Ultracold Polar KRb Molecules in Optical Lattices
byBrian Neyenhuis
B.S., Brigham Young University, 2006

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2012This thesis entitled:
Ultracold Polar KRb Molecules in Optical Lattices
written by Brian Neyenhuis
has been approved for the Department of Physics

Deborah Jin

Prof. Jun Ye

Date

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

iii
Neyenhuis, Brian (Ph.D., Physics)
Ultracold Polar KRb Molecules in Optical Lattices
Thesis directed by Prof. Deborah Jin

The creation of a gas of ultracold polar molecules with a high phase space density brings new
possibilities beyond experiments with ultracold atomic gases. In particular, long-range, anisotropic,
and tunable dipole-dipole interactions open the way for novel quantum gases, with applications
including strongly correlated many-body systems, and ultracold chemistry. This thesis will present
the final steps to complete control over both internal and external degrees of freedom of the molecule
which allows us to control, and even completely suppress, the chemical reactions between molecules.
First, the control over internal states has been achieved through coherent state transfer to thero-vibronic ground state and coherent manipulations of the hyperfine and rotational states withmicrowave radiation. Second, external degrees of freedom are controlled by loading the gas into
an optical lattice. With the molecules loaded into a one-dimensional lattice, the orientation of
the molecular collisions is controlled by manipulating both internal (hyperfine states) and external(motional states in the direction of tight confinement) degrees of freedom. Most striking is thatby preparing the molecules all in the lowest band of the lattice in the same internal state, themolecular collisions can only occur in a “side-by-side” orientation, where the chemical reaction rateis suppressed by the repulsive dipole-dipole interactions. The chemical reaction can be suppressed
completely by further constraining the motion in the trap in a strong 3D lattice. Here we see
lifetimes longer than 20 s, limited by off-resonant light scattering. Finally, the ac polarizabilityof the molecules is explored and controlled. The different rotational states of the molecule havedifferent polarizabilities and will experience a different trapping force in both the optical dipole
trap or lattice. We show that there is a “magic angle” between the quantization axis and thepolarization of the trapping laser at which the polarizabilities of two different rotational states
can be matched, eliminating dephasing and allowing for coherent manipulations between rotational

iv
states.

Dedication

To my family. I started this adventure with my wife Marisa and we are finishing it with three
beautiful children: Cameron, Liesel, and Ivy. Thank you for all your understanding and support.

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Acknowledgements

One of the many fantastic things about working at JILA is that you get to associate with a
lot of great people. I used to tell people that we had a small army working on the KRb project,and now that I look back on the list of all the people on the project, I realize that this is not too
far from the truth. I had the great pleasure of working with (listing only the experimentalists who
worked directly on KRb) Josh Zirbel, Kang-Kuen Ni, Silke Ospelkaus, Avi Pe’er, Dajun Wang,
Marcio de Miranda, Amodsen Chotia, Steven Moses, Bo Yan, and Jacob Covey.
An experiment as complicated as the KRb machine wasn’t built in the space of a single
graduate career. Josh and Kang-Kuen started the KRb project in an empty lab and by the time Joshgraduated they had created ultracold Feshbach molecules. We were able to continue to build onto
an already complex experiment because of Josh’s attention to detail in designing the experiment.
Even though Josh was close to graduation when I joined the experiment, he took the time topatiently explain the experiment to me. Avi also had a gift for explaining complex ideas in a very
simple way and was never too busy to stop and answer my questions or listen to my dumb ideas.
I learned a lot from working with Kang-Kuen and Silke. They showed me the ropes and thentaught me how to diagnose problems, troubleshoot the experiment, and made me dive into every
subsystem of the experiment in great detail. They expected a lot from me when I was a young
graduate student, and as a result I learned a lot.
There was about six months when we had enough man power that we ran the experiment in
shifts. Dajun and I ran the early shift, and then handed the experiment off to Silke, Kang-Kuen andMarcio in the afternoon. Dajun is one of the hardest working people I have ever met and I learned

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a great deal from him as we worked together on the direct imaging of KRb. Marcio taught me themystical ways of the octave spanning frequency comb. We were the two graduate students who“inherited” the experiment after Silke and Kang-Kuen left, and together with Dajun and Amodsen
we were able to bring the experiment beyond just creating and observing ground state molecules,
and onto manipulating them. On one of the papers we published together there is a star next toeach one of our names denoting that we “contributed equally.” That is an accurate description of all
the work that went on in the lab at that time. We all worked together and contributed equally on
all parts of the experiment and were able to see fantastic results. Working with Amodsen, Marcio,
and Dajun was a real pleasure. They put up with my odd sense of humor and made working in
the lab very enjoyable.
Now the experiment is being passed on to Steven, Jake, and Bo and I see a very bright
future. They are about to move the experiment into a new lab and make some major changes tothe apparatus, and having worked with them for the last few years, I have complete confidence intheir ability to succeed. I can’t imagine what it would be like to work on an experiment all bymyself and I feel very privileged to have worked with such great people.
In addition to those who worked directly on the experiment with me I had a lot of help
from theorists. Among them John Bohn and Goulven Quéméner stand out and warrant special
mention. John and Goulven have helped guide the KRb experiment theoretically for the majorityof my graduate school career. Goulven has always been willing to take time out of his day toanswer my questions and explain the finer points of scattering theory. I have also had the great
pleasure of working with Michael Foss-Feig, Kaden Hazzard, Salvatore Manmana, Jose D’Incao,
Chris Greene, Paul Julienne, and Svetlana Kotochigova. Their theoretical guidance allowed us toalways be moving forward.
I would like to acknowledge all the members of the bi-group especially Eric Cornell. Our
weekly meetings showed me a much broader view of atomic physics and provided great advice.
Jun’s group also provided countless solutions to technical problems. I of course need to thank the
JILA staff in both the electronics and instrument shop as well as Krista Beck. I am also gratefulviii
to my running partner Ryan Thalmann for getting me up at 6am every weekday to run and talkphysics, chemistry, and politics.
Lastly, I need to acknowledge Debbie and Jun. I could not have asked for better advisors.
I was not technically Jun’s student, it was Debbie who hired me my first year before the project
had fully merged into a joint venture. But he always took time to guide me and teach me. Jun’s
expertise in precision measurement has helped me learn how to do things right (as opposed to mynormal mode of operation which is “do it fast”), and more importantly taught me how to know
when it matters. I was always impressed that after all the amazing discoveries that Jun has been
apart of he would still get so excited about about our progress both big and small. He is always
driving the experiment along, thinking about the next steps and pushing forward.
I have never met anyone with as much physical intuition for cold atom gasses as Debbie has,and I hope some of that has rubbed off on me. Debbie takes great pleasure in solving problems. It
doesn’t seem to matter how big or small they are, and she isn’t afraid to dive in and get right into
the details. She has a gift for looking past all the noise and getting right to the very essence of aproblem. I have learned to trust and appreciate her advice and I consider her a true friend.

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Contents

Chapter
1 Introduction1

1.1

Why should I care about polar molecules at all, let alone this thesis? . . . . . . . . .

1

1.2

So what did you do that is such a big deal? . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2 Optical Lattice

9

2.1

Optical dipole force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2

Optical lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3

Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4

Experimental realization and charaterization of the lattice . . . . . . . . . . . . . . . 17

2.5

Diffraction of a BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6

Parametric heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.7

Band Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.8

An easier way to detect atoms/molecules in higher bands . . . . . . . . . . . . . . . 29

2.9

How to load just one band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Universal chemical reactions in KRb

34

3.1

Bimolecular chemical reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2

Two models to understand the loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3

Chemical reactions between distinguishable molecules . . . . . . . . . . . . . . . . . 41

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3.4

Control of chemical reactions with a DC electric field . . . . . . . . . . . . . . . . . . 41

3.5

Anti-evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 KRb in 2D

55

4.1

Controlling chemical reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2

What is 2D?

4.3

Stereodynamics and quantized collisions . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4

Preparation of the 2D molecular system . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.5

Measurement of the molecular loss rate

4.6

Comparison to theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.7

2D vs. 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Molecules in a 3D lattice. . . . . . . . . . . . . . . . . . . . . . . . . 63

76

5.1

Preparation of ground state molecules in a 3D lattice . . . . . . . . . . . . . . . . . . 77

5.2

Ground state molecule lifetime in the 3D lattice . . . . . . . . . . . . . . . . . . . . . 78

5.3

2D to 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.4

Feshbach molecules in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.5

100% conversion of atoms into Feshbach molecules . . . . . . . . . . . . . . . . . . . 87

6 Anisotropic Polarizability of KRb

89

6.1

Anisotropic polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.2

A more complete model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.3

Shift in transition frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.4

Rotational dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.5

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7 Conclusions and Outlook

104

7.1

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.2

Future work and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

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7.2.1

Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.2.2

Enhanced molecular formation in a 3D lattice . . . . . . . . . . . . . . . . . . 109

7.2.3

Many-body physics with long-range interactions . . . . . . . . . . . . . . . . 109

Bibliography

112

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TablesTable
3.1

Relevant molecular binding energies involved in possible chemical reactions. The
binding energies are given with respect to the threshold energy for free atoms in theabsence of magnetic field. The 87 Rb2 and 40 K2 binding energies include the isotope
shifts from the data in the respective references.

6.1. . . . . . . . . . . . . . . . . . . . 36

The experimentally measured rotational-hyperfine transition frequencies of the lowest vibrational level of the X1 Σ+ potential of KRb at zero laser intensity (i.e. without
trapping light) and a bias magnetic field with strength B = 545.9 G. Transitions startRb
at the |N = 0, mN = 0, mK
I = −4, mI = 1/2i state and go to three hyperfine states|ji within the N = 1 manifold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

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FiguresFigure2.1

Band structure of an optical lattice. Energy of the different bands in the lattice isplotted vs. quasimomentum q for 0, 2, 5, and 10 Erec . Only the first Brillouin zone
is shown.

2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

The Wannier functions and their corresponding probability density for a 3 Erec (toppanels) and a 10 Erec (bottom panels) lattice. The lattice potential is shown in red
to indicate the location of the sites. The probability density for three adjacent sites
is shown to illustrate the wavefunction overlap which determines the tunnelling rate.

2.3

16

A schematic of the vertical lattice. The lattice light comes out of the fiber is columnated by lens L1. Lens L2 focuses the beam onto the atoms/molecules in the centerof the cell. L3 is 2f away from the atoms and 2f away from the retroreflection
mirror RRM. This 2f-2f imaging system refocuses the lattice onto the atoms while
minimizing the sensitivity to small drifts in both the angle and position of RRM. . . 18

2.4

The diffraction of a BEC in a vertical optical lattice.

Momentum components

p = 0, ±2~k, ±4~k, ±6~k are shown after a diabatic release from the trap. The
populations oscillate as a function of hold time in the lattice. . . . . . . . . . . . . . 20
2.5

The relative populations in the p = 0 (blue), p = ±2~k (green), and p = ±4~k (red)diffraction orders. The solid lines are a fit to the data using a numerical solution to
Rb . 22the time dependent Schrodinger equation to extract the lattice depth V0 = 43(4) Erec

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2.6

KRb lattice. The amplitude
A parametric heating resonance for KRb in a 94.3(3) Erecof the lattice is modulated sinusoidally for 4 ms with a peak to peak amplitude 10%of the total intensity. The size is measured after 3 ms time of flight. A gaussian is
fit to the observed resonance to find the center of 49.37(9) kHz. . . . . . . . . . . . . 23
2.7

The transition from the ground band to the second excited band (in Erec ) as afunction of the trap depth. The solid black line is the transition at the center of theband (q = ~k/2). The dashed black lines represent the transition at the edges of theBrillouin zone (q = 0 and q = ~k). Twice the trapping frequency is plotted in red. . 25

2.8

The parametric heating resonance divided by twice the trap frequency as a functionof trap depth. The dashed lines are the transitions at q = 0 (bottom) and q = ~kand the solid line is the transition frequency at q = ~k/2. . . . . . . . . . . . . . . . 26

2.9

The depth of the lattice versus the parametric heating resonance. From this graphwe can read off the trap depth from our resonance. Units are given in Erec to keep
the plot general for particles of any mass. . . . . . . . . . . . . . . . . . . . . . . . . 27

2.10 Band mapping K atoms in a 1D lattice to measure the population in higher bands.
In a and b 28% of the atoms are in the second band. By holding in a weak lattice
K for 1 ms we allow these atoms in higher band tunnel out of thewith depth 3.2 Ereclattice leaving only atoms in the lowest band as shown in c and d. The black traceis the OD of the cloud integrated over the vertical direction (perpendicular to thelattice) to increase signal to noise. The fit is shown in red. . . . . . . . . . . . . . . . 30
2.11 The size of a gaussian fit to a simulated with a thermal population of higher bands,
imaged after release from the lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.12 The size of a gaussian fit to a K cloud released from the lattice. The population ofhigher bands was measured using band mapping. . . . . . . . . . . . . . . . . . . . . 33

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3.1

A sample molecular density decay vs time for inelastic collisions between indistinguishable fermionic molecules in the ro-vibronic ground-state of 40 K87 Rb. HereRb = 1/2i,
the molecules are prepared in a single hyperfine state, |mK
I = −4, mI

and the molecular density-weighted density decays slowly with a rate coefficient of
3.3(7) × 1012 cm3 /s at T = 250 nK. Figure reproduced from reference [1]. . . . . . . 36
3.2

Inelastic collision rate coefficient vs temperature for fermionic molecules in the rovibronic ground state of 40 K87 Rb. For the lower loss coefficients, the collision rate
coefficients were measured for molecules prepared in either |4, 1/2i (closed circles) or
the lowest energy state |4, 3/2i (open triangles). We observe the loss rate increases
linearly with temperature for spin-polarized molecules, which verifies that the dominant collision channel is p-wave. A linear fit (solid line) to the |4, 1/2i data yields
the temperature-dependent loss rate to be 1.2(3) × 105 cm3 /s/K. For the |4, 3/2i
case, where the collisional loss can only be due to chemically reactive scattering, theloss rate is similar. The dotted and dashed lines are theoretical predictions from the
QT model and MQDT (describe in the text), respectively. In contrast, when themolecular sample is prepared in a mixture of two hyperfine spin states, |4, 1/2i and|4, 3/2i (filled squares), s-wave collisions dominate. Here, we observe a temperatureindependent decay rate that is 10 − 100 times higher than for the spin polarized case.
Figure reproduced from reference [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3

Molecular density decay vs. time for induced dipole moments of d = 0.08 D (opentriangles) and d = 0.19 D (filled circles) at T0 = 300 nK. Figure reproduced fromreference [2].

3.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Rate coefficient β/T0 vs. induced dipole moment, d. The dashed line shows a fit to
the simple model based on the quantum threshold behavior for tunnelling through adipolar-interaction modified p-wave barrier. The solid line shows a result of a more
complete quantum scattering calculation. Inset, the calculated dependence of d on
the applied electric field E. Figure reproduced from reference [2]. . . . . . . . . . . . 44

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3.5

Schematic showing the effective intermolecular potential for fermionic molecules atzero electric field. At intermediate intermolecular separation, two colliding molecules
are repelled by a large centrifugal barrier for p-wave collisions. For a relatively small
applied electric field the spatially anisotropic dipolar interactions reduce the barrier for “head-to-tail” collisions and increase the barrier for “side-by-side” collision.
Figure reproduced from reference [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.6

Barrier height vs. effective dipole moment, d. The solid lines are calculated using theadiabatic energies considering partial wave mixing up to L = 9. The mL = ±1 barrierheights rise as the effective dipole moment increases. However, when d > 0.15D, thebarrier height reduces due to mixing of higher partial waves. The mL = 0 barrierheight reduces as the dipole moment increases. The dashed lines are the diabatic
curves without partial wave mixing. Figure reproduced from reference [2]. . . . . . . 48

3.7

Normalized fractional heating rate Ṫ /(T02 n) as function of dipole moment. The
heating rate is extracted using a linear fit to the initial temperature increase and isthen normalized by the initial density and temperature of the ensemble. The solidline is the expected heating rate given by Ṫ /(T02 n) = (β/T0 )/12. Typical conditions
for these data are n = 0.3 · 1012 cm−3 and T0 = 0.5µK s−1 at our highest electricfields. Figure reproduced from reference [2]. . . . . . . . . . . . . . . . . . . . . . . . 51

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4.1

(a) A quasi-2D geometry for collisions is realized for polar molecules confined in a
1D optical lattice. An external electric field is applied along the tight confinement
axis. (b) Schematic showing the three lowest adiabatic potentials for collisions as afunction of the intermolecular separation, R. These three channels are ordered withincreasing magnitude of the centrifugal barrier. The arrows indicate the change inthe potential for an increasing external electric field, and hence a growing induced
dipole moment. (c) Schematic showing each individual case for the three lowest
collision channels. The lowest-energy collision channel occurs when two molecules areprepared in different internal states (indicated here by the colors of the molecules).
The second channel is realized when two identical molecules are prepared in different
vibrational levels v for their ẑ motions. The third case as a much reduced loss rate
as a consequence of an increased centrifugal barrier when the two identical molecules
are prepared in the same vibrational level along ẑ. Figure reproduced from reference[3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2

We measure the relative population in each lattice vibrational level using a bandmapping technique. The results for the initial distribution of molecules (a) and fora non-thermal distribution created by parametric heating in ẑ (b). The two images
use the same colour scale for the optical depth (OD). The images are an averageof five shots (a) and seven shots (b), taken after 10 ms of free expansion. Beloweach image we show a trace along ẑ that corresponds to the OD averaged over thetransverse direction. A fit (red line) to the trace, which takes into account boththe size of the Brillouin zones and our imaging resolution, is used to extract therelative populations, nv /ntot , in each lattice level v. The horizontal axis corresponds
to momentum in ẑ and is marked in units of the lattice momentum ~k, where k is
the lattice wave vector. Figure reproduced from reference [3]. . . . . . . . . . . . . . 64

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4.3

The molecule distribution in the ẑ direction. Initially the molecule distribution follows a gaussian distribution (black), but after 500 ms (red) the most dense pancakesat the center of the cloud have decayed the most resulting in a distribution muchflatter than the initial gaussian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4

The calculated number loss due to two-body decay in a one-dimensional opticallattice. The exact numerical solution solving for the time dependent density in eachpancake and then summing over all pancakes to find the total number is shown inblue. If we use the initial number of layers, ξinitial , to scale the total number weoverestimate the loss (black curve). In the red curve we use ξ as a fit parameter tofind the time averaged value of ξ = 30. . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.5

Measurement of 2D loss rates. A fit (solid lines) to the measured loss curves, with(red circles) and without (black squares) 0.3ms of parametric heating in ẑ, is used
to extract the loss-rate coefficients β|3i and β|2i . Figure reproduced from reference [3]. 68

4.6

The extracted loss-rate coefficient for collisions of molecules in the same lattice vibration level (black squares) and from different lattice vibrational levels (red circles)plotted for several dipole moments. Measured loss-rate coefficients for molecules
prepared in different internal states are shown as green triangles. For comparison
with each of these three measurements, we include a quantum scattering calculationfor νz = 23 kHz, T = 800 nK (solid lines). The potentials corresponding to the dominant loss channel for the three cases are shown in matching colors in figure 4.1b,c.
Figure reproduced from reference [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

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4.7

The effective initial loss rate, βinitial , for polar molecules confined in a 2D geometry
depends on the fractional population (n0 /ntot ) in the lowest harmonic oscillator
level in ẑ, which for a gas in thermal equilibrium depends on the ratio kB T /hvz .
The measured initial loss rates for a dipole moment of 0.174 D are shown for two
different thermal distributions (solid triangles), a non-thermal sample created byparametric heating (the top open triangle) and an extracted pure β|3i for the limitof the entire population residing in the lattice ground vibrational level (the bottom
open triangle). The experimental results agree well with a simple model (black curve)
described in the text. The top line indicates the value of β|2i as given in figure 4.6.
Figure reproduced from reference [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.8

The extracted intralevel loss rate for identical fermionic KRb molecules in two dimensions (black circles) compared with the loss rate in three dimensions (blue triangles).
The 3D data for T = 300 nK are from Ref. [2]. The 2D data were taken at T = 800√
nK and are converted to 3D rates by multiplication with πaho , where aho is the
harmonic oscillator length in ẑ. Figure reproduced from reference [3]. . . . . . . . . . 75

5.1

Loss of ground-state KRb molecules as a function of time in a 3D lattice with depthsof 56, 56, and 70 Erec in x, y, and z, respectively, where Erec = ~2 k 2 /2m is the
KRb recoil energy, k is the magnitude of the lattice beam wave vector, and m themolecular mass. Neglecting the very short time points (red solid circles), the numberof molecules for times larger than 1 s (black solid circles) are fit to a single exponential
decay, yielding a 1/e lifetime of 16.3±1.5 s. Figure reproduced from reference [4]. . . 79

5.2

Comparison of the molecule lifetime in an isotropic lattice with a depth of 50 Erecin each direction, with (blue open squares) and without (black squares) an applied
electric field. The 1/e lifetime for molecules with an induced dipole moment of 0.17
Debye, 15±4 s, and the lifetime from molecules without dipole-dipole interactions,
16.2±1.5 s, agree within uncertainty. Figure reproduced from reference [4].

. . . . . 80

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5.3

Measurement of the lifetime of KRb ground-state molecules in an optical lattice as
the confinement is changed from a 2D lattice to a 3D lattice. Black circles: Theradial confinement by the x and y lattice beams is set at 56 Erec per beam, while
the potential along z is varied from 0 to 136 Erec (1 Erec corresponds to a latticeintensity I = 0.025 kW/cm2 ). The lifetime reaches a maximum of 25±5 s when
the z lattice depth is 34 Erec (point b). For higher lattice intensities, the lifetime
decreases, and we find that the lifetime depends on the total intensity of the light,
rather than just on the lattice strength, which is consistent with loss due to offresonant light scattering. The open circles correspond to 3D lattices where the radialconfinement was also varied. The red squares correspond to lifetimes measured withan additional traveling-wave beam at 1064 nm illuminating the molecules in the 3D
lattice. Point c (d) corresponds to the 3D lattice of point a with an intensity of 3.2
kW/cm2 (b with 3.7 kW/cm2 ) plus the additional beam intensity of 2.3 kW/cm2(3.5 kW/cm2 ). Figure reproduced from reference [4]. . . . . . . . . . . . . . . . . . . 82

xxi
5.4

The lifetime of Feshbach molecules and confinement-induced molecules measured asa function of the B-field. A purified sample of Feshbach molecules is held in an
isotropic 3D optical lattice (50 Erec per beam, 20 kHz trap frequency). Near the
Feshbach resonance, the loss rate due to photon scattering can be modeled (solidline) as a weighted sum of the free atom loss rate Γatom and a higher loss rate fortightly bound molecules Γmolecule . The grey shaded area indicates the single atom
lifetime, and its uncertainty, measured for the same experimental conditions. Inset:
The lifetime of Feshbach molecules in a 3D lattice as a function of the trap intensity,
at 545.8 G (blue stars) and 543.18 G (green diamonds). A small change in theintensity initially increases the lifetime from 150 ms in the ODT to 9 s in the 3D
lattice at 5 Erec per beam (blue stars). The lifetime then decreases as the intensity
is further increased, consistent with a lifetime limited by photon scattering. For the
more deeply bound Feshbach molecules (green diamonds), the lifetime is shorter. A
fit to the data gives a scattering rate of 15.9±1.6 MHz/(W/cm2 ) at 545.8 G and
30±3 MHz/(W/cm2 ) at 543.18 G. Figure reproduced from reference [4]. . . . . . . . 86

6.1(a) Experimental schematic. The lattice beam propagates along x̂, the magnetic
field points in the ẑ direction, and the polarization of the lattice light makes anangle θ with the magnetic field in the y-z plane. (b) Schematic of rotational energy
states. The degeneracy of the N = 1 level is split in a magnetic field. (c) A
sketch of the optical dipole potentials for the |0, 0i and |1, 0i states. A Gaussian isoverlaid to show the distribution of the molecular cloud in the trap. When the twostates are connected by a 2.22 GHz microwave drive, there is effectively a spatiallyvarying detuning across the cloud due to the difference in the trap potentials. Figure
reproduced from reference [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

xxii
6.2

Parametric heating resonances for θ = 57 degrees. The y-axis shows the rms size inx̂ of an expanded gas of KRb (Rb) after 3 ms (21 ms) of time of flight. The curves
have been offset vertically for clarity. Using Gaussian fits (lines) we determine thecenter of the parametric heating resonances for (from bottom to top) Rb (magenta
diamonds), and KRb in the |0, 0i (black squares), |1, 0i (blue circles), |1, 1i (redinverted triangles), and |1, −1i (green triangles) states. The resonant frequency
allows us to extract the trap depth for each state. Figure reproduced from reference[5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.3

The AC polarizability of KRb at 1064 nm for the |0, 0i (black squares), |1, 0i (bluecircles), |1, 1i (red inverted triangles), and |1, −1i (green triangles) states. Error
bars are from the Gaussian fit uncertainty in the center of the parametric heating
resonances and correspond to ±1 standard deviation. Theory lines are a simultaneous
fit with Eqn. 6.1 and the polarizabilities from the solution of H with three freeparameters θm , α⊥ , and αk . Open circles represent a separate measurement where
the polarizability is extracted from the shift in the microwave transition frequency.
Figure reproduced from reference [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.4

The angle-dependent polarizability for the “exact” model (dashed line) and the approximate 3 × 3 Hamiltonian. Figure reproduced from reference [5]. . . . . . . . . . 100

6.5

The Ramsey coherence time measured in the one-dimensional optical lattice as afunction of angle. A sharp increase in coherence time is observed at the “magic”
angle where the polarizabilities of the |0, 0i and |1, 0i states are matched. Inset: A
Ramsey oscillation fit to a damped sine wave to extract the coherence time at θ = 51
degrees. Figure reproduced from reference [5]. . . . . . . . . . . . . . . . . . . . . . . 102

xxiii
7.1

The loss rate coefficients for both reactive (inelastic) and elastic collisions as a function of the collision energy for an induced dipole moment of 0.2 Debye, a quasi 2D
trap with a tight trapping frequency of 23 kHz, and indistinguishable molecules allin the lowest vibrational state of the lattice. The elastic to inelastic ratio (the rationof good to bad collisions) is over 100 in the entire region of interest. Figure courtesyof G. Quéméner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.2

Preliminary data for direct evaporation of molecules. The number vs. temperaturefor different evaporation cuts is plotted on a log-log plot. The evaporation sequenceis the same for all points, with only the final depth of the optical trap being varied.
We see a slope of 1.2(1). Any slope less than 2 in 2D corresponds to an increase inphase space density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Chapter 1

Introduction

1.1

Why should I care about polar molecules at all, let alone this thesis?
In August 2006 when I arrived at the University of Colorado, the quest for ultracold polar

molecules was already well on its way. I walked onto a working experiment that had been builtwith the express purpose of creating and studying ultracold polar 40 K87 Rb molecules. In many
senses, ultracold polar molecules are a natural extension of work done with ultracold atomic gas
systems, which have been a real workhorse in atomic physics. I will define ultracold (as opposed
to just cold) as being sufficiently cold such that atoms interact via a single partial wave. The
actual energy scale or temperature at which this ultracold regime starts depends upon the atom
or molecule of choice, but is usually in the range of a few microKelvin [6]. (I should point out
here that this definition falls apart for polar molecules because the dipole-dipole interaction mixes
partial waves together, but this is a subtlety best left for later in this thesis.) Here in the ultracold
regime, atom gases become incredibly versatile and controllable quantum laboratories with diverse
applications including quantum information, atomic clocks, precision tests of fundamental physics,
and quantum simulation. The high degree of control over both internal (hyperfine, rotation, andelectronic states) as well as external (motion in the trap) states makes them ideal for studying
quantum many-body phenomena. Ultracold polar molecules build on the success of ultracold
atoms but with two important additions: they have a richer internal structure and, perhaps mostimportantly, they have long-range interactions.
Atoms in an ultracold gas (with the notable exception of a few atoms with large magnetic2
dipole moments) have short-range interactions. They act much like billiard balls in that they onlyinteract when they come in direct contact with each other. This contact interaction can be writtenwith a Dirac delta function. Even near a Fano-Feshbach resonance where the strength of theseinteractions can be tuned arbitrarily large, it is only the prefactor that sits in front of the delta
function that changes, and one still has short-range, contact interactions. Molecules, on the other
hand, have long-range, anisotropic interactions arising from the dipole-dipole interaction. In thecase of two identical electric dipoles aligned to an external electric field, the dipole-dipole interaction2

is 4πd0 r3 (1 − 3 cos2 (θ)) where d is the dipole moment, 0 is the electric permittivity of free space, ris the distance between dipoles, and θ is the angle between the external electric field and the line
between the two molecules. The anisotropy that comes from the 1 − 3 cos2 (θ), which means thattwo dipoles lined up “head-to-tail” attract, while two dipoles lined up “side-by-side” repel eachother. This anisotropy can be a nuisance as it ruins the symmetry of our system, but I will show
later that we can escape the anisotropy by moving to an optical lattice that confines the gas to twodimensions, and even exploit the anisotropy of the dipole-dipole interaction to explore quantum
stereodynamics of ultracold chemical reactions.
The 1/r3 dependence gives rise to an interaction that is long range for sufficiently large dipole
moments. When talking about the range, it is convenient to define the dipole length, which is the
distance at which the dipole-dipole interaction becomes stronger than the van der Waals interaction.
2d µ
The dipole length is defined as dl = 4π~
2  where µ is the reduced mass of the two molecules and ~0is Planck’s constant divided by 2π. The dipole length of KRb when it is fully polarized (d = 0.566
D) is 5740 a0 (where a0 = 5.29 × 10−10 m is the Bohr radius). Due to experimental constraints on
the maximum electric field we can apply, the dipole length of KRb in an experiment (d = 0.2 D)is currently limited to 720 a0 . For comparison, a much more polar molecule such as OH (d = 4 D)
has a dipole length of 4 × 104 a0 , and LiCs (the most polar of the bialkali molecules with d = 5.2
D) has a dipole length of 5.4 × 105 a0 . I should also mention that there are several atoms with large
magnetic dipole moments that have been cooled to the ultracold regime: among them are Cr, Dy,
and Er with magnetic moments of 6, 10, and 8 Bohr magnetons, respectively, which corresponds

3
to dipole lengths of 23, 195, and 130 a0 .
For a gas of ultracold polar molecules, these dipole lengths are already on the order of theinterparticle spacing, which opens the possibility of strongly correlated many-body systems. In
the limit that the dipole length becomes much larger than the interparticle spacing, the molecules
can self assemble into a crystal [7]. The dipole length is also on the same order as the spacing
between lattice sites in a 1064 lattice (1000 a0 ), which allows us to build on the success of cold
atom systems in realizing Bose and Fermi Hubbard Hamiltonians, but now with the ability to havenearest neighbor and next-nearest neighbor interactions. This opens the possibility of realizing
many interesting spin systems where massive entanglement can occur and quantum magnetism canbe explored.
Although perhaps not quite as exciting as strong long-range interactions, the extra complexityof a molecule is certainly a feature worth mentioning. The rotation states of the molecule are quite
useful because of their long lifetime and convenient transitions in the microwave. The long lifetime
allows transitions between rotation levels to be used as a sensitive spectral probe, allowing us tomeasure small energy shifts with great accuracy. This could be used to measure small energy shifts
resulting from interactions. It has also been proposed that polar molecules can be used for precision
measurements of fundamental constants, including the permanent electron electric dipole moment[8], time variation of the fine-structure constant [9], and the electron-to-proton mass ratio [10]. We
could use microwaves to couple the ground and rotationally excited states to tailor the adiabatic
collision potential and potentially even prevent short-range collisions altogether [11].
I should mention that although I have grown quite fond of KRb over the last six years, thespecific molecule is not too important at this point in the story. K and Rb were used simply
because they were convenient to cool down to quantum degeneracy. Although the details of KRb
and our choice of isotopes will play a very important role later in this thesis when I discuss ultracold
chemistry, the two main driving reasons to create ultracold polar molecules (long-range interactions
and complex internal structure) are general to any heteronuclear molecule.
At this point I could say something about how we could use polar molecules to simulate

4complex condensed matter systems such as high-temperature super conductivity in cuprates [12]. I
could say that although these systems don’t necessarily interact through dipole-dipole interactions,the physics of strongly correlated systems with long-range interactions should have some generalfeatures that our highly controllable and tunable polar molecule system should be able to explore,
and perhaps shed light on how to understand these even more complicated condensed mattersystems. But to be perfectly honest, although I’ve certainly used these arguments before, they don’t
really resonate with me. I think that a strongly interacting gas of polar molecules in the quantum
regime with tunable long-range interactions is interesting all by itself. In the right situations, orwith the proper manipulation, these systems may indeed have a Hamiltonian that looks a lot likeone of your favorite quantum magnetism spin models [13], or a Hubbard Hamiltonian that might
support anyonic excitations [14]. But studying strongly correlated many-body systems is interesting
simply because, despite the fact that we can control these systems with amazing precision, we still
don’t have a good way of predicting what should happen in the limit of a strongly interacting
many-body system. They are both amazingly simple (just a handful of interacting dipoles, how
hard could that be? I can write down that Hamiltonian, no problem.) and complex all at the sametime. We are taking the power of cold atom gas physics and adding long-range interactions andthe novel quantum gasses that arise are fascinating.

1.2

So what did you do that is such a big deal?
In this section, I provide a brief overview of the JILA KRb work that I contributed to, anddescribe the focus of this thesis. I was fortunate to start graduate school right as things in the
JILA KRb experiment were getting really interesting. When I arrived at JILA, Feshbach molecules
had just been created, but had not yet been studied. A difficulty in the experiment was a short
lifetime of the Feshbach molecules because our multi-mode optical dipole trap laser happened to
be on resonance with a molecular bound-bound transition and was driving the Feshbach molecules
into an electrically excited state within a few microseconds of their creation. It took quite a bitof time to figure out this problem, and before we did, we were very worried that the molecules

5were too collisionally unstable to be studied, let alone transferred to the absolute ground state of
the molecular potential. At this time, we were also joined by Silke Ospelkaus who, in her Ph.D.
work, had studied KRb Feshbach molecules in an optical lattice [15]. They did not see the sameloss, because they managed to avoid the bound-bound transition that our laser was driving, but, inany case, an optical lattice also shields the molecules from collisions. Although a quick calculationof density and collision rate should have convinced us that the problem was unlikely caused bycollisions, we were still worried, and for this reason, my first project in the lab was to get a laser for
an optical lattice and investigate how we should go about implementing it. This laser was used toreplace our multi-mode laser for the optical trap instead and an optical lattice was not implemented
in our experiment for another three years, but it did end up being the main focus of my Ph.D.
work when I became the senior graduate student and finally got “the keys to the car”.
After a brief detour to study the collisional properties of KRb Feshbach molecules, the lab
turned its attention to the main goal of creating ground-state molecules. Surprisingly, there wasquite a bit of information already known about the KRb molecular potential [16] and with thetheoretical guidance of Svetlana Kotochigova and Paul Julienne, we were able to pin down thelocation of the relevant states for transferring the molecules to deeper bound states. There wasquite a bit of feedback back and forth from theory to experiment, and every state we measured
helped pin down the molecular potential and the predictions got better and better. Because of thisfeedback process, our understanding of the molecular potential came gradually and our path tothe absolute ground state took several steps. First, to show that a coherent transfer via STIRAP(STimulated Raman Adiabatic Passage) [17] was possible, we transferred our Feshbach molecules
to a molecular state in the singlet potential bound by 10GHz [18]. We then transferred Feshbach
molecules to the lowest energy state of the ground triplet potential (3 Σ). And then in 2008, wefinally transferred our molecules to the absolute ro-vibronic ground state (in the singlet potential)[19]. With the molecules in the ground state, we applied an electric field to demonstrate the polar
nature of the molecules by measuring the Stark shift. We found that with with electric plates
outside of the vacuum chamber, we could only partially polarize our molecules to an effective

6
dipole moment in the lab frame of 0.2 Debye. We also mapped out the hyperfine structure of
the ro-vibronic ground state and demonstrated the ability to transfer our molecules to any desired
hyperfine state [20].
With complete control over the internal state of the molecule, we turned our attention to thecollisional properties of the ground-state molecules, and studied both atom-molecule and moleculemolecule collisions. We observed density-dependent loss for for K + KRb, which we attribute tochemical reactions. We also observed density-dependent loss for Rb + KRb when either Rb or
KRb was in an excited hyperfine state, which we attribute to hyperfine-state-changing collisions[1]. Most importantly for the work that will be discussed in this thesis, we found evidence ofchemical reactions between two KRb molecules at ultralow temperatures. This is both a feature
and a challenge. Energetically, the bimolecular atom-exchange reaction KRb + KRb → K2 + Rb2
is exothermic. Moreover, as is generally true for chemical reactions between alkali atoms [21], thischemical reaction has no potential barrier and therefore requires no activation energy.
However, even without a chemical reaction barrier, for ultracold, spin-polarized, fermionicmolecules, such as KRb, there is a centrifugal p-wave barrier that appears due to the requirementthat the wavefunction of two indistinguishable fermions be antisymmetric with respect to exchangeof the particles. With only a centrifugal barrier to prevent the molecules from chemically reacting,the chemical reaction rate is set by the rate at which molecules tunnel through the p-wave barrier.
Once inside the p-wave barrier, the molecules chemically react with near unit probability. We
verified this simple model by varying the temperature and observe that the chemical reaction rate
scales linear with temperature as predicted by the Wigner threshold laws [22].
Although the p-wave barrier is spatially symmetric, the dipole-dipole interaction is not, andso turning on an external electric field to polarize the molecules results in anisotropic collisions,which drastically change the chemical reaction rate. In simple terms, two dipoles colliding “side-byside” have a repulsive interaction so the p-wave barrier is enhanced and the chemical reactions aresuppressed, but two dipoles colliding “head-to-tail” have an attractive interaction that decreases
the p-wave barrier and enhances the chemical reaction loss rate. In a three-dimensional optical

7
dipole trap, both processes will occur and the chemical reaction rate is determined by the faster of
the two rates. As a result, we see a dramatic increase in the chemical reaction rate as an external
electric field is applied to polarize the molecules [1].
We decided to exploit the spatial anisotropy of the dipole-dipole interaction to create acollisionally stable gas in 2D. We can turn off the “head-to-tail” collisions by loading the molecules
into a one-dimensional optical lattice. In this two-dimensional trap geometry, we demonstratethat with an electric field applied along the lattice beam direction, “head-to-tail” collisions can besuppressed and strong dipolar interactions and collisional stability can be enjoyed simultaneously.
Even further suppression of the chemical reactions was achieved by loading the gas into a threedimensional optical lattice. In the limit of a deep lattice where the tunneling was suppressed andthe molecules never touched, we found that the molecules essentially didn’t chemically react at all[3].
Although I have been fortunate enough to be involved in the spectroscopy, transfer to boththe triplet and singlet ground state, as well as all the studies on the ground state molecules, theexperiments focused on controlling the molecules’ internal degrees of freedom (electronic, vibration,
rotation, and hyperfine) are already described well in the thesis of Kang-Kuen Ni [23], and I will
not focus on them in this thesis. Instead, this thesis will focus on the studies done to control
the molecules’ external degrees of freedom (motion in the trap). This includes our studies of thepolarizability, progress towards evaporation, and the work done in an optical lattice, where I will
discuss the role reduced dimensionality plays in controlling the molecular interactions. But topaint a complete picture of the physics involved I will necessarily include some overlap with theinformation in the theses of Kang-Kuen Ni [23] and Marcio de Miranda [24].

1.3

Outline of this thesis
Chapter 2 will introduce the optical lattice. I will give an overview of optical lattices, relevantto the discussion of our experiments in subsequent chapters. I will introduce the band structure
and corresponding Bloch wavefunctions, which are useful in describing weak lattices, as well as

8the Wannier functions which describe deep lattices. The second half Chapter 2 will be focusedon describing the experimental techniques we use to load, probe, and characterize the lattice. In
Chapter 3 I will present a brief overview of our experimental observations regarding the chemical
reactions between two KRb molecules. I will show that the chemical reaction rates are strongly
influenced by the dipole-dipole interactions. Chapter 4 will present the exploration of quantum
stereodynamics in a one-dimensional lattice. I will show how we can control the chemical reaction
rate between KRb molecules by controlling both the internal and external quantum states of themolecules in a 1D lattice. Chapter 5 will present our studies of both the ground-state molecules
and the Feshbach molecules in a 3D lattice. In Chapter 6 I will present our measurements of theac polarizability and discuss its effects on the coherence between different rotational states of themolecule. Finally, Chapter 7 will give conclusions and outlook.
The work described in this thesis has been published in several papers [18, 19, 25, 2, 1, 20, 3,
4, 26]. In particular, Chapter 3 closely follows Ref. [1] and [2], Chapter 4 closely follows Ref. [3],Chapter 5 follows Ref. [4], and Chapter 6 follows Ref. [5].

Chapter 2

Optical Lattice

Optical lattices have been an incredibly useful tool in atomic physics both because they allow
for very precise control over the trapping potential, and they provide a periodic potential similar
to those found in crystalline condensed matter systems [27]. But unlike the periodic potentials in a
crystal, the potentials in an optical lattice are almost entirely defect free. Moreover, the strength ofthe lattice (a parameter which cannot be tuned continuously in a solid state crystal) can be easilyadjusted in real time by controlling the intensity of the lattice. This control gives the ability tochange the tunneling rate in the lattice.
In 1998 it was pointed out by D. Jaksch and coworkers that ultracold atoms loaded into an
optical lattice could realize the Bose-Hubbard Hamiltonian [28]. This proposal was realized by M.
Greiner and coworkers in 2002 with the observation of the superfluid to Mott-Insulator quantum
phase transition in a BEC loaded into an optical lattice [29]. Since this initial observation, there
have been many exciting experiments both with bosonic and fermionic atoms in an optical lattice(for a review see [30]). Additionally the ability to independently control the scattering length with
Feshbach resonances adds the ability to control the onsite interaction, including changing fromattractive to repulsive interactions [31, 32, 33].
Although the ability to simulate condensed matter physics in a lattice is extremely interesting,especially with the addition of the long-range interactions between polar molecules, for most of theexperiments discussed in this thesis the optical lattice is just a well controlled potential with tightconfinement. We can combine multiple lattices in orthogonal directions to change the dimensionality

10of the system. A single lattice beam (a 1D lattice) gives an array of two-dimensional pancake shaped
traps. Two lattice beams (a 2D lattice) provides an array of 1D tubes. Three lattice beams (a3D lattice) gives an array of 0D, point-like traps. We exploit these geometries to control thestereodynamics of chemical reactions in 2D [3], or to suppress them entirely in a 3D lattice byholding each molecule in its own trap, unable to chemically react with molecules in adjacent sites[4].
To obtain these reduced geometries we need to load all the molecules into a single band,
which requires that the spacing between energy levels in the direction of tight confinement be muchlarger than the energy of the gas. This requires tight confinement and cold temperatures. The steep
potential gradients in an optical lattice allows these criteria to be met with reasonable experimental
parameters. To give an illustration, if we focused a 1064 nm laser down to a beam waist of 5 µm
(a small but reasonable beam waist) by 250 µm (to match our lattice geometry) it would take 1.7
kW to get a trapping frequency equivalent to our lattice with 1 W of power.
In this chapter I will give a quick theoretical background to understand the origin of thelattice potential, and how molecules interact with it. This will introduce the band structure andcorresponding Bloch wavefunctions, which are useful in describing weak lattices, as well as theWannier functions that describe deep lattices. The second half of this chapter will be focused on
the experimental techniques we use to load, probe, and characterize the lattice.

2.1

Optical dipole force
Atoms or molecules in a laser field will experience a shift in energy due to the AC stark shift.

Here the oscillating electric field of the laser light induces an AC electric dipole moment in theatom or molecule. Just as with a classical oscillator, if the laser light drives the atom or molecule
at a frequency below its resonance frequency (the laser is red-detuned) the atom/molecule will
respond in phase, such that the dipole is always aligned opposite of the oscillating electric field andwill be attracted to the most intense part of the laser beam. Conversely, if the laser field drives
the atom/molecule at a frequency higher than its resonance frequency (a blue-detuned laser), the11atom/molecule responds out of phase and will be repelled from the most intense part of the laser
beam.
For atoms, the potential from a far detuned optical trap is easy to derive and can be calculated
quite accurately using only the closest atomic resonances [34]. For molecules, the large number ofenergy states, and the lack of closed cycling transitions makes calculating the potential more difficult[35]. But all the interactions with all the excited states can be included into a single parameter,
the polarizability, which is defined by:
U (ω, ~r) = α(ω)I(~r),(2.1)

where ω = 2πc
λ is the frequency of the light, c is the speed of light, λ is the wavelength of the light,
U (ω, ~r) is the potential energy, I(~r) is the spatially dependent intensity of the laser, and α(ω) is
the frequency dependent polarizability. Although α(ω) can be quite complicated, in practice weonly care about the polarizability at the frequency of our optical trap. To avoid off-resonant lightscattering we use an optical trap at λ = 1064 nm, which for the absolute ground state of KRb isdetuned by more than 30 nm from the lowest excited state potential.

2.2

Optical lattice
In our experiment we form our optical lattice with a single retro-reflected gaussian laser beam.

The interference between the incoming and retroreflected laser beams forms a standing wave witha periodicity of λ/2. The optical potential is given by:
U (r, z) = −α4I0 exp−2r2
w2 (z)

cos2 (kz)where I0 = πw2P
2 (z) , P is the laser power of the incoming beam, w(z) = w0

(2.2)
p
1 + (z/zR )2 is the beam

waist at z, w0 is the beam waist at the focus, zR = πw02 /λ is the Rayleigh length, and k = 2π/λ isthe wave number. Notice that the potential at z = r = 0 has a depth four times that of a single
dipole beam with the same intensity I0 (U = αI0 ) due to the constructive interference of the two
beams.

12
Although the sinusoidal shape of the lattice potential is important to understand the motionof particles as they move from site to site in the lattice, for deep lattices it is often useful to thinkof an optical lattice as an array of traps each approximated with a harmonic potential. Here wecan define trapping frequencies
s

s
 2

∂ U (r, z)
1
16αI0
ωr =
−
=
,
2
m
∂r
mw02
r=z=0
s 
r

1
∂ 2 U (r, z)
8αI0 k 2
−
ωz =
=
.
m
∂z 2
m
r=z=0

(2.3)

(2.4)

These equations are only valid at the center of the lattice beam. At the edges of gaussian profileof the lattice beam, the intensity will be lower, resulting in a lower trapping frequency. In our
experiment, the beams are large (w0 = 250 µm) compared to the average cloud size (30 µm) so
this inhomogeneity can generally be ignored.
Higher dimensional lattices can be obtained by adding additional laser beams. Although there
are many different possible ways to arrange the lasers resulting in many different lattice geometries[36], in the experiments described in this thesis we have up to three orthogonal retroreflected laser
beams resulting in a square lattice geometry. To avoid interference between orthogonal beams thefrequency of each beam is shifted by tens of MHz relative to the other beams. The interference
between beams has a resultant potential that moves at tens of MHz, which is fast enough that themolecules only see the time-averaged potential.

2.3

Band structure
The solutions to a particle moving through a periodic potential were introduced by Felix

Bloch in the 1920s to describe the motion of electrons in crystalline solids [37]. For clarity wewill look at these solutions in the simplest case of a particle moving in one dimension through ahomogeneous potential given by V = V0 cos2 (kx). Although we will restrict ourselves to 1D, thepotential for a three-dimensional square lattice is completely separable and can be treated in thesame way with no additional complications. The motion of the particles in a lattice potential is13given by Schrodinger’s equation in one dimension:~2 ∂ 22−+ V0 cos (kx) Ψ(x) = EΨ(x).
2m ∂x

(2.5)

Ψ(x) is the Bloch wavefunction, which must follow Bloch’s theorem:


iqλ
Ψ(x).
Ψ(x + λ/2) = exp
2~

(2.6)

Here we introduce the quasimomentum q, which gives the phase difference between adjacent latticesites. To satisfy Bloch’s theorem, Ψ(x) is given by the product of a plane wave with quasimomentum
q multiplied by a function with the same periodicity as the potential:
Ψn,q (x) = exp (iqx/~)un,q (x),(2.7)

where un,q (x) is the periodic envelope function. Because of the periodicity of un,q (x), the Bloch
wavefunction is delocalized over the entire lattice. The quasimomentum, q, is only unique up to
the reciprocal lattice vector, so we restrict ourselves to the first Brillouin zone and work with−~k < q < ~k. For each value of q there is an infinite set of eigenstates, which we label with a band
index n. The solutions to equation 2.5 are well known and are easily obtained in Mathematica
with the “MathieuC” and “MathieuCharacteristicA” functions. In figure 2.1 we show the band
2 2kstructure for increasing lattice depth, where the lattice depth is given in units of Erec = ~2m. Withno lattice the band structure is just the energy for a free particle mapped onto the first Brillouinzone. As the lattice depth increases the bands separate and become flatter. For deep lattices the
spacing between bands approaches ~ω, where ω is the trap frequency of a single lattice site given
in equation 2.4.
Although the delocalized Bloch wavefunctions are the eigenstates of the lattice potential for all
lattice depths, in the limit of deep lattices, tunneling is strongly suppressed and atoms/molecules
are effectively localized on the timescales of the experiments. Here it is useful to switch to a
localized basis set given by the Wannier functions [38], which are defined through the following
Fourier transform:
1 X
wn (x − xi ) = √exp (−iqxi )Ψn,q (x),
M q

(2.8)

14

0 Erec122 Erec125 Erec

12

10

10

10

8

8

8

8

6

6

6

6

4

4

4

4

2

2

2

2

energy (𝐸rec )

10

0

0
1

0

𝑞 (ℏ𝑘)

10
1

0

𝑞 (ℏ𝑘)1

1

0

𝑞 (ℏ𝑘)

10 Erec

12

1

0
1

0

𝑞 (ℏ𝑘)

1

Figure 2.1: Band structure of an optical lattice. Energy of the different bands in the lattice isplotted vs. quasimomentum q for 0, 2, 5, and 10 Erec . Only the first Brillouin zone is shown.

15
where M is the total number of lattice sites, and the sum is over all quasimomentum in the first
Brillouin zone. The Wannier functions and the corresponding probability density are shown infigure 2.2 for 2 Erec (top panels) and 10 Erec (bottom panels). For 2 Erec , the wavefunction has alarge oscillation in the tail extending into the adjacent lattice sites, and as a result overlaps wellwith the Wannier functions on adjacent lattice sites. For 10 Erec , the oscillations in the tail is justbarely visible and the wavefunction overlap between adjacent sites is much smaller.
We can use these localized Wannier functions to look at the probability for an atom/molecule
to hop from one site to an adjacent site. We calculate the tunneling matrix element, J, by finding
the wavefunction overlap with:
Z
J=



~ ∂2
2
w1 (x − xi ) −+ V0 cos (kx) w1 (x − xj )dx,
2m ∂x2(2.9)

which relates directly to the band width of the first band [39]:J=

Eq=0,n=0 − Eq=~k,n=0.
4(2.10)J has units of energy and is related to the tunneling time, t = ~/J
In the limit of deep lattices, the Wannier function for the ground band can be approximatedwith a gaussian. This seems perfectly reasonable because the deep lattice potential is well approximated by a harmonic potential (and the ground state wavefunction of the harmonic oscillator isa gaussian). A gaussian wavefunction is often used when looking at the momentum distribution,or when calculating the onsite interaction, but it fails when trying to calculate the tunneling rate.
Even though the small oscillations in the tail of the Wannier function are barely visible at highlattice depths, they correspond to a small probability of the molecule being in the adjacent site,which is exactly what gives the tunneling probability. So, although a gaussian is a very good
approximation for the wavefunction, it underestimates the tunneling probability by an order ofmagnitude [39].

16

Wannier function w(x) Ulat = 2 Erec

2

2

1

0

1

Density |w(x)|2 Ulat = 2 Erec

2

2

1

0

1

x

x

Wannier function w(x) Ulat = 10 Erec

Density |w(x)|2 Ulat = 10 Erec

1

0

x

1

2

2

1

0

1

2

2

x

Figure 2.2: The Wannier functions and their corresponding probability density for a 3 Erec (toppanels) and a 10 Erec (bottom panels) lattice. The lattice potential is shown in red to indicate
the location of the sites. The probability density for three adjacent sites is shown to illustrate thewavefunction overlap which determines the tunnelling rate.

17

2.4

Experimental realization and charaterization of the lattice
The experimental realization of our 1D lattice is illustrated in figure 2.3. We have a single

gaussian beam with a beam waist w0 = 250 µm that is retroreflected to interfere with the incoming
beam to form the lattice. This large beam waist has a Rayleigh range of 18 cm, which makes
focusing of the lattice beam onto the atoms quite easy. Additionally, we use a 2f-2f imaging system(see figure 2.3) to retroreflect the beam, making the alignment of the retroreflected beam insensitive
to both the mirror position and angle. We tilt the lattice with a slight angle of 2.5 degrees relative tothe normal of both the glass cell and electric field plates to avoid optical interference with unwanted
reflections.
To load the 1D lattice, we start with atoms in a crossed optical dipole trap. The earlier
stages of the experiment (MOT, forced RF evaporation in the IP trap, evaporation in the crossed
beam optical dipole trap, ect.) are described well in the theses of Josh Zirbel [40] and Kang-Kuen
Ni [23]. The intensity of the lattice beam is ramped up over 150 ms. This ramp time is chosen to
ensure adiabaticity between bands and minimize heating. The experiments described in chapter 4
are preformed in a combined 1D lattice + crossed optical dipole trap. The combination of the two
traps is used to provide tighter radial confinement than would be obtained in the 1D lattice alone.
Because of our limited optical access, we currently have a single set of beams that act both asa crossed dipole trap and as the two horizontal lattice beams in our 3D lattice. Just like the vertical
optical lattice beam, each optical dipole trap beam is retro-reflected using a 2f-2f imaging system,
but the retro-reflection mirror is placed behind a shutter, which allows us to switch between using
the beam as an optical dipole trap or an optical lattice. To load the 3D lattice, we first load ouratoms into the 1D lattice as described above and then ramp down the intensity of the two optical
dipole trap beams. With the atoms confined entirely in the vertical lattice and the dipole trap
beams fully off, we open the shutter such that the optical dipole trap beam is now retroreflected
to form a lattice. The intensity of the two horizontal lattice beams is then ramped up over 150 ms
to ensure adiabaticity and avoid heating the cloud.

18

L1

L2

atoms

L3

RRM

Figure 2.3: A schematic of the vertical lattice. The lattice light comes out of the fiber is columnatedby lens L1. Lens L2 focuses the beam onto the atoms/molecules in the center of the cell. L3 is 2f
away from the atoms and 2f away from the retroreflection mirror RRM. This 2f-2f imaging system
refocuses the lattice onto the atoms while minimizing the sensitivity to small drifts in both theangle and position of RRM.

19
We align all three lattice beams to the position of the atoms in the magnetic trap (see thethesis of Josh Zirbel [40] for details on the atom preparation in the magnetic trap). For the twohorizontal beams that make the crossed dipole trap, this is done by maximizing the load from themagnetic trap into a single dipole trap beam. The alignment of the incoming vertical lattice beamis adjusted one axis at a time by scanning the vertical lattice beam through a cloud of atoms held injust one of the two crossed dipole trap beams. When the vertical lattice beam intersects the dipole
trap, the lattice beam provides trapping along the direction of the beam’s propagation. We simply
minimize the in-trap size in this direction to get the best alignment and then repeat the processusing the other dipole trap beam. The retroreflection of all three beams is aligned by pulsing thelattice on for a few µs and maximizing the diffraction of a BEC.

2.5

Diffraction of a BEC
A convenient way to characterize the strength of our lattice is to see the diffraction of a BEC

in the lattice [41]. The lattice is a standing wave with periodicity λ/2, so we can think of this simply
as diffraction off of a grating. Physically the atoms in the BEC absorb and then reemit photons
from the lattice beam and receive a momentum kick of 2m~k, where m is an integer. We work in
the Raman-Nath regieme where the time in the lattice is much shorter than the trap frequency, so
the atoms are stationary during the interaction. When the lattice is suddenly pulsed on the BEC,
which has zero momentum, is projected into the even bands of the lattice. Each of these bands
has a different energy and will evolve in time according to Ψ(t) = exp (−iEn (q)t/~)Ψ(0). Aftersome hold time in the lattice, the lattice is suddenly tuned off and the atoms are projected backinto momentum states. The phases acquired by the different Bloch bands during the lattice holdleads to interference in the momentum projections. These interferences produce oscillations in the
populations of the momentum components as a function of the hold time as seen in figure 2.4.
To find the lattice depth, we solve the time dependent Schrodinger equation numericallyusing the first 5 momentum components. Here all the parameters are known except for the lattice
depth. We vary the lattice depth and compare to the measured oscillations to minimize χ2 . The

20

1

2

3

4

5

6

7

8
9
10
time (µs)

11

12

13

14

15

16

Figure 2.4: The diffraction of a BEC in a vertical optical lattice. Momentum components p =0, ±2~k, ±4~k, ±6~k are shown after a diabatic release from the trap. The populations oscillate asa function of hold time in the lattice.

21
fit is shown in figure 2.5.

2.6

Parametric heating
Although Kapitza-Dirac scattering (diffraction of a BEC) allows us to align and calibrate

our lattice, it relies on the coherence of a BEC. We would like to have another measure of thetrap frequency and trap depth that may be used with fermions or thermal clouds. For this, we
can probe the band structure of the lattice directly with parametric heating. Here we modulate
the intensity of the lattice, and when the modulation frequency is equal to the energy gap between
the ground and second excited band of the lattice, the atoms or molecules will be transferred tothe higher band. The added energy from the atoms/molecules in the second excited band heats
the cloud, allowing us to identify the parametric heating resonance by looking at the momentum
distribution of the cloud. After the modulation, the cloud is released from the trap and the size ismeasured after time of flight expansion. A typical parametric heating resonance for KRb is shownin figure 2.6.
If the potential was perfectly harmonic, we would only be able to excite transitions fromthe ground state to other even states (n = 2, 4, 6...) because of the even parity of the drive. But
due to the anharmonicity of the sinusoidal potential of the lattice it is possible to drive transitions
from the ground state to odd excited states, but the coupling is much smaller [42]. By knowing
the polarizability, beam waist, intensity, and the attenuation of the light as it passes through theglass cell and electric field plates, we can predict the trapping frequency to within 10%, which
prevents us from misidentifying the parametric heating resonance. A small amplitude (a peak-topeak modulation < 15% of the total lattice intensity) should be used for the drive. If the transitionis driven too hard, the molecules will be excited to n = 2 and then again up to n = 4, which dueto the anharmonicity of the potential is slightly lower in frequency. If the transition is overdriven,
the observed resonance will be broadened and systematically shifted towards lower frequency.
It is common to say that the parametric heating measures the trap frequency. In the deep
lattice limit, where the potential can be approximated by a harmonic potential, the transition from

fraction in nth diffraction order

22

1
0.8
0.6
0.4
0.2
0
0

5

10

15

time (µs)
Figure 2.5: The relative populations in the p = 0 (blue), p = ±2~k (green), and p = ±4~k (red)diffraction orders. The solid lines are a fit to the data using a numerical solution to the timeRb .
dependent Schrodinger equation to extract the lattice depth V0 = 43(4) Erec

23sigy (pixels)

8.0
7.5
7.0
6.5
6.0

40

45

50

55

60

Freq (kHz)KRb lattice. The amplitude
Figure 2.6: A parametric heating resonance for KRb in a 94.3(3) Erecof the lattice is modulated sinusoidally for 4 ms with a peak to peak amplitude 10% of the total
intensity. The size is measured after 3 ms time of flight. A gaussian is fit to the observed resonanceto find the center of 49.37(9) kHz.

24
the ground band to the second excited band occurs at two times the trap frequency. However, forthe lattice depths accessible in our experiment we find that the anharmonicity of the trap causes a
significant deviation from this simplistic expectation, even when exciting only the n = 0 → n = 2transition. In figure 2.7, twice the trap frequency (red) along with the frequency of the n = 0 →n = 2 transition (black) are plotted versus the lattice depth. The solid black line is the transition
at q = ~k/2 and the black dashed lines are the transition at the edges of the Brillouin zone (q = 0
and q = ~k). A thermal gas (or Fermi gas) will have atom/molecules across much of the Brillouinzone, which for lattice depths less than 25 Erec will increase the width of the parametric heating
resonance. More importantly, care must be taken when measuring trap depths below 25 Erec withparametric heating. The frequency measured with a BEC, which only populates q = 0 (the higher
dashed line in figure 2.7), and a thermal cloud, which will populate all the quasimomentum with athermal distribution (spanning the entire range between the two dashed lines in figure 2.7), will bedifferent.
Figure 2.8 shows ratio between the parametric heating resonance and twice the trapping
frequency. Even at 100 Erec , there is still a 10% difference. In practice we sidestep this potential
problem by comparing our measured parametric heating resonance to an exact solution of the band
energies (shown in figure 2.9) to extract the trap depth. This trap depth (U = 4αI) can then be
inserted in to equation 2.4 to get the trap frequency.

2.7

Band Mapping
Once the atoms or molecules are loaded into the lattice, we would like to know the distributionof the atoms/molecules in the different bands of the lattice. The population of the bands canbe directly measured by adiabatically ramping down the lattice and measuring the free space
momentum distribution [43, 44]. Usually time-of-flight imaging is done after the trap is rapidly
switched off. This diabatic turn off projects the wavefunction onto momentum space. However if
we ramp down the lattice slowly compared to the spacing between bands, the quasimomentum inthe lattice is mapped onto the free momentum measured in time-of-flight imaging.

25

40

0 ® 2 transition HErecL

30

20

10

0
0

20

40
60
trap depth HErecL

80

100

Figure 2.7: The transition from the ground band to the second excited band (in Erec ) as a functionof the trap depth. The solid black line is the transition at the center of the band (q = ~k/2). The
dashed black lines represent the transition at the edges of the Brillouin zone (q = 0 and q = ~k).
Twice the trapping frequency is plotted in red.

parametric heating resonance  2 ´ trap frequency

26

1.0

0.9

0.8

0.7

0.6

0.5
0

20

40
60
trap depth HErecL

80

100

Figure 2.8: The parametric heating resonance divided by twice the trap frequency as a function oftrap depth. The dashed lines are the transitions at q = 0 (bottom) and q = ~k and the solid lineis the transition frequency at q = ~k/2.

27

trap depth (Erec)

100
80
60
40
20
15

20

25

30

35

0 to 2 transition (Erec)
Figure 2.9: The depth of the lattice versus the parametric heating resonance. From this graph we
can read off the trap depth from our resonance. Units are given in Erec to keep the plot general forparticles of any mass.

28
The speed for the ramp down must be chosen carefully to ensure the distribution in time offlight does indeed match the distribution in the lattice. The first criteria is that the ramp should be
adiabatic with respect to the atomic motion in a single lattice site. This means the ramp must be
slow compared to the band spacing, which ensures that we avoid changing the distribution between
the different bands. But this criteria cannot be perfectly fulfilled because the band gap goes to zeroas the lattice is ramped off. This problem is most evident at the edge of the Brilloiun zone where
the bands spacing is the smallest. Here the nonadiabaticity causes a “smearing” of the sharp edgesof the Brillouin zone. To further complicate matters, if the ramp is too slow the atoms will tunnel
out of the lattice during the last part of the ramp due to the force of gravity. By looking both at
the asymmetry of the background due to atoms that leaked out of the lattice during the ramp andthe rounding of the Brillouin zone edges, we empirically choose a ramp down time of 500-1000 µs.
The momentum distribution after 18 ms time of flight for a K cloud after band mapping isshown in figure 2.10. A hot K cloud (T=400 nK) was loaded into the lattice to ensure population ofboth the 0th and 1st bands. We integrate across the image perpendicular to the lattice direction toincrease the signal to noise ratio. This trace is shown in figure 2.10b. Because fermions uniformly
fill the 0th band, we observe a flat distribution for −~k < p < ~k. There are also atoms almost
uniformly distributed across the 1st excited band of the lattice which appears on either side of
the 0th band at −2~k < p < −~k and ~k < p < 2~k. However, the 1st band is not full, so we
see “shoulders” on either side of atom in the 0th band. To extract the relative populations, we
simply count the number of atoms in each of the zones. In practice this is done by fitting a seriesof Heaviside functions (which give the flat topped distributions for each band) convoluted with agaussian (which accounts for the smearing at the edge of the zones which is a result of not being
perfectly adiabatic on the ramp down). The only free parameters in the fit are the amplitude for
the 0th and 1st bands and the width of the momentum smearing. The fit to figure 2.10b reveals
28% of the atoms in the 1st excited band.
We can remove the atoms in higher bands by lowering the lattice depth and waiting for theatoms in higher bands to tunnel out of the lattice. We ramp down the lattice in 20 ms, hold

29
K for 1 ms, and then ramp the lattice the rest of the way down in 500 µs to map theat 3.3 Erec

quasimomentum into real momentum. The momentum distribution and its corresponding fit after
filtering are shown in figure 2.10c,d. Here we are left with only the lowest band and the sharp edgesof the Brillouin zone are revealed. This image also reveals a major limitation of our band mapping
technique, which is that we are unable to accurately fit populations in the exited bands that aresmaller than 5%. This is largely due to the slight asymmetry in the band mapping image thatis a result of atoms tunneling out of the trap during the ramp. This asymmetry is not currently
accounted for in our simple fitting program, and including it may increase our sensitivity. Even so,
it seems unlikely that band mapping can distinguish between an empty band or a band with a few
percent population.

2.8

An easier way to detect atoms/molecules in higher bands
In the deep lattice limit, the wavefunctions of the molecules in the optical lattice are welldescribed by the eigenstates of the simple harmonic oscillator. The excited bands will not only
have a larger extent in real space, but in momentum space as well. Because the wavefunctionof the excited bands has a larger momentum spread, if higher bands are populated, it will beevident in the size of the cloud observed in time-of-flight imaging. Molecules in higher bands will
increase the width of the fitted gaussian. To quantitatively extract a percentage in higher bands
I simulated the cloud in Mathematica using a thermal distribution to populate the eigenstates ofa harmonic oscillator. The probability density for all the atoms is summed up and then fit with asingle gaussian. The fit width as a function of percentage in higher bands is plotted in figure 2.11.
We see that for 10% of the atoms in higher bands a 10% increase in the momentum is observed.
In figure 2.12, we plot the measured size of a K cloud in time of flight vs. the populationin higher bands measured with band mapping. The population in higher bands was varied bychanging the initial temperature of the K cloud before it was loaded into the optical lattice. The
theory curve from figure 2.11 is overlaid with no free parameters. The size with no population in
the excited bands is given by the harmonic oscilator length in momentum space multiplied by the30

integrated OD (arb.)

a

30

b

20

10

0

c

integrated OD (arb.)

-4

30

-2

0

2

4

-2

0

2

4

quasimomentum q (ħk)

d

20
10
0
-4

quasimomentum q (ħk)

Figure 2.10: Band mapping K atoms in a 1D lattice to measure the population in higher bands.
In a and b 28% of the atoms are in the second band. By holding in a weak lattice with depth 3.2
K for 1 ms we allow these atoms in higher band tunnel out of the lattice leaving only atoms in
Erecthe lowest band as shown in c and d. The black trace is the OD of the cloud integrated over thevertical direction (perpendicular to the lattice) to increase signal to noise. The fit is shown in red.

31
expansion time, t:r
σn=0 (t) = t

~ω
.
2m

(2.11)

Equation 2.11 assumes the cloud is fully in momentum space, which means the in-trap cloud sizeis negligible compared to the expanded cloud size. This criteria is easily met with our verticallattice where the in-trap cloud size is less than one pixel on our camera (one pixel is 5.44 µm at theatoms). It is evident in figure 2.12 that the simple theory underestimates the size for more than
20% of atoms in excited bands. The higher bands have a larger momentum spread than would
be expected from the simple picture of a harmonic oscillator, because of the anharmonicity of thelattice potential. This makes the size in expansion difficult to use as a exact calibration of the band
populations, but it is still very useful. Most of the time we simply want to check that all the atoms
are in the ground band, and a single time-of-flight measurement to ensure that the size matcheswith equation 2.11 is often the quickest and easiest way to get at the band population.

2.9

How to load just one band
To understand the loading of the lattice, it is instructive to look back at the structure of the

Bloch bands with no optical lattice in figure 2.1a. Without an optical lattice, the band structure is2pthat is reflected at the edge of the Brillouin zone.
simply the energy of a free particle given by 2m

As soon as the lattice is turned on any particle with p > ~k will be transported into an excited
band of the lattice. This sets an upper limit on the energy of a gas that can be loaded into a2 2ksingle band of E = ~2m= 1Erec . A gas with a temperature larger than Erec /kB , or even a Fermi

temperature larger than this, will have atoms in the excited Bloch bands. This means that for a
fixed temperature, it is easier to load lighter particles into the lattice because Erec scales inversely
K = 210 nK and E KRb = 66 nK. For this reason we load our atoms
with the mass, for example Erecrecinto the lattice and then create Feshbach molecules in the lattice from a gas of atoms in the lowest
band of the lattice.

32

2.2

fitted size  size of lowest band

2.0

1.8

1.6

1.4

1.2

1.0
0.0

0.1

0.2
0.3
0.4
total fraction in all excited bands

0.5

0.6

Figure 2.11: The size of a gaussian fit to a simulated with a thermal population of higher bands,
imaged after release from the lattice.

33

size (pixels)

50

45

40

35
0.00

0.05

0.10

0.15

0.20

0.25

fraction in higher bands
Figure 2.12: The size of a gaussian fit to a K cloud released from the lattice. The population ofhigher bands was measured using band mapping.

Chapter 3

Universal chemical reactions in KRb

The creation of ultracold KRb in the absolute ground state was exciting for many reasons.
Not only was it the first realization of polar molecules in the quantum regime, which opened theway for studies of novel quantum gasses with long-range interactions, but KRb is also chemicallyreactive giving us the ability to study ultracold chemistry, which up until this point was completely
unexplored. At these ultra-low temperatures, chemical reactions might seem counter-intuitive, but
KRb’s chemical reactions are remarkably easy to understand for two main reasons. First, bimolecular chemical reactions for KRb have no reaction barrier and require no activation energy (thisis generally true of all the bialkali molecules with energetically favorable chemical reactions)[21].
Without a reaction barrier, the chemical reaction will always take place (regardless of temperature)
if the two molecules come sufficiently close together (∼ 10a0 ). Secondly, ultracold collisions, whereparticles scatter only in the lowest angular momentum channel, are governed entirely by quantum
statistics, so the rate at which the molecules enter the short-range part of the potential, where
chemical reactions happen with near unit probability, can be understood from the Bethe-Wigner
threshold laws [45, 22, 6]. Moreover, we observe that KRb’s bimolecular chemical reactions areuniversal in the sense that the details of the short-range potential do not influence the reaction
rates, which can be determined using only knowledge of the long-range potentials [46].
In addition, the amount of control that we have over the preparation of our system gives
us unique abilities to control the chemical reaction rate. By controlling the internal state of themolecule, we can prepare the molecules in a single quantum state, or an incoherent or coherent

35
mixture of our choosing. This modifies the quantum statistics of the collision partners and determines whether or not there is an angular momentum barrier that the molecules must tunnel
though to chemically react. We can modify the long-range potential directly by applying a DC
electric field to orient the molecules and “turn on” the dipole-dipole interaction between molecules.
And we will show in Chapters 4 and 5 that we can control the orientation of the molecules duringa collision (stereodynamics) and even suppress chemical reaction all together by controlling theexternal motion state of the molecule in an optical lattice.
In this chapter I will present the studies of the chemical reactions of KRb confined in athree dimensional optical dipole trap. Much of this chapter and its figures will be take directlyfrom references [2, 1] of which I am an author. This chapter will also necessarily overlap with thediscussions in the theses of K.-K. Ni [23] and M.H.G. de Miranda [24].

3.1

Bimolecular chemical reactions
In references [20, 19] we showed that we could transfer KRb molecules to the absolute lowest

energy state of the molecular potential, namely, the ground electronic, vibrational, rotational, andhyperfine state. One might think that in this absolute ground state the molecules would be assured
to be collisionally stable. But for molecules, chemistry is an possible inelastic collision mechanism.
In table 3.1, I show the binding energies of all possible combinations of K and Rb. It turns outthat only KRb + KRb → K2 + Rb2 is a exothermic reaction. By chemistry standards it’s not very
exothermic, as it only releases 10.3(4) cm−1 worth of energy. In fact the binding energies of thethe reactants and products were so similar that it was not known for certain if the reaction would
be exothermic or endothermic until after the transfer to the ground state nailed down the binding
energy of KRb [19]. But compared to the energy scales in our experiment, the released energy inthis reaction is enough to easily give the products enough kinetic energy to escape the trap.
Experimentally, the chemical reaction manifests itself as molecules disappearing from our
optical dipole trap. Once we have prepared a gas of ground-state molecules and removed all
unpaired atoms, we see the molecules disappear on a timescale of a few seconds. Figure 3.1 shows a

36
Molecule
87 Rb
2
40 K87 Rb
40 K
2
40 K87 Rb
2
40 K87 Rb2

v = 0 binding energy (D0 )3965.8(4) cm−1
4180.417 cm−14405.389(4) cm−1
6080(±300) cm−1
5900(±300) cm−1

reference
[47]
[19]
[48]
[49]
[49]

Table 3.1: Relevant molecular binding energies involved in possible chemical reactions. The binding
energies are given with respect to the threshold energy for free atoms in the absence of magnetic
field. The 87 Rb2 and 40 K2 binding energies include the isotope shifts from the data in the respective
references.

Figure 3.1: A sample molecular density decay vs time for inelastic collisions between indistinguishable fermionic molecules in the ro-vibronic ground-state of 40 K87 Rb. Here the molecules areRb
prepared in a single hyperfine state, |mK
I = −4, mI = 1/2i, and the molecular density-weighted
density decays slowly with a rate coefficient of 3.3(7)×1012 cm3 /s at T = 250 nK. Figure reproduced
from reference [1].

37
typical loss curve. We see that the molecular density-weighted density, n, decays follows a two-bodyloss curve that is the solution to the following differential equation,dn= −βn2 − αn
dt

(3.1)

The first term describes two-body loss of molecules. Here, β is the two-body loss coefficientand has units of cm3 /s. As the molecules are lost, we also observe an increase in temperature, which
decreases the density without molecule loss. For each data set we fit the measured temperature toc, wherea line to obtain the heating rate c with units of K/s, which we use to calculate α = 32 T0 +ct

T0 is the initial temperature. This heating can be understood from the dynamics of the loss (andwill be explained in section 3.5), but for now we enter it in as an empirically measured decreasein density. The data fits well to this two-body decay, and moreover, does so with the same lossRb
coefficient for data taken with molecules in the the lowest hyperfine state(|mK
I = −4, mI = 3/2i)

as well as for data taken with the molecules in an excited hyperfine state (| − 4, 1/2i).
Although we do not have the ability to measure the product states of the chemical reaction,
for a pure gas of molecules in the lowest hyperfine state, there is only one energetically allowed
reaction channel that will result in two-body loss, KRb + KRb → K2 + Rb2 . It turns out that KRb+ KRb → K2 + Rb2 is a barrierless chemical reaction, meaning that it requires no activation energy
to chemically react. With no chemical reaction barrier, the rate at which the molecules react will
be set entirely by the rate at which they come sufficiently close to chemically react (∼ 10a0 ). At theultralow temperatures in our experiment, collisions with large impact factors and correspondingly
large centrifugal barriers are frozen out and the collisions are dominated by a single partial wave.
Because 40 K87 Rb is fermionic, the collisional wavefunction has to be antisymmetric with respect
to particle exchange. So, if our molecules are all prepared in a single hyperfine state, the lowest
allowed partial wave is the L = 1 p-wave. The height of the L = 1 angular momentum barrier for
KRb-KRb collisions is kB · 24µK. With the height of this barrier a full order of magnitude higher
than the collision energies, the rate at which the molecules tunnel through this barrier and into

38
the short range part of the potential where chemistry occurs will be given by the Wigner thresholdlaws. Specifically, for identical fermions tunneling through a p-wave barrier, the rate should scalelinearly with the collision energy [6].
We show the measured chemical reaction rate β as a function of temperature in figure 3.2.
We fit the data to a power law β(T ) ∝ T L to find L = 1.1(2) in agreement with the predictedp-wave threshold law. We observe that the loss rates for the | − 4, 1/2i and | − 4, 3/2i states are thesame to within the error. So although hyperfine changing collisions are possible for the | − 4, 1/2i
state, the loss rate is not significantly increased.

3.2

Two models to understand the loss
To understand the loss rates, we use two models: a simple quantum threshold model (QT)

developed by Goulven Quéméner and John Bohn, and a model that uses the formalism of multichannel quantum defect theory (MQDT) developed by Paul Julienne. The inelastic collisional crosssection for indistinguishable particles is given by
in
σL,m
=
L

~2 π in 2|T| ,µEc L,mL(3.2)

inis the inelastic transition matrix, L is the relative
where Ec is the collisional energy and TL,mL

angular momentum of the two molecules, and mL is its projection onto the quantization axis. The
inelastic rate coefficient for a given temperature T is given by
in
in
βL,m
= hσL,m
vi =
L
L

Z ∞

in
σL,m
vf (v)dv,L(3.3)

v 2 exp[−(µv 2 )/(2kB T )]

(3.4)

0

where

f (v) = 4πµ2πkB T3/2

is the Maxwell-Boltzmann distribution for the relative velocities for a given temperature.
in
In the QT model, we assume |TL,m|2 = 1 for all collision energies, Ec , equal to or above theL

p-wave barrier (this is just the classical Langevin capture rate [50]). At collision energies less thanthe p-wave barrier we scale the loss rate according to the Bethe-Wigner threshold laws. Here the39

Figure 3.2: Inelastic collision rate coefficient vs temperature for fermionic molecules in the rovibronic ground state of 40 K87 Rb. For the lower loss coefficients, the collision rate coefficients were
measured for molecules prepared in either |4, 1/2i (closed circles) or the lowest energy state |4, 3/2i
(open triangles). We observe the loss rate increases linearly with temperature for spin-polarizedmolecules, which verifies that the dominant collision channel is p-wave. A linear fit (solid line) tothe |4, 1/2i data yields the temperature-dependent loss rate to be 1.2(3) × 105 cm3 /s/K. For the|4, 3/2i case, where the collisional loss can only be due to chemically reactive scattering, the loss
rate is similar. The dotted and dashed lines are theoretical predictions from the QT model and
MQDT (describe in the text), respectively. In contrast, when the molecular sample is preparedin a mixture of two hyperfine spin states, |4, 1/2i and |4, 3/2i (filled squares), s-wave collisions
dominate. Here, we observe a temperature-independent decay rate that is 10 − 100 times higher
than for the spin polarized case. Figure reproduced from reference [1].

40
c (L+1/2)|T |2 = ( E, where Vb is the height of the barrier. Using this assumption, we approximateVb )the inelastic cross section and rate coefficient by
in
σL,m
=
L

in
βL,m
=
L

~2 π
L+1/2

µVb
√

EcL−1/2

2~2 π
L+1/2

µ3/2 Vb

(3.5)

hEc i

(3.6)

where hEc i is the average collision energy.
We solve for the height of the barrier of the potential given by
V (R) =

~2 L(L + 1) C6
− 6
2µR2
R

(3.7)

and obtain(~2 L(L + 1))3Vb =54µ3 C6


1/2
.

(3.8)

Inserting Eq. 3.8 in Eq. 3.6 and using hEc i = 3kB T /2 we get 13 3 3 1/4
3 µ C6
kB T.
4
~10

π
in
βL=1,M=L(3.9)

This is the rate for each ML channel, to get the overall rate we must multiply by 3. Using avan der Waals dispersion coefficient of C6 = 16130 a.u. for KRb-KRb with an uncertainty of ±10%[51], the slope of the rate coefficient is predicted to be 1.5(1) × 10−5 cm3 /s/K, which agrees withthe experimental measurements of 1.2(3) × 10−5 cm3 /s/K (| − 4, 1/2i) and 1.1(3) × 10−5 cm3 /s/K(| − 4, 3/2i).
In the MQDT model, the loss rate coefficient is found directly by calculating the quantum
tunneling rate through the p-wave barrier [52], and gives β = 0.8(1) × 10−5 cm3 /s/K, which also
agrees with the measured value within mutual uncertainties. Both of these calculations know
nothing of the short-range potential yet give excellent agreement with the experiment, indicatingthat these chemical reactions are universal in the sense that they do not depend on the details of
the short-range potential and can be estimated using only knowledge of the long-range interactions
[46].

41

3.3

Chemical reactions between distinguishable molecules
The p-wave barrier that set the chemical reaction rate is a consequence of the two colliding

molecules being indistinguishable spin-polarized fermions. If however, the gas is prepared in a
50-50 incoherent mixture of two different hyperfine states, then s-wave collisions are allowed andthe chemical reaction rate will be very different. Here, the molecules are still identical fermions, so
the wavefunction must still be antisymmetric under particle exchange, but because the molecules
can be distinguished by their hyperfine state, the antisymmetry can be incorporated in the internal
state, which allows the collision to proceed via a symmetric s-wave collision.
With the gas in a incoherent 50-50 mixture of the | − 4, 3/2i and | − 4, 1/2i states, we measure
the time-dependant number density of the molecules for both hyperfine states. We observe thesame loss rate for both states, consistent with loss dominated by collisions between distinguishable
molecules in different spin states. As shown in figure 3.2, we obtain a rate coefficient of 1.9(4)×10−10
cm3 /s. The data are consistent with a rate that is independent of temperature as predicted by the
Bethe-Wigner threshold laws.
The QT model is unable to make any predictions for the s-wave rates, but the collision
rates can be estimated using the MQDT model. Here, it is assumed that when the molecules
approach each other within the van der Waals length a, chemical reactions occur with near-unity
probability. The universal loss rate coefficient β = 2(h/µ)a [46] predicts a loss rate of β = 0.8×10−10
cm3 /s, which is a factor of two lower than the experimentally observed rate. This suggests thatperhaps the reaction is not entirely universal and that there is some probability of reflection of thecolliding species back into the entrance channel, which can provide either constructive or destructive
interference, depending on the details of the short-range potential.

3.4

Control of chemical reactions with a DC electric field
In the previous section, I showed that the chemical reaction rate of two indistinguishable

molecules is determined by the long-range potential. This suggests that the chemical reaction rate

42
could be modified, and even suppressed with appropriate control over the long-range potential.
Fortunately polar molecules give us exactly this ability to modify the long-range potential by applying an external electric field, which induces a long-range, 1/r3 , dipole-dipole interaction between
molecules. We can control the strength of these dipole-dipole interactions, which is characterized byan effective induced dipole moment d, by applying an external electric field to align the moleculesalong the axis of the electric field. As the applied field strength is increased, the molecules become
more aligned and have stronger interactions.
When we apply an electric field, we see the lifetime of the molecule gas decrease as shown infigure 3.3. Just as before in the zero electric field case, we fit the data to a two-body loss curve witha linear heating rate (Eqn. 3.1) At higher induced dipole moments, the observed heating rate canbe quite large, but we limit our data to a range where the temperature increases by at most 50%.
Figure 3.4 shows a summary of our experimental data in a plot of β/T0 as a function of the induced
dipole moment d. The Bethe-Wigner threshold laws for inelastic loss, which we verified in the caseof zero electric field, still hold here even in the presence of the 1/r3 dipole-dipole interaction [6]. At
low electric field, d < 0.1 D, there is no significant modification of the loss rate compared to zero
electric field, but for higher electric fields we observe a rapidly increasing loss rate, which is morethan 40 times larger at 0.2 D than at 0.1 D. We fit the data to a power law dependance, β/T0 ∝ dp ,and obtain p=6.1(8).
This loss rate and its scaling on dipole moment can still be understood with the simple QT
model, but the simple picture of single partial wave ultracold scattering breaks down when an
electric field is applied to a polar molecule. A DC electric field breaks the rotational symmetry of
the molecule and mixes rotational states of opposite parity to align the molecule’s internuclear axis
along the external electric field. The aligned molecules now experience a dipole-dipole interaction,
which is a long-range interaction falling off as 1/r3 . As is true for any long-range interaction, thedipole-dipole interaction will mix partial waves. For example, the lowest adiabatic potential for two
identical fermionic KRb molecules colliding in the presence of an electric field with be a mixtureof odd partial waves L = 1, 3, 5.... Strictly speaking with multiple partial waves mixed together,

43

Figure 3.3: Molecular density decay vs. time for induced dipole moments of d = 0.08 D (opentriangles) and d = 0.19 D (filled circles) at T0 = 300 nK. Figure reproduced from reference [2].

44

Figure 3.4: Rate coefficient β/T0 vs. induced dipole moment, d. The dashed line shows a fit
to the simple model based on the quantum threshold behavior for tunnelling through a dipolarinteraction modified p-wave barrier. The solid line shows a result of a more complete quantum
scattering calculation. Inset, the calculated dependence of d on the applied electric field E. Figure
reproduced from reference [2].

45
our molecules are no longer ultracold, in the sense that they no longer collide in only the lowest
allowed partial wave, but they do collide via a single adiabatic potential curve which asymptoticallyconnects to the lowest allowed partial wave at long-range.
In addition to being long-range, the dipole-dipole interaction is also anisotropic, and breaks
the degeneracy between the different projections of the L = 1 partial wave. This is simply understood by the fact that two dipoles colliding in a head-to-tail orientation (an mL = 0 collision) will
have an attractive interaction, while two dipoles colliding in a side-by-side orientation (an mL = ±1
collision) will have a repulsive interaction. The head-to-tail attractive interaction effectively lowers
the p-wave barrier, while a side-by-side repulsive interaction effectively increases the height of thep-wave barrier. This is schematically shown in figure 3.5.
More quantitatively, the long-range potential in equation 3.7 is modified by the dipole-dipole
interaction and becomes0

hLML |V (R)|L ML0 i = 2
~ L(L + 1) C6
C3 (L, L0 ; ML )
0
0
−
δ
δ
−
δML , ML0
L,L ML ,ML
2µR2
R6
R3

(3.10)The −C3 /R3 term is usually written as the classical dipole-dipole interaction Vdd (R, θ) = d2 (1 −3 cos2 (θ)/(4π0 R3 ), but it is often useful to write it in the partial wave basis set, where C3 is given
by

√
√
 L 2
C3 (L, L0 ; ML ) = 2(−1)mL 2L + 1 2L0 + 1 
0 0

L0
0





 
×

L

2

L0

−ML 0 ML


 d2
.

4π0

(3.11)The large brackets denote the 3 − j coefficients. Notice that in an electric field the interaction
potential matrix is no longer diagonal. The long-range dipole-dipole interaction mixes different L
levels while preserves ML , the projection of angular momentum on the quantization axis, so ML is
still a good quantum number. For small dipole moments (small electric fields), we can neglect these
couplings and make a diabatic approximation by using only the diagonal elements of the matrix,
and ignoring the off-diagonal couplings between different values of L. With this approximation, we
can solve for the height of the now modified centrifugal barrier as we did in section 3.2. In figure
3.6, we show the barrier height for M0 and ML = ±1 for both the simple diabatic approximation

46

Figure 3.5: Schematic showing the effective intermolecular potential for fermionic molecules atzero electric field. At intermediate intermolecular separation, two colliding molecules are repelled
by a large centrifugal barrier for p-wave collisions. For a relatively small applied electric field thespatially anisotropic dipolar interactions reduce the barrier for “head-to-tail” collisions and increase
the barrier for “side-by-side” collision. Figure reproduced from reference [2].

47(dashed lines) along with the adiabatic solution (solid lines), where the off-diagonal coupling terms
were included and the interaction matrix was diagonalized using five partial waves L = 1, 3, 5, 7, 9.
Although the diabatic approximation works reasonably well for the ML = 0 partial wave, the effectof the couplings can be seen quite dramatically for d > 0.15 D where couplings with higher partial
waves reduce the adiabatic barrier as the dipole increases.
Although we can use the height of the adiabatic barrier (shown in figure 3.6) in equation 3.6
to get the inelastic rate coefficient, we would like to have an intuitive sense of how these barriersand chemical reaction rates scale with induced dipole moment. For zero electric field, we wereable to get an analytic expression for the barrier height, but in the presence of an electric field
the addition of the dipole-dipole interaction, which can be either attractive or repulsive, makes
this more difficult. We can however look in the two limits of both weak and strong dipole dipole
interactions.
For small dipole moments, the centrifugal barrier and van der Waals interaction dominate
and we can recover the zero-electric field limit derived above by setting C3 = 0. For large dipole
moments, the dipole-dipole interaction dominates and we ignore either the repulsive centrifugal
barrier if C3 is negative (corresponding to a repulsive “side-by-side” dipole-dipole interaction), or
we ignore the attractive van der Waals interaction if C3 is positive (corresponding to an attractive“head-to-tail” dipole-dipole interaction). For positive C3 coefficients (as for the ML = 0 partialwave), the long-range potential is a balance between the repulsive centrifugal barrier and theattractive dipole-dipole interaction. The height of the barrier is given byVb =[~2 L(L + 1)]3
∝ d−4 .
2
3
54µ C3(3.12)

Plugging this barrier height into equation 3.6, we get
3πin
βL=1,M
=
L =0
4



69 µ6
56 ~14

1/2 

d2
4π0

3
kB T.

(3.13)We see that for the ML = 0 at large dipole moment (d > 0.1D), the inelastic loss rate increases as
d6 , consistent with the measured power law in figure 3.4.

48

Figure 3.6: Barrier height vs. effective dipole moment, d. The solid lines are calculated using theadiabatic energies considering partial wave mixing up to L = 9. The mL = ±1 barrier heights riseas the effective dipole moment increases. However, when d > 0.15D, the barrier height reduces
due to mixing of higher partial waves. The mL = 0 barrier height reduces as the dipole moment
increases. The dashed lines are the diabatic curves without partial wave mixing. Figure reproduced
from reference [2].

49
If C3 is negative (the ML = ±1 partial waves) then the long-range potential is a balance
between the repulsive dipole potential and the attractive van der Waals potential. Here the heightof the barrier isVb =|C3 |2∝ d4 ,
4C6(3.14)

which again is put into equation 3.6 to obtain the loss rate3πin
βL=1,M
=
L =±1
4



50~4 /3C6µ

3/2 

d2
4π0

−3
kB T.

(3.15)

Unfortunately, the analytic equation for L = 1, ML = ±1 only holds true when the repulsive dipoledipole interaction is much larger than the centerfugal barrier, which for KRb requires d > 0.186
D, but at this large dipole moment the diabatic approximation breaks down and couplings tohigher partial waves become important, and start to reduce the barrier height such that the p-wave
barrier again becomes significant. Eqn. 3.15 does however confirm our “hand waving” picture thathead-to-tail collisions reduce the barrier height and lead to faster loss, while side-to-side collisions
increase the barrier height leading to a reduced loss rate.
In a three dimensional trap, the molecules experience both ML = 0 and ML = ±1 collisions,but at large dipole moment the loss rate is completely dominated by the ML = 0 channel. Although
the derivation above was for the diabatic potential where couplings to higher partial waves were
neglected, we can include them now with a simple scaling factor. For d > 0.1 D, it can be shownthat the coupling with L = 3, ML = 0 lowers the diabatic barrier of L = 1, ML = 0 by a factor of
in
0.76, yielding a correction of 0.76−3/2 = 1.51 for βL=1,M
[53].
L =0

Although we only have the rates in analytic form in the limits of no electric field and highelectric field, we see good agreement with the experiment by simply adding the two rates from thetwo limits together. This gives us the simple equation" # 9 11 6 1/2
17 µ3 C 3 1/4
6
π
3
2
3
µ
d
in
6
βL=1
=
p1
+ 1.51p2
×
kB T.
4
~10
56 ~14(4π0 )3

(3.16)

Here p1 (p2 ) is the loss probability at the height of the barrier for zero (nonzero) electric field. The
QT model assumes that p1 = p2 = 1, but they become fitting parameters for comparison to our50experimental data. By fitting the data in figure 3.4, we obtain p1 = 0.7(2) and p2 = 0.35(8). We
can alternatively fit a single loss probability p for all values of electric field and a factor b thatmultiplies the C6 coefficient. In this parameterization our fit gives p = 0.35(8) and b = 2.4(9).
We also compare our data to a more complete quantum scattering calculation. This calculation
employs a strong absorptive potential at short range, but captures the long-range physics anduses C6 as a single fit parameter. This fit also agrees well with the experimental data, and gives
C6 = 2100 ± 7000 a.u. which is consistent with the predicted value of C6 = 16130 a.u.[51].

3.5

Anti-evaporation
In addition to an increased loss rate for increasing d, we observe an increased heating rate

for the polar-molecule gas. In figure 3.7, we plot the fractional heating rate normalized by density,Ṫ /(T02 n), as a function of the induced dipole moment. The heating rate Ṫ = c is extracted using
a linear fit to the temperature of the molecular cloud measured as a function of time over a period
sufficiently long to allow T to increase by 20-30%.
This heating rate arises purely from inelastic loss and can be understood as “anti-evaporation”,
where the collisions preferentially remove particles with lower than average energy and leave thegas with slightly more energy per particle than before the collision. To describe this, we include
two competing effects. First, two-body loss preferentially removes molecules from the center of
the trap where the density is the highest and the potential energy is lowest, leading to heating.
Second, the loss rate due to p-wave collisions scales linearly with temperature, and therefore preferentially removes the particles with large relative kinetic energy, leading to cooling. We model bothcontributions to determine the rate of heating or cooling. To accommodate the spatial anisotropyof the interaction and the possible anisotropic energy distribution in the trapped gas, we define“temperatures” along three spatial directions using
Ti =

Ei
,
kB N

(3.17)

where the index i = x, y, z. As the trap shape is a flat disk perpendicular to the vertical (z) axis,

51

Figure 3.7: Normalized fractional heating rate Ṫ /(T02 n) as function of dipole moment. The heating
rate is extracted using a linear fit to the initial temperature increase and is then normalized by the
initial density and temperature of the ensemble. The solid line is the expected heating rate given
by Ṫ /(T02 n) = (β/T0 )/12. Typical conditions for these data are n = 0.3 · 1012 cm−3 and T0 = 0.5µK
s−1 at our highest electric fields. Figure reproduced from reference [2].

52which is the direction of the electric field, we assume that Tx and Ty are equal. The energies Ei
are defined as
X 1
1( mvz2 + mωz z 2 )
2
2
X 1
1
2
=
( mvx + mωx x2 ),
2
2

Ez =

(3.18)

Ex(3.19)

where the sum is over all particles. The heating rates are then given by the derivative of Eqn 3.17
dTi
Ti dN
1 dEi
=−
+
dt
N dt
kB N dt(3.20)

Each two-body inelastic collision removes two particles, so the loss rate is given bydN= −2N Γcoll ,dt(3.21)

where Γcoll is the inelastic collision rate per particle. Similarly,dEi= −∆Ei N Γcoll ,dt(3.22)

where ∆Ei is the average Ei for a pair of particles that is lost from the trap. Putting this together,
we have
dTi
=
dt



∆Ei
2Ti −
Γcoll .
kB

(3.23)

To find the heating caused by inelastic collisional loss, we need to calculate ∆Ez and ∆Ex . To
make things easier for ourselves, we consider the energy in center-of-mass and relative coordinates.
The total energy of a colliding pair of particles will be the sum of four terms: kinetic and potential
energy of the center of mass, and relative kinetic and potential energy. In any of the three directions,the ensemble average for each of the four terms is 1/2kB T . However, the average energy for pairs ofparticles that undergo inelastic collisions is, in general, different from the ensemble average because
the collision rate depends on the relative position and relative velocity. Because collisions are only
sensitive to relative motion, the kinetic and potential energy of the center-of-mass motion of thetwo colliding particles will be on average the same as the ensemble average, (1/2kB T ). The relative
potential energy is zero for colliding particles (because they must be at the same position to collide).

53
So even if the loss had no dependence on density or temperature, a colliding pair has only 3/4 ofthe average energy of a pair of molecules in the gas.
For p-wave inelastic collisions, the collision rate increases linearly with the relative energyof the colliding particles. Furthermore, the spatial anisotropy of the dipole-dipole interaction can
result in different rates for ML = 0 and ML = ±1 collisions. Therefore, we calculate the heatingfor ML = 0 and ML = ±1 collisions separately and then add the heating rates in the final result.
For ML = 0 collisions, we find the average relative kinetic energy for two colliding particles using
R∞4
1 −∞ f (vz , vr )vz dvz
0Ez,rel =µR ∞(3.24)
2 −∞ f (vz , vr )vz2 dvz
3
=
kB Tz
(3.25)
2 R∞
1 0 f (vz , vr )vr2 (2πvr )dvr0=
µ R∞
Ex,rel(3.26)
4
0 f (vz , vr )(2πvr )dvr
1
=
kB Tx
(3.27)
2
where we assume a Maxwell-Boltzmann distribution of relative velocities given byf (vz , vr ) ∝ exp −

µvr2
µvz2
exp −
2kB Tr
2kB Tz(3.28)

with vr2 = vx2 + vy2 . A factor of vz2 in both the numerator and denominator of equation 3.24 accounts
for the fact that the ML = 0 p-wave inelastic collision rate scales as collision energy ∝ vz2 . Similarly,
for ML = ±1 collisions, the collision rate scales as vr2 . We find the average relative kinetic energy
for two colliding particles using
1
Ez,rel==1
Ex,rel=R∞2
1 −∞ f (vz , vr )vz dvzµ R∞
2−∞ f (vz , vr )dvz
1
kB Tz
2 R
∞
1 0 f (vz , vr )vr4 (2πvr )dvr
µR ∞4 0 f (vz , vr )vr2 (2πvr )dvr

= kB Tx .

(3.29)(3.30)(3.31)(3.32)

Adding the center of mass energy and putting ∆Ei into the expression for the heating rate (equation
3.23), remembering that Γcoll = Γ0coll + 2Γ±1coll we find that
dTz
dt

1
= − Tz Γ0coll + Tz Γ±1
coll
2(3.33)

54
dTx
dt

1
= + Tx Γ0coll
2(3.34)

Remembering that β = 2Γncoll and assuming thermal equilibrium T = Tz = Tx we get an overall
heating rate of
dT
dt

=
=

1 dTz
2 dTx
+
3 dt
3 dt
n
[Tz (2β1 − β0 ) + 2Tx β0 ]12(3.35)(3.36)

Assuming thermal equilibrium, T = Tz = Tx we get
1dT= T nβdt
12(3.37)dT
β/T/(T 2 n) =.
dt
12(3.38)

or equivalently,

In figure 3.7, we plot equation 3.38 with no free parameters (solid line) and see that it fits well toour data.

Chapter 4

KRb in 2D

In the previous chapter we showed that atom-exchange chemical reactions cause loss andheating in our gas of 40 K87 Rb. We showed that the chemical reaction rate could be changed by either
changing the spin polarization of the particles (making them distinguishable or indistinguishable
fermions by manipulating the internal states of the molecule) or by introducing a long-range dipoledipole interaction by applying an external electric field. Unfortunately, either method of changing
the chemical reaction rate made the rate increase compared to the rate for spin polarized molecules
at zero electric field. Even more discouraging is the fact that the chemical reaction rate is tied to thestrength of the dipolar interaction, such that whenever there are strong dipole-dipole interactions(one of the main motivations for creating polar molecules) there is also strong inelastic loss. Thiscan make it difficult to study dipolar interactions in the ultracold gas.
Although the chemical reaction rate increased as the induced dipole moment was increased,
it was due to a single adiabatic channel corresponding to “head-to-tail” collisions. In this “headto-tail” orientation, the dipole-dipole interaction is attractive and effectively lowers the p-wave
centrifugal barrier that the molecules have to tunnel through to get to short range where chemical
reactions take place. But there are two other collision channels that correspond to “side-by-side”
collisions, where the dipole-dipole interaction is repulsive and effectively raises the p-wave centrifugal barrier and lowers the chemical reaction rate. In 3D, all three collision channels take place andthe loss is dominated by the “head-to-tail” collisions, but by reducing the dimensionality of thesystem, we can use Fermi statistics to suppress the “head-to-tail” collisions in much the same way

56
that Fermi statistics were used to suppress s-wave collisions in 3D.
In this chapter I will show how we can gain control over the orientation of the collision by
going to a quasi-2D system. This control over the stereodynamics lets us control the chemical
reaction rate. Specifically, we can suppress the M = 0 “head-to-tail” collisions, allowing us tocreate a gas of polar molecule with suppressed inelastic loss (leading to a long lifetime) and strong
dipolar interactions. Much of this chapter and its figures will be taken directly from Ref. [3] of
which I am an author. This chapter will also necessarily overlap with the thesis of Marcio de
Miranda [24] as we worked on these experiments together.

4.1

Controlling chemical reactions
In chapter 3 I showed that in the quantum regime KRb’s bimolecular chemical reaction rate

could be controlled. This control came about in two ways. First, an electric field could be applied
to change the long-range potential. Second, is that the quantum statistics of the molecules dictated a particular symmetry of the total wavefunction with respect to exchange of two molecules.
This couples the internal degrees of freedom of the molecules with their motional degrees of freedom, resulting in a strong modification of the collision process through control of internal states.
Exploiting this coupling relies on the fact that ultracold collisions, including ultracold chemical
reactions, will be dominated by the allowed collision channel with the lowest centrifugal barrier. In
three dimensions, ultracold fermions, such as our molecules, that are spin-polarized (in the sameinternal state) collide in the partial-wave channel with L = 1 (p wave), where ~L is the quantized
relative angular momentum. On the other hand, fermions in different internal states can collide
in s waves (L = 0). This behaviour is familiar from the example of ultracold fermionic atoms[54]. For the case of dipoles, the dipole-dipole interaction mixes states with different L, such thatone should replace L = 0 with the lowest-energy adiabatic channel with even L, which does nothave a centrifugal barrier. Similarly, L = 1 becomes the odd-L adiabatic channel with the lowest
centrifugal barrier. The approach of controlling the relative motion through control of the internal
molecular states has a serious limitation, which is that we do not have control over the projection

57of the relative angular momentum of the colliding particles on the ẑ axis, M . This is especially
important for dipoles, where M describes whether the dipoles approach ‘side-by-side’ (for example,
L = 1,M = 1), and therefore experience a repulsive dipole-dipole interaction, or ‘head-to-tail’ (forexample,L = 1,M = 0), where the dipole-dipole interaction is attractive.
By confining polar molecules in a two dimensional plane, we gain control over M . Withextremely strong confinement and large dipole moments oriented perpendicular to the 2D plane,
one could, in principle, achieve a fully 2D geometry where the molecules could approach each other
only side-by-side (odd M ) simply because of the repulsive part of the dipole-dipole interaction[55, 56]. Alternatively, in our present experimental situation of quasi-2D trapping, we can again
use the overall symmetry of the two-molecule wavefunction for fermionic molecules to control M . In
quasi-two dimensions, L is no longer a good quantum number for describing how molecules approachq~each other at intermolecular separations that are much larger than aho , where aho = 2mω
and isthe extent of the molecular cloud in the lowest harmonic oscillator state in ẑ. In 2D (and quasi-2D)
the relative motion is described by M . In addition, we have a new (compared with the 3D case)quantum number that describes the harmonic oscillator motional state in ẑ for each molecule. Thisquantum number, which we label v, acts as a new internal degree of freedom for the molecules as
they approach each other in the long range, in the sense that we can relatively easily control theoccupation in the different v states and this will in turn give us control over the relative motion
described by M . Specifically, fermionic molecules in the same internal state and the same v must
collide with odd M , whereas fermions in the same internal state but different v can collide witheven M .

4.2

What is 2D?
The spatial geometry of the confining potential can influence collisions in a trapped dipolargas. This effect has been explored for elastic collisions of magnetic atoms [57]; here, we exploit
spatial geometry to control chemical reactions of polar molecules. In particular, a 2D trap geometry,with the dipoles oriented parallel to the tight confinement direction ẑ, is well matched to the spatial

58
anisotropy of the dipole-dipole interaction [58]. We can realize such a geometry using a 1D opticallattice (see figure 1a and Chapter 2), where the trapped molecules are divided among several isolated
layers. In each of these layers, the lattice potential provides tight harmonic confinement in ẑ suchthat only the lowest few quantized motional states in ẑ are occupied. Consequently, within eachisolated layer, colliding molecules approach each other in two dimensions. At long intermoleculardistances, the molecules do indeed behave as if they are confined in a 2D geometry and collisions
are best described in a cylindrical symmetry with quantum numbers M and v. But at shorter
intermolecular separations this description breaks down.
The extent of the cloud in the direction of tight confinement (given by the harmonic oscillator
length aho ) is larger than both the van der Waals length and the dipole length. At these smallintermolecular separations the collisions are still best described in 3D with quantum numbers L
and ML . Put another way, just as the cylindric symmetry and tight harmonic confinement in ẑ
makes L a bad quantum number at large intermolecular separations, the spherical symmetry of
the van der Waals potential makes v a bad quantum number at shorter intermolecular separations.
The energies involved in the collision around the p-wave barrier are large enough to mix many v
levels together.
Because collisions can be treated in 2D at large intermolecular separation, R, but are 3D
at small R, our system is quasi-2D. But the molecules are oblivious to our need to change fromcylindrical to spherical coordinates midway through a collision. In both regions, M is a goodquantum number that is strictly conserved for all R. So, molecules that start in an adiabatic
potential curve at long range with M = 0 will smoothly connect to a potential curve at short rangethat is ML = 0. In this sense, it does not matter that the system are quasi-2D, just being 2D atlong-range is good enough to select out a single M state. Moreover, being 2D at long range also
provides us with an easily controlled quantum number v that allows us to control the symmetry of
the wavefunction which chooses the orientation of the collision.
I will note here that although we rely on the quantum statistics of our fermionic molecules
to provide a “statistical suppression” of collisions that arises from the centrifugal repulsion of

59
indistinguishable fermions, suppression that relies directly on a strong confining potential and a
repulsive dipole-dipole interactions can also be realized, but it requires both tighter harmonic
confinement and larger dipole moments [55, 56]. Although this regime is currently unreachable
for our experimental apparatus, the required lattice strengths and dipole moments are not entirely
unreasonable, and this suppression has the advantage of not relying on quantum statistics, whichshould allow it to work for bosons, fermions, and even distinguishable particles.

4.3

Stereodynamics and quantized collisions
We now consider the quantized collision channels that define the stereodynamics in this quasi-

2D geometry. We identify three relevant collision channels, and we label these |1i, |2i, and |3i,
in order of increasing centrifugal barrier heights. Figure 4.1b shows schematically the adiabatic
potentials for these three lowest-energy collision channels. Collision channel |1i corresponds tospatially isotropic collisions, collision channel |2i is the quantum analogue of ‘head-to-tail’ collisions,and collision channel |3i is the quantum analogue of ‘side-by-side’ collisions.
Channels |1i and |2i become increasingly favourable for chemical reactions as the dipole→−
dipole interaction strength is increased, for example by increasing the external electric field | E |.
In contrast, channel |3i has a centrifugal barrier whose height increases for higher dipole moment,→−
within the | E | range considered in this work. This barrier hence continues to prevent moleculesfrom reaching short range.
The low-energy collision channels |1i |2i, and |3i can be labelled with four relevant quantum
numbers, η, L, γ and M . The quantum number γ identifies the exchange symmetry of the partof the two-molecule wavefunction that describes the relative motion in ẑ; for the symmetric case,γ = 1 and for the antisymmetric case, γ = −1. For two molecules in the same ẑ harmonic oscillator
state, γ = 1, whereas both γ = 1 and γ = −1 are possible for two molecules in different harmonic
oscillator states. Similarly, we use a quantum number η to keep track of the exchange symmetryof the part of the wavefunction that describes the internal states of the two molecules. For two
molecules in the same internal quantum state, η = 1 whereas η = ±1 for molecules in different

60internal states. In two dimensions, three quantum numbers (M , γ, η) are sufficient to describe thequantum stereodynamics. However, because the interactions at short range must be described inthree dimensions, the quantum number corresponding to the 3D angular momentum, L, as well as
M becomes relevant. With collisional channels described by quantum numbers η, L, γ, and M ,the fermionic symmetry can be concisely stated in the following relations:η(−1)L = −1 short range, 3Dηγ(−1)M = −1 long range, 2D(4.1)

Each of the collision channels satisfies both the short-range and long-range symmetry requirements in Eqn. 4.1. Figure 4.1c shows how these different collision channels can be accessed
through control of the internal molecular states and the ẑ motional states. In figure 4.1c, molecules
in different internal states are shown in different colours and the harmonic oscillator states in ẑ
are labelled by v. In case (1) for two molecules in different internal molecular states and in any
combination of v levels, η = −1 allowing for L = 0 barrierless collisions in channel |1i. In case(2), the molecules are prepared in identical molecular states (so η = 1), but in different v levels
so the lowest-energy collision channel vibration state is |2i with γ = −1. At short-range there isa L = 1 p-wave barrier, but to fulfill the symmetry requirements at long-range M = 0, which isa “head-to-tail collision” where the barrier height decreases as the dipole moment is increased. Incase (3) where the molecules are prepared in the same internal state and the same v level, η = 1
and γ = 1, so at short range L = 1 and at long range M = ±1. This provides a p-wave barrier with“side-by-side” dipolar collisions where the barrier height increases as the dipole moment increases.
This case is the least favorable for bimolecular chemical reaction, and should allow for the creationof a gas that has strong dipole-dipole interactions without fast loss from chemical reactions.

4.4

Preparation of the 2D molecular system
For the set of experiments described in this chapter, we start with a mixture of 40 K and

87 Rb in a crossed-beam optical dipole trap, with a harmonic trapping frequency of 180 Hz along

61

(a)

(b)

V

|3>
z

|2>


E

v1=0 , v2=1

y



x

|1>

(c)
(1)

(2)

v1=0, v2=0
R

(3)

v=2

v=1


v=0

Figure 4.1: (a) A quasi-2D geometry for collisions is realized for polar molecules confined in a 1D
optical lattice. An external electric field is applied along the tight confinement axis. (b) Schematic
showing the three lowest adiabatic potentials for collisions as a function of the intermolecularseparation, R. These three channels are ordered with increasing magnitude of the centrifugal
barrier. The arrows indicate the change in the potential for an increasing external electric field,
and hence a growing induced dipole moment. (c) Schematic showing each individual case for thethree lowest collision channels. The lowest-energy collision channel occurs when two molecules areprepared in different internal states (indicated here by the colors of the molecules). The second
channel is realized when two identical molecules are prepared in different vibrational levels v for
their ẑ motions. The third case as a much reduced loss rate as a consequence of an increased
centrifugal barrier when the two identical molecules are prepared in the same vibrational level
along ẑ. Figure reproduced from reference [3].

62
the vertical direction ẑ and 25 Hz in the transverse directions. We add an optical lattice alongẑ, which is formed by a retro-reflected beam with a 1/e2 waist of 250 µm and a wavelength of1064 nm. Both optical dipole trap beams and the optical lattice beam are linearly polarized
and their polarizations are mutually orthogonal. Each layer of the optical lattice trap is tightly
confining in ẑ with a harmonic trapping frequency of νz = 23 kHz for the molecules, whereas in
the transverse directions, the combined trap has a harmonic trapping frequency of 36 Hz. The
tunnelling rate between lattice layers is negligible and, therefore, each layer realizes an isolated
trap for the molecules. Initially, 34,000 ground-state molecules are confined in roughly 23 layers,
with the center layer having 2,200 molecules and a peak density of 3.4 × 107 cm2 .
We start by loading ultracold 40 K and 87 Rb atoms from the crossed-beam dipole trap into
the combined trap by turning up the intensity of the optical lattice beam in 150 ms. We thencreate molecules in the lattice by first forming extremely weakly bound molecules through magnetoassociation of atom pairs and then coherently transferring these molecules into their rovibrational
ground state using optical transitions [19]. The temperature of the molecular gas, T, in the combined optical dipole plus lattice trap can be varied between 500 and 800 nK by varying the initialatom gas conditions.
To completely freeze out the motion of the molecules along ẑ requires that kB T /hνz . Fora gas at T = 800 nK in our lattice, kB T = hνz = 0.72 and we expect 25% of the molecules will
occupy higher v levels. As discussed above, to control the stereochemistry of bimolecular reactionsin the ultracold gas, we need to control both the internal state and the harmonic oscillator level,v, of the molecules. We create the molecules in a single internal quantum state. If desired, we
can subsequently create a 50/50 mixture of molecules in the ground and the first excited rotational
states by applying a resonant microwave π/2-pulse [20]. The occupation of lattice levels v can becontrolled by varying T ; alternatively, we can prepare a non-thermal distribution of molecules usingparametric heating (for details see section 2.6). Here, the lattice intensity is modulated at twiceνz , and, as a result, molecules initially in the v = 0 level are excited to the v = 2 level, as discussed
in chapter 2.

63
We determine the population in each lattice level using an adiabatic band-mapping technique[43, 44] described in section 2.7. As the lattice potential is ramped down slowly (compared toh/Ebg where Ebg the band gap), the quasimomentum in the lattice is mapped into real momentum,
and molecules in different vibrational levels of the lattice are mapped onto Brillouin zones. The
measured molecule momentum distribution following this ramp is shown in figure 4.2a for a T = 800
nK molecular gas. The measured fraction in v = 0 matches well with the expected thermal
distribution. In contrast, figure 4.2b shows the measured non-equilibrium occupation of lattice
vibrational levels following parametric heating.

4.5

Measurement of the molecular loss rate
We measure the bimolecular reaction rate by monitoring the loss of trapped molecules as afunction of time. To image the molecules, we reverse our coherent transfer process to bring themolecules back to a weakly bound state where we can detect the molecules with time-of-flightabsorption imaging [19]. The molecules are imaged after free expansion from the combined optical
dipole plus lattice trap. From the images, we obtain the total number of molecules and the radial
cloud size.
As we do not resolve the individual layers of the optical lattice, we obtain an average 2Ddensity per layer by dividing the total number by the cross-sectional area of the cloud, and by aneffective number of layers ξ. To determine ξ, we assume that as we load the optical lattice theextent of the cloud in the ẑ direction does not change, but the cloud is simply sliced into pancakes.
This is a reasonable assumption because the lattice provides no additional harmonic confinementin the ẑ direction. To measure the in-trap distribution we first load the lattice with atoms, create
molecules, transfer to the ground state, remove the residual atoms, and then load the molecules
back into the optical trap by ramping down the optical lattice with a 10 ms linear ramp. We
hold the molecules in the optical trap for 20 ms and then measure the size of the cloud in time offlight. Although the cloud size in the ẑ direction is below our imaging resolution, we can extract
the in-trap size by measuring the temperature and trap frequency. In this manner we are able to

64

(a)

(b)
gZ

gZ
(B)

n0 = 76 %0.12

n1 = 24 %0.08

n0 = 47 %n1 = 21 %n2 = 32 %

0.08

0.04

0.04

0.00
-6

(D)

AveragedOD OD

Averaged
OD OD

0.12

0.00
-4

-2

0

2

4

Quasimomentum (k)

6

-6

-4

-2

0

2

4

Quasimomentum (k)6

Figure 4.2: We measure the relative population in each lattice vibrational level using a bandmapping technique. The results for the initial distribution of molecules (a) and for a non-thermal
distribution created by parametric heating in ẑ (b). The two images use the same colour scale forthe optical depth (OD). The images are an average of five shots (a) and seven shots (b), taken
after 10 ms of free expansion. Below each image we show a trace along ẑ that corresponds to the
OD averaged over the transverse direction. A fit (red line) to the trace, which takes into accountboth the size of the Brillouin zones and our imaging resolution, is used to extract the relative
populations, nv /ntot , in each lattice level v. The horizontal axis corresponds to momentum in ẑ
and is marked in units of the lattice momentum ~k, where k is the lattice wave vector. Figure
reproduced from reference [3].

65
measure the 1/e2 gaussian radius of the cloud in-trap (σz ). For a typical cloud we measure σz = 3.8µm.
On a single pancake the density-weighted density, n, is given by Ni /(4πσr2 ), where Ni is thenumber of molecules on the ith lattice site and σr is the size of the cloud in the radial direction.
2
We also know that on a single pancake the molecules chemically react following dn(t)dt = −βn(t) .

Integrating over x̂ and ŷ and summing over the different layers in ẑ we get∞β X
Ni (t)2Ṅtotal (t) = −
4πσr2
i=−∞

∞ 
βNtotal (t)2 X
Ni (t) 2= −4πσr2
Ntotal

(4.2)

(4.3)

i=−∞

i (t)
where NNtotal
is the percentage of molecules in the ith pancake and

2
∞ 
P
Ni (t)
i=−∞

Ntotal= 1/ξ(t), theeffective number of layers in our system. We can rewrite Eqn. 4.3 in the more familiar form of thetwo-body loss equationn2D
˙ = −βn22D(4.4)

where n2D = Ntotal /(4πσr2 ξ) is the average 2D density of the system and Ntotal /ξ is the numberweighted average Ni , averaged over the occupied layers. This result relies on the assumption that
ξ is time independent, which is not correct. For the conditions used in this experiment ξinitial = 23,
but the loss is density dependent so the layers at the center of the cloud, with high density, will
decay faster than the layers on the edge of the cloud, with low density. ξ increases for longer timeas the initial gaussian distribution across the layers become much flatter as illustrated in figure
4.3. For our analysis we use a time averaged value ξ = 30 that was determined by comparing ananalysis based on a uniform layer density with a numerical simulation of the loss in each layer,
shown in figure 4.4.
In figure 4.5, we show the average 2D density as a function of time. For these data, the→−molecules are all prepared in the same internal state and | E | is 4 kV/cm, which gives an induced
molecular dipole moment in the laboratory frame of 0.158 Debye. The two data sets in figure 4.5
correspond to an unperturbed T = 800 nK gas (black squares) and a parametrically heated gas

66

Figure 4.3: The molecule distribution in the ẑ direction. Initially the molecule distribution follows
a gaussian distribution (black), but after 500 ms (red) the most dense pancakes at the center ofthe cloud have decayed the most resulting in a distribution much flatter than the initial gaussian.

67

Figure 4.4: The calculated number loss due to two-body decay in a one-dimensional optical lattice.
The exact numerical solution solving for the time dependent density in each pancake and thensumming over all pancakes to find the total number is shown in blue. If we use the initial numberof layers, ξinitial , to scale the total number we overestimate the loss (black curve). In the red curve
we use ξ as a fit parameter to find the time averaged value of ξ = 30.

68

n0/ntot = 0.76(3)n0/ntot = 0.49(3)n1/ntot = 0.23(3)n1/ntot = 0.15(3)n2/ntot= 0.01(3)n2/ntot = 0.36(3)-2ntot (10 cm )

9

6

6

3

0
0.00

0.25

0.50

0.75

1.00

time (s)Figure 4.5: Measurement of 2D loss rates. A fit (solid lines) to the measured loss curves, with(red circles) and without (black squares) 0.3ms of parametric heating in ẑ, is used to extract theloss-rate coefficients β|3i and β|2i . Figure reproduced from reference [3].

69
(red circles). For the case where parametric heating was used to increase population in v > 0 levels,
the data show a faster initial loss of molecules. This indicates that the initial loss is predominately
due to interlevel collisions as described in case (2) of figure 4.1c, whereas intralevel collisions (case(3) of figure 4.1c) give a slower loss of molecules at longer times. It should be pointed out herethat because of the separation of timescales between intralevel and interlevel loss, the system is self
purifying, meaning that if the gas is initially prepared with a small fraction in excited v states thefast interlevel loss will quickly get rid of these molecules and leave a gas entirely in a single band
harmonic oscillator state. However, this self purification is not without a cost, every unwanted
molecule in a higher v state will react with a molecule in the ground v state, and thus reduce thenumber and density of the molecules in the ground state.
We fit the data using a simple model, which assumes two loss-rate constants: one for interlevel
collisions, β|2i , and a second one for intralevel collisions, β|3i (with the subscripts referring to theadiabatic channels labelled in figure 4.1b). Here,
dn0
dt
dn1
dt
dn2
dt

= −β|3i n20 − β|2i n0 n1 − β|2i n0 n2= −β|2i n0 n1 − β|3i n21 − β|2i n1 n2(4.5)= −β|2i n0 n2 − β|2i n1 n2 − β|3i n22

where nv is the 2D density of molecules in a particular lattice vibrational level v. To fit themeasured time dependence of the total 2D density, ntot (t), we use ntot (t) = n0 (t) + n1 (t) + n2 (t).
We input the measured initial populations nv /ntot (see figure 4.2) at t = 0, and we fit the data
to the numerical solution of Eqn. 4.5. We obtain β|2i and β|3i from a simultaneous fit to the two
measured ntot (t) curves shown in figure 4.5.
→−
By repeating this procedure for different values of | E |, we measure the chemical reaction
rate constants, β|2i and β|3i , as a function of the induced dipole moment. In figure 4.6, we show
the intralevel (black squares) and interlevel (red circles) chemical rate constants as a function of
the dipole moment. Also shown as green triangles in figure 4.6 are the results of two measurements for a 50/50 mixture of molecules in different rotational states (case (1) of figure 4.1c). The

70
mixture of distinguishable molecules is achieved with a single π/2 pulse between |0, 0, −4, 1/2i
Rb
and |1, 1, −4, 1/2i (using the notation |N, mN , mK
I , mI i). Although the microwave pulse puts themolecules in a coherent superposition of the two rotational states (where the molecules would still
be indistinguishable), we observe that this coherence is lost (making the molecules distinguishable)in less than 1 ms due to the difference in polarizabilities between the two rotational states (seechapter 6). Here, we fit the loss of molecules in the ground rotational state to the solution of
dntot
2dt = −β|1i ntot to extract a single loss-rate constant.

4.6

Comparison to theory
We compare our measurements to quantum scattering calculations performed by GoulvenQuéméner and John Bohn using a time-independent quantum formalism based on spherical coordinates with cylindrical asymptotic matching to describe the molecular collisions in quasi-two
dimensions [59]. They use an absorbing potential at short distance to represent chemical reactions[53, 52]. This technique showed excellent agreement with previous experimental data for KRb
bimolecular reactions in three dimensions (see chapter 3 and Refs. [1, 2]). They computed theloss-rate coefficients βv1 ,v2 for molecules in different initial lattice vibrational states v1 , v2 , at acollision energy of 800 nK. When the induced dipole moment is still small (< 0.2 D), the measured temperature is a good approximation for the mean collision energy. The loss rates of thedifferent processes can be separated into fast loss rates (β0,1 , β0,2 , β1,2 ) ≈ β|2i and slow loss rates(β0,0 , β1,1 , β2,2 ) ≈ β|3i . The black theoretical curve in figure 4.6 corresponds to an average of theslow rates weighted by the initial populations n0 , n1 , n2 . The red curve corresponds to the same
average, but for the fast rates. The green curve corresponds to the loss rate of molecules in different
internal states.
The three measured reaction rate constants shown in figure 4.6 are consistent with the quantum scattering calculations for the collision channels shown in matching colors in figure 4.1b,c.
Molecules in different rotational states (green triangles in figure 4.6) have the highest rate forchemical reactions, consistent with the fact that they can collide in channel |1i, which corresponds

71

-5

2

-1

2D (cm s )

10

10

-6

10

-7

0.00

0.05

0.10

0.15

0.20

Dipole moment (D)Figure 4.6: The extracted loss-rate coefficient for collisions of molecules in the same lattice vibrationlevel (black squares) and from different lattice vibrational levels (red circles) plotted for several
dipole moments. Measured loss-rate coefficients for molecules prepared in different internal states
are shown as green triangles. For comparison with each of these three measurements, we include
a quantum scattering calculation for νz = 23 kHz, T = 800 nK (solid lines). The potentials
corresponding to the dominant loss channel for the three cases are shown in matching colors infigure 4.1b,c. Figure reproduced from reference [3].

72
to spatially isotropic collisions with no centrifugal barrier. On the other hand, molecules preparedin the same internal molecular state (red circles and black squares in figure 4.6) have suppressed
reaction rates because the lowest-energy collision channel is no longer allowed. Instead, identicalmolecules in different lattice levels (red circles in figure 3b) react predominantly through collisions
in channel |2i, or “head-to-tail”, whereas identical molecules in the same lattice level (black squares
in figure 4.6) react through collisions in channel |3i, or “side-by-side”. The importance of stereodynamics on the reaction rate for polar molecules is manifest in the very different dipole-moment
dependence of the reaction rates in these two collision channels. In particular, for the case where
the molecules are prepared both in the same internal quantum state and in the same v level, thereaction rate is suppressed even as the dipole moment is increased.

4.7

2D vs. 3D
Figure 4.7 shows how the initial loss rate in a gas of identical molecules depends on thefractional occupation of the lowest lattice level, n0 /ntot . The fractional molecular populationf (v, T ) in a vibration level v at temperature T is obtained from a Boltzmann distribution, and theeffective βinitial is then calculated as
βinitial = β|3i

Xv

f (v, T )2 + β|2i

X

f (v1 , T )f (v2 , T )(4.6)

v1 6=v2

As n0 /ntot increases, the calculated initial loss-rate constant for a molecular gas in thermal equilibrium (solid black line) changes from close to β|2i (the red line indicating the measured value at
0.174 D from figure 4.6) to β|3i (open symbol at n0 /ntot = 1). On the top axis of figure 4.7, we givethe corresponding values of the scaled temperature (kB T /hνz ). The filled triangles in figure 4.7
correspond to the measured initial loss rate at different temperatures (500 nK and 800 nK), andthe open symbol at n0 /ntot ≈ 0.5 corresponds to the initial loss rate for the parametrically heated,
non-thermal molecular gas.
We also directly compare the suppressed chemical reaction rate in quasi-two dimensions withthat of the 3D case in figure 4.8. Here, we compare data for a 3D geometry from chapter 3 againstinitial(cm2 s-1)

73

10

-5

10

-6

10

-7

1.5

1

0.5

0.6

kBT/ hz

0.7

0.8

0.5

0.9

0

1.0

n0/ntot
Figure 4.7: The effective initial loss rate, βinitial , for polar molecules confined in a 2D geometry
depends on the fractional population (n0 /ntot ) in the lowest harmonic oscillator level in ẑ, which
for a gas in thermal equilibrium depends on the ratio kB T /hvz . The measured initial loss rates fora dipole moment of 0.174 D are shown for two different thermal distributions (solid triangles), a
non-thermal sample created by parametric heating (the top open triangle) and an extracted pureβ|3i for the limit of the entire population residing in the lattice ground vibrational level (the bottom
open triangle). The experimental results agree well with a simple model (black curve) describedin the text. The top line indicates the value of β|2i as given in figure 4.6. Figure reproduced fromreference [3].

74
the suppressed loss-rate constant measured in quasi-two dimensions. For the comparison, the 2D√
loss rate is scaled to three dimensions using β3D = πaho β2D [58, 60, 61]. At d = 0 the loss rate
for 2D is 2/3 of the 3D rate because only two of the three L = 1 channels are allowed in 2D forindistinguishable molecules in the same v state. For a dipole moment d, greater than 0.1 D, the3D loss-rate constant increases markedly as d6 [2, 53], whereas the scaled loss-rate constant for thequasi-2D case remains close to the value at zero electric field. At a dipole moment of 0.174 D, themeasured suppression in quasi-two dimensions is a factor of 60.

75

10

-9

3 -1

3D(cm s )

3D
2D

10

-10

10

-11

10

-12

0.00

0.05

0.10

0.15

0.20

Dipole moment (D)Figure 4.8: The extracted intralevel loss rate for identical fermionic KRb molecules in two dimensions (black circles) compared with the loss rate in three dimensions (blue triangles). The 3D data
for T = 300 nK are from Ref. [2]. The 2D data were taken at T = 800 nK and are converted to√
3D rates by multiplication with πaho , where aho is the harmonic oscillator length in ẑ. Figure
reproduced from reference [3].

Chapter 5

Molecules in a 3D latticeControllable long-range and anisotropic dipole-dipole interactions can enable novel applications of quantum gases in investigating strongly correlated many-body systems [62, 7, 63, 13, 64, 65].
Recent experiments have realized an ultracold gas of polar molecules in the ro-vibrational groundstate [19] with high-resolution, single-state control at the level of hyperfine structure [20]. However,an obstacle to creating long-lived quantum gases of polar molecules was encountered with the observation of bimolecular chemical reactions in the quantum regime [1]. Even with the demonstrated
strong suppression of the reaction rate for spin-polarized fermionic KRb molecules, the lifetime of
a 300 nK sample with a peak density of 1012 /cm3 was limited to ∼1 s. Furthermore, when an
external electric field is applied to polarize the molecules in the lab frame, the attractive part ofthe dipole-dipole interaction dramatically increases the ultracold chemical reaction rate, reducing
the lifetime of the dipolar gas to a few ms when the lab-frame molecular dipole moment reaches 0.2
Debye [2]. A promising recent development was the demonstration that confining fermionic polar
molecules in a 1D optical lattice could suppress the rate of chemical reactions even in the presenceof dipolar interactions. Here, the spatial anisotropy of the dipolar interaction was exploited by confining a gas of oriented KRb molecules in a two-dimensional geometry to suppress the attractive
part of the dipolar interaction and thus achieve control of the stereodynamics of the bimolecular
reactions [3]. In this regime, the lifetime of a trapped gas of polar molecules with a lab-frame dipole
moment of approximately 0.2 Debye, a temperature of 800 nK, and a number density of 107 cm−2 ,was approximately 1 s.

77
In this chapter, we study KRb molecules confined in 2D and 3D optical lattice traps, where
we explore the effects of the lattice trap confinement on the lifetime of the ultracold gas. We notethat a lifetime of 8 s has been achieved for homonuclear Cs2 molecules in a 3D lattice [66]. In
our work, we also find that long lifetimes are achieved for the molecules in a strong 3D lattice
trap, even when there is a significant dipole moment in the lab frame. In addition, we observethat adding a weak axial corrugation to a 2D lattice can result in long lifetimes for the trappedmolecules. In contrast to our previous experiments where ground-state polar molecules were createdstarting with ultracold atoms in a harmonic trap, here we report on the creation of ground-state
KRb molecules starting from ultracold K and Rb atoms loaded into a 3D lattice. To aid in future
efforts to optimize the ground-state molecule creation process, we have also examined the creation
efficiency and lifetime for the weakly bound Feshbach molecules that provide the starting point forcoherent two-photon optical state transfer to the molecular ground state. Much of this chapter isreproduced from reference [4] of which I am an author.

5.1

Preparation of ground state molecules in a 3D lattice
The experiments start with an ultracold mixture of 2.9 × 105 40 K atoms and 2.3 × 105 87 Rb

atoms in a crossed optical dipole trap (ODT) at 1064 nm, at a temperature that is twice the Rb
condensation temperature Tc . The trap frequencies for Rb are 21 Hz in the horizontal (x, y) planeand 165 Hz in the vertical (z) direction; the trap frequencies for K are 1.37 times larger. The atoms
are transferred into a 3D optical lattice in three steps. We first turn on a retro-reflected verticalbeam in 150 ms to create a weak 1D lattice. In the second step, the ODT is ramped off in 100 ms sothat the two beams used for the ODT (which propagate along x and y) can be converted to lattice
beams by allowing them to be retro-reflected. The intensities along all three directions are thenramped to their final values in 100 ms. The three lattice beams are derived from a common laser
but individually frequency shifted to give a 160 MHz frequency offset between the x and y beams
and an 80 MHz frequency offset between the z beam and each of the horizontal beams. The x andy beams are elliptical with a 200 × 40 µm waist and are linearly polarized orthogonal in the x-y

78
plane; the z-beam has a circular waist of 250 µm and is linearly polarized along x. We calibrate thelattice strength using Rb atoms and then account for the differences in mass and ac polarizability
to get the lattice strength for KRb molecules. We use two different calibration methods, whichyield consistent results. The first method consists of fitting the time-dependent oscillations seen in
the diffraction pattern of a BEC after a Kapitza-Dirac scattering pulse [41], while in the second
method, the lattice intensity is modulated to drive atoms in a reasonably deep lattice to higher
lattice vibrational states. The values reported herein for the lattice depth in each direction are
expressed in units of the molecule recoil energy, Erec , and have an estimated 10% uncertainty.
Once the atoms are loaded in the 3D lattice, we ramp an external magnetic field across an
s-wave Feshbach resonance at 546.78 G to form loosely bound 40 K87 Rb molecules with an efficiency
of about 10%. With the Feshbach molecules at B = 545.8 G, where their binding energy is h×400
kHz, we use two-photon stimulated Raman adiabatic passage (STIRAP) to coherently transfer
the Feshbach molecules to the ro-vibrational ground state [19], with a typical one-way transfer
efficiency of 80%. All the molecules are in a single nuclear spin state in the rotational groundRbstate, |N = 0, mN = 0, mK
I = −4, mI = 1/2i, following the notation defined in [20]. We thenremove unpaired K atoms using a resonant light pulse on the S|F = 9/2, mF = −9/2i → P |F =11/2, mF = −11/2i transition, where S and P denote electronic ground and excited states of theatoms, while F and mF denote the total atomic spin and its projection. We remove unpaired
Rb atoms using a microwave adiabatic rapid passage from S|1, 1i −→ S|2, 2i followed by a lightpulse on the S|2, 2i −→ P |3, 3i transition. To measure the number of ground-state molecules in
the lattice, we reverse the STIRAP process and then image the resultant Feshbach molecules using
absorption of a probe beam that is tuned to the S|9/2, −9/2i → P |11/2, −11/2i transition for K
atoms.

5.2

Ground state molecule lifetime in the 3D lattice
Figure 5.1 shows a time-dependent evolution of the ground-state molecule population in the

3D lattice. In the first few hundreds of ms, the measured number of molecules exhibits relatively

79

35

3

number (10 )

30
25
20
15
10
5
0

0

5

10

15 20
time (s)25

30

Figure 5.1: Loss of ground-state KRb molecules as a function of time in a 3D lattice with depths of56, 56, and 70 Erec in x, y, and z, respectively, where Erec = ~2 k 2 /2m is the KRb recoil energy, k
is the magnitude of the lattice beam wave vector, and m the molecular mass. Neglecting the very
short time points (red solid circles), the number of molecules for times larger than 1 s (black solid
circles) are fit to a single exponential decay, yielding a 1/e lifetime of 16.3±1.5 s. Figure reproduced
from reference [4].

80

3number (10 )

15
10
5
0

0 Debye0.17 Debye
0 5 10 15 20 25 30
time (s)Figure 5.2: Comparison of the molecule lifetime in an isotropic lattice with a depth of 50 Erec ineach direction, with (blue open squares) and without (black squares) an applied electric field. The
1/e lifetime for molecules with an induced dipole moment of 0.17 Debye, 15±4 s, and the lifetime
from molecules without dipole-dipole interactions, 16.2±1.5 s, agree within uncertainty. Figure
reproduced from reference [4].

81
large variations in repeated iterations of the experiment and is consistent with some fast initial decay
in the number of trapped ground-state molecules. In all our measurements of ground-state molecules
in deep 3D lattices (for example, in the data for figure 5.3), we observe a similar feature. One
possible explanation for this fast decay is collisions of the ground-state molecules with impurities,
such as molecules in excited internal states that might be produced in the STIRAP process. Fitting
the data for times greater than 1 s to an exponential decay, which is consistent with a single-bodyloss mechanism, gives a 1/e lifetime of 16.3±1.5 s. This is much longer than previously measured
lifetimes of trapped ultracold polar molecules of about 1 s in an ODT [1] or in a 1D lattice [3]where bimolecular reactions are the dominant loss mechanism.
The long lifetime for ground-state molecules in a reasonably deep 3D lattice can be understood
simply from the fact that the optical lattice trap localizes the molecules and therefore prevents
bimolecular reactions. It was previously seen that an applied electric field, which induces a labframe dipole moment in the molecules, strongly increased the chemical reaction rate [2]. However,for molecules individually isolated in a 3D lattice, we expect no dependence of the lifetime of thetrapped molecules on the strength of an applied electric field. In figure 5.2, we show that indeed
we do not observe any decrease of the lifetime for polarized molecules with a dipole moment of 0.17
Debye.

5.3

2D to 3D
To understand what limits the lifetime of the molecules in the 3D lattice, we investigate its

dependence on the lattice strength as summarized in figure 5.3. First, we explore the transition
from a 2D lattice (an array of one-dimensional tubes) to a 3D lattice. The x and y lattice beams are
kept at an intensity corresponding to 56 Erec and the intensity of the lattice beam along z is varied
from 0 to 136 Erec . In this configuration, the harmonic trapping potential due to the gaussian
spatial profiles of the x and y beams can support the molecules against gravity even for the caseof no lattice in z. For a molecular gas confined in the tubes with no lattice in z, we find a lifetimeof ∼1 s, similar to that observed in the ODT [1] or in a 1D lattice [3]. However, as soon as a small

82

30

119

lifetime (s)2520

total Erec
200
277

753

b

a
c

15

d

10
5
0
2

4
6
19.0
2
total intensity (kW/cm )19.5

Figure 5.3: Measurement of the lifetime of KRb ground-state molecules in an optical lattice as theconfinement is changed from a 2D lattice to a 3D lattice. Black circles: The radial confinementby the x and y lattice beams is set at 56 Erec per beam, while the potential along z is varied
from 0 to 136 Erec (1 Erec corresponds to a lattice intensity I = 0.025 kW/cm2 ). The lifetime
reaches a maximum of 25±5 s when the z lattice depth is 34 Erec (point b). For higher latticeintensities, the lifetime decreases, and we find that the lifetime depends on the total intensity of
the light, rather than just on the lattice strength, which is consistent with loss due to off-resonant
light scattering. The open circles correspond to 3D lattices where the radial confinement was also
varied. The red squares correspond to lifetimes measured with an additional traveling-wave beamat 1064 nm illuminating the molecules in the 3D lattice. Point c (d) corresponds to the 3D latticeof point a with an intensity of 3.2 kW/cm2 (b with 3.7 kW/cm2 ) plus the additional beam intensityof 2.3 kW/cm2 (3.5 kW/cm2 ). Figure reproduced from reference [4].

83
lattice potential is added along z, the lifetime is dramatically increased, reaching 5 s at 12 Erec and
20 s at 17Erec (point a in figure 5.3). To verify that bimolecular reactions are the dominant loss
mechanism, we have checked that the lifetime in uncorrugated tubes decreases significantly (to 0.1s) when we apply an electric field (oriented along the tubes) that gives an induced dipole momentof 0.17 Debye. In addition, the fact that we can place an upper limit of 10% of the initial number
remaining at long times puts a limit on the contribution to our signal from tubes that are occupied
with only one molecule.
We consider a number of factors to understand the rapid suppression of loss as a function of
the lattice strength in ẑ. In general, Pauli blocking for identical fermions and dissipation blockade
effects (suppression of loss when the loss rate for particles on the same site is much larger than thetunneling rate [67]) can play a role in the lifetime of KRb molecules in an optical lattice. However,it should be noted that in the measurements reported here, the optical lattice is sparsely filled. For
our identical fermionic molecules, we expect Pauli blocking would give rise to a rapid suppressionof loss as a function of the lattice strength in z, with an even steeper function than we observe inthe measurements. Moreover, we find that for the case of a 5 Erec lattice in z, the lifetime of KRb
molecules in the tube does not change significantly in the presence of an applied electric field. Thisobserved insensitivity of the lifetime to a change of the on-site loss rate suggests that an incoherent
process, such as heating of the trapped gas, is responsible for limiting the lifetime in this regime of
weak lattice confinement. Instabilities in the optical phase of the lattice beams directly give rise totranslational noise of the lattice, which can promote molecules to higher bands of the lattice, where
they have increased mobility and could then collide with other molecules. Using a simple model
employing a single parameter representing a constant heating rate, and considering that moleculesin higher bands will collide on a timescale much shorter than the heating time, the theoretically
predicted trend of loss suppression (represented by two solid lines in figure 5.3, corresponding toheating rates of 1 Erec /s (66 nK/s) and 2 Erec /s, respectively) is consistent with the experimental
observation.
The lifetime reaches a maximum of 25±5 s when the lattice potentials are 56, 56, and 34 Erec84in x, y, and z, respectively. As the intensity is increased further, the lifetime starts to decrease,consistent with off-resonant photon scattering becoming the dominant loss mechanism. The richinternal state structure of molecules ensures that each off-resonant photon scattering event has ahigh probability of causing the loss of a molecule from the ground state. To explore this effect, weadded an additional traveling-wave beam with a wavelength of 1064 nm; this increases the photon
scattering rate without increasing the trap depth. In figure 5.3, we compare the measured lifetimes
with (red squares c and d) and without (black circles a and b) this extra light beam and observe asignificant reduction of the lifetime due to the additional light. We can extract the imaginary partof the polarizability of 40 K87 Rb molecules by fitting the lifetime as a function of the light intensity
to 1/(αI). Here, α is the imaginary part of the polarizability at 1064 nm, which we determine tobe (2.052 ± 0.009) × 10−12 MHz/(W/cm2 ); this is consistent with a theory estimate for KRb [68].
Light scattering has also been found to limit the lifetime of ground state Cs2 in a 3D lattice [66].

5.4

Feshbach molecules in 3D
We have also explored the lifetime of KRb Feshbach molecules in the 3D optical lattice. In

addition to intrinsic interest in these exotic weakly bound molecules, understanding their lifetime isimportant because the Feshbach molecules are the starting point for the coherent Raman transfer
that produces ground-state KRb molecules. The Feshbach molecules only exist near a magneticfield-tunable Feshbach scattering resonance and their size can be controlled with the detuning of
the magnetic field from the resonance. It has been shown that these weakly bound molecules can be
rapidly lost from an ODT due to collisions with atoms [69]. Our conversion of atoms to Feshbach
molecules typically has an efficiency of only 10 to 20%, and therefore removal of the remaining
atoms is important to prevent loss due to atom-molecule collisions. Even with removal of the Rb
atoms (as described above) and the RF transfer of the K atoms to a different hyperfine state, allpreviously measured lifetimes for KRb Feshbach molecules were less than 10 ms [18]. However,with the ability to create ground-state molecules, which do not scatter light that is resonant withthe single-atom transitions, we can more efficiently use light pulses to remove any residual atoms.

85
In particular, removal of K atoms, where light on the cycling transition is absorbed by the Feshbach
molecules as well as by the atoms, is facilitated by hiding the molecules in the ground state. A
few ms after this atom removal, we reverse the STIRAP process and re-create a Feshbach moleculegas. For the case of molecules in the ODT (no lattice), we find that this extends the lifetime of the
Feshbach molecules to 150 ms. Therefore, we can conclude that previous lifetime measurements
were likely limited by collisions with residual atoms.
When we perform the procedure described above for KRb molecules in an optical lattice, wefind that the purified gas of Feshbach molecules can have lifetimes as long as 10 s. We explore thelifetime of the Feshbach molecules in the 3D lattice as a function of the magnetic-field detuningfrom the resonance (B0 = 546.78 G), where varying the magnetic field changes the binding energy
and the size of the Feshbach molecules. We start by forming a purified sample of the Feshbach
molecules in a strong 3D lattice with an intensity of 50 Erec per beam at a magnetic field B =545.8 G. The magnetic field is then ramped to its final value in 1 ms. At the end of the hold timein the 3D lattice, B is ramped back to 545.8 G where we image the molecules.
The Feshbach molecule lifetime measurements are shown in figure 5.4. Above the Feshbachresonance, the lattice potential allows for the existence of confinement-induced molecules that do
not exist in free space [70]. We find that confinement-induced molecules have a lifetime (25 s)that is comparable to that for K or Rb atoms in the same trap. Below the Feshbach resonance,
the molecule lifetime decreases quickly when the magnetic field is ramped to lower values. Several
Gauss away from the resonance, the Feshbach molecule lifetime is reduced to approximately 1 s,
which is still significantly longer than in the ODT. Overall, these results represent a two-orders-ofmagnitude improvement in the lifetime of Feshbach molecules compared to a previous measurementof KRb molecules in an optical lattice [71].
To understand the dependence of the lifetime on the magnetic field as shown in figure 5.4, we
can assume that the lifetime is limited by off-resonant photon scattering from the lattice light andconsider two limiting cases. For B  B0 , the photon scattering limit is simply that for free atoms
Γatom ; for B  B0 , we have a higher photon scattering rate Γmolecule due to a larger wavefunction

86

100
8

atom

545.8G

lifetime (s)lifetime (s)

6

10

4

1

543.18G

2

0.1
0.01
1

ODT

10
lattice depth per beam (Erec)1001
543 544 545 546 547 548 549
magnetic field B (Gauss)
Figure 5.4: The lifetime of Feshbach molecules and confinement-induced molecules measured as afunction of the B-field. A purified sample of Feshbach molecules is held in an isotropic 3D opticallattice (50 Erec per beam, 20 kHz trap frequency). Near the Feshbach resonance, the loss rate dueto photon scattering can be modeled (solid line) as a weighted sum of the free atom loss rate Γatom
and a higher loss rate for tightly bound molecules Γmolecule . The grey shaded area indicates the
single atom lifetime, and its uncertainty, measured for the same experimental conditions. Inset:
The lifetime of Feshbach molecules in a 3D lattice as a function of the trap intensity, at 545.8 G(blue stars) and 543.18 G (green diamonds). A small change in the intensity initially increases the
lifetime from 150 ms in the ODT to 9 s in the 3D lattice at 5 Erec per beam (blue stars). The
lifetime then decreases as the intensity is further increased, consistent with a lifetime limited byphoton scattering. For the more deeply bound Feshbach molecules (green diamonds), the lifetimeis shorter. A fit to the data gives a scattering rate of 15.9±1.6 MHz/(W/cm2 ) at 545.8 G and 30±3MHz/(W/cm2 ) at 543.18 G. Figure reproduced from reference [4].

87
overlap with electronically excited molecules and therefore an increased photon scattering rate [72].
In a two-channel model of the Feshbach resonance [73], the Feshbach molecule wavefunction canbe written as an amplitude Z 1/2 times the bare “closed-channel” molecule wavefunction plus anamplitude (1 − Z)1/2 times the “open channel” wavefunction that describes the scattering stateof two free atoms. We then take the total photon scattering rate to be given by ZΓmolecule +(1 − Z)Γatom . With pairs of atoms confined in an optical trap with a known depth, Z can becalculated straightforwardly with a coupled-channel theory [73, 46]. Using the measured loss rates
for the limiting cases, Γatom and Γmolecule , this simple theory (solid line in figure 5.4) without
any additional adjustable parameters describes very well the experimental results (filled circles)
across the entire range of B. We note that the rate of atom-molecule collisions properties has been
analyzed in a similar way [71][74].
The assumption that the lifetime of the purified gas of Feshbach molecules in a 3D lattice islimited by only the photon scattering can be checked by varying the lattice beam intensity. This isshown in the inset of figure 5.4 for two values of B; these magnetic-field values are indicated in themain part of figure 5.4 with a blue star and a green diamond. In the inset, the lifetime is plotted
as a function of the intensity per lattice beam in an isotropic lattice, expressed in units of Erec .
Similar to the case of ground-state molecules, we observe a rapid initial increase in the lifetime
going from no lattice (only the ODT) to a weak lattice. Following this initial rise, we observea decrease in the Feshbach molecule lifetime as the lattice intensity is increased, consistent withloss due to off-resonant scattering of the lattice light. The dashed and solid lines are fits to one
over a constant times the total light intensity, and we extract an imaginary part of the Feshbach
molecule’s ac polarizability at 1064 nm of 15.9±1.6 MHz/(W/cm2 ) for B = 545.8 G and 30±3MHz/(W/cm2 ) for B = 543.18 G.

5.5

100% conversion of atoms into Feshbach molecules
In a simple model of the conversion of atoms to Feshbach molecules in a 3D lattice, one could

assume 100% conversion efficiency for individual lattice sites that are occupied by exactly one K

88
atom and one Rb atom, with the overall conversion efficiency then determined by the filling of thelattice and the number of such “pre-formed pairs” of atoms. The ability to prepare a purified sampleof Feshbach molecules in the 3D lattice after a round-trip STIRAP sequence gives the opportunity totest the assumption of 100% conversion between pre-formed pairs of atoms and molecules. Startingwith a purified sample of Feshbach molecules prepared with round-trip STIRAP and atom removal
as discussed above, we dissociate the molecules in the lattice by ramping the magnetic field above
the Feshbach resonance to 548.97 G. This should ideally produce only pre-formed pairs of atoms.
We then ramp the magnetic field back down to 545.8 G and measure the molecular conversion
efficiency. The result is (87 ± 13)%, where the uncertainty is dominated by fluctuations in STIRAP
efficiency in successive runs of the experiment. This high efficiency is far above the maximum of
25% observed in an ODT [75]; this indicates that optimizing the loading procedure in order to havea larger number of sites with exactly one Rb and one K atom would optimize the efficiency of theconversion from atoms to Feshbach molecules. For heteronuclear Bose-Fermi mixtures, optimizing
the number of pre-formed of atom pairs in a lattice remains a challenge. Recent progress in thisdirection includes the characterization of a dual Bose-Fermi Mott insulator with two isotopes ofYb [76], as well as proposals to use interaction effects to optimize the lattice loading [77, 78, 79].

Chapter 6

Anisotropic Polarizability of KRb

Molecules have complex internal structure with many more internal degrees of freedom thanatoms. In particular, the rotational degree of freedom provides a set of long-lived excited states thatare easily coupled to the ground state with microwaves. Because of the accessible frequency in the
microwave domain and narrow intrinsic linewidth, the transition between rotational states could be
ideal for use as a spectroscopic probe of the system. For example, such a narrow transition could
be used to measure small energy shifts due to dipolar interactions [80]. In addition, dipole-dipole
interactions can be realized without applying a DC electric field but instead by directly coupling
the two lowest rotational states with a microwave field. Within the rotating frame of the microwave
transition, there is a strong dipolar interaction effect, which can be used to model novel quantum
many-body Hamiltonians [13]. Additionally, the microwave detuning and power can be varied tomodify the collision dynamics [56]. This has been proposed as a way to achieve a topologicalsuperfluid of paired fermionic polar molecules [81].
A prerequisite for such experiments is long coherence times for the interaction between a microwave field and the rotational states. However, for molecules confined in an optical dipole trap,
the difference in AC (or dynamic) polarizability between different rotational states must be considered [25]. A difference in polarizability leads to different trap frequencies and spatially dependent
variations in the rotational transition frequency, which can lead to dephasing and decoherence [82].
For atoms, if the trapping light is far detuned compared to the energy splitting between two states,
their AC polarizabilities will be nearly equal. In contrast, for molecules, the ground and the first90
rotationally excited state with the same angular momentum projection onto the quantization axis
have different parity and will therefore couple to different electronic excited states. This can resultin AC polarizabilities that differ by more than 30%, even when the light is far detuned [83].
In atomic systems, it is possible to adjust the wavelength of the trapping light such that thepolarizabilities of two states of interest (often clock states in alkaline earth atoms) are the same[82]. Although, in principle, one could find such a “magic” wavelength trap for molecules [84], thelarge number of additional states from rotation and vibration makes it difficult to find a suitablewavelength that is sufficiently detuned to avoid off-resonant light scattering. However, molecules
provide a different way to adjust the polarizability; the AC polarizability of a molecule depends on
the relative orientation of the molecule and the polarization of the trapping light [83].
In this Chapter, we explore the interaction between the trapping light and the molecules by
examining the real part of the AC polarizability. In particular, we determine how the polarizabilityof the rotationally excited states depends on the relative orientation of the quantization axis, ẑ(which in these experiments is given by a magnetic field of 545.9 G) and the polarization of thetrapping light. There is a “magic” angle where the polarizability of the |N = 0, mN = 0i (where
N is the rotation quantum number, and mN is its projection onto the quantization axis) and|N = 1, mN = 0i states match, making the AC Stark shift the same for these two internal state of
the molecule. This “magic” angle is expected at cos2 θ = 1/3, or θ ≈ 54 degrees [85]. We observe
this magic angle both through measurements of the AC polarizability and through measurementsof the coherence time for driving the |0, 0i to |1, 0i transition. We compare our experimental results
to the analytic results of an approximate Hamiltonian, which mixes the three projections of thefirst rotationally excited state. Much of this chapter is reproduced from reference [5] of which I am
an author.

6.1

Anisotropic polarizability
Molecules have a definite orientation that strongly affects their polarizability. In the frameof a diatomic molecule such as KRb, the polarizability along the internuclear axis is much stronger

91
than the polarizability perpendicular to the internuclear axis. This has been successfully exploited
to align molecules with intense laser pulses in order to study the stereodynamics of chemical reactions [86, 87, 88]. The N = 0 rotational ground state is spherically symmetric and therefore
its AC polarizability has no dependence on the relative orientation of the laser polarization andthe quantization axis. However, the rotational wavefunction of the three projections (mN ) of theN = 1 rotationally excited state has a well-defined orientation relative to the quantization axis. For
example, the |1, 0i state corresponds to a pz orbital that is aligned along the ẑ-axis, and therefore
the polarizability is the largest when the polarization of the AC field is along the ẑ-axis.
The degeneracy of the three projections of the N=1 state is broken by the hyperfine interaction, specifically, the interaction between the nuclear quadrupole moment and the rotation of
the molecule [20, 89]. In the rotationally excited states, this coupling gives a specific nuclear spinstate up to a 5% admixture of other hyperfine states; however, we will ignore these contributions
Rb
and will work only with states whose dominant hyperfine character is mK
I = −4, mI = 1/2 unlessRbotherwise noted, where mK
I and mI are the nuclear spin projections onto the ẑ axis. At a field of

545.9 G and no light, we find that the |1, 1i state is 58 kHz above the |1, −1i state and the |1, 0i
state is 268 kHz above the |1, 1i state (figure 6.1b). The splitting between the mN = ± 1 states is of
the same order of magnitude as the AC Stark shifts induced by the trapping lasers. Consequently,
we calculate the angle dependence of the dynamic polarizability of the three rotationally excitedstates with perturbation theory that includes the light-induced couplings between the bare states(no light).
Using the general formalism of Ref. [90], the complex dynamic polarizability for the |0, 0i
state is given by1
α|0,0i = (αk + 2α⊥ )
3

(6.1)

where the “reduced” polarizabilities αk and α⊥ are the parallel and perpendicular (with respect tothe intermolecular axis) polarizabilities that describe the averaged contributions from ro-vibrational
states of all electronically excited 1 Σ+ and 1 Π potentials. Because of its spherically symmetric

92

𝑘𝐿

(a)

𝑥

𝐵

θ 𝑧

𝑦
(b)
|1,-1>

𝑘𝐿

(c)

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

Figure 6.1: (a) Experimental schematic. The lattice beam propagates along x̂, the magnetic field
points in the ẑ direction, and the polarization of the lattice light makes an angle θ with the magnetic
field in the y-z plane. (b) Schematic of rotational energy states. The degeneracy of the N = 1
level is split in a magnetic field. (c) A sketch of the optical dipole potentials for the |0, 0i and|1, 0i states. A Gaussian is overlaid to show the distribution of the molecular cloud in the trap.
When the two states are connected by a 2.22 GHz microwave drive, there is effectively a spatiallyvarying detuning across the cloud due to the difference in the trap potentials. Figure reproduced
from reference [5].

93
rotational wavefunction, the polarizability of the |0, 0i state is independent of the angle θ betweenẑ and the polarization of the 1D optical lattice used to trap the molecules (see figure 6.1a).
For the |1, 0i and |1, ±1i states, the dressed or mixed polarizabilities at laser intensity I are
E (I)−Ej (0), where h is Planck’sIgiven by the total Stark shift divided by the total intensity, αj = − jconstant, and Ej , with j = 1, 2, 3, are the eigenvalues of the 3 × 3 Hamiltonian|1, 0i|1, −1i|1, 1i−α12 I−α13 I −α11 I + 1





H =  −α12 I,−α22 I + 2−α23 I−α13 I−α23 I−α33 I + 3
where
3αk + 2α⊥αk + 4α⊥sin2 θ +cos2 θ
55
2αk + 3α⊥αk + 4α⊥= α33 =
sin2 θ +cos2 θ
55√ αk − α⊥= −α13 = 2
sin θ cos θ51=(α⊥ − αk ) sin2 θ ,
5

α11 =
α22
α12
α23

and 1 , 2 , and 3 are the energies for states |1, 0i, |1, 1i and |1, −1i, respectively, at I = 0.
We measure the AC polarizability of the molecules in a one-dimensional optical lattice witha peak intensity of 2.3 kW/cm2 and a wavelength of λ = 1064 nm [3]. Using a single microwave
pulse, we can selectively transfer population from the rotational ground state, |0, 0i, to any ofthe projections of the rotational excited states with near 100% efficiency [20]. We measure thetrap depth, UKRb , from which we can extract the AC polarizability, by measuring the parametricheating resonance (see figure 6.2). We modulate the intensity of the optical lattice for 4 ms with
an amplitude of approximately 10% of the total depth. The modulation frequency is varied tofind the resonant frequency where molecules are excited from the lowest band of the lattice to thesecond excited band. In the deep lattice limit, this resonant frequency is twice the trap frequency.
However, we operate our lattice in an intermediate intensity regime where the relationship between

94
the trap depth and the resonant frequency must be extracted from a numerical solution to thelattice potential. We determine the polarizability without having to characterize the optical beam
parameters such as power or beam waist by comparing the molecular result to a similar measurementof the trap depth for Rb. The polarizability of KRb is then given by αKRb = αRb UKRb /URb ,where αRb /h = 3.242 × 10−5 MHz/(W/cm2 ) at 1064 nm [91].
In figure 6.3, we show the polarizabilities of the N = 0 ground state as well as the threeprojections of the N = 1 rotationally excited state as a function of θ, which is varied by adjusting ahalf-waveplate in the lattice beam. Although the rotation of the half-waveplate allows us to chooseθ with a precision better than one degree, the absolute alignment relative to the magnetic field
has an estimated systematic uncertainty of ±3 degrees. We fit Eqn. 6.1 and the polarizabilities
from the eigenenergies of H to the experimental data with three free parameters, θm , αk and α⊥ .
From the best fit, we determine that the “magic” angle θm = 48(4) degrees, αk /h = 10.0(3) × 10−5
MHz/(W/cm2 ), and α⊥ /h = 3.3(1) × 10−5 MHz/(W/cm2 ).
Hyperfine couplings between the |1, 0i and |1, ±1i states result in a small change of the magic
angle from the expected value of 54 degrees. For the polarizabilities and intensity given above wewould expect θm = 52 degrees, which agrees with our fit to within the error. For comparison to themeasurement, theoretical values of the polarizabilities αk and α⊥ are obtained using non-relativistic
potentials and dipole moments described in Ref. [83], and are h × 12.2 × 10−5 MHz/(W/cm2 ) andh × 2.01 × 10−5 MHz/(W/cm2 ), respectively. In principle, αk and α⊥ depend on the ro-vibrational
state of the molecule, however, we find in the calculation that for small N , the polarizabilities αkand α⊥ are independent of N to better than 0.01%.
Because the bare states are mixed by the lattice light, the dressed state polarizabilities dependon the intensity of the light. This effect is strongest for the |1, ±1i states, where the energy splittingof the bare states is relatively small. However, we experimentally verify that at the intensity usedhere this effect is small. For example, for a 50% increase in intensity with θ = 93 degrees, we see
only a 3.9(8)% decrease in the polarizability of the |1, 1i state.

95

18
16

size (arb.)

14
12
10
8
6
4
2

35

40

45

50

55

60

modulation frequency (kHz)

Figure 6.2: Parametric heating resonances for θ = 57 degrees. The y-axis shows the rms size in x̂
of an expanded gas of KRb (Rb) after 3 ms (21 ms) of time of flight. The curves have been offset
vertically for clarity. Using Gaussian fits (lines) we determine the center of the parametric heating
resonances for (from bottom to top) Rb (magenta diamonds), and KRb in the |0, 0i (black squares),|1, 0i (blue circles), |1, 1i (red inverted triangles), and |1, −1i (green triangles) states. The resonant
frequency allows us to extract the trap depth for each state. Figure reproduced from reference [5].

96

 (10-5MHz/(W/cm2))

8.0
7.5|1,-1

7.0
6.5
6.0

|0,0

5.5

|1,1

5.0
4.5
4.0

|1,0
0

15

30

45

60

75

90

measured angle (degrees)
Figure 6.3: The AC polarizability of KRb at 1064 nm for the |0, 0i (black squares), |1, 0i (bluecircles), |1, 1i (red inverted triangles), and |1, −1i (green triangles) states. Error bars are fromthe Gaussian fit uncertainty in the center of the parametric heating resonances and correspond to±1 standard deviation. Theory lines are a simultaneous fit with Eqn. 6.1 and the polarizabilities
from the solution of H with three free parameters θm , α⊥ , and αk . Open circles represent a separate measurement where the polarizability is extracted from the shift in the microwave transition
frequency. Figure reproduced from reference [5].

97

6.2

A more complete model
As was shown in Ref. [20], the nuclear quadrupole moment couples to the rotation of themolecule and mixes rotationally excited states with different hyperfine character. It is precisely
this coupling that breaks the degeneracy of the three projections of the N = 1 state. These energy
shifts move the magic angle from its expected value of 54 degrees to 52 degrees. However in theprevious section, other than to include this broken degeneracy, we completely ignored the couplings
between different hyperfine states. To justify this assumption, we now compare the results fromthe approximate 3 × 3 Hamiltonian in the previous section to an “exact” model, which couples thenuclear spin to the rotation of the molecule.
We extend the ideas of mixing of rotational levels due a static external electric field of Ref. [83]to include mixing of rotational-hyperfine levels due to the intrinsic nuclear electric-quadrupole
interactions introduced in Ref. [20]. First, we construct the full Hamiltonian for the rotationalRb
hyperfine levels labeled by |N, mN , mK
I , mI i, where N is the rotation quantum number, mN is

its projection onto the quantization axis, mK
I is the projection of the K nuclear spin onto themagnetic-field axis, and mRb
is the projection of the Rb nuclear spin onto the magnetic-fieldI~ for atom a = K or
axis. The Hamiltonian includes the nuclear Zeeman interaction, −ga µN I~a B
Rb, where ga is nuclear g-factor and µN is the nuclear magneton of atom a. It also includes thenuclear quadrupole interaction, proportional to (eqQ)a /(Ia (Ia −1))C2 (θφ)·T2 (Ia , Ia ), with coupling
constants (eqQ)a for each atom a that couples its nuclear spin to rotational states. Here, Clm (θφ)
is a spherical harmonic and T2m (Ia , Ia ) is a rank-2 tensor created from the spin Ia . Finally, we
include a polarizability interaction Hamiltonian, −(αk Ok + α⊥ O⊥ )I, with strengths αk and α⊥ ,tensor operators Ok and O⊥ that depend on light polarization and rotational angular momentum~ , and I is the peak laser intensity of the trapping light.
N
Rb
This Hamiltonian couples (1 + 3) × 9 × 4 = 144 channels |N, mN , mK
I , mI i and has fourparameters: the quadrupole interaction constants for each of the two atoms and the “reduced”
polarizabilities αk and α⊥ , which are the vibrationally-averaged v = 0 parallel and perpendicular

|ji|1, 0, −4, 1/2i
|1, 1, −4, 1/2i
|1, −1, −4, 1/2i

Frequency (kHz)2 228 110(1)2 227 842(1)2 227 784(1)98Table 6.1: The experimentally measured rotational-hyperfine transition frequencies of the lowest
vibrational level of the X1 Σ+ potential of KRb at zero laser intensity (i.e. without trapping light)
and a bias magnetic field with strength B = 545.9 G. Transitions start at the |N = 0, mN =Rb0, mK
I = −4, mI = 1/2i state and go to three hyperfine states |ji within the N = 1 manifold.

polarizabilities that include contributions from all excited 1 Σ+ and 1 Π potentials, respectively.
We find transition energies by diagonalizing the Hamiltonian and analyzing its eigenfunctions toconnect to the states that have been observed experimentally. Eigenstates are identified by thechannel state with the largest contribution. Typically this contribution is more than 90% and weRb
are justified in labeling eigenstates by one |N, mN , mK
I , mI i.

The two quadrupole parameters were estimated in Ref. [20] based on measurements of transition energies between sub-levels of the N = 0 and N = 1 states in the optical dipole trap. However
when these measurements were made, the state-dependent polarizability and the effects of the trapping potential on the transition frequency were not yet understood, and the measurements for thedifferent states were not taken at a constant intensity. Here we improve these constants using newmeasurements of the transition energies for three hyperfine levels of the N = 1 rotational state.
We remove the effects of the trapping potential by measuring the transition energy directly after
turning off the trapping light. The transition energies are given in Table 6.2.
We optimize the quadrupole constants to fit the experimental energies of Table 6.2. We findthat Be /h=1.1139514(5) GHz, (eqQ)K /h=0.452(9) MHz, and (eqQ)Rb /h = −1.308(9) MHz. The
first two values are in good agreement with the previous estimate of [20]. The (eqQ)Rb coupling
constant has changed by ≈ 7%.
The polarizability of eigenstate j with energy Ej (I) is defined as the total Stark shift divided
E (I)−Ej (0).
Iby the total intensity, αj = − j

For comparison to the approximate 3 × 3 Hamiltonian,

we use the fitted values of αk and α⊥ in the “exact” model. Figure 6.4 shows the polarizability as a99function of θ based on the “exact” Hamiltonian (dashed lines), and the approximate Hamiltonian(solid lines) for the four states studied here and laser intensity I=2.35 kW/cm2 . From this figure weconclude that the agreement between the two models is good and use of the simplified approximate
Hamiltonian is justified.

6.3

Shift in transition frequency
We can also use an independent measurement of the shift in the microwave transition fre-quency to determine the polarizability as a function of θ. The difference in polarizability between
the |0, 0i and |1, 0i states results in different trap depths for the two states, resulting in a shift inthe microwave transition frequency:f = f0 + (α|0,0i − α|1,0i )I0 /h +ω|1,0i ω|0,0i
−
+ ∆f
4π
4π

(6.2)

where f is the measured transition frequency, f0 is the transition frequency from |0, 0i to |1, 0iqwith I = 0 (measured after release from the lattice), ω/(4π) = 2λαI
2 m is the zero point energy inthe lattice divided by h, m is the mass, ∆f =α|1,0i −α|0,0i
kB T /h is the shift in the center of the
α|0,0itransition frequency caused by the spatially dependent detuning, kB is Boltzmann’s constant, andT = 400 nK is the temperature. The transition frequency is only sensitive to the polarizability
difference between the |0, 0i and |1, 0i states, but can be compared with the polarizability fromthe direct measurement of the trap depth by fixing α|0,0i and then solving Eqn. 6.2 for α|1,0i . The
resultant polarizabilities are shown in open symbols in figure 6.3. We see good agreement between
the two methods.

6.4

Rotational dephasing
To study the effect of the magic angle, we measure the rotational excitation coherence time asa function of angle, and the results are shown in figure 6.5. We measure the coherence time between
Rb
Rb
K
the |N = 0, mN = 0, mK
I = −4, mI = 1/2i and |N = 1, mN = 0, mI = −3, mI = 1/2i stateswith Ramsey spectroscopy. The probe pulse is 40 µs long, and the probe frequency is detuned from (10-5MHz/(W/cm2))

100

7.5
7.0

|1,-1

6.5
6.0
|0,0
5.5
|1,1
|1,0

5.0
4.5

0

15

30

45

60

75

90

angle (degrees)
Figure 6.4: The angle-dependent polarizability for the “exact” model (dashed line) and the approximate 3 × 3 Hamiltonian. Figure reproduced from reference [5].

101
the resonance by 3 to 12 kHz. We fit the Ramsey oscillation as a function of time to a damped sine
wave to extract the coherence time (see inset of figure 6.5).
The coherence time due to the mismatch in polarizability should scale as one over the difference in ∂E/∂I, which because of the intensity dependence of the polarizability is not the sameas the difference in polarizability. Technically, the ∆f term in Eqn. 6.2 should also use the “local
polarizability”, ∂E/∂I, instead of α because the trapped gas experiences only a small range ofintensities. However, in Eqn.6.2 the difference between α and ∂E/∂I gives a small correction to aterm that accounts for less than 10% of the total frequency shift, and is therefore negligible. On
the other hand, because the coherence time depends critically on this polarizability difference, the“local polarizability” must be used.
We fit to the data with a simple model that includes the angular dependence:s2∂E|1,0i /∂I − ∂E|0,0i /∂I2
τ = 1/ (1/T2 ) +∆E/~∂E|0,0i /∂I(6.3)

where T2 is the coherence time from all sources of decoherence other than the polarizability, ∆E is
the spread of energy across the cloud in the |0, 0i state, and |∂E|0,0i /∂I − ∂E|1,0i /∂I| depends upon
θ, αk , α⊥ , I, 1 , 2 , and 3 . Using the measured values of I, 1 , 2 , and 3 and the fitted valuesof αk and α⊥ from figure 6.3, we use Eqn. 6.3 and obtain T2 = 1.5(2) ms, ∆E/~ = 3.3(4) × 104
s−1 , and θm = 46.5(5) degrees. The expected value of θm for our values of I, α⊥ , and αk is 48
degrees, which agrees with the fitted value to within the systematic error. Note that the expectedvalue of θ is different when probing the difference in ∂E/∂I rather than α. Possible sources of T2
range from technical noise to resonant dipole-dipole interactions [80], and further study is required
to understand this limit to the coherence time.

6.5

Conclusion
In conclusion, we have measured the angle dependence of the AC polarizability of ultracold

KRb and observed the “magic” angle where the polarizability of the |0, 0i and |1, 0i states match.
Using this angle, we are able to increase the coherence time between the two rotation states by an

102

fraction in |0,0>

coherence time (ms)

1.0

1.5

0.5

0.0
0.0

1.0

0.6

1.2

75

90

time (ms)

0.5

0.0

0

15

30

45

60

measured angle (degrees)
Figure 6.5: The Ramsey coherence time measured in the one-dimensional optical lattice as a function of angle. A sharp increase in coherence time is observed at the “magic” angle where thepolarizabilities of the |0, 0i and |1, 0i states are matched. Inset: A Ramsey oscillation fit to adamped sine wave to extract the coherence time at θ = 51 degrees. Figure reproduced fromreference [5].

103order of magnitude. This opens the way for the use of the rotational state for precision spectroscopy
as well as the modification of the collision dynamics.

Chapter 7

Conclusions and Outlook

7.1

Conclusions
In this thesis I presented our studies of ultracold KRb molecules that have led to complete

understanding of, and control over, both the single-particle physics as well as the two-body collisional physics of ultracold polar KRb molecules. The first step towards this goal was presented in
the thesis of Josh Zirbel [40] with the creation of ultracold Feshbach molecules in a single quantum state. This was followed by the coherent transfer of the Feshbach molecules to the rovibronic
ground state as described in the thesis of Kang-Kuen Ni [23]. Combined with the transfer between hyperfine states [20], this represents full control over all internal states of the molecule. We
can populate the absolute rovibronic ground state and coherently transfer to other rotational or
hyperfine states of our choosing with over 90% efficiency. With a single internal quantum state
populated, we observed bimolecular chemical reactions at ultracold temperatures [1]. Not only is
it striking that chemical reactions can still proceed at temperatures of a few hundred nK, but thechemical reaction rate is set entirely by the long-range physics, which makes it universal (in thesense that the rate does not depend on the specific short-range potential of KRb). Additionally
we showed that the chemical reaction rate was strongly influenced by the long-range dipole-dipole
interactions [2]. With the application of an external field to polarize the molecules, we could change
the chemical reaction rate by almost two orders of magnitude.
Unfortunately, applying an electric field can only increase the chemical reaction when themolecules allowed to collide in three dimensions. By loading the molecules into a one-dimensional

105
optical lattice, we confine the molecules in 2D and gain full control over the chemical reaction rate.
By changing the quantum states of the molecules, either in their internal states or their externalmotional state in the lattice, we are able to change the orientation of the collision, which changes
the chemical reaction rate. We showed that we could exploit this quantum stereodynamics tosuppress the chemical reaction rate in 2D if we put all the molecules in the lowest motional stateof the lattice as well as a single internal quantum state [3]. This allows for the creation of a dipolargas with a lifetime over a second.
The next step in demonstrating full control over the molecules was to further constrain theirmotion by creating molecules in a three dimensional optical lattice. Here, in the deep lattice limit
we were able to show that the chemical reactions between molecules could be completely suppressed[4]. With each molecule isolated on its own lattice site and tunneling between sites turned off, themolecules can no longer collide and therefore can’t chemically react. Although this may not seem
like a very surprising result, it demonstrates an important step towards realizing many theoretical
proposals to use polar molecules in a lattice to study exotic phenomena such as quantum magnetism[13] and fractional quantum hall effect [81]. We showed that the lifetime of the gas was limited
only by the off-resonant light scattering from our lattice and lifetimes of up to 25 s are possible.
We also demonstrated long-lived Feshbach molecules with a lifetime limited only by off-resonant
light scattering. Additionally, we showed that a pair of atoms on a single site in the 3D latticemake a molecule with unit efficiency. This suggests a way forward to get a lattice with high fillingof molecules.
Finally, we studied the AC polarizability of the molecule. We showed that the polarizabilities for the ground state and the three projections of the first rotationally excited state aredifferent. Moreover, the polarizabilities of the rotationally excited states depend on the angle between the polarization of the trapping light and the quantization axis. By scanning this angle weshowed that we could find a “magic” angle where the polarizabilities of the |N = 0, mN = 0i and|N = 1, mN = 0i states match. Here we showed an increase in the coherence time when the twopolarizabilities match. This understanding of the polarizability, and the ability to get rid of the106dephasing from the spatially dependent detunings between rotational states allows us to prepare
arbitrary superpositions of rotational states and preform precision spectroscopy on the rotationaltransition.

7.2

Future work and outlook
I see a very bright future for KRb, and many exciting things are going on in the lab as I write

this thesis. With the chemical reactions suppressed and the single-particle physics understood, it is
now just about time to turn our attention to studying many-body physics. This is extremely exciting
because this is many-body physics arising from long-range and potentially strong interactions. But
rather than give a review of the many theoretical proposals on what to do with polar molecules, I
would like to focus this last section of my thesis on the shorter term goals in the lab, which focus
on getting colder, denser, and more strongly interacting gases, along with suggesting some of thefirst places we plan to look for signals of interesting dipolar many-body physics.

7.2.1

Evaporation

One very exciting area is direct evaporation of dipolar fermions. Although we currently create
the world’s densest and coldest gas of ground-state polar molecules, we are still unable to produce
a quantum degenerate gas. Currently there is a limit to how cold we can create our molecules
because of the atoms from which we make the molecules [75]. Although there are many parameters
to vary, and our optimization is by no means exhaustive, the best conditions we have ever achieved
have T /TF = 1. We would like to go much colder and into the quantum degenerate regime.
Because we are unable to create a degenerate gas of molecules directly from atoms, we must
look to increase the phase space density of the molecules after their creation. One promising route tohigher phase space density is evaporative cooling. Evaporation needs two things: rethermalization
and a way to selectively remove the hottest particles. Additionally, this must happen on a timescale
faster than the heating rate or loss rate. This makes KRb seem particularly challenging because
it is chemically reactive, and the timescale for the loss due to chemical reactions will limit the107efficiency of evaporative cooling. We can suppress this loss by preparing our gas in a single quantum
state, preventing s-wave collision that would otherwise lead to extremely fast loss due to chemical
reactions.
With a spin polarized gas of fermions, we must rely on dipolar interactions to rethermalize
the gas. As shown in chapter 4, we must go into a quasi-2D geometry to have both strong dipolarinteractions and a long lifetime. Although this evaporation might sound difficult, we have someencouraging theoretical results from Goulven Quéméner that show the elastic to inelastic collision
ratio is favorable for evaporation. The elastic and inelastic rate coefficients are shown as a functionof collision energy in figure 7.1. At our starting temperature of 200 nK, we would have a good-tobad collision ratio over 100. Furthermore, the ratio increases as the temperature goes down because
the inelastic rate scales linearly with temperature.
To remove the hottest particles from the trap, we tilt the trap using an electric-field gradientsimilar to how a magnetic-field gradient was used in reference [92]. The gradient points along
one of the radial directions of the 1D optical lattice and tilts the molecules out of the pancakeshaped traps. Although we are still in the process of designing the best way to get repeatable field
gradients, we have seen some preliminary signs of evaporation, which are shown in figure 7.2. Here
we held the molecules in a combined lattice plus optical dipole trap. We ramp on an electric fieldof 4 kV/cm in 100 ms. This induces a dipole moment of 0.158 D and we estimate the electric-field
gradient at the molecules to be 125 V/cm2 . Once the field was ramped on, we ramped down the
optical trap in 100 ms. This lowers the trap depth and allows the electric-field gradient to “tilt”
the lattice and spill the most energetic molecules out the side. The trap was then ramped back to
its original strength in 100 ms and the electric field was ramped off in another 100 ms. We plot
the number of molecules remaining at the end of this sequence versus the final temperature (asmeasured in time of flight) on a log-log plot for various ramp depths (figure 7.2). We fit a line tothe data and see a slope of 1.2(1), which for fermions in 2D corresponds to an increase in phasespace density. This data must be repeated and optimized to verify that the gas is in equilibrium
at the end of the evaporation ramp and evaporation is indeed taking place. But, both theoretically108

Figure 7.1: The loss rate coefficients for both reactive (inelastic) and elastic collisions as a functionof the collision energy for an induced dipole moment of 0.2 Debye, a quasi 2D trap with a tighttrapping frequency of 23 kHz, and indistinguishable molecules all in the lowest vibrational state of
the lattice. The elastic to inelastic ratio (the ration of good to bad collisions) is over 100 in theentire region of interest. Figure courtesy of G. Quéméner.

109
and experimentally evaporation is looking promising.

7.2.2

Enhanced molecular formation in a 3D lattice

It was demonstrated in Chapter 5 that any lattice site with one K and one Rb atom will be
converted into a Feshbach molecule with 100% efficiency. This suggests an appealing way to createa dense gas of molecules with low entropy. If we were able to start with a perfect Mott insulator of
Rb with one Rb atom per lattice site, and have a band insulator of K with one K atom per lattice
site, then after ramping through the Feshbach resonance we should create a gas of molecules with
one molecule per lattice site, on every lattice site, all in the lowest band.
Many studies have been done exploring the interaction between K and Rb on a lattice [93,94, 95]. These studies all treated the K atoms as an impurity, which because of the interspecies
attraction, would localize the Rb and prevent a proper Mott insulator from forming. This suggeststhat an effective way to create a dual Mott/band insulator of K and Rb is to tune the interspecies
scattering length to zero. This was demonstrated in the realization of a dual Mott insulator ofbosonic and fermionic Yb [76].
We are currently working on simplifying the 3D lattice setup to minimize the heating during
the load so a Mott insulator can be created.

7.2.3

Many-body physics with long-range interactions

There are a lot of theoretical proposals for all sorts of interesting things to do with ultracold
polar molecules [96, 13, 97, 11, 98, 99, 100, 101, 102, 103, 81] . But really the bottom line isI’m not sure what the most interesting experiments are. So before we start engineering collisional
potentials [11] or start looking for topological phases with anyonic excitations [81], we should startby looking for mean-field-like energy shifts due to the dipolar interactions. With the dephasing
between rotational states understood, the transition to the rotationally excited state can serve asa good spectroscopic probe to measure energy shifts from dipole-dipole interactions [80].
I won’t try to, and don’t think I can, predict the physics that we will measure, but that isnumber (103)

110

6
5
4
3
2

200

300

400 500

temperature (nK)
Figure 7.2: Preliminary data for direct evaporation of molecules. The number vs. temperature fordifferent evaporation cuts is plotted on a log-log plot. The evaporation sequence is the same for all
points, with only the final depth of the optical trap being varied. We see a slope of 1.2(1). Any
slope less than 2 in 2D corresponds to an increase in phase space density.

111
partially what makes it so exciting.

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