Control of dipolar collisions in the quantum regime
byMarcio H. G. de Miranda
B.Sc., Federal University of Pernambuco, 2002
M.Sc., Federal University of Pernambuco, 2004
M.A., University of Colorado at Boulder, 2008

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
2010This thesis entitled:
Control of dipolar collisions in the quantum regime
written by Marcio H. G. de Miranda
has been approved for the Department of Physics

Dr. Jun Ye

Dr. Deborah S. Jin

Date

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

iii
Miranda, Marcio H. G. de (Ph.D., Physics)
Control of dipolar collisions in the quantum regime
Thesis directed by Professor Dr. Jun Ye

The preparation of ultracold polar molecular gases close to quantum degeneracy opens
novel research prospects ranging from dipolar quantum many-body physics to ultracold
chemistry. With a near quantum degenerate gas of fermionic 40 K87 Rb polar molecules,
this thesis presents studies on dipolar collision and chemical reaction dynamics, exhibiting
long range interactions and spatial anisotropy. With full control over the internal quantum
state of the molecules, we show how quantum statistics of the molecule determines the rateof chemical reactivity in the limit of vanishing collisional energy. Manipulating the interaction potential between indistinguishable polar molecules by means of control over the dipole
moment of the molecules, we study the dramatic influence of the dipolar interaction on thechemical reaction rate. In particular, we show that the chemical reaction rate increases
steeply with the dipole moment following a characteristic power law. This power law reflects
the long-range character of the dipole-dipole interaction. Studies on thermodynamics in themolecular quantum gas reveal the anisotropic properties of the dipolar interaction. Finally,
combining control over the molecular dipole moment and the dimensionality of the spatialconfinement, we suppress inelastic collisions between polar molecules by up to two ordersof magnitude. The suppression of inelastic collisions is achieved by changing the geometryof the confinement from three-dimensional to two-dimensional optical trapping. With the
combination of a sufficiently tight 2D confinement and Fermi statistics of the molecules,
two polar molecules approach each other only in a “side-by-side” collision, where the inelastic collisions are suppressed by the repulsive dipole-dipole interaction. This suppression
requires quantum state control of internal (electronic, vibrational, rotational and hyperfine
states) and external (harmonic oscillator levels of the optical lattice) degrees of freedom of

iv
the molecules. This is a fundamental advance in stabilizing a polar molecular gas for futureapplications in quantum many-body systems.

Dedication

To my mother Leda, to my brother Marcelo and to my dear wife Fabiana for all loveand support.

vi

Acknowledgements

To be a physicist is not an easy thing. If you want to be a fully realized physicist one
day, you have to go through a Ph.D. program, for most of the average people. I chose tofollow this path in 1998 when I started as a student in the Federal University of Pernambuco
in my home town Recife. After spending 4 years to get bachelors and 2 years to get mymasters, I decided to do a Ph.D. when my application was accepted at the University ofColorado at Boulder.
My Ph.D. work would be impossible without help of many people. I will start bythanking Jun Ye for giving me the chance to work in his lab, for providing such an amazing
scientific environment, and for being on my side in all decisive moments. I thank Deborah
Jin for her attention and patience. I thank John Bohn for his patience and for answering mydumb questions with a good mood. Thanks to Ana Maria Rey for stimulating discussions.
Thanks to John Hall for providing a stable laser source and lots of nice conversations about
lasers, physics, and life in general.
Thanks to Seth Foreman (“A” as in America), Daren Hudson, Michael Thorpe, Matt
Stowe, and Thomas Schibili for interesting discussions and frequency comb secrets that theyshared with me.
Thanks to the KRb group that I was part of. Thanks to Avi Pe’er who is an incredibly
patient person, very sharp and fun to work with. Thanks to Silke Ospelkaus for all shetaught me about cold atoms and molecules and for all that she did for me. Silke is theadvisor that any student would want to have. Thanks to Kang-Kuen Ni who taught me soviimuch and sided with me most of the time despite of some disagreements. Thanks to Dajun
Wang (“Dajunio”) who is a hard working and sharp scientist. Thanks to Brian Neyenhuis
who is a very sharp guy and patient. Thanks to my French friend Amodsen Chotia who
is smart, has a nice sense of humor, and was a pleasure to work with. Thanks to anotherFrench friend Goulven Quéméner (Goulvinho “vote em mim”) for many good discussions
and explanations, for his theoretical support, and for his nice sense of humor with no French
accent. Thanks to Steve Moses who is going to be in charge of our frequency comb fromnow on and who is also an excellent reinforcement for our group.
Thanks to my frequency comb friends Dylon Yost, Kevin Cossel, Travis Briels, Ticijana Ban, Thomas Allison, Aleksandra Matyba, Arman Cingoz, Florian Adler, and Piotr
Maslowski.
Thanks to my Sr friends Sebastian Blatt, Andrew Ludlow, Martin Boyd, Matt Swallows, Travis Nicholson, Michael Martin, Michael Bishof, Gretchen Campbell, Tanya Zelevinzky, Thomas Zanon, Jan Thomsen, Yige Lin, Ben Bloom, and Jason Williams.
Thanks to my OH friends Brian Sawyer, Benjamin Stuhl, Eng Hiang Yeo, Matthew
Hummon, and Yong Xia.
Thanks to my friends from theory and soccer games such as Murray Holland, Javier
Von Stecher, Dominic Meiser, Brandon Peden, David Tieri, and Ron Pepino.
Thanks to the electronic shop staff Terry Brown (the master Yoda of microwave electronics), James Fung-A-Fat, Michael J. Whitmore, David J. Tegart, Carl Sauer and Paul
Beckingham for the excellent development and support with electronics.
Thanks also to J.R. Raith, Michael Page and Alan Dunwell for the excellent support
on computers.
Thanks to the machine shop staff Hans Green, Kim R. Hagen, Paul Ariel, Todd R.
Asnicar, Thomas P. Foote, Blaine P. Horner, Tracy Keep and David Alchenberger for highquality of technical assistance.
Thanks to the supply office staff Jeff Sauter, Brian Lynch, Randall Holliness, andviii
William Blue for providing such efficient support. Thanks to Diana Moreland and Krista
Beck for being so organized and helpful. Thanks to Agnieska Lynch for taking care offinancial paper work. Thanks to Jeanne M. Nijhowne for her efficient work and kindness.
Thanks to my friends Fan-Chi Lin, Yu-Li Liang, Oliver Lin, Ricardo Jimenes, Sungsoo
Choi, Eduardo Calleja, Quan Zhang and Qing Li, Max Brown, Shannon Sankar, Chester
Rubbo, and Qing Chao.
Thanks to Mike Dubson for help me with my comps II exam.
Thanks to my my dear Brazilian friends Paulo Berton, Leila Grieger, Helena Berton,
Doris Berton, Felipe Nievinsky, Giovana Vazatta, Jose D’Incao, Valeria Gabarra, Eloisa
D’Incao, Nick McCulloch (the one who has a Brazilian spirit) and Daiana McCulloch forbeing so entertaining, nice and extremely helpful in hard times.
Thanks to professor Lúcio Hora Acioli for assisting me with my masters and helping
me come to JILA.
Thanks to my mother Leda and to my brother Marcelo for their love, dedication, careand sacrifice that they did in order to raise me like a honest, educated and hard working
person. Thanks to my dear wife Fabiana for all her love, patience, sacrifice, care and forbeing so understanding when I had to spend so much time working and not with her.
Thanks to CAPES and Fulbright for financial support.

Contents

Chapter
1 Introduction

1

1.1

From ultracold atoms to ultracold molecules . . . . . . . . . . . . . . . . . .

1

1.2

Dipolar interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Overview of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72 Experimental Apparatus
2.1

2.2

2.3

9

From ultracold K and Rb atoms to ultracold ground-state molecules . . . . .

10

2.1.1

Preparation of a near quantum-degenerate mixture of 40 K and 87 Rb

10

2.1.2

Feshbach molecule creation . . . . . . . . . . . . . . . . . . . . . . . .

11

2.1.3

Two-photon coherent transfer - STIRAP . . . . . . . . . . . . . . . .

12

2.1.4

Preparation of ground-state polar molecules . . . . . . . . . . . . . .

23

Essential manipulation techniques for ground-state polar molecules . . . . .

26

2.2.1

Electric dipole moment of the molecules . . . . . . . . . . . . . . . .

27

Tuning the geometry of the external confinement - 1D Optical Lattice . . . .

29

3 Dipolar 40 K87 Rb Molecules in 3D geometry

32

3.1

Bimolecular chemical reactions at ultracold temperature . . . . . . . . . . .

33

3.2

The role of quantum statistics in determining the chemical reaction rate . . .

34

3.3

The experimental system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

x
3.4

3.5

Bimolecular chemical reactions at vanishing dipole moment . . . . . . . . . .

36

3.4.1

p-wave collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.4.2

s-wave collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

Dipolar collisions with ultracold 40 K87 Rb in 3D geometry . . . . . . . . . . .

42

3.5.1

Dipoles and chemical reactions . . . . . . . . . . . . . . . . . . . . . .

42

3.5.2

The experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.5.3

Controlling chemical reactions by means of the dipole moment of themolecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.5.4

Thermodynamics of dipolar collisions . . . . . . . . . . . . . . . . . .

48

3.5.5

Anisotropy of heating . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

4 Polar Molecules in an 1D Optical Lattice: Going to 2D Geometry
4.1

Atoms in an one-dimensional optical lattice54. . . . . . . . . . . . . . . . . .

55

4.1.1

Optical dipole traps for neutral atoms and molecules . . . . . . . . .

55

4.1.2

One-dimensional optical lattice . . . . . . . . . . . . . . . . . . . . .

57

4.1.3

Experimental realization of a 1D optical lattice . . . . . . . . . . . .

59

4.1.4

Quantum mechanical description of a particle confined in a periodic
potential: Bloch bands . . . . . . . . . . . . . . . . . . . . . . . . . .

60

Characterizing the optical lattice . . . . . . . . . . . . . . . . . . . . . . . .

65

4.2.1

Measuring the trapping frequencies by parametric heating . . . . . .

65

4.2.2

Kapitza-Dirac Scattering . . . . . . . . . . . . . . . . . . . . . . . . .

67

Loading atoms into the optical lattice . . . . . . . . . . . . . . . . . . . . . .

69

4.3.1

Loading atoms in the optical lattice . . . . . . . . . . . . . . . . . . .

72

4.4

Probing the quasimomentum distribution of atoms in optical lattices. . . . .

74

4.5

40

79

4.2

4.3

K87 Rb molecules in the optical lattice . . . . . . . . . . . . . . . . . . . . .

5 Suppression of Inelastic Collisions in 2D geometry
5.1

Chemical Reaction Rates and Stereodynamics . . . . . . . . . . . . . . . . .

85
86

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5.2

Collisions between Polar Molecules in Quasi-2D Geometry . . . . . . . . . .

87

5.3

Band Mapping of Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

5.4

Suppression of Inelastic Collisions in the Optical Lattice . . . . . . . . . . .

95

6 Conclusion and Future Work

103

6.1

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2

Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Bibliography

105

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FiguresFigure1.1

p-wave centrifugal barrier for dipolar collisions between fermionic polar molecules.
(A), The effective intermolecular potential for fermionic molecules at zero electric field. At intermediate intermolecular separation, two colliding molecules
are repelled by a large centrifugal barrier for p-wave collisions. (B), For a
relatively small applied electric field, the spatially anisotropic dipolar interactions reduce the barrier for head-to-tail collisions and increase the barrier forside-by-side collisions. From [40]. . . . . . . . . . . . . . . . . . . . . . . . .

2.1

8Potential curves for 40 K-87 Rb molecules and energy levels used for STIRAP
transfer of Feshbach molecules to ground-state. The Feshbach molecules areformed in the initial state |ii = a3 Σ. The intermediate state |ei = (v 0 = 23)
is in the electronically excited potential of 23 Σ, and the final state |gi is in
the ro-vibronic ground state N = 0, v = 0 of X 3 Σ. The initial and the final
states are about 125 THz apart from each other. This figure is reproduced
from reference [26]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

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2.2

The schematic diagram of STIRAP pulse sequence. In (a), the molecules arein the initial state. In (b), the intensity of laser 2 is adiabatically ramped
up, coupling states |ei and |gi. In (c), the transfer of molecules in state |ii
to state |gi happens by adiabatically ramping the intensities of laser 2 down
and laser 1 up. In (d), laser 1 intensity is ramped down, and (e) is the end of
the process, where about 90% of the Feshbach molecules are tranfered to theground state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

Schematic picture of the spectrum of the frequency comb. The two CW lasers
at frequencies ν690 and ν970 are phase-locked to the comb. . . . . . . . . . . .

2.4

16

18

Frequency comb laser set up. This schematic diagram shows the Ti:S cavity(red box), the f-2f interferometer (blue box), the heterodyne beats for frep
(dashed blue box), and the ν690 (dashed red box) and ν970 (orange box) phaselocks. The the light from the Ti:S is then spectrally broadened in the photonic
crystal fiber (PCF) to cover a range from 500 nm to 1100 nm. The input power
to the PCF is 680 mW. fceo is stabilized via a f-2f interferometer. frep is phaselocked to a narrow YAG laser at 1064 nm. The several optical components
are used to deliver the light with the right wavelength to their respective
heterodyne beat set ups. F2, F3, F4, F5, and F6 are all band pass filters with
FWHM 10 nm, at 532, 690, 1064, 1064, and 970 nm respectively. F4 also isused as a “mirror” to reflect 970 nm light. F1 is a high pass filter with R(532nm)/T(600 nm - 850 nm). BS1 and BS2 are dichroic beam splitters where thereflection and transmition of the first are R(950 nm - 1100 nm)/T(500 nm 920 nm), and for the second are R(532 nm)/T(920 nm - 1100 nm). . . . . . .

19

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2.5

Locking electronics for frep . The beat signal of frep and the Nd:YAG laser
is amplified, goes to a band pass filter (BPF), and then to a home built
digital phase detector, where the frequency is divided and the signal is mixed
to a direct digital synthesiser (DDS). The DDS is referenced to a 10 MHz
commercial quartz clock (Wenzel Associates). The error signal goes to ahome built loop filter, where the signal is divided and sent to a PZT driver toa fast PZT (≈ 50 kHz bandwidth), and to a slow integrator box (with time
constant of ≈ 45 seconds), to a high voltage driver to a slow PZT. Using both
fast and slow PZTs guarantee short and long term stability for frep . . . . . .

2.6

20

fceo detection via f-2f interferometer. From a broad spectrum, for fceo detection, the important wavelengths are 532 nm (high frequency portion) and
1064 nm (low frequency portion). The low frequency portion is doubled in a
BBO crystal, and overlapped with the high frequency portion. The difference
between the doubled low frequency portion and the high frequency portion is
fceo . Figure reproduced from reference [55]. . . . . . . . . . . . . . . . . . . .

2.7

21

Locking electronics for fceo . The fceo beat signal is amplified, fed to a band
pass filter (BPF), and then to a home built digital phase detector, where thefrequency is divided and the signal is mixed to a direct digital synthesiser(DDS). The DDS is referenced to a 10 MHz commercial quartz clock (Wenzel
Associates). The error signal goes to a home built loop filter, to a bias box,
which puts an offset in the output voltage going to an AOM driver, feedingback the AOM that controls the intensity of the Verdi 10W laser. . . . . . .

2.8

22

Raman laser system for 970 nm transition. This system has three outputs:
one for a wavemeter, one for the frequency comb and one for the STIRAP
transfer. The power of the laser is amplified by using a tapered amplifier. . .

23

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2.9

Raman laser system for 690 nm transition. This system has three outputs:
one for a wavemeter, one for the frequency comb and one for the STIRAP
transfer. The power of the laser is amplified by using another diode laser(slave laser) seeded with ≈ 1 mW of power from the master laser. The slavelaser is locked via injection lock. . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.10 Timing diagram of the coherent two-photon transfer (STIRAP) from Feshbach molecules to the absolute rovibrational ground state X 1 Σ (v = 0). (a)STIRAP pulse sequence, where I1 and I2 are the intensities of laser 1 and laser
2. (b) Measured population in the initial Feshbach state during the STIRAP
pulse sequence. Starting with 4·104 Feshbach molecules, the molecules are coherently transferred to the rovibrational ground state X 1 Σ (v = 0) by the first
pulse sequence (t = 15 to 20µs). The rovibrational ground state molecules areinvisible to the detection light. Reversing the pulse sequence, X 1 Σ (v = 0)molecules are converted back to weakly bound Feshbach molecules (t = 45 tot = 50µs). This figure is reproduced from reference [48]. . . . . . . . . . . . .

25

2.11 Comparison of the kinetic energy of the Feshbach molecules before STIRAP
transfer (blue circles) and after a round-trip STIRAP process (red circles).
The latter can be interpreted as an upper limit on the kinetic energy of therovibrational ground state molecules. The temperature of both clouds is extracted by time of flight expansion analysis. We extract T = 400(15) nK for
the Feshbach molecules and T = 430(20) nK for rovibrational ground-state
molecules. The analysis shows that the transfer process does not cause any
noticeable heating on the molecules. This figure is reproduced from reference [48]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

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2.12 Stark spectroscopy of the singlet v = 0 molecules. The bottom panel shows
the Stark shift of the rovibrational ground-state of the singlet potential (v =0, N = 0 of X 1 Σ), and the top panel shows the shift of the v = 0, N = 2 state.
The systematic error in the applied electric field is 3% (horizontal error bars).
The level difference between N = 0 and N = 2 is 6.6836(5) GHz, which yields
a rotational constant B of 1.1139(1) GHz. Given the measured B, the fit of the
Stark shift (line in lower panel) gives a permanent electric dipole moment of
0.566(17) D. The theory curves for N = 2 for different |mN i projections (lines
in upper panel) are calculated using the measured B and the dipole momentderived from the N = 0 fit. This figure is reproduced from reference [26]. . .

27

2.13 Calculated dipole moment versus applied DC electric field. In our experiment
we can go to electric fields as high as 5.0 kV/cm. . . . . . . . . . . . . . . .

28

2.14 Schematic figure of the electric field plates and 1D optical lattice set up. The
maximum output power of the fiber is 1.7 W. The polarization is set to beparallel to the optical table. The intensity of the optical lattice is stabilized
via an AOM, which also shifts the frequency by +80 MHz. . . . . . . . . . .
3.1

30

Hyperfine structure of rovibronic ground-state 40 K87 Rb molecules at a magnetic field of 545.9 G. Here we have 36 nuclear spin states by their spin projecKtions, mRb
I and mI . The energy spacing between hyperfine states is h·130 kHz¯¯¯¯for ¯mK ¯ = 1 and h · 760 kHz for ¯mRb ¯ = 1. By comparison, at a temperatureI

Iof 300 nK, the molecules thermal energy is equivalent to h · 6 kHz, which ismore than an order of magnitude smaller than the spin flip energy. Molecules
can be prepared in either a single state or in a mixture of The |−4, 1/2i
and the lowest-energy state |−4, 3/2i (open ellipses). Figure reproduced fromreference [39]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

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3.2

Transfer from the rovibrational ground state |−4, 1/2i to the absolute groundstate |−4, 3/2i via coherent Rabi transfer. The intermediate state is in therotational level N = 1 and it is a mixed state with g≈ 5 %. . . . . . . . . . .

3.3

38

Inelastic collisions between spin-polarized (indistinguishable) or different spinstate (distinguishable) fermionic molecules in the rovibronic ground state of40

K87 Rb. (A) Data showing the time dependence of the molecule number den-sity. Here the molecules are prepared in a single hyperfine state, |−4, 1/2i, andthe molecular density decays slowly with a rate coefficient of 3.3(±0.7) · 1012
cm3 s−1 at T = 250 nK. (B) Loss rate coefficient as function of the temperature. The loss rate increases linearly with temperature for spin-polarizedmolecules, which collide via p-wave [inset in (A)] at low temperature. Data
were taken for molecules prepared in either |−4, 1/2i (solid circles) or thelowest-energy state |−4, 3/2i (open triangles). A linear fit (solid line) to the|−4, 1/2i data yields the temperature-dependent loss rate to be 1.2(±0.3) · 105
cm3 s−1 K−1 . 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 anddashed lines are the predictions from the QT and MQDT models, respectively.
When the molecules are prepared in a mixture of the |−4, 1/2i and |−4, 3/2i
states (solid squares), we observe a temperature-independent decay rate thatis 10 to 100 times that for the spin-polarized case. Figure reproduced fromreference [39]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4

41

p-wave barrier for 40 K87 Rb dipolar collisions “head-to-tail” (solid curve inblue) and “side-by-side” (solid curve in red). The dashed line represents thep-wave in the absence of electric-field. . . . . . . . . . . . . . . . . . . . . . .

43

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3.5

Two-body inelastic loss for fermionic polar molecules. (a), We extract theinelastic loss rate coefficient, β, from a fit (solid lines) to the measured time
evolution of the trapped molecular gas density. Data are shown here forinduced dipole moments of d = 0.08 D (open triangles) and d = 0.19 D (filled
circles), and T0 = 300 nK. (b), Data points show β/T0 plotted as a functionof d. The dashed line shows a fit to a simple model based on the quantum
threshold behavior for tunneling through a dipolar-interaction-modified pwave barrier. The solid line shows the result of a more complete quantum
scattering calculation. Inset, the calculated dependence of d on the applied
electric field, E. Figure reproduced from reference [40]. . . . . . . . . . . . .

3.6

45

p-wave barrier for dipolar collisions between fermionic 40 K87 Rb molecules.
(a), The effective intermolecular potential for fermionic molecules at zero electric field. At intermediate intermolecular separation, two colliding molecules
are repelled by a centrifugal barrier for p-wave collisions. (b), For an applied electric field, the spatially anisotropic dipolar interactions reduce thebarrier for “head-to-tail” collisions and increase the barrier for “side-by-side”
collisions. (c), Height of the p-wave barrier as a function of dipole moment.
Dipolar interactions lower the centrifugal barrier for M = 0 collisions (V0 )and raise the barrier for M = ±1 collisions (V1 ). The lowering of the M = ±1
barrier at very large dipole moments is due to mixing with higher L partial
waves (L = 3, 5, 7, ). Figure reproduced from reference [40]. . . . . . . . . . .

46

xix
3.7

Normalized fractional heating rate, Ṫ /T02 n, versus the dipole moment. The
heating rate is extracted from 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 experimental initial conditions are n = 0.3·1012 cm−3 and T0 = 0.5µK,
and the absolute heating rate ranges from 0.5µK s−1 at zero electric field to
2µK s−1 at our highest electric fields. Figure reproduced from reference [40].

3.8

50

Cross-dimensional rethermalization in the polar molecule gas. Shown as afunction of dipole moment, d, for Tz > Tx (a-c) and Tz < Tx (d-f). The experimental data shows difference between heating the gas in the vertical direction(a-c) and heating it in the horizontal directions (d-f). This provides evidence
for the strong anisotropic characteristic of dipolar interactions. The electricfield is applied along the ẑ direction. Figure reproduced from reference [40]. .

4.1

52

Illustrative contour plot of one-dimensional optical lattice potential. (a) Normalized potential in false color for w0 = λ. (b) Potential for w0 = 250λ. The
point with maximum density are the anti-nodes where, for a red-detuned optical lattice, the particles will be trapped. In our case, the lattice beam waist
is w0 ≈ 250µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

58

Optical lattice’s trap frequencies versus the trap depth for 87 Rb . This twoplots show the axial frequency νz compared to the radial frequency νr . The
confinement in the axial direction is much tighter than the radial direction.
The optical lattice parameters are: λ = 1064 nm, and the beam waist is 250µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

xx
4.3

Bloch bands in the one-dimensional optical lattice. Energy versus quasimomentum in the first Brillouin zone for different lattice depths. The lattice
depths are 0, 10, 15, and 20 Erec . As the lattice height increases, the separation between the Bloch bands also increases and the energy levels flattens
out.

4.4

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

62First four energy levels versus the lattice depth with quasimomentum q = 0,
for an atom in an optical lattice. For lattice depths above 30Erec , the spacing
between the energy levels approaches ~ωz like a harmonic oscillator energy. .

4.5

63

Semi-log plot of tunneling time in an 1D optical lattice versus lattice depth.
Dotted blue line is for 40 K and red dashed line is for 87 Rb . For our typicalK
conditions, the tunneling time for 40 K is 300 ms (for 19 Erec
) and for 87 Rb isRb
1 s (for 42 Erec).

4.6

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

65

Trap frequency measurement for Feshbach molecules 40 K-87 Rb via parametric heating. σ is the molecular cloud width in the ẑ direction. The center
frequency is 44 kHz, which indicates that the axial frequency of the opticallattice is 22 kHz. The ramp procedure for this measurement is similar to theone described in chapter 5 in fig. 5.2 except with a sudden turn off of the endof the ramp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7

67

Ramp procedure to measure trap depth of the optical lattice. The depth
measurement is done with a 87 Rb BEC. The optical lattice is diabaticallypulsed on and briefly switched off in order to excite 87 Rb atoms to higher
bands. The hold time is varied on the order of tens of microseconds. The

4.8time of flight is 5 ms, and the lattice height is 42 Erec . . . . . . . . . . . . .

69

87

70

Rb BEC matter-wave diffraction into discrete momentum components. . .

xxi
4.9

Extracted relative population of momentum components versus pulse duration. The lines show a fit to eq. 4.19. The optical lattice depth as a freeparameter, and the extracted one from the data is 42 Erec . The optical lattice
parameters are λ = 1064 nm, beam waist of 250 µm, and laser power of 1
W. The three curves show the 0~k (solid blue), ±2~k (solid green), and ±4~k(solid red) momentum components. . . . . . . . . . . . . . . . . . . . . . . .

71

4.10 Energy gap between lowest and first excited Bloch bands for different quasimomenta. Solid blue line is for q = 0, dashed green line is for q = 0.5 ~k, andred dotted line is for q = ~k. . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

4.11 Ramping procedure to adiabatically load 40 K and 87 Rb in the optical lattice.
Rb
K
The optical lattice depth for 87 Rb is 42 Erecand for 40 K is 19 Erec.

. . . . .

73

4.12 Lifetime of 87 Rb and 40 K atoms and respective heating rates in the opticallattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4.13 Calculated functions for the distribution in quasimomentum space at the first
Brillouin zone in the optical lattice. Red dotted curve is a gaussian function,
green dashed curve is a Heaviside function, and solid blue line is the convolved
curve that is used to fit the atomic cloud along the lattice axis. . . . . . . . .

75

4.14 Band mapping with filtering of 40 K in higher bands. The optical lattice is adiabatically ramped up exponentially in 150 ms to 17.7 Erec . At the beginning
the optical lattice and the optical trap are combined to hold the atoms against
the gravity. The filtering is done for two different lattice heights: 3.2 Erec ,and 14.5 Erec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

xxii
4.15 Band mapping for the filtering procedure with 40 K atoms. (a), and (c) are theatomic cloud images taken with time of flight of 18 ms. The initial temperatureof the atoms in (a) is 350 nK and in (b) is 400 nK. The vertical profiles areshown in (b) and (d). In (b) all atoms are in the first Brillouin zone, whilein (d) about 28% of the atoms are in the second Brillouin zone. However, asignificantly difference in population of bands is observed in the two cases due
to a different optical lattice heights (3.2 Erec , and 14.5 Erec ) applied for thefiltering proceedure(see fig. 4.14). . . . . . . . . . . . . . . . . . . . . . . . .

79

4.16 Ramping procedure for formation of ground state 40 K87 Rb molecules in theKRboptical lattice. The final lattice depth is 60 Erec
. . . . . . . . . . . . . . . .

81

4.17 Heating rate for 40 K87 Rb in the optical lattice. The optical lattice depth is
KRb
60 Erec
depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.18 Ramp procedure for forming/loading ground state molecules in the opticallattice. The number 1 indicates where the ground state molecules are formed.
Then the molecules are loaded in the optical lattice. The number 2 indicateswhere the ground state molecules are formed. In this case the molecules
are formed in the fully ramped up optical lattice. The arrow with the note
“Variable Height” means that the final value of the lattice is changed. The
KRbmaximum height (depth) value that the lattice can go is 120 Erec
. . . . . .

82

4.19 Measurement of the number of ground state molecules versus the opticallattice depth. The black open circles show the molecule number when themolecules are formed in the fully ramped-up optical lattice. The red open
squares show the molecule number when the molecules are formed in the partially ramped up optical lattice. . . . . . . . . . . . . . . . . . . . . . . . . .

83

xxiii4.20 Ramp procedure for forming/loading ground state molecules in the opticallattice. The number 3 indicates where the ground state molecules are formed.
The molecules are formed in a partially ramped up optical lattice. The arrow
with the note “Variable Height” means that the partial value of the lattice is
KRbchanged. The final height (depth) value of the lattice is 120 Erec
. . . . . . .

5.1

84

Quantized stereodynamics of ultracold chemical reactions in quasi-2D. (A) A
quasi-2D geometry for collisions is realized for polar molecules confined in a
one-dimensional optical lattice. An external electric field is applied along thetight confinement axis. (B) The lowest three adiabatic potentials for collisions
are shown schematically as a function of the intermolecular separation, 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) Threedifferent cases are shown schematically for each of the three lowest collision
channels. The lowest energy collision channel occurs when two molecules
are prepared in different internal states (indicated here by the colors of themolecules). The second channel is realized when two identical molecules areprepared in different vibrational levels v for their ẑ motions. The third casehas 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 [74]. . . . . . . . . . . . . . .

91

xxiv
5.2

Ramp procedure for transfering atoms from the dipole trap to an optical lattice
as well as for band mapping for molecules. Molecules are formed after an 150
ms adiabatic ramp. A non-thermal distribution is created by using parametricheating by pulsing a sinosoidal wave with 300 µs. The band mapping is doneby linerly ramping down in 500 µs. To image the molecules, we reverse ourcoherent transfer process to bring the molecules back to a weakly bound state
where we can detect the molecules with time-of-flight absorption imaging [26]. 93

5.3

Relative population of molecules in the lattice vibrational levels.We measure
the relative population in each lattice vibrational level using a band-mapping
technique. The results are shown for (A) a thermal distribution of moleculesand (B) a non-thermal distribution created by parametric heating in ẑ. The
two images use the same color scale for the optical depth (OD). The images
are an average of 5 shots and 7 shots for (A) and (B), respectively, taken
after 10 ms of free expansion. Below each image we show a trace along ẑ that
corresponds the OD averaged over the transverse direction. A fit (red line) tothe trace, which takes into account both the size of the Brillouin zones andour 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
wavevector. Figure reproduced from reference [74].

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

95

xxv
5.4

Ramp procedure for measurement of chemical reaction rates. The 40 K87 Rb molecules
are produced when the optical lattice is fully formed at ≈ 100 Erec , after 150
ms ramp time. The molecules can be excited to higher vibrational levels of
the optical lattice via parametric heating. The parametric heating is done via
a sinosoidal modulation of the lattice intensity for 300 µs. The electric field
is turned on together with the parametric heating and it is turned off 5 ms
before we switch off the dipolar trap and the optical lattice. The molecules
are held in the combined dipolar trap and the optical lattice for a variable
time. After the molecules are released from the trap, the image is taken after
10 ms time of flight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.5

97

Measurements of 2D loss rates and comparison with theory. (A) A fit (solidlines) to the measured loss curves, with (red circles) and without (blacksquares) 0.3 ms of parametric heating in ẑ, is used to extract the loss rate
constants β|3i and β|2i . (B) The extracted loss rate constants for collisionsof molecules in the same lattice vibrational level (black squares) and fromdifferent lattice vibrational levels (red circles) are plotted for several dipole
moments. Measured loss rate constants for molecules prepared in different internal states are shown as green triangles. For comparison with each of thesethree 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 in fig. 5.1 B.
Figure reproduced from reference [74]. . . . . . . . . . . . . . . . . . . . . . 101

xxvi
5.6

Loss rates from 3D to 2D. 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 thermalBTequilibrium depends on the ratio khν. The measured initial loss rates for az

dipole moment of 0.174 D are displayed for two different thermal distributions(solid triangles), a non-thermal sample created by parametric heating (thetop open triangle), and an extracted pure β|3i when the entire population isresiding in the lattice ground vibrational level (the bottom open triangle). The
experimental results agree well with a simple model (black curve) described inthis section. The top line indicates the value of β|2i as measured in fig. 5.5 B.
(Inset) The intralevel loss rate for identical fermionic 40 K87 Rb molecules in
2D (black circles) is compared with the loss rate in 3D (blue triangles). The
3D data for T = 300 nK are borrowed from reference [40]. The 2D data weretaken 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 [74]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Chapter 1
Introduction

1.1

From ultracold atoms to ultracold molecules
Over the last decades, the field of ultracold atoms has progressed at a tremendous pace.

Initial experiments focused on the development of techniques to cool and trap atoms [1, 2, 3],
including the decisive techniques of laser cooling and evaporative cooling. Based on thesetechniques very exciting and important milestones have been achieved: The preparation of
Bose-Einstein condensates (BEC) [4, 5, 6, 7] and quantum degenerate Fermi gases in diluteatomic gases [8] opened new avenues in atomic physics. Fano-Feshbach resonances allowed
control of atom-atom interactions [9]. This created novel opportunities such as the realizationof strongly interacting systems, the study of BEC-BCS crossover physics [10, 11, 12], and the
preparation of ultracold weakly bound Feshbach molecules [13, 14] from quantum degenerategases of atoms.
Compared to atomic quantum systems, molecular quantum systems promise to open
new research frontiers. However, the molecules’ complex internal structure makes molecularsystems very rich but also really challenging. Due to numerous vibrational and rotational
quantum degrees of freedom, traditional cooling methods for atoms have not worked formolecules. Experimental efforts towards the creation of quantum gases of molecules havetherefore followed two distinct paths. The first approach is to directly cool molecules to
low translational temperatures. This can be achieved by Stark deceleration or buffer gas
cooling [15, 16, 17]. Unfortunately, these techniques have so far been limited to low densities2(108 molecules/cm3 ) and relatively high temperatures in the miliKelvin range. These parameters correspond to a phase-space density of 10−13 and therefore many orders of magnitude
away from quantum degeneracy. The second way is to cool and trap atoms first and thenassociate pairs of cold atoms to tightly bound molecules. A challenge here is the efficientconversion of free atoms to molecules while preserving density and temperature of the initialatomic ensemble. In 2005, Sage et al. [18], demonstrated the formation of vibrational groundstate RbCs molecules formed via photoassociation from free atoms. Starting from a laser
cooled atomic ensemble of 2 × 108 Rb and 3 × 108 Cs atoms at temperatures of 100 µK
and phase space densities on the order of 10−14 , they obtained a conversion efficiency offew percent [18]. These experiments elucidated that to achieve better conversion efficiencies and higher phase space densities, the initial conditions of the atomic ensemble needs tobe far closer to quantum degeneracy and the conversion process from free atoms to deeply
bound molecules needs to be very efficient. In our lab, this was achieved by combining two
techniques. The first is Feshbach molecule creation in ultracold quantum degenerate atomic
gases, which allows the efficient conversion of atomic ensembles into weakly bound molecules
prepared in a well-defined quantum state.
The next challenge was to transfer the weakly bound molecules into their absolute
ground state. Our goal is to transfer to a single internal quantum state without heating thesample of molecules. Therefore, a fully coherent conversion process was needed. The conversion efficiency is also essential to preserve the initial high phase-space density. A techniqueknown as STImulated Raman Adiabatic Passage (STIRAP) [19], involving an electronically
excited intermediate state, can be envisioned for the transfer process between the initialand final vibration levels. Because of a nearly complete mismatch between the vibrational
wave-functions of the weakly bound and the absolute ground-state molecules, it was largely
believed to be impossible to find a suitable intermediate state that could provide sufficienttransition strengths for both the upward and downward transitions. A proposal was madeto use a train of two-color, phase-coherent pulses that would allow coherent accumulations

3of a pump-dump process to implement a fully coherent molecular conversion process [20].
However, systematic and detailed single photon spectroscopy ensued [21], connecting the initial Feshbach state to specific electronically excited states with a CW laser referenced to an
optical frequency comb. An intense theory-experiment collaboration led us to the realizationthat we could use a single stationary intermediate state instead of dynamic wave-packets,
which a key point is to have an intermediate state that provides favorable Franck-Condon
factors to both the initial weakly bound Feshbach state and the rovibrational ground state.
Coherent conversion of weakly bound molecules into a more deeply bound states has
been demonstrated in several experiments [22]. In the work done by our group, Ospelkaus
et al. [23], STIRAP is used to convert an ensemble of weakly bound 40 K-87 Rb molecules,
with binding energy of a few hundred kilohertz, into an ensemble of molecules in a vibrational state bound by more than 10 GHz1 . In 2008, we demonstrated that a single stepof a coherent transfer can even be used to convert an ensemble of heteronuclear Feshbach
molecules into an ensemble of rovibrational ground state polar molecules [26]. In these experiments, the energy difference between the initial and final states is about 125 THz andtherefore it is necessary to establish a fixed phase relation between lasers of extremely different frequencies. This task was accomplished by referencing the two lasers to individual
teeth of an optical frequency comb laser [27, 28, 29]. Using this technique, we demonstrated
the creation of an ultracold high phase space density molecular gas prepared in its lowest
internal quantum state. The resulting ground state molecular ensemble has a peak densityof 1012 molecules/cm3 , a temperature of 200 nK, and a phase-space density of 0.06, and istherefore prepared close to quantum degeneracy. Also, the molecules have an electric dipole
momentum of 0.566 Debye. This opens exciting perspectives for the study of quantum gaseswith strong dipole-dipole interaction.
1

Related work has been done for homonuclear molecules of Cs2 and Rb2 by Danzel et al. [24] and Langet al. [25] respectively.

4
1.2

Dipolar interactions
Interactions between particles determines many of the observed phenomena in ultracold

quantum degenerate atomic gases. Interactions profoundly modify the static and also the
dynamic properties of the system [30]. Short-range and isotropic “contact” interactions
dominate the properties of most quantum gases of ultracold atoms, whereas dipole-dipole
interactions play only a negligible role. However, recent developments in the manipulationof cold atoms and molecules have opened the way for the study of dipole-dipole interparticle
interactions in ultracold quantum gases.
Dipole-dipole interactions have interesting properties. First, the dipole-dipole interaction is spatially anisotropic, which means that the strength and sign of the interaction
depends on the relative orientation of the dipoles. Second, the dipole-dipole interaction islong range, where the interaction decays with 1/r3 , r being the interparticle distance. Third,
the electric-dipole moment of molecules is tunable by means of an externally applied field.
Pioneering work on the study of magnetic dipole-dipole interactions in ultracold quantum gases was done in Tilman Pfau’s group [31, 32]. Pfau and coworkers managed to preparea Bose-Einstein condensate (BEC) of 52 Cr atoms. Chromium has a large magnetic dipole
moment of 6 Bohr magnetons - a factor of 6 larger than the magnetic-dipole moment ofalkali atoms. By enhancing the dipole-dipole interaction by the choice of the atomic species
and making use of Fano-Feshbach resonances in chromium to reduce the isotropic contact
interaction, the group was able to achieve a regime where the anisotropic magnetic dipoledipole interaction between 52 Cr in the BEC dominates over the short-range interaction. This
allowed the study of the anisotropic character of the dipole-dipole interaction in collision andexpansion dynamics of the atomic gas [32].
However, interactions between magnetic dipole moments of atoms are typically weaker
than those between electric dipole moments of polar molecules. For example, the electric dipole moment of a typical polar molecule is on the order of 1 Debye, where 1 Debye

5is ≈ 3.34·10−30 C·m. The interaction energy between polar molecules is then typically afactor of 10000 times larger than the interaction energy between atoms interacting via magnetic dipole-dipole interactions with a typical dipole moment of 1 Bohr magneton (µB ),(1 Debye)2 c2≈ 104 .
(1 µB )2

Building on the preparation of an ultracold polar gas of fermionic 40 K-87 Rb moleculesin our group in 2008 [26], a variety of novel experimental possibilities have opened up basedon the control and use of the dipole-dipole interaction between polar molecules. Theoretical proposals range from the study of quantum phase transitions [33] and quantum gas
dynamics [34] to quantum simulations of condensed matter spin systems [35] and schemesfor quantum information processing [36, 37, 38].
1.3

Overview of this thesis
The main work of this thesis is about the confinement of an ensemble of ground-statepolar 40 K87 Rb molecules in a quasi-2D geometry to control chemical reactions at ultralow
temperature. As a background for my thesis work, I review previous work in ultracold
chemistry of 40 K87 Rb [39] and dipolar interactions in 3D [40]. These subjects are also discussed in Kang-Kuen Ni Ph.D. thesis [21] to which I contribute. The chemical reaction
KRb+KRb→K2 +Rb2 is exothermic and proceeds without a chemical reaction barrier at
short-range. Starting from fermionic polar molecules close to quantum degeneracy, we study
this atom-exchange chemical reaction in a regime where the motion of the molecules is strictly
quantized. When the ensemble of 40 K87 Rb molecules is created in a single quantum state- that means the molecules are indistinguishable fermions - we observe chemical reactions
to be strongly suppressed by a long-range p-wave barrier of 24 µK that effectively hinders
the molecules from coming into short-range [39]. However, changing the quantum statisticsof the molecular ensemble by preparing a 50:50 mixture of two spin states, we observe thecollisional rate to be enhanced by a factor of 10-100. For these distinguishable molecules,
collisions will proceed via the s-wave channel, which means that there is no repulsive colli-

6sional barrier at long-range. The two molecules easily get within short-range where reaction
loss proceeds with almost unity probability.
With this basic understanding of how chemical reactions occur in the quantum regime,
we make use of the large tunable electric dipole moment to control the long-range potential
between two colliding molecules and therefore the chemical reaction rate. Applying an
external electric field induces a dipole moment in the molecule. The dipole-dipole interaction
will then strongly determine the collisional processes occurring in the sample. Of particularimportance is the anisotropic character of the dipole-dipole interaction. As shown in fig. 1.1,
depending on whether the molecules are colliding in a “head-to-tail” (attractive) or “sideby-side” configuration (repulsive), chemical reactions will either be enhanced or suppressed.
This is because the long range dipole-dipole interaction significantly modifies the height ofthe centrifugal barrier and thus changes the inelastic collision rate. The attractive natureof dipole-dipole interactions for polar molecules colliding “head-to-tail” lowers the barrier,whereas the repulsive dipole-dipole interaction for polar molecules colliding “side-by-side”
raises the barrier. In a three-dimensional geometry, both “side-by-side” and “head-to-tail”
collisions are allowed and will occur in the molecular sample. In our experiment, we showthat for high electric fields, “head-to-tail” collisions dominate the loss process and we observe
the exothermic bimolecular reactions to steeply rise with the 6th power in the dipole moment
due to attractive “side-by-side” collisions [40, 41]. The larger the electric dipole-moment thefaster will be the inelastic decay of the molecular ensemble.
These experiments show that due to the long inelastic collisional loss process in a 3D
geometry, it will not be possible to prepare a sample of quantum degenerate polar moleculesin such geometry. In particular, evaporative cooling of the molecules towards quantum degeneracy does not seem to be feasible. This challenge is addressed in this thesis, where
we prevent “head-to-tail” dipole-dipole interactions. “Head-to-tail” dipole-dipole interactions can be suppressed if these molecules are confined in a 2D geometry [42, 43]. We use
an one-dimensional optical lattice to confine the fermionic polar molecules in a quasi-two-7dimensional, pancake-like geometry. The dipoles are oriented along the tight axis of confinement. The combination of tight confinement (restricting the movement of the moleculesalong the axis of strong confinement to a single motional state) and Fermi statistics of themolecules strictly forbids molecules to approach in a “head-to-tail” configuration. Two polar molecules can approach each other only in a “side-by-side” collision, where the chemical
reaction rate is suppressed by the repulsive dipole-dipole interaction. This quantum stereodynamics of the ultracold collisions can be exploited to suppress the chemical reaction rate bynearly two orders of magnitude. The suppression of chemical reactions for polar molecules ina quasi-two-dimensional trap opens the way for investigation of a dipolar molecular quantum
gas.
1.4

Outline of the thesis
Chapter 2 gives a short overview of the experimental set up: the apparatus for the

preparation of a two-species quantum degenerate ensemble of 40 K and 87 Rb , the creationof Feshbach molecules, and the laser system for the realization of the STIRAP transfer.
Also, I will review how to realize a two-dimensional geometry by means of a one-dimensional
optical lattice. In Chapter 3, I will discuss basic properties of the dipole-dipole interactions
in 3D geometry, which sets the stage for 40 K87 Rb molecules in 2D geometry. In Chapter 4, I
present the basic concepts of a 1D optical lattices, adiabatic loading of 40 K and 87 Rb atoms
into the lattice, and formation of 40 K87 Rb molecules in the optical lattice. In Chapter 5, I
discuss and present the suppression of inelastic collisions in the tight confinement regime. In
Chapter 6, I summarize the thesis and discuss future work.

8

Figure 1.1: p-wave centrifugal barrier for dipolar collisions between fermionic polar molecules.
(A), The effective intermolecular potential for fermionic molecules at zero electric field.
At intermediate intermolecular separation, two colliding molecules are repelled by a large
centrifugal barrier for p-wave collisions. (B), For a relatively small applied electric field, thespatially anisotropic dipolar interactions reduce the barrier for head-to-tail collisions andincrease the barrier for side-by-side collisions. From [40].

Chapter 2
Experimental Apparatus

The basis for the experiments presented in this thesis is a reliable apparatus for the
preparation of high-phase space density gases of ultracold polar 40 K-87 Rb molecules in theirrotational and vibrational ground state. In this chapter, I will give an overview of theexperimental techniques and the experimental apparatus for the preparation of ground statepolar molecules. As already mentioned in the introduction, our approach for the preparationof a high-density gas of ultracold ground state polar molecules is to cool and trap theconstituent atoms 40 K and 87 Rb first and then implement a controlled chemical reaction
at ultracold temperature. The latter is done via a two-step process, where in the first step,
weakly bound molecules with a binding energy of h·300 kHz are formed in the vicinity of a
Fano-Feshbach resonance [44]. These molecules are also called Feshbach molecules (FBM).
Once these FBMs are formed, we transfer the molecules to the electronic, vibrational androtational ground state via a single step of STImulated Raman Adiabatic Passage (STIRAP).
In the following, we will call the resulting molecules ground state molecules (GSM). The
heteronuclear 40 K87 Rb GSMs have an electric-dipole moment of 0.56 Debye [26], whichallows the study of dipolar collisions controlled by external electric fields.
In the following, I will review our experimental path from the preparation of ultracold
40

K and 87 Rb atoms to the manipulation of ground state molecules. I will also discuss basic

manipulation techniques for the ground-state molecules such as precise control over the dipole
moment of the molecules by means of external electric fields and techniques for controlling10
the confinement geometry of the molecules via dipole and optical lattice potentials.
2.1

From ultracold K and Rb atoms to ultracold ground-state molecules

2.1.1

Preparation of a near quantum-degenerate mixture of 40 K and 87 Rb

The starting point for the formation of a high-phase space density gas of ground state
molecules is the preparation of a near quantum-degenerate mixture of two atomic species.
In our case, we are working with the alkali atoms 40 K and 87 Rb . Both atoms are nowadays
workhorses for atomic physics experiments in the quantum degenerate regime, and coolingand trapping techniques are therefore well established. In the following, I will briefly summarize the main experimental steps. For more details please refer to the Ph.D. thesis of Josh
Zirbel [45].
Cooling and trapping of 40 K and 87 Rb starts with a two-species magneto-opticaltrap (MOT). Typical atom numbers in the MOTs are 2 − 4 × 109 for Rb and 107 for 40 K
, respectively. After a short sub-Doppler cooling, optical pumping and compression state,these atoms are loaded into a quadrupole magnetic trap, which is then subsequently moved
to a second part of the vacuum apparatus - the so-called science chamber. In the sciencechamber, the atoms are transfered into a Ioffe-Pritchard (IP) magnetic trap, where both87

Rb and 40 K are cooled to temperatures on the order of 1 µK. This is performed byevaporative cooling of Rb atoms via microwave radiation, which drives high energy 87 Rb
atoms in the stretched |F = 2, mF = 2i hyperfine state to an untrapped state |1, 1i. HereF is the total atomic spin number and mF is the projection. 40 K atoms prepared in thestretched |9/2, 9/2i state are sympathetically cooled in the bath of 87 Rb atoms. At the endof the evaporation in the IP trap, we typically achieve atom numbers of 6 − 7 × 105 for 40 K
and 2 − 3 × 106 for 87 Rb , respectively, at temperature of 1 µK.
Magnetic trapping is restricted to low-field seeking atomic states. However, for the
preparation of Feshbach molecules, often high-field seeking states are needed to access a

11Fano-Feshbach scattering resonance. Also, it is desirable to be able to choose the magnetic
field freely, independent of any trapping potentials. In a next step, we therefore transfer theatomic mixture into an optical trap (OT). In our case, the OT is realized using a crossed
beam configuration. In the experiment, we have chosen two elliptically shaped Gaussian
beams with beam waists of 40 µm in the vertical and 200 µm in the horizontal direction,
respectively. These two beams intersect at the center of the IP trap and have a wavelengthof 1064 nm. The potential is pancake-shaped with an aspect ratio of 1:5.
The OT is loaded with 40 K and 87 Rb in the |9/2, 9/2i and |2, 2i states, respectively. We
then transfer these atoms to their lowest hyperfine states, |9/2, −9/2i and |1, 1i performed
via two adiabatic rapid passages, by sweeping RF (for 40 K ) and microwave frequencies (for87

Rb ) at a magnetic field of 31.29 G. Choosing these two atomic quantum states has two

distinct advantages. First of all, these quantum states are collisionally stable. Second, they
allow access to a relatively broad Feshbach resonance between 40 K and 87 Rb at about 546
G [46, 47]. In preparation for Feshbach molecule creation, the magnetic field is ramped to553.3 G and the atoms are evaporatively cooled closer to quantum degeneracy by decreasing
the depth of the optical trapping potential. This procedure achieves temperatures of 200−300
nK, with 40 K and 87 Rb numbers of 2.5 × 105 and 3.5 × 105 - an ideal starting point for the
preparation of 40 K87 Rb Feshbach molecules.
2.1.2

Feshbach molecule creation

For Feshbach molecule creation, we use a Fano-Feshbach resonance at a magnetic fieldof 546.7 G [46, 47]. This Feshbach resonance has a width of approximately 3 G and occurs
via an avoided crossing between the open scattering channel K |F = 9/2, mF = −9/2i +Rb |1, 1i and the closed molecular scattering channel K |7/2, −7/2i + Rb |1, 0i. Startingfrom a mixture of approximately 2.5 · 105 40 K atoms and 3.5 · 105 87 Rb atoms at 200 nK
prepared in the optical dipole trap at a magnetic field of 553 G, we ramp the magnetic field
through the Feshbach resonance to a field of 545.90 G in 4 ms. This results in the formation

12of approximately 6 · 104 Feshbach molecules with an expansion energy of kB · 300nK. At themagnetic field of 545.9 G, the Feshbach molecules are very weakly bound with a binding
energy of h × 240 kHz and a size of approximately 300 Bohr radii.
In the experiment, we observe conversion efficiencies from free atoms to Feshbach
molecules of about 15 − 25%. The conversion efficiency is somewhat limited by a trade-off
between high phase-space density of the initial atomic gases and good spatial overlap between
the two species. To achieve good spatial overlap, the different quantum statistical characterof 40 K (fermion) and 87 Rb (bosons) requires a temperature of the atomic gases just at theonset of quantum degeneracy - instead of starting deep in the quantum degenerate regime.
Another limiting factor for the conversion efficiency is strong inelastic losses between 87 Rb
and 40 K87 Rb in the vicinity of the Feshbach resonance.
2.1.3

Two-photon coherent transfer - STIRAP

To prepare a high phase-space density ensemble of ground-state polar molecules, theensemble of Feshbach molecules has to be transfered into the rotational and vibrational
ground-state. The transfer has to be done coherently to preserve density and temperature and therefore phase-space density - of the initial Feshbach molecule gas.
We make use of a two-photon coherent transfer to transfer Feshbach molecules, prepared in the least bound vibrational state of the electronic ground-state molecular potential,
to the rotational and vibrational ground state. The rotational and vibrational ground stateis bound by approximately h · 125 THz. A sketch of the scheme is shown in fig. 2.1. The
Feshbach molecule state (|ii) is coupled via a laser field to a vibrational state in an electronically excited molecular potential (|ei). This electronically excited state is then coupled
via a second laser field to the rotational and vibrational ground state (|gi). Together, thetwo laser fields provide an effective coupling between the Feshbach state and the rotationaland vibrational ground state. As the transfer scheme is fully coherent, the weakly boundFeshbach molecules are directly driven to the rovibrational ground state. This avoids any

13heating of the molecules due to spontaneously emitted photons.
In the experiment, we make use of a very robust two-photon coherent transfer scheme- called STIRAP (STImulated Raman adiabatic passage). This specific scheme results in arobust transfer of the initial quantum state |ii to the final quantum state |ei without everacquiring population in the lossy electronically excited state |ei. The latter just provides abridge between the initial and the final state [19]. Fig. 2.2 shows the details of the specificpulse sequence for the STIRAP transfer. The experiment starts with the entire molecularensemble prepared in the initial state |ii. Then, via the intensity ramp sequence shown infig. 2.2, a coherent superposition is established between the intermediate state |ei and thefinal state |gi with laser 2 field. The intensity of laser 2 is adiabatically ramped down in
5 µs, and at the same time, the intensity of laser is ramped up, performing transfer to thefinal state |gi, without populating |ei.
However, the actual implementation of the single-step of coherent two-photon transfer
is challenging. First, it is very critical to achieve good coupling between the initial Feshbachstate and the final ground molecular state. But these two wavefunctions are vastly different
in size and the direct wavefunction overlap is very small. It is therefore necessary to identify
a bridge in the electronically excited state to achieve the best possible coupling between theinitial and the final state. In short, we need to choose an intermediate state with sufficient
wave function overlap for both initial and final states, i.e. Franck-Condon factor (FCF).
Second, it is critical to have phase coherence between the two lasers involved in the transfer
scheme. This is challenging, because the two lasers bridge a frequency gap of 125 THz andhave therefore vastly different wavelengths. In the following, we will discuss how we addressand solve these experimental challenges.
2.1.3.1

The choice of the intermediate state

As pointed out in the previous paragraph, an appropriate choice of the electronically
excited intermediate molecular state is critical for the preparation of a high-phase density gas

14

Figure 2.1: Potential curves for 40 K-87 Rb molecules and energy levels used for STIRAP
transfer of Feshbach molecules to ground-state. The Feshbach molecules are formed in theinitial state |ii = a3 Σ. The intermediate state |ei = (v 0 = 23) is in the electronically excitedpotential of 23 Σ, and the final state |gi is in the ro-vibronic ground state N = 0, v = 0 of
X 3 Σ. The initial and the final states are about 125 THz apart from each other. This figure
is reproduced from reference [26].

of ground state polar molecules. In our experiments, we make use of the scheme sketchedin fig. 2.1 [26]. In this scheme, two cw lasers with wavelength of 970 nm and 690 nm
couple the Feshbach state in the 3 Σ electronic ground-state molecular potential and therovibrational ground-state in the 1 Σ electronic ground state molecular potential to a common
intermediate state. The chosen intermediate state with good Franck-Condon factor for both
FBM and GSM is the v 0 = 23 state of the 23 Σ electronically excited potential. This state

15
has a small admixture of singlet spin character due to the proximity of the 1 Π molecularpotential, which allows the transfer of triplet spin character Feshbach molecules to singlet
spin character rovibrational ground state molecules. The upward transition strength fromthe Feshbach molecule state to this intermediate state was determined to be 0.005(2) ea0 [48].
The downward transition strength from the intermediate state to the rovibrational groundstate has been determined to be 0.012(3) ea0 [48].
2.1.3.2

Phase and frequency stabilization of the coherent transfer lasers

In order to implement an efficient coherent transfer between two quantum states using
STIRAP, it is essential to realize good phase coherence between the lasers involved in thetransfer scheme. Loosely speaking, this means that the relative frequency jitter between the
two lasers has to be as small as possible during the time window of the coherent transfer.
In our specific application, achieving good phase coherence is a challenging task, since thetwo lasers involved in the transfer scheme are operating at vastly different wavelengths,
corresponding to a frequency difference of 125 THz. A stable two-photon beat between
these two lasers can be maintained by referencing each laser individually to a stable opticalfrequency comb.
The frequency comb
Frequency combs are powerful tools for several applications [49, 50, 51, 52, 53]. In ourcase, it is used as a stable “frequency ruler”, as shown in fig. 2.3, with 300 THz of spectral
range (from 500 nm to 1100 nm). This frequency ruler is used to phase-lock two Raman
lasers with a frequency difference of 125 THz [26].
The frequency comb used in our experiments is a solid state, ultracompact femtosecond
titanium-sapphire (Ti:S) laser, pumped by a 532 nm Verdi 10W laser from Coherent. The
pump power is 4.9 W (where the maximum power pumped to the Ti:S crystal should be
no higher than 5.5 W). The laser has a ring cavity, with a 750 MHz repetition rate. The
spectrum is centered at 800 nm and is 30 nm wide. The laser was modified to be actively

16





:















:

:

:

 









:




















 : Figure 2.2: The schematic diagram of STIRAP pulse sequence. In (a), the molecules are in
the initial state. In (b), the intensity of laser 2 is adiabatically ramped up, coupling states|ei and |gi. In (c), the transfer of molecules in state |ii to state |gi happens by adiabatically
ramping the intensities of laser 2 down and laser 1 up. In (d), laser 1 intensity is ramped
down, and (e) is the end of the process, where about 90% of the Feshbach molecules are
tranfered to the ground state.

17stabilized by controlling two degrees of freedom that determine the frequency of each comb
tooth: the repetition rate, frep , and the carrier-envelope-offset frequency, fceo . The frequency
for each comb tooth is given by the equation:

νn = nfrep ± fceo ,(2.1)

where n is a integer number that represents the number of the comb tooth. frep = 750 MHz,and fceo is in the range of 0 ≤ fceo ≤ frep /2. To be useful as a “frequency ruler”, both frep
and fceo have to be stabilized. In the following, I will detail the stabilization of these twodegrees of freedom.
Frequency comb stabilization
As already mentioned above, the frequency of the teeth of the optical frequency comb
are determined by two degrees of freedom: frep , and fceo . These two degrees of freedom need
to be controlled precisely. The stabilization of frep is realized by controlling the Ti:S laser
cavity length via two piezoelectrics (PZTs) used as actuators. Using these knobs, frep is thenstabilized by referencing a certain frequency tooth in the spectrum of the comb to a stable
1064 nm Nd:YAG. The Nd:YAG laser itself is locked to a stable cavity in John Hall’s lab andhas a linewidth lower than 1 kHz. The locking electronics of frep are shown in fig. 2.5, where
the error signal is fed back to the phase-lock loop electronics to the slow and fast PZTs.
The stabilization of fceo is a bit trickier. It requires a full octave spanning spectrum.
To expand the Ti:Sa laser spectrum, a Photonic Crystal Fiber (PCF) [54], FEMTOWHITE
800 (polarization maintaining fiber and encapsulated from Crystal Fibre) is used [21]. The
detection of fceo is done via a so called f-2f interferometer. This interferometer basicallybeats the low (1064 nm) and the high (532 nm) frequency portions of the spectrum to extract
fceo [28]. The low frequency portion is doubled using a beta-barium borate (BBO) crystal
and beat against the high frequency portion. The essentials of this scheme are schematicallyshown in fig. 2.6. Eq. 2.2 shows a simple relation that the f-2f interferometer uses to extract

18

I(Q)

Qn= n fREP + fCEO
fREP

Q

fCEO

Q970

Q690

Figure 2.3: Schematic picture of the spectrum of the frequency comb. The two CW lasers
at frequencies ν690 and ν970 are phase-locked to the comb.

fceo :

2νn − ν2n = 2(nfrep + fceo ) − (2nfrep + fceo ) = fceo .

(2.2)

The error signal of fceo , is fed back via the phase-lock electronics to a Acousto-Optical
Modulator (AOM), as shown in fig. 2.7. The AOM controls the Verdi-10W laser intensity
by dumping a variable amount of pump power, between 0% to 5%, to the first diffraction
order. The zeroth order is used for pumping the Ti:S laser.
Realizing the CW Raman lasers

19

Diode
beat
From CW
Diode laser,
970 nmAOM

Verdi
532 nm, 10 W

F6

O/2

O/2

1st order
O/4
F4

750 MHz Ti:S comb laser
O/2

Fast PZT

Slow PZTBS 2Beam dump
F2

O/2
BBO
CrystalOC
PCF
L~30 mmTi:S Crystal

Fceo
beat

O/4

BS 1

F1

Mirror underneaththe beam

From CW Stabilized
YAG laser, 1064 nm

Frep
beat

O/2

From CW Diodelaser, 690 nm

Diode
beat

O/2
O/2

F5

O/2
F3

Figure 2.4: Frequency comb laser set up. This schematic diagram shows the Ti:S cavity(red box), the f-2f interferometer (blue box), the heterodyne beats for frep (dashed bluebox), and the ν690 (dashed red box) and ν970 (orange box) phase-locks. The the light fromthe Ti:S is then spectrally broadened in the photonic crystal fiber (PCF) to cover a range
from 500 nm to 1100 nm. The input power to the PCF is 680 mW. fceo is stabilized via
a f-2f interferometer. frep is phase-locked to a narrow YAG laser at 1064 nm. The several
optical components are used to deliver the light with the right wavelength to their respective
heterodyne beat set ups. F2, F3, F4, F5, and F6 are all band pass filters with FWHM 10
nm, at 532, 690, 1064, 1064, and 970 nm respectively. F4 also is used as a “mirror” to reflect
970 nm light. F1 is a high pass filter with R(532 nm)/T(600 nm - 850 nm). BS1 and BS2
are dichroic beam splitters where the reflection and transmition of the first are R(950 nm 1100 nm)/T(500 nm - 920 nm), and for the second are R(532 nm)/T(920 nm - 1100 nm).

20

Figure 2.5: Locking electronics for frep . The beat signal of frep and the Nd:YAG laser isamplified, goes to a band pass filter (BPF), and then to a home built digital phase detector,
where the frequency is divided and the signal is mixed to a direct digital synthesiser (DDS).
The DDS is referenced to a 10 MHz commercial quartz clock (Wenzel Associates). The error
signal goes to a home built loop filter, where the signal is divided and sent to a PZT driver toa fast PZT (≈ 50 kHz bandwidth), and to a slow integrator box (with time constant of ≈ 45
seconds), to a high voltage driver to a slow PZT. Using both fast and slow PZTs guarantee
short and long term stability for frep .

In the experiment, we make use of grating feedback stabilized laser diodes to provide
single frequency laser light at 970 nm and 690 nm. Details on the setup can be found in fig. 2.8
and fig. 2.9, respectively. For both lasers, we use commercially available AR-coated diodelasers from Eagleyard in an external cavity in Littman configuration. We have chosen thislaser configuration as opposed to a Littrow configuration to allow for a large frequency tuning

21

Figure 2.6: fceo detection via f-2f interferometer. From a broad spectrum, for fceo detection,
the important wavelengths are 532 nm (high frequency portion) and 1064 nm (low frequency
portion). The low frequency portion is doubled in a BBO crystal, and overlapped with thehigh frequency portion. The difference between the doubled low frequency portion and thehigh frequency portion is fceo . Figure reproduced from reference [55].

range without changing the beam pointing direction [56, 57]. However, at the same time,some output power is sacrificed as compared to the more common Littrow configuration.
The output of the laser is split into three separate beams: one beam is used for monitoring
the wavelength with a commercial wavemeter, one for the heterodyne beat with the comb forphase-locking, and the last one is sent to the molecules to perform STIRAP. To achieve large
Rabi frequencies on the order of MHz for both the up transition and the down transition of
STIRAP, we need to amplify the power of both lasers. The 970 nm light is amplified using a

22

Figure 2.7: Locking electronics for fceo . The fceo beat signal is amplified, fed to a band passfilter (BPF), and then to a home built digital phase detector, where the frequency is divided
and the signal is mixed to a direct digital synthesiser (DDS). The DDS is referenced to a 10
MHz commercial quartz clock (Wenzel Associates). The error signal goes to a home built
loop filter, to a bias box, which puts an offset in the output voltage going to an AOM driver,feeding back the AOM that controls the intensity of the Verdi 10W laser.

tapered amplifier from Eagleyard. The 690 nm light is amplified by an injection locked diodelaser. The amplified light is then sent to the experiment via a 10-meter long optical fiber.
Typically, we deliver about 50 mW for the 970 nm laser, and 20 mW for the 690 nm laser
to the experiment. the Raman laser beams are then focused down onto the atoms. In theexperiment, we prepare a spot size of 40 µm × 200 µm. This configuration results in Rabi
frequencies for the up and down transition of 0.3 MHz and 3.0 MHz, for dipole strengths of230.005(2) ea0 and 0.012(3) ea0 respectively [26].

     ! "#$ %#& '(          O   

O,-.  /012  




O


O

  

)*+     3 
OFigure 2.8: Raman laser system for 970 nm transition. This system has three outputs: onefor a wavemeter, one for the frequency comb and one for the STIRAP transfer. The powerof the laser is amplified by using a tapered amplifier.

2.1.4

Preparation of ground-state polar molecules

With these experimental prerequisites in place, we can transfer Feshbach molecules
to the rovibrational ground state. Our experimental data are shown in fig. 2.10. In the

24    !" #!$ %&       O   

+,-  ./01O   

O

   

O

   2 '()  *  
3    O

4   Figure 2.9: Raman laser system for 690 nm transition. This system has three outputs: onefor a wavemeter, one for the frequency comb and one for the STIRAP transfer. The powerof the laser is amplified by using another diode laser (slave laser) seeded with ≈ 1 mW ofpower from the master laser. The slave laser is locked via injection lock.

experiment, we start with about 6 · 104 Feshbach molecules. These molecules are thentransfered to the rovibrational ground state. However, the ground-state molecules are dark
to our imaging light. To image these molecules, we then reverse the STIRAP sequence andmap the molecules back onto the Feshbach state, where we image them via direct absorption

25

Figure 2.10: Timing diagram of the coherent two-photon transfer (STIRAP) from Feshbach
molecules to the absolute rovibrational ground state X 1 Σ (v = 0). (a) STIRAP pulse
sequence, where I1 and I2 are the intensities of laser 1 and laser 2. (b) Measured populationin the initial Feshbach state during the STIRAP pulse sequence. Starting with 4·104 Feshbach
molecules, the molecules are coherently transferred to the rovibrational ground state X 1 Σ(v = 0) by the first pulse sequence (t = 15 to 20µs). The rovibrational ground state molecules
are invisible to the detection light. Reversing the pulse sequence, X 1 Σ (v = 0) molecules
are converted back to weakly bound Feshbach molecules (t = 45 to t = 50µs). This figure is
reproduced from reference [48].

imaging. From fig. 2.10, we can extract an efficiency of the entire process of 80% round triptransfer. This suggests a a creation efficiency of ground state polar molecules of 90% [26].
We also demonstrated a scheme for direct absorption imaging of GSM [58].
Note that the transfer process from Feshbach molecules to ground-state polar molecules
proceeds without introducing heat into the molecular cloud. This is illustrated in fig. 2.11
where we compare the expansion dynamics of Feshbach molecules to that of ground-state
molecules. The expansion dynamics gives us precise information about the temperatureof the molecular clouds. From the data, we extract for both ground-state molecules andFeshbach molecules a temperature of 400 nK. The two temperatures agree within the error

26bars. This shows that the huge amount of binding energy of the ground state molecules is
not released to the molecular ensemble during the transfer but instead carried away by thetwo light fields.

Figure 2.11: Comparison of the kinetic energy of the Feshbach molecules before STIRAP
transfer (blue circles) and after a round-trip STIRAP process (red circles). The latter canbe interpreted as an upper limit on the kinetic energy of the rovibrational ground state
molecules. The temperature of both clouds is extracted by time of flight expansion analysis.
We extract T = 400(15) nK for the Feshbach molecules and T = 430(20) nK for rovibrational
ground-state molecules. The analysis shows that the transfer process does not cause any
noticeable heating on the molecules. This figure is reproduced from reference [48].

2.2

Essential manipulation techniques for ground-state polar molecules
In the experiments presented in this thesis, two manipulation techniques of the groundstate molecules are of large importance: accurate control over the dipole moment of themolecules and accurate control over the external confinement. These two experimental knobs
allow us to demonstrate a large degree of control over the quantum dynamics of the collisional
processes.

27
2.2.1

Electric dipole moment of the molecules

Figure 2.12: Stark spectroscopy of the singlet v = 0 molecules. The bottom panel shows the
Stark shift of the rovibrational ground-state of the singlet potential (v = 0, N = 0 of X 1 Σ),and the top panel shows the shift of the v = 0, N = 2 state. The systematic error in theapplied electric field is 3% (horizontal error bars). The level difference between N = 0 and
N = 2 is 6.6836(5) GHz, which yields a rotational constant B of 1.1139(1) GHz. Given themeasured B, the fit of the Stark shift (line in lower panel) gives a permanent electric dipole
moment of 0.566(17) D. The theory curves for N = 2 for different |mN i projections (lines inupper panel) are calculated using the measured B and the dipole moment derived from the
N = 0 fit. This figure is reproduced from reference [26].

One of the most interesting aspects of ultracold polar 40 K-87 Rb molecules in the groundstate is their electric dipole moment. Using two-photon Stark spectroscopy, we measured
the dipole moment of the ground state 40 K87 Rb molecules to be 0.566 Debye [26], as shownin fig. 2.12. However, this dipole moment has to be induced an applying an external electric field. This electric field induces a dipole moment in the molecule by mixing different rotational states with opposite parity. Fig. 2.13 shows how the dipole moment of the40

K87 Rb molecule develops as function of the applied external electric field. The dipole

moment of the molecules is zero at zero electric field, then initially increases linearly with

28

Dipole moment (Debye)

increasing electric field and finally saturates to a value of about 0.56 Debye.

0.5
0.4
0.3
0.2
0.1
0.0
0

5

10

15

Electric field (kV/cm)

20

Figure 2.13: Calculated dipole moment versus applied DC electric field. In our experiment
we can go to electric fields as high as 5.0 kV/cm.

In the experiment, we implement the external electric field by means of two electrodes
located outside of the glass cell (“science chamber”) as shown in fig. 2.14. The plates are
made from #CH-50IN-S209 transparent plates, coated with indium tin oxide from Delta
Technologies. This material transmits 80-90% of the light in the wavelength range from 600
nm to 1100 nm and allows us to shine all necessary laser beams through the electrodes. The
plates dimensions are 4 cm length, 2.37 cm width, and 0.09 cm thickness. The distance
between the two plates is 1.35 cm. The thickness of the glass cell wall is 0.125 cm and it
is made with a type of pirex glass with dielectric constant ≈ 4.8. The electric field in thecenter of the glass cell is calculated from the geometry after taking into account the dielectric
constants. For each 1 volt applied across the plates, the electric field in the center of theglass cell is:29

1V
1.35 cmV= 0.87 .
1.35 cm 1.35 cm − 2 · 0.125 cm + 2 · (0.125/4.8) cm
cm

(2.3)

Since the electrode plates are outside of the vacuum chamber, dielectric breakdowns
at relatively low applied voltages can happen. This limits the maximum voltage difference
that can be applied across the plates to 5.6 kV and therefore limits the accessible dipole
moment range to 0.20 Debye. However, within this range, we can tune the dipole momentof the molecules at will (from 0 to 0.20 Debye), therefore controlling collisional processes in
the quantum regime (see chapter 5).
2.3

Tuning the geometry of the external confinement - 1D Optical LatticeDue to the anisotropic nature of the dipole-dipole interactions, collisional properties ofthe molecules change drastically as a function of the geometry of the confinement (2D/3D)
and the orientation of the dipole moment of the molecules with respect to the axis of tightconfinement. This fact provides the motivation to implement versatile confinement geometries in our experiment. The crossed dipole trap mentioned above realizes a pancake geometry
with an aspect ratio of 1:4. Although this geometry resembles somewhat a two-dimensional
confinement geometry, this trap is providing a 3D geometry in terms of physics that canbe studied in the confined system. This is the case since all energy scales of the confining
geometry (~ωz ) are still much smaller than the kinetic energy in the system (kB T ).
However, we can realize a 2D geometry by making use of a 1D optical lattice. The
1D optical lattice will result in a stack of pancake-like trapping structures, where the energy
scale of tight confinement (~ωz ) is much larger than the kinetic energy in the system. This
restricts the molecules to a single quantum state in the tight confinement direction andtherefore changes drastically the collisional properties of dipolar molecules as compared toa 3D system (see chapter 5).
In the experiment, we construct a one-dimensional lattice by establishing an optical

301064 nm lightfrom Verdi laser

Lens f=50 cm

O/2><
Breadboard
Positive
E-field plate20
gravity

1.35 cmGlass cell

1 cm

Mirror with
~2% transmition

Negative
E-field plate
Lens f=50 cm><

Photo detector
for intensity
stabilization>
Optical Table

Figure 2.14: Schematic figure of the electric field plates and 1D optical lattice set up. The
maximum output power of the fiber is 1.7 W. The polarization is set to be parallel to theoptical table. The intensity of the optical lattice is stabilized via an AOM, which also shifts
the frequency by +80 MHz.

standing wave. A schematic figure of the optical lattice’s experimental set up is shown infig. 2.14. The optical lattice potential will then divide the cloud in the optical dipole trap
into a series of pancakes with tight confinement along the vertical ẑ axis (in the gravity
direction). The optical lattice is realized by a single retroreflected laser beam. The beam
waist of the optical lattice matches the beam waist of the dipole optical trap, in order toavoid heating as the intensity of the optical lattice is ramped up. The beam waist of the
optical lattice is 250 µm. For the optical lattice we use a single-mode Verdi-IR laser with

31
wavelength of 1064 nm, and power up to 25 W. We send light via a polarization-maintaining
optical fiber with coupling efficiency of 40 to 50 %. At the output of the optical fiber, we
get a maximum power of 1.7 W going to the molecules. The laser beam passes through theelectric-field plates and the glass cell and then is reflected back, forming the optical latticepotential inside the cell. Due to losses in the electric field plates and the glass cell (the glass
cell in not AR coated), the beam power after the rectroreflection is 40 % lower than theinitial power. The optical lattice beam is tilted from the vertical direction by an angle of 2
degrees to avoid back reflections inside of the glass cell impinging the molecular cloud andcausing corrugations in the potential. The stabilization of the optical lattice depth is doneby controlling the laser beam intensity. Approximately 2 % of the light leaks through theretroreflection mirror. This light is detected and sent to a servo that feeds back to an AOM
that controls the amount of light going to the optical fiber.

Chapter 3
Dipolar 40 K87 Rb Molecules in 3D geometry

The preparation of a near quantum degenerate gas of all ground-state polar molecules
opens tremendous opportunities for the study and control of molecular collisions in the limitof vanishing collisional energy. For chemically reactive molecules such as 40 K87 Rb, it opens
the unique opportunity to study chemical reaction dynamics in the completely unexplored
limit where both the internal quantum states and external quantum states of motion are
well defined. In this regime, collisions can be analyzed by simple quantum mechanical rules
such as quantum statistics, single partial wave scattering, and quantum threshold laws.
In this chapter, I present the studies of ultracold chemical reactions in the quantumgas of polar molecules when molecules are confined in a 3D geometry as background material for my thesis work. These studies were already discussed in the thesis work of K.-K.
Ni [21]. Starting with an optically trapped near-quantum-degenerate ( TTF ≈ 1.4) gas of polar
40

K87 Rb molecules prepared in their absolute ground state, we observe experimental ev-

idence for exothermic atom-exchange chemical reactions. When these fermionic molecules
are prepared in a single quantum state at a temperature of a few hundred nanokelvin, we
observe p-wave dominated quantum threshold collisions arising from tunneling through anangular momentum barrier followed by a short-range chemical reaction with a probability
near unity. When these molecules were prepared in two different internal states, the reaction
rates were enhanced by a factor of 10 to 100 as a result of s-wave scattering, which does nothave a centrifugal barrier. Inducing an electric dipole moment in the molecules results in

33
tuning of the chemical reaction rate over several orders of magnitudes. Finally, I present thethermodynamics of the molecular cloud under the influence of strong collisional loss revealing
the anisotropic nature of the dipolar interaction. Throughout this chapter, I closely follow
references [39, 40], which describe our work.
3.1

Bimolecular chemical reactions at ultracold temperature
Molecular quantum gases of 40 K87 Rb molecules offer the unprecedented opportunity

to study chemical reactions in the limit of vanishing collisional energy. 40 K87 Rb molecules
are not stable against atom exchange chemical reactions. Two 40 K87 Rb molecules collide
and form with a high probability different chemical species, namely K2 and Rb2 [39]. This isbecause the chemical reactions KRb+KRb→K2 +Rb2 is exothermic and moreover proceeds
without chemical reaction barrier at short range [39].
However, chemical reactions have never been studied in the limit of vanishing collisional energy. Even basic questions such as how a chemical reaction proceeds at ultralow
temperature and what quantum mechanical rules are responsible for molecular reactivity
have never been addressed.
That chemical reactions could occur at ultralow temperatures seems at first glancecounterintuitive. However, ultracold collisions, where particles scatter only in the partial
wave with lowest angular momentum, are governed by quantum statistics and quantum
threshold behaviors described by the Bethe-Wigner laws [59, 60]. In this regime, particles
are represented by their de Broglie wavelength, which increases with reduced temperature.
This wave nature of particles replaces our intuitive classical picture of collisions. The wave
manifestation of tunneling through reaction or angular momentum barriers may play a dominant role in dynamics, and scattering resonances can have dramatic effects on reactions [61].
In addition, any barrierless chemical reactions will always take place when two reactants
are sufficiently close together [62]. In this case, chemical reaction rates will be determined
to a large extent by collisional properties at large intermolecular separations, and thus by34
how the two molecules approach each other. Once their separation reaches a characteristic
length scale ( 10a0 , where a0 = 0.53 · 10−10 m), a chemical reaction happens with a near
unity probability. Therefore, chemical reactions can be surprisingly efficient even at ultralow
temperatures. Indeed, this model for barrierless reactions predicts loss rates that are universal in the sense that they do not depend on the details of the short-range interactions, butinstead can be estimated using only knowledge of the long-range interactions [63].
3.2

The role of quantum statistics in determining the chemical reaction rate
For molecules prepared at a temperature of a few hundred nanokelvin, the collision rateis largely determined by the quantum statistics of the molecular ensemble. In the regime ofvanishing collisional energy, collisions proceed in a single partial wave, because higher orderangular momentum scattering channels are frozen out due to large centrifugal barriers. Thismeans that depending on quantum statistics, molecules will collide in an s-wave L = 0
or p-wave L = 1 angular momentum scattering channel. In our experiments, we prepare
fermionic 40 K87 Rb molecules. For spin-polarized molecules all prepared in the same internal
quantum state, the p-wave collisional channel is then the lowest energy symmetry-allowed
collision channel. The height of the centrifugal barrier for the p-wave (L = 1) KRb-KRb
collision is kB · 24 µK. It is determined by the molecule-molecules long-range potential:
VLongRange (R) =~2 L(L + 1) C6
− 6,
2µR2
R

(3.1)

where L = 1, µ is the reduced mass of the 40 K87 Rb and C6 = 16, 130Eh · a60 (a.u.) [64],where Eh = 4.36 · 10−18 J. This barrier height is more than an order of magnitude larger thankB T , where T is the translational temperature of the molecular gas. Thus, collisions of spinpolarized molecules are expected to proceed predominantly via tunneling through the p-wave
barrier. If two molecules make it through the barrier to short range, chemical reactions or
hyperfine state-changing collisions can take place, leading to a loss of the entrance channel
population. We note that even a single nuclear spin flip corresponds to a released quantity of35
energy that is above the trap depth, and this would contribute to loss of trapped molecules.
Preparing a 50:50 mixture of two different spin states allows molecules to collide in an s-wave
collisional channel. Thus, there is no centrifugal energy barrier and chemical reactions will
therefore proceed at a much higher rate.
3.3

The experimental system
To study chemical reactions of molecules in the quantum regime, we prepare a gaseous

molecular sample at densities on the order of 1012 to 1013 cm−3 and temperatures of around
200 to 300 nK. The fermionic 40 K87 Rb molecules are prepared in a single hyperfine level ofthe rovibronic ground state (N = 0, v = 0 of X 1 Σ+ ) [26, 65] and in particular in the absolute
lowest internal quantum state. At the ultralow temperature of a few hundred nK, even thetiny molecular hyperfine-state energy splitting of ≈ 760 kHz at B = 545.9 G [65], is muchlarger than the translational energy (300 nK corresponds to 6 kHz). The preparation ofa well-defined hyperfine state is possible and is of essential importance for the experiments
presented in this chapter. Complete control over the internal quantum state of the molecules(including hyperfine quantum degrees of freedom) permits direct observation of the role ofquantum statistics in determining the molecular interactions and the chemical reaction rate.
Details on the precise and coherent manipulation of the molecular population in different hyperfine states of the rovibrational ground state X 1 Σ+ of 40 K87 Rb can be foundin [65]. The X 1 Σ+ state has zero total electronic angular momentum, so that the hyperfine
structure is basically the Zeeman effect of the nuclear spins I K = 4 and I Rb = 3/2 [63, 66]
at the applied magnetic field. The hyperfine structure is depicted in fig. 3.1, where a total
40
of 36 states are labeled by their projections of the nuclear spins of 87 Rb , mRb
K , mK
I and
I

onto the external magnetic field. For the study of chemical reactions, we produce molecules¯®
Rb, or in a mixture of two spin states. The hyperfine
either in a single spin state ¯mK
I , mI
states used are an excited state |−4, 1/2i and the lowest-energy spin state |−4, 3/2i. These
are marked by the two ellipses in fig. 3.1. The |−4, 1/2i state is populated directly by the36
two-photon Raman transfer starting from Feshbach molecules [26, 65]. The spin state ofthese molecules can be further manipulated coherently. For example, the entire |−4, 1/2i
population can be transferred into the lowest hyperfine state, |−4, 3/2i, using two successive
microwave π-pulses through a rotationally excited N = 1 intermediate level as shown infig. 3.2. Within the N = 1 manifold, coupling between rotation and the nuclear electric
quadrupole moment of 87 Rb or 40 K atoms lead to mixing of different nuclear spin characters.
Due to this mixing, it is possible to use the rotationally excited state as a bridge between
different hyperfine states in the rotational ground state [65, 66]. We can probe molecules
in any particular hyperfine state by reversing the entire transfer process and putting the
population back into the initial weakly bound state. We use absorption imaging to measure
the molecular gas number and temperature. Alternatively, we can also do direct image ofground state molecules by using a open optical transition [58].
3.4

Bimolecular chemical reactions at vanishing dipole moment
In first experiments, we studied the chemical reaction KRb+KRb → K2 +Rb2 in thelimit of vanishing electric field. In this limit, the 40 K87 Rb molecules do not have an electric
dipole moment in the lab frame. The long-range interaction potential between two molecules
is then entirely set by the competition between the attractive van-der-Waals interaction andthe centrifugal barrier of the collisional process. Since the centrifugal barrier depends on L,
the quantum statistics of the molecules has a important role for the long range interaction
potential.
3.4.1

p-wave collisions

When the molecules are prepared in a single hyperfine state within the rotational andvibrational ground state, they are indistinguishable from each other. Since these molecules
are fermions, the collision can only happen through odd angular momentum channels. In thelimit of vanishing collisional energy, the dominating collisional channel is then the p-wave

37

Figure 3.1: Hyperfine structure of rovibronic ground-state 40 K87 Rb molecules at a magnetic
Kfield of 545.9 G. Here we have 36 nuclear spin states by their spin¯ projections,mRb
I and mI .
¯
¯ K¯The¯ energy¯ spacing between hyperfine states is h · 130 kHz for mI = 1 and h · 760 kHz
Rbfor ¯mI ¯ = 1. By comparison, at a temperature of 300 nK, the molecules thermal energy isequivalent to h · 6 kHz, which is more than an order of magnitude smaller than the spin flip
energy. Molecules can be prepared in either a single state or in a mixture of The |−4, 1/2i
and the lowest-energy state |−4, 3/2i (open ellipses). Figure reproduced from reference [39].

collisional channel and the competition between the van-der-Waals interaction and the finite
centrifugal energy, which results in a p-wave collisional barrier given by eq. 3.1, with theangular momentum L = 1 for p-wave collisions. Here C6 = 16, 130 a.u. [64], and the heightof the centrifugal barrier is 24 µK.
As the collisional energy of molecules is in the sub µK regime, and much smaller thanthe height of the centrifugal barrier, chemical reactions can only proceed through quantum
mechanical tunneling through the p-wave barrier. On the other hand, once molecules get
within short range of each other, chemical reactions proceed with near-unity probability [41,67]. We therefore expect the chemical reaction rate to be set by the tunneling rate of themolecules through the p-wave barrier.

38

N=1, |-4, 3/2> +g|-4, 1/2>

N=0, |-4,3/2>

N=0, |-4,1/2>

Figure 3.2: Transfer from the rovibrational ground state |−4, 1/2i to the absolute groundstate |−4, 3/2i via coherent Rabi transfer. The intermediate state is in the rotational level
N = 1 and it is a mixed state with g≈ 5 %.

Fig. 3.3A shows the decay of 40 K87 Rb molecules in a single quantum state |−4, 1/2i ona slow time scale of a few seconds. Analysis of the decay can be performed using a simple
two-body collisional loss model where the density decay is given by:dn(t)= −βn2 .
dt(3.2)

Here, β is the two-body decay coefficient.
The quantum nature of the collisions can be observed in the temperature dependenceof loss rates. The Bethe-Wigner threshold law predicts that the p-wave inelastic (reactive)
collision rate is directly proportional to temperature (β ∝ T ) [59, 60]. To observe thisbehavior, we prepared spin-polarized molecules in the single hyperfine state |−4, 1/2i forseveral values of T ranging from 200 to 900 nK. The temperature of the molecular gas iscontrolled by changing the initial temperature of the 40 K and 87 Rb atoms during the evaporative cooling. The temperature is measured from the expansion energy of the molecular
gas after releasing it from the optical trap. For each initial temperature, we look at the time-

39
dependent molecular loss and extract a two-body loss rate β (which is twice the collisional
event rate) by fitting the decay of the molecular gas density n versus time t (fig. 3.3a) todn= −βn2 − αn.
dt

(3.3)

Here, the first term on the right takes into account the number loss and the measured β
can be compared to theoretical predictions. The second term allows density change due
to heating of the trapped gas during the measurement. Within a single measurement, weobserve an increase in temperature that is at most 30%. In the analysis for each data set,
we fit the measured temperature to a linear heating rate and obtain a constant slope c. In
eq. 3.3, α = (3/2)[c/(T + ct)], where T is the initial temperature. At the lowest temperaturethat we can reach, 250 nK, the heating was (7±1) nK s−1 and β = (3.3±0.7)· 10−12 cm3
s−1 . The measured dependence of β versus T is summarized in fig. 3.3b (solid circles).
Here, we fit the data to a power law β(T ) ∝ T L and find that L = 1.1(±0.2), which agrees
with the predicted p-wave threshold law. This result demonstrates that indistinguishable
40

K87 Rb molecules at ultralow temperatures collide via tunneling through a p-wave barrier

followed by an inelastic collision in the short range. A linear fit to the data (L = 1) yields aslope of the decay rate coefficient of (1.2 ± 0.3) · 10−5 cm3 s−1 K−1 .
We repeated this measurement for molecules in the lowest hyperfine state |−4, 3/2i
(open triangles in fig. 3.3b). The data again show β ∝ T with a slope of (1.1 ± 0.3) · 10−5
cm3 s−1 K−1 , similar to that measured for molecules in the |−4, 1/2i state. However, in thecase of |−4, 3/2i molecules, hyperfine state-changing collisions are no longer possible and theonly possible loss channels are the chemical reactions discussed above. Thus, we find thatthe rate of chemical reactions is determined by the p-wave angular momentum barrier, andthe chemical reaction barrier must be below the collision energy. This suggests that thesereactions are barrierless and can thus occur freely at ultralow temperatures. Meanwhile,the fact that the same loss rate is observed for both |−4, 1/2i and |−4, 3/2i state moleculessuggests that chemical reactions dominate the loss in these ground-state molecular collisions.

40
In order to understand the loss rates, two models are used: a quantum thresholdmodel (QT) [41], and a model that uses the formalism of multichannel quantum defecttheory (MQDT) [67]. In the QT model, the loss rate for collisions with energy equal toor above the height of the p-wave barrier is determined by the Langevin capture rate [68],which assumes that the probability for chemical reactions and/or hyperfine state-changing
collisions is unity. For collision energies lower than the height of the p-wave barrier, in
this model we assume that the loss follows the Bethe-Wigner threshold law [59, 60]. Using
this assumption, we have a analytical expression for the p-wave loss rate coefficient of two
indistinguishable molecules, which scales linearly with T [41]:πβ=4µ 17 3 3 ¶3 µ C6
kB T,
~10

(3.4)

where µ is the reduced mass. Using a van der Waals dispersion coefficient of C6 = 16, 130 a.u.
for KRb-KRb with an uncertainty of ±10%, the slope of the rate coefficient is predicted tobe 1.5(±0.1)·10−5 cm3 s−1 K−1 , which agrees well with the experimental data.
In the MQDT model, the loss rate coefficient is calculated directly by the quantum
tunneling rate through the p-wave barrier [69]. From this calculation β = 0.8(±0.1) · 10−5
cm3 s−1 K−1 , which agrees with the experiment within mutual uncertainties. This β can also
be derived analytically from the properties of the long-range potential to give β = (11.48ā)3(kB T /h), where ā = 0.4778(2µC6 /~2 )1/4 = 6.3 nm is the characteristic length of the vander Waals potential [63]. This fully quantum calculation can be put in the same form
as the QT model (eq. 3.4) and gives a β that is smaller by a factor of 0.528. With twomodels, the agreement with our molecule-molecule collisional loss measurements suggeststhat the chemical reaction rates are strongly influenced by the long-range interactions. Thisobservation opens intriguing control possibilities because the long-range interaction can becontrolled by selecting quantum states and tuning collision energies via applied electric andmagnetic fields.

41

Figure 3.3: Inelastic collisions between spin-polarized (indistinguishable) or different spinstate (distinguishable) fermionic molecules in the rovibronic ground state of 40 K87 Rb. (A)Data showing the time dependence of the molecule number density. Here the molecules areprepared in a single hyperfine state, |−4, 1/2i, and the molecular density decays slowly with arate coefficient of 3.3(±0.7)·1012 cm3 s−1 at T = 250 nK. (B) Loss rate coefficient as functionof the temperature. The loss rate increases linearly with temperature for spin-polarizedmolecules, which collide via p-wave [inset in (A)] at low temperature. Data were taken formolecules prepared in either |−4, 1/2i (solid circles) or the lowest-energy state |−4, 3/2i
(open triangles). A linear fit (solid line) to the |−4, 1/2i data yields the temperaturedependent loss rate to be 1.2(±0.3) · 105 cm3 s−1 K−1 . For the |−4, 3/2i case, where thecollisional loss can only be due to chemically reactive scattering, the loss rate is similar. The
dotted and dashed lines are the predictions from the QT and MQDT models, respectively.
When the molecules are prepared in a mixture of the |−4, 1/2i and |−4, 3/2i states (solid
squares), we observe a temperature-independent decay rate that is 10 to 100 times that forthe spin-polarized case. Figure reproduced from reference [39].

3.4.2

s-wave collisions

Chemical reaction rates are very different if molecules are prepared in a mixture ofdifferent hyperfine states. In this case the molecules will interact via the s-wave collisional
channel. The long range potential between two-colliding molecules is then dominated by
the attractive van der Waals interaction, due to the vanishing centrifugal energy of the swave collisional channel. The collision can happen without any collisional barrier and are
expected to proceed much faster at ultralow temperature than p-wave collisions studied in
the previous section.

42
In order to confirm this expectation, we measure the inelastic collision rates for rovibronic ground-state molecules that were prepared in a roughly 50/50 incoherent mixture of
the two hyperfine spin states |−4, 3/2i and |−4, 1/2i. The time-dependent number densityof trapped molecules was measured for both spin states. We measured the same loss rate
for both states, consistent with loss due to collisions between distinguishable molecules indifferent spin states. The rate coefficient is determined to be 1.9 (±0.4) · 10−10 cm3 s−1 , independent of temperature (solid squares in fig. 3.3b). Comparing to the for p-wave collisions
measurements between spin polarized molecules, the s-wave collision rate between molecules
in different hyperfine states is higher by a factor of 10 to 100 for a similar temperature range.
Dipolar collisions with ultracold 40 K87 Rb in 3D geometry

3.5

Up to now, we have only studied collisions between 40 K87 Rb molecules in the limit ofvanishing dipole moment. In the following, we will now study the unique properties of thedipole-dipole interaction as they manifest in collisions between two polar molecules. Precise
control over the dipole moment of the molecules is given by an external electric field. The
field mixes different rotational states with opposite parity within the vibrational ground state
manifold and therefore induces an electric dipole moment in the rovibrational ground state.
3.5.1

Dipoles and chemical reactions

The dipolar interaction between molecules will strongly affect the chemical reaction
rate between polar molecules. This is, because the dipolar interaction adds to the longrange potential between two colliding molecules, which is then given by:
V (R) =

~L(L + 1) C6 (1 − 3cos2 (θ))C3
+ 6+
.
2µR2
R
R3

(3.5)

Here, C3 is the relevant coefficient describing the dipole-dipole interaction between two
molecules. Due to the anisotropic nature of the dipole-dipole interaction, this interaction
can be either attractive or repulsive depending on the collisional process. This has im-43Potential energy ( K)
Potential energy ( K)

30
30
20
20

p-wave

10
10
00

-10
-10
00

200 400
400 600
600 800
800 1000
1000
200

Inter-moleculardistancedistance(a(aInter-molecular)0)00

Figure 3.4: p-wave barrier for 40 K87 Rb dipolar collisions “head-to-tail” (solid curve in blue)and “side-by-side” (solid curve in red). The dashed line represents the p-wave in the absenceof electric-field.

portant consequences on the chemical reaction rate for molecular ensembles, as illustrated
in fig. 3.4. Preparing an ensemble of indistinguishable fermionic molecules, molecules will
collide predominantly in the p-wave collisional channel which has a collisional barrier of
24µK. In this case, the dipole-dipole interaction will either reduce or enhance the height ofthe collisional barrier, depending on whether collisions are attractive (“head-to-tail” collisions) or repulsive (“side-by-side collisions”). The dependence of the height of the collisional
barrier as a function of the dipole moment of the molecules for both collisional channels
is shown in fig. 3.6c. In the limit of small dipole moments relevant to our experiments,
the collisional barrier will go down with the fourth power of the dipole moment for “headto-tail” attractive collisions and will rise with the fourth power of the dipole moment for
repulsive “side-by-side” collisions. We therefore expect the rate of attractive “head-to-tail”

44
collisions to increase dramatically with the dipole moment, whereas we expect at the sametime repulsive “side-by-side” collisions to be strongly suppressed with increasing dipole moment. In the experiments performed in 3D, we expect the loss mechanism of the interactions“head-to-tail” to dominate our measurement.
3.5.2

The experiment

In order to study dipole-dipole interactions in collisions between molecules, we preparea near quantum degenerate ensemble of 40 K87 Rb molecules in a single nuclear hyperfine
state |−4, 1/2i within the rovibronic ground state. The gas is confined in a pancake-shaped
optical dipolar trap. For this experiment, the harmonic trapping frequencies are ωx =2π × 40 Hz and ωz = 2π × 280 Hz in the horizontal and vertical directions, respectively.
The 40 K87 Rb molecules have a permanent electric dipole moment of 0.57 Debye. However,
the effective molecular dipole moment in the laboratory frame is zero in the absence of an
external electric field. When an external electric field is applied, the molecules begin to alignwith the field and have an induced dipole moment, d, that increases as shown in the inset offig. 3.5b. In our set-up, the external electric field points up (in the ẑ direction), parallel tothe tight axis of the optical dipole trap. Thus, the spatially anisotropic dipolar interactions
will be predominantly repulsive for molecules colliding in the horizontal direction (“side-byside”) and predominantly attractive for molecules approaching each other along the verticaldirection (“head-to-tail”). However, the molecular confinement in this pancake shaped trap
is 3D and hence neither of the collisional channel is suppressed.
3.5.3

Controlling chemical reactions by means of the dipole moment of themolecules

Using a single quantum state fermionic gas of rovibrational ground state polar molecules,
we study the effect of electric dipolar interactions on collisions and find an unexpectedly large
effect even for our relatively modest range of applied electric fields. We measure the molec-

45

Figure 3.5: Two-body inelastic loss for fermionic polar molecules. (a), We extract theinelastic loss rate coefficient, β, from a fit (solid lines) to the measured time evolution of thetrapped molecular gas density. Data are shown here for induced dipole moments of d = 0.08
D (open triangles) and d = 0.19 D (filled circles), and T0 = 300 nK. (b), Data points show
β/T0 plotted as a function of d. The dashed line shows a fit to a simple model based on
the quantum threshold behavior for tunneling through a dipolar-interaction-modified p-wave
barrier. The solid line shows the result of a more complete quantum scattering calculation.
Inset, the calculated dependence of d on the applied electric field, E. Figure reproduced
from reference [40].

ular loss rate by monitoring the time evolution of the average number density of trappedmolecules, n. We fit the data to the solution of eq. 3.3, as shown as solid lines in fig. 3.5a.
Fig. 3.5b shows the experimental data in a plot of β/T0 as a function of d, where T0 is46

Figure 3.6: p-wave barrier for dipolar collisions between fermionic 40 K87 Rb molecules. (a),The effective intermolecular potential for fermionic molecules at zero electric field. At intermediate intermolecular separation, two colliding molecules are repelled by a centrifugal
barrier for p-wave collisions. (b), For an applied electric field, the spatially anisotropic dipolar interactions reduce the barrier for “head-to-tail” collisions and increase the barrier for“side-by-side” collisions. (c), Height of the p-wave barrier as a function of dipole moment.
Dipolar interactions lower the centrifugal barrier for M = 0 collisions (V0 ) and raise thebarrier for M = ±1 collisions (V1 ). The lowering of the M = ±1 barrier at very large dipole
moments is due to mixing with higher L partial waves (L = 3, 5, 7, ). Figure reproduced
from reference [40].

the initial temperature of the molecules. We plot the ratio β/T0 because the Wigner threshold law for p-wave scattering predicts that β is directly proportional to T , a temperaturedependence verified at d = 0 in fig. 3.3. For the data in fig. 3.5, T0 ranged from 250 nK

47
to 500 nK. In fig. 3.5b, we observe that dipolar interactions have a pronounced effect on
the inelastic collision rate. At low electric field, where d < 0.1D, we observe no significant
modification to the loss rate at zero electric field (which is plotted at d = 0.01D due to thelogarithmic scale). However, for higher electric fields, we observe a fast increasing loss rate,with an increase in β/T0 of well over an order of magnitude by d = 0.2D. Fitting the datafor d > 0.1D, we find that the inelastic loss rate coefficient has a power-law dependence ond: β/T0 ∝ dp , where p = 6.1 ± 0.8.
In order to understand the behavior for both low and large dipole moment regime, we
assume that the collision rate follows the Wigner threshold law for p-wave inelastic collisions,that is, β ∝ T /V 3/2 . For dipole moments d < 0.1 Debye, the barrier height V is set by thecompetition between the attractive van-der-Waals interaction and the centrifugal energy of
the collisional channel. Dipole-dipole interactions are negligible in this regime. However,when the dipole moment increases above 0.1 Debye, the barrier height for attractive “headto-tail” collisions V0 is significantly smaller than the barrier V1 for repulsive “side-by-side”
collisions and the loss will proceed predominantly through “head-to-tail” attractive collisionsof the polar molecules. In this regime, V0 scales as d−4 and the model predicts that β/T0
will increase with a characteristic dependence on the sixth power of d for d > 0.1D [41].
This prediction is in good agreement with our measured dependence of the loss rate on d for
d > 0.1D (fig. 3.5b).
To give a quantitative description of the inelastic collisional rate over our full rangeof experimentally accessible dipole moments, we include both contributions from both attractive “head-to-tail” collisions (M = 0) and repulsive “side-by-side” collisions (M = 1),
where M is the quantum number that describes the projection of the relative orbital angularmomentum (quantum number L) onto the electric field direction. We write the inelastic loss
rate coefficient as β = K0 Tz + 2K1 Tx , the sum of two terms corresponding respectively to
M = 0 and M = ±1 scattering, and we assume that Tz = Tx = T . The dipole moment

48
dependent coefficients, K0 and K1 , are obtained using
K = γp3π~22µ3 V 3/2

kB ,

(3.6)

where K and V respectively equal K0 and V0 or K1 and V1 , and µ is the reduced massof the colliding molecules. The barrier heights, V0 and V1 , are taken to be the respective
maximum energies of the long-range adiabatic potential, V (R), calculated in a basis set ofpartial waves, |L, M i, for M = 0 and M = ±1. The potential V (R)V (R) =

(1 − 3cos2 (θ))C3 bC6 ~2 L(L + 1)
− 6 +
,
R3
R
2µR2

(3.7)

where the first term represents the dipolar interaction, the second term represents an attractive isotropic van der Waals interaction, and the last term is a repulsive centrifugal potential.
b and γ are the only fit parameters, when fitting this model to the measurements of β/T0
versus d.
We fit the prediction of the QT model to the experimental data using two fit parameters: a scaling factor, γ, that can be interpreted as the loss probability when the collisionenergy equals the height of the barrier; and another scaling factor, b, that multiplies thecoefficient of the van der Waals interaction, C6 . The final theoretical prediction (fig. 3.5b,
dashed line) agrees well with the data (fig. 3.5b, open circles); from the fit we extract
γ = 0.35 ± 0.08 and b = 2.4 ± 0.9. For comparison with the QT model, fig. 3.5b also
shows (solid line) the result of a more complete quantum scattering calculation. This calculation uses a strong absorptive potential at short range but captures the long-range physics
and uses C6 as the single fit parameter. This fit also agrees well with the experimental
data, and gives C6 = 21, 000 ± 7, 000 a.u., which is consistent with the calculated value of
C6 = 16, 130 a.u. [64].
3.5.4

Thermodynamics of dipolar collisions

The increased inelastic loss rates with increasing dipole moment d is accompanied by adramatic rise of the heating rate for the polar-molecule gas. In fig. 3.7, we plot the fractional

49
heating rate, Ṫ /T0 , normalized by the initial density n and temperature T0 , as a function ofd. 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 byapproximately 20-30%. We have developed a simple thermodynamic model for heating that is
directly caused by the inelastic loss. We consider the energy lost from the gas when molecules
are removed in inelastic collisions, and assume that the gas stays in thermal equilibrium.
In this model, the heating arises solely from density-dependent loss of particles from thetrap, where the particles removed by inelastic collisions have, on average, lower energies
than typical particles in the gas. This is also called “anti-evaporation” mechanism [70].
One way to understand this anti-evaporation is to note that inelastic collisions preferentiallyremove particles from the center of the trap, where the number density is the highest andthe particles have the lowest potential energy from the trap. In our model, we also include acompeting, “cooling”, effect that comes from the fact that the p-wave inelastic collision rateincreases linearly with the collision energy. Including these two competing effects, our model
bassed on the analysis shows that the initial heating rate of the molecules is proportional tothe inelastic loss rate β by a factor of 1/12. When we plot Tβ0 /12 in fig. 3.7 (solid line) we
can see that our model explains the heating well, where we obtain T02 n = (β/T0 )/12.
3.5.5

Anisotropy of heating

The anisotropy of the dipole-dipole interaction is shown directly in the energy distribution of molecules in the trap. The average energy per particle, which we measure fromthe expansion of the gas following a sudden release from the trap, can be different in thevertical and horizontal directions. In the following, we present measurements of the time
evolution of the expansion energy in these two directions for different values of d. To probe
the spatial anisotropy of dipolar collisions, we start by adding energy along one directionof the cylindrically symmetric trap using parametric heating. We modulate the power ofboth optical trapping beams at twice the relevant harmonic trapping frequency, for 50 ms (ẑ50

Figure 3.7: Normalized fractional heating rate, Ṫ /T02 n, versus the dipole moment. The
heating rate is extracted from a linear fit to the initial temperature increase and is thennormalized 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 experimental initial conditions
are n = 0.3 · 1012 cm−3 and T0 = 0.5µK, and the absolute heating rate ranges from 0.5µK
s−1 at zero electric field to 2µK s−1 at our highest electric fields. Figure reproduced fromreference [40].

direction) or 100 ms (x̂ and ŷ directions). Then we wait 100 ms before increase the electricfield (in less than 1 ms) to the desired final value and measuring the time dependence of thevertical and horizontal “temperatures” of the cloud, respectively denoted Tz and Tx . These
two quantities are associated with the measured expansion energies in the two directions.
Note that this type of measurement is used in experiments on ultracold atoms, to measure
the elastic collision cross-section [71].
In fig. 3.8 we show the experimental data from these rethermalization experiments
for three different values of dipole moment, d, and under two initial conditions: Tz > Tx
(fig. 3.8a-c) and Tz < Tx (fig. 3.8d-f). For d = 0D (fig. 3.8a, d), Tz and Tx reaches equilibrium
slowly, in approximately 4 s. Because d = 0D, there are no dipolar interactions and the dataagrees with our expectation of very slow equilibration for spin-polarized fermions. Thisis the longest rethermalization time we observed in our trap, and therefore the data are51
consistent with there being no elastic collisions and only technical imperfections such as asmall cross-dimensional coupling in the trapping potential.
In an applied electric field case, the elastic collision cross-section due to long-range
dipolar interactions is predicted to increase in proportion to d4 [72]. For the case in which
initially Tz > Tx (fig. 3.8b, c), the data show that Tz and Tx approach each other in what
seems on casual inspection to be cross-dimensional rethermalization. The time scale forthis apparent rethermalization even decreases steeply as d increases, as might be expected.
However, we note that in fig. 3.8c the temperatures cross each other, which is inconsistent
with rethermalization driven by elastic collisions. Even more striking is the fact that thethermodynamic behavior of the gas is completely different when the gas initially has Tz < Tx .
In this case, Tz and Tx do not equilibrate during the measurement time (fig. 3.8e, f).
To explain these observations we take into account the spatially anisotropic natureof inelastic dipole-dipole collisions and the fact that the molecular gas undergoes number
loss. We have observed (fig. 3.7) that loss due to inelastic collisions heats the gas. This
heating rate can be quantitatively understood by considering the effect of molecule loss on
the average energy per particle. We can modify the model of heating and inelastic collisions
described above to allow the average energy per particle, or “temperature”, to be different in
the two trap directions. The model then predicts that the dominant “head-to-tail” collisions(M = 0) will lead to heating in the x̂ and ŷ directions but cooling in the ẑ direction. “Sideby-side” collisions (M = ±1), however, should contribute to heating in the ẑ direction but
produce no temperature change in the x̂ and ŷ directions. Comparing this model with theexperimental data, we fix the d-dependent β using the fit to our data in fig. 3.5 (solid line).
This fixes both the time evolution of the molecule number as well as the heating rates in the
two trap directions. Then we include a possible elastic collision effects in the model by adding
a term that would exponentially drive the energy difference between the two directions tozero. In fig. 3.8 we also present a comparison between the results of the model (solid lines)
and our experimental data. Although the model uses few free parameters (only the elastic

52collision cross-section, σel , in addition to the initial values of n, Tx and Tz ), it provides good
agreement with the experimental data.

Figure 3.8: Cross-dimensional rethermalization in the polar molecule gas. Shown as a function of dipole moment, d, for Tz > Tx (a-c) and Tz < Tx (d-f). The experimental datashows difference between heating the gas in the vertical direction (a-c) and heating it in thehorizontal directions (d-f). This provides evidence for the strong anisotropic characteristicof dipolar interactions. The electric field is applied along the ẑ direction. Figure reproduced
from reference [40].

The differential equations used as model are shown as it follows. The solid lines in
fig. 3.8 are a fit of the measured time evolution of n, Tz and Tx to the numerical solution of
the following three differential equations [40]:
n dTz
n dTx
dn
= −(K0 Tz + 2K1 Tx )n2 −−,dt
2Tz dt
Tx dt
n
2Γel
dTz
= (−K0 Tz + 2K1 Tx )Tz −(Tz − Tx ) + cbg ,
dt
4
3
n
Γel
dTx
= K 0 Tz Tx +(Tz − Tx ) + cbg .
dt
4
3

(3.8)
(3.9)
(3.10)

53
Here we have allowed for a difference in the average energies per particle in the two trap
directions, Tz and Tx . For the fits, we fix the d-dependent coefficients K0 and K1 using theprevious fit to the inelastic loss rate data in fig. 3.5. In addition to heating due to inelastic
loss, we include a measured background heating rate of cbg = 0.01 µK s−1 . The elastic collip
sion rate in eqs. 3.9 and 3.10 is given by Γel = nσel v/Ncoll , where v = 8kB (Tz + 2Tx )/3πµand the constant Ncoll is the mean number of collisions per particle required for rethermalization. We use Ncoll = 4.1, which was computed for p-wave collisions [73]. However, wenote that Ncoll depends on the angular dependence of the scattering and may be different
for dipolar elastic collisions.

Chapter 4
Polar Molecules in an 1D Optical Lattice: Going to 2D Geometry

The anisotropy of the dipolar interaction allows one to control scattering by controllingthe orientation of the molecular dipole moment and the geometry of the external confinement.
In this context, 2D geometries are of particular interest. Orienting the dipole moment of themolecules along the axis of tight confinement will result in a strong suppression of attractive“head-to-tail” collisions, if the confinement along that direction is sufficiently strong. This
reduces inelastic scattering events in the molecular gas by a factor of 60 [74] and opens thepath towards the study of long-lived ultracold polar molecular quantum gases and quantum
degenerate molecular ensembles.
In our experiment, we realize a two-dimensional trapping geometry by means of a 1D
optical lattice. This results in a stack of two-dimensional trapping geometries (pancakes)
for the molecules. In this chapter, I will review the well-known concepts of optical dipole
trapping and the realization of a 1D optical lattice. I will discuss how to describe particles
in a periodic potential in terms of Bloch bands or Wannier functions. I will then introduce
basic experimental techniques such as loading of molecular gases into the optical lattice andcharacterization techniques such as measurement of the momentum and quasimomentum
distributions of the molecules in the optical lattice. Finally, I will discuss how to prepare
ground-state polar molecules in the 1D optical lattice and how we achieve population control
in selected vibrational levels of the optical lattice.

55
4.1

Atoms in an one-dimensional optical lattice

4.1.1

Optical dipole traps for neutral atoms and molecules

Atoms placed in an inhomogeneous light field will experience a force. This force relieson the interaction of the induced AC electric dipole of an atom with a far-detuned monochromatic laser field. The electric field E of the laser will in general induce a dipole momentd = α(ω)E in the atom, where α(ω) is the complex a.c. polarizability and ω is the laser
frequency.
The real part of the polarizability will then determine the so called dipole force of theinhomogeneous light field on the atom, given by [75, 76]:

Fdip = d · ∇E(r) = <[α(ω)]E(r)∇E(r),(4.1)

This force is associated with a corresponding dipole potential Vdip given by:

Vdip =1<[α(ω)]I(r).
2²0 c

(4.2)

where I(r) is the intensity of the far-detuned light at the position r of the atom.
The complex atomic polarizability is given by [75]α(ω) = 6π²0 c3

Γ/ω02,ω02 − ω 2 − i(ω 3 /ω02 )Γ

(4.3)

where Γ is the width of the relevant atomic transition, c is the light velocity, ²0 is thepermittivity constant in the vacuum, ω0 is the resonant frequency of the atomic transition,and ω is the frequency of the light. By inserting eq. 4.3 in eq. 4.2 for the dipole potential we
obtain:3πc2Vdip =
2ω03

µ

Γ
Γ
+
ω − ω0 ω + ω0

¶
I(r),

(4.4)

This expression can be simplified by applying the the rotating wave approximation in thelimit of large detuning:

56

3πc2
Vdip =
2ω03

µ

Γ
∆

¶
I(r),

(4.5)

where ∆ ≡ ω − ω0 is the detuning.
Eq. 4.5 explains the basic physics of dipole trapping in a far-detuned laser field. The
potential scales as I(r)/∆. The potential depth therefore increases linearly with the intensityof the light field and decreases linearly with increasing detuning of the light field. Whether
the potential is attractive or repulsive depends on the detuning of the light field relative
to the relevant atomic transitions. For red detuned light, ∆ < 0, the dipole potential is
attractive, and for blue detuned light, ∆ > 0, the potential is repulsive.
The simplest configuration for the realization of a confining potential by means of dipole
potentials is therefore a single Gaussian laser beam with a frequency red-detuned from therelevant atomic transitions and focused down to a spot size of typically about 10-100 µm.
The atoms will then be confined in the focus of this dipole trap with a strong confinement
perpendicular to the axis of propagation of the laser light and a relatively weak confinement
along the axis of propagation.
According to eq. 4.5, the depth of the confining potential is entirely set by the intensityof the light field and the detuning of the light frequency relative to the atomic transitions.
At first sight, it might therefore seem favorable to operate dipole traps at small detunings toachieve a maximal depth at low intensity. However, atoms interacting with light fields will
also scatter photon. Photon scattering results in heating of the atomic clouds. The photon
scattering rate is determined by the imaginary part of the polarizability:

Γsc =

1
=[α(ω)]I(r),
~²0 c

(4.6)

This expression can be simplified in a similar formalism as applied above to
3πc2
Γsc =
2~ω03

µ

Γ
∆

¶2
I(r).

(4.7)

57
As can be seen in eq. 4.7, the scattering rate scales ∝ I(r)/∆2 . In particular, thephoton scattering rate decreases much faster with increasing detuning (with the square of
the detuning) as compared the potential depth (decreasing linearly with detuning). It istherefore favorable to operate optical dipole traps as far detuned as practically possible to
avoid photon scattering processes that lead to heating of the confined atomic ensemble.
For more details please refer to reference [75] by R. Grimm et al.
4.1.2

One-dimensional optical lattice

A one-dimensional (1D) periodic lattice potential can be created by retro-reflecting a
monochromatic laser beam and therefore super-imposing two counter-propagating light fields
with a fixed phase relation with each other. This configuration will result in the formationof a standing wave with alternating intensity maxima and intensity minima. Neighboring
intensity maxima/minima will be separated by λ/2, where λ is the optical wavelength forthe lattice beam.
In the experiment, the optical lattice will be formed by two counter-propagating Gaussian laser beams with a typical waist of a few hundred µm. The resulting optical potential
is then given by

V (r, z) = V0 e−2r2
w2 (z)

8P α
where V0 ≡ πw
2 (z) , P is the laser power, w(z) = w0

cos2 (kz),

(4.8)

p

1 + (z/zR )2 is the beam waist at z, w0

is the beam waist at the focus, k = 2π/λ is the wave number, λ is the wavelength of thelaser light, α is the polarizability, and zR = πw02 /λ is the Rayleigh length.
For red detuned laser light, the atoms will be trapped in the anti-nodes of the standing
wave (optical lattice), as shown in fig. 4.1. The one-dimensional optical lattice results in atrapping potential consisting of a stack of pancake-like traps. The confinement perpendicular
to the plane of the pancake is much tighter than the confinement in the plane of the pancake

58with trapping frequencies given by:
s
ωr =

1
m

s
r
¶
µ 2
∂ V (r, z)
4Erec
V0
=
,
2
2
∂r
mw0 Erec
r,z=0

(4.9)

r
µ 2
¶
∂ V (r, z)
2Erec
V0
=
,
∂z 2
~
Erec
r,z=0

(4.10)

and
s
ωz =

1
m

2 2

respectively. Here, Erec = ~2mk is the recoil energy associated with the lattice photon. Atoms
in a single pancake are trapped in 2D-geometry if the optical lattice potential is sufficiently
deep.
w0 = Λ

w0 = 250Λ

1

1

0
0

rw0

0
0

rw0

1

2

1

2

-1

-1

-2
-4

-2
-4

-2

0
zΛ

(a)

2

4

-2

0
zΛ

2

(b)

Figure 4.1: Illustrative contour plot of one-dimensional optical lattice potential. (a) Normalized potential in false color for w0 = λ. (b) Potential for w0 = 250λ. The point withmaximum density are the anti-nodes where, for a red-detuned optical lattice, the particles
will be trapped. In our case, the lattice beam waist is w0 ≈ 250µm.

4

59
4.1.3

Experimental realization of a 1D optical lattice

In the experiment, we realize a 1D optical lattice by retroreflecting a Gaussian laser
beam with a beam waist of 250 µm. The optical lattice is operated red-detuned to all atomic
and molecular transitions in the 40 K , 87 Rb and 40 K87 Rb system, with wavelength of 1064
nm. The one-dimensional optical lattice is overlapped with the crossed beam optical dipole
trap described in chapter 2. The optical lattice beam propagates along the tight axis of thepancake shaped optical dipole trap and therefore, cuts a single loose pancake into a stack ofabout 30 tightly confining pancakes.
Using the optical lattice, it is possible to trap atoms and molecules in a two-dimensional
trapping geometry. Here we define 2D trapping by the criterion ~ωz >> kB T . In this regime,
the motion of particles will be restricted to a single quantum state along the axis of tightconfinement.
As an illustrative example, the trapping frequencies for 87 Rb in the radial and axial
direction respectively are shown as a function of optical lattice depth in fig. 4.2. As can beseen in the diagram, the confinement along the axial direction - that means the confinement ina single pancake - is much tighter than the confinement in the radial direction. In particular,
it is possible to enter a regime of 2D confinement where ~ωz >> kB T . Note that 2 kHz trap
frequency correspond to an equivalent temperature via T = hν/kB T of about 100 nK. At atemperature of around 100 nK, the regime of 2D confinement for 87 Rb sets in at a lattice
depths around 1 Erec . A very similar criterion can be derived for 40 K87 Rb molecules. This is
the regime that is of particular interest for the control and study of dipolar collisions. It is in
this regime that scattering properties between fermionic polar molecules can be significantly
altered as compared to a three-dimensional geometry (see chapter 5).
By increasing the optical lattice depth, it is possible to continuously tune the trapping
geometry of particles confined in the optical lattice from 3D (~ωz << kB T ) to a 2D configuration (~ωz >> kB T ) and, therefore control molecular scattering processes by means of60
external confinement.
4.1.4

Quantum mechanical description of a particle confined in a periodic
potential: Bloch bands

The quantum mechanical picture of a particle moving in a periodic potential is a wellknown problem. The description of atomic motion is captured by the formalism of Bloch
wavefunctions and Bloch bands [77]. The periodic potential in this case is provided by theinterference pattern of light. A single particle of mass m moving in this periodic potential
V (z) = V0 cos2 (kz) is described by the time-independent Schrödinger equation:
¶
µ
1 2
2
−
p + V0 cos (kz) Ψq,n (z) = Eq,n Ψq,n (z),2m

(4.11)

where n is the quantum number relative to the internal state of the optical lattice. The
eigenfunctions of this Schrödinger equation are called Bloch waves. These Bloch functions
can be written as the product of the plane waves eiqz/~ with quasimomentum q times anΝx HHzLΝz HHzL

30 00025 000
20 000
15 000
10 000
5000
0
0

10

20

30 40
V0 HErec L
(a)

50

60

70
60
50
40
30
20
10
0
0

10

20

30
40
V0 HErec L

50

60

(b)

Figure 4.2: Optical lattice’s trap frequencies versus the trap depth for 87 Rb . This two plotsshow the axial frequency νz compared to the radial frequency νr . The confinement in theaxial direction is much tighter than the radial direction. The optical lattice parameters are:λ = 1064 nm, and the beam waist is 250 µm.

61envelope function uq,n (z) with the periodicity of the confining potential:Ψq,n (z) = eiqz/~ uq,n (z).

(4.12)

These wavefunctions are delocalized over the whole lattice structure.
Due to the periodicity, we can expand both the external potential and the wavefunctionsin terms of a Fourier sum:

V (z) =X

Vj ei2jkz ,(4.13)X (q,n)cl ei2lkz .

(4.14)janduq,n (z) =lSubstituting eq. 4.13 and eq. 4.14 into eq. 4.11 and after some algebraic manipulations, theSchrödinger equation can be written in a matrix form given by [77]:
¸
½·³
¾
q ´2
V0
V0
(q,n)
(q,n)
2l +
Erec −
δl,l0 − δl,l0 ±1 cl= Eq,n cl ,~k2
4(4.15)

where −~k ≤ q ≤ ~k, limited to the first Brillouin zone. Eq,n represents the eigenenergies of
the nth Bloch band for a given q. The eigenstates c(q,n) are the Bloch wavefunctions.
Note that we have introduced an additional quantum number, the band index n. The
quasimomentum q is unique only up the addition of integer multiples of the reciprocal lattice
vector k. It is therefore convenient to restrict the quasimomenta to the first Brillouin zone
given by −~k < q < ~k. This means that a multiple number of eigenfunctions and eigenstates
exists for a given quasimomentum q. These eigenstates are denoted by the band index n.
Fig. 4.3 shows the energy spectrum of particles confined in optical lattices for various
lattice depths. In the limit of vanishing optical lattice potential, the quasimomentum q isequal to the free particle momentum and the energy spectrum is given by the free particle
energy dispersion Eq = ~2 q 2 /2m. The band structure in this case appears due to the artificial

62
mapping of quasimomenta onto the first Brillouin zone. With increasing lattice depth, anenergy gap appears between the different energy bands. This gap increases with increasing
lattice depth whereas the width of the Bloch bands becomes smaller. Finally, the Blochbands converge to harmonic oscillator states in a single lattice well in the limit of very deep
optical lattices.
0 Erec20

20
10Energy HErecL

Energy HErecL

10
0

-10

0

-10

-20
-1.0

-0.5
0.0
0.5
Quasimomentum qHÑ kL

1.0

-20
-1.0

15 Erec

20

20

-0.5
0.0
0.5
Quasimomentum qHÑ kL

1.0

20 Erec

10
Energy HErecL

10Energy HErecL

10 Erec

0

-10

0

-10

-20
-1.0

-0.5
0.0
0.5
Quasimomentum qHÑ kL

1.0

-20
-1.0

-0.5
0.0
0.5
Quasimomentum qHÑ kL

1.0

Figure 4.3: Bloch bands in the one-dimensional optical lattice. Energy versus quasimomentum in the first Brillouin zone for different lattice depths. The lattice depths are 0, 10, 15,
and 20 Erec . As the lattice height increases, the separation between the Bloch bands also
increases and the energy levels flattens out.

This is illustrated in fig. 4.4, where we show behavior of the first four energy levels
versus the lattice depth at quasimomentum q = 0. Above 30Erec , energy levels flatten out
and the gap between energy levels converges towards the energy splitting between harmonic

63
oscillator states in a harmonic trapping potential ~ωz . The eigenstates for deep lattices isgiven by localized wavefunctions called Wannier functions.

20
Eigenenergies HErecL

10
0

-10
-20
-30
-40
-50

0

10

20
30
40
Lattice Depth HErecL

50

60

Figure 4.4: First four energy levels versus the lattice depth with quasimomentum q = 0,
for an atom in an optical lattice. For lattice depths above 30Erec , the spacing between theenergy levels approaches ~ωz like a harmonic oscillator energy.

4.1.4.1

Wannier functions

Bloch wavefunctions are the common quantum mechanical description of particles moving in a periodic potential. However, these wavefunctions are delocalized over the lattice.
Sometimes it is more convenient to approach the description of particles in the periodic potential by means of localized wavefunctions, with interactions of particles on a single latticesite and tunneling of particles from one lattice well to the next. In this case, it is appropriate to think of the optical lattice as an infinite chain of harmonic oscillator wells coupled toeach other through tunneling processes. The wavefunctions used in this formalism are called
Wannier functions and they approach harmonic oscillator states in the limit of very deep
optical lattices. A good review is provided in Ana Maria Rey’s Ph.D. thesis [78].

64
The Wannier functions are the Fourier transform of the Bloch functions and therefore
defined as:
1 X −iqzi /~wn (z − zi ) = √e
Ψq,n (z).
A q(4.16)

where A is a normalization constant and Ψn,q (z) is the Bloch function eq. 4.12.
To calculate the probability of a particle tunneling from site i to a neighboring site j,
we need to know the tunneling energy for 1D optical lattice. This energy is given by [77]
Z
Jz =

µ

¶
~2 ∂ 22
w1 (z − zi ) −+ V0 cos (kz) w1 (z − zj ).
2m ∂z 2

(4.17)The tunneling matrix element J, which describes the tunnel coupling between latticesites, is directly related to the width of the first Bloch band [77]:J=(max(Eq,0 ) − min(Eq,0 )).
4(4.18)The tunneling energy corresponds typically to a timescale for particles tunnel from one latticesite to the next. The time scale is given by t = ~/J.
In fig. 4.5, we show the tunneling time for 40 K and 87 Rb atoms versus the optical lattice
depth. The tunneling times for 40 K atoms and 87 Rb atoms are significantly different. 40 K
atoms are lighter than 87 Rb atoms and in the same lattice depth therefore have a higher Erecand a higher probability to tunnel from one lattice site to the next than 87 Rb . At a latticeK
Rbdepth of 19 Erecfor 40 K and 42 Erecfor 87 Rb , the tunneling time for 40 K is 300 ms andfor 87 Rb is 1 s. Note that, these two lattice depths are different when they are expressed interms of the recoil energies of 40 K and 87 Rb , which are different because of the differencein mass of each atom (given that they are in the same optical lattice, and they also have
similar polarizabilities).

65ÑJ HsL

1
0.1
0.01
0.001
0

10

20
30
40
Lattice Depth HErecL

50

60

Figure 4.5: Semi-log plot of tunneling time in an 1D optical lattice versus lattice depth.
Dotted blue line is for 40 K and red dashed line is for 87 Rb . For our typical conditions, theK
Rbtunneling time for 40 K is 300 ms (for 19 Erec
) and for 87 Rb is 1 s (for 42 Erec).

4.2

Characterizing the optical latticeExperimental studies of atoms or molecules in the optical lattice rely on an accurate

knowledge of the depth of the optical lattice and corresponding trapping frequencies of atoms
or molecules in a single well of the optical lattice. In the following, we will introduce two
experimental techniques for the characterization of the depth of the optical lattice. The firstrelies on parametric heating of atoms or molecules in the optical lattice - therefore providing
a direct measurement of the trapping frequencies and the energy splitting between bands inthe limit of strong confinement. The second technique relies on Kapitza-Dirac scattering ofatoms off a periodic potential.
4.2.1

Measuring the trapping frequencies by parametric heating

One very common method for the calibration of the optical lattice depth is to parametrically heat the ensemble in the optical lattice. For parametric heating the system, is driven
periodically at twice the trapping frequency of the particles confined in the potential, for ex-

66
ample, by modulating the lattice beam intensity. This will deposit energy in the system andin particular drive transitions from e.g. the lowest harmonic oscillator state to the second
excited harmonic oscillator state (n = 2). In the band picture of the optical lattice, this isequivalent to driving transitions from the first to the third band of the optical lattice andtherefore allows to study the energy difference between these two bands. Due to the finite
width of the involved bands, calibration of the optical lattice depth via parametric heating
has to be performed either with an atomic cloud occupying only the lowest quasimomentumof the optical lattice (q = 0) or in the limit of a very deep optical lattice where the widthof the involved bands tends to zero. As an illustrative example of the intrinsic uncertainty
for this measurement, let us consider an optical lattice of 40 Erec . The energy difference
between the first and the third band is then > 20 Erec with a width of each band of < 0.3
Erec , corresponding to a relative uncertainty of < 0.015.
In order to know the trap frequency of the optical lattice in the vertical direction forthe molecules, we perform the calibration procedure with 40 K-87 Rb Feshbach molecules. A
typical calibration procedure for the optical lattice using parametric heating is performedin the following way: a sample of Feshbach molecules, 40 K-87 Rb, is prepared in the optical
dipole trap. The lattice is then adiabatically ramped up in 150 ms to a lattice depth of 100
Erec . The optical lattice is then modulated with a relative amplitude of approximately 20-30% for about 300 µs. This time scale corresponds to approximately 10 cloud oscillations.
Fig. 4.6 shows a typical resonance feature for parametric heating of atoms in the opticallattice. As a function of the frequency of the drive, we observe the width of the cloud, σ,after 5 ms time of flight, and therefore the temperature (T ∝ σ 2 ) of the cloud to vary by
about a factor of four in the vicinity of the resonanc! e. The resonance feature has typicallya gaussian rms width of 5 kHz and a resonance frequency of 44(5) kHz. A more systematic
study of the dependence of the trap frequency, νz , versus the amplitude of the drive would
be necessary to determine the trap frequency with a smaller error. In chapter 5 we also use
the parametric heating idea to excite molecules to higher bands in a controlled way.

67

(pixels)16
f =44(5) kHz

14

c

12
10
8
30

35

40

45

50

55

60

Frequency (kHz)

Figure 4.6: Trap frequency measurement for Feshbach molecules 40 K-87 Rb via parametricheating. σ is the molecular cloud width in the ẑ direction. The center frequency is 44 kHz,
which indicates that the axial frequency of the optical lattice is 22 kHz. The ramp procedure
for this measurement is similar to the one described in chapter 5 in fig. 5.2 except with asudden turn off of the end of the ramp.

4.2.2

Kapitza-Dirac Scattering

Kapitza-Dirac scattering is the diffraction of matter waves off an off-resonance lightfield. The diffraction results in a coherent mixing of momentum modes [79]. Starting froma BEC of e.g. 87 Rb atoms, the atoms occupy a single collective state with momentum nearq = 0, where the momentum spread is much smaller than |~k|. The dynamics of the matterwave diffracting off a standing light wave can then be nicely observed. While interactingwith a standing wave interference pattern of light the atoms will constantly absorb andimmediately reemit photons. This changes the atomic momentum along the standing wave
axis by either zero or two-photon momenta. The momentum distribution of atoms after
the interaction with the standing light field can then be probed by projecting the atomic

68
quasimomentum onto free space momentum in time of flight expansion.
As a particular case of Kapitza-Dirac scattering, we operate in the Raman-Nath regime
where the atom motion during the interaction with the light field can be neglected. The
population of the diffracted states then only depends on the area of the applied pulse [79],i.e., ωτ ¿ 1, where τ is the pulse duration and ω is the harmonic oscillator frequency. In
our experiment, τ is on the order of tens of µs and ω = 2π × 28 kHz.
In our experiment, we use Kapitza-Dirac scattering of a 87 Rb BEC for an accurate
determination of the depth of the optical lattice. The number of atoms , Pn , in the nthdiffraction order observed in momentum space as a function of interaction time τ is given
by a Bessel function of the first kind Jn [79]:Pn = Jn2 (

V0 τ
).
2~

(4.19)

where, V0 is the lattice depth. The dynamics of the atomic population during the pulse indifferent diffraction orders then gives accurate information on the optical lattice depth.
Fig. 4.7 shows the experimental procedure for measuring the depth of an optical latticeby means of matter wave diffraction. First, a BEC is prepared in an optical dipole trap.
The optical dipole trap is then suddenly turned off and at the same time the optical lattice
is flashed on for a variable amount of time τ - typically on the order of tens of µs. Fig. 4.8
shows the dynamics of the diffraction pattern of a 87 Rb BEC matter-waves versus the holdtime in the lattice. From these images we extract the fractional amplitudes of different ~kmomentum diffraction order and fit the data to a numerical solution of eq. 4.11 following
reference [79]. Here, only the trap depth is a free parameter. In the particular example, we
extract a depth of 42 Erec .
The discrepancy for the optical lattice’s trap frequency determined via parametricheating and via Kapitza-Dirac scattering is 15%, where the fist is smaller than the second.
For Kapitza-Dirac scattering with 87 Rb , we know the frequency of the optical lattice for 87 Rb

69

Image

Dipole Trap

Optical Lattice
Hold TimeTime

Figure 4.7: Ramp procedure to measure trap depth of the optical lattice. The depth measurement is done with a 87 Rb BEC. The optical lattice is diabatically pulsed on and briefly
switched off in order to excite 87 Rb atoms to higher bands. The hold time is varied on theorder of tens of microseconds. The time of flight is 5 ms, and the lattice height is 42 Erec .

atoms. Then we obtain the optical lattice frequency for 40 K87 Rb by doing a mass scaling
between 87 Rb and 40 K87 Rb. For the parametric heating we have a direct measurement of thetrap frequency for 40 K87 Rb. This is the reson why we trust it more than the Kapitza-Dirac
scattering to determine the optical lattice trap frequency.
4.3

Loading atoms into the optical lattice
The loading of ultracold atoms or molecules into the optical lattice has to ensure acontrolled transformation of a harmonic optical dipole potential into a periodic potential. In
particular, it is important that atomic or molecular motional states in the initial harmonic
oscillator potential are adiabatically transfered into motional states of the final periodic
potential.
An illustrative example is given by the loading of a Bose-Einstein condensate into anQuasimomentum (³k)

70

6
4
2
0
-2
-4
-6

3

7

10

12

15

20

25

30

35

Time (Ps)

Figure 4.8: 87 Rb BEC matter-wave diffraction into discrete momentum components.

optical lattice. Starting from a Bose-Einstein condensate in the lowest harmonic oscillator
state in the harmonic optical dipole potential, the loading procedure should ensure that theatoms in the condensate remain in the lowest possible energy state. In the case of the opticallattice, this lowest energy state is given by the Bloch state with quasimomentum q = 0. The
previous Kapitza-Dirac scattering discussed in section 4.2 provides an example when thetransfer is not adiabatic.
The adiabaticity condition with respect to band population for loading atoms withquasimomentum q in the lowest Bloch state, |0, qi is given by [80]:
¯
¯¿
¯
À¯
¯
¯ ∂H ¯¯2¯¯ 1, q ¯¯¯ ∂t ¯ 0, q ¯ ¿ ∆E (q, t)/~,¯(4.20)

where |1, qi is the first excited Bloch state, and ∆E is the energy difference between |0, qi
and |1, qi.

71
1
0.9
0.8
0.7

fraction

0.6
0.5
0.4
0.3
0.2
0.1
0

0

5

10

15
20
time (µs)

25

30

35

Figure 4.9: Extracted relative population of momentum components versus pulse duration.
The lines show a fit to eq. 4.19. The optical lattice depth as a free parameter, and theextracted one from the data is 42 Erec . The optical lattice parameters are λ = 1064 nm,
beam waist of 250 µm, and laser power of 1 W. The three curves show the 0~k (solid blue),±2~k (solid green), and ±4~k (solid red) momentum components.

For a 87 Rb bosonic condensate, where the atoms all accumulate in the q = 0 state atzero lattice depth, the splitting between the two lowest bands at q = 0 is ∆E = 4Erec forlattice height V0 = 0. If q = 0, eq. 4.20 reduces to [81]:√ Erec1 d
V0 ¿ 32 2.
Erec dt~(4.21)

4Erec
A linear ramp of the optical lattice should occur with rate no higher than 100µs
to ensureadiabatic loading.
Fig. 4.10 shows the energy difference ∆E versus lattice depth V0 for three different
quasimomenta. With increasing quasi momentum, the energy gap decreases for small optical lattice depth. The loading of a fermionic gas occupying a continuous quasimomentum
distribution up to a characteristic energy scale given by the Fermi energy then poses a much

72
more stringent criterion for the timescales for adiabatic loading than a BEC.
10DE HErecL

8
6
4
2
0

0

5

10
15
20
Lattice Depth HErecL

25

30

Figure 4.10: Energy gap between lowest and first excited Bloch bands for different quasimomenta. Solid blue line is for q = 0, dashed green line is for q = 0.5 ~k, and red dotted line
is for q = ~k.

4.3.1

Loading atoms in the optical lattice

In this section, I present the experimental procedure for loading the optical lattice with40

K and 87 Rb atoms. In order to load the atoms in the lattice and avoid heating them, theadiabaticity criterion, eq. 4.20, has to be satisfied. For 40 K and 87 Rb atoms, the ramp rate
4Erec
. We exponentially ramp up the lattice in 150 ms as shown in
has to be lower than 100µs

fig. 4.11. The dipole trap is ramped down adiabatically within the same time scale. Note that
we choose 150 ms as ramping time arbitrarily, just to be safe with respect to the adiabaticity
criterion.
Note that the lifetime of atoms in the optical lattice is much longer than the timescales
associated with adiabaticity. For 87 Rb , we observe a lifetime of 5(1) s and for 40 K is the
lifetime 8(1) s, as shown in fig. 4.12 (a) and (c).
We have also measured the heating rates of atoms in the optical lattice and find them
to be comparable to heating observed in our dipole trapping potentials [21]. For 87 Rb , weobserve a heating rate of 6.1(5) nK/s and for 40 K 5(1) nK/s, as shown in fig. 4.12 (b) and

73
(d).

150 msImage

Dipole Trap

150 ms

Optical Lattice
Hold TimeTime

Figure 4.11: Ramping procedure to adiabatically load 40 K and 87 Rb in the optical lattice.
Rb
K
The optical lattice depth for 87 Rb is 42 Erecand for 40 K is 19 Erec
.

74
4.4

Probing the quasimomentum distribution of atoms in optical lattices.
For our experiments it is very important to know the distribution of atoms andmolecules over different quasimomentum states and Bloch bands in the optical lattice. This
distribution can be probed by adiabatically ramping down the optical lattice potential with

4

Rb number (10 )

100
80
60

=5 (1) s

40
20
0
0

5

10

15

20

Rb temperature ( K)

respect to the band separation τ À ωz−1 . For the experiment described in this section, τ =0.30

Heating Rate = 6.1 (5) nK/s

0.25

0.20

0.15

0.10
0

5

10

15

Time (s)Time (s)

(a)

(b)

20

30
25
= 8 (1) s

20
15
10
5
0
0

10

20

30

40

50

K temperature ( K)4

K number (10 )35
0.6
Heating Rate = 5 (1) nK/s

0.5

0.4

0.3

0.2
0

10

20

30

Time (s)Time (s)

(c)

(d)

40

50

Figure 4.12: Lifetime of 87 Rb and 40 K atoms and respective heating rates in the opticallattice.

75
500 µs and ωz−1 ≈ 4 µs, for lattice depth of 17.7 Erec . Note that, we choose 500 µs to ramp
our lattice down, because if we go slower the atoms will free fall deu to the gravity. This procedure results in an adiabatic conversion of quasimomentum into free-space momentum. The
resulting free-space momentum distribution can then be probed in time of flight absorption
imaging.
Fermionic atoms always occupy a continuous distribution of quasimomenta. This is incontrast to a BEC where all atoms accumulate in the zero quasimomentum state. Dependingon the quasimomentum distribution of the fermionic atoms, the lowest Bloch band can thenbe partially or entirely populated. Increasing the atom number or the temperature of thefermionic cloud can result in a population of fermions in several Bloch bands.

Amplitude Ha. u.L

1.0
0.8
0.6
0.4
0.2

0.0
-0.0001-0.00005

0.00000.00005zsize per pixel

0.0001Figure 4.13: Calculated functions for the distribution in quasimomentum space at the first
Brillouin zone in the optical lattice. Red dotted curve is a gaussian function, green dashedcurve is a Heaviside function, and solid blue line is the convolved curve that is used to fit
the atomic cloud along the lattice axis.

By converting the quasimomentum distribution into a free-space momentum distribution adiabatically, we can then accurately determine the fraction of atoms/molecules indifferent bands. This is performed by fitting the atomic momentum distribution after time offlight absorption to a “wedding cake”-like structure [82, 83]. Here, the edges of the wedding
cake occur at integer multiples of ~k and correspond to the edges of the nth Brillouin zone.
However, due to limitations in our image resolution, we are not able to resolve the sharp

76

1 ms

500 PsImage

Dipole Trap
20 ms

Optical Lattice

18 ms

Variable Height

150 ms

50 ms
130 msHold TimeTime

Figure 4.14: Band mapping with filtering of 40 K in higher bands. The optical lattice isadiabatically ramped up exponentially in 150 ms to 17.7 Erec . At the beginning the opticallattice and the optical trap are combined to hold the atoms against the gravity. The filtering
is done for two different lattice heights: 3.2 Erec , and 14.5 Erec .

edges in the momentum distribution. To model this, we convolute a single step function(Heaviside function) with a Gaussian function accounting for the finite imaging resolution:¶µ¶¸
Z ∞· µ−(x−z)2∆ − 2x∆ + 2x−Θe 2σ2 dx =
Θ
2σ
2σ
−∞
r ·
µ
¶
µ
¶¸
∆ + 2z∆ − 2z
π
√
√
σ
Erf
+ Erf
,
2
2 2σ
2 2σ

(4.22)

where x is a variable which is integrated, z is a variable representing the vertical coordinate
at ẑ direction, Erf is the error function, σ is the rms width of the gaussian model of ourimage resolution, and ∆ is the size of the expanded cloud corresponding to the first Brillouinzone, 2~k. Since both z and σ are conveniently measured in units of pixels, we convert ∆ tounits of pixels as well:

∆=

TT OF
2~k
,
mK 2 × 5.4µm/pixel(4.23)

77
where TT OF = 18 ms is the time of flight, mK is the atomic mass of 40 K , σ is the image
resolution in pixels, and 5.4 µm/pixel is the conversion from micrometers to pixels at theimage. The factor of 2 multiplying the 5.4 µm/pixel is because of the binning of the image
was 2. Fig. 4.13 shows the Heaviside (dashed green), the Gaussian (dotted red) and theresulting convoluted (solid blue) functions.
We multiply the result of eq. 4.22 by A/2 where A is an amplitude scaling factor, andalso by shifting the center of the function by zc . We replace the amplitude because we want
to extract the relative atom (molecule) population in the lattice levels. We then fit the datawith three free parameters: the amplitude A, the image resolution σ, and the center zc (zcis also in units of camera pixels). The fitting function for atomic distribution in the first
Brillouin zone is given by:·µf (z) = A/2 Erf∆ + 2(z − zc )
√
2 2σ

¶

µ
+ Erf

∆ − 2(z − zc )
√
2 2σ

¶¸
,

(4.24)and the fitting function for a distribution in both first and second Brillouin zones is given
by:

·

µ

¶
µ
¶¸
∆ + 2(z − zc )∆ − 2(z − zc )√√f (z) = A/2 Erf
+ Erf
2 2σ
2 2σ
·
µ
¶
µ
¶¸
−∆ + 2(z − zc )−∆ − 2(z − zc )√√+B/2 Erf
+ Erf
2 2σ
2 2σ
·
µ
¶
µ
¶¸
2∆ + 2(z − zc )2∆ − 2(z − zc )√√+B/2 Erf
+ Erf
.
2 2σ
2 2σ

(4.25)

where B is the amplitude of the distribution in the second Brillouin zone.
Fig. 4.15 shows two illustrative examples how we extract the band population using
the procedure described above. In fig. 4.15(a), all 40 K atoms have been loaded into thelowest Bloch band only and we therefore observe a single sharp momentum distribution
in the vertical direction with the largest momentum given by ~k. Fig. 4.15(b) shows thecorresponding radially integrated momentum distribution of 40 K atoms in the first Brillouin

78zone. Although the edges of the momentum distribution are blurred due to the finite imaging
resolution, it can clearly be seen that all atoms occupy momentum states within the first
Brillouin zone.
Heating the atomic 40 K ensembles to temperatures in the 400 nK range, atoms will not
only occupy the first but also the second Brillouin zone. This is shown in fig. 4.15(c) and (d).
Now, atoms also occupy quasimomenta −2~k < q < 2~k and we can extract the fraction of
atoms in higher bands by fitting eq. 4.25 to the wedding cake structure and comparing thetwo areas underneath the fitting curve. From this, we determine the fraction of 40 K atoms
in the second Brillouin zone to be approximately 28%.
It is known that collisions between atoms in higher bands and atoms in the lowest
band induce very fast collisional loss in the sample, due to a two-body decay process [84].
To circumvent this issue and ensure the preparation of atoms in the lowest Bloch band only,
we introduce a filtering procedure during the optical lattice loading. The filtering will resultin fast tunneling of atoms in higher optical bands out of the optical lattice. Atoms in thelowest Bloch band will stay in the lattice. Our ramping procedure is shown in fig. 4.14,
where the optical lattice is adiabatically ramped up in 150 ms to the height of 17.7 Erec .
The atoms are held in the lattice for 130 ms and the dipole trap is adiabatically ramped
down in 50 ms. The filtering is accomplished by ramping down the lattice in 20 ms holding
it for 1 ms and then ramped down in 500 µs to 0 Erec , for the band mapping. In fig. 4.15a
and b, I show the 40 K atoms all in the first Brillouin zone due to the filtering procedure. In
order to filter atoms in the second band, the optical lattice height was reduced to 3.2 Erec .
In fig. 4.15c and d, I show the 40 K atoms in the first and second Brillouin zones. The opticallattice height was reduced to 14.5 Erec . The filtering procedure was not used for moleculesin the optical lattice.

79
0.30
0.25

OD

0.20
0.15
0.10
0.05
0.00
-3

Z direction

-2

-1

0

1

2

3

quasimomentum (hk/2 )

(a)

(b)
0.30

OD

0.25
0.20
0.15
0.10
0.05
0.00
Z direction

-3

-2

-1

0

1

2

3

quasimomentum (hk/2 )

(c)

(d)

Figure 4.15: Band mapping for the filtering procedure with 40 K atoms. (a), and (c) are theatomic cloud images taken with time of flight of 18 ms. The initial temperature of the atomsin (a) is 350 nK and in (b) is 400 nK. The vertical profiles are shown in (b) and (d). In(b) all atoms are in the first Brillouin zone, while in (d) about 28% of the atoms are in thesecond Brillouin zone. However, a significantly difference in population of bands is observedin the two cases due to a different optical lattice heights (3.2 Erec , and 14.5 Erec ) applied forthe filtering proceedure(see fig. 4.14).

4.5

40

K87 Rb molecules in the optical latticeThis thesis describes the control of collisional processes between polar molecules bymeans of control over the dipole moment and the confinement of the polar molecules inversatile geometries - in particular in a 2D geometry. As mentioned previously, tuning of the80trapping geometry from 3D to 2D is achieved in the experiment by gradually increasing thedepth of an optical lattice and transferring molecules from a dipole trap to a 1D lattice.
In this section, we will describe how we prepare molecules in the lowest band of the
optical lattice and how we control the population in different bands. Polar molecules areformed in the lattice following the ideas already outlined in chapter 2 and earlier sectionsof chapter 4. In the first step, we load atoms from the optical dipole trap into the opticallattice as described above. The optical lattice is ramped up adiabatically in 150 ms following
a exponential ramp, then the dipole trap is ramped down within 150 ms. In the next step,
we prepare Feshbach molecules in the optical lattice by ramping an external magnetic field
through a Fano-Feshbach resonance. We subsequently transfer the Feshbach molecules tothe rotational and vibrational ground state using STIRAP.
After preparing molecules in the optical lattice, we measure their heating rate. The
result is shown in fig. 4.17. We observe a heating rate of 105(12) nK/s. The origin of thisheating, which is an order of magnitude larger than what we measure for atoms is unknown.
We speculate that extra heating might be induced by imperfections in the potential, formation of molecules in higher bands, anti-evaporation induced by number loss or non-adiabatic
heating due to a sudden change of the external trapping potential.
For our experiments, we need a maximum number of molecules in the lowest band of the
optical lattice. In order to optimize this number, we have experimentally compared different
procedures for forming molecules in the optical lattice. The first procedure is to prepare
molecules in the optical dipole trap first and then in a subsequent step load these molecules
into the optical lattice as shown in fig. 4.18. To evaluate this procedure, we measured thenumber of molecules in the optical lattice versus the lattice depth. However, we observed avery fast loss after loading the molecules into the lattice. Presumably, this loss of molecules
was due to a large fraction of the molecules accumulating in other than the lowest band of
the optical lattice and fast s-wave two-body losses [84].
In the second experiment, we load atoms in the optical lattice and then convert atoms

81
into ground state molecules. The experimental sequence is shown in fig. 4.18. Fig. 4.19 (black
open circles) shows the number of molecules prepared in the lowest band of the optical latticeversus the lattice depth at which the molecules are formed. As can be seen from fig. 4.19,
the molecule number seems to be almost independent of the optical lattice depth. In thethird measurement, we formed ground state molecules at different optical lattice heights
and then finished to ramp the optical lattice up in 7 ms to a fixed final lattice height of
120 Erec as shown in fig. 4.20. The measurement is shown in fig. 4.19 (red open squares).
We implemented this procedure to maximize the number of molecules in the lowest band of
the optical lattice. The number of molecules created in both cases are very similar to one
another. We decided to prepare molecules by forming them in the optical lattice when its
intensity is fully ramped up.

 !  " # 

     

Figure 4.16: Ramping procedure for formation of ground state 40 K87 Rb molecules in theKRb.
optical lattice. The final lattice depth is 60 Erec

82

Temperature ( K)

0.6

heating rate = 105(12) nK/s

0.5

0.4

0.3
0.00

0.25

0.50

0.75

1.00

Time (s)Figure 4.17: Heating rate for 40 K87 Rb in the optical lattice. The optical lattice depth is 60
KRb
depth.
Erec

150 ms

Image
5 ms
10 ms

Dipole Trap
2

Variable Height

Optical Lattice
1
Time

Figure 4.18: Ramp procedure for forming/loading ground state molecules in the opticallattice. The number 1 indicates where the ground state molecules are formed. Then themolecules are loaded in the optical lattice. The number 2 indicates where the ground state
molecules are formed. In this case the molecules are formed in the fully ramped up opticallattice. The arrow with the note “Variable Height” means that the final value of the latticeKRb.
is changed. The maximum height (depth) value that the lattice can go is 120 Erec

83

4

KRb number (10 )

3.0
2.5
2.0
1.5
1.0
0.5
0.0
0

20

40

60

80

100

Optical Lattice Depth (E

120

rec

)

Figure 4.19: Measurement of the number of ground state molecules versus the optical lattice
depth. The black open circles show the molecule number when the molecules are formed in
the fully ramped-up optical lattice. The red open squares show the molecule number whenthe molecules are formed in the partially ramped up optical lattice.

84

3 ms
150 ms

7 ms Image
5 ms
10 ms

Dipole Trap

Optical Lattice3

Variable HeightTime

Figure 4.20: Ramp procedure for forming/loading ground state molecules in the opticallattice. The number 3 indicates where the ground state molecules are formed. The molecules
are formed in a partially ramped up optical lattice. The arrow with the note “Variable
Height” means that the partial value of the lattice is changed. The final height (depth)KRbvalue of the lattice is 120 Erec.

Chapter 5
Suppression of Inelastic Collisions in 2D geometry

As we have shown in chapter 3, atom-exchange chemical reactions dominate the collisional dynamics in the ultracold molecular gas when the gas is confined in three dimensions(3D). The inelastic rate increases with increasing dipole moment and leads to a very fast lossof molecules from the gas and strong heating [40]. The inelastic loss processes dominate overelastic collisional processes, thereby preventing evaporative cooling of the molecular sample
into quantum degeneracy. However, the chemical reactions are dominated by “head-to-tail”
collisions between polar molecules where the attractive interactions between the dipoles significantly reduces the height of the p-wave collisional barrier. “Side-by-side” collisions should
however be suppressed with increasing dipole moment. In this chapter, I report the suppression of inelastic “head-to-tail” collisions in the molecular sample by confining the molecules
in a 2D geometry. This suppression of inelastic collisions by about 2 orders of magnitudes ascompared to that in a 3D geometry promises to open the way towards evaporative coolingof polar molecules into quantum degeneracy.
In the following, I will discuss how attractive “head-to-tail” collisions can be controlled
and suppressed by means of 2D spatial confinement. I will define the concept of stereodynamics and explain how the quantum statistics and the quasi-two-dimensional confinement cometogether to exclude certain collisional decay channels. I will then discuss our experimentswith polar molecules in a 2D geometry. I will explore collisions between polar molecules
in distinguishable and indistinguishable quantum states of motion and in particular discuss

86
the suppression of chemical reactions for 40 K87 Rb molecules confined in the optical lattice.
Throughout this chapter, I closely follow reference [74], which describe our work.
5.1

Chemical Reaction Rates and Stereodynamics
Chemical reaction rates often depend strongly on stereodynamics, namely the orienta-tion and tranlational movement of molecules in three-dimensional space [85, 86, 87]. An ultracold molecular gas, with a temperature below 1 µK, provides an unusual regime for chemistry,where polar molecules can be oriented using an external electric field and where, moreover,the motion of two colliding molecules is strictly quantized. As explained in chapter 3, atomexchange reactions were observed in a trapped ultracold gas of 40 K87 Rb molecules [39]. In an
external electric field, these exothermic and barrierless bimolecular reactions, KRb+KRb→
K2 +Rb2 , occur at a rate that rises steeply with increasing dipole moment to the power ofsix [40]. The quantum stereodynamics of the ultracold collisions can be used to suppress thechemical reaction rate by nearly two orders of magnitude [74]. We use an optical lattice toconfine fermionic polar molecules in a quasi-two-dimensional, pancake-like geometry, withthe dipoles oriented along the tight confinement direction [33, 88], perpendicular to eachpancake. With the combination of sufficiently tight confinement and Fermi statistics of themolecules, two polar molecules can approach each other only in a “side-by-side” collision,
where the chemical reaction rate is suppressed by the repulsive dipole-dipole interaction.
The suppression of the reaction rate requires quantum-state control of both the internal andexternal degrees of freedom of the molecules.
Two colliding polar molecules interact via long-range dipole-dipole forces well before
they reach the shorter distance scales where chemical forces become relevant. Therefore, thespatial anisotropy of the dipolar interaction can play an essential role in the stereochemistryof bimolecular reactions of polar molecules. In general, one expects the attraction between
oriented dipoles in a “head-to-tail” collision to be favorable for chemical reactions, while
the repulsion between two oriented polar molecules in a “side-by-side” collision presents an87
obstacle for reactions. Up to now, however, large center-of-mass collision energies have precluded the direct control of chemical reactions via dipolar interactions. In a cold collision
regime, where tens of scattering partial waves contribute, one can begin to exert modifications of intermolecular dynamics through the dipolar effect [89]. An ultracold gas, however,provides an optimum environment in which to investigate dipolar effects [40, 90, 91]. Here,the 40 K87 Rb molecules are prepared in identical internal quantum states, with the dipolesoriented using an external electric field, and the molecular gas confined in external potentials created using light. In the limit of vanishing collision energies, the stereodynamics is
described by only a few quantized collision channels, and, moreover, for indistinguishable
molecules, the states of translational motion are coupled to internal molecular states dueto the fact that the quantum statistics of the molecules (fermions or bosons) dictates aparticular symmetry of the total wavefunction with respect to exchange of two molecules.
In this quantum regime, we have an opportunity to suppress or enhance reaction rates by
understanding and precisely controlling the stereodynamics of colliding polar molecules.
5.2

Collisions between Polar Molecules in Quasi-2D Geometry
The spatial geometry of the confining potential can influence collisions in a trapped gasof polar molecules. In particular, a two-dimensional (2D) trap geometry, with the dipoles oriented parallel to the tight confinement direction ẑ, is well-matched to the spatial anisotropyof the dipole-dipole interaction [92, 42, 43]. We can realize such a geometry using a onedimensional optical lattice (see fig. 5.1 A), where the trapped molecules are divided among
several isolated layers. In each of these layers, the lattice potential provides tight harmonic
confinement in ẑ such that only the lowest few quantized motional states in ẑ are occupied.
Consequently, within each isolated layer, colliding molecules approach each other in 2D. However, the range of the van der Waals interaction (and, for that matter, the range of dipolarinteractions at our largest external electric field) is still smaller than the spatial extent of
the cloud in the direction of tight confinement, aho , and, therefore, at short intermolecular

88
distances a collision still must be treated in three dimensions (3D).
We now consider the quantized collision channels that define the stereodynamics inthis quasi-2D geometry. For intermolecular separations that are much larger than aho , therelative motion of the two molecules is described by a quantized angular momentum, ~M ,around the ẑ axis, as well as by a quantized relative motion along ẑ. As discussed above, thestereodynamics of ultracold collisions of indistinguishable molecules is strongly influenced
by the fact that the two-molecule wavefunction must obey an overall symmetry with respect
to the exchange of the identical molecules. Following the formalism from reference [93], asmolecules are identical, a overall wavefunction Ψ is constructed in a way that the molecularpermutation operator P gives:
P Ψ = ²P Ψ,(5.1)

where ²P = 1 for bosonic molecules and ²P = −1 for fermionic molecules. The overall
wavefunction Ψ, has an internal wavefunction |α1 α2 , ηi, where α1 and α2 represent theelectronic, vibrational, rotational and nuclear spin degrees of freedom of molecules 1 and
2, and the external wavefunction representing the confinement in one dimension, i.e., theconfinement in the plane perpendicular to the ẑ direction, due to the optical lattice. The
symmetrized states of the external wavefunction are given by:1[|n1 n2 i + γ |n2 n1 i] ,2(1 + δn1 ,n2 )|n1 n2 , γi = p(5.2)

where n1 and n2 are the harmonic oscillator states for molecules 1 and 2. For molecules in
identical harmonic oscillator states n1 = n2 , γ = 1, δn1 ,n2 = 1; for molecules in different
states n1 6= n2 , δn1 ,n2 = 0, and P |n1 n2 , γi = γ |n1 n2 , γi. An essential aspect of relative
motion in the plane perpendicular to the ẑ direction is the exchange symmetry of this partof the two-molecule wavefunction, which we identify with a quantum number γ. For thesymmetric case, γ = 1 and for the antisymmetric case, γ = −1. For two molecules in thesame ẑ harmonic oscillator state, γ = 1; while both γ = 1 and γ = −1 are possible fortwo molecules in different harmonic oscillator states. For the internal wavefunction, the89symmetrized states are given by:1[|α1 α2 i + η |α2 α1 i] ,2(1 + δα1 ,α2 )|α1 α2 , ηi = p(5.3)

where for molecules in the same internal states α1 = α2 , δα1 ,α2 = 1, for molecules in different
internal states α1 6= α2 , δα1 ,α2 = 0. Similarly, we use a quantum number η to keep trackof the exchange symmetry of the part of the wavefunction that describes the internal statesof the two molecules. For two molecules in the same internal quantum state, η = 1, whileη = ±1 for molecules in different internal states.
In 2D, these three quantum numbers (M, γ, η) are sufficient to describe the quantum
stereodynamics. However, because the interactions at short range must be described in
3D, 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 [93]:η(−1)L = −1,

Short range, 3D(5.4)

ηγ(−1)M = −1.

Long range, 2D(5.5)

For ultracold collisions where long range interactions play decisive roles, the chemical
reaction rate will be dominated by the allowed collision channel with the lowest centrifugal
barrier. Combining this fact with the relations above, we identify three collision channels
relevant to the stereodynamics, and we label these |1i, |2i, and |3i, in order of increasing
centrifugal barrier heights. The dipole-dipole interaction mixes states with different L. However, for convenience, we will refer to the lowest energy adiabatic channel, which does nothave a centrifugal barrier, as L = 0. Similarly, we will use L = 1 to denote the odd-L adiabatic channel with the lowest centrifugal barrier. Collision channel |1i has η = −1, L = 0,
γ = 1, and M = 0, and corresponds to spatially isotropic collisions. Collision channel |2i hasη = 1, L = 1, γ = −1, and M = 0, and is the quantum analog of “head-to-tail” collisions.
Collision channel |3i has η = 1, L = 1, γ = 1, and M = ±1, and is the quantum analog of

90“side-by-side” collisions. Fig. 5.1 B shows schematically the adiabatic potentials for thesethree lowest energy collision channels. Channels |1i and |2i become increasingly favorable
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 thiswork. This barrier hence continues to prevent molecules from reaching short range.
Fig. 5.1 C shows how these different collision channels can be accessed through controlof the internal molecular states and the ẑ motional states. In fig. 5.1 C, molecules in different
internal states are shown in different colors and the harmonic oscillator states in ẑ arelabeled by v. In case (1), for two molecules in different internal molecular states and inany combination of v levels, channel |1i is allowed when η = −1, resulting in no centrifugal
barrier. In case (2), when the molecules are prepared in identical internal molecular states
but in different v levels, the lowest energy collision channel is |2i (“head-to-tail”), which isallowed when γ = −1. In case (3), where the molecules are prepared in the same internal
state and the same v level, the two lower energy collision channels are no longer allowed,
and reactions can only proceed through channel |3i (“side-by-side”). This case is the least
favorable for chemical reactions.
5.3

Band Mapping of Molecules
We create a trapped, ultracold gas of 40 K87 Rb molecules, in their lowest energy ro-

vibrational level and in a single hyperfine state. To confine the molecules, we start witha crossed-beam optical dipole trap, with a harmonic trapping frequency of 180 Hz along
the vertical direction (ẑ) and 25 Hz in the transverse directions. We ramp up the opticallattice along ẑ in 150 ms as shown in fig. 5.2. Both dipolar trap and optical lattice beams
are linearly polarized and their polarizations are mutually orthogonal. The structure of the
optical lattice is like a “stack of pancakes”. Each pancake (layer) of the optical lattice istightly confining in ẑ with a harmonic trapping frequency of νz = 23 kHz for the molecules;

91
V

(A)

(B)
3>

z

E

2>
v1=0 , v2=1

y

³Z

x

(C)

1>
(1)

(2)

(3)

v1=0, v2=0

R

v=2

v=1
³Z

v=0

Figure 5.1: Quantized stereodynamics of ultracold chemical reactions in quasi-2D. (A) A
quasi-2D geometry for collisions is realized for polar molecules confined in a one-dimensional
optical lattice. An external electric field is applied along the tight confinement axis. (B) The
lowest three adiabatic potentials for collisions are shown schematically as a function of theintermolecular separation, R. These three channels are ordered with increasing magnitudeof 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) Three different
cases are shown schematically for each of the three lowest collision channels. The lowest
energy collision channel occurs when two molecules are prepared 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 has 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 [74].

while in the transverse directions, the combined trap has a harmonic trapping frequency of 36
Hz. The tunneling rate between lattice layers is negligible and, therefore, each layer realizes
an isolated trap for the molecules. We begin with 34,000 ground-state molecules confined inroughly 23 layers, with the center layer having 2200 molecules and a peak density of 3.4×107
cm−2 . The temperature of the molecular gas, T , in the combined dipole trap and optical

92
lattice can be varied between 500 nK and 800 nK by changing the initial atom gas conditions.
To completely freeze out motion of the molecules along ẑ requires that kB T ¿ hνz , whereBT
kB is Boltzmann’s constant. For a gas at T = 800 nK in our lattice, khν
= 0.72 , and wez

expect 25% of the molecules will occupy higher v levels.
In order to control the bimolecular reactions in the ultracold gas, we need to control
both the internal state and the harmonic oscillator level v of the molecules. We form themolecules in a single internal quantum state. The occupation of lattice levels v can becontrolled by changing the temperature T . We can also prepare a non-thermal distributionof molecules using parametric heating. The parametric heating is done by modulating thelattice intensity at twice νz in a 300 µs time window as shown in fig. 5.2. As result, a fractionof molecules initially in the v = 0 level are excited to the v = 2 level.
The population in each lattice level is determined by using an adiabatic band-mapping
technique [94, 82]. As shown in fig. 5.2, the lattice potential is ramped down slowly, molecules
in different vibrational levels of the lattice are mapped onto Brillouin zones, with corresponding quasi momenta of ±k~, ±2k~, etc. The measured molecule momentum distribution
following this ramp is shown in fig. 5.3 A for a T = 800 nK molecular gas. The measured fraction in v = 0 matches well with the expected thermal distribution. The thermal
distribution is calculated using the Boltzmann distribution:f (v, T ) = Pz)exp(− 0.5hν
kB T

v exp(−(v+0.5)hνz
)
kB T

,

(5.6)

where v is the vibrational level number. In contrast, Fig. 5.3 B shows the measured nonequilibrium occupation of lattice vibrational levels following parametric heating.
As discussed in chapter 4, the density distribution of the molecules in the verticaldirection can be recovered from the image by fitting with the result of a convolution between
a Gaussian function and a Heaviside function. It is given by:

93

Parametric Heating
Modulation

300 Ps500 PsImage
10 ms

Dipole Trap

Optical Lattice
150 ms

10 ms

Time

Figure 5.2: Ramp procedure for transfering atoms from the dipole trap to an optical lattice
as well as for band mapping for molecules. Molecules are formed after an 150 ms adiabatic
ramp. A non-thermal distribution is created by using parametric heating by pulsing asinosoidal wave with 300 µs. The band mapping is done by linerly ramping down in 500 µs.
To image the molecules, we reverse our coherent transfer process to bring the molecules backto a weakly bound state where we can detect the molecules with time-of-flight absorption
imaging [26].

·

µ

¶
µ
¶¸
∆ + 2(z − zc )∆ − 2(z − zc )√√f (x) = A/2 Erf
+ Erf
+
2 2σ
2 2σ
·
µ
¶
µ
¶¸
−∆ + 2(z − zc )−∆ − 2(z − zc )√√B/2 Erf
+ Erf
+
2 2σ
2 2σ
·
µ
¶
µ
¶¸
2∆ + 2(z − zc )2∆ − 2(z − zc )√√B/2 Erf
+ Erf
+
2 2σ
2 2σ
·
µ
¶
µ
¶¸
3∆ + 2(z − zc )−2∆ − 2(z − zc )√√C/2 Erf
+ Erf
+
2 2σ
2 2σ
·
µ
¶
µ
¶¸
−2∆ + 2(z − zc )3∆ − 2(z − zc )√√C/2 Erf
+ Erf
,
2 2σ
2 2σ

(5.7)

94
where A, B, and C, are the amplitudes of the 1st , 2nd , and 3rd Brillouin zones, d is the centerof the distribution, σ is the image resolution, and x is the variable of the function. ∆ is theTtofsize of the distribution in the Brillouin zone, ±~k. ∆ = 2~k, where Ttof is the time
m 5.44µm/pixof flight, m is the mass of 40 K87 Rb, and 5.44 µm is the corresponding value in micrometers
per pixel at the image. A, B, and C, are given in units of OD (optical density). Since theimage coordinates are given in pixels, the fit gives, zc , z, ∆, and σ are all given in pixel units.
Then, the axial coordinate is converted to quasimomentum units ~k. The free parametersof the fitting function are A, B, C, zc , and σ.
The relative population in each vibrational level of the optical lattice is given by thearea of the distribution in a momentum zone divided by the total area of the distribution.
Since these zones have the same width along the quasimomentum axis, we can use only
the amplitudes to calculate the relative population on each band. The relative molecules
population in the optical lattice are given by:

n0
A
=
,
ntot
A+B+C
n1
B
=
,
ntot
A+B+C
n2
C
=
,
ntot
A+B+C

(5.8)
(5.9)
(5.10)

where, n0 , n1 , and n2 are the initial populations in the three first bands, and ntot is the total
initial population.
The traces in fig. 5.3 were obtained by averaging the images in the transverse direction
within one rms width of the Gaussian distribution. We fit the traces with eq. 5.7. The
uncertainty in the relative population is 3%, and is dominated by systematic errors arisingfrom the variation of the imaging resolution within the range of 1 to 2 pixels.

95

(A)

(B)
^
Z

0.20
0.15

^
Z

0.20

n0/ntot=0.76(3)n0/ntot=0.49(3)n1/ntot=0.23(3)n1/ntot=0.16(3)0.15

n2/ntot=0.35(3)

0.10

OD

OD

n2/ntot=0.01(3)

0.05

0.05
0.00
-6

0.10

-4

-2

0

2

Quasimomentum (³k)

4

6

0.00
-6

-4

-2

0

2

4

6

Quasimomentum (³k)

Figure 5.3: Relative population of molecules in the lattice vibrational levels.We measure
the relative population in each lattice vibrational level using a band-mapping technique.
The results are shown for (A) a thermal distribution of molecules and (B) a non-thermal
distribution created by parametric heating in ẑ. The two images use the same color scale forthe optical depth (OD). The images are an average of 5 shots and 7 shots for (A) and (B),respectively, taken after 10 ms of free expansion. Below each image we show a trace along ẑ
that corresponds the OD averaged over the transverse direction. A fit (red line) to the trace,
which takes into account both the size of the Brillouin zones and our imaging resolution, isused 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 wavevector. Figure reproduced from reference [74].

5.4

Suppression of Inelastic Collisions in the Optical Lattice
We measure the chemical reaction rate by monitoring the loss of trapped molecules

as a function of time. We follow the ramp procedure shown in fig. 5.4. The molecules
are imaged after being released and freely expanding from the combined dipolar trap andoptical lattice. From the images, we obtain the total number of molecules and the radial
cloud size. Since we do not resolve the individual layers of the optical lattice, we obtain

96
an average 2D density per layer. The average 2D density is obtained by dividing the total
number of molecules, N , by an effective area, 4πασr2 , where σr is the rms cloud size in thetransverse direction and N/α is a number-weighted average over the occupied lattice layers.
We calculate α(t = 0) = 23 assuming an initial discrete Gaussian distribution in ẑ with anrms width that we measure after transferring the molecules back to the optical dipole trap.
However, α increases at longer times because of the density dependence of the loss. For ouranalysis, we use a time-averaged value α = 30 that was determined by comparing an analysis
based on a uniform layer density to a numerical simulation of the loss in each layer. The
uncertainties for the loss rate coefficients are dominated by statistical uncertainties in thefits to ntot (t).
In fig. 5.5 A, 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 of 0.158 Debye (D), where 1D = 3.336 ×10−30 C·m. The
two data sets in fig. 5.5 A correspond to an unperturbed T = 800 nK gas (black squares)
and a parametrically heated gas (red circles). For the case where parametric heating wasused to increase population in v > 0 levels, the data show a faster initial loss of molecules.
This suggests that the initial loss is predominately due to interlevel collisions as describedin case (2) of fig. 5.1 C, while intralevel collisions (case (3) of fig. 5.1 C) give a slower loss ofmolecules at longer times.
We fit the data using a simple model, which assumes two loss rate constants: one forinterlevel collisions, β|2i , and a second one for intralevel collisions, β|3i (with the subscripts
referring to the adiabatic channels labeled in fig. 5.1 B). Here,dn0= −β|3i n20 − β|2i n0 n1 − β|2i n0 n2 ,dtdn1
= −β|2i n0 n1 − β|3i n21 − β|2i n1 n2 ,dtdn2
= −β|2i n0 n2 − β|2i n1 n2 − β|3i n22 ,dt(5.11)

where nv is the 2D density of molecules in a particular lattice vibrational level v. To fit the

97measured 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 fig. 5.3) at t = 0, and we fit
the data to the numerical solution of eq. 5.11. We obtain β|3i and β|2i from a simultaneous
fit to the two measured ntot (t) curves shown in fig. 5.5 A.

Parametric Heating
Modulation

300 PsImage
10 ms

Dipole Trap

Optical Lattice
150 msVariablehold time
5 ms
E-field on
Time

Figure 5.4: Ramp procedure for measurement of chemical reaction rates.
The
40 87
K Rb molecules are produced when the optical lattice is fully formed at ≈ 100 Erec ,after 150 ms ramp time. The molecules can be excited to higher vibrational levels of the
optical lattice via parametric heating. The parametric heating is done via a sinosoidal modulation of the lattice intensity for 300 µs. The electric field is turned on together with theparametric heating and it is turned off 5 ms before we switch off the dipolar trap and theoptical lattice. The molecules are held in the combined dipolar trap and the optical lattice
for a variable time. After the molecules are released from the trap, the image is taken after
10 ms time of flight.
−→
By repeating this procedure for different values of | E |, we measure the chemical reac-

98tion rate constants, β|3i and β|2i , as a function of the induced dipole moment. In fig. 5.5 B,
we show the intralevel (black squares) and interlevel (red circles) chemical rate constants asa function of the dipole moment. Also shown as green triangles in fig. 5.5 B are the resultsof two measurements for a 50/50 mixture of molecules in two different rotational states (case(1) of fig. 5.1 C) |0, 0, −4, 1/2i and |1, 1, −4, 1/2i. The 50/50 mixture is created by using a
π/2 microwave pulse when the molecules are in the ground state. The notation used here is¯®
¯N, mN , mK , mRb , where N is the rotational quantum number, mN is the projection of the
I
I
Rb/K

rotational quantum number on the applied electric field direction, and mI

is the nuclear

spin quantum number as in reference [65]. Here, we fit the loss of molecules in the groundrotational state to the solution of dndttot = −β|1i n2tot to extract a single loss rate constant.
For comparison with these measurements, Goulven Qumner and John Bohn perform
quantum scattering calculations using a time-independent quantum formalism based on
spherical coordinates with cylindrical asymptotic matching to describe the molecular collisions in quasi-2D [93]. We use an absorbing potential at short distance to represent chemical
reactions [42, 67]. This technique showed excellent agreement with previous experimental
data for 40 K87 Rb bimolecular reactions in 3D [39, 40]. We computed the loss rate coefficients βv1 ,v2 for molecules in different initial lattice vibrational states v1 , v2 , at a collisionenergy of 800 nK. When the induced dipole moment is still small (0 - 0.2 D), the measured
temperature is a good approximation for the mean collision energy. The loss rates of the different 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 fig. 5.5 B corresponds to an average ofthe slow rates weighted by the initial populations n0 , n1 , n2 :βblack = β0,0 − (β0,0 − β1,1 )ζ12 ,(5.12)

where ζ1 = n1 /ntot is the percentage of the initial relative population in the lattice vibrational
state v = 1. The red curve corresponds to the same average but for the fast rates. The green

99
curve corresponds to the loss rate of molecules in different internal states.
βred =

β0,1 ζ0 ζ1 + β0,2 ζ0 ζ2 + β1,2 ζ1 ζ2.
η0 ζ1 + ζ0 ζ2 + ζ1 ζ2(5.13)The three measured reaction rate constants shown in fig. 5.5 B are consistent with
the quantum scattering calculations for the collision channels shown in matching colors in
fig. 5.1 B and fig. 5.1 C. Molecules in different rotational states (green triangles in fig. 5.5 B)
have the highest rate for chemical reactions, consistent with the fact that they can collide
in channel |1i, which corresponds to spatially isotropic collisions with no centrifugal barrier.
On the other hand, molecules prepared in the same internal molecular state (red circles andblack squares in fig. 5.5 B) have suppressed reaction rates because the lowest energy collision
channel (|1i) is no longer allowed. Instead, identical molecules in different lattice levels (redcircles in fig. 5.5 B) react predominantly through collisions in channel |2i, or “head-to-tail”,
while identical molecules in the same lattice level (black squares in fig. 5.5 B) react throughcollisions 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 thereactions 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, the reaction
rate is suppressed even as the dipole moment is increased.
Fig. 5.6 shows how the initial loss rate in a gas of identical molecules depends on thefractional occupation of the lowest lattice level, n0 /ntot . 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 fig. 5.5 B) to
β|3i (open point at n0 /ntot = 1). In thermal equilibrium, the fractional occupation of thelowest vibrational level is given by the Boltzmann distribution. For the solid black line in
fig. 5.6, the fractional molecular population f (v, T ) in a vibrational level v at temperature
T is obtained from a Boltzmann distribution, and the effective βinitial is then calculated as
βinitial = β|3i

Xv

f (v, T )2 + β|2iX
v1 6=v2

f (v1 , T )f (v2 , T ).

(5.14)

100
On the top axis of fig. 5.6, we give the corresponding values of the scaled temperature
kB T
.
hνz

The solid triangles in fig. 5.6 correspond to the measured initial loss rate at different

temperatures (500 nK and 800 nK), while the open symbol at n0 /ntot ≈ 0.5 corresponds tothe initial loss rate for the parametrically heated, non-thermal molecular gas.
We directly compare the suppressed chemical reaction rate in quasi-2D to the 3D casein the inset to fig. 5.6. Here, we compare data for a 3D geometry from reference [40] against
the suppressed loss rate constant measured in quasi-2D. For the comparison, the 2D loss rate√is scaled to 3D using β3D = πaho β2D [95, 96, 43], where aho is the harmonic oscillator lengthq~in ẑ, aho =. For a dipole moment d greater than 0.1 D, the 3D loss rate constantmωzincreases dramatically as d6 [40, 41], whereas the scaled loss rate constant for the quasi-2Dcase remains close to the value at zero electric field. At a dipole moment of 0.174 D, themeasured suppression in quasi-2D is a factor of 60.

101

(B)

3x10

n0/ntot = 0.49(3)n1/ntot = 0.15(3)n2/ntot= 0.01(3)n2/ntot = 0.36(3)

6

10

-5

10

-6

10

-7

-1

n0/ntot = 0.76(3)n1/ntot = 0.23(3)

2

6x10

6

6

0
0,00

D

-2

ntot (cm )9x10(cm s )

(A)

0,25

0,50

Time (s)

0,75

1,00

0,00

0,05

0,10

0,15

0,20

Dipole moment (D)Figure 5.5: Measurements of 2D loss rates and comparison with theory. (A) A fit (solidlines) to the measured loss curves, with (red circles) and without (black squares) 0.3 ms ofparametric heating in ẑ, is used to extract the loss rate constants β|3i and β|2i . (B) The
extracted loss rate constants for collisions of molecules in the same lattice vibrational level(black squares) and from different lattice vibrational levels (red circles) are plotted for several
dipole moments. Measured loss rate constants 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 inmatching colors in fig. 5.1 B. Figure reproduced from reference [74].

102

-5

kBT/ h z

0.5

0

-6

10

-9

3 -1

(cm s )

10

-7

10

-10

10

-11

10

D

2

-1

(cm s )
initial

10

1

1.5

-12

10

-13

10

0.00

0.05

0.10

0.15

0.20

Dipole moment (D)

0.5

0.6

0.7

0.8

0.9

1.0

n0/ntot
Figure 5.6: Loss rates from 3D to 2D. 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 ratiokB T. The measured initial loss rates for a dipole moment of 0.174 D are displayed for twohνzdifferent thermal distributions (solid triangles), a non-thermal sample created by parametricheating (the top open triangle), and an extracted pure β|3i when the entire population isresiding in the lattice ground vibrational level (the bottom open triangle). The experimental
results agree well with a simple model (black curve) described in this section. The top line
indicates the value of β|2i as measured in fig. 5.5 B. (Inset) The intralevel loss rate for
identical fermionic 40 K87 Rb molecules in 2D (black circles) is compared with the loss rate in
3D (blue triangles). The 3D data for T = 300 nK are borrowed from reference [40]. The√ 2Ddata 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 [74].

Chapter 6
Conclusion and Future Work

6.1

Conclusion
In this thesis I have presented experimental advances of a dense, ultracold, and nearquantum degenerate polar gas of 40 K87 Rb. For dipolar molecules in a 3D geometry, thechemical reaction loss was seen scale with the dipole moment to the power of six [40], due
to attractive dipole-dipole interactions via “head-to-tail” collisions. The main result demonstrated in this thesis is that the inelastic collisions can be suppressed when the molecules
are confined in a quasi-2D geometry in a one-dimensional optical lattice. Compared todipole-dipole collisions in 3D geometry, collisions in quasi-2D geometry under appropriate
state preparations can suppress the inelastic loss by about two orders of magnitude [74]. We
control the population in the optical lattice of the motional states, which makes it possible
to observe collisions with single components of the p-partial wave.
6.2

Future work
For future work with these 40 K87 Rb molecules there are many possibilities for new

exciting experiments:• Anisotropic rethermalization: observe rethermalization of the molecular cloud between the two transverse directions (x̂ and ŷ) in this 2D geometry.
• Measurement of the elastic cross section of molecules: when the molecular cloud

104
rethermalizes. We can directly measure the rate of rethermalization as a function of
the dipole moment. This will let us determine the elastic cross section for dipolar
collisions as proposed by Bohn et al. [72].
• Realization of quantum degenerate polar gas of 40 K87 Rb: With the first two steps
accomplished, it is possible to reach quantum degenerate regime for a gas of polar
molecules via evaporative cooling.
• Detection of chemical reaction products: so far, we only detect molecular loss. However, the vacuum chamber could be designed with an ion detector to enable detectionof ultracold chemical reactions products.
• Detecting novel quantum phases in the geometry of stacked 2D traps [33, 34, 35, 97,
88, 98].

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