Ultracold molecules for the masses: evaporative cooling andmagneto-optical trappingby
B. K. Stuhl
B.A., Cornell University, 2005
M.S., University of Colorado Boulder, 2010

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
2012This thesis entitled:
Ultracold molecules for the masses: evaporative cooling and magneto-optical trapping
written by B. K. Stuhl
has been approved for the Department of Physics

Jun YeEric Cornell

Date

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

iii
Stuhl, B. K. (Ph. D., Physics)
Ultracold molecules for the masses: evaporative cooling and magneto-optical trapping
Thesis directed by Prof. Adj. Jun Ye

While cold molecule experiments are rapidly moving towards their promised benefits of precision spectroscopy,
controllable chemistry, and novel condensed phases, heretofore the field has been greatly limited by a lack of methods
to cool and compress chemically diverse species to temperatures below ten millikelvin. While in atomic physics
these needs are fulfilled by laser cooling, magneto-optical trapping, and evaporative cooling, until now none of thesetechniques have been applicable to molecules. In this thesis, two major breakthroughs are reported. The first is theobservation of evaporative cooling in magnetically trapped hydroxyl (OH) radicals, which potentially opens a path all
the way to Bose-Einstein condensation of dipolar radicals, as well as allowing cold- and ultracold-chemistry studies
of fundamental reaction mechanisms. Through the combination of an extremely high gradient magnetic quadrupole
trap and the use of the OH Λ-doublet transition to enable highly selective forced evaporation, cooling by an order ofmagnitude in temperature was achieved and yielded a final temperature no higher than 5 mK. The second breakthroughis the successful application of laser cooling and magneto-optical trapping to molecules. Motivated by a proposal
in this thesis, laser cooling of molecules is now known to be technically feasible in a select but substantial poolof diatomic molecules. The demonstration of not only Doppler cooling but also two-dimensional magneto-opticaltrapping in yttrium (II) oxide, YO, is expected to enable rapid growth in the availability of ultracold molecules—justas the invention of the atomic magneto-optical trap stimulated atomic physics twenty-five years ago.

to Deena

Acknowledgements

A doctoral degree is a long process, and is always a team effort. I want to thank Jun Ye for accepting me into
his group: it’s a great set of people, and I’ve had a lot of fun doing great science with all of you. I especially wantto thank Brian Sawyer for his years of leadership on the collision experiments and for passing on his intuition for themachine he helped build. I also overlapped briefly with Eric Hudson and Benjamin Lev, who both helped at the very
beginning to introduce me to the world of dipolar molecules.
More recently, I have had the pleasure of working with Mark Yeo, Matt Hummon, and Alejandra Collopy —and just this year, Hao Wu and David Reens. Mark has been here almost as long as I have, and he is a true lasers man:the YO laser-cooling experiment would not have gotten anywhere without him. Matt’s leadership of the laser-coolingexperiment has been exemplary, and I have been delighted to see my dream from 2008 be demonstrated under his
watch. Alejandra has grown into a strong team member on YO in only a year, and I am sure that Hao and David willrapidly do the same as they take over the OH experiment.
Over the years, we have shared a very fruitful collaboration with the theory group of John Bohn. In particular, his former student Ed Meyer was an invaluable resource on molecular structure calculations, and his recently
departed postdoc Goulven Quéméner’s calculations of OH–OH scattering cross-sections were a major motivation in
our undertaking the evaporative cooling experiment.
Our ability to perform experimental science at JILA is also strongly supported by the excellent technical staff,
in both the instrument and electronics shops. I particularly want to thank Terry Brown and Carl Sauer in the electronics
shop for their guidance in analog electronics and the vagaries of extremely high voltages, and Hans Green, Tracy Keep,and Todd Asnicar in the instrument shop for always having time to answer my design and machining questions.
Beyond JILA, I want to thank several people for their support during the many-year project that is a doctorate.

vi
I want to thank my parents Sheila Kojm and Louis Stuhl for their encouragement of my love of science from a young
age and for their more focused encouragement in recent years, even when laboratory progress seemed to be goingbackwards. Most of all, I want to thank my wife Deena Rosenberg for her love, her support, and her tolerance of thetrials of being married to a graduate student!

Contents

Chapter
Symbols used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi
1

2

3

Introduction1

1.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3Side note: on moving an experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7Background9

2.1

Molecular structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92.2

Angular momenta and Hund’s cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.2.1

Hund’s case a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.2.2

Hund’s case b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.2.3

Hund’s case c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.2.4

Hund’s cases d and e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.3

Molecules in external fields: Stark and Zeeman effects . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.4

Detection techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Stark deceleration and magnetic trapping

21

3.1

Creating a supersonic beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

3.2

Stark deceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

3.3

Magnetic trapping: a story in 2 21 acts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

viii

4

6

7

Version 1.0: the electromagnet trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3.3.2

Version 2.0: switching to permanent magnets . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3.3.3

Versions 2.2 and 2.5: Variations on a theme of fixturing . . . . . . . . . . . . . . . . . . . . .

34

Cold beam-trap collisions

40

4.1

OH–He and OH–D2 collisions at 77 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

4.1.1

Cryogenic supersonic expansions for tunable-velocity He and D2 beams . . . . . . . . . . . .

41

4.1.2

Collision results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

OH–ND3 collisions at 4 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

4.2.1

An introduction to buffer-gas cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

4.2.2

Buffer-gas-cooled ND3 and the challenges thereof . . . . . . . . . . . . . . . . . . . . . . .

48

4.2.3

Collision results and ND3 density calibration . . . . . . . . . . . . . . . . . . . . . . . . . .

52

4.2

5

3.3.1

Microwave manipulation of trapped OH

60

5.1

Avoided crossings in mixed electric and magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . .

60

5.2

Trap losses from Landau-Zener transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

5.3

Microwave coupling circuit design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

5.4

Numerical simulations of ARP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

OH–OH inelastic collisions

72

6.1

Two-body loss of | f i-state molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

6.2

Electric-field dependence of the loss rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Evaporative cooling of OH

82

7.1

Forced evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

7.2

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

7.2.1

Stark deceleration and magnetic trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

7.2.2

Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

7.2.3

Spectroscopic thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

ix
7.2.4
8

93

Laser cooling molecules: from off-beat dream to experimental reality

96

8.1

A modest proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

8.1.1

How to achieve optical cycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

8.1.2

TiO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

8.1.3

State remixing in a J 0 = J 00 − 1 MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

8.1.4

Monte-Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

8.2

9

Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

An experimental implementation, YO! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.2.1

An introduction to YO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

8.2.2

Polarization modulation and the RF-MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

8.2.3

Experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

8.2.4

Doppler cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.2.5

Two-dimensional magneto-optical trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Visions for the future
9.1

9.2

119

Towards a hydroxyl BEC (and maybe a Fermi gas?) . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9.1.1

Improved detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

9.1.2

Ballistic expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

9.1.3

Magnetic dipole evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

9.1.4

Preventing Majorana losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

9.1.5

Fermionic OD? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Enhanced 3-D magneto-optical trapping of YO, and what to do with it . . . . . . . . . . . . . . . . . 124
9.2.1

Dark-spot trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

9.2.2

Closing the A02 ∆ leak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

9.2.3

Narrow-line cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

9.2.4

Collisions and evaporation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

9.2.5

Fermionic 89 Y17 O? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

xBibliography

128

xi

List of Tables

Table

8.1

The wavelengths, Franck-Condon factors, and saturation intensities of the cooling and repump transitions of TiO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

List of Figures

Figure1.1Electrostatic control of chemistry in the OH–H2 CO hydrogen abstraction reaction. In order forthe reaction to proceed, the reactants must align to form the transition state TS. This state has the twomolecular dipole moments (indicated by the arrows in the diagram) at an angle of greater than 90◦ . In
the presence of a polarizing electric field, this orientation is disfavored by the Stark shift of the OH
molecule as it attempts to anti-align to the electric field. As the reaction may be otherwise barrierless,even small changes in the energy cost of state TS could enable large changes in the ultracold reaction
rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

Comparison of the theoretically predicted versus experimentally testable ranges of the e-EDM .
(Courtesy of D. Gresh.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3

3

3

Temperature and density ranges relevant to cold and ultracold molecule experiments. The x-axes
are density in a loosely logarithmic scale from 104 cm-3 to 1013 cm-3 and the y-axes are temperature ina similar scale from 10 K at the top to 1 nK at the bottom. (a) Temperatures and densities necessary forsome of the experiments discussed in section §1.1. (b) Temperatures and densities available from the
major known techniques to produce cold and ultracold molecules. Until the developments of molecular laser cooling and evaporative cooling (described in part in this thesis) there were no techniques
available to bridge the green region and thereby produce chemically diverse ultracold molecules: as ofthis writing, RbK remains the only heteronuclear molecule available at high densities from ultracold
atomic association techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

xiii
2.1

A comparison of atomic line spectra versus molecular band structures. (a) Atomic spectra of H,
He, and Ne. (b) One vibronic band in the spectrum of OD. Note the difference in wavelength scales
between (a) and (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

10

Vector diagram for Hund’s case a. The electronic spin and orbital angular momenta, S and L, arestrongly coupled to the internuclear axis by spin-orbit and electrostatic interactions respectively. Theirprojections along the internuclear axis, Σ and Λ, couple to form Ω which is the projection of the total
angular momentum J = R + Ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

12

Vector diagram for Hund’s case b. The electronic spin S has weak or zero coupling to the internucear
axis; it instead couples to the intermediate vector N = R+L to form J, where R is the rotational angularmomentum. If L = 0, N and R are equivalent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

13

Vector diagram for Hund’s case c. Neither L nor S are coupled to the internuclear axis, but spin-orbit
interactions couple the two together to form the total spin-orbit angular momentum Ja with projectionΩ along the internuclear axis. Ja couples with the rotational angular momentum R to form J. . . . . .

2.5

Vector diagram for Hund’s case d. L is more strongly coupled to the rotation R than to S, and soforms the intermediate vector N. N then couples to the spin S to form J. . . . . . . . . . . . . . . . .

2.6

15

Vector diagram for Hund’s case e. L and S are very strongly mixed to from Ja which is coupled to
the rotation R to form J. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.7

14

16

Illustration of molecular detection techniques. (a) Resonance fluorescence, where a single molecule
scatters many photons which are directly detected. (b) Frequency-resolved single-photon laser-induced
fluorescence, where only one photon is scattered per molecule but a Stokes shift enables efficient
rejection of laser background photons. (c) Absorption detection, where the fractional loss of photons
from the pumping laser is detected rather than the fluorescence photons. Optical cycling enhances thesensitivity but is not required. (d) Resonance-enhanced multi-photon ionization (REMPI), where themolecule is photo-ionized through a resonant intermediate level and the daughter ion is detected. . . .

3.1

18

Schematic view of a supersonic expansion. Gas expands from the reservoir at left through the converging nozzle, then reaches supersonic speeds throught the zone of silence. . . . . . . . . . . . . . .

22

xiv
3.2

Stark deceleration. (a) A molecule in a weak-field-seeking state (dipole moment opposing ~E) flies
into a region of large electric field ~E. The increase in the molecule’s Stark potential energy U results
in a reduction of its kinetic energy and therefore its forward velocity. (b) When the molecule has
converted a selected amount of kinetic energy into potential, the electric field is turned off. This
permanently removes U from the molecule, leaving it moving slower than it started. This cycle may
be repeated by many stages in a linear decelerator. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

24

Phase angle in a multi-stage decelerator. The solid green and dashed blue curves indicate theapproximate shape of the Stark potential for the two alternate switching states; the heavy black
curve shows the Stark energy of the synchronous particle as it traverses the decelerator. (The total
kinetic-plus-Stark energy decreases stepwise at each switching point.) The phase angle is defined as
ϕ = Zswitch /2lstage · 180◦ so that maximal slowing occurs at ϕ = 90◦ . . . . . . . . . . . . . . . . . . .

3.4

25

The electromagnet trap. The molecules exit the Stark decelerator (yellow and gray rods) and thenfly into the trap region between the electromagnets (blue rings). The electric field rods (red) permit theapplication of electric fields with either uniform or quadrupole geometry. Fluorescence is collected
out the side by a lens held in the green lens mount. . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

27

Loading sequence for the electromagnet trap. Molecules entered the trap after Stark deceleration
with only the downstream magnet powered; after the molecules were stopped by the magnetic field
gradient, the upstream magnet was powered up to create the confining quadrupole potential. . . . . .

3.6

27

Performance of the electromagnet trap. (a) Fluorescence versus time curve showing the loading of
the trap. The red line is a Monte Carlo simulation of the loading dynamics. (b) Lifetime measurementsin the electromagnet trap. Solid points were taken with a background pressure of 1 × 10−6 torr of N2 ,while open points had 4 × 10−8 Torr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.7

28

The first-generation permanent magnet trap. Molecules exit the Stark decelerator from the left,
and are loaded into the trap between the two magnets (small black rings at center of the larger green
and cyan mounting fixtures). The mounting fixtures also provide a slot which holds a lens mount (notshown) for fluorescence collection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

xv
3.8

Details of the permanent magnet trap. (a) Contours of equal B (spaced by 25 mT), showing thegeometry of the trapping potential created by a pair of permanent ring magnets. (b) The electrostatic
stopping sequence used to load the permanent magnet trap. . . . . . . . . . . . . . . . . . . . . . . .

3.9

32

Performance of the permanent magnet trap. (a) Fluorescence versus time curve showing the loading of the trap. The solid red line is a Monte Carlo simulation of the loading dynamics. (b) Lifetime
measurement in the permanent magnet trap demonstrating vacuum-limited lifetimes of almost half asecond. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

3.10 The cryogenic permanent magnet trap. The trap magnets (black rings) are supported by thermallyconductive stainless steel arms, which are in turn supported by a complex mounting stack. From thebottom, the stack consists of a steel base-plate, a thermally and electrically insulating Macor ceramic
fixture (white), and a thermally conductive but electrically insulating alumina standoff (gray). The
alumina standoff allows the steel arms and trap magnets to be cooled to 80 K by a liquid nitrogencooling loop (not shown). The steel mounting arms also provide a transmission-line geometry topermit coupling microwave fields across the trapped molecules. . . . . . . . . . . . . . . . . . . . . .

36

3.11 The present permanent magnet trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

4.1

Experimental system for OH–He and OH–D2 collision experiments. OH molecules exit the decelerator (yellow, rear-left) and are electrostatically stopped and loaded into the permanent magnet trap
(blue rings). The molecules are allowed to settle before a supersonic pulse of collision gas (He or D2 )is fired from the solenoid valve (green cylinder) through the trapped cloud. . . . . . . . . . . . . . . .

4.2

42

Cryogenic solenoid valve circuit and performance. (a) Circuit diagram of the single-ended solenoid
drive circuit, including the transient voltage suppressor (TVS) used to achieve fast turn-off. (b)Solenoid current profile achieved using the circuit of (a), including the two-pulse time structure described in the text. (c) Pressure versus time recorded from impact of the supersonic beam on a miniature microphone mounted in line-of-sight with the skimmed helium beam, showing a 155 µs FWHM
gas pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

xvi
4.3

Supersonic expansion performance of the cryogenic solenoid valve. (a) Beam velocity versus valve
temperature for D2 . (b) Relative density versus valve temperature for pulsed He and D2 supersonic
expansions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

4.4

Experimental sequence for OH–He and OH–D2 collision experiments. . . . . . . . . . . . . . . .

46

4.5

OH fluorescence versus time showing the impact of a helium packet. The collision promptly removes ∼47% of the trapped molecules, but there is no visible change in the background-gas collisionalloss rate after the helium packet completes its transit through the trap. . . . . . . . . . . . . . . . . .

4.6

Absolute OH–He and OH–D2 trap-loss cross-sections versus collision energy. Note that the y-axis
scales differ between the two cases. Error bars are one standard error. . . . . . . . . . . . . . . . . .

4.7

46

47

Experimental system for OH–ND3 collision experiment. OH molecules are Stark-decelerated andloaded into the magnetic quadrupole trap, and then illuminated by the 4 K ND3 molecular beam. This
beam is formed by a capillary-filled buffer-gas cell, and then electrostatically guided around a 90◦
bend to separate it from the helium carrier gas and to provide a rapid turn-on/turn-off capability. The
relative OH density in the trap is detected by laser-induced fluorescence; the absolute ND3 beam
density is determined by resonance-enhanced multiphoton ionization, calibrated against a hot-cathodeion gauge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8

47

Buffer-gas-cooled 4 K ND3 beam source. Warm (200–250 K) ND3 gas flows down the central capillary fill line, and is mixed with 4 K helium to produce a beam of 4 K ND3 . The beam is guided bya hexapole (see figure 4.7) to separate the ND3 from the helium carrier gas. The ND3 fill line enters
the cell through a G-10 fiberglass thermal standoff, allowing the line to be maintained at 250 K using
only 200 mW of heater power. The fiberglass standoff is designed to maximize the path length fromany 4 K surface to the tip of the ND3 line and therefore maximize the run time available before an ice
bridge is formed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.9

49

The rotational and parity spectra of OH and ND3 . The parity splittings are exaggerated for clarity.
The red-highlighted parity states are those used for the collisions described in this section. . . . . . .

49

xvii
4.10 ND3 beam parameters. (a) Relative density versus velocity, quoted as OH–ND3 collision energy.
(b) Rotational state distribution for two different buffer-gas flow rates: (upper, red) 2.0 sccm He and2.5 sccm ND3 and (lower, black) 3.5 sccm He and 1.0 sccm ND3 . The unparenthesized labels indicate
the J and K quantum numbers of the ground rotational state probed by each line, while the parenthesized labels indicate (J, K) for the intermediate rotational state in the excited B̃(v0 = 5) level. Due totheir lesser confinement within the hexapole guide, states with K < J are not observed. . . . . . . . .

51

4.11 Experimental sequence for OH–ND3 collision experiment. . . . . . . . . . . . . . . . . . . . . . .

53

4.12 OH fluorescence versus time showing the effect of the ND3 beam. The black points are a plot of
number versus time with no impinging ND3 , while the red points are with the ND3 beam illuminating
the trapped OH molecules. Solid lines are one-body loss-rate fits to the data; quoted uncertainties are
one standard error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

4.13 ND3 density calibration sequence. (a) The REMPI ion signal of the guided 4 K beam is measured.
(b) The guide is turned off, and the chamber is filled with room-temperature ND3 gas to a sufficientdensity as to reproduce the REMPI ion signal. (c) The density of the room-temperature ND3 gas
is measured using a residual gas analyzer calibrated against a Bayard-Alpert type ionization gauge.
(d) The true density of the guided beam is calculated using the results of (a)–(c) combined with calculated spectroscopic overlap factors, the RGA gas-correction factor, and the Boltzmann distribution. . .

57

4.14 The REMPI spectrum of ND3 at 295 K. Solid red line is a model fit. Solid blue line denotes thecalculated contribution to the line strength from the |J, Ki = |1, 1i state used in the collision experiments. 58
4.15 OH–ND3 trap loss cross-sections without and with a polarizing electric field. The combined average cross-sections are stated in black (unpolarized) and red (polarized) text below their respective data
points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1

58

The Zeeman structure of OH with bias electric fields. (a) 0 V/cm, (b) 500 V/cm at a 90◦ magneticto-electric field angle θEB , (c) 5000 V/cm at 65◦ , and (d) 5000 V/cm at 90◦ . The crossings XM of thee; + 23 state with the three f ; M = {− 23 , − 12 , + 12 } states are labeled in (a). . . . . . . . . . . . . .

62

xviii
5.2

Magnetic trap geometry relevant to microwave control of OH. To scale. (a) The magnetic field
distribution in the permanent magnet quadrupole trap. (b). The electric field distribution created byapplying a potential difference to the trap magnets. (c) The spatial location of the avoided crossings
XM . (d) The relative angle θEB between electric and magnetic fields in the trap. White denotes 0◦
while black denotes 180◦ , implying that the avoided crossings are widest in the gray regions. . . . . .

5.3

64

Post-ARP e-state population versus applied bias field. ARP chirp widths of 10 (green triangles), 20(blue squares), and 40 MHz (red diamond). Solid lines are Monte-Carlo simulations of the populationtransfer using realistic bias and RF field distributions from finite-element calculations, a semiclassicaltreatment of the Stark detunings, and a homogeneous stray field as a free parameter. (inset) Time-offlight of e-state population immediately after ARP, with no applied bias field. The rapid loss through−1
the X+1/2 crossing is clearly visible: the solid line is a fit to a single-exponential loss at a rate of τdyn
..

5.4

66

Landau-Zener loss dynamics of magnetically trapped e; M = + 23 OH molecules. (a) Loss rateof e; M = + 32 molecules versus applied electric field. The solid red line is a fit to a simple LandauZener model; the green shaded stripe denotes the range of observed loss rates for f ; + 23 molecules.
The bias field is turned on 5 ms after the ARP pulse is completed, and thus well after the prompt loss
seen in the inset of Fig. 5.3 is completed. (inset) Widths δM of the avoided crossings between e; + 32
and | f ; Mi versus applied electric field, at their respectively maximizing values of θEB . δ+1/2 is linearwith field, while δ−1/2 and δ−3/2 are cubic. (b) Time-of-flight trace of trap oscillations, observed
by loading the trap with a non-zero center-of-mass velocity. The solid red fit yields a trap frequency
νtrap = 1350 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

5.5

A schematic of the microwave system use to transfer OH between Λ-doublet levels. . . . . . . . .

69

5.6

ARP spectra at several different bias E-fields. Points are data while solid lines are simulation results.
The Stark-Zeeman broadening due to the avoided crossings is clearly visible. . . . . . . . . . . . . .

70

xix
6.1

Two-body loss of magnetically trapped OH molecules. A comparison of pure one-body loss withzero electric field (blue circles), and two different two-body rates with 1500 V/cm (red diamonds) and3000 V/cm (green squares) applied fields. The solid lines are fits to the data: pure one-body loss forthe zero-field curve, and combined one- and two-body loss (using the zero-field one-body rate) for thetwo non-zero-field curves. Error bars are one statistical standard error. . . . . . . . . . . . . . . . . .

6.2

73

Testing two-body loss by varying the starting density. (a) Four sets of number-versus-time measurements, with different initial densities due to one-body losses before beginning the measurement.
A 3000 V/cm DC electric field was turned on at t = t0i , where t0i is the time of the first point for eachseries. All four measurements yield statistically consistent two-body rate constants. Error bars are one
statistical standard error. (b) A schematic timeline of the experiments. . . . . . . . . . . . . . . . . .

6.3

74

Variation of the fitted two-body rate constant β with applied electric field. Arrows indicate thatthe error bars are consistent with a zero rate constant; error bars without points indicate upper bounds.
Error bars are one standard error.

6.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

Variation of the fitted two-body loss rate constant β with induced dipole moment. This is thesame data as figure 6.3, but with the electric field parametrized as the effective dipole moment µ
given the maximum available dipole moment d, the electric field E, and the Λ-doublet splitting ∆.
Arrows indicate that the error bars are consistent with a zero rate constant; the five points of lowestµ are presented as upper bounds rather than specific points. The point of highest µ is at a field of
9200 V/cm. Error bars are one standard error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.5

76

The two uppermost potential surfaces of OH in mixed E and B fields. The radial coordinate is
the magnetic field strength in Gauss, and the angular coordinate is θEB , the angle between ~E and~B. Electric field strengths are (a) 500 V/cm, (b) 1000 V/cm, and (c) 3000 V/cm. Note the classical

6.6

Landau-Zener shape near 90◦ at the larger electric field strengths. . . . . . . . . . . . . . . . . . . . .

78| f i-state Landau-Zener loss rates versus electric field. . . . . . . . . . . . . . . . . . . . . . . . .

79

xx
6.7

Variation of the two-body loss constant β after deconvolution of the Landau-Zener losses. (a)Variation with electric field. (b) Variation with induced dipole moment. Blue shaded regions indicate
the values of β found by varying the Landau-Zener loss rate by a factor of 2 up or down. Smaller
values of the Landau-Zener rate produce larger values of β . . . . . . . . . . . . . . . . . . . . . . . .

7.1

81

Ground-state structure and scattering theory of OH. (a) The ground-state Λ-doublet and Zeeman
structure of OH. Dashed black lines are energy levels in the absence of any electric field. In thepresence of an electric field, the level crossings become avoided (solid orange lines demonstrate a
500 V/cm field orthogonal to the magnetic field). Xi label the crossings of the e; 32 state with the| f ; MJ = ii states; these crossings allow e; 23 molecules to escape the trap in the presence of anelectric field. Molecules are loaded into the magnetic trap in the bolded f ; 32 state. (b), Zero electricfield cross sections as a function of collision energy: elastic cross section in a 50 mT magnetic field(solid), inelastic in 50 mT (dotted), and inelastic in 150 mT (dashed). The red dash-dotted line is asemi-classical expression given by Eq. (7.2). Inset: elastic (solid) and inelastic (dotted) cross sections
as a function of magnetic field at Ec = 50 mK. (c), Adiabatic energies as a function of the intermolecular distance at B = 50 mT for the M` = 0 components of the ` = 2 · {0 to 6} partial waves forevery combination of initial states. The inset zooms in on the repulsive van der Waals interaction for
the experimentally relevant case of two colliding f ; 32 molecules (red lines). . . . . . . . . . . . . .

84

xxi
7.2

Trap system and inelastic collisions. (a) Schematic of the Stark decelerator (left) and magnetic trap
(center rings) system. The DC-blocking capacitor (rear) decouples the high voltage used for trap
loading from the microwave system used for spectroscopy and evaporation, while the laser and lensprovide fluorescence detection of the trapped molecules. Inset: an artist’s impression of evaporative
cooling. As the trap depth is successively lowered by the RF knife, the hottest molecules escape andthe remainder rethermalize to a colder temperature. (b) Relative molecule number versus trap holdtime, demonstrating electric field-induced collisions at 45 mK. Black squares, consistent with pureone-body loss, are with no applied electric field; red circles are well-fit by pure two-body inelastic
loss induced by a large applied electric field of 3040 V/cm. (The field used in the RF knife is only
240 V/cm, for comparison.) Error bars are one standard error.

7.3. . . . . . . . . . . . . . . . . . . . .

86

Microwave spectra illustrating six different final temperatures. Black squares are data (error bars
are 1 standard error), and solid lines are fits to the sensitivity-corrected Boltzmann form of Eq. (7.3):(a) the unperturbed 45 mK distribution, (b) anti-evaporation to 54 mK, (c)-(f) forward evaporation to(c) 18 mK, (d) 12 mK, (e) 9.8 mK, and (f) 5.1 mK. Dotted lines reproduce the fit from (a), shown forcomparison. Note that the x-axis scale differs between (a)-(b) and (c)-(f); all curves have been shiftedvertically to have a zero baseline.

7.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

Scaling relations observed in the evaporative cooling of OH. (a) Final observed temperature versus
effective trap depth at evaporation ramp end. (b) Final temperature (solid black, left axis) and calculated relative phase-space density (dashed red, right axis) versus remaining relative molecule number.
Error bars are 1 standard error; phase space density is calculated assuming a 3-D quadrupole trap
geometry. (c) Molecular Boltzmann distributions implied by the fit curves of Fig. 7.3(a) and (c)-(f)(right to left). Distributions are normalized so that the area under each curve is proportional to the total| f i-state fluorescence observed at that temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.5

89

Technical sketch of the trapping and evaporation apparatus. The evaporation synthesizer is an
SRS SG384 broadband synthesizer, chosen for its ability to perform octave-spanning frequency sweeps

7.6

at rates of up to 100 Hz. The spectroscopy synthesizer is an HP 8644A. . . . . . . . . . . . . . . . .

92

Experimental sequence for evaporation and spectroscopic thermometry. . . . . . . . . . . . . . .

94

xxii
8.1

Level structure of TiO. (a) The electronic level structure of TiO and the transitions of interest for laser
cooling. The X3 ∆ ground state is split by the spin-orbit interaction into the three X3 ∆1−3 sublevels, ofwhich the X3 ∆1 level is the lowest. Each sublevel contains a vibrational ladder, while each vibrational
level contains a ladder of rotationally excited states (not shown). 48 Ti16 O has zero nuclear spin andthus there is no hyperfine structure. The ground-state Λ−doublet (not shown) is much less than thenatural linewidth of the E3 Π ← X3 ∆ transition. The solid arrow denotes the v0 = 0 ← v00 = 0 P(1)branch cooling laser, and the dashed arrows denote the v0 = 0 ← v00 = 1 and v0 = 0 ← v00 = 2 P(1)branch repump lasers. The squiggly lines depict the dipole-allowed decays, with the associated FranckCondon factor q next to each decay. (b) The rotational and Λ−doublet structure of the E3 Π0 electronic
excited state. The states are interleaved, as the rotational splitting is smaller than the Λ−doublet
splitting; a and b denote the parity states. Both the cooling and repump lasers address the J 0 = 0, a
state. Not to scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.2

98

Comparison of a traditional MOT with an ER-MOT. (a) The level structure of a traditional MOT.
The local magnetic field strength and orientation combined with the Doppler shift enhance the scattering from the laser beam that provides the damping and restoring forces and suppresses scatteringfrom the counter-propagating beam. Since J 0 > J 00 , the ground state(s) are always able to scatter fromevery beam. (b) The level structure of the ER-MOT. The local magnetic field still governs which laser
is preferentially scattered, but angular momentum conservation forbids some ground states from interacting with the preferred beam. To overcome this, the ground-state magnetic sublevel populations are
remixed by pulsed electric fields, as represented by the dashed lines. (c) A sample ER-MOT design.
A pair of electromagnet coils are aligned in anti-Helmholtz fashion to produce a quadrupole field.
Six beams of the cooling laser are converged on the center with their polarizations oriented as usual
for a MOT, but a set of four open-mesh grids are added. The grids are pulsed in pairs (e. g., first the
X-axis pair and then the Y-axis pair) to produce the dipole electric fields needed to remix the magnetic
sublevels. The center is also illuminated by the repump lasers (not shown). . . . . . . . . . . . . . . . 101

xxiii
8.3

Number (upper, black) and temperature (lower, red) time-of-flight plots for the loading of amolecular packet into a simulated TiO ER-MOT. The initial spike on the number plot is the molecular packet flying through the ER-MOT volume; the broad hump is the actual captured molecules.
The decay of the molecule number is due to radiation pumping of the captured population into excitedv00 ≥ 3 states, and yields an ER-MOT lifetime of 80 ± 5 ms. Error bars represent statistics over multiple
simulation runs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

8.4

Fractional capture vs electrostatic remix rate after 160 ms of simulation time for the TiO ERMOT. Error bars represent statistics over multiple simulation runs. The curve is only a guide to theeye. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

8.5

Level structure of YO and operation of the RF-MOT. (a) The vibronic states relevant to laser cooling, with Franck-Condon factors q listed for each relevant transition. (b) The rotational and hyperfine
structure of the X2 Σ and A2 Π states. (c) The Zeeman spectrum of the X2 Σ ground state. The A2 Π1/2
excited state has nearly zero magnetic moment. (d) Operation of the YO RF-MOT. As the laser polarization is switched to eliminate dark states, the direction of the magnetic field must also be flipped. . . 106

8.6

Variation of the Doppler-cooled YO beam temperature with the polarization chopping rate.
Since the final temperature is limited purely by the number of photons scattered during the molecules’transit of the cooling region, lower temperatures directly imply higher scattering rates. . . . . . . . . 108

xxiv
8.7

Experimental setup for a 2-D YO MOT. (a) A general layout of the experimental system, shown with
only one dimension of cooling for clarity. YO molecules (light blue) are produced by laser ablation ofsintered Y2 O3 pellets by the green 532 nm ablation laser. The molecules are buffer-gas cooled by 4 K
helium gas before escaping the buffer-gas cell (left) as a molecular beam. The beam is then collimatedand passes through the MOT region inside the RF magnetic quadrupole coil (orange rectangles). The
molecular beam is cooled and compressed by the multipass laser light, before being repumped outof excited vibrational states (red laser) and imaged by its resonance fluorescence in the final orange
beam. (b) The laser system used to produce the 614 nm cooling light and the 648 and 649 nm repump
light. The v = 0 cooling light is produced by a dye laser, while the v = 1 and 2 repumps are createdusing grating-stabilized external-cavity diode lasers. Each master diode injection-locks two highpower slave diodes to produce sufficient optical power in each hyperfine component. All three master
oscillators are locked to a self-referenced, octave-spanning Er:fiber frequency comb. . . . . . . . . . . 110

8.8Doppler cooling results from the YO experiment. (a) CCD beam profiles of the YO beam for fourcases: (i) the unperturbed beam, (i) the beam with only v = 0 light applied, which fully pumps themolecules into v = 1 and 2, (iii) a Doppler-heated beam, and (iv) a Doppler-cooled beam. (b) The
detuning dependence of the final beam x-temperature for a uniform detuning δ of all three hyperfine
components of the cooling light. The asymmetry is due to the close level spacings of the G = 1
manifold. Black points are the unperturbed beam temperature, as imaged using light of the give
detuning and red points are Doppler-cooled or heated. The red line is a fit using a multi-state rateequation to determine the optical forces and a Monte-Carlo molecular simulation to determine thefinal temperature achieved. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.9

Comparison of in- versus out-of-plane cooling. (a) When the cooling lasers are in the same plane
as the illumination for fluorescence imaging, the width of the image is reduced. (b) When the coolinglasers are transverse to the illumination, the image width does not change, but the entire image isintensified due to the increase in on-axis density from the cooling. . . . . . . . . . . . . . . . . . . . 112

xxv
8.10 1-D MOT results from the YO experiment. (a) 1-D fluorescence images showing the effects of the
MOT for three cases: (i) the unperturbed beam, (ii) the magnetic field in phase with the optical polarization modulation (ϕMOT = 0◦ ), and (iii) the magnetic field anti-phase to the polarization modulation(ϕMOT = 180◦ ). (b) The variation of the final beam temperature with ϕMOT . The red points are thetemperature with the magnet coils turned off, and are insensitive to the absolute phase of the polarization modulation. The blue points have the magnetic field on, and demonstrate the oscillation between
confining and anti-confining magneto-optical forces as ϕMOT varies. The solid blue curve is a purecosinusoidal line to guide the eye. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.11 A photograph of the completed RF magnetic quadrupole coil. The large brown bars are the coilitself, embedded within its epoxy potting. The copper U-loops above and below the coil provide watercooling; the spacers between the U-loops and the coil are thermally conductive boron nitride. The
toroidal coupling transformer can be seen near the upper-lefthand side of the coil, against the flange. . 115
8.12 Circuit used for stabilization of the YO MOT magnetic field phase. (a) The servo loop layout. (b)The variation of the final beam temperature with the RF drive power applied to the coil. The saturation
may be due to thermal detuning of the coil rather than true magneto-optical effects. . . . . . . . . . . 116
8.13 Power dependencies of Doppler cooling and the 2-D MOT. (a) Final temperature versus relative
cooling laser (614 nm) optical intensity for Doppler cooling (red) and magneto-optical trapping (blue).
(b) Final temperature versus relative repump (648 and 649 nm) laser intensity. Solid curves are fits toa multi-level rate-equation model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Symbols used

Roman Symbols
CP

charge-parity symmetry, that a system is invariant if the charges of all the particles are reversed and the system
is put through an inversion transformation

PMT

photo-multiplier tube

Z

atomic number

Fundamental Constants
α

the fine structure constant, numerically ∼ 1/137cthe speed of light in a vacuum, 299 792 458 m/s

D

the Debye, a unit of electric dipole moment equal to 3.336 × 10−30 C·me

the charge of an electron, 1.602 × 10−19 C

kB

Boltzmann’s constant, 1.38 × 10−23 J/Kme

the mass of the electron, 9.109 × 10−31 kg

mp

the mass of the proton, 1.673 × 10−27 kgµB

the Bohr magneton, 9.274 × 10−24 J/TµN

the nuclear magneton, 5.05 × 10−27 J/TAngular Momenta

xxviiF

total angular momentum vector (including rotational and nuclear angular momenta), with quantum number F
and projection MF

G

coupled nuclear and electronic spin angular momentum vectors, with quantum number G and projection MG

I

nuclear spin angular momentum vector, with quantum number I and projection MI

J

total electronic angular momentum vector, with quantum number J and projection M

K

the projection of J along the molecular symmetry axis in a symmetric top moleculeLtotal electronic orbital angular momentum vector, with quantum number L and projection Λ on the internuclear
axis

N

molecular total angular momentum minus electron spin vector, with quantum number N and projection MN

R

molecular rotational angular momentum vector, with quantum number R and projection MR (for moleculeswith L = 0, R and N are equivalent)S

total spin angular momentum vector, with quantum number S and projection Σ on the internuclear axis

Molecular ConstantsA

the spin-orbit splitting constantBrotational constantD

quadratic rotational constantdthe total electric dipole moment of a moleculem

the molecular massωeequilibrium vibrational constantωe xequadratic vibrational constant

Assorted Symbols

xxviii
H

a Hamiltonian

ϕdecelerator phase angle

Chapter 1
Introduction

1.1

Motivation
The current excitement over cold molecules is, in many ways, the natural progression from the past twenty-fiveyears of cold atom physics. Just as the development of first Doppler cooling[1, 2] and then magneto-optical trapping[3]led to such diverse results as Bose-Einstein condensation[4, 5], exploration of the BEC-BCS crossover regime indegenerate Fermi gases[6, 7, 8, 9], the creation of strongly-coupled atom-cavity quantum optical systems[10, 11],
measurements of parity non-conservation in Cs and its nuclear anapole moment[12, 13], and optical clocks[14, 15, 16,
17, 18, 19], so too are cooled molecular species expected to enable a new set of experimental frontiers. This section
describes some of the major arenas which cold molecules are making possible.
In molecular physics, again as in atomic physics, experiments of interest can be loosely divided into two
categories: those which care about the specific (chemical, isotopic, mass, etc.) nature of the particle under study,
and those which do not. The former encompasses both the obvious studies of chemical or nuclear properties, butalso — especially in molecules — those where some detail of the particle’s specific structure enables or enhances
the experiment. One example from atomic physics is the Cs anapole experiment, where the high Z number of Cs
enhances the spectroscopic couplings[12] over the lighter alkali-metals. Another example is alkaline-earth opticalclock experiments: here it is the electronic and hyperfine structure of the species which creates the clock transition,and there are great differences in character between the clock transitions in Sr[14, 17], Al+ [18, 19], and Yb[15, 16].
By contrast, there are also a great many experiments which could in principle be done with any atom providing areasonable approximation of a generic two- or three-level system. While alkali metals, especially Rb, are commonly2
used for these experiments, it is a choice of convenience rather than necessity. Some atomic examples of the latter
category include strong-coupling experiments, electromagnetically-induced transparency (EIT) demonstrations, andvapor-cell squeezed light tests.
Chemical studies of atomic gases, however, face a profound limitation: two-body collisions of isolated atoms
cannot undergo chemistry. Conservation of momentum requires a third partner in the interaction, and even then the
only possible reaction is recombination (A + B + M → AB + M). (M is usually another atom, although a photon canalso serve in excited-state collisions.) Molecules, by contrast, can undergo exchange (2AB → A2 + B2 ), abstraction(2AB → A2 B + B), or dissociation (2AB → AB + A + B) interactions in a simple two-body collision. Study of bondbreaking as well as formation requires molecules. The first major motivation for cooling molecules is therefore
chemistry: cold systems are clean (and more easily calculable) systems. Moreover, electric dipole moments, aswell as possibly Feshbach resonances, enable control of the intermolecular potential surface and therefore control
of chemistry as well — if the sample is cold enough that thermal averaging does not wash out the induced perturbationof the potential surface. One example of a possible route to electric-field-based control of chemistry can be seen in the
abstraction reaction OH + H2 CO → H2 O + HCO [20], as illustrated in figure 1.1. The application of a static electricfield opposes the relative reorientation required to form the reaction’s transition state, and therefore enables control of
the reaction rate. Another reaction which might be field-controllable is 2OH → H2 O + O. In this case the mechanism
would be more subtle and is discussed in chapter 6. More generally, since electric fields couple together all statesof opposite parity, including those of differing electronic configurations, they may be capable of controlling not only
external chemical interactions, but even internal dynamics like conical intersections[21] and spin-dependent chemistry[22].
Beyond direct chemical studies of a molecular sample, there are many other experiments which require a
specific species. One of the most well-known is the search for the electron’s theoretical electric dipole moments (thee-EDM). Even in the Standard Model of particle physics, the fact that weak interactions do not preserve charge-parity(CP) symmetry means that the electron’s charge distribution is not precisely spherical: it has a nonzero electric dipole
moment. Extensions to the Standard Model, including both supersymmetric theories and technicolor, in general imply
a much larger and potentially experimentally observable e-EDM.
Of course, since the electron itself is charged, it is impossible to simply hold it still, apply an electric field,

3

Figure 1.1: Electrostatic control of chemistry in the OH–H2 CO hydrogen abstraction reaction. In order for thereaction to proceed, the reactants must align to form the transition state TS. This state has the two molecular dipole
moments (indicated by the arrows in the diagram) at an angle of greater than 90◦ . In the presence of a polarizing
electric field, this orientation is disfavored by the Stark shift of the OH molecule as it attempts to anti-align to theelectric field. As the reaction may be otherwise barrierless, even small changes in the energy cost of state TS could
enable large changes in the ultracold reaction rate. Figure reproduced from [20].

Figure 1.2: Comparison of the theoretically predicted versus experimentally testable ranges of the e-EDM .
(Courtesy of D. Gresh.)4and look for a change in the spin-flip transition frequency. Instead, one must probe an electron confined to a neutral
system, an atom or a molecule. Until less than two years ago, the best limit on the e-EDM was set by an experiment
done in atomic Tl[23]. Now, however, an experiment in a warm YbF beam[24] has beat that limit by a factor of
two. Moreover, in the coming years, several experiments in ThO[25], HfF+ [26], WC[27], and YbF are expected to
improve the detection sensitivity by over an order of magnitude and directly probe e-EDMs of the size predicted bysupersymmetry.
This vast improvement in sensitivity is due entirely to using molecules rather than atoms as the test systems.
Specifically, by using polar molecules with a high-Z atom, polarizing the molecule in the laboratory from also applies
an enormous (tens of GV/cm, far larger than the hundreds of kV/cm directly achievable in the lab) effective electricfield to a valence electron. Moreover, a careful choice of electronic state provides direct cancellation of some of themost important sources of systematic errors.
Another area where molecules can provide much higher sensitivity than atoms lies in tests of fundamental
constants. Some theories of cosmic inflation, for instance, suggest that the fine-structure constant α or the proton-toelectron mass ratio m p /me might vary in time or space. The Λ-doublet transition of the OH radical provides strong
rejection of systematic errors in cosmological hunts for α̇, since the main (∆F = 0) and satellite (∆F = ±1) linetransition frequencies have very different dependencies on α[28, 29]. This means that observers can use one set oflines to determine the red-shift (i. e., the distance) to an astrophysical source, and then use the other set to determine thevalue of α when the source emitted the radiation. Since OH Λ-doublet lines are radiated by OH mega-masers[30, 31],
there are very bright sources available at substantial red-shifts.
Alongside α̇, molecular vibrations enable very direct time-constancy tests for me /m p . A simple long-term
comparison of an atomic optical clock (whose frequency, being an electronic transition, depends mostly on α) with amolecular vibrational clock allows testing the stability of both α and me /m p . While molecular vibrational spectroscopy
does not intrinsically require cold molecules, it is only the advent of optical lattice-based atomic clocks which hasprovided sufficiently high precision to make the atomic side of such a comparison feasible, and so some similarly low
temperature is needed for the molecular side as well.
A final area of measurement enabled by molecules is the nuclear weak charge or anapole moments of a substantial breadth of elements. While the anapole moment of Cs was measured directly, no other atom has been so

5measured: it was only the extreme sensitivity of the parity non-conservation measurement in laser-cooled Cs whichenabled its detection. However, the availability of closely-spaced, opposite-parity levels in dipolar molecules is expected to greatly enhance the sensitivity of these measurements[32]. A reasonably diverse set of anapole momentmeasurements would provide a much-needed test of nuclear structure theories.
Parallel to the chemical nature of molecules, there is also great interest in them as approximately “atoms
with an electric dipole moment.” While atom-atom interactions are almost purely isotropic, the dipole moment’s
intrinsic p-wave character[33, 34, 35] is expected to create fascinating and in some cases entirely novel phases ofmatter[36, 37, 38, 39] like the so-called “checkerboard lattice” [40, 41], as well as potentially never-before-seen
Cooper-pairing symmetries in fermionic species[42]. Atoms, of course, may have magnetic dipole moments, but these
moments are relatively weak compared to molecular electric dipole moments. For scale, the interaction energy (whichscales as µ 2 /r3 for a generic dipole moment µ and distance r) is roughly 1000X stronger for a 1 D electric dipole
moment than for a 1 µB magnetic moment at a given distance r. Atomic magnetic dipole moments max out at 10µB (inTb and Dy), while molecular electric dipole moments go up to roughly 10 D. Moreover, the atomic magnetic moment
is fixed, while the molecular electric moment can be tuned by an externally applied field. While the molecular momentis generally referred to as “permanent,” molecules in the absence of an electric field are in eigenstates of parity andtherefore have zero net electric dipole moment. Application of a field of increasing strength mixes molecular states of
opposite parity to reveal more and more of the full dipole moment. Varying the electric field strength thus tunes the
dipole moment.
The dipole moment’s strength and tunability also have led to suggestions of using molecules as qubits inquantum information applications. Dipole-dipole collisions provide a long-range way of coupling qubit-molecules,
while a spatially-varying Stark shift enables spectroscopic addressing of individual qubits[43].
Given all these exciting applications for cold molecules, why does the field seem to be roughly twenty years
behind its atomic analogue? While it may seem a touch glib to say, the truth is just that molecules really are thatmuch harder to work with. Until roughly a decade ago, the coldest molecules available were limited by the reach of
cryogenic helium dilution refrigerators — many orders of magnitude hotter than laser-cooled atoms. This thesis is
primarily about several breakthroughs in applying cold atom techniques to molecules.

(a)(b) Production techniques

Applications

Astrophysics
Precision measurement

Buffer gas cooling
Magneto-opticaltrapping
Stark deceleration
Evaporative coolingTemperatureDensity

Association ofultracold atoms

Temperature

Cold chemistry
Cold collisions
Molecular lattice clocks
Controlled chemistry
Ultracold chemistry
Quantum info/sim
Dipolar quantum gases

6

Density

Figure 1.3: Temperature and density ranges relevant to cold and ultracold molecule experiments. The x-axes
are density in a loosely logarithmic scale from 104 cm-3 to 1013 cm-3 and the y-axes are temperature in a similar scalefrom 10 K at the top to 1 nK at the bottom. (a) Temperatures and densities necessary for some of the experiments
discussed in section §1.1. (b) Temperatures and densities available from the major known techniques to produce cold
and ultracold molecules. Until the developments of molecular laser cooling and evaporative cooling (described in partin this thesis) there were no techniques available to bridge the green region and thereby produce chemically diverse
ultracold molecules: as of this writing, RbK remains the only heteronuclear molecule available at high densities fromultracold atomic association techniques.

1.2

History
When I first joined the group of Jun Ye in early 2006, cold molecules still felt like a backwater within AMO

physics, and Stark deceleration in particular still had an air of novelty to it. At that point, there were only a handful oftechniques known for producing molecular gases at temperatures below 4 K, and only two of them (Stark deceleration
and buffer-gas cooling) had demonstrated any reasonable final density of cold molecules. While the concept of Stark
deceleration was proposed over forty-five years before[44, 45], the first functioning demonstration had only been madein 1999[46]. As far as I know, in 2006 there were only two labs in the whole world with operational decelerators[46, 47,48]. The Ye group had two: the original machine was being used to finish a precision spectroscopy project on the OH
Λ-doublet line[28, 29], while the second had recently demonstrated slowing of formaldehyde[20] and was currently
working on a detailed study of deceleration dynamics[49] in preparation for a drive towards magnetic trapping of OH
molecules. (The first trapping experiments are discussed more fully in chapter 4.)
Since then, working with molecules has become much less fringe, if not yet truly mainstream. The availabilityof near-quantum-degenerate dipolar RbK molecules[50, 51] made it clear that there was hope for experimental tests
of dipolar Fermi gas and lattice physics. Our proposal for the practical laser cooling of selected molecules (chapter 8)
has been demonstrated in SrF[52] at Yale and in YO in this group[53]. Most recently, we have observed evaporative

7
cooling of magnetically trapped OH molecules. Since evaporation is the single technique which enabled BEC in
atomic systems, its availability in molecules is potentially revolutionary. At a broader level, many more groups aroundthe world have taken up molecule projects since 2006, and our sessions at the annual Division of Atomic, Molecular,
and Optical Physics conference are now much better attended!

1.3

Side note: on moving an experiment
The work described in this thesis also faced an additional complication: the entire experiment was moved in

January 2012 from its original location in JILA S-1B05 (now the corridor connecting the S- and X-wings) to X-1B10.
While the hiatus this imposed was not appreciated, the move did have its positive points. The new space in X-1B10 isa substantial improvement over the old lab: it has more space in general, more storage space in particular, and the Xwing service corridor is a great thing. Moreover, the move offered an opportunity to finally correct some long-standing
defects in the experimental layout.
The Stark decelerator is not a small system: the vacuum chamber is roughly 1.5 m long and over 1 m wide
(at its widest point); it also has no less than five turbomolecular pumps on it, plus the rough vacuum system backing
those pumps. In S-1B05, the entire system was installed sitting on the optics table. While this made optical accesssimple, it also ruined both the vibration and electrical properties of the table for anything else. The turbopumps drive
high-frequency vibrations, and the decelerator switches tens of kilovolts at roughly ten kilohertz; the switching drives
enough electronic noise onto the high-voltage ground as to contaminate any nearby signals. (To combat this noise, allsensitive cables are shielded by an extra electrically-isolated layer of ground braid.)
When the experiment was moved to X-1B10, several changes were made. The first is that the decelerator is now
mounted on a free-standing support frame. This both eliminates the coupling of turbopump vibration to the optical
table and also vastly improves access to the side of the decelerator facing the optics table. (When the decelerator
was on the table, it was an exercise in contortionism to access the rough vacuum lines and the feed-throughs forthe negative polarity decelerator rods, but no longer!) In combination with the mechanical decoupling, the control
system was audited for connections between the decelerator’s high-voltage ground and the controls’ signal grounds.
By isolating the two, the experiment is now able to take advantage of the star-ground system built into the X-wing.
The high-voltage ground is tied to one of the building star-ground panels, while the general signal ground is tied to8a different panel. Putting the two grounds on different branches of the star has vastly reduced the pickup from thedecelerator switching.
Another area in which the setup in X-1B10 is a major improvement is cable routing. Cable (and in manylabs, optical) routing is the bane of any lab’s existence: cables are added for “a quick test” and never removed; a new
instrument needs a connection, and it is added by the path that the cable will reach, rather than a path that is reasonably
unobstructing. In the X-wing, we have made an investment for the future: we installed a set of proper cable trays andarranged a reasonable routing of all of the signals needed to run the current experiment. At least for now, the clutter iscontained.
The new space is not an unblemished improvement, however: there were and are some “teething troubles.”The most egregious problem was definitely the chilled-water system incidents. While the exact details will never beknown, Facilities Management somehow managed to sufficiently overpressure the system as to mechanically breakthings – twice! Most likely, they accidentally induced a water hammer effect while adjusting valves and pumps. The
temperature stability of the labs has for the most part been excellent, but there remains some work to be done to smoothout the switch-over between building heating and cooling in the spring and fall.

Chapter 2
Background

2.1

Molecular structure
The pioneering atomic physicist Arthur Schalow once defined a diatomic molecule as being a molecule with

one atom too many. To a person who is used to the simple, discrete spectra of isolated atoms, this may be true.
Both the richness and the difficulties of molecular physics arise from the distributed nature of molecules. Even if
one thinks of a diatomic molecule as simply two atoms magically held on opposite ends of a stick, it is immediately
obvious that there is now a rotational degree of freedom which neither atom posessed separately. Taking the model
one step closer to reality, by changing the stick into a spring with a finite elastic modulus, makes a vibrational mode
evident. The recognition that the equilibrium length and spring constant of the interatomic bond is set by the electronic
configuration of the system makes it clear that electronic transitions can change the rotational and vibrational state ofa molecule as well; such transitions are referred to as rovibronic.
At the most rigorous level, molecular structure requires the full power of quantum field theory, as electrons may
acquire relativisitic speeds near high-Z nuclei[54]. However, in the context of this thesis, several major simplifications
may be made. The first is that for relatively light nuclei relativistic effects can be treated perturbatively, adding theelectronic and nuclear spins and their couplings to each other and to the motion-generated currents to the effective
Hamiltonian by hand. This simplifies the problem to a form
1 
e 2
1 
e 2
p̂i + Â + ∑p̂i − Â + Hrelativisticccnuclei 2mi
electrons 2me

Hmolecule = ∑where p̂i is the momentum operator of the i’th particle and Â is the electromagnetic vector potential operator. A further
vast simplification comes from the Born-Oppenheimer approximation, in which it is assumed that the electronic and10(a) 400 nm

700 nmH
HeNe(b) 270 nm

280 nmOD

Figure 2.1: A comparison of atomic line spectra versus molecular band structures. (a) Atomic spectra of H, He,
and Ne. (b) One vibronic band in the spectrum of OD. Note the difference in wavelength scales between (a) and (b).

11
nuclear parts of the wavefunction are separable,|Ψmolecule i = |Φnuclear i ⊗ |Ψelectronic i, and moreover that the electronic wavefunction depends only on the positions of the nuclei. This results in the
nuclear part acquiring an effective internuclear potential, while the electronic part consists of discrete configurations
which each yield a different internuclear potential. Separation of the nuclear motion into components parallel andperpendicular to the internuclear axis then yields an approximate Hamiltonian
Hmolecule ≈ Helec + Hvib + Hrot
where the electron terms and relativistic corrections have been combined into the effective electronic Hamiltonian
Helec .
To first order, Hvib and Hrot are both conceptually simple:


1
1
0
Hvib
= kr̂2 = ωe v +
2
2
0
Hrot
=

1 2 2
h̄ N = BN (N + 1)
2I

. Hvib is a harmonic oscillator, where k is the effective spring constant due to the electronic configuration and r̂ is theradial position operator; Hrot is the rotation perpendicular to the internuclear axis with an effective moment of inertia
I . By convention, molecular theory and spectroscopy work with the simplified parameters
1
ωe =
· h̄
hc
B=

s

k
mred

1 h̄2
·
hc 2I

which are generally reported in cm-1 . Of course; the parallel and perpendicular motions are not truly separable: B
may be perturbatively expanded in powers of v + 21 to account for the change in mean bond length with increasing
vibrational excitation; there also exist higher order terms like1Hvib
= −ωe xe



1 2
v+
2

1
Hrot
= −D [N (N + 1)]2

12

J

R

L

S

Λ

Σ

Ω = Λ+Σ
Figure 2.2: Vector diagram for Hund’s case a. The electronic spin and orbital angular momenta, S and L, are strongly
coupled to the internuclear axis by spin-orbit and electrostatic interactions respectively. Their projections along theinternuclear axis, Σ and Λ, couple to form Ω which is the projection of the total angular momentum J = R + Ω.

to account for the anharmonicity in the molecular potential. With sufficient signal-to-noise ration, molecular spectra
may be fitted with coefficients out to third or fourth order in powers of v + 12 , J (J + 1), or a mix of the two. However,that level of spectroscopic detail is only relevant when attempting to describe the structure of an entire set of molecular
bands. For the purposes of this work, it suffices to know the positions of only a few low-J, low-v states in any given
molecule, and therefore the second-order expansions are more than sufficient.

2.2

Angular momenta and Hund’s cases
While the previous section treated the rotation of the molecule as though it were the only source of angularmomentum in the system, for general molecules that is untrue. Molecules may posess net electronic orbital angularmomentum L, net electron spin S, and nuclear spin I as well as rotational angular momentum. Since all of theseangular momenta posess magnetic moments, they all can interact. The relative strength of the various couplings leadto several qualitatitively distinct spectral forms, known as the five Hund’s cases[55, 56]. Each case is characterized
by a specific hierarchy of levels (e. g. many rotational levels within an apparent spin-orbit manifold for case a); therotational Hamiltonian has a different natural expression in each case.

13

S
J
N

R
L

Λ
Figure 2.3: Vector diagram for Hund’s case b. The electronic spin S has weak or zero coupling to the internucear
axis; it instead couples to the intermediate vector N = R + L to form J, where R is the rotational angular momentum.
If L = 0, N and R are equivalent.

2.2.1

Hund’s case a
Case a (figure 2.2) occurs when spin-orbit interactions are much weaker than the electrostatic interactions

which split different electronic states, but are much stronger than the spin-rotation or orbit-rotation interactions, thatis when Eelec  AΛ  BJ. This results in a hierarchy where widely-separated electronic levels are split into a setof spin-orbit manifolds, each of which contains many closely-spaced rotational levels. This commonly occurs inmolecules with relatively low Z and non-zero electronic angular momentum. The natural expression of the rotational
Hamiltonian is0
Hrot= B(R) (J − L − S)2

where B(R) emphasizes the dependance of B on the internuclear spacing. This form makes it clear that N is not a goodquantum number in this case.

2.2.2

Hund’s case b
If the spin-orbit coupling is weak or nonexistant (most commonly, L = 0 for close-shell molecules), low-Z

molecules are generally in case b. In this case (figure 2.3), electronic levels directly contain a manifold of rotational
levels, each of which may be broken into a small spin-orbit multiplet. The rotational Hamiltonian
Hrot = B(R) (N − L)2

14

R
J
S
L

Ja
Ω

Figure 2.4: Vector diagram for Hund’s case c. Neither L nor S are coupled to the internuclear axis, but spin-orbit
interactions couple the two together to form the total spin-orbit angular momentum Ja with projection Ω along theinternuclear axis. Ja couples with the rotational angular momentum R to form J.

has a simple first-order form if orbit-rotation couplings are ignored:
Hrot = B(R) N (N + 1) − Λ2
. Case b is appropriate when Eelec  BJ  AΛ.

2.2.3Hund’s case c
The last common case for non-Rydberg molecules is case c, which occurs in molecules with high-Z nuclei andis an analog to j j-coupling in atomic spectra[57]. In this case, the spin-orbit interactions are sufficiently strong as
to necessitate their incorporation into Helec , and neither L nor S are well-defined (figure 2.4). Instead, there occur
electronic levels (which now fully include the spin-orbit character of the level, rather than forming a multiplet as incase a) with well-defined total electronic angular momentum Ja and its projection along the internuclear axis Ω. The
rotational Hamiltonian in this case is similar to that of case a,0
Hrot= B(R) (J − L − S)2= B(R) J2 − 2J · Ja + J2awhere the J · Ja term couples together states of different electronic configurations and ∆Ω = 1. This case is appropriate
for AΛ ' Eelec  BJ.

15

S
L

J
N

R

Figure 2.5: Vector diagram for Hund’s case d. L is more strongly coupled to the rotation R than to S, and so forms
the intermediate vector N. N then couples to the spin S to form J.

2.2.4

Hund’s cases d and e
Cases d and e (figures 2.5 and 2.6) are much less common—in fact, they almost never occur outside of Rydberg

systems, where the valence electron is almost completely decoupled from the ion core[56]. Case d is an analog to caseb, where spin-orbit coupling is weak but the rotational splitting is much larger than the splittings between adjaced
Rydberg levels—that is, BJ  Eelec . Case e is analagous to case c, where the spin-orbit couplings are stronger than theelectrostatic level splittings, and so the effective hierarchy is AΛ  BJ  Eelec . (Note that Λ is not a good quantum
number in case e, merely a statement that there is nonzero spin-orbit coupling.)

2.3

Molecules in external fields: Stark and Zeeman effects~ generates interaction matrix
In general, the interaction between a dipole moment ~µ and an external field F~ (where F~ represents either a magnetic field ~B or an electric field ~E). In order toelements of the form Hi = −~µ · F
evaluate these moments it may be necessary to transform between the molecule-fixed frame in which the appropriate~ is defined. The correct form for this transformation
dipole moment ~µ is defined and the laboratory frame in which F
depends on the Hund’s case character of the molecule in question. For instance, in ground-state OH, both the electricand magnetic dipole moments are approximately constrained to follow the internuclear axis in a Hund’s case a configuration, at least for very low J. By contrast, ground-state YO has a Λ = 0 and therefore case b structure. This meansthat the magnetic moment due to the coupled electron and nuclear spins is free to align directly in the laboratory frame
and therefore YO’s magnetic moment does not require any coordinate transformation.

16

Ja
J
R

Figure 2.6: Vector diagram for Hund’s case e. L and S are very strongly mixed to from Ja which is coupled to therotation R to form J.

For molecules which require transformations, we follow Lara et al. [58] and use the Wigner rotation matrix
1∗ (ω) to write
elements[59] Dqk!
q~ = − ∑ (−1)
Hi = −~µ · F
q

∑

1∗
Dqk
(ω) µkF−q .

kWhile OH is not fully a Hund’s case a molecule, its J = 32 state is nonetheless close enough that both the electric andmagnetic dipole moments may be considered to lie along the internuclear axis and so Hi simplifies to
1∗
Hi = − ∑ (−1)q Dq0(ω) µk=0 F−q .
q

If ~B and ~E are not parallel, it is further necessary to rotate the laboratory fields to a combined frame. Without loss ofgenerality, we take the laboratory Z axis to be along ~B and transform ~E using the reduced rotation matrix elements1∗ (θ ) to write
dqk
EB

(B)

(E)

1∗
1∗
1∗
Hi = −D00(ω) µk=0 |B| − ∑ Dq0(ω) µk=0 d−q0
(θEB ) |E| .
q

Using Lara et al.’s results for the JM Ω̄ε parity basis yields the matrix elements

17JM Ω̄ε Hi J 0 M 0 Ω̄0 ε 0

=

J

1

−Ω̄

0

Ω̄0


J+J 0 +2Ω̄

!

√

√
2J + 1 2J 0 + 1 


×







J0

1 + εε 0 (−1)

2








 (−1)M−Ω̄


J

1

J0

−M

0

M0

J

1

J0

+

1 − εε 0 (−1)
2

!


∑

q

−M

q









J+J 0 +2Ω̄
 µB B (Λ + ge Σ) 1∗
 d−q0 (θEB ) µe E



0

M

where ge is the electron Landé g-factor and µe is the molecular electric dipole moment.
Evaluating the matrix elements described previously for the OH J = 32 , M, Ω̄ = 32 , ε ground state manifold
yields an 8 × 8 matrix which can be diagonalized to find the Stark-Zeeman spectrum of OH for an combination offields sufficiently large that hyperfine structure does not appreciably contribute. From left-to-right and top-to-bottom,
the state ordering used to defined the matrix begins with e; − 32 and increases first M and then ε:




























√
− 53 µe E sin θEB001
5 µe E cos θEB− 25 µe E sin θEB

0

0

−2
5 µe E sin θEB6−∆
2 + 5 µB B

0

0

−1
5 µe E cos θEB√− 53 µe E sin θEB0∆ − 6µ B
2
5 B

0

0

0

0

0

0

0

√
− 53 µe E sin θEB∆ − 2µ B
2
5 B

0

0

∆ + 2µ B
2
5 B

0

−3
5 µe E cos θEB

0

0

0

∆ + 6µ B
2
5 B

6
−∆
2 − 5 µB B

0

0

0

0

2
−∆
2 − 5 µB B

00

3 µ E cos θEB
5 e√− 53 µe E sin θEB

0

0

2
−∆
2 + 5 µB B

0

0

0
√
− 53 µe E sin θEB00

1 µ E cos θEB
5 e− 25 µe E sin θEB0− 52 µe E sin θEB00− 15 µe E cos θEB√− 53 µe E sin θEB3
5 µe E cos θEB√− 53 µe E sin θEB√− 53 µe E sin θEB−3
5 µe E cos θEB




























.

2.4

Detection techniques
Due to their lack of cycling transitions, most molecules are substantially harder to detect than the commonly-used atoms. (Laser-coolable molecules, such as YO, SrF, and TiO, are exceptions to this statement.) The lack of
optical cycling means that fluorescence generally yields at most a few photons per molecule—independent of exposure
duration—while absorption techniques rapidly bleach the sample. Moreover, just as many atoms are rarely used dueto the inconvenience of the laser wavelengths required for their spectroscopy, so too do many molecules only havedeep-UV transitions. The difference is that the choice of molecule for an experiment is generally set by considerations
other than simplicity of the laser system (e.g. the mass-to-dipole-moment ratio is a major motivation for using OH),and so one makes whatever sacrifices of convenience are required to get the necessary light.
In practice, there are four major families of detection techniques used in our experiments. The first, and mosttrivial, is the resonance fluorescence imaging used in the YO laser cooling experiment and illustrated in figure 2.7(a).

18

(a)

(b)

(c)

(d)
+

Figure 2.7: Illustration of molecular detection techniques. (a) Resonance fluorescence, where a single molecule
scatters many photons which are directly detected. (b) Frequency-resolved single-photon laser-induced fluorescence,
where only one photon is scattered per molecule but a Stokes shift enables efficient rejection of laser backgroundphotons. (c) Absorption detection, where the fractional loss of photons from the pumping laser is detected rather
than the fluorescence photons. Optical cycling enhances the sensitivity but is not required. (d) Resonance-enhanced
multi-photon ionization (REMPI), where the molecule is photo-ionized through a resonant intermediate level and thedaughter ion is detected.

19
Since each molecule can scatter hundreds or thousands of photons, simply illuminating the sample with a resonant
laser produces a bright enough image to directly record it with a CCD camera.
The second technique, used for some YO spectroscopy and for detection of OH, is single-photon, frequencyshifted, laser-induced fluorescence. Just as before, a laser excites a transition in the molecule. However, in this casethe molecule fluoresces on a different transition and wavelength than the excitation laser, as schematically shown infigure 2.7(b). While this necessarily means that the molecule is now dark to the exciting laser, a sharp optical filter canbe used to descriminate between the excitation light and the fluorescence. In combination with temporal gating and asingle-photon-sensitive PMT, this technique allows extremely low backgrounds for the detection of small samples. It
is, however, difficult to calibrate. Not only the fluorescence-collection efficiency, but also (when using a pulsed laser)
the excitation volume are hard to calculate.
The third possibility, shown in figure 2.7(c), is absorption detection. As in the previous two techniques, a laser
is tuned to a molecular resonance; however, instead of detecting the scattered photons, the signal here is the loss ofphotons from the laser beam. The fractional loss can be immediately converted into an optical depth, and if the opticalabsorption cross-section is known the optical depth can then be converted into an absolute (mean) column density. Fortrue density measurements, absorption is the gold standard. Unfortunately, it is a libright-background technique: theabsorption dip is a small change on a large baseline. If photon shot-noise is the only limit, absorption can be quitesensitive—but in the hostile environment of the Stark decelerator technical noise can be a severe issue. As of thiswriting, there is effort underway to demonstrate absorption detection of the magnetically trapped OH molecules, but
the laser stability requirements are substantial.
The last family is photo-ionization, in the form of resonance-enhanced multi-photon ionization (REMPI).
REMPI (figure 2.7(d)) comes in many forms, depending on how many photons and how many laser wavelengths
are used. In the OH–ND3 collision experiments, we used single-color 2+1 REMPI to determine the density of ourcold ND3 beam. This means that two laser photons combined to drive a two-photon transition to an intermediate level,
and then a third laser photon ionized the excited molecule. Once ions are created, they can be extracted by an electricfield and detected with a multi-channel plate (MCP). The MCP has near-unit detection efficiency, and so REMPI canbe extremely sensitive.
The sensitivity of REMPI, however, does not make it any easier to determine an absolute density calibration.

20
Similar to pulsed-laser LIF, there is a large uncertainty associated with the ionization volume and the ion extraction
efficiency. Moreover, since the ionization is a nonlinear process, perturbations in the laser mode structure are amplified
in the effective mode volume.

Chapter 3
Stark deceleration and magnetic trapping

All of results reported in this thesis, other than laser-cooling, were achieved using a Stark decelerator[46, 47, 20]
and magnetic trap[60, 61] machine. The premise of the system is simple: supersonic expansions yield gas packets
which are very cold (∼ 10 K) in their co-moving frame, but have a laboratory center-of-mass velocity of ∼ 300–
1000 m/s. If there was some way to collectively slow that packet down, the supersonic expansion would be an excellent
source of relatively cold molecular samples. This slowing is what the Stark decelerator provides. Moreover, the Stark
decelerator is capable of bringing a packet all the way to rest, so that the packet can then be captured in a magnetostatic(or electrostatic[62], or in principle microwave[63] or optical) trap.

3.1

Creating a supersonic beam
No Stark decelerator has colder or denser output than the beam source from which it is fed. Since the decelera-tion is a conservative (rather than dissipative) process, this statement is rigorously true, and therefore implies that it ispotentially worth spending a great deal of effort to develop a high-brightness, low-temperature initial source. Unfortunately, while there is a large body of both research and theory on the behavior of supersonic expansion sources, theactual development of any real source continues to be a substantially empirical endeavor. (Consider, for instance, thatthe title of the chapter on source development in the thesis of E. Hudson[64] is “Sourcery.”)
A basic picture of a supersonic expansion is shown in figure 3.1. The gas begins at some initial temperature T0
and pressure P0 , with a very small mean velocity. Driven by the differential between P0 and the external background22

background pressure Pb

M>1
hock
rel s
bar

T0, P0M<<1

ZONE OFSILENCE

M<1

M>>1
gas flow

M
ach
dis

k

bo

un

da

ry

sho

ck

Figure 3.1: Schematic view of a supersonic expansion. Gas expands from the reservoir at left through the converging
nozzle, then reaches supersonic speeds throught the zone of silence. Styled after Fig. 2.1 of [65].

23pressure Pb , the gas flows out the nozzle of the expansion. If the pressure ratio exceeds a critical value
P0
>G≡
Pb



γ +1
2

γ/(γ−1)

where γ = cP /cV is the ratio of the gas’s heat capacity at constant pressure to that at constant volume, the flow will reachsonic velocity (Mach number M = 1) at the nozzle’s narrowest point. (Note that γ = 5/3 for an ideal monatomic gas
and is less than 2.1 for all gases.) In this case, the pressure at the throat becomes independent of Pb and is equal to P0 /G.
Since there still exists a pressure differential between the gas exiting the throat and Pb , the gas expands supersonically.
This expansion means that any given parcel of gas performs isentropic work on the parcels downstream of it, and so
the gas continues to both cool and accelerate away from the nozzle. Information cannot propagate upstream against thesupersonic flow, so within the expansion the dynamics are independent of any boundary conditions, like the vacuum
chamber walls. Of course, the expansion cannot propagate forever, so it is eventually terminated by a set of shocks.
The region inside all of the shocks, where the expansion is unhindered, is sometimes called the “zone of silence” —referring to the fact that disturbances cannot propagate against the supersonic flow[65]. The expanding gas can reachtransverse and rotational temperatures below 10 K with forward velocities of hundreds of m/s.
To acquire a cold beam from the supersonic expansion, it is necessary to somehow extract a region of the gas
without permitting it to be slowed by any shocks. The usual way of doing this is to use a skimmer projected into
the zone of silence. The skimmer is sharp-edged and carefully shaped so that it does not form a shock across its
aperture, but rather allows the cold, expanded gas to transit through into a region of vacuum sufficiently high as topermit ballistic travel for the packet. In our machine, the skimmer has a 3 mm diameter aperture and is located 12 cm
downstream from the valve nozzle. Once the packet has been skimmed, it is focused by an electrostatic hexapole into
the first stage of the Stark decelerator.

3.2

Stark deceleration
The high-level description of Stark deceleration is simple, and illustrated in figure 3.2: a particle in a weak-

field-seeking state feels a repulsion from a region of increasing electric field, and therefore slows down as it flies into
one. If the field is turned off while the particle is inside it, the Stark potential energy that the molecule possessedis permanently removed from the system. Since this is a purely single-body effect and is independent of the initial

24

(a)

(b)

E
d

Figure 3.2: Stark deceleration. (a) A molecule in a weak-field-seeking state (dipole moment opposing ~E) flies intoa region of large electric field ~E. The increase in the molecule’s Stark potential energy U results in a reduction of its
kinetic energy and therefore its forward velocity. (b) When the molecule has converted a selected amount of kinetic
energy into potential, the electric field is turned off. This permanently removes U from the molecule, leaving it moving
slower than it started. This cycle may be repeated by many stages in a linear decelerator.

velocity of the particle, it is a conservative process.
In principle, with a large enough electric field, a single stage of Stark deceleration can suffice to stop any
molecular packet. In practice, however, laboratory fields are limited to a few hundreds of kV/cm, and so multiple
stages of deceleration must be used. (For scale, our machine can remove a maximum of ∼1 cm-1 of kinetic energy per
stage, and has 143 stages.)
In a multi-stage decelerator, the total degree of slowing is commonly quantified by the “phase-angle” ϕ, which
is defined asϕ=Zswitch· 180◦
2lstagewhere Zswitch is the nominal position of the “synchronous molecule” (the notional molecule for which the switch
timings are designed) at the time of switching and lstage is the distance between adjacent rod pairs (i. e., a “stage” ofthe decelerator). Switching at 0◦ , or when the molecule is at the electric-field minimum directly between a pair ofgrounded rods, leads to pure bunching: molecules which are ahead of the the synchronous molecule are decelerated
and molecules which are behind are accelerated, leading to spatial confinement of a distributed packet. Switching at
90◦ , when the molecules are at the field maximum between a pair of charged rods, removes the maximal amount ofenergy from the synchronous molecule but provides no bunching. The advantage of using the phase angle, rather than
the switching time or position, is that it enables comparison of different decelerator geometries and is independent of
the current speed of the packet.

25

2lstage

ZswitchFigure 3.3: Phase angle in a multi-stage decelerator. The solid green and dashed blue curves indicate the approximate shape of the Stark potential for the two alternate switching states; the heavy black curve shows the Stark energyof the synchronous particle as it traverses the decelerator. (The total kinetic-plus-Stark energy decreases stepwise
at each switching point.) The phase angle is defined as ϕ = Zswitch /2lstage · 180◦ so that maximal slowing occurs at
ϕ = 90◦ .

26
The use of multiple stages substantially complicates the dynamics in a real decelerator. Since molecules
experience repeated, quasi-periodic kicks in both the longitudinal and transverse dimensions, resonances may be
excited[49, 66]. These resonances, and the longitudinal/transverse couplings needed to understand them, are treatedfully in the theses of E. Hudson[64] and B. Sawyer[67].

3.3Magnetic trapping: a story in 2 12 acts
There are many reasons to seek to magnetically trap molecules. Electrostatic trapping[68] is, in many ways,

less challenging since it enables extremely deep traps—but theory predicts that polar molecules will invariably suffer
rapid inelastic losses in the large electric fields required to trap molecules at 50 mK temperatures[69]. (Our own observations of electric-field-induced inelastic collisions are discussed in chapter 6.) Optical trapping of polar molecules has
been demonstrated at microkelvin temperatures for molecules created from laser-cooled atoms[51], but the kilowattscale optical powers required to confine molecules at 10 mK have so far precluded the use of optical trapping on
molecules directly cooled from room temperature.

3.3.1

Version 1.0: the electromagnet trap
The first magnetic trap for OH molecules was a electromagnet-created magnetostatic quadrupole[60], shown

schematically in figure 3.4. Each trap magnet consisted of four turns of hollow, Kapton-insulated square wire, throughwhich cooling water was circulated. The laser for LIF propagated longitudinally through the magnets and up thedecelerator, and fluorescence was imaged out the side. A set of six large rod electrodes permitted the application ofeither a homogeneous or quadrupole electric field to the trapped molecule sample.
Trap loading was accomplished magnetically (figure 3.5): as the molecules exited the decelerator and flew
into the trap region, only the downstream magnet was turned on. The half of the molecules in a magnetic weak-fieldseeking state1 were stopped by the magnetic field, and then the front magnet coil was also turned on to create thequadrupole trapping potential.
In order to achieve magnetic fields of sufficient strength with magnets of only four turns, exceedingly highcurrents of up to 2000 A were required. The molecular packet was slowed to 20 m/s, at a phase angle ϕ = 47.45◦ .
1 The Stark decelerator slows molecules in thef ; ± 32 states, but only the f ; 23 state is magnetically trappable.

27

ρ

z
Figure 3.4: The electromagnet trap. The molecules exit the Stark decelerator (yellow and gray rods) and then fly into
the trap region between the electromagnets (blue rings). The electric field rods (red) permit the application of electricfields with either uniform or quadrupole geometry. Fluorescence is collected out the side by a lens held in the green
lens mount.

valve fire
discharge
deceleration
trap loading
molecules trapped
LIF
downstream magnet only (2000 A)
both magnets (1500 A, declining to 1160 A)
0

10

20
Time (ms)tLIF

Figure 3.5: Loading sequence for the electromagnet trap. Molecules entered the trap after Stark deceleration with
only the downstream magnet powered; after the molecules were stopped by the magnetic field gradient, the upstream
magnet was powered up to create the confining quadrupole potential.

28
1.2

(a)

(b)

1.0

.025

LIF signal (arb. u.)

LIF signal (arb. u.)

.033

.0166
.0083

0.8
0.6
0.4
0.2

0

τ = 16 � 3 ms
(τc = 20 � 5 ms )

0.0

0

2
4
6
8
10
Time from decelerator exit (ms)12

τ = 98 � 5 ms
(τc = 500 � 130 ms )05

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80

Trap Time (ms)Figure 3.6: Performance of the electromagnet trap. (a) Fluorescence versus time curve showing the loading of the
trap. The red line is a Monte Carlo simulation of the loading dynamics. (b) Lifetime measurements in the electromagnet trap. Solid points were taken with a background pressure of 1 × 10−6 torr of N2 , while open points had 4 × 10−8Torr.

The downstream coil was driven with 2000 A, yielding a maximum stopping velocity of 22 m/s. Switching on the
upstream coil (in series with the downstream coil) to close the trap reduced the current to 1500 A over 800 µs due tothe increase in resistance; the current then decayed at a rate of 2000 A/s down to a steady-state level of 1160 A due tothe high-current supplies’ internal regulation circuitry. At 1500 A, the trap operated with a 6700 G/cm gradient with acenter-to-center magnet spacing of 1.5 cm.
Some typical results from the electromagnet trap are shown in figure 3.6. A fluorescence versus time curve(often referred to as a time-of-flight trace) of the trap loading is shown in figure 3.6(a). The first peak shows the packet’s
flight through the trap center before it is stopped, while the second reveals the pile-up of the stopped molecules. The
sharp decrease in mean number between the second peak and the later steady-state is due in large part to the fast
25% reduction in the total trap current. figure 3.6(b) shows two measurements of the one-body-loss limited lifetime ofmolecules confined in the trap. The red and blue fitted lines yield lifetimes of 16 and 98 ms, respectively. Deconvolving
out the losses due to the ramping of the trap current from 1500 down to 1160 A yields background-gas-limited lifetimesof 20 and 500 ms.
Driving the electromagnet coils presented serious technical difficulties. At the high currents used, ordinary #00gauge welding cable is insufficient: we used water-cooled cables usually sold for robotic arc-welding equipment. The
currents were supplied by a pair of 500 A programmable supplies operating in parallel. Fast coil turn-off was achieved

29by the use of a hockey-puck-sized 6000 A high-power diode, although this in turn created large dI/dt voltages dueto the inductance of the magnet coils. Safely switching these currents required a water-cooled bank of high-current
MOSFETs and a custom board to drive the gate of each MOSFET individually. (The circuit diagram for this board
may be found in [67].)

3.3.2

Version 2.0: switching to permanent magnets
Given both the technical difficulties of handling 2000 A and the low densities achieved using electromagnets,

after the initial demonstration our focus rapidly turned to searching for a new trap design. The low trap density(then estimated at 103 cm-3 , but possibly as high as 106 cm-3 given the densities implied by chapter 7) had threeobvious causes. The first was the volumetric mismatch between the decelerator packet and the quadrupole geometry.
The decelerator has a rod-to-rod stage spacing of 5.461 mm; at a phase angle on the order of 45◦ , that implies a
2.5 mm longitudinal length of the packet at the last switch of the decelerator. The transverse size is likewise set by
the decelerator dimensions, in this case the 2 mm × 2 mm square profile created by the alternating rod pairs. The
second cause builds upon the first: the packet freely expands as it traverses the distance from the last stage to the
trap. This means that there is great value in getting the trap as close to the decelerator as mechanics and high-voltage
considerations allow. The third cause is the trap loading velocity. Lower loading velocities hurt twice, first from therapid drop in decelerator efficiency as velocities go below about 50 m/s, and then a second time because slow velocities
mean more time for the packet to expand before it enters the trap.
These three issues clearly motivate the design criteria for an optimal magnetic trap. The ideal trap would be
as close to the decelerator exit as possible, tightly confining with a size matched to the packet’s own size at trap
loading, and would be able to stop and confine a very fast packet. Since it took 2000 A to stop a 20 m/s packet using
electromagnets, magnetic stopping did not seem to be a promising avenue — although the use of high-field, short-pulse
magnets was considered. The coupled issues of the size of the trapped packet and the size of the trap magnets (thelatter setting how close the trap can get to the decelerator) implied a need for extremely large field gradients, typicallyachievable only through either permanent magnets or superconductors. The latter would pose tremendous technicalchallenges, and so we focused on permanent magnets.
The key insight of the permanent magnet trap was that the stopping potential, to load the trap, could still be30

Figure 3.7: The first-generation permanent magnet trap. Molecules exit the Stark decelerator from the left, andare loaded into the trap between the two magnets (small black rings at center of the larger green and cyan mounting
fixtures). The mounting fixtures also provide a slot which holds a lens mount (not shown) for fluorescence collection.

31
electrostatic (i. e., Stark stopping) if we used the surface plating of the magnets as electrodes. The strongest permanent
magnets available are of neodymium-iron-boron (NdFeB) composition, and can provide remnant magnetizations Br
of up to 1.48 T (grade N52) at temperatures up to 80◦ C [70], or slightly less at higher temperatures. The magnets,
however, are vulnerable to rapid corrosion by water vapor and so are almost always encased in some non-permeable
coating. The most common surface coating is a three-layer stack of electroplated nickel, copper, and then nickel again.
Electric fields do not affect the magnetic properties of the material, and testing indicated that the nickel plating issufficiently clean and smooth as to support high voltages in a vacuum, so the combination of a permanent magnet
quadrupole with electrostatic stopping became our design target.
The actual design process consisted of two parallel tracks. The first was the optimization of the magnet geometry. In order to maximize the optical (or molecular-beam) access to the trap, we decided on a ring magnet geometry
with the Stark-decelerated packet entering through the bore of one magnet. In addition to providing 360◦ opticalaccess, this ensured that the longitudinal velocity width of the packet (expected to be substantially larger than thetransverse width) would be matched by the stronger confinement direction of the trap. (Magnetic quadrupoles have
twice the gradient and depth in their longitudinal direction as in the transverse.) Moreover, the electric field generated
between the decelerator and the upstream magnet during the trap loading is mildly focusing in the transverse direction,
which may provide a further enhancement to the loading efficiency.
The actual geometry optimization was an iterative process, alternating between calculating field geometries fora given magnet shape using a finite-element package (COMSOL) and estimating the resulting trap’s loading efficiency
using Monte-Carlo dynamics simulations. The final choice consisted of two identical magnets of Grade N42SH
material (Br = 1.24 T with a maximum temperature of 150◦ C to allow baking the vacuum chamber), arranged antiHelmholtz with a 3 mm gap between the magnets. Each magnet is a square cross-sectioned ring, with an ID of 4 mm,
an OD of 12 mm, and a thickness of 4 mm. This geometry appears to provide the maximum effective trap depth, limitedby the shoulder of the trap on the surface of each magnet facing the gap as can be seen in figure 3.8(a). The shoulder
appears because of the hole in the center of the magnet: field lines can either exit through the bore or out the sides, andso a secondary, toroidal minimum appears between the faces of the magnets in addition to the 3-D quadrupole at thecenter of the trap. This toroidal minimum runs into the surface of the magnets at a lower field magnitude than either
the longitudinal or transverse saddle points.

32

(a)

(b)
valve fire
discharge
deceleration
trap loading
molecules trapped
LIF
magnets charged to +-12 kV
magnets grounded
0

10

20
Time (ms)tLIF

Figure 3.8: Details of the permanent magnet trap. (a) Contours of equal B (spaced by 25 mT), showing the geometryof the trapping potential created by a pair of permanent ring magnets. (b) The electrostatic stopping sequence used toload the permanent magnet trap.

33
The second track in the trap design focused on the mechanical mounting of the magnets. While the magnets
are strong and mounted in a repulsive orientation, the forces between them are not that large — no more than 10 N or
so. However, the magnets need to be able to support a 30 kV potential difference between them, and also be safely
isolated from the ±12.5 kV on the Stark decelerator rods. This requirement is in direct conflict with the desire to get
the trap center as close to the end of the Stark decelerator as possible. In addition, the trap needs to be very accurately
registered to the decelerator: the magnets must both be parallel and coaxial to the decelerator bore. The obvious way
to achieve this is to mount the trap directly to the decelerator, and so we designed a set of dielectric holders for themagnets, machined from Ultem polyetherimide plastic. The holders are illustrated in figure 3.7: the upstream mount
was as thin as possible, allowing the trap center to be only 10 mm from the center of the last decelerator rod. The
downstream mount, however, had a truncated-conical structure so that the supports between the two mounts could
be lengthened to reduce surface currents between the magnets[71]. The upstream mount was attached directly to thedecelerator, with precision-machined seats to reference it to the four large decelerator support rods and holes so that it
could be screwed onto the tapped holes available in two of those rods. The magnets were seated so that their repulsion
would push them further into their seats, and high-voltage connections were made through a set-screw also used toanchor the magnets.
We originally electropolished the magnet plating using currents of a few amps for roughly 45 seconds, but
determined over the course of the experiment that such polishing was at best unnecessary and possible detrimental tothe actual voltage-handling performance of the magnets. Certainly the risk of scratching the magnet plating outweighs
the gain from the polishing. However, while the magnets themselves show excellent high-voltage characteristics, thesame cannot be said for the hardware used to connect the high-voltage feed wires. We use Kapton-insulated wire,
mainly for its convenience: the Kapton is not thick enough to provide any insulation above a kilovolt or so. (Kaptonwrapped wire with a rating of 10 kV is commercially available, but the thickness of the insulation makes the wire
unmanageably stiff.) As long as the wire maintains a clearance of ∼ 5 mm from any other conductor, there are noissues. If the wire brushes the vacuum chamber wall, however, it will short out in operation. Moreover, since themagnets are only 3 mm apart, so too are the connectors — which would not work. To mitigate this issue, we oriented
the connectors so that the upstream magnet’s connector is on top, while the down-stream one is on the bottom. Isolationof the decelerator rods was also initially an issue due to surface-currents across the Ultem, but that was fixed by the34

OH signal (photons/shot)(b)

OH signal (photons/shot)(a)Time from decelerator exit (ms)

τ = 432 ± 47 ms

Trap time (ms)Figure 3.9: Performance of the permanent magnet trap. (a) Fluorescence versus time curve showing the loadingof the trap. The solid red line is a Monte Carlo simulation of the loading dynamics. (b) Lifetime measurement in thepermanent magnet trap demonstrating vacuum-limited lifetimes of almost half a second.

1
in. ceramic (Macor) caps on the ends of the decelerator support rods. The ceramic has surface-current
insertion of 16

properties which are so much better than the plastic that even such a tiny thickness was enough to completely end thesurface-tracking issues.
The performance of the permanent magnet trap is shown in figure 3.9. While the fluorescence-collection
geometry is somewhat different (the permanent magnet trap slotted a lens into the trap mounts, allowing a f = 5 cm
lens to be mounted 5 cm from the trap center), the difference in scale between figures 3.9 and 3.6 is illustrative.
The permanent magnet trap provides OH densities at least two orders of magnitude larger than the electromagnet.
Moreover, without the current-sag issues described in section 3.3.1, only a bake-out to 100◦ C was required to achievea 400+ ms trap lifetime.
Loading of the trap used voltages of ±12 kV to stop a 36 m/s packet from a decelerator phase angle ϕ =50.352◦ .

3.3.3

Versions 2.2 and 2.5: Variations on a theme of fixturing
The great success of the original permanent magnet trap has meant that we have not yet made any further

changes to the magnetic field geometry or the choice of magnets. Indeed, the trap described in the previous section
was used almost unchanged for three years during the experiments described in chapter 4. However, more recently we35
have experimented with changes to the magnet support structures in pursuit of several goals.
The first variation in trap mounting was the “cryogenic trap,” shown in figure 3.10. This mounting design
had two goals. The first was to permit the application of a microwave field across the trapped molecules, to enable
control of their Λ-doublet state and thus permit collision studies with |ei-state molecules. (See chapters 5 and 7.) The
second design goal was an attempt to improve the vacuum-limited trap lifetime by using the trap itself as a cryopump
to remove background krypton gas from the supersonic expansion. We have known since the electromagnet trap that,once the residual water vapor has been remove by either a 100◦ C bake-out or simply enough pumping time, thevacuum is dominated by diffuse, 300 K krypton. The clearest demonstration of this fact is that the OH lifetime varies
inversely with the experiment repetition rate: at a 5 Hz rate, the lifetime is twice as long as at 10 Hz. By lowering therepetition rate to 1 Hz, we have observed lifetimes of over a second.
Since longer lifetimes would provide improved signal-to-noise at a fixed repetition rate, we decided to try to
reduce the background krypton density. We were unable to design a support fixture which mounted directly to thedecelerator and had a large enough thermal resistance to enable cooling to 77 K, so this fixture was free-standingon the bottom ConFlat flange of our 6-inch cube chamber. The fixture consisted of first a thermal-break stage madeof Macor machinable ceramic, to minimize heat flow from the vacuum wall, and then an electrically insulating but
thermally conductive alumina block which provided the thermal contact between the liquid nitrogen flow line andthe 316L stainless steel arms which held the magnets. (316L was chosen due to it having a lower rate of acquiring
ferromagnetic lattice impurities than 304SS when cooled to cryogenic temperatures.) Each arm was referenced to theprecision-ground alumina block by a shallow, three-sided rectangular cup on the end of the arm; connection force wasprovided by a gold-plated screw which penetrated a clearance hole in the alumina and anchored into a tapped hole in
the end of the arm.
The magnets in this fixturing system were in blind cups, with a thin stainless-steel plate facing the trap center.
The concept was that a well-electropolished steel surface might permit higher stopping voltages than the magnet
plating. However, on the first installation of the trap, a problem with this mounting design became immediately
obvious: the magnets’ mutual repulsion actually pushed them out of the fixture. Originally, the magnets had been heldin by a single set-screw; after this problem became clear, the arms were modified to take a thin retainer plate affixedby two screws, to keep the magnets confined.

36

Figure 3.10: The cryogenic permanent magnet trap. The trap magnets (black rings) are supported by thermallyconductive stainless steel arms, which are in turn supported by a complex mounting stack. From the bottom, the stackconsists of a steel base-plate, a thermally and electrically insulating Macor ceramic fixture (white), and a thermallyconductive but electrically insulating alumina standoff (gray). The alumina standoff allows the steel arms and trap
magnets to be cooled to 80 K by a liquid nitrogen cooling loop (not shown). The steel mounting arms also provide atransmission-line geometry to permit coupling microwave fields across the trapped molecules.

37
The goal of applying microwaves to the trap was accomplished easily. The actual circuitry is described inchapter 5; the in-vacuum transmission line feeds were attached to the gold-plated screw, so that microwaves could
propagate down the fixture’s arms in a continuation of the transmission line geometry. Measurements using an ultrahigh-speed oscilloscope indicated that microwave fields of several V/cm (across the 3 mm gap) were easily attainable
using only ∼ +15 dBm of applied power.
The goal of cryopumping krypton, however, was a total failure. No difference in either vacuum gauge readings
or trap lifetime was observable when the trap was cooled to 77 K with flowing liquid nitrogen. The equilibrium vapor
pressure of krypton at 80 K is still 5 mbar[72], so any pumping would have to occur as result of surface cryosorption.
The binding energy of krypton on clean, polished steel or copper is presumably low, so the lack of pumping is notentirely surprising. It remains possible that krypton could be pumped using a high-surface-area charcoal or zeolite
cryosorption system, but this trap fixture did not contain any.
One other difficulty of this fixturing system was alignment. The original system mechanically referenced themagnets directly to the decelerator, and the fluorescence collection lens to the magnets. (The second lens which
focuses the fluorescence through the iris has always been free-standing, however.) With this new system, however, theentire stack had to be aligned by hand to the decelerator. The system was designed with three axes of translational
adjustment and one axis of rotation (vertical), so that a set of feeler tools and centering pins could be used to achieve
proper alignment. This process was slow, and apparently trouble-prone: after the initial alignment and several months
of use, on a subsequent vacuum break the magnets appeared to be rotationally misaligned by possibly as much as 15◦ .
The fluorescence lens alignment also proved to be challenging. With the lens now supported from a 2.75” CFflange at the base of the cube, the tip-tilt alignment of the lens appears to be responsible for at least factor-of-threevariations in the fluorescence collection efficiency. The solution I used is that the flange was originally connectedusing the minimum compression of its copper gasket needed to achieve a seal. Then, with the system operational, thebolts were selectively tightened to maximize the fluorescence signal on the bunching peak. This may not have been
the truly optimal solution, either, since the narrow time structure of the bunched peak is convolved with the optimal
longitudinal position of the focal point. In hindsight, optimization using the trapped signal would have been better —but the 20X reduction in signal-to-noise between bunching and trapping would have made the process much slower
still.

38

Figure 3.11: The present permanent magnet trap.

39
The final trap variation described in this thesis (figure 3.11) is in many ways a rejection of the cryogenic
design. Instead, it is a return to the original fixturing method with some major optimizations. The goals for this newest
system were a return to mechanically referencing the trap directly to the Stark decelerator, a continuation of the easy
application of microwave fields, and a continuation of the previous trap’s success in using only metals and ceramics
for its construction (rather than the plastic used in the original mount). The resulting design is simple and elegant:each magnet is held in a steel bar supported diagonally by two of the four decelerator rods. In order to provide theelectrical isolation between the decelerator rods and the trap supports, a Macor stand-off is used. The fastening of thestand-off to the decelerator support rod is the true elegance of this system. A titanium sleeve fits over the end of eachrod, and has an L-slot which locks over the last decelerator rod. The sleeve has a shallow, internally-threaded cup on
its end, into which the Macor stand-off is screwed. The stand-off then has a blind, tapped hole in its center, which isused to anchor the magnet support.
Applying the lessons of the previous mount, the magnets are again inserted so that their plated surfaces face thetrap center and their mutual repulsion serves to keep them seated. The microwave transmission line is now connectedhorizontally to each mount by a titanium set-screw. The set-screw was drilled out, and a rigid copper wire soldered into
it using vacuum-compatible Ag-Sn solder, with a ring-terminal soldered to the other end of the wire. The wires were
hand-bent to create the transmission-line geometry, with several rounds of iteration required to get a nice, parallel,
symmetric geometry and a good fit. As with the original trap, one high-voltage feed is connected at the top of the
upstream mount, and one is on the bottom of the downstream mount. The fluorescence-collection lens, however,remains supported from the vacuum flange.

Chapter 4
Cold beam-trap collisions

Following the achievement of magnetically trapping OH, one immediately obvious use for a trapped sample isthe study of molecule-atom and molecule-molecule collision cross-sections. The long trap lifetime allows for even avery weak source of collision-partner particles to be used, since loss may be integrated over hundreds of milliseconds.
The signal of interest is the fractional loss of trapped OH molecules, so an absolute density calibration of the trap is notneeded. However, to determine true, absolute collision cross-sections, the velocity and density of the impinging probe
beam are needed. Since these two quantities are generally much harder to measure than the loss of OH molecules, theirsystematic uncertainties therefore dominate the overall uncertainty of the measured cross-sections. The fractional lossof trapped molecules χ may be considered as effective “absorption” by the colliding beam, and so the Beer-Lambert
law saysχ = e−nσ vτ
where n is the density of the collision partner, v is its velocity, τ is the duration of the collision packet, and σ is theeffective cross-section.
It is important to emphasize that the observed σ in a beam-trap experiment is only an effective cross-section[73].
Specifically, it is the “cross-section for any scattering process which causes disappearance of fluorescence.” In the caseof OH, this means that it includes inelastic scattering to the |ei-state (since LIF is state-sensitive), inelastic scattering
to lower, untrapped levels of the | f i manifold, and any collision which transfers enough momentum to eject the OH
molecule from the trap. Unfortunately, the last type of collision may only be a fraction of the total elastic crosssection, particularly in the presence of strongly anisotropic scattering. This leads to a greater reliance on theoretical
calculations to back out the true, total elastic cross-section than might otherwise be desired.

41

4.1

OH–He and OH–D2 collisions at 77 K
The great success of the first permanent magnet trap (described in section 3.3.2) prompted an immediate desire

to use it for real measurements[61]. An ongoing theme in our study of OH has been its relevance to astrophysics,
and its collisional properties are no exception. Astrophysical masers emitting on the OH Λ-doublet transitions havebeen attributed to collisional pumping by H2 [74, 75]. However, molecule-molecule collisions remain substantiallyintractable to calculate using ab initio methods, prompting theorists to try to approximate the cross-section using OH–
He calculations. As there was no scattering data available at the temperatures of interest below the lowest OH rotational
excitation, these properties were a prime target for a demonstration of the power of beam/trapped-molecule collisionexperiments. Unfortunately, our turbomolecular pumps do not efficiently remove H2 from our vacuum system due toits exceptionally low mass. Therefore, we chose to compare OH–He collisions to OH–D2 instead, since the pumps
are equally effective upon the two gases. While the different mass and rotational constant of D2 as opposed to H2
may give it substantively different scattering properties, if OH–D2 collisions are noticeably different from OH–He it
is likely that OH–H2 collisions are as well.

4.1.1

Cryogenic supersonic expansions for tunable-velocity He and D2 beams
The apparatus used for this first collision experiment was extremely simple. A stainless solenoid valve (General

Valve Series 99) was outfitted with a custom Kel-F poppet tip, with the material chosen so that the valve would continue
to seal down to cryogenic temperatures[76]. The valve was driven with a high-voltage supply (generally 140-300 V),
enabling a very fast opening time. The switching circuit (figure 4.2(a)) is single-ended, with a transient-voltage
suppressor (TVS) connected to ground. When the supply switch was opened, the reactive voltage from the solenoid’s
inductance is above the TVS’s clamping level, and so it becomes conducting and very rapidly drains the current fromthe coil to achieve a fast shut-off of the solenoid. The valve’s true closing speed, however, is not set by the solenoid
but by a spring mechanically pushing the poppet into place to seal. The high opening forces from the fast solenoid
turn-on cause the poppet holder to have enough momentum that the poppet bounces several times before the valve
fully seals. To prevent the bounces, we fired a second, shorter current pulse through the solenoid to act as a brakeagainst the spring forces slamming it closed, as illustrated in figure 4.2(b). With a two-pulse sequence, we were able

42

Figure 4.1: Experimental system for OH–He and OH–D2 collision experiments. OH molecules exit the decelerator(yellow, rear-left) and are electrostatically stopped and loaded into the permanent magnet trap (blue rings). The
molecules are allowed to settle before a supersonic pulse of collision gas (He or D2 ) is fired from the solenoid valve(green cylinder) through the trapped cloud.

43
to achieve gas pulses as short as 160 µs FWHM (figure 4.2(c)).
In order to achieve the desired low collision energies, the entire solenoid valve was immersed in liquid N2 ,in a custom vacuum-insulated vessel designed to mate with the vacuum feed-through system used by the fast pulsed
valves sold by Jordan TOF Products, Inc. A custom nozzle flange was made for the solenoid valve which was wrappedwith 25 turns of manganin wire to allow control of the nozzle temperature between 77 and 330 K, as measured bya cryogenic silicon diode. The resulting supersonic beam velocities were measured by a two-position time-of-flighttechnique, and showed that the velocity scaled smoothly as T 1/2 as shown in figure 4.3(a). The valve nozzle was
1 mm in diameter, and opened in a 40◦ cone to enhance the beam brightness. The valve was operated with a 2.0 bar
backing pressure for He and 2.7 bar for D2 ; this enabled minimum center-of-mass collision energies of∼ 60 cm-1 and∼ 145 cm-1 respectively. We observed an 8% velocity width in the He beam, implying an energy resolution of 9 cm-1
at the lowest collision energy. The beam density was also temperature-dependent, but easy to calibrate using a fastionization gauge as shown in 4.3(b).

4.1.2Collision results
The experimental sequence for this experiment (figure 4.4) was as simple as the equipment: OH molecules

were decelerated and trapped, allowed to settle for 24 ms, and then the pulsed valve was fired. The fluorescencesignal was measured alternately before and after the gas packet had flown through the trap, to determine the fractional
scattering. Determination of the total cross-section for collisions which cause trap loss then required knowledge of thegas pulse’s density n and velocity v.
It is easy to determine the velocity of a gas pulse: simply record its arrival time at two distinct positions anddivide the distance between them by the difference in times. Density, while more complex, was not a major difficultyeither. We took advantage of the temperature control available on the valve to measure the OH–He collision crosssection near a collision energy of 250 cm-1 twice: once using a velocity-tuned packet from the valve, and a second timeusing thermal, 298 K helium leaked into the chamber. The latter experiment allowed the helium density to be measured
using a calibrated ionization gauge, and the OH–He cross-section to be determined from the OH molecules’ lifetimein the magnetic trap. In combination, these two measurements established an absolute scale for the measured crosssections; relative measurements of the beam density at each temperature using a fast ionization gauge then enabled the

44

(a)
Valve solenoid

+HV

TVS

2.5
30

10x Amplified Mic Signal [mV]General Valve Current [A]

(b)
2.0

1.5

1.0

0.5

(c)

20

10

0

0.0
-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0

1.2

0.5

1.0

1.5

Time from Valve Trigger [ms]Time [ms]Figure 4.2: Cryogenic solenoid valve circuit and performance. (a) Circuit diagram of the single-ended solenoid
drive circuit, including the transient voltage suppressor (TVS) used to achieve fast turn-off. (b) Solenoid current
profile achieved using the circuit of (a), including the two-pulse time structure described in the text. (c) Pressure
versus time recorded from impact of the supersonic beam on a miniature microphone mounted in line-of-sight withthe skimmed helium beam, showing a 155 µs FWHM gas pulse.

(a)

(b)
1600

Beam density (arb. u.)

D2 velocity (m/s)

1700

1500
1400
1300
1200
1100

1.0

He

0.8
0.6
0.4

D2

0.2

1000
60

80

100 120 140 160 180 200 220 240

Valve temperature (K)

0.0
50

100

150

200

250

300

Nozzle temperature (K)Figure 4.3: Supersonic expansion performance of the cryogenic solenoid valve. (a) Beam velocity versus valve
temperature for D2 . (b) Relative density versus valve temperature for pulsed He and D2 supersonic expansions.

45transfer of that absolute scale to all other measurements. The resulting calibration is that the OH–He cross-section is
127 ± 18 2 at 250 K; all other cross-sections may be interpreted relative to this value. It should be emphasized that thissort of absolute calibration was only possible because the OH molecules were trapped.
A sample time-of-flight trace of the He packet’s impact upon the trapped OH sample is shown in figure 4.5.
There is a rapid loss of trapped OH over roughly a millisecond — longer than the He pulse’s intrinsic duration, but
only slightly more than a single trap oscillation period — and then the OH population remains effectively constant over
the remainder of the time plotted. Even with the added gas load from the solenoid valve we observed trap lifetimesof over 50 ms, and so the prompt loss can safely be considered to be truly beam-trap collisions rather than loss due tomultiply-scattered He or D2 particles.
The results of these experiments are shown in figure 4.6, and prompt several conclusions. The strong drop
in both the OH–He and OH–D2 cross-sections below 90 cm-1 is suggestive of the closing of the J = 52 ← J = 32
rotational-excitation channel in OH at 84 cm-1 . It should be noted that the absolute magnitude of the He–OH crosssection is more than a factor of 5 smaller than that of OH–D2 : we attribute this in part to the quadrupole moment
possessed by J = 1 orthohydrogen, whose population is frozen into the supersonic expansion at roughly the room
temperature 3:1 ratio with J = 0 parahydrogen. The quadrupole moment can interact with the OH dipole moment andtherefore adds both strength and anisotropy to the scattering interaction.
There is also the apparent peak in the OH–D2 cross-section near 305 cm-1 to consider. It is difficult to attributea cause to it without substantial theoretical work, but it is suggestively located near the 300 cm-1 J = 3 ← J = 1
excitation in orthohydrogen.

4.2

OH–ND3 collisions at 4 K
Prompted by the success of the OH–He/D2 collision experiment and the impressive results then being demon-

strated by buffer-gas cooling[77, 78], we began a collaboration with the group of John Doyle at Harvard University,and particularly with Dave Patterson. The goal of this collaboration was to use a buffer-gas cooled beam of deuterated
ammonia, ND3 , to look for dipolar OH–ND3 interactions at a 4 K collision energy[79, 67].
Figure 4.7 shows a sketch of the experimental setup for this experiment. OH molecules were decelerated andtrapped as before — Stark deceleration is now a tool, rather than a research project. The new ingredients are the46

valve fire
discharge
deceleration
trap loading
molecules trapped
LIF
magnets charged to ±12 kV
He valve opening pulse
He valve braking pulse
He packet transits trap region
5

0

10
15
Time (ms)20

25

OH signal (photons/shot)

Figure 4.4: Experimental sequence for OH–He and OH–D2 collision experiments.

Time from He valve trigger (ms)s
Figure 4.5: OH fluorescence versus time showing the impact of a helium packet. The collision promptly removes∼ 47% of the trapped molecules, but there is no visible change in the background-gas collisional loss rate after the
helium packet completes its transit through the trap.

47
180

800

160
140
120

600

100
80
60

400

40
20
0

0

100

200

300

400

500

Total D2-OH Cross Section (Å2)

Total He-OH Cross Section (Å2)

200

200
600

Ecm (cm-1)
Figure 4.6: Absolute OH–He and OH–D2 trap-loss cross-sections versus collision energy. Note that the y-axis
scales differ between the two cases. Error bars are one standard error.

Hexapole guideS
dec tark
ele
rat
or

Magnetictrap

OH fluorescence,
OH fluorescence
ND3 ionization lasers collection lens

ND+
3 ToFMSassembly

He, ND3 buffer
gas source

MicroChannel
PlatesFigure 4.7: Experimental system for OH–ND3 collision experiment. OH molecules are Stark-decelerated andloaded into the magnetic quadrupole trap, and then illuminated by the 4 K ND3 molecular beam. This beam is formed
by a capillary-filled buffer-gas cell, and then electrostatically guided around a 90◦ bend to separate it from the helium
carrier gas and to provide a rapid turn-on/turn-off capability. The relative OH density in the trap is detected by laserinduced fluorescence; the absolute ND3 beam density is determined by resonance-enhanced multiphoton ionization,
calibrated against a hot-cathode ion gauge.

48
buffer-gas source producing a 4 K ND3 beam and the REMPI detection system to measure the ND3 beam density.

4.2.1

An introduction to buffer-gas cooling
The basic principle of buffer-gas cooling sounds trivial: cold, 4 K helium gas is mixed with some warm species

X in a cell to produce a 4 K sample of X. If the cell has an opening in it, a beam of X in a helium carrier comes out(figure 4.8). Of course, the actual implementation has substantially more subtleties. In the limit of low helium density,
the “beam” exits the cell effusively: there is no actual beam, just a flux of particles escaping with a thermal, isotropic
momentum distribution. The downside is that since the X particles are transported diffusively, the fraction of the initialsample which actually exits the cell is very small. This fraction, usually termed the extraction efficiency, is just theratio of the exit aperture area to the cell’s total internal surface area — usually on the order of 0.1–1 percent.
At much higher helium densities, there exists a substantial pressure differential between the cell and the vacuum
space outside it. In this case, the helium hydrodynamically entrains the X particles, and pulls a large fraction of themout the aperture. However, the trade-off is that the helium undergoes multiple collisions as it exits the cell, and forms
a supersonic expansion. While this can yield internal state distributions even colder than the cell temperature, it also
gives the beam a mean center-of-mass velocity on the order of 150 m/s.
In between the effusive and supersonic regimes, there exists a crossover region commonly referred to as thehydrodynamic regime. In this part of parameter space, there is enough helium density to entrain the X particles, but
the density is low enough that the gas only undergoes a couple of collisions as it exits the cell. The result is a beamwith a much smaller center-of-mass boost, but still good extraction efficiency.

4.2.2

Buffer-gas-cooled ND3 and the challenges thereof
The ND3 system operated most likely in the hydrodynamic regime. The helium flow rate of 3.5 sccm (that is,

atm.-cm3 /minute; 1 sccm= 4.4 × 1017 molecules/s) is considered relatively low, while the cell aperture’s large 6 mmdiameter reduced the pressure differential. The mean velocity of the ND3 beam after the bent-hexapole guide was
110 m/s (see figure 4.10). The forward velocity acceptance of the guide can be calculated using the guided species’
maximum Stark shift and the guide’s radius of curvature. For this guide with ±5 kV and ND3 in its |J = 1, K = 1i
state, we predicted a cutoff velocity of 150 m/s.

49

4K Pulse Tube

ND₃ ﬁll line

Mixing cell

He ﬁll line
Figure 4.8: Buffer-gas-cooled 4 K ND3 beam source. Warm (200–250 K) ND3 gas flows down the central capillary
fill line, and is mixed with 4 K helium to produce a beam of 4 K ND3 . The beam is guided by a hexapole (see figure 4.7)
to separate the ND3 from the helium carrier gas. The ND3 fill line enters the cell through a G-10 fiberglass thermal
standoff, allowing the line to be maintained at 250 K using only 200 mW of heater power. The fiberglass standoff isdesigned to maximize the path length from any 4 K surface to the tip of the ND3 line and therefore maximize the runtime available before an ice bridge is formed.

Figure 4.9: The rotational and parity spectra of OH and ND3 . The parity splittings are exaggerated for clarity. The
red-highlighted parity states are those used for the collisions described in this section.

50
The Stark shift spectrum of ND3 is qualitatively very similar to that of OH; both are shown in figure 4.9. ND3
possesses a 1.5 D dipole moment, and a low-energy parity doubling of 1.5 GHz due to the so-called “umbrella-mode”
inversion vibration. (The N atom can be either above or below the three D atoms, so the ground state consists ofsymmetric and antisymmetric combinations of the two possibilities. The splitting is set by the tunneling rate between
the two possibilities — in NH3 , with its 2x lighter H atoms and exponentially higher tunneling rate, the splitting is23.8 GHz.) As a symmetric-top molecule, ND3 is characterized by three rotational quantum numbers, J, K, and M. J
is the total angular momentum and M its projection on a laboratory space-fixed axis, as before. New, however, is thefact that ND3 possesses two distinct moments of inertia. We define B as the rotational constant associated with rotationtransverse to the 3-fold symmetry axis and C as the constant associated with rotation about the symmetry axis. This
produces the rotational Hamiltonian
Hrot= B(Jˆx2 + Jˆy2 ) +CJˆz2= BJ2 + (C − B)Jˆz2= BJ(J + 1) + (C − B)K 2

where we define K as the projection quantum number of J along the three-fold symmetry axis of the ND3 molecule.
Formally, the inversion doublet is the symmetric and antisymmetric combinations of K = |K| and K = −|K|;
K = 0 states have no doublet. Since the rotational averaging term for symmetric top molecules for a dipole moment µisµeff = µMK
J (J + 1)

and K and M both vary between −J and J, molecules in high-JK states are guided even more effectively than low-J
molecules by an electrostatic guide. To separate the ND3 from its He carrier and also to permit switching the beamon and off, we used a two-segment electrostatic hexapole guide. The first segment was a simple straight hexapole,consisting of six 3 mm diameter electropolished stainless steel rods with an inner diameter of 6 mm and running from2 mm after the buffer-gas cell exit to a gate valve 20 cm away. The gate valve allowed the pulse tube cryostat to beisolated so that the helium released from the cryosorption pumps when the system warmed up did not contaminate thetrap and decelerator vacua. The gate valve was custom built for us by the JILA instrument shop, and enabled us tominimize the gap between between the first and second hexapoles to only 3 mm. The second hexapole contained a 90◦

a)

1

2

ND3 Signal [arb. u.]

ND3 Population [arb. u.]

51

3

4

5

6

7

8

Collision Energy [cm-1]

9

1,1(2,0)b)2,2(2,1) 3,3(3,2)
4,4(4,3)31498 31500 31502 31504 31506 31508

Frequency [cm-1]

Figure 4.10: ND3 beam parameters. (a) Relative density versus velocity, quoted as OH–ND3 collision energy. (b)Rotational state distribution for two different buffer-gas flow rates: (upper, red) 2.0 sccm He and 2.5 sccm ND3 and(lower, black) 3.5 sccm He and 1.0 sccm ND3 . The unparenthesized labels indicate the J and K quantum numbersof the ground rotational state probed by each line, while the parenthesized labels indicate (J, K) for the intermediate
rotational state in the excited B̃(v0 = 5) level. Due to their lesser confinement within the hexapole guide, states withK < J are not observed.

bend with a 13.5 cm radius of curvature and terminated 1 cm away from the center of the magnetic trap. This bend iswhat accomplished both the separation of the ND3 from the buffer gas and the gating of the ND3 beam.
Since the buffer-gas cell is fed by long, narrow gas lines, turning off the gas flows outside of the cryostat does
not immediately cut off the molecular beam. Instead, the beam decays on a timescale of tens of seconds. However,
without the electrostatic guide, effectively no ND3 molecules are able to enter the trap chamber. With the guide off,
there is no ND3 to the sensitivity limit of our REMPI detection, and even with both the helium and ND3 gas flows on
there is no detectable change in the OH trap lifetime versus the pure background vacuum level.
The temporal gating ability of the hexapole guides allowed us to characterize the forward velocity distributionof the guided ND3 beam. The guide was turned on for increasing lengths of time and the peak signal was measured.
The derivative of this density-versus-guidance time curve yields the velocity profile of the beam, which is plottedin figure 4.10(a). (For computational simplicity, the measured curve was fit with a 6th order polynomial, whose
derivative is plotted here.) The rotational temperature of the beam was determined using REMPI spectroscopy. The
relative strengths of the different rovibronic lines can be calculated using the Hönl-London factors; dividing the line
strengths out yields the relative densities of the different ground rotational states and hence the rotational temperature.
We found that the rotational temperature varied strongly with the gas flow rates. Unfortunately, the flow rates with52
the highest absolute beam densities also possessed the highest rotational temperatures. Spectra of two different sets offlow rates, are plotted in figure 4.10(b). The final choice of a 1.0 sccm flow rate for ND3 and 3.5 sccm for He was acompromise yielding a rotational temperature of 5.4 K and an acceptable beam intensity.
The majority of the technical effort involved in the OH–ND3 experiment was spent on developing a stablebuffer-gas source. While it took longer than expected to observe any hexapole-guided ND3 , a further fourteen months
elapsed between the first guided signal and the beginning of the actual collision experiment. Those six months werespent trying to get the source to work for more than five minutes at a time.
The symptom we observed was that after some short period of running, there would occur a simultaneous spikein the cell temperature and in the ND3 signal in both REMPI and on the residual-gas analyzer (RGA). As the RGA does
not have a line-of-sight view of the guided beam, a pressure jump on it indicates a very large flux of unguided, warm
ND3 . We went through several theories involving temperature/pressure oscillations in the helium fill line, insufficient
cryosorption pumping speed, or poor thermalization of the helium to the cell, before this problem was solved. The
true issue lies in the fact that ND3 freezes at 195 K. Our original cells were simply too small: the cell filled withammonia ice, and when the ice growth reached the (warm) ammonia fill line the ice flash-evaporated. This spiked thecell temperature, and released the observed large puff of both ammonia and helium. Moreover, if enough helium wasreleased to thermally connect the 40 K radiation shield to the 4 K cryosorption pump plates, it is possible to get thermal
runaway. The ’sorbs release their adsorbed helium as they warm up, which further increases the thermal conduction to
the 40 K shield—a positive feedback loop which is only broken once the ’sorbs have released their entire helium load.
The solution to this icing problem is simple once it is understood: just make the cell bigger! While there arelimits due to the in-cell gas dynamics (as described in section 4.2.1), up to a point increases in cell volume can becompensated by increasing the exit aperture size. Since there is no substantial loss to having the cell aperture matchthe hexapole guide aperture, we arrived at a 6 mm aperture and a cell with approximately 1 cm of free space on all
sides of the ND3 fill line.

4.2.3

Collision results and ND3 density calibration
Once the properties of the ND3 beam were understood, the actual collision experiment was undertaken. The

sequence for each shot (illustrated in figure 4.11) was similar to that of the OH–He experiment. OH molecules

53

valve fire
discharge
deceleration
trap loading
settling timecollisions
LIF
magnets charged to ±12 kV
magnets optionally charged to Vpolarize
ND3 guide hexapole charged to ±5 kV
0

10

40
Time (ms)200

Figure 4.11: Experimental sequence for OH–ND3 collision experiment.

54
were Stark decelerated and trapped, then allowed to settle. If a polarizing field was applied, it was turned on at the
beginning of the settling period, as well. After ∼ 25 ms of settling time, the ND3 guide was turned on for durationsof up to 160 ms and the relative OH population was measured as a function of time to yield a one-body loss rate Γon .
The difference between Γon and the loss rate with the ND3 beam turned off (Γoff ) gives the collisional loss rate due to
OH–ND3 scattering. Each experimental shot measured the relative number after a single period of ND3 bombardment;
many shots were averaged together and the bombardment duration was varied shot-to-shot to build op the numberversus-time curve. A fresh baseline (ND3 off) curve was taken each day, since the vacuum quality appears to vary ontimescales of a day or two. A typical pair of lifetime curves with the ND3 beam off and on is shown in figure 4.12
While the OH-ND3 collision rate is easy to measure, it is much more complicated to convert that to a true
cross-section. Just as with the OH–He experiment, a method to determine the true density of the impinging ND3
beam is required. Since REMPI has large systematic uncertainties related to laser mode shape, absolute ionization
efficiency, and ion extraction and detection efficiency, REMPI ion signals must be calibrated against some standard.
For this experiment, we again turned to an ionization gauge as the reference standard.
The scheme for calibrating the ND3 beam density is shown schematically in figure 4.13. Each day, a pair of
ND3 off and ND3 on collision measurements were performed, with the pulsed dye laser using Rhodamine 6G dye
and tuned to 282 nm for laser-induced fluorescence detection of OH. Then, working quickly so that the buffer-gas cell
did not accumulate too much ice before the density could be calibrated, we swapped a different set of dye cuvettes
into the laser (this time using DCM Special, a functionalized variant of the normal DCM with increased solubility inmethanol) and retuned the laser to produce 317 nm light for REMPI. An f = 30 cm final focus lens was inserted into
the beam path and the laser was realigned to go through the magnetic trap.
Extracting ions across the enormous magnetic field gradients in the permanent magnet trap required large
acceleration voltages so that the ion cyclotron period of ∼ 1 µs was much longer than the time required for the ionsto escape the trap region. We applied a +950 V voltage pulse to the decelerator-side magnet (while keeping the thedownstream magnet grounded) to push ND3 + ions out of the trap and into a time-of-flight mass spectrometer (TOFMS).
The TOFMS consisted of a pair of grids held at a uniform potential of -1.1 kV to create a 2 cm free drift region, before
the ions are detected on a pair of chevron-oriented microchannel plates. To block stray electric fields (due to charging
of dielectric surfaces by electrons or ions emitted by the Stark decelerator), the drift region was fully enclosed in55ϭτĐŽůůсϮϲϰ±ϱϰŵƐ

Ϭ͘ϵ
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Ϭ͘ϳ

K,^ŝŐŶĂů΀Ăƌď͘Ƶ͘΁

Ϭ͘ϲ
Ϭ͘ϱ

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Ϭ͘Ϯ

EϯKĨĨ͗ϮϮϳ±ϯϳŵƐ
EϯKŶ͗ϭϮϮ±ϭϱŵƐ
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dƌĂƉdŝŵĞ΀ŵƐ΁

Figure 4.12: OH fluorescence versus time showing the effect of the ND3 beam. The black points are a plot of
number versus time with no impinging ND3 , while the red points are with the ND3 beam illuminating the trapped OH
molecules. Solid lines are one-body loss-rate fits to the data; quoted uncertainties are one standard error.

56
a metal tube; a metal ground shield was also mounted to the (plastic) support part for the downstream magnet, andelectrically tied to that magnet. The +950 V extraction voltage was only pulsed on after the laser ionization step, so thatthe ionization would occur with parity being a good quantum number. Since the B̃ state in ammonia possesses planar
symmetry, its vibrational levels alternate in parity. Therefore, in the absence of an electric field, the two ground-state
umbrella-mode parity states connect to different, spectroscopically-resolved B̃, v0 levels. We wanted to measure thedensity of only one parity level, so it was important to ensure that there was no contamination by the opposite parity
due to an electric field. (The ND3 guide hexapole was also turned off immediately prior to the ionization laser trigger.)Once the laser was aligned, the ion signal due to the cold ND3 beam was measured. The gate valve was thenclosed, and a precision leak valve was used to back-fill the trap chamber to achieve an equal ion signal. It took several
tens of minutes for the ND3 density to stabilize, since the gas needed to re-passivate the chamber walls versus proton
exchange (which converts ND3 to NDx H3-x ); a similar passivation was performed every day on the buffer-gas cell
before cooling it to 4 K. When the ND3 density as measured by the RGA was stable and the ion signal level was
well-matched to that recorded from the buffer-gas-cooled beam, a series of measurements of the exact RGA reading
and ion signals were taken; these measurements were used to determine the local calibration between ion signal and
RGA-measured density. It was important to only perform this calibration using very similar ion currents, because themultichannel plate is susceptible to both ion saturation and gain shifts due to the background pressure.
While REMPI is a spectroscopically-selective technique, at room temperature the REMPI spectrum of ND3
suffers a great deal of spectral confusion as can be seen in figure 4.14. At a 5 K rotational temperature, it is certain thatthe entire ion signal from the buffer-gas-cooled beam is due to the population in the |J = 1, K = 1i state. At 295 K,
however, that is not true. Spectral simulations (detailed in chapter 4 of [67]) indicate that only 16% of the 295 K
ion signal is due to the |1, 1i state. Correcting for this line-strength factor and using the Boltzmann distribution todetermine the the relative fraction in the weak-field-seeking |1, 1i state then gives the final density calibration factor.
The final results of the collision experiments are shown in figure 4.15. Two sets of cross-section measurements
were taken: electric-field-free, and with an electric field large enough to polarize both OH and ND3 . The polarizing
case consists of two different electric field configurations: one with the magnets themselves charged, and one where
the ground-screen behind the downstream magnet mount had electrically disconnected and was therefore floating at a
substantial voltage. (This was determined by the appearance of strong-field-seeker lines in the buffer-gas-cooled beam

57

NaD

NbD
RG
A

RG
A

ND₃

excita�on laser

vacuum chamberﬁlled with ND3

ND₃⁺ ions

MCP

RG
A

MCP

NdD

1

.

0

0

.

8

0

.

6

0

.

4

0

.

2

0

.

0

I

o

n

S

i

g

n

a

l

[

a

r

b

.

u

.

NcD

ND₃⁺ ions]

excita�on laser

1

4

8

0

3

1

4

9

0

3

1

5

0

0

3

1

5

1

0

3

-

F

6

r

e

q

u

e

n

c

y

[

c

m

1

5

2

0

3

1

5

3

0

1

]

P22 / P11
0.21
0.08

6.0/2.0 sccm He/ND
3
6.0/0.5

5

REMPI Signal [nVs]vacuum chamberﬁlled with ND3

3

4
3
2
1
0
31480

31490

31500

31510

31520

31530

-1
Frequency [cm]Figure 4.13: ND3 density calibration sequence. (a) The REMPI ion signal of the guided 4 K beam is measured.
(b) The guide is turned off, and the chamber is filled with room-temperature ND3 gas to a sufficient density as toreproduce the REMPI ion signal. (c) The density of the room-temperature ND3 gas is measured using a residual
gas analyzer calibrated against a Bayard-Alpert type ionization gauge. (d) The true density of the guided beam is
calculated using the results of (a)–(c) combined with calculated spectroscopic overlap factors, the RGA gas-correction
factor, and the Boltzmann distribution.

58
1.0

Ion Signal [arb. u.]

0.8

0.6

0.4

0.2

0.0
31480

31490

31500

31510

31520

31530

-1

Frequency [cmTrap Loss Cross Section [10-12 cm2]

Figure 4.14: The REMPI spectrum of ND3 at 295 K. Solid red line is a model fit from [79, 67]. Solid blue line
denotes the calculated contribution to the line strength from the |J, Ki = |1, 1i state used in the collision experiments.

1.0
0.8
0.6
0.4
0.2
0.0

|E| = 0

|E| > Epol

Figure 4.15: OH–ND3 trap loss cross-sections without and with a polarizing electric field. The combined averagecross-sections are stated in black (unpolarized) and red (polarized) text below their respective data points.

59
spectrum. Since the hexapole efficiently rejects strong-field-seeking molecules, any ionization through those line must
be due to an electric field breaking parity symmetry. With a correction due to the different Franck-Condon factors forthe different B̃, v0 levels, the relative strengths of the strong- and weak-field-seeking |J 0 , K 0 i ← |J 00 , K 00 i lines indicate
the fractional parity state mixing and therefore the actual electric field.)
The two different electrically polarized conditions yield cross-sections in excellent agreement with each other,while the unpolarized case is 2σ (stat) lower. The absolute magnitudes of the cross-sections are in reasonable agreement with theory[79], although the corrections required to determine the (measurable) trap-loss cross-section from thetrue differential elastic and inelastic cross-sections are large.

Chapter 5
Microwave manipulation of trapped OH

The success of the OH–ND3 collision experiment prompted us to turn our attention to taking control of theinternal state of the trapped OH molecules. In particular, a look at a pure Zeeman spectrum of OH suggests that justas the f ; 23 state is magnetically trappable, so too should be the e; 32 state. The collisions of section §4.2 were
performed with OH in f ; 23 and ND3 in the weak-field-seeking parity of its |J, Ki = |1, 1i inversion doublet. The
OH Λ-doublet is split by 1.67 GHz, while the ND3 inversion doublet is split by 1.5 GHz. This near-resonance suggeststhat the collision of OH and ND3 molecules in opposite parity states could have a substantial resonant enhancement,
due to the near-elasticity of the parity-swap (i. e. flip both parities) channel.
Since 1.67 GHz is a easy frequency to generate, the obvious way of controlling the OH state is to directly drive
the electric-dipole-allowed Λ-doublet transition. While some technical ingenuity was required (see section §5.3) tocouple the microwaves to the molecules, this can be made to work very well. Unfortunately, the trap dynamics ofe; 23 molecules are much less simple than suggested above when electric fields are added to the magnetic trapping
field — as one would wish to do to enable dipolar collisions, for instance.

5.1

Avoided crossings in mixed electric and magnetic fields
While the combination of electric and magnetic dipoles enables fascinating new phenomena, it also comes with

substantial complications. In particular, simple linear Zeeman spectra are converted to tangles of interwoven avoided
crossings by the application of transverse electric fields, even at strengths much smaller than the full polarizing field.
In this chapter, I report the experimental observation of these crossings in the polar radical OH and their dramatic
impact on dynamics in a magnetic trap. In its 2 Π3/2 ground electronic state, stationary OH has a magnetic dipole

61
moment of 2µB (where µB is the Bohr magneton) and an electric moment of 1.67 D (0.65 a. u.); rotational averaging(e f f )reduces each of these in the laboratory frame to µi

M Ω̄= µi J(J+1), where J is the total angular momentum, M is its

lab-fixed projection, and Ω̄ is the magnitude of J’s projection on the internuclear axis. Each rotational level of OH
contains two Zeeman manifolds of opposite parity, |e; Mi and | f ; Mi; with no applied fields, these states are split bya Λ-doublet coupling of 1.667 GHz. In an applied electric field, |ei and | f i are not strictly good quantum numbers.
However, they are still useful as labels of a state’s Stark character: |ei states are strong-electric-field-seeking while | f istates are weak-field-seeking.
A representative Hamiltonian for Λ-doublet molecules in mixed electric and magnetic fields has been derived
in [58] and was presented in section §2.3. While realistic systems must be solved numerically, the basic structure of the
combined Htotal = Hzero− f ield + HStark + HZeeman Hamiltonian is simple: the zero-field part generates diagonal matrix
elements in a suitable basis (e. g. the Hund’s case a fine-structure plus parity basis JM Ω̄ε for OH, where ε = e or f
is the J-relative parity label), as does the Zeeman part if one takes the space-fixed Z axis to lie along the magnetic field~B. The off-diagonal elements are then purely from the Stark portion. The component of the electric field ~E parallel to~B generates matrix elements which preserve M and change the parity ε, while the transverse components change M
by ±1 as well as ε. As can be seen in the Zeeman spectrum of OH (Fig. 5.1(a)), all states in the lower parity manifold
except for the absolute ground state have Zeeman-induced crossings with one or more of the upper parity states: inan E-field, these real crossings are coupled by the off-diagonal Stark elements and become avoided. Working in theperturbative limit, the crossings XM of Fig. 5.1 have widths δ+1/2 ∝ |E sin θEB |, δ−1/2 ∝ E 3 sin2 θEB cos θEB , andδ−3/2 ∝ E 3 sin3 θEB , where θEB is the angle between ~E and ~B. Since the off-diagonal elements only couple states of
opposite parity, the total perturbation theory order must be odd: crossings with total ∆M even such as X−1/2 require
one extra ∆M = 0 coupling order so that the total coupling changes ε as it must. M-changing couplings have a sin θEBdependence, while M-preserving couplings go as cos θEB , so this leads to a (sin θEB )∆M cos θEB angular dependence
in the coupling strengths for these even crossings, while the angular dependence for crossings with ∆M odd is simply(sin θEB )∆M . Perturbation theory begins to fail, however, at relatively small E-fields (much smaller than the polarizing
E-field at zero B-field) because of the strength of the electric dipole coupling and the fact that the Zeeman crossingsare gapless at zero E-field. Therefore, for all calculations described in this chapter we diagonalize the full 8 × 8 groundstate fine-structure Hamiltonian of OH described in section §2.3 and also in [67].

62

0.3

Energy (cm-1)

0.2

(a)

X+1/2

X-1/2

0.1
0
−0.1

X-3/2
−0.2
−0.3
0.3

Energy (cm-1)

0.2|f; +32>|e; +32>|f; +12>|e; +12>|f; -121>|e; -2>|f; -32>|e; -32>

(c)

(b)

(d)

0.1
0
−0.1
−0.2
−0.3
0

0.1

0.2

0.3

Magnetic field (T)

0.4 0

0.1

0.2

0.3

0.4

Magnetic field (T)Figure 5.1: The Zeeman structure of OH with bias electric fields. (a) 0 V/cm, (b) 500 V/cm at a 90◦ magnetic-toelectric field angle θEB , (c) 5000 V/cm at 65◦ , and (d) 5000 V/cm at 90◦ . The crossings XM of the e; + 32 state withthe three f ; M = {− 23 , − 21 , + 12 } states are labeled in (a).

63
We observe these avoided crossings by their dynamical effects on a sample of magnetically trapped OH
molecules. Our Stark deceleration and magnetic trapping system has been described elsewhere [61]. Briefly, OH
molecules are created in an electric discharge through water vapor in a supersonic expansion and then Stark decelerated and brought to rest in a permanent magnetic quadrupole trap (Fig. 5.2(a)) at a temperature of ∼ 70 mK, in theirf ; + 32 state. By applying chirped microwave fields to the trap magnet surfaces, we can then transfer the trappedmolecules to the e; + 23 state, wherein we can probe the three crossings labeled in Fig. 5.1(a); application of a DC
voltage to the magnets results in a bias E-field with the distribution shown in Fig. 5.2(b). The spatial loci of the threecrossings in the actual field gradient of our trap are shown in Fig. 5.2(c). The multiple-well structure is a result of ouruse of permanent magnets: the hole in the center of the magnets allows flux to exit both inside and outside the magnet
ring, yielding a toroidal field minimum between the faces of the magnet, as well as the central quadrupole trap. The
distribution of θEB angles generated by charging the magnets is plotted in Fig. 5.2(d).

5.2

Trap losses from Landau-Zener transitions
As our trapped molecules are moving at velocities . 10 m/s, only a very small energy gap (δ ∼ 10−5 cm−1 )is needed for each molecule to have some chance of following the adiabats rather than the diabats of the potential
surfaces. As can be seen in Fig. 5.1, the adiabats of all the levels below the “doubly-stretched” f ; + 32 state aresubstantially less trapped than the e; + 32 diabat – or in some cases, are outright antitrapped. Thus, if molecules aretransferred to the e; + 23 diabat in zero field and then a bias field is applied, some fraction of the molecules willleak out of the trap as they follow the adiabats rather than Landau-Zener hopping [80] to follow the trap diabat. At
low fields, the Landau-Zener hopping probability is still substantial, and so it takes multiple trap oscillations for amolecule to leak out. At higher fields, however, effectively all molecules follow the adiabats and so the loss rate is−1
simply limited by the trap dynamical time τdyn = 14 νtrap
where νtrap is the trap oscillation frequency.

In order to transfer population from f ; + 32 to e; + 32 , we use an Adiabatic Rapid Passage (ARP) technique[81] via microwave fields applied directly to the trap magnets. As the magnets also experience ±14 kV potentials as
part of the trap loading sequence, the microwave system is AC-coupled to the trap by a pair of in-vacuum high voltage
capacitors as described in section §5.3. This capacitive coupling stage isolates the microwave system by roughly 40
dBV from the ±14 kV magnet bias; two 1.2 GHz high-pass filters further block both the residual (few-volt) near-DC

64

(a)

(b)

(c)

(d)

Χ-3/2
Χ-1/2

Χ+1/2

Figure 5.2: Magnetic trap geometry relevant to microwave control of OH. To scale. (a) The magnetic field distribution in the permanent magnet quadrupole trap. (b). The electric field distribution created by applying a potential
difference to the trap magnets. (c) The spatial location of the avoided crossings XM . (d) The relative angle θEB between electric and magnetic fields in the trap. White denotes 0◦ while black denotes 180◦ , implying that the avoided
crossings are widest in the gray regions.

65
bias and the switching transients from the microwave. By chirping our microwave frequency through the entire rangeof differential Zeeman shifts [82] in our trap over a period τARP  τdyn , we are able to achieve transfer of ∼ 75 %of the observable trap population. (Since the transfer is a π transition, only the fraction of ~ERF k ~B is useful. Thus,
molecules in the gray regions of Fig. 5.2(d) see much smaller effective intensities than molecules in the white or black
regions.)
In addition to causing dynamical trap loss, the avoided crossings XM also reduce the efficiency of our ARP
pulses: in the vicinity of the avoided crossings, not only are the states shifted away from the nominal 1.667 GHztransition frequency, but also the two parity states are mixed by the electric field. The net result of this mixing is thatmolecules transferred near said crossings may not in fact be transferred to the e; + 32 diabat but rather may end up on
the | f ; Mi diabat and thus be invisible to our state-selective detection. Figure 5.3 displays the e; + 32 population as
a function of the DC bias field applied during the ARP. The bias field leads to a rapid drop in transferred population
as the f ; + 32 → e; + 23 transition is Stark-shifted out of resonance; the bias field required for this shift increases
with the frequency width of the ARP pulse. We have performed simulations using a numerical solution of the optical
Bloch equations to calculate a position dependent ARP transfer probability with zero electric field. We then calculate
the position-dependent Stark-Zeeman shift by diagonalizing the OH Hamiltonian and convolve the resulting frequency
distribution with the ARP bandwidth to determine the total fraction transferred. The simulation results shown in Fig.
5.3 give a good match to the population measurements, showing that the ARP process is well-understood; a moredetailed description of the simulation code is given in section §5.4. The simulations suggest the presence of a stray
E-field in the trap region on the order of 100–150 V/cm.
Applied E-fields also have a profound effect on trap dynamics. A loss of ∼ 40% of the e-state population over
τdyn is observed immediately after the ARP pulse (Fig. 5.3 inset), almost certainly due to prompt loss of molecules
following an adiabat through the X+1/2 crossing (which is fully avoided in just the observed stray E-field) and exiting−1
the trap. At larger bias fields, loss rates up to τdyn
are observed with critical fields again on the order of 200-300 V/cm,

corresponding to a unit probability of loss through either X−1/2 or X−3/2 . The observed loss rates (Fig. 5.4(a)) are
well-fit by a probabilistic model

66

0.75

0.5

0.25

OH signal (arb. units)e-state OH signal (arb. units)

1

0.2
0.15
0.1
0.05
0
0.1

1

10

Time after ARP (τdyn)
0
1

10

100

1000

Applied bias field (V/cm)
Figure 5.3: Post-ARP e-state population versus applied bias field. ARP chirp widths of 10 (green triangles),20 (blue squares), and 40 MHz (red diamond). Solid lines are Monte-Carlo simulations of the population transfer
using realistic bias and RF field distributions from finite-element calculations, a semiclassical treatment of the Stark
detunings, and a homogeneous stray field as a free parameter. (inset) Time-of-flight of e-state population immediately
after ARP, with no applied bias field. The rapid loss through the X+1/2 crossing is clearly visible: the solid line is a fit−1
to a single-exponential loss at a rate of τdyn
.

67

4

τ-1dynGap width (cm-1)

total loss rate Γ (s-1)

(a)10

103
102

10−3
10−6

δ+1/2 (90๐)δ-1/2 (54๐)δ-3/2 (90๐)

10−9
1

10

100

1000

Electric field (V/cm)

101

Γvacuum
100
1

10

OH signal(arb. units)

(b)

100

1000

Applied bias field (V/cm)
1

0

0

2

4

6

8

10

Time after trap loading (ms)Figure 5.4: Landau-Zener loss dynamics of magnetically trapped e; M = + 23 OH molecules. (a) Loss rate ofe; M = + 23 molecules versus applied electric field. The solid red line is a fit to a simple Landau-Zener model; thegreen shaded stripe denotes the range of observed loss rates for f ; + 32 molecules. The bias field is turned on 5 ms
after the ARP pulse is completed, and thus well after the prompt loss seen in the inset of Fig. 5.3 is completed. (inset)
Widths δM of the avoided crossings between e; + 32 and | f ; Mi versus applied electric field, at their respectively
maximizing values of θEB . δ+1/2 is linear with field, while δ−1/2 and δ−3/2 are cubic. (b) Time-of-flight trace oftrap oscillations, observed by loading the trap with a non-zero center-of-mass velocity. The solid red fit yields a trap
frequency νtrap = 1350 Hz.

68




−1
× 1 −
Γ = Γvacuum + τdyn


∏3 P
LZ
1
(M)

M∈{− 2 ,− 2 }where Γvacuum is the background vacuum loss rate and
PLZ

(M)

 !
2πξ δM2 Ee f f
= exp −h̄ ∆µe f f (M) dB
dt

is the Landau-Zener probability that the molecule stays on the e; + 23 diabat through the crossing XM . Angle brackDE
~∇B ·~v is the rate of change of the magnetic field and δM Ee f f =ets denote population averages: dB
=
dt
E
D q
2
is the width of the crossing XM as a function of applied electric field E and stray field Estray ,δM
E 2 + Estray


3 3
3
3
while ∆µe f f (M) = 3//22··5//22 − 3//22··M
5/2 · 2µB is the difference between the effective magnetic moments of the f ; + 2
and |e; Mi states. The model contains only 3 free parameters: Γvacuum , Estray , and ξ ≈ 0.17, which incorporates theerrors in our estimates of the population-averaged parameters. (While we can estimate
Monte-Carlo trap dynamics simulations, δM Ee f f



dB
dt

reasonably well fromhas a very fast variation with hθEB i and so is hard to accurately

calculate or population-average. It is therefore responsible for most of ξ ’s deviation from unity.) Although we hap−1
pened to observe experimental agreement between the visible trap oscillation frequency νtrap and 4τdyn(Fig. 5.4(b)),

our magnetic quadrupole trap’s coupling of oscillation magnitude and frequency means that these two quantities would
not necessarily have to be equal in every experimental realization. Γvacuum can be independently observed from thef -state lifetime, while Estray can be derived from Fig. 5.3; both of these independent measurements are consistent withthe fit. As both δ−1/2 and δ−3/2 are third-order in E, the most striking feature of the Landau-Zener model is that itexplains the extremely steep rise in Γ: Γ ∼ 1 − exp −αE 6 where α represents the Landau-Zener parameters.
The success of this model prompts two conclusions. The first is that magnetic trapping of the lower parity
doublet in any molecule is fraught with difficulties: in OH, even just the fields from stray charge accumulating on
the trap mounts is enough to cause unity loss through the X+1/2 channel. (We are currently implementing a new trap
mount design to minimize the patch charges in our future work.) In general, if one wishes to explore polar radicals
in mixed fields in any states except the doubly-stretched ones, either optical trapping or an Ioffe-Pritchard trap witha sufficiently large offset B-field as to be above all the crossings is necessary; this implies that polar radicals in theirlower parity doublet are extremely challenging candidates for slowing with a magnetic coilgun [83, 84]. More broadly,in precision measurement applications it is imperative to take a close look [85] at the exact Stark-Zeeman spectrum

69
+HV

Vacuum region-HV
SMA
feedthru

Magnets
HV isolating
capacitors

VHF-1200+
high-pass filter

VHF-1200+
high-pass filter

8644A Synthesizer

ZFDC-10-5-S+
coupler

GRF5016A
amplifierRF monitor
FM

PM

Figure 5.5: A schematic of the microwave system use to transfer OH between Λ-doublet levels.

of the system of interest, as the spectral distortions from the avoided crossings can persist over very large ranges inmagnetic field.

5.3

Microwave coupling circuit design
In order to couple microwave power on to our trap magnets while protecting the microwave system from the±14 kV applied to the magnets during the trap loading sequence, we use a pair of custom in-vacuum capacitors toblock the DC high voltage plus two more high-pass filters (Mini-Circuits VHF-1200+1 ) to block the AC transients
generated by the magnet switching.
The microwave system layout is depicted in figure 5.5. Tunable microwave power is generated by an HP 8644A
synthesizer, which is programmed to form the ARP pulse by voltages supplied to its frequency and pulse modulation
inputs. A Mini-Circuits ZFDC-10-5-S+ directional coupler allows monitoring of the microwave signal before it isamplified by a GTC GRF5016A power amplifier. The microwaves are then coupled through the protection filters andinto the vacuum system via an SMA feedthrough. Typical power levels are 30.4–35.4 dBm (1–3 W) at the feedthrough.
As the ARP chirp time τARP must be much shorter than τdyn , τARP is generally between 20 and 80 µs.
1 Manufacturer and part names are listed for completeness only and do not constitute an endorsement by NIST.

OH signal (arb. units)

70

0 V/cm157.3 V/cm
301 V/cm
405 V/cm
536.7 V/cm

1
0.8
0.6
0.4
0.2

0
1650 1660 1670 1680 1690 1700 17101720

νcenter (MHz)Figure 5.6: ARP spectra at several different bias E-fields. Points are data while solid lines are simulation results.
The Stark-Zeeman broadening due to the avoided crossings is clearly visible.

5.4

Numerical simulations of ARP
As mentioned in section §5.2, we have performed numerical simulations of the ARP population dynamics. The

simulation code works as follows:(1) For each molecule in a Monte Carlo distribution, the optical Bloch equations are solved using the local magnitude of the microwave field (the microwave field is assumed to follow the DC field distribution generated
by charging the magnet surfaces) and its angle against the local magnetic field. Both field distributions were
calculated using a finite-element package. The final probability of the molecule transferring to the lower
Λ-doublet state is then recorded.
(2) For each molecule, the local static E-field is calculated and then combined with the local B-field to find thetotal Stark-Zeeman detuning. This yields a transfer probability density as a function of microwave frequency.
(3) The probability density function is then integrated over the range of frequencies the ARP chirp sweeps throughto get a final transfer probability.
Figure 5.6 shows several ARP spectra at different bias E-fields. The agreement between data (points) and simulation(solid lines) is good; in particular, the simulations capture the Stark-Zeeman broadening visible at larger fields. In

71
a homogeneous field and the absence of avoided crossings, the application of an E-field would merely push the two
Λ-doublet states apart without broadening the transition. The field in the trap region is not fully homogeneous, which
contributes some broadening, while the curvature of the e; M = + 23 state near its avoided crossings further increases
the Stark-Zeeman broadening. In order to fit the simulations to the data, a background stray E-field on the order of
120 V/cm was required, consistent with the field required for the Landau-Zener model.

Chapter 6
OH–OH inelastic collisions

6.1

Two-body loss of | f i-state molecules
With the recognition that e; 23 molecules cannot be magnetically trapped in the presence of an electric field,

our goal of observing mixed-parity OH–ND3 collisions turned towards using f ; 32 molecules and somehow parityflipping the ND3 molecules as they fly into the magnetic trap. What we observed, however, was that in the presenceof a DC electric field, even f ; 32 molecules experienced loss from the trap. We had seen this previously, while doing
the original OH–ND3 experiment, but as the loss terminated after some tens of milliseconds, we simply conducted thecollision experiment after the trap population had time to resettle after the application of the polarizing field.
Now, however, we undertook to more closely examine the phenomenon and realized that it appeared to have a
two-body loss profile. In the presence of a pure two-body loss rate, the total number N obeys a differential equationdN= −β N 2
dtwhich has the solution
N (t) =

N0
1 + β N0twhere N0 = N (0) is the initial number. For two-body losses which are fast compared to any other loss rates, thisform is sufficient. When, however, the timescale β N0 is comparable to the one-body loss rate Γ it becomes necessary
to use a combined one- and two-body loss modeldN= −β N 2 − ΓNdt

73Figure 6.1: Two-body loss of magnetically trapped OH molecules. A comparison of pure one-body loss with zero
electric field (blue circles), and two different two-body rates with 1500 V/cm (red diamonds) and 3000 V/cm (green
squares) applied fields. The solid lines are fits to the data: pure one-body loss for the zero-field curve, and combined
one- and two-body loss (using the zero-field one-body rate) for the two non-zero-field curves. Error bars are one
statistical standard error.

orN (t) =

ΓN0
(Γ + β N0 ) eΓt − β N0.
Of course, the above only holds true in the case of a flat density distribution: if the density is spatially-varying,
the local density n(~x,t) must be substituted for N (t). At the simplest level, this implies that two-body loss will
selectively remove particles from the region of maximum density. However, if the sample is in thermal equilibriumand β is independent of collision-energy, the density distribution will remain self-similar. In this case, the use of
N to describe the entire distribution will remain approximately correct, although the resulting β will be an effective
averaged value rather than the true microscopic loss rate constant.
Figure 6.1 shows some representative results of our two-body loss experiments. OH molecules are magneticallytrapped, allowed to settle for a few milliseconds, and then a DC bias electric field is applied between the trap magnets.
The bias field of strengths of . 10 kV/cm are not enough to substantially distort the effective trap potential: loading
the magnetic trap involves fields of over 60 kV/cm. It is clear from the figure, however, that applying an electric field
has a profound effect upon the evolution of the trapped molecules. Moreover, the curvature of the number-vs.-time
curve is inconsistent with one-body loss.

74

-a[
1

|f>-state signal [photons/shot]

β= 41.2± 4.0

β= 48.4± 8.1

β= 45.4± 5.5

β= 40.4
± 3.0

0.1

0

25

50

75

100

125

150

175

Time [ms]-b[valve fire
discharge
deceleration
trap loading
settling timecollisional evolution
LIF
magnets charged to ±12 kV
magnets charged to Vcollide
0

10

20
Time -ms[tLIF

Figure 6.2: Testing two-body loss by varying the starting density. (a) Four sets of number-versus-time measurements, with different initial densities due to one-body losses before beginning the measurement. A 3000 V/cm DC
electric field was turned on at t = t0i , where t0i is the time of the first point for each series. All four measurements yieldstatistically consistent two-body rate constants. Error bars are one statistical standard error. (b) A schematic timelineof the experiments.

75
A second, more formal test of the two-body nature of the losses is shown in figure 6.2. By delaying the turn-ontime of the DC bias field, the initial density at the onset of two-body losses is varied. For four different densities overa factor of two in total number we observe statistically consistent values of β while the slopes on a semi-logarithmicplot are clearly different.
By independently measuring the zero-electric-field one-body loss rate (due to collisions with background gas),
it is possible in principle to use the combined one- and two-body loss equation to measure two-body rates almost as
low as the one-body loss rate. However, as will be described in the following section, vacuum loss is not the only
one-body loss rate in the problem.

6.2

Electric-field dependence of the loss rate
In order to attain an understanding of the inelastic collision phenomena, the most obvious parameter to varyis the electric field strength. A large set of measurements were taken over three distinct epochs of data, separated by
vacuum breaks and realignments of the LIF optics. The changed optics alignments cause the starting signal level tobe different between the three eras, and therefore the β value; to combine the three data sets, the 3040 V/cm points
were taken as a fixed reference, and two of the three data sets were linearly scaled so that their 3040 V/cm points
matched the third data set. 3040 V/cm is an ideal matching point, because β is large enough to have good statisticaluncertainty but the field is small enough that E-field-induced one-body losses are still (hopefully) fractionally small.
The combined data are plotted in figure 6.3
It is also possible to plot β as a function of the induced dipole moment,d2εµ=q(dε)2 + ( ∆2 )2where d = 1.67 D is the total dipole moment of OH, ε is the electric field strength, and ∆ is the Λ-doublet splitting.
The result of this scaling is shown in figure 6.4. With this scaling, there is a clear change in behavior as the dipole
moment saturates: at µ . 0.9 D, β scales roughly as µ 2 , while at higher µ the slope rapidly steepens.
This abrupt change in character around µ ≈ 0.9 D (E ≈ 3500 V/cm) suggests that two-body loss may not be thewhole story: the loss rate continues to increase with electric field, even though the dipole moment is saturating. Theory
suggests that the two-body rate should be a pure (although potentially complex) function of the dipole moment—

76

Figure 6.3: Variation of the fitted two-body rate constant β with applied electric field. Arrows indicate that theerror bars are consistent with a zero rate constant; error bars without points indicate upper bounds. Error bars are one
standard error.

Figure 6.4: Variation of the fitted two-body loss rate constant β with induced dipole moment. This is the same data
as figure 6.3, but with the electric field parametrized as the effective dipole moment µ given the maximum available
dipole moment d, the electric field E, and the Λ-doublet splitting ∆. Arrows indicate that the error bars are consistent
with a zero rate constant; the five points of lowest µ are presented as upper bounds rather than specific points. The
point of highest µ is at a field of 9200 V/cm. Error bars are one standard error.

77
perhaps there is another one-body loss rate contaminating the fits?
An examination of the two highest potential surfaces of OH molecules in a uniform electric field superimposedover a quadrupole magnetic field (figure 6.5) reveals a fascinating structure. At large values of E, there is a very
narrow avoided crossing between the surfaces at ~B ⊥ ~E and small B. Just as changes in the direction of ~B can cause
Majorana loss in very small magnetic fields, here small changes in θEB can allow molecules to Landau-Zener hop fromthe upper-most, trapped state to the lower, untrapped state. We can apply the Landau-Zener formula PLZ (r) = e−2πΓ(r)with
Γ(r) =

δ (E, B(r), θEB = π2 )2h̄( 1r vϑ )(dδ /dθEB |r, π )2where δ (E, B(r), θEB ) is the splitting between the two surfaces at the radial coordinate r and vϑ is the azimuthal
velocity of the molecule. To determine an absolute loss rate γ, we can then integrate the probability function PLZ overthe molecular Boltzmann distribution:
ˆ ∞
γLZ =

4πr2 nr (r) dr

ˆ ∞

0

0

2

1 vϑ −2πΓ(r) 1 − 2kmvϑT·e· e B dvϑπ rZ

´ ∞ − mvϑwhere nr (r) is the 3-dimensional relative density, Z = 0 e 2kB T dvϑ is the partition function, m is the molecular2

mass, kB is Boltzmann’s constant, and T is the temperature. The factor π1 vrϑ occurs because each molecule encountersθEB = π2 + nπ twice per orbit. For a quadrupole magnetic trap of gradient B0 in the r̂ direction,nr (r) =
with

ˆ ∞
Zn =

1 − µkm BT0 r
e B
Zn

0

− µkm BT r

4πr2 eB

dr0.
Given a calculable Landau-Zener loss rate, the collision data can then be refitted to deconvolve the added onebody loss rate. By numerically integrating over a 45 mK thermal distribution and assuming a 1 T/cm gradient in the
direction parallel to the plane θEB = 90◦ , I calculate the Landau-Zener rates shown in figure 6.6.
Unfortunately, while the integral is not difficult, it is difficult to assign a systematic confidence interval to thissort of simplistic theory. For this work, I take an uncertainty factor rather than an absolute range: I assume that thecalculation is correct to within a factor of two in either direction. Deconvolution of the calculated Landau-Zener loss

78

(a)

(b)

(c)

Figure 6.5: The two uppermost potential surfaces of OH in mixed E and B fields. The radial coordinate is themagnetic field strength in Gauss, and the angular coordinate is θEB , the angle between ~E and ~B. Electric field strengths
are (a) 500 V/cm, (b) 1000 V/cm, and (c) 3000 V/cm. Note the classical Landau-Zener shape near 90◦ at the larger
electric field strengths.

79

Figure 6.6: | f i-state Landau-Zener loss rates versus electric field.

80
rate — that is, using the combination of one-body loss rates
Γ = γLZ + γvacuum
yields the two-body rates shown in figure 6.7. The upturn at high induced dipole moment µ is much less pronounced
when the Landau-Zener losses are removed. In fact, if one supposes that the calculation underestimates the LandauZener rate by a factor of two (the lower limit of the shaded blue regions in figure 6.7), the fitted β actually levels off athigh electric fields and fits a single, rather than broken, power law.

81

(a)

(b)

Figure 6.7: Variation of the two-body loss constant β after deconvolution of the Landau-Zener losses. (a) Variation with electric field. (b) Variation with induced dipole moment. Blue shaded regions indicate the values of β foundby varying the Landau-Zener loss rate by a factor of 2 up or down. Smaller values of the Landau-Zener rate produce
larger values of β .

Chapter 7
Evaporative cooling of OH

Atomic physics was revolutionized by the development of forced evaporative cooling: it led directly to theobservation of Bose-Einstein condensation[4, 5], quantum-degenerate Fermi gases[86], and ultracold optical latticesimulations of condensed matter phenomena[87]. However, until the work reported in this chapter, evaporative coolingof molecules had not yet been achieved, due to unfavorable elastic-to-inelastic ratios[88] and the impractically slow
thermalization rates in the available trapped species. In this chapter I report the observation of microwave-forced
evaporative cooling of hydroxyl (OH) molecules loaded from a Stark-decelerated beam into an extremely high-gradientmagnetic quadrupole trap. We demonstrated cooling by at least an order of magnitude in temperature and three ordersin phase-space density, limited only by the low-temperature sensitivity of our spectroscopic thermometry technique.
With evaporative cooling and sufficiently large initial populations, much colder temperatures are possible, and even aquantum-degenerate gas of this dipolar radical – or anything else it can sympathetically cool – may now be in reach.

7.1

Forced evaporation
Evaporative cooling of a thermal distribution[89] is, in principle, very simple: by selectively removing particles

with much greater than the average total energy per particle, the temperature decreases. In the presence of elastic
collisions, the high-energy tail is repopulated and so may repeatedly be selectively trimmed, allowing the removalof a great deal of energy at low cost in particle number. This process may be started as soon as the thermalization
rate is fast enough to be practical and continued until its cooling power is balanced by the heating rate from inelastic
collisions. It generally yields temperatures deep into the quantum-degenerate regime (far below the recoil limit of
optical cooling), and in a few cases has reached quantum degeneracy without requiring optical pre-cooling[90, 91].

83
The key metric for evaporation is therefore the ratio of two timescales. The first is the rate of elastic collisions,which rethermalize the distribution, while the second is the rate at which particles are lost from the trap for reasons
other than their being deliberately removed, e. g. the rates of inelastic scattering and background gas collisions. For along time, both theoretical[92, 93, 94] and experimental work[95, 88, 96] seemed to show a generically poor value of
this ratio across multiple molecular systems; this has led to a general belief that evaporative cooling is unfavorable inmolecules[97]. As no trapped molecular system has achieved sufficiently rapid thermalization, there has been a lackof experiments to test this expectation—although some recent theoretical results have challenged it[98, 99].
Hydroxyl would not, at first glance, seem to be a promising candidate for evaporative cooling. Its open-shell2Π

3/2 ground state and its propensity towards hydrogen bonding create a large anisotropy in the OH–OH interaction

potential, which would intuitively motivate a large inelastic scattering rate. Chemical reactions are also possible, via
the OH+OH→H2 O+O pathway; it is unclear whether this reaction has an activation energy barrier[100, 101]. It isthus perhaps surprising that the elastic collision rate actually exceeds the inelastic rate, allowing evaporative cooling.
However, our experimental observation is unambiguous, and is further supported by quantum scattering calculations
based on the long range dipole-dipole interaction between the molecules[69, 34] considering all of the fine-structurestates of the rotational ground state. This analysis considers only elastic collision or inelastic relaxation to lower energy
states, as the long-range interactions appear to fully dominate over short-range effects such as chemical reactions.
In its ground state, OH has a 2 Π3/2 electronic character, with the lowest rotational level having total non-nuclear
angular momentum J = 23 . The electronic orbital angular momentum couples to the rotational angular momentum tosplit the two opposite-parity states within J = 32 by a Λ-doubling of ∼ 1.667 GHz; the upper parity state is labeled | f i
and the lower |ei. Hydroxyl is both paramagnetic, with a molecule-fixed moment of 2µB (µB is the Bohr magneton),and electrically polar with a dipole moment of 1.67 Debye (5.57 × 10−30 C·m). The Zeeman spectrum of OH is shownin Fig. 7.1a. Our magnetic trap[61] is loaded with molecules in the uppermost | f ; MJ = + 23 i state, where MJ is thelaboratory projection of J.
The results of our scattering calculations for f ; 32 molecules are shown in Fig. 7.1b. The elastic cross section
dominates the inelastic one for low energies: at a collision energy of Ec = 50 mK, the ratio of elastic over inelastic is
R = 5 in a 50 mT magnetic field (B) and R = 18 in 150 mT, while at lower energy Ec = 5 mK the ratio increases to
R = 23 and 137 respectively. Because the collisions occur in a quadrupole magnetic trap where B is inhomogeneous,

84

Energy (cm-1)

0.2

a|f; +32>|e; +32>

X+1/2

X-1/2

|f; +12>0.1|e; +12>0|f; -12>|e; -12>−0.1|f; -32>
−0.2

X-3/2

|e; -32>

0

0.1

0.2

0.3

0.4

Magnetic field (T)Cross-section (cm2)

10−11

b
10−12

10−13

10−12

0

0.15

0.3

0.45

B (T)
10−13

10−14 −6
10

10−5

10−4

10−3

0.01

0.1

Collision energy / kB (K)
1000
100

Adiabatic energy (mK)

c

80
60
40

500

20
0
100

1000

0

−500

100

1000

Intermolecular distance (Bohr radii)

Figure 7.1: Ground-state structure and scattering theory of OH. (a) The ground-state Λ-doublet and Zeeman
structure of OH. Dashed black lines are energy levels in the absence of any electric field. In the presence of an electricfield, the level crossings become avoided (solid orange lines demonstrate a 500 V/cm field orthogonal to the magnetic
field). Xi label the crossings of the e; 32 state with the | f ; MJ = ii states; these crossings allow e; 32 molecules toescape the trap in the presence of an electric field. Molecules are loaded into the magnetic trap in the bolded f ; 23
state. (b), Zero electric field cross sections as a function of collision energy: elastic cross section in a 50 mT magnetic
field (solid), inelastic in 50 mT (dotted), and inelastic in 150 mT (dashed). The red dash-dotted line is a semi-classicalexpression given by Eq. (7.2). Inset: elastic (solid) and inelastic (dotted) cross sections as a function of magnetic
field at Ec = 50 mK. (c), Adiabatic energies as a function of the inter-molecular distance at B = 50 mT for the M` = 0
components of the ` = 2 · {0 to 6} partial waves for every combination of initial states. The inset zooms in on therepulsive van der Waals interaction for the experimentally relevant case of two colliding f ; 32 molecules (red lines).

85
the inset of Fig. 7.1b shows the cross sections as a function of B at Ec = 50 mK. This demonstrates that inelastic
processes are even further suppressed at B > 50 mT, in agreement with previous analysis [34].
These scattering results can be interpreted by the emergence of an effective repulsive van der Waals interaction
between the two molecules. In zero electric field, the effective interaction in a scattering channel m can be evaluatedin second-order perturbation theory by|hm|Vdd (r)|ni|2 C6∼ 6,
Em − Enrn

VvdW (r) = ∑(7.1)

where Vdd is the electric dipole-dipole interaction between different scattering channels, and is non-vanishing only
between molecular states of distinct parity; and Em and En are the asymptotic energies of the relevant scatteringchannels. For the initial molecule-molecule channel of interest f ; 32

f ; 32 , the other fine-structure channels arelower in energy (En < Em ) and repel this energy upward. Thus, the C6 coefficient is positive, as illustrated in Fig. 7.1cfor B = 50 mT, for the highest energy channel. The contribution from the next rotational state J = 25 of the 2 Π3/2
manifold is too high in energy (Em − EJ=5/2 ≈ −100 K) to give an appreciable attractive contribution at long-range.
Hence, for low collision energies, the scattering of OH molecules is dominated by the long-range interaction rather
than the short-range structure of the potential surface.
Given an effective, repulsive C6 coefficient in the incident channel, the elastic cross section can be approximatedsemi-classically at energies above the threshold regime[102], as
σ

el

=

π 11/5 (Γ(5/2)/Γ(3))2/5
sin (π/5) Γ(2/5)



C̄6
h̄

2/5 

2Ec
mred

−1/5
(7.2)

where mred = 15497 atomic units (a. u.) is the reduced mass of two OH molecules and C̄6 ≈ 9 × 104 a. u. is the calculated isotropic C6 . (We contrast this C̄6 with values of 2–8×103 a. u. for the alkali metal atoms[103].) Equation (7.2),
plotted as a red dash-dotted line on Fig. 7.1b, only slightly overestimates the numerical results for the elastic cross−1/5

section but gives a proper trend in Ec.

We observed inelastic scattering in the presence of a large electric field (Fig. 7.2b), demonstrating the presenceof two-body collisions in our trapped sample. Motivated by the prediction of a favorable R, we then undertook an
experiment to implement microwave-forced evaporative cooling. The f ; 23 → e; 32 microwave transition has asmall but nonzero differential Zeeman shift[82], red-shifting by 26.6 kHz/mT. In the presence of a small electric field,e; 32 molecules can escape the trap through the avoided crossings labeled Xi in Fig. 7.1a[104], while inelastic losses

86

Relative number

a

1.25
1
0.75
0.5
0.25
0
0

b

25

50

75

100

125

Time (ms)Figure 7.2: Trap system and inelastic collisions. (a) Schematic of the Stark decelerator[47] (left) and magnetic
trap (center rings) system. The DC-blocking capacitor (rear) decouples the high voltage used for trap loading from themicrowave system used for spectroscopy and evaporation, while the laser and lens provide fluorescence detection of thetrapped molecules. Inset: an artist’s impression of evaporative cooling. As the trap depth is successively lowered by the
RF knife, the hottest molecules escape and the remainder rethermalize to a colder temperature. (b) Relative molecule
number versus trap hold time, demonstrating electric field-induced collisions at 45 mK. Black squares, consistent withpure one-body loss, are with no applied electric field; red circles are well-fit by pure two-body inelastic loss induced
by a large applied electric field of 3040 V/cm. (The field used in the RF knife is only 240 V/cm, for comparison.) Error
bars are one standard error.

87of | f i molecules remain unmeasurably slow. A brief microwave pulse to selectively transfer f ; 32 -state molecules to
e; 32 , followed by a longer period with an electric field present to eject e; 32 molecules from the trap, is therefore
a field- (position-) selective method to remove molecules from the magnetic trap (see Methods). This yields what iscommonly called an RF knife (Fig. 7.2a). A Zeeman depletion spectrum can be acquired by using the knife to remove
molecules at a set of fixed frequencies and measuring the fractional depletion ζ ; this yields a histogram of relative
molecule number versus B. We fit this spectrum with a modified thermal distributionµBζ (B)dB = ζ0 B2 dB × exp[−
]×
kB T




1,
if B > 49.6 mT×exp{− µ[(49.6 mT)−B] }, otherwise
kB T

(7.3)

where ζ0 is a fitting coefficient, B2 dB ∝ r2 dr is the volume element for a 3-D quadrupole trap, µ = 1.2 µB is themagnetic moment of the f ; 32 state, B is the magnetic field strength implied by the microwave frequency, kB is
Boltzmann’s constant, and T is the fitted temperature. The first term in the product is the simple Boltzmann expression
for the molecule number as a function of B, while the second is a correction for the fact that e; 32 molecules only
disappear when they go through one of the Xi crossings. Specifically, while molecules transferred at fields above
49.6 mT (the known location of X−3/2 ) are always energetically able to reach one of the Xi crossings and thus disappear,of the molecules transferred at lower fields only those with enough kinetic energy to fly up the trap potential toX−3/2 can escape the trap. This implies an additional Boltzmann factor exp[−µ∆B/kB T ] in the probability of thosemolecules’ disappearance.
With a B-selective technique for removing molecules, we easily implement forced evaporation by moving theedge of the RF knife from some large initial value of B towards zero, at a rate slow enough that the distribution remains
in thermal equilibrium. Six different Zeeman spectra are shown in Fig. 7.3, demonstrating both forced evaporationand forced anti-evaporation, where for the latter the knife is held fixed at some low Bknife (µBknife < kB T ). The
trap is initially loaded with a temperature of 51 mK; left unperturbed, it free-evaporates down to 45 mK. We haveforced evaporative cooling by an order of magnitude down to 5.1 mK, while forced anti-evaporation can overcome
the free evaporation and increase the temperature to 54 mK. The limit of 5.1 mK is attained approximately when the
RF knife edge reaches X−3/2 : while further reductions in temperature are possible, the exponential suppression of the88

Relative number

0.75

a

0.5

b

0.25
0
−0.25
−0.5

0

0.2

0.4

0

0.2

0.4

Relative number

0.75
0.5

c

d

e

f

0.25
0
−0.25

Relative number

−0.5
0.75
0.5
0.25
0
−0.25
−0.5

0

0.1

0.2

0

0.1

0.2

Implied magnetic field (Tesla)
Figure 7.3: Microwave spectra illustrating six different final temperatures. Black squares are data (error bars are
1 standard error), and solid lines are fits to the sensitivity-corrected Boltzmann form of Eq. (7.3): (a) the unperturbed
45 mK distribution, (b) anti-evaporation to 54 mK, (c)-(f) forward evaporation to (c) 18 mK, (d) 12 mK, (e) 9.8 mK,
and (f) 5.1 mK. Dotted lines reproduce the fit from (a), shown for comparison. Note that the x-axis scale differs
between (a)-(b) and (c)-(f); all curves have been shifted vertically to have a zero baseline.

105

10

5

0
0

25

50

75

100

Evaporation endpoint(millikelvin)

125

b

104

10
100
10

1
0.05

1
0.1

0.2

Relative number

0.5

Relative number

Temperature (millikelvin)

a
15

89

0.5

Calculated relative PSD

Temperature (millikelvin)

20

c

0.4
0.3
0.2
0.1
0
0

0.05 0.1 0.15 0.2 0.25

Magnetic field (Tesla)

Figure 7.4: Scaling relations observed in the evaporative cooling of OH. (a) Final observed temperature versus
effective trap depth at evaporation ramp end. (b) Final temperature (solid black, left axis) and calculated relative phasespace density (dashed red, right axis) versus remaining relative molecule number. Error bars are 1 standard error; phasespace density is calculated assuming a 3-D quadrupole trap geometry. (c) Molecular Boltzmann distributions impliedby the fit curves of Fig. 7.3(a) and (c)-(f) (right to left). Distributions are normalized so that the area under each curve
is proportional to the total | f i-state fluorescence observed at that temperature.

spectroscopic signal below X−3/2 renders our current system unable to measure temperatures lower than this. The
plots of Fig. 7.3 are all normalized so that the area under the spectroscopic curve is proportional to the total | f istate fluorescence signal. Thus, the increase in signal height at low B in Fig. 7.3c-f is direct evidence of increasingphase-space density.
The apparent negative signal in Fig. 7.3e-f can be fully fitted by assuming the presence of accumulated, trapped|ei-state molecules in thermal equilibrium with the visible | f i-state ones. Since e; 32 molecules are totally trapped ifthey do not have enough kinetic energy to reach X−3/2 , they will appear, rather than disappear, during the microwave
spectroscopy and contribute to the total depletion signal with a negative sign. Fitting the curves in this fashion gives
even colder temperatures of 6.8 and 3.5 mK for Figs. 7.3e and f, respectively. As the appearance of low-energy,
trapped |ei-state molecules would also constitute direct evidence of evaporation, we undertook a direct search andindeed detected them in laser-induced fluorescence.
We make several comments on the observed evaporation. The first is on the evaporation timescale: it is fast.
In comparison to typical cooling rates of d(log T )/dt ∼ 0.5 s−1 , we cool by an order of magnitude in only 70 ms, for arate of 33 s−1 . This implies elastic collision rates on the order of 100–1000 s−1 , comparable to our trap frequency[104]of ∼ 1400 s−1 . As we are able to set a bound on the inelastic loss rate of < 2 s−1 (as shown in Fig. 7.2b), this implies
a distribution-averaged R & 50-500, consistent with the B-field dependence of R shown in Fig. 7.1b.
We quantify the behavior of the evaporation by a set of power-law scaling relations[89], three of which are90plotted in Fig. 7.4. The average energy removed per molecule, η ≡ (trap depth)/kB T , is the first metric of interest: we observe a ratio η = 5.6, as shown in Fig. 7.4a. (For scale, η =5–10 is considered reasonable in atomic
evaporation[89].) Such a relatively low value of η is unsurprising, given both the possible low (< 1000) value ofthe elastic-to-inelastic ratio R and the fact that molecules are only actually lost when they cross the spatial regions
corresponding to the Xi crossings. This reduces the selectivity of the RF knife, as molecules transferred to the |ei-state
may recollide and rethermalize before they find their way to a crossing. The other metrics are the behavior of temperature and relative phase-space density (PSD) as function of remaining molecule number, shown in Fig. 7.4b. The
efficiency of evaporation is determined by the number of molecules remaining at a given temperature and PSD, that isby α ≡ d(log T )/d(log N) and γ ≡ −d(log PSD)/d(log N). We measure α = 1.26 and so using the fact that density
scales as T13 for our 3-D quadrupole trap we find γ = 4.7. Since R is worse at 10 mK than any other temperature down
to 1 µK, linear extrapolation suggests that it would take roughly a factor of 200 reduction in number to increase PSD
by 1010 , while the repulsive s-wave barrier would serve to stabilize the cloud.
We calculate both α and γ assuming that our fluorescence signal is linear with total molecule number, justified
by the optical power broadening being larger than the Zeeman broadening of the trap. It is very difficult to determine
the sensitivity of pulsed-laser-induced fluorescence in situ, so we use the observed collision rate to estimate a lower
bound on the density in our trap: assuming a unitarity-limited scattering rate βu with a maximum collisional angularmomentum of 12h̄ (and the elastic scattering rate in Fig. 7.1b is only a factor of 3 below this value), an elastic collision
rate of βu n0 = 300 s−1 implies a peak density n0 ≈ 5 × 1010 cm−3 . (Integration of a Boltzmann distribution withthis peak density over the trap suggests a total number of ∼ 106 molecules in the free-evaporated trap sample and an
initial peak PSD of 3 × 10−10 .) This density is high enough to permit the use of absorption imaging techniques todirectly visualize the trap distribution in the future. Since imaging allows direct, non-spectroscopic measurement ofboth density and temperature, it will enable us to quantify further reductions in temperature. With the addition of amechanism to prevent Majorana loss[89], the favorability of R down to microkelvin temperatures indicates that evenBose-Einstein condensation of OH may be feasible.

91

7.2

Methods

7.2.1

Stark deceleration and magnetic trapping
Our Stark decelerator and permanent magnet trap have been described elsewhere[47, 61] and are illustrated bothin Fig. 7.2a and with more technical detail in figure 7.5. The molecular beam is formed by a supersonic expansion ofa saturated mixture of water vapor in 150 kPa of krypton through a pulsed valve; a fraction of the water is cracked by
a 1250 V electric discharge (seeded by a hot-cathode ionization gauge) to produce OH. The gas packet is skimmed,
focused by an electrostatic hexapole, and Stark-decelerated to 34 m/s. The slowed packet is then stopped at the centerof the magnetic quadrupole trap by a high-voltage field applied between the permanent magnets; turning off that fieldleaves the molecules with zero center-of-mass velocity and confined by the magnetic trap. The magnetic trap field iscreated by a pair of NdFeB permanent magnets (grade N42SH), each of which is a (4 mm ID)×(12 mm OD)×(4 mmthick) ring; this yields a maximum gradient of ∼ 2 T/cm in the strongly-confining direction. The high voltage forloading the trap, the low voltage DC bias for ejecting |ei-state molecules, and the microwave fields are all directly
applied to the surface plating of the permanent magnets.

7.2.2

Detection
We observe the presence of OH molecules through pulsed-laser induced fluorescence (LIF). The molecules are

excited on the A2 Σ1/2 (v = 1) ← X 2 Π3/2 (v = 0) transition at 282 nm, and subsequently decay with ∼ 70% efficiency
by A2 Σ1/2 (v = 1) → X 2 Π3/2 (v = 1) around 313 nm. This large Stokes shift enables strong rejection of the excitation
light by an optical filter stack on the photomultiplier tube (PMT) detector. The fluorescence is collected in-vacuum
by a single collimating lens located one focal length from the trap center, and is subsequently focused onto the PMT.
Unfortunately, neither the fluorescence collection efficiency nor the laser excitation efficiency are easily calculable, so
LIF yields only a relative number measurement rather than an absolute one. LIF is parity-state selective, since the upper
A2 Σ1/2 state does not have a parity doublet: the |ei and | f i ground states must couple to different, spectroscopicallyresolvable excited rotational levels.

92

-HV
+HV

pulseddischarge

pulsedvalve

focusing
hexapole

skimmer

in-vacuumDC-blockcapacitors
Stark decelerator

magnetic
trap
SMA feed-throughvacuum chamberFMRF synthesizer
for evaporation-10 dB
coupler

FM
AM

power
amplifierhighpass
filterhighpass
filter

RF synthesizer
for spectroscopy
to monitoring
instruments

Figure 7.5: Technical sketch of the trapping and evaporation apparatus. The evaporation synthesizer is an SRS
SG384 broadband synthesizer, chosen for its ability to perform octave-spanning frequency sweeps at rates of up to
100 Hz. The spectroscopy synthesizer is an HP 8644A.

93
7.2.3

Spectroscopic thermometry
In order to determine the temperature of the trapped distribution, we use a microwave depletion spectroscopy

technique. A series of adiabatic rapid passage (ARP)[104] pulses are repeated over a single narrow (50-500 kHz)band to transfer Zeeman-selected molecules from f ; 32 to e; 23 , alternated with longer DC bias pulses (390 V/cm;
Vspectroscopy = 160 V) to eject the transferred |ei-state molecules from the trap. The ARP pulse duration is generally
40 µs, while the DC bias pulses vary from 200 to 1000 µs over a typical 10 ms, 17-repetition sequence. Multiple ARP
pulses are used to enhance the contrast of the depletion, as a single ARP with +10 dBm peak power at the vacuum
feed-through (limited by the maximum acceptable power-broadening of the microwave transition) yields less than
10% peak depletion. Since the ARPs are all at a single, fixed center frequency, there is virtually no net evaporation or
anti-evaporation from the repetitions; the only effect is to enhance the contrast.
In order to accumulate a full spectrum, many repetitions of the experiment are required: each instantiation
involves a new packet, which is slowed, trapped, and evaporated, then spectroscopically depleted and finally detectedby LIF, which destroys the sample. Each data point in the plots of Fig. 3 is an average of many repetitions of theexperiment, normalized by many repetitions which omitted the spectroscopic depletion step. Specifically, the relative
depletion Drel is given by
Drel = 1 −7.2.4

molecules with depletion ON
molecules with depletion OFF

Evaporation
Once trapped, the molecules are allowed to settle briefly (on the order of 5 ms) before evaporation begins.

The evaporation sequence consists of alternating microwave (80 µs with 0 dBm at the vacuum feed-through) and DC
bias field (∼ 650 µs at 240 V/cm; Vevaporation = 100 V) pulses, though in this case with no substantial frequency chirp
withing any single pulse. The microwave frequency is ramped along an exponential curve towards zero trap depth,
truncated at the desired final depth. The entire evaporation sequence was completed in 70 ms for all of Fig. 3c-f, so
the effective ramp rate varies between the different final temperatures. An illustration of the complete evaporation andspectroscopy sequence is shown in figure 7.6.
It is common in atomic experiments to use a magnetic-dipole transition to directly couple magnetically trapped

94

valve fire
discharge
deceleration
trap loading
settling time
evaporation
spectroscopy
LIF
magnets charged to ±12 kV
evaporation RF pulses (quasi-single-frequency)
magnets charged to Vevaporation
spectroscopy ARP pulses
magnets charged to Vspectroscopy
0

10

80

90

Time (ms)Figure 7.6: Experimental sequence for evaporation and spectroscopic thermometry.

95
states to untrapped ones, but that is not an option at the initial high temperatures in our trap. At 50 mK, hot molecules
sample magnetic fields of over 300 mT, implying that the differential Zeeman shift between the f ; 32 and the | f ; + 12 istates is over 3.3 GHz as can be seen in Fig 7.1a. Since this is much larger than the Λ-doublet splitting, attempts touse drive this transition will also drive | f i → |ei transitions at other locations in the trap, removing the specificity of
the RF knife. The f ; 23 → e; 32 electric-dipole transition, by contrast, can be driven with low enough power as tomake parasitic magnetic-dipole transitions a non-issue. Once the sample is cold enough that the RF knife has passed
through X+3/2 and the magnetic-dipole transition is far red-detuned from any electric-dipole line, it should be possible
to switch to direct pumping to untrapped | f i states. We have not implemented this yet, due to the technical complexityof doing so.

Chapter 8
Laser cooling molecules: from off-beat dream to experimental reality

8.1

A modest proposal
When we wrote our original proposal on laser-cooling molecules with a technically reasonable number oflasers[105], the only ultracold molecules available at substantial density were RbK[106] — and four years later, that isstill true! Incoherent photo-association techniques could reach the X1 Σ (v = 0) ground state at 100 µK[107], but withvery low densities — and did nothing to increase the chemical diversity of the available molecular species. Arbitrary
species could be cooled to the kelvin regime through buffer-gas cooling[108, 109], while Stark deceleration[110, 60]reached the tens of millikelvin level for selected light molecules. However, there was no demonstrated technique tofurther compress and cool the lukewarm molecular clouds resulting from either technique. Cavity-mediated schemes
for molecular laser cooling[111, 112, 113, 114], while in the abstract highly attractive methods for cooling a broad,chemically interesting set of molecules, even now remain unable to cool these lukewarm samples, due to the schemes’
low scattering rates[113], small cavity mode volumes[112], and requirement of multiparticle collective effects[111,
114].

8.1.1

How to achieve optical cycling
Direct, free-space laser cooling and trapping would be the ideal method for producing ultracold molecules, justas it is for atoms. Unfortunately, atoms are in general much easier to laser cool than molecules, due to the latter’s
glaring lack of cycling transitions. Laser cooling generally requires electronic transitions, as vibrational and rotationaltransitions have impractically long excited state lifetimes unless a cavity is used[Andre2006]. Unfortunately, these“electronic” transitions are never purely electronic. Rather, they are rovibronic, and decay into various rotational,

97
vibrational, or hyperfine excited states, as well as the original ground state[115].
The branching ratios of these rovibronic decays, however, are governed by the molecular structure and thedipole selection rules. This implies that a clever choice of molecule can greatly reduce the number of possible decays.
Decays into excited hyperfine states are impossible in molecules with zero nuclear spin, as these molecules have nohyperfine structure. The Franck-Condon ratios for decay back to the ground vibrational level can be quite good (99+%for selected molecules[116]). However, the only constraint on decays to rotationally excited levels is that all decays
satisfy the total angular momentum selection rule ∆J = 0, ±1[110]. Thus, even without hyperfine structure, it may
take up to three lasers per vibrational level to repump the three possible rotational decays. However, if the ground state
angular momentum J 00 is greater than the excited state angular momentum J 0 , two of these three decays are forbidden.
In this case, the molecule must follow the angular momentum cycle J 00 → J 0 = J 00 − 1 → J 00 , and so only one laser isrequired per relevant vibrational level — making laser cooling of these molecules truly practical.
Thus, by combining these various transition closure criteria, we can identify a class of molecules that are
exceptionally good candidates for laser cooling: they have no net nuclear spin, good Franck-Condon overlaps, andtheir ground or lowest metastable state has a higher angular momentum than the first accessible electronically excitedstate. For non-singlet molecules, the excited electronic level must also not be a Σ state, as the lack of spin-orbit splittingin Σ states means that the excited state can decay across the spin-orbit ladder.
We have identified a number of molecules that satisfy all of the above requirements. TiO and TiS are bothsatisfactory in their absolute ground states. Metastable FeC, ZrO, HfO, ThO, SeO, and the like are also promising[118].
We expect that some other, as-yet uncharacterized metal oxides, sulfides, and carbides should also have the necessary
electronic structure. If one is willing to accept some hyperfine structure as the price of chemical diversity, some metal
hydrides and metal halides may also be suitable candidates.

8.1.2

TiO
Of these candidates, we chose to focus on TiO, due to its viability in its absolute ground state and the breadthof spectroscopy and theory available in the literature[119, 120, 117, 121, 122, 123, 124, 125, 126]. A simplified
level structure of TiO is shown in Fig. 8.1. The lowest ground state of TiO is the X3 ∆1 , with spin-orbit constant3

3

A(X ∆) = 50.61 cm−1 and rotational constant B(X ∆) = 0.534 cm−1 [126]. The lowest excited state is the E3 Π0 with

98

Figure 8.1: Level structure of TiO. (a) The electronic level structure of TiO and the transitions of interest for laser
cooling. The X3 ∆ ground state is split by the spin-orbit interaction into the three X3 ∆1−3 sublevels, of which theX3 ∆1 level is the lowest. Each sublevel contains a vibrational ladder, while each vibrational level contains a ladderof rotationally excited states (not shown). 48 Ti16 O has zero nuclear spin and thus there is no hyperfine structure. The
ground-state Λ−doublet (not shown) is much less than the natural linewidth of the E3 Π ← X3 ∆ transition. The solid
arrow denotes the v0 = 0 ← v00 = 0 P(1)-branch cooling laser, and the dashed arrows denote the v0 = 0 ← v00 = 1 andv0 = 0 ← v00 = 2 P(1)-branch repump lasers. The squiggly lines depict the dipole-allowed decays, with the associatedFranck-Condon factor q[117] next to each decay. (b) The rotational and Λ−doublet structure of the E3 Π0 electronic
excited state. The states are interleaved, as the rotational splitting is smaller than the Λ−doublet splitting; a and b
denote the parity states. Both the cooling and repump lasers address the J 0 = 0, a state. Not to scale.

99
Table 8.1: The wavelengths, Franck-Condon factors, and saturation intensities of the cooling and repump transitionsof TiO.
ν 00

λ0,ν 00 [nm]0844.7227 [121]

1
2

9233
1017c

3

Franck-Condon
factor q0ν 00 1
0.845; 0.872
[119]
0.144
0.0107

estimated
Isat [µW/cm2 ]2
8
48
645

3

A(E Π) = 86.82 cm−1 and B(E Π) = 0.515 cm−1 [121]. (While the d1 Σ+ and a1 ∆ level are energetically below theE3 Π level, the inter-system branching ratio is expected to be very small.) As Fig. 8.1 shows, the Franck-Condon
factors[117, 119] for the E3 Π − X3 ∆ band are quite favorable, yielding a population leak of 3 × 10−4 scatter−1 (or amean of ∼ 3300 scatters before going into a dark state) with two repump lasers. The laser wavelengths, saturation
intensities (Isat ), and Franck-Condon factors for the cooling and repump lines are summarized in Table 8.1. Notethat these transitions are all accessible with diode lasers. The saturation intensities are extremely low, as the naturallinewidth γ of the E3 Π − X3 ∆ transition is on the order of 2π×32-40 kHz[122, 123]. This is about 5 times weaker
than the intercombination line used to build a Yb magneto-optical trap (MOT) in[127]. However, with the use of acryogenic buffer-gas-cooled TiO source (similar to [128]), the scattering rate is still large enough to work with.

8.1.3

State remixing in a J 0 = J 00 − 1 MOT
The prospect of building a TiO MOT is tantalizing, given this quasi-closed transition. Traditional MOTs work

using a J 0 = J 00 +1 transition and a magnetic field to break the degeneracy between the excited-state magnetic sublevels.
The MOT beams are polarized so that the local orientation and strength of the quadrupole magnetic field causes theatom to preferentially scatter from the laser beam providing a position-dependent restoring force and a velocitydependent damping force. The fact that J 0 > J 00 means that the atom can always scatter from the correct beam, as
shown in Fig. 8.2(a). The standard MOT will therefore not work for molecules using the aforementioned J 0 = J 00 − 1transition [Fig. 8.2(b)]. While a magnetic field can break the degeneracy of the ground-state magnetic sublevels andthus provide beam selectivity, the |mJ 00 | = J 00 stretched states are effectively dark states, as they can only interact with
one of the laser beams, not both[129].
What is needed, then, is a way to continually remix the ground-state sublevels so that all the molecules spend

100some fraction of their time in bright states. Fortunately, polar molecules provide just the handle needed to accomplish
this: the effective magnetic (B) and electric (E) moments of a polar molecule depend in different ways on mJ . Thus,
applying a sudden (i.e. nonadiabatic) electric field orthogonal (or at least nonparallel) to the local magnetic field
reprojects the total angular momentum against a new axis, randomizing mJ (and the Λ−doublet state) by coupling
the two Λ−doublet manifolds together. At high remix rates and high laser saturations, the molecules’ time is equallydivided across the 2 (2 J 00 + 1) ground and 2 J 0 + 1 excited states (the factor of 2 in the ground state is due to theelectrostatic mixing of the Λ−doublet), but they can only decay while they are in an excited state. Thus, while the0

2 J +1molecules are effectively always bright, the maximum photon scattering rate is only 2 (2 J 00 +1)+(2J 0 +1) γ. Such remixingof the ground-state magnetic sublevels allows the building of a new kind of trap, the electrostatically remixed magnetooptical trap (ER-MOT). The ER-MOT operation is shown in Fig. 8.2(b). Note that, as the local direction of thequadrupole B-field spans all of 4π steradians over the MOT volume, a single E-field pulse will be parallel to the local
B-field in some region and therefore ineffective at remixing the mJ 0 s there. This hole can, however, easily be closed
by applying a second E-field pulse, nonparallel to the first. A basic ER-MOT design is shown in Fig. 8.2(c).
To build an ER-MOT with TiO, there is a minor technical complication. To leading order, the molecularΩmagnetic moment can be written as µ = µB mJ (gL Λ + gS Σ) J(J+1)
Since gL ≈ 1 and gS ≈ 2, the magnetic moment of

the X3 ∆1 (Λ = 2 and Σ = −1) state is small, likely on the order of α µB , or the fine-structure constant times the Bohrmagneton. In contrast, while Ω = 0 in the E3 Π0 state, the large Λ−doublet splitting indicates strong mixing withhigher electronic excited states, and so by analogy with the B3 Π0 optical Zeeman measurements of [130], we estimate
the magnetic moment to be ∼ 10 times that of the X3 ∆1 . This, combined with the narrowness of the E3 Π ↔ X3 ∆transition, implies that the dynamics of a TiO ER-MOT will have more in common with narrow-line alkaline-earth
MOTs [131] than normal alkali metal MOTs. Given these predicted magnetic moments, the magnetic gradient in a
TiO ER-MOT must be . 100 G/cm. This gradient can be easily achieved with water-cooled electromagnets [60] orrare-earth permanent magnets [61]. In contrast, the large (≈ 3 Debye[125]) electric dipole moment of TiO and its
extremely small ground-state Λ−doublet spacing[124] mean that electric fields of only 1 V/cm will give Stark shiftsof about 50γ — far more than the Zeeman shift within the ER-MOT and thus sufficient to reproject mJ . These small
fields can easily be switched with rise times on the order of 10 ns (a frequency of 2800γ and 56 times the Larmor
frequency due to the electric field), and thus nonadiabaticity is assured.

101

Figure 8.2: Comparison of a traditional MOT with an ER-MOT. (a) The level structure of a traditional MOT. The
local magnetic field strength and orientation combined with the Doppler shift enhance the scattering from the laser
beam that provides the damping and restoring forces and suppresses scattering from the counter-propagating beam.
Since J 0 > J 00 , the ground state(s) are always able to scatter from every beam. (b) The level structure of the ER-MOT.
The local magnetic field still governs which laser is preferentially scattered, but angular momentum conservation
forbids some ground states from interacting with the preferred beam. To overcome this, the ground-state magnetic
sublevel populations are remixed by pulsed electric fields, as represented by the dashed lines. (c) A sample ER-MOT
design. A pair of electromagnet coils are aligned in anti-Helmholtz fashion to produce a quadrupole field. Six beamsof the cooling laser are converged on the center with their polarizations oriented as usual for a MOT, but a set of fouropen-mesh grids are added. The grids are pulsed in pairs (e. g., first the X-axis pair and then the Y-axis pair) to produce
the dipole electric fields needed to remix the magnetic sublevels. The center is also illuminated by the repump lasers(not shown).

102

Figure 8.3: Number (upper, black) and temperature (lower, red) time-of-flight plots for the loading of a molecular packet into a simulated TiO ER-MOT. The initial spike on the number plot is the molecular packet flying
through the ER-MOT volume; the broad hump is the actual captured molecules. The decay of the molecule number
is due to radiation pumping of the captured population into excited v00 ≥ 3 states, and yields an ER-MOT lifetime of
80 ± 5 ms. Error bars represent statistics over multiple simulation runs.

A final concern regarding the viability of the TiO ER-MOT is that either the electrostatic or magnetic fields
might somehow cause population loss by mixing in rotationally excited J 0 > 0 states, which could then decay to
J 00 > 1 states and be lost. Fortunately, this loss is inhibited by the ∼ 1 cm−1 rotational splitting [Fig. 8.1(b)]. Neither
the Zeeman or Stark shifts within the ER-MOT are anticipated to be larger than ∼ 100 MHz, and so the perturbative
probability to leave the desired J 0 = 0 state is
100 MHz 2≈ 10−5 . This is much smaller than the 3 × 10−4 scatter−1
30 GHzloss rate from decays to v00 ≥ 3 and so is of no importance.

8.1.4

Monte-Carlo simulations
To verify the feasibility of building an ER-MOT with TiO, we performed a set of 3-D semiclassical Monte-Carlosimulations. We conservatively assumed a natural linewidth of γ = 2π × 32 kHz, a magnetic dipole moment of α µB ,and an electric dipole moment of 3 D. We used a 1/e2 laser waist diameter of 6 cm. Our code treated photon scatteringand molecule kinematics semiclassically and approximated the electrostatic remixing as a sudden reprojection of
the diagonalized Zeeman Hamiltonian wavefunction against a new Stark + Zeeman Hamiltonian. In addition, thesimulations used a set of 60 additional red-detuned frequency components (4.1γ spacing, 7.8 MHz total bandwidth)
within the cooling beams to increase the capture velocity of the ER-MOT, similar to the approach used in the Yb MOTof [127]. For ease of simulation, we used a discrete molecular packet rather than a continuous source. The packet was

103
a 3 mm sphere initially centered at −4.2 cm on the X-axis in the coordinate system of Fig. 8.2(c). It had a flat velocity
distribution of 8 ± 3.5 m/s centered in orientation around x̂, with an opening half-angle of 0.3 rad. The magnetic field
gradient was 51 G/cm and the electric field was pulsed to 4 V/cm for 100 ns at a rate of 50 kHz. The cooling laser was
detuned by 3.5γ to the red, with a saturation parameter of scool = 10.5 per frequency component. The repump lasers
had saturations srepump = 6.1.
These simulation parameters yield the loading and temperature curves shown in Fig. 8.3(a) and 8.3(b), respectively. The final temperature was approximately 10 µK. The ER-MOT lifetime was limited to about 80 ms by radiation
pumping into vibrationally-excited dark states. The estimated capture velocity was 5.7 m/s, which when taking therotational distribution into account allows the capture of about 0.02% of a 4.2 K thermal distribution. If one assumes a
4.2 K source flux of 1010 s−1 [128], the calculated capture velocity, lifetime, and ER-MOT radius predict an ER-MOT
number of 105 and a density of 109 cm−3 .
In addition, we studied the importance of the electrostatic remixing to the ER-MOT operation. Figure 8.4
plots the molecule number after 160 ms against the electrostatic remix frequency Γremix . The plot clearly shows theγimportance of the electrostatic remixing. At Γremix  2π, no molecules are captured, since the molecules are opticallyγ
the capture efficiency
pumped out of the bright state much faster than they are remixed back into it. For Γremix . 2πγrises with increasing Γremix , and then the efficiency saturates around Γremix = 2π
, as the molecules become evenly

divided among the various ground sublevels. As additional validation checks on the simulation code, we verified thatturning off the repump lasers does indeed inhibit the formation of an ER-MOT by pumping the entire population into
vibrationally excited states. We also verified that we could reproduce the experimental Yb MOT of [127] by modifying
the code to simulate a J 0 = J 00 + 1 setup with the correct atomic parameters.
In summary, we have shown that molecules whose lowest ground- or metastable level-intersecting electronictransition has J 0 = J 00 − 1 constitute good candidates for direct laser cooling. We have found several molecules thatsatisfy this requirement and have no hyperfine structure. We have proposed a method to use the electric dipole momentof these molecules to remove dark states in the ground and thus build an electrostatically remixed magneto-optical trap.
We have validated these ideas through Monte-Carlo simulation for a specific molecule, TiO, and verified the necessityand efficacy of the electrostatic remixing.
We thank M. Yeo, E. Hudson, and D. DeMille for valuable discussions and thank NIST, DOE, and NSF for104

Figure 8.4: Fractional capture vs electrostatic remix rate after 160 ms of simulation time for the TiO ER-MOT.
Error bars represent statistics over multiple simulation runs. The curve is only a guide to the eye.

105
support.

8.2

An experimental implementation, YO!
There are almost alwayssubstantial changes between a neat idea and its implementation; molecular laser cool-ing has been no exception. Instead of using TiO, we expanded and reevaluated our list of candidate molecules andchose yttrium (II) oxide (YO) instead. TiO’s exceedingly narrow transition would only make the low scattering rates
found in molecules (due to the large number of ground states) even worse. YO has an excited-state lifetime of only
33 ns; it also actually has superior Franck-Condon factors to TiO[132, 133, 134, 135, 136]. Only two vibrational
repump lasers on v00 = 1 and 2 (in addition to the main cooling laser) are required to close the vibrational ladder to theorder of 10−6 . YO is also even more polar than TiO, with a dipole moment of 4.5 D[125]. However, YO possess asmall parasitic leak through its low-lying, metastable A02 ∆ level on the order of 10−4 .

8.2.1

An introduction to YO
The level structure of YO is shown in figure 8.5. While the proposal in the previous section achieved rotational

closure through the use of an Ω00 > Ω0 electronic structure, YO follows a slightly different method. The 2 Σ groundstate of YO does not have a parity doublet: every rotational level has a distinct parity. The J 0 = 32 , + excited state isthen constrained against decays to N 00 ≥ 3 by angular momentum selection rules, and against decays to N 00 = 0, 2 byparity. (This is the same scheme used in SrF[52].)
The hyperfine structure of YO is more complex than that of TiO[137, 138]. 16 O possesses no nuclear spin,but 89 Y (the only stable isotope of yttrium) has I = 12 . Since the valance, unpaired electron is in a σ orbital largely
centered on the 89 Y nucleus, there is a strong Fermi-contact interaction of approximately 750 MHz. Since there is no
spin-orbit interaction (due to the 2 Σ state) and the spin-rotation interaction is much smaller than the Fermi contact,
YO has a somewhat unusual Hund’s case bβ level structure. The electronic spin S is coupled first to I to yield anintermediate quantum number G = S + I; G only then couples to the rotational angular momentum N to give F. The
rotational-hyperfine structure of YO is shown schematically in figure 8.5(b). Each rotational manifold N is split into a
G = 0 singlet and G = 1 pseudo-triplet state. The G = 1 state is then slightly split into three states F = N − 1, N, N + 1;
for N 00 = 1, the F 00 = 0 state is roughly 20 MHz above the unresolvable F 00 = 1, 2 states.

106

λ=
m

n
649

N" = 1, -G" = 1F" = 1764 MHzF" = 020 MHzF" = 1,2

G=I+S

F=G+N

N" = 0, +

6

8

10

σ+
σ-

F' = 0,1

G = 1, F = 0

-180

,F=
G=1

-190

σ- / σ +σ+ / σ -1

m=+1/-1

Coupling of Zeeman states leads

G=

-200

-210

0

1, F

2

G" = 1=2
4

6

8

Magnetic Field (G)10

B field

Energy (MHz)

1

0.0

038

m
-4
8n
6e λ= 643.

λ = 614 nm

0.0v" = 2

G" = 0d)
4

2+, F' = 0,1-, F' = 0,1

J' = 1/2

N" = 2, +G = 0, F = 1
0

A2Π1/2 ,v'

q=

=

q=

577

v" = 1v" = 0

582

-170

q

q = 0.9944

X2Σ+

q = 0.984

A Π1/2 v' = 0c)b)v' = 12

X2Σ+,v"

a)

Time

m=0
m=-1/+1

Position

Figure 8.5: Level structure of YO and operation of the RF-MOT. (a) The vibronic states relevant to laser cooling,with Franck-Condon factors q listed for each relevant transition. (b) The rotational and hyperfine structure of the X2 Σ
and A2 Π states. (c) The Zeeman spectrum of the X2 Σ ground state. The A2 Π1/2 excited state has nearly zero magnetic
moment. (d) Operation of the YO RF-MOT. As the laser polarization is switched to eliminate dark states, the directionof the magnetic field must also be flipped.

107
The strong Fermi contact and weak spin-rotation couplings also mean that YO has a simple Zeeman structure,
shown in figure 8.5(c). The singlet G = 0 state has almost zero magnetic moment, while the G = 1 state acts like asimple triplet at magnetic fields down to ∼ 1 G. This structure leads to a conceptually simple MOT operation, withmuch smaller fields and gradients than would be required for TiO.

8.2.2

Polarization modulation and the RF-MOT
Like any molecule, a YO MOT would be a so-called Type II MOT with its cycling transition requiringJ 0 = J 00 − 1. Since the ground state has a higher degeneracy than the excited state, for any single laser polarization there are always two Zeeman sublevels which are not coupled to any excited state. To prevent optical pumping
into these dark states from quenching the optical cycling, either the populations must be remixed between the states(e. g. electrostatically as in the TiO proposal or with a skewed magnetic field as in [139]) or the laser polarization must
be varied[140]. Since the G00 = 0, F 00 = 1 level is approximately magnetically insensitive, magnetic remixing is infeasible — as well as incompatible with the goal of a MOT. The fact that optical cycling in YO requires parity to remain
a good quantum number prohibits the use of electrostatic remixing. The remaining option is then to vary the laser
polarization: this is accomplished with the use of a Pockels cell (a electro-optical polarization modulator) switchedat a rate comparable to the natural linewidth. A demonstration of the importance of this polarization modulation isshown in figure 8.6.

8.2.3

Experimental apparatus
The apparatus used for this experiment is illustrated in figure 8.7(a). YO molecules were produced by laser

ablation of sintered Y2 O3 tablets with ∼ 3 mJ of 532 nm light from a frequency-doubled nanosecond Nd:YAG laser.
(Y2 O3 tablets are commercially available, as the material is commonly used in optical coatings.) The ablation targets
are inside a 4.5 K liquid-helium-cooled buffer-gas cell. A gas flow of ∼ 6 sccm and a 3 mm cell aperture produce amolecular beam with a mean forward velocity of ∼ 120 m/s and a FWHM width of 40 m/s (determined by Doppler
spectroscopy), corresponding to a longitudinal temperature of ∼ 3.3 K.
The molecular beam is collimated by an aperture to a transverse temperature of 25 mK before crossing thecooling lasers. The laser cooling is accomplished over a 10 cm region crossed by 11 round-trip passes of the 3 mm

108

15

Tx (mK)

10

5

0

2

4

6

8

10

Polarization chopping rate (MHz)Figure 8.6: Variation of the Doppler-cooled YO beam temperature with the polarization chopping rate. Since
the final temperature is limited purely by the number of photons scattered during the molecules’ transit of the cooling
region, lower temperatures directly imply higher scattering rates.

109(1/e2 diameter) cooling beams. For 2-D Doppler cooling, a second orthogonal multipass is applied. The molecularbeam is also on the axis of the magnet coil which produces the 2-D quadrupole field used for magneto-optical trapping.
After the cooling region, the beam is allowed to ballistically expand over a 30 cm free-flight region. It then passes
through a “clean-up” laser multipass, which optically pumps the molecules in v00 = 1 and 2 back into v00 = 0 before themolecular beam is illuminated by a single retro-reflected probe beam and imaged in fluorescence by a CCD camera.
Optical cycling in YO requires six laser frequencies, produced by three lasers and three acousto-optical modulators (AOMs). While the details of the laser system are out of the scope of this thesis (and will be in the thesis
of Mark Yeo), a schematic of the system is shown in figure 8.7(b). The cooling light at 613.7 nm is produced by adye laser, pumped by a frequency-doubled, diode-pumped Nd:YAG laser. The repump colors at 648 and 649 nm areproduced by two identical diode laser systems. In each system, a grating-stabilized master oscillator injection-locks
a high-power diode to produce one hyperfine component. A few milliwatts of the amplified light is frequency-shiftedby an AOM and used to injection-lock a second high-power diode. The two hyperfine components of each repump are
combined on a polarizing beam splitter; the two repump colors are then combined on a non-polarizing beamsplitter.
Both of the combined repump beams are used: one is combined with the cooling light on a dichroic mirror, while theother becomes the clean-up beam. The dye laser and the two master diodes are frequency-stabilized through direct
optical heterodyne beats with a frequency-doubled, self-referenced, octave-spanning Er:fiber frequency comb.

8.2.4

Doppler cooling
The first major experiment of the YO system was 1-D Doppler cooling. The small spacings between the|G00 = 1, F 00 = 0, 1, 2i states lead to spectral confusion and complex dependence of the cooling rate on the laser detunings. Some typical results from the Doppler cooling experiment are shown in figure 8.8(a). Each curve shows thetransverse (parallel to the cooling laser beams) spatial distribution of the YO beam after the free-flight region for a
distinct set of laser parameters. The black curve (i) shows the initial, unperturbed beam with the entire cooling region
dark. The green curve (ii) has the v00 = 0 cooling light turned on, but the v00 = 1, 2 repump lasers off. The coolinglight efficiently optically pumps the entire beam into v00 > 0 over a wide band of detunings. The red (iii) and blue(iv) curves show Doppler cooling and heating, respectively. The former was taken at a detuning of δ = −5 MHz,
while the latter was taken with δ = +5 MHz. Both curves were taken with three hyperfine components in the cooling

110

Ga2

CCD
cle

an

pr o
- up

cryogenic
beam source

be

be

am

be

am

MOT coil

co
m
lli
at
g
in
ap
er

co

y

tu
re

ol
in
g

be

ablation beam

v=k

ECDL
(-8nmB h mW

v=k
G=g

Diode Laser

v='
G=g

Diode Laser

v='
v=k
G=k

Diode Laser

v='
G=k

Diode Laser

v=g

Gb2

to Er frequecy comb
for stabilization

AOM)(- MHz

OI(-9nmB 8g mW

ECDL

8g MHzOI(-8nmB 8g mW(-9nmB + mW

AOM

OI

z

x

am

OI

to Er frequecy comb
for stabilization

λp'

PBS

OI

(-8nmB 8g mW

PBS

OI

(-9nmB 8g mW

λp'
Dye laser

AOM(k-nmB -gg mW)(- MHzto Er frequecy comb
for stabilization

to probe
regionλp'
PBSdichroic()p++
to interaction region

hgphg

Pockel1s Cellh MHz �PpS modulation

to probe region

Figure 8.7: Experimental setup for a 2-D YO MOT. (a) A general layout of the experimental system, shown with
only one dimension of cooling for clarity. YO molecules (light blue) are produced by laser ablation of sintered Y2 O3pellets by the green 532 nm ablation laser. The molecules are buffer-gas cooled by 4 K helium gas before escaping
the buffer-gas cell (left) as a molecular beam. The beam is then collimated and passes through the MOT region
inside the RF magnetic quadrupole coil (orange rectangles). The molecular beam is cooled and compressed by themultipass laser light, before being repumped out of excited vibrational states (red laser) and imaged by its resonancefluorescence in the final orange beam. (b) The laser system used to produce the 614 nm cooling light and the 648
and 649 nm repump light. The v = 0 cooling light is produced by a dye laser, while the v = 1 and 2 repumps are
created using grating-stabilized external-cavity diode lasers. Each master diode injection-locks two high-power slave
diodes to produce sufficient optical power in each hyperfine component. All three master oscillators are locked to aself-referenced, octave-spanning Er:fiber frequency comb.

8

(a)

60

iii

4

iv

2
0

(b)

i

6

-5

0

40

20

ii

-10

Tx (mK)

Fluorescence signal (arb. u.)

111

5

Position (mm)

10

0

-8

-4

0

4

8

Laser detuning δ/2π (MHz)Figure 8.8: Doppler cooling results from the YO experiment. (a) CCD beam profiles of the YO beam for fourcases: (i) the unperturbed beam, (i) the beam with only v = 0 light applied, which fully pumps the molecules into
v = 1 and 2, (iii) a Doppler-heated beam, and (iv) a Doppler-cooled beam. (b) The detuning dependence of the final
beam x-temperature for a uniform detuning δ of all three hyperfine components of the cooling light. The asymmetry is
due to the close level spacings of the G = 1 manifold. Black points are the unperturbed beam temperature, as imaged
using light of the give detuning and red points are Doppler-cooled or heated. The red line is a fit using a multi-state rateequation to determine the optical forces and a Monte-Carlo molecular simulation to determine the final temperatureachieved.

112

(a)

(b)

Figure 8.9: Comparison of in- versus out-of-plane cooling. (a) When the cooling lasers are in the same plane as theillumination for fluorescence imaging, the width of the image is reduced. (b) When the cooling lasers are transverseto the illumination, the image width does not change, but the entire image is intensified due to the increase in on-axis
density from the cooling.

laser: one detuned by δ from the |G00 = 0, F 00 = 1i line, one detuned by δ from |G00 = 1, F 00 = 0i, and one detunedby δ from the unresolved |G00 = 1, F 00 = 1, 2i manifold. A detuning (temperature-versus-δ ) spectrum is shown infigure 8.8(b). The strong asymmetry is due to the fact that the gap between |G00 = 1, F 00 = 0i and |G00 = 1, F 00 = 1, 2i
is only 20 MHz. When the |G00 = 1, F 00 = 0i laser is blue-detuned by 8 MHz from |G00 = 1, F 00 = 0i, it is also reddetuned from |G00 = 1, F 00 = 1, 2i by only 12 MHz. The red curve is a fit to a model using multi-level rate-equations
to calculate the optical forces and a Monte-Carlo particle-tracking simulation to determine the resulting final temperature.
The extension from one to two-dimensional Doppler cooling is relatively simple. The laser polarization modulation retains its effectiveness in two (or, presumably, three) dimensions; the only technical change is to add a laser
multipass along the second dimension. Since our imaging remains purely one-dimensional, the effectiveness of thesecond dimension cannot be directly judged. However, with the 2-D cooling active we observe a reduction in the
one-dimensional cooling rate by a factor of two and an increase in the total (integrated) molecular number. Both ofthese effects are expected: the addition of the second dimension means that only half of the total photons scattered
come out of the imaging dimension lasers and contribute to visible cooling, while the (invisible) cooling in the crossed
dimension reduces the divergence perpendicular to the imaging direction and therefore increases the total number ofmolecules in the image. These effects are illustrated in figure 8.9.

8

(a)

15

ii

6

iii

4

Tx (mK)

Fluorescence signal (arb.u.)

113

i

2
0

-10

-5

0

Position (mm)

5

10

(b)

10

φMOT

5

σ+
σ-

0
−180 -120 -60

MOT Field

0

60 120 180

φMOT(degrees)

Figure 8.10: 1-D MOT results from the YO experiment. (a) 1-D fluorescence images showing the effects of the
MOT for three cases: (i) the unperturbed beam, (ii) the magnetic field in phase with the optical polarization modulation(ϕMOT = 0◦ ), and (iii) the magnetic field anti-phase to the polarization modulation (ϕMOT = 180◦ ). (b) The variationof the final beam temperature with ϕMOT . The red points are the temperature with the magnet coils turned off, andare insensitive to the absolute phase of the polarization modulation. The blue points have the magnetic field on, anddemonstrate the oscillation between confining and anti-confining magneto-optical forces as ϕMOT varies. The solid
blue curve is a pure cosinusoidal line to guide the eye.

114
8.2.5

Two-dimensional magneto-optical trapping
The addition of magneto-optical forces is technically non-trivial. While the use of a quadrupole magnetic

field and circular laser polarization to induce a spatially-dependent scattering force is well known[3], the polarization
modulation required to eliminate optical dark states complicates the picture. It has been shown that the magnetooptical force can be time-averaged, with the magnetic field direction and optical polarizations switched in unison atacoustic frequencies[141]. We, however, require the magnetic field to oscillate at radio frequencies, on the order of
the transition linewidth.
To achieve magnetic field gradients on the order of 10 G/cm over a 1 cm scale, we used a “baseball” geometry 2D quadrupole coil, illustrated in figure 8.7(a). The coil was wound using 25 turns of 350/46 gauge Litz wire (meaning
350 strands of AWG 46 wire; in Litz wire, each strand is individually insulated, and the wire is wound such that eachstrand spends on average the same amount of length in the center of the bundle as at the outside), so that it maintained
high conductivity and current-carrying capacity at 2 MHz. The entire coil served as the inductor in an LC-resonator,
tuned by a fixed, high-voltage RF capacitor. The resonator achieved a quality factor Q ≈ 75. In order to achievethe ∼ 1.5 A currents required to produce the requisite magnetic field gradients, 25 W of RF power were inductively
coupled through a custom-wound impedance-matching transformer. These currents resulted in a peak voltage on theorder of 4 kV across the coil, or about 200 V between each turn and 800 V between layers. The coil was wound as
five layers of five turns each, with a coating of Q-dope (basically pure polystyrene dissolved in methyl ethyl ketone;
dissolving polystyrene in acetone also works in a pinch) between layers for enhanced insulation.
The coil was wound on an aluminum form consisting of a center block matching the 34 × 34 inch inner spacingof the coil, and four more blocks in a cross pattern to create the form; the principle is that each of the four blocks canbe detached and removed and then the center pulled out to extract the coil from the form. The form was sprayed witha Teflon-based mold-release agent before winding was begone. The coil was wound in a set of five layers. Each turnwas tacked to the previous turns and layers using superglue (Loctite 401, activated by brushing the Litz wire with adamp Q-tip) thrice on each straight section and once on each end turn. Once a full layer was completed, it was coatedwith Q-dope. The cross-overs, where the wire wrapped from the outside of the previous layer to the inside of the nextlayer, were staggered around the coil and Q-doped both below and above the wire, since there is an 800 V potential

115

Figure 8.11: A photograph of the completed RF magnetic quadrupole coil. The large brown bars are the coil itself,
embedded within its epoxy potting. The copper U-loops above and below the coil provide water cooling; the spacers
between the U-loops and the coil are thermally conductive boron nitride. The toroidal coupling transformer can beseen near the upper-lefthand side of the coil, against the flange.

Kab
σ± modulation

116

Kbb
to Pockels cell

10

KPM inb

RF drive
20 Watts

pickupcoil

Tx KmKb

8
Loop Filter

6
4
2

LMOT
Ctune

0

20 25

30

35

40 45

RF drive power KdBmb

Figure 8.12: Circuit used for stabilization of the YO MOT magnetic field phase. (a) The servo loop layout. (b)The variation of the final beam temperature with the RF drive power applied to the coil. The saturation may be due tothermal detuning of the coil rather than true magneto-optical effects.

difference between the innermost and outermost wires.
Once wound, the coil was potted in Epo-Tek H70E epoxy, which is both thermally conductive (0.9 W/m·K)
and meets NASA low-outgassing specifications. Copper cooling water lines were attached above and below the coil,using boron nitride (BN) standoffs. BN is thermally conductive but electrically insulating, and therefore does not haveeddy current losses. If the copper tubes were places closer than ∼ 0.5 in. away from the coil, a measurable drop in Q
was observed, presumably due to eddy currents. The resonator (secondary) side of the transformer was also woundusing Litz wire, while the 50 Ω impedance (primary) winding used standard Kapton-insulated solid copper wire. The
transformer, like the coil, was mounted in-vacuum to minimize the length of the leads between it and the resonator,
and to allow the use of a 50 Ω coaxial feed-through. A photo of the complete, mounted coil is shown in figure 8.11.
In use, the coil reached temperatures of 32◦ C, leading to thermal drifts in the resonant frequency. For simplicity, the modulation frequency (and therefore the coil drive frequency) was held fixed; instead the magnetic phase wasstabilized using an in-vacuum pick-up coil to servo the relative phase of the synthesizer, as illustrated in figure 8.12.
The final temperature of the beam was almost always limited by its ∼800 µs interaction time with the multipass
cooling beam — that is, by the total number of photons scattered, and therefore by the maximum scattering rateachieved during the cooling process. This can be seen in the temperature-versus-power plots of figure 8.13. In thepure Doppler cooling case, the temperature actually reaches a minimum as the cooling laser intensity is increased andthen begins to rise. This rise is not the power-broadened Doppler temperature, but indicates that the lines were being

117

(a)

(b)

25

20

Tx (mK)Tx (mK)

30

10

20
15
10
5

0

0

4

8

12

Relative cooling
power

0

0

0.2

0.4

0.6

0.8

Relative v'' = 1,2
repump power

Figure 8.13: Power dependencies of Doppler cooling and the 2-D MOT. (a) Final temperature versus relative coolinglaser (614 nm) optical intensity for Doppler cooling (red) and magneto-optical trapping (blue). (b) Final temperature
versus relative repump (648 and 649 nm) laser intensity. Solid curves are fits to a multi-level rate-equation model.

118
sufficiently power-broadened as to produce spectral confusion in the |G00 = 1, F 00 i manifold and reduce the cooling
effectiveness. This fact can be seen from the fact that in the MOT case there is no such increase in final temperature:
the Zeeman-split G00 = 1 manifold has less spectral confusion. The importance of scattering rate is further emphasized
by the continual reduction in final temperature with increasing v00 = 1, 2 repump power. Since less than 0.5% of thetotal photons scattered are from the repump lasers, their optical power contributes no direct cooling. The repump rate,
however, is the major limitation on the effective time-averaged scattering rate.
These relatively low average scattering rates are entirely consistent with both the expectations of the previous
section and also with the results of Shuman et al.[52]. YO possesses 12 states per vibrational level in the ground electronic configuration (two F 00 = 1 Zeeman manifolds, and one each F 00 = 0, 2) and 4 in the A2 Π1/2 , v0 = 0, F 0 = 0, 1
levels. There are three ground vibrational levels in the total optical cycle, although the effective scattering rate iscomplicated by the fact that the v00 = 2 level is connected to the A2 Π1/2 , v0 = 1, F 0 = 0, 1 level rather than v0 = 0.
If all three ground vibrational levels were connected to the same excited state, the long-term average scattering ratewould saturate to4γγ=≈ 5 × 105 s−1
3 × 12 + 4
10, substantially lower than is typical in an atomic MOT. This result means that as a practical matter, it is worth a great
deal of technical effort to maximize the scattering rate. For instance, while a 3-D MOT geometry formally does nothave any dark Zeeman sublevels as there is always a transverse beam with a bright polarization component, it is still
worth the effort of using an RF magnetic field and polarization modulation. In a DC MOT, the MG = 1 state is always
detuned by twice the local Zeeman shift of the molecule. Since an RF magnetic field swaps the MG = ±1 states, thefarthest detuned state in an RF MOT is the MG = 0, with only a single power of the Zeeman shift. As scattering rate
goes inversely as the square of detuning, the RF MOT will have on average a scattering rate four times higher than a
DC MOT.

Chapter 9
Visions for the future

9.1

Towards a hydroxyl BEC (and maybe a Fermi gas?)With the resounding success of evaporative cooling and its predicted applicability down to microkelvin tem-

peratures, it seems appropriate to consider just what would be required to actually evaporate down to Bose-Einsteincondensation.

9.1.1

Improved detection
The first major challenge is most certainly detection. While we have always known that LIF does not havesingle-molecule sensitivity, the density number implied by evaporation indicate that it is in fact spectacularly bad:probably three orders of magnitude worse than a naive estimate would suggest. If that is true, then direct absorption
detection is not only possible, but should be substantially more sensitive than LIF.

9.1.2

Ballistic expansion
Continuing on the theme of detection, it should be considered just how one could detect the BEC state. The

usual technique in atomic experiments is to turn off the trap, allow the cloud to undergo ballistic expansion, and thenimage its time-of-flight. The permanent magnet trap, however, would seem to render that impossible. (Since it is
only the high gradient available from permanent magnets which makes evaporation possible at all, switching back toelectromagnets is infeasible.)
If we cannot turn off the trap fields, can we turn off the molecules’ Zeeman shift? As OH is bosonic, one
natural idea is to somehow rapidly transfer the entire population to the mF = 0 Zeeman level. However, in the large

120
magnetic fields found in our trap, the correct quantum number is J, not F. Since J is half-integer, there do not exist
any states in the X2 Π3/2 manifold which are field-insensitive over any substantial volume in our trap.
That is not the end of the story, however: there is also the X2 Π1/2 spin-orbit-excited manifold. Since themagnetic moment for a Hund’s case (a) molecule goes as gS Σ + gL Λ, the X2 Π1/2 state has gS · 12 + gL · (−1) ≈ 0
moment. (gL ≈ 1, gS ≈ 2 + ganom where ganom ≈ 2 × 10−3 is the anomalous part of the electron spin g-factor.) Thissuggests that trap-release could be accomplished by collective excitation of the entire population to this level. OH,
however, is not a pure case (a) molecule: even at J ≥ 23 , the X2 Π1/2 manifold has substantial intermediate-casebehavior[142]. Therefore, this near-zero moment exists only for the lowest J = 12 state of the X2 Π1/2 manifold. If themolecules are excited to the X2 Π1/2 , J = 12 , M = 12 level, they will have a moment of orderµ = µnonrotating

1 1
·
MΩ
∼ (.002µB ) 21 23 ≈ 1.3 µK/mT
J(J + 1)2·2

which is small enough to let the expansion be quasi-ballistic over 10 µm scales at 1 µK.
How might the molecules be transferred up to the X2 Π1/2 state? Using the data of Varberg and Evenson[143],
we find that the P12 ( 32 ) transition is at approximately 3.786 THz (79.2 µm, 126.3 cm-1 ) – a remarkably inconvenient
wavelength, although quantum cascade lasers may be possible. If a direct light source is available, with sufficient
power one could in principle use a simple adiabatic rapid passage (ARP) to transfer the entire population. However, polarization issues add a complexity: only the σ − polarization connects the f ; 23 trap state to the desiredX2 Π1/2 , J = 21 , M = 12 excited state, and so illumination from multiple directions would be required in the context
of a magnetic quadrupole trap.
Another possibility, which trades two UV light sources for one terahertz-band source, is stimulated Raman
adiabatic passage (STIRAP). This technique’s power and precision are well-demonstrated[51], and the long (∼730 ns)
lifetime of the A2 Σ state in OH may make transfers through it feasible with low optical powers. The differential
Zeeman broadening between the 2 Π3/2 and 2 Π1/2 levels may prevent the use of STIRAP, however.
A third possibility, which would use a single, high-power mid- or near-IR source would be to perform thespin-orbit transfer as part of a rovibrational transition to v00 = 1 or 2. The source would require a great deal of power,
since the spin-orbit transition is forbidden in the X2 Πi ground state of OH—but it might still be easier to get a lot of
power on the v00 = 2 ← v0 = 0 vibrational overtone near 1.4 µm than a little power anywhere else. If enough power is121available, ARP could be used on this transition as well.

9.1.3

Magnetic dipole evaporation
At magnetic fields below 49.6 mT (the location of the X−3/2 avoided crossing of chapters 5 and 7), the e; 32

state is totally trapped, and therefore no longer useful for evaporation. Therefore, it will be necessary to at somepoint cease using the electric-dipole-allowed f ; 32 → e; 32 transition and switch to driving the magnetic-dipole| f ; Mi → | f ; M − 1i transitions. It is impossible to do this for magnetic field values above the location of X+1/2 : themicrowave field will also be resonant with some electric-dipole-allowed transition, and will lose its magnetic field
selectivity. The ideal procedure is probably to electric-dipole evaporate down to at least∼ 90 mT below X+1/2 , so thatthe magnetic-dipole radiation will be detuned by O(100 MHz) from any electric-dipole transition.
Technically, there is a complication. The largest issue is that the magnetic trap has almost zero coupling formagnetic transitions: to a good approximation, the trap is a parallel-plate capacitor. The magnetic field due to thedisplacement current inside a circular capacitor of radius R goes as
B(t) = µ0 I(t)a
2πR2for a charging current I(t) and a radial position a. Since the molecules are near the center of the magnets, they havea  R and so see only a very small magnetic field. It will be necessary to add a coil outside the trap to couple
to the magnetic transitions. Fortunately, both the steel and the NdFeB magnet material have relative permeabilities
close to unity, so they should not present any screening issues. Applying the magnetic-dipole microwaves through anelectrically isolated coil also removes any issues regarding high-voltage protection for the microwave system as thefrequencies drop down towards DC.

9.1.4

Preventing Majorana losses
The largest issue on the path to lower temperatures is Majorana loss. Majorana loss occurs due to a loss ofadiabaticity in a quadrupole trap when the molecules pass through the trap minimum. In order for the molecules toremain trapped, they must adiabatically follow the magnetic field direction, so that they remain in the magnetic weakfield-seeking state. Near the trap center, the Larmor frequency ΩL = gµB B0 r/h (where g is the Lande g-factor and B0

122
is the magnetic field gradient) goes to zero with the the radial coordinate r. The rate of change of the magnetic field
angle, however, does not go to zero. Since mean velocity is proportional to temperature,
dθB
vθ
=
∼
dt
r

p

T /m
r

for particles of mass m. Therefore, there always exists someprdeath ∼T /mgµB B0!1/2

B
for which ΩL ≤ dθ
dt ; since the mean radius rT of the cloud simply scales as T for a 3-D quadrupole trap,

rdeath∼ T −3/4
rT.
Semi-empirical scaling from atomic systems suggests that Majorana loss will become a major issue in OH
around 1 mK—only a factor of 5 colder than was demonstrated in chapter 7. Therefore, it is critical to find a way tosuppress this loss mechanism.
In atomic gases, the standard way to avoid Majorana loss in magnetic traps is to use a trap geometry which does
not have a true magnetic field zero at its center: an Ioffe-Pritchard (IP) trap[144] or a time-orbiting potential (TOP)trap[145]. The former uses a large magnetic bias gradient to convert the linear quadrupole trap to a harmonic trapwith a nonzero central magnetic field, while the latter uses an AC magnetic bias to shift the field zero of a quadrupole
trap out from under the trapped atoms. If the zero describes a ring trajectory outside the trapped cloud, and does sofaster than the dynamical timescale of the cloud, the cloud will remain trapped at the center since the time-averaged
magnetic field is still lower there than over the trajectory of the field zero.
Both of these trap geometries work well, although the IP geometry’s technical simplicity has lead to it being
far more common than the TOP trap. In the context of OH, however, both traps present major technical challenges.
The major issue is that the OH trap requires both immense depth, so that it can be loaded directly from the Stark
decelerator, and very tight confinement—i. e. large magnetic gradients. Even with the use of permanent magnets, itappears that switching to an IP trap implies a factor of two reduction in the trap depth. Adding TOP coils to the current
permanent magnet quadrupole entails on the order of 5000-10000 amp-turns of current, which present enormous
thermal challenges.

123
There may be another option, however—at least for one to two orders of magnitude in temperature, at which
point using optical dipole forces to plug the hole in the quadrupole trap becomes eminently reasonable. Since OH
molecules are polar with a substantially Hund’s case a level structure, an applied electric field should also preserve
the molecules’ spin projection. Specifically, the electric field maintains the laboratory projection M of the total angularmomentum (excluding nuclear spin) J, since the molecule will precess around the electric field vector with an (electric)
Larmor frequency
(e)

ΩL =

µE
h̄

. This suggests that a homogenous DC electric field will act like the bias field in an IP trap and therefore will prevent
Majorana loss.
However, there is a complication to this story: DC electric fields can also induce Landau-Zener loss, as described in section §6.2. Numerical simulations suggest that sufficiently small electric fields (a few V/cm) are safe
down to ∼ 100 µK.

9.1.5

Fermionic OD?
While OH is bosonic, due to the IH = 12 spin of the hydrogen’s proton coupling with the unpaired valence

electron, OD (ID = 1) is fermionic. OD can be made by an electric discharge through D2 O, and Stark deceleration andmagnetic trapping would all proceed almost identically to OH. (There is a correction to the timing sequences required
for OD’s higher mass of 18 amu, but the net change needed is small, on the order of 50◦ ×

18
17 − 1≈ 3◦ degreesof phase angle.) The uncertainties only begin once the OD molecules are trapped. There are two questions whose
theoretical investigation may be warranted before undertaking experimental efforts:(1) What is the |ei-to-| f i differential magnetic g-factor in OD? The electronic and hyperfine structure of OD havebeen studied experimentally[142, 146, 147], but the differential g-factor has not been measured in OD.
(2) What are the collisional properties of OD? Does the large change in reduced mass substantially change theelastic-to-inelastic ratio (e. g. through a shape resonance), or does the repulsive barrier continue to dominate
in f ; 23

f ; 23 collisions?

124

9.2

Enhanced 3-D magneto-optical trapping of YO, and what to do with it

9.2.1

Dark-spot trapping
While I believe that it will be possible to demonstrate a 3-D MOT of YO without closing the leak to the A02 ∆level, such a MOT will have a very short lifetime, on the order of 1 ms, due to optical pumping through the leak into
N 00 = 0, 2. However, simply reducing the scattering rate for molecules that have already been cooled and trapped into
the MOT will go a long way to improving the MOT lifetime. At a Doppler-limited temperature of 120 µK, a moleculetakes on the order of 5 ms to traverse a 1 mm sphere. If that sphere is a hole in the v = 1 repump beam, the molecule
will rapidly cease to scatter once it enters the sphere. At the most naive level, one might suppose that the molecule
will scatter ∼ 200 photons every time it exits the the dark spot, i. e. that it will go dark as soon as it decays into v = 1,
since the 200 photons will be plenty to push it back into the dark spot. If one assumes that it takes ∼ 2500 photons toscatter through the A02 ∆ level, the dark spot would extend the MOT lifetime from 1 ms to 5 × 2500/200 ≈ 62 ms – avery substantial improvement!

9.2.2

Closing the A02 ∆ leak
The other avenue to improving the MOT lifetime is to close the leak through the A02 ∆ state[148] entirely. Thiscan conceivably be done in at least 3 ways:(1) Repumping out of the N 00 = 0, 2 states through the A02 ∆ state: as a ∆ state, the A0 level presumably has avery small Λ-doublet – most likely on the MHz scale or smaller. If one assumes a 3 D dipole moment anda 50 MHz Λ-doublet (both likely to be conservative estimates), it only takes 33 V/cm to reach the transition
into the linear Stark regime. If the goal is simply to give strength to the parity-forbidden |A02 ∆, J = 32 , −i →|X2 Σ, N = 1, −i transition, much smaller fields are possible. However, this state mixing also makes the|A02 ∆, J = 23 , −i → |X2 Σ, N = 3, −i transition allowed, which adds a requirement for a third laser.
(2) Repumping out of the N 00 = 0, 2 states through a X2 Σ, v > 0 manifold: while direct microwave remixing of|v00 = 0, N 00 = 0, 2i into |v00 = 0, N 00 = 1i is undesirable due to the large reduction in scattering rate it entails,
molecules spend little enough time in v > 0 that adding a bit more due to stimulated pumping between|v00 = 0, N 00 = 0, 2i and some |v0 > 0, N 0 = 1i is immaterial. The concern with this idea is acquiring light125
sources – although at least a few, and possibly tens of milliwatts may be available from quantum cascadelaser sources. (Repumping through v0 = 1 requires light at near 867 cm-1 (11.5 µm), while the v0 = 2 overtone
transition is at a more accessible 1740 cm-1 (5.75 µm).) The scattering rates required are low — a mean
scattering rate of only 2 kHz (and therefore in principle a 2 ∆ repump rate of 1 s-1 in a dark spot MOT) is
enough to balance gravity. BaF2 is still substantially transparent even out to 11.7 µm.
(3) Directly repumping out of the A02 ∆3/2 , J 0 = 23 state before it decays: this can in principle be achieved by
optical pumping to the B2 Σ, N 0 = 0 level with 1.6 µm light. The B2 Σ level has a very strong propensity to
decay directly to the X2 Σ level, and has good Franck-Condon overlaps with it (95+% decay to v00 = 0, 1, 2).
However, the 1.6 µm line has never been observed – and as an excited-to-excited transition, the spectroscopy
may be quite challenging. Moreover, the preferential decay of the B2 Σ necessarily implies that the repump
line out of the A02 ∆ must be weak, and possibly very weak.
All three of these options have their drawbacks; the second one seems at the moment to be the best. Although it
requires two lasers rather than the one needed for third option, the laser power is likely to be much easier to achieve.
The laser control is also the easiest of the three: the spectroscopy can probably be done with in-cell absorption
measurements, and the stabilization requirements can be met with an optical cavity. None of the three techniques offer
an immediate solution to molecules which decay through the A02 ∆ to v00 = 1+, but if the lasers are sufficiently agile
it may be possible to dither them at kilohertz rates between the repump lines for at least v00 = 0 and 1. Given the∼ 90% Franck-Condon overlap to v00 = 0 for decays from the A02 ∆, just repumping that level would give a ten-foldimprovement in the number of available scatters for a total of 3 × 104 .

9.2.3

Narrow-line cooling
The development of lasers addressing the A0 –X transition offers another benefit: a narrow-line second-stage

MOT for cooling. The estimated 15 µs lifetime of this transition gives a Doppler limit of 1.2 µK. Since the vibrational
repumps contribute less than 10% of the total scatters, the repumps through the A–X transition may be used without
their contributing substantial heating.

126
9.2.4

Collisions and evaporation?
At least thermodynamically, YO should be stable against chemical reactions: while yttrium’s preferred oxida-

tion state is +3, Y2 O3 cannot be formed in a two-body YO–YO collision. Y2 O + O and YO2 + Y are both energeticallydisfavored relative to 2YO. However, in the context of a laser-cooled MOT, the relevant issue is probably inelastic scattering rather than chemical loss. Since the MOT population is in the excited N 00 = 1 state, it is free to collisionallyde-excite, leading to the molecule going dark – and also releasing 2B ≈ 0.78 cm-1 of energy. That comes out to 14 m/s,
sufficient to eject the molecule from the MOT. Of course, collisions would be exciting to see in their own right, but if
the ultimate goal is to reach a regime where evaporative cooling can be used to reach quantum degeneracy it is clearthat the molecules must be transferred to the N 00 = 0 level to accumulate. This may be possible using microwaves or
one of the optical transfer methods of section 9.2.2; the ideal option would be to simply let decays through the A0 –X
transition accumulate narrow-line-cooled molecules into an optical dipole trap by applying a tiny electric field to allow
decay to N 00 = 0, 2 after scattering some hundreds of photons. However, if the MOT works well it seems likely thatits density may be collision-limited even without narrow-line cooling, and so some more active accumulation method
would be required. That said, a static DC electric field could be used to allow direct decays from the A2 Π level to
N 00 = 0, 2 at any rate desired. Roughly 2 kV/cm is sufficient to enter the linear Stark regime, implying > 25% mixingof the excited-state Λ-doublet.

9.2.5

Fermionic 89 Y17 O?
The vast majority (99.95%) of YO is bosonic, since yttrium has only one stable isotope and both 16 O and 18 O

have I = 0. 17 O does exist, however. It is quite expensive, but if it is possible to resurface ablation targets rather than
discard them after a single usage cycle there should be no major issues with producing an 89 Y17 O beam. Isotopicallyenriched Y2 O3 does not exist commercially, but combining yttrium metal with isotopically-enriched 17 O2 in a vacuum
furnace should not pose any major difficulty. The larger concern actually lies with the laser system. 17 O has I = 52 ,leading to five hyperfine levels F 00 = 12 , 32 , 25 , 72 , 29 . Since the valance electron of the 2 Σ state has a low overlap withthe oxygen nucleus, its Fermi contact interaction should be small. The magnetic moment of the 17 O nucleus is 1.8938 µN , and as an I = 25 state it also possesses an electric quadrupole moment of -0.026 e-barns[149]. Without
detailed structure calculations it is impossible to predict the actual hyperfine structure of 89 Y17 O, although in the best127
case it would preserve the G00 = 0, 1 structure of bosonic YO and only have small splittings due to the 17 O nucleus.
Calculations of this hyperfine structure would be of great interest.

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