factor and nuclear magneton respectively and θ is the relative
angle between the nuclear and electron spins. We thus see that c is proportional to theelectron-nuclear dipolar interaction and b is a linear combination of c and the Fermi contact
interaction. Thus, we define two new parameters: bF , the Fermi contact term and t, theaxial dipolar hyperfine term as
bF = b +t=c3(2.15)c3(2.16)The effective Hamiltonian can now be rewritten in tensor notation as
Hef f = γSO S · N + bF T 1 (I) · T 1 (S) −√30tT 2 (C) · T 2 (S, I)(2.17)

While a ”typical” Hund’s case (b) case molecule first couples S and N to form J via
spin orbit coupling, our X 2 Σ+ state has an unusual angular momentum coupling scheme dueto the large Fermi contact interaction. First, I and S couple to form the angular momentum
G. Then G couples with N to form the total angular momentum F. The angular momentum
states are written as |((I, S)G, N )F i and this is known as the bβS basis.

14
Section 10.3.1 pages 745-749 of [92] gives the matrix elements for each of the terms of
the Hamiltonian. The electron spin-rotation is given by
hI, S, G, N, F, MF | γSO T1 (S) · T1 (N ) |I 0 , S 0 , G0 , N 0 , F 0 , MF0 i=γSO δ(F, F 0 )δ(MF , MF0 )δ(N, N 0 )0(−1)G +F +N ((2G0 + 1)(2G + 1)) N 0 G0 F 
1/2

 G

(−1)G+I+1+S



 G

S



I 

N


1 

(S(S + 1)(2S + 1)N (N + 1)(2N + 1))1/2(2.18) S G0 1 
the Fermi-contact interaction is independent of N
hI, S, G, N, F, MF | bF T1 (I) · T1 (S) |I 0 , S 0 , G0 , N 0 , F 0 , MF0 i S I G 
I+G+S=bF (−1)(I(I + 1)(2I + 1)S(S + 1)(2S + 1))1/2
 I S 1 

(2.19)and the most complicated term, the Dipolar Hyperfine Interaction,√hI, S, G, N, F, MF | 6gs µB gN µN (µ0 /4π)T2 (S, I) · T2 (C) |I 0 , S 0 , G0 , N 0 , F 0 , MF0 i√=t 30δ(F, F 0 )δMF , MF0 N 0 G0 F 
G0 +F +N(−1)

 G N 2 
1/2((2G0 + 1)(2G + 1)I(I + 1)(2I + 1)S(S + 1)(2S + 1))





0


G G 2 




0


1/2  N 2 N (−1)N ((2N + 1)(2N 0 + 1)) 

I
I
1




0
0
0



 S S 1 

(2.20)

In zero field and ignoring matrix elements off diagonal in N , we obtain the reduced
Hamiltonian for each rotational state.

15
0

0

(G , F )(G, F )

(1, N + 1)

(1, N)(0, N)(1,N-1)(1, N+1)

γSO N+ b4F − t 4NN+62

0

0(1, N)

00(0, N)0− γSO+ b4F + 2t2pγSON (N + 1)
2

0
p
γSO
N (N + 1)2− 3b4F0(1, N-1)

0

0

0

+1)
N +1
− γSO (N+ b4F − t 4N2
−2

Diagonalizing Equation 2.17, we obtain the hyperfine structure for N = 0, 1, 2 as
depicted in Fig. 2.3. The large Fermi contact interaction, bF , creates a ∼760 MHz splitting
between the G = 0, 1 states. For N > 0, the coupling of N with G further splits the G = 1
state into states with total angular momentum F = G ± 1.
To calculate the Zeeman structure, we add the following Hamiltonian to Equation.
2.17 by only consider the coupling of electron spin S to the magnetic field Bz (see Brown
and Carrington [92] sections 9.5 and 5.5.4).

HZeeman = h((I, S)G, N ), F, mF | gS µB Bz T01 (S) |((I, S)G0 , N ), F 0 , m0F i


0
1 F 
 F
=gS µB Bz (−1)F −mF −mF 0 m0F



0
0

 G F N 
F +N +1+G(−1)
 F G 1 
 S G0 I 
G+I+1+S(−1)
 G S 1 (S(S + 1)(2S + 1)(2G0 + 1)(2G + 1)(2F 0 + 1)(2F + 1))1/2(2.21)We thus obtain the Zeeman structure as shown in Figure 2.4. A salient feature of the Zeeman
spectrum is that at low fields, the energy splitting for the G = 0 states is almost zero, forexample, at a magnetic field of 1 G the Zeeman splitting between the mF = ±1 states is16579.4 MHzG = 0, F = 2

46.526 GHz

193.4
MHz

N = 2 (-)
F=1

18.7 MHz F = 2

G=1

F=3

7.5 MHz

579.3 MHzG = 0, F = 1

193.3
MHz

N = 1 (-)
F=0

18.8 MHz F = 1
F=2

F=G=0

579.3 MHz

23.263 GHz

G=1

193.1
MHz

N = 0 (+)

F=G=1

Figure 2.3: Hyperfine structure of the 2 Σ1/2 state of YO.

3.5 MHz17

Figure 2.4: Zeeman structure of the X 2 Σ+ , N = 1 state of YO.

18≈30 kHz. Because of this low Larmor frequency, using an off-axis magnetic field to remix
the Zeeman sub-levels [69] becomes dreadfully slow and would limit the maximum optical
cycling rate. We will discuss how we address this issue in Chapter 5.
For the G = 1 states, from Equation 2.21 we can calculate the low-field effective
magnetic moment to be g4s µB mF . This a good approximation for magnetic fields up to
2 G. This is when the Zeeman interaction energies become roughly comparable to the spinrotation (γSO ) and the dipolar interaction (t). From 10 to 50 G, the Zeeman shift is again
close to linear and can be approximated with an effective moment of g2s µB mG where mGis the projection of G on the lab frame. Finally, the third linear regime occurs when the
Zeeman interaction becomes much larger than the Fermi contact bF , i.e for fields greaterthan 1000 G. The effective magnetic moment is that of the unpaired electron and reverts tothe familiar form of gs ms where ms is the lab projection of the electron spin. It is in thisregime where the G = 0 state finally picks up an appreciable Zeeman shift. While we neversubject YO molecules to such strong fields in this work, this regime can be of importance if
we were to attempt magnetic guiding.
2.4

Hyperfine and Zeeman Structure of the A2 Π1/2 state.
The state that we excite to in our optical cycling transition is the A2 Π1/2 , J = 1/2 state.

It is a Hund’s case (a) state with |Λ| = 1, |Σ| = 1/2 and |Ω| = 21 . The lowest rotational state
has J = |Ω| = 12 . J combines with nuclear spin I to form the total angular momentum F.
F can take on values of 0 or 1 but is spectroscopically unresolved. In zero field, the state is
split into two parity states via Λ- type doubling. Mulliken and Christy’s Λ-doubling p and q
parameters [96] were measured in Bernard and Gravina [81] to be p = −0.150343 cm−1 and
q = −0.1331 × 10−3 cm−1 . This yields the zero field lambda doublet splitting frequency of∆νef = −(p + 2q) = 4.51 GHz.

(2.22)

19
A general form for the Λ-type doubling frequency for higher J levels can be found in [97]but is beyond the scope of this thesis.
For Hund’s case (a) states, the effective magnetic moment is given by [98]µef f = µB

(gL Λ + gs Σ)Ω
MF
J(J + 1)

(2.23)

Where gL = 1. For the A2 Π1/2 , J = 1/2 state, Λ = 1 and Σ = −1/2, which implies thatthe first term in the numerator vanishes to first order. However, a small mixing of the
nearby B 2 Σ1/2 state creates a Zeeman shift [73]. This Zeeman shift can be calculated bydiagonalizing the full Zeeman Hamiltonian (Eq. 9.71 of [92]) and turns out to be
10
EZeeman,± = ± (gl0 − gre )µB Bz MF
3(2.24)
0where ± refers to the parity of the Λ-doublet. The Zeeman parameters gl0 and gre depend on
Mulliken and Christy’s Λ-doubling p and q parameters as follows [99], [100]
p
2BΠ
q
0
gre = −
BΠ
gl0 =(2.25)

where BΠ is the rotational constant for the A2 Π1/2 state. Bernard and Gravina measured
this to have value of 0.385785 cm−1 [81] which yields a magnetic moment of −0.065µB mF .
The maximum Zeeman shift in the A2 Π1/2 , J = 1/2 occurs for the F = 1, |mF | = 1 stateand is a factor of 15 smaller than the largest Zeeman in the X 2 Σ+ state.
2.5

Vibrational structure of YO
As was pointed by Di Rosa [67], laser cooling molecules requires very diagonal vibra-

tional branching in order to laser cool. As such, we need to calculate the Franck-Condon
factors, i.e. the vibrational state overlap
qeg = | hψg |ψe i |2

(2.26)

20
where e and g are the vibrational quantum numbers of the excited state and ground state
respectively. The rovibrational energy for YO can be expressed as quadratic in terms of(v + 1/2) [98], i.e.
111
E = ωe (v + ) − ωe xe (v + )2 + Be J(J + 1) + De (J 2 (J + 1)2 ) − αe J(J + 1)(v + ) (2.27)
2
2
2
A method used to calculate the vibrational wave functions from spectroscopically derived
equilibrium rovibrational molecular parameters was developed by Rydberg [101], Klein [102]and Rees [103] and is known as the RKR method. For each vibrational energy U, we cancalculate the classical turning points,1

r1,2 = (f /g + f 2 ) 2 ± f(2.28)

where the parameters f and g in units of cm and cm−1 respectively are given bys l2 − 4mA h√f [cm] =ln(2.29)
8π 2 cµωe xe
l − 4mAs
 l2 − 4mA √
2π 2 µc√] (2.30)g[cm−1 ] =
[α
4mA
+
(2mB
h
+
4mD
J(J
+
1)
−
α
l)
ln
e
e
e
e
h(ωe xe )3
l − 4mA
Here, h, c are expressed in CGS units and the molecular parameters defined in Equation 2.27
are in cm−1 . Here µ is the reduced mass in grams and the parameters A, l and m are definedbyA = h(U − Be J(J + 1) − De (J 2 (J + 1)2 )(2.31)l = ωe − αe J(J + 1)

(2.32)m=ω e xeh(2.33)The molecular parameters were measured by Bernard and Gravina [81] and relevant parameters are tabulated in Table 2.2
We can then calculate the vibrational potential using the RKR method for the X 2 Σ+and A2 Π1/2 states. For this calculation, we step U in increments of 0.01 cm−1 and calculateParameter
Te
ωe
ωe xe
Be
αe
De

X 2 Σ+
0861.472
2.843
0.388920.00173
0.318 × 10−6

A2 Π1/2
16315.57
820.095
3.346
0.38680
0.00201
0.344 × 10−6

21

Table 2.2: Equlibrium rovibrational molecular parameters as measured in [81]. All quantitiesin cm−1 .

the classical turning points r1,2 as defined in Eq. 2.28 and thus obtain the potential curves
shown in Fig.2.5
With the potentials we can then numerically calculate the vibrational eigen-functions.
The time-independent Schrödinger equation is:−

h2 d2
ψ(r) + U (r)ψ(r) = Eψ(r)2µ dr2(2.34)We resample the vibrational potential along a regular grid in r with an increment d =2 × 10−3 Å. We can then approximate Eq. 2.35 with the discrete equation−h2(ψ(r − d) − 2ψ(r) + ψ(r + d)) + U (r)ψ(r) = Eψ(r)2µd2(2.35)

Writing ψ as a column vector whose index indicates r, we can then solve the resulting eigenvalue problem and obtain the vibrational wave functions shown in Fig. 2.6. We can seethat the v = 0 state for the X 2 Σ+ and A2 Π1/2 states are very similar. The calculated
Franck-Condon factors are tabulated in Table 2.3.
The most striking feature is that the q00 = .995, therefore without any vibrational
repumps, the YO molecule can endure 200 optical cycles before falling into higher vibrational
state. Furthermore, by repumping the v = 1 and 2 levels, we can obtain more than 106
scatters, which is sufficient for slowing and trapping.
Since it turns out that J is not a good quantum number in the relevant X 2 Σ+ , N 00 = 1
state (see Section 2.7), we calculate the Franck-Condon factors for J = 1/2 and 3/2 and

q00
q01
q02
q03
q04

J” = 1/2 (RKR)
0.99514.672 × 10−3
2.454 × 10−4
4.282 × 10−7
5.633 × 10−8

J 00 3/2 (RKR)
0.99514.680 × 10−3
2.453 × 10−4
4.279 × 10−7
5.624 × 10−8

22
Morse
.9944
.0038
-

Table 2.3: Columns 1 and 2: Franck-Condon (FC) factors calculated using the RKR method.
Column 3: Franck-Condon factors calculated in [81] using a Morse potential.

we find negligible difference between the two. Also tabulated are Franck-Condon factors
calculated using a crude Morse potential in reference [81]. We note that the q01 FranckCondon calculated by RKR is twice as large as that calculated using a the Morse. This isbecause the high similarity of the two vibrational potentials makes the calculated values ofoff-diagonal Franck-Condon factors very sensitive to the shapes of the vibrational potential.

23

4

−1

Vibrational Potential Energy [cm ]

4.5

x 10

4
3.5
3
2.5
2
1.5

2

Σ

2

1

Π3/2

0.5
0

1.4

1.6

1.8

2
2.2
2.4
2.6
Internuclear distance [Å]2.8

3Figure 2.5: Vibrational energy curves for the 2 Σ and 2 Π1/2 states. The equilibrium internuclear distance for the states are similar, 1.788 Å and 1.793 Å , hence the diagonal FC factors.
The 16000 cm−1 electronic energy offset is not shown for the A2 Π1/2 state.

24

0.015
2

Σ, v = 02

Σ, v = 1

0.01

2

Σ, v = 22

Π1/2, v = 0

ψ

0.005
0
−0.005−0.01−0.015
1.4

1.6

1.8
2
2.2
Internuclear distance [Å]2.4

Figure 2.6: Vibrational wave functions of states used in optical cycling.

25
2.6

Rotational Closure.
As we have seen in section 2.5, by repumping on just the first two vibrational excitedstates, we can close the vibrational branching to better than 1 part in 106 . Multiplying thisby the photon recoil velocity
vrecoil =~k= 6 mm/s
mY O(2.36)

we can see that in the absence of other loss mechanisms, we would be able to slow 6000 m/s
molecules. Hence this is not a limiting factor in our laser cooling. In this section, we will
discuss how we address losses due to rotational branching. This method was inspired byour theoretical proposal [68] and the demonstration of optical cycling in SrF in the Yalegroup [104].
The ground state is a X 2 Σ+ as shown in Fig 2.7. The integer parity of N is also thestate’s parity, indicated by p = ±. In panel (a), we show the way by which YO [71] can bemade to optically cycle. Here, only the first excited rational state (N = 1) is excited to theA2 Π1/2 , J = 1/2 state. The electric dipole (E1) transition selection rules can only link statesof opposite parity and so we may only excite to the p = + in the Λ-type parity doublet. As
we will see in Eq. 2.38a in Subsection 2.7.1, each excited rotational level (N ≥ 0) in theX 2 Σ+ state is a superposition of J = N ± 1/2. Therefore, by the ∆J = 0, ±1 selection rule,the excited state with J 0 = 1/2 may only decay to N < 3. The same E1 parity selection rule
prevents decays to N = 0, 2 and thus only decays to N = 1 are allowed. This closes off therotational branching. Clearly, it would be most desirable to cool on the N = 0 rotational
ground state. However, the same E1 transition selection rules show that rotational closure
is not preserved as shown in Figure 2.7(b). For such a scheme, the X 2 Σ+ , N = 0, + state
will be excited to the A2 Π1/2 , J = 1/2, − state. While the ∆J = 0, ±1 selection rule still
restricts decays to states with N < 3, parity selection populates both the N = 0 and N = 2
levels and thus rotational branching opens up.
Finally, YO has an intermediate electronic state, A02 ∆3/2 . As we will see in Chapter

26

(a)
4.51 GHz

A2Π1/2 ,J' = 1/2+
p(b)4.51 GHz

A2Π1/2 ,J' = 1/2+
pA2Π1/2 ,J' = 1/2
+
p

(c)
2

A' Δ3/2,v' = 0,J' = 3/2 (-)

N'' = 2 (+)

N'' = 2 (+)

N'' = 2 (+)

N'' = 1 (-)
N'' = 0 (+)

N'' = 1 (-)
N'' = 0 (+)

N'' = 1 (-)
N'' = 0 (+)

X2Σ+, v''= 0X2Σ+, v''= 0X2Σ+, v''= 0

Figure 2.7: (a) Depiction of main optical cycling transition. Via angular momentum selection
rules, the YO molecule can only decay back to the X 2 Σ+ , N = 1 state. Hence preventing
branching. (b) Depiction of the X 2 Σ+ , N = 0 non rotationally closed transition. Selection
rules allow decay to both N = 0, 2. The latter decay breaks the optical cycling. (c) Depictionof decay through intermediate electronic state. Selection rules allow decays into N = 0, 2,
which breaks the optical cycling.

27
6, there is a small branching (≈ 3 × 10−4 ) to this state. This is a loss mechanism as the fullcycle starting from and returning to the ground state is a three-photon (1 pump and 2 decay)(Fig .2.7 (c)), thus after this process, the ground state changes parity. On occasion, insteadof decaying back to the X 2 Σ+ , N = 1 state, the molecule relaxes to the A02 ∆3/2 , J = 3/2, −state. The J = 3/2 is the lowest rotational level here and the ∆J selection rules prevents
decays to higher rotational levels. The ∆J selection rules permits decays to states with
N < 4 and from these permitted rotational levels, parity allows decays to only N = 0, 2. In
sum, this process pumps the N = 1 to the N = 0, 2, thus shutting down optical cycling.
2.7

Calculating branching ratios.

2.7.1

Converting the X 2 Σ+ state to the Hund’s case (a) basis.

To calculate branching ratios to each of the hyperfine states in molecules, it is necessary
to convert all the states in question into one Hund’s case. For the YO molecule, thesecalculations will be done in the (a) basis.
However, the X 2 Σ+ state is not only a Hund’s case (b) state but due to the large
Fermi contact, the YO molecule has the unusual bβS angular momentum coupling. For
more conventional Hund’s case bβJ basis, the angular momentum coupling follows the form|(I, (S, N )J)F i. Thus there is a 2 step process to transform to the (a) basis.
First, to transform from the bβS to the bβJ basis, we need to apply Wigner 6-j symbols
as specified in [92], p 156.

|((I, S)G, N )F i =X[(2G + 1)(2J + 1)]1/2 (−1)I+S+F +N
J



 I



S G 


 N F

|(I, (S, N )J)F i

J (2.37)

28
For the N = 1 state, the four hyperfine states are written as
r
 r

2
1
3
1
|G = 0, F = 1i =J = ,F = 1 −J = ,F = 132
32|G = 1, F = 0i = |J = 1/2, F = 0i
r
 r

1
2
3
1
|G = 1, F = 1i =J = ,F = 1 +J = ,F = 1
3
2
3
2

3
|G = 1, F = 2i = J = , F = 22(2.38a)(2.38b)(2.38c)(2.38d).
It turns out that in general, for other values of N,
r

1
1 √
|G = 0, F = N i =N + 1 J = N + ,F = N
2N + 12
√
1− N J = N − ,F = N
2
1|G = 1, F = N − 1i = J = N − , F = N − 1
2
r


√
1
1
|G = 1, F = N i =N J = N + ,F = N
2N + 12√
1+ N + 1 J = N − ,F = N
2
1|G = 1, F = N + 1i = J = N + , F = N + 1
2(2.39a)(2.39b)(2.39c)(2.39d). where for the N = 0 case, only Eq. 2.39 a,d are to be used.
In Hund’s case (a), the quantum numbers are Λ, S, Σ, J and Ω and following from p
230 of [92], the conversion is:|Λ, N, S, Ji =

+S
X





√
N 
 J S
(−1)J−S+Λ 2N + 1  |Λ; S, Σ, J, ΩiΩ −Σ −ΛΣ=−S(2.40)

where the array in parentheses is a Wigner 3-j symbol. We see that from equation 2.39,
for each rotational manifold, we have superpositions of states with J = N ± 21 . Applying29
Eq. 2.40, we get:



1
1 
1
1
1
1 
N, J = N +=√Σ = ,Ω =− Σ = − ,Ω = −
2
2
2
2
2
2




1
1
1
1
1 
1
N, J = N −=√+ Σ = − ,Ω = −Σ = ,Ω =
2
2
2
2
2
2

(2.41a)(2.41b)Finally, this lets us write out the all four of the hyperfine states in the bβS in terms of the abasis.

s1
2(2N + 1)
√1
1
N + 1 J = N + ,Σ = Ω = ,F = N
2
2

√
1
1
− N + 1 J = N + ,Σ = Ω = − ,F = N
2
2

√
1
1
− N J = N − ,Σ = Ω = ,F = N
2
2

√
1
1
− N J = N − ,Σ = Ω = − ,F = N
2
2
r 

1
1
1
J = N − ,Σ = Ω = ,F = N − 1|G = 1, F = N − 1i =
2
2
2

1
1
+ J = N − ,Σ = Ω = − ,F = N − 122s1|G = 1, F = N i =2(2N + 1)√11
N J = N + ,Σ = Ω = ,F = N
2
2

√
1
1
− N J = N + ,Σ = Ω = − ,F = N
2
2

√
1
1
+ N + 1 J = N − ,Σ = Ω = ,F = N
2
2

√
1
1
+ N + 1 J = N − ,Σ = Ω = − ,F = N
2
2
r 

1
1
1
|G = 1, F = N + 1i =J = N + ,Σ = Ω = ,F = N + 1
2
2
2

1
1
− J = N + ,Σ = Ω = − ,F = N + 1.
2
2|G = 0, F = N i =(2.42a)(2.42b)(2.42c)(2.42d)30
Again for the special case of N = 0, we only use Eq. 2.42 a,d. Specifying for N = 1, theoptical cycling ground state,

r

 r

1
1
1
1
3
3
|G = 0, F = 1i =J = ,Σ = Ω = ,F = 1 −J = ,Σ = Ω = − ,F = 1
3
2
2
3
2
2
r
 r

1
1
1
1
1
1
J = ,Σ = Ω = ,F = 1 −J = ,Σ = Ω = − ,F = 1
−
6
2
2
6
2
2
r 


1
1
1
1
1
|G = 1, F = 0i =J = ,Σ = Ω = ,F = 0 + J = ,Σ = Ω = − ,F = 0
2
2
2
2
2
r
 r

3
1
3
1
1
1
|G = 1, F = 1i =J = ,Σ = Ω = ,F = 1 −J = ,Σ = Ω = − ,F = 1
6
2
2
6
2
2
r
 r

1
1
1
1
1
1
+
J = ,Σ = Ω = ,F = 1 +J = ,Σ = Ω = − ,F = 1
3
2
2
3
2
2
r 


3
1
3
1
1
|G = 1, F = 2i =J = , Σ = Ω = , F = 2 − J = , Σ = Ω = − , F = 2 (2.43)
2
2
2
2
2
2.7.2

Calculating dipole moments.

There are three transitions for which we wish to calculate the hyperfine branchingratios. The electric dipole (E1) transition between the A2 Π1/2 , J = 12 state and the X 2 Σ,
N = 1 state; the E1 transition between the A2 ∆3/2 , J = 32 state and the X 2 Σ, N = 1; andthe magnetic dipole (M1) transition between the A2 Π1/2 , J = 12 state and the X 2 Σ, N = 1
state.
We are going to calculate transition dipole moments
hdi = hΨe | Tp1 (d) |Ψg i(2.44)and square them to obtain the branching ratios in question. Here Tp1 (d) is the first rank
tensor denoting the electric or magnetic dipole moment and p denotes the polarization oflight. Following the treatment in section 5.5 of [92], in particular, applying the Wigner-Eckart

31
theoremhdi = hJ 0 , I 0 , F 0 , MF0 | Tp1 (d) |J, I, F, MF i


0
F
1 F 
0
0 
= (−1)F −MF −MF0 p MF J F I p
F +J 0 +I+1
0(−1)
(2F + 1)(2F + 1)

 F 0 J0 1 hΛ0 , S 0 , Σ0 , Ω0 , J 0 | |T 1 (d)| |Λ, S, Σ, Ω, Ji .

(2.45)

We can further reduce the final term by transforming into the molecule’s body fixed frame:
0

0

0

0

0

1

hΛ , S , Σ , Ω , J | |T (d)| |Λ, S, Σ, Ω, Ji =

1
X

0

0

(−1)J −Ωp
(2J 0 + 1)(2J + 1)

q=−1





0

1 J  0 0 0 1
 J
el/mag.
 hΛ , S , Σ | T (d) |Λ, S, Σi ˆq
0−Ω q Ω(2.46)The term hΛ0 , S 0 , Σ0 | T 1 (d) |Λ, S, Σi would require knowledge of the electronic wavefunctions
but as it is common to all branches, it is not needed to calculate the relative branchingel/magratios. The term ˆq

is a vector that indicates the projection of the electric or magnetic

dipole moment on the body fixed axis and from the appendix of [58]:ˆel±1 = i(2.47)

ˆmag±1 = ∓1(2.48)

el/magˆ0= 0.

(2.49)

These vectors are responsible for the parity selection rules for electric (E1) and magnetic(M1) dipole transitions.

32
2.7.3

X 2 Σ+ , N = 1 ← A2 Π1/2 , J = 1/2, E1 transition.

The optical cycling connects to the state of positive parity in the A2 Π1/2 , J = 21 stateas the X 2 Σ, N = 1, has negative parity. From p.251 of Brown and Carrington [92], statesof good parity for Hund’s case (a) are given by

1 
J−S
|Λ; S, −Σ; J, −Ω, MJ i
|Λ; J, MJ ; +i = √ |Λ; S, Σ; J, Ω, MJ i + (−1)
2

1 
J−S
|Λ; J, MJ ; −i = √ |Λ; S, Σ; J, Ω, MJ i − (−1)|Λ; S, −Σ; J, −Ω, MJ i
2(2.50)The A2 Π1/2 , J = 21 (+) state is can be rewritten as:


1
1 
1
1
1
1
Π; J = , MJ ; + = √Λ = 1; S = , Σ = − ; J = , Ω = , MJ
2
2
2
2
2
2

1
1
1
1
+ Λ = −1; S = , Σ = + ; J = , Ω = − , MJ
22
22(2.51)

A computer program was written in Mathematica to evaluate all of the branchingratios. Due to the angular momentum selection rules for the electric dipole transition, only
decays to the N = 1 level in the ground Σ state are allowed from the A2 Π1 /2, J = 1/2 (-)
state are allowed. As a consistency check, the branching ratios in each column of Table 2.4
add up to 1. In table 2.5, we sum up the branching ratios over MF and average over MF0 .

33
0(F , MF0 )(G, F, MF )(0,0)(1,-1)(1,0)(1, 1)(0,1,-1)

8/27

1/54

1/54

0

(0,1,0)

8/27

1/54

0

1/54

(0,1,1)

8/27

0

1/54

1/54

(1,0,0)

0

2/9

2/9

2/9

(1,1,-1)

1/27

25/10825/1080(1,1,0)1/27

25/10825/1080(1,1,1)1/27

025/10825/108(1,2,-2)

0

1/6

0

0

(1,2,-1)

0

1/12

1/12

0

(1,2,0)

0

1/36

1/9

1/36

(1,2,1)

0

0

1/12

1/12

(1,2,2)

0

0

0

1/6

Table 2.4: Branching ratios for the X 2 Σ, N = 1 ← A2 Π1 /2, J 0 = 1/2, electric dipole (E1)transition, i.e. the main optical cycling transition. Columns are denoted by the excitedstate’s hyperfine numbers (F 0 , MF0 ) and the rows are denoted by the ground states (G, F, MF )numbers.

34(G, F )

0

0

F =0

F =1

(0,1)

8/9

1/27

(1,0)

0

2/9

(1,1)

1/9

25/54

(1,2)

0

5/18

Table 2.5: Branching ratios for the X 2 Σ, N = 1 ← A2 Π1 /2, J = 1/2, electric dipole transition, i.e. the main optical cycling transition. The branching ratios are summed over MF
and averaged over MF0 . Columns are denoted by the excited state’s hyperfine numbers F 0
and the rows are denoted by the ground states (G, F ) numbers.

2.7.4

X 2 Σ, N = 0, 2 ←A02 ∆3/2 , J = 1/2, E1 transition.

Branching to the A02 ∆3/2 state occur at a part in three thousand level from the A2 Π1/2
state. Such decays are deleterious to the optical cycling as this decay requires 3 photons andthus the YO molecule decays to (+) parity states in the ground electronic state. This is the
opposite parity from where the YO molecule and would stop the optical cycling as different
rotational states are now populated. The rotational ground state of the A02 ∆3/2 state has
J = 23 and by the ∆J ≤ 1 selection rule, is the only state that can be populated by decays
from the A2 Π1/2 , J = 1/2 state. For the A0 ∆ state, we optically pump into the odd parity
state since it requires a two-photon process from the ground state and from Equation 2.50,
is expressed as


1
1 
3
1
1
3
∆; J = , M ; + = √Λ = 2; S = , Σ = − ; J = , Ω = , M
2
2
2
2
2
2

1
1
3
3
+ Λ = −2; S = , Σ = + ; J = , Ω = − , M
.
2
2
2
2

(2.52)

The calculated branching ratios for the X 2 Σ+ , N = 0 state are shown in Table 2.6 andfor N = 2 is shown in Table 2.7. In Table 2.8, branching ratios are summed over MF andaveraged over MF0 . By summing over all the G and F levels in each rotational manifold, we35
see that the total branching ratio to N = 0 and to N = 2 are both 21 .
∆3/2 , J 0 = 3/2, (F 0 , MF0 )(G, F, MF )N =0(1,-1)

(1,0)

(1,1)

(2,-2)(2,-1)

(2,0)

(2,1)

(2,2)

(0,0,0)

1/3

1/3

1/3

0

0

0

0

0

(1,1,-1)

1/12

1/12

0

1/2

1/4

1/12

0

0

(1,1,0)

1/12

0

1/12

0

1/4

1/3

1/4

0

(1,1,1)

0

1/12

1/12

0

0

1/12

1/4

1/2

Table 2.6: Branching ratios for the X 2 Σ, N = 0 ← A02 ∆3/2 , J 0 = 3/2, electric dipole transitions. Columns are denoted by the excited state’s hyperfine numbers (F 0 , MF0 ) and the rows
are denoted by the ground states (G, F, MF ) numbers.

36(G, F, MF )∆3/2 , J00= 3/2, (F , MF0 )N =2(1,-1)

(1,0)

(1,1)

(2,-2)(2,-1)

(2,0)

(2,1)

(2,2)

(0,2,-2)

1/250

0

0

49/37549/75000

0(0,2,-1)

1/500

1/500

0

49/75049/1500

49/5000

0(0,2,0)1/15001/375

1/1500049/500049/5000(0,2,1)

0

1/500

1/500

0

0

49/50049/1500

49/750(0,2,2)

0

0

1/250

0

0

0

49/75049/375(1,1,-1)

1/6

1/6

0

1/25

1/50

1/150

0

0

(1,1,0)

1/6

0

1/6

0

1/50

2/75

1/50

0

(1,1,1)

0

1/6

1/6

0

0

1/150

1/50

1/25

(1,2,-2)

12/12500

128/1125

64/112500

0(1,2,-1)

6/125

6/125

0

64/1125

32/1125

32/37500(1,2,0)

2/125

8/125

2/125

0

32/375032/3750(1,2,1)

0

6/125

6/125

0

0

32/375

32/1125

64/1125(1,2,2)0012/12500064/1125

128/1125(1,3,-3)

0

0

0

1/15

0

0

0

0

(1,3,-2)

0

0

0

1/45

2/45

0

0

0

(1,3,-1)

0

0

0

1/225

8/225

2/75

0

0

(1,3,0)

0

0

0

0

1/75

1/25

1/75

0

(1,3,1)

0

0

0

0

0

2/75

8/225

1/225

(1,3,2)

0

0

0

0

0

0

2/45

1/45

(1,3,3)

0

0

0

0

0

0

0

1/15

Table 2.7: Branching ratios for the X 2 Σ, N = 2 ← A02 ∆3/2 , J 0 = 3/2, electric dipole transition. Columns are denoted by the excited state’s hyperfine numbers (F 0 , MF0 ) and the rows
are denoted by the ground states (G, F, MF ) numbers.

37
0

0

∆3/2 , J = 3/2, (F )(N, G, F )

(1)

(2)

(0,0,0)1/3

0(0,1,1)1/6

1/2(2,0,2)

1/150

49/250(2,1,1)

1/3

1/25(2,1,2)

4/25

64/375(2,1,3)0

7/75

Table 2.8: Branching ratios for the X 2 Σ, N = 0 ← A02 ∆3/2 , J 0 = 3/2, electric dipole transitions. Branching ratios are summed across MF and averaged over MF0 . Columns are denotedby the excited state’s hyperfine numbers (F 0 ) and the rows are denoted by the ground states(N, G, F ) numbers. Summing across G and F , the branching ratio to N = 0 and N = 2 areboth 21 .

2.7.5

X 2 Σ, N = 0, 2 ← A02 Π1 /2, J 0 = 1/2, M1 transition.

The optical cycling scheme for molecules relies on the fact that an electric dipole
transition can only connect states of opposite parity. However, M1 transitions connect statesof the same parity, hence, if the YO molecule relaxes via an M1 transition, it will end up in
the wrong parity and hence we opent up decays to other rotational states and the cyclingcomes to a stop.
While it is commonly assumed that magnetic dipole transitions are suppressed by afactor α2 ≈ 5 × 10−5 compared to E1 transitions, where α is the fine structure constant,quantum chemistry calculations and experimental measurements in OH show a factor of
7 [58] higher than expected magnetic dipole transition strength. This may lead to larger
than expected rotational branching and become a limiting factor in optical cycling.
In the optical cycling transition, the YO molecules relaxes from the J = 1/2, + state

38
and by angular momentum and parity selection the N = 0, 2 levels are populated. The
branching ratios are tabulated for N = 0 in Table 2.9 and for N = 2 in Table 2.10. The
branching ratios are summed up over MF for each F value and averaged over MF0 andtabulated in Table 2.11. By summing over all the G and F levels in each rotational manifold,
we see that the total branching ratio to N = 0 is 32 and to N = 2 is 13 .
(G, F, MF )

Π1/2 , J 0 = 1/2, (F 0 , MF0 )

N =2

(0,0)

(1,-1)

(1,0)

(1,1)

(0,0,0)

0

2/9

2/9

2/9

(1,1,-1)

2/9

2/9

2/9

0

(1,1,0)

2/9

2/9

0

2/9

(1,1,1)

2/9

0

2/9

2/9

Table 2.9: Branching ratios for the X 2 Σ, N = 0 ← A2 Π1/2 , J = 1/2, magnetic dipole (M1)transitions. Columns are denoted by the excited state’s hyperfine numbers (F 0 , MF0 ) and therows are denoted by the ground states (G, F, MF ) numbers.

39(G, F, MF )

Π1/2 , J00= 1/2, (F , MF0 )

N =2

(0,0)

(1,-1)

(1,0)

(1,1)

(0,2,-2)

0

1/15

0

0

(0,2,-1)

0

1/30

1/30

0

(0,2,0)

0

1/90

2/45

1/90

(0,2,1)

0

0

1/30

1/30

(0,2,2)

0

0

0

1/15

(1,1,-1)

1/9

1/36

1/36

0

(1,1,0)

1/9

1/36

0

1/36

(1,1,1)

1/9

0

1/36

1/36

(1,2,-2)

0

1/10

0

0

(1,2,-1)

0

1/20

1/20

0

(1,2,0)

0

1/60

1/15

1/60

(1,2,1)

0

0

1/20

1/20

(1,2,2)

0

0

0

1/10

Table 2.10: Branching ratios for the X 2 Σ, N = 2 ← A2 Π1/2 , J = 1/2, magnetic dipole
transitions. Columns are denoted by the excited state’s hyperfine numbers (F 0 , MF0 ) and therows are denoted by the ground states (G, F, MF ) numbers. F = 3 states are omitted sincethe ∆F ≤ 1 selection rule prevents decays to that state.

40
0

0

∆3/2 , J = 3/2, (F )(N, G, F )

(0)

(1)

(0,0,0)0

2/9(0,1,1)

2/3

4/9(2,0,2)0

1/9(2,1,1)

1/3

1/18(2,1,2)0

1/6(2,1,3)0

0

Table 2.11: Branching ratios for the X 2 Σ+ ← A2 Π1/2 , J 0 = 1/2, magnetic dipole transitions.
Branching ratios are summed across MF and averaged over MF0 . Columns are denoted by
the excited state’s hyperfine numbers (F 0 ) and the rows are denoted by the ground states(N, G, F ) numbers. Summing across G and F , the branching ratio to N = 0 is 23 and to
N = 2 is 13 .

Chapter 3
Laser setup.

Due to the more complex internal structure of molecules, the laser cooling of YO
requires several laser systems. At the heart of the optical setup is a 100 MHz self-referenced
Er doped fiber comb [105]. This provides a frequency ruler that references all of the otherlasers that are used to manipulate the YO molecules. These lasers include a dye laser, several
external cavity diode lasers (ECDLs) and a fiber based Raman amplifier. In this chapter wewill describe these laser systems and the schemes used to frequency stabilize them.
3.1

An introduction to frequency combs
Mode-locked pulsed lasers have been powerful tools in frequency metrology [106,107] byallowing easily generated and measured radio frequencies to control a wide comb of regularly
spaced optical frequencies. A mode-locked laser in the time domain is a set of regular spaced
intensity pulses separated in time by trep , the repetition period; the inverse of this is known−1
as the repetition frequency frep = trep
(Fig. 3.1 (a)). The width of the pulse is on the orderof the frequency bandwidth of the gain medium (shown schematically in blue in (b).
As depicted in (a), the electric field also has a phase offset with respect to the amplitudeenvelope. For a mode-locked laser, between consecutive pulses, there is a constant phaseoffset φco , known as the carrier offset phase. In the frequency domain the mode-locked
laser’s spectrum is a regular comb with frequency spacing frep offset by the so called carrier

42

(a)
E

tFWHM

φco

2 φco

t

1/tFWHM

(b)

I

fco

frep

ν

Figure 3.1: Illustration of a mode-locked laser. (a) Mode-locked laser electric field (E) in thetime domain. (b) Mode-locked laser in the frequency domain.

offset frequency, fCO :fCO = frep ×

φco
2π

(3.1)

The phase slip between the amplitude envelope and the electric field is due to the difference
between the group and phase velocities and an excellent description may be found in Seth
Foreman’s thesis [108].

43
3.2

Erbium Doped Fiber Frequency Comb
The laser stabilization scheme relies upon an 100 MHz repetition rate Erbium (Er3+ )mode-locked fiber comb . The comb is remarkably robust and can maintain mode-lock forseveral months at a time. Technology for the Erbium fiber laser has pushed forward due tothe telecommunications industry, which makes the laser fairly inexpensive and parts readily
available.
The relevant energy levels for Er3+ are shown in Fig. 3.2. The pump is at 980 nm
and the lasing transition occurs between the lowest two lowest levels at 1530 nm. A twopump photon transition also occurs, leading to radiation at 550 nm. This gives rise to thecharacteristic green glow from the Erbium doped fibers.
When doped into the silica fiber host, a combination of homogenous and inhomogeneous
broadening [109] results in a broad absorption band which makes it suitable for a mode-locked
laser.
This laser is passively mode-locked using the technique known as polarization additive
pulse mode locking (P-APM) [110, 111]. In brief, elliptically polarized light entering thegain fiber is rotated by an angle that is intensity and ellipticity dependent. By applying
the opposite polarization rotation at high intensity with the freespace waveplates as shownin figure 3.3(a), the laser will passively mode-lock. The resulting mode-locked spectrum isshown in figure 3.3 (b).
Figure 3.3 (a) shows a schematic of the comb oscillator. The pump light is provided bya fiber Bragg grating stabilized 980 nm pump diode (part number : JDSU 29-800-500) thatprovides up to 500 mW of power. The pump laser is coupled into the cavity via a wavelength
division multiplexer (WDM) in a counterpropagating manner. This is advantageous as thelaser pulse propagates towards regions of greater gain inversion, thus increasing the overall
gain. Also, because of the counterpropagating topology, no pump light pollutes our output
due to the free space optical isolator (OI). The cavity has a net dispersion of 0.022 ps2 . This

44

4

F7/2

S3/2

4

I11/2

4

I13/2

4

I15/2

980 nm4

490 nm
550 nm

980 nm1530 nm
550 nm1530 nm

980 nmFigure 3.2: Level structure of Er3+ . The lasing transition occurs at 1530 nm and is the laser
is pumped by 980 nm light.

relatively high dispersion was chosen to achieve longer intra-cavity pulses (∼ 100 fs), whichleads to higher output power. A polarizing beam splitter (PBS) cube is used to outcouple∼20% of the light, and we thus obtain ≈6 mW output power with ≈35 mW of circulating
power.
Stabilization of the repetition rate is achieved with an rf phase lock to the 10th harmonic of frep . Approximately 4% of of the mode-locked laser’s output is picked off and sent
to an InGaAs photo-diode and band-passed filtered. This detection scheme is shown in Figure 3.4. A 1 GHz signal generated by a ultrastable Wenzl crystal oscillator is mixed with an
adjustable frequency frep,synth ≈5 M Hz signal that is generated by an Analog Devices Direct

45
Digital Synthesizer referenced to the same crystal oscillator source. The 10th harmonic isthen phase locked to the lower sideband with frequency ≈995 MHzm which allows us to use
frep,synth to tune the entire frequency comb.
frep =

1GHz − frep,synth
10

(3.2)

Changes in frep are actuated with an in-fiber electro-optic modulator (General Photonics Fiber Phase Shifter) with a bandwidth of ∼12 kHz. A slower long throw actuatorconsists of a collimating lens mounted on a PZT driven mount.

-ahFiber/
φ-shifter
OI

46

980/nm/pump/diodeEr3+/doped/
fiber
Lyot/
PBS
λ/4 λ/2
Filterto/EDFAWDM
λ/4

InGaAs/PD-bh

Figure 3.3: (a) Schematic of 1550 nm mode-locked fiber comb. (b) Mode-locked Spectrum.

47

Figure 3.4: Schematic of frep detection and stabilization.

48
3.2.1

Octave spanning supercontinuum

As can be seen in Fig. 3.3 (b), the comb oscillator spectrum only covers about 100 nm.
This is insufficient for our purposes for 2 reasons. One, this does not overlap with the lasersthat we wish to stabilize and two, a much larger spectrum is required in order to measure thecarrier-offset frequency fco . The key to broadening the spectrum is a highly nonlinear fiber.
For our wavelength range, such fibers are Germanium doped, have a reduced core diameter of
3.7 µm and have a zero dispersion wavelength of ≈ 1520 nm [112,113]. The small core (whichis less than half the diameter of SMF-28) ensures a high intensity to ensure the non-linear
spectral broadening effects. An advantage of this fiber is that can be spliced onto regularfiber (e.g. SMF-28) using standard techniques. In our system ≈10 cm of non-linear fiberis spliced onto ≈10 cm of SMF-28, as the comb light is fiber-coupled into the much larger
core diameter fiber, it is much less sensitive to misalignments and makes the supercontinuum
generation much more robust. This is in contrast to the small core diameter micro-structure
photonic crystal fiber (PCF) used to broaden mode-locked titanium-sapphire lasers where
the microstructure is destroyed during the splicing process.
The 6 mW that we produce from the oscillator is insufficient to induce the spectral
broadening. As such, we amplify the light using an Erbium Doped Fiber Amplifier (EDFA)[114] as shown in Figure 3.5. The EDFA is a ≈10 cm length of the same Er3+ fiber, pumped
on both ends with 980 nm fiber Bragg grating diode lasers coupled via WDMs. Each pump
diode produces 700 mW (Part number: World Star Tech LC96UF74-20R) and we obtain
180 mW of infra-red power after amplification. The light is then sent through a free spacesilicon prism pair that recompresses the pulse. Fine tuning of the dispersion compensation isprovided by moving one of the prisms mounted on a micrometer stage. The light is then fiber
coupled into the non-linear fiber, which produces 80 mW total power in the super-continuum.
A typical octave spanning supercontinuum spectrum is shown in Fig. 3.6. The comb
light used to stabilize the various lasers and for measuring fCO are indicated by grey dots.

49

980LnmL
pumpLdiodes

Er3+LdopedL
fiber
Isolator

Si-LprismL
compressor
HalfLMirror

FromLModeL
LockedLComb

supercontinuum

SMF-28
NonLlinearL
fiber

Figure 3.5: Schematic of supercontinuum generation. 6 mw light from the mode-locked
comb is amplified with the EDFA to 180 mW. A silicon prism pair is used compress thepulse before being spectrally broadened by the highly non-linear fiber.

50
Light at 1050 nm and 2100 nm (not shown) is used for self-referencing, while the comb teeth
at 1228 and 1298 nm are used to stabilize the various cooling and repump lasers.

51

Figure 3.6: Super-continuum Spectrum. The wavelengths used for various heterodyne beats
are indicated by grey dots. The 1228 and 1298 nm light is frequency is used to stabilize
the lasers used in the experiment . Light at 1050 nm and 2100 nm (not shown) is used tomeasure fCO .

52
3.2.2

Measuring fCO with an f-2f interferometer

The determination of the carrier offset frequency fco is more difficult to measure than
the repetition frequency. For this, we use the technique of self-referencing of an octave
spanning comb [115–117].
The frequency of the n-th comb tooth can be expressed as
fn = nfrep + fco(3.3)

To perform the self referencing measurement, we frequency double the portion of the supercontinuum at ∼2100 nm yielding a new frequency comb with the following frequency comb
teeth,
f2f −arm = mfrep + 2fco

(3.4)

where m is another integer. The term 2fco comes from the fact that each new comb toothis generated by the sum frequency of two of the original comb teeth. We may mix thefrequency-doubled comb, i.e. the 2f arm, and the undoubled light at ∼1050 nm and thelowest frequency component in the heterodyne signal is the offset frequency fco .
The optical scheme is shown in figure 3.9. To generate the frequency-doubled light,
we used a magnesium oxide doped periodically poled lithium niobate (MgO:PPLN) crystal(part number: Covega SHG 7 crystal). The pulse intensity is sufficiently high that we may
use the crystal without cavity enhancement and obtain an approximate conversion efficiencyof 10 %. The f-2f interferometer is set up with a Michelson interferometer topology. The
retroreflecting mirror on the undoubled arm is mounted on a micrometer stage; this allows
fine adjustments in order to temporally overlap the pulses from the two arms. Both arms are
split and recombined with a 1500 nm dichroic mirror and are both sent through the doubling
crystal. A 10 nm bandpass filter at 1050 nm filters out the unneeded spectral components inthe undoubled arm and the resulting heterodyne beat is detected on an InGaAs photodiode.
Figure 3.9 shows a schematic for the fCO stabilization. First the heterodyne signal issent through a 50 MHz low pass filter, thus filtering out all frequency components except

531050 nmband pass
MgO: PPLN
crystalλ/4

Fromsuper-cont nuum

1500 nm
Dichroic

Figure 3.7: Schematic of fco detection. Blue arrows indicate light at ∼1050 nm and red
arrows indicates light at ∼2100 nm. A dichroic mirror at 1500 nm splits the light between
the 2 arms of the interferometer. The MgO:PPLN frequency doubles the 2100 nm portionof the spectrum.

for fCO . This signal is mixed with a 43 MHz signal and high-pass filtered such that only the
43 MHz+fCO sideband passes through. This signal is cleaned up with an rf tracking filter, i.e.
a VCO phase locked to the input signal. The phase difference between the tracking filter’s
output and a tunable synthesizer, fsynth,CO is measured with a home built phase frequency
detector (note that the tracking filter’s frequency is divided by 4 in the phase frequency
detector). To close the loop, feedback is applied to the pump diode current of the comb

54
oscillator. This affects the in-cavity dispersion, which in turn affects the group and phasevelocity mismatch and therefore changes fCO . We may express the carrier offset frequency
as:fCO = 4 × fsynth,CO − 43M Hz

(3.5)

VCO

~12 MHz43 MHzFigure 3.8: fco stabilization.

3.3

Light for the main cooling transition at 614 nm

3.3.1

Dye Laser

To provide the 614 nm light required for the main cooling transition, we use a commercial Matisse DR dye laser from Sirah Lasertechnik [118]. To attain this awkward wavelength,
we use Rhodamine 610 dye mixed in ethylene glycol with a concentration of 0.57 g/L. The
dye is forced through a slit nozzle at a pressure of 13.5 bar, creating a free-space dye jet lessthan a 1 mm thick. Care needs to be taken to ensure that the dye jet drains correctly; if
not, the dye will spray uncontrollably, and a tedious clean up job awaits.
The laser is pumped with 6.5 W of 532 nm from a Verdi V-10 Nd:YAG laser. Two

55
concave mirrors (FM) set the cavity mode and ensure the cavity is stable. Rhodamine 610
has a lasing range of several 10s of nm while the free spectral range (FSR) of the overalllasing cavity is ∼160 MHz. Therefore in order to ensure that only one of these modes lases,
we require three frequency selection elements. The coarsest frequency selection element isthe birefringent filter [119],(Bi-Fi), consisting of a stack of three birefringent plates having
thicknesses ratio 1:3:15. This narrows the possible lasing range to ∼200 GHz. The thinetalon (TE) and the thick etalon (PZETL) have FSRs of 250 GHz and 20 GHz respectively.
The PZETL’s length is dither locked to coincide with the maximum transmission of theappropriate longitudinal laser cavity mode.
To ensure lasing only occurs in the counter-clockwise sense, there is Faraday rotator(FR), which consists of a Terbium-Gallium-Garnet crystal plate placed between two rareearth magnets. The out of plane mirror on the right of the laser cavity induces a small
polarization rotation that is cancelled by the FR for counter-clockwise propagation and adds
for clockwise propagation. Brewster losses off the dye jet, FR, TE and Bi-Fi prevents lasingin the clockwise sense; this forms an optical diode. Despite this optical diode, care must
be taken to slightly vertically misalign the PZETL as back reflections off it can lead to bidirectional lasing. This leads to intensity and frequency instability and the laser becomesunworkable at that point. To stabilize the frequency, the overall cavity length can be changedby two mirrors mounted on PZTs. The first, labelled the FPZ mirror has a bandwidth of∼ 20 kHZ. A slow long throw PZT on the mirror labelled SPZ has bandwidth of ∼10 Hz
and takes care of long-term drifts. Typically, we can obtain ≈400 mW of power on fresh dyewhich lasts for 2-3 weeks.

3.3.2

Raman Amplifier

For the implementation of the 3D MOT where longitudinal slowing is required, an
independent laser at 614 nm became very desirable for the independent frequency tuning andfor more power. Therefore, we have a Raman fiber amplifier from MPB-Communications

56
SPZ mirror

PZETL

TE

FR

FPZ mirrorFM

λ-metermon.

Dye jetBiFI

Figure 3.9: Photograph of the Matisse DR dye laser system.

Inc [120]. Here, a seed laser, in our case at 1228 nm, is amplified in a P2 O5 doped fiber
pumped with a Ytterbium fiber laser. It turns out that for these so called P-doped fibers,
a strong stimulated Raman scattering peak occurs at ≈40 THz [121]. This therefore allows
us to bridge the gap between the seed wavelength at 1228 nm and the pump wavelengthof ≈1050 nm. The input seed light is generated with an ECDL diode (Innolume GC-1230TO-200) that is directly frequency stabilized to the comb. 20 mW of seed diode power isfiber-coupled into the Raman amplifier. The 1228 nm light is amplified to ∼10 W and isthen frequency doubled and we can obtain greater than 1 W of 614 nm light.
As the entire system, sans the seed laser, is fiber coupled, operation of this laser isrobust and mostly turn-key. The wavelength range of such Raman-based amplifier systems
is limited by the Yb fiber laser pump and ranges from 515-670 nm [120]. Other wavelengths
should be accessible by using a different pump laser. With the more robust optical alignment,
similar wavelength range and the elimination of (mostly toxic) laser dye, this can be an
excellent replacement for a dye laser. The seed diode is used in a Littrow ECDL configuration[122] and frequency stabilization is performed by changing the pump current.

57
3.3.3

Stabilization of 614 nm lasers.

To stabilize the seed diode, the comb light and ∼ 100 µW of the seed diode are
mixed using a 50-50 beam splitter cube as shown in Figure 3.10. As the optical heterodyne
signal from the two exit ports of the PBS have a π phase shift between them, we use two
photodiodes and subtract the signals in order to obtain the heterodyne signal. In addition,
this subtraction mostly eliminates the seed diode’s technical noise in the heterodyne signal.
We note that this is the same configuration as used to generate the heterodyne between thevibrational repumps and the comb.
For the 1228 nm seed laser, we applied frequency feedback by changing the diodepump current, and we obtain an in-loop linewidth of 140 kHz (Fig 3.11(b)) (for 1 second
averaging). The light at 614 nm will have twice that linewidth but is still well within thenatural linewidth of the YO transition.
Relative to the comb frequencies, the output frequency of the Raman amplifier’s frequency doubler (at 614 nm) is thus
framan = 2(nraman frep + fco + fHet,RamanComb )(3.6)

where fHet,RamanComb = 21.5M Hz is the difference between the seed laser’s frequencies andthe nearest comb tooth and nraman is an integer on the order of 2.5 × 106 .
With the Raman amplifier stabilized, we then phase lock the dye laser to the doubled
portion of the Raman amplifier. A schematic of the dye laser to Raman amplifier locking
is shown in Figure 3.11 (a). Most of the Raman amplifier power is sent through a 60 MHz
AOM, where the -1 order is used to provide the light for radiation pressure slowing. This
allows us to perform rapid switching of the slowing beam. If we were to address the stationary
molecule, the frequency at the output of the Raman amplifier’s doubler, framan , will be givenbyframan = fY O + 60M Hz(3.7)

where fY O is the resonance frequency of the YO transition. It turns that in the course of

58
the longitudinal slowing, we will be chirping framan to gradually address slower molecules(the chirping will be discussed in section 3.6). The peak of the velocity distribution is at
70 m/s, which implies a doppler shift of 112 MHz. In contrast, as the dye laser provides the
MOT radiation, we do not need to have any frequency scanning. This allows us to leave thedye laser at the optimal MOT detuning at all times and also lessens the workload of the dyelaser’s frequency stabilization. Therefore, the offset frequency between the Raman amplifier
and the dye laser is controlled by a VCO, which has a tuning range of 130 MHz, allowing
us to compensate for the full slowing chirp. Therefore, the control voltage on our VCO has
two components, 1) the voltage to apply the correct frequency offset for MOT loading and2) the voltage to cancel the Raman amplifier’s frequency chirps.
A small portion of the doubled Raman amplifier light is sent through a 107.5 MHz
double-passed AOM. This upshifts the frequency by 215 MHz which gets us in the range ofthe VCO. The two 614 nm optical frequencies [in MHz] are related to fY O byframan + 215M Hz = fY O + 60M Hz − 1.6 ∗ vlong
fdye = fY O + δ

(3.8)
(3.9)

where fY O is the resonance frequency of YO, vlong , is the longitudinal velocity class that
we wish to address and δ is the desired MOT detuning. We can then solve for the VCO’s
frequency to be

fV CO = framan + 215[M Hz] + 60[M Hz] − δ − 1.6 ∗ vlong(3.10)

which for the case of δ = −Γ = −5M Hz, the slowing laser can address molecules
ranging from 0-81 m/s. In contrast to the Raman-comb lock, the heterodyne components
here have much more power (∼ 100 µW each), and we can easily obtain signal to noise ratiosof 40 dB. This results in an in-loop linewidth with respect to the doubled Raman amplifierof 400 kHz (Figure 3.11 (c)).

59
3.4

Vibrational repumps
The light for each vibrational repump is provided with a master diode that injection

locks a string of slave diodes and tapered amplifiers. Like the seed laser for the Raman
Amplifier, the master laser for the vibration repumps are set in a Littrow configuration [122].
The diode used is an Eagleyard Photonics EYP-RWE-0650. The diode has a nominal central
wavelength of 653 nm.
The diode mounting is shown in Fig. 3.12 and is based on a design from [123]. The
diode is mounted in a standard Thorlabs laser diode collimation tube. The grating is aholographic reflecting grating with 1800 lines/mm and is mounted on a vertical flexure, thatcan be adjusted by a micrometer actuator. This can be used to maximize the feedback of
the diode. The mount is cooled down to ≈16 ◦ C to reach the desired wavelength of 648 or
649 nm. Optical isolation of 60 dB is required and we can obtain 11 mW of output power.
Approximately 1 mW of the diode light is picked off and frequency shifted by a single pass
AOM by ≈100 MHz. This light is then heterodyned with the doubled comb light as shown
schematically in Fig. 3.10. Here the doubling of the comb light is performed with a custom
MgO:PPLN crystal from HC Photonics.
The frequency of the repump light is therefore
fv=x = nv=x frep + 2fco + fHet + |fao |(3.11)

here x = 1, 2 refers to the vibrational repump in question. The exact frequency of the offset
AOM can be set to position the locked heterodyne beat fHet at a convenient location. The
factor of 2 fco is due to the frequency doubling of the comb-light. The frequency correctionis actuated by the PZT that changes the grating angle; this leads to an in-loop linewidth
relative to the comb of 770 kHz.

60

Figure 3.10: Schematic of comb-laser heterodyne beat generation. Optical PPLN crystal
doubles near infrared to the visible and is used in locking vibrational repumps. For the
Raman laser at 614 nm no PPLN crystal is required.

61

a)

b)

c)

Figure 3.11: a) Schematic of Dye-Raman Amp phase lock. (b) In-loop linewidth measurement of the Raman Amplifier seed laser at 1228 nm with respect to the frequency comb. (c)In-loop linewidth of the dye laser with respect to the doubled Raman amplifier output at
614 nm.

62

Laser Diode PZT

(b)

(a)

Grating
FlexureRelative rf power

(b)

(c)

(b)

Figure 3.12: (a) Top view of Littrow mount. (b) Side view of partially assembled mount.
(c) In-loop linewidth of the comb to master diode lock.

63
3.5

Frequency to Voltage Convertor
In the YO experiment, all laser locks to the comb use a home built frequency to voltage

convertor. Commercial frequency to voltage convertors are generally limited to acoustic
frequencies and are not fast enough to measure the optical heterodyne beats that can go up
to frep /2 = 50 MHz. We therefore adapted a circuit designed by John Hall that can measurefrequencies of up to 80 MHz.
An illustration of the f to V convertor is shown in fig 3.13. A TTL convertor converts
the incoming heterodyne frequency into a TTL signal. This is then sent through a bistableflip flop, which divides the TTL frequency by two and helps to sharpen the rising andfalling edges of the signal. Based on the sign of the TTL signal, a high stability current
supply will charge the charge pump capacitor with either plus or minus 10 mA. Two diodes
upstream of the charge pump capacitor are tied to two voltage rails set at ±6.95 V. Therefore,on each half TTL cycle, the charge pump capacitor is alternately charged to the voltage±VRail = ±(6.95 V + Vf orward ), where Vf orward is the diode forward velocity drop.
Thus every time the TTL signal switches, the charge on the capacitor changes by
Q = 2 ∗ VRail /CCP

(3.12)

where CCP is the capacitance of the charge pump capacitor. The diode pair downstream of
the charge pump capacitor acts as a current rectifier and one obtains a current I = Q × fT T L
where fT T L is the frequency of the TTL signal. A transimpedance amplifier then converts
the frequency to a voltage. Thus we obtain a voltage that is proportional to the input
frequency. It turns out that we get a frequency to voltage conversion of 0.022 V/MHz andthe full electronic schematic may be found in Appendix 3.

64

Figure 3.13: Illustration of frequency to voltage convertor. The incoming heterodyne beat isconverted into a TTL signal. This TTL controls a high precision current source, providing
either ±10 mA. This current then alternately charges the charge pump capacitor to ±VRail .
A diode rectifier shunts the current to ground on one sign the TTL signal and to the transimpedance amplifier to the other. This yields the frequency dependent voltage which weuse as an error signal to stabilize our lasers.

65
3.6

Frequency scanning
For certain applications, we will be chirping the frequencies of our lasers. For example,

this is used in radiation pressure slowing as we wish to gradually address slower molecules.
The doppler shift is given by∆f∆vlong=−f0
c(3.13)

where f0 is the resonant frequency at zero velocity and vlong is the velocity that we wish toaddress. Thus, the fractional frequency change is the same regardless of the laser in question.
It turns out that this chirp can be achieved by chirping frep and ensuring that all
the v 00 = 0, 1, 2 lasers frequencies follow the comb. The terms in Equations. 3.6 and 3.11
proportional to frep dominates over all of the other frequency terms due to the multiplicationby the comb tooth integer. As an example, the frequency of the main cycling transition is488.492 THz. Therefore, the red-shift to address a 100 m/s molecule is -163 MHz or afractional change of −3.3 × 10−7 . Therefore, we need to add 330 Hz to frep,synth in order toshift the laser frequencies be resonant with the 100 m/s molecules.
It turned out that to ensure the robustness of the lock while frequency chirping, weneed to apply feed-forward to the scanned lasers. Feed-forward helps minimize the corrections that the feedback has to perform. Figure 3.14 illustrates this concept. The repetition
frequency is changed by tuning frep,synth by applying a voltage Vchirp to the frequency modulation input of the synthesizer. This changes the frequency lock-point of frep and drags allof the other optical lock points with it as well. The feed-forward is performed by applying a
voltage proportional to Vchirp to the laser’s frequency actuator. When the sign and magnitude of the scaling is tuned correctly, the feed forward voltage makes the laser perform thedesired frequency scan and the frequency error signal will be minimized. The feed-forward isapplied to the stabilization loops for frep , the Raman amplifier, and the v 00 = 1, 2 vibrational
repumps.

66
3.7

Putting them together.
In order to address all of the hyperfine manifolds, each vibrational frequency (v 00 =0, 1, 2) requires two additional frequency sidebands. The carrier frequency is tuned to theG00 = 0 transition whilst the additional sideband at +760 MHz addresses the G00 = 1, F 00 = 0
manifold and the +774 MHz sideband addresses the G00 = 1, F 00 = 1, 2.
For the 614 nm lasers, as we have sufficient power, these sidebands are generated witha single passed, two toned high speed AOM. However, as the ECDLs for the vibrational
repumps have insufficient power, we end up with a string of injection locked diodes andtapered amplifiers. Each master diode injection locks a slave diode to provide light for theG00 = 0 hyperfine manifold. A small pickoff (∼ 5 mW) of this light is then shifted up by adouble passed AOM with a frequency dithered between 380 MHz and 387 MHz in order forthe light to cover the G00 = 1 hyperfine manifold. The output of the double passed AOM
is then used to injection lock a second slave diode. The slave diodes are made by Opnext
with part number HL6385dg and typically provide ≈ 80 mW of power each. Both hyperfine
components are combined and fiber-coupled to seed a tapered amplifier from Sacher (partnumber TPA-0650-0250) providing a total power of 200 mW for each vibrational repump.
An added benefit is that all of the hyperfine components in each vibration now have thesame polarization. Therefore, we can combine the v = 1 and v = 2 vibrational repumps
with PBS cubes without loss. Combining the v 00 = 0 light with the vibrational repumps isachieved with dichroic mirrors.
Figure 3.15 illustrates the lasers used in the longitudinal slowing configuration. In
addition to the aforementioned sideband generation to address hyperfine states, a 5 MHz
Pockel cell rapidly chops the polarization between σ + and σ − . This is used to disrupt the
optical pumping into dark Zeeman states and will be discussed in Chapter 5. Also resonant
electro-optic modulators at 10 MHz are used to spectrally broaden the slowing light in orderto aid with the efficiency of the longitudinal radiation pressure slowing (see chapter 6.

67

Figure 3.14: Example schematic of feed-forward applied to a laser locked to the comb. VChirpchanges frep , pulling the laser lock point. Vchirp is scaled by a factor A in order to follow thechanging lock point. A ∗ VChirp is added to the feedback voltage Vf eedback .

68

100

Raman Amp

Figure 3.15: Longitudinal slowing laser setup for YO. Abbreviations: ECDL, external cavity diode laser; OI, optical isolator; AOM, acousto-optic modulator; EOM, electro-optic
modulator; λ/2, half-wave retardation plate; PBS, polarizing beam splitter.

Chapter 4
Generating a Cold YO Beam

The two main sources to create a chemically diverse set of polar molecules are supersonic expansion jets and buffer gas cooled beams. While supersonic beams are easily able
to generate rotationally cold and vibrationally cold samples, such beams have forward velocities of several hundreds m/s. This turns out to be fast for MOT trapping. Therefore,
we produce a YO beam in a buffer gas beam formed in a cryogenic cell [48, 71, 124]. This
generates a bright source of molecules in the correct quantum state with a modest forward
velocity. Figure 4.1(a) shows a general schematic of the concept of the buffer gas beam. In
this system, the buffer gas is fed into a cold copper cell and via collisions with the cell wall,
the buffer gas comes to thermal equilibrium. The species of interest in turn thermalizes withthe cold buffer gas. As rotational constants are on the Kelvin scale, temperatures on theorder of 10 K are required in order to ensure sufficient population in the low rotational levels.
The fractional population in each rotational state can be expressed as:
P (N ) =BN (N +1)
1−
e kB T
Z(8N + 4)(4.1)

where Z is the partition function. The resulting rotational population distribution for room
temperature and 3.5 K are shown in 4.1(a) At 3.5 K, this yields a third of the populationin the desired N = 1 state, which is a factor of 200 more than for a room temperaturesample. Because vibrational states thermalize about 100-1000 times slower than rotational
states [125], the buffer gas cooling is not as effective in themalizing these states. However, as

70
the vibrational spacing is on the order of a 1000 K, which is roughly the temperature of theablation plume [126], there is still a significant amount of population in the desired v = 0
state
4.1

The Cryostat
The first ingredient to generating the YO beam is the cryostat. In the 2D MOTexperiments, the system was cooled down via liquid helium and liquid nitrogen, whilst in the
3D-MOT, the system was upgraded to a pulse tube cooling system (CP2800 Cryomech) [127].
The pulse tube has good cooling power (600 mW at 4 K), has lower base temperature, 2.8 K
as opposed to 4.2 K for a Helium-4 system, and will eventually pay for itself as it does notconsume expensive liquid helium. The pulse tube has two cold heads, one at ≈3 K and theother at ≈40 K.
The main heat load on the cell is due to black body radiation. The power absorbed isgiven by:
PBB = σSB T 4 A(4.2)

where σSB is the Stefan-Boltzmann constant, T is the temperature of the environment andA is the surface area of the object. We therefore apply two layers of radiation shielding. The
inner shield layer is thermally anchored to the 3 K cold head and the outer shield layer is tied
to the 40 K cold head. The cell has an approximate area of 100 cm2 . If this were exposed to
the room temperature outer vacuum can, we would put on a heat load of 4.5 W and would
thus overwhelm the cooling power of the pulse tube. The inner shield at 40 K reduces theheat load to 1.4 mW, which the pulse tube can handle easily and is also negligible comparedto the ∼100 mW of heat load due to the ablation and radiation pressure slowing lasers.
In contrast, the inner shield at 4 K’s primary purpose is not to reduce the heat load but
to ensure that the exiting buffer gas first bounces off the coldest surface possible. Otherwise,
He atoms may bounce off the 40 K shield or a room temperature surface, heat up and destroy

71
the YO beam. Also, by coating the inner shield with activated charcoal, the helium buffer
can be effectively cryo-pumped . We note that from work in other groups [128, 129] andexperience from our own apparatus (see Section 4.3), the relative position of the shields andcharcoal can make a significant difference to the beam’s brightness, velocity and robustness.
There are some general construction techniques that we found useful. Everything in
the cryostat that needs to be cold is made out of oxygen-free high thermal conductivity(OFHC) copper. As the name suggests, this grade of copper has high thermal conductivity
at cryogenic temperatures: κ = 630 W/(m K) at 4 K and κ = 1470 W/(m K) at 40 K [130],
which is more than an order of magnitude better than “regular” copper [131]. Indium is used
for seals as they do not crack when cold or thermal cycled. For mating copper surfaces thatneed to be removable, the low temperature vacuum grease Apiezon N is liberally applied
to the mating surfaces to increase the thermal conductivity [130]. Due to the differing
thermal expansion coefficients, care must be taken when bolting the apparatus together as
the differential expansion of different materials can pull the apparatus apart. To combat this,we use Belleville washers, which are conically shaped washers with a considerable amount ofspring to them. By cranking down tightly, the washers spring out if the apparatus contracts
and thus keep the pieces together. Thus we do not have to worry about building stacks of
washers with different materials to compensate for differing thermal expansion.
4.2

The CellFigure 4.2 (a,b) shows photographs of the cryogenic buffer gas beam source. The 1ststage of the cell is made out of a 1.5”×1.5”×1.25” long OFHC copper rectangular cuboid.
The cell cavity is formed by two through holes of 0.85 inch diameter. The total volume of
r3 = 18.8 cm3 . The last term is the volume of
the main cell is thus, V = πr2 ∗ (l + w) − 163
the so called mouhefanggai shape formed by the intersection of the two cylinders of radiusr. The beam exits through a 3 mm aperture.
YO molecules are formed via laser ablation of sintered Y2 O3 pellets from Alfa-Aesar.

72
A Continuum Minilite I Q-switched laser produces ∼4 ns long pulses with energy 3 mJ at a
10 Hz rate at 532 nm. The ablation laser is focused onto the target by a 35 cm lens. The
30 mW average power is sufficient to raise the temperature of the main cell by about 0.5 K.
From in cell absorption measurements, we estimate a YO density in the N 00 = 1 state of1010 cm−3 [71].
The helium buffer gas is continuously fed into the back of the cell via a 2.5 mm diametercopper tube. The buffer gas is pre-cooled by wrapping and soldering ≈60 cm of the feed line
around a 1 inch diameter copper bobbin that is bolted to the 3 K cold plate. The helium
flow rate is measured in units of standard cubic centimeter per minute (sccm), (i.e. the flowrate that would raise the pressure of a 1 cm3 volume by 1 bar at 273 K). This corresponds
to an number flow rate of 1 sccm = 4.4 × 1017 atoms/s. We typically run the experiment at
5-6 sccm which is controlled by a mass flow controller from Alicat Scientific.

73

1st5Stage
Cell

2nd5Stage
Cell5(Slower)(a)Buffer5Gas
Fill5Line

2nd5Stage5
Aperature
1st5Stage5Aperature

Gap

Vent

0

10

(b)

Fractional5Population

3005K
3.55K

−1

10

−2

10

−3

10

0

0.5

1

1.5

2
N

2.5

3

3.5

4

Figure 4.1: (a) Schematic of slowing cell. The YO molecules are themalized to the cold
copper cell walls via He buffer gas. Higher helium density is represented with darker blue
shading. The density in the slower cell is lower due to the gap and the vents. (b) Calculated Boltzmann rotational distribution for 300 K and 3.5 K. At the lower temperature, thefractional population in N = 1 is more than 200 greater than for room temperature.

74

3.5CKCcoldCplate

3.5CKCcoldCplate

Saw

Sbw copperC
bobbinY2O3CCablationC
targets

3.5CKCcoldCplate

MeshCcovered
2ndCaperture
1stCstageC SlowerC
cell
cell

braidedCCuC
heatClink

30CKCradiationCshield

2.5
Scw

CharcoalCsorb

fluorCSarb.u.w

2
1.5
1
0.5
0

0

0

20
0

40

60
80
TimeC[min]

100

120

Figure 4.2: (a) Front view of slower cell and cryostat. The 9 mm diameter 2nd stage apertureis covered with a 30% transmissive copper mesh. The charcoal sorb is used to cryo-pump
He buffer gas. (b) Side View of slower cell. Sintered Y2 O3 pellets are laser ablated to form
YO. The 1st stage cell is directly connected to the 3.5 K cold plate while the 2nd stage isconnected via a copper braid heat link and thermally isolated from the 1st stage. (c) YO
fluorescence measured ≈30 cm from the cell exit.

75
4.2.1

Gas flow regimes

Based on the helium flow rate and thus the stagnation buffer gas pressure, we canaccess three different gas flow regimes. These regimes can be characterized by the Knudsen
number Kn
Kn =

λ
d

(4.3)

where d is the aperture diameter and λ is the mean free path given by
λ= √12nHe σ(4.4)

where σ is the He-He elastic cross section.
In the effusive regime, Kn > 1, implying that collisions are rare near the aperture. As
a result the effusive beam is essentially a sampling of the thermal distribution in the cell.
The forward velocity has the following distribution [49]fbeam (v) =1  m 2 3 − 2kmv2Tv e B
2 kB T

Integrating to find the exact velocity, we obtainrZ ∞3π 8kB Tv̄ef f =vfbeam (v)dv =
8
πm
0

(4.5)

(4.6)

On the other extreme, for Kn less than ∼ 10−2 , inertial forces dominate over viscous forces
and many collisions occur with the buffer gas near the aperture. As a result, the YO
molecules experience a hydrodynamic boost as they exit the cell and take on the velocity of
the carrier gas. This is known as the full hydrodynamic regime or in room temperature applications, the supersonic regime [47]. Here, enthalpy is converted into the forward velocity andis therefore able to cool the rotational degrees of freedom. Assuming that the temperatureof the expansion is much lower than the in-cell temperature, the terminal forward velocity
is [47, 132]sv∞ =2kB T0 γ=
mHe γ − 1

r

5kB T
mHe

(4.7)

76
vef f
37 m/s
29 m/s5K
3Kv∞
228 m/s
177 m/svmeas
120 m/s
48 m/sTable 4.1: Calculated forward beam velocities in effusive and supersonic regimes compared
to measured velocities.

We note that since YO is the minority species in the expansion, the supersonic beam’s
properties are set by the buffer gas. As a result, since mY O /mHe = 26, the velocity ofthe supersonic beam is much greater than that of the effusive beam given the same celltemperature. This is summarized in Table 4.1. It turns out that our original YO beam [71]
was actually run in the so called partially hydrodynamic regime [133] as our measured
center velocity was 120 m/s , which is intermediate between vef f and v∞ . Due to the large
forward velocity, buffer gas cells are not used in the hydrodynamic regime. In the effusive
regime, the extraction efficiency is poor as any non-helium species will stick to the cell
walls. Therefore buffer gas cells are typically run in the intermediate regime or partiallyhydrodynamic regime [129] with Kn between 0.01 and 1. However, for the 3D MOT, whose
capture velocity is only ∼10 m/s, a slower beam is highly desirable and this is the subjectof the next subsection.

4.2.2

Slower Cell

In order to further slow the beam, we attached an additional slowing cell [133, 134] tothe 1st stage. This 2nd stage cell is deliberately made to be very leaky as shown schematically
4.1 (a). The 2nd stage cell is stood off 2.8 mm away from the main stage’s aperture face.
In addition, two rectangular vents on the side of the cell provide additional leakage andthe beam exit aperture is 9 mm in diameter. The ratio of the slowing cell leak area to themain leak area (i.e. the aperture area) in our system is 50:1 . These leakages creates apressure region intermediate between the cell and the low vacuum region of the rest of thecryostat. This yields a region with a mean free path on the order of several mm. Therefore,

77
the molecules experience several collisions on its way out, shedding some of their forward
velocity. Copper mesh of various transparencies was placed on the side vents (40%) and theaperture (30%) in order to fine tune the intermediate pressure. The longitudinal doppler
spectrum was measured 89 cm downstream of the cell and was fit using a Gaussian multipliedby the square of the velocity in order to account for transverse spreading of the beam.
f (v) ∝ v 2 e−m(v−vc )2
2kB T

(4.8)

The result is a 48 m/s beam with a tempature of 8 K as shown in 4.3. This is more than afactor of 2 reduction in the beam velocity,
Photographs of the slowing cell are shown in Figure 4.2. The slowing cell is made ofan OFHC copper block that is 0.38” long with the same 1.5” square face as the main cell.
The aperture to aperture distance is 12.5 mm. A machine drawing of the cell is shown inAppendix 2. In order to prevent unnecessary heating of the slowing cell from the ablation
laser, the slower is stood off by stainless steel washers (κ = 0.2 W/m.K) and is cooled
via a flexible copper heat link, which is described in section 4.3.2 of Wesley Campbell’s
thesis [135]. This keeps the temperature of the slower cell ≈0.5 K lower than the main cell
during operation. Our slowing cell was based very closely on the a design from the Doylegroup at Harvard University and was presented in reference [134].

4.2.3

In-cell buffer gas density

To estimate the density of the helium buffer gas, we need to know the conductance of
the aperture. In the viscous regime, the mean free path is much smaller than the aperture
diameter, and the conductance in Liter/s is given by [136]:
s
γ−1
2γ T
9.13Aap γ1ξ(1 − ξ γ )Cvisc =
1−ξ
γ −1M

(4.9)

where Aap is the area of the aperture in cm2 , T is the temperature in K, γ = 53 is the ratioof the specific heat of He at constant pressure to the specific heat at constant volume and78
M is the atomic weight of He. The variable ξ = PPout, the ratio of the pressure outside vsininside the cell. However, for helium, for PPout
below the critical value of ≈0.49, ξ saturates tointhis critical value. This is indeed the regime that we are in as the pressure outside the cellshould be much lower than inside.
Plugging these numbers into Equation 4.9, the conductance is√
Cvisc = 6.5Aap T(4.10)

which gives a value of 1.6 liter/s where we haveused values of T = 3.5 K, and A =π(0.15cm)2 = 0.07cm2 . The exponential pumpout time is then given by
τpumpout = Vcell /Cvisc = 11.8ms.

(4.11)

Therefore at steady state, at a flow rate of 5 sccm, the number density of He atoms in thecell isnHe = τpumpout × 5 × 4.4 × 1017 /V = 1.4 × 1015 cm−3

(4.12)

where V is the volume of the 1st stage cell.
4.3

Charcoal Cryopumping
Pumping out the buffer gas by a mechanical pump is difficult as the cell is enclosed

two layers of radiation shielding, which makes the pumping speed very small. In addition,
turbomolecular pumps has a poor compression ratio for helium gas. For example, for the
Varian Turbo-V 70 [137], the ratio is 4 × 103 . This makes it difficult to have a sufficiently
good vacuum to make the YO beam.
With the temperatures that we can attain with our pulse tube refrigerator, cryopumping for any other gas is automatic. For example, the vapor pressure for neon at 4 K
is less than 10−9 torr [49]. In contrast, at 4 K the vapor pressure of helium is still several
hundred Pascal, and hardly any cryopumping occurs. However, it turns out that activatedcharcoal cooled to less than 10 K becomes an excellent cryopump for helium [131]. With

79≈100 cm2 of 4 K, several hundred l/s worth of pumping power can be attained. In addition,
the charcoal, due its porous nature, can adsorb several percent of its weight in helium at
4 K. This is of key importance due to the relatively high flow rates that we use.
To implement this, we glued the charcoal to the 4 K radiation shield with the Stycast
2850 epoxy [130]. In addition, to utilize the additional volume inside the cryostat, we built
a set of 12 copper fins shown in Figure 4.2(a). These were connected to the 4 K cold plate
via 1 inch diameter brass rods. As compared to only coating the inner 4 K radiation shieldwith charcoal, these additional fins increased the charcoal sorb area by a factor of 5 with asorb area of 900 cm2 . The temperature of the charcoal attached to the fins is 8 K. We notethat without these additional fins, the overall adsorption capacity was initially a limiting
factor to the run time to the experiment, where we initially obtained a continuous run timeat 6 sccm of approximately 15 minutes. The addition of the fins gave us a run time of about
2 hours as shown in Figure 4.2 (c) making it feasible to run the experiment.
As the beam quality would degrade after about 2 hours of running, particularly in thelower end of the velocity distribution, we have to regenerate the charcoal sorb. To do this,we let the cryostat warm up by turning off the pulse tube. This process is accelerated as
the helium pressure rises as it is desorbed and acts as a convective exchange gas to room
temperature. The cryostat temperature rises to about 40 K very rapidly and we use theTurbo-V 70 turbo molecular pump at low speed to pump away the excess helium. A gate
valve installed between the cryostat and the rest of the apparatus allows us to isolate thetwo vacuum regions and protects the MOT apparatus from this burst of helium. This allows
for ablation target changes without exposing the MOT region to atmosphere.

fluor. (arb.u.)

80

8
6
4
2
0

0

25
50
75
forward velocity (m/s)

100

Figure 4.3: Forward velocity distribution of YO as measured 89 cm from the cell exit via
light induced fluorescence. The solid line is a fit to a Gaussian (with center longitudinalvelocity of the beam at 49 m/s and a temperature of 8 K) multiplied by the square of theforward velocity to account for the transverse velocity spread.

Chapter 5
Magneto-Optical Trapping of YO

5.1

Introduction to Laser CoolingLaser cooling of atoms and ions has had a long and venerable history starting with thefirst proposals for Doppler cooling in 1975 [138,139] and its first demonstration in 1978 [140].
Laser cooling atoms has become a cornerstone of atomic physics. However, it was not until
2010 that the first molecule was laser cooled [70].
Building on laser cooling, the magneto-optical trap (MOT) was first demonstrated in
1987 in Bell Laboratories [141] by using inhomogeneous magnetic fields to use the same large
forces available from the photon cycling already exploited in Doppler cooling to spatiallyconfine neutral atoms.
To gain an quantitative understanding of these two processes, we consider a moleculemoving with velocity v. Following the notation of [73], the molecule has a set of stateslabelled with ψl and ψu (for lower and upper) and the transitions have angular frequency
ωu,l and natural linewidth Γ. Multiple circularly polarized laser beams, each of which may
contain several frequency sidebands, illuminate the molecule with frequency ωp , intensity Ip
and propagate in the p̂ unit vector direction . For each transition, there is a set of detunings
associated with each laser beam or frequency side band δu,l,p . The saturation intensity Is foreach transition is given by
Is =

πhcΓ
3λ3u,l(5.1)

82
where λu,l are the transition wavelengths. The fractional population in each state Nl , Nu willthen be governed by the following rate equationsṄl = ΓX

rl,u Nu +uXRl,u,p (Nu − Nl )(5.2a)u,pṄu = −ΓNu +X

Rl,u,p (Nl − Nu )(5.2b)

l,p

where rl,u are the branching ratios calculated in Sections 2.7. The excitation rates Rl,u,p are
given by [1, 73]:
Rl,u,p =Γ
2

1+4

rl,u sp δ −2πk p̂·v−∆Zeeman 2
p

p

(5.3)

u,l

Γ

are the laser’s angular wavenumber ∆Zeeman= (µu − µl )B/~ is the differential
where kp = 2πu,lλZeeman shift between the two states. (The magnetic moments, µu − µl are calculated in
Section 2.3) and the on-resonance saturation parameter sp is given by:
sp =

Ip
Is

(5.4)

To provide the inhomogeneous magnetic field, a quadrupolar magnetic field is used andthe spatially dependent field can be written as:yx
B = A( x̂ + ŷ − zẑ)
2
2

(5.5)

The on-resonance saturation parameter sp is related to the Rabi frequency Ωu,l by rl,u sp =2|Ωu,l,p |2 /Γ2 .
Because spontaneous emission is isotropic in free space, this process leads to no force on
average. Therefore, the force is the sum of the excitation due to each laser beam multipliedby the photon momentum, i.e.
F=~X

kp p̂Rl,u,p (Nl − Nu )

(5.6)

l,u,p

It is the imbalance of these scattering rates from counter-propagating beams as a function of velocity and magnetic field that leads to the damping and restoring forces associatedwith doppler cooling and magneto-optical trapping.

83
5.1.1

2 Level Atom

For a qualitative understanding of how laser cooling works, we assume that the magnetic field is zero and the molecule is a 2-level system. We first apply a single frequency laser
counterpropagating to the velocity v of the atom.
The force exerted on the atom is given by the stimulated absorption rate multipliedby the momentum of the photon and can be written as
F = Γ~kp N̄u = ~kp

Γ
2

sp

2
1 + sp + 4 δ+kΓ p v(5.7)

where N̄u is the steady state population obtained by solving the rate equations. As expected,
this force saturates to F = ~k Γ2 at large sp .
If we added an additional laser beam to the system that is copropagating with the
motion of the atom, we can solve the rate equations (Equation 5.2) to obtain the opticalmolasses force [142]:
F1D,OM =8sΓkp vδ4
2
2
16(kp v) /Γ + 8(kp v) (1 + s − 4δ 2 /Γ2 + (1 + 4δ 2 )((1 + 2s)Γ2 + 4δ 2 )(5.8)

As the denominator is positive definite, the sign of this force is only dependent on kv
and so we see that if δ is negative, this yields a damping force for all velocities. This isthe basis of doppler cooling. In figure 5.1 we plot this damping force for various saturation
parameters as a function of velocity. We note that with increasing saturation parameter,as the overall scattering rate increases, the cooling force increases and due to the power
broadening, and the overall width of the cooling force also increases.
To first order in kp v the force is linear with respect to v and we can define the damping
coefficient β/m as:
8~sΓkp2 δ
F
β
≈− v=
v
m
m
m(Γ2 + 4δ 2 )(1 + 2s + 4δ 2 /Γ2 )(5.9)

In the limit of s  1 this is consistent with conventional laser cooling theory [1, 143], which
is derived by adding the forces from each of the counterpropagating beams. This turns out

84
to be
F ≈

5.1.2

8s~kp2 δv
mΓ(1 + s + 4δ 2 /Γ2 )2(5.10)MOT Forces

The doppler shift kp v and the Zeeman shift ∆Zeeman affect the populations and hence
the forces (Equation 5.2) in a mathematically equivalent manner. Therefore, we may replace
kv with A(µu~−µl ) . Applying the same linearization process for a stationary molecule about z=0, the MOT force is therefore approximated asF
8~sΓkp (µu − µl )Aδ≈ −ωz2 z =z2m~m(Γ + 4δ 2 )(1 + 2s + 4δ 2 /Γ2 )(5.11)

Therefore, if the ωz2 term is positive, the molecule will be confined about z = 0 and will
perform harmonic oscillation with secular frequency ωz /(2π). For the quadrupole geometry,the other two trapping frequencies are given by1
ωx = ωy = √ ωz .
2

(5.12)

Similar to doppler cooling, for small displacements we may linearize this restoring force andobtain in the z direction,
Fz = −βv − mωz2 z(5.13)

where the secular square of the secular frequency can be expressed as
ωz2 z =(µu − µl )A)β~k(5.14)

This is the familiar first order differential equation describing a damped harmonic oscillator.
To ensure that the MOT forces confine and not blow up our sample we must also pay
attention to the polarization of the light. Figure 5.2 (a) illustrates a typical MOT transition,(a so called type I MOT [141]) . The upper state has more total angular momentum thanthe lower state, i.e. J 0 = J + 1. A laser beam with σ − circular polarization propagates
towards the left and may only drive the mJ → m0J = mJ − 1 transition. The σ + laser beam

85
propagating to the right only drives the mJ → m0J = mJ + 1. As both lasers are red detuned
to facilitate doppler cooling, an atom at a position of positive B-field will preferentially absorb
from the leftwards propagating beam as shown in 5.2 (a). Similarly, an atom experiencing anegative B field will preferentially absorb from the rightward propagating beam. This resultsin the net confining force in Equation 5.11.
Figure 5.2 (b) shows a schematic of the so called type-II MOT. Such MOTs operate
on either J → J 0 = J − 1 or J → J 0 = J transitions. Due to the requirement of maintaining
rotational closure, all previously laser cooled molecules [70, 71, 75] have used such Type-II
MOT transition. The type-II MOT is complicated by optical pumping into Zeeman sublevels
dark to the restoring beam leading to zero confinement. For example in panel b, it turns
out that this optical pumping leads to a population imbalance between the mJ = ±1 levels
that exactly cancels the ratio of the absorption rate between the confining and anti-confining
beam thus leading to zero confinement [74]. (The mJ = 0 while dark to both beams in 1D
is remixed via orthogonal MOT beams.)
However if the ground state Zeeman sub-levels are aggressively remixed, then the combination of the red detuning and polarization selection rules will lead to the MOT confining
force similar to the Type-I case as shown in Figure 5.2 (b). In this work, this remixing isachieved by rapidly switching the circular polarization of the MOT beams in phase with thedirection of the magnetic quadrupole field [71]. This is described in further detail in sections
5.2.3 and 5.2.6. We note that for polar molecules, this remixing can also be achieved by reprojecting the molecular’s angular momentum by applying a pulsed electric field non-parallelto the magnetic field [68].
Despite the expectation of zero confinement in type-II MOTs without additional remixing mechanisms, this trapping scheme has been demonstrated in several atomic systems[144–147] and was also demonstrated in the SrF 3D MOT system [72, 74]. A theoreticalexplanation was put forth by Tarbutt in 2014 [73] for which the non-zero Zeeman shift inthe upper state breaks this symmetry and yields the weak confining forces.

86

Figure 5.1: Force on a 2 level system by two red detuned counterpropagating lasers.

'

'

0

B

0

B

Figure 5.2: (a) Illustration of Type I MOT, where transitions have the form of J → J 0 = J+1.
Illustration of Type II MOT, where transitions are of type J → J 0 = J − 1.

87
5.1.3

Doppler Temperatures

Equation 5.13 would imply that the temperature of the sample would converge toabsolute zero and is clearly nonphysical. The finite temperature is due to random momentum
kicks due to spontaneous emission. The momentum drift due to this process is zero as the
spontaneous emission is spatially isotropic. The evolution of the momentum can be thoughtof as a random walk, thus the variance of the momentum scales linearly with the numberof emission events. Following from reference [148], the rate of change of the momentum
variance isX
d < p2x >= 2~2 k 2
p̂Rl,u,p (Nl − Nu )
dt
l,u,p

(5.15)The heating can then expressed as
1 d < p2x >
dQ
=
dt
2m
dt

(5.16)

Balancing the cooling rate −βv · v at low saturation and for δ = −Γ/2, the final
temperature is given by
Tdoppler =~Γ
2kB(5.17)

Tdoppler represents the minimum temperature attainable by Doppler cooling and for the YO
molecules’ main cooling transition X 2 Σ+ → A2 Π1/2 , this turns out to be 116 µK.
The root mean squared velocity this corresponds to is
vdoppler =~Γ= 9.5cm/s
2m(5.18)

Two other special laser cooling temperature can be derived [1]. The first is the so called
Capture Limit. This corresponds to the velocity at which the doppler shift is equivalent to
the natural linewidth. I.e. vc = Γ/k, this yields a temperature of

Tc =

mΓ2
kB k 2(5.19)

This yields values of vc = 3 m/s and Tc = 100 mK. This yields an order magnitude estimateof the maximum trappable velocity.

88
The much colder limit is given by the recoil velocity due to a single photon, vr = ~k/m.
This temperature is given byTr =~2 k 2kB m(5.20)

This evaluates to Tr = 500 nK and vr = 6.2 mm/s. Reaching sub-Doppler temperatures
involve Sisyphus type cooling [1,149], distinct from the Doppler cooling described above andwas observed in laser cooling experiments in SrF in [70].
5.2

A 2D MOT for molecules.
In this section, we will describe the experiments on the transverse laser cooling andmagneto-optical trapping of the a cryogenically produced buffer gas beam of YO [71].

5.2.1

Recap of YO structure

We briefly recap the salient features of the YO structure used in optical cooling. The
main optical cycling transition is the X 2 Σ+ → A2 Π1/2 at 614 nm. The natural lifetime of thistransition Γ−1 = 33 ns allows for rapid cycling. Due to the highly diagonal Franck-Condon
factors as calculated in section 2.5, vibrational branching into the A2 Π1/2 state is limited.
We therefore apply repump lasers at 648 nm and 649 nm to repump the X 2 Σ+ , v 00 = 1 →A2 Π1/2 , v 0 = 0 and the X 2 Σ+ , v 00 = 2 → A2 Π1/2 , v 0 = 1 as shown in Figure 5.3 (a).
Fig. 5.3 (b) shows the rotational and hyperfine structure. Parity and angular momentum selection rules prevent decays to all rotational states except the N 00 = 1 state (seeSection 2.6). To address all of the hyperfine states in the ground electronic state (see section 2.3, in addition to the frequency tuned to the X 2 Σ+ , G00 = 0 → A2 Π1/2 transition, 2
additional frequencies ∼ 750 MHz are used to address the G00 = 1, F 00 = 0 and the spectroscopically unresolved G00 = 1, F 00 = 1, 2 states. The hyperfine structure is similar across v 00 =0-2 (see Table 2.1), hence the same frequency sideband structure can be used to address all
vibrational states.

89

(a)

(b)
-4
~4e

A Π1/2

F' = 0,1

0.994
614 nm 4
q01 = 0.0038
648 nmq00 =e-4

q 02+ = 3.6

A'2Δ3/2

690 nmA2Π1/2 ,J' = 1/2(+)

2

G''=0
v" = 0v" = 1

X2Σ+N'' = 1 (-)
G''=1
X2Σ+

G=I+S

Figure 5.3: (a) Rovibronic structure. (b) Hyperfine structure of the X 2 Σ+ state. Orange
solid arrows show the three frequency components used to excite the hyperfine components.

90
5.2.2

Beam Apparatus

The initial YO molecule beam is produced via a cryogenic buffer gas beam source, whichis thermally anchored to a 4.5 K liquid helium reservoir described in Chapter 4. The YO
density in the N 00 = 1 state is measured to be ∼ 1010 cm−3 with ∼ 1010 molecules produced
per pulse. The forward velocity of the beam is measured via doppler shift fluorescence
spectroscopy with mean velocity of 120 m/s and FWHM of 40 m/s. A 2.5 mm aperture isplaced 130 mm away from the buffer gas cell aperture. This collimates the beam and results
in a transverse temperature of 25 mK.
After the collimating aperture, the YO molecules fly through a 10 cm interaction region
where they interact with the cooling lasers, set up in a retroreflected multipass configuration.
The molecules fly through the ∼3 mm (FWHM) diameter cooling lasers 11 times, whichcorresponds to a 275 µs interaction time. The multipass setup in each transverse dimension
consists of a pair of mirrors and λ/4 waveplates. This ensures the correct polarization for
MOT confinement. We use a rectangular baseball geometry coil to provide the quadrupole
magnetic field. This will be discussed in further detail in Section 5.2.5.
After the interaction region, the molecules are allowed to free fly a further 30 cm tothe detection region. This detection region consists of a multipass clean-up beam consistingof the v 00 = 1, 2 repumps followed by a retroreflected detection laser consisting of the samelight used for the laser cooling. The fluorescence is imaged onto a CCD camera in the ydimension. The temperature is calculated via
Tx = σv2x mY O /kB .

(5.21)

Due to the long free flight region after the interaction region, the spatial profile of the YO
beam is dominated by the variance of the velocity, σv2x . Monte-Carlo calculations using themagneto-optical forces are used to determine the spatial and velocity profile of the YO beam;
this allows us to assign a temperature based on the beam profile.

91

CCD
pro

cle
a

n-u

be

pb

cryogenic
beam source

eam

be

am

MOT coil

llim
co
g

in

at
er

ap

y

re
tu
ablation beam

co
o

lin

g

be

x

z

am

Figure 5.4: Depiction of the MOT/Laser Cooling apparatus in the 1D implementation. An
additional multipass in the y direction is implemented for the 2D MOT. (Image created by
Brad Baxley)

92
5.2.3

Polarization chopping and Doppler cooling

The last piece in the puzzle to obtain optical cycling is the repumping of Zeeman
dark states. For the J 00 → J 0 = J 00 + 1 type II MOTs which are required for rotational
closure, optical pumping into dark Zeeman states is guaranteed. The first molecular coolingexperiment [69, 70] used an off axis magnetic field to Larmor precess the molecules out ofthe dark Zeeman sublevel. However, YO is not amenable to this technique as the G00 = 0
hyperfine has a very small Zeeman shift (Figure 5.8 (a)) at the relevant magnetic fields.
Therefore, we use the method of polarization chopping in order to repump the Zeeman
sublevels.
From the analysis of Berkeland and Boshier, it turns out this technique works efficiently when the polarization is modulated at frequency of order the natural linewidth of thetransition [150]. The polarization chopping is done with a Pockel’s cell and a λ/4 waveplate.
This yields alternately σ + and σ − light. The cold YO beam is sent through the coolingmultipass as described in the previous section, with each v 00 = 0 laser frequency component
detuned by δ = −2π × 5 MHz. We see that the final transverse temperature 1 monotonicallydecreases with increasing chopping rate as shown in Figure 5.5. The cooling rate settles at
a chopping rate ∼5 MHz consistent with the analysis of Reference [150].

5.2.4

Doppler cooling of YO

With all the pieces in place for optical cycling, we consider the Doppler forces that the
YO molecule experiences. To calculate the cooling and trapping forces fully, we need to take
into account 44 zeeman states, 15 optical frequencies and time dependant polarizations. andsolve Equation 5.2. However, certain simplifications turned out to satisfactorily model thesystem and match experimental data. The X 2 Σ+ state was modelled with only 5 states. The
first state represents the v 00 = 1, 2, the second represent the v 00 = 0, G00 = 0. For this only the1

The initial transverse temperature is 15 mK here due to a different collimating. All subsequent experimental data in this chapter has an unperturbed temperature of 25 mK

93
614 nm light with carrier frequency, fY O addresses this state. The last three states represent
the G00 = 1, F 00 = 0, 1, 2 manifolds respectively. These three states are addressed by the two
frequency components at fY O + 760 MHz and fY O + 774 MHz to address the F 00 = 0 state
and the F 00 = 1, 2 respectively. We note that the two-toned AOM also puts on unintended
frequency sidebands at fY O + (2 × 760 − 774) MHz and at fY O + (2 × 774 − 760) MHz. These
sidebands have a factor of 10 less power than the intended sidebands but must be taken
into account in order to correctly model the cooling and MOT forces. With a saturation
parameter of 20 we can calculate the optical molasses forces. In panel a, we see that coolingis least maintained for all velocities. Due to the spin-rotation and the dipolar hyperfine
interaction, there is a ∼ 4 Γ splitting between the F = 0, and F = 1, 2 manifolds, leading to
the more complex shape of the optical molasses force compared to a simple 2 level system.
In panel (b), the damping coefficient mβY O is plotted as a function of the laser detuning.
Here we see an even greater departure from the typical two level system’s behaviour where
there are regions in the frequency space for which negative detuning leads to heating andpositive detuning leads to cooling, the opposite of the expected behaviour. In the case of reddetuning, this occurs when the frequency sideband used to originally address the F 00 = 1, 2
manifold is detuned by more than 1.5 Γ, it ends up being closer to the F = 0 transition.
As the laser frequency is now in effect blue-detuned with respect to this transition, it yields
a net heating effect.
Figure 5.7 (a) shows the transverse YO beam spatial profile under various conditions.
The fluorescence curve (i) shows the unperturbed beam with a transverse temperature of
25 mK. Turing off the v 00 = 1, 2 repumpers in both the clean up beam and the interaction
beam curve(ii) leads to severe depletion as expected. Curves (iii) shows doppler coolingwhere we red detune all of the v 00 = 0 laser frequencies by δ = −2π × 5 MHz and as a result
the spatial distribution narrows and the temperature decreases to 10 mK. In order for thevibrational repumps to work at maximum efficiency, they are all left at resonance. Curve(iv) shows the effect of blue detuning the lasers to δ = +2π ×5 MHz and we as expected heat

94
the beam to 60 mK. (b) Shows the final temperature Tx as a function of laser detuning. We
find the optimal cooling occurs at δ = −2 × 5 MHz. In addition, due to the closely spaced
hyperfine levels, cooling can occur at apparent positive detunings.

95

transverse temperature

15

10

5

0

0

1

2

3
4
5
6
7
Polarization modulation rate (MHz)

8

9

10

Figure 5.5: Temperature after Doppler cooling as a function of polarization chopping rate.

96

(a)

(b)

Figure 5.6: (a) Acceleration due to optical molasses on YO molecule with saturation parameter = 20 and laser detuning for each level of -Γ = −5 MHz. (b) β/mY O as a function oflaser detuning. Note that there are regions of blue (red) detuning which cool (heat).

8

(a)
i

6
iii

4

iv

2
0

(b)

60

ii

-10

-5

0

5

Position0(mm)10Tx0(mK)

Fluorescence0signal0(arb.0u.)

9740
Γpol (MHz)

20

0

-8

-4

0

4

8

Laser0detuning0δ/2π0(MHz)Figure 5.7: 1-D Doppler cooling. (a) Spatial profile of YO beam under the conditions: (i)unperturbed, (ii) Doppler cooling turned on and v 00 = 1 repumper turned off, (iii) Doppler
cooling (δ = −2π× 5 MHz) , (iv) Doppler cooling (δ = 2π× 5 MHz) (b) Temperature of 1-D
doppler cooled YO beam in terms of laser detuning.

98
The MOT

5ab 2582 G2=2T/2F2=21

Energy25MHzb

2577
T
+17T

4

2

5bb
6

8

1T

φ

F92=2T/1

G2=21/2F2=2T

+18T

21

σ"Pσ+2

/2F2=

G2=21

+19T

Coupling2of2Zeeman2states2leads+2TT G2=21

r

/2F2=

+21T

T

2

G42=21

22

4

σ+Pσ"

6

8

Magnetic2Field25Gb

1T

m2=2"1P+1
m2=2T
m2=2+1P"1σ"
σ+

B2field

5.2.5

Time

Position

Figure 5.8: (a) Zeeman energy of the X 2 Σ+ state. (b) Schematic of polarization chopping.

5.2.6

A YO MOT

While the polarization chopping does solve the problem of optical pumping into dark
Zeeman sublevels, it does create a significant problem to realize a MOT. In essence, thehandedness of the polarization of the incoming laser determines whether the MOT forces
confine or blows up the YO beam, hence, with the polarization chopping, each cycle willconsist of equal times with confining and anti-confining forces averaging to zero. To solve
this problem, we therefore had to switch the direction of the magnetic field in phase withthe polarization switching as shown in figure 5.8 (b). Such a scheme was shown to work at
acoustic frequencies by Harvey and Murray [151], however in our case, we need to switch
our fields at radio frequencies.
In order to match the aspect ratio of our interaction region, we built a rectangularbaseball geometry quadruple coils [152] as shown in figure 5.9 (a). The coil is wound with 25
turns of 350/46 Litz wire. The Litz wire was chosen as each wire consists of a bundle of 350
individually insulated 46 gauge wires. The skin depth at ∼ 1 MHz is 50 microns, which is

99
greater than the diameter of each wire, therefore we can maintain both high current carrying
capacity and conductivity at rf frequencies. The coils dimensions are 5 × 5 × 15 cm.
The coil itself acts as the inductor in a resonant LC circuit configuration as shown infigure 5.9(b). We were able to achieve a quality factor of Q ≈77 at a frequency of 2 MHz.
In order to achieve the required field gradient, we required a peak current 1.5 A which isprovided by a inductively coupling 25 W of RF power through a home-built impedance
matching transformer. Assuming perfect rf power coupling, we should obtain a root meansquare value of the magnetic field gradient to Arms = 6 G/cm.
As the expected voltage drop across the coil is on the order of 4 kV, for voltage
management, the coils were wound in layers of 5 and glued on to the previous layer with
Loctite 401 epoxy. Each layer is then coated with Q-Dope. After winding, the coils are
potted in Epo-Tek H70E epoxy. It combines low outgassing with decent thermal conductivity(0.9 W/(m.K)). In-vacuum cooling water lines are attached to the coils with 1 inch thick
Boron Nitride standoffs. This is because that bulk metal placed less than a cm from the coils
would noticeable reduce the Q (by ≈ 10%) (presumably due to Eddy currents). Boron nitride
was used as it has good thermal conductivity (30 W/(m.K)) but is electrically insulation.
During the running of the experiment, the coils would heat by ≈ 10 K, thus leading toshifts in the phase of the magnetic field with respect to the supply voltage. For simplicity,
we held the polarization switching frequency and phase fixed. The magnetic field is detectedwith a pickup coil and is phase stabilized to the polarization chopping as shown in figure
5.9(b). This phase locking also allows us to place an arbitrary phase offset φ between thepolarization chopping and the magnetic field.
Figure 5.10 shows the effect of applying the MOT cooling forces in the x̂ dimension.
The laser detuning was set at δ = −2 × 5 MHz and the v 00 = 1, 2 repumps are on resonance..
In panel (a) the YO beam spatial profiles under various conditions. Curve (i) shows theuncooled beam while (ii) shows the purely doppler cooled beam for comparison. Turning on
the MOT forces with the correct polarization switching (φ = 0) further cools the beam down

100
to Tx ∼2 mK. There is also a clear enhancement of the molecular signal at x=0, signifying
an increase in density in that region. In panel (iv) we switch the magnetic field out of phasewith respect to the polarization switching and the beam is only cooled down to 15 mK.
While the MOT forces are anti-confining, the doppler cooling is still in effect.
Figure 5.10 (b) shows the dependance of the final transverse temperature as a functionof the phase offset between the polarization switching and the magnetic field. As expected,
the final temperature is coldest when in phase and least cold when out of phase. The final
temperature can be modelled with a MOT force varying by cos(φ).
While we have achieved a greater than 10 fold decrease in transverse temperature, wenote that we are still a factor of 20 above TDoppler which is due to the short interaction time.
From this reduction in temperature, we can calculate the doppler cooling rate 2β/mY O to
be 5×103 s−1 . Which is in good agreement with the theoretical model of 8×103 s−1 . In
addition, using equation 5.14the MOT frequency ωx is found to be 2π × 155 Hz which isgood agreement with the calculated value of 2π × 155 Hz as derived from a field gradientof Arms = 6G. We note that the YO beam’s interaction time is much less than the MOT
oscillation period but is well explained by the rate equation model and provides a clear
brightening of the beam. We then extend the MOT action to two dimensions. For this,we set up an identical multipass in the ŷ dimension. Figure 5.11 (a) shows the effects of
the MOT and doppler cooling in only the ŷ dimension while still measuring the spatial
profile along the x̂ dimension. From this we see an enhancement of the signal and this isbecause the spatial spread of the unperturbed molecules in the ŷ dimensio is larger than thedetection beam. Therefore, the doppler cooling (curve (ii)) and MOT (curve (iii)) brightens
the beam towards the center and thus more molecules are detected as compared to theuncooled beam (i). Note that Tx is unchanged as we do not manipulate the molecules inthis dimension. Figure (b) shows a comparison between the 1D and the 2D MOT. Curve(i) is the unperturbed case and curve (ii) shows the spatial profile for a 1D MOT. The
2D MOT action (curve iii) does a similar overall beam enhancement as in (a). However,101
the final temperature is higher than of the 1D MOT and is consistent with damping and
MOT forces one half of the 1D case. This is to expected qualitatively as when driven above
saturation, the scattering rate is limited to in our case Γ/7. As half of these scatters are due
to absorptions from the ŷ cooling beams, the force along the x̂ dimension has to be halved.
This result also comes out be solving equations 5.2 and eq: Multilevel Force.
We also observe a ∼ 10% decrease in the total molecular signal when performing thedoppler cooling and MOT experiments. This is due to decays through the intermediate
A02 ∆3/2 electronic state that leads to optically pumping into dark rotational states. Due tothe short ∼275 µs interaction time, this is not a problem in this experiment but presents
a serious challenge for the loading of of a 3D MOT. The solution to this problem will be
discussed in Chapter 6.

102Transformer(a)Water Cooling

Boron Nitride
Standoffs

Litz Wire
Baseball Coils(b)

Figure 5.9: (a) Photograph Baseball MOT coils resonance circuit. (b) Schematic of resonant
MOT coil. The phase of the magnetic field is monitored with a pick-up coil and is locked tothe polarization switching phase.

8 (a)

15

iii

(b)

ii
10

6

iv

4

i

TxK(mK)

FluorescenceKsignalK(arb.u.)

103

5

σ+
σK-

2
0

-10

-5

0

5

PositionK(mm)100−180 -120 -60

φ

MOTKField

0

60 120 180

φK(degrees)

Figure 5.10: 1-D MOT and Doppler cooling. YO beam spatial profiles for (i) unperturbed
beam, (ii) a Doppler cooled beam (iii) a 1D MOT (φ = 0) (iv) a 1D anti-MOT (φ = 180◦ )(b) (Purple) Tx as a function of φ, the phase offset between the polarization switching andthe magnetic field. (Red) Doppler cooling only.

104

6

8 Unperturbed

5

iii
ii

Signal (arb.u.)Signal (arb.u.)

Doppler (1D y)

7 MOT on (1D y)
6
5

i

4
3
2

Unperturbed
MOT on (1D x)
MOT on (2D)

iii

4

ii

3

i

2
1

1
0

-10

-5

0

5

10

0

-10

x position (mm)Observed Tx

30

-5

0

5

x position (mm)10Transverse Cooling
Unperturbed
Anti-MOT
Doppler
MOT

25
20
15
10
5
0

One dimension (x)Two dimensions

Figure 5.11: (a) YO beam profiles in the x̂ dimension with cooling action only in the ŷdimensions. (i) An unperturbed beam, (ii) Doppler Cooling only and (iii) 1D MOT. (b) YO
spatial profiles for (i) Unperturbed, (ii) 1D MOT in x̂ an (iii) 2D MOT. (c) Summary oftransverse temperature Tx for Doppler and MOT cooling in 1 and 2 dimensions.

Chapter 6
Microwave remixing and longitudinal slowing.

6.1

Introduction
There are two closely related problems that we need to solve to extend the 2D MOT

technology to the realization of a 3D MOT for YO. The first is to sufficiently slow the
YO beam to MOT trappable velocities, and the second is to stop the leakage through theintermediate A02 ∆3/2 electronic state. In this chapter, we will make an estimate for the MOT
trapping velocities, which will turn out to be much slower than what is emitted from thecryogenic beam source. Branching through the A02 ∆3/2 state presents a problem as decays
through this state lead to decays into the N 00 = 0, 2 rotational dark states. This presents a
problem to both the final MOT lifetime and the generation of sufficiently slow YO molecules
to be loaded into the trap.

6.1.1

Capture Velocity Estimation

A standard method to estimate capture velocity is to simulate the trajectory of themolecule and look for the highest velocity for which it stays within the cooling laser beam
radius [1], which we shall set to 5 mm.
For the laser and MOT cooling along 1 dimension, we determined that the doppler
cooling rate isΓD =2β= 5000s−1
m

(6.1)

106
and the trap oscillation frequency isωM OT =p
κ/m = 2π × 155Hz(6.2)

For the 3D MOT, the cooling forces must be derated by a factor of three from the
1D case as we now have 3 pairs of counter-propagating beams. We therefore use the force
equation,mβFx = − v − (ωM OT )2 x
3
3

(6.3)

In figure 6.1, the trajectories are plotted in terms of position and velocity. Eachmolecule starts at position -5 mm with varying velocities. The trajectory for the 11 m/s
molecule is shown in red and as it travels past +5 mm it would no longer be trapped.
Therefore by this criterion we estimate 10 m/s as be the capture velocity of the MOT.

107

12

10

Velocity [m/s]

8

6

4

2

0

−2
−5

−4

−3

−2

−1

0
1
Position [m]

2

3

4

5

6
−3

x 10

Figure 6.1: Simulation of capture process for 3D MOT. Above 10 m/s, the molecule does
not turn around by x = 5 mm and is lost.

108
6.1.2

MOT Loading in a cell

Due to the relatively large longitudinal velocity of the beam compared to the MOT
trapping velocity, it may be tempting to perform the MOT inside the cryogenic cell and notworry about trying to slow the molecular beam. However, this approach turns out to not be
feasible.
We simulate the MOT action 1 cm diameter laser beams on all three cartesian axes
with the MOT forces. The new 3 D MOT coils are designed to have the same magnetic field
gradient along the weak axes as the baseball coil in the 2D MOT. Therefore, we simulate
the MOT using the expression for the forceβmFx = − v − (ωM OT )2 x
33mβ
Fy = − v − (ωM OT )2 y33β2m
Fz = − v −(ωM OT )2 z
3
3

(6.4)
(6.5)
(6.6)

We assume that the MOT is in a uniform density of helium, and we model the YO
molecule’s trajectories as well as their internal rotational states. Fluorescence is estimatedby counting the number of molecules within all 6 cooling beams and in the N 00 = 1 state.
Further details of the simulations can be found in the Appendix 1. The estimated collision
rate can be expressed in terms ofγcol = nHe σvcol(6.7)

where nHe is the buffer gas number density, σ is the collision cross section and vcol is themean relative velocity of the He and YO molecules. As YO is heavier than helium, thisvelocity is roughly the rms thermal velocity of helium. While the YO-He collisions crosssection is not known, the collision cross section between trapped CaH and He was measured
to be ∼ 10−14 cm2 and we use this value to estimate the He-YO collisions rate from a known

109
helium density.
Figure 6.2 shows three helium density regimes. The high helium density regime isneeded in order to thermalize the ablation plume and in our cell that density is ∼ 1015 cm−3 .
In this regime, the YO molecules are confined due to the high He density and concomitantlong diffusion time constant. At low pressures the MOT forces dominate and lifetimes are on
the order of several 10s ms. In between these two extreme density regimes, there is a regionof minimum lifetime. This is shown in Figure 6.3. This minimum lifetime in our system∼1 ms and occurs at a helium number density of 1013 cm−3 . This presents a problem as thepump out time of the cell as calculated in Chapter 4 is ∼12 ms, which is much longer than
this minimum lifetime. Therefore, there is no way to move between the high and low density
regimes quickly enough without destroying the YO MOT.

110

5

10

4

10

nHe = 1017 cm-3
γcol = 1.8e7 s-1

3

10nHe = 109 cm-3
γcol = 1.8e-1 s-1

2

10

1

nHe = 1013 cm-3
γcol = 1.8e3 s-1

10

0

10

0

0.002

0.004

0.006

0.008

0.01

Figure 6.2: Simulation of expected fluorescence signal for three different buffer gas density
regimes.

111

40
35
1/e MOT LifeTime [ms]nHe~1013 cm-3

30
25
20
15
10
5
0
2
10

3

10

4

10

5

10

6

10

7

10

8

10

γcol [s −1]
Figure 6.3: 1/e MOT lifetime as a function of He buffer gas density. The minimum lifetime
is at 0.8 ms corresponding to an approximate density of 1013 cm−3 .

112
6.2

Microwave mixing and slowing

6.2.1

Introduction

While directly loading a buffer-gas cooled atomic beam into a three-dimensional (3D)
magneto-optic trap (MOT) is possible [153], for molecules with much lower photon scattering
rates compared to typical laser cooled atomic species, longitudinal optical cooling is necessary
for loading into a 3D MOT. Typically 104 photons must be scattered to slow molecules towithin the capture velocity of a MOT. During the slowing process, the molecular transition
used for optical cooling experiences a Doppler frequency shift of tens or hundreds of resonancelinewidths, resulting in inefficient photon scattering for a monochromatic unchirped laser
beam. To date, molecules that have been laser cooled all have the magnetic sub-levels in
the ground electronic state continually remixed by the multi-level optical cycling process.
This is consequence of the type-II MOT transition required for rotational closure. Hence,
the widely used atomic Zeeman slower [154], which relies on the Zeeman shift of a single
magnetic sublevel to compensate for the Doppler shift during deceleration is not applicable.
To maintain a sufficiently large optical scattering rate throughout the slowing process, twotechniques have been employed. Chirping the laser frequency was shown to reduce thevelocity of a supersonic beam of CaF by 30 m/s [75]. For SrF, broadband laser radiation
was used to slow molecules to be loaded directly into a 3D MOT [72].
Optical cycling in molecules [68, 70] is critical for the radiation pressure force to directly slow and cool molecular beams. However, achieving optical cycling in a large class ofmolecules remains difficult due to optical pumping into dark states. While vibrational darkstates can be addressed with repump lasers [67], maintaining rotational closure has heretofore been ensured with a mixture of angular momentum and parity selection rules. However,there are several loss mechanisms that break the rotational closure in current molecular optical cycling schemes and they arise from additional decays via intermediate states or higher
order transition moments.

113
The first mechanism is due to decays between vibrational levels in the ground electronic
states. In the case of YO, this loss mechanism occurs due to decays from the v 00 = 1, N 00 = 1
state down to the v 00 = 0 state. Decays from higher vibrational states are suppressed by
the harmonic oscillator selection rules, ∆v = ±1. Due to parity selection rules, this process
will end up populating the N 00 = 0, 2 levels in the case of YO for example. From [155], thetransition dipole moment is given byrµv+1,v =

v+1
2

r

~
mY O ωe



dµ
dR


(6.8)
R=Re

where v here is the vibrational quantum number, ωe = 861cm−1 is the vibrational constant,dµand dR
is the derivative of the electric dipole moment function evaluated at Re , theR=Reequilibrium internuclear distance. The derivative’s value is calculated to be 3.5 Debye/a0 [82]where a0 is the Bohr radius. The resulting transition dipole moment, µ1,0 , is .090 Debye.
The linewidth is then given byΓvibrational =1∗ ωe3 µ21,0
3π0 ~c3(6.9)

which yields a value of 1.6 s−1 or a 1/e lifetime of 620 ms.
Magnetic dipole moments connect states of equal parity, in contrast to electric dipole
transitions which connect states of opposite parity. Decays via magnetic dipole transitions
are typically suppressed by a factor of α2 ≈ 5 × 10−5 compared to electric dipole transitions
where α is the fine structure constant [156]. However recent measurements in OH indicatethat they can be a factor of ∼10 faster than expected [58], making this a potentially large
population leakage mechanism.
For some molecules, maintaining optical cycling can be more complex due to decays
through intermediate electronic states, which are present in, for example, YO [78] andBaF [157]. Such decays are normally suppressed by the relatively long transition wavelengths. For YO the suppression factor is ∼3×10−4 compared to the main cycling transition.
However, the decay is sufficient to cause population leakage to rotational dark states whena large number of photon scatters are needed for cooling.

114
For the rest of this section, we discuss the demonstration of microwave mixing ofrotational states to close these additional loss channels. This enables us to slow a beam of
YO emitting from a two-stage buffer gas cell to molecular velocities < 10 m/s, at which
point the YO molecules can be loaded into a 3D MOT.
The optical cycling transition in YO was discussed in sections 2.6 and 2.5. In brief,
highly diagonal Franck-Condon factors limit vibrational branching [81] such that a single
repump laser limits branching loss to 3.6 × 10−4 (Fig. 6.4(a)). Rotational branching is
suppressed by a combination of parity and angular momentum selection rules [68, 69] whendriving the X 2 Σ, N 00 = 1 level, where N is the rotation quantum number. Figure 6.4(b) shows
the hyperfine structure of the X 2 Σ states, where G is the coupled nuclear and electronic spin,
and F is the total angular momentum. For the A2 Π1/2 and A02 ∆3/2 states, J is the total
angular momentum excluding nuclear spin. For all electronic states, p = ± represents theparity.
Figure 6.4(a) shows the additional loss channel where the A2 Π1/2 , J 0 = 1/2 state decays
to the A02 ∆3/2 , J 0 = 3/2 state. Decays to states of higher J in the A02 ∆3/2 are forbidden due
to angular momentum selection rules. The A02 ∆3/2 state has a radiative lifetime of ∼1 µs [78]and will rapidly decay back to the X 2 Σ state. Since the full cycle starting from and returning
to the ground state is a three-photon process, the parity at the end of this process mustchange sign and therefore allow decays only to even rotational states (Fig. 6.4(c)). In theX 2 Σ state, while J 00 is not a good quantum number, each rotational state can be expressedin terms of states with J 00 = N 00 ± 1/2. The ∆J = 0, ±1 angular momentum selection rule
prevents decay to N 00 ≥ 4. This is because the lowest value of J 00 for these states is 7/2.
Therefore this process optically pumps into the N 00 = 0, 2 states. The rotational constant,B, for the X 2 Σ state is 11.634 GHz. Microwave radiation tuned to 2B (4B) can mix the
N 00 = 0 (2) levels with the N 00 = 1 level. This mixing effectively removes the rotational darkstates; however the maximum optical scattering rate is lowered as the number of states in
X 2 Σ involved in optical cycling increases. We estimate the maximum photon scattering rate

115
as follows,Γmax,cycle = Γnene + ng(6.10)

where ne and ng are the number of zeeman states in the excited and ground state respectively.
In the case of the optical cycling described in chapter 5, ne = 4; the number of states ineach rotational level in the X 2 Σ+ state is given by 8N 00 + 4, as such ng is 24 where we note
this value also includes the zeeman states in v 00 = 1, N 00 = 1 as the repumping drives to thesame A2 Π1/2 , J 0 = 1/2 state as the main cycling transition. This yields a maximum cycling
rate of Γ/7.
Once microwaves are applied, ng increases to 48. This is because the v 00 = 0 state now
involves states from the N 00 = 0, 1, 2 levels. As we will see shortly, the number of states inv 00 = 1 is unchanged at 12 as the microwave radiation that we produce is not sufficiently
power broadened to mix the v 00 = 1 rotational states. This leads to a maximum cycling rateof Γ/13 once microwave mixing is applied, nearly a two-fold decrease.
We employ this rotational state microwave mixing scheme for longitudinal slowing ofa YO beam generated from a cryogenic buffer gas beam source as described in Chapter 4.
We laser ablate sintered Y2 O3 pellets in a copper cell filled with helium buffer gas at 3.5 K(Fig. 6.5(a)). The hot YO thermalises rotationally and translationally to the cold helium
and the two stage cell [49] produces a YO beam with a mean forward velocity of 70 m/s asmeasured in the detection region. The YO beam then travels 89 cm to the detection region,
during which it may be illuminated by a 6 mm diameter, counter-propagating slowing beam.
The 614 nm v 00 = 0 laser has 70 mW of power and the 648 nm v 00 = 1 laser has 80 mW(Fig. 6.4(a)). The slowing laser has three frequency components to address the hyperfine
manifold (Fig. 6.4(b)). To provide the rotational mixing, we use two pairs of microwave
horns located along the beam path. Each pair produces ∼1 mW of either the 23 GHz or
46 GHz microwaves.
For velocity sensitive detection, we measure the fluorescence induced by a low intensity

116
4 mW/cm2 , 614 nm beam aligned 45◦ to the molecular beam. This probe laser has asaturation parameter of ∼0.5 (Fig. 6.4(b)). This eliminates mechanical and optical pumping
effects from the detection beam. In contrast to the slowing beam, we note that the doppler
sensitive detection beam only consists of the frequency components to address the G00 =0, F 00 = 1 and G00 = 1, F 00 = 1, 2. Due to the small 15 MHz hyperfine spacing, molecules in
the G00 = 1, F 00 = 1, 2 manifold travelling ∼15 m/s faster than the intended velocity classend up being resonant with the G00 = 1, F 00 = 0 frequency component, thus this frequency
component is not used as it causes overestimation of the slow molecule number. The detection
laser can be scanned over a range of ∼100 MHz at a rate of 1 kHz, corresponding to a velocity
detection range of ∼100 m/s.

117

(a)

(c)

(b)
-4

F~3e

2

A Π1/2A Π1/2 ,J' = 1/2(+)

q00 F=F
0.994
F614Fnm 4
q01F=F0.0038
648FnmF'F=F0,1

A'2Δ3/2,Fv' = 0,J' = 3/2 (-)

6e-4

q 02+FFF=F3.

A'2Δ3/2

F690Fnm2

N''F=F2F(+)
G''=0vMF=F0vMF=F1

X2Σ+N''F=F1F(-)
G''=1
X2Σ+

GF=FIF+FS

46FGHz760FMHz

N''F=F1F(-)
20FMHz N''F=F0F(+)

23FGHz2 +X Σ ,Fv''= 0

Figure 6.4: (Color online) (a) YO vibronic structure. Dashed arrows indicate decay paths
with corresponding Franck-Condon factors, q [81]. Solid arrows indicate cooling and repump
laser transitions. (b) Hyperfine structure. Solid arrows indicate the three hyperfine pumping components used in each vibrational manifold. (c) Rotational mixing. Dashed arrows
indicate the decay paths to X 2 Σ through the A02 ∆3/2 state and black solid arrows show the
microwave rotational mixing.

118

y
e

x

ob

YaO

microwave
horns

pr

ablation
laser

PMT
YbO

fluor.nYarb.u.O

Y2O3ntargetsnn

v=n0

v=1

8

YcO

6
4
2

exitn
0
bufferngasn
aperture 0
fillnline 1stn 2ndn
stage stage

25
50
75
forwardnvelocitynYm/sO

100

Figure 6.5: (a) Depiction of the beam apparatus. The YO molecular beam interacts witha counter-propagating slowing laser beam along x. A 45◦ probe beam is used for Doppler
sensitive detection. A PMT is used to collect fluorescence induced by either the slowing
or probe beam. (b) Two-stage buffer gas cell. (c) Velocity distribution measured at thedetection region. The solid line is a fit to a Gaussian (with center longitudinal velocity ofthe beam at 49 m/s and a temperature of 8 K) multiplied by the square of the forward
velocity to account for the transverse velocity spread.

119
6.2.2

Optical cycling enhancement via microwave mixing.

We study the rotational mixing by observing the fluorescence induced by the slowing
beam in Fig. 6.6(a). We apply the microwaves in four binary combinations. For case i, no
microwaves are applied. In contrast, when microwave mixing is applied to both N 00 = 0, 2,(curve iv), there is a factor of four increase in fluorescence. Applying only the N 00 = 0 ↔
N 00 = 1 or N 00 = 2 ↔ N 00 = 1 mixing (Fig. 6.6(a) ii, iii respectively), we see a factor of two
increase in fluorescence compared to case i.
These measurements in panel (a) can also be used to identity the predominant opticalpumping mechanism. As calculated in Section 6.2.1, the vibrational lifetime is ∼600 ms,
which is much longer than the optical pumping time scale. The branching ratios, calculated in
Section 2.7, allow us to eliminate the M1 transitions as the branching ratios from the A2 Π1/2
state is 2/3(1/3) for the N 00 = 0(2) levels in the X 2 Σ state. On the other hand, electric dipole
transitions from the A02 ∆3/2 state have equal branching ratios to the N 00 = 0, 2 rotational
levels, consistent with curves ii and iii. Hence, we identify decays through the intermediate
A02 ∆3/2 as the dominant loss process.
To verify that the mixing microwaves are sufficiently strong to address all hyperfine
states without the need of additional frequency sidebands, we simultaneously vary the frequency of the N 00 = 1 ↔ N 00 = 0(2) mixing microwave by δ(2δ) and measure the corresponding enhancement in fluorescence (Fig. 6.6(a) inset). This enhancement follows a
24 MHz full-width half-maximum Lorentzian lineshape. This width is larger than the hyperfine variations of <5 MHz (see Section 2.3) among the three relevant rotational manifolds.
The mixing for higher vibrational states is suppressed because the rotational constant B forv 00 = 1 is 50 MHz smaller than that for v 00 = 0. This results in a 100 (200) MHz detuning
for the N 00 = 0 (2) transition in v 00 = 1. Hence, we do not suffer any further reduction in the
maximum optical cycling rate.
In Fig. 6.6(b), we study the time dependence of the optical pumping into v 00 = 1, and120
also into rotational dark states via a previously unexplored branching route from the A2 Π1/2
state to the A02 ∆3/2 state. Again we measure the fluorescence induced by the slowing beam.
For case i, the slowing and repumping light as well as microwaves are turned on throughout
the experiment. For case ii, the v 00 = 1 repump laser is rapidly shut off in ∼10 µs at 14
ms and rapidly switched back on at 19 ms after ablation. This leads to an abrupt decreasein the fluorescence as the YO molecules are rapidly pumped into the v 00 = 1 state and nolonger scatter light. To extract this pumping rate, we divide curve (ii) by curve (i) and fit
an exponential with a constant background ratio of 0.25, resulting in a 1/e time constant of
620 µs. Combining this with the calculated Franck-Condon factor q01 = 3.8×10−3 [81] yields
a photon scattering rate of 4.3×105 s−1 . This is smaller than the maximum possible scattering
rate by a factor of ∼5 [71] which may be due to inefficient depletion due to highly Doppler
detuned molecules, insufficient power in the wings of the slowing beam, or an overestimateof the q01 Franck-Condon factor. Curve iii shows the fluorescence when only the rotational
mixing microwaves are turned off between 14 − 19 ms. The 1/e decay due to pumping into
rotational dark states has a 7.4 ms time constant (inset curve iii). By combining the ratioof the two loss rates and q01 , we estimate the branching ratio from A2 Π1/2 to A02 ∆3/2 tobe ∼3 × 10−4 . This branching ratio agrees with the value derived from transition dipole
moments calculated in Ref. [82] and is similar to the combined branching ratio to v 00 ≥ 2.
At 19 ms after ablation, the vibration repump (curve ii) and mixing microwaves (curve iii)
are turned back on and the fluorescence immediately recovers.

121

Figure 6.6: (Color online) (a) Optical cycling enhancement due to rotational mixing. The
fluorescence is induced by the high power optical slowing beam and v 00 = 1 repump: (i)No mixing; (ii) N 00 = 0 ↔ N 00 = 1 mixing only; (iii) N 00 = 2 ↔ N 00 = 1 mixing only;(iv) simultaneous N 00 = 2 ↔ N 00 = 1 and N 00 = 0 ↔ N 00 = 1 mixing. The inset shows
the fractional enhancement of optical scattering vs. microwave detuning from the rotational
resonance. (b) Comparison of branching losses between v 00 = 1 and A02 ∆3/2 : (i) Microwaves
and v 00 = 0, 1 are on throughout; (ii) v 00 = 1 repump is shut off from 14-19 ms after ablation;(iii) Microwave mixing is switched off from 14-19 ms. The inset shows curves (ii) and (iii)
divided by curve (i). Solid green lines are exponential fits.

122
6.2.3

Radiation pressure slowing

We estimate the capture velocity of a 3D MOT for YO with a laser beam diameter of
1 cm to be 10 m/s. To slow some fraction of the molecules to below this capture velocity,
we apply longitudinal radiation pressure slowing with protocols shown in Fig. 6.7(a). To
increase the fraction of molecules addressed by the slowing beam, we spectrally broaden thelight [158] via a resonant electro-optic modulator (EOM) with a modulation frequency of
10 MHz and a phase modulation index of 3.3. This spectrally broadens each laser frequency
to ±30 MHz, corresponding to a velocity spread of ±20 m/s. Polarization switching used fordestabilizing Zeeman dark states [71] further adds 5 MHz sidebands and we obtain frequency
components spaced by the natural linewidth of the main cooling transition. This ensurescontinuous deceleration across a ∼ 50 m/s velocity range (Fig. 6.7(b)).
As the range of the slowing force is smaller than the velocity spread of the YO beamitself, we implement frequency chirping on the slowing light. In Fig. 6.7(c) and (d), theslowing sequence is designed to enhance the number of molecules at 30 m/s. The molecules
propagate freely for the first 12 ms after ablation, then the slowing beam is turned on from
12-22 ms. During this time, the center of the broadband radiation is linearly swept frombeing resonant with 64 m/s to 44 m/s molecules. In panel (c), we see a molecular population
enhancement between 20 − 40 m/s (negligible for below 20 m/s), and depletion for velocities
greater than 40 m/s. In panel (d), we maintain the probe laser’s frequency to be resonantwith the 30 m/s molecule and observe an enhancement in the number of molecules in thisvelocity class. Due to off-resonant excitations, some faster molecules (35 m/s) contribute to
the detected signal at 27 ms for the unslowed case.
To produce even slower molecules, we increase the time during which the slowing laser
is turned on and perform a frequency chirp to address lower velocity molecules. The slowing
light is turned on 2 ms after ablation and is left on for the next 24 ms. During the first 20 ms,
the center frequency of the broadband slowing laser is linearly increased from being resonant

123with 64 m/s to 34 m/s molecules (Fig. 6.7(a)). Figure 6.7(e) shows the velocity distribution
after the slowing sequence compared to the original velocity distribution. Slowing enhances
the molecular population below 20 m/s and depletes molecules with velocities higher than
20 m/s. We tune the probe laser frequency to be resonant with the 10 m/s molecule andwe see a clear enhancement in the number of molecules in this velocity class (Fig. 6.7(f)).
Unslowed molecules leaving the cell with 10 m/s forward velocity will arrive 89 ms after theablation, which is well after the detection time. Hence, we attribute the non-zero 10 m/s
signal in the unslowed case to off-resonant excitations from fast molecules.
For the detection time windows shown in Fig. 6.7(d,f), we estimate the percentage ofmolecules in the 30 m/s and 10 m/s velocity classes with a Doppler resolution of ±3 m/s tobe 2% and 0%, respectively for the unslowed beam, based on Fig. 6.5(c). The laser slowing
enhances the corresponding population fractions to be 3% and 0.1% of the total original
distribution. We note that without microwave mixing, the radiation pressure slowing stops
working due to the fast optical pumping to the dark states.
We demonstrate enhanced optical cycling in YO via microwave mixing of rotational
states in the ground electronic state. This enabled us to slow longitudinally the YO beam
via radiation pressure, producing molecules that can be loaded into a 3D MOT. Because
this technique maintains rotational closure with less stringent requirements on the cyclingtransition, it becomes relatively straightforward to perform laser cooling of molecules on
J 00 → J 0 = J 00 + 1 transitions, which would avoid the complications of dark states thatare present in J 00 → J 0 = J 00 − 1 molecular MOTs [71, 73], as was originally proposed [68].
This opens the door to laser cooling a greater variety of molecules with more choices ofcooling transitions. To further close the vibrational branching to < 10−6 , an additionalv 0 = 1 ← v 00 = 2 repump laser at 649 nm can be used [71, 81]. By turning off the microwave
mixing for N 00 = 1 ↔ N 00 = 0 at an appropriate time, we can accumulate trapped cold YO in
the ro-vibrational ground state. The A02 ∆3/2 state lifetime of ∼1 µs opens up the possibilityof a narrow line MOT [90, 159] and we will discuss this in the next chapter.

124

white7lightvel.7center

347m/sdopplerdetection

fluorescence7(arb.u.)

0 2

500
500

20
20

80
100
40
60
80
100
forward7velocity7(m/s)
4060
forward velocity (m/s)

(d)

6

10

4

5

2

0

0

20
40
60
forward7velocity7(m/s)

1

250
250
0

(c)

12

307m/s

20

80

3

2

750

00

22 26
time7(ms)

150fluorescence7(arb.u.)

750

2647m/s(b)

1000

deceleration (m/s )slowing

deceleration7(m/s2)

1000

relative7sideband7power(a)ablation

30
40
time7after7ablation7(ms)

50

2
(e)

8

107m/s

1

4

(f)

0

0
0

20
40
60
forward7velocity7(m/s)

80

-1

20

30
40
time7after7ablation7(ms)50

Figure 6.7: (Color online) (a) Timing sequence used to slow molecules to below 10 m/s. (b)Example of spectrally broadened slowing light with center velocity of 50 m/s. Black lines
show the sidebands generated from an EOM and rapid polarization switching. The red curveshows molecular deceleration with an optical saturation parameter of 3. (c-f) Comparisons
between no slowing in red and with slowing in blue. Solid lines are guides to the eye. (c)Doppler spectra for a 10 ms long slowing sequence. (d) Time of flight for 30 m/s molecules.
(e) Doppler spectra for a 24 ms long slowing sequence as depicted in (a). (f) Time of flightfor 10 m/s molecules.

Chapter 7
Prospects for a 3D YO MOT.

7.1

3D MOT Coils
As the interaction volume is now only limited to the intersection of the 6 MOT beams,

as opposed to the 10 cm long interaction region required for the 2D MOT, the MOT coil
design is somewhat simplified. Again in order to maintain sufficient field gradients with thehigh switching rates, we require the MOT coils to be in vacuum. The coils are made froma commercial composite laminate board designed for rf applications [160]. This consistsof a 1.5 mm dielectric material, known as 6035 HTC. The laminate board has thermal
conductivity of 1.44 W/(m.K) and has decent outgassing properties. We obtain vacuum
pressures on the order of 10−8 torr, much smaller than the partial pressure due to the He
buffer gas (10−6 torr). Rogers boards have been gaining traction as a construction materialof surface ion traps (see for example reference [161]). On both sides of the dielectric materiala 35 µm layer of copper is electro-deposited.
The MOT coils are set up in the standard anti-Helmholtz configuration. On both sideof the Rogers Board, a spiral track is cut into the copper surface, winding 7 times froman outer diameter of 3.4 cm to an inner diameter of 1.9 cm. A 1.8 cm diameter hole isdrilled through in the center of the spiral. To maximize the magnetic field gradient, eachcoil consists of two Roger’s boards stacked on top of each other with a thin polyimide film
to insulate between the two closest copper faces. The two coil stacks are mounted 2.3 cm
apart. This set of coils is run in a similar LC tank configuration as the 2D MOT coils and

126

(a)

(b)

Figure 7.1: 3D MOT apparatus. (a) Machine drawing of 3D MOT apparatus. (b) Photograph of PCB MOT Coils.

127
we obtain a Q of 70 and a resonance frequency of 5.01 MHz. By coupling in 20 W of rf
power at resonance, we expect a magnetic field gradient of 12 G/cm along the cylindricallysymmetric quadrupole axis and 6 G/cm perpendicular to this axis.
We expect that the helium pressure limited lifetime to be limited to the He-YO collision
rate. The helium pressure rises by 10−6 torr when the buffer gas is turned on. Estimating
the lifetime using:τM OT = (nHe σvHe )−1(7.1)

where vHe is the thermal velocity of helium at 4 K, ∼ 18 × 103 cm/s, nHe is the number
density which we estimate from the partial pressure to be ∼ 3 × 1010 cm−3 , and σ is thecollision cross section, which is estimated to be 10−14 cm2 , the collision cross section measured
in previous He-polar molecule collision experiments [56]. This yields an estimated helium
density limited MOT lifetime of ∼160 ms.
We have set up a new imaging system for the MOT, allowing us to capture images
with a CCD camera. This is illustrated in Figure 7.2 (a). An optical band pass filter system
is placed in one of the regions where the light is collimated in order for best performance ofthe system. An iris is placed at the first image plane to block off scattered light from thein-vacuum MOT coils. Panel (b) depicts an image of the unslowed molecules illuminated by
the MOT beams. The camera was taken 10-30 ms after ablation, thus allowing us to observe
almost all of the free flying YO packet. As a quick estimate of the molecule number, weestimate the photon collection efficiency ηη = ηq × ηBP × ηΩ

(7.2)

where ηq ∼ 0.9 is the quantum efficiency for the Apogee Alta U47 CCD camera, ηBP ∼ 0.93
the transmission of the Edmund Optics NT84-118 bandpass filter at 614 and ηΩ ∼ 0.04 thefractional solid angle captured by the imaging system. This results in η ∼ 0.02. Figure 7.2(b) we subtract the image with molecules from one with no molecules, thus to first ordersubtracting the background scatter from the MOT beams. In panel (b) the image was taken

128

(a)
f = 150 mm

f = 76 mm
BP

150 mm

150 mm

150 mmCCD

76 mm 76 mm 76 mm(b)

CountsPixel bin number

iris

Pixel bin number
Figure 7.2: (a) Imaging set up with a CCD camera. BP stands for optical bandpass filter.
(b) Image of YO molecules in the MOT detection region. The size of the image corresponds
to a 1 × 1 cm object size.

129
by averaging 400 shots in each state. The total number of counts per shot was found to be
2400 counts, which corresponds to ∼ 7 × 104 emitted photons.
The conversion to a molecule number is complicated by the fact that the fluorescenceis transit time limited. The average 70 m/s molecule travels through the 1 cm beam in 14 µs,
giving it enough time to only scatter ∼10 photons. This corresponds to approximately 7000
YO molecules. Of course, absolute measurement of molecule number from light induced
fluorescence is notoriously difficult [162], as such the preliminary estimate of molecule number
should be used with caution.
7.2

A narrow line YO MOT
So far in this thesis, the A02 ∆3/2 has been cast in a villainous light as it was shownthat decays through this state turned out to be the main loss mechanism in our laser coolingscheme. However, it turns out that we can turn this state to our advantage as the much lower
linewidth for this state, 2 × π160 kHz, allows a much colder MOT with Tdoppler ∼ 5 µK [90].
7.2.1

The A02 ∆3/2 state’s linewidth

We use two methods to estimate the linewidth of the X 2 Σ+ → A02 ∆3/2 transitions.
First as the A2 Π1/2 → X 2 Σ+ transition linewidth is well known, we can compare the absorption in the cryogenic cell for the respective transitions. The identified transitions fromthe G00 = 1 are tabulated in Table 7.1 and was consistent with the observed transitions inreference [85].

130Transition

Frequency (THz)

N 00 = 0, G00 = 1 → A02 ∆3/2 (−)

434.70310(3)N 00 = 1, G00 = 1 → A02 ∆3/2 (+)

434.67989(3)N 00 = 2, G00 = 1 → A02 ∆3/2 (−)

434.63333(3)Table 7.1: Frequencies of X 2 Σ+ to A02 ∆3/2 transitions.

To verify that rotational level assignment in the X 2 Σ+ state, the frequency difference
between the N 00 = 0 and 2 states is indeed 6 times B, the rotational constant in the X 2 Σ+state. Therefore we identify the transition at 434.67989(3) THz as the one from the N 00 = 1
state and compare that absorption to the absorption from the main cooling transition (whichof course is from the same N 00 = 1 state.) We thus obtain a lifetime estimate of τ∆ = 425 ns.
The second method to estimate the lifetime involves calculating the state mixing of theA2 Π3/2 into the A02 ∆3/2 state. This is because the A02 ∆3/2 → X 2 Σ+ is E1 forbidden. Fromthe procedure outlined in [78] we can obtain the A02 ∆3/2 state in terms of Born-Oppenheimer
states as
E
ΨA02 ∆3/2 ,v0 =0 ≈ 0.9865 A02 ∆3/2 , v 0 = 0 − 0.1582 A2 Π1/2 , v 0 = 0(7.3)−0.0369 2 Π3/2 , v 0 = 1 + 0.0176 A2 Π1/2 , v 0 = 0 + ...

(7.4)
(7.5)

The dominant mixing term is a 2.5% mixing with the2

Π3/2 , v 0 = 0 and with the known∼39 ns lifetime of the A2 Π3/2 state, we estimate from this method a lifetime of 1200 ns.
These two methods give similar order estimates of the lifetime, and therefore, we use anintermediate lifetime of 1 µ s for further calculations of the MOT. Table 7.2 compares thecharacteristic laser cooling temperatures and rms velocities [1] between the main cooling andthe new narrow line transition. Most noticeably, we see that Doppler temperature is now afactor of 20 smaller compared to the main cooling transition.

131
Transitionvc

Tc

vDoppler

TDopplervrTr

X 2 Σ+ → A2 Π1/2

3 m/s

140 mK

9.5 cm/s

116 µK

6.2 mm/s

500 nKX 2 Σ+ → A02 ∆3/2

10 cm/s

150 µK

2 cm/s4 µK

5.5 mm/s

380 nK

Table 7.2: Comparison of characteristic laser cooling temperature between the main optical
cycling and the narrow line transition. Here vc and Tc are the capture velocity and temperatures respectively, vdoppler and Tdoppler are the Doppler velocity and temperatures respectively
and vr and Tr are the recoil velocity and temperatures respectively

We may derive the vibrational branching from equation 7.3. As transitions between
the leading term and the A2 Π1/2 is dipole forbidden, it is the other states that provide thetransition strength and hence the vibrational branching. One may perform the RKR [103]method to calculate the Franck-Condon factors for the A2 Π1/2 → X 2 Σ+ and A2 Π3/2 → X 2 Σ+as described in section 2.5. This results in the branching ratios shown in Table 7.3. We notethat the branching ratios from the A02 ∆3/2 , v 0 = 0 state are not as diagonal as the main
cooling transition, however this is not a problem as both the scattering rate and the total
photon numbers needed for the cooling are much smaller than for the main transition.
X 2 Σ+ , v 00 =Branching ratio

00.9422

1

0.0568

2

0.011

3

< 10−4
Table 7.3: Vibrational branching ratios form A02 ∆3/2 , v 0 = 0 to X 2 Σ+ .

132
7.3

Narrow-line cooling on the X 2 Σ+ to A02 ∆3/2 transition
Due to the lower scattering rate, this narrow line MOT transition can only be used

after the molecules have already been trapped with the normal 614 nm (orange) MOT.
Figure 7.3 (a) shows the YO level structure in the context of a possible narrow line
MOT. We choose to excite from the N 00 = 0 transition in contrast to the orange MOT as thiswould be a Type-I MOT and the formation of dark Zeeman states is no longer a problem. In
the main cooling transition, the N 00 = 0 rotational is pumped to the A02 ∆3/2 , J 0 = 3/2, p = −state and the usual angular momentum selection rules restrict decays back to the N 00 = 0
and N 00 = 2 states. Microwave radiation tuned to twice and four times the rotational
constant B are used to mix the lowest three rotational states [89], identical to the previous
chapter. In addition, the v 00 = 1 laser used to repump the orange transition can be reused
without alteration for the narrow line transition. However, as the vibrational branching inthis transition is much larger than that of the A2 Π1/2 → X 2 Σ+ transition, microwave mixingof higher vibrational also needs to be taken care of. However, as the rotational constants
are known [77, 163] (and tabulated in Table 2.1) this should be fairly straightforward withadditional microwave frequencies only several 100 MHz smaller than that already used forthe orange MOT. The saturation intensity is expected to be ∼60µW/cm2 and the wavelength
at 690 nm is generated with commercially diode lasers.
Figure 7.3 (b) shows two possible type I MOT transitions from the N 00 = 0 state.
The first transition is between the X 2 Σ+ , N 00 = 1, G00 = 1, F 00 = 1, m00F = −1 and theA02 ∆3/2 , J 0 = 3/2, F 0 = 2, m0F = −2 states. Here the Landé g factors are shown in the figureand leads to a differential magnetic moment between the upper and lower states of 0.4 µB .
The other transition occurs between the G00 = 1, F 00 = 0, m00F = 0 and the F 0 = 1, m0F = −1
and differential magnetic moment of 0.2µB .

133
7.4

Narrow-line MOT simulation
Monte-Carlo simulations were performed (predominantly by my colleague Alejandra

Collopy [90] for this MOT scheme starting from 1 mK YO sample with a trap oscillation
frequency of ωM OT = 2π×155 Hz [71], to mimic the conditions of the orange MOT. In
addition as vc is much smaller than the initial root mean square velocity we put on additionalfrequency sidebands in order address the entire velocity distribution [159,164]. We therefore
require 5 frequency sidebands each separated by 2π×1 MHz with the detuning of the bluest
sideband denoted as δ1 . The counter-propagating circular polarized light have two mainfrequencies to drive both transitions shown in figure 7.3(b). As the hyperfine splitting
between G00 = 1 and G00 = 0 is approximately 760 MHz, only optical branching from theA02 ∆3/2 state may redistribute population between the two manifolds and these are simulatedusing the branching ratio of 1:3 into the G00 = 0 and G00 = 1. The MOT forces are calculatedwith the methods in Chapter 5, however, as YO is predominantly in the N 00 = 1, 2 levels,
which are not addressed optically, the calculated MOT force must be derated by a factorof 9 due to the ratio of the total number of states in N 00 = 0 vs the number of states in
N 00 = 0, 1, 2. In addition, heating is simulated by adding a random velocity kick of vr whena photon is spontaneously emitted. Figure 7.4 shows the results of simulations with δ1 set at
2π × 240 kHz (1.5 times the natural linewidth) and a saturation parameter of 3. Panel (a)shows the acceleration vs velocity profile for the optical molasses, i.e. zero B field whilst (b)shows the MOT acceptation on a stationary YO molecule as a function of displacement from= 950 Hzthe quadruple center. At the low velocity limit, the damping rate is ΓD = m2βYOwhile the secular frequency ωM OT is found to be 2π × 33 Hz for a magnetic field gradient of
3 G/cm. The characteristic trapping time then found to be ΓD /(ωM OT )2 = 20 ms. Panel (c)and (d) shows the results of the Monte Carlo simulation. In (c) the initial phase space of
the YO molecules is indicated in blue and the phase space distribution of the YO molecules
after 20 ms of narrow line MOT action. We see that root mean square velocity decreases by134
a factor of 10, implying a decrease in temperature of 100. The spatial distribution becomes∼2 more narrow. This is because while the temperature decreases by a factor of 100, theMOT forces also decrease by a factor of 25 by going to the narrow line. The simulation
indicates a photon scatter rate of 104 s−1 and cooling to the Tdoppler requires 200 photons.
2
To extrapolate to the full 3D case, again β and ωM
OT must be derated by a factor of three

and we require 600 photons to cool, which is much less than the inverse of the v 00 ≥ 3 total
branching ratio. The v 00 = 3 lifetime is expected to be on the order of 1s and the expected
MOT accelerations are 5 times earth’s gravitational acceleration.

135

A2Π1/2 v'=0
J'=1/2(+)

A'2Δ3/2

mf'=2
mf'=1

F'=1,2A'2Δ3/2
δ0J'=3/2(-)

J'=3/2(-)
v'=0

mf'=1
mf'=0 mf'=0
mf'=-1
mf'=-1
mf'=-2

G"=0,F"=0
N"=2(+)
N"=2(+)

X2Σ+

mf"=0

N"=0(+)

mf"=1

N"=1(-)
mf"=0

N"=1(-)
N"=0(+)2 +X Σ

N"=0(+)v"=1
v"=0

G"=1,F"=1
mf"=-1Figure 7.3: Level structure for the narrow-line MOT. (a) Rovibronic structure of the narrow
line transition. The maroon arrow indicates the main transition used in the narrow line
MOT. Gray lines indicate microwave mixing of rotational states and the red arrow indicatesthe v 00 =1 repump, which is the same frequency as used to repump the orange MOT. (b)Zeeman structure and MOT scheme. g represents the corresponding magnetic Landé factor.
The large arrows indicate counter propagating circularly polarized laser beams. Both F 00 =0→ F 0 =1 and F 00 =1→F 0 =2 transitions can effect a MOT action (maroon arrows).

(a)
200
0

−200
−1

Acceleration4(m/s2)

Acceleration4(m/s2)

136

(b)
100
0

−100

0

1

0
-0.5
0.5
Position4(mm)Velocity4(m/s)
(d)

0.4
0

-0.4

0
-0.5
0.5
Position4(mm)Temperature4(K)Velocity4(m/s)

(c)

−4

10

−5

10

−6

10

0

20

Time4(ms)40

Figure 7.4: Narrow line 1D MOT simulations. (a) Damping acceleration as a function ofvelocity at zero magnetic field. (b) Restoring MOT acceleration as a function of displacement
at zero velocity. (c) Phase space distribution of YO molecule for (blue dots) X 2 Σ+ → A2 Π1/2
MOT and (red) X 2 Σ+ → A02 ∆3/2 MOT after 20 ms of cooling of this transition. (d) Narrow
line MOT temperature as a function of time.

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Appendices150

Appendix 1
He-YO collisions simulations

This appendix describes the simulation technique used to model the He-YO collisions.
Much of this work is inspired by Robert Michniak’s thesis [165]First, we look at elastic collisions between the buffer gas and the YO molecules. In ourtrap, we will tend to get a wide variety of impact parameters as well as collision energies.
Since we do not know the differential cross section (DCS) a priori, we will assume an isotropic
one. In this simulation, the helium atoms’ trajectories are not tracked.
Since the density of YO is much smaller than the density of Helium, the elastic crosssection rate is given by
γ = nHe σel vrel

(1.1)

where nHe is the buffer gas density, σel is the elastic cross section between He and YO and vrelis the relative velocity between the two species. The reciprocal of this gives a characteristic
time of collisions. YO molecules are assumed to not interact with each other. The collisiontime is assumed to follow a Poissonian process with mean inter-collision time set to beτcol = 1/γ. The probability distribution of collision times is proportional to e−tcol /τcol . To
generate this we use the following prescription [166]:tcol = τcol ln(1 − rand)(1.2)

where rand is uniformly distributed random number between 0 and 1.
Therefore, each YO molecule will evolve for the time tcol according to the MOT forces.
When a collision occurs, we do the following: a helium atom is generated with a random

152
velocity consistent with the Maxwell-Boltzmann distribution. We transform to the He-YO
center of mass frame and randomize the YO velocity direction as we assume an isotropic
collision cross section. We then transform back to the lab frame and generate a new collisiontime and repeat the process.
1.1

Simulation Results.
We perform these simulations in FORTRAN 95, using a quasi-fixed step symplectic

integrator [167]. In order to correctly simulate the MOT forces, time steps are restricted to
be no more than 100 ns. The time steps are reduced when the randomly generated tcol drops
to below the 100 ns time step.
The first test is to turn off collisions and compare the analytic solution of the damped
harmonic oscillator with our numerical solution. The results are shown in figure A1 and areself consistent.
The next step is to add in elastic collisions. As a consistency check, we compare oursimulation with no MOT forces and at various helium densities and compare to the analytic
diffusion time constant for a cylinder as given by [168]:
τ = ((π1
2.4 2) + ( )2 )−1
R
L
D

(1.3)

where D is the diffusion constant for a binary gas given by the Chapman Enskog formula [169]:s
32πkB TD=(1.4)
16nHe σdµwhere σd is the momentum transfer cross which in the case of an isotropic DCS, is the sameas the elastic collision cross section. We can calculate the diffusion with Equation 1.3 fora cylinder with 1.5 cm radius and 3 cm height. We obtain a diffusion time constant of
2.8(5.6) ms for a helium density of nHe = 5 × 1015 (1016 ) cm−3 .
We performed a simulation of the YO molecules in these two density regimes as shownin figure A2. Here a uniform density of YO molecules is initialized and allowed to collide

153

−3

6

x 10

4

x [m]

2

0

−2
Fortran Solution
Exact Solution (10 m/s)
Exact Soltuion (5m/s)
−4

−6

0

0.001

0.002

0.003

0.004

0.005 0.006
Time [s]

0.007

0.008

0.009

0.01

Figure A1: Simulation of YO trajectory with no helium collisions. Solid lines are analytic
solution while points are calculated using the Fortran symplectic integrator.

154
elastically with the 4 K molecules. Upon striking the boundary of the aforementioned cylinder, the YO molecule is lost. The resultant time constants are 2.4(4.3) ms for helium densityof 5 × 1015 (1016 )cm−3 , consistent with the analytic prediction from Equation 1.3. Therefore
we conclude that our elastic collision protocol is indeed working.
Finally, we consider rotational collisions. We assumed that the collisions may change
the rotational state by ∆N 00 = ±1 or the molecule remains in the same rotational state. The
algorithm proceeds as follows.
1.) Upon collision, we calculate the center of mass collision energy Ecol given by2
Ecol = 21 µred vrel.

2.) In order for the collision to excite the molecule to the next highest rotational state,conservation of energy dictates that Ec ol > 2(N 00 + 1)~B where B is the rotational constant.
If the equality is satisfied, we proceed to 3a, otherwise we jump to step 3b.
3a) The number of states in each rotational level is given by 8N 00 +4. If there is sufficientenergy for rotational excitation, we then excite to N 00 +1 with probability (3+2N 00 )/(3+6N 00 ),relax to N 00 − 1 with probability (−1 + 2N 00 )/(3 + 6N 00 ) or stay put at N 00 with probability
1/3.
3b) In the case where we cannot excite, the rotational state relaxes to N 00 − 1 withprobability (2N 00 − 1)/(4N 00 ) or stays put at N 00 with probability (2N 00 + 1)/(4N 00 ).
4.) The magnitude of the relative COM velocity is reduced or increased appropriately
in order to conserve energy if a rotational transition does occur.
Using this crude approximation for rotational collisions, we calculate the rotational distribution and as shown in figure A3, we can still obtain good agreement with the Boltzmannrotational distribution.
Putting these pieces together we were able to calculate the expected lifetimes as discussed in section 6.1.2 for the various helium densities.

155

nHe=5e15 cm−3n−3

=1e16 cm

He

4

Molecule Number

10

3

10

1

2

3

4

5
Time [s]

6

7

8

9
−3

x 10

Figure A2: Simulation of YO number in a cylinder with two different densities with no
MOT force and with elastic helium collisions. The extracted diffusion time constants are
2.4(4.3) ms for helium density of 5 × 1015 (1016 ) cm−3

156

Rotational Distribution With Angular Momentum Conservation
0.35
Monte Carlo
Boltzmann

0.3

Molecule Fraction

0.25
0.2
0.15
0.1
0.05
0

0

5

10
N

15

20

Figure A3: Simulation of rotational distribution. Blue dots are derived from simulation
using density of states and conservation of to assign probabilities for changes in rotationlevel. Green dots indicate the Boltzmann distribution at 4 K.

Appendix 2
Beam Source Machine Drawings

PRODUCED BY AN AUTODESK EDUCATIONAL PRODUCT

4

3

2

1

.14

.15

D

D

.850

.380
C

C

1.50

B

B

PRODUCED BY AN AUTODESK EDUCATIONAL PRODUCTPRODUCED BY AN AUTODESK EDUCATIONAL PRODUCT

1.50

1.25
DRAWN

YO Data Taker

1/29/2015

CHECKED

A

QA

TITLE

A

MFG
APPROVED

4

3

2

PRODUCED BY AN AUTODESK EDUCATIONAL PRODUCT

REV

SIZE

DWG NOC

Buffer_gas_cell_full

SCALE

SHEET

1

1 OF 1Figure A4: Machine drawing of 2 stage cell. Diameters of 1st stage aperture is 3 mm andaperture diameter of the slower stage is 9 mm.

158

Area
Aperture
Gap
Side Vents2
π × 9mm= 0.64cm22

4×1.25 inch×2.5 mm
0.42 inch×0.1 inch×2

T ransmission
0.3
1
0.4

Effective Area
0.19 cm2
3.81 cm2
0.216 cm2

Table A1: Table of effective leak areas of the second stage cell

Appendix 3
Electronic Schematics

A

B

C

D

J1

R4 C1
62
0.1uF

C2
0.1uF

Figure A5: Electronic schematic of frequency to voltage convertor.

3

2

1

8

J4

8

+

+

C3
1uF

C32
0.1uF

-15V

10UF

C30

C31
0.1uF

+15V

10UF

C29

R2 150R1 150
1

3

2

2

1

4

OUT

OUT

1
GND

GND
2

LM7905CT
U8

IN

IN

U7
LM7805CT

MC10125
U1-A

3

3

11

12

3

2

7

-5V

C33
0.1uF

C34
0.1uF

+5V

7

S
Q

Q

+5V

13

R

CLK Q

D

S

10

+5V

+5V

1

R

CLK Q

D

4

+5V

8

9

U2-B
74AS74

6

U2-A
74AS74
5

3

2

1

M1

M2

M3

/4 - 80MHz

J5/2 - 40MHz

6

M4

-15V

R6
2.2k

R5
820

Q1

13V

R7
100

B
C

E

R14
200

12.3V

E

C

C

E

B

C6
1nF

B

10.5V

B

C5
1nF

Q3 Q2

C

E

Q4
2N3904

9.95V

2N3906

20mA

R3
510

+15V

9.24-10.8V

C4
0.1uF

6

-15V

C8
0.1uF

5

C7
1nF

R11
5.1k

R10
2.7k

R13
2.7k

R12
12k

9.8V

R9
12k

R8
2.7k

5

D3

D4

-15V

R16
7.5k

4

3
+

R15
7.5k

+15V

2

1

4

+
3

1

2

3

C11
2pF

OMIT
R21

R22
OMIT

10k
R20

U6
8

C35
0.1uF

1

-15V

4

+

3

7
-

CW

L1
362uH

R27
10K

OP-07CP

6

R23
10k

C36
0.1uF

+15V

C12
1nF

C13
270pF

2

D6

D5

C37
0.1uF

+5V

3

R25
100K

C28
1uF

C14
1.5nF

L2
283uH

R28
5.1k

CW

R24
OMIT

C15
510pF

R17
220

R27,28 and C37 are not on board and must be added.

C10
0.1uF

4

C9
0.1uF

6.95V
LM399H
U4

U3
LM399H
6.95V

4

C24

1uF

330nF

42.2k
R18

C16 10pF

C25

1

5

6
10
R26

J3

SHEET 1 OF 1

2

1

DRAWN BY: C. SAUERJN: YJ132A1

DATE: 4/2011

BUILT FOR: M. YEO

HIGH SPEED F-V CONVERTERJILA ELECTRONICS LAB

LF356N

C26
0.1uF

U5

-15V

4

+

-

J2-2

BUILT BY:

3

2

1

2

0.1uF

33nF

10nF

3.3nF

1nF

330pF

100pF

7 C27 0.1uF

+15V

11 CW

3

4

5

6

7

8

9

10

1

DESIGN BY: J. HALL / CS

220
R19

J2-1

C23

C22

C21

C20

C17

C19

C18

2

A

B

C

D

160