Exploring Few-body Scattering Resonances in a Bose-Fermi
Mixture: from Feshbach Dimers to Efimov Trimers
byTyler Dunnington Cumby
B.A., Washington State University, 2004

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2012

iii
Cumby, Tyler Dunnington (Ph.D., Physics)
Exploring Few-body Scattering Resonances in a Bose-Fermi Mixture: from Feshbach Dimers to
Efimov Trimers
Thesis directed by Prof. Deborah Jin

This thesis presents experiments exploring Feshbach molecules and Efimov trimers in an
ultracold Bose-Fermi atomic gas mixture of 40 K and 87 Rb. These bound states are accessed via
scattering resonances, called Fano-Feshbach resonances, which arise as a consequence of a boundstate being tuned through degeneracy with the threshold energy of colliding atoms. In addition
to allowing the tunability of interactions, the Fano-Feshbach resonance allows for a weakly boundmolecule state, called the Feshbach molecule state, to be populated. The efficiency of conversion
to the Feshbach molecule state depends upon the rate with which the bound state energy is swept
through degeneracy with atomic threshold energy. In regions of large scattering lengths, threebody bound states, called Efimov states, also influence inelastic Bose-Bose-Fermi collisions andboson+molecule collisions in our system. The work presented in this thesis utilizes a magneticallytunable Fano-Feshbach resonance to study Feshbach molecule creation in the perturbative limit(the limit of fast magnetic-field sweeps) and the saturated limit (the limit of slow sweeps), where aphenomenological model is applied to understand the conversion efficiency as well as the quantum
degeneracy of the molecules. In addition, I present a study of boson+molecule collision rates
and Bose-Bose-Fermi collision rates in order to observe signatures of Efimov states and compare
against results in the gas mixture 41 K-87 Rb to address the question of universality of the three-bodyparameter in heteronuclear gas mixtures.

Dedication

To my father, Clifton L. Cumby, 1950-2008.

Acknowledgements

Although the work in this thesis involves the manipulation of atoms at ultralow vacuum
pressures, my time here was not spent in a vacuum. During my years here I have had the privilege
to work with many excellent scientists. First, I would like to thank my advisor, Debbie Jin, for givingme the opportunity to join one of her labs and be taught how to be an effective physicist. Debbie’s
labs are also closely associated with Eric Cornell and his labs, and so I’ve been the benefactor of
his questions and thoughtful and thought-provoking insights. When I joined the lab, Michele Olsen
was still in the process of rebuilding the experiment after the K sources were depleted. Her efforts
produced an apparatus that is still going strong today, albeit with a few upgrades. Along the way a
post-doc, John Perreault, joined us and contributed tremendously to the lab with his experimental
and theoretical knowledge, as well as his infectious inquisitiveness.
Later John and I were joined by Ruth Shewmon. Ruth has always impressed me with her
determined quest to understand the world at large. This drive, coupled with her keen wit, led tonumerous insights, both scientific and otherwise (e.g. she once hypothesized that the price of soy
milk was driven by the fact that it is quite challenging to milk a soy bean) while we were manningthe experiment. Next we had the luck to have Ming-Guang Hu join the lab after he was lured awayfrom Condensed Matter theory to AMO experiment. Our lab benefitted greatly from Ming-Guang’s
quick wit and remarkable ingenuity. With Ruth and Ming-Guang at the helm of this apparatus, I
am confident that many great experiments are in store.
I would be remiss to not thank the many other graduate students that I had the fortune ofassociating with during my time at JILA. In my early years, I benefited greatly from the friendly

vi
advice of the more senior grad students in the group, Scott Papp, Josh Zirbel, Cindy Regal, andKang-Kuen Ni, as well as the then post-doc Marcus Greiner. I have also the good fortune to work in
the Jin labs with fellow graduate students and friends Jayson Stewart, John Gaebler, Tara Drake,
Rabin Paudel, Brian Neyenhuis, Steven Moses, Jacob Covey, Juan Pino, Rob Wild, Phil Makotyn,
and Cathy Klauss. Given the strong ties of Jin and Cornell groups, I’ve also had the pleasure
to associate with many of Eric Cornell’s students: Russell Stutz, Laura Sinclair, Huanqian Loh,
Shih-Kuang Tung, and Dan Lobser. I also had the good fortune to associate with several greatpost-docs: Yoav Sagi, Amodsen Chotia, Bo Yan, and Lin Xia.
In addition to the scientists at JILA, I have to give many thanks to the support staff. Tremendous thanks also goes to Tracy Keep from the Instrument Shop for designing and machining our
new Ioffe-Prichard trap, which will enable the great work to come. Hans Green answered my assorted machining questions and helped with various projects with unbridled enthusiasm. Terry
Brown of the Electronics Shop taught me a tremendous amount about electronics in the numerous discussions I had with him. Other members of the Electronics Shop were also of great help:
James Fung-A-Fat, Carl Sauer, Mike Whitmore, and David Tegart. Mike Paige and J.R. Raith of
the Computing Team provided great assistance and helped prevent several computer calamities.
Krista Beck also deserves numerous thanks for all the help during my years here, as well as for her
bureaucratic ninjutsu skills that saved my lab darkness.

vii

Contents

Chapter
1 Introduction
1.1

1.2

1

Bose-Fermi interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.1.1

Two-body bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.1.2

Three-body bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5Contents of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2 Creating an ultracold Bose-Fermi mixture

9

2.1

New Rb and K laser setups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Bose-Fermi mixture: onward and downward to the ultracold regime . . . . . . . . . . 12

2.3

Characterizing the Fast B coil circuit time response . . . . . . . . . . . . . . . . . . . 14

2.4

2.5

2.3.1

Fast B closed-loop transfer function . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.2

Measuring the Fast B magnetic-field sweep rate in situ . . . . . . . . . . . . . 16

Imaging and state control improvements . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1

Measuring imaging magnification . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.2

The new and nimble Rb microwave setup . . . . . . . . . . . . . . . . . . . . 23

Calibrating the Rb atom number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5.1

Calibrating K atom number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Feshbach molecules in the perturbative and saturated limits
3.1

9

30

Basic scattering theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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3.2

Feshbach molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.1

The 546.7 G Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3

Magneto-association of Feshbach molecules . . . . . . . . . . . . . . . . . . . . . . . 39

3.4

Molecule creation in the perturbative limit . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5

Molecule creation in the saturated limit . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.5.1

Semi-classical simulation of molecule formation . . . . . . . . . . . . . . . . . 46

3.5.2

Molecule creation efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.5.3

Experimental concerns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.5.4

Theoretical optimization of Feshbach molecule degeneracy . . . . . . . . . . . 53

4 The search for Efimov Physics in 40 K-87 Rb

58

4.1

The Efimov effect and discrete scaling . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2

Observation of Efimov resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3

Analysis of loss data to extract α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.1

Identifying the feature at 547.4 G . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4

Extracting β from atom-molecule inelastic collisional loss . . . . . . . . . . . . . . . 71

4.5

Comparison with universal predictions . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.6

Three-body parameter and comparison with 41 K −87 Rb . . . . . . . . . . . . . . . . 76

5 The “Fabrergé egg” Ioffe-Pritchard trap
5.1

83

Magnetic trap potential and design considerations . . . . . . . . . . . . . . . . . . . 85
5.1.1

Modeling the magnetic field from the coils . . . . . . . . . . . . . . . . . . . . 87

5.1.2

Design principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2

Water cooling concerns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3

Magnetic trap details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6 Conclusions and future directions
6.1

103

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

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6.2

Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Bibliography

107

Appendix
A Myriad details of the SPSS model

118

A.1 Accept-reject Monte Carlo algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
A.2 Fermionic atom generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
A.3 Bosonic atom generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
A.4 Verifying Monte Carlo atom density and momentum distributions . . . . . . . . . . . 123
A.5 Fitting the SPSS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
A.6 Other details of the SPSS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

xTablesTable
3.1

Comparison of the 40 K – 87 Rb Feshbach resonance parameters for the |9/2, −9/2i +|1, 1i resonance at ≈546.7 G as reported in various publications. In this table, B0 is
the measured resonance center, ∆ is the zero-crossing width, and abg the backgroundscattering length. All quantities with an asterisk (∗ ) are theoretical predictions. . . . 37

4.1

Classes of collisions and whether or not Efimov states exist when one, two, or all
three inter-particle scattering lengths (aij , ajk , aki ) are large compared to rvdW , thelength scale of the two-body potential. . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2

Comparison of Efimov resonance parameters between 41 K – 87 Rb and 40 K – 87 Rb. . 78

4.3

Comparing the Feshbach resonances between 40 K – 87 Rb and 41 K – 87 Rb . . . . . . 81

5.1

Derivatives of the magnetic field on axis at the center in between a coil pair separated
by a distance d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.2

Calculated magnet coil properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.3

Measured and predicted magnet coil resistances. Measurement of the trap coils was
subject to a ∼ 1mΩ due to uncertainties in the lead length, which is represented
by the second error bar. The measured values of the coils resistances agree withthe calculated values within the error bars with the exception of the bias coils.
The resistances were calculated from the resistivity of copper (1.71 µΩ-m) and thedimensions of the tubing and coils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

xiFiguresFigure1.1a) A simplified model of a Feshbach resonance where atoms colliding in the entrance,or open, channel threshold are brought into degeneracy with a bound state in theclosed channel where the energy of the colliding atoms differs from the bound stateenergy by Ecc . b) A plot of the scattering length as a function of magnetic field neara Feshbach resonance located at B = B0 . The resonance width is characterized by∆, which is the difference in magnetic field between the field at which a = 0 and B0 .

1.2

3a) A series of discretely scaled leaves with a scaling factor of 1/2 between leaves. b)
Energy levels for the three-body system shown as a function of the two-body scattering length. The three-body bound states predicted by Efimov are shown as the thicksolid curves. The three-body bound states begin at the three-atom collision threshold (Energy = 0) at negative scattering lengths and then asymptotically approaches
the atom+molecule threshold given by the molecule energy (dashed line). The scattering lengths at which the three-body states reaches the three-atom threshold are
related by a discrete scaling factor, which was artificially chosen to be two for thisdiagram.

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

6

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2.1

Diagrams of Rb and K laser setup and locking scheme. Light is supplied by thevarious lasers and tapered amplifiers (TA) and the light frequencies are shifted byacousto-optic modulators (AOM)(yellow squares). All of the light for the MOT
and all probe beams are coupled into optical fibers (black loops) to be sent to theapparatus, whereas the optical pumping beams are propagated through free-space.
Frequencies for the AOMs and the offset lock are shown for the various stages of theexperiment.

2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

CAD rendering of the apparatus without the Ioffe-Pritchard trap (IP trap), whichsurrounds the science cell. At the start of an experiment, the Quadrupole-trap coils(Q-pole trap) are located at the collection cell where they supply the magnetic-field
gradient for the MOT. After the MOT stage, the atom clouds are compressed andcooled and then loaded into the Q-pole trap. The Q-pole trap coils are mounted ona movable cart (cart track is 1.2 m long) which whisks them away to the science cell
where the atoms are loaded into the IP trap for further cooling.

2.3. . . . . . . . . . . 13a) Calculated closed-loop transfer function for the Fast B servo from measurementsof the input waveform and measured response. The bandwidth of the servo, asmeasured by the -3dB frequency, is 39.2 kHz (dashed line). The peaks in the transfer
function are likely resonances in our Hall-effect current transducers (LEM Ultrastab866). b) Measured servo control voltage (lower plot, denoted f(t) below) and servo
Hall-effect current transducer (upper plot, denoted g(t) below). Both plots have been
undersampled from the actual data by a factor of 100. Note that the servo’s small
response compared to the control voltage is a signature of its limited bandwidth. . . 15

2.4a) Energy diagram for a coupled two-level system as discussed in the text. b) Timing
diagram for a magnetic field ARP where the rf (red sinusoidal curve) is turned on
while the magnetic field is initially far from resonance and the magnetic field later
sweeps linearly down through the resonance. . . . . . . . . . . . . . . . . . . . . . . . 18

xiii
2.5

a) On resonance Rabi flopping between |9/2, −9/2i and |9/2, −7/2i at a synthesizer
power of 0.1 mW and an rf frequency of 80.0378 MHz. b) Calibration of the Rabi
frequency versus synthesizer power. The fit of Ω2 versus power to a line returns
a slope of 4.97(13) × 1011 rad2 s−2 mW−1 . c) Sample measurement of the 10 µs/G
ramp with a 10 G span. The number of atoms remaining in the |9/2, −9/2i Zeeman
state are plotted as a function of the rf synthesizer power and then fit to an decaying
exponential of the form N = N0 exp(−p/p1/e ) + y0 . From the fitted value of p1/e andEqn. (2.3.2), the magnetic-sweep rate is measured, where δµ = h × 66kHz/G near
550 G. d) Collection of measurements such as c) plotted against the inverse ramp
rate requested of the servo and a line with a slope of 1. The measured ramp rate
was calculated from c) using a calibration of the Rabi frequency measured from Rabi
flopping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6a) Fractional correction of the measured OD (ODmeas ) due to OD saturation effects
caused by unabsorbed light. b) Fractional correction of the ODstray due to intensity
saturation effects.

2.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

L
Plot of Eqn. (2.4.4) using Eqn. (2.4.3) for ODsat = 4 and IIsat= 1/10 (dashed line)Land IIsat= 1/20 (solid line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.8

Sample image of the back-lit test target taken using our imaging system. . . . . . . . 24

2.9

Circuit diagram for the new Rb microwave setup. We mix the stable microwave
source (Agilent E8251A) with a pre-amplified (Mini-circuits ZFL-1000LN+) rf sourcefrom a DDS (Toptica VFG-150) using a single-sideband mixer (Marki Microwave
SSB-0618LXW-1) that suppresses the higher frequency sideband power by 23 dB.
The pulse length of this source is controlled by an absorptive solid state switch (Narda
SP123DHTS) which in turn feeds a high-power (41 dBm) microwave amplifer (AML
L0608-41-T479). Lastly, the microwaves travel through an isolator (Narda 4914) andto our double stub-tuned microwave antenna that was built in-house. . . . . . . . . . 25

xiv
2.10 Measured condensate fraction after corrections for the various effects listed in the
text. Fitting to Eqn. (2.5.3) with a scaling parameter for the horizontal axis returns
the solid curve for which predicts the transition temperature to be T = 1.025(6)(20)
Tc where Tc is defined in Eqn. (2.5.1) and the first and second errors are the statistical and systematic errors, respectively. The dashed line is a plot of the condensate
fraction for a non-interacting system (Eqn. 2.5.2). . . . . . . . . . . . . . . . . . . . 27
2.11 Plot of the molecule number measured using the bound K and bound Rb transitionsin the same experiment run. The data were fit to a line (solid line) with a horizontal
offset which gave a slope of 1.00(5) and a horizontal offset of 1.3(7) × 103 . If we fit
the data without an offset, the fit returns a slope of 0.83(3). For comparison, thedashed line is a line with slope of 1 and no offset. . . . . . . . . . . . . . . . . . . . . 28
3.1

Two-channel model of a Feshbach resonance. . . . . . . . . . . . . . . . . . . . . . . 31

3.2

S-wave scattering-length near a Feshbach resonance. . . . . . . . . . . . . . . . . . . 32

3.3

Binding energy, Ed , of a Feshbach molecule in the vicinity of the Feshbach resonance.
The zero energy axis represents the energy of the colliding atoms in the entrance channel 35

3.4Energy diagram for a Landau-Zener avoided crossing. The two bare states φ1 and φ2(bare-state energies 1 and 2 are red dashed lines) are coupled together at a systemparameter value η = η0 . Because of the coupling, φ1 and φ2 are no longer the
eigenstates; the eigenenergies of the coupled system are given by E1 and E2 (thickblack curves). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.5

Magneto-association of atom with zero relative kinetic energy. The magnetic field
begins with a < 0 where no weakly bound state exists. As the magnetic field isramped through the Feshbach resonance at B = B0 , atom pairs can be bound into
a Feshbach molecule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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3.6

Molecule creation measured after sweeping the magnetic field through the FanoFeshbach resonance. The molecules were produced from 3.4×105 Rb atoms and2.6×105 K atoms at T/Tc = 1.5 and T/TF = 0.6. To reduce the density, theatom clouds were released from the trap 0.4 ms before the magnetic field crossed
the Fano-Feshbach resonance. The data were fit to Eqn. (3.3.2) returning Γ =8.5(3) × 10−3 G/µs and f0 = 0.171(4), as defined in the text. . . . . . . . . . . . . . . 42

3.7

Initial molecule creation rate compared with theory from Eqn. (3.4.8). The initialmolecule creation rate in the limit of fast magnetic-field sweeps is plotted against the
average majority-species density, hn> i. For the Fano-Feshbach resonance parameters
given in the text, the theoretically predicted slope of Γ vs. hn> i is 2.99(12) × 10−15
G/µs/cm−3 (dashed line). The solid line is a weighted least-squares fit to the data,
which gives a slope of 3.06(13)×10−15 G/µs/cm−3 . The fit has a reduced chi-squared
parameter of 0.7.

3.8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Molecule creation efficiency measured in trap as a function of the initial atomic
T/TF . The molecules were formed from 1.2×105 Rb atoms and 1.6×105 K atoms.
The initial temperature was varied by parametrically heating the gas before sweeping
the magnetic field to create molecules. The solid line is the results of a least-squares
fit of the SPSS model to the five points with the largest initial atomic T/TF returning
γ = 0.352(6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.9

Measurement of the molecule temperature defined by their expansion energy as afunction of the initial atomic temperature. The solid curve is the simulation resultsof the SPSS model and the dashed line is a line with a slope of one. . . . . . . . . . 49

xvi
3.10 (a) Simulated molecule creation efficiency with an axial momentum offset between
the Rb and K clouds, and no added position offset other than gravitational sag.
The simulation was run for NK = NRb = 105 and T/Tc = 1.3 with the assumptionthat the K and Rb clouds were in thermodynamic equilibrium. For conditions in
the experiment, f < 0.2 and thus we conclude that slosh is a negligible concern.
(b) Measured molecule creation efficiency as a function of the axial offset between
the Rb and K clouds before molecule production. The atom clouds were separated
by applying an axial magnetic-field gradient that was abruptly turned off before
the magnetic-field was swept across the Feshbach resonance. The solid line is thesimulation results using the atom conditions and the dashed line is a fit of the data
to a gaussian.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.11 Simulated molecule creation after varying periods of ballistic expansion after releasing from a harmonic trap with aspect ratio λ = ωz /ωr = 0.0131. The atom clouds
were generated for NK = NRb = 3 × 104 with T/Tc = 1.5 and the assumption ofthermodynamic equilibrium between Rb and K clouds. Note that the discrepancy
in molecule creation efficiency between a) and b) is due to greater significance ofgravitational sag in the weaker trap of a). . . . . . . . . . . . . . . . . . . . . . . . . 53
3.12 a) Molecule fraction as a function of the K atom number to Rb atom number ratio
for fixed NRb = 105 and T/Tc =1.05. b) Average molecular potential energy per
particles (PE), kinetic energy (KE), and total energy (PE+KE) compared to themolecular Fermi energy as a function of NK /NRb . c) Molecular E/EF versus trap
frequency for fixed atomic T/Tc and T/TF . d) Molecular E/EF versus trap frequency
aspect ratio.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.13 a) Axial rms cloud size versus the Rb to K mass ratio. b) Axial rms velocity spreadversus the Rb to K mass ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

xvii
3.14 a) Axial rms cloud size for K atoms (blue triangles)), Rb atoms (red circles), andmolecules (violet diamonds) versus the frequency of the compressed K trap. b)
Axial velocity spread for K atoms, Rb atoms and molecules. c) Molecule conversion
fraction. d) Molecular potential energy (PE), kinetic energy (KE), total energy(PE+KE), and Fermi temperature (TF ). e) Ratios of PE, KE, and (PE+KE) to the
Fermi temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1

Schematic of the binding energy of the lowest two Efimov states (solid orange curves)plotted along with the dimer binding energy (dashed black curve) for s0 = 0.65317.
The inset is a magnified view of the origin showing where the lowest Efimov state (n(0)= 0) reaches the atom threshold (a = a− ) as well as the entirety of the next-to-lowest
Efimov state(n = 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2

Three-body recombination event rate coefficient as a function of scattering length.
The solid line corresponds to a value for the inelasticity parameter η∗ = 0.01. The
dashed and dash-dotted correspond to η∗ = 0.1 and η∗ = 1, respectively. Measured
values of η∗ span the range from 0.016[165] to 0.251[65]. . . . . . . . . . . . . . . . . 67

4.3(a) Diagram of the magnetic-field sequence for measuring α. The magnetic fieldstarts far from the Feshbach resonance and is ramped towards resonance where itis held for a duration thold . After the hold, the remaining atoms are released fromthe optical trap (vertical dotted line) and the magnetic field is ramped away fromresonance. (b) Measurement of α at 548.695 G (a = -460 a0 ) with initial in-trap
density-weighted densities of 1.2 × 1013 cm−3 and 3.3 × 1012 cm−3 for Rb and K,
respectively. The solid lines are from a dual-species fit described below. The fit
returned α = 15.3(6) × 10−27 cm6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.4

Measured three-body event rate coefficient, α, as a function of magnetic field. . . . . 70

xviii
4.5

Comparison of negative scattering length data from Fig. 4.4 to a fit of Eqn. (4.2.10)(solid curve) and a plot of Eqn. (4.2.10) assuming η∗ = 0.017 (dash-dotted curve).
(1)

The fit was restricted to |a| < 890a0 with a− = −3 × 104 a0 and returned η∗ =0.0167(7).
4.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Axial cloud sizes for the Rb (red circles) and molecule (violet diamonds) clouds as afunction of the hold duration in Fig. 4.7a.

4.7. . . . . . . . . . . . . . . . . . . . . . . 71(a) Magnetic-field sequence for measuring atom-molecule collisional loss. First molecules
are generated by sweeping slowly through the Fano-Feshbach resonance (dash-dottedline). Next, unwanted remaining atoms are heated out of the trap with resonant light.
For removing K, we merely apply light resonant with |9/2, −9/2i − |11/20 , −11/2i,
but for Rb we ARP to the |2, 2i hyperfine state and apply light resonant with|2, 2i − |30 , 3i. The magnetic field is then ramped to another field and held for a duration thold after which optical trap is extinguished. While the atoms and molecule
are expanding, the molecules are dissociated by ramping back across the resonance.
(b) Measured Rb (red circles) and molecule number (violet diamonds) after holding
at 545.14 G (a = 200 a0 ) with an initial temperature of 450 nK and a Rb densityweighted density of 2 × 1012 cm−3 . The molecule number is determined from imaging
K atoms after dissociation. The measured Rb number after the dissociation of themolecules is the sum of the molecule number and Rb atoms that were never bound,which is the reason for the 2:1 Rb number to molecule number loss.

. . . . . . . . . 73

xix
4.8

Measured atom-molecule collision event-rate coefficient, β, for various collision partners with KRb molecules. The data shown as circles correspond to collisions between
molecules and Rb atoms in the |1, 1i hyperfine state. The Rb-molecule collision canbe influenced by Efimov physics, unlike the resonant K-molecule collisions, shown intriangles. The resonant feature also is not seen in collisions between distinguishable
atoms (K atoms in |9/2, −7/2i) and molecules, shown as squares. The open circles,
squares, and triangles occur at scattering lengths where molecules can additionallydissociate due to their finite temperature. . . . . . . . . . . . . . . . . . . . . . . . . 74

4.9

Replotting of the data from Figs. 4.4 and 4.8 in terms of scattering length. The
scattering length was calculated by averaging the scattering lengths produced by the
Feshbach parameters of Refs. [86] and [117].a) Atom-molecule event rate coefficientfrom Fig. 4.8. The Rb+molecule collisions indicate a resonance centered at 250 a0 .
b) Overlaid over the three-body recombination data are predictions of α using theuniversal relations (Eqns. (4.1.11) and (4.1.12)) to relate a− and a+ to a∗ . The solid(2)

curves assume that the atom-molecule feature in a) is a∗and the dashed curves(1)

assume the feature is a∗ . The data near the d-wave Feshbach resonance have been
omitted from the plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77(−)

4.10 Plot of the data from Fig. 4.9b with the universal theory for αd(Eqn. (4.2.10))

assuming a− = −246 a0 as in the 87 Rb – 87 Rb – 41 K reported in Ref. [7].

. . . . . . 79

xx
4.11 a) Quantum Defect calculation of the scattering length and effective range, reff , for
40 K in the |9/2, −9/2i hyperfine state and 87 Rb in the |1, 1i hyperfine state courtesyof Brandon Ruzic and John Bohn [136]. The scattering length plot shows both thebroad s-wave Feshbach resonance as well as a narrow d-wave Feshbach resonance at547.35 G. The effective range exhibits divergences at the two zero crossings of thescattering length. The divergence in reff at the s-wave resonance location is perhaps
a numerical artifact. For a length-scale comparison, ±rvdW = ±71.9a0 is shownas the horizontal dotted lines. b) Ratio of the effective range term (1/2 reff k 2 )inthe effective range expansion of k cot δ0 defined below to the scattering length term(-1/a) at 250 nK.
5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

View of the ‘Science cell’ end of the vacuum chamber with the IP trap removed.
The Science cell (small cyan square) is flanked by the Quadrupole trap coils (orange
rectangles) which restricts the size of the new IP trap. . . . . . . . . . . . . . . . . . 84

5.2

Simplified diagram of a Ioffe-Pritchard trap where the arrows indicate the directionof current flow through each coil. Axial curvature is provided largely by the pinch
coils (red coils) and the bias coils (blue coils) serve to largely cancel the offset field
produced by the pinch coils. Radial confinement is provided by the radial quadrupolar field produced by the Ioffe bars (horiz. arrows). . . . . . . . . . . . . . . . . . . . 85

5.3a) Diagram for the rectangular coil used to model the Ioffe bars where the unit
vector ŷ points out of the page and the current flows in the direction indicated by
the arrows. b) Diagram of the winding tilt model showing the individual loops edgeon (solid lines) for two layers of windings. As a consequence of the tilt, the magnetic
field from each layer acquires a vertical component which does not completely cancel
when the coil is considered as a whole. . . . . . . . . . . . . . . . . . . . . . . . . . . 88

xxi
5.4

a) - c) Cross-sections of the calculated magnetic field magnitude for the operating
currens Ibias = 193.75A, Ipinch = 186A, and IIof f e = 186A. The field from the bias
and pinch coils we calculated from Eqn. 5.1.6 and the field from the Ioffe bars arecalculated from Eqns. (5.1.8-5.1.10), which includes the contribution from the endsof the Ioffe bars. The scale for the magnetic field in G is shown in c). d) A simplified
diagram of the coils showing the coordinate system. In this diagram the bias coils
are shown as the gold rings, the pinch coils are the lavender rings, the Fast B coils
are red rings, and the Ioffe bars are the salmon square rings. . . . . . . . . . . . . . . 92

5.5

Definition of the dimensions of a helix with pitch p, coil diameter D, and inner wetted
diameter d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.6

Diagram of the test setup for measuring the water pressure drop and fluid flow
through a test coil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.7

Measured pressure drops and flow rates through the test setup. Initially the ends oftest coil had burrs left after cutting the copper tubing. After filing the burrs, themeasured pressure drops more closely follow the phenomenological theories of Ali [1]and Ito [81]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.8a) Setup for measuring heat transfer from the coil to the cooling water. Thermocouples (indicated by Ti ) were attached to brass tubing connectors, and covered withinsulation, to measure fluid temperature. The temperature increase of the fluid wasmeasured from T4 − T3 , which were isolated thermally from the coil with a segmentof plastic tubing, while passing current through the test coil and measuring the fluid
flow rate with the flow meter. b) Measured fluid temperature increase versus thepower/flow rate ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

xxii
5.9

a) CAD rendering of the vaccuum chamber with the new trap installed and alignedwith the Quadrupole coils on the cart track. b) Photograph of the assembled IoffePritchard trap with the visible coils labeled. The trap structure is made from Garolite
G-10, which is the light green material in the photograph. The coils are made from
Kapton coated square tubing (the amber-colored tubing in the photo), which ismated to standard round tubing by crimping and soldering the tubes together andthen the round tubing is covered in heat shrink resulting in the black tubes shownin the foreground.

6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Interspecies scattering length for the 545.9G Feshbach resonance between K in the|9/2, −9/2i hyperfine state and Rb in the |1, 0i state (a90 , dash-dotted curve) andthe 546.618 G resonance between K in the |9/2, −9/2i and Rb in the |1, 1i state (a91 ,solid curve). In the region between the vertical lines, a90 is the dominant interspecies
scattering length, whereas to the right of the right-most vertical line, a91 is dominant. 105

A.1 Boxes in position and momentum whose probability of occupation, Ψ, is shown. . . . 119
A.2 Comparison of a Bose distribution generated with the Monte Carlo (red points)
with the ideal Bose gas distribution (black curve). The distributions were generate
for Nboson = 105 and T/Tc = 1.05 for a harmonic trap with radial trap frequency
ωr = 2 π 448 Hz and trap aspect ratio λ = 0.013. The momentum and spatial density
distributions are the result 24 runs of the Monte Carlo simulation with maxp = 0.05.
c) and d) Fractional residuals for a) and b), respectively, where the fractional residual
is defined as the Monte Carlo result divided by Eqn. (A.4.1) or (A.4.2) minus 1. . . 123

xxiii
A.3 Comparison of a Fermi distribution generated with the Monte Carlo (blue points)
with the ideal Fermi gas distribution (gray curve). The distributions were generate
for Nf ermion = 105 and T/TF = 0.11 for a harmonic trap with radial trap frequency
ωr = 2 π 627 Hz and trap aspect ratio λ = 0.013. The momentum and spatial
density distributions are the average of 24 runs of the Monte Carlo simulation withmaxp = 0.05. c) and d) Fractional residuals for a) and b), respectively. . . . . . . . . 125
A.4 Checking the Virial theorem for Bose and Fermi distributions generated by theMonte-Carlo code. Each point is an average of the ratio hU i / hKi for twenty distributions at a given temperature. The error bars at the standard error of the mean. . 126
A.5 a) Fitting results where the system was scaled such that the product ω 3 N was kept
constant. b) Fitting results for the model where the system was scaled such that the
product N λ was kept constant. The vertical dashed line indicates where the aspectratio λ = 1. c) Surface area to volume ratio for a spheroid.

. . . . . . . . . . . . . . 128

A.6 a) Simulated molecule fraction calculated by varying N and λ such that the productNλ was fixed. b) Molecule fraction calculated by varying the atom number with allother input parameters fixed. c) Attempt to calculate the molecule fraction in thethermodynamic limit where N→ ∞, ω → 0, but Nω 3 is constant. . . . . . . . . . . . 129

Chapter 1

Introduction

The matter that we interact with in our day-to-day lives (which, as a whole, astoundingly
constitutes only 4.6 % [91] of the matter-energy in the universe1 ) can be classified into two categories2 according to the statistics of the particles. The first class of particles, which possess
half-integer spins, obey Fermi-Dirac statistics [58, 44] and are named fermions. Belonging to thisclass are electrons, protons, neutrons, and all atoms with an odd number of neutrons (e.g. 3 He,
6 Li, 40 K, and 87 Sr). A consequence of Fermi-Dirac statistics is that two identical fermions cannot

occupy the same quantum state. This restriction, called the Pauli exclusion principle [125], createsa pressure in the system and is responsible for preventing neutron stars from collapsing to a blackhole and allowing for chemistry and life as we know it. In contrast, the second class of particles
called bosons, obey Bose-Einstein statistics [16, 51, 52], which imposes no such restriction. To thisclass belong the force-carrying particles such as photons, W bosons, Z bosons, as well as composite
bosons (i.e. atoms with an even number of neutrons). Examples of bosonic atoms include 4 He, 7 Li,
23 Na, 87 Rb, and 133 Cs.

For an ensemble of atoms, the statistics of the atoms only comes to the forefront when theensemble is cold and sufficiently dense (sufficiently dense for our atom clouds is still 5-6 ordersof magnitude less dense than air) such that the extent of the atoms’ wavefunctions becomes oforder of the interparticle spacing. This criterion is expressed in terms of the phase-space density
1
2

Discovery and explanation of the remaining 95.4% will be left as an exercise for the reader.
In reduced dimensionalities quasi-particles which do not belong in either category are theorized to exist [97].

2
defined as D = n λ3dB , where λdB is the thermal de Broglie wavelength of the atom3 As theatoms’ wavefunctions begin to overlap, the statistics of the particles becomes important as it
dictates how the ensemble wavefunction is symmetrized. When the ensemble reaches D ∼ 1,many interesting phenomena are laid bare. For bosons, a macroscopic number of atoms pile into
the ground state creating a coherent form of matter know as a Bose-Einstein Condensate (BEC),
which was theoretically predicted in 1924 [16, 51, 52] and then realized experimentally in 1995 [3, 31]
in ultracold atomic gases. In following years, researchers were able to demonstrate the coherence
and long-range correlations of this new form of matter [4] as well as the production of vortices[106] with quantized circulation and shortly thereafter superfluidity in a BEC of a Na atomic gas[118, 131]. These discoveries with weakly interacting BECs further cemented the utility of atomicgas experiments as they were able to delve into idealized aspects of physics of superfluid liquid 4 He
without the strong interactions inherent in 4 He that make it theoretically challenging to describe.
Similarly, atomic Fermi gas experiments showed similar excitement with the production ofa degenerate (large value of D) Fermi gas [34] followed by the creation of degenerate bosonicmolecules by binding two fermions together [144, 61], and later the mapping of the BCS-BEC
cross-over [133], which is a theory that supplies a possible interpretation for the pseudogap phasein high-Tc superconductors [22]. These early fermion experiments relied upon cooling two spinstates of the fermions to ensure that the atoms always have a collision partner and can thermalize.
However, another method to cool the fermions emerged: evaporate a bosonic atom gas in contact
with the fermionic gas and sympathetically cool the fermionic gas [148, 139]. While the Bose-Fermi
mixture was initially another method to create a degenerate Fermi gas, they became interesting intheir own right.
hwhere h is Planck’s constant, m is
2π m kB T
the atomic mass, kB is Boltzmann’s constant, and T is the temperature of the atoms.
3

The thermal de Broglie wavelength [124] is defined as λdB = √

3
a(B)

E

Closed channel
ECC

B-B0

Entrance channel

Internucleus separation r

(a)

(b)

Figure 1.1: a) A simplified model of a Feshbach resonance where atoms colliding in the entrance,or open, channel threshold are brought into degeneracy with a bound state in the closed channel
where the energy of the colliding atoms differs from the bound state energy by Ecc . b) A plot of
the scattering length as a function of magnetic field near a Feshbach resonance located at B = B0 .
The resonance width is characterized by ∆, which is the difference in magnetic field between thefield at which a = 0 and B0 .

1.1

Bose-Fermi interactions
Many of the experiments since the first generation of quantum degenerate atom gases have

involved the controllable interactions possible in these systems. The source of this novel control
are Fano-Feshbach resonances, which are scattering resonances most commonly controlled with amagnetic field. Fano-Feshbach resonances were discovered first in Bose gas systems [78, 26, 154]and later in Fermi gas systems [100, 40, 115, 82] and in 40 K – 87 Rb Bose-Fermi mixture [79]. These
Fano-Feshbach resonances occur when a molecular state (denoted the ‘closed channel’ in Fig. 1.1a
) that can couple to the threshold collision state of the atoms (denoted the ‘open channel’ in Fig.
1.1a) is tuned into degeneracy with the threshold state. The introduction of a new bound state in
the system causes a divergence of the scattering length, which, at ultracold temperatures, is often
the sole parameter characterizing the interaction strength between two atoms.
Near a Feshbach resonance, the scattering length has the form a = abg [1 − ∆/(B − B0 )] [109]where abg is the background scattering length (i.e. the scattering length far from resonance), ∆ thezero-crossing resonance width, and B0 is the resonance location. A plot of the scattering length isshown in Fig. 1.1b for ∆ < 0 and abg < 0. As can be seen, the scattering length can take positive

4and negative values, which correspond to repulsive and attractive interactions, respectively. As thedivergence in the scattering length is due to the presence of a molecular state of the appropriate
character to interact with the open channel, there is also an associated two-body bound state thatcan be populated by the atoms from the open channel.
When the search for Feshbach resonances commenced in heteronuclear Bose-Fermi systems, it
was noted that with overlapping Feshbach resonances involving the same boson but different fermion
states, these systems could emulate boson-mediated Cooper pairing [14, 70, 50]. Additionally, thepossibility of producing ground state polar molecules was raised [6, 167, 30], which were ultimately
created and spurred a new field of inquiry. However, the first step of the creation of ground-statepolar molecules required the creation of Feshbach molecules.

1.1.1

Two-body bound states

The new two-body bound state that appears near a Fano-Feshbach resonance is called the
Feshbach molecule state and only exists on the side of the resonance with positive scattering length.
Near the Feshbach resonance, the Feshbach molecule has a binding energy Ed = ~2 /2µ2b a2 whereµ2b is the reduced mass of the two atoms in the molecule. Feshbach molecules were first generated
in Na [169] by sweeping the magnetic field from the negative to the positive scattering lengthside of the resonance. Feshbach molecules were also created in 6 Li [144] and 40 K [134], where a
Bose-Einstein condensate was formed when starting with a strongly degenerate Fermi atom gas[61]. A phenomenological model (later called the ‘Stochastic Phase-Space Sampling’ (SPSS) model[164]) was developed to explain the governing physics in molecule formation for adiabatic magnetic
field sweeps [75]. The basic assumption was that in order for an atom pair to efficiently associate
into a molecule, the atoms would need to be ‘close enough’ in relative phase space because theadiabatic magnetic field sweep, by its very nature, would not change the occupation in phasespace. The cutoff for atoms being ‘close enough’ was determined by the sole fitting parameter inthe model. This model was applied to formation of 85 Rb2 Feshbach molecules and 40 K2 molecules,
where experiments demonstrated that the Feshbach molecule formation efficiency (i.e. the ratio of

5
molecules to atoms) was governed solely by the phase-space density of the constituent atoms [75].
Furthermore, the fitting parameters for the 85 Rb molecules and the 40 K molecules agreed, hintingthat this model could be applicable also to heteronuclear molecules [75].
Heteronuclear molecules were first created in 40 K – 87 Rb [119] and later in many other systems as the desire for other polar molecule candidates surged 4 . The SPSS model was applied to
85 Rb −87 Rb [123], 40 K – 87 Rb [174], and 6 Li −40 K [142] using the parameter values from Ref. [75]and found to accurately describe the molecule creation efficiency in the limit of slow magnetic fieldsweeps. However, unlike in the homonuclear molecules, the creation efficiency for 40 K – 87 Rb isquite small for the typical conditions used at the time. An important question was whether or notthere was a fundamental limit for fermionic Feshbach molecule creation efficiency and also whether
a higher phase-space density of molecules could be made with the available conditions. A separate
question was whether the molecule creation efficiency in the perturbative limit(the limit of fast
sweeps) followed the predictions developed for single-species gases [25].

1.1.2

Three-body bound states

Two-body bound states are not the end of the story, though. In 1970 Vitaly Efimov predicted,
for resonantly interacting systems5 , that three-body bound states would also exist [48]. Unlike thetwo-body case, there was predicted to be a semi-infinite ladder of states bounded by the energy
corresponding to the length scale of the two-body potential. The three-body bound states also
were predicted to remain bound even for negative scattering lengths where the two-body systemdoes not support a bound state. The three-body bound states exhibit a discrete scaling symmetry
between the different ‘rungs’ of the ladder of states (see an example of discrete scaling in Fig. 1.2a).
The scattering lengths at which these bound states appear, a− , demonstrate this scaling through(n+1)

the relation between adjacent bound states a−(n)= ζ a− , where ζ is the scaling factor [48]. The

three-body binding energies also exhibit the discrete scaling as seen in Fig. 1.2b, where the solid
4

To date heteronuclear Feshbach molecules have been reported in 85 Rb −87 Rb [123],6 Li −40 K [150, 142] ,Rb −133 Cs [146], and 23 Na −40 K [168]5
These are systems for which the scattering length is large in magnitude compared to the two-body potential
length scale.
87

6Energy
1/a

(a)

(b)

Figure 1.2: a) A series of discretely scaled leaves with a scaling factor of 1/2 between leaves. b)
Energy levels for the three-body system shown as a function of the two-body scattering length.
The three-body bound states predicted by Efimov are shown as the thick solid curves. The threebody bound states begin at the three-atom collision threshold (Energy = 0) at negative scattering
lengths and then asymptotically approaches the atom+molecule threshold given by the moleculeenergy (dashed line). The scattering lengths at which the three-body states reaches the three-atomthreshold are related by a discrete scaling factor, which was artificially chosen to be two for thisdiagram.

curves are the different trimer states, the dashed line is the two-body bound state energy, and thezero energy level corresponds to the three-atom collision threshold energy.
Although Efimov originally derived this result in the context of bound states within nuclei6, they were first observed in an ultracold Cs gas system in 2005 [92]. Observation of these trimer
states was made from the influence of the trimer states on three-body inelastic collisions: at scattering lengths corresponding to where the trimer binding energies meet the three-atom collisionthreshold, a− , there is a resonant enhancement in three-body recombination. Following this was
the observation of resonances in a gas of 7 Li [64, 129] and in a three spin-state mixture in fermionic
6 Li [122, 165, 77]. Efimov resonances were observed in a gas of 39 K [170], where the discrete scalingof the Efimov states was confirmed. In 6 Li [101] and 7 Li [105] the trimer binding energy was directly
measured by rf-association into the trimer state and observing the rapid loss.
One looming question in this line of research was ‘what sets the location of the lowest Efimov
6Specifically three-nucleon bound states within 3 H and three α-particle bound states within 12 C.

7resonance’ ? Initially it was thought that the three-body parameter, which sets the location ofa− , is a mixture of two-body and three-body physics and is not knowable a priori for a particularatomic system. Experiments in 7 Li located Efimov resonances in two different spin states whose
locations agreed [66]. Furthermore, in 133 Cs, four Efimov resonances were found using a variety
of Feshbach resonances (two were a consequence of a g-wave resonance that couples to the s-wave
scattering length) [12]. The locations of these four Efimov resonances clustered about a meanvalue of a− = −9.1(4) rvdW [12] (the van der Waals lengths, rvdW , is the characteristic length scale
for the two-body potential) with a spread that was substantially smaller than the scaling factor
between different Efimov resonances in the same series [12]. This suggested that the location of thefirst Efimov resonances were universally determined for broad Feshbach resonances such as thoseused in 133 Cs. The ansatz was made that the location of the first Efimov resonance was universally
determined by the length scale of the two-body potential, a view which was corroborated by Efimov
resonances in 6 Li, 85 Rb, and 7 Li, where the locations were also close to -9.1 rvdW [23]. Theoreticalexplanations of this observation soon followed for the case of broad Feshbach resonances [155]. With
the observation of Efimov resonances in the heteronuclear Bose-Bose system 41 K – 87 Rb [7, 8], it
is natural to ask the question does a similar universality apply to heteronuclear systems as well.

1.2

Contents of this thesis
In this thesis I address the questions described above with measurements in a ultracold

Bose-Fermi mixture of fermionic 40 K atoms and bosonic 87 Rb atoms.
Chapter 2 discusses the apparatus and the process by which we create the ultracold mixture.
Two previous theses discussed this experiment in detail, and so I make the discussion brief, but
include discussion of the various upgrades and improved understanding that have happened along
the way.
Chapter 3 contains our work on Feshbach molecule creation. Following a short overview ofthe physics behind Feshbach resonances, I discuss our work to understand molecule creation in
the perturbative limit, where the magnetic field is swept quickly through the resonance, and in

8
the saturated limit, where the magnetic field is swept slowly enough that the molecule number
has saturated. Additionally, we look at the expansion energy of the molecules, and the chapter
concludes with predictions on how to create molecules with the minimum temperature and T/TF .
Chapter 4 discusses our efforts to measure Efimov resonances in the 40 K – 87 Rb systemthrough measurements of inelastic collision rates. In particular, we compare our results with the
published 41 K – 87 Rb resonances and address the question of universality of the three-body parameter in heteronuclear systems. The chapter begins with a overview of the relevant physics anddetails how we analyze our data to extract the three-body event rate coefficients. Our measurements of three-body recombination do not uncover any Efimov resonances, unlike in 41 K – 87 Rb,
but we do find a feature in Rb+molecule collisions. The chapter concludes with a comparison of
the 40 K – 87 Rb and 41 K – 87 Rb systems to address possible reasons as to why the Efimov spectra
are so strikingly different.
In Chapter 5 I discuss the design and construction of a new Ioffe-Pritchard magnetic trap forour experiment. I discuss basic scaling laws for coils to motivate the coil geometry as well as thevarious physical restraints imposed on the design by the apparatus. The size of the trap coils wasmore restricted by the apparatus compared to others in JILA, and so it necessitated using smaller
diameter tubing to build the magnets than used previously. This poses the question of whether
or not water flow sufficient to cool the magnets is possible, and so I discuss the phenomenologicalmodels used to describe fluid flow through helical tubes as well as some sample data to verify theequations. The chapter ends with construction details and other information about the trap.
Chapter 6 ends the thesis with a conclusions and offers possible future directions for theexperiment.

Chapter 2

Creating an ultracold Bose-Fermi mixture

While it is often taken for granted in the day-to-day operation of experiments such as thisone, it is an amazing feat to be able to catch a cloud of atoms with sufficient isolation fromthe room-temperature environment to be able to produce a sample so cold and dense that thequantum mechanical nature of the sample is brought to light. Through steady efforts to advancethe field technically and scientifically, we, as a field, can delve deeper and explore more complicatedphenomena. The subject of this chapter is the process by which we generate our ultracold mixtureof bosons and fermions as well as the various improvements we have made to the apparatus along
the way. In Sec. 2.1 we discuss the upgraded laser setup, which is an integral part of any modern
atomic physics lab. Following in Sec. 2.2, we discuss briefly how our atom clouds are generated
and in Sec. 2.3 we discuss how we characterize the performance of our fast magnetic coils system,
which we rely upon for the measurements presented in Chapters 3 and 4. The chapter ends with a
discussion of a few improvements we have made to our imaging and state control in Sec. 2.4 anda calibration of atom number in Sec. 2.5 using the BEC transition for Rb number calibration, andcomparing Rb and K atom numbers in bound molecules to calibrate the K atom number.

2.1

New Rb and K laser setups
Nearly all experiments like this one begin and end with lasers. Diagrams of our setup andfrequency locking schemes for K and Rb lasers are shown in Fig. 2.1. The K laser system has beenrevamped since Michele Olsen’s thesis [116] to eliminate the need for the K trap slave laser, which

10
was run at −40◦ C, and to replace the K trap master, which was run at −10◦ C. The K slave laser
was particularly painful to use because of the need to warm the laser up to room temperature toclean out ice and then cool the laser back down slowly such that the diode does not stress fracture.
In their stead, we have a K trap External Cavity Diode Laser (ECDL) and a K repump ECDL,
which both use the AR-coated diode from Eagleyard (EYP-RWE-0790-04000-0750-SOT01-0000).
These two lasers supply the power to directly inject the K tapered amplifier (K TA in Fig. 2.1,
Eagleyard EYP-TPA-0765-01500-3006-CMT03-0000) as well as for the various probe and opticalpumping beams. We find that the K repump to trap power ratio for the MOT needs to be at least
1:2. This is in rough agreement with what one would expect from off resonant transitions to otherexcited states due to the trap laser. We run these lasers at a balmy 17◦ C, which is both below theroom temperature and above the dew point of the majority of the year. Although these lasers area definite improvement over the old setup, their mode-hop free tuning range does degrade over thetime scale of a few years. This seems to indicate that the AR coating on the diodes degrades for K
frequencies, which is a view supported by the manufacturer.
The Rb laser system has been partially upgraded as well with the replacement of the Rb trap
master with a laser that uses the same AR coated diode as the K lasers1 . The Rb trap mastersupplies the power for the optical pumping and probing (including the new Rb high B imaging
system with the 400 MHz double-passed AOM (Brimrose TEF-400-100-.780) in place of the triple
pass system [116]) as well as injection locking the Rb slave laser. The Rb slave laser, in turn injects
the Rb tapered amplifier (Rb TA in Fig. 2.1). The other new laser in the setup is the Rb repump
DBR (Distributed Bragg Reflector) from Photodigm (PH780DBR). Over the several months of use
we have had from this laser, it has proven to be remarkably robust and rarely falls out of lock2 .
1

The mode-hop free tuning range of this laser curiously does not degrade as do the K trap and repump lasers.
This stability is due in part to the feedback the diode receives from the Bragg grating built into the diodestructure itself. However, we find that the laser requires at least 60 dB of optical isolation to block light that travels
back to the laser from the TA and various reflections2

11
69 MHzRb trap

Rb low-1+2

MOT: 73.5 MHz+1
CMOT: 96.2 MHz
Rb repump

M

hi

0

to Rb
opticalpumpingHgh z

to MOT

Rb TA

to Rb
opticalpumping

Rb

Rb trapslave+2

lock to f=1 to
f’ =1,2 crossover

to Rb low
B imagingto Rb highB imaging

K TA

K repump
ECDL

Rb ZP

+2
lock to f=2 to
f’ =2,3 crossover

Rb repumpDBR

105.3 MHz38

Rb trap
master ECDL

C/MOT: 55.7 MHz
molasses: 49.1 MHz

to MOT

212.2 MHzK sat.
abs.
+2

to K
opticalpumpingK
243 MHzhigh B

offset lock:
lock to
MOT: δf = 796.1 MHz 39K f=2
CMOT: δf =809.8 MHz

K trap
ECDL

193.8 MHzK trap+2
to K high B
imaging-1

ZP: 128.3 MHz

to K
opticalpumping+2

K ZP/probe
probe: 112 MHz

to K low B
imaging

Figure 2.1: Diagrams of Rb and K laser setup and locking scheme. Light is supplied by the various
lasers and tapered amplifiers (TA) and the light frequencies are shifted by acousto-optic modulators(AOM)(yellow squares). All of the light for the MOT and all probe beams are coupled into opticalfibers (black loops) to be sent to the apparatus, whereas the optical pumping beams are propagatedthrough free-space. Frequencies for the AOMs and the offset lock are shown for the various stagesof the experiment.

12

2.2

Bose-Fermi mixture: onward and downward to the ultracold regime
The details of how we generate an ultracold Bose-Fermi mixture from the atomic vapor in

our vacuum chamber is discussed in John Goldwin’s thesis [60] and Michele Olsen’s thesis [116],and so are only briefly summarized here. A CAD rendering of the apparatus is shown in Fig.
2.2. The experiment begins by slowing a portion of the atomic vapor in the collection cell usinga magneto-optical trap. The atomic vapor comes from a commercial Rb alkali dispenser and ahome-made K alkali dispenser [35] (the two ‘rabbit ears’ on the collection cell) wherein an electricalcurrent drives a redox reaction which in turn releases alkali into the collection cell. The MOT isable to capture ∼ 109 Rb atoms and ∼ 1.4 × 108 K atoms, with the higher power produced by the
K TA (¿100 mW after fiber coupling), compared to ∼ 5 × 107 with the old K slave laser (≈40 mW
after fiber coupling).
The next stage involves loading the atom clouds into the Quadrupole trap (Q-pole trap),
whose coils are the same as used to supply the magnetic-field gradient for the MOT. In order toreduce the amount of potential energy the atoms would gain from the load into the Q-pole trap,
we must reduce the sizes of the Rb and K clouds before releasing the MOT. The Rb cloud iscompressed by reducing the radiation pressure due to radiation trapping of the trap light. This isaccomplished by allowing Rb atoms to go ‘dark’ to the trap light by reducing the intensity of therepump light as well as detuning the repump light further from resonance. Following the releasefrom the MOT is a stage of Doppler cooling for K, a stage of molasses cooling for Rb, and then
optical pumping to pump Rb into the |2, 2i state and K into the |9/2, 9/2i state. The atoms are thenloaded into the nearly full-strength Q-pole trap (147 G/cm in direction perpendicular to the planeof the coils), whose coils are mounted on a movable cart. The cart is attached to a motor-driven
linear positioning stage whose motion is governed by Aerotech’s Soloist HLe 10-60-A controller 3. The cart transports the Q-pole trap to the science cell in 1.25 s for the roughly 1 m of travel.
3
This new controller has a linear power supply to drive the motor, as well as optically isolated inputs. Whereas
the old cart controller nearly brought the experiment to its knees due to the amount of rf it polluted the lab with,this new controller is whisper quiet in the rf.

13

TOP VIEW

SciencecellCollection cell

Q-pole trap
coils

Cart track

Figure 2.2: CAD rendering of the apparatus without the Ioffe-Pritchard trap (IP trap), whichsurrounds the science cell. At the start of an experiment, the Quadrupole-trap coils (Q-pole trap)
are located at the collection cell where they supply the magnetic-field gradient for the MOT. Afterthe MOT stage, the atom clouds are compressed and cooled and then loaded into the Q-pole trap.
The Q-pole trap coils are mounted on a movable cart (cart track is 1.2 m long) which whisks themaway to the science cell where the atoms are loaded into the IP trap for further cooling.

Before the atoms enter the science cell, the magnetic-field gradient is lowered to 29 G/cm (30.6
G/cm is necessary to support |2, 1i and |1, −1i atoms at rest) for 200 ms to spin purify the clouds
such that only the Rb atoms in the |2, 2i state are supported against gravity; K in the |9/2, −9/2i,|9/2, −7/2i states also remain trapped.
Once in the science cell, the atoms are transferred into a Ioffe-Pritchard [80, 130] magnetic
trap in which the bulk of the cooling is performed. In this trap, the Rb is cooled by forced rf
evaporation and K is cooled sympathetically through collisions with Rb. The process begins with
5 × 108 Rb atoms and 4 × 106 K atoms4 at 600 µK and ends with 1 − 2 × 106 atoms of each species
at 1 µK (fradial = 122 Hz, faxial = 25 Hz). Since the atomic spin states that are needed for the work4

Because of the low density of K atoms, it is difficult to measure the K atom number at early stages in our
experiment; this number is based on what is measured after the evaporation has commenced.

14
described in Chapters 3 and 4 are not able to be trapped magnetically, the atoms are transferred
to an far off-resonance trap (FORT), which, by the nature of the large detuning from the atomic
transitions, is insensitive to the hyperfine state of the atom. This trap is formed from the focus(waist = 22 µm) of a single beam from a laser that produces light at 1090 nm (SPI SP-10C). To load
the atoms into the FORT, the IP trap current is slowly ramped off in 100 ms after the FORT is at
its full load strength. This new loading procedure results in substantially less heating than the oldprocedure where the IP trap was abruptly shut off5 . Next, Rb and K are transferred into the |1, 1i
and |9/2, −9/2i spin states via adiabatic rapid passage (ARP) pulse sequences discussed below inSec. 2.3. The atoms then undergo the last stage of cooling as the large bias field 6 ramps close to
the field needed for the work described in Chapters 3 and 4. To further cool the atoms, the FORT
power is lowered, allowing hot atoms to spill out of the trap. To bias the evaporation towards losing
more Rb in order to preserve as much K in the FORT as possible, a small magnetic-field curvature(from the pinch coils of the IP trap) is added that preferentially enhances the trapping frequency,
and thus trap depth, of K compared to Rb, which in |1, 1i has half the magnetic moment that K
in the |9/2, −9/2i state possesses. At the end of the evaporation, the FORT power is ramped up
by a factor of ∼ 2 to ensure that the atom temperature is not at the trap depth. This results in asample of several 105 of each species at 200-500 nK in a trap with axial and radial trap frequenciesof 5 Hz and 420 Hz, for Rb, and 6 Hz and 550 Hz for K.

2.3

Characterizing the Fast B coil circuit time response
The next stage of our experiment utilizes a low inductance coil pair, called the Fast B coils,

to rapidly change the magnetic field on microsecond to tens of milliseconds time scales. The Fast
B system is integral to the experiments detailed in Chapters 3 and 4, which required the ability to
rapidly sweep the magnetic field through a magnetically-tunable scattering resonance, known as a5Previously atoms were heated all the way to the trap depth (4 µK) with the old loading procedure. By ramping
the IP trap off slowly, the load temperature is 2.3 µK, which is nearly what one would expect due to adiabatic
compression alone (1.8 µK).
6This field is provided by the Q-pole trap coils with one of the coils having its current direction switched.

0 .6 0tr a n s fe r fu n c tio n a m p litu d e

0 .5 7
0 .5 4
0 .5 10 .1

0 .0 1
1

0 .1
0 .0
-0 .1
-0 .2
-0 .3
-0 .4
-0 .5

1 0
1 0 0
fre q u e n c y (k H z )
(a)

-4 0

0

2 8 0

3 2 0

s e rv o c o n tro l (V )1s e r v o H a ll p r o b e ( V )

15

3 6 0

t i m e ( µs )(b)

Figure 2.3: a) Calculated closed-loop transfer function for the Fast B servo from measurements of
the input waveform and measured response. The bandwidth of the servo, as measured by the -3dB
frequency, is 39.2 kHz (dashed line). The peaks in the transfer function are likely resonances inour Hall-effect current transducers (LEM Ultrastab-866). b) Measured servo control voltage (lower
plot, denoted f(t) below) and servo Hall-effect current transducer (upper plot, denoted g(t) below).
Both plots have been undersampled from the actual data by a factor of 100. Note that the servo’s
small response compared to the control voltage is a signature of its limited bandwidth.

Fano-Feshbach resonance [78]. Our system has the added complication of strong eddy fields (withdecay times of order 1 ms) due to the copper cooling blocks onto which the IP trap and Fast B
coils are epoxied 7 . The details of our open-loop compensation of these eddy fields are provided inMichele Olsen’s thesis [116]. In view of these requirements and constraints, we need to thoroughly
characterize how well the Fast B system performs.

2.3.1

Fast B closed-loop transfer function

The first step is to characterize how the full Fast B system (i.e. the servo electronics andhigh-current path) responds to time-varying control signals. This response is more often analyzed
7
Although the cooling blocks do have slots that cut into them to stave off the largest eddy current paths, theamount of copper left unobstructed is sufficient to be a problem.

16in the frequency domain by measuring the transfer function for the system [96]. The transfer
function, H, is defined as the impulse-response function for a system (in Physics parlance this is a
Green’s function for the system). Suppose we input a time varying function f(t) to our system andmeasure its response, g(t). The output is the convolution of f(t) with our transfer function, H. In
the frequency domain the input and output functions can be related using linear response theory.
The relation between the input and output isg̃(ω) = H(ω)f˜(ω)(2.3.1)

where g̃(ω) and f˜(ω) are the Fourier transforms of g(t) and f(t), respectively. Therefore the system
can be characterized by measuring input and output response and then numerically calculatingg̃(ω). We can perform this measurement in either a spectrally narrow manner where f(t)f˜(ω)=sin(ω0 t) and f˜(ω) ∝ δ(ω − ω0 ) − δ(ω − ω0 ) and map out the response versus frequency or in aH(ω) =

spectrally broad manner8 with f(t) =δ(t) and f˜(ω) ∝ 1.
Shown in Fig. 2.3a is the amplitude of our closed-loop transfer function calculated frommeasured servo input signal and servo Hall-effect current transducer (Hall probe) output shown in
Fig. 2.3b. For values of the transfer function amplitude near unity, the servo is able to faithfully
reproduce the control signal. The -3dB frequency, which is a measurement of the system bandwidth,is f−3dB = 39.2 kHz. The transfer function amplitude also exhibits two peaks, which are likely due
to resonances in the Hall probe that is used to measure the current.

2.3.2

Measuring the Fast B magnetic-field sweep rate in situ

Now that we have a good measurement of how the Fast B system performs, we need tomeasure the limit on how fast we can sweep the magnetic field. To measure the sweep rate we
8

A true δ function cannot realized and in reality the output is more correctly a Heaviside Π function f(t) =Π(at) (width of the pulse is 1/a) such that f˜(ω) ∝ sinc(ω/a). The function sinc(ω/a) sports zeros at ω = 2πna,
for n = 1, 2, 3, . . ., and so the spectral width is limited to ω = 2πa. At this point the calculated transfer function
could exhibit numerical instabilities as we are dividing by zero and so the pulse length should be short compared
to any frequencies of interest in the system. For the data shown in Fig. 2.3 a 1 µs pulse was used, which limits
the measurement to frequencies below 1000 kHz. In this regard, driving the system with a step function might be abetter input waveform as its Fourier transform does not have any finite zeros.

17
rely on our atoms for the magnetometery signal by using the magnetically-sensitive |F, mF i =|9/2, −9/2i → |9/2, −7/2i transition in K and looking at the adiabaticity of a Landau-Zener sweep.
By applying rf resonant with this transition at a given field, the |9/2, −9/2i and |9/2, −7/2istates are coupled together and the true eigenstates of the system then exhibit a Landau-Zener
avoided-crossing [172]. This problem is a classic two-level problem with a Hamiltonian given by


~Ω/2
 1
(2.3.2)
H=

~Ω/2
2
where Ω is the on-resonance Rabi frequency, i is the energy of the bare (uncoupled) state i. With
the applied rf, the eigenenergies of this Hamiltonian are no longer that of the bare state energies,
and are given byE± =1 + 2 1 p±(1 − 2 )2 + (~Ω)2 .
2
2(2.3.3)A diagram of the coupled system is shown in Fig. 2.4a where we have assumed that 1 from
Eqn. (2.3.2) varies linearly with magnetic field with an assumed value for Ω. In this diagram, thebare states 1 and 2 (dashed lines) represent the uncoupled |9/2, −9/2i and |9/2, −7/2i states.
With the application of rf resonant with the |9/2, −9/2i → |9/2, −7/2i transition at B0 , the twostates become coupled and the eigenstates of the system are E+ and E− (solid curves). If we
begin with the magnetic field infinitely far away from B0 , where the bare states and eigenstates
are identical, and with the system initialized in 2 , for example, then as we ramp the magnetic
field slowly down through B0 , then the system will adiabatically follow E− ultimately ending withthe system transferred to state 1 once the magnetic field is infinitely far from B0 . This process isknown as Adiabatic Rapid Passage (ARP). If the ramp is not slow, then the system with end as a
mixture of atoms in E+ and E− . The probability of the system undergoing a transition between

18

ε1
E+

E-

ε1-ε 2Ω

B0

Fast B

ε2

B

t

(a)

(b)

Figure 2.4: a) Energy diagram for a coupled two-level system as discussed in the text. b) Timing
diagram for a magnetic field ARP where the rf (red sinusoidal curve) is turned on while the magnetic
field is initially far from resonance and the magnetic field later sweeps linearly down through theresonance.

E− and E+ as the magnetic field transits through B0 is [172]π~Ω2
P1→2 = exp −d
2 | dt(1 − 2 )|!
π ~Ω2= exp −
2 |δµ dB
dt |!

(2.3.4)

where δµ is the magnetic-moment difference between states 1 and 2 . The Rabi frequency is
proportional to a dipole matrix element, and so Ω2 is proportional to the rf power.
To measure the sweep rate, we begin with a polarized sample of K in the |9/2, −9/2i statewith the magnetic field 5 G away from field at which the rf is resonant. Since we are most concernedwith the ramp rate at the center, we fix the rf frequency to be resonant with the |9/2, −9/2i →|9/2, −7/2i transition at this field. As shown in Fig. 2.4b, we perform the magnetic field ARP by
applying the rf at a constant frequency and sweeping the magnetic field, after which we measure theremnant |9/2, −9/2i population. These measurements were performed at fields near 550 G where
the imaging transition for the |9/2, −7/2i is roughly 13 natural line-widths detuned from the probe
light for the |9/2, −9/2i imaging. As a consequence the atoms in the |9/2, −7/2i do not create asignificant imaging background. The magnetic-field sweep rate can be measured by performing thesame field sweep for various rf powers and independently measuring the Rabi frequency for each19
power from on-resonance Rabi flopping. An example of such a field ARP measurement is shownin Fig. 2.5c for an inverse field sweep rate of 10 µs/G for a ramp that spans 10 G. For this data,
the current ramp sent to the Fast B servo includes an eddy current correction. The data were fit
to an decaying exponential of the form N = N0 exp(−p/p1/e ) + y0 . From the fitted value of p1/e
and Eqn. (2.3.2), the magnetic-sweep rate is measured, where δµ = h × 66kHz/G near 550 G. Fig.
2.5d shows the fidelity of our Fast B system for a small range of sweep rates, showing that we cantrust inverse sweep rates as low as 3 µs/G.

2.4

Imaging and state control improvements
The last stage of the experiment involves imaging the atoms, and perhaps molecules, using

standard absorption imaging. This process, although simple in principle, requires careful calibrationof experimental parameters and is also subject to various systematic errors [135, 98, 15]. In thisSection I focus on the various systematic effects in order to understand how well we know our atom
number and I include discussion of various improvements to apparatus.
To take an image, the gas of atoms and molecules is released from the optical trap andallowed to expand. We take an image of the atoms9 by shining a probe beam resonant with either
the |9/2, −9/2i → |11/20 , −11/2i transition when imaging K, or the |2, 2i → |30 , 3i transition when
imaging Rb10 . The atoms absorb light from the probe beam casting a shadow onto the camera
which records the image. Additionally, we take an image of the intensity profile without atoms andan image of the dark counts on the camera when no probe beam is applied. From these images we−IDcalculate the optical depth of the cloud at each pixel from OD = ln IILS −I
where IL , IS , and ID areD

the intensities of the light image, the shadow image, and the dark image, respectively. The opticaldepth of the cloud is proportional to the column density (i.e. the result integrating the full threedimensional density through the probe direction) [107], and from the cloud sizes and column density
9

Molecules are imaged by first dissociating the molecules and then imaging their constituent atoms. This requiresthat we remove unpaired atoms of the type we will be imaging in the molecules. If we image the dissociated K atoms,
we first shift the unpaired K atoms into the |9/2, −7/2i state. Imaging of dissociated Rb involves shifting unpaired
Rb into |2, 2i and then blasting them away with light resonant with |2, 2i → |30 , 3i.
10This requires that we transfer Rb atoms from |1, 1i to |2, 2i, which is achieved with a microwave ARP.

20
5

4 x 1 05

2 x 1 05

1 x 1 05

1 1

1 .0 x 1 0

1 1

8 .0 x 1 0

1 0

6 .0 x 1 0

1 0

4 .0 x 1 0

1 0

2 .0 x 1 0

1 0Ω (ra d

2

N

-9 /2

2

s

-2

)

3 x 1 01 .2 x 1 0

0 .0

0
0

2 0

4 0

6 0

8 0

1 0 0

0 .0 0r f d u r a t i o n ( µs )5

4 x 1 05

3 x 1 05

2 x 1 05

1 x 1 05(b)

M e a s u r e d r a m p r a t e ( µs / G )5 x 1 0

N

-9 /2

(a)

0
0 .0 00 .0 50 .1 00 .1 50 .2 0r f s y n th e s iz e r p o w e r ( m W )(c)

0 .0 5 0 .1 0 0 .1 5 0 .2 0 0 .2 5r f s y n th e s iz e r p o w e r ( m W )

1 0
8
6
4

0

2
0

2
4
6
8
1 0
R e q u e s t e d r a m p r a t e ( µs / G )(d)Figure 2.5: a) On resonance Rabi flopping between |9/2, −9/2i and |9/2, −7/2i at a synthesizer
power of 0.1 mW and an rf frequency of 80.0378 MHz. b) Calibration of the Rabi frequency
versus synthesizer power. The fit of Ω2 versus power to a line returns a slope of 4.97(13) ×
1011 rad2 s−2 mW−1 . c) Sample measurement of the 10 µs/G ramp with a 10 G span. The numberof atoms remaining in the |9/2, −9/2i Zeeman state are plotted as a function of the rf synthesizer
power and then fit to an decaying exponential of the form N = N0 exp(−p/p1/e ) + y0 . From thefitted value of p1/e and Eqn. (2.3.2), the magnetic-sweep rate is measured, where δµ = h×66kHz/G
near 550 G. d) Collection of measurements such as c) plotted against the inverse ramp rate requestedof the servo and a line with a slope of 1. The measured ramp rate was calculated from c) using acalibration of the Rabi frequency measured from Rabi flopping.

we can calculate the atom number and temperature, as well as other thermodynamic quantities[33].
There are various systematic errors that impact the number extracted from an absorption
image. Here I discuss two systematic effects due to stray light in the image and intensity saturation

21
2 .0

1 .6
O D

1 .6

s tra y

3

1 .4

/ O D

4

c o rre c te d

5
1 .4
1 .2
1 .0

1 /2
1 /5
1 /1 01 /2 0

2

1 .2

0

O D

s tra y/ O D

m e a s

1 .8

O D

IL / Isat =
=

s a t

1

2
O D

3

1 .0
4

0

1

2
O D

m e a s(a)34

s tra y(b)

Figure 2.6: a) Fractional correction of the measured OD (ODmeas ) due to OD saturation effects
caused by unabsorbed light. b) Fractional correction of the ODstray due to intensity saturation
effects.

of the probe transition. First let’s consider the situation where a portion of the probe light is notabsorbed by the atoms, either because it has the wrong frequency or polarization to be absorbed,
or the light simply scatters off of some structure in the vacuum chamber and hits the camera. The
measured optical depth is then
ODmeas = ln

IL + IN A
IS + IN A(2.4.1)

where we have ignored the dark image for simplicity and IN A is the intensity of the unabsorbed
light. The amount of unabsorbed light imposes an upper limit on the ODmeas , denoted ODsat ,given byODsat = ln

IL + IN A.
IN A(2.4.2)This OD saturation can severely distort the imaged profile of dense samples, but also causes one
to underestimate the atom density and number. The significance of this effect can be determinedby solving Eqn. (2.4.1) for the true optical depth [98], which results in
ODstray = ln

1 − e−ODsat,
e−ODmeas − e−ODsat(2.4.3)where ODstray represents the OD after correcting for the effects of stray, or un-absorbed, light.
Fig. 2.6a consists of plots of Eqn. (2.4.3) divided by ODmeas for various values of ODsat . For

22

1 .1 2
1 .0 8
1 .0 4O D

c o rre c te d

/ O D

m e a s1 .1 6

1 .0 0

0 .0

0 .5

1 .0
O D

1 .5

2 .0

m e a sL
= 1/10 (dashed line)
Figure 2.7: Plot of Eqn. (2.4.4) using Eqn. (2.4.3) for ODsat = 4 and IIsat
IL
and Isat = 1/20 (solid line).

small values of ODsat the correction for the OD is significant 11 and so it is desirable to eliminate
sources of unabsorbed light. After upgrading our imaging system with higher quality polarizing
beam splitter cubes and waveplates from CVI, our ODsat increased to ODsat = 4 from 3.
An additional correction is a consequence of saturation of the probe transition due to highprobe intensity. After integrating the Beer-Lambert law [107], the corrected OD is
ODcorrected = ODstray + (1 − e−ODstray )

IL
Isat

(2.4.4)

under the assumption that the probe laser is on resonance with the atomic transition with Isat
denoting the saturation intensity for the probe transition12 . Fig. 2.6b shows the extent of thiscorrection for various values of IL /Isat . In most circumstances it is advisable to image with IL /Isat ≤1/10. Fig. 2.7 shows the combination of these two systematic effects assuming ODsat = 4 for ourLtypical range of IIsat= 1/10 (dashed line) to 1/20 (solid line). For the range of OD we typicallymeasure (OD = 0.2 to 1.5) these corrections imply that our measured OD is low by 7% to 11 %.
1.
In the limit of low OD, the ratio ODstray /ODmeas =1 − e−ODsat
I
L12
The low OD limit for ODcorrected /ODstray = 1 +
.
Isat
11

23
2.4.1

Measuring imaging magnification

Since an accurate measurement of the atom number relies on knowing the physical size ofthe cloud in the OD image, we need an accurate and precise measure of the imaging system’s
magnification. Previously, we had relied upon mapping the position of a falling cloud as a functionof time to calibrate the magnification, but this is complicated by the magnetic curvatures present
while our clouds drop. Efforts were also made to measure the magnification from a knife-edge
diffraction pattern, but results between different probe beams were inconsistent, which hinted thatthe measurement was being influenced by the phase curvature of the beams. The method whichgave the best result, which also agreed with our model of the optics, was suggested to us by a visiting
scientist in the Cornell labs. It was to image a commercial resolution target onto the camera. Afterthe camera was focused onto the atom clouds, the test target (Melles Griot 04 TRP 003) was backlit with a halogen lamp. The image was projected through our optics setup with an added mirror
and onto the camera. A 780nm notch filter was placed on the camera to block the broad spectrumof the halogen lamp and thus avoid having to correct for the dispersion of the achromatic lenses in
the imaging system. A sample image is shown in Fig. 2.8 from which we extract a magnificationof M = 2.51(3) using the manufacturer’s measurements of the line spacing13 . The error bar in
the measured magnification is due to the width of the edges of the features. The magnification
measured from the test target agrees with magnification measured from watching Rb atoms in the|2, 0i state fall at low fields (M = 2.49(3)).

2.4.2

The new and nimble Rb microwave setup

For some of the work described in Chapter 4, we needed to be able to quickly shuttle Rb
atoms to different hyperfine states at multiple fields in rapid succession using ARPs. The microwave
synthesizer we use (Agilent E8251A) is programmed over GPIB and so takes of order 100 ms to
reprogram the center frequency of a frequency sweep. Additionally, the center frequency differenceat the different fields was large enough that it could not be accommodated by increasing the13This test target has the standardized USAF 1951 resolution target pattern

24

Figure 2.8: Sample image of the back-lit test target taken using our imaging system.

frequency span of the ARP. What was needed was a more nimble microwave system.
Our solution to this was to mix our microwave source with a Direct Digital Synthesizer(DDS), as shown in Fig. 2.9. This DDS (Toptica VFG-150) can be programmed with multiple rf
pulse sequences and rapidly triggered between them and so performs the nimble tasks needed of
the system. The microwave and rf are mixed with using a single-sideband mixer (Marki Microwave
SSB-0618LXW-1) for which one frequency sideband is suppressed by 23dB through destructive interference. The microwaves then pass through a microwave solid-state switch (Narda SP123DHTS)
and then through a high-power microwave amplifer (AML L0608-41-T479, -1 dB compression point
at 41 dBm=12.5 W). Lastly, the microwaves pass through an isolator and then to a doubly stubtuned microwave antenna (resonant at 6.8 GHz and 8 GHz) that was built in-house and tuned
in situ. The system operates with the carrier (supplied by the E8251A) far detuned and allows
us to perform some additional checks on K number calibration discussed below. With this new
microwave setup we can switch frequencies as fast as 100 ns according to the manufacturer, which
is substantially faster than our typical ARP durations of 100 µs to 2 ms.

25
solid-stateswitch

SSB mixer
Agilent
E8251Ahigh-poweramp.

μ-waveisolatorstub-tuned
μ-waveantennapre-amp.

VFG-150

Figure 2.9: Circuit diagram for the new Rb microwave setup. We mix the stable microwave source(Agilent E8251A) with a pre-amplified (Mini-circuits ZFL-1000LN+) rf source from a DDS (Toptica
VFG-150) using a single-sideband mixer (Marki Microwave SSB-0618LXW-1) that suppresses thehigher frequency sideband power by 23 dB. The pulse length of this source is controlled by anabsorptive solid state switch (Narda SP123DHTS) which in turn feeds a high-power (41 dBm)
microwave amplifer (AML L0608-41-T479). Lastly, the microwaves travel through an isolator(Narda 4914) and to our double stub-tuned microwave antenna that was built in-house.

2.5

Calibrating the Rb atom number
Measurement of the condensate transition temperature provides a means to estimate theaccuracy of other measured quantities in the system (e.g. atom number, temperature). For an
ideal Bose gas in a harmonic potential, the transition temperature in the thermodynamic limit,(Tc ), is given by
kb Tc = ~ ω (N/ζ(3))(1/3)(2.5.1)

where kb is Boltzmann’s constant, ω the geometric mean trap frequency, and ζ(3) ≈ 1.20206 theRiemann zeta function. The ideal Bose gas condensation temperature in a harmonic potential isrelated to the condensate fraction by
N0
=1−
N



T
Tc

3
,

(2.5.2)where N0 is the number of atoms in the condensate and T the temperature of the cloud. The
strong dependence of condensate fraction on T/Tc yields a sharp signature of the transition, which
is useful for calibration purposes.
This ideal Bose gas treatment ignores the importance of atom-atom interactions in 87 Rb (aswell as finite number effects [29]). Although the Rb-Rb background scattering length is small (abg ≈26
100a0 ), the sharpness of the condensation transition is “smoothed” by the repulsive interactions
between the condensed and thermal atoms. This effect was modeled in Ref. [112] where thecondensate atoms were treated in the Thomas-Fermi limit and the potential for the thermal atoms
was a sum between the harmonic potential and mean-field repulsion of the condensate atoms. The
authors derived an implicit expression for the condensate fraction in terms of T/Tc , which, to thefirst non-trivial order in the interaction parameter η, is given by [112]
N0
=1−
N



T
Tc

3

ζ(2)
−η
ζ(3)



T
Tc

2 

N0
N

2/5
(2.5.3)

with η defined as

2/5
a
1
1/3
1/6
N
,
(2.5.4)
η = ζ(3)2aHOpwhere a is the scattering length and aHO = ~/mω is the harmonic oscillator length for an atomof mass m in a harmonic trap of frequency ω.
To use Eqn. (2.5.3) as a test of how well the atom number is being counted, we first preparea series of Rubidium clouds with varying degrees of condensation. The clouds are imaged using
standard absorption imaging and the images of partially condensed clouds are processed into an
optical depth (OD) image and then fit to a sum of a Thomas-Fermi parabola and a Bose distribution
with the fugacity forced to one. The OD image fit returns peak OD’s and cloud sizes, from which
the atom number and temperature can be calculated from the following equations:2π
N0 =(1 + 4δ 2 )ODpk,BEC σy,BEC σx,BEC
5 σ0


2π
pix 2
2
Nth =
(1 + 4δ )ODpk,th σy,th σx,th
σ0
M

2
2
σi,th
pix
Ti = mRb ωi2
1 + ωi2 t2 M
where the photon scattering cross-section σ0 =

3λ2Rb
2π



pix
M

2
(2.5.5)(2.5.6)(2.5.7)for photons with wavelengths resonantwith the Rb transition, λRb , δ is the probe light detuning from resonance, σi,BEC the BEC size inthe i-th direction, mRb is the Rubidium atom mass, t the time of flight for the cloud, ωi the harmonic
14 (pix) to the imaging
trap frequency in the i-direction, and pix
M is the ratio of the camera pixel size
14

The pixel size for our Princeton Instrument Pixis camera is 13 µm.

27

N0  N

0.3
0.2
0.1
0.0
0.85

0.90

0.95

1.00

1.05

TTc

Figure 2.10: Measured condensate fraction after corrections for the various effects listed in the text.
Fitting to Eqn. (2.5.3) with a scaling parameter for the horizontal axis returns the solid curve forwhich predicts the transition temperature to be T = 1.025(6)(20) Tc where Tc is defined in Eqn.
(2.5.1) and the first and second errors are the statistical and systematic errors, respectively. The
dashed line is a plot of the condensate fraction for a non-interacting system (Eqn. 2.5.2).

system magnification, M . From the calculated number and temperature, Tc can be calculated from
Eqn. (2.5.1) and then TTc can be calculated. As can be seen from Eqns. (2.5.5−2.5.7,2.5.1), thecalculated quantities depend strongly on the magnification, as well as the harmonic trap frequency
in the case of Tc .
Shown in Fig. 2.10 is a single parameter fit of Eqn. (2.5.3) (solid blue curve) to the measured
condensate fraction (blue points) compared to the ideal Bose gas prediction (dashed black curve).
The fit of Eqn. (2.5.3) assumed a rescaled condensation temperature Tc0 = b Tc , where Tc is
the measured condensation temperature assuming the experimental calibrations and b is the sole
fit parameter to account for a miscalibration of the atom number. The BEC transition occurs
at T /Tc0 = 1 and therefore the measured transition point is TTc = b in terms of experimentallycalibrated quantities. The fit returns b = 1.025(6)(20), after correcting the data for the nonnegligible frequency width of the probe laser 15 . The first error is purely statistical and thesecond a combination of trap frequency calibration error, imaging magnification, and probe laser
frequency width error. This value of b implies that the number is being under-counted by 7(2)(5)%,in agreement with Section 2.4. The Rb numbers presented in this thesis are left uncorrected for15
Our Rb probe lineshape are generally 6.6 MHz wide, compared to the natural linewidth of 6.0666*(18) [143].
This amounts to a 10% correction on the Rb number [15].

28

b y K tr a n s itio n

9 .0 x 1 0 3

m o l

1 .2 x 1 0 4

3 .0 x 1 0 3

N

6 .0 x 1 0 3

0 .0

0 .0

3 .0 x 1 0 3 6 .0 x 1 0 3 9 .0 x 1 0 3 1 .2 x 1 0 4
N

m o l

b y R b tr a n s itio n

Figure 2.11: Plot of the molecule number measured using the bound K and bound Rb transitionsin the same experiment run. The data were fit to a line (solid line) with a horizontal offset whichgave a slope of 1.00(5) and a horizontal offset of 1.3(7) × 103 . If we fit the data without an offset,
the fit returns a slope of 0.83(3). For comparison, the dashed line is a line with slope of 1 and nooffset.

the 7% systematic error.
The dominant sources of atom number measurement uncertainty can be separated into two
categories: uncertainties in the properties of the apparatus and uncertainties intrinsic to the imaging
process. From Eqns. (2.5.5−2.5.7, 2.5.1), the trap frequency and the ratio pix/M are the mostcritical parameters from the apparatus. The linewidth of the probe laser also can reduce themeasured OD if it is a significant fraction of the natural linewidth of the transition (see discussion
in Ref. [15]). Additional mechanisms include radiation trapping of probe light within the cloud andDoppler shifting of atoms’ transition due to probe photon recoil [135], which are a small correction( 1% OD correction) for typical conditions of this experiment.

2.5.1

Calibrating K atom number

Calibration of the K atom number is more complicated, as the K gas does not have a sharp
phase transition. However, when we generate KRb molecules, as in Chapter 3, we know that asingle K atom has bonded with a single Rb atom. The K atom number can be calibrated by imaging
the bound K and bound Rb in the same experiment run. This required a new Rb microwave setup

29
as we have to quickly remove unbound Rb with a series of ARPs and kill pulses and then ARP
the once-bound Rb into the imaging state after the molecules are dissociated shortly thereafter.
Shown in Fig. 2.11 are the results of this measurement performed by generating a variable numberof KRb molecules using methods discussed in Chapter 3. Fitting the data to a line with horizontal
offset returns a slope of 1.00(5) and offset of 1.3(7) × 103 . The offset was due to an imperfect killpulse that left unbound Rb atoms in the |2, 2i state, which were imaged with the bound Rb16 .
Therefore our K and Rb number calibration agree to within 5 %.

16
The imperfect kill pulse was confirmed by running the same magnetic field sequence with Rb present. After thekill pulse, atoms remaining in the |2, 2i state were observed.

Chapter 3

Feshbach molecules in the perturbative and saturated limits

Using ultracold samples of atoms, atomic physics has answered fundamental questions inphysics, including some that cross over into particle physics and condensed matter. This was madepossible by the years of work put into learning how to create ultracold atomic sample, as well as
the more recent ability to control the sign and strength of atom-atom interactions. Tuning theatom-atom interaction has led to new understanding of the BCS-BEC crossover, observation of afermionic superfluid, new understandings of few-body collisions, and many others.
This control of the atomic interaction strength is made possible by a Feshbach resonance,
which results from coupling colliding atoms to one or more bound states in other inter-atomic
potentials. In Fig. 3.1, a simplified picture of the potentials involved in a Feshbach resonance isshown. Only a single bound state is considered as represented by the solid orange line within theclosed channel in Fig. 3.1. Initially the atoms are colliding at a low energy above the dissociationthreshold of the entrance channel (denoted by the dashed blue line). If the closed channel has
the correct quantum numbers, then the bound state will couple to the entrance channel and thus
‘dress’ the entrance channel. The energy of the bound state is nominally separated by an amount
ECC relative to the threshold of the entrance channel. When the experimentalist is able to vary
ECC , then as ECC passes through zero the presence of a new bound state in the dressed state of theatoms causes a divergence in the dominant low-energy interaction-strength parameter, the s-wave
scattering length.
One method of manipulating the relative energies of the entrance and closed channel is to use

31E

Closed channel
ECC

Entrance channel

Internucleus separation r

Figure 3.1: Two-channel model of a Feshbach resonance.

a homogeneous magnetic field and take advantage of the differential Zeeman shifts when the closed
channel has a magnetic moment that differs from that of the open channel. The s-wave scattering
length near a Feshbach resonance has a simple relation to the magnetic-field given by [109]a = abg 1 −∆B − B0,(3.0.1)

where abg is the non-resonant scattering length associated with the entrance channel, ∆ is thezero-crossing resonance width, and B0 is the location of the Feshbach resonance. Fig. 3.2 is a plotof (3.0.1) for the case of abg , ∆ < 0. By varying the magnetic field in the vicinity of a Feshbachresonance, the interaction strength can be varied over a large range.
The atoms can also be transferred into the bound state creating an extremely weakly boundmolecule called a Feshbach molecule. In this chapter, Feshbach molecules created via a magneticallytunable Feshbach resonances will be discussed. In this context, the entrance and closed channels
have different magnetic moments allowing ECC to be tuned by applying a homogeneous magnetic
field of varying strengths. Feshbach molecules have been generated in numerous homonuclear andheteronuclear systems 1 . Bosonic Feshbach molecules composed of two fermions [134] have been
made to Bose condense [61]. Coherent oscillations between atoms and molecules have been observedin a homonuclear [46, 145] and a heteronuclear system [117]. Feshbach molecules have also proven1

To date Feshbach molecules have been observed in 6 Li [144, 82], 23 Na [169], 39 K [37], 40 K [134], 52 Cr [9], 85 Rb[75], 87 Rb [47], 133 Cs [74], 40 K-87 Rb [119], 41 K-87 Rb [159], 85 Rb-87 Rb [123], 87 Rb-133 Cs [32], 6 Li-40 K [150, 142],
23
Na-40 K [168], and 23 Na-6 Li [73].

32
a(B)

B-B0

Figure 3.2: S-wave scattering-length near a Feshbach resonance.

to be an efficient starting point for the generation of ultracold polar molecules in the absolute
molecular ground state [113].
However, the Feshbach molecule association is interesting in its own right. This chapter details research aimed at increasing the understanding of Feshbach molecule creation in the 40 K-87 Rb
mixture system. The work described in this chapter is the subject of a manuscript in preparation
for publication [27]. In Sec. 3.1, basic scattering physics will be discussed to orient the reader.
This will be followed by a more detailed description of Feshbach resonances in section 3.2 andmagnetoassociation of Feshbach molecules in Sec. 3.3, where molecules are generated by sweeping
the magnetic field across the Feshbach resonance to generate molecules. Next I will discuss the research performed to understand molecule formation in the perturbative (fast magnetic-field sweep)
limit and lastly the saturated (slow magnetic-field sweep) limit. Our approach to understanding
the magnetic-field sweep rate dependence differs with previous studies, which characterized thesweep rate dependence with a e−1 inverse sweep rate [75]. This previous approach, in effect, mixes
information from the two limits. Instead we characterize the molecule creation in terms of the twolimits, which have different dependences on the gas parameters. In Subsection 3.4, we show thatthe number of molecules made in the perturbative limit is governed only by the Feshbach resonanceparameters and the spatial overlap of the atom clouds. In contrast, the number of molecules associated in the saturated limit depends upon the phase-space density, as will be discussed in Section

33
3.5. To understand this limit, we use a phenomenological model which has successfully describedhomonuclear[75] and heteronuclear systems [174, 142]. While this model has been previously applied to our system, 40 K – 87 Rb, for rf association [174], we fit the model to molecule data taken
by magneto-association. We examine various experimental concerns, including relative offsets between the atom clouds in position and center-of-mass velocity. Finally, we theoretically explore thelimitations that prevent the creation of a quantum degenerate gas of fermionic Feshbach moleculesin the absence of a BEC.

3.1

Basic scattering theory
Two-body scattering is a standard problem addressed in many quantum mechanics text books[20, 137]. This problem is generally expressed in the center-of-mass frame of the two colliding
particles, 1 and 2, with a two-body reduced mass µ2b = m1 m2 /(m1 + m2 ), relative separation
vector ~r, and a relative momentum ~ ~k. As the two particles approach each other from far away,
they begin to interact through an inter-particle potential V (~r). In the case of the work discussed
in this thesis, the inter-atomic potential is isotropic and thus only depends on the magnitude of~r. Because of this symmetry, the Hamiltonian commutes with the angular momentum operators~ 2 and L~ z and thus the wavefunction is separable into radial and angular parts. The scatteringL
problem has an additional azimuthal symmetry, which yields spherical harmonics, Yl,m (θ, φ), as thesolution to the angular part. The rest of the wavefunction is an expansion in partial waves of order`, which is a product of a radial wavefunction R` (k, r) and Legendre polynomial P` (cos(θ)). The
partial wave orders take on values ` = 0, 1, 2, . . . which are termed s-, p-, d-wave,. . . contributions.
The radial part of the wavefunction is found by solving the Schroedinger equation of the form



~2 d2
~2 `(` + 1)
~2 k 2
−
+
+
V
(r)
−
2µ dr22µ12 r22µ12after making the substitution u` (k, r) = r R` (k, r)[20].

u` (k, r) = 0(3.1.1)

34
The solution to Schroedinger equation has the following asymptotic formR` (k, r) −−−→r→∞A`sin (kr − `π/2 + δ` (k)) eıδ` (k)kr(3.1.2)where all the details of the two-particle interaction are carried in scattering phase shift, δ` (k). At
ultracold temperatures, scattering events are generally cold enough to only involve s-wave scattering. The details of the scattering can then be described solely by the s-wave scattering phase shift.
The s-wave scattering phase shift is related to the s-wave scattering length, a, and effective range,
reff , by the following [13]:1
k cot(δ0 (k)) = −1/a + reff k 2 .
2(3.1.3)

However, at ultracold temperatures the effective range term can often be neglected [24] andthen the sign and magnitude of the scattering length fully parameterizes the interaction between thetwo particles. The strength of the interaction between the particles increases with the magnitudeof the scattering length and the interaction is attractive for a < 0 and repulsive for a > 0.
For neutral alkali atoms scattering in their electronic ground states, the leading term in thelong-range part of the interatomic potential is a van der Waals term given by [89]
V (r)

=

r→∞

−

C6
r6

(3.1.4)

where the quantity C6 is known as the van der Waals coefficient and can be calculated from firstprinciples [36]. This coefficient also defines a length scale for Eqn. (3.1.4) by equating an energyof the form −~2
to Eqn. (3.1.4) and solving for r. The length scale is then given by [89]2µ2b r2
1
rvdW =
2

2 µ2b C6~2

(1/4)(3.1.5)after dividing the result of the previous sentence by two for definitional purposes. This quantity, thevan der Waals length, is not the only way to define a length scale for the van der Waals potential.
An alternative length scale is the mean scattering length given by [63]ā = 4 π/Γ(1/4)2 rvdW ≈ 0.956 rvdW

(3.1.6)

B-B0

35

-Ed

Figure 3.3: Binding energy, Ed , of a Feshbach molecule in the vicinity of the Feshbach resonance.
The zero energy axis represents the energy of the colliding atoms in the entrance channel

where Γ(x) is the Gamma function. For s-wave collisions at ultracold temperatures, interatomic
separations on order of the van der Waals length mark the transition between the short-range andlong-range aspects of the atom pair wavefunction [83].

3.2

Feshbach molecules
The Feshbach molecule state exists on the positive scattering-length side of the Feshbachresonance (B − B0 < 0 in Fig. 3.2) and corresponds to a weakly bound molecule with a consequentially large spatial extent. Shown in Fig. 3.3 is a plot of the universal molecule binding energy
in the vicinity of the Feshbach energy, where Ed is referenced to the open-channel threshold. These
weakly bound molecules exhibit a universal binding energy given by
Ed = ~2 /(2µ2b a2 )(3.2.1)close to the Feshbach resonance, where by “close”, we mean that the scattering-length that satisfies
2
Ed  ~2 /(2µ2b rvdW
) (or a  rvdW ).

As Köhler et al. explain in Ref. [89], the universality of the molecules close to the Feshbachresonance is a consequence of their large spatial extent. In the vicinity of a Feshbach resonance the1 e−r/a, which has a mean inter-atomic separa2π a rtion of hri = a2 . Thus, for large scattering lengths (a  rvdW ) much of the wavefunction exists at
molecule wave function has the form φd (r) = √large inter-atomic separations well outside of the inter-atomic potential, creating what is known as

36
a halo dimer [89]. For this asymptotic limit, inter-atomic interaction makes a negligible energetic
contribution and the Schroedinger equation reduces to Ed φd ∼ −~2∇2 φd . The molecule wave2µ2bfunction is then insensitive to the details of the inter-atomic interaction and the molecule exhibits
universal behavior determined solely by the scattering length.
The range in magnetic field over which the molecule exhibits universal behavior depends
upon the details of the Feshbach resonance. A convenient way to categorize resonances into two
limiting cases is to use the resonance strength parameter given by [24]sres =abg ∆ δµā Ē(3.2.2)where Ē = ~2 /(2µ2b ā2 ) is the energy scale associated with ā and δµ is the difference between
the entrance- and closed-channel’s magnetic moments [24]. Resonances for which sres  1 arecalled open-channel dominated. Open-channel dominated resonances are generally broad (∆ >1 G) and the spin character of the wavefunction for threshold collisions, as well as the molecule
wavefunction near the resonance, are largely that of the open channel. These resonances exhibit
universal behavior over a large fraction of their resonance width, ∆. Closed-channel dominatedresonances correspond to the limit sres  1. Generally closed-channel dominated resonances are
narrow2 (∆ < 1 G). Additionally, closed-channel resonances exhibit universal behavior over asmall fraction of their resonance width, ∆.
The molecule binding energy only follows (3.2.1) for small detunings from the Feshbachresonance. Further away from the universal regime, the molecule binding energy must be corrected
for the non-zero range of the inter-atomic potential. The lowest-order correction is given by [24]
Ed =

~2
,
2µ2b (a − ā + R∗ )2(3.2.3)where ā term is a correction for finite-range of the van der Waals interaction and R∗ = ā/sres isan additional term introduced in [128].

37
Ref.

B0 (G)∆ (G)abg (a0 )Olsen et al.(2009) [117]546.71(1)-3.34(5)-185(7)∗

Klempt et al. (2008) [86]546.618(5)(6)-3.04(2)-187∗

Zirbel et al. (2008) [174]546.76(5)-3.6(1)-185(7)∗

Simoni et al.(2008) [140]546.75∗

-3.1∗

-189∗

Deuretzbacher et al. (2008) [38]546.669(24)(2)-2.92-185(4)∗

Klempt et al. (2007) [87]546.71-3.07∗-185(7)∗a

Zaccanti et al. (2006) [171]546.65(2)-3.3(2)-185(4)∗

Ospelkaus et al. (2006) [121]546.8(1) G

-2.9∗

-189∗

Ferlaino et al.(2006) [55]

546.7

-2.9∗

-185(7)∗aRef. does not quote a value, so one is assumed from its references.

Table 3.1: Comparison of the 40 K – 87 Rb Feshbach resonance parameters
for the |9/2, −9/2i + |1, 1i resonance at ≈546.7 G as reported in various
publications. In this table, B0 is the measured resonance center, ∆ isthe zero-crossing width, and abg the background scattering length. All
quantities with an asterisk (∗ ) are theoretical predictions.

3.2.1

The 546.7 G Feshbach resonanceOne of the more challenging aspects of working with a Feshbach resonance is having an
accurate understanding of how the scattering length depends on magnetic field, which relies on
having an accurate and precise measurement of the Feshbach resonance parameters. The s-wave
Feshbach resonance employed for the work detailed in this chapter has been the subject of numerous
theoretical [55, 87, 140, 79] and experimental investigations [55, 140, 117, 86, 38, 87, 171, 79]. The
more recent results for the Feshbach resonance parameters are summarized in Table 3.1. Thisparticular Feshbach resonance is between |f, mf i = |1, 1i hyperfine state of 87 Rb and the |9/2, −9/2istate of 40 K, with the dominant closed-channel contribution being from the second bound state
below the scattering threshold in the |1, 0i and |7/2, −7/2i interatomic potential [174, 84]. Here f
2
One exception to this rule-of-thumb is the 737 G resonance in 7 Li: sres = 0.80 even though it has a width of∆ = −192G.

38

ε1
E1

E2

ε1-ε2
δ

η0

ε2

η

Figure 3.4: Energy diagram for a Landau-Zener avoided crossing. The two bare states φ1 and φ2(bare-state energies 1 and 2 are red dashed lines) are coupled together at a system parametervalue η = η0 . Because of the coupling, φ1 and φ2 are no longer the eigenstates; the eigenenergiesof the coupled system are given by E1 and E2 (thick black curves).

is the total atomic angular momentum and mf is the projection along the magnetic field.
Many of the early studies of this resonance relied on three-body recombination to locate theresonance [79, 55, 171, 87] and rethermalization between Rb and K to measure the zero-crossing
width, ∆ [171]. Most of the more recent characterizations of this resonance involve spectroscopic
measurements of the molecule binding energy to characterize the Feshbach resonance. Many measurements of the Fano-Feshbach resonance have used rf-association spectroscopy to map out themolecule binding energy [174, 86, 38]. In Ref. [117] (of which this author is a co-author), Olsenet al. observed coherent atom-molecule quantum beats whose frequency was determined by themolecule binding energy. Using a Ramsey-type experiment, the authors were able to map out themolecule binding energy in the vicinity of the Feshbach resonance. Fitting the measured binding
energies to the universal molecule binding energy formula returned values for the resonance position
and width. The Feshbach parameters from this last measurement (Olsen et al. [117]) are used in
calculations for this chapter.

39

3.3

Magneto-association of Feshbach molecules
There are several different ways to use a Feshbach resonance to bind atoms together. In

Feshbach-assisted radio-frequency association, the atoms begin in a non-resonant state with themagnetic field on the a > 0 side of resonance. Applying a pulse of rf of the frequency corresponding to atom-atom transition plus the molecule binding energy causes some of the atoms to
be bound into molecules [86, 174, 38, 9]. In a similar fashion, atoms can be bound by sinusoidally
modulating the magnetic field at the frequency corresponding to the binding energy of the timeaveraged magnetic field [62, 147, 123, 68, 94]. A third method, called magneto-association, involves
ramping the magnetic field linearly in time from the negative scattering-length side of the Feshbachresonance to the positive scattering-length side. This widely used method has been used to create bosonic Feshbach molecules from fermionic atoms in 40 K[134], 6 Li [144], and more recently in
6 Li-40 K[150, 142].

Bosonic atoms have also been associated to create bosonic Feshbach moleculesin 23 Na[169], 85 Rb[75], 87 Rb[47], 85 Rb-87 Rb[123], 133 Cs[74], and very recently in 87 Rb-133 Cs[32].
Fermionic Feshbach molecules have previously been generated by magneto-association in 40 K-87 Rb
in Refs. [119, 174, 117] and recently in 23 Na-40 K [168] and 23 Na-6 Li [73].
The magneto-association process can be qualitatively understood in terms of a Landau-Zenermodel3 , whose schematic energy level diagram is shown in Fig. 3.4. The Landau-Zener model
begins with two bare states φ1 and φ2 of energies 1 and 2 , which approach each other energeticallyas a function of a system parameter η, which in the case of this chapter is the magnitude of themagnetic field. Due to the coupling between the two states, they form an avoided crossing at someparameter value η = η0 [172]. This coupling results in a mixing of the two bare states such that they
are no longer eigenstates of the system. The eigenvalues of the coupled system are the black curves
shown in Fig. 3.4 labeled E1 and E2 , which asymptotically approach 1 and 2 far from η = η0 .
If the system is initialized infinitely far away from η0 , in one of the two bare states, and then ηis swept linearly in time through, and infinitely far past, η0 , then the system can adiabatically be3Many-body treatments of magneto-association predict an approach to saturation that is slower than for a
Landau-Zener model[45]

E

40
1/a

-E d

Figure 3.5: Magneto-association of atom with zero relative kinetic energy. The magnetic field
begins with a < 0 where no weakly bound state exists. As the magnetic field is ramped throughthe Feshbach resonance at B = B0 , atom pairs can be bound into a Feshbach molecule.

transferred from one bare state to another. The probability of the system adiabatically following
the eigenstate is given by the following:
PLZ = 1 − e−2πδLZ .

(3.3.1)d
The Landau-Zener parameter is given by δLZ = δ 2 /( 4~| dt(1 −2 )|) for a relative energy separationof the two eigenenergies of δ at η = η0 [172]. Thus, the probability of starting in φ1 and adiabaticallydtransferring into φ2 is higher for slower ramps (i.e. smaller | dt
(1 − 2 )|).

For the case of magneto-association, the energy diagram resembles Fig. 3.5. The systemenergy is controlled by the magnetic field; the two bare states correspond to a scattering state in
the entrance channel and a bound state in the closed channel, and the energy separation of thetwo states is controlled by the magnetic field. To associate atoms, the magnetic field begins on thenegative scattering-length (unbound) side of resonance (the left side of Fig.3.5) and sweeps linearlyin time to the positive-scattering-length (bound) side of the Feshbach resonance.
Shown in Fig. 3.6 is an example of magneto-association with ultracold 40 K-87 Rb, where themolecule creation efficiency (f = Nmol /N< , where N< is the number of atoms of the species with thelower atom number) as a function of the inverse magnetic-field sweep rate (|Ḃ|−1 ). We prepared
the system with 3.4 × 105 87 Rb atoms in the |1, 1i state and 2.6 × 105 40 K in the |9/2, −9/2i state at

41
a quantum degeneracy of T /Tc = 1.5 in an optical trap with a radial trap frequency of 630 Hz (450Hz) and axial trap frequency of 8.2 Hz (5.9 Hz) for K (Rb). The magnetic field began 5 G above
the heteronuclear Feshbach resonance between Rb |1, 1i and K |9/2, −9/2i at approximately 546.7
G. Next, we ramped the magnetic field linearly down through resonance to 5 G below resonance on
the bound side where the molecules have a binding energy of roughly h·12 MHz (cf. Fig. 5 of [174]).
To reduce the density of the atom clouds before molecule association, we released the atoms fromthe optical trap by abruptly shutting off the trap. While the atom clouds expanded, we swept themagnetic field through the Feshbach resonance such that the magnetic field crossed the Feshbachresonance 0.4 ms after trap release. We imaged the molecules as in Refs. [117, 174] where unbound
K atoms were transferred to the |9/2, −7/2i state using an rf adiabatic rapid passage (ARP) with a
1.6 MHz frequency span. Once the unbound K atoms were transferred to |9/2, −7/2i, the magnetic
field was then ramped to 0.7 G below resonance and the molecules were imaged using the K probe
light4 ; the first probe photon breaks the molecule and subsequent photons excite the once-bound
K atom on the |9/2, −9/2i → |11/20 , −11/2i cycling transition. The ARP used to clear out theunbound K is only ≈ 99% efficient, which leaves adds a background of ≈ 2000 to our moleculesignal, which has been subtracted from Fig. 3.6. This background is measured after the samemagnetic-field sweep used for magneto-association, but without Rb present in the trap.
In order to extract information about the molecule formation, the data in Fig. 3.6 were fit
to an exponential (similar to Ref. [75]) given byΓf = f0 1 − exp − |Ḃ|−1
f0(3.3.2)where the quantity Γ is the initial slope of Fig. 3.6 and f0 is the saturated molecule fraction in
the limit of slow magnetic-field sweeps. This parameterization of the fit allows for information to
be extracted from two regimes. In contrast, in Ref. [75] the magnetic-field sweep dependence wascharacterized in terms of a 1/e inverse sweep rate whose fitted value did not agree with theory,
even after thermally averaging the Landau-Zener coefficient over open-channel continuum states
4

K in the |9/2, −7/2i at this field is roughly 13 natural linewidths detuned from the probe transition and it does
not significantly perturb the molecule imaging.

42
6x104
0.20

Saturated limit

3x104

0.10
Perturb
ative
limit

2x104

0.05
0.00
0

Nmol

4x104

0.15

Nmol / N<

5x104

1x104
0
50 . 100 150|B|-1 (µs/G)

200

Figure 3.6: Molecule creation measured after sweeping the magnetic field through the FanoFeshbach resonance. The molecules were produced from 3.4×105 Rb atoms and 2.6×105 K atoms
at T/Tc = 1.5 and T/TF = 0.6. To reduce the density, the atom clouds were released from thetrap 0.4 ms before the magnetic field crossed the Fano-Feshbach resonance. The data were fit toEqn. (3.3.2) returning Γ = 8.5(3) × 10−3 G/µs and f0 = 0.171(4), as defined in the text.

[90]. With this new parameterization, the saturated molecule creation efficiency, f0 , only contains
information about the saturated limit. The molecule creation efficiency in the saturated limit, thelimit of slow magnetic-field sweeps, is governed by the quantum degeneracy of the atom clouds[75, 164]. This will be the subject of Subsec. 3.5.1. As will be discussed in Sec. 3.4, Γ contains information on the perturbative limit. In this limit, which is the limit of fast magnetic-field sweeps, wewill show that the molecule creation efficiency is determined by the Feshbach resonance parameters
and the spatial density overlap of the two clouds.

3.4

Molecule creation in the perturbative limit
For the perturbative limit, or fast magnetic-field sweep rate limit, theoretical treatment ofmagneto-association is discussed in Refs. [25, 89], where the approach begins in the context of a
Landau-Zener model [172]. For the case of two isolated atoms in a box of volume V, the probability of a molecule being created after the magnetic-field sweep through the Feshbach resonanceis P2−body = 1 − exp(−2πδLZ ). The two-body Landau-Zener coefficient for atoms interacting via a43
Feshbach resonance is given by [89]δLZ =1 4π~ abg ∆V 2µ2bḂ

(3.4.1)in the quantization volume V for reduced mass µ2b = (1/m1 + 1/m2 )−1 , background scattering
length abg , and at an inverse magnetic-field sweep rate Ḃ −1 [89]. In the limit of high ramp speed,
the two-body association probability reduces to P2-body ≈ 2πδLZ .
A simple classical probability theory can be used to extend this two-body probability to
an expected number of molecules formed in a many-particle system [25]. In this fast-sweep limit
the probability of creating a molecule from a single atom-pair is small and one can think of theatom populations as large reservoirs of atoms from which a perturbative number of molecules areformed. These reservoirs of atoms enhance the probability of a given atom associating in that each
N1 atoms of one species has N2 possible partners, where N2 is the number of atoms of the otherspecies. Since the Bose-Fermi mixture system is an incoherent mixture of bosons and fermions, thefraction of molecules generated in the perturbative limit is then the incoherent sum of all possible
two-particle combinations of dissimilar atoms (N1 N2 ) multiplied by P2−body and divided by N< :f=

N1 N2 8π 2 ~ abg ∆N1 N2P2−body ≈for f  f0
N<N< V 2µ2bḂ

(3.4.2)Using a local density approximation, we can extend this to the more realistic situation of an
inhomogeneous density distribution. In this approximation, the clouds are divided into boxes ofdifferential volume dV. The number of atoms of species-i in a given volume is ni (~rj ) dV, whereni (~rj ) is the number density of particles of species i at the location ~rj . The expression for themolecule fraction in a fast magnetic-field sweep becomesf==

8π 2 ~ abg ∆ 1 X
1n1 (~rj ) dV n2 (~rj ) dV
2µ2b
N
dV
Ḃ
< j
8π 2 ~ abg ∆ 1 X
n1 (~rj ) n2 (~rj )dV
2µ2b
N<
Ḃ
j

for f  f0(3.4.3)

44,which becomes the following after taking the continuum limit of (3.4.3):
8π 2 ~ abg ∆
hn> i
2µ2b
Ḃ
Γ
=
|Ḃ|

for f  f0f=(3.4.4)(3.4.5)2~where Γ = 8π2µ2b |abg ∆| hn> i and the average majority-species density is hn> i =R

n1 n2 /N< d3 r.

Eqn. (3.4.4) can be further simplified by evaluating the integral assuming gaussian atom
clouds. In general, the two atom clouds can have disparate sizes as well as a relative position offset.
The density product integral is a product of three gaussian integralsZ3

n1 n2 d x = n1,0 n2,0

3 Z
Y

e

2 )
−(xj −δj )2 /(2σ1,j

2 )−x2j /(2σ2,j

e


dxj

(3.4.6)j=1= N1 hn2 iδ=0

3
Y
j=1

s

2
2
2
2
e−δj /[2(σ1,j +σ2,j )]2
2
σ1,j /σ2,j + 1!
,(3.4.7)

where σi,j is the rms width of the species-i cloud in the j -th direction and δi is the relative cloudseparation and hn2 iδ=0 denotes the density overlap integral in the case of perfect cloud overlap.
The expression for Γ from Eqn. (3.4.5) then becomes3Y
8π 2 ~| abg ∆|hn> iδ=0
Γ=
2µ2b

j=1

s

2
2
2
2
e−δj /[2(σ1,j +σ2,j )]2
2
σ1,j /σ2,j + 1!
(3.4.8)where it is important to note that this expression is symmetric upon exchange of species-1 withspecies-2.
Figure 3.7 shows a comparison of our measurements with the theoretically predicted Γ asa function of hn> i. Ultracold clouds of K and Rb were prepared with a range of conditions (1.3
≤T/Tc ≤ 1.7, 0.6 ≤ T/TF ≤ 0.7) in the optical trap with NK ≈ 2 × 105 and NRb ≈ 3 − 5 × 105 .
To vary the atomic density, the clouds were abruptly released from the optical trap and allowed
to expand for 0.1 ms to 0.5 ms before the magnetic field crossed the Fano-Feshbach resonance.
Assuming ballistic expansion, the density overlap integral is reduced from in-trap conditions by up
to a factor of four. The experimental values for Γ were extracted from least-squares fits of molecule
association data as in Fig. 3.6 using Eqn. (3.3.2). In Fig. 3.7 the vertical error bars show the45
0 .0 4

Γ ( G / µs )0 .0 30 .0 20 .0 10 .0 00246〈n > 〉(1 0

8
1 2

1 0
c m

-3

1 2

1 4

)

Figure 3.7: Initial molecule creation rate compared with theory from Eqn. (3.4.8). The initialmolecule creation rate in the limit of fast magnetic-field sweeps is plotted against the average
majority-species density, hn> i. For the Fano-Feshbach resonance parameters given in the text, thetheoretically predicted slope of Γ vs. hn> i is 2.99(12) × 10−15 G/µs/cm−3 (dashed line). The solidline is a weighted least-squares fit to the data, which gives a slope of 3.06(13) × 10−15 G/µs/cm−3 .
The fit has a reduced chi-squared parameter of 0.7.

statistical error (one standard deviation) in Γ from the fit (e.g. from Fig. 3.6) and the horizontal
error bars show the statistical uncertainty in the density.
Using the Fano-Feshbach resonance parameters from Ref. [117] (abg = −185(7) a0 [54] and ∆= -3.34(5) G), the theoretical prediction for the slope of Fig. 3.7 is 2.99(12) × 10−15 G/µs/cm−3 .
The dashed line in Fig. 3.7 shows the theoretical prediction of the slope. A linear fit to the data in
Fig. 3.7 (with an intercept of 0) returned 3060(200) for the slope, where the error includes statistical error and systematic error due to uncertainties in trap frequencies and the imaging system’s
magnification. The data agree well with the theory, which provides evidence that, in the perturbative limit, the number of Feshbach molecules created depends only on the Feshbach resonanceparameters and the density overlap of the two atom clouds. Furthermore, this demonstrates thatby using the parameterization of Eqn. (3.3.2), we can indeed separately examine the perturbative
limit.

46

3.5

Molecule creation in the saturated limit
To understand the molecule formation process in the saturated limit, we employ a phe-nomenological model that successfully described the Feshbach molecule formation efficiency formagneto-association of 40 K-40 K [75], 85 Rb-85 Rb[75], 85 Rb-87 Rb[123], and 6 Li-40 K [142], as well as
the rf-association efficiency of 40 K-87 Rb[174] molecules. This model will be the subject of the next
subsection. Although this model does not include any time-dependent quantities, adiabatic formation of the molecules is assumed. Adiabatic formation requires slowly sweeping the magnetic
field through the Feshbach resonance, which entails spending substantial time at large scattering
lengths where the inelastic losses of atoms and molecules are large. These inelastic losses are notincluded in the model.

3.5.1

Semi-classical simulation of molecule formation

In order to simulate the fraction of Feshbach molecules formed in the saturated, or adiabatic,
limit, the so-called “Semi-classical Phase-Space Sampling” (SPSS) model was used [75, 164]. Thismodel was originally based on the argument that efficiency of molecule formation should be closely
connected to the atoms’ relative proximity in phase space5 . In this model, only pairs of atoms
that are sufficiently close together in phase space can transform into a molecule. The authors ofRef. [75] used the defining relation for the modelµ2b |~vrel | | ~rrel | < γh,2(3.5.1)

where µ2b is the reduced mass of the atom pair, ~vrel their relative velocity, ~rrel their relative spatial
separation, and h is Planck’s constant. As part of the model, a “monogamy” clause is enforced: asingle atom of type 1 is only allowed to bind to a single atom of type 2 and, once the atoms arebound, both are removed from the pool of available atoms. The parameter γ was determined byfitting the model to molecule association data for 40 K-40 K and 85 Rb-85 Rb returning γK = 0.38(4)5Theoretical underpinning has since been provided for the homonuclear case in Ref. [164] and the heteronuclear
case in Ref. [158], but the assumption of thermodynamic equilibrium is made in the theory, which is not the case forour molecules.

47and γRb = 0.44(3), respectively. For this model, the governing parameter is the degeneracy of thesystem, i.e. T/TF and T/Tc for Fermi and Bose systems, respectively. The agreement between thevalues for γ, as well as the good agreement with the model for homonuclear systems, suggests that
this model could be applied to heteronuclear systems. This model was applied to heteronuclear
molecule formation for the 85 Rb-87 Rb system [123], 40 K-87 Rb [174, 116], and more recently in
6 Li-40 K[142].

For this thesis, the model was implemented in a Matlab code and, after calibrating γ, wasused to address the importance of relative atom cloud offsets and provide guidance for improving
the molecules quantum degeneracy. This code was optimized to decrease the computation timethrough vectorizing loops, using vectorized conditional logic, preallocating memory for arrays, andusing static sized arrays. This speed increase allowed for simulations using realistic atom numbers,
instead of scaling up calculations for smaller atom numbers. Details of this simulation and its
implementation are discussed in Appendix A.

3.5.2

Molecule creation efficiency

To look at the saturated molecule fraction, we generated molecules with a range of initial
T/TF values. Parametric heating was employed to vary the cloud temperature with the atom
numbers fixed. The atoms were prepared with the magnetic field 5 G above the Feshbach resonanceon the positive scattering-length side. The magnetic field is then swept linearly down through theresonance at an inverse ramp rate of 200 µs/G and as the field crosses the resonance, the atoms andmolecules are abruptly released from the optical trap and allowed to expand from an optical trapwith a 627 Hz (448 Hz) radial trap frequency and 8.2 Hz (5.9 Hz) axial trap frequency for K (Rb).
The magnetic field continues sweeping down until it is 5 G below the resonance, at which point
the unbound K atoms can be transferred to the |9/2, −7/2i state using a 1.6 MHz wide Adiabatic
Rapid Passage (ARP, c.f. Subsection 2.3.2) 6 . After the ARP, the magnetic field is ramped up to
0.7 G below the Feshbach resonance where the molecules are imaged 1.7 ms in expansion using the6

The molecules are ≈12 MHz bound at this magnetic field and are not addressed by the ARP.

48
0 .4

0 .2

N

m o l

/ N

<

0 .3

0 .1

0 .0

0 .0 00 .2 5

0 .5 00 .7 5
In itia l a to m ic T /T F

1 .0 01 .2 5Figure 3.8: Molecule creation efficiency measured in trap as a function of the initial atomic T/TF .
The molecules were formed from 1.2×105 Rb atoms and 1.6×105 K atoms. The initial temperature
was varied by parametrically heating the gas before sweeping the magnetic field to create molecules.
The solid line is the results of a least-squares fit of the SPSS model to the five points with the largest
initial atomic T/TF returning γ = 0.352(6).

K probe light on the |9/2, −9/2i → |11/20 , −11/2i transition.
We fit the molecule creation efficiency data to the SPSS model using a full-scale simulationthat uses the same atom numbers and trap frequencies as in the experiment. This eschews thefinite-system size corrections 7 employed previously by the authors of Ref. [75]. The SPSS model
is semi-classical in nature and neglects the fermionic statistics of the molecules 8 . Figure 3.8 shows
the result of the fit (solid line) overlaid with our data, where the fit returned γ = 0.352(6). This
result is in agreement with the fermionic atom result of Ref. [75], γF = 0.38(4), but not the bosonicatom result γB = 0.44(3). We fit only the data with the atomic T/TF > 0.75 where the effect of
inelastic collisions between atoms and molecules are smaller because of the lower density of theatom gas. Overall, the model shows good agreement, with the exception of the systematic offset at
low T/TF . At low T/TF the systematic offset is likely due to the effects of inelastic collisions with
unpaired atoms.
7

We find that this correction amounts to a ∼10% correction in γ, which was correctly accounted for by the authorsof Ref. [75].
8Given the temperatures (initial K atom T/TF > 0.4) and molecule numbers, this effect is unlikely to be importantfor our data.

49
1 2 0 0E m o ls / k B ( n K )1 0 0 0
8 0 0
6 0 0
4 0 0
2 0 0
0
0

2 0 0

4 0 0

6 0 0

8 0 0

T a to m s ( n K )Figure 3.9: Measurement of the molecule temperature defined by their expansion energy as afunction of the initial atomic temperature. The solid curve is the simulation results of the SPSS
model and the dashed line is a line with a slope of one.

Another interesting question to ask of this model is how does temperature, or expansion
energy, relate to the initial atomic temperature? This question is particularly important to experiments that use Feshbach molecules to generate absolute ground-state molecules because theground-state molecules are often generated from Feshbach molecules [113]. We find that if we allow the SPSS model code to pair atoms purely randomly by not enforcing Eqn. (3.5.1), then themolecular temperature is equal to the initial atomic temperature. However, when Eqn. (3.5.1) is
enforced, then the molecular temperature is higher than the atomic temperature, as shown in Fig.
3.9.
The data shown in Fig. 3.9 is from the same data set as Fig. 3.8. While the molecular distribution is not thermalized, we use Emols = kB Tmols to express the expansion energy in temperatureunits. The theory curves qualitatively describes the expansion energy of the molecules, but thetheory curve (solid curve) predicts a systematically lower expansion energy. In addition, we plot
the simulation results where the trap frequencies of the Rb and K atoms are forced to be equal(dash-dotted curve). In this case only, the two-atom system can be separated into relative andcenter-of-mass coordinates. As can be seen, the the lack of separability is not a significant factor

50
in determining the molecule expansion energy. This prediction of lower expansion energy by themodel may be due to interaction energy between the molecules and atoms, collisional heating of themolecules, or merely a failure of the SPSS model defined by Eqn. (3.5.1) to accurately describe themolecule momentum distribution. Further investigation is needed to sort out these possibilities.

3.5.3

Experimental concerns

This model of molecule formation serves another use for the experimentalist: determining theimportance of various experimental problems and techniques. Given the dependence on relative
occupation in phase-space, it is reasonable to assume that the molecule creation efficiency will
be decreased by relative offsets in position or momentum of the atom clouds. Adding relative
momentum and positions offsets into the simulation allows one to estimate the importance of these
offsets. Additionally, the effect of ballistic expansion before molecule formation can be modeled.

3.5.3.1

Relative position and momentum offsets

The optical trap utilized for this experiment has a large aspect ratio with weak confinementin the axial direction (typical axial trap frequencies are 5-10 Hz). In the optical trap, Rb and K
experience different trapping frequencies9 . The atoms also have different magnetic moments10 ,which allows for magnetic-field gradients to cause relative position offsets between the atom clouds,
or momentum offsets (i.e. slosh) if the magnetic-field gradient is changed too rapidly. To model
the significance of the effect of axial slosh on the molecule creation efficiency, a momentum offset
was added to the simulated Rb cloud, and none to the simulated K cloud, before the molecule
formation was simulated. The slosh kinetic energy was parameterized as a fraction f of the cloud2
temperature such that mRb vslosh= f kB T , where kB is Boltzmann’s constant and mRb is the Rb

atom mass. Similarly, the significance of position offsets was modeled by offsetting the Rb cloud
before the molecule formation.
9This is due to dissimilar masses, but similar ground-state electric dipole polarizabilities[36, 76]
Rb in the |1, 1i hyperfine state has a magnetic moment of µ = −1/2µB and K in the |9/2, −9/2i has µ = −µB ,where µB is the Bohr magneton.
10

51
0 .4
0 .2 50 .3

0 .1 5/ N

0 .2

0 .1 0

N

N

m o l

m o l

/ N

<

<

0 .2 00 .1
0 .0 50 .0

0 .0 0
0

2

4

6

f = m

v s lo s h / k b T

82(a) Momentum offset

1 0
0

1

2

3

C l o u d s e p a r a t i o n ( σR b )(b) Position offset

Figure 3.10: (a) Simulated molecule creation efficiency with an axial momentum offset between the
Rb and K clouds, and no added position offset other than gravitational sag. The simulation was
run for NK = NRb = 105 and T/Tc = 1.3 with the assumption that the K and Rb clouds were inthermodynamic equilibrium. For conditions in the experiment, f < 0.2 and thus we conclude thatslosh is a negligible concern. (b) Measured molecule creation efficiency as a function of the axial
offset between the Rb and K clouds before molecule production. The atom clouds were separated
by applying an axial magnetic-field gradient that was abruptly turned off before the magnetic-field
was swept across the Feshbach resonance. The solid line is the simulation results using the atom
conditions and the dashed line is a fit of the data to a gaussian.

Results for simulating the effect of axial slosh on the molecule creation efficiency are shownin Fig. 3.10a. The simulation was performed for 105 atoms of each species at T/Tc = 1.3. As canbe seen, the reduction in molecule creation efficiency does not become significant until the sloshkinetic energy is on the order of the temperature. Thankfully the amount of slosh that typicallyoccurs in the experiment corresponds to2
mRb vslosh. 0.2, which has negligible impact on thekB T

molecule creation efficiency.
Heteronuclear systems can also be subject to relative position offsets between the atom clouds
due external forces acting differently on the two atom species. To simulate the importance of thiseffect, distributions with 2.00 × 105 atoms of K and 2.85 × 105 were generated with T/Tc = 1.46
for a K (Rb) radial trap frequency of 627 Hz (448 Hz) with an aspect ratio of λ = 0.0131. Before
the molecule formation code is run, a relative position offset is added to the Rb cloud that is52
a fraction of the calculated Rb gaussian cloud size. The solid curve in Fig. 3.10b is the resultof this simulation. To realize this in the experiment, 2.85(12)×105 Rb atoms in the |1, 1i stateand 2.00(7)×105 K atoms in the |9/2, −9/2i state were prepared at T/Tc = 1.46(5) in an opticaltrap with the same frequencies as the simulation. Because the |1, 1i and |9/2, −9/2i states havedifferent magnetic moments, an axial offset between the two clouds could be produced by applying
an axial magnetic-field gradient across the cloud. The magnetic-field gradient was abruptly shut
off shortly before the magnetic-field sweep to associate the Feshbach molecules. The measured
molecule fraction was then fit to a gaussian (dashed line) returning a rms width of 1.8(1) σRb .
Given that the measured cloud offset in this trap is 30 µm = 0.17 σRb , the reduction in molecule
creation efficiency is likely a small effect in our measurements. In other heteronuclear alkali systems
with large mass ratios (e.g. Li-Cs), this might not be the case.

3.5.3.2

Generating molecules in expansion

The SPSS model does not include the three-body effects. Three-body recombination, as themagnetic field transits through the Feshbach resonance, results in either a weakly or deeply boundmolecule being formed, with its binding energy being converted into the kinetic energy of the third
atom and the newly bound dimer. Strong losses and heating can occur from crossing through the
Feshbach resonance. Since three-body recombination rates in a two-species gas scale as n21 n2 ,reducing the atom density will ameliorate the heating and losses. This reduction in density can be
had by a combination of confining the atoms more weakly , reducing the atom number, working
at higher temperatures, etc. These methods have the drawback of creating a lower signal-to-noise
ratio for measuring molecules. Alternatively, the clouds can be diabatically released from the trap
and allowed to expand. Assuming ballistic expansion, the peak atom densities would decrease byroughly a factor of ten compared to the in-trap density after a millisecond of expansion for the trap
frequencies considered in Fig. 3.11.
Shown in Fig. 3.11 are the results of simulating the dependence of the molecule formation
efficiency on ballistic expansion. Ballistic expansion was simulated by propagating the individual

53(a) ωr,K = 2 π 400Hz(b) ωr,K = 2 π 970Hz

Figure 3.11: Simulated molecule creation after varying periods of ballistic expansion after releasingfrom a harmonic trap with aspect ratio λ = ωz /ωr = 0.0131. The atom clouds were generated
for NK = NRb = 3 × 104 with T/Tc = 1.5 and the assumption of thermodynamic equilibrium
between Rb and K clouds. Note that the discrepancy in molecule creation efficiency between a)
and b) is due to greater significance of gravitational sag in the weaker trap of a).

atom positions according to their velocity for an amount of time (‘time-of-flight’ in Fig. 3.11). As
can be seen in Fig. 3.11, the molecule creation efficiency decreases slowly, even though the peakdensities vary by a factor of roughly 60 (30) for K (Rb) in the case of Fig. 3.11a. One thought as
to why there is such a weak dependence is that as the clouds expand the atoms begin to segregate
in terms of velocity. This creates a “local cooling” effect where atoms in a given location in thecloud have very similar momenta, which prolongs the efficiency of molecule creation. Since Rb
and K atoms have different masses, Rb and K atoms with similar momenta will gradually movefurther apart as the cloud expands thus making association less likely to occur. This simulation
indicates that molecule creation efficiency should remain high well into expansion, although theactual experiment is a bit more complicated due to the finite duration of the magnetic-field sweeps(which is not taken into account by the SPSS model) and interactions during expansion.

3.5.4

Theoretical optimization of Feshbach molecule degeneracy

The SPSS model can also provide guidance for the experimentalist who desires to create
the most degenerate sample of Feshbach molecule given the constraints of their apparatus. This

54
1 .0 04

P E / k B T F , m o ls
K E / k B T F , m o ls

0 .7 5E m o ls / k B T F ,m o ls

R b

/ N

0 .5 0m o lsN( P E + K E ) / k B T F , m o ls

3

0 .2 50 .0 0
0

2

4
N

6
/ N
K

2

0

1

8

0

2

4
N

R b

(a)

6
/ N
K

8

R b

(b)

5

5
P E / ( k B T F ,m o ls )P E / ( k B T F ,m o ls )4

K E / ( k B T F ,m o ls )4

K E / ( k B T F ,m o ls )( P E + K E ) / ( k B T F ,m o ls )( P E + K E ) / ( k B T F ,m o ls )3

E / ( k B T F ,m o l)

E / ( k B T F ,m o l)

3
2
1

2

0

1
0

2

4

6

8

1 0

0

0 .0 10 .1

1f / f0λ = ωz / ωr

(c)

(d)

1 0

Figure 3.12: a) Molecule fraction as a function of the K atom number to Rb atom number ratio for
fixed NRb = 105 and T/Tc =1.05. b) Average molecular potential energy per particles (PE), kinetic
energy (KE), and total energy (PE+KE) compared to the molecular Fermi energy as a function of
NK /NRb . c) Molecular E/EF versus trap frequency for fixed atomic T/Tc and T/TF . d) Molecular
E/EF versus trap frequency aspect ratio.

goal is complicated in mixture systems by the inelastic collisions with unbound atoms. As will beshown in the next chapter, near the Feshbach resonance Rb is the dominant collision partner with
the Feshbach molecules; the collision event rate between Rb and Feshbach molecules is roughly anorder of magnitude higher than K and Feshbach molecules. The general strategy is then to lookfor conditions where the K atom number greatly outnumbers the Rb atom number.

55
1 .6

4 0 0

K
K
R b

R b

1 .2

m o le c u lem o le c u leσv , z ( c m / s )σz ( µm )

3 0 0
2 0 0
1 0 0
0
0

2

4
m

6
R b

/ m

8

1 0

K

(a) σz

0 .8
0 .4
0 .0
0

2

4

6
m

R b

/ m

8

1 0

K

(b) σv,z

Figure 3.13: a) Axial rms cloud size versus the Rb to K mass ratio. b) Axial rms velocity spreadversus the Rb to K mass ratio.

First we consider the case of fixed Rb number, trap frequencies, and T/Tc = 1.05, whileallowing the K atom number to vary. In this simulation, the K and Rb atom distributions are inthermal equilibrium. The radial and axial trap frequencies for K are 627 Hz and 8 Hz, respectively;
Rb has radial and axial trap frequencies of 448 Hz and 6 Hz, respectively. The results of thissimulation are shown in Fig. 3.12 for a range of fixed Rb numbers. As can be seen in Fig. 3.12a,
the Rb conversion fraction favors high K atom numbers, which leaves little Rb left over after
association. This simulation also predicts that the molecule E/EF saturates at its lowest value forhigh K to Rb number ratios, as shown in Fig. 3.12b.
In Figs. 3.12c and 3.12d are the simulation results where the geometry of the trap was varied
for fixed atom numbers (NK = 10 × NRb = 105 ), T/Tc = 1.05, and T/TF = 0.18, in the absenceof the relative vertical offset between the Rb and K clouds due to gravitational sag. In Fig. 3.12c
the trap frequency aspect ratio, λ = ωz /ωr , was fixed, but the trap frequencies for both Rb andK were scaled, whereas for Fig. 3.12d the radial trap frequencies were fixed, but λ was varied. As
can be seen, the molecular E/EF is independent of the frequencies and aspect ratio of the trap.
The result in Fig. 3.12b leaves the question of what is setting the energy of the molecules.

56
To address this question we look at the velocity spread, as well as the cloud sizes, for the atom andmolecule clouds by fitting the momentum and spatial density distributions to gaussian distributions
to extract rms widths. For the case of thermal clouds the velocity and spatial rms widths are
q
q
kB T
kB T
σz = mω2 and σv =m , respectively, for particles of mass m and trap frequency ω. If wevary the Rb atomic mass, while keeping the K atomic mass, trap frequencies, temperature, and
T/Tc fixed with NK = 10 × NRb = 105 , the simulation returns the results shown in Fig. 3.13
for the axial cloud width, σz , and axial velocity spread, σv,z . As can be seen in both Fig. 3.13a,
the molecule widths (violet diamonds) closely follow whichever width is smallest. This result isunderstandable given that the molecule association criterion (Eqn. (3.5.1)) is expressed entirely interms of relative coordinates of velocity and distance. This restriction on the relative separation inposition and velocity means that the distribution with the smaller width will dictate the subsequent
width for the molecules.
One possible way to gain better cloud overlap, and hopefully a more degenerate Feshbach
cloud, might be to adiabatically compress the fermionic atom cloud such that the overlap with thebosonic cloud is better. The results simulating this experiment are shown in Fig. 3.14 using thetrap frequencies listed above for NK = 10 × NRb with T/Tc = 1.05 and T/TF = 0.18 at f0 =627 Hz for K. In this simulation the adiabatic compression is assumed to progress faster than therethermalization time scale with the Rb cloud such that only the K cloud heats as a consequenceof the compression. The reduction of the K cloud size can be seen in Fig. 3.14a, but as thecloud is compressed further, the cloud is heated and the velocity spread increases. As the K cloudis compressed further and heats more, the number of K atoms able to pair with the Rb atoms
decreases and the molecule fraction is lower as a consequence, which can be seen in Fig. 3.14c.
The resulting average energies of the molecules is shown in Fig. 3.14d, where the ensemble
average potential energy (PE) is shown as black squares, average kinetic energy (KE) as red circles, the average total energy (PE+KE) as green upwardly pointing triangles, and the computed
molecule TF as blue downwardly pointing triangles 11 . As the K cloud is compressed further, the11

For fr,K /f0 < 0.5 relative gravitational sag between the Rb and K clouds becomes important.

57
1 .2 52 5 0

1 .0 0R b

2 0 0

m o le c u le

R b

0 .7 5m o le c u le

/ N

1 5 0

<

0 .7 5

1 0 0

0 .5 0m o lsσv , z0 .2 55 0
0

0 .0 00 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0
f r,K / f 0

Nσz ( µm )1 .0 0KK

0 .2 50 .0 00 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0
f r,K / f 0(a)

0 .5 00 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0
f r,K / f 0

(b)

(c)

5
P E / k B

1 2 5 0K E / ( k B T F , m o ls )4

(P E + K E ) / k B

1 0 0 0( P E + K E ) / ( k B T F , m o ls )T F ,m o l

3

E / ( k B T F ,m o l)

E / k B (n K )P E / ( k B T F , m o ls )K E / k B

7 5 0
5 0 0

2
1

2 5 0
0

0 .0

0 .5

1 .0

1 .5
2 .0
f r,K / f 0
(d)

2 .5

3 .0

0

0 .0

0 .5

1 .0

1 .5
2 .0
f r,K / f 0

2 .5

3 .0

(e)

Figure 3.14: a) Axial rms cloud size for K atoms (blue triangles)), Rb atoms (red circles), andmolecules (violet diamonds) versus the frequency of the compressed K trap. b) Axial velocity
spread for K atoms, Rb atoms and molecules. c) Molecule conversion fraction. d) Molecularpotential energy (PE), kinetic energy (KE), total energy (PE+KE), and Fermi temperature (TF ).
e) Ratios of PE, KE, and (PE+KE) to the Fermi temperature.

Fermi temperature of the molecules increases since the reduction in molecule creation efficiency
is weak. While the average molecular kinetic energy is only weakly affected by the compression,
the potential energy, and the total energy, increases dramatically with the compression. Hence theratio (PE+KE)/kB TF,mols increases with the trap frequency and never approaches the value for adegenerate Fermi gas: PE+KE = 43 kB TF,mols [132]. Therefore, even if the molecules were able tothermalize with a bath and survive the collisions, compression would not lead to a more degenerate
molecular gas.

Chapter 4
The search for Efimov Physics in 40 K-87 Rb

In addition to the two-body bound state induced by the Feshbach resonance, in many atomic
systems there also exist three-body (and 4-body [153, 56, 170, 129] and perhaps even higher n-body[67, 152, 151]) states in the vicinity of the Feshbach resonance. These other bound states are aconsequence of the Efimov effect, which was originally discussed in terms of bound states amongst
three nucleons (i.e. the 3 H nucleus) and three α particles (i.e. the 12 C nucleus) [48]. The Efimov
states arise from strong two-body interactions near the Feshbach resonance. Curiously, the Efimov
bound states exist on both sides of the resonance, for positive and negative two-body scattering
lengths. Another remarkable feature is that the Efimov states exist not as a single bound state,
but as a theoretically semi-infinite ladder of states with a discrete scaling symmetry between the
different rungs.
Signatures of Efimov trimers were not observed until 35 years after Efimov’s prediction; theobservation was made in an ultracold gas of cesium by the Innsbruck group [92] through resonantly
enhanced three-atom inelastic collisions. Following the Innsbruck group, Efimov resonances wereobserved in 39 K by the LENS group [170], in 7 Li by the Rice [129] and Bar-Ilan [64] groups, in athree spin-state mixture of 6 Li by the Penn State [165] and Heidelberg [163] groups, and recently
in a 41 K – 87 Rb mixture by the LENS group [7]. In 39 K, and later in 7 Li, the log-periodic scaling of
the Efimov spectrum in 39 K was measured from the locations of the first two Efimov features. The
authors of Ref. [170] measured a scaling of 25(4) in agreement with theory (22.7). Additionally,
the energy of the lowest Efimov state has been mapped out for positive scattering lengths through59
rf-association in a three spin-state mixture of 6 Li[101, 111] as well as in 7 Li[105].
One of the remarkable discoveries of this line of research is with regards to the location of thefirst Efimov bound state. Originally it was thought that the three-body parameter that sets the firststate’s location was a mixture of two- and three-body physics and therefore a priori calculation of
the location was not possible. In 7 Li, Efimov resonances where found near two Feshbach resonancesthat involved different atomic hyperfine states and the locations of the associated Efimov resonanceswere found at the same location [65]. Additionally, the Innsbruck group measured the locations of
the first Efimov resonance at four Feshbach resonances and found their locations to clustered about
an average far more tightly than the scaling factor of 22.7 in scattering length between different
rungs of the Efimov state ladder [12]. These two observation suggested that the location of thefirst Efimov resonance was in fact determined by two-body physics (i.e. the length scale of thetwo-body potential, the van der Waals length rvdW ) and the authors of reported the locations to(1)

be a− = −9.1(4)rvdW . This view of the three-body parameter being universally determined by
two-body physics was corroborated in other homonuclear gas systems [23] and provided theoreticalsupport for the case of broad Feshbach resonances [23, 155]. One remaining question is whether
the three-body parameter in heteronuclear systems is also universally determined by rvdW . Since
40 K – 87 Rb and 41 K – 87 Rb have very similar values of rvdW ,our system is well positioned toaddress the question of a universal three-body parameter.
In this chapter we will discuss our observations of Efimov resonances in a mixture of 40 K
and 87 Rb atoms. The contents of this chapter are the subject of a manuscript in preparation [28]for publication. Section 4.1 will review fundamental physics of Efimov resonances and Section 4.2
will give a brief overview of how Efimov resonances are observed through inelastic collisions rates.
In Sections 4.3 and 4.4 discusses our analysis methods to extract the inelastic collision rates fromthree-body recombination and atom+molecule relaxation, respectively. Next the observations will
be compared against theoretical relations between different features in Section 4.5 and lastly we willcompare the Efimov resonances in 41 K-87 Rb and 40 K-87 Rb in Section 4.6 to address the questionof a universal three-body parameter in heteronuclear systems.

60

4.1

The Efimov effect and discrete scaling
In 1970 Efimov derived the effective attractive three-body interaction in the limit of strong

two-body interactions and the associated ladder of three-body bound states that bears his name.
Efimov derived the effective potentials for the three identical bosons, where the lowest-energy
potential is an attractive 1/R2 and the infinite number of higher potentials are repulsive 1/R2
potentials [18]. The ladder of states is a consequence of the attractive 1/R2 three-body potentialthat arises for certain collision partners (e.g. three identical bosons, more discussion below) in
the limit of strong repulsive or attractive interactions. For the general case of three particles withunequal masses, with at least two particles interacting strongly, the effective three-body potentialis [18]Vef f (R) = −(s20 + 14 ) ~2 (m1 + m2 + m3 ), rvdW  R  |a|.
2(m1 m2 + m2 m3 + m3 m1 ) R2

(4.1.1)

where R is the hyperradius1 and the parameter s0 characterizes not only the potential strength,
but also the periodicity of the Efimov ladder. The values that s0 assumes depend strongly on themasses and spin statistics of the particles2 . An interesting aspect of the Efimov effect is that thethree-body system in the limit of strong two-body interactions exhibits a discrete scaling symmetry.
One part of this symmetry is that in the limit a→ ±∞, the trimer binding energies corresponding
(n)

(n+1)

to different bound states are related by ET /ET= exp (−2π/s0 ). This would seem to indicatethat the trimer binding energies are unbounded, but as part of the derivation for the Efimov effectthe assumption that |a|  rvdW is made. This enforces a minimum length scale given by rvdW , or2alternatively, an energetic cutoff ~2 /(2µ2b rvdW
) of the Efimov trimer binding energy ladder; anytrimer states past the energetic cutoff would no longer follow the discrete scaling symmetry.
This energetic cutoff terminates the ladder of trimer states from below. The energy of thelowest Efimov state is defined by the three-body parameter, κ∗ , which has units of wavenumber.
1

The hyperradius is rms size of the three-body system.
For the general three-body system R2 =2
2
2
m1 m2 r12 + m2 m3 r23 + m3 m1 r31[18].
m1 m2 + m2 m3 + m3 m1
2
For example for three strongly interacting equal mass particles, s0 ≈ 1.00624. However, for two identical bosons
colliding with a third equal mass non-identical particle with only two of the three scattering lengths being large, thens0 = 0.4137.

61
Collision typeaij large

aij , ajk large

aij , ajk , aki large

BBB

N/A

N/A

Yes

BBX

No

Yes

Yes

XYZ

No

Yes

Yes

FFX

No

Yesa

N/A

a

This is true iff mF /mX > 13.61 [71, 41, 127].

Table 4.1: Classes of collisions and whether or not Efimov
states exist when one, two, or all three inter-particle scattering
lengths (aij , ajk , aki ) are large compared to rvdW , the lengthscale of the two-body potential.

The three-body parameter is defined such that(n)ET

→ exp (−2nπ/s0 )~2 κ2∗
as a → ±∞.
2µ2b(4.1.2)where µ2b is the two-body reduced mass.
In Ref. [18], the authors derive the trimer binding energy for the three identical bosons
case. The general three-body system consists of three atoms with masses m1 , m2 , and m3 with any
mixture of bosons, fermions, and distinguishable particles. We can classify three-body systems into
those with two identical bosons and a distinguishable particle (BBX), two identical fermions anda distinguishable particle (FFX), three identical bosons (BBB), three identical fermions (FFF),
and three distinguishable particles (XYZ). The general three-body system then has three interparticle scattering lengths (a12 , a23 , and a31 ). The Efimov effect occurs only when two or more of
the inter-particle scattering lengths are large compared to the two-body potential range [2, 49], assummarized in Table 4.1. In our system we have both FFX and BBX collisions, but, given thatmK /mRb = 0.46 < 13.61, only the BBX collision channel supports Efimov states.
The trimer binding energy ET for three identical bosons of mass m is given by [18]
(n)

ET +

~2
~2 κ2∗−2π/s0 n=(e)exp[∆(ξ)/s], n = 0, 1, 2, . . .
0
m a2
m

(4.1.3)62
where the angular argument, ξ, to Efimov’s universal function, ∆(ξ), is defined bytan(ξ) = −(m ET /~2 )1/2 a.

(4.1.4)

The angular argument ξ spans the range from ξ = −π, corresponding to three-atom thresholdat ET = 0, to ξ = − π4 , where the trimer energy reaches the atom+dimer threshold for which2

ET = m~a2 . The function ∆(ξ) is not known to have an analytical form and has been calculated
numerically for identical bosons in Ref. [19] where ∆(ξ) is found to vary monotonically from∆(−π) = −0.89 to ∆(−π/4) = 6.04. There does not exist a similar published calculation for the
BBX system, but to get a qualitative picture of what the Efimov trimer energies look like in oursystem we will adapt Eqn. (4.1.3) by replacing the mass m with twice the two-body reduced mass,
2µ2b . Eqn. (4.1.3) then becomesET +~2~2 κ2∗−2π/s0 n, n = 0, 1, 2, . . .
=
(e
)
exp
[∆(ξ)/s]
02µ2b a22µ2b(4.1.5)The value for s0 can be calculated for the BBX system from a transcendental equation:[18]2 sinh[s0 (π/2 − γ12 )]=0(4.1.6)s0 sin(2γ12 )1/2
1 +m2 +m3 )where the angle γ12 satisfies tan γ12 = m3 (mm. For this system m1 = m2 = mRb , m3 =1 m2
cosh(s0 π/2) −mK , a23 = a31 = aKRb = a, and a12 = aRbRb ; Eqn. (4.1.6) gives s0 ≈ 0.65317 for 87 Rb – 87 Rb – 40 K.
The trimer binding energy for the lowest two Efimov states is shown in Fig. 4.1 for 87 Rb – 87 Rb – 40 K
using the published calculation of ∆(ξ)[19, 18] for identical bosons3 . In the figure the lowest twotrimer energies (solid orange curves) are plotted as a function of 1/a. For this figure, E = 0 represents the three-atom threshold, above which the system consists of three colliding atoms. The
universal dimer binding energy (black dashed curve) demarcates the atom-dimer threshold, where
between the atom-dimer threshold and E = 0 the system consists of a mixture of dimers and atoms.
(n)

(n)

At a scattering length a− < 0, the n-th trimer state terminates at ET = 0. ET reaches the atom(n)dimer threshold at a scattering length a∗(n)> 0. Additionally at scattering lengths a+> 0quantum mechanical interference occurs, which will be discussed below in Section 4.2.
3

Although this does not give the correct scalings between features on opposites sides of the Feshbach resonance,
it does give the qualitative features of the Efimov states.

63
E

(1)

1/a*

(1)

(1)

1/a-

n=1

1/a

1/a*
n=2

n=1

Figure 4.1: Schematic of the binding energy of the lowest two Efimov states (solid orange curves)plotted along with the dimer binding energy (dashed black curve) for s0 = 0.65317. The inset isa magnified view of the origin showing where the lowest Efimov state (n = 0) reaches the atom(0)
threshold (a = a− ) as well as the entirety of the next-to-lowest Efimov state(n = 1).

Figure 4.1 shows the discrete scaling symmetry characteristic of Efimov states. This scaling
symmetry involves a particular value of κ∗ where the system is unchanged by transformations of
the formκ∗ → κ∗ , a → ζa, ET → ζ −2 ET(4.1.7)

where the scaling factor ζ = exp(π/s0 ) ≈ 122.7 for 87 Rb – 87 Rb – 40 K. This symmetry creates a
relation between the n-th and (n+1)-th trimer state of the form
(n+1)

(n)

as well as for a∗

(n)

ET

(a) = ζ −2 ET (ζa)

a∗

(n+1)

= ζa∗

(n+1)

= ζa− ≈ 122.7a− .

(4.1.8)(n)

and a−

a−(n+1)

The relation between a∗

(n)

(n)

(n)

(n)

(4.1.9)(n)(4.1.10)≈ 122.7a∗

(n)

(n)

and a− , as well as a+ and a− , was calculated in Ref. [72] to be
(n+1)

a∗

(n)

(4.1.11)(n)(4.1.12)/|a− | = 0.51(n)|a− |/a+ = eπ/2s064using a zero-range (i.e. rvdW = 0) effective field theory.
As one might expect, there are limitations to the number of Efimov states we can observe due
to a finite range of accessible scattering lengths. The lower limit is imposed by the aforementioned
limit of strong interactions, namely a  rvdW . The upper limit is essentially set by the point where
the scattering length becomes of order of the interparticle spacing. This limit corresponds to theppoint kthermal a ∼ 1, where kthermal = 2m3body kB T /~2 [43] where the three-body reduced massq
1 m2 m3
is defined as m3body = mm[41]. For the data shown below, the temperature is typically
1 +m2 +m3
250 nK, which means that the scattering lengths we can span are limited to rvdW = 72 a0 < |a| <3000 a0 . Given this limitation and the scaling factor of 122.7, we will at best be able to measure
one feature each from the a− , a∗ , and a+ series.

4.2

Observation of Efimov resonances
In order to address the question of whether the three-body parameter for heteronuclear system

is universal, we set out to look for Efimov resonances. Observation of Efimov resonances can bemade through their influence on inelastic collision rates [10, 17, 53, 114, 18] since the presenceof the trimer state increases the likelihood of several particles being close together. One suchinelastic process is three-body recombination where three atoms collide resulting in the creation ofa molecule. In atomic systems, there exist both shallowly bound molecule states (i.e. a Feshbach
molecule, for a > 0 only) as well as deeply bound molecule states, and so three-body recombinationof particles A, B, and C in general has two possible paths:
A + B + C → (AB)∗ + C(4.2.1)→ (AB) + C(4.2.2)where (AB) denotes a deeply bound molecule and (AB)∗ a shallowly bound molecule. This collision
releases an amount of energy equal to the binding energy of the molecule, which is carried awayin the kinetic energy of the atom and molecule. Additionally, atom-molecule relaxation can occur
where the shallowly bound dimer can collide with an atom and vibrationally relax into a more65
deeply bound state. This results in(AB)∗ + C → (AB) + C,

(4.2.3)where again the binding energy difference between the deeply and shallowly bound molecule iscarried away by the kinetic energy of the molecule and atom.
For the BBX system, the three-body recombination event rate per unit volume is of the form
αn2B nX , where the nB is the density of identical bosons and nX is the density of the distinguishable
species. The event rate coefficient, α, is strongly dependent upon the scattering length and is definedby density loss rate equations4 .
dd
nB = 2 nX = −2α n2B nX .
dt
dt

(4.2.4)

Similarly, the atom-dimer relaxation rate is defined in terms of an event rate constant, β,ddnA = nD = −βnA nD
dt
dt(4.2.5)where nA denotes the density of atoms and nD the density of weakly bound molecules.
The authors of Ref. [72] derive expressions for the recombination rate for positive and negative
scattering length in the BBX system with the assumption that the molecule binding energy has
the universal form ED = ~2 /(2µ2b a2 ), where µ2b is the two-body reduced mass. The recombination
coefficient is different for positive and negative scattering lengths in part because recombination
into shallow dimers is possible only for positive scattering lengths. The three-body recombination
coefficient, α, has the piece-wise formα= αs + αd(+)

a>0(4.2.6)(−)

a<0(4.2.7)



αd

(±)

with contributions from shallow (αs ) and deep dimers (αd ). The coefficient for recombination
into shallow dimers is [72]αs = C(δ)
4

D {sin2 [s0 ln(a/a+ )] + sinh2 η∗ } ~ a4
sinh2 (π s0 + η∗ ) + cos2 [s0 ln(a/a+ )] mX(4.2.8)This expression assumes that two bosonic atoms are lost per distinguishable atom. The ratio of loss can vary incases where secondary collisions result in further loss. For example, if three-body recombination results in the weakly
bound molecule remaining trapped, but the third atom not, then the molecule can collide with another atom andvibrationally relax with the loss of another atom. The loss ratio in this case can be between 1:1 to 3:1.

66√
with D = 128π 2 (4π −3 3 ) and C(δ) is a coefficient that depends upon the mass ratio δ = mX /mB .
For 87 Rb – 87 Rb – 40 K collisions, δ ≈ 0.46 and C(δ)≈ 0.037 [72]. Eqn. (4.2.8) exhibits minima5 at
(n)

(n)

(n−1)

scattering lengths a = a+ with the log-periodic spacing of a+ = exp(π/s0 ) a+

. The expression

for recombination into deeply bound dimers [72] is(+)

αd= C(δ)

D coth(π s0 ) cosh(η∗ ) sinh(η∗ )~ a4,
2sinh (π s0 + η∗ ) + cos2 [s0 ln(a/a+ )] mX(4.2.9)which exhibits weak local extrema since sinh2 (π s0 )  1 ≥ cos2 [s0 ln(a/a+ )]. The coefficient forrecombination into deep dimers for negative scattering lengths is [72]
(−)

αd

=

C(δ)
D coth(π s0 ) sinh(2η∗ )
~ a4
.
2
2
2 sin [s0 ln(a/a− )] + sinh (η∗ ) mX

(4.2.10)

A plot of α is shown in Fig. 4.2 for a variety of values for η∗ assuming a− = −1000a0 and therelation between a− and a+ from Eqn. (4.1.12). The measured values of η∗ span the range from
0.016 (in 6 Li [165]) to 0.251 (in 7 Li [65]).). From this plot, one can see the log-period scaling factor(−)

of 122.7 as well as the higher sensitivity of αd(+)

to η∗ compared to αdand αs .

The other inelastic collision process that exhibits strong Efimov features is atom-dimer relaxation. The atom-molecule relaxation coefficient for a BBX system is
β = 2πC2 (δ)~aδ(δ + 2)sinh(2η∗ ),δ + 1 sin2 [s0 ln(a/a∗ ) + sinh2 η∗ mX(4.2.11)

where C2 (δ) = 2.08 for 87 Rb – 87 Rb – 40 K [72]. The coefficient β exhibits strong peaks at a =(n)a∗ , where the trimer binding energy intersects the atom-dimer threshold. Even though α is(n)expected to lack sharp features at a = a∗ , various experiments have observed peaks in threebody recombination on the a > 0 side, which were attributed to secondary relaxation events where
recombined shallow molecules relaxed into deeply bound molecules through collisions with a third
atom [129, 170, 7, 104, 105]. Although recent theoretical work questions this interpretation [95],in 7 Li the location of this secondary collision peak agrees with a∗ measured by mapping out thetrimer binding energy via rf-association [105].
5
The source of these minima are an interference effect. Essentially, the incoming hyperradial wave function is asuperposition between a state that feels the lowest repulsive 1/R2 potential and an atom-molecule state, which is in
the attractive 1/R2 potential. The part of the wavefunction in the repulsive 1/R2 potential scatters at short distancesand makes a nonadiabatic transition at long distances (R ∼ a) into the atom-molecule state where it destructively
interferes with the part of the wavefunction in that was originally in that state[18].

67η* = 0 . 0 1

1 0 -1 4

1 0 -1 4

η* = 0 . 1
η* = 1a (- 2 )1 0 -1 8

α(c m

6

/s )

1 0

a (+ 2 )-1 8

1 0 -2 2

1 0 -2 6

1 0 -3 0 2
1 0

1 0 -2 2

a (+ 1 )

1 0 3

1 0 4

1 0 5

-1 0 5-1 0 4a (- 1 )1 0 -2 6-1 0 3

1 0 -3 0-1 0 2a (a 0)Figure 4.2: Three-body recombination event rate coefficient as a function of scattering length. The
solid line corresponds to a value for the inelasticity parameter η∗ = 0.01. The dashed and dashdotted correspond to η∗ = 0.1 and η∗ = 1, respectively. Measured values of η∗ span the range from
0.016[165] to 0.251[65].

4.3

Analysis of loss data to extract α
The process to measure α at a given magnetic field is fairly simple. We begin with a mixtureof 87 Rb atoms in the |1, 1i hyperfine state and 40 K atoms in the |9/2, −9/2i hyperfine state withthe magnetic field far from the Feshbach resonance where the inelastic loss time scales are longcompared to our optical trap lifetime. The magnetic field is then ramped closer to resonance andheld for a duration thold , as shown in Fig. 4.3a. At this field, the atoms undergo enhanced inelastic
collisional loss. After the hold, the atoms are released from the optical trap (indicated by thevertical dashed line) and the magnetic field is ramped away from the Feshbach resonance. The
remaining atoms are then imaged using standard absorption imaging, from which we extract theatom numbers and cloud sizes.
Fig. 4.3b shows a measurement of α at 548.695 G (a = -460 a0 ), for initial in-trap densityweighted densities of 1.2 × 1013 cm−3 and 3.3 × 1012 cm−3 for Rb and K, respectively. The curves

68

thold
Feshbachresonance

N u m b e r o f r e m a in in g a to m smagnetic fieldB(t)4 x 1 0 5
3 x 1 0 5
2 x 1 0 5
1 x 1 0 5
0
0

1

2

3

4

t h o ld ( s )time(a) Magnetic-field sequence(b) Number loss measurement

Figure 4.3: (a) Diagram of the magnetic-field sequence for measuring α. The magnetic field starts
far from the Feshbach resonance and is ramped towards resonance where it is held for a duration
thold . After the hold, the remaining atoms are released from the optical trap (vertical dotted line)
and the magnetic field is ramped away from resonance. (b) Measurement of α at 548.695 G (a =-460 a0 ) with initial in-trap density-weighted densities of 1.2 × 1013 cm−3 and 3.3 × 1012 cm−3 for
Rb and K, respectively. The solid lines are from a dual-species fit described below. The fit returned
α = 15.3(6) × 10−27 cm6 .

correspond to a fit to the model developed below. We extract α from the atom number andcloud sizes using methods similar to those outlined in Refs. [21, 120]. To develop the threebody recombination model let’s consider the K loss rate first. To begin we make the local density
approximation for Eqn. (4.2.4) and divide through by NK (t0 ) yielding
n2Rb (r, t0 )nK (r, t0 )1 dnK (r, t0 )=−α.
NK (t0 )dt0
NK (t0 )(4.3.1)

Next we integrate both sides over all space
1 dNK (t0 )d ln(NK (t0 ))== −αNK (t0 ) dt0
dt0

Z

d3 r

n2Rb (r, t0 )nK (r, t0 )NK (t0 )(4.3.2)and then integrate with respect to time, which results ind ln(NK (t0 ))
dt
= ln
dt0
0

Z t



NK (t)NK (0)

Z t


= −α

0

dt
0

Z

d3 r

n2Rb (r, t0 ) nK (r, t0 ).
NK (t0 )(4.3.3)

69
Similarly for the Rb loss rate we findlnNRb (t)NRb (0)

Z t


= −2α

dt

0

0

Z

d3 r

n2Rb (r, t0 ) nK (r, t0 ).
NRb (t0 )(4.3.4)The spatial integrals on the right-hand side of Eqns. (4.3.3) and (4.3.4) are evaluated assuming a
gaussian distribution. The expression for the spatial integral on the right-hand side of Eqns. (4.3.3)
and (4.3.4) is thenZ

d3 r n2Rb (r, t0 ) nK (r, t0 ) =

2
4
σz,Rb
(2π)3 σr,Rb

2 (t0 )NK (t0 )NRbq,2
22 /σ 2
2σr,K /σr,Rb + 1
2σz,Kz,Rb + 1(4.3.5)where σr,i and σz,i are the radial and axial rms cloud widths at time t0 , respectively, for species
i. The time integrals are then discretely evaluated by a linear interpolation of the spatial integral
at each measurement. For a function f (t), the discrete evaluation of its integral with respect to t
from t1 to tm is
Z tm
t1

dt0 f (t0 ) ≈

m−1
X
i=1

f (ti+1 ) + f (ti )(ti+1 − ti ).
2(4.3.6)

A value of α is then extracted by performing a simultaneous least-squares fit of Eqns. (4.3.3) and(4.3.4) to a line, using Eqn. (4.3.6) to evaluate the time integrals with the measured atom number
and sizes as inputs.
The results of these measurements are shown in Fig. 4.4 as a function of the magnetic field.
For this Fano-Feshbach resonance, magnetic fields above 546.6 G or below 543.5 G correspond tonegative scattering lengths. The data were taken with an initial temperature of T ≈ 250 nK as
measured before the atoms were brought to the hold field. Other than the Fano-Feshbach resonanceat 546.6 G, the only dramatic feature is the peak at 547.35(10) G 6 .

4.3.1

Identifying the feature at 547.4 G

The feature at 547.35(10) G could be due to a narrow Fano-Feshbach resonance or an Efimov
resonance. Ref. [84] predicts a narrow d-wave Feshbach resonance (which couples to the s-wave
scattering length) near the s-wave resonance, although the predicted location is roughly 1 G higher
6

The 0.1 G width of this feature is probably due to the gas temperature and the experimental magnetic-field
resolution.

70

1 0 -2 1

αR b R b K ( c m

6

/ s )

1 0 -2 3
1 0 -2 5
1 0 -2 7
1 0 -2 9
5 4 5

5 4 6

5 4 7

5 4 8
5 4 9
m a g n e tic fie ld ( G )

5 5 0

5 5 6

Figure 4.4: Measured three-body event rate coefficient, α, as a function of magnetic field.

than where we observe the peak in α. Zaccanti et al., in studies of interactions between 40 K and
87 Rb, also observed a peak in number loss at 547.4(1) G [171] and they state that their model ofthe KRb potentials predicts a narrow Feshbach resonance at this field.
To see if the feature could be consistent with an Efimov resonance, we can also compare theshape of the observed feature in three-body recombination to the prediction of Eqn. (4.2.10), as
shown in Fig. 4.5. In Fig. 4.5, the points correspond to the B > 546.6 G data replotted in termsof scattering length, a(B), is given bya(B) = abg 1 −∆B − B0(4.3.7)and is calculated by averaging the scattering lengths from the Feshbach resonance calibrationsof Refs. [86] (abg = −187 a0 , B0 = 546.618 G, and ∆ = −3.04 G) and [117] (abg = −185 a0 ,B0 = 546.71 G, and ∆ = −3.34 G). The hollow points are for scattering lengths where a > 1/kthermal
and are excluded from the fit (solid curve). The least-squares fit of the data for |a| < 890 a0 to Eqn.
(4.2.10) returns η∗ = 0.0167(7). For this fit, the log-periodic series of resonances in Eqn. (4.2.10)(1)

was truncated such that for scattering lengths |a| ≤ e−π/(2s0 ) |a− |, α = Ca4 . The constant C was

71
1 0 -2 1

αR b R b K ( c m

6

/ s )

1 0

a x ia l c lo u d s iz e ( p ix e ls )

2 5 0
-2 3

1 0 -2 5
1 0 -2 7

fit, a -(1 )= -3 x 1 0

4

a 0

η* = 0 . 0 1 7 , a - ( 1 ) = - 1 0 2 4 a 0

1 0 -2 9-1 0 4-1 0 3a (a 0)-1 0 2

Figure 4.5: Comparison of negative scattering length data from Fig. 4.4 to a fit ofEqn. (4.2.10) (solid curve) and a plot of Eqn.
(4.2.10) assuming η∗ = 0.017 (dash-dottedcurve). The fit was restricted to |a| < 890a0(1)
with a− = −3 × 104 a0 and returned η∗ =0.0167(7).

2 0 0
1 5 0
1 0 0
5 0
0

0 .0

0 .5

1 .0

1 .5
t h o ld ( m s )

2 .0

2 .5

Figure 4.6: Axial cloud sizes for the Rb (redcircles) and molecule (violet diamonds) clouds
as a function of the hold duration in Fig. 4.7a.

(1)

determined by matching to Eqn. (4.2.10) at a = e−π/(2s0 ) a− . For comparison Eqn. (4.2.10) is
plotted (dash-dotted line) with the same value of η∗ , but with a− = −1024a0 , which is centeredon the points at 547.36 G. As can be seen, the peak in the data is substantially narrower than thepeak of Eqn. (4.2.10) and attempts to use lower values of η∗ in order to make the peak narrower
only result in worse agreement at low scattering lengths (i.e. |a| < 890 a0 ). Thus, we see that thethree-body recombination feature at 547.36(10) G is not consistent with the expected shape of an
Efimov feature. Therefore, we conclude that this peak is likely to be due to a narrow Feshbachresonance.

4.4

Extracting β from atom-molecule inelastic collisional loss
We extract values for β in a similar fashion to what was done for α. The analysis has to bechanged slightly since when we image the molecules, we dissociate the molecules first by ramping
the magnetic field to the unbound side. The molecule number is measured from the previously
bound K atoms, but the Rb atom signal includes both bound and unbound atoms. Eqn. (4.2.5)

72
needs to be slightly modified: nRb , the unbound Rb density, is replaced by nRb = ñRb − nmol . The
quantity ñRb is the density of bound and unbound Rb atoms, which what our Rb absorption images
measure. The density rate equations for a homogeneous gas then becomeddñRb = 2 nmol = −2β nmol (ñRb − nmol ).
dt
dt(4.4.1)After making a local density approximation integrating over space, Eqn. (4.4.1) results in
dNmol
= −β
dt

Z

d3 r nmol (ñRb − nmol ).

(4.4.2)

We now make the approximation that the atom and molecule density distributions are gaussians
at all times. The integrals on the right-hand side of Eqn. (4.4.2) becomes


√
Z


N 2 (t)
8 NRb (t)/Nmol (t) . (4.4.3)r−
1
d3 r nmol (ñRb − nmol ) = 3/2 2 mol
228π σr,mol (t)σz,mol (t)  σr,Rb
σz,Rb
(t)
(t)
+
1
+
1
2
2
σ
σ
(t)
(t)
r,mol

z,mol

where the NRb is the total Rb number (i.e. both bound and unbound Rb) and the radial andaxial rms sizes for species i are σr,i and σz,i , respectively. The radial and and axial rms cloud sizes
are measured and change in time due to effects such as anti-evaporation 7 . We observe that thecloud sizes initially vary linearly with time (c.f. Fig. 4.6); we fit the initial slope of σ vs. t andincorporate this into the numerical solution of Eqn. (4.4.2). With the cloud sizes and atom andmolecule numbers as inputs, we fit the data to Eqn. (4.4.2), using Eqn. 4.4.3, to extract a value ofβ.
A sample measurement and the associated magnetic field trajectory are shown in Figs. 4.7b
and 4.7a, respectively. We create molecules by sweeping the magnetic field across the Feshbachresonance. The magnetic field continues to 2G below the Feshbach resonance where the molecules
are so deeply bound that they can be distinguished from atoms. The molecules at this field havea binding energy of approximately 2 MHz [174]. This allows for unwanted atoms to be cleared
7

Given the strong dependence on density, three-body recombination preferentially removes atoms at the highdensity region near the trap center. Three-body recombination depletes the cloud of atoms with low potential energy,
compared to the ensemble average, and therefore the new ensemble average potential energy is higher than before[160]. This process is termed “anti-evaporation”.

73
1 .5 x 1 0 5
1 .2 x 1 0 5
9 .0 x 1 0 4

thold

n u m b e r

magnetic fieldB(t)6 .0 x 1 0 4
3 .0 x 1 0 4"kill"
pulse

0 .0
0 .0

time
(a) Magnetic-field sequence

0 .5

1 .0

1 .5
t h o ld ( m s )

2 .0

2 .5

(b) Rb+molecule collisions

Figure 4.7: (a) Magnetic-field sequence for measuring atom-molecule collisional loss. First molecules
are generated by sweeping slowly through the Fano-Feshbach resonance (dash-dotted line). Next,
unwanted remaining atoms are heated out of the trap with resonant light. For removing K, wemerely apply light resonant with |9/2, −9/2i − |11/20 , −11/2i, but for Rb we ARP to the |2, 2i
hyperfine state and apply light resonant with |2, 2i − |30 , 3i. The magnetic field is then ramped toanother field and held for a duration thold after which optical trap is extinguished. While the atoms
and molecule are expanding, the molecules are dissociated by ramping back across the resonance.
(b) Measured Rb (red circles) and molecule number (violet diamonds) after holding at 545.14 G (a= 200 a0 ) with an initial temperature of 450 nK and a Rb density-weighted density of 2×1012 cm−3 .
The molecule number is determined from imaging K atoms after dissociation. The measured Rb
number after the dissociation of the molecules is the sum of the molecule number and Rb atoms
that were never bound, which is the reason for the 2:1 Rb number to molecule number loss.

out of the trap with a combination of resonant light and rf. We can then selectively examine
collisions between molecules and Rb atoms in the |1, 1i hyperfine state, K in the |9/2, −9/2i state,or distinguishable atoms, where 40 K atoms in the |9/2, −7/2i state served as our distinguishable
atoms.
Shown in Fig. 4.8 are the results of measuring the atom-molecule loss rates for a wide rangeof magnetic fields. The data represented as circles are Rb ( in the |1, 1i state) colliding with themolecules, the triangles are K (in the |9/2, −9/2i) colliding with the molecules 8 , and the squares
are distinguishable atoms (K in the |9/2, −7/2i state, whose scattering length is not resonant witheither Rb atoms in |1, 1i or K atoms in |9/2, −9/2i) colliding with molecules 9 . For the hollow
8

Unbound Rb are removed by a “kill” pulse.
To create this collision partner, unbound K atoms are transferred to |9/2, −7/2i with an ARP while unboundRb is removed with a “kill” light pulse. After the dissociated molecules are imaged these K atoms are transferredback to the |9/2, −9/2i and imaged.
9

74

1 0 -1 0

β(c m

3

/s )

1 0 -9

8 7

1 0

-1 1

R b + 40K 87R b
K + 40K 87R b
X + 40K 87R b

4 0

5 4 4

5 4 5
5 4 6
m a g n e tic fie ld ( G )5 4 7

Figure 4.8: Measured atom-molecule collision event-rate coefficient, β, for various collision partners
with KRb molecules. The data shown as circles correspond to collisions between molecules andRb atoms in the |1, 1i hyperfine state. The Rb-molecule collision can be influenced by Efimov
physics, unlike the resonant K-molecule collisions, shown in triangles. The resonant feature also is
not seen in collisions between distinguishable atoms (K atoms in |9/2, −7/2i) and molecules, shownas squares. The open circles, squares, and triangles occur at scattering lengths where molecules
can additionally dissociate due to their finite temperature.

data points, the temperature of the atoms and molecules are are equal to, or greater than, themolecule binding energy such that spontaneous dissociation is possible. In this figure, there isclearly a resonant feature at 545.3 G in the Rb+molecule relaxation coefficient, which is absent
in collisions with distinguishable atoms as well as K atoms. Recall from Section 4.1 that FFX
collisions in our system do not exhibit Efimov states. Additionally, collisions with non-resonant
distinguishable atoms would not exhibit an Efimov feature, but any effects that are independentof spin-statistics of the atom would become apparent. The fact that this resonance only occurs inresonant Rb+molecule collisions is consistent with this feature being an Efimov resonance.

75

4.5

Comparison with universal predictions
One remaining question is whether or not the resonant feature we observe in Rb+molecule

collisions is an Efimov resonance, albeit one subject to finite-range corrections since the scattering
length at which the feature is centered (a = 250 a0 ) is only about three times the van der Waals
length (rvdW = 71.9a0 ). Shown in Fig. 4.9 is the same data as in Figs. 4.8 and 4.4, but replottedin terms of the scattering length. The scattering length is determined by the average of those givenby the Feshbach resonance parameters of Refs. [86] and [117]; the scattering length error bars in
the plot are half the difference between the two scattering lengths. The answer to the question ofwhether this feature is an Efimov resonance has three possibilities, which we can partially addressusing the universal formulae for α as well as the relations between a∗ , a− , and a+ .
The first possibility is that the resonant feature in Fig. 4.9a is the second lowest atom(2)molecule relaxation feature, a∗ . The Rb-molecule feature is centered at 250 a0 and if, for the
moment, we assume the universal relations are applicable, then from Eqn. (4.1.11) we would expect(1)

a peak in three-body recombination at a= a− = −490 a0 . The three-body recombination spectra(1)using Eqns. (4.2.10), (4.2.9), and (4.2.8) for this result, and the accompanying value a+ = 44 a0(1)

from Eqn. (4.1.12), are shown as the solid curves in Fig. 4.9b. The feature at a= a− = −490 a0
disagrees strongly with what we measure. However, given that a+ = 250 a0 ≈ 3.5 rvdW , it is
(n)

(n+1)

unlikely that the universal relations between |a− | and a∗will hold in our case. The other(1)

possibilities are that the Rb-molecule resonance is either the a∗resonance (shown as the dashedcurves in Fig. 4.9b) or is not an Efimov resonance at all. Given the lack of features in α, these two(1)possibilities cannot be distinguished because, if the Rb-molecule resonance is a∗(1)= 250 a0 , then(1)a− ≈ −60, 000 a0 and a+ = 5430 a0 10 , which are both outside of the range of realizable scattering
10

For these two theory curves, the theory prediction for α was modified to truncate 
the Efimov
series at the(1)lowest resonances. We assumed simple a4 scaling of α for scattering lengths a ≤ |a− exp − 2sπ0 | on the negative(1)side and for a ≤ a+ exp − 2sπ0 on the positive; the prefactor for the a4 was matched to the universal theory




(1)
(1)
at a = a− exp − 2sπ0 on the negative side and a = a+ exp − 2sπ0 on the positive. The matching points at




(1)
(1)
a = a− exp − 2sπ0 and a = a+ exp − 2sπ0 were chosen because they are half of a log-period away from theirassociated extrema where the universal theory is at the opposite extrema and is purely a4 in nature.

76
lengths. Thus the nature of this Rb-molecule feature is uncertain due to the lack of corroborating
features in the three-body recombination data. However, the three-body recombination data is
(1)

(1)

most consistent with the a− and a+ resonances occurring at scattering lengths well past what we
can realize experimentally. This is the scenario that a recent theory paper predicts[157].

4.6

Three-body parameter and comparison with 41 K −87 Rb
Now we return to the question of universality for the three-body parameter in heteronuclear

systems. Recall that the observation that sparked this question involved measurements of Efimov
resonance positions from four Feshbach resonances in Cesium. The authors measured that theEfimov resonances on the a < 0 (a− ) corresponded to very similar scattering lengths. The averagevalue for the a < 0 Efimov resonance positions was found to be a− = −9.1(4) rvdW [57]. Similar
observations have been made in 6 Li, 7 Li and 85 Rb [23].
Recent theoretical work supports the notion that for open-channel dominated Feshbach res(1)onances, the locations of a− resonances are universally determined by rvdW . In Ref. [23] it isargued that a strong quantum reflection of the three-body wavefunction at the point where thevan der Waals potential begins to dominate the 1/R2 three-body potential (c.f. Eqn. (4.1.1)), sets(1)

a boundary condition at a hyperradius R ≈ rvdW which in turn sets the location of a− . Similar
results are voiced in Ref. [155] where they find that, for model single-channel two-body potentials,
the effective three-body potentials (akin to Eqn. (4.1.1)) converge to a single curve as the numberof two-body bound states is increased. These effective potentials have the sharp drop-off as in Ref.
[23] as well as a strong repulsive barrier at R ≈ 2rvdW 11 . As a consequence of this barrier, they findthat the location of the first Efimov resonance converges to a value consistent with experimental observations as the number of two-body bound states in their model potentials is increased to roughly
10, which is a reasonable number for atomic potentials [140]. The authors of Ref. [138] extend theapproach to a two-channel model in order to address the question of universality in systems where
11This repulsive barrier restricts the three-body wavefunction from having substantial weight at length scales
comparable to the length scale of the non-additive three-body potential [141].

77
rvdW

1 0 -1 0

β(c m

3

/s )

1 0 -98 7

1 0

R b + 40K 87R b
K + 40K 87R b
X + 40K 87R b

-1 1

4 0

1 0

1 0 0
a (a 0)1 0 0 0αR b R b K ( c m

6

/ s )

(a)

1 0 -2 2

1 0 -2 2

1 0 -2 4

1 0 -2 4
a _ (1 )= 4 9 0 a 0

1 0

a + (1 )= 4 4 a 0-2 6

1 0 -2 8 2
1 0

1 0

3

1 0

4

a _

(1 )

= 6 0 1 4 8 a 0

a +

(1 )

= 5 4 3 0 a 0-1 0a (a 0)4

1 0 -2 6

-1 0

3

-1 0

1 2 0 -2 8(b)Figure 4.9: Replotting of the data from Figs. 4.4 and 4.8 in terms of scattering length. The
scattering length was calculated by averaging the scattering lengths produced by the Feshbachparameters of Refs. [86] and [117].a) Atom-molecule event rate coefficient from Fig. 4.8. The
Rb+molecule collisions indicate a resonance centered at 250 a0 . b) Overlaid over the three-bodyrecombination data are predictions of α using the universal relations (Eqns. (4.1.11) and (4.1.12))(2)
to relate a− and a+ to a∗ . The solid curves assume that the atom-molecule feature in a) is a∗ and(1)
the dashed curves assume the feature is a∗ . The data near the d-wave Feshbach resonance havebeen omitted from the plot.

78
40 K 87 Rb 87 Rb

41 K 87 Rb 87 Rb

41 K 41 K 87 Rb

s00.6532 [72]0.6444 [72]0.2462[72]eπ/s0122.7 [72]131.0 [72]a− /a0None (for |a| < 3000a0 )-246(14) [7]348 000 [72]3
−22 +4−6 × 10 [7]a∗ /a0250667(1)[7]η∗
rvdW /a071.9

0.12(1)[7]0.017(10) [7]

72.2

72.2

Table 4.2: Comparison of Efimov resonance parameters between
41 K – 87 Rb and 40 K – 87 Rb.

the Feshbach resonance is not necessarily open-channel dominated 12 . They find that the location
(1)

(1)

of a− asymptotes to a constant value for sres  1 whereas in the sres  1 limit a− approaches
infinity 13 . All of these treatments consider only collisions between three-identical bosons, however
it is likely that the universality in the three-body parameter extends to heteronuclear systems.
The LENS group in Florence published observations of Efimov resonances in 41 K – 87 Rb,
where they used measurements of the atom number after holding at a fixed field for a fixed time
[7, 8]. They located a− features in the 41 K – 41 K – 87 Rb and the 87 Rb – 87 Rb – 41 K loss channels,
as well as an Rb+molecule resonance observed via atom loss at fixed hold. The summary of theirobservations is shown in Table 4.2, as well as the predicted log-periodicity of their Efimov features
and the van der Walls length, rvdW , for both 41 K – 87 Rb and 40 K – 87 Rb. Note that the FFX
collision channel in 40 K – 87 Rb does not support Efimov states and therefore only the locations offeatures in 87 Rb – 87 Rb – 40 K and 87 Rb – 87 Rb – 41 K can be compared.
As can be seen in Table 4.2, the van der Waals lengths for the 41 K – 87 Rb and the 40 K – 87 Rb
are very similar and under the assumption of universality in the three-body parameter, we would
12

Recall from Eqn. (3.2.2) that sres  1 denotes open-channel dominated Feshbach resonances and sres  1
denotes closed-channel dominated resonances.
(1)
(1)
13
For 39 K (sres = 2.1 [23]), the authors of Ref. [138] predict a− ≈ −13 rvdW , compared to a− = −8.27 rvdW for
open-channel dominated resonances.

79η* = 0 . 1

αR b R b K ( c m

6

/ s )

1 0 -2 2

1 0 -2 2

η* = 0 . 0 1

1 0 -2 4

1 0 -2 4

1 0 -2 6

1 0 -2 6

1 0 -2 8 2
1 0

1 0

3

1 0

4

a (a 0)

-1 0

4

-1 0

3

-1 0

(−)

Figure 4.10: Plot of the data from Fig. 4.9b with the universal theory for αdassuming a− = −246 a0 as in the 87 Rb – 87 Rb – 41 K reported in Ref. [7].

1 2 0 -2 8

(Eqn. (4.2.10))

have expected to see features at nearly the same scattering lengths as in 41 K – 87 Rb. Perhaps
our most surprising result is the lack of an observed a− feature in our system compared to thefeature observed at a = -246 a0 in 41 K – 87 Rb; this is in stark contrast to the three-body parameter
universality seen in homonuclear systems14 . The magnitude of this discrepancy is shown in Fig.
4.10, where the theoretical curves from Eqn. (4.2.10) are for a− = −246 a0 . This begs the question:
what is different between our system and LENS’ that would explain this discrepancy in observed
features? One large difference is in the properties of the two Feshbach resonances. As can be seenin Table 4.3, the Feshbach resonance used by the LENS group is substantially broader than oursin terms of magnetic field, as well as being more clearly open-channel dominated.
The Feshbach resonance we employ is only starting to be open-channel dominated withsres ≈ 2. In systems with narrow Feshbach resonances (i.e. sres  1), the three-body physics is
14

The disagreement between our observed a∗ features, while large, is not too alarming. The locations of theatom+molecule resonances are not expected to be universal because of non-universal behavior in the molecule state[138]. Furthermore, in homonuclear systems, the atom+molecule resonances have not agreed with the universal
predictions [88, 129, 170].

80
1 0 1

| ( 1 / 2 r e ff k 2 ) / ( - 1 / a ) |1 0 0
1 0 -1
1 0 -2
1 0 -3
1 0

d -w a v e
F R-4s -w a v e
F R

1 0 -5
1 0 -6
5 4 0

5 4 2

5 4 4

5 4 6

5 4 8

5 5 0

m a g n e tic fie ld ( G )(a) Scattering length and effective range(b) Fraction difference in scattering phase due to effective
range

Figure 4.11: a) Quantum Defect calculation of the scattering length and effective range, reff , for 40 K
in the |9/2, −9/2i hyperfine state and 87 Rb in the |1, 1i hyperfine state courtesy of Brandon Ruzic
and John Bohn [136]. The scattering length plot shows both the broad s-wave Feshbach resonanceas well as a narrow d-wave Feshbach resonance at 547.35 G. The effective range exhibits divergencesat the two zero crossings of the scattering length. The divergence in reff at the s-wave resonancelocation is perhaps a numerical artifact. For a length-scale comparison, ±rvdW = ±71.9a0 is shownas the horizontal dotted lines. b) Ratio of the effective range term (1/2 reff k 2 )in the effective range
expansion of k cot δ0 defined below to the scattering length term (-1/a) at 250 nK.

modified significantly resulting in different scattering-length scaling laws for α [128, 156] because
the scattering length is the dominant length scale for only a small region of the resonance. However,a counter example to this concern of the width of our Feshbach resonances are the observed Efimov
spectrum in 7 Li [64] (sres = 0.80, rvdW = 32.5a0 [24]). Even though the Feshbach resonance used inRef. [64] is a borderline closed-channel dominated resonance, they find the first Efimov resonance(1)

to occur at |a− |/ā = 8.5, which is close to what is seen in Cs[12].
Questions of the width of the Feshbach resonance boil down to considerations of the effective
range. Recall that the effective range expansion of the two-body s-wave scattering phase, δ0 , isgiven by k cot(δ0 (k)) = −1/a + 12 reff k 2 (Eqn. (3.1.3)), where reff is the effective range. The
effective range is a function of scattering length [24] and diverges at a = 0 [24, 64]. In order tosettle questions about the width of our Feshbach resonance, we need to examine the effective range.

81
40 K 87 Rb

41 K 87 Rb

B0 (G)

546.71(1) [117]38.40(3) [159]∆(G)-3.34(5) [117]37 [140]

δµ(µB )

2.38 [174]1.65 [24]abg (a0 )-185(7)[54]284 [140]

sres

2.15

25.8 [24]Table 4.3: Comparing the Feshbach resonances between 40 K – 87 Rb and 41 K – 87 Rb

Brandon Ruzic and John Bohn were kind enough to calculate the locations and widths of Feshbachresonances in our system using a Quantum Defect theory and provided a calculation of the effective
range as well. The results of Brandon’s calculations are shown in Fig. 4.11a. The α data in thischapter corresponding to scattering lengths below thermal saturation, spans the magnetic-field
ranges 545.08 G < B < 546.33 G and 546.90 G < B < 556.10 G. Generally we want to work in
the regime where the scattering length is the dominant length scale and the effective range is notsubstantially larger than rvdW 15 (e.g. reff < 10 rvdW ).
To determine how significant the effective range is at a given magnetic field is to consider how
the effective range changes the s-wave phase shift, δ0 . By examining the significance of the effective
range term to the s-wave phase, we are asking if the scattered wavefunction of the two atoms isaffected by the effective range term. Recall from scattering theory that the scattered wavefunction
amplitude for central forces can be expressed by a partial wave expansion
f (k, θ) =∞X

fl (k)Pl (cos θ)

l=0

where Pl (cos θ) are Legendre polynomials. The partial wave amplitudes relate to the scattering
15

For bosons near a narrow Feshbach resonance, a large effective range is predicted to modify the equations for
the Efimov log-periodic scaling factor [156].

82
phases by [20]fl (k) =2l + 12l + 1[exp 2ıδl (k) − 1] =2ıkk cot δl − ık

where δl is the phase shift for the l-th partial wave. For the temperatures that we operate at(100nK-1 µK) we only need to concern ourselves with s-wave scattering16 . Recall from Eqn. (3.1.3)
that the first two terms in the effective range expansion of the s-wave scattering phase shift is1
k cot(δ0 (k)) = −1/a + reff k 2 .
2
We can estimate the wave number, k, from the average speed of two colliding thermal atoms(seeqBTRef. [60]); the wave number is then k = 8µk, where µ is the two-body reduced mass. We canπ~2
now indicate regions where the effective range is an issue by calculating the ratio

1/2reff k2−1/a .

The

results of this ratio, using the results in Fig. 4.11a and assuming T = 250 nK, are shown in Fig.
4.11b. The region in magnetic field covered with the α data (below thermal saturation) is indicatedas the diagonally hashed regions and the horizontally hashed region indicates the magnetic field
span covered in the β data. As can be seen the effective range amounts to a < 1.5% correction to thevalue of k cot δ0 over the hashed regions, excluding the region near the d-wave Feshbach resonance.
The scattering length over the range excluding the d-wave Feshbach resonance is the dominant
length scale. Therefore, the fact that our resonance is not as clearly open-channel dominated as
the resonance in 41 K – 87 Rb, is unlikely to be an issue significant enough to explain the fact that
we do not observe Efimov features at the locations seen by LENS.

16
The p-wave threshold for 40 K-87 Rb is roughly 110 µK [60] below which the p-wave scattering cross section falls
off as T2 as per the Wigner threshold law.

Chapter 5

The “Fabrergé egg” Ioffe-Pritchard trap

Near the end of August 2010, the south Fast B coil developed a break in its insulation insideof the existing Ioffe-Pritchard (IP) trap. The Fast B is a capacitor driven system (C = 6.8 mF)
that is initially charged to ≈150 V. Due to the break in the coil’s insulation, a violent arc within
the IP trap would result discharging ≈1 Coulomb into the IP trap’s copper cooling forms. We wereable to fix the issue after partially disassembling the IP trap and coating the areas of obliterated
insulation with an electrically insulating enamel. This incident, combined with the previous failuresof two out the four rf antennas (we need two antennas minimum), which are epoxied inside of the
IP trap, prompted the design of a replacement magnetic trap.
In designing the new magnetic trap, we desired to make a few improvements over the existing
IP trap. The existing IP trap coils are made from solid core wire, which is wrapped around copper
heatsinks. Cooling water flows through these heat sinks to carry away the heat generated by
the trap. Although slots were cut through the copper heat sinks to suppress eddy currents, thecontribution of eddy currents to magnetic field at the atoms is not insignificant. While we cancorrect the total magnetic field for these eddy currents (c.f. the eddy current correction section ofMichele Olsen’s thesis [116]), it is desirable to eschew the copper cooling forms and instead build
the coils out of hollow-core copper tubing. The cooling water then flows directly through the coils
and eliminates the need for extra copper near the atoms. Another design goal was to improve theoptical access for improved imaging as well as future lattice experiments. Lastly, we also want to
use the trap’s bias coils to provide the large field necessary for reaching our Feshbach resonances

84

Figure 5.1: View of the ‘Science cell’ end of the vacuum chamber with the IP trap removed. The
Science cell (small cyan square) is flanked by the Quadrupole trap coils (orange rectangles) which
restricts the size of the new IP trap.

(and beyond to 800 G) and therefore we require this coil pair to have a small amount of fieldcurvature.
Unfortunately, there are fairly strict space constraints on the size of the IP trap due to theorientation of Quadrupole trap coils on the moving cart system as shown in Fig. 5.1. Due to these
constraints we cannot simply borrow the hollow tubing based designs from the other labs in JILA
as they require more space in the horizontal direction than the Quadrupole coils will allow. The
need to miniaturize the design will influence the coil geometry, as will be discussed in Sec. 5.1. The
space constraints will also necessitate building the trap coils out of narrower tubing than previous
designs. This raises the concern that the cooling water flow through the coils will be insufficient
to remove the heat generated by the trap. Addressing this concern is the subject of Sec. 5.2. This
chapter concludes with a summary of the magnetic trap specifications. However, as of the writingof this chapter, the trap has not yet been installed on the apparatus.

85

Figure 5.2: Simplified diagram of a Ioffe-Pritchard trap where the arrows indicate the direction ofcurrent flow through each coil. Axial curvature is provided largely by the pinch coils (red coils) andthe bias coils (blue coils) serve to largely cancel the offset field produced by the pinch coils. Radialconfinement is provided by the radial quadrupolar field produced by the Ioffe bars (horiz. arrows).

5.1

Magnetic trap potential and design considerations
Rb and K, as with the other alkalis, possess magnetically sensitive hyperfine states in their

ground electronic states. This magnetic sensitivity occurs as a consequence of the coupling between the magnetic moment µ of the atom, which is caused by the angular momentum of thealkali atom’s unpaired electron and nuclear spin, to a magnetic field via the Zeeman effect. The
Zeeman effect couples eigenstates of the atom to an applied magnetic field B with an interaction~ = mF gF µB |B(~~ r)| [20], where mF is the projection of the totalenergy of the form U = −~µ·B
atomic angular momentum along the magnetic field, gF is the Landé g-factor, and µB is the Bohrmagneton, which determine the strength of the magnetic moment, µ. As shown in Refs. [166, 85]
the magnitude of the magnetic field in free space cannot have a maximum due to Gauss’ law, but
local minima are possible. Therefore we can trap states for which gF mF > 0 (so-called ‘low-fieldseeking’ states) in a local minimum [149, 69, 93, 108]. For 40 K the Landé g-factors in the groundelectronic states are g7/2 = −2/9 and g9/2 = 2/9, and the states |F, mF i = |7/2, −7/2i, |7/2, −5/2i,|7/2, −3/2i, |7/2, −1/2i, |9/2, 9/2i, |9/2, 7/2i, |9/2, 5/2i, |9/2, 3/2i, and |9/2, 1/2i can all potentially be trapped. For 87 Rb the g-factors are g2 = 1/2 and g1 = −1/2 and the list low-field seeking
states is significantly shorter: |2, 2i, |2, 1i, and |1, −1i can be trapped.
The magnetic trap style that we chose to implement is the Ioffe-Pritchard trap [80, 130]

86
because of its ease of construction and modelling. The trap is formed by fields from three sets ofcoils, as shown in Fig. 5.2. The pinch coils (small red coils) provide an axial curvature as well as
an offset field. This offset field is largely canceled by the bias coils (large blue coils) whose current
flows the opposite orientation as the pinch coils. Lastly, radial confinement is provided by the radial
quadrupolar field produced by the Ioffe bars (green wires). The magnetic field produced by thisconfiguration at the center of the trap is approximately [130]~B(x,y, z) =









1
1
1
x2 + y 2β x − γ z x x̂ − β y + γ z y ŷ + B0 + γ z 2 −
ẑ
2
2
2
2

(5.1.1)

where β the radial field gradient of the Ioffe bars, γ is the total axial field curvature from thepinch and bias coils, and B0 is the sum the pinch and bias offset fields. If we now approximate
the potential at the center of the trap as harmonic, the trap frequency in the i-th direction can becalculated frommωi2 = µ

~
∂|B|
∂xi

(5.1.2)x=y=z=0

where m is the atomic mass and µ the atomic magnetic moment. The trap frequencies from Eqn.
(5.1.1) are given by

s
ωx =
s
ωy =
r
ωz =

µ
m



µ
m



β2
1
− γ
B0 2



β2
1
− γ
B0 2



µ
γ.
m

(5.1.3)(5.1.4)(5.1.5)Since evaporation of atoms is aided by having large trap frequencies1 , some design principles arise
from Eqn. (5.1.3). In addition to desiring large field gradients and curvatures, we also wish to havethe lowest offset field possible. However, the offset field cannot be set to zero because the magnetic
moment of the atom will not be able to adiabatically follow the magnetic field direction as it travels√
The elastic collision rate between K and Rb is proportional to ω̄ 3 N/T [60] where ω̄ = ωx ωy ωz . This collision
rate sets the time scale by which the system rethermalizes and thereby repopulates the high-momentum tail of the
distribution which had just been truncated by the rf-evaporation knife. By having a larger trap frequency, theevaporation is able to proceed faster and reduce the loss due to background collisions.
1

87
through the center of the trap. In that case, the atom will undergo Majorana spin flips resulting
in loss of atoms from the trap [126]. Additionally, the campus AM radio station broadcasts at 1.19
MHz which potentially could drive spin flips in |9/2, 9/2i K at 3.8 G and so we designed the trapwith an offset field of 4 G in mind.

5.1.1

Modeling the magnetic field from the coils

To design the trap we need to model the magnetic field from the coils, which are in reality
multiple layer helices made from hollow tubing (square outer cross-section and round inner crosssection) with a non-zero thickness. For the bias and pinch coils, we model each turn as a closed
loop of current with a separation between loops equal to the tube thickness. For a coil of radius R
centered on the axial location z = z0 , the magnetic field components from the coil are [11]R2 − r2 − (z − z0 )22
2
Bz (r, z, z0 ) = p
K(k ) +E(k )(R − r)2 + (z − z0 )2 )(R + r)2 + (z − z0 )2
0.2
R2 + r2 + (z − z0 )2
z − z02p
Br (r, z, z0 ) =−K(k 2 ) +E(k)
r(R − r)2 + (z − z0 )2(R + r)2 + (z − z0 )2
0.2(5.1.6)(5.1.7)

where Bz and Br are the axial and radial magnetic-field components, respectively, in G/A, withall spatial dimensions in cm. The functions K(k 2 ) and E(k 2 ) are complete elliptic integrals of thefirst and second kind, respectively, where the argument is defined as k 2 =4Rr(R + r)2 + (z − A)2.

The bias and pinch coils are then modeled by summing up the fields of individual loops of theappropriate radius and axial location. To simulate the real coils, which are multiple layered helices,tthe loops are tilted by an angle θ = (−1)n+1 arcsin 2R
for a turn in the n-th layer of radius R for
tubing of thickness t (see Fig. 5.3b). The winding tilt creates a non-axial field component which iscanceled at the trap center by winding each bias and pinch coil in the coil pair as a mirror imageof the other where the mirror plane is parallel to the plane of the ideal (i.e. winding tilt free) coil2 .
2
The non-axial field component for a coil pair separated by a distance d, is roughly Bz (r, z, d/2) sin(θ) +tBz (r, z, −d/2) sin(−θ) = sin(θ)(Bz (r, z, d/2) − Bz (r, z, −d/2)) = 2R
(Bz (r, z, d/2) − Bz (r, z, −d/2)). Similarly the
axial
componentisreducedtoBz (r, z, d/2) cos(θ) + Bz (r, z, −d/2) cos(−θ) = cos(θ)(Bz (r, z, d/2) + Bz (r, z, −d/2)) =
q

t 2
t
1 − 2R(Bz (r, z, d/2) − Bz (r, z, −d/2)). The ratio 2R= 0.04 for the innermost layer of the bias coils and

88

t
^

y

^

x
^

z

Lz

B

2R

B

Lx
(a)

(b)

Figure 5.3: a) Diagram for the rectangular coil used to model the Ioffe bars where the unit vector ŷpoints out of the page and the current flows in the direction indicated by the arrows. b) Diagram of
the winding tilt model showing the individual loops edge-on (solid lines) for two layers of windings.
As a consequence of the tilt, the magnetic field from each layer acquires a vertical component which
does not completely cancel when the coil is considered as a whole.

The Ioffe bars are modeled as sets individual rectangular loops of current to incorporate theeffect of the ends of the Ioffe bars. The magnetic field from each rectangular loop is the sum of themagnetic fields from four finite-length wires, whose magnetic field can be calculated from the BiotSavart law. The geometry of this coil is shown in Fig. 5.3a where the ŷ direction points out of theApage. To simplify the equations, let’s introduce the function f (A, B, C) = p.
2
A + 4(y − yc )2 + B 2
For a rectangular coil centered at y = yc of dimensions Lx and Lz , the magnetic field componentsq
t 2= 0.08 for the innermost layer of the pinch coils; for the bias coils 1 − 2R
≈ 0.999 and for the pinch coilsqt 2
1 − 2R≈ 0.997, which indicates that the winding tilt effect is not a large concern for our coils.
t
2R

89
quasi-Helmholtz (d2R)

Helmholtz (d=R)

2πµ0 I B(z = 0)2
3d2
R − 4R3 + O

√16
5 5R

2π ∂ 2 Bµ0 I ∂z 2 z=0− R63 + 45d+ O R−74R50

2π ∂ 4 Bµ0 I ∂z 4 z=0

290− 1575d
+O
R5
4R7



√
− 62555296
5 R5

R−5



2

R−9

Table 5.1: Derivatives of the magnetic field on axis at the centerin between a coil pair separated by a distance d.

(in SI units) areµ0 I
f (Lz − 2z, Lx − 2x) + f (Lz + 2z, Lx − 2x)(y − yc ) −(5.1.8)4π(−Lx /2 + x)2 + (y − yc )2
f (Lz − 2z, Lx + 2x) + f (Lz + 2z, Lx + 2x)+(Lx /2 + x)2 + (y − yc )2"(2((−1)n L2z + z)(f (Lx − 2x, (−1)n Lz − 2z) + f (Lx + 2x, (−1)n Lz − 2z))
µ0 I X(−1)n−1By =4π(y − yc )2 + ((−1)n Lz /2 + z)2
Bx =

n=1

(5.1.9)#)((−1)n L2x + x)(f (Lz − 2z, (−1)n Lx + 2x) + f (Lz + 2z, (−1)n Lx + 2x))+((−1)n Lx /2 + x)2 + (y − yc )2µ0 I
f (Lx − 2x, Lx − 2z) + f (Lx + 2x, Lz − 2z)
Bz =(y − yc ) −4π(y − yc )2 + (−Lz /2 + z)2
f (Lx − 2x, Lz + 2z) + f (Lx + 2x, Lz + 2z)+(y − yc )2 + (Lz /2 + z)2(5.1.10)where I is the current flowing in the orientation show in Fig. 5.3a. With Eqns. (5.1.6) and(5.1.85.1.10) the full trap can now be designed.

5.1.2

Design principles

Design principles can be found by first modeling the coils with simpler components 3 . First
let’s consider the bias and pinch coils, which we treat each as a simple loop of current. The magnetic
3

Ref. [110] is great read for when you are designing a magnetic coil, but be wary of the formulae for cooling fluid
flow as they only apply for straight tubes. The fluid flow through helical tubes is substantially reduced compared tostraight tubes.

90field on axis with a simple loop of current is, in SI units,R2~ = µ0 IBẑ
2π (z 2 + R2 )3/2

(5.1.11)

where µ0 is the vacuum permeability, z the axial distance from the coil, and R the radius of the loop.
The bias and pinch coils are constructed from coil pairs whose current flows in the same orientation.
Shown in Table 5.1 are the even-order derivatives for a coil pair separated by a distance d withthe z axis centered at z=0 in between the pair. An alternative way to examine this problem is todefine a new variable, η = d/R, which will serve to characterize the geometry of the coil pair. The
magnitude of the magnetic field for the coil pair, and its even partial derivatives, at z=0 becomesµ0 I
16
2π (4 + η 2 )3/2 R(5.1.12)∂2Bµ0 I 768(η 2 − 1)=∂z 2 z=0 2π (4 + η 2 )7/2 R3(5.1.13)∂4Bµ0 I 92160(η 4 − 6η 2 + 2)=
∂z 4 z=0 2π(4 + η 2 )11/2 R5(5.1.14)B(z = 0) =which allows for separate analysis of the scaling of the magnetic field on the system size, characterized by R, and the system geometry, characterized by η. For a fixed value of η and current, themagnetic field scales as 1/R and the axial field curvature (Eqn. 5.1.13) scales as 1/R3 . A special
case is the Helmholtz configuration (η = 1) for which the field curvature vanishes. The higher-ordereven derivatives of the magnetic field do not vanish at η = 1, and so there is a finite range overwhich the field is flat to within some fraction of the central field value (this range scales as R).
Ideally the bias coils would have the Helmholtz configuration, but since the coils are made fromseveral layers of turns, this is not possible. Therefore, to reduce the field curvature contribution
from the turns which are not in the Helmholtz configuration, the bias coils should have a large
radius and the separation should be adjusted to minimize the field curvature. From the pinch coils
we desire a large field curvature, and so they should have a radius smaller than the bias coils andη > 1 so that the field curvature is positive.
Scaling laws can also be derived for the Ioffe bars from the magnetic field from an infinite

91
Bias

Pinch

Ioffe bars

Fast Bturns / layer

4

4

2

2

number of layers

4

4

1

2

inside diameter (cm)

7.26

3.66

1.4, 4.66

3.68

outside diameter (cm)

10.02

6.42

2.76, 6.02

4.28

B0 (G/A)

3.32

3.48

β (G/cm/A)
γ (G/cm2 /A)

1.65
1.00

1.03

0.11

Table 5.2: Calculated magnet coil properties

wire. The magnetic field at a radius R from the wire with a current I is
~ = µ0 I φ̂B
2πr

(5.1.15)

where field points in the azimuthal direction, φ̂. As shown in Fig. 5.2, the wires for the Ioffe bars
are arranged on a the vertices of a square with the current flowing in the same direction for wires
on opposite corners of the square. If we assume that adjacent wires are separated by a distance,d, then the magnetic-field gradient at the center of the wires scales as 1/d2 . Although this scaling
is not as strong as the scaling of the pinch coil curvature, given that the radial trap frequencies
depend roughly linearly on the Ioffe bar gradient and axial trap frequency depends on the squareroot of the curvature, it is advantageous to have the Ioffe bars fit within the pinch coils.
The best configuration we came up with for the coils given our space constraints is the one
shown in Fig. 5.4d with the bias and pinch oriented colinear with the Quadrupole trap coils in Fig.
5.1 and the Ioffe bars fitting within the radial coils4 . Cross-sections of the calculated magnetic
field are shown in Fig. 5.4 for the coordinate system shown in Fig. 5.4d using Eqn. 5.1.6 to model
magnetic-field from the bias and pinch coils and Eqns. (5.1.8-5.1.10) to model the field from the4

Other configurations with the pinch coils within the Ioffe bars, or with the orientation of the bias and pinch
perpendicular to the Quadrupole trap, were also considered, but when configured to produce the appropriate fieldstrengths, gradients, and curvatures, the traps simply did not fit within our space constraints.

92

(a)

(b)

^

y
^

x
^

z

(c)

(d)

Figure 5.4: a) - c) Cross-sections of the calculated magnetic field magnitude for the operating
currens Ibias = 193.75A, Ipinch = 186A, and IIof f e = 186A. The field from the bias and pinch
coils we calculated from Eqn. 5.1.6 and the field from the Ioffe bars are calculated from Eqns.
(5.1.8-5.1.10), which includes the contribution from the ends of the Ioffe bars. The scale for themagnetic field in G is shown in c). d) A simplified diagram of the coils showing the coordinate
system. In this diagram the bias coils are shown as the gold rings, the pinch coils are the lavenderrings, the Fast B coils are red rings, and the Ioffe bars are the salmon square rings.

93
Ioffe bars (including the contribution from the ends of the Ioffe bars). As can be seen from Figs.
5.4a and 5.4b, the minimum of the magnetic field is shifted by 0.6 mm in the z-direction, whichis small compared to the 1 cm x 1 cm interior dimensions of the science cell. This shift is dueto the field gradients produced by the ends of the Ioffe bars as well as the coil leads and can becorrected by either shifting the Quadrupole trap potential over before loading into the IP trap, orby applying a modest gradient of 10 G/cm in the z-direction.
The details of the coils are listed in Table 5.2. In order to allow for the coil leads to exit
the trap as a pair, and thus reduce the gradient produced by the leads, the bias and pinch were
designed with an even number of layers, and likewise Ioffe bars were designed for an even numberof turns. In designing the trap, it became apparent that we would need to use smaller tubing thanhas been used previously in our labs, so we would need to think carefully about what size tubing
to use.

5.2

Water cooling concerns
In order to miniaturize the design of the coils, the channels through which cooling waterflows must also be reduced compared to previous designs [173]. This raises the immediate concernthat restricting the water to flow through narrower pipes will result in the coils overheating. To
assuage these concerns before purchasing the tubing, we first predicted the water flow rate througha helical coil for a given pressure drop.
While a more complete description of fluid flow requires numerically solving the NavierStokes equation, in this case it is sufficient to rely on simpler phenomenological equations. As fluid
flows through a tube, it suffers frictional loss of energy due to roughness of the tube walls, sharp
obstructions to flow, tight bends, etc. The Darcy-Weisbach equation describes the pressure loss asa fluid flows through a pipe. The pressure loss is [81] 5∆P = f5

In the notation of Ref. [81], ∆P = ρw gh.

L1
ρw v 2d 2(5.2.1)

94

Figure 5.5: Definition of the dimensions of a helix with pitch p, coil diameter D, and inner wetted
diameter d.

for a fluid of density ρw , mean speed v, in a tube of length L, and inner diameter d. The Darcy
friction factor f is a dimensionless factor that depends on the characteristics of the fluid flow ( e.g.
whether the flow is turbulent or laminar) as well as the pipe roughness.
The degree of turbulence in a given fluid flow is encapsulated in a single parameter, theReynolds number, Re. The Reynolds number is related to the fluid’s kinematic viscosity ν (in unitsof m2 /s), mean speed, and tube diameter d, by
Re =

vd
ν

(5.2.2)and is proportional to the ratio of the drag force to the viscous force of the fluid. In a sense,
the Reynolds number sets the minimum size of eddies possible in the fluid. Large values of Re
correspond to turbulent flows which support small eddies in the flow. Conversely, low values of Re
indicate smooth laminar flow.
The pressure loss when flowing through a helical tube can be significantly different from theflow through straight pipes. Although helical pipes have been used as heat exchangers for sometime, the problem of understanding the flow and pressure drop is still an active area of research[39, 1]. The author of Ref. [1] measured the pressure drop and flow rates through a large setof helical tubes with different pitches p, coil diameters D, and inner diameters d (cf. Fig. 5.5).
From the pressure drop data, the author models the pressure drop with an equation akin to the95
Darcy-Weisbach equation:
2∆PAli = 2 αRe−β G−1
rhc ρw v(5.2.3)where the quantity Grhc is the geometrical group that characterizes fluid flow through a regularhelix such that all the pressure drop data from the disparate group of helices falls on the same
curve. The volumetric flow rate, Q, in cm3 /s is thensπd2
πd2 ∆PAli Grhc
Q=
v=
.
4
4
2 αRe−β ρw(5.2.4)

The author found the best fit for Grhc is
0.15
Grhc = d0.85 Deq/Lcwhere the equivalent diameter Deq =(5.2.5)pp2 + πD2 /π and Lc is the linear length of the coiled section.

Upon fitting separate sections of the curve that appear to have constant values of α and β, theauthor of Ref. [1] finds the following piecewise definition:

21.88 Re−0.9 ,Re < 500, low laminar








500< Re <6300, laminar 5.25 Re−2/3 ,αRe−β =
0.56 Re−2/5 ,6300 < Re < 10000, mixed flow






0.09 Re−1/5 ,Re > 10000, turbulent(5.2.6)(5.2.7)(5.2.8)(5.2.9)where the lowest regime (Re<500) is termed the low laminar region, the next ( 500< Re <6300) thelaminar flow region, the mixed flow region, and the last the turbulent region. To prevent vibrations
within the coils due to turbulent flow, the IP trap coils are connected hydraulically such that thefluid flow stays within the laminar flow region.
In order to verify the results of Ref. [1], we constructed a simple test setup as illustrated
diagrammatically in Fig. 5.6. The coil consisted of 6 turns of copper refrigerator tubing (3/16”
OD tubing from Mueller Industries) with a coil diameter of 9.4 cm and 3.24 mm inner diameter.
Water pressure was monitored before and after the coil with pressure gauges and the flow rate wasmeasured with a Proteus Industries 804PN2 flow meter that was manually calibrated. The water
source was domestic water at an initial pressure of approximately 45 psi. The flow rate was varied96

pressure
gauge
coil
flow
meter

Figure 5.6: Diagram of the test setup for measuring the water pressure drop and fluid flow througha test coil.

5 0
d a tath e o ry - Itoth e o r y - A li

4 0

d a tath e o ry - Itoth e o r y - A li

5 0
4 0

( p s i)3 0

∆P

( p s i)

3 0

2 0

∆P

2 0
1 0

1 0
0
0
0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

flo w r a te ( L /m in )flo w r a te ( L /m in )(a) with burrs(b) burrs removed

3 .0

3 .5

Figure 5.7: Measured pressure drops and flow rates through the test setup. Initially the ends of test
coil had burrs left after cutting the copper tubing. After filing the burrs, the measured pressure
drops more closely follow the phenomenological theories of Ali [1] and Ito [81].

by throttling a valve before the coil. Shown in Fig. 5.7 are two attempts to verify the results of Ref.
[1]. Fig. 5.7a shows marked disagreement with the predictions of Ref. [1] and another prediction
by Ref. [81]. In the case of Fig. 5.7a, both ends of the coil had small burrs that resulted in a muchlarger pressure drop. Fig. 5.7b is after the burrs were removed and exhibits better agreement withthe predictions of Refs. [81, 1] and so the predictions can be trusted.
Another issue is the question of heat transfer from the current carrying coil to the cooling97
1 1∆T = W

1 0
9
8

T1 T2
T4

plastic
tubing3

δT = 0 . 4 1 ( 1 4 ) ° C

7

T o u t - T in ( ° C )plastic
tubingT3/ ( c p Q ) + δtc p = 4 .1 9 (1 3 ) J /K /c m

6
5
4
3
2
1
0
0

1 0

2 0

3 0

d is s ip a te d p o w e r /flo w r a te ( W /c m

(a)

4 0
3

/m in )(b)

Figure 5.8: a) Setup for measuring heat transfer from the coil to the cooling water. Thermocouples(indicated by Ti ) were attached to brass tubing connectors, and covered with insulation, to measure
fluid temperature. The temperature increase of the fluid was measured from T4 − T3 , which wereisolated thermally from the coil with a segment of plastic tubing, while passing current throughthe test coil and measuring the fluid flow rate with the flow meter. b) Measured fluid temperature
increase versus the power/flow rate ratio.

water. The test setup for measuring heat transfer is shown in Fig. 5.8a. The test coil used in thiscase was a bias coil used in an old magnetic trap from S1B05; the coil is roughly 9.5 cm in diameterwith four turns per layer and four layers and a total resistance of ≈ 11mΩ. The coil had been in use
for an extended period of time with domestic water and so the inside of the tubing was oxidized.
Water and electrical power were connected to the coil through two copper cooling plates drawn
as the red hollow squares in Fig. 5.8a. Water connections were made with brass connectors onto
which thermocouples were attached (denoted T1 through T4 ) and then covered with insulation.
Thermocouples T3 and T4 were isolated thermally and electrically from coil by small segments ofplastic tubing and therefore measure the incoming and outgoing water temperature, respectively.
The flow rate was measured using the manually calibrated flow meter from above.
The fluid temperature increase (T4 − T3 ) as a result of power dissipated in the test coil isshown in Fig. 5.8b for currents up to 200 A. The temperature increase was fit to ∆T = cW
+ δT ,
pQ
where W is the dissipated power in Watts, Q the volumetric flow rate, cp the specific heat capacity

98
including reduction due to power heat transfer 6 , and δT an offset due to calibration offsets between
the two thermocouples. The value of the specific heat capacity from the fit, shown in Fig. 5.8b, is
cp = 4.19(13) J/K/cm3 (error bar is purely statistical), which is consistent with water (cp = 4.186
J/K/cm3 [110]) indicating that the heat conduction from the coil to the water is not substantiallyhampered by the oxide layer on the inner wall of the tubing.
Now that we know what flow rate to expect and that the oxide layer inside the copper tubing
does not significantly impair heat transfer for coils of the size used for the trap, we can decide on
an inner diameter for the tubing. The upper limit on the temperature for the coils is 45◦ C, which is
the glass-liquid transition temperature 7 for the epoxy used to bind the turns of the coils together[99]. The chilled water that will be used to cool the trap is maintained at a temperature of 10◦ anddifferential pressure of 65 psi. We want to choose the inner diameter for the tubing such that westay well below the glass-liquid transition even in a fault condition where the pressure is drasticallyreduced.
The coils which dissipate the most power are the bias coils which, at their design field of 800G
(Ib ias = 241 A), dissipate roughly 500 W each. Assuming the coils are in parallel hydraulically suchthat the ∆P = 65 psi for each coil, and using Eqn. (5.2.3) to calculate the flow rates for the bias coils
and then calculating the temperature increase, we find that an inner diameter of 0.08 inches should
provide a sufficiently safe temperature increase. For ∆P = 65 psi the coils should pass 0.4 L/mineach (corresponding to Re = 3200 and thus laminar flow), which will limit the temperature increase
to ∆T = 13◦ C. If the water pressure drops to 32 psi, the then the temperature increase is predicted
to be ∆T = 22◦ C, which keeps the total temperature well below the maximum temperature forthe epoxy. Similarly for the pinch coils, which dissipate 310 W total while the trap is running, thecalculated flow rates are 0.5 L/min and 0.3 L/min for 65 psi and 32 psi, respectively, assumingthat the coils are in series hydraulically. For these flow rates the predicted temperature increases
6

If the heat transfer from the coil to the water is poor, then the measured heat capacity will deviated substantiallyfrom the published values for water.
7
At the glass-liquid transition temperature the epoxy will transition from a glass state to a molten state[5] with
reduced mechanical strength [99].

99
are ∆T = 9◦ C and ∆T = 15◦ C for 65 psi and 32 psi, respectively. Therefore the coils should be
able to safely operate under normal and some abnormal conditions with an inner tubing diameterof 0.08 inches.

5.3

Magnetic trap details
From the considerations in the previous sections we designed bias and pinch coils and the

Ioffe bars with the ability to reach 800 G as a design goal for the bias coils in order to reachhigh magnetic field Feshbach resonances and trap frequencies comparable to our existing IP trap:
fradial = 125 Hz and faxial = 25 Hz. At the currents Ibias = 193.75A, Ipinch = IIof f e = 186A the
new trap is calculated to achieve fradial = 118Hz and faxial = 18Hz with the field minimum of
the trap B0 = 4 G. The coils were wound from 0.125” oxygen-free copper square tubing with an
inner circular diameter of 0.080”. The tubing was purchased from Small Tube Products and thencoated in Kapton polyimide tape by S&W Wire. The details of the coils and their field strengths
are shown in Table 5.2 and the measured coil resistances are shown in Table 5.3.
For constructing the coils as well as designing and constructing the structure of the magnetic
trap, I have Tracy Keep to thank. The trap structure was built from Garolite G-10, a glass phenolic8. The bias coils were each wound directly onto each half of the trap structure applying Araldite 2011
epoxy, which has been found to adhere well to Kapton [173]. The trap structure is bolted together
with titanium bolts. Given the low thermal expansion of G-10 (it is comparable to titanium) the
spacer between the two halves is also made from G-10. Nestled inside the trap structure are theFast B pair, which consist of two layers with two turns per layer of 17 gauge magnet wire. The
pinch coils were each wound onto a separate G-10 structure, which bolts inside of the trap structure.
This means that the pinch coils can be removed in the future if the full IP trap is no longer needed.
Lastly, the Ioffe bars were wound onto a separate form and inserted through the pinch structure.
A photograph of the trap is shown in Fig. 5.9b as well as a CAD rendering of the new trap with
8

Earlier traps were constructed from Garolite G-11, but, as noted by Tracy Keep, this material tends to delaminate
when it is machined.

100
coil

Rmeasured (mΩ)

Rexpected (mΩ)

North Bias

9.3(2)(10)

10.8South Bias

9.2(2)(10)

10.8

North Pinch

6.1(2)(10)

6.2South Pinch

6.1(2)(10)

6.2

Upper Ioffe bar

1.50(2)(100)0.7Lower Ioffe bar

1.38(2)(100)0.7North Fast B

15.1(6)

16

South Fast B

16.4(6)

16

Table 5.3: Measured and predicted magnet coil resistances. Measurement of the trap coils was subject to a
∼ 1mΩ due to uncertainties in the lead length, whichis represented by the second error bar. The measured
values of the coils resistances agree with the calculated
values within the error bars with the exception of thebias coils. The resistances were calculated from theresistivity of copper (1.71 µΩ-m) and the dimensionsof the tubing and coils.

our existing vacuum chamber and Quadrupole trap coils in Fig. 5.9a. In Figs. 5.9b and 5.9a thesliding stage onto which the trap attaches is visible. This stage will allow the trap to be easily
positioned over the science cell (not shown) and then locked into position.
Since the square tubing has a relatively small inner diameter, which impedes water flow, thesquare tubing is mated to standard round copper tubing as soon as it can be parted from the trap
structure. The square tubing was mated to the round tubing by crimping the round tubing onto
the square tubing and then soldering the joint. The square tubing was strain relieved against thetrap structure to reduce stress on the joints. The round tubing was then coated in heat shrink
tubing to provide electrical insulation and then routed to an electrical power and water junction.
Optical access to the atoms is greatly improved compared to our existing IP trap on two fronts.

101
The first improvement is the increased access for imaging lenses. A 25mm diameter aperture wascut between the bias coils (it can be seen in Fig. 5.9b) allowing for optics to be inserted as close
as 2.7 cm from the center of the trap. This aperture limits the numerical aperture9 to NA = 0.26.
Smaller optics (up to 14 mm in outer diameter) can be inserted in this direction with unobstructed
access to the science cell. The maximum numerical aperture is then limited by the dimensions of
the science cell to NA = 0.62. Additionally, small, custom-made lens tubes can be bolted into thetrap in between the Ioffe bars from either side, or from below the trap, and allows for 1/2 inch
optics to be positioned as close as 1.35 cm from the IP trap center allowing a maximum numericalaperture of 0.39.
The openings also allows horizontal beams to enter the science cell through the pinch coils(in the z-direction from Fig. 5.4d) at up to 22 ◦ in the zx-plane (limited by the pinch coil form) off
of normal to the science cell and 13 ◦ (limited by the Ioffe bars) in the yz-plane. For beams entering
the science cell from above between the bias coils in the xy-plane, the system accommodates beams
up to 1 mm in diameter at the science cell at an angle 45◦ off of the normal to the science cell.
Currently the new IP trap is assembled and being tested. Control electronics for the IP trap
current have been built and will allow for the high current transistors to be tested to check thatthey perform as specified. Once the electronics are ready, and the experiment is at a good point
for a major upgrade, then the new IP trap will be installed.

9
The numerical aperture is defined NA = n sin(θ) where n is the index of refraction and θ is half of the maximum
angle of light that transmits through the aperture from a point source at some distance.

102

(a)

(b)

Figure 5.9: a) CAD rendering of the vaccuum chamber with the new trap installed and alignedwith the Quadrupole coils on the cart track. b) Photograph of the assembled Ioffe-Pritchard trapwith the visible coils labeled. The trap structure is made from Garolite G-10, which is the lightgreen material in the photograph. The coils are made from Kapton coated square tubing (theamber-colored tubing in the photo), which is mated to standard round tubing by crimping andsoldering the tubes together and then the round tubing is covered in heat shrink resulting in theblack tubes shown in the foreground.

Chapter 6

Conclusions and future directions

6.1

Conclusions
In this thesis I have described the various upgrades to the apparatus and our understandingthat enabled the study of Feshbach molecule formation as well as Efimov physics in our ultracold
atomic mixture, 40 K – 87 Rb. The overall stability of the system was aided with the replacementof the old K laser system, which replaced a K slave laser maintained at −40◦ C and a master laser
maintained at −10◦ with a tapered amplifier and master laser maintained at temperatures justunder room temperature. Studies of the performance of the Fast B were detailed, which was anintegral part of the apparatus for the studies of Feshbach molecule formation and Efimov physics.
Additionally I detailed the new Rb microwave setup and systematics and calibrations of our atom
number measurements.
I have presented measurements of Feshbach molecule magneto-association in our Bose-Fermi
mixtures system in terms of formation in the limit of fast magnetic-field sweep (perturbative limit)
and the limit slow magnetic-field sweeps (saturated limit). In the perturbative limit, the initialmolecule creation rate was shown to be determined the Fano-Feshbach resonance parameters andthe density overlap of the boson and fermion atom clouds. In the saturated limit, the governingparameter is the phase-space density of the clouds and not just the spatial density. Here I presented
measurements of the molecule creation efficiency as a function of the atomic T/TF and fit the measurements to the phenomenological SPSS model. The SPSS model only allows molecule formation
between two atoms that are “close enough” in phase-space, where “close enough” is defined by the104sole adjustable parameter of the model. The molecule creation efficiency data were fit to the SPSS
model, and the value of the adjustable parameter agreed with similar measurements in homonuclear
systems. Measurements of the molecular expansion energy were also presented and were found to
be in excess of what the SPSS model would have predicted. The task of minimizing the molecularquantum degeneracy was explored using the SPSS model in the limit of the majority atom being
the fermion where no method to create a degenerate molecular gas was uncovered.
Measurements of three-body recombination rates for Bose-Bose-Fermi collisions, as well as
atom-molecule collision rates, were presented in order to study Efimov physics in 40 K – 87 Rb. This
study was prompted by the discovery of universality in the three-body parameter(which sets thelocation of Efimov resonances) for Efimov physics in homonuclear systems and the discovery of
Efimov resonances in 41 K – 87 Rb. In homonuclear systems, the three-body parameter was found(1)

to set the location of the first Efimov resonance in three-body recombination at a− = −9.1rvdW ,and since the van der Waals lengths between 40 K – 87 Rb and 41 K – 87 Rb are very similar, we
expected to find resonances at nearly the same scattering lengths as 41 K – 87 Rb. However, we did
not find evidence for Efimov resonances in three-body recombination within the accessible rangeof scattering lengths, in stark contrast with the results of 41 K – 87 Rb. A feature in atom-molecule
collisions was observed in the the Rb+molecule collisions that was absent in collisions between K
atoms and molecules, as well as distinguishable atoms and molecules, in agreement with Efimov
physics. Since we did not observe features in the same locations as 41 K – 87 Rb, we found noevidence for universality in the three-body parameter for heteronuclear systems. Observations inother heteronuclear systems will help resolve this issue.

6.2

Future work
The field of Efimov physics and many-body collisions has a few options for further study.

One important question to answer is why did we not observe an a− feature in 87 Rb – 87 Rb – 40 K
collisions, whereas in 87 Rb – 87 Rb – 41 K one was observed? This can be addressed by either repeating the measurements done in 41 K – 87 Rb or by measuring three-body recombination rates in105

scattering length (a0)4000

|a90| > a91

2000|a90| < |a91|

0- 2000- 4000

543

544

545

546

547

548

549

magnetic field (G)Figure 6.1: Interspecies scattering length for the 545.9G Feshbach resonance between K in the|9/2, −9/2i hyperfine state and Rb in the |1, 0i state (a90 , dash-dotted curve) and the 546.618 G
resonance between K in the |9/2, −9/2i and Rb in the |1, 1i state (a91 , solid curve). In the region
between the vertical lines, a90 is the dominant interspecies scattering length, whereas to the rightof the right-most vertical line, a91 is dominant.

other isotopic mixtures. Amongst the published Feshbach resonances in all the other KRb isotopic mixtures, the only two that are borderline entrance-channel dominated are in 41 K-85 Rb: thetheoretically predicted resonances at 185.2 G and 672.19 G [140] have sres = 3.6 and sres = 5.5,
respectively. However, this mixture would be quite challenging to realize at ultracold temperatures without a third atomic species in the mixture given 41 K’s small background scattering length(abg ≈ 60a0 [103]) and 85 Rb’s large negative background scattering length (abg = −443a0 [24]). If we
expand our search to include Feshbach resonances which are borderline closed-channel dominated,then one candidate is the 317.9 G resonance in 39 K-87 Rb [140]. This resonance has a magnetic-field
width of 7.6 G and sres ≈ 0.7. Another is the 78.9 G resonance in 41 K – 87 Rb used by LENS which
has sres = 0.8. Both of these resonances are at easily accessible magnetic fields.
Another interesting theoretical proposal involves measuring three-body recombination in unequal mass systems with overlapping Feshbach resonances [42]. This reference predicts the scattering length dependence for various three-body collision processes for the case of three distinguishable
particles with only two resonant scattering lengths. In 40 K – 87 Rb there is another 3 G wide s-wave
Feshbach resonance at 545.9 G [140] between the spin states |9/2, −9/2i in K and |1, 0i in Rb. The

106
degree to which this Feshbach resonance overlaps with the one used in Chapters 3 and 4 is shownin Fig. 6.1. Ref. [42] predicts that the event rate coefficient for recombining into deeply boundmolecules changes from simple a4 scaling to a290 a291 in the limit where one scattering is much larger
than the other. The recombination rate into Feshbach molecules is also predicted to scale as a290 a291 .
Other topics in many-body collisions to explore involve exploring Efimov physics with closedchannel dominated s-wave Feshbach resonances, where reff becomes of considerable magnitude compared to rvdW . For systems of three identical bosons in the recombination rates are predicted todepend upon reff as well as the scattering length: for |a|  reff , three-body recombination scales as
1/2

a4 , but for |a|  reff recombination scales as a9/2 reff [128]. It is also predicted that near closedchannel dominated resonances, inelastic processes leading to deeply bound molecules are suppressed
for bosons and enhanced for fermions [156] in contrast to what is seen with entrance-channel dominated resonances.
Since our three-body recombination spectra in the range of accessible scattering lengths was
devoid of Efimov resonances, we can also consider venturing into many-body physics territory.
One promising direction is to examine the effects of strong Bose-Fermi interactions on the boson
condensation temperature, Tc . Ref. [59], the authors predict a reduction in Tc as a consequence ofthe boson-fermion interaction leading to the vanishing of Tc at a small positive scattering length.
Examining the quantum phase diagram of a strongly interacting Bose-Fermi mixture also is arich avenue to take. Ref. [102] predicts the presence of first- and second-order quantum phasetransitions as well as path dependent (in terms of varying the scattering length) quantum phases.
This hysteretic effect would manifest itself in terms of a jump in the condensate number. An
additional route of research is to examine the BEC-BCS cross over in the presence of bosonic
impurity; this creates the possibility of new physics as well as a reliable thermometer.

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A, 78(1):013416, 2008.

Appendix A

Myriad details of the SPSS model

In this Appendix we discuss the finer details of our implementation of the SPSS model. We
begin in Secs. A.1, A.2, and A.3 with discussion of the Monte Carlo algorithm implemented togenerate random Bose-Einstein and Fermi-Dirac distributions in position and momentum space,which are used as the atom distributions inputted into the SPSS model. Following this, we showthat the generated distributions do indeed faithfully reproduce the Bose-Einstein or Fermi-Dirac
distributions in Sec. A.5. Next we discuss scaling the atom numbers and their effects on the fit of
the SPSS to our data in Sec. A.5. Lastly, in Sec. A.6 we discuss signatures of edge-effects in themodel and the dependence of simulated molecule conversion fraction on factors other than T/Tc
and T/TF .

A.1

Accept-reject Monte Carlo algorithm
Monte Carlo algorithms are a widely used computation method for calculating probability

distributions using random numbers and some knowledge of the desired probability distribution. In
the case of the Fermi-Dirac, or Bose-Einstein, probability distribution, we can directly sample theprobability distribution, which makes the generation reasonably simple. This process generates alist of particle locations and momenta that has either a Bose-Einstein probability distribution, or a
Fermi-Dirac distribution. Our atom generation loops through position space and momentum spacevolumes (shells to be exact) and the Monte Carlo algorithm decides whether a particle should beplaced within a particular shell. This accept-reject algorithm adds a particle at the given location

119

vp
vr

Ψ=1/2 Ψ=1/4
Ψ=1/4 Ψ=1/8

Figure A.1: Boxes in position and momentum whose probability of occupation, Ψ, is shown.

ifu ≤ Ψ(vr , vp )(A.1.1)

where u is a pseudo-random number drawn from a uniform distribution whose possible values span
0 to 1, and Ψ(vr , vp ) is the known probability of a particle existing in the position space volume
vr and momentum space volume vp (defined in Sec. A.2). Once a particle is added, its location inposition and momentum is randomly offset within the position and momentum volume.
To understand why this algorithm can faithfully reproduce the Bose-Einstein or Fermi-Dirac
distribution we first focus on the uniform distribution. For this uniform distribution with u ∈ [0, 1],
the conditional probability that a number randomly chosen is less than or equal to a particularvalue, X, isZ X
P rob(u ≤ X) =dy = X.

(A.1.2)0Now, imagine we have a series of boxes whose probability of being populated is known apriori, as shown in Fig. A.1. For the lower right box, we know that that box will be populated
1/8 of the time. Additionally, if we draw a uniform random number whose value u ≤ 1/8, we also
know that the probability of that happening is also 1/8. Therefore, if u ≤ Ψ, then a particle should
be added at that location in position and momentum space. The only restriction is that we must

120
make small steps in position and momentum space such that the probability of a given box being
occupied is Ψ  1 in order faithfully reproduce the Bose-Einstein or Fermi-Dirac distribution.

A.2

Fermionic atom generation
This Monte Carlo algorithm requires that we know what the probability is for a fermion

existing at a given shell in position and momentum space. To calculate this probability, we begin
with the number of fermions as determined by the Fermi-Dirac distribution [124]:
NF =

1
h3

Z

Z
···

1
z −1 e(KE+P E)/(kB T ) + 1

d3 x d3 p(A.2.1)

where h is Planck’s constant, z is the fugacity, kB is Boltzman’s constant, and T is the temperature.
The kinetic energy (KE) and potential energy (P E) for a harmonically trapped gas is1
1
1
P E = m ωr2 x2 + m ωr2 y 2 + m ωz2 z 2
2
2
2
p2y
p2z
p2x
+
+
KE =
2m 2m 2m

(A.2.2)(A.2.3)for a particle of mass m. Here we assume that the trap frequencies in the x- and y-direction are
given by ωx = ωy = ωr and the trap frequency in the z-direction is ωz . Next, we make the following
coordinate system change

s

r

s

mωr2
x=X
2kB T

1
px = Px
2mkB T

r

mωr2
y=Y
2kB T

1
py = Py
2mkB Tsrmωr2
λz = Z
2kB T

1pz = Pz
2mkB T

where λ = ωz /ωr , and then the fermion number isZZ
2kB T 3/2
1
3/2
d3 X d3 P(2mkB T )···2 +Y 2 +Z 2 +P 2 +P 2 +P 2
2
X
x
y
z + 1
mωr
z −1 e
Z Z(4π)2 2kB T 3 1 ∞ ∞R2 P 2=dR dP(A.2.4)h3
ωr2
λ 0
z −1 eP 2 +R2 + 1
01
NF = 3hwhere in the last line we have change to spherical coordinates with R2 = X 2 + Y 2 + Z 2 , P 2 =Px2 + Py2 + Pz2 and evaluated the angular integral, which yielded the (4π)2 factor. The probability,121
Ψ, of a fermion existing at a given value of R and P within a 6-d shell of thickness ∆R∆P isdetermined by the integrand of the previous equation:
16π 2
Ψ= 3
h



2kB T
ωr

3

R2 P 2
1∆R ∆P.
λ z −1 eP 2 +R2 + 1(A.2.5)

We prefer to express Ψ in terms of T/TF and atom number. The fermion atom number
integrals can be solved analytically when rewritten in terms of density of states and energies. The
number of fermions relates to the Fermi temperature by:
1
NF =
6



kB TF
~ωr

3

1
λ

(A.2.6)

If we use this to rewrite our expression for Ψ, we arrive at the expression used in the Monte Carloafter a few steps:
R2 P 2
NF 16π 2 2kB T 3 1
Ψ=
∆R∆P
NF h3
ωr
λ z −1 eP 2 +R2 + 1
3
1
16π 2 2kB T
3ωr
λ
R2 P 2
h
∆R∆P
= NF
3

−1 P 2 +R2 + 1
1 kB TF
1 z e
6

= NF

96
π

~ωr



T
TF

3

λ

R2 P 2∆R∆P.
z −1 eP 2 +R2 + 1(A.2.7)

where in the Monte Carlo code, the parameter T 1 = T/TF . The volume variables that the code
loops over are defined as:
1
vr = R 3
31
vp = P 3
3(A.2.8)∆vr = R2 ∆R∆vp = P 2 ∆P

(A.2.9)where the last line is the volume of the position space and momentum space shells.
Finally, we’d like to have a sensible way to set the step size in phase-space such that we avoid
the situation where our step size is large enough to have allowed for multiple particles. In fact
we would actually prefer to have the maximum probability, maxp  1 such that the Monte Carloalgorithm more faithfully reproduces the Fermi-Dirac distribution1 . If we let R = P , we can find1

A value of maxp = 0.05 is sufficient for most calculations with fermions or bosons.

122
the maximum with the following:
3 z z − e2 P 2 (P 2 − 1)
4P
d
=0
=
2
dP z −1 e2P 2 + 1
e2 P 2 + z
2⇒ z − e2 P (P 2 − 1) = 0.
P4

The value of P which maximizes the probability depends roughly on z as√ln z and varies between

P = 1 for z = 0 and P ≈ 7 for z = 2.6 × 1043 (T/TF = 0.01). So, we arbitrarily chose to set P = 1
and arrive at an expression for maxp
96
maxp = NF
π



T
TF

3

1
∆R∆P
z −1 e2 + 1(A.2.10)

Solving for the shell thickness ∆R∆P gives the step size (in vr and vp ) for the algorithm as
maxp π(step size) = ∆R∆P =
NF 96
2

A.3



T
TF

−3

(z −1 e2 + 1).

(A.2.11)

Bosonic atom generation
The derivation for the boson algorithm expressions follows a similar logic as the fermions.

The number of bosons can be calculated from the Bose-Einstein distributions, which yields
ZZ
1
1d3 x d3 p
NB = 3 · · ·(KE+PE)/(kB T ) − 1
−1
h
z e


Z Z
(4π)2 2kB T 3 1 ∞ ∞R2 P 2
=
dR dP
h3
ωr2
λ 0
z −1 eP 2 +R2 − 1
0(A.3.1)after making the same coordinate system change as in the fermionic case. The probability of agiven shell in position and momentum space is then
16π 2
Ψ= 3
h



2kB T
ωr

3

1
R2 P 2∆R ∆P.
λ z −1 eP 2 +R2 − 1(A.3.2)Using the definition of Tc , we can rewrite Ψ in terms of NB and T/Tc , which yields
16
Ψ = NB
πζ(3)



T
Tc

3

R2 P 2∆R ∆P,z −1 eP 2 +R2 − 1

where ζ is the Riemann zeta function, for which ζ(3) ≈ 1.20206.

(A.3.3)

p2 PHpL H 1030 kg-1 m-1 sL

r2 nHrL H107 m-1 L

123
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0

1

r HΜmL

2

3

4

5

6

7

7
6
5
4
3
2
1
0
0.0

0.5

0.02
0.00
-0.02
-0.04
0

1

2

r HΜmL

3

4

5

1.5

2.0

p H 10-27 kg msL

2.5(b) Momentum distribution

fractional residual

fractional residual(a) Spatial density distribution

1.0

6

0.02
0.00
-0.02
-0.04
0.0

7

(c) Spatial density residual

0.5

1.0

1.5

2.0

p H 10-27 kg msL

2.5(d) Momentum distribution residual

Figure A.2: Comparison of a Bose distribution generated with the Monte Carlo (red points) with
the ideal Bose gas distribution (black curve). The distributions were generate for Nboson = 105 andT/Tc = 1.05 for a harmonic trap with radial trap frequency ωr = 2 π 448 Hz and trap aspect ratio
λ = 0.013. The momentum and spatial density distributions are the result 24 runs of the Monte
Carlo simulation with maxp = 0.05. c) and d) Fractional residuals for a) and b), respectively, where
the fractional residual is defined as the Monte Carlo result divided by Eqn. (A.4.1) or (A.4.2) minus
1.

Following the analysis of the fermionic case, the step size for loops in the Monte Carloalgorithm is then:(step size)2 = ∆R∆P =A.4

maxp πζ(3)
NB 16



T
TC

−3

(z −1 e2 − 1).

(A.3.4)

Verifying Monte Carlo atom density and momentum distributions
Before the SPSS model can be applied, an ensemble of atoms with the correct density andmomentum distributions is generated. To demonstrate that our Monte Carlo code produces dis-124
tributions that, on average, agree with either the Bose-Einstein or Fermi-Dirac distribution, weaverage an ensemble of Monte Carlo distributions and compare with the ideal distributions. In
Figs. A.2 and A.3 the Monte Carlo spatial (solid circles) and momentum (open circles) density
distributions are shown for the Bose and Fermi gas simulations, respectively. We also plot theharmonically trapped ideal Bose and Fermi gas spatial and momentum density distributions. The
harmonically trapped Bose gas spatial2 (nBoson (r)) and momentum densities (ΠBoson (p)) are given
by


λN
−r2 /2σr2
Lize/Li3 (z)3/2(2π)3/2 σr3N−p2 /2σp2/Li3 (z)Πboson (p) =
Li
ze
3/2
(2π)3/2 σp3
nboson (r) =where z is the fugacity, Lin (z) is the polylogarithm function3 , σr2 =(A.4.1)(A.4.2)

kB T
for atomic mass m,mωr2

σp2 = mkB T ,and r2 = x2 + y 2 + λ2 z 2 . Similarly for a harmonically trapped fermions, the spatial
and momentum densities areλN−r2 /2σr2
Li−ze/Li3 (−z)3/2(2π)3/2 σr3N−p2 /2σp2/Li3 (−z) .
Πfermion (p) =
Li
−ze
3/2
(2π)3/2 σp3nfermion (r) =(A.4.3)(A.4.4)The plots of Eqn. (A.4.1) and Eqn. (A.4.2) for Bosons and Eqn. (A.4.3) and Eqn. (A.4.4)
for Fermions used the same harmonic trap parameters and fugacities as with the Monte-Carlocalculation. The Monte-Carlo distributions were generated for a ‘cigar-shaped’ harmonic trap withan aspect ratio of λ = = 0.013, where ωz is the axial trap frequency, and with radial trap frequency
ωr =2π 448 Hz and 627 Hz for the bosons and fermions, respectively. For both the boson andfermion distribution, approximately 105 atoms were generated at a moderately high degeneracy(T/Tc = 1.05, T/TF = 0.11) in order to look for any problems with undersampling. The
2
These densities are found by integrating the right-hand side of Eqn. A.3.1 over just momentum to calculate
nBoson (r) or just position to calculate ΠBoson (p).
3
The polylogarithm is related to the Bose-Einstein integrals on the right-hand side of Eqn. A.3.1 by [161]
R ∞ ks dk= Γ(s + 1) Li1+s (z), where Γ(z) is the gamma function. The relation for the integrals on the right0 z −1 ek − 1
R ∞ ks dk
hand side of Eqn. A.2.1 by 0 −1 k= −Γ(s + 1) Li1+s (−z).
z e +1

p2 PHpL H 1030 kg-1 m-1 sL

r2 nHrL H107 m-1 L

125

4
3
2
1
0
0

1

2

r HΜmL

3

4

5

14
12
10
8
6
4
2
0
0.0

(a) Spatial density distribution

fractional residual

fractional residual

0.02
0.01
0.00
-0.01
-0.02
1

2

r HΜmL

3

0.4

0.6

0.8

1.0

p H 10-27 kg msL

1.2(b) Momentum distribution

0.03

0

0.2

4

5

(c) Spatial density distribution

0.04
0.03
0.02
0.01
0.00
-0.01
-0.02
-0.03
0.0 0.2 0.4 0.6 0.8 1.0 1.2
p H 10-27 kg msL(d) Momentum distribution

Figure A.3: Comparison of a Fermi distribution generated with the Monte Carlo (blue points) with
the ideal Fermi gas distribution (gray curve). The distributions were generate for Nf ermion = 105
and T/TF = 0.11 for a harmonic trap with radial trap frequency ωr = 2 π 627 Hz and trap aspectratio λ = 0.013. The momentum and spatial density distributions are the average of 24 runs of theMonte Carlo simulation with maxp = 0.05. c) and d) Fractional residuals for a) and b), respectively.

excellent agreement between the Monte-Carlo results and the ideal Bose and Fermi gas distributions
indicates that the Monte-Carlo simulation reproduces the appropriate distributions faithfully.
An additional check of the distributions generated by the Monto-Carlo code is to check that
the Virial theorem is upheld. The Virial theorem is a relation between the average kinetic energyof a sample and the forces acting on each particle in the system. Classical systems, as well as
quantum mechanical systems, follow the Virial theorem. The Virial theorem is given by [124]*+Xri ṗi = −2 hKi ,(A.4.5)iwhere the brackets represent an ensemble average, K is the kinetic energy, and ṗi is the time-

126
derivative of the momentum of particle i at the position ri . If the particles are assumed to be
non-interacting and trapped in a potential of the form U (r) = A rn (A is a constant), then the
Virial theorem simplifies ton hU i = 2 hKi ,(A.4.6)

where hU i is the ensemble average potential. This result simplifies to hU i = hKi for the harmonically trapped case (n=2).
To test that the Virial theorem is obeyed, twenty distributions were generated by the MonteCarlo over a range of temperatures for both bosons and fermions and the average potential andkinetic energies were computed. The results are plotted in Fig. A.4, where the error bars are thestandard errors of the mean of the twenty distributions for each temperature. The good agreementwith the Virial theorem provides further evidence that the distributions generated by the MonteCarlo are faithfully reproducing the ideal Fermi and Bose distributions.

(a) Bosons(b) Fermions

Figure A.4: Checking the Virial theorem for Bose and Fermi distributions generated by the MonteCarlo code. Each point is an average of the ratio hU i / hKi for twenty distributions at a given
temperature. The error bars at the standard error of the mean.

A.5

Fitting the SPSS model
Fitting the SPSS model to atom data where the atom numbers are of order 105 to 106

is a computationally intensive process. One way to speed up the fitting process is to scale the127atom number down in a sensible fashion such that the T/TF (or T/Tc ) axis is preserved. From
Eqn. (2.5.1) Tc ∝ ω̄N 1/3 , where ω̄ = (ωr2 ωz )1/3 for radial and axial trap frequencies, ωr andωz . The Fermi temperature also has the same dependence on atom number and trap frequency:
TF ∝ ω̄N 1/3 . One scaling method is to scale the atom number and trap frequencies (with fixed
aspect ratio λ = ωz /ωr ) such that ω̄N 1/3 is fixed and fix the temperature, as well. The results ofthis scaling method is shown in Fig. A.5a where the model was fit to the data in Fig. 3.8 for thevalue of γ from Eqn. (3.5.1) using a range of scaling factors. The solid horizontal lines are thefit results for 85 Rb2 molecules (upper solid line) and 40 K2 molecules (lower solid line) from Ref.
[75]. As can be seen, the fitted value of γ depends strongly on the value of the scaling factor andtherefore scaling the problem in this manner would require corrections to account for the sensitivity
to the scaling factor.
An alternative method is to scale the aspect ratio, λ, with the atom number such that
TF ∝ ω̄N 1/3 = ωr (λN )1/3 is fixed. This method, in addition to fixing T/TF and T/Tc also
fixes the rms radial cloud size. The results of this scaling method are shown in Fig. A.5b as afunction of the scaling factor. For this figure, a scaling factor of 1 corresponds to λ = 0.013, which
describes a prolate spheroid (λ < 1). As can be seen, this results for γ from the fits asymptoticallyapproach a constant value for low values of the scaling parameter corresponding to prolate spheroids.
Additionally, the deviation from the zero scaling factor limit is small upto where the simulation
corresponds to a spherical atom cloud (vertical dashed line where λ = 1). This means that we canreduce the atom number in the simulation by a factor of 1/0.013 = 77, which will decrease thesimulation time by4 772 without a substantial loss in accuracy.

A.6

Other details of the SPSS model
While the scaling of λ and N provides a substantially better way to scale the model than themethod described previously, it may also be a signature of edge effects in the model where atoms at
the edges of the cloud are associated at a different efficiency compared to the bulk. If we consider
4

The molecule association algorithm, which is the part of the simulation of longest duration, scales as N 2 .

128
0 .3

8 5

0 .2

4 0

R b

F i t t e d v a l u e o f γ/2

F i t t e d v a l u e o f γ/2

0 .3

K

0 .1

0 .0

0 .2

0 .1

c ig a r s h a p e ds p h e r ic a l

8 5

R b

4 0

K

s k ittle s h a p e d

0 .0
1

1 0

1 0 0
1 0 0 0s c a l i n g f a c t o r ( N s i m = N / s c a l e , ωs i m = ω* s c a l e 1 / 3 )

1
1 0
1 0 0
1 0 0 0s c a l i n g f a c t o r ( N s i m = N / s c a l e , λs i m = λ* s c a l e )

(a)

(b)

s u r fa c e a r e a / v o lu m e

2 0
1 6
1 2
8
4
0
0 .0 10 .1

λ = ωz / ωr

1

1 0

(c)

Figure A.5: a) Fitting results where the system was scaled such that the product ω 3 N was kept
constant. b) Fitting results for the model where the system was scaled such that the product N λ
was kept constant. The vertical dashed line indicates where the aspect ratio λ = 1. c) Surface area
to volume ratio for a spheroid.

the case of constant density, n, distributed throughout a volume, then the number of atoms on thesurface at a radius R is NS = SA(R) n dR, where SA is the surface area and dR is the differential
thickness of the surface. The number of atoms in the volume, V , is simply NV = n V and so theratio NS /NV ∝ SA/V . The surface area to volume ratio tells us the the number of molecules on
the edges of the cloud compared to the bulk. In Fig. A.5c the surface area to volume ratio fora spheroid is shown as a function5 of λ for a fixed radial size of one, and therefore radial trap
5

The trap frequency aspect ratio λ is related to the axial and radial cloud sizes (σz and σr , respectively) by
λ = σr /σz .

129

0 .2 50 .3

0 .1 5m o ls

/ N

R b

0 .2

0 .1

0 .1 0N

N m o ls .N R b

0 .2 00 .0 50 .0

0 .0 10 .1
1
λ = ωz / ωr ( N λ f i x e d )0 .0 0

1 0

1 0 2

1 0 3
N

1 0 4

(a)

K

, N

1 0 5

R b

(b)

0 .4

0 .2

N

m o ls

/ N

R b

0 .3

0 .1
0 .0

1 0 2

1 0 3

1 0 4
1 0 5
3
N R b , N K ( N ω fix e d )

1 0 6

(c)

Figure A.6: a) Simulated molecule fraction calculated by varying N and λ such that the productNλ was fixed. b) Molecule fraction calculated by varying the atom number with all other input
parameters fixed. c) Attempt to calculate the molecule fraction in the thermodynamic limit whereN→ ∞, ω → 0, but Nω 3 is constant.

frequency, from equations in Ref. [162].
Shown in Fig. A.6a is the simulated molecule creation fraction where the atom number,N, and λ was varied with the product Nλ fixed for γ = 0.34. For this simulation NK = NRb ,T/Tc = 1.05, T/TF = 0.54, and the radial trap frequencies were 627 Hz for K and 448 Hz for Rb.
As a consequence of all these fixed parameters, the atom temperatures in this simulation is also
fixed, as well as the radial cloud sizes. In this figure the solid points have the Nλ product for which
λ = 0.01 corresponds to N = 105 ; for the hollow points λ = 0.01 corresponds to N = 104 . This

130
model predicts, in addition to the dependence on T/Tc and T/TF , that the molecule conversion
efficiency also depends on the aspect ratio of the trap. Additionally, the model predicts that themolecule fraction also depends on the atom number even for fixed ratio of Rb to K atoms, as canbe seen from Fig. A.6a, as well as in Fig. A.6b where the molecule fraction was computed forvariable atom number (NK = NRb ) and fixed trap frequencies, λ = 0.013, T/TF , and T/Tc . While
the atom number in our data range from several 104 to a few 105 and thus the size of this effect isbelow our number noise, it is a bit disconcerting that the model predicts a molecule fraction thatstrictly increases with atom number. If we examine the thermodynamic limit to address the number
dependence concern (N → ∞, N ω 3 = const, i.e. a fixed average density but increasing system size)for fixed T/Tc and T/TF , we find that the molecule fraction does not reach an asymptotic value foratom numbers up to NK = NRb = 5 × 105 , as shown in Fig. A.6c. From a theoretical stand-point
this is a bit alarming, but perhaps over a narrow range of conditions it is not too terrible of an
approximation.