Ultracold Fermionic Feshbach MoleculesbyJoshua Zirbel
B.S. Physics, University of Missouri-Rolla, 2001

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
2008This thesis entitled:
Ultracold Fermionic Feshbach Molecules
written by Joshua Zirbel
has been approved for the Department of Physics

Dr. Jun Ye

Date

J

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

iii
Zirbel, Joshua (Ph.D., Physics)
Ultracold Fermionic Feshbach Molecules
Thesis directed by Professor Dr. Deborah S. Jin

This thesis presents experiments creating heteronuclear diatomic moleculesfrom ultracold mixtures of atomic bosons and fermions. The work presented
takes advantage of a type of scattering resonance, known as a magnetic Feshbachresonance, to efficiently and selectively create the fermionic molecules. I present
studies of the weakly bound, highly vibrationally excited molecules created near
the Feshbach resonance. These molecules have inelastic collisions that are affectedby the particular atoms involved in the collisions. Inelastic collisions are enhanced
or suppressed depending on whether the colliding atoms are bosonic, fermionic,
or distinguishable when compared to the molecule’s constituent atoms.

iv

Acknowledgements

This section is devoted to the long list of people who have helped me along
my journey through graduate school. This list is made longer by the fact that I
have had an especially long graduate career.
Of course, I must first thank my advisors. From 2001 until 2006, Carl Wieman was my first thesis advisor. I am continually impressed by his ability to findinteresting problems and then find solutions to them. Deborah Jin took over our
project and its students after Carl left. Even though Debbie has many obligations
she always managed to find the time to help us, either directly by coming into
the lab and taking data with us, or indirectly by asking probing questions. She
has impressed me with her ability to cut through a problem and partition it into
manageable sizes. Although we’ve forgiven her for being (somewhat) responsible
for breaking our magnetic trap, we haven’t forgotten and are ready to bring it up
if ever we need a get-out-of-jail-free card! She has been incredibly patient and I
can’t thank her enough for all that she has done for me. Of course, I can’t forgetthe final “Tri-group” advisor, Eric Cornell. Although he was not a direct thesis
advisor to me, I thank him for giving suggestions and encouragement during mytime at JILA.
The KRb molecule project was, from the beginning, a joint effort of Carl,
Debbie, and Jun Ye. Jun has recently become much more involved with theexperiment and it has been my pleasure to work with him. He has an infectious

v
optimistic attitude in spite of daunting challenges and has brilliant suggestions
when it comes to almost any problem. Jun possesses a unique way of building
confidence in his students by focusing on their positive attributes.
I’ve worked directly with a few graduate students and post-docs. Scott Papp
and I worked together building up the 87 Rb-85 Rb experiment from 2001-2003 until
I moved to the 40 K -87 Rb experiment. Scott has an unmatched intensity for physics
and getting things to work. After I moved downstairs, he was always around to
give his advice, solicited or otherwise! I thank him for always listening to myincessant questions regarding pretty much anything from lab stuff to life stuff. I
look forward to our continued friendship. In 2003, Kang-Kuen Ni joined me inbuilding the 40 K -87 Rb experiment and, as a result, has contributed significantly tothis thesis work. Kang-Kuen has grown into a capable researcher and, I’m proud
to say, she has succeeded into my former role in the lab while doing a better job.
I thank her for putting up with both my ups and downs throughout the years
and for being a good friend. Travis Nicholson was our undergraduate for the
early years and is responsible for building many parts of our apparatus. He is a
hard worker so I suspect he’ll have many successes in the future. Recently, Brian
Neyenhuis has joined us. He is very talented and his pleasant style has made it easy
working with him. I look forward to the seeing his future accomplishments. Silke
Ospelkaus joined our lab as a post-doc in early 2007. She immediately injected
her enthusiasm into the lab. With Silke’s help, anything can be accomplished.
This thesis would not be possible without her. It has been a pleasure becoming
friends with her and her husband, Christian.
A visitor to our lab, Andrew Wilson, was particularly helpful around the lab
during the early stages. He also has become a good friend I always look forward
to see at various conferences and when he is in the U.S. His attempts to instill
confidence in myself are greatly appreciated.

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Indirectly, I’ve worked with many talented students and post-docs. Here is
the short list, graduate students: Neil Claussen, Kurt Miller, Sarah Thompson,
Jon Goldwin, Cindy Regal, Michele Olsen, Lisheng Chen, Volker Schweikhard,John Obrecht, Juan Pino, Russell Stutz, Jayson Stewart, Laura Sinclair, John
Gaebler, Shihkuang Tung, Robert Wild, Tyler Cumby and post-docs: Elizabeth Donley, Stephan Dürr, Eleanor Hodby, Peter Engels, Shin Inouye, Markus
Greiner, Aaron Leanhardt, Avi Pe’er, John Perreault, and Giacomo Lamporesi.
JILA has been a wonderful place to work. Part of what has made it great is
how well it helps the students and faculty do tasks that might otherwise distractfrom science. Since I have helped build large portions of two apparatuses, I haveworked closely with two shops that have helped immensely: the JILA electronic
and machine shops. In particular, Terry Brown, Carl Sauer, Mike Whitmore,
Blaine Horner, Hans Greene, Tracy Keep, Todd Asnicar, Tom Foote, and Kim
Hagen have been helpful with suggestions and in building many components of
our apparatus. In addition to the technical expertise at JILA, the support staff
has been invaluable. These staff include those working in the supply office or
those assisting the fellows. In particular, Krista Beck has provided me with muchassistance over the years while she was an assistant to Carl and Debbie.
My family has been instrumental in developing my ability to make it to thispoint. I thank my parents for cultivating my desire to ask questions and helping
me find some of the answers. My mother, Donna Zirbel, always entertained us
with simple head-to-head strategy games. Using these games she would get us
to work by having the loser do a chore. I wish work could always be as fun now
as it was then. Having a father, James Zirbel, who tried and succeeded in fixing
nearly everything gave me many opportunities to see the insides of mechanicalthings, from boilers, cars, and washing machines to surplus He:Ne laser bar-code
scanners. His fix-it-all mentality gave me a sense that almost everything can bevii
understood and then repaired with the right tools. I thank my sister, Angela
Pulfer, for putting up with her kid-brother and giving him the chance to sponge
off of her knowledge. My grandmother, Jeanette West, always encouraged me
to pursue an education and tried to teach me that almost anything is achievablewith adequate preparation and a healthy body & mind. I thank my grandfather,
Theodore Morris, for always giving me a history lesson about his experiences
during WWII and for teaching me what he could about airplanes, finance, andfriendship.
I would also like to thank my teachers and professors before graduate school.
In particular I thank: Karen Harrison, Don Sparlin, Gerald Wilemski, Ron Bieniek
and Paul Parris.
Finally, I thank all of my friends. In particular I thank Anne Kellerman forgiving good advice, sharing her good cooking, & being my biking buddy and Zane
Altman for providing encouragement & companionship while writing this thesis.

Contents

Chapter
1 Introduction

1

1.1 KRb Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2 Why Ultracold Molecules? . . . . . . . . . . . . . . . . . . . . . .

2

1.3 Making Ultracold Molecules . . . . . . . . . . . . . . . . . . . . .

3

1.4 History of Ultracold Mixtures of Alkali Atoms . . . . . . . . . . .

4

1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52 Apparatus

7

2.1 Introduction and Overview . . . . . . . . . . . . . . . . . . . . . .

7

2.2 Laser Systems for the Atoms . . . . . . . . . . . . . . . . . . . . .

9

2.2.1

The MOTs . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.3 Vacuum Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.3.1

MOT Region . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.3.2

Transfer Tube . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.3.3

Science Cell . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.4 Magnetic Traps and Fields . . . . . . . . . . . . . . . . . . . . . .

20

2.4.1

Quadrupole and Transfer Trap . . . . . . . . . . . . . . . .

21

2.4.2

Ioffe-Pritchard Trap . . . . . . . . . . . . . . . . . . . . .

21

2.4.3

Oscillating Magnetic Fields . . . . . . . . . . . . . . . . .

25

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2.5 Optical Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

2.6 Imaging System . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

2.7 Computer Control

28. . . . . . . . . . . . . . . . . . . . . . . . . .

3 Methods

29

3.1 Introduction to Methods . . . . . . . . . . . . . . . . . . . . . . .

29

3.2 The MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.3 The Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

3.4 IP Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.5 Optical Trap Evaporation . . . . . . . . . . . . . . . . . . . . . .

37

3.6 Absorption Imaging . . . . . . . . . . . . . . . . . . . . . . . . . .

39

4 Creation of Feshbach Molecules

40

4.1 Feshbach Resonance . . . . . . . . . . . . . . . . . . . . . . . . .

40

4.2 Our Early Attempts . . . . . . . . . . . . . . . . . . . . . . . . .

42

4.3 Molecule Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

4.4 Creating Molecules: New Techniques and Tools . . . . . . . . . .

54

4.5 Driving Bad Bound-Bound Transitions . . . . . . . . . . . . . . .

56

4.6 Molecule Association Process . . . . . . . . . . . . . . . . . . . .

58

4.7 Ultracold, Trapped Molecules . . . . . . . . . . . . . . . . . . . .

62

4.8 Binding Energy of the Feshbach Molecules . . . . . . . . . . . . .

64

4.9 Properties of the Feshbach Molecule . . . . . . . . . . . . . . . . .

70

4.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

5 Inelastic decay of the molecules

76

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

5.2 Collisional Stability of Fermionic Feshbach Molecules . . . . . . .

77

5.3 Isolating Collision Partners . . . . . . . . . . . . . . . . . . . . . .

79

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5.3.1

Distinguishable Atoms . . . . . . . . . . . . . . . . . . . .

79

5.3.2

Bosonic Atoms . . . . . . . . . . . . . . . . . . . . . . . .

80

5.3.3

Fermionic Atoms . . . . . . . . . . . . . . . . . . . . . . .

81

5.4 Three-Body Recombination . . . . . . . . . . . . . . . . . . . . .

82

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

6 Conclusions and future directions

88

6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

Bibliography

95

Appendix
A Magnetic Trap Trouble

102

TablesTable
2.1 Major impurities of the enriched KCl. Source: Trace Sciences International. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

5.1 Scattering length, a, dependence of the loss coefficient for various
collision partners. ‘BF’ refers to the Feshbach molecule, ‘B’ to theboson, ‘F’ to the fermion, and ‘X’ to a distinguishable K atom.
P (a) and M (a) are log periodic functions in a with modulations
resulting from the formation of Efimov states. . . . . . . . . . . .

78

Figures

Figure2.1 General schematic for the laser system (same for Rb and K lasers).
We use saturated absorption on atomic references to detect and stabilize the repump laser frequency. Using the frequency-stabilized
repump light we “offset lock” the trap laser frequency by electronically detecting the beatnote between the trap and repump lasers
and stabilizing the beatnote to an rf source. Light from both lasers
are amplified for the MOTs. . . . . . . . . . . . . . . . . . . . . .

11

2.2 Construction of the injection locked laser for amplifying K trap
light. On the left, an exploded view of only the collimation tube
holder and the cooling stages. On the right, the interior view of
the IjLL. A laser diode designed for room-temperature operation
at 780 nm was cooled to -45 degrees Celsius to shift the wavelength
down to 767 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

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2.3 Schematic of our vacuum chamber. From right to left: Atomic
sources supply atoms to the MOTs and keep the MOT cell at higher
pressures. We limit migration of contaminants from the MOT cellusing the differential pumping tube. The bump region has a borosilicate glass slide to block atomic line-of-sight to the science region.
A gate valve can be closed to separate the MOT region from thescience region of the vacuum chamber. . . . . . . . . . . . . . . .

15

2.4 A zoomed-in overhead view of the MOT region of the vacuum chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.5 A schematic of the coils of the Ioffe-Pritchard magnetic trap. The
large circular pair of coils are the bias coils. The small circular pairof coils are the pinch coils. Finally, the elongated, “racetrack” pairof coils are the Ioffe coils. Electrically, the Ioffe and pinch coils areconnected in series. The bias coils are connected in series with eachother but are electrically separate from the other coils. . . . . . .

22

2.6 A computer generated model of the magnetic trap and its holder.

23

xiv
2.7 Contour plots of the magnitude of the magnetic field produced by a
Ioffe-Pritchard Trap. The data is obtained from a computer modelof the magnetic field using our coil geometry and dimensions. The
horizontal direction of the plot is along the direction of the bias
magnetic field. The vertical direction of the plot corresponds to
the radial direction of the trap. The color scale is the same foreach plot with purple corresponding to a low magnetic field andred corresponding to a high magnetic field. For each plot the current in the Ioffe coils and pinch coils are fixed and have the samevalue. The difference between the plots comes from increasing thecurrent through the bias coils from low current (top) to high current (bottom). By increasing the bias current, the magnetic trap’s
depth increases and the radial trap frequencies increase.

. . . . .

24

2.8 Circuit diagram of the rf coil that is impedance matched to 50 Ω
at DC-10MHz and near 80 MHz. . . . . . . . . . . . . . . . . . .

25

2.9 Imaging system used to detect the atoms and molecules. . . . . .

27

3.1 Number of 87 Rb atoms recaptured into the MOT after loading aquadrupole magnetic trap. The atoms are held at the indicatedgradients for 200 ms. The two data sets are before (4) and after (•)
optimizing the molasses stage before the quadrupole load. Opticalpumping is not optimized. . . . . . . . . . . . . . . . . . . . . . .

32

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3.2 Number transferred past the “bump” as a function of current in
the bump coils. The optimized transfer efficiency across only thebump is about 97%. This efficiency is calculated by comparing thenumber transferred past the bump to the number transferred tojust before the bump. It is important that the speed of the track isless than 20 cm/sec for the best transfer efficiency across the bump.
The pyrex obstruction totally blocks the cloud’s path for currents
below 4.5 amps and the vacuum chamber wall totally blocks thecloud’s path for currents above 8 amps.

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

34

3.3 Effect of forced evaporative and sympathetic cooling on the temperature and number of both atomic species. The clouds’ temperature
decreases at the expense of 87 Rb atom number (solid circles). The40

K atom number (solid triangles) is only slightly affected. . . . .

36

3.4 Breit-Rabi diagram for 87 Rb at low magnetic field. We evaporatively cool the 87 Rb gas by driving the |2, 2i → |1, 1i transition.
This removes 87 Rb atoms from the magnetic trap. We also drive
the |2, 1i → |1, 0i transition to remove impurity states of 87 Rb thatinelastically collide with 40 K.

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

38

4.1 Magnetic field can be used to tune a closed molecular state andthe open atomic states into degeneracy. The different magnetic
moments of the two states allow the tunability. If these states
are coupled, a Feshbach resonance will occur at a magnetic fieldin the vicinity of the crossing between the two states. The greyarrow indicates how the molecule state can be populated with aslow magnetic-field sweep. . . . . . . . . . . . . . . . . . . . . . .

41

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4.2 Loss of K atoms in the presence of Rb resulting from an interspeciess-wave Feshbach resonance. The atoms are in the |9/2, −9, 2iK and|1, 1iRb states and are held at the indicated field for one second
before abruptly turning off the magnetic field and imaging them.
For this measurement the number of 40 K atoms was about 105 andthe number of 87 Rb atoms was about 5 · 105 . Both clouds were
initially in thermal equilibrium at a temperature of about 500 nK.

44

4.3 Loss of K atoms, in the presence of Rb, resulting from an interspecies p-wave Feshbach resonance. The atoms are in the |9/2, −9, 2iK
and |1, 1iRb states and are held at the indicated field for one second
before abruptly turning off the magnetic field and imaging them.
For this measurement the number of 40 K atoms was about 105 andthe number of 87 Rb atoms was about 5 · 105 . Both clouds were
initially in thermal equilibrium at a temperature of about 500 nK.

45

4.4 Solid lines show the time dependent ramps near the Feshbach resonance. The top figure shows a ramp that should not create molecules.
The lower figure shows a ramp direction that should create molecules.
46
4.5 The peak optical depth of K after ramping across the Feshbach resonance. The time to complete the ramps of Fig. 4.4 was adjusted.
The atomic loss is consistent with molecule formation after ramping slowly from high to low magnetic field (solid circles). The timespent near the resonance is not a significant factor for atomic loss
since little loss is observed after ramping slowly from low to highmagnetic field (solid triangles). The dashed line indicates the initial
optical depth of the K cloud without ramping across the resonance.
The optical depth was measured after 12ms of free expansion. . .

47

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4.6 Solid line shows a typical time-dependent magnetic-field ramp near
the Feshbach resonance. The hold magnetic field was changed foreach data point of Fig 4.7. The ramps up to and away from thehold field were adiabatic for creating molecules. . . . . . . . . . .

49

4.7 Finding the resonance location using time-dependent magnetic-fieldramps. The transition to more loss occurs when the hold fieldof Fig. 4.6 is above the resonance position. This loss is due tothe creation of molecules which, in this experiment, were quickly
removed by the optical trap light. The magnetic field was heldfor 300 µs and the ramps toward and away from the resonance had
inverse ramp rates of 0.139 ms/gauss. The solid line is a fit, using
an offset error function, to the data. B0 is taken as the center of thefitted error function. The 10% to 90% width of the error function
is taken as the uncertainty in B0 . Using these criteria, we obtain
B0 = 546.47 ± 0.12 gauss. The initial 40 K and 87 Rb atom numbers
were about 1.1 · 105 and 2.9 · 105 and T = 240 nK = 0.8Tc = 0.6TF .
We detected the atoms at low magnetic field after turning off not
only the optical trap but also the magnetic fields. . . . . . . . . .

50

4.8 Magnetic-field ramp near the Feshbach resonance to create molecules,
probe their lifetime, and reconvert the molecules to atoms for low
magnetic-field imaging. The vertical jumps represent diabatic crossings of the resonance. The ramps are adiabatic for creating anddissociating molecules. . . . . . . . . . . . . . . . . . . . . . . . .

52

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4.9 A measurement of the lifetime of molecules. We used magnetic-fieldramps similar to Fig. 4.8 to reconvert the molecules into atoms.
The hold magnetic field for this measurement was 546.2±0.2 gauss.
The solid line is the results of an exponential fit to the data. The
fitted decay time is 220 ± 7 µs.

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

53

4.10 Rf association spectrum of heteronuclear Feshbach molecules at
B = 546.04 G. The molecules are created starting from an ultracold gas of 1.7×105 87 Rb atoms at T /Tc = 1 in thermal equilibriumwith 8×104 40 K atoms. The plot shows the detected atom/molecule
number as a function of the detuning ∆ν of the rf from the atomic
resonance frequency. We observe a large atomic peak at zero detuning and a molecular peak at E/h = 161 kHz due to the associationof atom pairs into the Feshbach molecule state. In this measurement the rf pulse duration was 600 µs, which is 50 times longer thanan atomic π-pulse. The grey band indicates the expected envelope
full width at half maximum for the power broadened atomic transition. The inset shows a simplified energy diagram for magnetic
fields below the Feshbach resonance. The upper solid line corresponds to Rb |1, 1i and K |9/2, −7/2i, while the lower solid line
corresponds to Rb |1, 1i and K |9/2, −9/2i. The molecule stateis represented by the dashed line. Depending on the detuning,
the rf is capable of driving an atomic hyperfine changing transition or magnetoassociation. This data was taken while the atoms
and molecules were held in an optical trap derived from a 1064 nm
single-frequency fiber laser.

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

57

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4.11 KRb molecular potentials. Svetlana Kotochigova has calculated
that the 1075nm optical trap laser predominately drives boundbound transitions from the Feshbach molecule state to vibrational
states of either the 3(1) or 2(1) molecular potentials. The molecule
potential energy data for this figure was obtained from Ref. [1]. .

59

4.12 Time dependence of the rf association process at B = 546.17 G,
where the measured molecule binding energy is E/h = 96 kHz. The
plot shows the measured number of molecules created as a functionof the 1/e2 gaussian rf pulse length. The solid line is a simple fit
used to estimate the impact of the observed decay on the maximum
number of molecules created. . . . . . . . . . . . . . . . . . . . . .

61

4.13 Temperature dependence of Feshbach molecule creation by rf association. Plotted is the molecule conversion fraction Nm /NK as afunction of T /Tc for the Rb atoms. The solid circles (•) are experimental data and the curves are results of a Monte Carlo calculation.
The dashed curves represent the uncertainty in the conversion fraction taking into account the uncertainty in the trap frequencies andatom numbers. The molecules are created starting from an atom
gas mixture with NRb = 3 × 105 and NK = 1 × 105 . We observe
maximum conversion efficiency of 25% at T /Tc = 1. . . . . . . . .

63

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4.14 (Top inset) Images of the molecule cloud after release from theoptical trap. (Top) Radial cloud size for the molecules after releasefrom the optical trap. (Bottom) Slosh of the molecular cloud in the
optical trap seen in images of the expanded gas. The expansion
time was 3 ms. The observed oscillation in the molecule cloud’s
position demonstrates that the molecules are indeed trapped. The
strong damping seen here is most likely due to anharmonicity ofthe optical trap.

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

65

4.15 Energy of the heteronuclear Feshbach molecules, relative to the Rb|1, 1i + K |9/2, −9/2i atomic threshold, plotted as a function ofmagnetic field. The data (solid circles) agree well with our calculation based on a full coupled-channel theory (solid line). Near the
Feshbach resonance one expects a universal relationship between
the s-wave scattering length and the binding energy. The black,dashed line shows the predicted binding energy using the universal
prediction [2]. Inset: Same as the main plot but looking at theregion close to the Feshbach resonance.

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

68

4.16 Calculated molecule energies near the K |9/2, −9/2i + Rb |1, 1i
atomic threshold. All levels have total spin projection quantum
number MF = −7/2. The level energies are calculated using a full
coupled-channel calculation and have been scaled so that the zeroin energy is the K |9/2, −9/2i + Rb |1, 1i atomic threshold. The
bold line is the adiabatic level associated with the 546.7 G Feshbachresonance.

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

69

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4.17 Estimated molecule size vs. magnetic field near the Feshbach resonance as extracted from the experimentally measured binding energy (•). We compare to the universal size prediction, a/2 (dashedline). Inset: Closed channel fraction extracted from the experimentally measured binding energy (•)(see Sec. 4.9). . . . . . . . . . .

71

4.18 The calculated open-channel molecular size, ro , as a function ofbinding energy. The solid red line is the universal prediction, a/2.
The dashed blue line is the classical outer turning point of the KRb
ground state potential.

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

74

5.1 Molecule loss coefficient β for collisions with distinguishable atoms(K in the |9/2, −7/2i state) as a function of the heteronuclear scattering length. For large a, we observe β to decrease with increasing
scattering length in agreement with numerical calculations (solid
red curve). We observe β saturating for small a, (a ∼ 100 a0 ) where
the molecular size, hrmol i, no longer has a strong dependence on a.
Inset: β vs. molecule size hrmol i as extracted from the measured
binding energy of the Feshbach molecules [3]. The green dashedcurve is a power law fit βfit ∝ hrmol ip to the experimental data; we
obtain p = −0.97 ± 0.16.

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

84

5.2 Molecule loss coefficient for collisions with indistinguishable bosons
versus heteronuclear scattering length. For large a, β increases
with increasing a. In this regime the molecular loss is enhanced
compared to loss due to molecule collisions with distinguishable
atoms (green dashed curve and Fig. 1). For small a, β increases
with decreasing scattering length (see Section 5.3.2).

. . . . . . .

85

xxii
5.3 Molecule loss coefficient β for collisions with indistinguishable fermions.
For large a we observe loss rates scaling as a−1.6±0.2 . The power law
dependence is consistent with numerical calculations for this scattering length regime (solid red curve). In addition, the molecularloss is suppressed compared to inelastic decay of molecules due tocollisions with distinguishable atoms (green dashed curve and Fig.
1). Inset: Molecule decay at a ≈ 1300 a0 . The 1/e-lifetime of themolecular cloud is τ = 100 ± 20 ms . . . . . . . . . . . . . . . . .

86

5.4 Three-body recombination near the heteronuclear Feshbach resonance. The measured K3 varies as a4 and saturates at large a dueto the unitary limit. The dashed curve is an a4 fit to the data for|a| < 4000 a0 . The solid line is the unitarity limit at 150 nK. . . .

87

6.1 Diagram showing the laser frequency we initially used to find excited molecular states. The laser was tuned energies lower than theD1 line of atomic Rb. This corresponds to wavelengths longer than
795 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

6.2 Observed molecule loss due to excitation to excited electronic statesof the KRb molecule. Here we start with Feshbach molecules andilluminate them with laser light having the indicated frequency.
The Feshbach molecules are lost when the laser light excites themto states where they spontaneously decay and are lost from the trap. 92

xxiii
6.3 Loss of molecules in the Feshbach state resulting from optical excitation to an excited, bound molecular state of KRb. All experimental parameters were the same as in Fig. 6.4 except here, theinitial state is the Feshbach molecular state instead of freely colliding atom pairs. Here, τ is shorter by a factor of 500 compared tofreely colliding atom pairs.

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

93

6.4 Loss of atoms resulting from photoassociation into an excited, boundmolecular state of KRb. The τ for this data is about a factor of
500 longer than the data shown in Fig. 6.3.

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

94

A.1 The resistance of the coil plotted against the temperature of thewater flowing through the coil. The slope of the curves is consistent
with the thermal coefficient of resistance of copper. The verticalresistance shifts are caused by mechanical perturbations of the coil.
These shifts are relatively small at about 1% of the total resistanceof the coil. For comparison, if a complete winding were shorted, itwould have a shift 13 times larger. . . . . . . . . . . . . . . . . . . 104
A.2 Correlation between the cloud’s measured position and percent resistance change. The data was taken after releasing from the magnetic trap for 15ms of free expansion. The slope is roughly consistent with the slope predicted by a numerical calculation. . . . . . 106
A.3 A view of the Ioffe coil’s problematic lead. The pressure between
the wires at this transition point caused an incomplete short toother windings through the polyimide coating. . . . . . . . . . . . 107

Chapter 1
Introduction

1.1

KRb Molecules
This thesis discusses some of the first experiments to make ultracold het-

eronuclear molecules. The motivation behind this work is to efficiently create
KRb molecules as a first step toward achieving a long-term goal of producing
a relatively dense gas of ultracold polar molecules. To accomplish this, we implemented well-established capturing, cooling and associating techniques used inother ultracold alkali atom experiments. In addition, we developed new techniquesthat are specific to the the mixture of atoms we use: fermionic 40 K and bosonic87

Rb. This thesis will present our experimental techniques with an emphasis on

the new techniques with the hope that this will prove useful to people either
using the existing apparatus or planning on building similar experiments. In addition to experimental techniques, this thesis will describe how loosely-bound KRb
molecules were associated and characterized. It will also present a study of inelastic atom-molecule collisions near a Feshbach resonance through measurementsof the molecules’ lifetime. The atom-molecule collisions are interesting because
the atoms involved can be either distinguishable or indistinguishable from themolecule’s constituent atoms. This means we were able to test how inelastic ratesof the molecules scaled when either fermions, bosons, or distinguishable atoms
were allowed to collide with the molecule.

2
1.2

Why Ultracold Molecules?
The major attractive feature of molecules, compared to atoms, is their richinternal degrees of freedom. Experiments with ultracold molecules can explore thevibrational and rotational structure of the molecule with much better precision
than previously attained with higher temperature gases. For example, improvements in precision can be used to test fundamental physics [4, 5, 6]. Additionally,
the molecule can offer unique properties to the experimenter that atoms, alone,cannot provide. For example, many molecules are easily affected by external electric fields. Compared to atoms, it is easier to affect the internal energy of themolecule using external electric fields if the molecule possesses a spatial asymmetry. In the simplest case, a molecule composed of two unlike atoms can be
manipulated by an external electric field because the two atoms share the electrons in a lopsided way which induces a permanent electric dipole moment.
A perfectly symmetric molecule, like the homonuclear H2 , will not posses a
dipole moment. However, virtually any molecule with an asymmetry will posses
a dipole moment of some degree. Remarkably, this even applies to HD, a nearlyhomonuclear molecule [7, 8]. Although the HD dipole moment is incredibly weak,
it played a role in cooling gas clouds in the early universe [9]. The dipole momentsof HD and LiH allowed these early gas clouds to radiatively cool and, therefore,
collapse into early stars. It is important to note that I am cavalier with thephrase “dipole moment”. My “polar” molecules do not posses a true permanent
dipole moment. Instead, they are easily polarized in relatively weak electric fields
making it appear as if they possessed a permanent dipole moment. There iscurrently no direct experimental evidence showing that atoms or molecule posses
a true permanent dipole moment but this may change in the near future.
We decided to attempt making ultracold polar molecules because the dipole

3
moment offered some exciting prospects for future studies at such low temperatures. In particular, because many heteronuclear diatomic molecules have relatively large dipole moments & 1 Debye, it seemed like it should be easy topartially polarize the molecules in an external electric field, observe long-range
dipole-dipole interactions between the molecules, or use the molecules as quantum bits for quantum information processing schemes [10].
1.3Making Ultracold Molecules
There are several strategies one could pursue to create ultracold molecules.

Starting with room temperature molecules one could obtain cold molecules using
Stark deceleration [11, 12], buffer gas cooling [13], or velocity selection [14]. The
challenge for these methods is obtaining large densities of cold molecules. Anothermethod is to start with an ultracold, dense atomic cloud and produce molecules
through association techniques. This method is hindered by the availability ofatoms that can be easily cooled. The alkali atoms enjoyed early laser coolingsuccess because of their nearly closed “cycling” transition. Additionally, alkali
atoms had been associated into molecules using optical techniques [15, 16, 17]. It
was because of these advances and our ability to cool the alkalis that we decided
to use ultracold alkali atoms as our starting point for ultracold molecules.
We began constructing the apparatus in the summer of 2003 in an empty lab
shared with another molecule experiment. Our plan was to create an apparatus
in which a mixture of atoms would be used to form molecules. For collecting
and cooling the gases and enhancing molecule production, the experience of Carl
Wieman’s and Deborah Jin’s groups would be important. Once the molecules
were produced, we would then take advantage of Jun Ye’s laser experience toproduce tightly bound molecules having an electric dipole moment.
In 2003, the techniques for trapping and cooling alkali atoms were well estab-4lished. In addition, at about the same time, experimenters had taken advantage
of Feshbach resonances to efficiently create loosely bound molecules. These resonances were magnetic-field tunable. So, by ramping the magnetic field near a
Feshbach resonance, one could efficiently convert atoms into molecules [18, 19].
These “Feshbach molecules” were well suited for further manipulation steps because their lifetimes could be relatively long and because their size was tunable
near the Feshbach resonance. The tunable size could allow for improved FranckCondon overlap with molecular vibrational states in the excited electronic potential.
1.4

History of Ultracold Mixtures of Alkali Atoms
A major limitation for precision studies using atoms and molecules has his-

torically been that they move at high speeds because of their thermal energy.
Such speeds induce uncertainties caused by Doppler broadening and short interaction times. It has, therefore, been a goal of researchers to slow the motion of
the atoms and molecules relative to the lab so these limitations are removed [20].
After slowing their motion, collisional interactions can become important and canthen be studied and controlled.
The major successes of controlling and cooling atoms’ motion came throughout the 1980’s with the advent of laser cooling. Based on suggestions of Hänsch,
Schawlow, Wineland, Dehmelt, and Ashkin, and success of laser cooling ions,
it became clear that cooling neutral atoms could be possible. In fact, coolingneutrals to and then trapping them at microKelvin temperatures was possible.
These techniques typically slow atoms from speeds of hundreds of miles-per-hour
to around one-tenth of a mile-per-hour. The work of cooling and understanding
the cooling mechanisms of neutral atomic gases was awarded a Nobel Prize in1997 and is summarized nicely in the corresponding Nobel lectures [21, 22, 23]

5
and references therein.
The success of laser cooling culminated in the experimental realization of
Bose-Einstein Condensation (BEC) in a dilute, weakly interacting atom gas in1995. This result was, in itself, a major achievement, which overcame manytechnical obstacles, and was also awarded a Nobel prize in 2001. Again, theNobel lectures [24, 25] and references therein are good sources for a summary of
the work and more historical details.
Following the creation of BEC, fermionic atoms were put into the cool glareof laser and evaporative cooling. These techniques were equally successful andresulted in fermionic atoms entering the quantum degenerate regime by the late
1990s [26, 27]. The resulting degenerate Fermi gas (DFG) was a clean, defect-freesystem having well known two-body interactions. A serious goal at that time
was to make a low-temperature two-component strongly-interacting Fermi gas
and explore the BCS-BEC crossover regime [28, 29, 30, 31, 32, 33]. The experimental realization of this goal in 2004 was aided by using a special tool, calleda Fano-Feshbach resonance[19], to control the interatomic interactions between
two different spin states of the fermions [34, 35]. Such control over the atoms has
given researchers a better understanding of the BCS-BEC crossover.
Although BECs, DFGs, and BCS-BEC physics are not directly related to theexperiments described in this dissertation, we utilize similar supporting techniques
to create heteronuclear molecules. This is accomplished by capturing and cooling
two types of atoms: bosons and fermions. The resulting mixture of atoms arecoaxed into forming fermionic diatomic molecules using a Feshbach resonance.
1.5

Thesis Outline
Chapter 2 will give details of the 40 K + 87 Rb apparatus. It will describe what

the apparatus was supposed to accomplish for our science goals. It will also include

6
a brief compilation of atomic parameters, a motivation for and layout of our opticalsystem, a description of the vacuum chamber and how we create ultracold samplesof K and Rb. Chapter 4 will describe how we created heteronuclear molecules using
Feshbach resonances. Chapter 5 will describe experiments that explored inelastic
collisions between various atomic species and the loosely bound molecules. Mostof Chapter 5 has been derived from a published article [36]. Finally, Chapter 6
will summarize the thesis and give a look toward future experiments.

Chapter 2
Apparatus

2.1

Introduction and Overview
The apparatus is the machinery responsible for capturing, cooling, control-

ling, and detecting the atoms and molecules that are used in the experiments
described in subsequent chapters. It incorporates a diverse set of tools including
computers and software, optics, lasers, digital and analog circuits, a mechanicalmotion stage and an ultra-high-vacuum chamber. Most of these tools had been
perfected and incorporated in other laboratories in JILA and elsewhere. It was
our goal to reproduce the good from other labs while avoiding the bad. I’ve foundthat the best way to reproduce results is to copy the methods. I will try to point
the reader in the direction of the references of the methods when I can. This
chapter is devoted to giving the reader technical details and a basic flavor of how
and why we chose certain elements of the apparatus. I will attempt to point out
whether these choices are either physically justified or aesthetically preferred. I
hope this will not only help the future users of the apparatus but will also help
guide others interested in building similar apparatuses.
Inside of an ultra-high vacuum (UHV) chamber, we use standard techniquesof laser cooling and trapping combined with evaporative cooling to obtain mixturesof ultracold 40 K and 87 Rb atoms suitable for Feshbach molecule formation. I will
discuss these techniques in greater detail in a subsequent chapter. Briefly, these8
techniques start by using a magneto-optical trap (MOT) to capture and cool atoms
from a room-temperature gas. The MOT is located in a section of the vacuum
chamber having relatively large partial pressures of K and Rb. Typically weselectively capture about 107 40 K and 109 87 Rb atoms in the MOT. These trapped
clouds have temperatures of a few hundred micro-Kelvin. Once the MOTs are fully
loaded, we compress the MOT and transiently cool the atoms using polarization
gradient cooling. After this preparation, the atoms are optically pumped into
the magnetically trappable stretched states and are ready to be transferred into aquadrupole magnetic trap. After all laser fields are turned off, the quadrupole trap
is suddenly turned on and captures K and Rb atoms. Using a mechanical motion
stage, the quadrupole trap is then physically moved from the high pressure regionof the vacuum chamber to a lower pressure region called the Science cell. Here,the atoms are transferred into a standard Ioffe-Pritchard (IP) magnetic trap. In
the magnetic trap we use microwave fields to force evaporative cooling of the 87 Rb
atom gas, which sympathetically cools the 40 K gas. Once the temperature is about
1 µK we load the atoms into a far off-resonant optical dipole trap. This trap canhold atoms in any of the possible hyperfine states rather than just the low-fieldseeking states that are trapped by the magnetic trap. We use rf and microwave
fields to transfer the 40 K and 87 Rb atoms into their energetically lowest hyperfine
states and then increase the local magnetic field to approximately 500 gauss. Here,we continue to evaporative cool the clouds of atoms inside the optical trap until
the gas reaches quantum degeneracy.
Since the apparatus is a marriage of two sub-apparatuses, one for 40 K and
one for 87 Rb, the overall complexity of the apparatus increases by at least afactor of two when compared to a single-species apparatus. The apparatus’ overall
reliability decreases with the addition of more lasers, electronics, and equipment todeal with the 40 K atoms, 87 Rb atoms, and molecules. Because of this, the intrinsic

9reliability of each component needs to be as high as possible for as long as possible
for the sanity of the graduate students and advisors. It is, therefore, unacceptableto build important components of the apparatus too hastily because any time thatwas saved will be eventually consumed (and then some) while debugging problems
later. As a simple example, assume I have an external cavity diode laser (ECDL)
whose frequency stabilizing servo stays locked for an average time of one hour. If
I rely on six similar ECDLs in the apparatus, the average time between failures
would be 10 minutes. This failure rate is unacceptable when collecting data at a
rate of a single point every 2 minutes!
2.2

Laser Systems for the Atoms
To accomplish the cooling and, later, detection of the ultracold gas, we

require light that is very nearly single frequency when compared to the natural
frequency width of the atoms’ electronic transition. For the case of K and Rb, thenatural widths are approximately 6 MHz. Diode lasers stabilized by an external
cavity are able to easily achieve linewidths down to about 100 kHz and they aresmaller and less expensive than other solid-state lasers. Fortunately, because of
the Compact Disc writer market, laser diodes (LD) with ample power at the Rb
transition wavelength of 780 nm have been readily available. However, these lasers
are becoming more obsolete with the current market shift towards higher density
discs and downloadable media.
Each atom species requires a laser system for cooling, optical pumping, anddetecting the atoms. An overly simple schematic is shown in Fig. 2.1. Eachsystem is composed of two major lasers, which we call repump and trap lasers fortheir roles in the MOT. The repump laser’s frequency is referenced to an atomic
source using saturated absorption spectroscopy. The trap laser’s frequency needs
to be changed during an experimental cycle, for example, from the frequency

10that is needed for loading the MOT to the frequency for imaging the atomic
cloud. It is because of these frequency changes that we use a fast photodiode anda phase/frequency detector to “offset” lock the trap laser to the repump laser.
More details related to this technique can be found in Scott Papp’s thesis [37].
The major advantage of this technique is the ease of changing the laser frequency
using a computer controlled voltage. We use a voltage controlled oscillator (VCO)
to provide the local frequency reference for the difference frequency between therepump and trap lasers. The VCO’s frequency is easily controlled with an external
voltage from the control computer.
We amplify the trap lasers using injection locked lasers (IjLLs)[38, 39, 40].
By coupling trap light into the rejection port of an optical isolator, we inject asmall amount (≈ 1 mW) of trap laser power into the IjLL. The injected beam into
the IjLL is aligned by measuring the output power of the IjLL after the output portof the optical isolator. The threshold current of the IjLL shifts to lower current
with proper alignment of the injecting beam. Using the IjLL reduces the operating
current of the trap lasers and gives plenty of laser power for detection of the atoms.
For both IjLLs, we use a 780 nm diode laser (Sharp model GH0781JA2C). It is
important that the non-injected, free-running wavelength of the IjLL is close to therequired operating wavelength. The IjLL’s free-running wavelength can be tuned
by adjusting the IjLL diode’s temperature. We use a diode that is approximately
at Rb’s D2 wavelength at room temperature and we cool the Rb IjLL only to 10
degrees Celsius and stabilize this temperature to about 0.001 degrees Celsius. To
reach the K wavelength at 767nm, the K IjLL is cooled to -45 degrees Celsius witha stability of 0.001 degrees Celsius. Because this temperature is well below thedew point, it is important that the K IjLL is properly sealed from the atmosphereto prevent moisture from depositing on the cold laser holder. We used Torr Sealr to seal the electrical leads and rubber gaskets to seal the lid and base of the1156"&':;!'<='/1>$?+'12!3/,1&$@$%33#$;,+2'!

!"#$%"&'!

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97

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Figure 2.1: General schematic for the laser system (same for Rb and K lasers). We
use saturated absorption on atomic references to detect and stabilize the repump
laser frequency. Using the frequency-stabilized repump light we “offset lock” thetrap laser frequency by electronically detecting the beatnote between the trap andrepump lasers and stabilizing the beatnote to an rf source. Light from both lasers
are amplified for the MOTs.

12
IjLL box. We also insulate the holder to reduce blackbody and convective heattransfer. Figure 2.2 shows a diagram of the K IjLL. A hole is cut into the bottom of
the aluminum box to allow direct contact between the Peltier cooler and chilledwater plate. Having this direct contact was critical to obtain sufficiently low
temperatures.
For both laser systems we combine light from the IjLL and the repump intoa Tapered Amplifier (TA). The TA provides 300 mW of optical power. Here weuse Eagleyard TAs controlled by Thorlabs current controllers (Model LDC 340).
Both repump and trap light are amplified by the TA and sent to polarization
maintaining (PM) fibers for spatial filtering and delivery to the MOTs at thevacuum chamber. After the fiber, there is 150mW of total power for the MOTs.
2.2.1

The MOTs

The MOT beams are formed after the polarization maintaining fibers by
combining the K and Rb light and enlarging the beams. The K and Rb lightare combined on a polarizing beam splitting (PBS) cube. After the cube, thepolarizations of K and Rb are perpendicular to each other but they need to beparallel for the MOT. So, we use a high-order waveplate (29.5 λ) to rotate theK light’s polarization to be parallel to the Rb light’s polarization. Subsequent
polarization manipulations for the MOT are carried out by large diameter 12 and1wave plates that affect both wavelengths almost equally. A telescope composed
4of three lenses is used to enlarge the MOT beams so they have a diameter of 3
centimeters.
2.3

Vacuum Chamber
We use a ultra-high vacuum (UHV) chamber to isolate the ultracold atoms

from the room temperature atmosphere. A simplified schematic of the vacuum13!""+,&'+!./0($1!"#()
6+)7'+89'
6"/,+-/,:!;<=+'9!/'$"+#>?/00()@&35('
2$3'&45(#$*("'+()!!"()3!"#$%&'()$*"&'(%+-#!=

Figure 2.2: Construction of the injection locked laser for amplifying K trap light.
On the left, an exploded view of only the collimation tube holder and the coolingstages. On the right, the interior view of the IjLL. A laser diode designed forroom-temperature operation at 780 nm was cooled to -45 degrees Celsius to shift
the wavelength down to 767 nm.

14chamber is shown in Fig. 2.3. This chamber contains our atomic sources of Rb
and K in a higher pressure region designed for the initial capture and cooling of
the atomic gases. This initial section of the vacuum chamber is called the “MOTcell”. A series of tubes connects the MOT cell to a lower pressure region, called
the “Science cell”. These transfer tubes limit the conductance of gas from the
MOT cell to the Science cell.
Although Rb and K are dominant in the MOT region, other contaminants
can pollute the rest of the ultra-high vacuum chamber. These other contaminants will limit our ultracold gas lifetimes and number of atoms after evaporation. Therefore, it was important to condition the stainless steel components
before assembling the vacuum chamber. We started by cleaning the components
using detergents, deionized water and alcohols to remove any machine oil or finger
prints. Once we had chemically cleaned the parts we treated the stainless steel
components to minimize Hydrogen outgassing. Typically this can be done by baking them either in vacuum or in air. We did an air bake at 400 degrees Celsius for
one day. Presumably, baking in air causes an oxide layer to form on the stainlesssteel and limits the migration of hydrogen off of the walls of the chamber [41, 42].
After assembly and initial pump-down, we baked the chamber at 250 degrees Celsius for a week. This helped to further purge the chamber of water andadsorbed gases. Since alkali atoms attack glass at high temperatures, we were
careful to keep the alkali sources from exceeding 250 degrees Celsius during thebake. In spite of this, we noticed a yellow/brown discoloration of the glass in theampoules of rubidium and potassium.
2.3.1

MOT Region

The MOT cell, shown in Fig. 2.4, is evacuated down to UHV pressures.
Atomic sources create a low-pressure vapor of Rb and K inside the cell. From

15!"#$%"$&;$44!"#$%"$ &'$(#)%>4#(%7$%2
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6/78'$(#)%?#..$+$%2#,4=/78#%(&*/0$>2)7#"&!)/+"$-

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Figure 2.3: Schematic of our vacuum chamber. From right to left: Atomic sources
supply atoms to the MOTs and keep the MOT cell at higher pressures. We limit
migration of contaminants from the MOT cell using the differential pumping tube.
The bump region has a borosilicate glass slide to block atomic line-of-sight to thescience region. A gate valve can be closed to separate the MOT region from thescience region of the vacuum chamber.

16these nearly room-temperature gases, ultracold clouds are captured and cooled toonly a few hundred microKelvin. Because the MOTs require optical access, thecell is made of borosilicate glass so it is transparent to the laser light for Rb andK. Electromagnetic coils are positioned near the MOT cell to provide the requiredmagnetic fields for the MOTs and a quadrupole magnetic trap. Surrounding thecell, an aluminium oven, heated by resistive heaters, keeps the borosilicate walls
at about 60◦ Celsius. This temperature helps keep the K atoms from sticking to
the walls of the cell.
The MOT cell is connected to the rest of the vacuum chamber through along narrow tube. Because of its size, the tube limits migration of atoms from
the MOT cell to the rest of the chamber. This tube has a conductance of about
1 liter of air per second. At the end furthest from the MOT, the tube opens into
the bump region. Here, a borosilicate glass obstruction is positioned perpendicular to the differential pumping tube. This obstruction was intended to preventline-of-sight atoms from readily entering the science region. The bump region is
pumped by a 20 liter per second ion pump (Gamma Vacuum TiTan 20S). The
tube connecting the bump region to the ion pump has a conductance of about
38 liters per second. This conductance creates an effective pumping speed in thebump region of about 13 liters per second and about 0.9 liters per second in the
MOT cell. This design ensures the pump always has a pressure that is at least
20 times lower than the MOT cell and helps prevent alkali buildup on the highvoltage cells inside the ion pump. Assuming the MOT cell is the only sourceof contamination for the entire chamber, the total pressure differential from the
MOT cell to the science cell should be about a factor of 180 if nitrogen were themajor contaminant. Magnetic trap lifetimes in both regions indicate the differential is about 80. Hydrogen probably contributes most to the discrepancy between
the calculated and measured conductances since it is desorbed from the stainless17

8&/.1+#1")#&&0@<

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Figure 2.4: A zoomed-in overhead view of the MOT region of the vacuum chamber.

18
steel walls of the vacuum chamber and it is not pumped efficiently by the ion
pumps.
2.3.1.1

Atom Sources

For our chamber, we use two different types of atomic sources. For Rb, weuse pure Rb contained inside of a glass ampoule. The Rb was purchased from
Alfa Aesar, who claims the purity of Rb is 99.75%. There are two ampoules, eachcontaining 1 gram ofl Rb attached to the MOT cell. One of the ampoules was
broken in-situ to expose the Rb metal to the interior of the MOT cell. Since there isno valve separating the ampoule from the MOT cell, the ampoule was broken after
the initial bake of the MOT cell. If future bake-outs of the MOT cell are required,it will be necessary to remove the opened ampoule of Rb otherwise Rb will be
evaporated from the ampoule and deposited throughout the vacuum chamber.
The ampoule was chosen because of its high purity and simplicity. There are noelectrical feed-throughs or extra materials to contaminate the vacuum. However,
we are unable to easily control the amount of Rb in the chamber. The flux of Rb
into the MOT cell is controlled by the conductance of the ampoule arm and Rb’s
saturated vapor pressure at room temperature.
There are also two ampoules containing enriched K attached to the MOT
cell. However, at best, only one milligram of K is contained in each ampoule.
During the bake of the MOT cell, the K appears to have reacted with the borosilicate glass, which caused a yellow/brown discoloration inside the ampoule. The
reaction appears to have depleted the metallic K so we chose not to open theK ampoules to the MOT cell. As a backup to the ampoules, we installed four
homemade enriched K dispensers. Each dispenser is made by combining 12 mg
of KCl salt, that has been enriched in 40 K, with shavings of calcium inside a
Nichrome “boat” [43]. The molar ratio of K to Ca is approximately 1:5. The KClElement
Cu
Cr
Na
Zn
Sr

ppm
9
<10
15
18
25

Element ppm
Fe
<50
Ca
<200
Rb
200
Mg
280
Ba
1100
Total
<1907

19

Table 2.1: Major impurities of the enriched KCl. Source: Trace Sciences International.

was purchased from Trace Sciences International. We purchased the KCl in 2003
and the enrichment was 5.2% in 40 K. Today, according to Trace Sciences’ website,
enrichments exceeding 13% in 40 K are available.
For improved MOT conditions and reduced background gases, it is important that impurities in the sources are minimized. The metallic calcium used in
the dispenser is 99.98% pure according to Alfa Aesar. The KCl contains impuritiesof other elements. Trace Sciences provided the impurity fractions and compositions, which are reproduced in Table 2.1. Rb is listed as an impurity at the 200
parts per million level. We empirically discovered this fact when we set up andcaptured Rb in our MOT before breaking the Rb ampoule!
2.3.2

Transfer Tube

Separating the MOT chamber and the science chamber is an in-line gate
valve (VAT Inc. Part number: 48124-CE01). The valve was included so changing
either side of the chamber would be easier. It has a conductance of 10 liters per
second. A bellows nipple is attached to the gate vale on the science cell side.
These bellows allow for easier alignment of the entire transfer tube and have aconductance of about 11 liters per second. Following the bellows, a 3 inch sectionof 0.5 inch OD tubing finishes the narrow section of the transfer tube. This small

20
section has a conductance of about 1.9 liters per second and is directly welded
into a larger diameter elbow.
2.3.3

Science Cell

The science cell is a borosilicate glass fluorometer cell (from Starna cells)with an interior height of 10 mm, width of 20 mm, and length of 45 mm. The cell is
attached to a glass-to-metal flange. This is attached to the elbow as shown in Fig.
2.3. This elbow is attached to a large diameter section containing the titanium
sublimation pump (TSP) and the science region’s ion pump. The TSP cartridge(Varian part number: 9160050) contains three filaments of titanium. After baking
the science cell, current was passed through a filament heating it until titanium
was evaporated and deposited onto the vacuum chamber’s walls. The large surface
area of titanium creates the pumping surface. Titanium absorbs hydrogen andreacts with nitrogen, oxygen, and other gases. These interactions create impressive
pumping speeds of a few liters per second for every square centimeter, meaning
our TSP’s speed could be as high as 1000 liters per second. Although impressive,these speeds are only possible for sufficiently reactive or absorbable gases. Noble
gases and methane are not pumped by the titanium and require the ion pump.
We apply a new titanium layer about once a year.
2.4

Magnetic Traps and Fields
In this section I will describe the magnetic traps and magnetic fields we use

to control the atoms. These fields are generated by passing current through coils
located near the vacuum chamber. The magnetic fields must have particular spatial geometries to trap the atoms and we use two typical geometries, a quadrupole
field and Ioffe-Pritchard field. The first type is used for the MOT and for transferring atoms to the science cell. The second type is used to trap and evaporatively

21
cool the atoms. While the atoms are in the second trap, we use microwave andradio-frequency (RF) oscillating magnetic fields for evaporative cooling. These
fields require a special set of coils designed to broadcast the microwave and RF
fields at the atoms.
2.4.1

Quadrupole and Transfer Trap

The quadrupole fields for the MOT and the transfer trap are generated by
the same pair of coils. The coils are wound from square hollow tubing have 20.5
turns each. They have an inside diameter of 3.2 inches and an outside diameterof approximately 4.9 inches and a 0.7 inch thickness. They are centered on thetransfer tube with a face-to-face separation of 4.25 inches. At the maximum
current of our power supply, 440 A, the coils can generate magnetic-field gradientsof about 180 G/cm. We move the coils using a linear positioner (Parker 404XR
series). This positioner has a total travel length of about 800 mm.
2.4.2

Ioffe-Pritchard Trap

The Ioffe-Pritchard trap [44] is similar to the trap for the 85 Rb-87 Rb experiment’s trap [37]. Tracy Keep from the JILA machine shop helped design the
trap. Figure 2.5 is a view of only the coils from Tracy’s computer model. It has
two elongated “racetrack” coils for the Ioffe coils that are electrically connected
in series with the pinch coils. The bias coils are electrically separate.
We control the fields produced by the coils in the Ioffe-Pritchard trap bycongrolling the current that passes through them. Since there are two electricallyseparate sets of coils, we have two servos for stabilizing the currents in each set.
This, in turn, stabilizes the magnetic field for the atoms.
All of the coils are made of square copper tubing. The tubing is wrapped ina polyimide coating to minimize electrical shorts. However, we discovered a weak

22

Figure 2.5: A schematic of the coils of the Ioffe-Pritchard magnetic trap. The
large circular pair of coils are the bias coils. The small circular pair of coils are
the pinch coils. Finally, the elongated, “racetrack” pair of coils are the Ioffe coils.
Electrically, the Ioffe and pinch coils are connected in series. The bias coils areconnected in series with each other but are electrically separate from the othercoils.

23

Figure 2.6: A computer generated model of the magnetic trap and its holder.

24$&'"()*+&',!-&*+!".(()'/

%

$
%

!"#

!"#

Figure 2.7: Contour plots of the magnitude of the magnetic field produced by a
Ioffe-Pritchard Trap. The data is obtained from a computer model of the magnetic
field using our coil geometry and dimensions. The horizontal direction of the plotis along the direction of the bias magnetic field. The vertical direction of the plotcorresponds to the radial direction of the trap. The color scale is the same for eachplot with purple corresponding to a low magnetic field and red corresponding toa high magnetic field. For each plot the current in the Ioffe coils and pinch coils
are fixed and have the same value. The difference between the plots comes fromincreasing the current through the bias coils from low current (top) to high current(bottom). By increasing the bias current, the magnetic trap’s depth increases andthe radial trap frequencies increase.

25

Figure 2.8: Circuit diagram of the rf coil that is impedance matched to 50 Ω at
DC-10MHz and near 80 MHz.

short in the Ioffe coils that was created in the coil winding process. See Appendix
A for more details. We circulate chilled water through the coils to dissipate theheat generated through resistive heating.
2.4.3

Oscillating Magnetic Fields

We use rf and microwave frequency magnetic fields to control the internal
states of the atoms. These fields are produced using two coils, one for rf and one
for microwave.
The rf coil is designed to operate below 100MHz. The coil is made from twoturns of 22 AWG wire with twisted-pair leads. The coil acts approximately like an
inductor below its self resonance of about 300 MHz. We optimize the frequency
dependent impedance of the coil to match 50 Ω at DC-10 MHz and at 80 MHz.
Figure 2.8 shows a circuit diagram of the coil.
The microwave coil is impedance matched at a slightly higher frequency than
the 87 Rb hyperfine splitting (6.8 GHz) for the frequencies used for evaporation.
It also is matched near 8 GHz for manipulation of the states of 87 Rb in a large
magnetic field. The microwave coil is positioned in close in proximity to the atoms
for improved coupling.

26
2.5

Optical Trap
In addition to trapping atoms with magnetic fields, it is also possible to trap

atoms in optical fields. Using optical fields enables trapping of states that aren’tmagnetically trappable. Travis Nicholson constructed our first optical trap andgeneral details of the trap can be found in his undergraduate honor’s thesis [45].
We used his design to take our initial molecule data using magnetic-field sweeps.
Later, we modified his design to create a trap with a waist of 50 µm. This
weaker trap was intended to lower the density of the atoms to reduce inelastic
losses. The design was further modified after discovering that the 1075 nm fiber
laser was causing significant loss of the Feshbach molecules. After this discovery,
we replaced the 1075 nm multi-mode laser with a 1064 nm single-mode laser.
2.6

Imaging System
To detect the atom and molecule clouds, we resonantly drive optical tran-

sitions of the atoms and molecules with laser light. The clouds cast shadows in
the laser beams. The shadows are imaged onto a CCD array of a digital camera.
We use a Princeton Instruments MicroMAX 1024B camera (Model: MicroMAX1024B). This camera has 16-bit readout resolution and 1024 x 1024 pixels witheach pixel having dimensions 13.0 x 13.0 µm. The quantum efficiency of the detector is about 75% at wavelengths of 780 nm.
We image the atom’s shadow onto the camera using achromat lenses. The
lens closest to the atoms has a focal length of 80 mm, a diameter of 25 mm andis placed about 80 mm from the atoms. The second lens has a focal length of
190 mm, a diameter of 50 mm and is placed about 270 mm from the first lens.
Finally, the camera is placed about 190 mm from the second lens. Figure 2.9
shows the imaging layout.

27!"#$%&"'()*'($
9#8:21:/;<$!(#

&

233

"3

667/68#(38+,-.##/0(1$+,45.##/0(1$Figure 2.9: Imaging system used to detect the atoms and molecules.

28
2.7

Computer Control
We use computers to control the experimental timing. The computer con-trols up to 64 digital output lines using a digital input/output(DIO) board. This
DIO board is made by Viewpoint Systems (model: 64-DIO64PCI). The DIO
board’s internal clock sets the timing for the entire experimental sequence. The
computer also has three digital to analong converters (D/AC) boards made by
National Instruments (model: NI-PCI6733). These D/AC boards are synchronized with the DIO board. Two of the PCI6733 boards generate 16 outputs ofcomputer controlled analog voltages. The other PCI6733 board generates thedetailed, time-dependent ramps for the optical trap.

Chapter 3
Methods

3.1

Introduction to Methods
In this chapter I will describe the techniques we use in preparing ultra-

cold mixtures of 40 K and 87 Rb. Although these techniques are similar to thoseused in single-species experiments, they become slightly more complex because ofinteractions between the species [46, 47].
The prepared mixture should have temperatures and numbers that are adequate for association of atoms into molecules. Efficient association of ultracold
atoms near a Feshbach resonance requires an ultracold atom gas near quantum
degeneracy [48]. Quantum degeneracy in both 40 K and 87 Rb gases requires thatthe temperature of the gases is near the Fermi temperature (TF ) for 40 K and the
BEC transition temperature (Tc ) for 87 Rb. These degeneracy temperatures depend on the harmonic trapping potential and the number of atoms, NK and NRb .
They are given in Eqns. 3.1 and 3.2.
¢1
~ω K ¡
6NK 3
kB

(3.1)

~ω Rb ¡ NRb ¢ 31
kB 1.202

(3.2)

TF =

Tc =

Here, ω is the geometric mean of the three angular trapping frequencies, kB
is the Boltzmann constant, and ~ is the Planck constant divided by 2π. Since we30want both gases to be near quantum degenerate, we need these temperatures to beapproximately equal, TF ≈ Tc . We typically prepare atom gasses with a number
ratio of NRb :NK about 3:1. The evaporation is stopped when the temperature of
the atom clouds reaches about T = 1.2 Tc and T = 0.6 TF .
3.2

The MOTAs mentioned previously, we use a two-species magneto-optical trap (MOT)

to capture and cool atoms from a room-temperature vapor. The MOT uses three
retroreflected laser beams that mutually intersect at the zero of a quadrupole
field with a 10 G/cm gradient along the vertical direction. The laser beams havefrequencies that are slightly lower than the 40 K and 87 Rb cycling transitions andare combined with light that is tuned to the repump transition for the atoms.
Since the exponential fill time in the MOTs is typically around three seconds, we
capture atoms in the MOTs for at least 15 seconds to ensure complete filling of
the MOTs. The densities within the MOTs are limited by radiation trapping in
the optically thick clouds. To increase the densities of the clouds for loading into
the quadrupole trap, we compress the MOT [49] by reducing the repump beams’
optical power and changing the trap lasers’ detuning. After compressing, we
further cool the 87 Rb gas using a molasses stage. The molasses stage is done in theabsence of the quadrupole field. Finally, the MOT laser beams are turned off and,in a weak magnetic field, we optically pump the atoms into the weak-field seeking“stretched state” for both 87 Rb and 40 K (|2, 2i and |9/2, 9/2i, respectively).
After optically pumping the atoms, the quadrupole magnetic field is turnedon again to capture the spin polarized atoms. First the clouds are loaded into aweak (30 G/cm vertically) trap for 200 ms. This trap only holds 87 Rb in the |2, 2i
state while possibly holding up to seven spin states of 40 K. After the 200 ms filter
time, the trap is then linearly ramped to 100 G/cm in 700 ms. Ramping to the31strong gradient compresses the atom clouds in preparation for the transfer to thescience cell.
Recapturing atoms into the MOT is simple and useful for diagnosing problems with the preparation steps. We do this by suddenly turning the MOT backon to its normal operating parameters. We can then sensitively detect the MOT
fluorescence and compare this to the fluorescence of a full MOT. For example,
Figure 3.1 shows MOT recapture after loading 87 Rb atoms into the quadrupole
trap versus the vertical trapping gradient. The atoms were held at each gradientfor 200 ms, which is long enough for untrapped states to be lost. The two data
sets were taken before (4) and after (•) optimizing the molasses cooling stages.
The sharp step-like transitions occur at about 15 G/cm and 30 G/cm represent
the gradient required to support different weak-field seeking states of 87 Rb against
gravity. The states have magnetic moments of µB and µB /2 respectively. For thisdata, the optical pumping was not optimized for the |2, 2i state as shown by thelarge increase in total number of recaptured atoms above 30 G/cm. We also use
MOT recapture to measure the lifetime of the atoms in the quadrupole trap inside
the MOT cell. Typically, this lifetime is about 2 seconds. As discussed in the next
section, the atom clouds can be transferred to other locations along the transfer
tube then back to the collection cell for MOT recapture. Using this technique, we
can measure the lifetime at different locations in the vacuum chamber.
3.3

The Transfer
We transfer the atoms from the MOT cell to the science cell using a linear

positioner (also known as “the track”) [50]. The relatively short magnetic-trap
lifetime of 2 seconds in the MOT cell limits the number of atoms we can efficiently
transfer from the MOT to the science cell. We begin the transfer from the MOT
cell 500 ms after the load of the weak quadrupole trap. This occurs while the trap

Number (arb)

32

0.3

0.2

0.1

0.0
0

20

40

60

80

100

Gradient (G/cm)
Figure 3.1: Number of 87 Rb atoms recaptured into the MOT after loading aquadrupole magnetic trap. The atoms are held at the indicated gradients for 200
ms. The two data sets are before (4) and after (•) optimizing the molasses stage
before the quadrupole load. Optical pumping is not optimized.

33
is ramping to 100 G/cm.
The positioner leaves the MOT’s position with an acceleration of 1 m/s2 for
340 ms until it reaches a speed of 34 cm/s. After traveling 15.25 cm, the cart slows
to 20 cm/s in 140 ms for the bump region. This speed was chosen by minimizing
losses past the bump. Here, the cloud’s position is shifted vertically by a static
bias magnetic field provided by a pair of coils. These coils are turned on wellbefore the positioner reaches the bump region. To avoid the pyrex obstruction,
the position of the cloud needs to shift downward by about 1 cm. A bias field of
about 100 G is needed for this shift. Figure 3.2 shows the result of transferring
past the obstruction as a function of current in the bias coils. The decrease inefficiency for large currents (large bias fields) is caused by too large of a verticalshift pushing the cloud into the wall of the vacuum chamber. After optimizing,
the efficiency of transferring atoms past the bump is about 97%. This efficiency isdetermined by comparing the number of atoms transferred past the bump to thenumber transferred to just before the bump. The overall efficiency to the sciencecell is about 10% of the full MOT. This is quite low relative to other track systems
and is caused, predominantly, by the transition into the first section of the transfer
tube (40% efficient) and the short lifetime of the trapped atoms in the MOT cell
(60% efficient). Additional contributions to the loss are the optical pumping (70%efficient) and losses in the bellows section of the transfer tube (70% efficient).
Except for the optical pumping, it should be possible to improve these losses.
For example, lengthening the initial section of transfer tube so that its opening
is closer to the MOT position would minimize the time that the atoms spend in
the high-pressure MOT cell. It could also make it easier to apply magnetic shimfields to efficiently “funnel” the atoms into the tube. The losses at the bellows
are probably caused by magnetic fields that can also be shimmed.

Number (arb)

34

7
6
5
4
3
2
1
0

4

6

8

Current (A)Figure 3.2: Number transferred past the “bump” as a function of current in thebump coils. The optimized transfer efficiency across only the bump is about 97%.
This efficiency is calculated by comparing the number transferred past the bumpto the number transferred to just before the bump. It is important that the speedof the track is less than 20 cm/sec for the best transfer efficiency across the bump.
The pyrex obstruction totally blocks the cloud’s path for currents below 4.5 amps
and the vacuum chamber wall totally blocks the cloud’s path for currents above
8 amps.

35
3.4

IP Evaporation
After transferring the mixture to the science cell, we load the a weak Ioffe-

Pritchard trap by turning on the pinch-Ioffe bar coils to 220 amps and the bias
coils to 20 amps. At these currents, the measured 87 Rb trapping frequencies in
the |2, 2i state are ωz = 2π · 20 Hz and ωr = 2π · 23 Hz. We then compress thetrap by increasing the current in the bias coils to 88 amps. At these currents,
the magnetic fields from the pinch coils and bias coils nearly cancel. The bias
field here is about 1.7 gauss and the trap frequencies are ωz = 2π · 20 Hz andωr = 2π · 156 Hz. At the start of evaporation, we typically have about 3 · 108 87 Rb
atoms and 2 · 106 40 K atoms at about 500 µK.
We evaporatively cool the 87 Rb gas by driving the |2, 2i → |1, 1i transition.
By sweeping the microwave frequency from high to low, the most energetic atoms
are removed leaving the remaining atoms at lower temperatures. The evaporation
efficiency depends on the microwave power, time-dependence of the frequency
sweep, and the collision rate in the gas. Others have discussed the intricacies of
evaporative cooling and details can be found in Ref. [51] and references therein.
The 40 K gas is sympathetically cooled by collisions with 87 Rb atoms. Figure 3.3
shows the effect of the forced/sympathetic evaporation on the atom clouds. The
87

Rb and 40 K clouds are cooled at the expense of 87 Rb atom number.
The first data we obtained on sympathetic cooling showed significant num-

ber losses in the 40 K cloud near the end of evaporation. If we didn’t load 40 K,
we discovered that there was an impurity of 87 Rb in the |2, 1i state at the endof evaporation. However, if we loaded 40 K, the |2, 1i state was depleted alongwith the 40 K. The microwaves do not depopulate this state and inelastic losses
in 87 Rb are low. Initially, we thought these atoms were being created throughout the evaporation by optical or rf fields but the number of atoms in the |2, 1i

Atom Number

36

107

106
1

cloud temp ( K)10

Figure 3.3: Effect of forced evaporative and sympathetic cooling on the temperature and number of both atomic species. The clouds’ temperature decreases atthe expense of 87 Rb atom number (solid circles). The 40 K atom number (solid
triangles) is only slightly affected.

37state remained constant throughout the experiment. Presumably, the atoms do
not adiabatically follow the local magnetic field during the transfer of atoms fromthe quadrupole trap into the Ioffe-Pritchard trap. To remove the |2, 1i atoms,
we continuously drive the |2, 1i → |1, 0i transition at the magnetic field of thebottom of the trap throughout the microwave evaporation.
3.5

Optical Trap Evaporation
After evaporation in the Ioffe-Pritchard magnetic trap we have 5 × 106 Rb

and 1 × 106 K atoms at 3 µK. These atoms are transfered into a far-off-resonanceoptical dipole trap with a 50 µm waist derived from a single-frequency fiber laser
operating at a wavelength of 1064 nm. The dipole trap is a convenient tool fortrapping atoms having Zeeman states that aren’t magnetically trappable. Thesestates are immune to two-body inelastic losses and have the Feshbach resonanceused to create the Feshbach molecules discussed in Ch. 4. Once the optical
dipole trap is loaded, we use rf adiabatic rapid passage to transfer the atoms
into the magnetically untrappable |1, 1i and |9/2, −7/2i states of 87 Rb and 40 K,
respectively. The gases are then simultaneously evaporated at a field of B=540 G
by lowering the optical power in the trapping beam until 3 × 105 Rb at T /Tc ∼ 1
and 1×105 K at T /TF ∼ 0.6 remain. Tc is the critical temperature for the onset of
Bose-Einstein condensation (BEC) and TF is the Fermi temperature. At the endof the optical trap evaporation we adiabatically increase the optical trap power
until the measured radial trapping frequencies are 211 ± 4 Hz and 136 ± 2 Hz for
K and Rb, respectively. The ratio of these frequencies is 1.55 ± 0.04. Assuming aq≈ 1.47. The discrepancy
harmonic potential, the ratio should be given by mmRb
K
is resolved because the potential is not harmonic. The disparity in atomic masscauses the two clouds to be pulled by gravity to different equilibrium positions in
the trapping beam. The 87 Rb cloud is in a lower and more anharmonic region of38

Frequency (MHz)

2650
|2,2>

2600

|2,1>
|2,0>

2550

|2,-1>|2,-2>

2500

f=6834.682MHz
@ B=0

-4250

|1,-1>
|1,0>
|1,1>

-4300
-4350

0

5

10

15

20

25

30

Magnetic Field (gauss)

Figure 3.4: Breit-Rabi diagram for 87 Rb at low magnetic field. We evaporatively
cool the 87 Rb gas by driving the |2, 2i → |1, 1i transition. This removes 87 Rb
atoms from the magnetic trap. We also drive the |2, 1i → |1, 0i transition toremove impurity states of 87 Rb that inelastically collide with 40 K.

39the trapping beam. Here the 87 Rb trap frequency should be lower than the one
calculated using the 40 K trap frequency and assuming a harmonic potential.
3.6

Absorption Imaging
We extract the number and temperature of the atoms from time-of-flight,

resonant absorption images. The gas is suddenly released from the trap andallowed to freely expand before imaging. Using a resonant light pulse of theappropriate frequency, we selectively image the shadow cast by K or Rb atoms ina particular spin state. The number and temperature are extracted by analyzing
the spatial variation of the optical depth of the cloud. We use similar imaging
techniques as other groups, see, for example, Jacob Roberts’s thesis [52](Sec.
4.11).

Chapter 4
Creation of Feshbach Molecules

4.1

Feshbach Resonance
Magnetic-field-tunable Feshbach resonances, also known as Fano-Feshbachresonances, are useful tools for controlling the interactions between ultracold
atoms. Their utility has been demonstrated in a variety of situations. For example, Feshbach resonances are used for the efficient evaporation of 85 Rb [53, 52]
and 133 Cs [54], which has led to BEC in those species. In 85 Rb, the interacting BEC has been studied under the influence of changing the scattering length[53, 55, 56, 57]. Feshbach resonances have also been used to efficiently evaporate Fermi gases and have allowed for detailed studies of the BCS-BEC crossover[34]. Finally, Feshbach resonances have been used to convert ultracold atoms intomolecules [58, 59, 48]
Feshbach resonances in ultracold alkali collisions has been reviewed elsewhere [19]. Briefly, the resonance results when a ro-vibrational molecule state ina closed channel is coupled to the open, scattering channel. In our case, the relative energies of the closed-channel bound state and the open-channel threshold
can be tuned with a magnetic field since the magnetic moments between the twochannels are different. This is shown schematically in Fig. 4.1.
Weakly bound molecules can be efficiently created from atoms if the magnetic field is swept slowly across the Feshbach resonance in a direction that takes

41,#$-".

/0*$'1*$(2%!%$345!%0/&'(2%!%$23645!"#$%&'()&$*+
Figure 4.1: Magnetic field can be used to tune a closed molecular state and theopen atomic states into degeneracy. The different magnetic moments of the twostates allow the tunability. If these states are coupled, a Feshbach resonance will
occur at a magnetic field in the vicinity of the crossing between the two states.
The grey arrow indicates how the molecule state can be populated with a slow
magnetic-field sweep.

42
the closed channel state from higher energy to lower energy relative to the open
channel [60, 18, 59, 48]. In Fig.4.1, this direction is shown by lowering the magnetic field through the crossing. Hodby et al. [48] has described this techniquein detail. Briefly, the crossing is analogous to an avoided crossing in a two-levelsystem. Like a two-level crossing, the rate of the magnetic-field sweep near the
Feshbach resonance controls the population that is converted into the molecular state. However, it is not exactly like a two-level crossing. In particular, theclosed-channel molecule state is coupled to the continuum of states connectedto the open-channel. When the closed-channel has a higher energy relative tothe open-channel threshold (the right side of Fig 4.1), this coupling causes theclosed-channel to have a finite lifetime. A further complication to the two-levelanalogy is that the molecule formation process depends on pairs of atoms thatare near to one another in phase-space. Therefore, the atom gases may not be
totally converted into molecules during a sweep if the atoms are not close enoughin phase-space.
In addition to magnetic-field sweeps, the closed channel can be populatedusing oscillating magnetic fields. This technique has been utilized in other experiments when crossing the resonance is not favorable [61, 46, 62]. Furthermore,this technique has been extended to the radio-frequency (rf) domain and used tocouple atoms of another spin state into Feshbach molecules [63, 3].
4.2

Our Early Attempts
When we began our search for the known resonance positions in 2006, there

was some uncertainty in the assignment of the ro-vibrational states that were
coupled to the open channel. JILA first reported 40 K and 87 Rb interspecies resonance locations in the energetically lowest states of 40 K and 87 Rb [64]. These
resonances were assigned particular partial wave character using a comparison

43with theory. However, later the LENS group repeated and expanded upon thismeasurement and determined that the assignment was incorrect[65]. Their assignment was based on both theory and the qualitative observation that the width ofa suspected p-wave resonance was smaller at lower temperatures.
We repeated these measurements not only to ensure our magnetic field wascalibrated but also to determine if the suspected p-wave resonance was in fact
p-wave. The measurements were performed starting with a cold mixture in an
optical trap with 87 Rb atoms in the |1, 1i state and 40 K atoms in the |9/2, −9/2istate. We ramped the magnetic field to a location near the resonance position andheld the atoms at this field for one second. During the hold time, inelastic losses
could deplete the clouds. Afterwards, the field was turned off and the clouds wereimaged to determine the loss. Figures 4.2 and 4.3 shows two of the loss curves.
The s-wave resonance is located near 546 gauss as shown in Fig. 4.2. The losses
are maximized at the resonance locations. The double peaked structure of Fig. 4.3
shows the p-wave resonance. Here we measured the positions to be 515.18 ± 0.06
gauss and 515.32 ± 0.06 gauss for the two peaks. The uncertainty comes from themagnetic-field calibration. The double-peaked structure is understood [66] andresults from the magnetic dipole-dipole interaction of the atoms. These results
confirmed that this was a p-wave resonance, as reported by the LENS group.
As mentioned previously, the magnetic field can be swept in the vicinity of
the Feshbach resonance. We ramped the magnetic field across the resonance asshown in Fig. 4.4. Figure 4.5 shows the atomic response to these ramps. The
data is consistent with molecule formation when the magnetic field is ramped
appropriately.
We also used the ramp experiments to find the resonance locations. In
principle, this should be more accurate than using the inelastic loss measurements
since the inelastic loss rate may not peak exactly on resonance. We prepared

0.6

5

K number (10 )

44

0.4

0.2

0.0
500

520

540

560

580

Magnetic Field (G)Figure 4.2: Loss of K atoms in the presence of Rb resulting from an interspeciess-wave Feshbach resonance. The atoms are in the |9/2, −9, 2iK and |1, 1iRb states
and are held at the indicated field for one second before abruptly turning off themagnetic field and imaging them. For this measurement the number of 40 K atoms
was about 105 and the number of 87 Rb atoms was about 5 · 105 . Both clouds were
initially in thermal equilibrium at a temperature of about 500 nK.

45

1.0

5

K Number (10 )

1.2

0.8
0.6
0.4
0.2
0.0
514.8

515.0

515.2

515.4

515.6

Magnetic Field (G)Figure 4.3: Loss of K atoms, in the presence of Rb, resulting from an interspecies
p-wave Feshbach resonance. The atoms are in the |9/2, −9, 2iK and |1, 1iRb states
and are held at the indicated field for one second before abruptly turning off themagnetic field and imaging them. For this measurement the number of 40 K atoms
was about 105 and the number of 87 Rb atoms was about 5 · 105 . Both clouds were
initially in thermal equilibrium at a temperature of about 500 nK.

46$%&'#(!)*+!#,-+#./0%)/*
1#.2'%')#!"#$%&'#(!)*+!#,-+#./0%)/*
1#.2'%')#!"#

Figure 4.4: Solid lines show the time dependent ramps near the Feshbach resonance. The top figure shows a ramp that should not create molecules. The lower
figure shows a ramp direction that should create molecules.

47

pk OD (OD)

0.30
0.25
0.20
0.15
0.10
0.05
0.00
0.0

Ramp wrong way
Ramp to make molecules

0.5

1.0

1.5

RampTime (ms)Figure 4.5: The peak optical depth of K after ramping across the Feshbach resonance. The time to complete the ramps of Fig. 4.4 was adjusted. The atomic
loss is consistent with molecule formation after ramping slowly from high to low
magnetic field (solid circles). The time spent near the resonance is not a significant factor for atomic loss since little loss is observed after ramping slowly from
low to high magnetic field (solid triangles). The dashed line indicates the initial
optical depth of the K cloud without ramping across the resonance. The opticaldepth was measured after 12ms of free expansion.

48
40

K and 87 Rb in the |9/2, −9/2i and |1, 1i states, respectively, and ramped themagnetic field in the vicinity of the Feshbach resonance as shown in Figure 4.6.
Here, the ramps are slow enough to ensure molecule creation can proceed efficiently
but not so slow that inelastic losses dominate [48]. By measuring where molecule
formation occurs, we can measure the resonance location. We varied the holdmagnetic field while observing the atom number. The results are shown in Figure
4.7. With this measurement, we measured the s-wave resonance location to be
546.47 ± 0.12 gauss. We fit the data using an error function. The uncertainty
given here corresponds to the 10%-90% width of the transition. This data wastaken with the atoms held in the optical trap where the atomic densities werehigh which could lead to losses near the resonance. Such losses could cause themeasured location of the Feshbach resonance to be shifted systematically to lower
magnetic fields. Since this data was taken using only ramps that approach the
Feshbach resonance from below, the losses near the resonance could be caused bythree-body recombination or two-body photoassociation from the trapping laser
instead of two-body Feshbach molecule formation.
At this point, we were able to create molecules but could only detect loss of
atoms and could not detect the molecules directly. Being able to directly detect
the molecules is important for further studies of molecular properties.
4.3

Molecule Lifetime
After creating the molecules, our goal was to find their lifetime and what

limited it. We had intentionally selected the lowest-energy Zeeman states for eachof the atoms. This choice eliminates inelastic two-body, or dipolar, losses since
they are energetically forbidden. However, three-body losses can occur througha process called vibrational quenching. This occurs when vibrationally excitedmolecules release their vibrational energy through collisions with other atoms

49$%&'#(!)*+!#,-+#./0%)/*1#.2'%')#
3%1!%0,#*/2,-*
4!#,-5%"6*"%7#.*"2,#)8,#.

!"#Figure 4.6: Solid line shows a typical time-dependent magnetic-field ramp near
the Feshbach resonance. The hold magnetic field was changed for each data pointof Fig 4.7. The ramps up to and away from the hold field were adiabatic for
creating molecules.

4

K Number (10 )

50

12
10
8
6
4
2
0
545

Initial K number

546

547

Magnetic Field (G)548

Figure 4.7: Finding the resonance location using time-dependent magnetic-fieldramps. The transition to more loss occurs when the hold field of Fig. 4.6 is above
the resonance position. This loss is due to the creation of molecules which, inthis experiment, were quickly removed by the optical trap light. The magnetic
field was held for 300 µs and the ramps toward and away from the resonance had
inverse ramp rates of 0.139 ms/gauss. The solid line is a fit, using an offset error
function, to the data. B0 is taken as the center of the fitted error function. The
10% to 90% width of the error function is taken as the uncertainty in B0 . Using
these criteria, we obtain B0 = 546.47 ± 0.12 gauss. The initial 40 K and 87 Rb atom
numbers were about 1.1 · 105 and 2.9 · 105 and T = 240 nK = 0.8Tc = 0.6TF .
We detected the atoms at low magnetic field after turning off not only the opticaltrap but also the magnetic fields.

51
and molecules. Since collisions are not suppressed between Bosons, we suspectedthat inelastic collisions with the bosonic 87 Rb would occur at a relatively fast
rate. We also suspected that, in analogy to other Fermi systems [67, 68], themolecule lifetime would be relatively long when it was limited by collisions withthe fermionic 40 K.
Using a technique developed by others [68, 69], we attempted to measure
the molecular lifetime by creating molecules, waiting a specific time for molecularloss to occur, and reconverting the molecules back to atoms. We accomplishedthis by ramping across the resonance as shown in Fig. 4.8. The first, verticalcrossing of the FR is diabatic and does not contribute significantly to atom loss.
The ramp crossing the FR from above creates molecules and the second ramp
breaks the molecules into atoms for low field detection. The final crossing of the
FR is diabatic. The hold field and duration of the hold define the interesting
parameter space.
By ramping across the Feshbach resonance to create molecules then ramping
back to convert the molecules back into atoms, we were able to measure themolecule lifetimes. Results of a lifetime measurement are shown in Fig. 4.9. The
measured lifetime is 220 µs at a hold magnetic field of 546.2 ± 0.2 gauss.
These measurements suffered from the fact that we were not directly detecting the molecules, but rather, we were reconverting the molecules back into atoms.
The lifetime measurements were strongly affected by even modest fractional atom
number noise. Also, the atoms were detected using absorption imaging at low
magnetic fields. This means we were turning off the ≈ 547 gauss magnetic field
while crossing the Feshbach resonance a final time. These limitations led us toconsider new ways to detect the molecules in the absence of a fluctuation background.
As discussed in the next section, we found that we were able to directly52$%&'#(!)*+!#,-+#./0%)/*
1#.2'%')#!"#
Figure 4.8: Magnetic-field ramp near the Feshbach resonance to create molecules,
probe their lifetime, and reconvert the molecules to atoms for low magnetic-fieldimaging. The vertical jumps represent diabatic crossings of the resonance. The
ramps are adiabatic for creating and dissociating molecules.

5

K Number (10 )

53

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0

0.0

0.5

1.0

1.5

2.0

Hold Time (ms)Figure 4.9: A measurement of the lifetime of molecules. We used magneticfield ramps similar to Fig. 4.8 to reconvert the molecules into atoms. The holdmagnetic field for this measurement was 546.2 ± 0.2 gauss. The solid line is theresults of an exponential fit to the data. The fitted decay time is 220 ± 7 µs.

54detect the molecules at high magnetic field using absorption imaging. Because
the atoms and molecules can both absorb the same frequency imaging light, it
was convenient to start with the atoms in a different Zeeman state. We thencould use spin-selective absorption imaging to see the molecules directly.
4.4

Creating Molecules: New Techniques and Tools
To directly detect the molecules, we adopted a procedure similar to thatused by the Hamburg group [63] and created the Feshbach molecules using radiofrequency (rf) oscillating magnetic fields. In this scheme, the 40 K atoms are prepared in a higher energy spin-state and driven to the Feshbach resonance statewith an rf field. In the work by the Hamburg group the atoms were confined ina 3D optical lattice potential with precisely one atom pair per site. Confinement
in an optical lattice has two advantages. First, the tight confinement of a pair ofatoms in each lattice site meant that the initial motional state of each atom pair
was the ground state of the trapping potential at that site. Magnetoassociation
can then proceed by driving a transition between two well defined quantum states.
Second, confinement of the atoms and molecules in the optical lattice protected
the Feshbach molecules from collisions. This protection is important because theextremely weakly bound molecules can suffer heating and loss through inelastic
collisions that cause vibrational quenching.
In our case, we created the fermionic heteronuclear 40 K87 Rb Feshbach molecules
in a far-detuned optical dipole trap. In this relatively weak trap, the atoms startin a near-continuum of motional states, and furthermore, atoms and molecules
are free to collide. Without using the confinement in an optical lattice, we foundsimilar overall efficiency in the radio frequency (rf) association of ultracold heteronuclear molecules and created as many as 25,000 molecules at a temperatureof about 300 nK. We studied the association process and measured the magnetic-

55
field dependent binding energy of the Feshbach molecules. Compared to previous
results [63], we extended our measurements to magnetic fields much farther fromthe Feshbach resonance and calculated basic properties of the molecules such as
their size and quantum state composition.
We extract the number and temperature of the atoms and molecules using
time-of-flight, resonant absorption images. The gases are suddenly released fromthe optical trap and allowed to expand in the homogeneous magnetic field before
imaging. Using a resonant light pulse of the appropriate frequency, we selectively
image K or Rb atoms in a particular spin state at high magnetic field.
To create molecules, we utilize a Feshbach resonance between Rb |1, 1i andK |9/2, −9/2i atoms at B0 =546.76 G [64]. Similar to previous work using thisresonance [63], we start with atoms at a magnetic field near the Feshbach resonance and apply a transverse rf magnetic field to convert atom pairs to molecules.
However, unlike the work of ref. [63], in our experiment the atom pairs are notconfined in an optical lattice. Therefore, the atoms are free to collide.
In Fig. 4.10 we show a typical rf association spectrum for a magnetic field
tuned 0.7 G below the Feshbach resonance. The Rb and K atoms are initiallyprepared in the |1, 1i and |9/2, −7/2i states, respectively, and we apply an rf
pulse with a frequency tuned near the atomic K Zeeman transition, |9/2, −7/2i→|9/2, −9/2i. For these data we applied a gaussian rf pulse with a 600 µs 1/e2
full-width and an rf strength that can drive a 12 µs π-pulse on the atomic K
Zeeman transition. The effect of the rf pulse is detected by counting the numberof K atoms transferred to the previously unoccupied |9/2, −9/2i state. For thispurpose, we implemented spin-state selective absorption imaging of the cloud inthe optical trap at B = 546 G. We observe two peaks in the rf spectrum: one at thefrequency of the atomic K Zeeman transition and one shifted to higher frequency.
The higher frequency peak corresponds to a stimulated transition from a pair of56
atoms into the Feshbach molecule state. The shift of this spectral feature fromthe atomic transition reveals the binding energy of the molecule. In comparing
the widths of the two features it is important to note that the atom transition is
power broadened by the rf pulse, which has a duration 50 times longer than themeasured time for an atomic π-pulse. The smaller width of the molecule feature
indicates that the coupling strength is much weaker for molecule association thanfor the atomic Zeeman transition. The full width at half maximum of the moleculefeature is 7 kHz. This is consistent with the thermal distribution of the relative
energies for atom pairs.
We are able to easily observe the molecular rf association feature because the
Feshbach molecules can absorb the imaging light that is resonant with the K atom
cycling transition at high magnetic field. Presumably, the first photon dissociates
the weakly bound molecule and subsequent photons scatter off the resulting K
atom. Compared to the case of homonuclear molecules, this direct imaging ispossible for a larger range of Feshbach molecule binding energies. This is because
the energy separation between the excited and ground electronic potentials varies
relatively slowly with internuclear distance, R, since the heteronuclear excited andground electronic potentials both vary as 1/R6 . In contrast, for the homonuclear
case, the excited electronic potential varies as 1/R3 and the ground potential
varies as 1/R6 .
4.5

Driving Bad Bound-Bound Transitions
After unambiguously creating and detecting KRb Feshbach molecules, wemeasured their lifetimes in the optical trap and discovered it was comparable tothe 200 µs we had measured previously using ramps. We began to suspect thatthe optical trap could be causing the short lifetime. To test this, we measured
the molecule lifetime without the presence of the optical trap light. After creating

57

40

3

Number (10 )

rf
30
E/h

20

E/h
10

0
-200

-100

0

100

200

(kHz)

Figure 4.10: Rf association spectrum of heteronuclear Feshbach molecules at B =546.04 G. The molecules are created starting from an ultracold gas of 1.7 × 105
87
Rb atoms at T /Tc = 1 in thermal equilibrium with 8 × 104 40 K atoms. The plotshows the detected atom/molecule number as a function of the detuning ∆ν ofthe rf from the atomic resonance frequency. We observe a large atomic peak atzero detuning and a molecular peak at E/h = 161 kHz due to the association of
atom pairs into the Feshbach molecule state. In this measurement the rf pulse
duration was 600 µs, which is 50 times longer than an atomic π-pulse. The grey
band indicates the expected envelope full width at half maximum for the power
broadened atomic transition. The inset shows a simplified energy diagram formagnetic fields below the Feshbach resonance. The upper solid line corresponds to
Rb |1, 1i and K |9/2, −7/2i, while the lower solid line corresponds to Rb |1, 1i andK |9/2, −9/2i. The molecule state is represented by the dashed line. Depending on
the detuning, the rf is capable of driving an atomic hyperfine changing transitionor magnetoassociation. This data was taken while the atoms and molecules were
held in an optical trap derived from a 1064 nm single-frequency fiber laser.

58
the molecules, the optical trap was turned off. We measured the molecule lifetime
under these conditions and discovered it was longer than a few milliseconds. The
longest lifetime that could be measured was limited by the expansion of the cloud.
Since the expansion lowered the density, it is possible that density dependent
collisional loss rates were also decreasing during the expansion. To check the effectof the optical trap light for a lower density cloud, we turned off the optical trap fora millisecond then turned the optical trap back on for the lifetime measurement.
Again we found the lifetime was about 200 µs in the presence of the opticaltrapping light.
The optical trap light came from a multi-longitudinal mode, 1075nm fiber
laser (model YLR-10-1075-LP from IPG photonics). We believe the laser was
exciting the Feshbach molecules to the lowest vibrational states of the 3(1) excitedstate potential. This is shown in Fig. 4.11. As discussed in Chapter 6, we would
later deliberately take advantage of this Feshbach enhancement for bound-boundtransitions. To eliminate the limit to the molecule lifetime, we switched to using
a single-longitudinal mode fiber laser operating at 1064 nm for the optical trap.
This new laser did not excite molecules in the Feshbach state. However, since thelong-term goal of this experiment is to create more deeply bound polar molecules,
we note that a similar problem may exist for other, more deeply bound vibrational
states of the ground-state potential.
4.6

Molecule Association Process
Because the atoms are free to move and collide in the trap, it is interesting

to look at the time dependence of the rf association. The number of molecules
observed as a function of the rf pulse duration is shown in Fig. 4.12. The gaussian
1/e2 width of the pulse is plotted on the horizontal axis. For these data, the peak
rf strength is fixed and only the duration of the rf pulse is changed. The number

59

15000
5(1)

-1

E/(hc) (cm )

4(1)

10000

3(1)

2(1)

5000

= 1075 nm-

1(0 ,1)

0
2

4

6

Å

8

10

R( )

Figure 4.11: KRb molecular potentials. Svetlana Kotochigova has calculated thatthe 1075nm optical trap laser predominately drives bound-bound transitions fromthe Feshbach molecule state to vibrational states of either the 3(1) or 2(1) molecular potentials. The molecule potential energy data for this figure was obtained
from Ref. [1].

60of molecules increases rapidly for the first few hundred microseconds and thenslowly decays. We do not observe Rabi oscillations. This is not surprising since
we are driving transitions from a near continuum of free atom states to the boundmolecule state. The observed decay results from atom-molecule collisions [36]. To
estimate the effect of this decay on the maximum number of molecules created
we fit the data to the product of a linear growth and an exponential decay (curvein Fig. 4.12). We find that the decay causes a 30% reduction in the maximum
number of molecules created at the peak of the curve in Fig. 4.12. Since themaximum molecule creation in Fig. 4.12 corresponds to approximately 10% of
the total number of K atoms, this raises the question of what limits the maximum
conversion fraction?
Previous studies of molecule creation by slow magnetic-field sweeps acrossa Feshbach resonance have shown that the maximum conversion fraction depends
on the phase space density of the atoms [48]. Not surprisingly, for a two-species
atom gas mixture, the molecule conversion efficiency also depends on the spatial
overlap of the two atom clouds [46]. We have measured the molecule conversion
fraction as a function of temperature for Feshbach molecules produced using rf
association at B=546.17 G. For this measurement, the gas mixture was initiallycooled to a temperature below the critical temperature, Tc , for the onset of BoseEinstein condensation of the Rb gas. The temperature is then varied simply
by waiting a variable amount of time in a sufficiently deep trap and letting thegas slowly heat up before creating Feshbach molecules. With this technique, thenumber of Rb and K atoms is kept nearly constant. To create Feshbach molecules
we use rf adiabatic rapid passage and sweep the rf frequency from 80.132 MHz to
80.142 MHz in 2 ms. The results are shown in Fig. 4.13.
We found that the maximum conversion fraction occurs at T /Tc ≈ 1.
Here, 25% of the K atoms are converted to molecules; this corresponds to 25,000

3Number (10 )

61

12
10
8
6
4
2
0
0

5

10

15

20

rf Time (ms)Figure 4.12: Time dependence of the rf association process at B = 546.17 G, where
the measured molecule binding energy is E/h = 96 kHz. The plot shows themeasured number of molecules created as a function of the 1/e2 gaussian rf pulse
length. The solid line is a simple fit used to estimate the impact of the observed
decay on the maximum number of molecules created.

62
molecules. It is interesting to compare the measured conversion fraction for rf
association to the prediction of a phenomenological model introduced in ref. [48].
This model, using experimentally determined parameters, has been successful in
predicting the molecule conversion efficiency for Feshbach molecules created byslow magnetic-field sweeps and slow magnetic-field oscillations [48, 46], but has
not previously been compared to data for rf association. Following the procedure
outlined in [48], Michele Olson calculated the predicted maximum conversion efficiency using realistic trap potentials and distributions of our atomic clouds. The
curves in Fig. 4.13 show the results of the calculation, multiplied by 0.70 to approximately include the effect of the molecule loss seen in Fig. 4.12. The dashedcurves represent the uncertainty in the conversion fraction due to the uncertainty
in the trap frequencies and atom numbers. The agreement between the model for
association by slow magnetic-field sweeps and the data for rf association is quite
good. The predicted dependence on temperature agrees with our experimental results for rf association. The peak in the molecule conversion fraction as a functionof T /Tc can be understood as a balance between two factors: phase-space density
and the spatial overlap of the two clouds. As T /Tc decreases, the phase-spacedensity increases but, below T /Tc = 1, the onset of Rb BEC reduces the spatial
overlap of the Rb cloud with the K cloud. The effect of gravitational sag also
reduces the spatial overlap of the two clouds and at T /Tc = 1 the model predicts
a 30% reduction in molecule conversion efficiency when compared to the case withno gravitational sag.
4.7

Ultracold, Trapped Molecules
A feature of using rf association to create heteronuclear Feshbach molecules

is that we can selectively image the Feshbach molecules using light resonant withthe appropriate K atom spin-state. The upper part of Fig. 4.14 shows the radialConversion Fraction

63

0.3

0.2

0.1

0.0
0.0
0

0.5

1.0

1.5
T/T

2.0

2.5

C

Figure 4.13: Temperature dependence of Feshbach molecule creation by rf association. Plotted is the molecule conversion fraction Nm /NK as a function of T /Tc
for the Rb atoms. The solid circles (•) are experimental data and the curves are
results of a Monte Carlo calculation. The dashed curves represent the uncertainty
in the conversion fraction taking into account the uncertainty in the trap frequencies and atom numbers. The molecules are created starting from an atom gas
mixture with NRb = 3 × 105 and NK = 1 × 105 . We observe maximum conversion
efficiency of 25% at T /Tc = 1.

64
root-mean-squared size, σRM S , of the molecular gas as a function of expansion
time after the optical trap is abruptly turned off. We fit the size to σRM S (t) =pp(1/ωr ) kB T /mKRb 1 + ωr2 t2 and obtain an expansion energy of T = 310±20 nK.
Here, T is the only fit parameter, kB is Boltzmann’s constant, mKRb is the massof the KRb molecule, and ωr is the radial angular trapping frequency for themolecule.
To confirm that the KRb molecules are trapped, we can look for sloshingof the molecule cloud after a perturbation to the trap potential. The lower partof Fig. 4.14 shows the position of the molecule cloud along a radial direction as
a function of time after the optical trap was abruptly shut off for 1 ms and thenturned back on. A fit to a damped sine wave (solid curve in the lower part of Fig.
4.14) gives a radial slosh frequency of 174 ± 3 Hz. To reduce collisional loss ofthe molecules, these data were taken after removing Rb atoms from the trap [36].
For comparison, using the same trapping fields, the measured frequencies for Rb
and K atoms are 136 ± 2 Hz and 211 ± 4 Hz, respectively. The trapping frequencyof the far-off-resonance optical dipole trap depends on the mass of the particle
and its polarizability. Within the measurement uncertainties, the ratios of the Rbatom, K atom, and KRb Feshbach molecule trap frequencies are consistent withthe weakly bound molecule having a mass equal to the sum of the atomic masses
and a polarizability equal to the sum of the atomic polarizabilities, which is areasonable assumption for these weakly bound molecules.
4.8

Binding Energy of the Feshbach Molecules
Rf association spectra, such as the one shown in Fig. 4.10, can be used

to determine the binding energy of the molecules. We take the rf frequency at
the peak of the molecule association feature and subtract the rf frequency of theatomic Zeeman transition to get a measure of the binding energy. The measured

65

60

1 ms

m)

3 ms
6 ms

Cloud Size (

40

20

0

0

3

6

9

12

Expansion Time (ms)Position ( m)

40
20
0

-20
-40
0

5

10

Time (ms)15

Figure 4.14: (Top inset) Images of the molecule cloud after release from the optical
trap. (Top) Radial cloud size for the molecules after release from the optical trap.
(Bottom) Slosh of the molecular cloud in the optical trap seen in images of the
expanded gas. The expansion time was 3 ms. The observed oscillation in themolecule cloud’s position demonstrates that the molecules are indeed trapped.
The strong damping seen here is most likely due to anharmonicity of the optical
trap.

66
binding energy can have a systematic shift because the initial atom pairs have arange of relative kinetic energies. We estimate that our measured binding energy
could be systematically high by 0.6kB T /h ≈ 4 kHz [61], where h is the Planck
constant. For binding energies larger than about 0.5 MHz, the molecules are
no longer resonant with the imaging light and we are no longer able to see therf association feature by directly imaging the molecules. However, we can stilldetect molecule creation by measuring the loss of K atoms from the |9/2, −7/2istate. As the magnetic-field detuning from the Feshbach resonance increases, we
also find that the maximum conversion fraction decreases, and a longer (as longas 20 s) rf pulse is required.
Figure 4.15 shows the measured molecule binding energy as a function of
the magnetic field. We plot the negative of the molecule binding energy, −E/h, toindicate that the molecule is bound below the atomic Rb |1, 1i + K |9/2, −9/2ithreshold. We observe a quadratic dependence of the binding energy at magnetic fields near the resonance. However, far from the resonance, the binding
energy becomes linear with magnetic field. The data in Fig. 4.15 are comparedwith two theoretical curves. The dashed black line shows the expected universal
behavior very near the resonance [2, 70]. Here, the binding energy is given by
E = ~2 /(2µKRb (a − ā)2 ), where a is the s-wave scattering length, µKRb is the 40 K
and 87 Rb reduced mass, and ā = 68.8 a0 is the mean scattering length determinedby the long-range behavior of the potential [2] and a0 is the Bohr radius. Neara Feshbach resonance, the scattering length is given by a = abg (1 − w/(B − B0 ))[71, 72], where abg = −185a0 and w = −2.9 G for this resonance [65]. To calculate
the molecule binding energy farther from the resonance, our collaborator, Paul
Julienne, performed a multiple-channel calculation. The calculated magnetic-field
location of the Feshbach resonance did not exactly match with our data. We therefore fit the experimental data using the coupled-channel calculation and one free67
parameter, which is the Feshbach resonance position, B0 . From this fit, we obtain
B0 = 546.76 ± 0.05 G. We find excellent agreement between the data and thecoupled-channel calculation (solid line in Fig. 4.15). This measurement of theresonance position disagrees with our previous measurements shown in Fig. 4.7
by nearly 0.3 G. The discrepancy could result from the relatively high atomic
densities that could cause systematic shifts of the measured resonance location.
See Sec. 4.2 for more details of the earlier measurement.
To provide a larger view of the Feshbach molecule state that we probe in theexperiment, we show the results of Paul Julienne’s coupled-channel calculation fora much wider range of magnetic fields in Fig. 4.16. A Feshbach resonance occurs
because of coupling between an atom scattering state and an energetically closed
bound state. The channels used in the calculation are those having a total spin
K
projection quantum number MF = mRb
f + mf = −7/2 and are therefore coupled

to free atoms in the |9/2, −9/2i and |1, 1i states for K and Rb, respectively. For
the s-wave Feshbach resonance at B0 = 546.76 G, the dominant closed channel
contribution comes from the K |7/2, −7/2i + Rb |1, 0i channel with vibrational
quantum number corresponding to the second level below the dissociation limit
in this channel.
We have seen that we can efficiently create heteronuclear Feshbach molecules
starting from an ultracold gas of 40 K and 87 Rb atoms. This could provide astarting point for future experimental efforts aimed at creating polar molecules.
Transfer of the Feshbach molecules to more deeply bound states could proceedusing light or microwave fields. Therefore, it is useful to consider how basic
properties of the Feshbach molecules, such as their typical size and their hyperfine
character, depend on the magnetic-field detuning from the resonance. We canestimate properties of the molecules from the measured binding energy curve
shown in Fig. 4.15.

68-E/h (MHz)

0
-5

-0.2

-10

-0.4

546.0

546.5

-15
-20
-25
-30
536

538

540

542

544

546

B (G)

Figure 4.15: Energy of the heteronuclear Feshbach molecules, relative to the Rb|1, 1i + K |9/2, −9/2i atomic threshold, plotted as a function of magnetic field.
The data (solid circles) agree well with our calculation based on a full coupledchannel theory (solid line). Near the Feshbach resonance one expects a universal
relationship between the s-wave scattering length and the binding energy. The
black, dashed line shows the predicted binding energy using the universal prediction [2]. Inset: Same as the main plot but looking at the region close to the
Feshbach resonance.

69

B

0

0.0

-E/h (GHz)

-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
-3.5

0

200

400

600

800

B (G)

Figure 4.16: Calculated molecule energies near the K |9/2, −9/2i + Rb |1, 1i
atomic threshold. All levels have total spin projection quantum number MF =−7/2. The level energies are calculated using a full coupled-channel calculationand have been scaled so that the zero in energy is the K |9/2, −9/2i + Rb |1, 1i
atomic threshold. The bold line is the adiabatic level associated with the 546.7 G
Feshbach resonance.

70
4.9

Properties of the Feshbach Molecule
The properties of the Feshbach molecule, including its size, are determinedby both the open channel and the dominant closed channel. The bare states
associated with these channels have different magnetic moments, and this is, ofcourse, why the molecule’s binding energy is magnetic-field tunable. The open
and closed channel contributions to the actual dressed-state molecule can then be
determined from the molecule’s magnetic moment relative to the open channel,dEwhich is simply the numerical derivative, dB, of the measured binding energy

shown in Fig. 4.15. Explicitly, we find the closed channel fraction, fc , from the
1 dEfollowing equation: fc = ∆µ. Here, ∆µ = 2.38 × µB is the difference of thedBatomic magnetic moments of the bare closed and open channels at a magnetic fieldnear the Feshbach resonance and µB is the Bohr magneton. The closed channel
fraction as a function of the magnetic field is shown in the inset of Fig. 4.17.
For magnetic fields more than 2 G below the Feshbach resonance, we find that
fc > 0.5 indicating that molecule is predominantly closed channel in character.
We can also estimate the molecule size as a function of magnetic field, B.
Near the resonance, where a is large, the molecule size is rmol = a/2. However,far below the resonance, the molecule becomes dominated by the closed channel
and the molecule size approaches that of the closed channel molecule. To capture
this behavior, we use a simple estimate of the molecule size given by a weighted
average of the closed and open channel sizes, rc and ro .
¡¢rmol = fc × rc + 1 − fc × ro .

(4.1)

rc and ro are calculated using a single-channel model incorporating the correct long-range behavior of the interatomic potential. Figure 4.17 shows our estimate of the size, rmol , as a function of magnetic field near the Feshbach resonance.

1000

Closed Channel Fraction

71

1.00
0.75
0.50
0.25
0.00

100

542

544

546

B (G)

541

542

543

544

545

546

B (G)

Figure 4.17: Estimated molecule size vs. magnetic field near the Feshbach resonance as extracted from the experimentally measured binding energy (•). We
compare to the universal size prediction, a/2 (dashed line). Inset: Closed channel
fraction extracted from the experimentally measured binding energy (•)(see Sec.
4.9).

72
To estimate rc and ro , we numerically solve the Schrödinger equation. Since
the loosely bound vibrational wavefunctions have large amplitudes at large internuclear separation, it is important that our model internuclear potential has thecorrect long-range form. The short-range details of the potential are not as critical and simply contribute to the overall phase of the wavefunction. To simplify
the calculation, we follow an approximation given by Gribakin and Flambaum [2]and use a model internuclear potential that incorporates the correct long-range
behavior and has an impenetrable hard core with a radius R0

U (r) =
∞r < R0¡¢10
− C66 + C88 + C10
r ≥ R0 .
r
r
r

(4.2)

For C6 , C8 , and C10 we use the values reported by Pashov et al. [73]. R0
controls the location of the bound states in the potential and the scattering length.
Since this potential is a single-channel approximation to a multichannel potential,
the value of R0 is chosen to be large to avoid the significant divergence between
the singlet and triplet potentials that occurs at small internuclear separation.
However, R0 must be small enough so the potential has an adequate number ofbound states. For R0 ≈ 22 a0 both criteria are satisfied.
Before calculating the size of the closed channel molecule, we first fine-tune
R0 so that the calculated background scattering length agrees with the known
40

K 87 Rb s-wave scattering length, abg = −185 a0 . We note that this procedureyields binding energies of the last three vibrational states that agree to within
7% with energies obtained using a much more accurate, multichannel potential.
The second-to-last vibrational state with binding energy E−2 ≈ h × 3.2 GHz is
responsible for the B = 546.7 G Feshbach resonance. We, therefore, use its
wavefunction to calculate the closed-channel size, rc = hri ≈ 40 a0 .
Using a similar procedure, we calculate the open-channel size, ro , which

73
depends on the magnetic field detuning from the Feshbach resonance. In thiscase, however, we vary R0 to control the binding energy, E, of the weakest boundmolecular state. For each E, we calculate the size, ro = hri, from the calculated
molecular wavefunction. For the smallest binding energies, the calculated size
agrees with the universal prediction. At larger binding energies the molecular size
is approximately the classical turning point. Figure 4.18 shows the behavior of ro .
4.10

Conclusions
In conclusion, we have used rf association to create heteronuclear Feshbach

molecules from an ultracold gas of 40 K and 87 Rb atoms confined in an optical
dipole trap. Unlike previous work [63], we do not isolate pairs of atoms in individual wells of a deep optical lattice potential. Consequently, the atoms and themolecules are free to collide and pairs of atoms are initially in a near-continuumof motional quantum states. Nevertheless, by optimizing the pulse duration andfrequency of the rf, as well as the initial atom gas temperature, we achieve comparable overall efficiency in converting atoms to weakly bound Feshbach molecules.
We also demonstrate that these molecules are ultracold and trapped.
A gas of ultracold, heteronuclear Feshbach molecules can provide a starting
point for optical manipulation to create a sample of ultracold polar molecules. In
anticipation of such experiments, we have measured the magnetic-field dependent
binding energy of the Feshbach molecules. We extend these measurements tomagnetic fields much farther from resonance than what is typically of interest for
ultracold atom experiments. We measure binding energies as large as 26 MHz
and find that for magnetic fields more than 2 or 3 G detuned from resonance, thebinding energy curve is linear. This means that by lowering the magnetic field
to a few gauss below the resonance, one can have Feshbach molecules that are
predominantly closed channel in character with the corresponding molecular size.

74

1000

100

10
0.01

0.1

1

10

E/h (MHz)
Figure 4.18: The calculated open-channel molecular size, ro , as a function ofbinding energy. The solid red line is the universal prediction, a/2. The dashedblue line is the classical outer turning point of the KRb ground state potential.

75
This is important to improve the Franck-Condon overlap with other molecularstates. Improved Franck-Condon overlap will increase the transfer efficiency tothese other states.
Although the rf association technique was a useful tool for directly detecting
the molecules and diagnosing problems with our apparatus, we no longer use rf
association to form Feshbach molecules. The rf causes offset errors in our control
electronics for the magnetic field so we have found that it is more convenient andstable to form the molecules using magnetic field sweeps across the resonance.

Chapter 5
Inelastic decay of the molecules

5.1

Introduction
In this chapter, I will present our study of inelastic collisions starting froma Bose-Fermi mixture of atoms near a heteronuclear Feshbach resonance. Using
the magnetic-field controlled Feshbach resonance, we created ultracold heteronuclear molecules and studied their inelastic decay. We found that the dominating
decay process arises from atom-molecule collisions and observed the effect of both
fermionic suppression and bosonic enhancement on the inelastic collisions. The
effects of quantum statistics arise because of indistinguishability of the colliding atom and one of the constituents of the molecule. With appropriate atomic
collision partners, we observe molecular lifetimes as long as 100 ms. For large
scattering length, a, experimental data are compared to theoretical predictions
based on a numerical solution of the three-body Schrödinger equation [74].
As mentioned previously, Feshbach molecules are atoms that are weakly
bound to one another [19]. These molecules have a large amount of vibrationalenergy that can be released through collisions with atoms and molecules in aprocess known as vibrational quenching. Often, this process leads to heating
and loss of the atomic and molecular clouds. In most previous experiments using
ultracold atoms to create Feshbach molecules the molecules are composite bosons.
These studies are particularly interesting when the original atoms are fermions. In

77this case, the vibrational quenching of the bosonic molecules is strongly suppressed
due to Fermi statistics [67]. In contrast, when bosonic molecules are created froman atomic Bose gas, this suppression is not present and the molecules tend to havemuch shorter lifetimes [75]5.2

Collisional Stability of Fermionic Feshbach Molecules
Understanding the loss mechanisms in Bose-Fermi mixtures near heteronu-

clear Feshbach resonances is essential for experiments attempting to explore interesting many-body effects in these systems [76]. Moreover, understanding andcontrolling these collisional loss processes is important for proposed experiments
using heteronuclear Feshbach molecules as the starting point for the creation ofultracold polar molecules (see [77] and references therein).
For the experiments discussed in this chapter, we prepared a mixture ofultracold 40 K and 87 Rb held in an optical trap at the end of evaporative cooling.
The mixture had 3 × 105 Rb at T /Tc ∼ 1 and 1 × 105 K at T /TF ∼ 0.6 in the
optical trap. The atoms are then converted into molecules using rf association
as described in Chapter 4. The molecules are imaged in absorption using lightresonant with the K S|9/2, −9/2i → P |11/2, −11/2i cycling transition at highmagnetic field [63]. We measure the lifetime of the molecules in the weakly boundFeshbach state. We characterize the decay of the cloud by its 1/e lifetime, τ ,and introduce a loss coefficient, β, which characterizes the strength of vibrational
relaxation. This coefficient is related to τ by β = nA1 τ , where nA is the initialRaverage density of the dominant atomic collision partner, nA = N1 n(r)2 d3 r.
For Feshbach molecules, β will, in general, depend on the magnetic-field
detuning from the Feshbach resonance, B − B0 . This detuning is related to theWheteronuclear scattering length, which is given by a = abg (1 − B−B). The back0

ground scattering length is abg = −185 a0 , where a0 is the Bohr radius, and theCollision partners
X+BF
B+BF
F+BF
B+B+F

Expected dependence for a > 0β(a) ∝ a−1 [78]β(a) ∝ P (a) × a [79, 80, 81]β(a) ∝ a−3.12 [79, 80]
K3 (a) ∝ M (a) × a4 [79, 80]

78

Table 5.1: Scattering length, a, dependence of the loss coefficient for various
collision partners. ‘BF’ refers to the Feshbach molecule, ‘B’ to the boson, ‘F’ tothe fermion, and ‘X’ to a distinguishable K atom. P (a) and M (a) are log periodic
functions in a with modulations resulting from the formation of Efimov states.

width of the resonance is W = −2.9 G [65]. In the limit of zero temperature andlarge a (a À rvdW , where rvdW = 72a0 is the van der Waals length [19]), β ispredicted to have a universal power law dependence on a [79, 80]. The power laws
are strongly dependent on the quantum statistics of particular collision partners(see Table 5.1). While our measurements are not fully in the limit of T = 0
and large a, we are able to observe the strong influence of the quantum statisticsof the particles involved in an inelastic atom-molecule collision. For large a, we
compare our experimental data with finite temperature numerical calculations.
Our theory collaborator, Jose D’Incao, has performed these calculations by solving the Schrödinger equation in the hyperspherical adiabatic representation [74].
He assumed a simple finite range potential model for the interatomic potentials
v(r) = −D sech2 (r/rvdW ), where r is the interatomic distance and D is adjusted
to produce the desired scattering length variations. We limit our comparison between experiment and theory to a scattering length regime where the finite-range
model can reproduce experimentally observed binding energies to within 20%.
The cut-off occurs at a = 700 a0 . For each comparison with the experimental data
the numerically calculated inelastic rate is scaled by a multiplicative factor. At
finite temperature (T ≈ 150 nK) there is collisional dissociation when the binding energy of the molecules is comparable to the collision energy, (a & 4500 a0 )

79[82, 79, 80]. We, therefore, restrict our measurements in a regime where collisional
dissociation is not a problem.
5.3

Isolating Collision Partners

5.3.1

Distinguishable Atoms

In measuring the loss rate β, ideally one would like to isolate a particularinelastic collisional process. Experimentally, we control the internal states of the
Rb and K atoms using applied rf fields. Combining this rf control with resonant
light pulses, we are able to control the atom densities through selective removalfrom the trap. We first consider the case of distinguishable atoms colliding withmolecules, where the quantum statistics of the colliding particles is not important.
The distinguishable atom is chosen to be K in the |9/2, −7/2i state because itis easily accessed experimentally. The small Zeeman energy of this state causes
it to be stable against inelastic two-body collisions with either K |9/2, −9/2i or
Rb |1, 1i atoms. A nearly pure mixture of K |9/2, −7/2i atoms and Feshbach
molecules is prepared by removing excess Rb atoms from the trap after creating
molecules. Using a combination of microwaves to drive Rb atoms to the |2, 2i state
and resonant light on the cycling transition, we remove 99% of the Rb atoms andleave the K |9/2, −7/2i atoms and molecules essentially unperturbed.
Figure 5.1 shows a plot of experimentally measured β as a function of theheteronuclear scattering length along with numerical calculations for this case. As
predicted [78], the loss rate, β, decreases near the Feshbach resonance (increasing a). Even though the quantum statistics for this case is not important, β is
expected to be suppressed as a−1 as a result of the repulsive character of the effective atom-molecule interaction. The suppression for smaller scattering lengths,
but reasonably large molecule sizes, can be qualitatively explained by an intu-

80
itive argument involving poor wavefunction overlap with deeply bound molecularstates. For our smallest scattering lengths, we find that β stops changing withdecreasing a. Here, the universal prediction for the Feshbach molecule size, a/2,
is no longer valid. To account for this effect we can instead plot β versus an
improved estimate of the molecular size, hrmol i, which takes into account bothclosed and open channel contributions [3]. The calculation of the molecule size isdiscussed in Chapter 4. Fitting the data in Fig. 5.1 (inset) we find a power lawof −0.97 ± 0.16.
5.3.2

Bosonic Atoms

Next, we present measurements of molecule collisions with bosonic atoms.
Here we measure the inelastic decay rates of Feshbach molecules when both Rb|1, 1i atoms and K |9/2, −7/2i atoms are present. The data was taken with the
Rb density more than a factor of three larger than the K density. The contributionof collisions with distinguishable K is known (Fig. 5.1) and subtracted from thedata to extract the loss coefficient, β, for collisions with Rb only.
Figure 5.2 shows a plot of β as a function of the heteronuclear scattering
length, a. For large a, a & 1000 a0 , we observe a distinct enhancement of β that
increases with increasing a. This agrees with the expectation that Bose statistics
should enhance the molecule decay rates due to the attractive character of theeffective atom-molecule interaction [79, 80]. In addition, Efimov three-body boundstates might induce enhanced resonant losses due to trimer formation [83, 82]. The
present data do not rule out Efimov states. However, we also do not observe a clear
signature of Efimov states for the range of scattering lengths and temperatures
carried out in this work.
The measured molecular loss coefficients increase for decreasing a for a .
1000 a0 . This trend is reminiscent of collisions with distinguishable atoms. One

81possible explanation is that the increasing closed channel fraction of the Feshbach
molecule far from resonance makes the colliding free atom effectively distinguishable from the molecule’s constituent atoms, and hence decreases the influence ofbosonic enhancement.
5.3.3

Fermionic Atoms

To study collisions between molecules and fermionic atoms, we prepare anearly pure mixture of K |9/2, −9/2i atoms and Feshbach molecules using thefollowing procedure: after creation of the molecules, we remove Rb atoms fromthe trap and drive the remaining K atoms to the |9/2, −9/2i state with an rf
π-pulse on the atomic transition. This rf pulse does not affect the molecules since
its frequency is detuned by about five linewidths. After waiting a variable time
for the molecule decay measurement, we drive the K atoms back into the dark
|9/2, −7/2i state and image the remaining molecules.
Experimental results are shown in Figure 5.3. These data agree with thetheoretical expectation that the decay is suppressed with increasing a. For our
measurements at finite a, we observe β scaling as a−1.6±0.2 . This trend qualitatively
agrees with numerical calculations (solid curve in Fig. 5.3). The suppression ofmolecular decay allows for molecule lifetimes as long as 100 ms near the resonance(see inset of Fig. 5.3). We note that the calculated β eventually scales as a−3.12
for larger a [79, 80].
For the fermionic case (Fig. 5.3), we observe a suppression of βF +BF whencompared to the decay rates due to collisions with distinguishable atoms (Fig.
5.1). In contrast, for the bosonic case (Fig. 5.2), we observe βB+BF to be larger
than the distinguishable atom case near the resonance. The generality of thisbehavior is an open question because β depends on short-range physics that will
be different in other systems.

82
Molecule-molecule collisions might also contribute to the inelastic loss ofmolecules, but these are expected to be suppressed due to the fermionic characterof the KRb molecules. In the analysis, we have assumed no molecular decay due tomolecule-molecule collisions. From our longest molecule lifetimes, an upper limiton the molecule-molecule loss coefficient at a scattering length of a = 1500 a0 is
βBF +BF ≤ 4 × 10−11 cm3 /s.
5.4

Three-Body Recombination
We also measured three-body recombination rates for atoms in a Bose-Fermi

mixture near a heteronuclear Feshbach resonance. We extracted the three-bodyloss coefficient, K3 , by analyzing measurements of the time-dependent loss of theatomic species as a function of a on both sides of the resonance (Fig. 5.4). Notethat finite temperature will result in a saturation of the collisional decay withincreasing scattering length. We find that near the Feshbach resonance, the loss
rate K3 is enhanced by many orders of magnitude. It follows the expected powerlaw dependence K3 ∝ |a|4 [79, 80] before saturating near the resonance due to theonset of unitarity for scattering lengths |a| > 4000 a0 [84]. Fitting the data to
K3 = η × (a/a0 )4 on both sides of the resonance, we find η = (9.4 ± 1.2) × 10−37
cm6 /s.
5.5

Conclusions
In conclusion, we have created weakly bound fermionic molecules near aheteronuclear Feshbach resonance and studied inelastic, atom-molecule loss processes. We observe that the quantum statistics of the particles strongly affects
vibrational quenching rates of the molecules near the Feshbach resonance. Ourdata are consistent with the expectation that collisions between identical fermionicmolecules are suppressed. This allows for lifetimes of the fermionic heteronuclear

83
molecules as long as 100 ms, which is at least ten times longer than required forcoherent manipulation steps and opens the possibility of production of ultracold
polar molecules. However, when bosonic atoms are available to collide with themolecules, we observe enhanced decay rates. Therefore, studies of the many-body
behavior of a Bose-Fermi mixture near a Feshbach resonance may prove challenging.

84

10

-9

10

10

-10
10

10

10

-9

-10

-11

10

2

10

3

-11

-30 -25

10

1

10

2

10

3

Figure 5.1: Molecule loss coefficient β for collisions with distinguishable atoms (Kin the |9/2, −7/2i state) as a function of the heteronuclear scattering length. For
large a, we observe β to decrease with increasing scattering length in agreementwith numerical calculations (solid red curve). We observe β saturating for small a,(a ∼ 100 a0 ) where the molecular size, hrmol i, no longer has a strong dependence
on a. Inset: β vs. molecule size hrmol i as extracted from the measured binding
energy of the Feshbach molecules [3]. The green dashed curve is a power law fit
βfit ∝ hrmol ip to the experimental data; we obtain p = −0.97 ± 0.16.

85

10

10

10

-9

-10

-11

10

2

10

3

Figure 5.2: Molecule loss coefficient for collisions with indistinguishable bosons
versus heteronuclear scattering length. For large a, β increases with increasing a.
In this regime the molecular loss is enhanced compared to loss due to molecule
collisions with distinguishable atoms (green dashed curve and Fig. 1). For smalla, β increases with decreasing scattering length (see Section 5.3.2).

86

1.5
4

(10 )

1.0

N

BF

10

-9

0.5
0.0
0

10

10

200

400

600

t (ms)

-10

-11

10

2

10

3

Figure 5.3: Molecule loss coefficient β for collisions with indistinguishable
fermions. For large a we observe loss rates scaling as a−1.6±0.2 . The power law
dependence is consistent with numerical calculations for this scattering length
regime (solid red curve). In addition, the molecular loss is suppressed compared
to inelastic decay of molecules due to collisions with distinguishable atoms (green
dashed curve and Fig. 1). Inset: Molecule decay at a ≈ 1300 a0 . The 1/e-lifetimeof the molecular cloud is τ = 100 ± 20 ms

87

10
10
10
10
10
10
10

-20

-21

-22

-23

-24

-25

-26

-6000-3000

03000 6000

Figure 5.4: Three-body recombination near the heteronuclear Feshbach resonance.
The measured K3 varies as a4 and saturates at large a due to the unitary limit.
The dashed curve is an a4 fit to the data for |a| < 4000 a0 . The solid line is theunitarity limit at 150 nK.

Chapter 6
Conclusions and future directions

6.1

Conclusion
This thesis has presented work that has led to the creation of ultracold het-

eronuclear molecules from atomic gases of K and Rb. We used a magnetic-field
tunable Feshbach resonance that affects interspecies collisions to associate atoms
into loosely bound molecules. By measuring the molecular cloud position andits oscillations in the trap, we determined that the molecules were trapped by
the optical trapping potential. Molecular properties such as binding energy, size,
and temperature were characterized. Additionally, the inelastic atom-molecule
collision properties were determined near the Feshbach resonance. It was shownthat inelastic atom-molecule collisions were dramatically affected by not only the
K+Rb scattering length but also by the quantum statistics of the atom involvedin the atom-molecule collision. In particular, if the atom was a boson, we observed
an increase in loss of molecules near the Feshbach resonance. If the atom was afermion, we observed a decrease in the loss of molecules near the resonance. Finally, if the atom was distinguishable from the atoms in the molecule, we observed
a decrease in the loss of molecules near the resonance but the suppression was notas pronounced as when the atom was a fermion.
This work is important for future studies because it characterized the Feshbach molecules that will be transferred to more deeply bound vibrational states of

89
the molecular potential. We also found that the molecules will need to be isolated
from collisions with atoms as early as possible. If the intermediate-state molecules
are allowed to collide with “distinguishable” atoms, their lifetimes will be greatly
reduced. However, once inelastic collisions have been eliminated, the prospects
look bright.
6.2

Future WorkAt the time this thesis was written, there has already been considerable

progress in our experiment toward the goal of producing more deeply boundmolecules using STIRAP techniques [85]. These techniques rely on transferring
the Feshbach molecules to a tightly bound molecular state using an intermediate,
excited state.
At the very end of my work in the lab, we began driving transitions from the
Feshbach molecule state to electronically excited states having an energy that was
lower than the atomic D1 line of Rb. This induces loss of the Feshbach molecules
when we tune the frequency of the laser to an excited vibrational state. A level
diagram is shown in Fig. 6.1. The number of Feshbach molecules remaining after
exciting them with different frequencies of light is shown in Fig. 6.2. Here we used
an optical wavemeter to determine the frequency. Each minimum corresponds todifferent hyperfine state of an electronically excited vibrational level.
Choosing one of the excited states, we measured the loss rates of the Feshbach molecules caused by illuminating the molecules with the laser light. The
results are shown in Fig. 6.3. The loss rates are high because of improved FranckCondon overlap between the Feshbach molecule and the excited state. To estimate
how much having Feshbach molecules has improved the Franck-Condon overlap,
it is useful to compare the bound-bound transition rate of Fig. 6.3 to a free-boundtransition rate. We measured the free-bound transition rate by preparing a mix-

90
ture of atoms under the same conditions as the Feshbach molecules except. We
then illuminate the atom mixture with the laser light. The results are shown in
Fig. 6.4. The ratio of the two time constants, τ , gives an estimate of the improvement in Franck-Condon overlap using Feshbach molecules instead of unboundatoms. The ratio is approximately 500.
In the future, STIRAP may be able to efficiently and coherently transfer
the Feshbach molecules into the tightly bound rovibrational ground-state. Then
one could consider looking for signatures of polar character in the final molecularstates.
Such polar molecules will allow many studies of novel quantum dynamics
that arise from anisotropic, long-range dipolar interactions between the molecules.
For example, molecule-molecule interactions can be affected by applying an external electric field [86, 87]. Using these molecules confined to individual opticallattice sites, there are proposals to create quantum simulators of condensed matter systems [88, 89, 90]. Along the same lines, using ultracold polar molecules
as qubits for a quantum computer has been proposed [10]. Ultracold, fermionic,
polar molecules will certainly have an exciting future.

91

-1

E/(hc) (cm )

15000

10000
= 795 nm

5000

f~377100 GHz

0
5

10

15

Å

20

25

30

R( )

Figure 6.1: Diagram showing the laser frequency we initially used to find excitedmolecular states. The laser was tuned energies lower than the D1 line of atomic
Rb. This corresponds to wavelengths longer than 795 nm.

92

Figure 6.2: Observed molecule loss due to excitation to excited electronic statesof the KRb molecule. Here we start with Feshbach molecules and illuminate themwith laser light having the indicated frequency. The Feshbach molecules are lostwhen the laser light excites them to states where they spontaneously decay andare lost from the trap.

93

3
Nmol/104

τ = (3.9 ± 0.7) · 10−3 msmol
2 Ṅt=0= (640 ± 120) · 107 /s

1
0
0

0.02

0.04

0.06

t/ms
Figure 6.3: Loss of molecules in the Feshbach state resulting from optical excitation to an excited, bound molecular state of KRb. All experimental parameters
were the same as in Fig. 6.4 except here, the initial state is the Feshbach molecular state instead of freely colliding atom pairs. Here, τ is shorter by a factor of
500 compared to freely colliding atom pairs.

94

Natom/104

16

87

Rb
40
K

12
8
τ = (1.9 ± 0.7) ms

4

atom
Ṅt=0
= (3.6 ± 1.2) · 107 /s

0

0

2

4

6
t/ms

8

10

Figure 6.4: Loss of atoms resulting from photoassociation into an excited, boundmolecular state of KRb. The τ for this data is about a factor of 500 longer thanthe data shown in Fig. 6.3.

Bibliography[1] S. Rousseau, A. R. Allouche, and M. Aubert-Frecon, Theoretical Study of
the Electronic Structure of the KRb Molecule, J. Mol. Spectrosc. 203, 235(2000).
[2] G. F. Gribakin and V. V. Flambaum, Calculation of the scattering length
in atomic collisions using the semiclassical approximation, Phys. Rev. A 48,
546 (1993).
[3] J. J. Zirbel, K.-K. Ni, S. Ospelkaus, T. L. Nicholson, M. L. Olsen, C. E.
Wieman, J. Ye, D. S. Jin, and P. S. Julienne, Heteronuclear molecules in an
optical dipole trap, Phys. Rev. A, in press, arXiv:0712.3889v1 (2007).
[4] P. G. H. Sandars, Measurability of the Proton Electric Dipole Moment, Phys.
Rev. Lett. 19, 1396 (1967).
[5] M. G. Kozlov and L. N. Labzowsky, Parity violation effects in diatomics, J.
Phys. B: At. Mol. Opt. Phys. 28, 1933 (1995).
[6] E. R. Hudson, H. J. Lewandowski, B. C. Sawyer, and J. Ye, Cold Molecule
Spectroscopy for Constraining the Evolution of the Fine Structure Constant,Phys. Rev. Lett. 96, 142004 (2006).
[7] S. M. Blinder, Dipole Moment of HD, J. Chem. Phys. 32, 105 (1960).
[8] M. Trefler and H. P. Gush, Electric Dipole Moment of HD, Phys. Rev. Lett.
20, 703 (1968).
[9] S. Lepp and J. Shull, Molecules in the early universe, Ap. J. 280, 465 (1984).
[10] D. DeMille, Quantum Computation with Trapped Polar Molecules, Phys.
Rev. Lett. 88, 067901 (2002).
[11] H. L. Bethlem, G. Berden, and G. Meijer, Decelerating Neutral Dipolar
Molecules, Phys. Rev. Lett. 83, 1558 (1999).
[12] B. C. Sawyer, B. L. Lev, E. R. Hudson, B. K. Stuhl, M. Lara, J. L. Bohn, andJ. Ye, Magnetoelectrostatic Trapping of Ground State OH Molecules, Phys.
Rev. Lett. 98, 253002 (2007).

96[13] J. D. Weinstein, R. deCarvalho, T. Guillet, B. Friedrich, and J. M. Doyle,
Magnetic trapping of calcium monohydride molecules at millikelvin temperatures, Nature 395, 148 (1998).
[14] S. A. Rangwala, T. Junglen, T. Rieger, P. W. H. Pinkse, and G. Rempe,
Continuous source of translationally cold dipolar molecules, Phys. Rev. A
67, 043406 (2003).
[15] H. R. Thorsheim, J. Weiner, and P. S. Julienne, Laser-Induced Photoassociation of Ultracold Sodium Atoms, Phys. Rev. Lett. 58, 2420 (1987).
[16] A. J. Kerman, J. M. Sage, S. Sainis, T. Bergeman, and D. DeMille, Production and State-Selective Detection of Ultracold RbCs Molecules, Phys. Rev.
Lett. 92, 153001 (2004).
[17] J. Kleinert, C. Haimberger, P. J. Zabawa, and N. P. Bigelow, Trapping of
Ultracold Polar Molecules with a Thin-Wire Electrostatic Trap, Phys. Rev.
Lett. 99, 143002 (2007).
[18] S. J. J. M. F. Kokkelmans, H. M. J. Vissers, and B. J. Verhaar, Formation ofa Bose condensate of stable molecules via a Feshbach resonance, Phys. Rev.
A 63, 031601(R) (2001).
[19] T. Kohler, K. Goral, and P. S. Julienne, Production of cold molecules via
magnetically tunable Feshbach resonances, Rev. Mod. Phys. 78, 1311 (2006).
[20] C. E. Wieman, D. E. Pritchard, and D. J. Wineland, Atom cooling, trapping,and quantum manipulation, Rev. Mod. Phys. 71, S253 (1999).
[21] W. D. Phillips, Nobel Lecture: Laser cooling and trapping of neutral atoms,
Rev. Mod. Phys. 70, 721 (1998).
[22] S. Chu, The manipulation of neutral particles, Rev. Mod. Phys. 70, 685(1998).
[23] C. N. Cohen-Tannoudji, Manipulating atoms with photons, Rev. Mod. Phys.
70, 707 (1998).
[24] E. A. Cornell and C. E. Wieman, Nobel Lecture: Bose-Einstein condensation
in a dilute gas, the first 70 years and some recent experiments, Rev. Mod.
Phys. 74, 875 (2002).
[25] W. Ketterle, Nobel lecture: When atoms behave as waves: Bose-Einsteincondensation and the atom laser, Rev. Mod. Phys. 74, 1131 (2002).
[26] B. DeMarco and D. S. Jin, Onset of Fermi degeneracy in a trapped atomicgas, Science 285, 1703 (1999).

97[27] B. DeMarco, Quantum behavior of an atomic Fermi gas, Ph.D. thesis, University of Colorado - Boulder, 2001.
[28] D. M. Eagles, Possible pairing without superconductivity at low carrier concentrationsin bulk and thin-film superconducting semiconductors, Phys. Rev.
186, 456 (1969).
[29] P. Nozieres and S. Schmitt-Rink, Bose condensation in an attractive fermion
gas: From weak to strong couplingsuperconductivity, J. Low-Temp. Phys.
59, 195 (1985).
[30] A. J. Leggett, Superfluidity, Rev. Mod. Phys. 71, S318 (1999).
[31] M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo, and R. Walser,
Resonance Superfluidity in a Quantum Degenerate Fermi Gas, Phys. Rev.
Lett. 87, 120406 (2001).
[32] M. L. Chiofalo, S. J. J. M. F. Kokkelmans, J. N. Milstein, and M. J. Holland,
Signatures of Resonance Superfluidity in a Quantum Fermi Gas, Phys. Rev.
Lett. 88, 090402 (2002).
[33] Y. Ohashi and A. Griffin, BCS-BEC Crossover in a Gas of Fermi Atoms witha Feshbach Resonance, Phys. Rev. Lett. 89, 130402 (2002).
[34] C. A. Regal, M. Greiner, and D. S. Jin, Observation of Resonance Condensation of Fermionic Atom Pairs, Phys. Rev. Lett. 92, 040403 (2004).
[35] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman,
and W. Ketterle, Condensation of Pairs of Fermionic Atoms near a Feshbach
Resonance, Phys. Rev. Lett. 92, 120403 (2004).
[36] J. J. Zirbel, K.-K. Ni, S. Ospelkaus, J. P. D’Incao, C. E. Wieman, J. Ye, and
D. S. Jin, Collisional Stability of Fermionic Feshbach Molecules, Phys. Rev.
Lett. 100, 143201 (2008).
[37] S. B. Papp, Experiments with a two-species Bose-Einstein condensate utilizing widely tunable interparticle interactions, Ph.D. thesis, University ofColorado-Boulder, 2008.
[38] S. Kobayashi and T. Kimura, Injection locking in AlGaAs semiconductor
laser, Quantum Electronics, IEEE Journal of 17, 681 (1981).
[39] B. Dahmani, L. Hollberg, and R. Drullinger, Frequency stabilization of semiconductor lasers by resonant optical feedback, Opt. Lett. 12, 876 (1987).
[40] T. Pawletko, M. Houssin, M. Knoop, M. Vedel, and F. Vedel, High power
broad-area diode laser at 794 nm injected by an external cavity laser, Optics
Communications 174, 223 (2000).

98[41] Y. Ishikawa, T. Yoshimura, and M. Arai, Effect of surface oxide layers on
deuterium permeation through stainless steels with reference to outgassing
reduction in ultra- to extremely high vacuum, Vacuum 47, 357 (1996).
[42] L. Westerberg, B. Hjrvarsson, E. Walln, and A. Mathewson, Hydrogen content and outgassing of air-baked and vacuum-fired stainless steel, Vacuum
48, 771 (1997).
[43] B. DeMarco, H. Rohner, and D. Jin, An enriched 40 K source for fermionic
atom studies, Rev. Sci. Instrum. 70, 1967 (1999).
[44] D. E. Pritchard, Cooling Neutral Atoms in a Magnetic Trap for Precision
Spectroscopy, Phys. Rev. Lett. 51, 1336 (1983).
[45] T. L. Nicholson, Optical Trapping in a Cold Molecule Experiement, 2006.
[46] S. B. Papp and C. E. Wieman, Observation of Heteronuclear Feshbach
Molecules from a 85 Rb-87 Rb Gas, Phys. Rev. Lett. 97, 180404 (2006).
[47] S. B. Papp, J. M. Pino, and C. E. Wieman, Studying a dual-species BEC
with tunable interactions, arXiv:0802.2591v1 (2008).
[48] E. Hodby, S. T. Thompson, C. A. Regal, M. Greiner, A. C. Wilson, D. S.
Jin, E. A. Cornell, and C. E. Wieman, Production Efficiency of Ultracold
Feshbach Molecules in Bosonic and Fermionic Systems, Phys. Rev. Lett. 94,
120402 (2005).
[49] W. Petrich, M. H. Anderson, J. R. Ensher, and E. A.Cornell, Behavior ofatoms in a compressed magneto-optical trap, J. Opt. Soc. Am. B 11, 1332
(1994).
[50] H. J. Lewandowski, D. M. Harber, D. L. Whitaker, and E. A. Cornell, Simplified System for Creating a BoseEinstein Condensate, J. Low Temp. Phys.
132, 309 (2003).
[51] H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Spring,
New York, 1999).
[52] J. L. Roberts, Bose-Einstein condensates with tunable atom-atom interactions: The firstexperiments with 85 Rb BECs, Ph.D. thesis, University ofColorado - Boulder, 2001.
[53] S. L. Cornish, N. R. Claussen, J. L. Roberts, and E. A. C. andC. E. Wieman, Stable 85 Rb Bose-Einstein condensates with widely tunable interactions,Phys. Rev. Lett. 85, 1795 (2000).
[54] T. Weber, J. Herbig, M. Mark, H.-C. Ngerl, and R. Grimm, Three-Body
Recombination at Large Scattering Lengths in an Ultracold Atomic Gas,
Phys. Rev. Lett. 91, 123201 (2003).

99[55] J. L. Roberts, J. P. Burke, Jr., N. R. Claussen, S. L. C. E. A. Donley, andC. E. Wieman, Improved characterization of elastic scattering near a Feshbach resonance in 85 Rb, Phys. Rev. A 64, 024702 (2001).
[56] J. L. Roberts, N. R. Claussen, S. L. Cornish, E. A. D. andE. A. Cornell, andC. E. Wieman, Controlled collapse of a Bose-Einstein condensate, Phys. Rev.
Lett. 86, 4211 (2001).
[57] N. R. Claussen, E. A. Donley, S. T. Thompson, and C. E. Wieman, Microscopic dynamics in a strongly interacting Bose-Einstein condensate, Phys.
Rev. Lett. 89, 010401 (2002).
[58] E. A. Donley, N. R. Claussen, S. T. Thompson, and C. E. Wieman, Atommolecule coherence in a Bose-Einstein condensate, Nature 417, 529 (2002).
[59] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Creation of ultracold
molecules from a Fermi gas of atoms, Nature 424, 47 (2003).
[60] J. Stenger, S. Inouye, M. R. Andrews, H.-J. Miesner, D. M. Stamper-Kurn,and W. Ketterle, Strongly Enhanced Inelastic Collisions in a Bose-Einstein
Condensate near Feshbach Resonances, Phys. Rev. Lett. 82, 2422 (1999).
[61] S. T. Thompson, E. Hodby, and C. E. Wieman, Ultracold Molecule Production via a Resonant Oscillating Magnetic Field, Phys. Rev. Lett. 95, 190404(2005).
[62] J. P. Gaebler, J. T. Stewart, J. L. Bohn, and D. S. Jin, p-Wave Feshbach
Molecules, Phys. Rev. Lett. 98, 200403 (2007).
[63] C. Ospelkaus, S. Ospelkaus, L. Humbert, P. Ernst, K. Sengstock, and K.
Bongs, Ultracold Heteronuclear Molecules in a 3D Optical Lattice, Phys.
Rev. Lett. 97, 120402 (2006).
[64] S. Inouye, J. Goldwin, M. L. Olsen, C. Ticknor, J. L. Bohn, and D. S. Jin,
Observation of Heteronuclear Feshbach Resonances in a Mixture of Bosons
and Fermions, Phys. Rev. Lett. 93, 183201 (2004).
[65] F. Ferlaino, C. D’Errico, G. Roati, M. Zaccanti, M. Inguscio, G. Modugno,
and A. Simoni, Feshbach spectroscopy of a K-Rb atomic mixture, Phys. Rev.
A 73, 040702 (2006).
[66] C. Ticknor, C. A. Regal, D. S. Jin, and J. L. Bohn, Multiplet structureof Feshbach resonances in nonzero partial waves, Phys. Rev. A 69, 042712(2004).
[67] D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, Weakly Bound Dimers of
Fermionic Atoms, Phys. Rev. Lett. 93, 090404 (2004).

100[68] C. A. Regal, M. Greiner, and D. S. Jin, Lifetime of Molecule-Atom Mixturesnear a Feshbach Resonance in K, Phys. Rev. Lett. 92, 083201 (2004).
[69] S. T. Thompson, E. Hodby, and C. E. Wieman, Spontaneous Dissociation of85
Rb Feshbach Molecules, Phys. Rev. Lett. 94, 020401 (2005).
[70] T. Kohler, T. Gasenzer, and K. Burnett, Microscopic theory of atom-molecule
oscillations in a Bose-Einstein condensate, Phys. Rev. A 67, 013601 (2003).
[71] A. J. Moerdijk, B. J. Verhaar, and A. Axelsson, Resonances in ultracold
collisions of 6 Li, 7 Li, and 23 Na, Phys. Rev. A 51, 4852 (1995).
[72] F. H. Mies, E. Tiesinga, and P. S. Julienne, Manipulation of Feshbach resonances in ultracold atomic collisions using time-dependent magnetic fields,
Phys. Rev. A 61, 022721 (2000).
[73] A. Pashov, O. Docenko, M. Tamanis, R. Ferber, H. Knockel, and E. Tiemann,
Coupling of the X 1 Σ+ and a 3 Σ+ states of KRb, Phys. Rev. A 76, 022511(2007).
[74] H. Suno, B. D. Esry, C. H. Greene, and J. P. Burke, Three-body recombination of cold helium atoms, Phys. Rev. A 65, 042725 (2002).
[75] V. A. Yurovsky, A. Ben-Reuven, P. S. Julienne, and C. J. Williams, Atomloss from Bose-Einstein condensates due to Feshbach resonance, Phys. Rev.
A 60, R765 (1999).
[76] D. V. Efremov and L. Viverit, p-wave Cooper pairing of fermions in mixturesof dilute Fermi and Bose gases, Phys. Rev. B 65, 134519 (2002).
[77] J. Doyle, B. Friedrich, R. V. Krems, and F. Masnou-Seeuws, Quo vadis, cold
molecules? - Editorial review, Eur. Phys. J. D 31, 149 (2004).
[78] J. P. D’Incao and B. D. Esry, Suppression of molecular decay in ultracoldgases without Fermi statistics, arXiv:0711.3238v1 (2007).
[79] J. P. D’Incao and B. D. Esry, Mass dependence of ultracold three-body collision rates, Phys. Rev. A 73, 030702 (2006).
[80] J. P. D’Incao and B. D. Esry, Scattering Length Scaling Laws for Ultracold
Three-Body Collisions, Phys. Rev. Lett. 94, 213201 (2005).
[81] E. Braaten and H.-W. Hammer, Enhanced dimer relaxation in an atomic andmolecular Bose-Einstein condensate, Phys. Rev. A 70, 042706 (2004).
[82] E. Braaten and H.-W. Hammer, Universality in few-body systems with large
scattering length, Phys. Rep. 428, 259 (2006).

101[83] T. Kraemer et al., Evidence for Efimov quantum states in an ultracold gasof caesium atoms, Nature 440, 315 (2006).
[84] J. P. D’Incao, H. Suno, and B. D. Esry, Limits on Universality in Ultracold
Three-Boson Recombination, Phys. Rev. Lett. 93, 123201 (2004).
[85] S. Ospelkaus, A. Pe’er, K.-K. Ni, J. J. Zirbel, B. Neyenhuis, S. Kotochigova,
P. S. Julienne, J. Ye, and D. S. Jin, Efficient state transfer in an ultracold
dense gas of heteronuclear molecules, Nat Phys advanced online publication, (2008).
[86] C. Ticknor and J. L. Bohn, Long-range scattering resonances in strong-fieldseeking states of polar molecules, Phys. Rev. A 72, 032717 (2005).
[87] C. Ticknor, Collisional Control of Ground State Polar Molecules and Universal Dipolar Scattering, Phys. Rev. Lett. 100, 133202 (2008).
[88] B. Sun and L. You, Observing the Einstein–de Haas Effect with Atoms in an
Optical Lattice, Phys. Rev. Lett. 99, 150402 (2007).
[89] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, Ultracold atomic gases in optical lattices: mimicking condensed matter physics
and beyond, Advances in Physics 56, 243 (2007).
[90] G. Pupillo, A. Griessner, A. Micheli, M. Ortner, D.-W. Wang, and P. Zoller,
Cold Atoms and Molecules in Self-Assembled Dipolar Lattices, Phys. Rev.
Lett. 100, 050402 (2008).

Appendix A
Magnetic Trap Trouble

In the summer of 2006 our new Ioffe-Pritchard trap failed. This failure
showed itself after we abused the trap by not cooling the Ioffe bars and pinch coils
during just one experimental cycle. The abuse itself wasn’t enough to burn thecoils but it was clear that at least one coil did not function properly afterwards.
The symptoms of failure were consistent with the Ioffe bar coils either having
an electrical short or moving vertically. This was because the atoms’ position wererandomly shifting vertically after every experimental cycle. The position shifts
had a standard deviation of about 10 µm. This could be caused either by anintermittent electrical short or by the coil itself was somehow shifting vertically
shot-to-shot.
We naively suspected that a short between two windings of a coil would
cause all of the current to flow to avoid the shorted winding. Such a short would’ve
caused our atom cloud to be shifted by about a millimeter. This large of a shift
was not observed. In addition, when we measured the resistances of the coils
during experimental cycles, they agreed with the expected resistance to muchbetter than the resistance of a single winding of the coil. This suggested that thecoil wasn’t shorted.
Another alternative was that the coil had a winding break free from thebulk of the coil and it was randomly shifting after each experimental cycle from103
the relatively strong magnetic forces. This explanation was nice because it agreed
with the qualitative observation that the atoms were shifting randomly. However,the position fluctuations of the atoms were much larger than we could expect fromreasonable movement of all of the windings. This made the movement requirementof just the one winding even larger. We measured that the entire coil moved byless than 6 µm during and after each experimental cycle.
As it turned out, the correct answer was probably a combination of thetwo. In the Ioffe coil below the cell, we had an intermittent short but not acomplete short. Such a short changed slightly after each experimental cycle due
to minute changes of the shorted windings’ relative positions caused by magnetic
forces. This explanation was supported by evidence that the coil’s resistance waschanging up to 0.2% after each experimental cycle. These shifts in resistancewere so small that they were initially masked by fluctuations in the resistancefrom temperature changes of the water. Since copper’s thermal coefficient ofresistance is about ηCu = 0.39%/K, only a small temperature shift would cause asimilar resistance shift. To account for this effect, we simultaneously measured theresistance and temperature of the coils. In Fig. A.1 I have plotted the resistance ofthe coil versus the water temperature. The family of lines all have slopes that areconsistent with the thermal coefficient of resistance for copper. Each transition
from one line to the other occurred after mechanical perturbations of the bottom
Ioffe coil such as moving the coil’s leads by hand or tightening retaining screws.
These resistance changes were about 5 times larger than when the coil’s current
was cycled normally in an experimental sequence.
We also measured the position of the cloud and found that it too was correlated to resistance shifts of the coil. Figure A.2 shows this correlation. The
effect of the coil’s temperature on the resistance has been accounted for and subtracted in this measurement. We measured the linear slope of the data to be

104

Bottom Ioffe

Resistance (Arb)

1068
1064
1060
1056
1052
1048
1044
1040
9.0

9.5

10.0

10.5

11.0

11.5

Water Temperature (C)Figure A.1: The resistance of the coil plotted against the temperature of the waterflowing through the coil. The slope of the curves is consistent with the thermal
coefficient of resistance of copper. The vertical resistance shifts are caused bymechanical perturbations of the coil. These shifts are relatively small at about
1% of the total resistance of the coil. For comparison, if a complete winding wereshorted, it would have a shift 13 times larger.

105
about 70 ± 15µm/% which is roughly consistent with the slope predicted by anumerical calculation.
These observations led to further inspection of the coils. We learned thatthe short was probably caused by the initial winding procedure for the Ioffe coils.
Because of the geometry of the Ioffe coil, there is a transition region for the coil’sinterior lead that circumvents the rest of the coil (See Fig. A.3). This region
causes the edges of adjacent copper tubes to slip past and press into one anotherduring winding. This pressure causes the polyimide (Kapton r) coating to rub
off at this region and is particularly bad for the tubing we used because of its
square cross-section, which has a relatively sharp edge.
It is remarkable and insidious how weakly the adjacent windings were shorted.
Presumably the polyimide’s adhesive was providing enough resistance to preventa full electrical short. Since the slippage past the two tubes caused the short, we
fixed this by wrapping the bad location with more polyimide after winding thecoil. After the fix, we performed the same set of tests on the bottom Ioffe coil andfound no problems. However, even though we checked all of the coils, they could
still have similar problems that have not yet presented themselves.

106

Cloud Position ( m)

60
50
40
30
20
10
0
0.0

0.1

0.2

0.3

0.4

0.5

Resistance Change (%)

Figure A.2: Correlation between the cloud’s measured position and percent resistance change. The data was taken after releasing from the magnetic trap for
15ms of free expansion. The slope is roughly consistent with the slope predictedby a numerical calculation.

107

Figure A.3: A view of the Ioffe coil’s problematic lead. The pressure between
the wires at this transition point caused an incomplete short to other windings
through the polyimide coating.