Experiments on Cold Molecules Produced via
Stark Deceleration
by
Eric R. Hudson
B.S., Morehead State University, 2000
M.S., University of Colorado - Boulder, 2004

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
2006

This thesis entitled:
Experiments on Cold Molecules Produced via Stark Deceleration
written by Eric R. Hudson
has been approved for the Department of Physics

Jun YeEric A. Cornell

Date

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

iii
Hudson, Eric R. (Ph.D., Physics)
Experiments on Cold Molecules Produced via Stark Deceleration
Thesis directed by Prof. Jun Ye

The work described in this thesis is the construction and operation of a Stark
deceleration apparatus. Specifically, this work represents only the second successful implementation of the method1 and the first deceleration of the hydroxyl radical (OH) andthe formaldehyde molecule (H2 CO). Experimentally, much of the work in the beginningof our experiments was the construction of a suitable molecular beam. Though molecularbeams have been used for decades in physical chemistry, the design goals of the typicalphysical chemistry experiment are very different than ours, and as such, much work was
required to adapt these sources for Stark deceleration. Since the maximum-attainable,
decelerated molecular density is set at this stage, the importance of a good molecularbeam source cannot be understated. Once a stable decelerated beam became routine in
our laboratory, we performed the first precision measurement on cold molecules leading
to the best ever values for the frequencies of the main lines in the lowest OH Λ-doublet.
These measurements, when coupled with the appropriate astrophysical measurements,
can be used to put the most stringent constraints on the variations of the fundamental
constants. Excitingly, we have recently been informed that the required astrophysicalmeasurements may be performed as early as the latter part of this year.
Our most recent work, not covered in this thesis,2 has been the construction andimplementation of a magnetic trap for OH molecules. To prepare decelerated molecules
for loading into this trap we have built a second-generation decelerator, which has
more stages than our first, to provide a higher flux of decelerated molecules. In theconstruction of this decelerator, several interesting phenomena, which currently limit
1
2

The Berlin group, whose collaboration has been fruitful (for both sides, I hope), was first.
This topic will likely be the crux of the dissertation of Brian Sawyer.

iv
the decelerator efficiency, have come to our attention. Specifically, these are transverse
over-focussing of the decelerated molecular packets in the decelerator and the possibilityof Majorana type transitions during deceleration. These effects (especially transverse
over-focussing) extremely limit the number of molecules decelerated to below 15 m/s in
our apparatuses. Since molecules of this speed or less are required for proper loading
into our magnetic trap, both of these phenomena must be addressed in order for the
optimal deceleration performance to be reached. Currently, as detailed in this thesis,
experiments are already underway to learn how to eliminate them. Thus, it is my
opinion that the technique of Stark deceleration is very close to blossoming into thebest technique for producing cold molecules,3 since addressing these issues will leadto several orders of magnitude gain in decelerated molecule number. Furthermore,with ideas for further cooling of trapped samples being developed, e.g. cavity-assisted
Doppler cooling, the possibilities for cold molecule physics are staggering.

3

Perhaps this argument could already be made.

Dedication

For Natasha

vi

Acknowledgements

As anyone who has worked in a research environment will tell you, success requiresa large number of great people working together. I have been fortunate enough to work
at JILA in Prof. Jun Ye’s group, which can be defined as a ‘large number of great people
working together’. In some sense, I feel I was in an environment where it was impossible
to fail because the people around me would not allow it to happen, and as such, I am
indebted to many. The help I received from the JILA electronics shop (Terry Brown,
Mike Whitmore, James Fung-a-Fat, Paul Beckingham, Carl Sauer, and David Tegart)
was instrumental in both the production and detection of our cold molecules. Likewise,
without the aid of the JILA instrument shop (Alan Pattee, Hans Green, Kim Hagen,Tom Foote, Tracey Keep, and Blaine Horner), our decelerator would never have been
constructed. These individuals are more than engineers and excellent technicians, they
are artisans who have mastered their craft. In the laboratory I have had the pleasure ofworking with many great scientists. I still remember my first visit to the lab (to see theempty table) when I met Jason Bochinski, the project’s first postdoc. Together, Jason
and I navigated the perils of switching high voltages and making a molecular beam, andthough Jason was clearly the leader, he never belittled my contributions and for that I
am grateful. Almost two years later Heather Lewandowski joined the effort as a postdoc,
and it was from a series of fruitful discussions with her that the work that comprises
Chapter 6 sprung forth. Brian Sawyer was next on the scene as the graduate student who
would take over my position. I have thoroughly enjoyed every moment I have worked

viiwith Brian, especially the construction of our second decelerator. His relaxed attitude
and sharp mind have greatly accelerated the progress of the project. At the beginningof the year Benjamin Lev started working with us as a postdoc. Benjamin quickly took
over the high resolution spectroscopy experiment, and I am completely impressed withhis wit and attention to detail. Last to join our group was Ben Stuhl, a graduate student.
While Ben and I have only worked together for a few short weeks, he has already taught
me much about coding, and I expect only the best things from him in the future.
Outside of the cold molecule laboratory I have had the benefit of a close relationshipwith the theory group of John Bohn. John, his posdocs, and his students (Alexander
Avdeenkov, Chris Ticknor, and Ed Meyer) have taught me molecular physics. I havealso been fortunate to have a close relationship with the group of Gerard Meijer inBerlin. The friendly competition/collaboration with this group, the originators of the
Stark deceleration technique, has not only accelerated the development of the field,
but has left me with many wonderful friends (Rick Bethlem, Floris Crompvoets, Basvan der Meeraker, Joop Gilijamse, Steven Hoekstra, Nicolas Vanhaecke, Joost Bakker,
Jacqueline van Vandhoven, and Gerard Meijer). Of course, none of this work would
be possible without the guidance of Prof. Jun Ye. It is impossible to overstate theimportance of Jun to the success of the experiment. In the difficult times Jun always
seems to know the right way to proceed. Furthermore, working for Jun is a joy because
he is as committed to the growth of his students as he is to doing excellent science. Jun
has allowed me to travel around the globe to further my development as a scientist,and I cannot begin to explain how important those trips have been to me. If I ever
have my own laboratory, I can only hope to approach the excellent leadership Jun has
demonstrated.
Beyond the amazing people at JILA and in Jun’s group, there is another element
to how easy it is to succeed here. This element, which is somewhere between an ideal
and an attitude, – maybe Zeitgeist is the right word – is a sense of family that pervades

viii
the institution. Whether it is loaning equipment or taking the time to talk with someone
about a problem on their experiment, this approach greatly decreases the time it takes
for good science to be done. More importantly though, it makes for a non-stressful
work environment. I am not sure how this culture comes to prevail at an institution.
Perhaps the foundation was laid by the early Fellows, or it has to do with living insuch a beautiful place as Boulder, e.g. even if the laser breaks, you can still go skiing
on the weekend. Regardless, it is, in my opinion, the gold standard for any working
environment and I hope to help recreate wherever I work.
I am extremely indebted to my parents, Raymond and Christine. Though theyknew the importance of education, they never pushed me into anything, preferring to
let me find my own way. I credit this relaxed atmosphere as the reason I was able to
make it this far. Because I enjoyed what I was doing, I was never ‘bogged-down’ by thework. Nonetheless, the most important gift I received from my parents was a strong
faith in Christ. In retrospect, it was my beliefs as a Christian that led me to physics. In
my opinion, there are scarce better ways to know God than to study his creation, andthe beauty and symmetry I have found in physics have only strengthened my faith. And
any reader who thinks that things like religion or prayer have no place in a laboratory
has never applied 30 kV across a couple of millimeters.
And thank Tasha for all the rest.
Eric R. Hudson
Boulder, 2006

ix

Contents

Chapter
1 Introduction to Cold Molecules1

1.1

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

2

1.2

Buffer Gas Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.3

Cold Atom Association . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.4

Slow Molecule Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2 Molecular Structure: A Primer

18

2.1

The Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.2

The Structure of OH . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.3

The Structure of H2 CO . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

3 Molecular Detection

37

3.1

Laser Induced Fluorescence Detection . . . . . . . . . . . . . . . . . . .

37

3.2

Hydroxyl Radical Detection . . . . . . . . . . . . . . . . . . . . . . . . .

40

3.3

Formaldehyde Molecule Detection . . . . . . . . . . . . . . . . . . . . . .

42

4 Sourcery: Supersonic Molecular Beams
4.1

4.2

43

Hydroxyl Radical Discharge Source . . . . . . . . . . . . . . . . . . . . .

47

4.1.1

Piezo-electric Transducer Actuated Valve . . . . . . . . . . . . .

57

Formaldehyde Beam Source . . . . . . . . . . . . . . . . . . . . . . . . .

58

x
4.3

Hexapole Focusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

4.4

Extracting Beam Parameters from Time-of-Flight Data . . . . . . . . .

67

5 Stark Deceleration

70

5.1

The Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

5.2

Decelerator Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . .

80

5.3

Decelerator Construction. . . . . . . . . . . . . . . . . . . . . . . . . .

80

5.3.1

Vacuum Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . .

81

5.3.2

Decelerator Electrode Construction . . . . . . . . . . . . . . . . .

82

5.3.3

Decelerator Electrode Conditioning . . . . . . . . . . . . . . . . .

87

5.3.4

Fast High Voltage Switching. . . . . . . . . . . . . . . . . . . .

92

Stark Deceleration of the Hydroxyl Radical . . . . . . . . . . . . . . . .

93

5.4.1

First-Generation Decelerator . . . . . . . . . . . . . . . . . . . .

95

5.4.2

Second-Generation Decelerator . . . . . . . . . . . . . . . . . . . 110

5.4.3

Transverse over-focussing . . . . . . . . . . . . . . . . . . . . . . 115

5.4

6 Acceptance Theory of a Stark Decelerator

118

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.2

Stark Deceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.3

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7 Alternative Deceleration Techniques

141

7.1

Decelerator Overtones . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7.2

Modified Decelerator Overtones . . . . . . . . . . . . . . . . . . . . . . . 151

7.3

Quadrupole Guiding Decelerator . . . . . . . . . . . . . . . . . . . . . . 153

8 Towards Cold Chemistry and Reaction Control: Deceleration of the Formaldehyde Molecule

168

xi
8.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

8.2

Reaction Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

8.3

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

9 High Resolution Microwave Spectroscopy of OH

186

9.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

9.2

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

9.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

10 Not Enough Hours in the Day

198

10.1 Polar Molecule Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . 198
10.1.1 Born Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 199
10.1.2 Eikonal Approximation . . . . . . . . . . . . . . . . . . . . . . . 200
10.2 Sympathetic Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
10.3 Para-electric Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
10.4 Cavity-Assisted Doppler Cooling . . . . . . . . . . . . . . . . . . . . . . 212
10.5 Capacitive Detection of Polar Molecules . . . . . . . . . . . . . . . . . . 218

Bibliography

225

xii

TablesTable
2.1

Relevant molecular constants in units of cm−1 for calculating the rovibrational spectrum of OH. Taken from Refs. [36, 83]. . . . . . . . . . .

2.2

25

Parameters for calculating the off-diagonal rotational contribution for an
asymmetric top molecule. . . . . . . . . . . . . . . . . . . . . . . . . . .

34

2.3

Direction cosine reduced matrix elements . . . . . . . . . . . . . . . . .

35

4.1

Working formulas for Tk [1]. . . . . . . . . . . . . . . . . . . . . . . . . .

46

10.1 Minimum attainable temperatures using para-electric cooling. . . . . . . 211

xiii

FiguresFigure1.1

Total collision rate of weak-field-seeking OH molecules as a function ofelectric field. Courtesy of John Bohn. . . . . . . . . . . . . . . . . . . .

3

1.2

Typical buffer gas loading experiment set-up. Taken from [41]. . . . . .

6

1.3

Representative potential curves for colliding alkali atoms along with thephoton frequencies involved in one-color photo-association. The laser
field, ωP , excites the colliding atoms to a bound state of the excitedpotential curve, which hopefully decays by ωB to a bound, ground state
molecule. Taken from [12].

1.4. . . . . . . . . . . . . . . . . . . . . . . . .

10

Two types of two-color photo-association. In panel (a), two-color photoassociation by a stimulated Raman process is shown, which populates
a single vibrational level. In panel b, two-color photo-association bya short range Rydberg state is shown, which populates several deeply
bound vibrational levels in the ground electronic state. Taken from [29].

1.5

12

The relevant information for Feshbach resonance association is shown. As
the magnetic field is tuned towards the resonance the scattering length
diverges, and it becomes possible to make a transition to the boundmolecule state (closed channel) from the initial atom states (open channel). Taken from [46]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

xiv
1.6

The oscillation in remaining condensate number is shown after the second
magnetic field pulse, the frequency of the Rabi oscillations match wellwith the energy of a bound molecule state indicating the production of acoherent superposition of atom and molecule states. Taken from [46].

1.7.

(a) Slow molecule filter experimental set-up. (b) Quadrupolar electricfield contour plot. Taken from [103]. . . . . . . . . . . . . . . . . . . . .

2.1

15

16

Schematic energy level structure of a molecule. The vibrational and rotational structure is drawn in the internuclear (Morse) potential for aground (X) and excited (A) electronic state. Labels to the right of thepotential curves are the rotational quantum number, R. . . . . . . . . .

19

2.2

Hund’s case (a) angular momentum coupling scheme for OH. . . . . . .

22

2.3

Mixing of the Hund’s case (a) basis. At low J, b(J) is nearly 1 and a(J)is nearly 0 making it possible to label the states in the Hund’s case (a)basis.

2.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25(a) Basic OH energy structure with both Λ- and ρ-doubling suppressed
for clarity. The relative energy scale is given on the left. (b) Energy
structure of 2 Π3/2 ro-vibrational ground state of OH.

2.5. . . . . . . . . .

26(a) The Stark energy shift for ground state OH molecules in low electricfields. The initial e (f) state is shown in the lower (upper) panel, where
the hyperfine levels are indicated next to the traces. (b) The Stark shift
for ground state OH molecules in high electric fields. The vertical dottedline indicates the regime where the Stark decelerator operates; the dashedhorizontal line denotes zero energy.

2.6. . . . . . . . . . . . . . . . . . . .

28(a) The Zeeman energy shift for ground state OH molecules in low magnetic fields. The initial e (f) state is shown in the lower (upper) panel,
where the hyperfine levels are indicated next to the traces.

. . . . . . .

30

xv
2.7

The absolute value of the of the OH dipole moment for the |2 ± 2±i state. 32

2.8Structure of formaldehyde (H2 CO). . . . . . . . . . . . . . . . . . . . . .

2.9

Stark shifts of the low-lying rotational states of H2 CO. States of interest
for Stark deceleration and trapping are shown in bold. . . . . . . . . . .

3.1

33

36

Gated photomultiplier schematic. The voltages on dynodes 1, 3, and 5
are reverse biased by -20 V shutting off the electron cascade. . . . . . .

39

4.1

Schematic of skimmed, supersonic molecular beam.

43

4.2

Diagram of the experimental apparatus and discharge assembly (not to. . . . . . . . . . .

scale). The system consists of two chambers individually pumped by
300 L/s turbo pumps. A differential pressure is maintained between thechambers by a mechanical skimmer. A discharge assembly is mounted
directly onto a pulsed current loop actuated valve in the source chamber. The discharge assembly consists of two disc electrodes separated byinsulating spacers. The second chamber contains an electric hexapole.
Molecule detection, by laser induced fluorescence, takes place in two regions marked by black circles. . . . . . . . . . . . . . . . . . . . . . . . .

49

xvi
4.3

(a) Longitudinal time-of-flight (TOF) profiles, acquired in the detection
region before the skimmer, for a pulsed discharge and a DC discharge at
1.4 kV and 1.9 kV. (b) TOF profiles, acquired in the detection region
before the skimmer, as a function of the time from the filament-assisted
discharge ignition to the LIF detection. The discharge duration is 2 µs for
all the data. The discharge ignition time, mean longitudinal packet speed,
and velocity width are listed in the legend for each trace. The velocity
width, ∆v, is the full-width half-max of the velocity distribution, which
is determine from the TOF profiles taken at both detection locations.
(c) Mean longitudinal speed (solid circles) and relative peak height (opencircles) of the TOF profiles as a function of time from the valve trigger
to the discharge ignition. For all traces, the lines serve as visual guides.

4.4

52

(a) TOF profiles, acquired in the detection region before the skimmer,
for several discharge voltages. The discharge pulse duration is 2 µs. The
solid lines are visual guides. (b) Rotational temperatures for different
discharge voltages. The number associated with each point is the fractionof the molecules in the lowest rotational state (J = 3/2).

4.5. . . . . . . .

55

Schematic of Piezo-electric Transducer Actuated Valve. The plunger isretracted by the PZT (∼200 µm travel) allowing a gas pulse (200 µs
typical) to escape from the nozzle. The Xe/H2 O mixture then experiencesa pulsed discharge. The discharge nozzle construction features a 40◦
opening angle, which produces a much more collimated beam. . . . . . .

4.6

4.7

59

Schematic of reliable, inexpensive high voltage push-pull switch. The
switch can operate reliably up to 1.5 to 2 kV. . . . . . . . . . . . . . . .

60

Formaldehyde cracking apparatus. . . . . . . . . . . . . . . . . . . . . .

62

xvii
4.8

Hexapole focusing curve for different OH molecular velocities. The symbols represent the data points for molecules with velocities of 350 m/s(squares), 385 m/s (diamonds), and 415 m/s (triangles), while the solidlines represent the corresponding simulation result. The inset depicts thecontribution of the 2 Π3/2 F = 2, |mF | = 2, 1 states and the 2 Π3/2 F =2, |mF | = 0 and 2 Π3/2 F = 1, |mF | = 0, 1 states to the observed signals
for the 385 m/s trace. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.9

Detection of a Gaussian molecular beam pulse at a position, L, with adetection window of width ∆L. . . . . . . . . . . . . . . . . . . . . . . .

5.1

66

69

System schematic of Stark decelerator, displaying the pulsed valve, themolecular beam skimmer, the electric hexapole, and the electrode stages.
The electrode stages alternate orientation (verticalto-horizontal) as shownin the figure to provide transverse guidance of the decelerated molecules.

5.2

72

Phase stable operation of the decelerator. (a) Longitudinal Stark energy
potentials generated by the two sets of electrodes, where the solid (dotted)
line represents the potential from the active (grounded) set of electrodes.
(b) Kinetic energy loss per stage (∆KE) experienced by molecules fromswitching between the two potentials given above. Solid (dashed) line
corresponds to a numerical calculation (sine function fit). . . . . . . . .

5.3

74

Phase stable area (separatrix area) versus phase angle for φo = 0◦ , 30◦ ,and 60◦ . As the phase angle is increased, the stable area is reduced;
equivalently, the stable longitudinal velocity width narrows and assuming
the source completely populates the phase stable areas, decreases thenumbers of molecules in the phase stable packet. . . . . . . . . . . . . .

76

xviii
5.4

Phase stable area (separatrix area) A versus slowing phase angles φo ,normalized to the phase stable area under bunching operation Ao . The
subsequent maximum allowed velocity spread of the phase stable packet∆vmax is shown with respect to the right axis.

5.5. . . . . . . . . . . . . .

Electron microscope pictures showing the effect of material and polishingchoice on surface quality. . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6

78

84

Pictures of Stark decelerator. (a) Hexapole mount. (b) Decelerator array.
In both pictures notice that in the Macor mounts grooves have been cut
to increase the surface path between oppositely charged electrodes. . . .

85

5.7

Schematic of fiber-optic isolated ammeter.

88

5.8

Typical current conditioning data. After each increase in voltage the ob-. . . . . . . . . . . . . . . .

served current is usually quite ‘noisy’ reflecting the quasi-random natureof the micro-discharges. However, after sometime the micro-discharges
burn away the emitter sites and the current decreases to a steady, lower
value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9

89

Current vs. Voltage curve after conditioning.(a) The exponential dependence of the current on the voltage is apparent, confirming the tunnelingtheory of Fowler and Nordheim. (b) Plotting the ln(I/V 2 ) versus 1/V
reveals a line whose slope is proportional to the electrode smoothness. .

91

5.10 Schematic of decelerator switches. The high voltage push-pull MOSFET
switches are shown schematically. The protection resistors RL (33 Ω) are
technically not needed if REM I (1kΩ) is in place, however, they provide
protection in the case of a breakdown that bypasses REM I . The buffer
capacitors, CB , provide the high current necessary to quickly energize
the electrodes.

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

94

xix
5.11 OH LIF TOF spectra under bunching (φo = 0◦ ) decelerator operation(a) before the 23rd stage, (b) before the 51st stage, and (d) within theelectrostatic trap region. The data points are drawn as squares, circles,
and triangles while the simulation results are shown as solid lines, with avertical offset for viewing clarity. Panel (d) shows both experimental andsimulation results on bunched peak areas, normalized to the 23rd stage;
the reduction in peak height with position is due to longitudinally phase
unstable molecules as well molecules with radial velocities ≥ 5 m/s being
lost from the packet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

5.12 OH packet of molecules in front of the 23rd stage under (a) transverse
guidance, (b) φo = 0◦ [385 m/s], (c) φo = 20◦ [365 m/s], (d) φo = 40◦ [344m/s], (e) φo = 60◦ [323 m/s], (f) φo = 67◦ [317 m/s], (g) φo = 80◦ [309m/s], and (h) φo = -45◦ [424 m/s], where the number in square brackets
indicates the mean velocity of the phase stable packet. At this spatial
location, under all operating conditions the spectra are dominated bysignals from still present phase unstable molecules. . . . . . . . . . . . .

98

5.13 3-D Monte Carlo simulation of the OH packet in front of the 23rd stage
under the bunching and slowing (60◦ ) conditions. Molecules are shownin longitudinal phase space as dots. Projections along the longitudinal
spatial and velocity coordinates are both shown. . . . . . . . . . . . . .

99

5.14 ToF LIF measurements of OH molecules before the 51st stage under (a)transverse guidance, (b) φo = 0◦ [385 m/s], (c) φo = 20◦ [337 m/s], (d) φo= 40◦ [283 m/s], (e) φo = 60◦ [220 m/s], (f) φo = 67◦ [198 m/s], (g) φo =80◦ [168], and (h) φo = -45◦ [470], where the number in square brackets
indicates the mean velocity of the phase stable packet. . . . . . . . . . . 102

xx
5.15 3D Monte Carlo simulation of the OH packet in front of the 51st stage
under the bunching and slowing (60◦ ) conditions. Molecules are shownin longitudinal phase space as dots. Projections along the longitudinal
spatial and velocity coordinates are both shown. . . . . . . . . . . . . . 104
5.16 Observations of OH molecules at the decelerator exit under (a) transverse
guidance, (b) bunching (φo = 0◦ , v = 385 m/s), (c) slowing (φo = 45◦ , v= 210 m/s), and (d) acceleration (φo = -67◦ , v = 541 m/s). Monte Carlosimulation traces are plotted above the data. . . . . . . . . . . . . . . . 105
5.17 Observations at the decelerator exit: numbers of molecules in the phasestable packet N relative to the number in the bunched packet No , forvarying slowing and accelerating phase angles. Data points (solid circles) and three-dimensional Monte Carlo simulations (open diamonds)
are shown. For comparison, theory traces generated from the simple
analytic expression (Eq. 5.5, solid line) and the numerical calculation(dotted line) for stable areas in phase space are shown. . . . . . . . . . . 107
5.18 Slowed, accelerated, and bunched molecular packages at the decelerator
exit under varying phase angles. Here, the downward-pointing arrow indicates solely the position of the bunched (trace (e) φo = 0◦ ) packet,
as all observed peaks are the result of phase-stable packets. Molecularpeaks arriving earlier in time result from accelerated OH packets withcorresponding negative phase-angles as denoted in the figure. Peaks arriving later in time are similarly the result of slowed molecules, generated
by using positive phase-angles. . . . . . . . . . . . . . . . . . . . . . . . 108
5.19 Schematic of second-generation Stark decelerator, showing increase sourcechamber volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

xxi
5.20 OH ToF Data (offset) taken at the exit of the second-generation Stark
decelerator. The results of 3D Monte Carlo simulations are also plotted.
All simulations are normalized by a numerical factor, which matches thesimulations results to the bunching pedestal.

. . . . . . . . . . . . . . . 112

5.21 Molecule Number Discrepancy between simulation and experiment. . . . 114
5.22 Simulations of the phase-stable molecule number as a function of stage
number in our second-generation decelerator. Note the dramatic decreasein number in the last several stages for φo = 30.45◦ . This decrease is due
to transverse over-focussing of these slow (10 m/s) molecules.
6.1. . . . . 117

Schematic of the Stark decelerator, displaying the pulsed valve and discharge plates, the molecular beam skimmer, the electric hexapole, andthe electrode stages. The electrode stages alternate orientation (verticalto-horizontal) as shown in the figure. The spatial locations indicated byarrows correspond to the locations where molecule number is measured
to determine decelerator efficiency. . . . . . . . . . . . . . . . . . . . . . 122

6.2

Phase stable operation of the decelerator. (a) Longitudinal Stark energy
potentials generated by two sets of electrodes, where the solid line represents the potential from one set of electrodes and the dotted line from theother set. Electrode pairs are centered at φ0 = -90, 90, 270. (b) Kinetic
energy loss per stage (∆KE) experienced by molecules due to switching
between the two potentials shown in (a). The dashed line corresponds toa sine function approximation of the potential energy loss. The shaded
region corresponds to the phase stable area for deceleration φ0 = 45◦ . . 123

xxii
6.3

Phase stable area (separatrix area) versus phase angle for φ0 = 0◦ , 30◦ ,and 60◦ . As the phase angle is increased, the stable area is reduced;
equivalently, the stable longitudinal velocity width narrows, and assuming the source completely populates the phase stable region, decreases
the number of molecules in the stable packet.

6.4. . . . . . . . . . . . . . 126

Longitudinal bucket loading. Panel (a) represents time-of-flight measurements taken at the decelerator input (location (a) in Fig. ??) for different
molecular beam operating conditions. Filled circles (filled squares) represent molecular beams with a mean speed of 415 m/s (350 m/s) and afull width at half maximum spread of 90 m/s (80 m/s). The solid line is
an idealized molecular beam of 300 m/s mean speed with a 10% velocity
spread. Panels (b), (c), and (d) represent the longitudinal phase spaceat the beginning of the Stark deceleration as derived from Monte Carlosimulations. In panel (b) the dashed separatrix represents bunching for415 m/s, while the larger solid line is the separatrix for bunching at 370
m/s. The separatrix for slowing a packet of molecules to rest is shown
inside the 370 m/s bunching separatrix. In panel (c) separatrix are shownfor bunching at 325 m/s and slowing to rest molecules from the 350 m/s
distribution, while panel (d) shows bunching at 300 m/s and slowing torest for the hypothetical molecular beam. . . . . . . . . . . . . . . . . . 127

6.5

Simple model for longitudinal bucket loading. Panel (a) shows the effectof varying the supersonic beam’s velocity spread on longitudinal efficiency
for several initial speeds, while panel (b) shows the longitudinal efficiency
as a function of initial pulse speed for several velocity spreads, where thespread is expressed as a fraction of the mean speed.

. . . . . . . . . . . 131

xxiii
6.6

Decelerator optimization condition. The optimum detuning of the design
speed from the pulse’s mean speed as a function of desired final speed
shown for three different supersonic beam distributions. . . . . . . . . . 134

6.7

Molecular packets at the decelerator exit (location (b) in Fig. 6.1) for
varying phase angles. Peaks arriving later in time are the result of slowed
molecules. From comparison of the number of molecules in these packets
to the total inputted molecular number, decelerator efficiency can be
determined. Solid lines are to guide the eye. . . . . . . . . . . . . . . . . 135

6.8

The stable molecule number normalized to φ0 = 0◦ for three cases ofoperation. Circles represent operation with a supersonic beam centeredat 415 m/s with a 90 m/s FWHM and a deceleration sequence designed formolecules at 415 m/s. Squares represent a deceleration sequence designed
for 370 m/s molecules from the same beam distribution. Triangles are fordeceleration designed for 325 m/s molecules from a distribution centeredat 350 m/s with an 80 m/s spread. Filled points represent data, while
open points represent results from the Monte Carlo simulation. The solidlines are the results of the model of Section 6.2. Typical molecule numbers
per stable packet range from 104 -106 , depending on the φ0 . . . . . . . . 137

6.9

Stark deceleration efficiency. Efficiency of the Stark decelerator as a function of final speed for slowing designed for an initial speed of 370 m/s
taken from the distribution centered at 415 m/s with a 90 m/s FWHM.
Filled squares are data points obtained from the results of Fig. 6.8 normalized by the results of trace (a) in Fig. 6.4. The solid line is the modelof Section 6.2 including both longitudinal and transverse efficiencies. . . 139

xxiv
7.1

Monte Carlo results for the Longitudinal phase space of bunched molecules.
The left column shows φo = 0◦ for S = 1 at several decelerator lengths,
while the right column shows φo = 0◦ for S = 3 for the same decelerator lengths. In the bottom left of each S = 3 figure is the value for themolecule gain S = 3 produces over S = 1. . . . . . . . . . . . . . . . . . 144

7.2

Deceleration Schemes. (a) Potential energy shift of polar molecules in
the Stark decelerator. The dotted (blue) curves show the potential energy shift when the horizontal electrodes are energized, while the dashed(red) curves show the potential energy shift when the vertical electrodes
are energized. Deceleration proceeds by switching between which electrodes are energized, and thus the corresponding potential energy curve.
In panels (a)-(c) the thick black line indicates which of the potentials is
experienced by the molecules. The (orange) circles indicates a switching
event. (b) Traditional (S = 1) operation at φo = 0◦ . For phase stability
the switching always occurs when the molecules are on an upward slope,and as such the molecules are never directly between an energized electrode. Thus, the maximum transverse guiding is never realized. (c) First
overtone operation (S = 3) at φo = 0◦ . By switching at 1/3 the S = 1
rate, the molecules are allowed to fly directly between an energized stage,
and thus, experience enhanced transverse guiding. (d) Optimized first
overtone operation (S = 3+ ) at φo = 0◦ . Operation proceeds identicallyto S = 3, except that the grounded set of electrodes is energized whenthe molecules are directly between the already energized electrodes. In
this way the molecules experience one more stage of maximum transverse
guiding for each slowing stage. Note that the grounded-set of electrodesis switched on when the molecules are directly between the energizedelectrodes to minimize the un-bunching effect.

. . . . . . . . . . . . . . 146

xxv
7.3

ToF data (offset) of OH molecules decelerated using S = 3. The resultsof 3D Monte Carlo simulations are also shown (red). . . . . . . . . . . . 148

7.4

Comparison of deceleration using S = 3 and S = 1. (a) Decelerated OH
packets. (b) De-convolved, integrated molecule number in the packets
shown in panel (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.5

Decelerated molecule number using S = 3 and φo = 20◦ for increasing
decelerator stage number. The number next to each data point is thenumber of stages used. Because of transverse over-focussing essentiallyno molecules survive below 100 m/s.

7.6

. . . . . . . . . . . . . . . . . . . 150(a) Decelerated OH packets produced using the S = 3+ modified overtone. Also, shown for comparison is the bunching packet for operation
at S = 1. (b) The de-convolved, integrated molecule number observed at
the decelerator exit for S = 3+, S = 3, and S = 1. . . . . . . . . . . . . 152

7.7

Decelerated molecule number using S = 3+ and φo = 20◦ for increasing
decelerator stage number. The number next to each data point is the final
velocity. Because of transverse over-focussing, essentially no molecules
survive below 100 m/s.

7.8

. . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Quadrupole guiding decelerator. (a) Schematic of QGD. (b) Electric fieldof quadrupole guiding stage energized to ±12.5 kV. (c) Switching scheme
for deceleration with the QGD. . . . . . . . . . . . . . . . . . . . . . . . 155

7.9(a) The Stark shift of an OH molecule in the |2 ± 2, +i state inside the
QGD. The solid curve is the Stark shift due to the slowing electrodes,
while the dashed curve is the Stark shift due to the electrodes that will beenergized at the switching time. (b) The change in the molecule’s kinetic
energy as a function of position is shown (solid) as well as a fit of Eq.
7.1, including up to n = 3. (dashed). This graph is calculated from thesubtraction of the two curves in panel (a).

. . . . . . . . . . . . . . . . 157

xxvi
7.10 The left column is stable molecule phase space of molecules decelerated
inside the QGD. The solid line is the separatrix predicted by the theoryof this section, while the points represent positions of molecule in the 3D
Monte Carlo simulations. The right column shows the ToF spectra of
OH molecules in the |2 ± 2, +i state at the exit of this decelerator whichhas 143 deceleration stage (286 stages in all). . . . . . . . . . . . . . . . 159
7.11 (a) The energy removed by deceleration per stage as a function of phaseangle for traditional deceleration and deceleration with a QGD. Bothcurves are calculated for OH in the |2 ± 2, +i state. (b) The phasestable area for deceleration as a function of phase angle for traditional
deceleration and deceleration with a QGD is plotted on the left axis,
while the gain of the QGD over traditional deceleration is plotted on theright axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.12 Comparison of the decelerated molecule number using traditional deceleration and deceleration using a QGD for increasing decelerator phaseangle. For both curves 143 stages of deceleration were used. The number
next to each data point is the phase angle used. Note that for operationof the QGD with ±2.2 kV on the guiding stages, essentially no molecules
survive below 100 m/s because of transverse over-focussing. If the voltage
on the guiding stages is lowered as the molecules are decelerated this effect is removed and the QGD supplies roughly 2 times as many molecules
as a traditional decelerator. . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.13 Optimization of the quadrupole guiding voltage for operation at φo = 0◦ . 163

xxvii7.14 (a)Simulations of the phase-stable molecule number as a function of stage
number in the QGD with the guiding voltages scaled as 1/v 2 . The traces
shown are (a) φo = 0◦ , (b) φo = 15◦ , (c) φo = 30◦ , (d) φo = 45◦ , (e)φo = 50◦ , and (f) φo = 53◦ . Note the dramatic decrease in number inthe last several stages for φo = 53◦ . This decrease is due to transverse
over-focussing of these slow molecules, which is still present, presumably
due to the over-focussing the decelerator electrodes. (b)Comparison of
deceleration at φo = 53◦ with 1/v 2 (f) and 1/v 0.875 (g) scaling of thequadrupole guiding voltage. Both panels are normalized to the φo = 0◦case.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.15 Decelerated molecule number using the QGD for increasing decelerator
stage number. The number next to each data point is the number ofdeceleration stages used. Because of transverse over-focussing there is a
sharp loss of molecules for the slowest speeds.
8.1. . . . . . . . . . . . . . 167(a) (Color online) H2 CO. (b-d) Stark shifts of the low-lying rotational
states of H2 CO. States of interest for this work are shown in bold. . . . 171

8.2

Schematic lowest-lying adiabatic potential curve for the scattering model.
For intermolecular distances R ≫ Rb , the ordinary dipole-dipole interaction holds, and the molecules attract head-to-tail. As the molecules
approach a potential barrier at R = Rb , the potential deviates from dipolar behavior, and the sign of V2 is switched so that, by R < Rb , themolecules attract in a tail-to-tail configuration to mimic chemical reaction described below. Also for R < Rb , the scattering channel is coupled
to an artificial channel (not shown), which represents chemical reactivity. 173

xxviii
8.3

Elastic (dashed) and spin-changing inelastic (solid) cross sections for cold
collisions of spin-stretched OH and H2 CO molecules. These cross sections
were computed at a collision energy of 1 mK. . . . . . . . . . . . . . . . 175

8.4

Calculated chemical reaction cross sections at a collision energy of 1 mK,
versus electric field, using the model in the text. . . . . . . . . . . . . . 177

8.5(Color online) The reaction mechanism for the H-abstraction channel
following Ref [48]. The molecules initially attract in the head-to-tail configuration forming the hydrogen bonded complex (HBC). The chemical
reaction proceeds via the transition state (TS) to the final reaction products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

8.6

ToF data for decelerated H2 CO, showing bunching and deceleration off
the main pulse (offset down for clarity) for both (a) the current-loop and(c) solenoid valves. Stark decelerated packets of H2 CO at the output ofthe decelerator for both the (b) current-loop and (d) solenoid valves. . . 181

8.7

Comparison of decelerated packet molecule number to the expected for(a) the current-loop valve and (b) the solenoid valve. . . . . . . . . . . . 183

9.1(Color online) a) OH ground Λ-doublet state. The arrows represent theeffect of the applied microwave pulses for the 2→2 (solid arrows) and 1→1(dotted arrows) transitions. b) Schematic of experiment (inset depicts
detection region).

9.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188(Color online) a) Rabi-flopping as a function of time. The contributionsof the individual magnetic sub-levels are shown near the bottom, while
the sum is shown as the dashed line. b) Rabi-flopping as a function ofvelocity for a fixed (spatial) length microwave pulse. The dashed line isthe expected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

xxix
9.3

Representative line shape of the 2→2 (a) and 1→1 (b) transitions. Bothmeasurements correspond to an interaction time of 0.5 ms (v = 200 m/s).
A center frequency is extracted from each fit (solid line).

9.4

Results of multiple measurements of the 2→2 transition (upper) and the
1→1 transition (lower).

9.5. . . . . . . . 193. . . . . . . . . . . . . . . . . . . . . . . . . . . 196

Ramsey spectroscopy for the 2→2 transition with 0.2 ms pulse separationtime.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

10.1 The Born and Eikonal approximations for OH-OH dipole-dipole collisions, in an external electric field high enough to saturate the OH dipole.
The Born approximation is valid to the left of the blue (left) verticalline, while the Eikonal approximation, strictly speaking, is valid to theright of the red (right) vertical line. Comparison, with more careful calculations show the predictions of the Eikonal approximation are actuallyquite good down to the temperature at which the cross-section of theBorn and Eikonal approximation are comparable. . . . . . . . . . . . . . 203
10.2 OH-OH cross-sections as a function of electric field for several temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
10.3 Schematic of the OH-Rb collision interaction.
10.4 OH-Rb collision cross-section versus C6 .

. . . . . . . . . . . . . . 205. . . . . . . . . . . . . . . . . 206

10.5 Simple model for describing the features of para-electric cooling. . . . . 208
10.6 (a) Entropy per unit mole of a sample of polar molecules. (b) Polarization
per unit mole of a sample of polar molecules. . . . . . . . . . . . . . . . 210
10.7 (a) Schematic of cavity cooling set-up. (b) Rayleigh scattered spectra ofmolecules with a finite temperature. (c) The Rayleigh scattered spectraof (b) in the presence of a high finesse optical cavity. . . . . . . . . . . . 214

xxx
10.8 3-level atom system. By letting state c have the properties of Raman
accessible final states, this simple model sufficiently represents a realmolecule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
10.9 (a) Capacitive detector assembly. (b) Results of a finite-element analysisof the deformation of the parallel plate capacitor for 5mG accelerationin the −x̂ direction. For this analysis, the detector assembly is held atfour points along its midplane. This mounting scheme reduces relative
vibration between the plates, since most of the acceleration results in acommon-mode movement of the plates. . . . . . . . . . . . . . . . . . . . 219
10.10The induced charge on a parallel plate capacitor spaced by 1 cm. . . . . 221
10.11Vibration spectrum of second-generation Stark decelerator. The vibration due to the turbo-molecular pumps is clearly visible, as is a harmonicof the 60 Hz line frequency. . . . . . . . . . . . . . . . . . . . . . . . . . 223

Chapter 1

Introduction to Cold Molecules

The study of cold and ultra-cold atoms has led to some of the most interesting
discoveries in the last quarter of the twentieth century. From the onset of the use of laser
cooling and magneto-optical trapping, the field has produced many of the most precise
and fascinating results in all of physics, culminating in the production of degenerate Boseand Fermi gases. Thus, it is not surprising that a new effort has begun to extend thesestudies to molecules, whose rich ro-vibrational spectra and lack of spherical symmetry
promise even more scientific reward. At present cold molecule research is poised to
revolutionize the face of molecular physics, as cold samples of molecules are becoming
available for study.
The topic of this dissertation is work which produced the first cold samples ofhydroxyl radicals (OH) and formaldehyde molecules (H2 CO) for study by Stark deceleration of molecular beams. In this thesis, the motivation for studying cold molecules
is detailed followed by a quick summary of the other successful techniques for producing cold polar molecules. Next, a brief introduction to molecular structure is given
before the technique of Stark deceleration is discussed. The remainder of this work details our results for production of cold polar molecules, culminating in a high-resolution
spectroscopy measurement of the Λ-doublet splitting in OH, which has important implications for constraining the fundamental constants. The final chapter contains outlines
of several ideas for future experiments that we have had over the course of this work.

2
It is the aim of this thesis to be accessible to the atomic physicist, and as such,the first four chapters provide an introduction to many of the ideas and techniquesof molecular physics. For those already familiar with these ideas, the remaining six
chapters describe our work on Stark deceleration, though chapters 3 and 4 may still
deserve attention since we have implemented several improvements to the standard
molecular detection and beam techniques.

1.1

Motivation
There are many compelling reasons for producing cold samples of polar molecules,

ranging from high-resolution spectroscopy and molecular clocks to long-range, collective
quantum effects and ultra-cold chemistry to the possibility for quantum computation.
The underpinning theme of nearly all these interesting areas of study is that they are
facilitated by the anisotropic electric dipole-dipole interaction present in polar molecules,
namely:13(µ1 · r)(µ2 · r)
µ1 · µ2 −,Vdd =34πǫo rr2(1.1)

where µi is the electric dipole moment of the colliding molecules and r is the vector
connecting the two molecules. This interaction is fundamentally different from the typesof interaction studied in ultra-cold atom research, where the interaction is isotropic andcomparatively short-ranged.
Theoretical work, pioneered by the group of John Bohn [18, 19, 122], has predicted new collision dynamics specific to polar molecules. Representative results of theircalculations are shown in Fig.1.1, where the total scattering rate for OH molecules in
the weak-field-seeking state is shown. The hallmark features of cold polar molecules
collisions are evident in this graph; the collision rate rises by nearly three orders ofmagnitude and then undergoes several large modulations with only a modest change inelectric field. Because the dipole-dipole interaction is anisotropic, molecules can either

3
10-8

K (cm3/sec)

T = 1µK

10-9

10 µK
100 µK

10-10
10-11
1

1 mK

2
3
4
5
3
Electric field (10 V/cm)

6

Figure 1.1: Total collision rate of weak-field-seeking OH molecules as a function ofelectric field. Courtesy of John Bohn.

repel or attract each other during a collision depending on their relative orientation.
Furthermore, the strength of the interaction is directly proportional to the dipole moment of the colliding molecules, and since the dipole moment is a function of the electricfield, the interaction is electric field dependent. Thus, the salient features of Fig. 1.1
can be understood as due to bound states forming at threshold, i.e. scattering resonances, in the attractive part of the inter-molecular potential curve as the strength of
the interaction increases with electric field.
Though currently there has been no experimental observation of cold polar molecule
collisions, there is considerable effort towards this goal and it will soon be realized. The
first detailed experiments will be extremely important since little is known about the
short-range structure of the inter-molecular potential curves and a better understanding
would afford more precise knowledge of chemical reaction dynamics, a necessary step
towards control of chemical reactions at the quantum level.
Other, possibly monumental, experiments are the investigation of the weak interaction in chiral molecules [106, 77] and the search for an electric dipole moment of
the electron [74]. A chiral (or handed) molecule is one that is not superposable on its
mirror image. A chiral molecule and its mirror image are classified as enantiomers.

4
While the usual physical properties of enantiomers such as density, melting point, andboiling point are identical within experimental error, the enantiomers of a given chiralmolecule can exhibit noticeable differences. For example, nature seems to prefer one
handedness to the other, e.g. all amino acids are left-handed chiral molecules. The
question then naturally arises as to why one mirror image is preferred over another.
The parity violating weak interaction could produce an energy difference between thetwo enantiomers leading to the selection observed in nature [106, 77]. With a cold
sample of chiral molecules, experiments may become sensitive enough to see this energy
difference (up to present such experiments have failed [40]). The existence of an electron
permanent electric dipole moment (EDM) would be equally interesting. If an electron
possesses an EDM this leads to a time reversal symmetry violation [74]. It is necessarythat any electron EDM would be fixed relative to the electron’s spin axis. Otherwise,
a separate quantum number is needed and twice as many electrons as observed could
be stored in each atomic shell. Thus, in an external electric and magnetic field, theEDM is manifested as an energy shift equal to the electric dipole moment multiplied by
the local electric field value. Preliminary work indicates that experiments with polar
molecules can be far more sensitive than current atom experiments, due to the larger
internal electric fields and ease of orientation in the lab frame [105, 63]. With a cold
sample of polar molecules, the sensitivity of the measurement can be enhanced and theresults used to filter the appropriate theories [49].
With the promise of enormous gain with molecule experiments and the considerable success of atom cooling and trapping, one may wonder why so little work has
been focused on creating cold molecular samples. The reason for minimal development
in molecule cooling is simply that molecules are a much more complicated system, andthus harder to control than atoms. Simply put, the techniques developed for atoms
cannot be easily applied in the production of cold molecules. For example, laser cooling, the foundation of cold atom research, is not amenable to molecules because of their

5complicated energy structure. Therefore new approaches are being developed to provide
physicists with samples of cold molecules. Currently there are only four techniques thathave been realized experimentally; buffer gas loading, association of cold atoms, slow
molecule filtering, and Stark deceleration - the topic of this dissertation. The implementation of these methods (save Stark deceleration) as well as their inherent advantages,
disadvantages, and outstanding issues will be discussed in the following sections.

1.2

Buffer Gas Loading
The loading of thermal molecules into a magnetic trap via cooling with a cold

Helium buffer gas, buffer gas loading, represents the conceptually simplest of all techniques for cold molecule production. Thermal molecules are sympathetically cooled
via collisions with a Helium gas to cryogenic temperatures and subsequently trappedin a minimum of an applied magnetic field. Because the collision cross-section of themolecules with the Helium buffer gas is essentially independent of the molecular energy level pattern and generally increases with decreasing temperature, the technique isapplicable to many species and should be successful for any molecule with a sufficientmagnetic dipole to be trapped at cryogenic Helium temperatures (∼250mK) [41]. Because at these low temperatures only Helium has an appreciable vapor pressure, it isnecessary to introduce the ‘molecule-to-be-trapped’ at thermal temperatures. The subsequent decrease of the molecule’s temperature can be easily modeled by hard sphereelastic scattering of two masses. From conservation of energy and momentum with averaging over a thermal velocity distribution it is easy to show the temperature changeof the ‘molecule-to-be-trapped’ due to one collision with the buffer gas is given as [41]:∆T =(MM ol + mHe )2−(Ti − T ),κ =κ
2MM ol mHe(1.2)

with Ti = initial temperature of the molecule, T = initial buffer gas temperature, andMM ol and mHe the mass of the ‘molecule-to-be-trapped’ and the helium, respectively.

6

Figure 1.2: Typical buffer gas loading experiment set-up. Taken from [41].

Integrating for the molecule temperature as a function of the number of collisions yields:nTiTn= ( − 1)e− κ + 1,
T
T

(1.3)

where Tn is the molecule temperature after n collisions. In order for the thermalmolecules to be cooled enough to fall below the trap edge before they collide withthe chamber walls, it is necessary for the buffer gas density to be sufficient to provide
enough collisions to lower the temperature below the trap edge in a volume on the orderof the chamber size (i.e. cm scale). Ultimately, this constraint sets the minimum attainable temperature since the buffer gas density is a strong function of temperature. If weassume the molecules are introduced at 1000 K (common for laser ablation) and have atypical mass of MM ol = 50 mHe , for the molecules’ temperature to fall within 10% of
the that of the buffer gas we see from equation 1.3 that approximately 300 collisions are
required. Assuming hard sphere elastic collisions the cross-section can be estimated as
10−14 cm2 , the required buffer gas density to thermalize the ‘molecule-to-be-trapped’
over the dimensions of the system is then 3(1016 ) cm3 . From the vapor pressure of 3 He,
the minimum buffer gas temperature is then 250 mK.
Experimentally, buffer gas cooling is achieved in a system similar to the one depicted in Fig. 1.2. The two main parts of the apparatus are the super-conducting

7
magnetic field coils and the science cell where molecules are trapped. The two NbTi
super-conducting coils are arranged in an anti-Helmholtz configuration and are immersed in 4 He at 4 K and can provide a magnetic field up to 3 T deep. The coils areheld within a Titanium cask that compensates the 105 N (11 tons!) repulsive force
between them. The science cell is thermally isolated from the relatively warm 4 K wallsof the chamber and is connected to the mixing chamber of a dilution refrigerator at
the top. Its temperature can be adjusted from 100 mK up to 800 mK via resistive
heaters. Inside the science chamber is a mirror that reflects the molecule detection laser
beams, which enter through the bottom of the cell. Typically, near the mirror is a solid
precursor to the ‘molecule-to-be-trapped’. In all buffer gas loading experiments to date,the ‘molecule-to-be-trapped’ was introduced into the buffer gas environment by laser
ablation of a metal precursor. To load the magnetic trap, the science chamber is filledwith a sufficient amount of Helium (typically 1017 cm−3 at 300 mK) for the sympathetic
cooling process, and the temperature reduced to ∼100 mK. This causes the Helium tocondense to the sides of the science cell. Next, the temperature of the cell is raised
via the resistive heaters to the appropriate temperature for the required Helium number density (typically ∼250 mK), and the laser ablation pulse is fired, introducing the‘molecule-to-be-trapped’ into the science chamber. The molecules thermalize with thebuffer gas and those in the weak field seeking state are trapped in the magnetic field,
while those whose energy decrease with increasing field (strong-field seekers) are pushed
towards the science cell walls where they are lost. Simultaneously, the temperature of
the science cell is reduced to begin the cryo-pumping of the Helium buffer gas to minimize its effect on the trap lifetime of the molecules. While this technique has been used
to cool several atomic species, it has to date only been implemented on one molecularspecies: Calcium Monohydride (CaH). Laser ablating a CaH2 metal precursor, it wasobserved that the strong-field seeking molecules were expelled from the trap, while theweak field seeking molecules were pulled toward the trap center. Approximately, 108

8
CaH molecules were trapped at a temperature of 400 ± 50 mK with a density of 8(107 )cm−3 [138]. These trapped weak-field seeking molecules can be observed in the trap
for approximately 2 s and decay from the trap exponentially with a time constant of
0.5 s. Further reduction of the CaH temperature by standard evaporative cooling was
not possible because the Helium buffer gas, which could not be pumped away quickly
enough, would warm the molecules.

1.3

Cold Atom Association
The association of cold atoms to make cold molecules has had several successful

manifestations including photo-association [79, 86], Feshbach resonance association [46,117], and three-body collision combination processes [118]. Of these, three-body collision
processes resulting in the formation of cold molecules has seen the least amount ofexperimental work and will be neglected here. However, it should be noted that recent
theoretical work holds much promise for this technique and new experiments may soon
follow [30]. Historically, the work in cold atom association started with one-color photoassociation, then two-color photo-association was developed with the aim of populating
only one ground ro-vibrational state. Feshbach resonance association was the latest of
the three to see use in producing cold molecules, but its result are certainly some of
the most exciting of the entire cold molecule field. One-color photo-association of cold
atoms aims to excite two colliding atoms to an excited, bound molecular state that will
hopefully spontaneously decay to a bound, electronic ground state molecule. The basic
ideas of photo-association were already apparent in the 1960s [47] in the context of freebound spectroscopy and were evidenced early in cold atom trapping where the process
was manifested as trap loss [67]. Photo-association of alkali atoms, as seen in Fig. 1.3,
occurs when two colliding S state atoms with total energy above the dissociation limit
are excited to a bound state of the S + P potential curve by the photo-association laser
ωP . Subsequent decay from this excited state is either to unbound S state atoms with9
the emission of a photon of frequency ωF , or decay to a bound, electronic ground state
molecule with emission of a photon of frequency ωB .
To date, this technique has been demonstrated for the homonuclear molecules
Li2 [12], Na2 [79], K2 [135], Rb2 [86], Cs2 [51], Sr2 [89, 145], and Ca2 [146], and forthe heteronuclear molecules NaCs [55], KRb [134] and RbCs [72]. These ground state
molecules usually have a temperature on the order of their parent atom’s temperature(∼100 µK). The production rates of ground state molecules can be estimated by firstfinding the rate, Rpa , at which the atoms in the colliding S state are photo-associatedinto bound excited molecules, which will roughly be: Rpa = Atom Density×Laser
Intensity×Photo-association probability. Pillet et al. [96], treat this problem from a
perturbative approach by calculating the overlap of approximate initial and final state
nuclear wave functions (essentially Franck-Condon factors) with an approximate dipole
matrix element for the electronic transition. They report the maximum value of thephoto-association rate as:(2J+2)/3

N π 2 Γ2 I µJ−1/2 T J−1 C3)
Rpa ≈ (V
2 Io∆(4J+7)/3(1.4)

Where N/V is the atomic density, Γ is the atomic spontaneous emission rate, I is
the laser intensity, Io is the transition saturation intensity, µ is the reduced mass, J
is the angular momentum of excited molecular state, C3 characterizes the long-range
potential of the excited molecule state (i.e. −C3 /R3 ), and ∆ is the binding energy ofphoto-associated state. From this result it is trivial to estimate the rate of productionof ground state molecules, Rgm , as:
Rgm = Rpa

X

Pi→f ,(1.5)

fwhere Pi→f is the probability of spontaneous emission to a final bound-state f, which
depends on the usual dipole matrix element, as well as the Franck-Condon factors introduced by the nuclear overlap integrals. While typical values for Rpa can be substantial,

10

Figure 1.3: Representative potential curves for colliding alkali atoms along with thephoton frequencies involved in one-color photo-association. The laser field, ωP , excites
the colliding atoms to a bound state of the excited potential curve, which hopefully
decays by ωB to a bound, ground state molecule. Taken from [12].

11the values of Pi→f are quite small leading to small molecular formation rates [96]1 .
Because colliding atoms do not spend much time near each other the absorption of
the associating photon, ωL , must occur at large intermolecular separation and thus theFranck-Condon factors are only significant to states of high vibration. Furthermore, thesubsequent decay to the bound, electronic ground state is typically to high vibrational
levels because, again, the Franck-Condon factors are only significant to high vibrational
levels of the ground state. The Franck-Condon factors do not provide strict transition
rules, and in the case of high vibrational level transitions they are usually quite weak.
This leads to the population of many different high-lying vibrational levels and theobserved low molecule production rate (typically 103 molecules/sec).
Since both high vibrational level molecules and a distribution of population complicate most experiments, two-color photo-association has been developed to circumventthe problems inherent to one-color photo-association. In two-color photo-association asecond laser field is applied with the aim of primarily populating a single, low vibrational level in the ground state. As seen in Fig. 1.4 there are two general types of
two-color photo-association. The more common type, shown in Fig. 1.4(a), is a stimulated Raman process. Here the two atoms absorb a photon from the laser field, ω1 , andemit a photon into a second laser field, ω2 , leaving the atoms as a bound, ground state
molecule. Because (ω2 − ω1 ) is only resonant with one vibrational level in the molecular
ground state, stimulated Raman photo-association only populates a single vibrational
state. Most notably, a variation of this process has been used to produce RbCs moleculesin the absolute ground state [72]. An important variation of this method, stimulated
Raman adiabatic passage photo-association (STIRAP PA), uses a counterintuitive pulse
sequence to provide near unit transfer of the initial free atoms to molecules in a single
vibrational state. By first applying the ω2 field, the ground and excited molecule states
1

Notable exceptions to this are the homonuclear alkaline earths and heteronuclear molecules, where
for the resonant dipole-dipole interaction (C3 ) is negligible and the overlap between the excited andground vibration wavefunctions can be large.

12

Figure 1.4: Two types of two-color photo-association. In panel (a), two-color photoassociation by a stimulated Raman process is shown, which populates a single vibrational level. In panel b, two-color photo-association by a short range Rydberg state isshown, which populates several deeply bound vibrational levels in the ground electronic
state. Taken from [29].

13
are coupled before the application of the ω1 field, which can then transfer the atoms into
a coupled state that, after the removal of ω2 , is parallel to the ground molecule state[21]. Unfortunately, it has been predicted that this type of photo-association should beabsent in all cold atom samples except a condensate [82]. One drawback to the stimulated Raman process (not STIRAP PA) is that the laser fields can excite the groundstate molecule back into free atoms. To combat this loss of molecules a variant of thistwo-color photo-association was developed. As seen in Fig. 1.4b, this method relies on
the irreversible process of spontaneous emission rather than the reversible process ofstimulated emission to populate the ground molecular states. The goal of this process
is to excite the free atoms to an excited bound molecule state that is then excited to ashort range molecular Rydberg state that will subsequently, primarily (the short range
Rydberg state has appreciable Franck-Condon overlap with the lower vibrational states)
decay by spontaneous emission to the low vibrational levels of the ground state molecule[20]. Using this technique researchers have been able to produce up to 105 molecules/sec
per vibrational level [91]. This method produces deeply bound ground state molecules(v < 10), but because it relies on spontaneous emission, and the Franck-Condon factors
do not provide strict transition rules, it populates many different vibrational levels.
The most recent method, Feshbach resonance association, used to produce cold
molecules is conceptually similar to the stimulated Raman photo-association in that theinteraction between atoms are drastically altered by the coupling of specific initial andfinal quantum states by an applied field. Fig. 1.5 represents the relevant informationin Feshbach resonance scattering, which occurs when the relative kinetic energy of twocolliding atoms is close to the energy of a bound molecule state. These bound molecularstates arise from the potential energy curve of a different hyperfine configuration of thecolliding atoms. Since the different hyperfine configurations of the atoms experience
different energy shifts in a magnetic field, the tuning of an external magnetic field canbring the colliding atoms in and out of resonance with the bound molecular state. As

14seen in Fig. 1.5(a) (for the specific case of 85 Rb) when the magnetic field is tuned
towards 155 Gauss (the position of a Feshbach resonance) the energy of the colliding
atoms moves towards that of the bound molecular state, and the scattering length diverges (Fig. 1.5(b)) [46]. During a Feshbach resonance scattering event, atoms colliding
on the open channel can make a transition to the closed channel and thus cold molecules
are formed. To date the technique of Feshbach resonance association has been used to
produce ultra-cold Li2 [117], Na2 [140], K2 [104], Rb2 [46], and Cs2 [58] molecules. Furthermore, heteronuclear molecules formed with feshbach resonance association should
become available in the near future as recent experiments have observed heteronuclear
feshbach resonances between K and Rb [65] and between Li and Na [115]. Typical experiments produce 103 -105 molecules at ∼100 nK, with the higher molecule production
numbers belonging to the experiments that employ fermions due to the suppression ofmolecule decay by Fermi statistics. The most notable success of these experiments isthe creation of a coherent atom-molecule superposition [46]. In this experiment, twomagnetic field pulses, slightly detuned from the Feshbach resonance, and separated bya constant but variable magnetic field significantly detuned from the resonance, Bevolve ,were applied. The first pulse converted some of the atoms to molecules and the second
pulse transformed the molecules back into atoms. Observations of the atom number
remaining in the condensate after the second pulse revealed dramatic oscillations as afunction of the time between the two pulses as seen in Fig. 1.6. This was interpreted
as evidence of a coherent superposition of atom and molecule states. The first pulse isbelieved to provide a rapid perturbation that leads to non-adiabatic mixing of the atomand molecule states that then evolves at the energy given by the magnetic field, Bevolve .
Thus, in analogy to Ramsey’s method of separated, oscillatory fields [102], the conversion of molecules back to atoms by the second pulse exhibits remarkable dependence on
the time between the two pulses.
The future of cold atom association is indeed bright with new ideas, such as

15

Figure 1.5: The relevant information for Feshbach resonance association is shown. As
the magnetic field is tuned towards the resonance the scattering length diverges, andit becomes possible to make a transition to the bound molecule state (closed channel)
from the initial atom states (open channel). Taken from [46].

Figure 1.6: The oscillation in remaining condensate number is shown after the second
magnetic field pulse, the frequency of the Rabi oscillations match well with the energyof a bound molecule state indicating the production of a coherent superposition of atomand molecule states. Taken from [46].

16

Figure 1.7: (a) Slow molecule filter experimental set-up. (b) Quadrupolar electric field
contour plot. Taken from [103].

association with coherent population accumulation via pulse shaping of a femtosecond
laser [32] and optical Feshbach resonances [145] being developed, the field may prove tobe the method of choice for cold molecule production – though it will likely always belimited to molecules composed of atoms that can be laser cooled.

1.4

Slow Molecule Filtering
The underlying principle behind slow molecule filtering is quite similar to the op-

eration of slow molecule spectroscopy in a cavity [141], namely: Slow molecules alreadyexist in a room temperature distribution. Thus, slow, cold molecules can be obtainedby distilling them out of a room temperature distribution. This idea was already apparent as early as 1956 [101], when it was (unsuccessfully) attempted to filter slow atoms
from a hot beam using gravity. The first successful implementation of the method camein 1999 [53], when slow Li atoms were selected out of a thermal source by a curved
magnetic guide that only the slowest atoms could follow.
The realization of this technique for polar molecules is nearly identical to thatof Ref. [53], except that the Stark effect associated with polar molecules is used forthe guiding force instead of the Zeeman effect. The basic experimental set-up, alongwith a contour plot of the guiding, quadrupole electric field, is shown in Fig. 1.7. The
molecule of choice (to-date H2 CO and ND3 ) is introduced into the electric-quadrupole

17
guide through an effusive nozzle, i.e. gas inlet. The molecules are transversely guided
by the quadrupole field until they reach a bend in the guide. The bend serves to limit
the maximum longitudinal velocity since fast molecules will not be guided around thebend and consequently, exit the guide. Typical molecular fluxes at the exit of the guideare 109 molecules/s and are composed of several rotational states since the gas inlet isnear room temperature. The quadrupole guide has also been operated in an alternate
gradient manner, opening the door for production of cold strong-field seeking molecules[68]. Recently, an electrostatic trap was placed at the exit of the quadrupole guide,
which held a sample of ND3 at a density of 108 cm−3 with an average lifetime of 130
ms [107].

Chapter 2

Molecular Structure: A Primer

2.1

The Basics
Unlike an atom, whose fine structure is completely determined by the behavior

of its electrons, a molecule’s structure largely depends on the degrees of freedom of thenuclei. The ability of molecules to rotate and vibrate makes their spectra fundamentallydifferent than that of atoms. Because of these extra degrees of freedom, the internalenergy of a molecule, EM , is the sum of three parts; rotation (ER ), vibration (EV ), andelectronic (EE ):EM = ER + EV + EE .

(2.1)

The relative scale of these energies is shown in Fig. 2.1. As seen in the figure, the threeenergy contributions represent three distinct energies scales - rotation, the smallest scale,adds energies on the order of 10 cm−1 , vibration contributes energies on the order of1000 cm−1 , and the electronic energy is on the order of 10,000 cm−1 . For this reason, it
is acceptable to treat vibration as a perturbation to the electronic energy and rotation
as a perturbation to vibration.
The electronic contribution to the energy, EE , is the same as for atoms. Namely,
it is the energy of the molecule’s electrons moving in the potential created by the
nuclei and the other electrons. While this contribution is considerably more difficult
to calculate for molecules than atoms, it is conceptually no different and all of one’s
intuition from atoms can be properly applied.

19

A

2
1
0

3
2
1
0

3

Energy

2
1
0

v = 2

X

3
2
1
0

v = 1

E

3

R

2

v = 0

V

E

1
0

3
2
1
0

R
Figure 2.1: Schematic energy level structure of a molecule. The vibrational and rotational structure is drawn in the internuclear (Morse) potential for a ground (X) andexcited (A) electronic state. Labels to the right of the potential curves are the rotational
quantum number, R.

20
For low vibration and rotation, the intermolecular potential of Fig. 2.1 is sufficiently approximated as a harmonic oscillator [59]. Thus, the expected vibrational
splittings are simply:
EV = hω(v + 1/2),(2.2)

where ω is the oscillator frequency and v is the vibrational quantum number. Clearly,
as the molecule is excited in rotation and vibration this approximation breaks down, i.e.
the vibrational potential becomes substantially anharmonic. To treat this, one typically
expresses the vibrational energy of the molecule as a power series of harmonic oscillator
energies with different oscillator frequencies,EV =Xhωn (v + 1/2)n .

(2.3)

n

Retaining the first three terms of Eq. 2.3 is typically enough to be accurate within 1
cm−1 . Though the values of the ωn ’s can be calculated, they are usually derived fromspectroscopic data.
The molecular rotation of a diatomic molecule can be approximated quite well bya rigid rotor model, which describes two masses m1 and m2 separated by a fixed distance,r. Solution of the rigid rotor Schrödinger equation yields the molecular rotation energies
as:
ER =

h2
R(R + 1),8π 2 I(2.4)

where I is the molecule’s moment of inertia and R is the rotational quantum number.
The prefactor of Eq. 2.4 is called the rotational constant and is usually written as B for
diatomic molecules. For a real vibrating molecule, the rigid rotor model breaks down
since the average internuclear distance depends on v. However, by defining a value of
B for each vibrational level, i.e. Bv , it is possible to use Eq. 2.4 to predict reasonably
accurate values for the rotational energy spacing.

21

2.2

The Structure of OH
The neutral hydroxyl free radical (OH), though extremely chemically reactive, isa stable, diatomic molecule. For low rotational levels, the angular momentum couplingof the 2 Π electronic ground state of the OH molecule is sufficiently described by Hund’scase (a) [59] as seen in Fig. 2.2. Under this interaction scheme, both the electron orbital
angular momentum L and electron spin S are coupled to the intramolecular axis, leading
to the definition of the total electron angular momentum as Ω = Λ + Σ, where |Λ| =1 (|Σ| = 1/2) represents the projection of L (S) onto this axis. This coupling resultsin two spin-orbit states, where the 2 Π3/2 state lies ∼126 cm−1 below the 2 Π1/2 state
as shown in Fig. 2.4(a). Strong spin-orbit coupling (ASO = -139 cm−1 ) is responsible
for the observed large splitting between the two Ω branches. The molecular angularmomentum, J, in the laboratory frame is defined as J = L + S + R, where R represents
the nuclear rotation angular momentum. The allowed values of J in the laboratory
frame are given as J = Ω, Ω + 1, Ω + 2, . . .
The fine structure energy in Hund’s case (a) for a diatomic molecule is primarily
given by the sum of the rotational energy described by the Hamiltonian HRot and thefine-structure energy described by the Hamiltonian HF S . The rotational Hamiltonianis clearly given as~2
HRot = B R= B[(J 2 − Jz2 ) + (L2 − L2z ) + (S 2 − Sz2 ) + (L+ S − + L− S + )−(J + L− + J − L+ ) − (J + S − + J − S + )],(2.5)

where subscript-z denotes the projection onto the space-fixed axis z (see Fig. 2.2).
While the fine-structure Hamiltonian is the sum of two parts describing the spin-orbit
and the spin-rotation coupling [144]~ ·S~ + γSR R~ ·S~
HF S = ASO L

22

Figure 2.2: Hund’s case (a) angular momentum coupling scheme for OH.

23
1= ASO Lz Sz + ASO (L+ S − + L− S + )2~ − S)~ · S,~+γSR (J~ − L(2.6)

where γSR is the spin-rotation constant. The total Hamiltonian can efficiently be written
in a Hund’s case (a) basis that respects parity
1|JM |Ω|ǫi = √ [|JM |Ω|i|ΛΣi + ǫ|JM − |Ω|i| − Λ − Σi],
2

(2.7)

with the rotational part of the wavefunction given by the Wigner rotation matrix elements [31]|JM Ωi =2J + 1
8π 2

1
2

∗

J
DM
Ω (θ, φ, χ).

(2.8)

In this basis, the matrix elements of the rotational Hamiltonian are given as11
1
1
E11 (J, ǫ) = hJM ǫ|HRot |hJM ǫi = − ASO + B(J + )2
2
2
2
2
1
1
1
+ (1 − ǫ(J + ))[p + q(1 − ǫ(J + ))]
2
2
2
3
3
1
1
E22 (J, ǫ) = hJM ǫ|HRot |hJM ǫi = ASO + B((J + )2 − 2)
2
2
2
2
1
1 2
+ q((J + ) − 1)2213E12 (J, ǫ) = hJM ǫ|HRot |hJM ǫi =
2
2
r
1
1
1
− [4B + p + 2q(1 − ǫ(J + ))] (J + )2 − 1,
4
2
2

(2.9)

(2.10)(2.11)

where p and q are the Λ-doubling parameters. Diagonalizing this Hamiltonian leads toan intermediate Hund’s case, i.e. mixture of Hund’s case (a) states, with eigenstates13|Ψ− i = −b(J)|JM ǫi + a(J)|JM ǫi
2
2
1
3
+
|Ψ i = a(J)|JM ǫi + b(J)|JM ǫi22qa(J) = 4(J + 12 )2 + ABSO ( ABSO − 4) + ( ABSO − 2)qb(J) = q

24(J + 12 )2 + ABSO ( ABSO − 4)4(J + 12 )2 + ABSO ( ABSO − 4) − ( ABSO − 2)q

2 4(J + 12 )2 + ABSO ( ABSO − 4)

(2.12)(2.13)
1/2


1/2


(2.14).

(2.15)

24
In Fig. 2.3, the values for the mixing coefficients, a(J) and b(J), are shown. At low J,
b(J) is nearly 1 and a(J) is nearly 0, and thus |Ψ+ i ≈ −|JM 12 ǫi and |Ψ− i ≈ |JM 32 ǫi,which means that for low rotation OH is sufficiently described by a Hund’s case (a)basis. For this reason, it is customary to label the OH states using a Hund’s case (a)basis – though it is technically correct only for the lowest J levels where Ω is still areasonable quantum number.
The energies of these eigenstates are given from the diagonalization of the 2 × 2
matrix of Eq. 2.11 as
E ± (J, ǫ) =E11 (J, ǫ) + E22 (J, ǫ) 1 q±(E11 (J, ǫ) + E22 (J, ǫ))2 − 4(E11 (J, ǫ)E22 (J, ǫ) − E12 (J, ǫ)2 ),2
2(2.16)

where the ± is the energy of the corresponding eigenstate in Eq. 2.15. From Eq.
2.16 the energies of the rotational states of OH can be calculated and are shown in
Fig. 2.4. The relatively large energy splitting between J states is due to the small
moment of inertia of the molecule and subsequent large rotational constant, B. The
coupling between the nuclear rotation and the electron angular momentum (projectionof L along the internuclear axis) results in Λ-type splitting (Ω-doubling) of each state,denoted as f and e states or in terms of the case a symmetry index as ǫ = +1 and ǫ = -1
(the parity of a state is given as ǫ(−1)J−1/2 ). In Fig. 2.4(b), these energy separations
are exaggerated for clarity. For the most abundant isotopomer (16 OH) with an intrinsic
nuclear spin, I of 1/2, the free radical molecule is a boson. The total angular momentum
is given by F = I + J. Each Λ-doublet component is split into hyperfine states that arecharacterized by a symmetry index p that defines the state’s parity (±). The hyperfine
energy separations have also been exaggerated for clarity of presentation in Fig. 2.4(b).
The vibrational structure of OH is the familiar spectrum of a diatomic molecule,which can be calculated from Eq. 2.3 using the values shown in Tab. 2.1.
The 2 Σ+1/2 first electronic excited state of the OH molecule has an electron spin

25

Eigenstate Composition
1

Relative Strength

b(J)
0.8

0.6

0.4

0.2

a(J)

0
0

2

4

J

6

8

10

Figure 2.3: Mixing of the Hund’s case (a) basis. At low J, b(J) is nearly 1 and a(J) is
nearly 0 making it possible to label the states in the Hund’s case (a) basis.

Table 2.1: Relevant molecular constants in units of cm−1 for calculating the rovibrational spectrum of OH. Taken from Refs. [36, 83].

B
p
q
ω1
ω2
ω3

v=0
18.531041
0.235214756
3.86786223(10−2 )3737.7613-84.881297
0.54090394

v=117.820121
0.22462668
3.692540(10−2 )v=217.118694
0.21387549
3.516101(10−2 )26Figure 2.4: (a) Basic OH energy structure with both Λ- and ρ-doubling suppressed forclarity. The relative energy scale is given on the left. (b) Energy structure of 2 Π3/2
ro-vibrational ground state of OH.

27
of 1/2 but no electron angular momentum; thus, the appropriate couplings are welldescribed by Hund’s case (b) [59]. For this angular momentum configuration, the only
important interaction is between the nuclear rotation of the molecule R and the electron
spin S – so-called ρ-doubling. This coupling separates every J state that has non-zero
R into two states of the same parity and like the Λ-doubling of the Π state has been
suppressed in Fig. 2.4(a) for clarity.
Because the Λ-doublet states possess opposite parity and small energy separations, adjacent levels of the ground state OH readily mix under the interaction of themolecule’s permanent electric dipole (µ = 1.67 D) with an external electric field. Since
the electric dipole moment, µ, must lie along the intermolecular axis, its projection on
the space fixed axis z, i.e. the electric field, is given as
MJ
µΩMJ
Ω
p
,
µef f = µ cos(Ω, J) cos(J, z) = µ p=
J(J + 1)J(J + 1) J(J + 1)

(2.17)

where cos(a, b) is the is the interior angle between vectors ~a and ~b. The matrix elementsof this dipolar coupling for a linear electric field, transformed into a hyperfine basis, aregiven by [87]~ ′ I ′ |Ω′ |F ′ M ′ ǫ′ i = − µE [1 + (−1)J+J ′ ǫǫ′ ](−1)F +F ′ +MF′ +I−|Ω|+1
hJI|Ω|F MF ǫ|~µ · E|JF2p× [J][J ′ ][F ][F ′ ] 


J

1



J′

−|Ω| 0 |Ω|





F′

1

−MF′

0



 J
F 
1



 −|Ω| 0
MF

J′








|Ω| 

.

Inspection of the above equation reveals the convenient fact that only states of the same
Ω and MF are coupled. With the large spin-orbit and rotational splittings present in
OH, this means that the Stark shifts of the relevant states are accurately calculatedwith a basis containing only the lowest Λ-doublet. For example, the Stark shift of fourof the main states of interest, |2 ± 2±i (|F MF pi) (the stretched states), is given fromthe Hamiltonian:

H Stark = 0− 35 µE− 35 µE

∆Λ




,


(2.18)

28

Figure 2.5: (a) The Stark energy shift for ground state OH molecules in low electricfields. The initial e (f) state is shown in the lower (upper) panel, where the hyperfine
levels are indicated next to the traces. (b) The Stark shift for ground state OH moleculesin high electric fields. The vertical dotted line indicates the regime where the Stark
decelerator operates; the dashed horizontal line denotes zero energy.

29
written in the basis |2 ± 2−i 
.


Ψ=


|2 ± 2+i±And thus, the Stark shift of these state is EStark= ∆2Λ ±

eigenvectors are given as1±|χ i = q±( 35 µE)2 + (EStark)2(2.19)q( ∆2Λ )2 + ( 35 µE)2 and the−3±µE |2 ± 2−i + EStark|2 ± 2+i .
5(2.20)The energy of these mixed states increases (weak-field seeking states) or decreases(strong-field seeking states) as a function of the electric field strength, as shown for
the ro-vibrational ground state of OH in Fig. 2.5. The Stark energy shift evolves from
quadratic to linear when the energy change due to the applied field exceeds the Λdoublet splitting value of 0.055 cm−1 . The ground state shift is unaffected by the next
higher rotational state due to the large energy separation of ∼84 cm−1 .
In the high electric field regime, the complex energy levels depicted in Fig. 2.5(a)emerge as four distinct fine-structure families. As shown in Fig. 2.5(b), for the Stark
slowing experiment discussed here, the weak-field seeking states are most relevant, withthe 2 Π3/2 F = 2, —mF — = 2,1 states experiencing approximately three times the Stark
energy shift of the 2 Π3/2 F = 2, mF = 0 and 2 Π3/2 F = 1, mF = 0,1 states. The
horizontal dashed line corresponds to the zero energy level while the vertical dotted line
represents the peak electric field magnitude under the normal operating voltages of the
Stark decelerator. We note molecules in strong-field seeking states do not survive thefirst state-selection device, namely the hexapole acting as a molecular focusing lens, andtherefore do not contribute to any observed signals thereafter.
From the eigenstates of Eq. 2.20 we calculate the expectation value for the electric
dipole moment as a function of an electric field as±±hχ |µ|χ i =

−1
±
3
2
( 5 µE) + (EStark
)2





3
±
µE EStarkh2 ± 2 + |µ|2 ± 2−i5(Energy - 1665401840 Hz) [MHz]

30

400

|22+>

200

|21+>

|20+>

|11+>

0

|2-1+>|10+>-200|1-1+>|2-2+>-400

400

Energy [MHz]

f

e

|22->

200

|21->

|20->

|11->

0

|2-1->|10->-200|1-1->|2-2->

-400
0

50

100

150

200

250

Magnetic Field [Gauss]

Figure 2.6: (a) The Zeeman energy shift for ground state OH molecules in low magnetic
fields. The initial e (f) state is shown in the lower (upper) panel, where the hyperfine
levels are indicated next to the traces.

31






3
±
µE EStarkh2 ± 2 − |µ|2 ± 2+i5!
±(µE)EStark
18
= −
µ.
±
25 (µE)2 + (EStark)2+(2.21)

A graph of the OH dipole moment according to this equation is shown in Fig. 2.7. The
signature of a ‘permanent-electric dipole moment’ is evident; as the field is increased
from zero the dipole moment increases from zero until it reaches its asymptotic value.
At this point the dipole moment is said to be saturated, and the Stark shift becomeslinear. Since the dipole-dipole potential, which depends on µ, is the dominant interaction between colliding polar molecules this leads to electric field dependent collision
properties as detailed in Chap. 10.
Similarly, the Zeeman shift of the lower Λ-doublet in OH can be calculated byconsidering a basis that contains only the lowest Λ-doublet states. Since a Hund’s case(a) molecule’s electronic spin and orbital angular momentum are strongly coupled tothe intermolecular axis, OH posses a magnetic dipole moment along its intermolecularaxis of µBo = µb (Λ+ ge Σ) (µb is the Bohr magneton and ge = 2.002). Like the case of anelectric dipole moment this leads to a projection on the space-fixed axis, i.e. magnetic
field, ofµBo,ef f =µb (Λ + ge Σ)ΩMJ.
J(J + 1)(2.22)The matrix elements of this dipolar coupling for a linear magnetic field, transformed
into a hyperfine basis, are given by [122]~ ′ I ′ |Ω′ |F ′ M ′ ǫ′ i = − (Λ+ge Σ)B [1 + (−1)J+J ′ +2|Ω| ǫǫ′ ](−1)J+J ′ +F +F ′ −MF +I−|Ω|+hJI|Ω|F MF ǫ|~µ · B|JF2p× [J][J ′ ][F ][F ′ ] 


J

1

J′

−|Ω| 0 |Ω|






F′
MF′

1
0



 F
F 



 J′
−MF

F′
J




1 


I 

Conveniently, the magnetic dipole operator does not couple states of the opposite parity
and the upper and lower Λ-doublet have essentially the same Zeeman shift as shown in
Fig. 2.6.

.

32

OH Dipole Moment [Debye]

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 2
10

10

3

4

10

5

10

6

10

7

10

Electric Field [V/m]Figure 2.7: The absolute value of the of the OH dipole moment for the |2 ± 2±i state.

33

Figure 2.8: Structure of formaldehyde (H2 CO).

2.3

The Structure of H2 COAs shown in Fig. 2.8, H2 CO is a near-symmetric prolate top molecule, with nearly

degenerate rotations about the b- and c- axes. The dipole moment, µa , along the a-axis
causes these nearly degenerate, opposite parity states to experience a large Stark shift
with an applied electric field. Calculations of the Stark shift are quite intense since
there is no analytical expression for the zero-field energies of asymmetric top molecules.
The rotational Hamiltonian for the asymmetric rotor can be written as [56]:H = AJ2a + BJ2b + CJ2c ,(2.23)

where A, B, and C are the rotational constants for rotation about the respective axes
and J2i is the angular momentum operator for the ith axis. Since the rotational energyof a symmetric top molecule is well known [123], it is convenient to recast Eq. 2.23 into
two terms – one of which is diagonal in the symmetric top basis and one that couples
different symmetric top states:
H = 1/2(A + C)J2 + 1/2(A − C)(J2a +2B − A − C 2
Jb − J2c ).
A−C(2.24)

Thus, the problem of calculating the rotational energies of an asymmetric top moleculehas been reduced to diagonalizing Eq. 2.24 in the symmetric top molecule basis of

34
Table 2.2: Parameters for calculating the off-diagonal rotational contribution for an
asymmetric top molecule.
Principal axis
F
G
H

a

b
0

1 2B−A−C− 1)
2 ( A−C

2B−A−C
A−C

1- 12 ( 2B−A−C
+ 1)
A−C

1

c
1 2B−A−C+ 1)
2 ( A−C-1
1 2B−A−C− 1)
2 ( A−C|JKM i, where K is the projection of J onto the symmetric top molecule’s symmetryaxis. Schematically, the rotational energy and eigenstates of the assymetric top moleculein the |Jτ M i (τ = Ka − Kc ) basis may now be written as:
EJτ M

=

|Jτ M i =11
eJ M
(A + C)J(J + 1) + (A − C)E
τ
2
2

X
K

aJKτ M |JKM i,(2.25)(2.26)eJ M are found from the eigenvalues and eigenvectors of
where both the values aJKτ M and Eτthe non-diagonal part of Eq. 2.24. The matrix elements of this part of the Hamiltonian
can be found in several reviews [56, 123], and are given as:2222hJKM |J2a + 2B−A−C
A−C Jb − Jc |JKM i = F [J(J + 1) − K ] + GK

H

q

2
2
hJ(K ± 2)M |J2a + 2B−A−C
A−C Jb − Jc |JKM i =1
4 [J(J + 1) − K(K ± 1)][J(J + 1) − K(K ± 1)(K ± 2)],where the values of F, G, and H depend on the molecule’s principal axis, i.e. themolecule’s axis of symmetry in the symmetric top limit, and are given in Tab. 2.2. Asseen in Fig. 2.8, H2 CO is a type-a molecule and the appropriate values for F,G, and H
are taken from the first column of Tab. 2.2.
Once the zero-field energies and eigenstates of the asymmetric top rotor are found,i.e. once eqns. 2.25 and 2.26 are solved, the Stark effect of an electric field, E, may be
calculated from the Hamiltonian:
HStark = EXiµi Φ i ,(2.27)

35
Table 2.3: Direction cosine reduced matrix elements
Matrix ElementhJ|Φi |J ′ i
hJM |Φi |J ′ M i
hJK|Φi |J ′ Ki

J′ = J + 1√ 1

J4(J+1)(2J+1)(2J+3)p
2 (J + 1)2 − M 2p2(J + 1)2 − K 2

1
4J(4J+1)

2M
2K

J′ = J − 1√1
4J 4J 2 −1√
−2 J 2 − M 2√−2 J 2 − K 2

where µi is the dipole moment along the ith axis and Φi is the direction cosine between
the electric field direction and the ith axis. The direction cosine matrix elements are
given in the |Jτ M i basis as:
hJτ M |Φi |Jτ′ ′ M i = hJ|Φi |J ′ ihJM |Φi |J ′ M i

X
K

J′ M

aJKτ M aKτ ′ hJK|Φi |J ′ Ki,(2.28)

with the values of the reduced matrix elements given in Tab. 2.3.
For H2 CO only the a-axis supports a dipole moment, and thus the Stark effectis conceptually similar to a diatomic molecule like OH. However, unlike the OH case,where different Λ-doublet states are coupled by the electric field, different J states arecoupled in H2 CO and thus a large basis of states is required to accurately approximate
the Stark shift. The results of a calculation that included up to J = 3 (16 basis states)
are shown in Fig. 2.9. Here, the states are labeled by their zero-field identity in the|Jτ mJ i basis. Of particular interest to this work is the |11 1i state, which is the uppercomponent of the lowest J level of ortho-formaldehyde and is hence well populated inour supersonic expansion. This state experiences a large Stark shift (1.32 cm−1 at 125
kV/cm), making it excellent for Stark deceleration. There are other states accessible
via, for example, stimulated Raman adiabatic passage that offer an improved Stark
deceleration efficiency (|22 2i) or a good candidate for implementing an AC Stark trap(|00 0i) [129].

36

Figure 2.9: Stark shifts of the low-lying rotational states of H2 CO. States of interest for
Stark deceleration and trapping are shown in bold.

Chapter 3

Molecular Detection

3.1

Laser Induced Fluorescence Detection
Potentially, background-free laser induced fluorescence (LIF) is a highly sensitivetechnique for detecting small numbers of molecules. In analytical chemistry and biology,
resonance fluorescence is used extensively for efficient detection and identification of
single molecules. The sensitivity of the fluorescence detection technique depends upon
the characteristics of the decay channels of the excited states, namely that the radiative
branching ratios are favorable and quenching processes are not significant. Further, formaximum detected signal, the decay photons should be collected over as large a solid
angle as possible; in our experiment, intra-vacuum optics provide solid angle efficiencies
of 0.5-4.5%, depending on the specific spatial location. With quantum efficiencies in
the range of 10% for the cathode surfaces of modern photomultiplier tubes (PMT),
one then expects a photoelectron event for every ∼1000 fluorescent decays. The darkcurrent of the quantum detector is a few to a few tens of counts per second. However,various phenomena may degrade the signal-to-noise ratio. Even with careful baffling andshielding, the most severe limitation on the detection sensitivity is often the scatteredexcitation laser photons, which may dominate stray background light, even after careful
imaging and spatial filtering.
To combat this scattered light noise, we have developed several useful techniques.
The most substantial of these is a PMT whose gain can be suppressed by 104 in 100 ns.

38
The key to switching the PMT’s gain is to reverse-bias several of the dynodes in theelectron multiplier chain. Experimentally, we have found that for most PMTs (circularand linear geometries) reverse biasing the dynodes 1, 3, and 5 by -20 V provided the most
suppression (see Fig. 3.1). Note, that because of the tube geometry linear PMTs usuallyexhibit a larger gain suppression. Another advantage of switching a small number ofdynodes is that the capacitive load of the switches is greatly reduced, thus speeding up
the attainable switching time. Typically, the gated photo-multiplier, which is normallyin the suppressed gain state, is turned ‘on’ ∼100 ns after the excitation laser fires andremains on for 5 to 20 µs, depending on the molecule’s fluorescence lifetime. Thus, thePMT and subsequent electronics do not ‘see’ the majority of the scattered excitation
laser light. This is particularly advantageous in situations when the scattered lightnoise dominates over the LIF signal, e.g. imaging within the decelerator array which
consist of highly polished stainless steel rods. Also, because the PMT is gated ‘on’
only for a few microseconds the contribution of dark counts is completely negligible.
Another useful technique for suppressing scattered light has been to coat the inside of
the vacuum chamber with a flat-black vacuum compatible paint, originally developedfor use in satellites [2]. This paint prevents multiple reflections of the scattered light andcan decrease the stray light signal as much as an order of magnitude, while still allowing
vacuum pressures of 10−9 torr. One final useful technique, which is often over-looked in
LIF detection experiments, is the use of a ‘red’ filter. The typical LIF experiment utilizes
an optical filter whose goal is to attenuate at the excitation laser frequency as much as
possible (typically OD ≥ 5), while transmitting as much as possible at the fluorescencewavelength (typically 50%). However, when using an ultra-violet wavelength for theexcitation frequency, as is required for most molecular detection schemes, the scattered
light can cause objects in its path (windows, vacuum chamber walls, etc.) to fluoresce.
This fluorescence is almost always red-shifted and can usually passes through the filter
used to block the excitation laser, leading to substantial noise backgrounds. By placing

39

Figure 3.1: Gated photomultiplier schematic. The voltages on dynodes 1, 3, and 5 arereverse biased by -20 V shutting off the electron cascade.

40
a filter in the fluorescence collection path, which blocks frequencies to the red of the laser(but obviously not the fluorescence wavelength) this source of noise can be completely
eliminated.
Using these three techniques, along with careful imaging and spatial filtering of
the collected fluorescence, we reduce the noise signal due to scattered/stray light to anegligible level. In optimum situations, we register ∼0.03 noise photons per pulse of
the excitation laser, while a typical bunching peak, containing of order 106 molecules,
registers 10 signal photons in a single pulse.
A key benefit of utilizing fluorescence detection, besides its relative ease of implementation, is, as opposed to a spatially fixed ionization-based scheme, the ability todetect in situ molecules at multiple locations within the decelerator, greatly increasing the experimental flexibility and level of understanding. OH signals are measured
within the decelerator during the actual slowing process, giving insight into the active
longitudinal phase-space manipulation of the molecules by the pulsed, inhomogeneouselectric fields. Specifically, in the experiment we employ LIF detection to monitor themolecules after the valve, observe the focusing effects of the hexapole on the molecules,
measure the molecular packet spectra at various positions within the Stark decelerator,and finally, detect the molecules at the decelerator exit.

3.2

Hydroxyl Radical Detection
A relatively favorable branching ratio (∼70%), a large difference between the

excitation and decay wavelengths, and a reasonable quantum efficiency in fluorescence
detection make laser induced fluorescence (LIF) a versatile approach for detecting thehydroxyl radical. The excitation source is tuned resonant with the A2 Σ+
1/2 (v = 1) ←2Π
3/2 (v = 0) electronic transition at 282 nm for subsequent detection of the red-shifted2
radiative decay around 313 nm A2 Σ+
1/2 (v = 1) → Π3/2 (v = 1). The separation inwavelength allows effective optical filtering to reduce the background scattering. The

41
characteristic fluorescence 750 ns decay lifetime provides a distinctive signature of thepresence of OH molecules.
Using a beta barium borate (BBO) crystal to frequency-double a 564 nm dye laser
output (∼10 ns pulse duration) creates the ultraviolet excitation light at 282 nm withhigh peak intensity. The excitation laser is counter-propagated to the molecular pulse
traveling down the central axis of the Stark decelerator. The relatively spectrally broadlight source (∼2 GHz) encompasses any Doppler-induced frequency shifts in the OH
resonance as the molecular longitudinal velocity changes during the deceleration process.
As aforementioned, reduction of the laser scatter is paramount for optimized signal
collection. First, due to the separation of the excitation and fluorescence wavelengths,
the carefully selected photomultiplier tube (PMT) has a photocathode responsivity thatis severely reduced at the laser wavelength versus the fluorescence wavelength. Next,
an interference filter selectively inhibits the laser light (105 suppression) versus thefluorescent photons (70% transmission). We also utilize a switched PMT whose dynodevoltages are quickly arranged during the laser pulse so as to actively repel photoelectrons
liberated by the scattered UV light on the cathode material.
The OH molecular signal is measured as a function of time at a particular spatial
location in the following manner. First, immediately prior to the measurement (∼1µs), all high voltage electrodes within the vacuum chamber are grounded to avoid Stark
shifting of the OH transition frequency. Subsequently, before the molecules have had
an opportunity to exit the specific detection region, the excitation laser pulse fires,
generating the OH fluorescence signal. The detected photoelectrons from the PMT arethen either counted with a photon counter or amplified, averaged, and integrated overa 5 µs time window to fully encompass the appropriate state decay lifetime. The laser
is stepped later in time relative to when the valve opens, and the entire measurement isrepeated. In this iterative manner, a spectrum consisting of the OH signal as a functionof time is generated.

42

3.3

Formaldehyde Molecule Detection
Laser induced fluorescence detection of the formaldehyde molecule is extremely

similar to the hydroxyl radical case. The molecules are excited from the |11 1i groundstate by photons at 353 nm generated from a frequency-doubled, pulsed-dye laser (PDL)
pumped by a Nd:YAG laser to the Ae1 A2 electronically excited state with one quantum inthe ν4 out-of-plane bending vibrational mode. Approximately 40% of the excited H2 CO
decays non-radiatively [57], while the remaining molecules emit distributed fluorescencefrom 353 nm to 610 nm [109]. This fluorescence is collected, filtered, and imaged onto a
photo-multiplier tube. The output of the photomultiplier tube is sent to a multi-channel
scalar/averager, which serves as a gated photon counter. Operating the decelerator at
5 Hz and the PDL system at 10 Hz allows a “lock-in” type data-collection, such thatevery other signal is subtracted from the total photon counts. By scanning the delay
between the PDL system and the triggering of the valve, a time-of-flight (ToF) curve isgenerated. A single data point is generated by collecting photon counts for 30 s to 2400s (depending on required signal to noise).

Chapter 4

Sourcery: Supersonic Molecular Beams

Supersonic expansion in molecular beam experiments is a widely used technique.
Under proper operating conditions, rotational and vibrational temperatures are significantly lowered along with the benefit of a reduced translational velocity spread in themolecular frame. Within the present experiments, collisions during the expansion represent the only true cooling mechanism. The maximum phase-space density achievablein the experiment is determined at this stage since during the subsequent deceleration
process the phase-space distribution of the molecules undergoes conservative rotation
without any enhancement in density. Therefore, as long as the relatively large translational speed of the molecular beam can be removed by the slowing capability of the Stark
decelerator, a supersonic expansion provides a very useful initial source for creation andexperimentation of cold molecules.
Anyone who has ever felt a leaky tire, and noticed the cool area around the leak,is familiar with the basic idea of a supersonic expansion. Namely, as a gas expands from
NozzlePR

PV

SkimmerMolecular
Beam

Reservoir

Figure 4.1: Schematic of skimmed, supersonic molecular beam.

44high to low pressure, it cools. This is akin to the adiabatic expansion process in a heatengine, where the gas cools by doing work on the piston. In this case, the piston is thegas itself1 . Furthermore, because the rotational and vibrational energies will equilibratethrough collisions with this local (moving frame) temperature, they are cooled as well.
Interestingly, the pressure differential necessary for the expansion accelerates the gas as
it expands, i.e. the piston is accelerated, leading to a beam of molecules with a highmean velocity and low spread about that mean. Therefore, one might expect that as thepressure of the reservoir (see Fig. 4.1) is increased the final speed of the molecular beamincreases. While this is true at low differential pressure, once the pressure difference
reaches a critical value the molecular beam is accelerated to the local velocity of soundand can no longer respond to the local boundary conditions. Thus, the pressure at
the nozzle exit is no longer given by the pressure in the vacuum chamber, PV , but israther some fraction of the pressure in the reservoir, PR , and further increase in thereservoir pressure does not result in any increase in the beam velocity. This final beam
velocity, v∞ , can easily be approximated from conservation of energy [11]. Conservationof energy for the expanding gas takes the form:
1
N kB To = M v 2 + N kB T,
2

(4.1)

where N is the number of molecules, kB is the Boltzman constant, To is the reservoir
temperature, M is the total mass of the expanding molecules, v is the speed of themolecular beam, and T is the final temperature of the expanded gas. Dividing Eq. 4.1
by M , utilizing the ideal gas law and the definition of enthalpy we have:
ho =

v2
+ h,
2

(4.2)

where ho and h are the enthalpy per unit mass of the gas in the reservoir and after
expansion, respectively. Assuming the specific heat, Cp , is constant with temperature
1

Perhaps more precisely, the piston is the gas in front of the expanding gas

45
and using its relation with enthalpy, i.e. dh/dT = Cp, the final beam velocity is given
as:
v∞ =

q

2Cp (T − To ).

(4.3)

For an ideal gas the specific heat can be expressed as Cp = (γ/(γ − 1))(R/m), where γ
is the ratio of the specific heats of constant pressure and volume (γ = 5/3 for an idealgas), R is the universal gas constant, and m is the mass of the expanding molecule.
Thus, the final beam velocity from a supersonic expansion can be written as:
v∞ =

s

2

γ R
(T − To ).
γ−1m

(4.4)

From this equation it is evident that typical beam velocities are on the order of 103 m/s,2and depend on the mass of the expanding molecule and the reservoir temperature.
Since the kinetic energy of a molecule is quadratic in velocity it is advantageous fordeceleration experiments to start with a molecular beam speed that is as low as possible.
Clearly, this can be accomplished by lowering the reservoir temperature or choosing alarge mass. Lowering the reservoir temperature is experimentally trivial and is usually
only limited by the vapor pressure of the expanding molecules. Choosing a large mass forthe expanding gas is often not possible since the choice of molecule is usually made basedon the desired experiment. Furthermore, the kinetic energy is linearly proportional tothe mass, so that overall decrease in beam energy is not substantial. Fortunately, it
is possible to dilute the molecule of interest into a heavy carrier gas, like Xenon (131amu), which sets the speed of the expansion. In Eq. 4.4 this requires replacing m by
the average mass of the mixture, m =Pi Xi mi , where Xi and mi are the molar fraction

and mass of the ith species of the gas mixture. Typical experiments use a mixture ofheavy carrier gas to molecule-of-interest on the order of 99:1, so that the final speed isessentially that of a beam of the carrier gas.
As aforementioned, the ability of a supersonic molecular beam to provide a beam
2

The approximation To ≈ 0 is usually sufficient in Eq. 4.4

46
Table 4.1: Working formulas for Tk [1].
Molecule
He
Ne
Ar
Kr
Xe
O2
HBr
CH3 F
SF6

γ
5/3
5/3
5/3
5/3
5/3
7/5
7/5
1.278
8/6
1.094
8/6

Tk working formulasTo
Tk = 6.1(Po d)−12/11
To
Tk = 10.4(Po d)−12/11
To
Tk = 24.3 (Po d)−12/11
To
Tk = 31.2(Po d)−12/11
To
Tk = 40.8 (Po d)−12/11
To
Tk = 6.1(Po d)−0.706
To
Tk = 8.4 (Po d)−0.7061
To
Tk = 4.3(Po d)−0.509
To
Tk = 6.5 (Po d)−0.6
To
Tk = 1.5(Po d)−0.182
To
Tk = 6.7 (Po d)−0.6

of molecules with a low velocity spread is crucial for Stark deceleration experiments.
Calculating the expected temperature, Tk , is non-trivial and calculations rarely agreewith experiments. Therefore, it is more beneficial to compare to empirical formulas forthe expected final temperature, like those shown in Tab. 4.1 [1]. The predictions ofthese formulas should be taken only as rough estimates of what to expect in an actual
apparatus. Nonetheless, it is clear the longitudinal beam temperature is usually on theorder of 1 K and as such, supersonic molecular beams are a good source of molecules
for input into a Stark decelerator, which can accept spreads up to a few 100 mK.
As anyone who has ever worked with molecular beams can attest, there is an art(or perhaps more correctly, a magic) to making a good molecular beam source. Real life
complications such as pumping speed, valve opening time, nozzle construction, skimmer
location, and velocity slip [11] can completely change the pulse speed and temperature.
Furthermore, as most modern molecular beams are pulsed to accommodate higher beam
intensities, the beam characteristics can be quite different than those predicted by theabove analysis, which, strictly speaking, is valid only for a continuous beam. In thefollowing sections, the pulsed molecular beam sources used in our Stark deceleration
experiments are detailed. It is important to remember, that these sources have been

47
optimized for producing a beam suitable for input into a Stark decelerator under theconditions imposed by our vacuum system and as such, should be thought of only as astarting point for future molecular beam use.

4.1

Hydroxyl Radical Discharge Source
There are several different techniques to produce a molecular beam of OH radi-cals. The four main methods for creating OH molecules are photolysis [15, 112], radiofrequency discharge [13], DC discharge [124, 64], and chemical reactions [121]. We chose
DC discharge because, of all of the methods, it is the simplest and most cost-effective
technique. As presented below, the system we have developed fulfills the goal of producing a large sample of cold molecules with a high phase-space density. We report
several key improvements to the standard DC discharge system, including a pulsed
high-voltage discharge to reduce heating of the molecular packet and to allow for control of the mean speed of the molecular packet. Also, the introduction of a hot filament
into the source chamber allows the discharge to operate more stably and at a lower voltage, thus reducing the heating of the OH molecules during their production. Throughcontrolled application of a high voltage discharge pulse, we are able to create packets
of OH molecules at reasonable densities that vary in mean speed from 265 to 470 m/s
with full-width half-maximum (FWHM) velocity spread as low as 16%3 . The resultsof an initial test experiment to characterize our pulsed-discharge source are presented
below.
A diagram of the test apparatus and discharge assembly is shown in Fig. 4.2. The
vacuum system consists of two chambers separated by a mechanical skimmer, which
maintains a differential pressure between the chambers. During operation the source(hexapole) chamber is at a pressure of 4 ×10−4 torr (1 ×10−6 torr). A current loop
3More recent work utilizing a Piezo transducer actuated valve produces spreads of 10%, as detailed
in a later subsection

48
actuated valve, commercially available from R. M. Jordan Company Inc., operates at 510 Hz to create a gas pulse ∼100 µs long. Directly in front of the 0.5 mm diameter valve
nozzle is a set of stainless steel disc electrodes, electrically isolated from one anotheras well as from the valve body by Boron nitride spacers. The relevant dimensions areshown in Fig. 4.2. The electrode closest to the valve has a 0.5 mm diameter hole tomatch the valve nozzle. The downstream electrode has an inner diameter of 4 mm toallow the gas to expand as it travels between the electrodes. For the test experiment,
the valve nozzle is placed ∼8 cm away from the downstream wall of the vacuum chamber
to ensure carrier gas atoms scattered from the wall do not interfere with the supersonic
expansion and beam propagation.
In the second chamber, a 13 cm long electric focusing hexapole is centered along
the beam path. The hexapole is used as a tool to determine the transverse velocity
spread of the molecular beam. The hexapole is formed by six, stainless steel, cylindrically shaped rods with rounded ends. They are 3.18 mm in diameter and set at
every 60◦ at a center-to-center radius of 4.6 mm. Alternate rods are charged to equalmagnitude but opposite polarity high voltage.
The experimental procedure begins with the pulsed valve opening for ∼100 µs,
thus creating a supersonically cooled pulse of Xenon (Xe) carrier gas seeded with afew percent water. The typical backing pressure of Xe is one to three atmospheres.
Xe is used instead of a lighter noble gas because of the resulting lower mean speedof the molecular beam, which is advantageous for our Stark-decelerator application.
At a variable time after the valve opens, a high-voltage pulse is applied to the discelectrodes. The duration of the high-voltage pulse can be varied from 1 to 200 µs. A
discharge duration greater than 150 µs is considered to be essentially DC because thedischarge duration is longer than the gas pulse. The polarity of the voltage applied issuch that electrons are accelerated against the molecular beam propagation direction,
which results in a more stable discharge than the opposite polarity. During the discharge

49

Turbo
Pump

Turbo
Pump

Filament
Skimmer

Hexapole

Pulsed valve
13.1 cm
4.94 cm 2.5 cm
Discharge assembly
0.7 mm thickstainless steel
Ø 0.5 mm
Ø 4 mm

2.3 mm thick
Boron nitride
Figure 4.2: Diagram of the experimental apparatus and discharge assembly (not toscale). The system consists of two chambers individually pumped by 300 L/s turbo
pumps. A differential pressure is maintained between the chambers by a mechanicalskimmer. A discharge assembly is mounted directly onto a pulsed current loop actuatedvalve in the source chamber. The discharge assembly consists of two disc electrodes
separated by insulating spacers. The second chamber contains an electric hexapole.
Molecule detection, by laser induced fluorescence, takes place in two regions marked byblack circles.

50
operation, ∼3 mA of DC current is passed though a tungsten filament, which is locatedinside the source chamber. The positive ions created by the filament are accelerated
towards the outer electrode and help to initiate a stable discharge at lower electrodevoltages and shorter discharge pulse durations, ultimately leading to a colder molecularbeam.
After the OH molecules are produced in the discharge, they are allowed to fly to
one of two detection regions, which are illustrated by black dots in Fig. 4.2. The densityof OH molecules in the detection region is determined by laser-induced fluorescence(LIF). The OH molecules are excited by a frequency-doubled pulsed dye laser on theA2 Σ1/2 (v = 1) ←X2 Π3/2 (v = 0) transition at 282 nm. The fluorescence from theA2 Σ1/2 (v = 1) →X2 Π3/2 (v = 1) transition at 313 nm (with a lifetime of 750 ns) is thenimaged onto a gated photomultiplier tube (PMT). An interference filter is placed in frontof the PMT to reduce the transmission of the excitation laser photons by > 103 , whilestill allowing 15% of the fluorescence photons to pass.4 . This spectral discrimination,
along with careful spatial filtering and imaging, greatly reduces the background signal
from scattered laser light. The signal from the PMT is averaged 300 times and integrated
over a 3 µs time window on a digital oscilloscope. The time from the discharge to thedetection is varied to obtain a time-of-flight (TOF) profile of the OH molecular packet(see Fig.4.3(a)).
Creating OH using a high-voltage discharge pulse shorter than the gas pulse
significantly reduces the translational and the rotational temperature, as well as permits
control of the mean speed of the OH packet. In our system, the applied voltage between
the discharging disc electrodes is controlled by a high-voltage MOSFET switch producedby Behlke Electronics GmbH. This device can switch up to 5 kV in well under 1 µs. Using
4

After this work the fluorescence filter was improved to 105 suppression at the excitation frequency
and 70% transmission at the fluorescence frequency through the combination of a colored glass filter(UG11 from Melles-Griot) and a bandpass filter centered at the fluorescence frequency (31BP10 from
Omega Optical)

51
this switch to pulse the discharging voltage, the velocity spread of the OH molecular
packet is greatly reduced. The TOF profiles in Fig. 4.3(a) show a dramatic narrowingof the longitudinal velocity distribution by reducing the duration of the discharge pulse
from DC to 2 µs. Also, the measured rotational temperature of the OH beam decreases
from 195 K to 28 K. The voltage on the electrodes is increased from 1.4 kV, for the
short discharge duration, to 1.9 kV for the DC case. For a DC discharge, very few OH
molecules are produced at 1.4 kV. To make a reasonable comparison between the twomodes of operation, we increased the voltage for the DC case until the peak signal of the
OH packet was approximately equal to that of the short discharge duration case. Whenthe discharge is allowed to occur during the entire gas pulse, there is a large amountof heating from the violent discharge process. Thus, shortening the discharge pulse
duration greatly reduces the temperature of the OH molecular packet and significantly
increases the molecular phase-space density. This effect can be understood as heating
only a small fraction of the expanding molecules, which then cool to a lower temperatureby collisions with the remaining (unheated) pulse.
A short discharge pulse duration also gives the freedom to produce OH molecules
at different stages during the supersonic expansion. OH molecular packets created at
different times in the expansion process are shown to have differing mean speeds andvelocity widths. Figure 4.3(b) is a plot of several example TOF profiles taken just before
the skimmer where the discharge durations is 2 µs for all the data. The time between
the signal triggering the valve to open and the discharge pulse, defined as “ignitiontime,” for each trace is listed in the legend. There is an ∼50 µs time delay between thevalve trigger and the valve opening. By timing the discharge correctly, OH molecular
packets can be created with a mean speed up to 465 m/s with the discharge ignition at
80 µs or down to 275 m/s with the discharge ignition at 190 µs. The mean speed of thepacket as a function of the discharge ignition time is summarized in Fig. 4.3(c) .
A likely explanation for this discharge ignition-time dependent beam velocity is

52

OH molecules (arb.)

1.0

2 ms, 1.4 kV
DC, 1.9 kV
DC, 1.4 kV

0.8
0.6
0.4
0.2
0.0
200

250

300

350

400

Time from valve opening (ms)

3.6

450

Ignition (ms) v±Dv/2 (m/s) Dv/v
110 422 ± 35 16.5 %120 398 ± 33 16.7%
130 371 ± 36 19.6 %140 338 ±41 24.0 %160 306 ± 48 31.2%

2.8
2.1
1.4
0.7
0.0
80

100

120

140

160

180

200

Flight time from discharge ignition (ms)

480

2.0

450
1.5

420
390

1.0

360
330

0.5

300
270
80

100

120

140

160

180

0.0
200

Relative peak height (arb.)

Mean speed of pulse (m/s)

OH denstiy(1010 molecules/cm3)

150

Time of discharge ignition (ms)Figure 4.3: (a) Longitudinal time-of-flight (TOF) profiles, acquired in the detection
region before the skimmer, for a pulsed discharge and a DC discharge at 1.4 kV and 1.9
kV. (b) TOF profiles, acquired in the detection region before the skimmer, as a functionof the time from the filament-assisted discharge ignition to the LIF detection. The discharge duration is 2 µs for all the data. The discharge ignition time, mean longitudinal
packet speed, and velocity width are listed in the legend for each trace. The velocity
width, ∆v, is the full-width half-max of the velocity distribution, which is determine
from the TOF profiles taken at both detection locations. (c) Mean longitudinal speed(solid circles) and relative peak height (open circles) of the TOF profiles as a functionof time from the valve trigger to the discharge ignition. For all traces, the lines serve as
visual guides.

53that our current-loop actuated valve heats the gas in an asymmetric way. The valve
opens by passing a few thousand amperes of current through two copper leaves forseveral microseconds, the leaves subsequently repel one another allowing gas to escapethrough a small hole in one of the leaves. Presumably, this high current pulse initiallyheats the leaves, and since molecules must collide with these leaves before escaping
through the small hole the expanding gas is heated. However, since the copper leaves
have a small thermal mass and are in good contact with the rest of the valve, theyquickly cool down. Thus, the later in the pulse a molecule exits the valve, the less it
is heated. We see evidence of this asymmetric operation of the valve using our pulsed
discharge to sample different parts of the expanding gas pulse. As seen in Fig. 4.3(c),the gas speed is large and constant over the first 15 µs of the pulse when the supersonic
expansion has reached a steady-state beam velocity while the number of molecules inthe beam is still steadily increasing. The speed of the gas gradually decreases as thevalve remains open. We note the peak signal size is reached (at ∼110 µs) only after
the mean speed of the supersonic expansion beam has already decreased. However, the
FWHM longitudinal velocity spread is still only 16.6%.
For the application of a cooled molecular beam as an input to a Stark decelerator,
we require a packet of OH molecules with a high phase-space density propagating at alow mean speed. Choosing to create the OH molecules towards the end of the gas pulse,
for example at an ignition time of 160 µs, produces a packet moving at an attractive
mean speed of only 306 m/s. However the amplitude of the packet is significantly
smaller and the velocity width is significantly larger than a packet created at 110 µs.
The variation of OH packet amplitudes for different ignition times can be seen in Fig.
4.3(c) . The optimum discharge ignition time for our application is around 110 µs. For
different applications, e.g. reactive collision dynamics, the tunability of the mean speedof the molecular packet could be advantageous.
The other important component in the improved discharge-based system is a

54
hot filament in the source chamber. The hot filament has two major effects on the
discharge. First, it allows the discharge to occur reliably and reproducibly even at
the shortest discharge pulse duration of 1 µs. The improvements from a short discharge
pulse duration are demonstrated in the previous section. Second, the hot filament allows
a stable discharge to occur at lower voltages on the disc electrodes. Without the hot
filament, the discharge is either not stable or does not even occur at an electrode voltage
less than 3 kV; using the hot filament, the discharge is stable down to 0.7 kV, which
results in a significantly colder molecular packet.
The longitudinal TOF profile and rotational temperature of the OH molecular
packet are measured for different discharge voltages (Fig. 4.4). For discharge voltages
below 1.9 kV, a single peak is observed in the TOF profile. However, for voltages at or
above 1.9 kV, the TOF profile starts to develop two distinct maxima and indicates aconsiderably larger velocity spread. We expect this heating arises from the higher energy
electrons created by a larger potential difference between the electrodes. As the voltage
is lowered from 1.6 kV to 1.2 kV, the velocity spread remains nearly constant, but thepeak number of molecules decreases as the electrons’ energy decreases and thus creates
OH molecules less efficiently. The rotational temperature also elucidates the heating
effect from the higher discharge voltages. The rotational temperature is determined bymeasuring the ratio of OH molecules produced in the J = 3/2 and 5/2 states. The
introduction of the hot filament permits the reduction of the discharge voltage from 3
kV to an optimized voltage of 1.4 kV, leading to almost a factor of four reduction inrotational temperature (Fig. 4.4(b)).
A Stark decelerator benefits from a molecular beam that has both a high phasespace density and a low mean longitudinal speed. The optimum configuration of thesource for this application uses a 2 µs discharge duration that is ignited 110 µs after thevalve is triggered to open. The filament-assisted discharge is created using a potential
difference between the electrodes of 1.4 kV. A molecular packet created under theseOH molecules (arb.)

55

1.0

1.2 kV
1.4 kV
1.6 kV
1.9 kV
2.0 kV

0.8
0.6
0.4
0.2
0.0

Rotational temperature (K)

80

120

90

100

110

120

130

140

150

Time from discharge ignition (ms)

100

0.67

80

0.74

60
40
20

0.93
0.98

0
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2

Discharge voltage (kV)

Figure 4.4: (a) TOF profiles, acquired in the detection region before the skimmer,
for several discharge voltages. The discharge pulse duration is 2 µs. The solid lines arevisual guides. (b) Rotational temperatures for different discharge voltages. The number
associated with each point is the fraction of the molecules in the lowest rotational state(J = 3/2).

56
conditions has a mean velocity of 422 m/s and a longitudinal velocity spread of 16.6%,
which corresponds to a translational temperature of 5 K. This is a significantly coldertranslational temperature than was reported by [124] of 26 K and [64] of 29 K.
The transverse velocity spread is determined through the use of the hexapole
focusing effect and detailed numerical simulations. The density of OH is measured
2 mm downstream of the hexapole for different hexapole voltages, thus producing a
focusing curve. From the comparison of the numerical simulations to the hexapole
focusing data the full-width transverse velocity spread is estimated to be 35 m/s, whichcorresponds to a transverse temperature of ∼1.3 K.
The density of OH molecules just before the skimmer tip is determined fromthe calibrated LIF signal. The peak density of molecules in the Ω = 3/2, J = 3/2,
f-component state created under these conditions is 3.5 ×1010 cm−3 measured at a distance of 5 cm from the valve nozzle. To compare with the density quoted in [124], we
assume 1/r2 position dependence, where r is the distance from the nozzle, and an equal
population in e and f parity states. Our calculated density at r = 2.3 cm in both parity
states is ∼3 ×1011 cm−3 , which is a factor of 2 less than [124]. This lower moleculardensity can be attributed to a longer flight time using Xenon versus Argon. The longer
flight time allows the molecular packet to spread in both the longitudinal and transverse
directions reducing the density detected at a specific location. To accurately compare
our results with the work of [124] and [64], we also performed the same experiments
using Argon as a carrier gas and measured a factor of five improvement in the numberof OH molecules produced.
In conclusion, we have developed and characterized a controllable discharge-basedsource of cooled OH free radicals. Through the use of a pulsed discharge we can tune
the mean velocity of the OH beam from 465 m/s down to 275 m/s, with a FWHM
longitudinal velocity spread as small as 16.6%. Also, the implementation of a hot
filament in the source chamber allows a stable discharge to occur for short discharge

57pulse durations and at low discharge voltages. We have shown that decreasing thedischarge pulse duration and voltage creates a colder packet of OH molecules.

4.1.1

Piezo-electric Transducer Actuated Valve

Though the current-loop valve offers an extremely reliable, stable source it presents
one large disadvantage for use with a Stark decelerator. Because the valve heats the gas
before it expands, the molecular beams produced by these valves are moving considerably faster than expected for a room temperature expansion making the deceleration
process harder. Thus, we have put forth a considerable effort in developing a beamsource that is as reliable and stable as the current-loop actuated valve, but does notheat the expanding gas. Our early efforts focused on using solenoid type valves (General
Valve Series 9 and 99), while these valves did not heat the pulse and resulted in lower
mean speeds (vOH ≈ 360 m/s) these valves are notoriously unreliable in both pulseto-pulse and long-term molecule production. More recently, we have implemented apiezo-electric actuated valve like the one shown in Fig. 4.5, and is basically identical tothe original design of Ref. [98]. By applying a few hundred volt pulse (600 V typically)
for a few hundred microseconds, the PZT retracts (200 µm travel) the plunger, allowing
the gas to escape. Typical pulse lengths are on the order of 100 µs and can be adjusted
through pre-tensioning of the plunger by screwing it into or out of the plunger holder.
After the gas escapes the valve, it undergoes a pulsed-electric discharge (in the case of
OH discharge production) as previously described. The discharge plates used for thisvalve are slightly modified from that described for the current-loop valve. While thehole in the first discharge plate still has a 0.5 mm hole to match the nozzle orifice, theboron nitride spacer between the plates opens with a 40◦ full-angle. This nozzle design
has been recently shown [60] to produce as much as factor of 8 gain in downstream
beam intensities by producing a more collimated molecular beam. In our experimentswith OH, we see only a factor ∼2 gain in post-skimmer signal, presumably because of

58
heating from the discharge.
For OH production, this valve produces molecular beams with a mean speed of
375 m/s with 10% longitudinal velocity spread. While, the rotational temperature issimilar to that of the current-loop valve, we observe slightly larger densities (less thana factor of 3). Furthermore, the produced molecular beam is extremely stable bothpulse-to-pulse and long-term (the valve has not needed adjustment since its installation
approximately 6 months ago). For these reason, we have replaced the current-loop valve
with the PZT valve in most of our recent experiments.
Because the PZT valve represents a capacitive load to the high voltage pulser
being used to drive it, it is important to employ a push-pull switch, such that the
PZT is actively charged and discharged, ensuring the shortest pulse. A schematic ofan inexpensive home-built push-pull switch (perfected by Brian Sawyer) is shown in
Fig. 4.6. This switch operates reliably up to ∼1.5 kV (VM ax ≈ 2 kV) and can be reconfigured to support negative polarity pulses by interchanging the high voltage input
port and ground on the output side of the switch.

4.2

Formaldehyde Beam Source
Because gaseous H2 CO molecules readily polymerize to most surfaces, and arethus not commercially available, it is necessary to make the H2 CO for the expansion.
We produce the H2 CO molecules by cracking formaldehyde polymer to produce themonomer, which is passed through a double u-tube apparatus for distilling [114](seeFig. 4.7). We normally heat the formaldehyde powder to 110 ◦ C, which is sufficiently
above the cracking temperature of 90 ◦ C. While the cracking process takes longer tocomplete at this temperature, we have found that cracking at higher temperatures leads
to increase polymerization in the apparatus, presumably because of the higher local
vapor pressure of H2 CO. While this polymerization does not prevent the monomer fromforming, it usually leads to much less total yield. After the monomer is released from59

Valve Body

PZT
Plunger+ Discharge
Plate
Plunger Mount- Discharge Plate

Figure 4.5: Schematic of Piezo-electric Transducer Actuated Valve. The plunger is retracted by the PZT (∼200 µm travel) allowing a gas pulse (200 µs typical) to escapefrom the nozzle. The Xe/H2 O mixture then experiences a pulsed discharge. The discharge nozzle construction features a 40◦ opening angle, which produces a much more
collimated beam.

60

8

7

6

5

4

3

2

1

HVIN

PCB INCLUDES THIS SECTION ONLY

R1
30-OHM

D

D

VOUT+12V
R9

+15v

+
C1
100nF

J1

R4
1K

C2
10uF

R7
U3

10K

D

6N137 8
1

2

8

7
4

7

6

2

C

180-OHM

1

6

3

5 MIC4451C
U2

8

2

Q1
G

IXBH15N160

7
4

5

6

S

5 MIC4451C
U5

C

C4
2.2uF

C7
100nF
+

C6
10uF

OUTPUT
U1

R2
11

1K

18
20
16

P+ +VO2
C2

E

-VO2

V+ +VO1
V-

C1
-VO1

10

J3

9
8

VOUT

1
2

OUTPUT PULSES3FROM 0V TO +V

722
+

B

C3
10uF

B

+12V
R10
180-OHM

+15V
R8

U4

10K

6N137 8
2

7

1

6

3

+

NOTE: NO CONNECTION!!

6
7

Q2
G

IXBH15N160

5 MIC4452C
4
U6

5

0V

D
8

2

C8
10uF

S

C9
100nF

THIS MAY BE BIASSED

0V

AWAY FROM GROUND!

AAJILA ELECTRONICS LABPULSE DRIVER
BOARD ONLYDESIGN BY: Sawyer/Hudson
BUILT BY:
DN:YJ070

8

7

6

5

4

3

BUILT FOR:DRAWN BY:

DATE:2/28/06

2

SHEET

OF

1

Figure 4.6: Schematic of reliable, inexpensive high voltage push-pull switch. The switch
can operate reliably up to 1.5 to 2 kV.

61
the powder, it passes through the first u-tube, which is held at dry ice temperatures (196K), and serves to remove contaminants such as water. The H2 CO then continues on to
the second u-tube, which is held at liquid-nitrogen temperatures. At this temperature
the H2 CO collects as a liquid in the bottom of the tube. Once all of the formaldehyde
powder has been cracked, the valves are closed and Xenon at 2 bar pressure is flowed
over the collected H2 CO, held at 196 K where H2 CO has ∼20 torr vapor pressure. In
this work, the Xe/H2 CO mixture was expanded through either a current-loop (60 µs
pulse length) or a solenoid supersonic valve (500 µs pulse length), producing beams witha mean speed (spread) of 470 m/s (10%) and 350 m/s (10%), respectively. In both cases
the density of the H2 CO in the |11 1i state was approximately 1010 cm−3 measured 5
cm from the nozzle. The higher mean speed of the current-loop valve pulse is due tothe aforementioned heating effects of these devices. It is interesting to note, that since
H2 CO has more modes to store internal energy than OH, the heating of the current-loop
valve leads to a much higher mean speed for H2 CO since this extra energy is converted
into forward velocity.
In our experiments with H2 CO, we have tried several different versions of theabove apparatus and have developed several practical rules for dealing with the H2 COmonomer. First, the polymerization rate to metals is much higher than to a clean
glass surface. Thus, all of the distilling apparatus should be made from glass. Second,
once H2 CO has polymerized to the glass the polymerization rate increases. Thus, it
is important to maintain a clean distilling apparatus. The easiest and most effective
way to do this is to bake the entire apparatus above the cracking temperature while
pumping. Typically, we cleaned the apparatus in this way every night after we produced
H2 CO. Third, as aforementioned, the highest yield in H2 CO production came when
the polymer was cracked at temperatures not-too-high above the cracking temperature.
Fourth, the mixture from the second u-tube was released directly into our valve reservoir
for expansion and pumped away every night. This was to prevent polymerization and62

Figure 4.7: Formaldehyde cracking apparatus.

63
eventual clogging of the valve. For the same reason, it was not possible to store the gas
for long term and we found it necessary to produce the H2 CO each day. By coating theinside of the apparatus with materials, which would reduce the polymerization rate, like
Teflon, it may be possible to construct a vessel, which could store the H2 CO for longer
term. Fifth, because H2 CO has a substantial vapor pressure even at low temperatures,
we attempted to cool the supersonic nozzle to produce a molecular beam with lower
forward velocity. While we had limited success with this technique, the polymerization
rate seemed to increase at low temperatures leading to a shorter lifetime of the gas.
However, a bake-able supersonic nozzle, which we did not have, should allow one totolerate the increased polymerization rate since it could be cleaned every night. Finally,
we also attempted to increase the H2 CO beam density by raising the temperature of theH2 CO reservoir (u-tube #2) which raised the H2 CO vapor pressure before the expansion.
This technique showed as much as an order of magnitude improvement in beam density,
however, it also led to an increased polymerization rate, which limited its usefulness in
our apparatus.

4.3

Hexapole Focusing
Electrostatic hexapoles have been widely used in beam experiments to perform

state-selected focusing and spatial orientation [56] of weak-field seeking, Stark-sensitive
molecules. In contrast to those experiments, the hexapole utilized here is quite shortin length and functions to increase the OH beam flux by matching molecules from thesource into the acceptance aperture of the Stark decelerator. The hexapole is formed bysix, hardened steel, 3.175 mm diameter, 50 mm long cylindrical shaped rods, set every
60◦ at a center-to-center radius of 4.39 mm, mounted to an insulated macor support disk.
The rods are mechanically polished and the ends are rounded to a smooth curvature.
Alternate rods are electrically connected, thus forming two sets of three rods. Each set
is charged to equal magnitude but opposite polarity high voltage. The hexapolar field

64
distribution in the transverse plane allows a weak-field-seeking molecule to be confinedand even focused in this plane, leading to transverse phase space “mode-matching”
between the supersonic nozzle and the Stark decelerator.
~ of an ideal hexapole is given as [22]:The electric field |E|~ =|E|

3Vo r 2ro3

(4.5)

where Vo is the absolute value of the symmetric, opposite polarity voltages applied
to each set of rods, ro is the radius of the hexapole (from the center of the hexapole
to the inner edge of the rods), and r is the radial spatial coordinate. A weak-fieldseeking molecule with a linear Stark shift will thus experience Stark potential energy~ We define an “effective” dipole moment of the molecule as |µef f | =as: W = |µef f E|.
µhcos θi, where µ is the magnitude of the electric dipole moment, θ the angle between the
moment and the electric field direction, and hcos θi represents the quantum mechanical~ is then written as:
expectation value. The radial force FF =−6Vo rµef f
r̂.
ro3

(4.6)

This linear restoring force results in radial harmonic motion of the weak-field seeking
molecules inside the hexapole field region. Thus, in analogy to ray tracing in optics, it
is possible to define the focal length f of the hexapole in the thin-lens limit (for l → 0asr

6Vo µef f l
mro3 v remains constant) as:f=

ro3
6µef f

!

mv 2
Vo l

!

(4.7)

where l is the longitudinal length of the hexapole, v is the molecule’s longitudinal velocity, and m is the molecular mass. Equation 4.7 demonstrates the focusing strengthof a given hexapole linearly increases (decreases) with applied voltage (molecule kinetic
energy). For a “real world” hexapole, the finite size of the rods can lead to deviationfrom Eq. 4.7 [16]. Accounting for this effect, as well as the full non-linear Stark shift as

65
treated in Chap. 2, detailed numerical simulations of the hexapole focusing effect are
shown in Fig. 4.8. The trajectory simulations deterministically map an initial volume
in phase-space to a final volume, which is then matched to the corresponding experimental data by proper weighting of initial molecular numbers. The figure shows threesets of focusing curves consisting of OH signals measured directly after the hexapole,
under conditions where the applied hexapole voltage has been pulsed to match to thecorresponding input molecules’ velocity. Operating the hexapole in such a switchedmanner is designed to give maximum benefit to a particular velocity class, wherein thehexapole voltages are controlled by fast switches that rapidly charge the rods when theselected molecules enter – and then terminate the voltages to ground when the molecules
exit – the hexapole region. Thus, molecules with speeds significantly different than thetargeted velocity class experience less focusing power of the hexapole, minimizing theaberration effect from the distribution of velocities in the molecular pulse. Symbols in
Figure 4.8 correspond to data points measured for 350 m/s (squares), 385 m/s (diamonds) and 415 m/s (triangles) velocity classes, while the solid lines joining the data
represent simulation predictions. From comparison of the simulations to the hexapole
focusing data, the transverse temperature of the OH beam is determined to be ∼4 K,
consistent with a supersonically cooled molecular beam. The upper inset in the figure
depicts the contributions from the two weak-field seeking states to the 385 m/s trace,
where molecules in the 2 Π3/2 F = 2, mF = 0 and 2 Π3/2 F = 1, |mF | = 0,1 states are
marginally focused (dashed line) by the hexapole fields, in contrast to the strong effectexperienced by molecules populating the 2 Π3/2 F = 2, |mF | = 2,1 states (dotted line).
The two weak-field seeking states are included in the simulations with equal weighting.
This figure demonstrates the powerful molecular-focusing capability of the electric
hexapole, as the OH molecules are observed only a few millimeters past the end of thehexapole rods. However, for the deceleration experiment, the hexapole is operated only
to provide efficient molecular coupling into the physical opening of the slower. This task

66

Figure 4.8: Hexapole focusing curve for different OH molecular velocities. The symbols
represent the data points for molecules with velocities of 350 m/s (squares), 385 m/s(diamonds), and 415 m/s (triangles), while the solid lines represent the corresponding
simulation result. The inset depicts the contribution of the 2 Π3/2 F = 2, |mF | = 2,
1 states and the 2 Π3/2 F = 2, |mF | = 0 and 2 Π3/2 F = 1, |mF | = 0, 1 states to theobserved signals for the 385 m/s trace.

67
requires simply matching the transverse velocity and spatial spreads with the decelerator
cross-sectional acceptance and the subsequent requisite focal length is thus longer thanthat needed to bring the molecules to a sharp spatial focus. Experimentally, we findutilizing ∼3 kV applied voltages results in sufficient transverse coupling. This empiricalresult agrees well with computer simulations of the process. Unless otherwise specified,
all subsequent data traces shown are taken with operating the hexapole in this described
pulsed manner.

4.4

Extracting Beam Parameters from Time-of-Flight Data
The primary source of information for molecular beam measurements (and Stark

deceleration experiments) is Time-of-Flight (ToF) data. Because a ToF signal observedon the data collection computer is a convolution of the true molecular pulse shape andthe detection region, it is necessary to de-convolve the ToF signal to extract the relevantbeam parameters, i.e. spatial and velocity spreads, from the ToF signal. Though thistreatment is likely done elsewhere (and is probably better done), this section details
the extraction of molecular beam parameters from ToF signals produced by a Gaussian molecular pulse. The importance of this procedure cannot be understated, since
analyzing the raw ToF data as an estimate of the molecular pulse always leads to an
overestimate of the relevant parameters, and proper operation of a Stark decelerator
relies on detailed knowledge of the input molecular beam.
If we assume that the molecular pulse in the longitudinal direction has a linear
density of

2

(4.8)

∆x2o + (∆vt)2 ,

(4.9)

−

ρ(x, t) = ρo e



x−vt
√
∆x/ ln 2

with
∆x =

q

where ∆xo is the pulse’s spatial spread at creation, ∆v is the pulse’s longitudinal velocity

68
spread x is the longitudinal coordinate, t is the time since the pulse creation, and ρo isthe peak linear density related to the total molecule number, N , as
N
ρo =
∆x

s

ln 2
.
π

(4.10)For traditional molecular beam experiments the time of creation is when the molecule
pulse is created at the nozzle (or at the discharge region), while for Stark deceleration
experiments the time of creation is when the Stark decelerator voltages are extinguishedand the pulse is allowed to free-fly into the detection region.
As seen in Fig. 4.9, ToF data is typically recorded in a window centered a specificdistance away from the source (L) with some non-zero width (∆L). Experimentally, L
is the distance from the source to the fluorescence collection optics, and ∆L describes
the width, about this center, over which fluorescence is collected. Thus, the observed
ToF data is given as the integral of ρ(x, t) over the detection region at time t ase

(L−∆L/2)



L + ∆L2√ − vt∆x/ ln 2!

S(t) = ρoorS(t) =r"π ρo ∆xErf
ln 2 2Z (L+∆L/2)

−

x−vt
√
∆x/ ln 2

− Erf

2

dx,

L − ∆L2√ − vt∆x/ ln 2(4.11)!#.

(4.12)

Since there are two unknowns, i.e. ∆x and ∆v, it is necessary to measure the ToF
profile at two spatially longitudinally separated locations. At each location, Eq. 4.12 is
fit to the data and a value for ∆x is found, i.e. the de-convolved spatial spread of thepulse. Using the values for ∆x with Eq. 4.9, ∆v is found as∆v =s∆x22 − ∆x21,
t22 − t21

(4.13)

where ∆xi and ti are the spatial spread and peak arrival time of the pulse at detection
region i. Once ∆v is known, Eq. 4.9 can be used to find the initial pulse length, ∆x.

69

Figure 4.9: Detection of a Gaussian molecular beam pulse at a position, L, with adetection window of width ∆L.

Chapter 5

Stark Deceleration

5.1

The Basics
Strong motivation for using the Stark deceleration approach to create cold molec-

ular samples originates from two convenient properties of supersonic expansions. Used
extensively in physical chemistry experiments, beam expansions populate primarily thelowest energy molecular rotational levels and provide an intense supply of molecules withvery narrow velocity widths, as compared with effusive sources. In the moving frameof the molecular pulse, the narrow velocity spread corresponds to a low temperature
distribution (∼1 K), i.e. the molecules have been cooled by the expansion versus room
temperature velocity distributions. However, while the beam expansion distribution is
sharper, the mean speed of the pulse is now actually faster than molecules generated bytypical effusive sources – characteristically, many hundreds of meters per second in thelaboratory reference frame. Stark deceleration enables reduction of the mean velocityof the pulse without disturbing the narrow velocity spread, resulting in a cold molecule
distribution in the laboratory frame. In practice, given attainable electric fields, typicalmolecular Stark shifts, and practical numbers of slowing stages, the deceleration process is more restrictive than the velocity width of the supersonically expanded pulse,
resulting ultimately in narrower velocity widths and correspondingly, colder molecularsamples than few Ks, typically < 100 mK. Finally, we note that the Stark deceleration
technique does not require the use of supersonic expansions; it will work with effusive

71
sources, too, at the cost of reduced initial phase-space density and therefore, ultimately
the numbers of cold molecules produced.
The conservative reduction of the pulse’s mean velocity is accomplished by exploiting the Stark effect associated with weak-field-seeking molecules. As such a polar
molecule moves from a low electric field into a high electric field region, its internalenergy increases at the expense of its kinetic energy; thus it slows down as it climbs
the potential energy hill created by the increasing electric field. If the electric field
is rapidly extinguished, the molecule will lose this Stark energy without regaining it
as kinetic energy1 . Utilizing the appropriate field geometry, this procedure may berepeated as many times as necessary to lower molecule’s speed to any arbitrary final
velocity. Experimentally, this is realized in what has come to be known as a “Stark
decelerator”, as shown schematically in Fig. 5.1. A more complete understanding of
the deceleration process is gained by careful examination of the electric field distribution produced by the spatially, alternating-oriented electrode stages. First, considering
the plane perpendicular to the molecular propagation direction, the electric field minimum at any longitudinal position always lies along the central axis of the slower; thus,
the alternating electrode geometry results in a stage-to-stage, on-axis focusing of the
weak-field seeking molecules. The guidance similarly alters its focusing axis every otherstage; thereby, molecules are always transversely guided down the slower simultaneously
as their longitudinal velocity is being modified by interacting with the inhomogeneouselectric fields. Second, in order to minimize the required number of expensive, fasthigh voltage switches, all the horizontal (vertical) electrodes of the same polarity areconnected to a single switch, thus allowing four switches in total to operate the entire
decelerator. A consequence of this practical switch minimization is that when the highvoltage at a particular slowing stage is removed and simultaneously turned on at the1

The molecule’s lost kinetic energy, which went into inducing a charge on the decelerator electrodes,
is dissipated in the resistors of the switching circuit when the electrode is grounded.

72

PulsedValve
Skimmer

Hexapole

Decelerator
Array
Figure 5.1: System schematic of Stark decelerator, displaying the pulsed valve, themolecular beam skimmer, the electric hexapole, and the electrode stages. The electrodestages alternate orientation (verticalto-horizontal) as shown in the figure to provide
transverse guidance of the decelerated molecules.

73
following (and thus preceding) stage, the local electric field does not go to zero but
rather to a value generated by the now active set of electrodes. While this effect detrimentally limits the maximum amount of energy per stage which can be extracted by
the decelerator, it beneficially maintains the quantum mechanical state identity of themolecules, as the molecules are phase-stably slowed.
Fig 5.2(a) shows the longitudinal potential energy distribution experienced by
OH molecules with our present experimental electrode geometry, generated under normal operating conditions of ±12.5 kV. The dark line corresponds to the active set ofelectrodes and the dashed line indicates the other (off) set, where the horizontal axis
(spatial phase angle) is defined below. Figure 5.2(b) demonstrates how switching theelectric field leads to a change in the molecular kinetic energy per slowing stage that
depends critically on the position of the molecules at the switching time. This dependence is conveniently characterized by utilizing a spatial coordinate φo , called the phaseangle, which is defined only at the instant of electric field switching and written as:
φo =z180◦ ,
L(5.1)

where z is the longitudinal position of the molecule and L is the stage-to-stage separation. We define z = 0 (φo = 0◦ ) as the point midway between two adjacent stages.
If a molecule has a phase angle 0◦ < φo < 180◦ at the switching of the fields, then it
will be slowed. To analyze the net effect of deceleration on a spatially distributed pulseof molecules, it is useful to define the synchronous molecule as one that is at exactly
the same phase angle, φo , at the moment of each switching of the electric fields; thus,
losing an identical amount of energy per stage. While the region 0◦ < φo < 180◦ does
lead to deceleration of the synchronous molecule since the kinetic energy loss ∆KE ispositive, only operating in the range 0◦ < φo < 90◦ results in phase stable decelerationof a molecular packet. This can be seen in Fig. 5.2(b), that only on the positive slope of
the kinetic energy loss curve do molecules which are slightly ahead (leading - i.e. faster)

74

Figure 5.2: Phase stable operation of the decelerator. (a) Longitudinal Stark energy
potentials generated by the two sets of electrodes, where the solid (dotted) line represents
the potential from the active (grounded) set of electrodes. (b) Kinetic energy loss per
stage (∆KE) experienced by molecules from switching between the two potentials given
above. Solid (dashed) line corresponds to a numerical calculation (sine function fit).

75
or behind (lagging - i.e. slower) the synchronous molecule experience a net restoring
force towards the synchronous molecule position. Physically, a faster (slower) nonsynchronous molecule travels higher (lower) up the hill than the synchronous moleculeand hence, has more (less) energy removed than the synchronous molecule. Detrimentally, operating on the negative slope of the hill, i.e. in the range 90◦ ≤ φo < 180◦ ,results in faster (slower) molecules becoming progressively faster (slower) relative tothe synchronous molecule, spreading the molecular pulse further and further apart. By
similar arguments, phase stable acceleration of a molecular packet is only possible on
the positive slope of the potential hill in the range 0◦ < φo < −90◦ . In Fig. 5.2(b),the phase stable ranges are indicated by the shaded portions. This restoring force
experienced by nonsynchronous molecules, when the slower is operated in the proper
regime, is the underlying mechanism for the phase stability of the molecular packet anda critical concept for understanding Stark deceleration experiments. Finally, operating
the slower with φo = 0◦ represents an interesting experimental condition. As shownin Fig. 5.2(a), this point is defined spatially as exactly between two adjacent stages.
While the molecules are accelerated then slowed each stage as they roll down and up
the hill, these two effects cancel and result in no net change in the longitudinal velocity
of a synchronous molecule; however, the molecular packet is phase stably bunched as
it propagates down the slower. This so-named “bunching” condition is a good test of
successful phase stable operation of the Stark decelerator.
Defining the excursion (at the switching time) of a non-synchronous moleculefrom the synchronous molecule position as ∆φ = φ − φo and with the help of a sinefunction fit to the change of kinetic energy per stage, it is straightforward to write forthe time evolution of ∆φ:
d2 ∆φ Wmax π d∆φ+(sin(∆φ + φo ) − sin(φ)) = 0
dt2
mL2 dt(5.2)

where Wmax is the maximum work done by a slowing stage on a molecule and m is

76

Figure 5.3: Phase stable area (separatrix area) versus phase angle for φo = 0◦ , 30◦ , and60◦ . As the phase angle is increased, the stable area is reduced; equivalently, the stable
longitudinal velocity width narrows and assuming the source completely populates thephase stable areas, decreases the numbers of molecules in the phase stable packet.

77
the molecular mass2 . Thus, non-synchronous molecules will oscillate around the synchronous molecule position inside an asymmetric oscillator potential. From numericalintegrations of Eq. 5.2 and the first time derivative of Eq. 5.1, solutions for the stableand unstable regions of phase-space are shown as a function of synchronous molecule
phase angle φo in Fig. 5.3(a). The most interesting and relevant feature of this figure is
the rapidly decreasing area of stable evolution, defined as the area inside the separatrix.
This decrease in stability is easily understood by analogy to the pendulum driven by aconstant torque. As the torque is increased (or equivalently, φo is increased), the equilibrium position of the pendulum is pushed toward the apex; therefore the maximum
amplitude that result in stable oscillatory motion is lowered. Reasonably assumingthat the molecular source homogeneously populates phase-space on the scales given
here, this reduction in stable phase-space area translates to a decrease in the numberof slowed molecules, setting a practical, observable limit to the ultimate temperatureof the molecular packet. From Figure 5.2(b), it is seen that for molecules ahead of thesynchronous molecule the maximum stable excursion, i.e. the separatrix upper spatial
bound, is given as:◦∆φ+max = 180 − 2φo ,(5.3)

since any non-synchronous molecule with φo < φ < 180◦ − 2φo will be slowed more than
the synchronous molecule. Furthermore, a molecule with zero velocity at this position,i.e. at φ = φo + ∆φ+
max , will have the maximum stable velocity at the φ = φo position,
thus the longitudinal acceptance of the slower in velocity as a function of synchronous
molecule phase is given as (see the next chapter for a more detailed derivation):s∆vmax (φo ) = 2
2

2Cmax
mπ





π
φ
cos φo −
1− ◦
2
90



sin φo(5.4)

This equation is the same as a harmonic oscillator with an offset equilibrium position, e.g. apendulum driven by a constant torque78

Figure 5.4: Phase stable area (separatrix area) A versus slowing phase angles φo , normalized to the phase stable area under bunching operation Ao . The subsequent maximum
allowed velocity spread of the phase stable packet ∆vmax is shown with respect to theright axis.

79
Figure 5.4 usefully summarizes the results of the calculation given above, giving
an intuitive expectation of performance under different modes of slower operation. The
solid square symbols represent stable areas, A, in phase space, plotted as a function ofslowing phase angle φo and normalized to that of the bunching case Ao . The open circles
are plotted along the right axis and depict the maximum longitudinal velocity spread,∆vmax , of the phase stable packet. As seen in the figure, operation at increasing phaseangles results in fewer but also cooler molecules in the phase stable packet. Hence, withan eye toward experimental goals, a careful balance between the required number ofmolecules in the phase stable packet and the desired longitudinal temperature must be
considered when using the Stark decelerator.
While Eq. 5.4 allows for the estimation of the longitudinal temperature of thestable molecular packet, the dependence of the transported molecule number on φomust be determined. For an initial velocity distribution of ∆vF W HM (full width at half
maximum), the number of molecules successfully loaded into the phase stable bucket
N (φo ), normalized to that for the bunching case, is shown to be (see the next chapter
for a more detailed derivation):)Erf ( ∆v∆vmax (φ√oln)
φoF W HMq 2
N (φo ) = (1 − ◦ ),2Wmax
90 Erf ()∆vF W HM

(5.5)

mπ ln 2where Erf is the Error function. In the above derivation, the area inside the separatrix isapproximated as square since we are only concerned with variation relative to molecularbunching. Also, we assume that at the input of the decelerator there is no spatial
variation of the molecular packet over the dimension of one slowing stage separation,
as well as ignore any convolution of the input distribution due to free flight from thecreation region. A more limiting assumption, made for the sake of a closed form solution,is that the lower separatrix bound relative to φo is assumed to vary in the same manner+as the upper bound, i.e. ∆φ−max = −∆φmax . Nonetheless, the results of Eq. 5.5 are ingood agreement with the measured molecule number as detailed in section 5.4, where a80more detailed numerical solution for Eq. 5.5 that accounts for the proper variation of∆φ−max is also shown, along with the results of a full 3D Monte Carlo simulation of theslowing process.

5.2

Decelerator Monte Carlo Simulations
The Monte Carlo simulations, presented throughout this thesis, are performed not

only for comparison of the expected stable molecule number, but also for assessment of
the detailed TOF spectra observed, as well as to aid in the understanding of the deceleration process. From the geometrical and electrical properties (sizes, distances, andapplied voltages) of the decelerator, a three-dimensional electric field map of the slower
is constructed, which when coupled with the understanding of the molecular energy
structure, gives complete information on the forces experienced by molecules within
the decelerator. The initial input distribution are assumed to be gaussian distributed
and the parameters (centers and full-widths) are determined from both experimental
dimensions (spatial parameters) and measurements on hexapole-focused, transversely
guided molecular pulses (velocity parameters). The trajectories of typcially ten-million
molecules sampled from the initial distribution are calculated for the appropriate Stark
manipulation process, i.e. bunching, slowing, or accelerating starting from the skimmer
entrance, then traversing the focusing hexapole and decelerator, and finally reaching
the appropriate detection region. Recently, we have implemented these simulations asa C++ class that can be run in a parallel processing mode. This, ever evolving, code isfreely distributed and can be obtained by contacting the author or Prof. Jun Ye.

5.3

Decelerator Construction
The most important practical issue in the construction of a Stark decelerator isthe stable operation of high voltages (∼25 kV) between electrodes spaced only 2 mmapart inside the decelerator. Discharge between electrodes will not only disrupt normal

81
operations of the decelerator, but could also inflict permanent damages to the electrodes,
as well as destroy the decelerated molecules through ionization or electron attachment.
There are several useful techniques for the stable application of large electric fields,
including a high vacuum system, proper electrode construction, and proper electrodeconditioning. This section details the methods of implementing these techniques ina Stark decelerator, as well as the implementation of the fast high-voltage switching,which is central to Stark deceleration experiments.

5.3.1

Vacuum Apparatus

Discharging between electrodes that support a high electric field occurs as anelectron/ion is accelerated to sufficient energies to ionize the nearby gas. Once the electron/ion ionizes a gas molecule, both are accelerated and can also ionize more neighboring gas particles. In this way, an exponentially-growing current path is created between
the high voltage electrodes by the (avalanche) gas mediated discharge. Clearly, placing the decelerator in high vacuum is helpful in terms of increasing the threshold forunwanted discharges in the apparatus. At pressures ≤ 10−7 Torr the mean free pathof the gas is larger than the electrode spacing and consequently the mechanism for gas
mediated discharge is removed, making the application of high voltages much easier.
Nonetheless, discharging can still occur via material ablated from the electrode surface.
This method of discharging, which depends on the pre-breakdown tunneling current,
will be treated in the next subsection.
Additionally, high vacuum is also useful for reducing collision events between
the decelerated molecules and background gas. This is especially important in themolecular beam source chamber where the beam density can be as much as 1012 cm−3 .
In fact, in our first-generation decelerator we found that the volume and pumping speed(250 L/s turbo-molecular pump) in our source chamber limited our molecular beam
intensity. We observed that for higher nozzle pressure and shorter nozzle to skimmer

82
distances scattering of the beam onto itself resulted in a collapse of the molecular beam,
evidenced by a sharp reduction in post-skimmer molecule number. In our secondgeneration decelerator we reduced this effect by using a larger volume source chamberand higher pumping speed turbo molecular pumps (1000 L/s). While a higher pumping
speed results in a lower average pressure, it does not not alleviate the problem of amolecular pulse scattering back onto itself. It is for this reason that making the sourcechamber larger in volume, so that molecules from the pulse cannot scatter into the beamduring the pulse time-scale, is important. As aforementioned, to separate the higher
pressure source chamber from the rest of the decelerator a molecular beam skimmer
is used, which acts as an effective differential pumping hole. During quiescence, thesource and slower chambers reach ∼10−8 and ∼10−9 Torr, respectively. Under active
conditions, the source and slower chamber are at ∼10−4 torr ∼10−7 , ensuring stable
operation of the Stark decelerator.

5.3.2

Decelerator Electrode Construction

As aforementioned, at the pressures inside a Stark decelerator gas mediated discharge is negligible, and the mechanism for discharge is via material ablated from theelectrode’s surface. The process responsible for the ablation of the electrode is known as
pre-breakdown tunneling current. This current is due to electrons that tunnel throughthe work function, φ, of the negative electrode and are accelerated towards the positive electrode. When these electrons strike the positive electrode they have sufficientenergy to ablate some material from the electrode’s surface (and sometimes produce
observable fluorescence). If the pre-breakdown current is high enough that the ablatedmaterial reaches a critical local pressure, a gas mediated discharge can occur. And assuch, understanding the characteristic of the pre-breakdown tunneling current is crucial
to proper high voltage electrode construction.
It is generally accepted that the pre-breakdown tunneling current originates from

83
localized regions on the electrode surface where the electrode’s work function is lower.
The work function at these emitter sites is lowered either by the presence of some
impurity in the metal or by the Schottky effect, i.e. the apparent reduction of the work
function at micro-protrusions on a surface in the presence of an electric field. From thetunneling theory of Fowler and Nordheim [52] it is possible to write the value of thetunneling current as [78]

3/2

Ae B1 E 2 − B2 φE
I = Ae j =
e
φ



,

(5.6)

where Ae is the total area of emitting sites, j the current density, E is the electric field magnitude, and B1 and B2 are constants given as 1.54 × 10−6 CΩ−1 and6.83 × 109 C −1 m−1 , respectively. From Eq. 5.6, we see that the parameters at theexperimenters control are φ and E. Furthermore, the electric field in Eq. 5.6 is theelectric field at the emitter site, which is typically written as E = βV /d, where V and d
are the voltage across and separation of the electrodes, respectively, and β is called thegeometrical enhancement factor which describes the local increase in electric field due tothe surface roughness at the emitter. For these reason, most electrodes are constructedfrom stainless steel since it has a relatively high work function (φ ≈ 4.5 ev) and can bepolished to high a surface quality, i.e. small β.
In our experiments, we first fabricated the high voltage electrodes from 304L
stainless steel, or hardened steel, and subsequently mechanically polished them to thebest possible surface finish we could achieve. Electron microscope pictures of these electrodes are shown in Fig. 5.5. Comparing the top two pictures at 70X magnification
clearly shows the advantages of mechanical polishing as the roughness of the raw material is replaced by a much more uniform surface. At higher magnifications, however,contamination on the surface becomes visible (left-most middle picture), which is due topolishing paste that has adhered to the surface. At even higher magnification, serrationsof the surface due to the ceramic beads in the polishing paste become visible (right-

84
Mechanically Polished
Raw Hardened Stainless 70X

Mechanically Polished
Hardened Stainless 1800X

Hardened Stainless 70X

Mechanically Polished
Hardened Stainless 2700X

Electro-polished 3 minutes

Electro-polished 3 minutes

316 Stainless 70X

316 Stainless 1800X

Figure 5.5: Electron microscope pictures showing the effect of material and polishingchoice on surface quality.

85

(a)

(b)

Figure 5.6: Pictures of Stark decelerator. (a) Hexapole mount. (b) Decelerator array.
In both pictures notice that in the Macor mounts grooves have been cut to increase thesurface path between oppositely charged electrodes.

86
most middle picture). Since both surface contamination and micro-roughness lead toa lowering of the electrode’s work function it is clear that mechanical polishing with apolishing paste in not optimal. It should be noted that our first-generation decelerator
utilizes electrodes polished in this way and they have not been problematic at current
voltages. However, test of these types of electrodes have revealed they should not be
used much above the current fields of 150 kV/cm.
To reach higher fields we have developed electrodes whose final polishing stepis electro-polishing. These electrodes are first mechanical polished in the same way
as previously described, but are then placed in an electro-polish bath, where microroughness features are removed atom by atom. Because 304L stainless steel does notelectro-polish very well, we now use electrodes constructed from 316 stainless steel,which has the convenient property of being the least magnetic of all steels. Electrodes
produced in this way are shown in the bottom two pictures of Fig. 5.5. It is clear
from these pictures that the contamination and serration has been greatly reduced.
Furthermore, as detailed in the next section these electrodes are stable to much higher
electric fields.
With great care taken to eliminate direct discharge between opposing electrodes,
surface currents and discharging to other structures become important. In our experiments, all high voltage elements are mounted on insulating materials (Macor, alumina,
or boron nitride) in such a manner to avoid close proximity to ground or opposite highvoltage polarities. When opposite polarity voltage elements are attached to the same
insulating support, grooves were fabricated into the insulating surface in order to create
longer surface path lengths to minimize current leakage between the components – agood ‘rule-of-thumb’ is that the surface path should be ten times longer than the vacuum gap. Shown in Fig. 5.6 are two pictures of our second-generation decelerator. In
Fig. 5.6(a) the hexapole mounting structure is shown, and in Fig. 5.6(b) the decelerator
array is shown. Notice that in both cases, how grooves are cut in the Macor mounts to87increase the surface path between electrodes.

5.3.3

Decelerator Electrode Conditioning

Even with highly polished decelerator electrodes and careful construction of thedecelerator apparatus,3 energizing a new decelerator to high voltage will most likely
result in catastrophic discharge. This is because there are always a few emitter sites
or contaminants on the decelerator array and only one is needed to start a discharge.
Experimentally, this problem is rectified by the method of ‘high voltage conditioning’,
which despite its various incarnations uses controlled ‘micro-discharge’ events that help
clean the surfaces and prepare the electrodes for high voltage operation. Typically, once
under vacuum, the electrodes are subject to a sequence of high voltage conditioning
sessions to ensure stable operation. We employ two related, but distinct methods: DC
current and glow-discharge conditioning modes [78].
Under DC current conditioning, all electrodes are gradually brought up to 1.2 2 times their nominal operating level in incremental voltage steps. A limiting resistor(∼0.1 GΩ) in series with the high voltage power supplies limits the possible discharging
current to small values (≤ 100µA) so that when discharges occur the electrodes are
not damaged, i.e. the resistor supplies negative feedback since as the current increases
more voltage drops across it and less voltage is applied to the electrodes. When amicro-discharge occurs it is always at the emitter site and works to either remove thecontaminant or smooth-out the micro-roughness, thus conditioning the electrodes. It
was empirically discovered that the Bayard-Alpert type ionization gauge that actively
monitored the chamber pressure (which rises during a micro-discharge) was also inducing
low level discharging during the conditioning process. UV lamps have also been utilized
to purposefully create these beneficial micro-discharges during the current conditioning
3

Because any contaminant on the decelerator, dust or otherwise, can cause discharges to occur, thedecelerator should be constructed in near clean room conditions

88

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89

Current ConditioningµPre-breakdown current [ A]50

40

±11 kV±10 kV

30

±9 kV
20
±8 kV

10
±6 kV

0

±7 kV
±5 kV
11:21 AM

2:52 PM

5:06 PM

7:19 PMTime

Figure 5.8: Typical current conditioning data. After each increase in voltage the observed current is usually quite ‘noisy’ reflecting the quasi-random nature of the microdischarges. However, after sometime the micro-discharges burn away the emitter sites
and the current decreases to a steady, lower value.

90
process. The series current can be actively monitored by floating an analog voltmeter orby a home-made floating ammeter that uses fiber-optic isolation to allow data-logging
with a computer. A schematic of this design is shown in Fig. 5.7, which uses voltage-tofrequency converting chip floating at high voltage to flash a light-emitting diode into afiber. At the other end of the fiber a photo-diode converts the light signal back into anelectrical signal that can be counted and logged on a data-taking computer. Recording
the pre-breakdown current as a function of applied voltage is an extremely useful toolfor diagnosing the quality of the electrodes. Shown in Fig. 5.8 is typically data forthe current conditioning process. As seen in this graph, when the voltage is increased
beyond a certain value the pre-breakdown current rises from zero to a few microamperes.
Further increase in the voltage results in an exponential increase in the current (seeEq. 5.6). Furthermore, after each increase in voltage the observed current is usuallyquite ‘noisy’ reflecting the quasi-random nature of the micro-discharges. However, aftersometime the micro-discharges burn away the emitter sites and the current decreases
to a steady, lower value. After the conditioning process, measuring the current versus
voltage results in graphs similar to Fig. 5.9. In Fig. 5.9(a) the current is plotted versus
the voltage and the exponential behavior predicted by Eq. 5.6 is apparent. A more
useful plot is given in Fig. 5.9(b), where the natural logarithm of the current divided
by the square of the voltage is plotted versus the reciprocal of the voltage. Note, using
E = βV /d, dividing Eq. 5.6 by V 2 , and then taking the natural logarithm leads toln(I/V 2 ) = ln(

Ae B1 β 2dB2 φ3/2 1
)
−
.
φd2
β
V

(5.7)

If this equation is plotted versus the reciprocal voltage it is clear is should be a linewith a slope proportional to the reciprocal of β. Thus, the smoothness of the electrodes
can be inferred simply from the slope of the line. As seen from these graphs the glow
discharge condition process produces much smoother electrode surfaces. Typically, theglow discharge conditioning process is reserved for situations when the apparatus fails

91

(a)

30
Current Conditioning,

β = 311.17

Current and glow discharge conditioning,

β = 137.88

25µ

I ( A)

20

15

10

5

10

12

14

16

18

20

V (kV)(b) -13.0
-13.2
-13.4

Ln(I/V

2

)

-13.6
-13.8
-14.0
-14.2
-14.4
-14.6

Current conditioning
Current and glow discharge conditioning

-14.8
0.05

0.06

0.07

0.08

0.09

0.10

1/kV
Figure 5.9: Current vs. Voltage curve after conditioning.(a) The exponential dependenceof the current on the voltage is apparent, confirming the tunneling theory of Fowler
and Nordheim. (b) Plotting the ln(I/V 2 ) versus 1/V reveals a line whose slope is
proportional to the electrode smoothness.

92
to DC current condition to satisfaction. The chamber is backfilled with a few tensof milliTorr of helium gas, and a Tesla coil is used to spark a discharge between therelevant electrodes for approximately ten minutes. The system is then vented and thisprocedure repeated with a backfill of nitrogen gas. Subsequently, the system is DC
current conditioned with significant improvements. The improved electrode quality ispresumably due to the more aggressive micro-discharging induced by the Tesla coil. For
this reason, glow discharging often or for more than the prescribed time can actuallylead to electrode degradation [78] and it should only be used when other methods fail.
It is interesting to note that another type of electrode conditioning, Spark-gap
conditioning, works by applying a high voltage, unlimited current pulse to the electrodes
for a few microseconds. In this way, a micro-discharge can be initiated and the emittersite ablated, however, the voltage is removed before an avalanche discharge can becreated, which would damage the electrode. Though we have never purposefully used
this technique for conditioning, it is exactly how a Stark decelerator is operated. Thus,
it is likely that the operation of a Stark decelerator is intrinsically stable, and perhaps
may be self-healing.

5.3.4

Fast High Voltage Switching

Successful operation of the Stark decelerator relies on thorough knowledge of theinitial position and velocity distribution of the molecules, as well as temporally accurate
control of the inhomogeneous electric fields in order to ensure stable manipulation. We
rely on machining precision in the manufacture of slower components (< ±50 µm statedtolerance) to ensure that any physical discrepancies from the specified dimensions areminimized. The applied electric fields are pulsed using commercially available, pushpull, fast high voltage switches with a specified 10 ns rise time (see Fig. 5.10). Externalcurrent limiting resistors, RL and REM I , are used both to protect the switches (Behlke
HTS 151-03-GSM with high frequency burst option) from excessive current and to slow

93
the rise time to 500 ns to minimize the generated electromagnetic interference (EMI).
The molecules are insensitive to the switching speed reduction, as even on the µstime scale, the fastest-moving molecules in the system are almost stationary. We notethe importance of reduction of the overall EMI from the system when high voltages arebeing switched actively. Two high voltage power supplies (one devoted to each polarity)
generate the requisite high voltage for the slowing stages. Each individual power supply
output is coupled in parallel with a 2 µF high voltage capacitor, CB , on the input sideof the fast switch in order to provide the large currents (12 A peak current) required
for the rapid charging of the slowing electrodes. A similar arrangement generates theslightly lower high voltage and current used for the pulsed electric hexapole operation.
All high voltage cabling is constructed from a corona-discharge suppressing cable
rated to 40 kV (Belden 8866 002), which is surrounded by Tygon tubing and finallyenclosed in a grounded, metallic braid shield. In this way the dielectric breakdown of
the cable is increased, while protecting anyone near the cable from electrical shock since
the discharge current should prefer to travel down the low impedance shield.
A digital delay generator acts as the stable master-clock for the experiment, controlling the firing of the pulsed laser and the supersonic valve, as well as triggering
a computer-based timing board. The timing board repetitively generates user-defined
timing signals to subsequently drive the fast high voltage switches that control the OHgenerating discharge, hexapole, and slowing stages. The timing pattern incorporated
into the timing board is derived from computer trajectory simulations of the Stark
deceleration process.

5.4

Stark Deceleration of the Hydroxyl Radical
In our experiments at JILA, we have constructed two Stark decelerators. The

first-generation decelerator was similar to the design of the Berlin group [24]. The first
production of cold OH radicals was achieved in this decelerator [27], as well as the

94

RL

+HV

-HV

CB

RL

RL

CB

R EMI

R EMI

RL
Figure 5.10: Schematic of decelerator switches. The high voltage push-pull MOSFET
switches are shown schematically. The protection resistors RL (33 Ω) are technicallynot needed if REM I (1kΩ) is in place, however, they provide protection in the case ofa breakdown that bypasses REM I . The buffer capacitors, CB , provide the high current
necessary to quickly energize the electrodes.

95
measurements of the microwave transition in the lowest OH Λ-doublet. The secondgeneration decelerator is fundamentally similar to our earlier design with the only major design changes being twice as many deceleration stages and a more suitable vacuum chamber for making a good molecular beam. The first production of cold H2 CO
molecules was achieved in this machine. Furthermore, OH has been decelerated in thismachine allowing for a comparison between the two decelerators, which shows the clear
advantage of a longer (but not-too-long) decelerator. The next two sections detail thedeceleration of the OH molecule in these decelerators.

5.4.1

First-Generation Decelerator

In-situ experimental observations of OH molecules undergoing longitudinal phasespace manipulation inside the properly timed decelerator provide firm confirmation to
the theoretical understanding. Time-of flight (ToF) spectra from three representativeand distinct spatial locations – before 23 stages, before 51 stages, and at the decelerator
exit – are selected to demonstrate the evolving character of the phase-stable molecular packet. Where present in all data figures that follow, a downward-pointing arrow
indicates the location of the phase-stable molecular packet.
Figure 5.11 depicts the OH ToF spectra when the decelerator is operated undera phase angle φo = 0◦ , i.e. the net longitudinal velocity of phase stable molecules
is not changed but the molecular packet maintains tight spatial confinement due tobunching. ToF spectra generated under properly timed and pulsed electric fields exhibit
peak and valley structures, demonstrating the velocity and spatial selectivity of theinhomogeneous electric fields acting on the evolving molecular packet. The experimental
data in Fig. 5.11 (a), (b), and (c) are plotted with symbols (filled circles) while the full3D simulation results are displayed as solid lines. The simulation traces are offset
vertically for clarity of presentation. Figure 10 (a) shows the spectra before 23 stages ofbunching (φo = 0◦ ). With such few stages of interaction, longitudinally phase unstable

96

Figure 5.11: OH LIF TOF spectra under bunching (φo = 0◦ ) decelerator operation (a)before the 23rd stage, (b) before the 51st stage, and (d) within the electrostatic trap
region. The data points are drawn as squares, circles, and triangles while the simulation
results are shown as solid lines, with a vertical offset for viewing clarity. Panel (d) shows
both experimental and simulation results on bunched peak areas, normalized to the 23rdstage; the reduction in peak height with position is due to longitudinally phase unstablemolecules as well molecules with radial velocities ≥ 5 m/s being lost from the packet.

97
molecules are still dominantly present and detected by the LIF technique, generating
the side peak features and contributing to the broad pedestal that the central peaksits on. By the time the molecular pulse has propagated in front of the 51st stage
under the same experimental conditions, the phase stable packet has become the only
distinct feature remaining, as shown in Fig. 5.11 (b). The molecules that correspond tountrapped orbits in phase-space generate the small, broad background on top of which
the phase-stable peak sits. Similarly, the bunched, phase-stable packet dominates the
spectrum in Fig. 5.11(c), where the molecules have exited the slower, propagated ashort free-flight distance, and passed through the center of the electrostatic trap region
where they are detected.
In principle, all molecules within the selected phase stable packet are transported
without loss down the decelerator; the difference in peak amplitude between Fig. 5.11(b)and (c) is a result of molecular loss due to molecules with too high transverse velocities
being progressively removed from the molecular pulse, inefficient coupling between the
decelerator and trap region, and scattering from background gas. Figure 5.11(d) depicts
the bunched packet peak areas at different spatial locations from both experimental
data and simulations, normalized to the peak before the 23rd stage. This figure, alongwith the understanding gained from simulations, demonstrates that the apparent loss of
signal is primarily due to the “boiling” away of longitudinal phase unstable molecules as
they propagate down the slower, as well as molecules possessing transverse velocities ≥5 m/s slowly escapes from the phase stable packet throughout the deceleration process.
Essentially, the focusing power of the hexapole is boosting the initial total number ofmolecules coupled into the slower. However, many of these molecules are ultimately
not useful as they do not enhance the population of the phase stable packet and aresubsequently lost. The simulation results provide excellent agreement with the observed
spectra, confirming good understanding and control of the decelerator.
We gain more insight into the slowing process by examining in detail the ToF

98

Figure 5.12: OH packet of molecules in front of the 23rd stage under (a) transverse
guidance, (b) φo = 0◦ [385 m/s], (c) φo = 20◦ [365 m/s], (d) φo = 40◦ [344 m/s], (e)φo = 60◦ [323 m/s], (f) φo = 67◦ [317 m/s], (g) φo = 80◦ [309 m/s], and (h) φo = -45◦[424 m/s], where the number in square brackets indicates the mean velocity of the phasestable packet. At this spatial location, under all operating conditions the spectra aredominated by signals from still present phase unstable molecules.

99

Figure 5.13: 3-D Monte Carlo simulation of the OH packet in front of the 23rd stage
under the bunching and slowing (60◦ ) conditions. Molecules are shown in longitudinal
phase space as dots. Projections along the longitudinal spatial and velocity coordinates
are both shown.

spectra obtained at a single specific spatial location as the phase angle is varied. Figure5.12 focuses on LIF measurements made before the 23rd stage. Figure 5.12(a) showsthe expected broad, basically featureless peak as the packet spreads longitudinally while
being transversely guided down the central axis of decelerator under steady-state highvoltages applied to all slowing stages. Figure 5.12(b) depicts the spectra under bunching
operation (0 = 0), creating the multi-peaked structure, as also shown in Fig. 5.11(a),wherein the spatial periodicity of the high voltage stages is imprinted onto the OH pulse.
The phase angle φo is then increased to 20◦ , 40◦ , 60◦ , 67◦ , and 80◦ in Fig. 5.12(c), (d),(e), (f), and (g), respectively.
Careful examination of the data traces reveals small changes in amplitudes andsubtle time shifts in the location of the peaks as φo increases. However, the basic
structure of the spectrum at this spatial location is mostly unchanged, as the signals
from phase unstable molecules dominate the spectra. The downward-pointing arrow
indicates the location of the phase stable packet – however, its presence is substantiallymasked by the signal from unstable molecules. This effect is expected and clearlyseen in Fig. 5.13, where the results of a 3D Monte Carlo simulation after 22 stages

100of deceleration, i.e. in front of 23rd stage, are shown for bunching and slowing with
φo = 60◦ . Figure 5.13 represents a snapshot of the longitudinal phase-space distributionof the molecular packet, with each dot corresponding to the longitudinal position andvelocity of a simulated molecule. Along the horizontal (vertical) axis is shown thespatial (velocity) projection of the distribution, and a horizontal (vertical) dashed line
represents the expected position (velocity) of the phase stable pulse at this instant.
Figure 5.13(a) shows the result of bunching to the 23rd stage, revealing that phase
unstable molecules are “boiling” away symmetrically around the stable molecule bunch.
On the other hand, when φo = 60◦ , Fig. 5.13(b) demonstrates that unstable molecules
are lost in only one direction as the phase stable packet is pulled to lower velocities(consistent with the non-synchronous molecule potentials shown in Fig. 5.3(b)). These
phase unstable molecules lead to dramatic effects in the observed ToF spectra of Fig.
5.12. In the bunching case (Fig. 5.12(b)), because the unstable molecules are lost
symmetrically the bunched peak is only broadened. However, in the case of slowing or
acceleration (Fig. 5.12(c) through (h)), the molecular loss is directional and the center ofthe observed peak is shifted from the expected location in addition to broadening. Fromthe projection onto the horizontal axis it is evident that a peak does indeed occur atthe spatial location expected for the phase stable molecule packet. However, because of
convolution with the detection window this peak is unresolved in the large central peakin the ToF data. Also in Fig. 5.13(b) the manipulation of the input velocity distribution(Vm = 385 m/s, ∆vF W HM ≈ 75 m/s) is beginning to become evident as molecules
are moved from the central region (385 m/s) towards smaller velocities. The stablemolecules have been incrementally slowed from the initial 385 m/s (Fig. 5.12(b)) to 309
m/s (Fig. 5.12 (g)), although this corresponds to only a ∼40 µs shift in the peak location
at this position. Generally speaking, the more stages used, the more straightforward
the interpretation of the spectrum will be, as contributions from the interesting thoughcomplicated dynamics of the longitudinally phase unstable molecules will be reduced.

101
For display purposes, Fig. 5.12(g) shows similar characteristics as we select φo = 80◦ .
At this aggressive slowing phase angle, the selected molecules will actually come to rest
before the end of the 69-stage Stark decelerator. Finally, demonstrating the flexibilityof this experimental technique, phase stable acceleration is depicted in Fig. 5.12(h),where a negative phase angle is utilized (φo = −45◦ ). The OH molecules have been
accelerated from 385 m/s to 424 m/s. However, similar to the slowing data traces, theacceleration spectrum is complicated by signals from molecules with open trajectories(not contained inside the separatrix) in longitudinal phase-space.
We next observe molecular packets after 50 stages of operation. Similar to above,
Fig. 5.14(a) shows the ToF spectrum solely under continuous, transverse guidance fields.
From Fig. 5.15 it is clear that after 50 stages of operation, i.e. observed in front of
the 51st stage, essentially all of the phase unstable molecules have boiled out of thephase stable region. This significantly simplifies the interpretation of ToF spectra since
only the well-characterized stable molecules remain in the peak of interest. Indeed,
under bunching operation experimentally (i.e., φo = 0◦ ), Fig. 5.14(b) shows a single tallcentral peak representing the phase stable molecules that are maintained together with
their net longitudinal velocity unchanged. The much smaller, lower-lying broad feature
again arises from the phase unstable molecules. As the slowing phase angle is further
increased, the behavior of the decelerated packet becomes manifestly clear in the data at
this spatial location unlike after 22 stages. The slowed peak corresponding to the phasestable packet evolves towards later arrival times, demonstrating its steadily decreasing
velocity. In Fig. 5.14(c), (d), (e), (f) and (g) , the slowed peak appears at later time
positions as the velocity of the OH molecules is reduced to 337 m/s, 283 m/s, 220 m/s,
198 m/s, and 168 m/s, respectively. As expected, the peak amplitude decreases as thephase angle gets larger, reflecting the reduced stable area in phase-space. Again, from
Fig. 5.15(b) for φo = 60◦ , we see this expected behavior in the snapshot projectionsof the longitudinal phase space distribution. In the horizontal projection we observe a102

Figure 5.14: ToF LIF measurements of OH molecules before the 51st stage under (a)transverse guidance, (b) φo = 0◦ [385 m/s], (c) φo = 20◦ [337 m/s], (d) φo = 40◦ [283m/s], (e) φo = 60◦ [220 m/s], (f) φo = 67◦ [198 m/s], (g) φo = 80◦ [168], and (h) φo= -45◦ [470], where the number in square brackets indicates the mean velocity of thephase stable packet.

103single phase-stable peak that has been moved towards the rear of the molecule pulse,
while in the velocity projection we see that the input velocity has been significantly
modified, resulting in a single packet of decelerated molecules traversing the slower.
Interestingly, there are distinctly different spectra features generated under bunching versus slowing conditions at this spatial location. In contrast to in front of the 23rdstage where the spectra maintain essentially the same characteristics at all operatingphase angles, after the 50th stage at any non-zero phase angle, the spectrum changes
from a single peak to a distinct triplet of peaks, where one of these structures corresponds to the phase stable packet of molecules that shifts to later (earlier) times forstable deceleration (acceleration) in the ToF observation. Within the presented data’stime resolution, the other two peaks remain essentially at the same temporal locations
for the same sign of phase angle, but these positions undergo a large time shift for opposite sign phase angles, as seen by comparison with Fig. 5.14(h). These structures areprimarily the result of molecules being removed from the center of the free-flight ToF
peak and decelerated (accelerated) to later (earlier) arrival times.
We now examine OH signals observed beyond the final (69th) stage of the decelerator. Here we present further ToF spectra of OH molecules observed moving withvarying velocities at the decelerator exit. Where appropriate, the simulation results areplotted as lines that have been offset vertically from the data trace for viewing clarity. Figure 5.16(a) shows the transversely guided pulse of OH molecules in the trap
region with a mean pulse velocity of 385 m/s. Under bunching conditions selected forthis mean velocity, the expected narrow, single, phase-stable packet shows up as a single
large peak depicted in Fig. 5.16(b), arriving at 1.37 ms. In Fig. 5.16(c) at a phase angle
φo = 45◦ , the slowed peak arrives later in time as the molecular speed is further slowed
to 210 m/s. In fact, the peak has been moved completely off of the residual signals fromphase unstable molecules that form the broad pedestal in Fig. 5.16(b). This complete
separation occurs at a smaller value of φo than that for 51 stages since the entire 69

104

Figure 5.15: 3D Monte Carlo simulation of the OH packet in front of the 51st stage
under the bunching and slowing (60◦ ) conditions. Molecules are shown in longitudinal
phase space as dots. Projections along the longitudinal spatial and velocity coordinates
are both shown.

105

Figure 5.16: Observations of OH molecules at the decelerator exit under (a) transverse
guidance, (b) bunching (φo = 0◦ , v = 385 m/s), (c) slowing (φo = 45◦ , v = 210 m/s),and (d) acceleration (φo = -67◦ , v = 541 m/s). Monte Carlo simulation traces areplotted above the data.

106
stages of the decelerator are utilized for kinetic energy extraction. Finally, phase-stableacceleration is shown in Fig. 5.16(d), wherein the molecular packet is accelerated to 541
m/s, arrives correspondingly earlier in the ToF spectrum and creates the narrow peak at
1.19 ms. Above the data traces, simulation spectra with higher temporal resolution areplotted. Consistency between simulation and measurement is quite good, with locationsof the phase stable packet and relative peak heights demonstrating excellent agreement.
We now focus with higher temporal resolution on measuring solely the phasestable packets generated by the decelerator moving through the trap region. Successful
operation of the decelerator relies on precise knowledge of the molecular packet’s velocity
and location in order to be able to stably slow the packet. Hence, we can easily predict
when the pulse of molecules will arrive in the trap region after it exits the decelerator.
We note, that at low phase angles, when the velocity change of the molecules is not thatlarge, the analytic expression, i.e. sine fit, to the potential energy hill presented in Fig.
5.2(b) enables sufficient accuracy to predict the peak arrival times. However, at higher
phase angles, the deviation between the analytical calculation and the actual shape of
the potential becomes a significant factor such that a numerical determination of thepotential must be used to generate the proper time sequence to operate the decelerator.
In fact, both simulations and measurements demonstrate that the molecular packet
actually becomes phase unstable once φo becomes greater than 40◦ , unless the moreaccurate calculation of the time sequence is used. Finally, as a minor detail, whenthe molecules are exiting the decelerator through the last stage, the voltage on thelast electrodes is switched off, while the voltage on the preceding stage is turned on.
The presence of this non-zero electric field results in a small accelerating force to themolecules. This effect needs to be taken into account to determine accurately how themolecules propagate from the last decelerator stage through the essentially field-free
region into the trap.
In Fig. 5.18, we detect the molecules with 2.5 µs timing resolution per point.

107

Figure 5.17: Observations at the decelerator exit: numbers of molecules in the phasestable packet N relative to the number in the bunched packet No , for varying slowing
and accelerating phase angles. Data points (solid circles) and three-dimensional Monte
Carlo simulations (open diamonds) are shown. For comparison, theory traces generated
from the simple analytic expression (Eq. 5.5, solid line) and the numerical calculation(dotted line) for stable areas in phase space are shown.

108

Figure 5.18: Slowed, accelerated, and bunched molecular packages at the decelerator
exit under varying phase angles. Here, the downward-pointing arrow indicates solely the
position of the bunched (trace (e) φo = 0◦ ) packet, as all observed peaks are the resultof phase-stable packets. Molecular peaks arriving earlier in time result from accelerated
OH packets with corresponding negative phase-angles as denoted in the figure. Peaks
arriving later in time are similarly the result of slowed molecules, generated by using
positive phase-angles.

109
Phase stable packets are measured over a broad range of operating parameters, bothin acceleration, bunching, and slowing for −70◦ ≤ φo ≤ 65◦ . Fig. 5.18 traces (a), (b),(c), and (d) represent OH radicals that have been accelerated from an initial velocityof 385 m/s to a maximum speed of 545 m/s. The phase stable molecules arrive earlier
in time relative to the bunched peak (Fig. 5.18 trace (e)) and create the observed
increasingly narrow structures. The bunched packet corresponds to ∼104 molecules
at a density of ∼106 cm-3. Traces (f), (g), (h), (i), (j), (k), (l), and (m) show phasestable OH packets which have been decelerated to correspondingly lower velocities, as
indicated in the figure. As the molecular packet is increasingly slowed, the molecules
arrive progressively later in time. For φo > 40◦ , the slowed peak is completely separated
from the broad background signals produced by residual phase unstable molecules. In
contrast to the accelerated molecules, the slowed, stable molecular packets broaden as
the phase angle increases, due the reduced molecular speed.
As the molecules slow down towards rest, there is an artifact in detection ofmolecular numbers due to the finite spatial size of the detection window. Basically,slow molecules are counted multiple times as they move through the detection region
where the LIF light pulses are shifted in fine time steps to obtain high resolution TOF
spectra. When this effect is properly de-convolved from the actual underlying signal size,
the number, N , of OH molecules in a phase stable packet as a function of phase angle
can be determined. The measured results (filled circles), normalized to the bunching
peak number No are plotted in Fig. 5.17. In addition, points generated under Monte
Carlo simulations (open diamonds) as well as theory curves from the simple analytic
expression of Eq. 5.5(solid line) and the more accurate numerical calculation (dottedline) are shown. The agreement between experimental results and simulation is quite
good. The main sources of error are peak reproducibility and accurate estimation of thepeak area of the bunched packet as it is temporally coincident with signals from phase
unstable molecules. Disagreement between the measured data points and the simulation

110

Hexapole
Valve Skimmer

Decelerator

Figure 5.19: Schematic of second-generation Stark decelerator, showing increase sourcechamber volume.

results are within experimental error. Deviation of the data points under acceleration
conditions (φo < 0◦ ) are primarily due to insufficient time resolution in the data points,
making peak area estimates difficult.

5.4.2

Second-Generation Decelerator

As briefly mentioned, our second-generation Stark decelerator contains roughly
twice the number of stages as our first (143 in all) and a more suitable vacuum chamber
for producing a good molecular beam as seen in Fig. 5.19. From the discussion in theprevious subsection and in Chap. 7, the advantages of a longer decelerator are clear.
With a longer decelerator one can take out less energy per stage, i.e. use a smaller
φo , and thus decelerate with a larger phase space acceptance, leading to an increasein decelerated molecule number. For example, the second-generation decelerator candecelerate a beam of OH from 380 m/s to rest (produced by the PZT valve) with φo= 30◦ , while the first-generation decelerator would require φo = 60◦ . From Fig. 5.4 we
see this constitutes an order of magnitude gain in decelerated molecule number.
However, with a longer decelerator the slowed molecule number becomes moresensitive to transverse focusing effects. As was seen in Fig. 5.11(d), the molecule number

111
in the bunching peak decrease monotonically as the pulse travels down the slower.
While initially this is mostly due to longitudinally un-stable molecules, the contribution
from transverse loss is non-negligible, especially at the end of the decelerator when themolecules are moving more slowly (in the case of slowing). Simulations for the secondgeneration decelerator show that for bunching after about 70 stages of decelerationof OH very few molecules are lost transversely, and the molecule number per stage
stabilizes 4 . It is important to note that this result depends on the Stark shift of themolecule being decelerated. Specifically, molecules like OH and NH whose Stark shift
becomes linear at small fields are transversely guided significantly better than molecules
like H2 CO whose dipole moment requires a higher field to saturate, i.e. reach the linear
Stark shift regime. The improved guidance is because the transverse guidance comes infront of the electrodes where the fields are often small. Thus, building a long decelerator
to decelerate molecules with a weak Stark effect will probably lead to extreme transverse
losses.
Shown in Fig. 5.20 are ToF spectra taken at several phase angles at the decelerator exit. Comparing the signal-to-noise ratio of this data with similar data in thefirst-generation decelerator shows the obvious molecule number enhancement. Simulation results, also shown in these figures, agree reasonably well with the observed data.
Specifically, all the modulations of observed signal are reproduced in the simulations,
showing that the deceleration dynamics are understood. For these graphs the simulations are normalized by a numerical factor, which matches the non-bunched molecule
number, i.e. the background pedestal, to the observed signal for φo = 0◦ . This same
numerical factor is used for all subsequent simulations. From the panels of Fig. 5.20
it is clear that the simulations properly predict the non-decelerated molecular pulse,
but overestimate the number of decelerated molecules. While this could be due to the4

For deceleration the situation is more complicated, as detailed in the next section, and a small
decrease of the number of molecules with stage number is observed for the entire deceleration process.

112

1.0

= 0 , 380 m/soo
OH LIF Signal [arb. u.]

OH LIF Signal [arb. u.]

1.0

o

o
0.8

0.6

0.4

0.2

0.0

0.8

0.6

0.4

0.2

0.0

2

3

4

5

2

Time from Discharge [ms]
1.0

3

4

5

Time from discharge [ms]
1.0

o

o

= 20 , 224 m/so
OH LIF Signal [arb. u.]o
OH LIF Signal [arb. u.]= 10 , 312 m/s

0.8

0.6

0.4

0.2

0.0

= 30 , 47.5 m/s

0.8

0.6

0.4

0.2

0.0
2

3

4

Time from discharge [ms]

5

2

3

4

Time from discharge [ms]Figure 5.20: OH ToF Data (offset) taken at the exit of the second-generation Stark
decelerator. The results of 3D Monte Carlo simulations are also plotted. All simulations
are normalized by a numerical factor, which matches the simulations results to thebunching pedestal.

5

113
poor modeling of the initial molecule distribution at the skimmer or improper modelingof transverse motion, similar simulations correctly predict the relative heights of thedecelerated peaks for our first-generation decelerator. Furthermore, comparison of ToF
data to simulation at location inside our second-generation decelerator (after 31 and106 stages of deceleration, where we have optical access) do not show the same discrepancies. For these reasons we suspect the decelerated molecule number discrepancy,plotted in Fig. 5.21, is due to an experimental effect present in the longer decelerator
and not an artifact of simulation.
There are several possible explanations for the unexplained molecule number loss,
such as collisions (background or OH-OH), discharge or other decelerator construction
problems, switching problems, and transitions to non-decelerated quantum states. Of
all these, transitions to non-decelerated quantum states appear the most likely since
we have verified proper construction and operation of the decelerator, and estimationof collision effects put the effect many orders of magnitude below the observed loss.
Furthermore, as seen in Fig. 5.21 the discrepancy between simulation and experimentinitially increases with deceleration, but becomes less at higher phase angles and allof these effects (except transitions) should be worse at slower speeds. There are two
types of transitions to non-decelerated states possible inside the decelerator; those thatare driven by the radiation generated at the switching of the fields and non-adiabatic(Majorana) transitions that occur when the molecules are in the relatively low fields orquickly changing fields present directly after and during a deceleration event. Becausethe Λ-doublet splitting is so large for OH these transition most likely occur between
the hyperfine components of the upper doublet. Specifically, a transition between anyof the strongest weak-field seeking states, e.g. |2 ± 2, +i and |2 ± 1, +i, to any of theless strongly decelerated states, e.g. |20, +i and |1 ± 1, +i, and |10, +i, will lead to theobserved loss. With current experimental data it is difficult to determine, which typeof transition (or if both) is responsible for the decrease in decelerated molecule number.

114

Measured Molecule Number

Simulated Molecule Number

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5
0

5

10

o

15

20

25

30

[Degrees]

Figure 5.21: Molecule Number Discrepancy between simulation and experiment.

115
Thus, two experiments are underway. One will test the contribution of Majorana flips byobserving the decelerated molecule number when switching the deceleration electrodes
between an adjustable voltage, V , and high voltage at 12.5 kV + V , instead of between
ground and 12.5 kV. In this way, the OH will remain completely polarized (for highenough V ) and any contribution from Majorana flips can be ruled out. The second
experiment will be to observe the decelerated molecule number for different switching
speeds. Currently, the high voltage is removed and applied to the electrodes in 500 ns.
By slowing down the switching speed via a charging resistor the frequency bandwidthof the generated radiation can be reduced below the minimum state separation.

5.4.3

Transverse over-focussing

As will be detailed in a later chapter (see Chap. 7) the transverse guiding motion
inside the Stark decelerator can play a large role in the total number of decelerated
molecules. One effect, detailed in Fig. 5.22, is the transverse over-focussing of a slow
molecular beam. In this figure, the number of phase-stable molecules predicted bysimulation is shown as function of stage number 5 for increasing phase angle. Here
we see that as expected a higher phase angle leads to a smaller number of decelerated
molecules. While initially the behavior more or less follows what is expected from thelongitudinal phase space dynamics at the highest phase (φ0 = 30.45◦ , 10 m/s) there is a
sharp loss of molecules in the last several deceleration stages. This loss is attributed totransverse over-focussing. That is, because the molecular beam is moving very slowly inthe last few deceleration stages the transverse guiding fields of the decelerator electrodes
have a greater focussing effect on the molecules (see Sect. 4.3) and focus the molecules
so tightly that they collide with the next deceleration stage and are lost. In the case of5

The phase-stable molecule number is determined by counting the number of molecules inside thebox determined by the maximum stable position and velocity excursion as defined in the next chapter.
Since the box does not exactly represent the phase-stable area, the increase in number shown early inthe decelerator is artificial and one expects the phase-stable number to be monotonically decreasing
with stage number.

116
our second-generation decelerator this leads to loss of 20% in the decelerated molecule
number between φo = 30◦ (48m/s) and φo = 30.45◦ (10m/s), which is significantly morethan the expected difference. Furthermore, this number is calculated directly after the
decelerator is switched-off and is thus the ‘best-case scenario’, since experimentation
or trapping requires the molecules to fly out of the decelerator where transverse losses
can be dramatic. This sudden decrease in molecule number has been experimentallyobserved in our experiments as well as in the work of the Berlin group [3] and is extremely
detrimental since trapping of polar molecules requires these low speeds.
To remove this undesired effect from deceleration the transverse focussing of thelast several decelerators stages needs to be reduced. This may be accomplished bychanging the electrode geometry, i.e. altering the electrode radius or moving the rods
transversely further apart, or lowering the voltage on the last deceleration stages. Furthermore, it may be possible to insert focussing elements into the deceleration beamline, whose voltage can be dynamically controlled to compensate, for the over-focussing.
These ideas are discussed in detail in Chap. 7.

o

o

=0

o

[arb. u.]

o

= 10

o

o

= 20

o

o

= 30

o

o

= 30.45

Log

10[Stable Molecule Number]

117

0

20

40

60

80

100

120

140

Stage Number

Figure 5.22: Simulations of the phase-stable molecule number as a function of stage
number in our second-generation decelerator. Note the dramatic decrease in number inthe last several stages for φo = 30.45◦ . This decrease is due to transverse over-focussingof these slow (10 m/s) molecules.

Chapter 6

Acceptance Theory of a Stark Decelerator

6.1

Introduction
The study of ultra-cold polar molecules is a rapidly emerging field as evidenced

by the large number of new experiments being undertaken [138, 24, 27, 72, 68, 119]. The
permanent electric dipole moment, µ, possessed by polar molecules allows for a new typeof interaction in the ultra-cold environment. This electric dipole-dipole interaction (andcontrol over it) should give rise to unique physics and chemistry including novel cold
collision dynamics [18], quantum information processing [42], and a second order phasetransition analogous to spontaneous magnetization of a ferromagnet. When the translational energy of colliding polar molecules becomes comparable to their dipole-dipole
interaction energy, the molecules can dramatically influence each other’s trajectory.
Lack of spherical symmetry in the interaction causes colliding molecules to be attracted
or repelled depending on their relative orientation. Thus, an external electric field,
which orients the molecules, may have a profound effect on the molecular interactions
leading to large changes in observed scattering rates [18]. Another interesting possibility is the observation of criticality in a two or three (3D) dimensional lattice of dipolar
molecules. For a typical polar molecule (∼2 Debye) in a 3D optical lattice with 1 µm
spacing, the critical temperature for the second order phase transition (spontaneouspolarization) is estimated by a 3D Ising model as Tc ∼ 3µ2 /(2πǫ0 kB R3 ) = 200 nK,
where R is the dipole-dipole separation, ǫ0 is the permittivity of free space, and kB is119Boltzmann’s constant [94]. By using extremely polar molecules such as those predicted
to occur from photo-association of free radicals with Rubidium ( µ ∼10 Debye) [111],
or by decreasing the dipole-dipole separation through simply trapping polar molecules
at a sufficient density, the critical temperature may be raised by at least one order ofmagnitude. It is interesting to note that at temperatures significantly higher than thisregime an analog to adiabatic demagnetization refrigeration [97] may become possible,
leading to a new method for further cooling polar molecules, so-called “paraelectriccooling” [71].
The technique of Stark deceleration is especially well-suited for production ofcold polar molecules since it utilizes the Stark shift associated with polar molecules toremove the mean translational energy from a supersonic beam of molecules. Supersonicmolecular beams, utilized extensively in physical chemistry, are capable of producing
intense pulses of molecules with mean velocities of a few hundred meters per second anda small velocity spread about this mean. Therefore, in the moving frame of the molecularpulse the associated velocity spread corresponds to a low temperature distribution (∼1 K). Stark deceleration conservatively removes this mean velocity, leaving behind acold molecule distribution in the laboratory frame. To date this technique has been
employed to produce slow, cold beams of CO [24], OH [27, 127], H2 CO [62] and ND3[23]. Slowing of YbF in an alternate-gradient decelerator has also been demonstrated
in a proof of principle experiment [119].
Due to practical limitations of both the physical dimensions of a Stark decelerator
and the applicable electric fields, only molecules within a specific range of velocities andpositions can be efficiently slowed and trapped. These constraints result in a restricted
phase space acceptance of the decelerator in directions both transverse and parallel tothe molecular beam axis. Thus, for efficient operation care must be taken to match thephase space distribution of the supersonic beam source (emittance) to the phase space
acceptance of the Stark decelerator. This chapter details work from both experiment

120
and model, describing the process of phase space matching between the supersonicbeam source and the molecular decelerator. The focus of the current chapter is topresent a clear understanding of the decelerator efficiency and describe explicitly theexperimental requirements for maximizing it. For the sake of the completeness, there
is some intentional overlap between this chapter and Chap. 5, though greater detail isgiven in the derivations of this chapter. For experimenters new to this field, this chapter
will serve as a useful and practical guide for the design of future Stark deceleration
experiments.
Our experiment centers on the deceleration of the neutral hydroxyl radical, OH.
For low rotational levels the 2 Π electronic ground state of OH is sufficiently described by
Hund’s case (a). Spin-orbit coupling results in the 2 Π3/2 state lying ∼139 cm−1 below
the 2 Π1/2 state. Because of the non-zero orbital angular momentum of the unpaired
electron in OH each total angular momentum, J, state is λ-type doubled, resulting in
two closely spaced (∼1.5 GHz), opposite parity levels in the ro-vibrational ground state.
The application of an electric field readily mixes these states and for the symmetric f
state results in an increase in energy with electric field [23]. The most polarized sublevel of this state, denoted as |2 Π3/2 , J = 3/2, mJ = 3/2, f i, is decelerated by our Stark
decelerator. We detect the presence of OH molecules through the technique of laser
induced fluorescence. A pulsed excitation laser tuned to 282 nm promotes the moleculesalong the A2 Σ1/2 (v = 1, J = 3/2) ←X2 Π3/2 (v = 0, J = 3/2) electronic transition. The
excited molecules subsequently decay primarily (73%) back to the ground electronic
state along the A2 Σ1/2 (v = 1) →X2 Π3/2 (v = 1) pathway at 313 nm. This fluorescenceis collected and imaged onto a photomultiplier tube. By varying the time of the laser
pulse, we sample the OH molecules at a single location at different times to extract
time-of-flight (ToF) information. Careful measurements of the total molecule numbers
before and after the Stark decelerator, and thus determination of the overall decelerator
efficiency as a function of decelerator/supersonic beam operation parameters, provides

121
information pertaining to the phase space matching of the source to the decelerator.
We have developed a simple model and Monte Carlo simulations that provide excellent
agreement with the observed behavior. The remainder of the chapter is organized asfollows. Section 2 briefly describes the operation of a Stark decelerator (for a more
complete review of Stark deceleration see [25, 28]) as well as develops a simple intuitivemodel for phase space matching. Section 3 details our experimental results and compares
them to the model of Section 2 to provide confidence in using it for optimization, while
Section 4 is reserved for conclusions and suggestions for optimal operation of a Stark
decelerator.

6.2

Stark Deceleration
Figure 6.1 shows a schematic diagram of our Stark decelerator, which has been de-scribed elsewhere [28]. Hydroxyl radical molecules produced by the discharge of H2 O in
Xenon (∼1:99 mixture) undergo a supersonic expansion and are subsequently skimmed
to separate the low vacuum source region (∼10−4 torr) from the high vacuum region(∼10−7 torr), which contains the decelerator electrodes and the applied high-voltage
electric fields. After skimming, the molecules are focused by an electrostatic hexapole
field to provide transverse coupling into the Stark decelerator. Once the molecules arecoupled into the Stark decelerator, the slowing process begins. The Stark decelerator isconstructed of 69 slowing stages spaced 5.475 mm apart with each stage comprised of
two cylindrical electrodes of diameter 3.175 mm separated axially by 5.175 mm and oppositely biased at high voltage (± 12.5 kV). Successive stages are oriented orthogonallyto each other to provide guiding of the molecular beam in both transverse dimensions.
The geometry of the slowing stages provides an electric field maximum between the electrodes with the field decreasing away from the electrode center. Therefore, a weak-fieldseeking molecule travelling longitudinally down the decelerator will be decelerated as
it moves into the region between the electrodes and will remain decelerated if the highpulsed valve skimmer+discharge… … …

hexapole

69 pairs of slower electrodes

1st

122
excitation
laser

69th

(a)

(b)

Figure 6.1: Schematic of the Stark decelerator, displaying the pulsed valve and discharge
plates, the molecular beam skimmer, the electric hexapole, and the electrode stages. The
electrode stages alternate orientation (vertical-to-horizontal) as shown in the figure. The
spatial locations indicated by arrows correspond to the locations where molecule number
is measured to determine decelerator efficiency.

voltage is removed before the molecule passes through the electrode’s field. In order tominimize the number of high voltage switches used, all the like-oriented electrodes of thesame polarity are connected to one switch (4 switches total for the entire decelerator).
As a consequence of this switch minimization, when the voltage at each slowing
stage is grounded and the next stage is turned on, the electric field is not completely
removed, but rather becomes the field generated by the alternate set of electrodes.
(The potentials associated with these fields are shown in Fig. 6.2(a). Electrode pairs
are centered at φ0 = -90◦ , 90◦ , 270◦ ) The potential energy experienced by the molecules
Thus, the decrease in the molecular kinetic energy, ∆KE, is given as the difference inpotential energy generated by the two sets of electrodes and is shown as a function ofsynchronous molecule phase angle, φ0 , in Fig. 6.2(b). The synchronous molecule phaseangle, φ0 , is defined as:
φ0 =

180◦
z,
L

(6.1)

where L is the distance between two adjacent stages and the z = 0 is defined to be
exactly between two adjacent stages. Thus, switching at φ0 = 0◦ corresponds to no net
change in molecular kinetic energy, while maximum deceleration (acceleration) occurs
at φ0 = 90◦ (-90◦ ). Though synchronous molecule phases between φ0 = 0◦ and 180◦ lead

123

-1

Potential Energy [cm ]

2.0

(a)
1.6

1.2

0.8

0.4

0.0
-270 -180 -90

0

90

180

270

360

450

phase angle φ 0 [degrees]
2.0-1∆KE [cm ]

1.5
1.0

(b)

∆KE numerical∆KE sine fit

phase stableacceleration

0.5
0.0

-0.5

phase stableslowing

-1.0
-1.5
-2.0
-90

0

90

180

270

phase angle φο [degrees ]Figure 6.2: Phase stable operation of the decelerator. (a) Longitudinal Stark energy
potentials generated by two sets of electrodes, where the solid line represents the potential from one set of electrodes and the dotted line from the other set. Electrode pairs
are centered at φ0 = -90, 90, 270. (b) Kinetic energy loss per stage (∆KE) experienced
by molecules due to switching between the two potentials shown in (a). The dashed line
corresponds to a sine function approximation of the potential energy loss. The shaded
region corresponds to the phase stable area for deceleration φ0 = 45◦

124
to deceleration, only 0◦ < φ0 < 90◦ results in stable slowing because it is only on thepositive sloping ∆KE curve that molecules ahead (behind) of the center of the bunch
are decelerated more (less) than the average. To further investigate the dynamics ofthe molecules within this stable packet, it is useful to define the excursion of a moleculefrom the synchronous molecule position as ∆φ = φ − φ0 . Then with the use of Newton’s
second law and a sine function fit to the change of kinetic energy as a function of φ0(Fig. 6.2(b)) it is trivial to write [139]:d2 ∆φ πCM ax+(sin(∆φ + φ0 ) − sin(φ0 )) = 0,
dt2
mL2

(6.2)

where CM ax is the maximum kinetic energy change per slowing stage, t is time, andm is the molecular mass. This equation is identical to a harmonic oscillator with its
equilibrium position (represented by φ0 ) offset from φ = 0. Thus one expects nonsynchronous molecules to oscillate around the synchronous molecule position inside an
asymmetric oscillator potential. From numerical integration of Eq. 6.2 and the firsttime derivative of Eq. 6.1 one can solve for the stable and unstable regions of phasespace, shown as a function of the synchronous molecule phase angle in Fig. 6.3 for
OH under typical decelerator operation. The most important feature of Fig. 6.3 isthe rapidly decreasing area of stable evolution (i.e. region bounded by the separatrix).
This decrease in stable area is easily understood by analogy to a pendulum driven bya constant torque. As the torque is increased (i.e. the synchronous phase angle isincreased) the equilibrium position of the pendulum is pushed toward the apex, which
reduces the amplitude of oscillation that results in stable oscillatory motion. Thisbehavior is responsible for the separatrix possessing only one cusp for non-zero φ0 . It
is clear that the number of molecules accepted by a decelerator operating at a specificsynchronous molecule phase angle is then given as an area integral inside the separatrix
weighted by the supersonic beam distribution at the decelerator’s entrance.
This phase space “bucket” loading is illustrated graphically in Fig. 6.4. In the125first panel of this figure, ToF data taken at the decelerator entrance (location (a) in
Fig. 6.1) are shown along with a hypothetical ideal supersonic beam of OH molecules(vz,center = 300 m/s, ∆vz = 30 m/s, where vz,center is the pulse mean speed and ∆vz
is the full width at half maximum (FWHM) of the distribution ). For this graph thesupersonic molecular beam parameters were varied by tuning the discharge initiation
time as described previously [80] resulting in the ability to input two vastly different
molecular beams into the decelerator. The first distribution utilized has vz,center = 415
m/s with ∆vz of ∼90 m/s, while the second distribution is centered about 350 m/s
with a spread of ∼80 m/s. For these two experimental distributions and the idealized
case, Monte Carlo simulation results are used to construct graphs of the longitudinal
phase space occupied by the molecules in the remaining panels of the figure. Eachgray point in these graphs represents the location in phase space of a sample molecule,as predicted by our simulations, at the time of loading into the decelerator. The twodashed curves on each graph are projections of the phase space distribution onto therespective axes. Overlaid on these graphs are the separatrix curves for operation at φ0= 0◦ , the so-called bunching condition, and for slowing the molecules to rest at the exitof the decelerator. From these graphs, the importance of creating a cold, i.e. small ∆v,
source of molecules as an input for a Stark decelerator is clear; with a smaller ∆v themolecular pulse spreads less in route from generation at the source to the decelerator
entrance, and thus is more efficiently coupled into the decelerator in both the spatial
and velocity dimensions. The beneficial effect of lowering the pulse mean speed is also
made evident; as the mean speed drops, the required phase angle for slowing is reduced
and thus the separatrix area increases.
As aforementioned, the effects of supersonic beam and Stark decelerator parameters on overall efficiency can be estimated from knowledge of the separatrix overlap withthe supersonic beam phase space. Approximating the area enclosed by the separatrix asthat of the corresponding rectangle whose height (width) correspond to the separatrix

126∆V [m/s]φ0 = 0oφ0 = 30oφ0 = 60oφ [Degrees]

Figure 6.3: Phase stable area (separatrix area) versus phase angle for φ0 = 0◦ , 30◦ , and60◦ . As the phase angle is increased, the stable area is reduced; equivalently, the stable
longitudinal velocity width narrows, and assuming the source completely populates thephase stable region, decreases the number of molecules in the stable packet.

1.00

415 m/s
350 m/s
300 m/s, Idealized

(a)

(b)
Velocity [m/s]

OH density [Linear scale]

127

0.75

0.50

0.25

0.00
300

375

450

525

Distance from skimmer [m]Time [ µ s](d)

Velocity [m/s]

Velocity [m/s](c)Distance from skimmer [m]Distance from skimmer [m]Figure 6.4: Longitudinal bucket loading. Panel (a) represents time-of-flight measurements taken at the decelerator input (location (a) in Fig. ??) for different molecularbeam operating conditions. Filled circles (filled squares) represent molecular beams
with a mean speed of 415 m/s (350 m/s) and a full width at half maximum spread of
90 m/s (80 m/s). The solid line is an idealized molecular beam of 300 m/s mean speedwith a 10% velocity spread. Panels (b), (c), and (d) represent the longitudinal phasespace at the beginning of the Stark deceleration as derived from Monte Carlo simulations. In panel (b) the dashed separatrix represents bunching for 415 m/s, while thelarger solid line is the separatrix for bunching at 370 m/s. The separatrix for slowing apacket of molecules to rest is shown inside the 370 m/s bunching separatrix. In panel(c) separatrix are shown for bunching at 325 m/s and slowing to rest molecules fromthe 350 m/s distribution, while panel (d) shows bunching at 300 m/s and slowing torest for the hypothetical molecular beam.

128
velocity (spatial) bounds and ignoring the minimal convolution due to the short free
flight of the molecules after creation, the overall longitudinal slower efficiency is givenas:ηLongitudinal = N [νz , ∆νz , νz,Bounds (φ0 )] × N [z, ∆z, zBounds (φ0 )],

(6.3)

with

N (x, ∆x, xBounds ) =R XU pperbound(φ0 )XLowerbound(φ0 ) f (x, ∆x)dx
R∞,−∞ f (x, ∆x)dx(6.4)

where z refers to the dimension along the decelerator’s axis (longitudinal axis) andf (x, ∆x) is the molecular distribution in the appropriate dimension. The integration
bounds correspond to the separatrix maxima along their appropriate dimension and are
an explicit function of the synchronous molecule phase angle. From Fig. 6.2(b) we seethat for φ0 = 45◦ , non-synchronous molecules with phase angles between 45◦ and 135◦
will be slowed more than the synchronous molecule (shaded region). In general, themaximum stable forward excursion is then given as:∆φM ax,+ (φ0 ) = 180◦ − 2φ0(6.5)

Unfortunately, a closed form solution for the maximum backwards excursion, ∆φM ax,− (φ0 ),does not exist, however, it can easily be found numerically or estimated by the negativeof Eq. 6.5 since both the maximum stable forward and backward excursions have thesame average derivative as a function of φ0 . From Eq. 6.2 we find the work done, W, inbringing a molecule starting at ∆φM ax,+ (φ0 ) with the same speed as the synchronous
molecule, i.e.at the separatrix cusp, back to the synchronous molecule position, i.e.
∆φ = 0◦ , is:−CM axW =π

Z 0∆φM ax,+ (φ0 )(sin(∆φ + φ0 ) − sin(φ0 ))d∆φ

(6.6)

129
Clearly, this molecule will possess the maximum stable velocity and we thus have theseparatrix velocity bound as:s∆νM ax (φ0 ) = 2

CM ax
π(cos(φ0 ) − ( − φ0 ) sin(φ0 )).
mπ
2

(6.7)

Now, assuming the molecular distributions are adequately described by the form:


2 

 x − xcenter   ∆xf (x, ∆x) = exp 

√

2

(6.8)

ln(2)

where ∆x refers to FWHM and xcenter to the value at which the distribution is centered.
Eq. 6.3 becomes:

1ηLongitudinal = [Erf
4
−Erf
×[Erf
−Erf

!

(vz,design − vz,center ) + ∆νz,M ax (φ0 ) q
ln(2)
∆νz
2

(6.9)

!

(vz,design − vz,center ) − ∆νz,M ax (φ0 ) q
ln(2) ]
∆νz2!

(zdesign − zcenter ) + Lπ ∆φM ax,+ (φ0 ) q
ln(2)
∆z
2

!

(zdesign − zcenter ) − Lπ ∆φM ax,+ (φ0 ) q
ln(2) ]
∆z2

Where Erf (x) refers to the Error function, and vz,design and zdesign have been introduced to account for the possibility of designing the decelerator pulse sequence to selectmolecules which are not at the peak of the molecular distribution. If molecules areloaded at the peak of the distribution, i.e. vz,design = vz,center and zdesign = zcenter , andwe approximate ∆φM ax,− (φ0 ) = −∆φM ax,+ (φ0 ), we find the simple result:

ηLongitudninal = Erf!

∆νz,M ax (φ0 ) q
ln(2)
∆νs
2





φ0 q
L(1 − π/2)× Erfln(2) ,∆x2(6.10)

which is useful for estimating expected decelerator efficiency. Plots of Eq. 6.10 withloading at the peaks of the distributions are shown in Fig. 6.5 versus both ∆vz (Fig.

130
6.5(a)) and vz,center (Fig. 6.5(b)), where the value of ∆φM ax,− (φ0 ) has been found numerically. In these graphs the slowing phase angle was chosen to bring the molecules torest at the decelerator’s exit. Again the importance of a low central velocity and small
spread about this mean is evident. The most striking feature of these graphs is theflattening of the longitudinal slower efficiency for small velocity spreads. In fact for a
sufficiently narrow velocity spread the mean speed of the pulse becomes to some degree“unimportant” as the supersonic beam emittance fits entirely inside the decelerator acceptance for most operating conditions. It is important to note that similar effects canbe achieved by changing the characteristics of the Stark decelerator. By increasing thestage-to-stage spacing one enlarges the spatial bounds of the separatrix resulting in more
decelerated molecules; however, increasing the distance without increasing the applied
electric field will result in less efficient transverse guiding and a smaller stable phasespace area transverse to the molecular beam axis. In contrast, increasing the applied
electric field enables deceleration of molecules with larger velocity spreads relative tothe synchronous molecule speed, and thus increases the overall slower efficiency. However, the technical challenges associated with going to voltages higher than the present
operating conditions are significant. Therefore a more practical alternative maybe toincrease the number of slowing stages to reduce the required phase angle.
Efficient operation of a given decelerator/supersonic beam combination occurs
when Eq. 6.10 is maximized at the desired final speed. Since the molecular distribution
and the desired final speed are assumed fixed the only parameter which can be variedis vz,design , the initial speed for which the decelerator pulse sequence is designed. By
decreasing vz,design the required slowing phase angle is decreased and the separatrix area
increased. However, if vz,design is lowered too far below vz,center the molecular beamphase space overlaps poorly with the decelerator input separatrix and slower efficiency
suffers. Thus, the most efficient slower operation will occur for a vz,design below vz,centerby an amount that depends on the ∆vz of the pulse and the desired final speed (typically

131

(a)

(b)

Figure 6.5: Simple model for longitudinal bucket loading. Panel (a) shows the effectof varying the supersonic beam’s velocity spread on longitudinal efficiency for several
initial speeds, while panel (b) shows the longitudinal efficiency as a function of initial
pulse speed for several velocity spreads, where the spread is expressed as a fraction of
the mean speed.

132
1 to 10 percent of vz,center for slowing to rest). Fig. 6.6 shows the result of maximizing
Eq. 6.10 for the given experimental distributions as well as the idealized source. In thisgraph the amount of velocity detuning from vz,center for most efficient operation is shown
as a function of desired final speed. For final packet speeds close to the initial pulse
speed (right part of graph) it is most efficient to utilize a phase angle of zero (bunching)because of its large acceptance and vary vz,design . However, as the final velocity isdecreased, this procedure eventually leads to a mismatch between the supersonic beamemittance and the decelerator acceptance, and it becomes more efficient to decelerate
molecules nearer the center of the supersonic beam distribution, leading to the sharp“transition” peaks. The position of this transition for our decelerator is at a detuningequal to approximately 15% of the pulse spread and is evident in the figure by thesudden drop in velocity detuning. As slower molecules are desired, the effect of thedecreasing acceptance of the decelerator (with increasing φ0 ) is combated by selecting
slower initial molecules, which require less slowing. This effect is evident by the increaseof detuning with slower final speeds. The gain in efficiency by optimizing decelerator
operation according to Eq. 6.10 depends sensitively on both the molecular beam anddecelerator parameters, but in the case of our experimental conditions ranges from only
a few percent to well over a factor of 100 improvement1 .
While Eq. 6.10 describes the efficiency of decelerator operation in the longitudinal
dimension, it says nothing about evolution transverse to the decelerator axis. In thisdimension the molecular distribution is centered about zero in both space and velocity
thus Eq. 6.3 becomes:

ηT ransverse = N [0, ∆νx , νx,Bounds (φ0 )]N [0, ∆x, xBounds (φ0 )]1

Since 415 m/s molecules cannot be slowed to rest in our decelerator, detuning the pulse design
speed to decelerate molecules with 405 m/s (the maximum speed which can be brought to rest in ourmachine) or less initial speed would, by definition, be an infinite gain in efficiency. Thus the usefulnessof this technique for hot sources is evident.

133
×N [0, ∆νy , νy,Bounds (φ0 )]N [0, ∆y, yBounds (φ0 )],(6.11)

where x and y refer to the dimensions perpendicular to the decelerator axis. Assuming
distributions given by Eq. 6.10 we have:ηT ransverse = Erf!

∆νx,M ax (φ0 ) q
ln(2) Erf
∆νx
2

Erf

!



∆x(φ0 ) ∆x q
ln(2)
×
2

∆νy,M ax (φ0 ) q
ln(2) Erf
∆νy
2



!

∆y(φ0 ) q
ln(2) .
∆y2

(6.12)The spatial acceptance of the decelerator is set by the decelerator rod spacing, thus
∆x and ∆y ∼ 1 mm. Because the transverse evolution of molecules through the Stark
decelerator is a complicated dynamical process that depends sensitively on the operating
conditions it is not possible to give a simple expression for ∆vx,M ax(φ0 ) and ∆vy,M ax(φ0 ) .
However, for most deceleration experiments to date, it can be estimated as ∼3 m/s withminimal error [28, 37]. The behavior of Eq. 6.12 is analogous to that of Eq. 6.10. As
the velocity (spatial) spread of the source is lowered or the maximum stable velocity(position) is increased by a decelerator change, the efficiency of the slower grows. It is
the aim of proper hexapole focusing to “mode-match” the supersonic beam into thistransverse decelerator acceptance .

6.3

Experiment
Time of flight measurements of OH molecules taken at the exit of the decelera-

tor (location (b) in Fig. 6.1) are shown in Fig. 6.7 for various decelerator operating
conditions. Open circles represent operation in transverse guidance mode, where thedecelerator array is biased at high DC voltage, resulting in a potential minimum along
the decelerator axis. This mode of operation does not affect the molecules’ longitudinal
speed, but provides transverse guidance through the length of the decelerator. The
remaining traces show the stable packet of molecules produced when switching the decelerator with non-zero phase angle. As the slowing angle is increased the packets arrive134(Vz, design - Vz, center) [m/s]-40415 m/s-30-20
350 m/s-10
Ideal source, 300 m/s
0
0

100

200

300

400

Desired Final Speed [m/s]
Figure 6.6: Decelerator optimization condition. The optimum detuning of the design
speed from the pulse’s mean speed as a function of desired final speed shown for threedifferent supersonic beam distributions.

135

OH density [Linear scale]

1.00(a) transverse guidanceo(b) φo = 0 , 370 m/s

(b)
(a)

(c)

o

(c) φo = 10 , 336 m/s

0.75

o

(d) φo = 20 , 300 m/so(e) φo = 30 , 259 m/s(d)o(f) φo = 40 , 211 m/s

0.50

o

(e)

(g) φo = 50 , 148 m/s
(f)

0.25

o

(h) φo = 60 , 33 m/s
(g)

(h)

0.00
1.2

1.6

2.0

2.4

2.8

Time [ms]Figure 6.7: Molecular packets at the decelerator exit (location (b) in Fig. 6.1) for
varying phase angles. Peaks arriving later in time are the result of slowed molecules.
From comparison of the number of molecules in these packets to the total inputtedmolecular number, decelerator efficiency can be determined. Solid lines are to guide the
eye.

136
later in time signifying a smaller mean speed. The spreading of the ToF pulse due tofree flight and convolution with the detection window is also evident. Integration of thedeconvolved ToF pulse reveals the total molecular number in each phase stable packet,
and is shown in Fig. 6.8, where the data points have been normalized to the bunchingcondition. In this graph the total molecular number is plotted versus the final speedof the molecular bunch for the three different operating conditions represented in Fig.
6.4. The filled circles represent data taken for slowing sequences designed for an initial
speed of 415 m/s. For this data the supersonic beam was operated to yield a distribution centered at 415 m/s with a 90 m/s FWHM. The filled squares in Fig. 6.8 represent
loading molecules from the same distribution into a phase bucket designed for 370 m/s
initial speed. For the data represented by the filled triangles, the supersonic beam wasoperated to yield a pulse centered at 350 m/s with an 80 m/s spread. In this data, thedecelerator was operated for a 325 m/s initial speed. The solid lines in this graph represent the simple theory of Section 2, while the open points correspond to the results of
our Monte Carlo simulations. Despite the assumptions made in deriving Eq. 6.10 we see
the theory of Section 6.2is quite accurate. The only noticeable discrepancy occurs forthe 415 m/s curve, where it appears that deviation in the experimental bunching point(which is used to normalize the graph) is responsible for the offset. From this graphwe see that operation at 325 m/s is clearly more efficient at producing slow molecules.
Again, the importance of developing a slow cold source as the input for efficient Stark
deceleration is clear.
By comparing the total molecule number observed in each packet to the total
number of molecules input into the decelerator (Fig. 6.4(a)) the overall efficiency of the
Stark decelerator can be determined and is shown in Fig. 6.9 for the 370 m/s case as
filled squares. Overlaid on this graph is the model of Section 2. For this theory thetransverse parameters ∆x and ∆y = 1.5 mm and ∆vx and ∆vy = 12 m/s were used in
Eq. 6.12 [?]. It is remarkable to note that this simple theory predicts the overall slowingRelative Stable Molecule Number

137

Design speed
415 m/s
370 m/s
325 m/s

1.00

0.75

0.50

0.25

0.00
400

300

200

100

0

Final Velocity [m/s]
Figure 6.8: The stable molecule number normalized to φ0 = 0◦ for three cases of operation. Circles represent operation with a supersonic beam centered at 415 m/s with a 90
m/s FWHM and a deceleration sequence designed for molecules at 415 m/s. Squares
represent a deceleration sequence designed for 370 m/s molecules from the same beam
distribution. Triangles are for deceleration designed for 325 m/s molecules from a distribution centered at 350 m/s with an 80 m/s spread. Filled points represent data, while
open points represent results from the Monte Carlo simulation. The solid lines are theresults of the model of Section 6.2. Typical molecule numbers per stable packet range
from 104 -106 , depending on the φ0 .

138efficiency quite accurately with no free parameters. Clearly, this level of agreement with
our model gives confidence in using it as an optimization tool.

6.4

Conclusions
We have developed a model of the process of phase space matching between asupersonic molecular beam emittance and a Stark decelerator acceptance. The abilityof this model to predict with accuracy the efficiency of the Stark deceleration process
gives confidence in its use as an optimization tool. While optimization according tothis model gives small gain for supersonic sources with low mean kinetic energy andsmall velocity spreads, it can lead to dramatic improvement for molecular beams thathave either a high kinetic energy, a large velocity spread, or both. As the technique ofStark deceleration is extended to more exotic molecular species, which, because of theirlarge mass, internal modes or process of creation result in non-ideal supersonic beams
for input into a Stark decelerator, maximization of the decelerator operation efficiency
will be essential.
Clearly, it is the aim of the experimenter to maximize the product of Eqs. 6.10
and 6.12 for the most efficient decelerator operation. The efficiency of the decelerator
depends sensitively on both the supersonic beam and Stark decelerator parameters. The
importance of developing a cold source as the input for any Stark deceleration experiment cannot be overstated since this is the only real cooling in the slowing experiment
and, for a given decelerator, sets the attainable efficiency. While increasing the dimensions of slowing stages and their separations to allow deceleration of more molecules isa viable way to increase the deceleration efficiency, it is met with increasing technicalchallenge as the electrode voltage must also be scaled. Perhaps a more reasonable alternative is to increase the number of slowing stages, thus dropping the amount of energy
removed at each stage, resulting in a larger phase space acceptance. A reasonable design
goal is to include enough stages in the decelerator so that molecules can be decelerated

139

Slower Efficiency [%]

2.0

1.5

1.0

0.5

0.0
400

300

200

100

0

Final Velocity [m/s]
Figure 6.9: Stark deceleration efficiency. Efficiency of the Stark decelerator as a function of final speed for slowing designed for an initial speed of 370 m/s taken from the
distribution centered at 415 m/s with a 90 m/s FWHM. Filled squares are data points
obtained from the results of Fig. 6.8 normalized by the results of trace (a) in Fig. 6.4.
The solid line is the model of Section 6.2 including both longitudinal and transverse
efficiencies.

140
to the desired final speed with a phase angle of φ0 ≤ 45◦ . Also, positioning the source as
close as possible to the input of the decelerator provides efficient spatial coupling in thelongitudinal dimension. Finally, for a given molecular beam/decelerator combination
the optimum phase angle and initial pulse design speed, vz,center , for slowing can befound from Eq. 6.10.

Chapter 7

Alternative Deceleration Techniques

In the work of previous chapters it was assumed that all motion in transverse
dimensions was stable up to some maximum excursion velocity from the beam center. While this is certainly a reasonable approximation for most cases, there are several
important instances where this assumption does breakdown. One of these cases is transverse over-focussing, detailed in Chap. 6, which occurs when the decelerated molecules’
speed becomes so low that the decelerating electrodes focuses the molecules too tightly(transversely) and they crash into the next deceleration stage. Furthermore, transverse
over-focussing appears to have a larger effect in long decelerators, where the molecules
may travel at low speeds for many stages. Nonetheless, there are several motivating factors for constructing a longer decelerator. First, a longer decelerator allows less energy
per stage to be removed and consequently a large longitudinal phase space acceptance.
Second, a longer decelerator may allow molecules to be decelerated with an unfavorable
Stark shift to mass ratio, i.e. molecules that have a small effective dipole moments or
are heavy. Thus, it is important for future deceleration experiments that phenomenonof over-focussing (and other transverse effects) be addressed.
The fact that the transverse guidance of the molecular beam arises from the same
electrodes that provide the deceleration means that the longitudinal and transverse
motions are necessarily coupled. While this phenomenon is well understood in the fieldof accelerator physics [139], it was first pointed out in the context of Stark deceleration

142
in an excellent paper from the Berlin group [126]. The basic effect can be seen in
the left column of Fig. 7.1, where the longitudinal phase space of a bunched OH
packet in the |2 ± 2+i state is shown for increasing number of deceleration stages.
In these graphs the dark lines represent the separatrix, separating stable decelerationfrom unstable motion, as calculated from Eq. 6.2 and each dot represents the position inphase space of a simulated molecule. In the absence of coupling between the longitudinal
and transverse motions, one would expect the entire area inside the separatrix to beoccupied. Therefore, the structure evident in these graphs is evidence of the importanceof the transverse motion.
In the left column of Fig. 7.1 the coupling of longitudinal and transverse motions
is responsible for two effects1 . First, in the center of the stable area, i.e. near thesynchronous molecule, the density of stable molecules is less than the surrounding area.
This is because molecules that spend time near the synchronous-bunched molecule experience very little of the transverse guiding forces. Clearly, this effect is by no means
dramatic and is only discernable for an exceedingly large number of stages. Furthermore, this effect is even less important for phase angles typically used for deceleration,
since for these switching sequences the synchronous molecule experiences more of thethe transverse guiding forces than it does during bunching. Nonetheless, it can still
be of moderate importance since the peak of the molecular beam is often matched tothis region. The second effect, which is much more evident, is the absence of molecules
at intermediate distances from the synchronous molecule as shown in the left columnof Fig 7.1. This so-called ‘halo’ is due to parametric amplification of the transverse
motion and is similar to the effects seen in cold molecule storage rings [37]. Essentially,
the longitudinal oscillation frequency of a molecule in this region is matched to thetransverse oscillation frequency, leading to amplification of the transverse motion, and1It should be noted that for these phase space simulations the input molecular beam had a flat
longitudinal position and velocity distribution, which helps to exaggerate the effects.

143
consequently, loss.
Clearly, the effects of the coupling between longitudinal and transverse motion
are detrimental to efficient operation of a Stark decelerator. Furthermore, these effects,
which reduce the total number of decelerated molecules, will be even more damaging
for decelerating many of the interesting molecules with an unfavorable Stark shift tomass ratio. Fortunately, the transverse and longitudinal motions can essentially be
decoupled by introducing a transverse focusing element to the deceleration beam linethat overwhelms the transverse focusing provided by the deceleration electrodes. This
technique also has the advantage of a larger stable region in the transverse phase space
further enhancing the decelerated molecule number. The remainder of this chapter deals
with methods of implementing a transverse focusing element to decouple the longitudinal and transverse motion. The first two subsections describe methods, which canbe implemented with current decelerators with moderate success, while the last subsection present an improved design for a Stark decelerator that solves the problem of
longitudinal and transverse coupling.

7.1

Decelerator Overtones
The simplest method of introducing a transverse focusing element to the decel-

erator beam line is to let the molecules fly through an energized deceleration stage
without removing the field. In this way, molecules experience the transverse focusingof the entire stage without having their longitudinal motion affected. Because traditional longitudinal phase stability requires the switching of the fields to occur on an
upward slope of the molecular potential energy, i.e faster molecules are slowed morewhile slower molecules are slowed less than the synchronous molecule, it is necessary to
de-sample the bunching switching rate by an odd factor (3,5,7...) – so called decelerator
overtones [125]. For convenience we define the quantity, S, (like in Ref. [125]) given as
S = vo /vSwitch , where vo is the synchronous molecule velocity and the switching speed,

144

420

143 Stages
S=1

143 Stages
S=3

Vz [m/s]

410
400
390
380
370
360
350
340

Gain = 1.18
0420

Vz [m/s]410

5300 Stages
S=1

420
410

400

400

390

390

380

380

370

370

360

360

350

350

340

Vz [m/s]

410

Gain = 1.26

340
0

420

300 Stages
S=3

5

600 Stages
S=1

0

420
410

400

400

390

390

380

380

370

370

360

360

350

350

340

600 Stages
S=3

Gain = 1.33

340

0z [mm]

55

0

5z [mm]

Figure 7.1: Monte Carlo results for the Longitudinal phase space of bunched molecules.
The left column shows φo = 0◦ for S = 1 at several decelerator lengths, while the rightcolumn shows φo = 0◦ for S = 3 for the same decelerator lengths. In the bottom left ofeach S = 3 figure is the value for the molecule gain S = 3 produces over S = 1.

145
vSwitch , is given as the stage spacing, L, divided by the switching time-interval. Though,here we expand the bunching case, that was considered in Ref. [125], to include actual
deceleration, the above definition of S is still valid. That is, neither vo nor vSwitch is aconstant for φo > 0◦ but, S is. With this definition we see that traditional deceleration
can be described by S = 1, while the method of de-sampling the switch rate by a factor
3 is described by S = 3. These two methods of deceleration can be seen in Figs. 7.2(b) and (c), where their respective switching schemes are shown for φo = 0◦ . In thisfigure, it is seen that by switching at 1/3rd the rate the molecule packet flies through a
deceleration stage that is energized, and thus, sees the enhanced transverse guiding.
Longitudinal phase space simulations of bunching for S = 3 are shown in the rightcolumn of Fig. 7.1. In these graphs we see that no structure due to couplings between
the longitudinal and transverse motions is present. Also evident in these simulations isthat the region of longitudinal phase stability for S = 3 is reduced compared to S = 1.
This is because the maximum stable velocity as calculated from Eqs. 6.2 and 6.6 depends
on the spacing between deceleration stages as L−1/2 and thus the separatrix velocity√
bound is reduced2 by 3. Nonetheless, the absence of coupling to the transverse motion
leads to a larger number of molecules for the φo = 0◦ case shown.
Fig. 7.3 shows time-of-flight (ToF) data (offset) taken in our second generation
decelerator for S = 3 with increasing phase angle along with the results of 3D Monte
Carlo simulations. Qualitatively, the ToF signal is no different than for S = 1. A
decelerated packet is evidenced by a peak in the ToF data that moves later in time witha reduced amplitude that is governed by the longitudinal phase stability. Shown in Fig.
7.4(a) are the decelerated molecular packets for S = 3 and S = 1 with increasing phaseangle. For each successive packet the S = 3 phase angle increases by 10◦ from φo = 0◦ ,while the S = 1 phase angle increases by 10/3◦ from φo = 0◦ . In this way the packets
2

Physically, this is because the molecules fly longer between deceleration stages, and thus, thevelocity mismatch can lead to a larger accumulation of spatial mismatch

146

(a)

(b)

(c)

(d)

Figure 7.2: Deceleration Schemes. (a) Potential energy shift of polar molecules in the
Stark decelerator. The dotted (blue) curves show the potential energy shift when thehorizontal electrodes are energized, while the dashed (red) curves show the potential
energy shift when the vertical electrodes are energized. Deceleration proceeds by switching between which electrodes are energized, and thus the corresponding potential energy
curve. In panels (a)-(c) the thick black line indicates which of the potentials is experienced by the molecules. The (orange) circles indicates a switching event. (b) Traditional(S = 1) operation at φo = 0◦ . For phase stability the switching always occurs when themolecules are on an upward slope, and as such the molecules are never directly between
an energized electrode. Thus, the maximum transverse guiding is never realized. (c)First overtone operation (S = 3) at φo = 0◦ . By switching at 1/3 the S = 1 rate, themolecules are allowed to fly directly between an energized stage, and thus, experienceenhanced transverse guiding. (d) Optimized first overtone operation (S = 3+ ) at φo =0◦ . Operation proceeds identically to S = 3, except that the grounded set of electrodes
is energized when the molecules are directly between the already energized electrodes.
In this way the molecules experience one more stage of maximum transverse guiding foreach slowing stage. Note that the grounded-set of electrodes is switched on when themolecules are directly between the energized electrodes to minimize the un-bunchingeffect.

147
are decelerated to roughly the same velocity – there is a slight difference because thetotal number of stages in our decelerator is not an exact multiple of 3. In Fig. 7.4
the de-convolved total molecule number for each of these packets is plotted versus final
speed. Here we see that while the S = 3 method dominates over S = 1 for small phaseangles 3 it quickly falls off as the deceleration becomes more aggressive – in fact by
224 m/s the S = 1 molecule number is already larger than the S = 3. This behavior is
expected since the phase angle used to decelerate to 224 m/s is 60◦ , while the required
S = 1 phase angle is only 20◦ . That is, though the S = 3 longitudinal phase bucket
does not exhibit any structure due to coupling to the transverse motion, it is so muchsmaller in enclosed area than the S = 1 that its total molecule number is less.
Since this decrease in the S = 3 molecule number is purely due to the longitudinal
phase space, it might be expected that building a longer decelerator, which will allow
operation at a lower phase angle, would allow one to enjoy the gain in molecule number
produced by S = 3. Simulations of this kind are shown in Fig. 7.5, where the numberof molecules present after deceleration at φo = 20◦ for an increasing number of stages
is shown4 . In this graph, the number next to a data point represents the numberof deceleration stages used. Initially the decelerated molecule number is relatively flatwith final velocity. However, after about 400 stages (224 m/s) the number of decelerated
molecules begins to decreases and dramatically falls off after 550 stages (150 m/s) withvery few molecules surviving below 100 m/s. This is because for S = 3 the decelerated
molecules must fly through an entire stage without seeing any guiding force (see Fig.
7.2(c)), and once the molecules are at slower speeds they can spread out transversely
or be over-focused and crash into the rods before being transversely guided again.
The transverse loss responsible for the extreme loss in molecule number for S = 3
3

The factor of ∼3 gain seen in the data is not reproduced in the 3D Monte Carlo simulations, which
predict a gain of roughly 1.5. The unexpected gain may be evidence of Majorana-like transitions in the
decelerator since S = 3 involves less switching of the fields.
4
In these simulations φo = 20◦ was chosen because it produced the most gain over S = 1 in our
deceleration experiments.

148

1.8

= 0o

2.0

(a)

2.2

2.4

2.6

2.8

3.0

3.2

OH LIF Signal [arb. u.]

Time from discharge [ms]
o

= 60o

o

OH LIF Signal [arb. u.]

OH LIF Signal [arb. u.]

o

1.8

= 30o

2.0

(b)

2.2

2.4

2.6

2.8

3.0

3.2

Time from discharge [ms](c)

1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8Time from discharge [ms]Figure 7.3: ToF data (offset) of OH molecules decelerated using S = 3. The results of
3D Monte Carlo simulations are also shown (red).

3.0

OH LIF Signal [arb. u.]

380m/s

2.5

S=1
S=3

358m/s(a)

336m/s2.0
1.5

312m/s286m/s1.0257m/s0.5
0.0

224m/s

2.5

2.6

2.7

2.8

2.9

3.0

Time from Discharge [ms]

3.1

3.2

Relative Molecule Number [arb.u.]

149

3.0

S=1
S=3

2.5

(b)

2.0
1.5
1.0
0.5
0.0

380 360 340 320 300 280 260 240 220

Final Velocity [m/s]

Figure 7.4: Comparison of deceleration using S = 3 and S = 1. (a) Decelerated OH
packets. (b) De-convolved, integrated molecule number in the packets shown in panel
(a).

150

Relative Molecule Number

1

142

200 300o400

o = 20 , S = 3

450
500

0.1

550
575

0.01
600
610
625

1E-3
300

250

200

150

100

50

0

Final Velocity [m/s]
Figure 7.5: Decelerated molecule number using S = 3 and φo = 20◦ for increasing
decelerator stage number. The number next to each data point is the number of stages
used. Because of transverse over-focussing essentially no molecules survive below 100
m/s.

151
deceleration occurs to a lesser degree in traditional deceleration, as detailed in Chap.
5. Because of this dramatic decrease in molecule number at low speeds, the usefulnessof slowing on S = 3 is likely limited to experiments that do not require the lowest
velocities, such as microwave spectroscopy and collision experiments. Nonetheless, byusing a combination of deceleration/bunching at S = 3 and S = 1 it may be possible toextend some of the gain of S = 3 to lower speeds.

7.2

Modified Decelerator Overtones
One possibility for extending the gain of overtone deceleration to lower speeds

is the use of what we have termed a ‘modified decelerator overtone’, denoted by anadditional plus sign, i.e. S = 3+. Deceleration in this manner is shown in Fig. 7.2(d)for φo = 0◦ . In this method, deceleration proceeds identically to the regular S = 3
except that when the synchronous molecule is directly between the energized guiding
stage the next stage is energized. While this does create a slight anti-bunching effect,
i.e. molecules in front of the synchronous molecule gain a small amount of energy, it
provides an extra stage of transverse guidance when compared to S = 3. Experimental
results of this method of slowing are shown in Fig. 7.6 for comparison with deceleration
using both S = 1 and S = 3. In Fig. 7.6(a), we see that, as consequence of the switchingsequence, even operation at φo = 0◦ leads to deceleration. Also, from the comparison
with the S = 1 bunching peak shown, we see the usual gain associated with overtone
deceleration. In Fig. 7.6(b), we see that operation with S = 3+ provides slightly more
molecules than S = 3, and decreases with increasing phase angle in accordance with thelongitudinal phase-stability dynamics. Presumably, the increase in molecule number
for S = 3+ over S = 3 is due to the extra transverse guidance, which for the higher
velocities packets we measured should lead to a larger transverse acceptance.
To determine whether the extra stage of transverse guidance would combat theOH LIF Signal [arb. u.]

2.5

(a)

o

= 0 , 380 m/s, S = 1
S = 3+ooo= 0 , 325 m/s

2.0o= 10o, 315 m/s

1.5

o

o

= 20 , 299 m/s
o

1.0

o

= 30 , 281 m/so

0.5= 40o, 259 m/so= 50o, 234 m/s
o

0.0

o

2.6

2.8

3.0

= 60 , 207 m/s

3.2Time from discharge [ms]3.4

Relative Molecule Number [arb.u.]

152

3.0

(b)

S=1
S=3
S = 3+

2.5
2.0
1.5
1.0
0.5
0.0

380 360 340 320 300 280 260 240 220

Final Velocity [m/s]

Figure 7.6: (a) Decelerated OH packets produced using the S = 3+ modified overtone.
Also, shown for comparison is the bunching packet for operation at S = 1. (b) The
de-convolved, integrated molecule number observed at the decelerator exit for S = 3+,
S = 3, and S = 1.

153
over-focussing effects5 we performed simulations of S = 3+ deceleration at φo = 20◦
for varying number of stages of deceleration. The results of these simulations, shownin Fig. 7.7, are similar to the results for S = 3. Namely, as the decelerator length
is increased, and the molecules’ speed reduced, there is a dramatic molecule number
loss for velocities below 100 m/s – in fact in our simulations we could not observe any
molecules below 50 m/s. Again, transverse over-focussing of the molecular packet by
the deceleration electrodes is responsible for large losses in the decelerator, and it seems
that more drastic measures are required to overcome the transverse focussing losses.

7.3

Quadrupole Guiding Decelerator
It may be possible to uncouple the longitudinal motion inside the Stark decelerator

from the transverse motion by redesigning the decelerator electrode geometry. One of
the simplest redesigns, which we call the quadrupole guiding decelerator (QGD), isshown in Fig. 7.8(a). In this decelerator, a quadrupole guiding stage (Fig. 7.8(b)) isinterleaved between each deceleration stage. While it may not be necessary to have aquadrupole guiding stage between each deceleration stage (especially in the beginningof the decelerator), it simplifies the analysis and will be used here. The switching of
the electric fields inside a QGD is similar to a traditional decelerator operated with
S = 1 as shown in Fig. 7.8(c), where the potential energy experienced by a molecule
decelerated at φo ≈ 45◦ is shown as a thick black curve. In this figure the quadrupole
guiding electrodes are omitted for clarity. Note that the quadrupole guiding stages arealways energized and their center coincides with the φo = 0◦ point, while φo = 90◦
occurs between the deceleration electrodes.
Clearly, deceleration with a QGD enjoys the same longitudinal phase-stability asa traditional decelerator. However, because the decelerating electrodes are most likely
5

While this may seem counter-intuitive, in some cases when the transverse over-focussing is not
too strong the addition of another focussing element can change the sign of the molecules transverse
velocity, keeping the beam from colliding with the electrodes.

154

Mol. Number Rel. to 3+=0o1

φ3+=20o301m/s256m/s203m/s0.1

0.01
151m/s

150

200

250

300

350

400

Number of Stages
Figure 7.7: Decelerated molecule number using S = 3+ and φo = 20◦ for increasing
decelerator stage number. The number next to each data point is the final velocity.
Because of transverse over-focussing, essentially no molecules survive below 100 m/s.

155

Decelerator
Array

(a)

Skimmer

Hexapole

PulsedValve(b) Quadrupole Guide Stage-V

3(107) V/m
2(107) V/m
1(107) V/m

+V

0 V/m
+V

-V

(c)

QG stage

QG stage

Figure 7.8: Quadrupole guiding decelerator. (a) Schematic of QGD. (b) Electric field ofquadrupole guiding stage energized to ±12.5 kV. (c) Switching scheme for deceleration
with the QGD.

156
farther apart6 in a QGD the maximum stable excursion position, ∆φmax , and velocity,∆vmax , will differ from that of the traditional decelerator. Furthermore, because thedependence of the decelerating force on φo is different (less steep) the shape of the stable
longitudinal phase space will change. Understanding the shape of the stable longitudinal
phase area is crucial to predicting the performance of the QGD, and can be derived fromexamining the longitudinal forces inside the QGD. Shown in Fig. 7.9(a) is the Stark
shift on-axis of an OH molecule in the |2 ± 2+i state inside the unit cell (3 deceleration
stages) of the QGD. The solid line is the Stark shift due to the slowing stage centeredat 11 mm, while the dashed line is the Stark shift of the stages which will be energizedat the switching of the fields. The subtraction of these two curves, shown in Fig. 7.9(b)as a solid line, is the amount of energy removed at each switching of the fields, ∆KE.
Similar to the previous section, we represent ∆KE as sum of sine-functions∆KE(φ) =Xan sin(nφ),

(7.1)

n=odd

where we have used the definition of the phase angle φ = 180◦ z/L. A fit of the first
three terms of this equation to the actual ∆KE for deceleration stages spaced by 11
mm is shown as a dashed line in Fig. 7.9 resulting in the fit values a1 = 1.221 cm−1 ,a3 = 0.450 cm−1 , and a5 = 0.089 cm−1 . Using this fit, along with Newton’s second
law, we have the equation of motion of the molecules about the synchronous molecule
position as
d2 ∆φ
π
+
(∆KE(∆φ + φ0 ) − ∆KE(φ0 )) = 0,
dt2
mL

(7.2)

where we have used the excursion of the molecule from the synchronous molecule ∆φ =φ−φo . The maximum stable forward excursion of a non-synchronous molecule is exactly
the same as a traditional decelerator and is given as◦∆φ+max (φo ) = 180 − 2φo .
6

(7.3)

To prevent high-voltage breakdown the quadrupole stages should be spaced from the deceleration
stages by the same amount as deceleration stages are spaced in a traditional decelerator. Thus, thedecelerating electrodes for a QGD will be twice as far apart as in a traditional decelerator.

157
2

U [cm-1]

(a)

1.5

1

0.5

0
0

10

20

30

40

z [mm]
(b)

2

1

0

0

5

10

15

20

25

z [mm]
Figure 7.9: (a) The Stark shift of an OH molecule in the |2±2, +i state inside the QGD.
The solid curve is the Stark shift due to the slowing electrodes, while the dashed curveis the Stark shift due to the electrodes that will be energized at the switching time. (b)The change in the molecule’s kinetic energy as a function of position is shown (solid) as
well as a fit of Eq. 7.1, including up to n = 3. (dashed). This graph is calculated fromthe subtraction of the two curves in panel (a).

158
Similar to the previous chapter we calculate the work done in bringing a molecule
starting at this position with zero velocity to the synchronous molecule position as
W (φo ) =

Z End
Start

F dx = −

1
π

Z 0

X

∆φ+
max (φo ) n=odds(an (sin(n(∆φ + φo )) − sin(nφo )))d∆φ.
(7.4)

Integrating this equation and setting it equal to the kinetic energy yields the maximum
stable excursion velocity asvu Xu∆vmax (φo ) = 2tanmπn=oddscos(nφo )π− ( − φo ) sin(φo ) ,
n
2

(7.5)

where φo is now in radians for convenience.
Using Eqs. 7.2-7.5 it is possible to solve for the longitudinal separatrix, which
separates stable deceleration from unstable motion inside the decelerator. These separatrices are shown (thick black lines) along with the results of 3D Monte Carlo simulationsof a QGD with 11 mm spacing between deceleration stages (5.5 mm spacing between
deceleration and quadrupole guiding stages) in the left column of Fig. 7.10 for increasing phase angle. The longitudinal phase space is shown with each dot representing theposition of a stable molecule. The lack of structure inside these separatrices is evidenceof a minimization of coupling between the transverse and longitudinal modes. The rightcolumn, which shows simulated ToF curves, reveals a single stable peak arriving at latertimes as the φo is increased. These simulations are for only the |2 ± 2+i state of OH,
and thus, do not exhibit the large background contribution of the other states of OH
present in experimental ToF data.
From the comparison of the 3D simulations with the separatrices, we see that thesimple theory of Eqs. 7.2-7.5 is quite accurate in describing the longitudinal performanceof the QGD. Thus, by numerically integrating the area inside these separatrices we canpredict the longitudinal performance of the QGD relative to the traditional decelerator.
As seen in Fig. 7.11(a), because the decelerator stage spacing (in this simulation) is
twice that of a traditional decelerator the energy removed per stage of the QGD is

159
1.00
410

400

0.75

vz [m/s]

390

0.50

380

370

0.25
360

350

0.00
0
z [m]

0.005

0.01

4

5

6

7

Time from discharge [ms]
1.00

400

390

0.75
vz [m/s]

380

370

0.50

360

0.25

350

340

0
z [m]

0.005

0.01

0.00

4

5

6

7

Time from discharge [ms]
1.00

350

340

0.75
vz [m/s]

330

320

0.50
310

0.25

300

290
0
z [m]

0.005

0.01

0.00

4

5

6

7

Time from discharge [ms]
1.00

240

230

0.75
220

0.50

z

v [m/s]

210

200

190

0.25
180

170

0.00
0
z [m]

0.005

0.01

4

5

6

7

Time from discharge [ms]Figure 7.10: The left column is stable molecule phase space of molecules decelerated
inside the QGD. The solid line is the separatrix predicted by the theory of this section,
while the points represent positions of molecule in the 3D Monte Carlo simulations. The
right column shows the ToF spectra of OH molecules in the |2 ± 2, +i state at the exitof this decelerator which has 143 deceleration stage (286 stages in all).

160
less steep with φo . Therefore, QGD deceleration with the same φo as a leads to afaster beam than one produced by a traditional decelerator, given the same number ofdeceleration stages. Thus, when comparing the longitudinal acceptance of the two typesof decelerators it is important to take the limit of high phase angles where the ∆KE’s
converge. Nonetheless, the QGD shows significant gain over traditional deceleration asshown in Fig. 7.11(b). This gain is primarily due to the increased physical size of thestable longitudinal phase space due to the larger deceleration stage spacing.
It is important, to note that since this graphs compares only the total area inside
the separatrix and S = 1 deceleration does not completely fill this area (due to thecoupling effects), this gain is actually an underestimate of the QGD longitudinal performance. Furthermore, these graphs do not include transverse focussing effects, which
can only be properly included through detailed simulation. The results of a 3D Monte
Carlo simulation, which includes these transverse effects, are shown in Fig. 7.12. In
this graph, the number of decelerated molecules versus the final speed is plotted forboth the traditional decelerator operating at S = 1 and the QGD decelerator. Here
we see that while the QGD initially delivers more molecules, once the molecules are
decelerated below 100 m/s the decelerated molecule number falls off abruptly. Thisbehavior is expected since for these simulations the voltage on the quadrupole guiding
stages was chosen to be ±2.2 kV, which maximizes the bunching signal (Fig. 7.13).
And, as detailed in Eq. 4.7, the focal length of a transverse guiding element is directly
proportional to the molecular kinetic energy. Therefore, as the molecule speed is decreased, the molecules are over-focused and crash into the decelerator electrodes. Bylowering the voltage on the quadrupole guiding stages during the deceleration process
this can be prevented. A simulation of deceleration with this dynamically controlled
voltage is shown in Fig. 7.12 as an open (red) circle. For this simulation, the voltages
were scaled as 1/v 2 after each deceleration stage. For ease of simulation, the transverse forces were scaled by 1/v 2 whenever the molecules were closer to a quadrupole

161

5

1.8

1.50

1.6

4

1.2
QGD

1.0
0.8
0.6
0.4
0.2

1.25
1.00

3

QGD

0.75

2

0.50

S=1

1

0.25
0.00

0.0
0

20

40
o

60

[Degrees]

80

100

0

20

Gain

S=1

1.4

Phase Stable Area

Energy Loss per Stage [cm-1]

2.0

40
o

60

80

0
100

[Degrees]

Figure 7.11: (a) The energy removed by deceleration per stage as a function of phase angle for traditional deceleration and deceleration with a QGD. Both curves are calculated
for OH in the |2 ± 2, +i state. (b) The phase-stable area for deceleration as a functionof phase angle for traditional deceleration and deceleration with a QGD is plotted on
the left axis, while the gain of the QGD over traditional deceleration is plotted on theright axis.

162

Molecule Number [arb. u.]

1

0o 15o
0o

30o

45o

10o

53o

20o

0.1

50o

0.01

1E-3

400

30o

Traditional Decelerator (S = 1)
2.2 kV Quadrupole Voltage
Dynamic Voltage (v2)
350

300

250

200

150

53o
100

50

0

Final Velocity [m/s]
Figure 7.12: Comparison of the decelerated molecule number using traditional deceleration and deceleration using a QGD for increasing decelerator phase angle. For bothcurves 143 stages of deceleration were used. The number next to each data pointis the phase angle used. Note that for operation of the QGD with ±2.2 kV on theguiding stages, essentially no molecules survive below 100 m/s because of transverse
over-focussing. If the voltage on the guiding stages is lowered as the molecules are decelerated this effect is removed and the QGD supplies roughly 2 times as many molecules
as a traditional decelerator.

163

2.00
1.75

Molecule Number

1.50
1.25
1.00
0.75
0.50
0.25
0.00
0

2

4

6

8

10

12

14

Guiding Stage Voltage [ kV]
Figure 7.13: Optimization of the quadrupole guiding voltage for operation at φo = 0◦ .

164
guiding stage than a deceleration stage. While this may be a poor approximation at
the lowest speeds, it will likely lead to an underestimate of the decelerated molecule
number since the transverse guidance of the quadrupole guiding stage extends beyond
this regime. However, even if it leads to an overestimate, it may be that by proper
control of the quadrupole voltages any over-focussing introduced by the decelerating
elements can be compensated. As seen from the slowest points in Fig. 7.12, dynamically controlling the voltage of the quadrupole guiding stages leads to nearly a factor
1000 increase in decelerated number for the QGD and delivers about 20 times more
decelerated molecules than traditional deceleration provides. Furthermore, because thevoltages on the quadrupole guiding stages is low, dynamic control of them should bepossible using an analog waveform generator and high-voltage amplifier. It should be
noted that scaling the voltages as 1/v 2 may not be the optimal scaling since the real
focal length depends sensitively on the electrode construction, and at low speeds thetransverse focussing of the decelerator electrodes becomes significant. In fact, simulations where the quadrupole guiding transverse forces were scaled as 1/v 0.875 show a gainof 1.8 in decelerated molecule number over the 1/v 2 scaling (see Fig. 7.14(b)). This is,presumably, because the over-focussing of the decelerator electrodes can be somewhat
compensated by inputting molecules which are already slightly over-focused, i.e. onelens focuses, but two can make a telescope. Thus, it may be possible to use evolutionary
algorithms to control the quadrupole voltage or change the design of the decelerating
electrodes so that they provide less transverse focusing, and maximize the number ofdecelerated molecules beyond what is reported here.
One important advantage of the QGD over traditional decelerators is its inherent
ability to support more deceleration stages. Because the molecules experience a tunabletransverse focussing element after each deceleration stage there is little loss in extending
the number of deceleration stages. In fact, as seen in Fig. 7.15, there is essentially no
loss until the molecules are slowed to the slowest speeds. And since this loss is due to

165

1
(a)
(b)
(c)
(d)

(e)

0.1

(f)

(a)
0

25

50

75

100

Stage Number

125

Molecule Number [arb. u.]

Molecule Number [arb. u.]

1

(g)

0.1

(f)

(b)
0

25

50

75

100

125

Stage Number

Figure 7.14: (a)Simulations of the phase-stable molecule number as a function of stage
number in the QGD with the guiding voltages scaled as 1/v 2 . The traces shown are(a) φo = 0◦ , (b) φo = 15◦ , (c) φo = 30◦ , (d) φo = 45◦ , (e) φo = 50◦ , and (f) φo = 53◦ .
Note the dramatic decrease in number in the last several stages for φo = 53◦ . Thisdecrease is due to transverse over-focussing of these slow molecules, which is still present,
presumably due to the over-focussing the decelerator electrodes. (b)Comparison of
deceleration at φo = 53◦ with 1/v 2 (f) and 1/v 0.875 (g) scaling of the quadrupole guiding
voltage. Both panels are normalized to the φo = 0◦ case.

166
the aforementioned transverse over-focussing, it is not a fundamental problem. Furthermore, even with this loss the QGD enormously outperforms both S = 3 (Fig. 7.5)andS = 3+ (Fig. 7.7). Thus, the QGD seems to be an ideal decelerator for producing large
numbers of currently decelerated molecules, or, perhaps more importantly, the ideal
decelerator for deceleration of molecules with poor Stark shift to mass ratios like H2 O
and SO2 .
In summary, the QGD solves the problem of coupling between the transverse andlongitudinal motions inside a Stark decelerator by introducing transverse focusing elements. By dynamically controlling the focal length (voltage) of these guiding elements,
enormous improvements in deceleration efficiency can be achieved7 . Furthermore, itappears that with dynamic control of the guiding stage focal length there should be nolimit to the length of decelerator that can be built. This opens the door for the deceleration of molecules with a poor Stark shift to mass ratio. Building upon the strong
correlation between simulation and experimental results, we are confident that the simulations presented in this chapter provide a solid guideline for future implementationsof Stark deceleration.

7

As detailed in Chap. 5 traditional decelerators appear to suffer from over-focussing effects in their
final stage. Thus, implementing dynamic voltage control ideas in current decelerators may lead to
immediate improvements in decelerator efficiency.

167

Molecule Number [arb. u.]

1

50

100

o
=
45
o

150
175

0.1

202

0.01

350

300

250

200

150

100

50

0

Final Velocity [m/s]
Figure 7.15: Decelerated molecule number using the QGD for increasing decelerator
stage number. The number next to each data point is the number of deceleration stages
used. Because of transverse over-focussing there is a sharp loss of molecules for theslowest speeds.

Chapter 8

Towards Cold Chemistry and Reaction Control: Deceleration of theFormaldehyde Molecule

8.1

Introduction
Recent exciting developments in ultracold matter research include the creationof ultracold molecules by magneto-association [46], leading to molecular Bose-Einsteincondensation [4]. Despite the rich physics demonstrated in these systems, all are characterized by spherically symmetric interactions whose effect is included through one
tunable parameter, namely the s-wave scattering length. By contrast, the permanentelectric dipole moment possessed by polar molecules permits long-range and anisotropicinteractions and enables new methods for external control in an ultra-cold environment[5]. The electric dipole-dipole interaction (and control over it) gives rise to unique
physics and chemistry including novel collision and chemical reaction dynamics. Lackof spherical symmetry in the interaction causes colliding molecules to be attracted or
repelled depending on their relative orientation. Thus, an external electric field, whichorients the molecules, will have a profound effect on the molecular interactions [6].
Furthermore, because the transition states of a chemical reaction often involve specificorientations of the molecular dipoles, an external electric field may shift the energy
barrier to reaction, making a particular reaction pathway more or less favorable. A
Stark decelerator [24, 27, 28, 127] producing cold polar molecules with tunable andwell-defined translational energy is thus an ideal tool for the study of low (or nega-

169tive) barrier chemical reactions. We note other experiments utilizing photo-association[134, 73, 84], helium buffer gas cooling [137], and billiard ball collisions [50] have succeeded in producing cold polar molecules. Furthermore, the possibility of producing
polar molecules through the use of a Feshbach resonance is also very exciting [115, 65]
because of the ultra-cold temperatures possible.
Here, we present an outline of the properties and possibilities for control of the
low temperature reaction of OH with H2 CO, followed by a detailed report on the production of cold H2 CO molecules via Stark deceleration, a necessary step in the study ofthese novel dynamics. Modeling suggests for the first time that chemical reactions andcollision cross sections can be altered orders of magnitude by varying either the molecular state, or external electric field strength. We specifically consider the H-abstraction
channel in the reaction of H2 CO and OH: H2 CO + OH → CHO + H2 O. This reaction
not only represents a key component in the combustion of hydrocarbons, but also plays
an important role in atmospheric chemistry where it is the primary process responsible
for the removal of the pollutant, H2 CO [88, 113, 116, 48, 90, 143, 81]. Near room temperature the rate of this reaction is weakly dependent on temperature, suggesting thebarrier to the process (if any) is very low [116]. Measurements of the thermal activationenergy, Ea , range from Ea /R = 750 K to -931 K [88, 113, 116, 48, 90, 143, 110], withthe most recent measurement giving a value of -135 K [110]. Here R is the universal
gas constant. The most accurate calculations [38] predict the energy of the transition
state for the abstraction to lie between -700 K and +60 K relative to the reactants.
Experimentally, both OH [27, 28, 127] and H2 CO (this work) molecules havebeen produced at low temperatures via Stark deceleration. In the case of spontaneous
reaction (Ea < 0) [88, 116, 90, 143], by magnetically trapping OH in the presence of atunable bias electric field and “bombarding” the trap with decelerated H2 CO packets,
the OH - H2 CO total scattering and reaction rates can be mapped as a function ofcollision energy and applied electric field. Thus Ea can be measured. If Ea > 0, as some170
measurements and theory suggest [113, 48, 81, 38], the collision energy tuning affordedby the Stark decelerator provides a direct way to measure the energy barrier to reaction.
Unlike thermal kinetics studies, which rely on fitting the Arrhenius formula to reaction
rates, the Stark decelerator can be used to directly tune the collision energy above andbelow the threshold energy.
Further motivation for pursuing this particular reaction comes from the prominent
position in molecular physics held by H2 CO. As a four-atom asymmetric rotor, H2 CO’s
rotational structure and six internal degrees of freedom give rise to the same complexities
of much larger molecules. However, as a relatively light molecule, H2 CO is still tractable
at a high level of theory, making it an ideal proving ground for molecular physics.
Additional incentive for pursuing H2 CO was that until this work an asymmetric rotor
had never been decelerated in a Stark decelerator; therefore this effort opens the door
for the production of the most general class of cold molecules.
As shown in Fig. 8.1(a), H2 CO is a near-symmetric prolate top molecule, withnearly degenerate rotations about the b- and c- axes. The dipole moment, µa , along thea-axis causes these nearly degenerate, opposite parity states to experience a large Stark
shift with an applied electric field, as shown in Fig. 8.1(b-d). Here, the states are labeledby their zero-field identity in the |Jτ mJ i basis, where τ = Ka − Kc (Ki is the projectionof J along the ith axis) [56]. Of particular interest to this work is the |11 1i state (Fig.
8.1(c)), which is the upper component of the lowest J level of ortho-formaldehyde and ishence well populated in our supersonic expansion. This state experiences a large Stark
shift (1.32 cm−1 at 125 kV/cm), making it excellent for Stark deceleration. There areother states accessible via, for example, stimulated Raman adiabatic passage that offer
an improved Stark deceleration efficiency (|22 2i) or a good candidate for implementing
an AC Stark trap (|00 0i) [129].

171

Figure 8.1: (a) (Color online) H2 CO. (b-d) Stark shifts of the low-lying rotational statesof H2 CO. States of interest for this work are shown in bold.

172

8.2

Reaction Control
The sensitive dependence of cold molecular interactions on internal state and elec-

tric field engenders new prospects for probing and controlling chemical reactions. However, a complete theoretical model of the scattering processes is a large task. Nonetheless, simple modeling can suggest the extraordinary influence that can be exerted overthe collisions by selecting appropriate initial states, or else by varying an electric field.
Much of this control arises from the dipolar interaction itself, as has been stressed in
the past [18, 19, 122]. We therefore construct a scattering model which is realistic withregard to the dipolar interaction, and deliberately schematic with regard to the chemicalinteraction dynamics. The Hamiltonian, H, used in the scattering calculations takes
the following form:
H = T + HOH + HH2 CO + HStark + Hdd + Hsr + Hchem.

(8.1)

Here T is the kinetic energy, HOH and HH2 CO are the Hamiltonians of the separated molecules, and HStark describes the action of the electric field on the individual molecules. Hdd represents the long-range dipole-dipole interaction between themolecules, which depends on the orientations of the molecules relative to the field axis,
as well as the distance R between them. This Hamiltonian will be expressed in a basisof molecular eigenstates (including the effect of the field), as was done in Ref. [18, 19].
In addition, to prevent the dipole-dipole interaction from overwhelming the short-range
interaction, we replace its 1/R3 variation with 1/(R3 + cdd ) for an appropriate constant
cdd [76]. The chemical reaction dynamics are modeled in a schematic way by the terms
Hsr and Hchem. The goal of Hsr is to mimic an anisotropic interaction with a barrier,
not necessarily to characterize the OH-H2 CO system. This approximation is justified
for the low reaction barrier where the dipole-dipole interaction dominates over internal
molecular dynamics. Thus, we construct a potential as an expansion into Legendre

173

Energy (a.u.)

0.004

0.000

-0.004Rb-0.008
4

5

6

7

8

9

10

R (a.u.)Figure 8.2: Schematic lowest-lying adiabatic potential curve for the scattering model.
For intermolecular distances R ≫ Rb , the ordinary dipole-dipole interaction holds, andthe molecules attract head-to-tail. As the molecules approach a potential barrier at
R = Rb , the potential deviates from dipolar behavior, and the sign of V2 is switched sothat, by R < Rb , the molecules attract in a tail-to-tail configuration to mimic chemical
reaction described below. Also for R < Rb , the scattering channel is coupled to an
artificial channel (not shown), which represents chemical reactivity.

174
functions,
Hsr = Vo (R) − 2V2 (R)C20 (θ, φ).

(8.2)

Here (θ, φ) represent the spherical angles of the relative coordinate between the molecules,
referred to the electric field axis, and C20 is a reduced spherical harmonic [31]. The
isotropic part of this potential carries the barrier at a radius R ≈ Rb ,C12
C6Vo (R) = 12 − 6 + DB exp
R
R−(R − Rb )2
wb

!

.

(8.3)

The height of the barrier depending on DB is fully adjustable with wb characterizing its
range. In the following, we set the barrier height equal to the threshold energy of theincoming molecule. The resulting isotropic potential has a depth that has arbitrarily
been set to 5 × 10−3 atomic units. The radial dependence of V2 is chosen to have amagnitude comparable to Vo , but to change sign as R crosses Rb , as seen in Fig. 8.2.
This is done to mimic the short-range physics responsible for the chemical reaction
as depicted in Fig. 8.5 and described in the accompanying text. Thus, the polar
molecules attract in a head-to-tail configuration for R > Rb , but attract in a tail-to-tailconfiguration upon crossing the barrier. Finally, chemical reactivity (Hchem ) is modeledby coupling to a purely repulsive artificial channel. The coupling to this channel isrepresented by a decaying exponential in R, which is of negligible size for R > Rb . To
complete a chemical reaction, the molecules must cross over the barrier, reverse theirrelative orientation, and find the artificial channel.
Collision cross sections for OH-H2 CO scattering in this schematic model are shownin Fig. 8.3. Here both molecules are assumed to be in spin-stretched, weak-fieldseeking states: |F mF , parityi = |22, f i for OH, and |11 1i for H2 CO. The collision crosssections are plotted as functions of electric field, with the collision energy fixed at 1
mK. The cross section for elastic (spin-changing) collisions is shown as a dashed (solid)line. These cross sections are large and comparable to one another, and exhibit strong
modulations as a function of an applied electric field. Modulations of this type have

175

10

-11

-12

[cm

2

]

10

10

10

-13

elastic
inelastic

-14

0.0

0.5

1.0

1.5

2.0

Electric field [kV/cm]
Figure 8.3: Elastic (dashed) and spin-changing inelastic (solid) cross sections for cold
collisions of spin-stretched OH and H2 CO molecules. These cross sections were computed at a collision energy of 1 mK.

176
been discussed previously, for the OH-OH scattering system, as arising from purely
long-range scattering by the dipole-dipole interaction [18, 19]. Thus, these results are
relatively robust and independent of the detailed form of the short-range interaction.
The influence of long-ranged dipolar forces should also extend to the chemical
reaction dynamics. To illustrate this, we show in Fig. 8.4 the chemical reaction crosssection versus electric field. The collision energy is again assumed to be 1 mK. (Notethat, because the reaction mechanism is schematic in our model we report only relativevalues for the cross sections.) The solid line is for collisions in which both OH and
H2 CO molecules are in their strong-field-seeking states. At this collision energy thefield has a profound influence on the reaction cross section, mostly by accessing a large
number of resonant states. Considering that each resonant state likely spans a different
region of configuration space, mapping out resonances such as these can in principle
serve as an extremely sensitive probe to transition states. The dashed trace in Fig. 8.4
shows the reaction rate for weak-field-seeking states of the reactants. In addition toprobing many resonances, this cross section rises by nearly four orders of magnitude as
the field increases over a range of 2 kV/cm. This rise is also a direct consequence ofthe dipole-dipole interaction. In Refs [18, 19, 122], it was shown that polar molecules
in weak-field-seeking states scatter at fairly long range at ultra-low energies (of ordertens to hundreds of atomic units), owing to avoided crossings in the relevant adiabatic
potential energy curves. In other words, the dipole-dipole interaction can “shield” thesemolecules from ever getting close enough to react chemically. Thus the cross section
is strongly suppressed relative to that for strong-field-seekers. However, as the field isincreased, the inner turning point of the relevant potential curve moves to smaller R,
making the shielding less effective, and hence, the reaction more likely.
We stress that these results are by no means quantitative predictions of the expected behavior of the OH-H2 CO scattering system. However, they are qualitatively
suggestive of the strong dependence of scattering on experimentally accessible parame-

177

-1

a

s

[ rb. unit ]

10
10
10
10

-3

strong-field-seekers-5

eak-field-seekers

w

-7

0.0

0.5

1.0

1.5

2.0

Electric field [kV/cm]
Figure 8.4: Calculated chemical reaction cross sections at a collision energy of 1 mK,
versus electric field, using the model in the text.

178
ters such as molecular internal state and applied electric field. Our confidence in thesequalitative conclusions is drawn from the fact that they originate largely in the dipolarinteraction, which can be treated accurately. More detailed theoretical studies, including
more realistic potential energy surfaces, are planned [7].
An important capability for current experiments (not included in our model) is the
direct control of the H-abstraction reaction barrier height through the application of an
external electric field. The basic H-abstraction reaction mechanism [48] is shown in Fig.
8.5. We note that Ref. [14] presents a slightly different H-abstraction configuration, butwith the same end-to-end dipole coupling scheme. During the H-abstraction process,the energy required to rotate one dipole moment (OH) versus the other (H2 CO) enables
our proposed control of reaction dynamics using an external electric field. Hence, thetwo abstraction configurations present no difference other than the magnitude of theshift we can create in the reaction barrier. The important characteristic to note isthat the hydrogen-bonded complex (HBC) in the second panel (Fig. 8.5) forms along
an attractive direction of the two electric dipoles, but in the transition state (TS) the
OH dipole has essentially flipped its orientation relative to the H2 CO dipole. Whilean external electric field, which orients the molecules, would allow (and perhaps evenencourage) the formation of the HBC; it would add an energy barrier to the formationof the TS, and thus the H-abstraction channel through the OH dipole-field interaction.
Based on the bond geometries shown in Ref. [48], the addition to the energy barrier in−→going from the HBC to the TS is ∼1.8µOH | E |, where µOH is the expectation value of
the OH dipole. For a high, yet attainable, electric field of 250 kV/cm, the additional
barrier energy would be 10 K. Assuming a reaction barrier at a H2 CO collision speed of
223 m/s (60 K [38]) the applied electric field would shift the required collision velocity
to 234 m/s, well within the Stark decelerator resolution.

179

Reactants

HBC

TS

O

O

O
C

C
C

H
H

H

.4º
122

O
H

Products

H

C
94.5º

H H

H
O

H

O
O

O
H
H

H

Figure 8.5: (Color online) The reaction mechanism for the H-abstraction channel following Ref [48]. The molecules initially attract in the head-to-tail configuration forming
the hydrogen bonded complex (HBC). The chemical reaction proceeds via the transition
state (TS) to the final reaction products.

180

8.3

Experiment
The decelerator used for H2 CO is similar to the apparatus in our previously

described OH experiments [27, 28, 61]. Molecules in a skimmed, pulsed supersonicbeam are focused by an electrostatic hexapole field to provide transverse coupling into
the Stark decelerator. The Stark decelerator is constructed of 143 slowing stages spaced
5.461 mm apart with each stage comprised of two cylindrical electrodes of diameter
3.175 mm separated axially by 5.175 mm and oppositely biased at high voltage (±12.5 kV). Successive stages are oriented at 90o to each other to provide transverse
guiding of the molecular beam. The geometry of the slowing stages provides an electricfield maximum between the electrodes with the field decreasing away from the electrodecenter. Switching the electric field when the molecules are directly between two adjacentstages (no deceleration) is denoted by a synchronous phase angle of φo = 0o (referred
to as bunching), while switching the electric field when the molecules are between theelectrodes (maximum deceleration, minimum stable area in phase space) is denoted by
φo = 90o .
The H2 CO molecules are produced from the cracking of the formaldehyde polymer
to produce the monomer, which is passed through a double u-tube apparatus [114].
Xenon at 200 kPa pressure is flowed over the collected H2 CO, held at 196 K where
H2 CO has ∼2.7 kPa vapor pressure. In this work, the Xe/H2 CO mixture was expanded
through either a current-loop (60 µs pulse length) or a solenoid supersonic valve (500µs pulse length), producing beams with a mean speed (spread) of 470 m/s (10%) and
350 m/s (10%), respectively. The higher mean speed of the current-loop valve pulse isdue to the documented heating effects of these devices [80].
H2 CO is detected using laser-induced fluorescence. The molecules are excitedfrom the |11 1 > ground state by photons at 353 nm generated from a frequency-doubled,
pulsed-dye laser (PDL) pumped by a Nd:YAG laser to the Ae1 A2 electronically excited

181

Current-loop
valve

Current-loop
valve

SolenoidvalvecSolenoidvalvedFigure 8.6: ToF data for decelerated H2 CO, showing bunching and deceleration off
the main pulse (offset down for clarity) for both (a) the current-loop and (c) solenoid
valves. Stark decelerated packets of H2 CO at the output of the decelerator for both the(b) current-loop and (d) solenoid valves.

182
state with one quantum in the ν4 out-of-plane bending vibrational mode. Approximately
40% of the excited H2 CO decays non-radiatively [57], while the remaining molecules emit
distributed fluorescence from 353 nm to 610 nm [109]. This fluorescence is collected,
filtered, and imaged onto a photo-multiplier tube. The output of the photomultipliertube is sent to a multi-channel scalar/averager, which serves as a gated photon counter.
Operating the decelerator at 5 Hz and the PDL system at 10 Hz allows a “lock-in”
type data-collection, such that every other signal is subtracted from the total photon
counts. By scanning the delay between the PDL system and the triggering of the valve,a time-of-flight (ToF) curve is generated. A single data point is generated by collecting
photon counts for 30 s to 2400 s (depending on required signal to noise).
Shown in Fig. 8.6 are ToF measurements taken ∼1 cm after the exit of thedecelerator, which is ∼0.9 m down-stream from the valve. Data taken using the currentloop valve is shown in the top two panels (a,b), while data for the solenoid valve is shownin the bottom two panels (c,d). In these figures the time axis is relative to the beginningof deceleration. In Fig. 8.6(a) when the decelerator is operated at φo = 0o , a large peakin the ToF curve denotes the arrival (marked by an arrow) of the phase stable packet
whose density is ∼106 cm−3 and contains a few million molecules. For these molecules
no deceleration has occurred [?]. Also shown in Fig. 8.6(a), but offset for clarity is
the ToF curve for deceleration at φo = 60o . Here the arrow denotes the arrival of aslowed, isolated H2 CO packet with a mean velocity of 269 m/s. In this curve the signalof deceleration is unmistakable, a peak in the ToF data arriving later than the main
pulse. Displayed in Fig. 8.6(b) are ToF curves for only the slowed molecular packets
as a function of φo . The effect of deceleration is evident, as φo increases and moreenergy per stage is removed, the phase stable packets arrive later in time signifying the
reduction in velocity. For φo ≤ 50o the phase stable packets have not been decelerated
off the main pulse and therefore their baseline values differ from that of the higher φo .
While the solenoid type valve is much less stable in pulse-to-pulse molecule num-

183

Figure 8.7: Comparison of decelerated packet molecule number to the expected for (a)the current-loop valve and (b) the solenoid valve.

ber than the current-loop valve, it supplies a slower molecular pulse with a mean speed
in agreement with a room temperature Xenon expansion. The benefit of the lower initial
speed is seen directly in Fig. 8.6(c), where only φo = 30o is required to decelerate apacket of molecules to after the main pulse. Because of the long pulse from the solenoid
valve, the associated ToF structure in Fig. 8.6(c) is more complicated than that of Fig.
8.6(a) due to the loading of many buckets at the beginning of the deceleration sequence.
Figure 8.6(d) further demonstrates the advantage of a lower initial speed, which like
8.6(b) shows only the decelerated phase stable molecular packets. As in Fig. 8.6(b)the baseline of the packets varies as they are decelerated off of the main pulse. Here it
is seen that with only φo = 50o , the molecular packet is decelerated down to 26 m/s(contrast this with 296 m/s for the current-loop valve). This slowed packet contains a
few thousand molecules at a density of ∼106 cm−3 and a translational temperature of∼100 mK as set by the phase space acceptance of the decelerator.
Based on the matching of the molecular phase space at the entrance of the decelerator to the decelerator acceptance, it is possible to predict the molecule number in agiven packet. Shown in Fig. 8.7(a,b) as open circles are the ToF-deconvolved, integrated
molecular numbers for the decelerated packets relative to operation at φo = 0o for the184current-loop valve and solenoid valve, respectively. Also, shown in these figures as solidlines are a simple theory for the dependence of the molecule number in a stable packet
on φo taken from Eqn. 9 of Ref. [61]. The agreement of the theory with the experimental data in Fig. 8.7(a) is excellent, with the only noticeable deviation at φo = 10o .
This is likely because we have neglected any effects due to the transverse phase space,and because φo = 10o experiences more of the slowing field than φo = 0o it has a larger
transverse acceptance, while the longitudinal acceptances are similar. The molecule
number instability of the solenoid valve is responsible for most of the deviations fromthe expected behavior in Fig. 8.7(b). For the data point at φo = 50o , however, the lowmean speed allows the pulse to expand spatially much more than φo = 0o to which it
is normalized as it exits the decelerator enroute to the detection region. Thus the data
point appears below the expected.
To observe the OH-H2 CO collision and reaction dynamics requires monitoring
both the OH population and the production of the formyl radical (CHO). The lowest temperature (228 K) measurement [116] of the reaction finds the reaction rate as
10−11 cm3 s−1 . Since Ea ∼ 0 the reaction rate at our temperatures of interest will likely
be similar. Thus, for a trap volume of 1 cm3 and reactant densities of 106 cm−3 weestimate the reaction rate as 10 molecule s−1 and the required signal-to-noise ratio for
OH detection at 105 . For cold H2 CO at 25 m/s colliding with a 1 cm long magnetic
trap of OH this leads to a production rate of CHO as one molecule per 250 pulses.
Clearly, observation of this reaction with current conditions would be heroic, however,further improvements such as multiple trap loading and secondary cooling of the molecular samples, which will increase the sample density, should make these observations
possible.
In summary, we have detailed new methods to study interspecies molecular collision and reaction dynamics, and have shown how control of these processes may beachieved. We have also demonstrated experimental preparation of both cold polar

185
molecules required for these studies.

Chapter 9High Resolution Microwave Spectroscopy of OH

9.1

Introduction
Current theories that attempt to unify gravity with the other fundamental forcespredict spatial and temporal variations in the fundamental constants, including the finestructure constant, α [93]. Compellingly, a space-time varying coupling constant, such
as α, implies violation of both Lorentz invariance and CPT symmetry [75]. Comparisonof different atomic clock systems [95] has provided tight constraints on the time variation of various fundamental constants, including α, during the modern epoch. While,observation of absorption lines in distant quasars [136, 99] provides conflicting results
about possible α variation over cosmological time. Since the largest α variation may
have occurred during the early universe [45], the inconsistent conclusions of the quasar
measurements are most interesting. Due to the use of spatially diverse absorbers, thesemeasurements are sensitive to relative Doppler shifts, and therefore an independentconfirmation of the variation of α is important.
Recently, there has been much interest in using OH megamasers in interstellar
space to constrain the evolution of fundamental constants [39, 34, 70] with several key
advantages. Most importantly, the multiple lines (that have different dependence on α)
arising from one of these localized sources differentiate the relative Doppler shift fromtrue variation in the transition frequency. Furthermore, if it is assumed that the only
fundamental constant to vary is α, it can been shown that the sum and difference of the

187∆F = 0 (F is total angular momentum) transition frequencies in the ground Λ-doublet of
OH depend on α as α0.4 and α4 , respectively [39]. Thus, by comparing these quantities asmeasured from OH megamasers to laboratory values it is possible to remove the Doppler
shift systematic and constrain α over cosmological time. Furthermore, because of theunique properties of the Λ-doublet, the ∆F = 0 transitions are extremely insensitive tomagnetic fields. However, as pointed out by Darling [39], for the current limits on ∆α/α
the change in the relevant measurable quantities is on the order of 100 Hz, which prior
to this work was the accuracy of the best laboratory based measurement. Moreover, anastrophysical measurement of OH megamasers, scheduled for later this year, expects aresolution better than 100 Hz [8], and thus better laboratory measurements of the OH
Λ-doublet microwave transitions are imminently needed to allow for tighter constraints
on ∆α/α.
Despite the prominence of the OH radical in molecular physics, the previous best
measurement of the OH ground Λ-doublet, performed by ter Meulen and Dymanus [120],has stood for over 30 years. This lack of improvement was due to the relatively slow
progress in the center-of-mass motion control of molecules, which limited the maximum
field interrogation time, and thus the spectroscopic resolution. The ability of a Stark
decelerator [24, 27] to provide slow, cold pulses of molecules makes it an ideal source
for molecular spectroscopy [131]. In this work, a Stark decelerator is used along withstandard microwave spectroscopy techniques to perform the best measurement to dateof the ∆F = 0, Λ-doublet microwave transitions in OH, which along with appropriate
astrophysical measurements can be used to constrain ∆α/α with a sensitivity of 1 ppm
over the last ∼1010 years.

9.2

Experiment
In its ro-vibronic ground state OH is a Hund’s case (a) molecule with a 2 Π config-

uration and total molecule-fixed angular momentum of Ω = 32 . For the most abundant

188

(a)

skim

PMT

Stark
d

eceler
ator

hexa
pole

hexa
pole

pulsed
valve

mer

(b)

micro
w
cavity ave

detec
ti
tube onFigure 9.1: (Color online) a) OH ground Λ-doublet state. The arrows represent theeffect of the applied microwave pulses for the 2→2 (solid arrows) and 1→1 (dottedarrows) transitions. b) Schematic of experiment (inset depicts detection region).

189
isotopomer (O16 H) the oxygen has no nuclear spin and the hydrogen carries a nuclear
spin of 12 leading to two total spin states with F = 1 and 2. Because the unpaired electronin OH has one unit of orbital angular momentum, these ro-vibronic ground states are‘Λ-doubled’, leading to the closely spaced opposite parity Λ-doublet states shown in Fig.
9.1(a) labeled as |F, mF , parityi. Though molecules are decelerated only in the |2, ±2, +i
and |2, ±1, +i states, the field-free region from the hexapole to the microwave cavity
leads to an equal redistribution of the population among the five magnetic sub-levels of
the F = 2 upper doublet state. In this work, both ∆F = 0 electric dipole transitions
between the upper and lower doublet states are studied. Though these transitions exhibit large Stark shifts (as is necessary for Stark deceleration), they show remarkably
small Zeeman shifts. In fact, because the magnetic dipole operator respects parity,
one expects the ∆F = 0 transitions of a pure case (a) molecule to show no magnetic
field dependence. Nonetheless, because the hyperfine splitting differs in the upper andlower doublet, and part of the |mF | = 1 magnetic dipole moment comes from mixing
with the other hyperfine component, there is a small quadratic shift of the |2, ±1, +i→ |2, ±1, −i transition frequency (150 Hz/Gauss2 ). Furthermore, because OH is notcompletely case (a) (i.e. the electron’s orbital angular momentum and spin are slightly
decoupled from the axis) the g-factor is slightly larger in the lower doublet [100]. Thus,
transitions between the mF < 0 (mF > 0) components are blue-shifted (red-shifted)
relative to the zero field value. Hence for experiments probing both positive and negative mF components such as this work, the g-factor difference leads to a broadening
and eventually a bifurcation in the lineshape for increasing magnetic field. This effect,
explored at relatively large fields (0.6 - 0.9 T), was found to shift the ∆F = 0 transition
frequencies of the |2, ±2, +i, |2, −1, +i, and |1, 1, +i states at a rate of -sgn(mF ) × 2.7
kHz/Gauss, and the |1, −1, +i and |2, 1, +i states at a rate of -sgn(mF )×0.9 kHz/Gauss[100]. For even population distribution this effect leads to a shift of 450 Hz/Gauss and-900 Hz/Gauss for the 2→2 and 1→1 transitions, respectively.

190

OH LIF Signal
Expected|mF| = 1|mF| = 2|mF| = 0

Figure 9.2: (Color online) a) Rabi-flopping as a function of time. The contributions of
the individual magnetic sub-levels are shown near the bottom, while the sum is shownas the dashed line. b) Rabi-flopping as a function of velocity for a fixed (spatial) length
microwave pulse. The dashed line is the expected.

191
Shown in Fig. 9.1(b) is a schematic of our experimental setup. OH molecules
seeded in Xenon are created in a pulsed discharge [80] producing a molecular pulse witha mean speed of 410 m/s and 10% longitudinal velocity spread. Following a skimmer
is an electrostatic hexapole used to transversely couple the molecules into our Stark
decelerator, which is described in detail elsewhere [27, 28, 61]. The Stark decelerator
is used in a variety of operating conditions, yielding pulses of molecules at a densityof 106 cm−3 with mean speeds chosen between 410 m/s and 50 m/s with longitudinaltemperatures between 1 K and 5 mK, respectively. In the current experiment, thepresent density of the OH beam limits any density related frequency shift to be below 1
mHz. Once the molecular pulse exits the decelerator it is focused by a second hexapole
to the detection region. Located between the second hexapole and the detection region isa 10 cm long cylindrical microwave cavity with its axis aligned to the molecular beam.
The microwave cavity is operated near the TM010 mode, such that the electric field
is extremely uniform over the region sampled by the molecules [66]. The microwave
cavity is surrounded by highly magnetically permeable material to provide shielding
from stray magnetic fields. From measurements of the magnetically sensitive F = 2→F= 1 transition frequency the residual field in the cavity was determined to be < 2milliGauss. This small field results in an absolute Zeeman shift of < 0.9 Hz and <1.8 Hz for the 2→2 and 1→1 transitions, respectively. Note that this is a cautious
upper-bound on the Zeeman shift, since it is based on work performed at ∼107 larger
magnetic field where the angular momentum decoupling is enhanced. Furthermore, theuse of all metal microwave cavity at the interaction region has removed all electric field
related frequency shifts, as an applied external field as large as 700 V/cm resulted in
no detectable shift in the measured transition frequencies.
Two independently switchable microwave synthesizers, both referenced to a Cesium standard, are used to provide the microwave radiation for driving the Λ-doublet
transitions. For probing the |2, mF , +i → |2, mF , −i Λ-doublet transition (2→2) only192
one synthesizer is needed since the molecules enter the cavity in the F = 2 state. Accordingly, two synthesizers are needed for probing the 1→1 transition. The first synthesizer
transfers molecules from the F = 2 upper doublet state to the F = 1 lower doublet
state. The second synthesizer then probes the 1→1 transition. After the molecules exit
the cavity the population of the upper doublet is probed by laser-induced fluorescence(LIF). For the LIF measurement the molecules are excited along the 2 Σ1/2 (v = 1) ←2Π
3/2 (v = 0) line at 282 nm by light produced from a doubled pulsed-dye laser.

This

excited state decays primarily along 2 Σ1/2 (v = 1) → 2 Π3/2 (v = 1) at 313 nm. The
red-shifted fluorescence is collected and imaged onto a photomultiplier tube coupled toa multi-channel scaler.
To characterize the performance of the microwave cavity the population in theupper doublet was recorded as a function of applied microwave pulse length as shownin Fig. 9.2(a) (so-called Rabi-flopping). For the data shown, the molecules were decelerated to 200 m/s and a microwave pulse resonant with the 2→2 transition was appliedsuch that its midpoint time coincided with the molecules being at the cavity center.
Thus, as the pulse length was increased the molecules encountered a pulse that grewsymmetrically about the cavity center. Because the LIF detection scheme is sensitive toall molecules in the upper doublet and the applied microwave field simultaneously drives
transitions between the different magnetic sub-levels, the Rabi-flopping signal is moreω′ t

′ is the effective Rabi frequency
complicated than the traditional sin2 ( R2 p ). Here ωR′ =given in terms of the detuning, δ, and Rabi frequency, ωR , as ωRq

2 , and t is
δ 2 + ωRp

the microwave pulse length. As shown in Fig. 9.1(a) for the 2→2 transition (solid arrows), both the |mF | = 2 and |mF | = 1 magnetic sub-levels are driven by the microwave
field (the |mF | = 0 level has a zero transition moment). Because the electric dipole transition moment of the |mF | = 2 is twice that of |mF | = 1 [18], the Rabi-flopping signal
exhibits beating. The calculated individual magnetic sub-level contributions are shownat the bottom of Fig. 9.2(a) with their sum represented by the dashed line plotted over

193
(a)

(b)

Figure 9.3: Representative line shape of the 2→2 (a) and 1→1 (b) transitions. Bothmeasurements correspond to an interaction time of 0.5 ms (v = 200 m/s). A center
frequency is extracted from each fit (solid line).

the data. The data points were determined by comparing the populations in the upper
doublet at the detection region with and without the microwave field applied. Clearly,
the behavior is exactly as expected until tp ≥ 300 µs when ωR of both the |mF | = 2
and |mF | = 1 transitions appears to decrease. This reduction in ωR is the result of theelectric field diminishing near the cavity end-caps [9].
Alternatively to fixing the molecular velocity, v, and varying tp , the Stark decelerator allows v to be varied while applying a fixed (spatial) length microwave pulse.
This allows a check of systematics associated with beam velocity. Data taken in thismanner is shown in Fig. 9.2b, where a microwave pulse resonant with the 2→2 transition was applied for the entire time the molecules were in the cavity. The microwave
power for this measurement was chosen such that molecules with a 400 m/s velocity
underwent one complete population oscillation (i.e. a 2π pulse for the |mF | = 1 anda 4π pulse for the |mF | = 2 transitions). This was done so that population revivals
occurred at velocities that were integer sub-multiples of 400 m/s. While the behavior
agrees well with the expected (dotted line), there are two noticeable deviations. First,
for 270 m/s ≤ v ≤ 400 m/s the fringe visibility is less than expected. This is because,194
as detailed in our earlier work [28, 61], molecules with these relatively high velocities
have not been decelerated out of the background molecular pulse. Thus, molecules witha large distribution of speeds (as compared to the decelerated molecules) are detected,
leading to reduced contrast. Second, substantial decoherence is observed for v ≤ 130
m/s. The source of this decoherence has been experimentally determined as the resultof microwave radiation leaking from the cavity and being reflected off the decelerator
back into the cavity. It is interesting to note that since the metal detection tube acts asa waveguide with a cut-off frequency much higher than that applied, no radiation leaks
from the rear of the cavity. Thus, future experiments should include a small waveguidesection on both sides of the cavity to prevent leakage. Furthermore, because this decoherence is due to reflected radiation it is microwave power dependent, and presents noproblem for the transitions frequency measurements, which use < 10% of the microwave
power used in Fig. 9.2, such that no noticeable decoherence occurs.

9.3

Results
Because two magnetic sub-levels with differing ωR ’s undergo the 2→2 transition,

the traditional Rabi-spectroscopy method of applying a π pulse over the length of thecavity is not optimal. Thus, for measurements of the 2→2 transition frequency the
microwave power was chosen such that for δ = 0 the maximum contrast was producedwith as small a microwave power as possible (i.e. transfer the molecules to the first dip
in Fig. 9.2(a) over the length of the cavity). Representative data for the 2→2 transitionis shown in Fig. 9.3(a) where the driving frequency, f, is varied under a fixed microwave
power. Data points in this graph were generated by comparing the population in theupper doublet at the detection region with and without the probing microwave field.
The fit to the data (solid line) is generated from the typical Rabi lineshape formula
except contributions from the different magnetic sub-levels are included. For probing
the 1→1 transition it is necessary to prepare the molecules in a F = 1 level since they

195
originate in the |2, mF , +i level from the Stark decelerator. This is accomplished byusing the first 70 µs the molecules spend in the cavity to drive them on the satellite
2→1 line, yielding molecules in the |1, ±1, −i and |1, 0, −i states (downward dottedarrows in Fig. 9.1). The remaining time the molecules spend in the cavity (dependingon v) is then used to probe the 1→1 transition frequency by applying a π pulse, whichtransfers only the |mF | = 1 molecules to the upper doublet (upward dotted arrows
in Fig. 9.1). Representative data for the 1→1 transition is shown in Fig. 9.3b. In
contrast to the 2→2, line the LIF signal is maximum on resonance because molecules
are being transferred into the detected upper doublet. Also, because only 40% of the
population participate in the transitions (i.e. the |mF | = 1) the contrast between on
and off resonance is reduced relative to the 2→2 transition, leading to a slightly larger
error for determining the center frequency. For both panels of Fig. 9.3 the molecules
were decelerated to 200 m/s yielding linewidths of 2 kHz. A typical fit determines thecenter frequency within 20 to 50 Hz, depending on the transition. The results of several
measurements of both the 2→2 and 1→1 center frequencies are displayed in Fig. 9.4.
Each point and its error bar represents the result of a fit to a measured lineshape likethose shown in Fig. 9.3. Using the standard error of each fit as a weight the meanand standard error of the transition frequencies are found to be (1 667 358 996 ± 4)
Hz and (1 665 401 803 ± 12) Hz for the 2→2 and 1→1, respectively. For comparison,
the lightly hatched boxes represent the bounds set on the transition frequency by theprevious best measurement [120], while the limits produced by this measurement are
displayed as darker cross-hatched boxes. The transition frequencies reported here arelimited only by statistical uncertainties, as we estimate the uncertainties due to theaforementioned systematic shifts as (< .001 (collision shift) + < 2 (magnetic field) +< .001 (electric field) + < 1 (doppler shift)) Hz.
The Ramsey technique of separated pulses inside the same microwave cavity was
also used to measure the transition frequencies as seen in Fig. 9.5 with a resolution

196

Upp r

e doublet population

Figure 9.4: Results of multiple measurements of the 2→2 transition (upper) and the
1→1 transition (lower).

1.0
0.9
0.8
0.7
0.6
0.5
-30

-20

-10

0

10

20

30

f - 1667358.996 [kHz]Figure 9.5: Ramsey spectroscopy for the 2→2 transition with 0.2 ms pulse separation
time.

197
comparable to the reported Rabi measurements. However, because the molecular pulses
are extremely mono-energetic, minimal gain was observed in the recovered signal-tonoise ratio. This technique will of course be critical for any future molecular fountainclock. The molecular clock could enjoy reduced systematic shifts, such as the magnetic
field insensitive transitions demonstrated in this work. It will be most interesting tocompare an atomic clock against a molecular one which depends differently on the finestructure constant.
In summary, microwave spectroscopy was performed on slow, cold molecularpulses produced by a Stark decelerator resulting in the most precise measurement of
the OH ∆F = 0, Λ-doublet transitions. These results along with appropriate astrophysical measurement of OH megamasers can be used to produce constraints on ∆α/αwith a sensitivity of 1 ppm over the last ∼1010 yr. At the same time the use of cold
molecules for the most precise molecular spectroscopy has been demonstrated. Specifically, by producing slow, cold molecular packets the Stark decelerator allows increased
interrogation time, while virtually eliminating any velocity broadening. With continued
improvement in precision of the astrophysical measurement of the OH megamaser transitions through either long period of data collection/averaging and/or discoveries of new
megamaser sources, measurement of this molecular system will provide an importantand independent constraint on the variation of fundamental constants that date backto the origin of the universe.

Chapter 10

Not Enough Hours in the Day

This chapter is a collection of several disjoint topics, which we have consideredover my stay at JILA. I feel these topics, which do not fit into any of the other chapters,
are important enough to future cold molecule work that I have included them here.

10.1

Polar Molecule Collisions
Understanding the collision properties of molecules at cold and ultra-cold is crit-

ical to their efficient production and storage. As mentioned in the Introduction, theelectric dipole moment possessed by polar molecules fundamentally alters the molecularcollision dynamics. Specifically, the long-range dipole-dipole interaction,

1
3(µ1 · r)(µ2 · r)Vdd =µ1 · µ2 −,34πǫo rr2(10.1)

where µi is the electric dipole moment of the colliding molecules and r is the vector connecting the two molecules, dominates over the other molecule-molecule interaction. The
group of John Bohn has pioneered the study of cold polar molecule collisions [18, 19, 122],
utilizing carefully calculated potential surfaces and coupled channels calculations. Fromtheir calculations, polar molecules are predicted to have elastic and inelastic scattering
rates that are large and depend dramatically on electric field. Interestingly, because
polar molecule collisions are dominated by the dipole-dipole interaction, it is possible toestimate their elastic collision rates with surprising accuracy from analytical methods.

199
The next two subsections presents the elastic cross-section predicted by two of thesemethods, the Born approximation and the Eikonal approximation, and discusses theirrespective range of validity.

10.1.1

Born Approximation

The exact formulation of the scattering problem, the Lippmann-Schwinger equation, is an integral equation of the scattering particle’s wavefunction and impossible
to solve. The Born approximation assumes that if the scattering particles’ interactionis ‘not-too-strong’ it is acceptable to replace the wavefunction in the integral with theincoming wavefunction. Doing this results in the famous result for the first-order Born
amplitude [108]1 2m
f (k, k ) = −
4π h2
′

Z

′

′

eı(k−k )·x V (x′ )d3 x,(10.2)

where m is the particle mass, k and k’ are the initial and final momenta, and V (x’) isdσ
dσ, is given as dΩ= |f (k, k’)|2 .
the scattering potential. The differential cross-section, dΩFor scattering via the long range dipole-dipole potential, Vdd (R), Eq. 10.2 becomesm µ2
f (k, k ) = 2h̄ 4πǫo
′

Z ∞
0

2

r dr

Z 1

eıqr cos θ−1

1 − 3 cos2 θ
mµ2
d(cos
θ)
=
,
r3
3πǫo h̄2(10.3)

where we have used the usual ~q = ~k − k~′ , ~q k ẑ, and taken the molecular dipole moments
as a parallel to an electric field pointing at an angle θ reltative to the ẑ-direction. Finally,
we have the cross-section as given by the Born approximation as
σBorn =

4m2 µ4.
9h̄4 πǫ2o(10.4)For OH in the |2 ± 2+i state, in an external electric field high enough to saturate its dipole moment, we have σBorn = 1.5 × 10−8 cm2 . Clearly, this large, elastic
cross-section is encouraging for the possibility of evaporative cooling, however, before
definite predictions can be made the range of validity of the Born approximation must

200
be understood. The condition that the interaction is ‘not-too-strong’ means that for
the Born approximation to be valid the incoming wavefunction cannot be that different
from the true out-going wavefunction. From the Lippmann-Schwinger equation thiscondition takes the form [108]Z ∞0drreıkr

Z 1eıkr cos θ−1

4πh̄2 4πǫo1 − 3 cos2 θ
d(cos
θ)
≪
.
r3
2m µ2

(10.5)

Using, h̄k = mv and T = mv 2 /3 we have for the validity of the Born approximation
T ≪ TU pper =m 12h̄3 πǫo 2() .
3kB m2 µ2(10.6)For OH in the |2 ± 2+i state, in an external electric field high enough to saturate its
dipole moment, this constraint reveals that the Born approximation is valid at temperatures less than 150 nK. This small range of validity for the Born approximation isa direct consequence of the strong, long-range nature of the dipole-dipole interaction.
Nonetheless, the Born approximation provides a useful upper bound on the elastic collision cross-section of polar molecules. Furthermore, at low temperatures it agrees quite
well with the more accurate calculations of the Bohn group.

10.1.2

Eikonal Approximation

The Eikonal approximation is valid when the interaction potential, Vdd (R), does
not vary considerably over the scale of the deBroglie wavelength of the scattering particle. This approximation complements the Born approximation since it is appropriate aslong as the collision energy, E, is much greater than the interaction strength. The basic
idea of the Eikonal approximation is that at these energies the idea of a semiclassicalpath is reasonable and the wavefunction can be replaced by a semiclassical wavefunction. The derivation proceeds using the usual approximations of WKB theory and canbe found in most any textbook [108] with the scattering amplitude given as′f (k, k ) = −ıkZ ∞0

dbbJo (kbθ)[e2ı∆(b) − 1]

(10.7)

201
with the phase given by∆(b) =

−m
2kh̄2

Z ∞

p

V ( b2 + z 2 )dz,−∞

(10.8)

where b is the well-known impact parameter and Jo is the zeroth Bessel function of thefirst kind. Using the optical theorem, the Eikonal scattering cross-section is given as
σEik =

4π
Im(f (k, k)) = 4π
k

Z ∞
0

dbb(1 − cos(2∆(b))).

(10.9)In the case of the anisotropic dipole-dipole interaction the integral in Eq. 10.8 iszero if the impact parameter is held constant during integration. While the anisotropic
dipole-dipole interaction can be treated with a ‘tilted’ impact parameter, it is moredifficult and the results are very similar to that of the isotropic dipole-dipole interaction
iso = 1 µ1 ·µ2 ), which is trivially calculated. Replacing V
iso(Vdd
dd with Vdd , the integrals
4πǫo r 3

of Eqs. 10.8 and 10.9 are straight-forward and yield the Eikonal cross-section as
σEik =

2π 2 µ2,4πǫo h̄v(10.10)where v is the velocity of the colliding particle. We can make a rough estimate of therange of validity of the Eikonal approximation by defining the length scale of the Vdd asequal to the value of r where the dipole-dipole interaction is equation to the incoming
kinetic energy
3µ2
kB T =
324πǫo rChar(10.11)

Note that this is akin to saying for a molecule at temperature T , Vdd does not vary
appreciable for r ≥ rChar . The requirement that the deBroglie wavelength be less than
this characteristic length scale can be written ash√<
3mkB Tµ2
6kB T πǫo!1/3.

(10.12)

Rearranging this equation leads to a constraint on the temperature, T , for the validityof the Eikonal approximation:T >4h6 π 2 ǫ2.
kB m3 µ4(10.13)

202
For OH in the |2 ± 2+i state, in an external electric field high enough to saturate its
dipole moment, this leads to the constraint that the Eikonal approximation is valid for
T > 1 mK.
The cross-section predicted by the Eikonal approximation for OH in the |2 ± 2+i
state, in an external electric field high enough to saturate its dipole moment,is shownin Fig. 10.1. For comparison the temperature independent Born approximation crosssection is also shown. The vertical blue (left) line denotes the high temperature boundfor the range of validity of the Born approximation, while the vertical red (right) line
denotes the low temperature bound for the range of validity of the Eikonal approximation. Comparisons of the results of Fig. 10.1 with the more careful calculations [19],show that the Born and Eikonal approximations do a remarkably good job of estimating
the cross-section. In fact, the real cross-section essentially follows the Eikonal approximation with temperature until it reaches the value given by the Born approximation.
After reaching the Born approximation value, the cross-section levels off and follows theBorn approximation at lower temperatures. This behavior is not too surprising, because
the lower temperature bound that we have placed on the Eikonal approximation is quite
cautious, since on the length scale of rChar the potential is still relatively flat.
Because the electric dipole moment of a polar molecule depends on the electricfield, the strength of Vdd (R) is also field dependent. Using Eq. 2.21 for the dipole
moment as a function of electric field, the cross-section is plotted in Fig. 10.2. In thisgraph, the Eikonal approximation is used for T > 150nK and the Born approximation
is used for T < 150nK. As the field is initially increased the dipole moment increases
and consequently, so does the cross-section. Likewise, once the dipole moment is saturated the cross-section no longer increases with field. For temperatures where the Born
approximation is valid, the cross-section does not depend on temperature and follows
the top-most curve.

203

Cross-section [cm2]

10 6
10 7
10 8

Born

10 9
10 10

Eikonal

10 11
10 12
10

10 8

10 6

10 4

10 2

100

10 2

Temperature [K]Figure 10.1: The Born and Eikonal approximations for OH-OH dipole-dipole collisions,in an external electric field high enough to saturate the OH dipole. The Born approximation is valid to the left of the blue (left) vertical line, while the Eikonal approximation,
strictly speaking, is valid to the right of the red (right) vertical line. Comparison, withmore careful calculations show the predictions of the Eikonal approximation are actuallyquite good down to the temperature at which the cross-section of the Born and Eikonal
approximation are comparable.

−510<100 nK
10 µK
1 mK

2

Cross−section [cm ]

−10

10

100 mK
−15

10

−20

10

−25

10

2

10

3

10

4

10

5

10

6

10

Electric Field [V/m]Figure 10.2: OH-OH cross-sections as a function of electric field for several temperatures.

204

10.2

Sympathetic Cooling
One possibility for further cooling of polar molecules is by elastic collisions witha colder gas, so-called sympathetic cooling. Like the buffer gas cooling described in
Chap. 1, sympathetic cooling of polar molecules would be performed by trapping thepolar molecule in a region of high density cold gas. Since current molecule production
techniques already produce molecules around 1 mK, and the cold gas for sympathetic
cooling must be colder than this, laser cooled atoms seem the only viable choice forthe buffer gas. Experimentally, sympathetic cooling with laser cooled atoms may be
implemented by co-trapping the laser cooled atoms with the polar molecules. Since
radical molecules, like OH, have large magnetic moments the simplest sympathetic
cooling scheme would be to confine both the polar molecule and laser cooled atom
in a single magnetic trap and allow them to thermalize. However, the laser cooled
atom will fill a much smaller volume in this trap than the warmer polar molecules,
and thus, the probability for cooling collisions is reduced. As such, it will likely bebetter to trap the polar molecules in an electric field. The strength of the confining
electric field can then be used to match the polar molecule volume to the volume of themagnetically trapped laser cooled atoms. Furthermore, by trapping the polar moleculesin their absolute ground state (in an AC Stark trap [130]) inelastic collisions are removed,
making each collision a cooling collision. Therefore, the only limit to the sympathetic
cooling process is the trap lifetime. And for successful sympathetic cooling, we require
the elastic collision rate to be sufficiently greater than the inverse trap lifetime, suchthat the cooling can be completed before the molecules are lost from the trap.
The van der Waals interaction will be the dominant potential between a polar
molecule and a laser cooled atom. To zeroth order this can be approximated as theresult of the polar molecule polarizing the atom and then interacting with the induced

205

H

Rb

O

+

Figure 10.3: Schematic of the OH-Rb collision interaction.

206

OH-Rb cross-section from C61(10)-13[cm2]

7(10)-14
5(10)-14
3(10)-14
2(10)-14
1(10)-14
50

100

200

500

1000

2000

C6 [a.u.]

Figure 10.4: OH-Rb collision cross-section versus C6 .

dipole moment, as shown in Fig. 10.3 for the case of OH and Rb,µpm−µpm (αatom 4πǫo r3 )−µpm µatom
C6
=
= 6,
V6 =
3
3
2πǫo r2πǫo rr(10.14)

where µpm and µatom are the dipole moments of the polar molecule and the atom, respectively, αatom is the polarizability of the atom, and C6 is the van der Waals coefficient.
From this simple estimation, we find C6 = 50 a.u for OH-Rb collisions.
A better estimation of C6 can be made from second-order perturbation theory
with the dynamic polarizability integral [44, 43]
C6 =

3
π

Z ∞
0

αpm (ω)αatom (ω)dω(10.15)X δn,i µ2n,i(10.16)withαi (ω) =n

2 + ω2δn,i,where µn,i and δn,i are the is the transition dipole moment and transition frequency,
respectively, of the nth transition in particle i. This method is quite accurate, and forcomparison it gives a C6 value for Rb-Rb collisions that is 90% of the measured value.
The extra 10% comes from atom core effects, which are clearly not accounted for inthis approximation. In the case of OH-Rb collisions this method gives C6 = 282 a.u.
[10]. From the C6 constant it is trivial to estimate the scattering cross-section from the207theory of Gribakin and Flambaum [54]. It should be noted that this theory is valid onlyin the semi-classical limit, where the idea of a trajectory is still a decent approximations,and may fail when the molecules are sufficiently cooled by the buffer gas. A graph ofthis result for the cross-section is shown in Fig. 10.4. From this graph, we see thecross-section for the range of predictions of C6 is around 4×10−14 cm2 . From Eq. 1.3
with 100 µK buffer gas and 100 mK molecular gas, 40 collisions are required to bringthe molecular gas temperature within 2% of the buffer gas. Using a buffer gas densityof ρ = 1010 cm−3 and 10 mK as the average temperature of molecular gas (calculated
from Eq. 1.3 for 40 collisions), and our estimated value of the cross-section, we havethe collision rate as Γ = ρσv = 0.002 Hz. And thus, the molecules would need to be
trapped for sim6 hours to come into thermal equilibrium with the cold buffer gas.
Though the outlook seems grim for sympathetic cooling of polar molecules there
may still be some possibilities. For example, using large electric fields to polarize thelaser cooled atoms would increase the collision rate (though simple estimates show
only a 10× increase in the cross-section). Furthermore, it is expected that the crosssection increases with decreasing collision energy [69], which would make the estimations
given here a little cautious. Perhaps the best chance at sympathetic cooling is to use
metastable laser cooled nobel gases like xenon or cold Rydberg atoms. These particles
are easily polarized and would have a larger collision cross-section with polar molecules,
however, their large internal energies may present a problem if inelastic collisions areprominent.

10.3

Para-electric Cooling
Because the entropy of a collection of polar molecules depends on the polarizationof the sample, para-electric cooling (PEC), an analog to adiabatic demagnetization
refrigeration (ADR) [97], is possible. PEC would proceed by using an external electricfield to polarize a sample of polar molecules in the presence of a thermal reservoir.

208

Energy [Delta]

6
4
2
E

0

0

2

4

E/

6

8

10

Figure 10.5: Simple model for describing the features of para-electric cooling.

The entropy of this sample, polarized at temperature Ti , is equal to the entropy of an
unpolarized sample at temperature, T < Ti . Thus, by removing the electric field andallowing the spin degree of freedom to equilibrate with the thermal degrees of freedom
the sample cools to temperature T.
The essential features of PEC can be described by the toy-model presented inFig. 10.5, where the energy difference between states due to the Stark shift is simply
given asE = 2ǫss∆+ (µE)2 = ∆ǫ 1 +
2



2µE
∆

2

.

(10.17)As seen in the above equation, the key difference between PEC and ADR is that the
Stark shift couples opposite parity states, which can be degenerate only for symmetric top molecules. Thus, even at low-field, the states representing parallel (strong-fieldseeking) and anti-parallel (weak-field seeking) to the electric field are usually not degenerate, which leads to a limit to the ultimate temperature. If interactions between
molecules are neglected, the behavior of a sample of these polar molecules is given by

209
the one-particle partition function
Q=

+
X

~
βE(ǫ,|E|)

e

ǫ=−



−β∆

= 1 + eq

2

1+( 2µE∆ ),(10.18)where β = 1/(kB T ). From the partition function we have the entropy, S, and polarization, P , as
S
R
P
R

=

=

s

δT ln(Q)= ln(Q) + β∆ 1 +
δT
+
X



−β∆

2µE
∆

q

2

Q−1
Q

1+( 2µE∆ )2(10.19)


s

1−e
1
~
q
µǫeβE(ǫ,|E|) = µ= µ tanh β∆ 1 +2
Q ǫ=−−β∆ 1+( 2µE
)
∆
1−e



2µE
∆

2



,

(10.20)where R is the universal gas constant and thus these quantities are per mole.
The basic scheme of PEC is shown in Fig. 10.6(a). Starting with a ‘warm’
sample in zero field (circle #1) an electric field is applied that polarizes the molecules.
Depending on the strength of the field the entropy is lowered (circle #2 or #2’). Because
the molecules are in contact with a thermal reservoir, e.g. a cloud of laser cooled atoms,
their temperature is not increased since any liberated energy from the dipole is absorbedby the reservoir. Next, the thermal bath and the electric field is removed, allowing thepolar molecule sample to depolarize, leading to a lower temperature. Effectively, theextra thermal energy has went into depolarizing the molecules, i.e. internal energy.
From the two possible paths in Fig. 10.6(a) and (b) it is seen that the degree of
initial polarization (or reduction in entropy) sets the final temperature. Furthermore,~ = 0 trace in Fig. 10.6(a) that it is impossible to get below a
it is seen from the |E|
certain temperature. This is feature is specific to PEC and is a direct result of the nondegeneracy at zero-field of weak- and strong-field seeking states. Because the cooling of
the translational degrees of freedom comes from putting energy in the internal degrees
of freedom, once the translational energy falls below the zero-field energy splitting the
depolarization ceases since there is not enough energy to flip a strong-field seeking

210

1.4

1

S/R

1.2

3

2

3

2

1

0.8

(a)
0.6
0

2

4

k T/

6

8

10

1

0.6

P/(

E ff

R)

0.8

0.4

0.2

(b)
0
0

2

4

k T/

6

8

10

Figure 10.6: (a) Entropy per unit mole of a sample of polar molecules. (b) Polarization
per unit mole of a sample of polar molecules.

211
Table 10.1: Minimum attainable temperatures using para-electric cooling.
Molecule
OH
NH
ND3
H2 CO
Symmetric Top

Minimum PEC Temperature
16 mK
1 µK
16 mK
46 mK
∼0

molecule to the weak-field seeking state. Because at a finite temperature there is avelocity distribution this minimum temperature can be significantly less than the zerofield energy splitting and from Fig. 10.6(a) is found to be around ∆/5. This minimum
temperature for a few important molecules is given in Tab. 10.1.
From this table it is clear that PEC may have limited usefulness for molecules likeOH and H2 CO, which have large zero-field energy splittings. However, the process could
be very useful for molecule like NH, which have a very small zero-field energy splitting.
Furthermore, symmetric top molecules, like CH3 CN, which can be degenerate in zero
field could be cooled to extremely low temperatures. In ADR, the only limit to the final
temperature is the second order phase transition of spontaneous magnetization. Oncethe magnetic dipoles spontaneously align they generate a magnetic field, which limits
the final temperature. For electric dipole moments at low energies the dipoles will want
to be anti-parallel and this limit should not exist. Therefore, this technique could be
used to get to extremely low temperatures, and perhaps, is a likely road to a degenerategas.
It is worth noting that several assumptions have been made in the previous analysis. First, by dealing with thermodynamical properties we have assumed that the gases
have reached equilibrium. Second, the gases move towards equilibrium through collisions. Thus, we require that collision rate between the polar molecules and the laser
cooled atoms, serving as a temperature reservoir, be large enough to reach equilibrium

212
within the trap lifetime. Furthermore, the polarizing and depolarizing collisions are
by definition inelastic and we thus require a state insensitive trap. From the previous
section on sympathetic cooling we note that the collision rates for current densities andtemperatures is probably too small for efficient PEC. Nonetheless, PEC may still be
useful for two reasons. First, since Stark decelerated molecules are in the weak-fieldseeking state, the first step in PEC of polarizing the molecules along the field could be
replaced by a microwave pulse, which transfers the molecules to the strong-field seeking
state with unit efficiency. Second, it may be that PEC can be used after some othertype of cooling has produced molecules in the microKelvin regime to reach even lower
temperatures, since in the microKelvin regime collisions should be much more efficientat depolarizing the sample.

10.4

Cavity-Assisted Doppler Cooling
While various forms of cavity-cooling for a single atom trapped in a cavity [142]have existed for some time [128, 85], recently a proposal for cavity-assisted Doppler
cooling by the Vuletic group has appeared [133, 132] that may hold promise for coolinglarge numbers of molecules. A schematic of the implementation of this method is shownin Fig. 10.7(a), where a cloud of molecules held at the center of a cavity (by sometrapping force) is irradiated with laser-light that is far detuned from any transition.
By sufficient detuning of the incident light, the probability of a molecule being in the
excited state is exceedingly small, and thus the problem of finding a closed transition
for molecular laser cooling is avoided. At these large detunings the time-scale on whichconservation of energy can be violated is small, and thus the scattering of photons isbest described by two-photon processes, i.e. Rayleigh scattering. The basic idea ofcavity-assisted Doppler cooling is that for a molecule cloud with finite temperature theRayleigh scattered light will be Doppler shifted to the red and blue of the incident
laser-light at ω (see Fig. 10.7(b)). Thus, if a cavity is placed around the molecules that

213
sufficiently modifies the electromagnetic mode density and its resonance is tuned to theblue of ω, the molecules will preferentially scatter blue-shifted Rayleigh light and will
be cooled (see Fig. 10.7(c)).
Of course, before definitive predictions can be made about the feasibility of cavityassisted Doppler cooling a detailed analysis of the scattering rates involved must be done.
We analyze the situation for the case of a 3-level molecule as shown in Fig. 10.8. The
molecules start in the ground state, a, and can scatter photons from the laser at ω via theexcited state b. Depending on detuning and saturation, there are several possibilities
for the atom-photon interaction; absorption followed by spontaneous emission, Rayleighscattering through b back to a, and Raman scattering through b to state c. Note thata scattering event that leaves the molecule in state c is undesirable since state c will
no longer interact appropriately with the cooling laser and will likely be ejected fromany trap. In the far detuned limit, we need only worry about the two-photon analog
to spontaneous emission, spontaneous Raman scattering. Raman scattering occurs via
the same process as Rayleigh scattering, and thus, depends on the laser detuning in thesame way. The Raman and Rayleigh rates are easily derived from Fermi’s golden rule.
Starting from Eq. 5 on page 525 of Ref. [35], where the Born expansion has been used
to evaluate the transition matrix to second order, we find the Rayleigh and Raman rates
as
Γaa =
Γac =
with

8π I 2π(h̄ω)2|Taa |2 3 3
3 h̄ω h̄h̄ c′ )28π I 2π(h̄ω|Tac |2 3 3
3 h̄ω h̄h̄ c√
µab µbi (2ωab + ω ′ − ω) ωω ′Tai =2ǫo (ωab + ω)(ωab − ω ′ )(10.21)(10.22)(10.23)where Γaa is the Rayleigh scattering rate, Γac is the Raman scattering rate, I is thelaser intensity, µxy is the transition dipole moment between states x and y, ω is thelaser and Rayleigh photon frequency, ωab is the resonance frequency between states a

214

(a)

ωL

(b)
ω
ωL

(c)
ω

Figure 10.7: (a) Schematic of cavity cooling set-up. (b) Rayleigh scattered spectra ofmolecules with a finite temperature. (c) The Rayleigh scattered spectra of (b) in thepresence of a high finesse optical cavity.

215
and b, ω ′ is the Raman photon frequency, and for i = a ω ′ is replaced by ω. Thus, wehave the Rayleigh and Raman scattering rates as
!2

Γaa =

I
ω4
µ4
h̄ω 6πh̄2 ǫ2o c4 ab

Γac =

I
ωω ′3
(2ωab + ω ′ − ω)
2 2
µ
µ
ab
bc
h̄ω 6πh̄2 ǫ2o c4(ωab − ω)(ωab + ω ′ )2ωab2
ωab − ω 2
(10.24)2.

(10.25)Since Raman scattering pumps the molecule to a dark state, we require Γaa ≫ Γac for
cavity-assisted doppler cooling to work as proposed in Ref. [132].
For a real molecule, the complex ro-vibrational structure means there are manystates accessible by Raman scattering. This can be accounted for in the above picture
by replacing the properties of state c by the net properties of all the Raman accessible
final states, that is
ebc =
µ
e′ =
ω

N
X

µbi

(10.26)N
1 X
ω′N i=1 i(10.27)i=1(10.28)In the case of the OH radical, the significant allowed Raman transition are to several rotational components of the ground and first-excited vibrational states. Since scattering
through the first electronic excited state requires a laser photon in the ultra-violet (308nm) and the ro-vibration spacings are at most in the infrared, one can assume ω = ω ′without loss of accuracy. With this assumption, the ratio of Raman and Rayleigh scattering rates becomes1
Γac
=
Γaa



ebc
µ
µac

2

= 6.5.

(10.29)Clearly, our requirement that the Rayleigh rate dominate over the Raman rate is notmet, and cavity-assisted doppler cooling as proposed in Ref. [132] will not work for OH.
In fact, this result should be quite general for most molecules since there are typically
1This ration is calculate for the R3/2 line, which is the lowest frequency stretched transition fromthe OH ground state.

216

Figure 10.8: 3-level atom system. By letting state c have the properties of Raman
accessible final states, this simple model sufficiently represents a real molecule.

217
many final states (other than a) to scatter into and the transition moment for each ofthese final states is usually comparable to µab .
Nonetheless, there is still some promise for cavity-assisted Doppler cooling ofa molecular gas. This hope lies in the so-called ‘collective effects’ observed in thecavity-assisted Doppler cooling of cold cesium atoms [26, 33]. In these experiments, twodistinct types of collective effects were observed. For smaller detunings, a non-negligible
intra-cavity photon number of blue-shifted Rayleigh light stimulated the molecules toemit into the cavity mode. For larger detunings (>1 GHz detuned), the cold cesium
atoms self-organized, under the intra-cavity field, into a regular lattice structure that
Bragg-scattered from the incident beams into the cavity. In both of these regimes,
the collective effects worked to increase the Rayleigh scattering rate significantly overother scattering processes – though in the latter case only center of mass damping isobserved, i.e. the atoms are only slowed, not cooled. A rule-of-thumb from the cold
atom work is that for collective effects to occur N η must be approximately one, where
N is the number of molecules in the cavity mode and η is the ratio of scattering into
free space versus scattering into the cavity mode. For a Gaussian transverse modewith waist wo and finesse F and cavity light with wave vector k = 2π/λ, η is given asη = 12F/(π(kwo )2 ) and is a measure of how strongly the cavity has influenced the local
electromagnetic mode density. For a molecule density in the range of 107 cm−3 , wo =100 µm, and 1 cm spacing, N ≈ 1000. Thus, one needs η ≈ 10−3 , which requires afinesse of approximately 1000 at 308 nm (for OH). While a cavity with finesse of 1000 in
the ultra-violet is non-trivial, it can be made with careful coatings, and the possibilityof seeing collective effects seem very real. Of course several factors such as, startingwith a warmer gas (the molecule temperature will be ∼ 100 mK, while the cesium work
started around 100 µK) and the effect of molecules moving in and out of the cavity
mode (any current molecule trap will be of order 1 cm, while the cavity waist is of order
100 µm), may work against the onset of the collective effects. For this reason detailed

218
quantum Monte-Carlo simulations are currently underway by Josh Dunn of the Chris
Greene group here at JILA that account for all of these effects, and we can hope for areasonable prediction for the onset of collective dynamics in the near future.

10.5

Capacitive Detection of Polar Molecules
The most serious challenge facing all of cold molecule research is the lack ofa sensitive, non-destructive detection technique. Currently, most molecule detection
schemes utilize laser-induced fluorescence or ionization detection. Typically, both ofthese schemes leave the molecule in a final state that differs from the initial state, i.e.
destructive detection, and are quite insensitive when compared with traditional atomic
fluorescence detection techniques. The reason for the lack of sensitivity is simple, in
molecular fluorescence detection the lack of a cycling transition means that with atypical solid-angle-of-collection one expects to collect on order 1 photon from every
1000 excited molecules, where atomic experiments expect roughly 1000 photons fromeach atom. It should be noted that, while ionization detection can be quite sensitive,
small ionization cross-sections usually translate to long integration times.
Techniques, like phase-contrast imaging [17], hold much promise for detectionof cold trapped samples of molecules. However, these techniques may not be suitablefor detecting a moving molecular beam. Here we present a new technique, capacitive
detection of polar molecules, which works by detecting the induced charge on a capacitor
when the dielectric constant of the space between the capacitor plates is changed, i.e.
when polar molecules fly between the plates. This technique is particularly amenable
to supersonic beam experiments, where the molecules are all moving with roughly thesame velocity.
A schematic of the detector is shown in Fig. 10.9(a). It consists of two parallel
stainless-steel plates held at constant voltage, V , ‘sandwiched’ around a Boron Nitride
spacer. The polar molecules pass through a hole drilled in the Boron nitride spacer and

219

(a)

Stainless Steel
Boron Nitride
Stainless Steel

Molecular propagation axis(b)

Figure 10.9: (a) Capacitive detector assembly. (b) Results of a finite-element analysisof the deformation of the parallel plate capacitor for 5mG acceleration in the −x̂ direction. For this analysis, the detector assembly is held at four points along its midplane.
This mounting scheme reduces relative vibration between the plates, since most of theacceleration results in a common-mode movement of the plates.

220
cause current to flow in the leads connected to the plates since the plates are held atconstant voltage. The amount of induced current is easily calculated by considering theextra amount of potential energy, ∆U , stored in the capacitive field due to the presenceof the polar moleculess

1∆U = ∆QV = N 
2

∆
2

2



∆
+ (µE)2 −  ,2(10.30)where N is the number of polar molecules, ∆Q is the charged induced by the presence ofthe polar molecules, ∆ is the energy separation between the states mixed by the Stark
interaction (see Chapter 2), µ is the expectation value of the dipole moment of the polar
molecule, and E is the electric field magnitude. Note that the right most term in Eq.
10.30 is simply the energy of the N polar molecules in an electric field of strength, E.
Solving for the induced charge, we have
2N
∆Q =

r 
2
∆
2

+ (µE)2 − ∆
2
V

!

.

(10.31)The predictions of this equation are shown in Fig. 10.10 for the case of OH molecules
flying through two parallel plates spaced by 1 cm. The induced charge reaches a maximum equal to 2µ/d, where d is the plate separation. This maximum represents theinduced charge for a linear Stark shift, µE, and is recovered in Eq. 10.31 in the limit
V → ∞.
Near the supersonic nozzle one can expect on order of N = 1012 cm−3 , andthus the total induced charge in a 10 kV/cm field is ∆Q = 0.6 fC. Since this charge
is induced in the time it takes a molecular packet to fly into the capacitor, which is∼100 µs at this beam location, we have the induced current as I = 6 pA. Detectionof current at this level is certainly possible (commercial ammeters with fA resolution
are available), though not trivial. Several techniques, such as applying a voltage ripple
to the capacitor and using lock-in detection techniques, using the knowledge of themolecular velocity to gate the detection in time, and using the fact that when the

221

7

x 10 286Q/N [C]

5
4
3
2
1
0
10

100

101

102

Applied Voltage [kV]

103

104

Figure 10.10: The induced charge on a parallel plate capacitor spaced by 1 cm.

222
molecules exit the capacitor there should be an equal-magnitude, opposite-sign current
pulse relative to the current pulse that occurred when they entered, may make highsignal-to-noise ratio detection of these pulses easier. Furthermore, with ever increasing
current measurement capabilities, detection of these currents and lower may soon makethis all-electronic detection technique trivial.
Voltage instability and mechanical vibration present the largest sources of noisefor these measurements. While time-gating will virtually eliminate them for measurements in a beam, where the signal persists for ∼100 µs, the eventual goal, measuring thecurrent in a capacitor situated around a trap of polar molecules as a means to monitor
the trap dynamics (lifetime, etc.), will require low DC noise. Since voltage lock-in detection should circumvent any voltage instability problems, mechanical vibration is themost serious problem. Specifically, if the distance between the capacitor plates moves
by an amount δ, due to vibrations, it generates a noise current off ∆QN oiseIN oise ≈=2

f Co V2δ,d(10.32)where Co is the vibration free capacitance and f is the frequency of the vibration. Depending on the speed of the signal, low-pass electronic filters can be used to remove alarge portion of the vibration-induced current spectrum. Nonetheless, if one desires tomonitor trap dynamics, say trap lifetime in a mixed molecule trap where chemical reactions may occur, a bandwidth of several kHz may be required. Shown in Fig. 10.11 is theacceleration spectrum of our second-generation decelerator taken by a piezo-electric accelerometer. As expected, the largest vibrations are due to the turbo-molecular pumps
and constitute ∼5 mG acceleration. From a static finite-element analysis of our prototype design, we find this vibration produces ∼0.5 pm plate displacement (see Fig.
10.9(b)). Which, produces a noise current of 0.1 fA. While this value is well below the
expected current for measurement near the beam source, it will be the limiting factor
for measurements in a trap where the density can be as much as 5 orders of magnitude

223

Vibration Spectra
6

Turbo-molecular Pumps

Acceleration [mG]

5

4

3

60 Hz multiple
2

1

0
0

500

1000

1500

2000

f [Hz]
Figure 10.11: Vibration spectrum of second-generation Stark decelerator. The vibration
due to the turbo-molecular pumps is clearly visible, as is a harmonic of the 60 Hz line
frequency.

224
smaller. However, with careful vibration isolation and mounting, as in the work of Ref.
[92], it appears this systematic can be controlled.

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[3] Sebastiaan Y. T. van de Meerakker, Private Communication.
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