Two-Component Bosons inState-Dependent Optical Lattices

A Dissertation Presented
byDaniel Alexander Pertot
to
The Graduate Schoolin Partial Fulfillment of the Requirements
for the Degree ofDoctor of Philosophy
inPhysics

Stony Brook University
December 2011

Stony Brook University
The Graduate School

Daniel Alexander Pertot
We, the dissertation committee for the above candidate for the Doctor ofPhilosophy degree, hereby recommend acceptance of this dissertation.

Dominik A. Schneble – Dissertation Advisor
Associate Professor, Department of Physics and Astronomy

Harold J. Metcalf – Chairperson of Defense
Distinguished Teaching Professor, Department of Physics and Astronomy

Jacobus Verbaarschot
Professor, Department of Physics and AstronomyPéter Petreczky
Scientist, Physics Department,
Brookhaven National Laboratory
This dissertation is accepted by the Graduate School.

Lawrence Martin
Dean of the Graduate Schoolii

Abstract of the Dissertation

Two-Component Bosons inState-Dependent Optical Lattices
byDaniel Alexander Pertot
Doctor of Philosophy
inPhysics
Stony Brook University
2011

Ultracold atoms in optical lattices provide a highly controllable environment for the clean experimental realization of various
model Hamiltonians from condensed matter and statistical physics.
For example, the two-component Bose-Hubbard model, which reduces to an anisotropic spin-1/2 Heisenberg model in a certain limit
and thus allows for the study of quantum magnetism, can be implemented by using bosons with two different internal states that
couple differently to an optical lattice potential. In this thesis, I
present our first experiments with two-component hyperfine-state
mixtures of ultracold 87 Rb atoms in a state-dependent optical lattice, both in the strongly correlated regime and in the context ofnonlinear atom optics.
For the production of 87 Rb Bose-Einstein condensates we havedeveloped a moving-coil transporter apparatus featuring a magnetic TOP trap which serves as a “phase-space funnel” to load acrossed optical dipole trap. The apparatus further incorporates aiii

three-dimensional optical lattice setup with simultaneously usable
hyperfine state-dependent and state-independent lattice beams ofdifferent spacing along the vertical axis. Internal state control is
performed via rf and microwave Rabi pulses and Landau-Zenersweeps.
As a first step towards studying strongly correlated twocomponent mixtures, we have realized a state-selective superfluidto-Mott insulator transition, where one component enters the Mott
insulator regime, while the other one stays superfluid. Using thestate-dependent lattice we can tune the second component’s properties from highly superfluid to strongly localized. At both extremes we find a reduction of the coherence of the primary component, i.e. a shift of the Mott transition to smaller values of theratio U/t of interaction to tunneling. We ascribe this to a polaronlike dressing on the one hand, and a “quantum emulsion” causing
a disordered atomic background potential on the other hand.
Further, we have observed and studied four-wave mixing of
two-component matter waves. Using state-selective Kapitza-Diracdiffraction of a two-component BEC, we prepare seed and pumpmodes differing both in momentum and internal state. A novel
collinear four-wave mixing process then populates the initiallyempty output modes. While this process can complicate studies
of bosonic mixtures loaded into state-dependent optical lattices,it might prove useful for possible applications in quantum atom
optics.

iv

Contents
List of Figuresviii

Acknowledgements

xii

1 Introduction

1

2 Fundamentals: Bose-Einstein condensation in dilute gases
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Bose vs. Fermi statistics . . . . . . . . . . . . . . . . . . . . .
2.3 The ideal Bose gas in a harmonic trap . . . . . . . . . . . . .
2.4 Taking into account interactions . . . . . . . . . . . . . . . . .
2.5 The Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . .
2.6 Density profile: Thomas-Fermi approximation . . . . . . . . .
2.7 Elementary excitations and collective modes . . . . . . . . . .

4
4
5
6
8
9
10
11

3 Fundamentals: Putting a handle on atoms
3.1 Atom-Light Interaction . . . . . . . . . . . . . . . . . . . . . .
3.2 Static external magnetic field . . . . . . . . . . . . . . . . . .
3.3 State-Dependent Optical Potentials . . . . . . . . . . . . . . .
3.3.1 Calculation of optical potentials . . . . . . . . . . . . .
3.3.2 Scattering rates and heating . . . . . . . . . . . . . . .
3.3.3 Experimental Lattice Depth Determination . . . . . . .
3.4 Optical lattice band structure and wave functions in terms of
the Mathieu equation . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Energy bands . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Bloch wave functions . . . . . . . . . . . . . . . . . . .
3.4.3 Localized basis states – Wannier functions . . . . . . .
3.5 Strongly-correlated atoms and the Bose-Hubbard model . . . .
3.5.1 Exact solutions for special cases . . . . . . . . . . . . .
3.5.2 Mean-field treatment and T = 0 phase diagram . . . .

12
12
15
16
17
19
22

v

25
25
26
27
29
32
33

4 Experimental Setup
4.1 Overview of our BEC machine . . . . . . . . . . . . . . . . . .
4.2 Cooling and Imaging Laser System . . . . . . . . . . . . . . .
4.3 Magnetic Traps and Transporter . . . . . . . . . . . . . . . . .
4.3.1 Movable Quadrupole Coil Pair . . . . . . . . . . . . . .
4.3.2 Time-averaged Orbiting Potential Trap . . . . . . . . .
4.4 Optical dipole trap . . . . . . . . . . . . . . . . . . . . . . . .
4.5 State-dependent lattice setup . . . . . . . . . . . . . . . . . .
4.6 RF Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Microwave Setup for Hyperfine State Manipulation . . . . . .
4.7.1 Microwave Generation . . . . . . . . . . . . . . . . . .
4.7.2 Hyperfine State Manipulation and Detection . . . . . .
4.8 Condensate production . . . . . . . . . . . . . . . . . . . . . .
4.8.1 Condensate production in the McTOP trap . . . . . .
4.8.2 Condensate production in the optical trap . . . . . . .
4.9 Demonstrating the Superfluid-to-Mott insulator Transition with
our Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . .

36
36
38
40
40
41
45
46
47
47
47
50
51
51
53

5 Interacting Bosonic Mixtures in Optical Lattices
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . .
5.3 Visibility and diffraction peak width as coherence measures . .
5.4 Delocalized background medium . . . . . . . . . . . . . . . . .
5.5 Strongly-localized background medium . . . . . . . . . . . . .
5.6 Varying the localization of the background medium . . . . . .
5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56
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58
59
61
65
65
67

6 Collinear Four-Wave Mixing of Two-Component
Matter Waves
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . .
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 FWM after release from a slowly-loaded lattice . . . . . . . . .
6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68
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77

7 Conclusion

78

Bibliography

54

80

vi

A Dipole Matrix Elements for Multi-level Atoms
A.1 Atom-light interaction . . . . . . . . . . . . . . . . . . . . . .
A.2 Optical Dipole Potential . . . . . . . . . . . . . . . . . . . . .
A.3 Factoring out the angular dependence of the dipole matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

92
92
93
95

List of Figures
Energies of the ground state hyperfine levels of 87 Rb as a function of an applied static magnetic field. . . . . . . . . . . . . .
3.2 Illustration of the basic principle behind state-dependent opticalpotentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 (a) Calculated lattice depths and (b) peak scattering ratesinΓpk
sc = 4Γsc for the two hyperfine states |ai = |1, −1i and|bi = |2, −2i for σ + and σ − polarized light as a function of
wavelength. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Measured lattice depths in a simple retro-reflected beam configuration for the two states |ai ≡ |1, −1i and |bi ≡ |2, −2i asa function of the lattice beam polarization at a given intensity
and at a wavelength of 785.1 nm. . . . . . . . . . . . . . . . .
3.5 Example of a Kapitza-Dirac (KD) diffraction pattern as seen inabsorption imaging after time of flight. . . . . . . . . . . . . .
3.6 (a) Sinusoidal lattice potential V (x) as created by an opticalstanding wave. (b) Relative energy of harmonic oscillator levels
fitting into a single well, as the lattice depth s is increased. . .
3.7 Allowed energy bands E(k) of solutions propagating with wave
vector k, here at s = 5. Shown are the reduced zone scheme,
the extended zone scheme, and the repeated zone scheme. . . .
3.8 The five lowest bands, i.e. the energy ranges where propagating
solutions exist, as a function of lattice depth s. In the gaps between these regions only localized and evanescent wave solutions
exist. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9 Ground state Wannier function |w0 (x)|2 localized at a single site
in a lattice potential V (x) of depth s = 50. . . . . . . . . . . .
3.10 (a) Size of the tunneling matrix elements as a function of s,
calculated as Fourier coefficients of the dispersion relation of thelowest band. (b) Magnitude of higher-order tunneling matrix
elements |tn | compared to nearest neighbor tunneling t. . . . .
3.1

viii

15
16

19

22
24

26

27

28
29

31

3.11 (a) Phase diagram of the Bose-Hubbard model at T = 0 according to the decoupling mean-field approach. (b) Atom number
per site along the dashed line in (a). . . . . . . . . . . . . . .
3.12 Shell structure in an isotropic harmonic trap. . . . . . . . . . .
4.1
4.2
4.3

33
35

Transporter apparatus. . . . . . . . . . . . . . . . . . . . . . .
Schematic illustration of the steps to BEC in our apparatus. .
Relevant level-structure of rubidium-87 and transitions used forlaser-cooling and imaging. . . . . . . . . . . . . . . . . . . . .
4.4 Schematic laser setup for laser-cooling and imaging. . . . . . .
4.5 Moving-coil TOP trap configuration (McTOP) around the science cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Closed-loop bias coil driving circuit. . . . . . . . . . . . . . . .
4.7 Illustration of the dipole trap and lattice beam configuration
with respect to the science cell. . . . . . . . . . . . . . . . . .
4.8 Schematic illustration of the experimental setup for the generation and delivery of microwave radiation to the atoms in thescience cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Superposition of absorption images taken at different times offlight 2 ms apart after a BEC mixture of the |1, −1i and |2, −2i
hyperfine states is released from the trap. . . . . . . . . . . . .
4.10 Example of Rabi oscillations between the hyperfine states |1, −1i
and |2, −2i as the microwave pulse duration τ is varied. . . . .
4.11 Condensate production in the McTOP trap: Phase transition
as seen in near-resonant absorption images (F = 2 → F 0 = 3)
after 16 ms time of flight along with vertical cuts through thedensity profiles. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.12 Condensate production in the McTOP trap: Reproducibility of
the condensate position in the horizontal plane. . . . . . . . .
4.13 All-optical trapping and manipulation of a condensate after
transfer from the McTOP trap. (a) Conversion of each of thetwo beams of the crossed dipole trap into an optical lattice beamby partial retro-reflection, using a double-pass AOM configuration. (b) Superfluid-to-Mott insulator transition and back in athree-dimensional state-independent optical lattice (1064 nm)3
as observed in absorption images after 18 ms time of flight. . .

37
39

5.1
5.2

57

Illustration of the two-component Bose-Hubbard model. . . . .
Ordered phases of the two-component Bose-Hubbard model inthe deep Mott-insulator limit at unit filling (na + nb = 1) at
T = 0 obtained via a simple mean-field ansatz. . . . . . . . . .

ix

41
42
43
44
45

48

49
50

51
53

54

58

5.3

5.4

5.5

5.6
6.1

6.2

6.3
6.4
6.5
6.6

State-dependent transition from the superfluid to the Mottinsulating regime. (A) Surfaces of equal probability density(1/e2 ) for non-interacting atoms. (B) Time-of-flight (TOF)
absorption image after release of a balanced mixture (fa ≈ fb ).
(C) TOF images after Stern–Gerlach separation. The bottom
graph illustrates the state-dependent z-lattice ramp. . . . . . .
Effects of a background medium on the superfluid coherence.
(A) Stern–Gerlach separated TOF images (for fa ≈ fb ) after agravitational phase shift. (B) Dependence of the visibility andthe peak width on the relative population of the backgroundmedium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(A) The delocalized background component forms a superfluid
bath for the foreground atoms. (B) The opposite case of astrongly localized background medium, now formed by the red
atoms, which act as static local impurities for the blue foreground component. . . . . . . . . . . . . . . . . . . . . . . . .
Dependence of superfluid coherence on the localization of thesecond component. . . . . . . . . . . . . . . . . . . . . . . . .
Experimental scheme. (A) State-selective Kapitza–Dirac diffraction of a two-component Bose–Einstein condensate. (B)Four-wave mixing with pump modes |b, 0i, |a, 2i and seed mode|a, 0i transfers |bi atoms to the output mode |b, 2i. . . . . . .
Typical absorption images taken after application of the stateselective Kapitza–Dirac pulse (τKD = 25 µs, Va = 6 ER ), 15 ms
time of flight and Stern–Gerlach separation along x, for thecase of (A) only |ai atoms present (fa = 1), (B) only |bi atoms(fa = 0), and (C) equal populations of both components (fa =0.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fraction of atoms with momenta ±2~kL in state |ai and |bi
after four-wave mixing. . . . . . . . . . . . . . . . . . . . . . .
Mode populations after four-wave mixing. . . . . . . . . . . .
Growth of the population in the output mode |b, ±2i following
the Kapitza–Dirac pulse for Vb = 0 and for Vb = Va . . . . . . .
Four-wave mixing effects for an adiabatically ramped-up opticallattice. (A) An |ai-selective lattice is ramped up to a depth of
6.0 ER within 100 ms onto an equal mixture (fa = 0.5). (B)
Growth of population in |b, ±2i as determined by blasting away
the |ai atoms after an evolution time τ . . . . . . . . . . . . . .

x

60

62

64
66

70

71
72
74
75

76

A.1 The two possible second order perturbation theory diagrams
to calculate the light shift. (a) The atom absorbs light andgets into an intermediate higher internal state, then reemits thelight, going back to the original state. (b) The opposite process
is also possible, i.e. the atom emits light and goes into a lower
internal state, and then reabsorbs a quantum from the radiation
field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi93Acknowledgements
The end of my journey as a graduate student is approaching, and there aremany people who have influenced my path and contributed in one way or
another to make this endeavor a successful one, after all.
First and foremost, I would like to thank my advisor, Dominik Schneble,
for giving me the possibility to explore the exciting field of ultracold atoms in
his group. He has been a constant source of inspiration and his enthusiasm
about physical concepts and experiments is contagious. With his guidance
and vision, Dominik has led us through all the ups and downs that come withbuilding a new experiment, and has kept us on track towards the bigger goals.
He is also a great mentor, takes good care of his students, and works to makethem thrive and prosper. It is hard to imagine a better advisor, and I havebeen extremely lucky to have been able to work with him.
I have also had the pleasure to work with great lab mates, of whom I firstwould like to thank David Sproles and Rebekah Schiller, who had done theinitial work in building the experiment together with Dominik. It has been
fun to design and build the TOP coils with David, and Rebekah has also
contributed a lot to our lab, before she left us to become a teacher, which I am
sure she is great at. Several generations of German exchange students havedone their Master’s thesis in our lab, and I am indebted to all of them. Stephan
Albert characterized and optimized our MOT and the magnetic transport into
the science cell, and together with Rebekah laid the foundations for achieving
BEC with our machine. This then was the privilege of Daniel Greif, whoworked out and optimized our evaporative cooling sequence, and characterized
our first BEC. It has been a great time working with Daniel and we also had
fun times outside the lab. René Reimann, another German exchange student,
contributed significantly to our state-dependent lattice setup by becoming themaster of our Verdi and Coherent 899 lasers, among other things. Moreover,René’s dark humor and his stories from his time as a student in Germany werevery entertaining during many lunch and dinner breaks.
I am very grateful to my fellow PhD student Bryce Gadway, who has beenxii

an invaluable addition to our group. Not only is he accompanied by an air ofbrilliance, but he is also a very talented, tireless and committed experimentalist, and it has been great to work with him. He has very interesting ideas, andI have learned a lot from our discussions about physics. Further, he can also
be quite a funny guy, especially after one or two beers (keyword “Tecate!”).
Additionally, I have to thank him for giving me quite a few rides in his car,when my car was making trouble. I am curious to see where Bryce will do his
postdoc, but wherever he will go I am sure he will do very well in the future.
Finally, although they have been working in the lab mostly after my active
time, I would like to thank Jeremy Reeves, our (meanwhile not so) new PhD
student, and Matthias Vogt, who had been the last German exchange student
during my time in Dominik’s group. In addition, I would like to thank thevarious undergraduate students who had done projects in our lab, among them
Azure Hanson, Bartosz Bogucki, who became our LIAD and microcontroller
expert, Justin Tian, and Gaku Nagashima.
I would also like to thank all the members of the other AMO groups in
Stony Brook and the other “inhabitants” of the basement, who helped withequipment, parts, or general advice and discussions. Among them are Tom
Bergeman, Tom Weinacht and his group, Hal Metcalf and his group, in particular Jason Reeves for his valuable help with connectorizing high-power fibers,
Marty Cohen, from whose experience and knowledge all the AMO groups are
benefitting, John Noe from the laser teaching center, and Peter Koch, who has
been lending us various pieces of equipment, notably a spectrum analyzer anda crucial SMA elbow.
Moreover, I would like to express my gratitude towards Hal Metcalf, Jac
Verbaarschot, and Péter Petreczky for agreeing to serve on my defense committee and for reviewing my dissertation despite the organizational complications involved. Many thanks also go to Michael Köhl and the members of hisgroup for their patience and support while I was working on completing thisdissertation.
Further, I am grateful to all the people from the Physics main office who
helped me navigate through the typical and less typical administrative issues
that can arise during a graduate student’s career. In particular, I would liketo mention Pat Peiliker, Maria Hofer, Pam Burris, Matt Gilbert, and of course
Sara Lutterbie, who somehow managed to make even the most complex administrative matters enjoyable. Regarding guidance in the more fundamental
decisions, I would like to thank all the Graduate Program Directors during mytime at Stony Brook, Peter Stephens, Laszlo Mihaly, and Jac Verbaarschot,
who have always been very supportive.
xiii

Also, I would like to say thank you to all my friends from Stony Brook,for the many lunches and dinners, the car rides, the board game nights, theweekend activities, and the occasional trips to the city. Without your company
and continued friendship, my time in Stony Brook would not have been thesame! Thanks also go to my friends from back home in Germany for keeping
in touch over the years.
Finally, I would like to thank my parents, Marianne and Carlo, who havealways taken good care of me and have always supported me in my pursuits.
Vielen Dank für Eure Unterstützung, und dafür, dass Ihr immer für mich dawart und seid!

This work was supported by the Research Foundation of SUNY, the Officeof Naval Research (ONR), and the National Science Foundation (NSF).
xiv

Chapter 1
Introduction
In 1925, inspired by Bose’s work [1], Einstein conjectured that at sufficiently
high phase-space density a gas of bosons saturates and any additional particles
have to condense into the ground state, which can thus become macroscopically
populated [2, 3]. The validity of this idea has been demonstrated in spectacularclarity in 1995 by the achievement of Bose–Einstein condensation (BEC) in
dilute atomic gases [4–7]. These gases, however, are not ideal Bose gases.
Binary collisions between pairs of atoms lead to weak interactions between thebosons, and the resultant deviations from the non interacting case were at thefocus of early experimental and theoretical research in the field of ultracold
atoms [8, 9].
About a decade ago, the focus shifted towards the experimental realizationand study of strongly interacting many-body systems which are characterized
by the competition of kinetic energy and interaction energy [10]. Such strongly
correlated many-body systems can be obtained either by tightly confining theatoms in optical standing waves [11], commonly called optical lattices, or bytuning their mutual interactions via Feshbach resonances [12]. Following thefirst approach and using a three-dimensional optical lattice, in 2002 Greiner et
al. [13] were able to observe the quantum phase transition from a lattice superfluid to a Mott insulator, realizing a theoretical proposal of Jaksch et al.
from 1998 [14]. This hallmark experiment constitutes an important achievement creating inter-disciplinary links between atomic physics, quantum optics,
condensed matter and statistical physics.
Many new, interesting phenomena arise when a second atomic species withproperties different from the first is added. Such mixtures allow for the study ofmany interesting topics, such as the two-component Bose-Hubbard model [15–18], with its connection to quantum magnetism [15, 19], and models for decoherence mechanisms [20, 21].
In this thesis, I describe a new versatile experimental setup to produce1two-component hyperfine-state mixtures of ultracold 87 Rb atoms in a statedependent optical lattice. Further, I discuss our first experiments, both in thestrongly correlated regime and in the context of nonlinear atom optics.
Bosonic mixtures in optical lattices
Compared to the single-component case, the description of mixtures presentsricher physics, but also depends on additional parameters, such as the ratio of
the tunneling rates, the interspecies interaction, and the relative atom numbers. The introduction of a second component allows for the investigation ofimportant phenomena such as polaron physics [22–24] and phonon-mediatedlong-range interactions [18, 23], as well as effects of impurities and disorder [25–29].
Here, I report on the observation of many-body interaction effects for ahomonuclear bosonic mixture in a three-dimensional optical lattice with variable state dependence along one axis. Near the superfluid-to-Mott insulatortransition for one component, we find that the presence of a second component can reduce the apparent superfluid coherence, most significantly when
the second component either experiences a strongly localizing lattice potential
or none at all. We examine this effect by varying the relative populations andlattice depths, and discuss the observed behavior in view of recent proposals
for atomic-disorder and polaron-induced localization.
Four-wave mixing and non-linear atom optics
In four-wave mixing, two waves form a grating from which a third wave
diffracts, thus generating a fourth wave. Four-wave mixing is a fundamental,
well-studied concept in nonlinear optics and spectroscopy [30]. Its matterwave analogue, based on binary collisions in ultracold atomic gases, was first
demonstrated experimentally a decade ago [31–33], establishing the field of
nonlinear atom optics [34]. Matter-wave four-wave mixing has been used forcoherent matter-wave amplification [35, 36], and for the generation of correlated atom pairs [34–38]. Energy and momentum conservation require themagnitudes of all atomic momenta in the center-of-mass frame to be equal
which, for atoms in a single internal state, necessitates a two-dimensional geometry [32, 33]. By modifying the dispersion relation with an optical lattice,non-degenerate four-wave mixing of a single species becomes possible also in
one dimension [39–41].
Despite considerable theoretical work on atomic four-wave mixing (FWM)with more than one internal state [42–45], experiments have only very recently
started to explore possible mechanisms for such FWM [38, 46]. The addi-

2

tional internal degree of freedom allows for degenerate FWM to occur in one
dimension, with pairs of waves in different internal states sharing the same momentum mode, opening up possibilities to generate non-classical matter-wave
states, e.g. with macroscopic spin entanglement [43, 44].
In this thesis, I demonstrate free-space collinear atomic FWM involving two
internal states with distinct, macroscopically populated momentum modes.
Starting from a single-species Bose–Einstein condensate, seed and pump modes
are prepared through microwave state transfer and state-selective Kapitza–
Dirac diffraction. Four-wave mixing then populates the initially empty output modes. Simulations based on a coupled-mode expansion of the Gross–
Pitaevskii equation are in very good agreement with experiment. We findthat for a homonuclear mixture of interacting superfluids, four-wave mixing
processes can alter the expected momentum-space distributions, masking oreven mimicking in-situ interaction effects. Moreover, our system should be ofinterest in the context of quantum atom optics [47].
Outline of the thesis
After summarizing the fundamentals of BEC in dilute atomic gases in chapter 2, I review important concepts from atomic physics in chapter 3. In chapter 4, a description of our experimental apparatus is given and experimental
techniques for the probing and the manipulation of the motion and the internal states of the BEC are discussed. Then, in chapter 5, I describe ourexperiments with two-component bosons in optical lattices. After a short lookat the two-component Bose-Hubbard model, our observations on the effects
of a second tunable “background” component on the SF-MI transition of afirst component are discussed and interpreted in terms of lattice polarons on
the one hand, and localization and percolation phenomena due to an atomic
disorder potential on the other. In chapter 6, I will switch to our experimentson two-component four-wave mixing, which, for the most part, are performed
on a trapped continuum BEC in three dimensions. Finally, a short conclusion
will be given in chapter 7.

3Chapter 2
Fundamentals: Bose-Einsteincondensation in dilute gases
2.1

Introduction

The idea of Bose-Einstein condensation has its origin in 1924, when Einstein
applied Bose’s new way of calculating Planck’s black-body radiation spectrum [1] to the ideal monatomic gas [2, 3]. He found that at a certain point
the density of the gas would saturate, and argued that any increase in density
beyond this point would lead to a “condensation” of the excess particles into
the state of lowest energy.
Initially, this “peculiar condensation phenomenon” had “rather got the reputation of having only a purely imaginary existence,” as London had put it
in 1938 [48], when he insightfully suggested that the λ-transition and thus
superfluidity in liquid Helium could be related to the concept of Bose-Einsteincondensation [48, 49]. Even though his idea was appealing, the strong interactions in this system considerably complicate the theoretical understanding
and obscure the basic physics of Bose-Einstein condensation. Therefore, formany decades a cloud of mystery was hanging over the topic and motivatedphysicists to find an experimental system in which the phenomenon could beclearly observed and studied.
Only 70 years after its prediction, Bose-Einstein condensation of weaklyinteracting particles could be unambiguously realized and observed in an ultracold dilute gas of neutral alkali atoms [4–7]. This involved cooling matter down to the lowest temperatures ever achieved, as the temperatures forcondensation are extremely low owing to the extreme diluteness necessary to
prevent the atoms from simply combining into molecules. The achievement of
BEC thus builds heavily upon the breakthroughs in laser cooling and atom4

trapping [50] and evaporative cooling [51] made throughout the 1980s.
Since the achievement of BEC in 1995, considerable theoretical and experimental progress has been made. Many textbook-like experiments not only
made it possible to accurately test 50-year-old predictions by Bogoliubov andothers, but also to explore entirely new questions. Detailed historical accounts
can be found, for example, in the Nobel lectures of Cornell, Wieman and
Ketterle [52, 53].
The phenomenon of Bose-Einstein condensation and its fundamental properties are described in detail in many statistical physics textbooks [54, 55], in
textbooks about many-body physics [56], and also in books that are devoted
to the particular case of BEC in the dilute alkali gases [57, 58]. Much knowledge can also be found in review articles [8, 9, 59]. In the following, the basic
properties which are relevant for the remainder of this thesis are introduced
and discussed.

2.2

Bose vs. Fermi statistics

In quantum mechanics the symmetrization postulate states that the wavefunction of a system of identical particles has to be either completely symmetricor completely anti-symmetric under the exchange of any two particles. In a
simple picture, this can be explained by considering the wave function of asystem of identical particles. Now, if we exchange the labels of two particles in
the wavefunction, measurements must not depend on this exchange. Assumingthat the wavefunction gets multiplied by a phase factor when two labels areinterchanged, the modulus of the wavefunction is unaffected only if the phasefactor is either +1 or −1. Particles of the first kind (completely symmetric)
are called bosons and they obey Bose-Einstein statistics, while particles of thesecond kind (completely anti-symmetric) are called fermions and they obey
Fermi-Dirac statistics. In all experiments performed to date, fermions havecorresponded to particles with half-integer spin, while particles with integer
spins have corresponded to bosons. In 1940, this connection between spin andexchange statistics could also be derived theoretically by Pauli [60] based onrelativistic quantum field theory (spin-statistics theorem). As an aside, more
exotic exchange statistics are possible in less than three spatial dimensions,for example anyonic statistics in two dimensions [61] with connections to thefractional quantum Hall effect.
The exchange statistics of atoms leads to fundamentally different ways of
how they behave at low temperatures.
We see this by comparing the thermal√de Broglie wavelength λT = h/ 2πmkB T and hence the spatial extent of thesingle particle wave function of an atom with the average particle separation5d = 1/n1/3 . Here T is the temperature, m is the mass, kB is Boltzmann’sconstant, h is Planck’s constant, and n is the number density. The thermalde Broglie wavelength becoming as large as the average inter-particle spacing
defines the degeneracy temperature TD around which exchange statistics start
to play a role, either bosonic or fermionic. Bose gases start to condense into
the ground state and Fermi gases fill up the Fermi sea. For T  TD theexchange statistics becomes more or less unimportant, and Fermi, Bose andBoltzmann statistics essentially lead to the same physical properties.
Neutral atoms are, of course, composite bosons or fermions made up of
more elementary particles. Therefore they have internal excitations and can,in principle, break apart into composites of different statistics. However, thetemperatures considered here are so low, that all internal excitations are frozen
out, and each atom can be regarded as a simple bosonic particle of mass m.

2.3

The ideal Bose gas in a harmonic trap

Let us first consider an ideal, i.e. non-interacting, Bose gas in a harmonic
trapping potential. The Hamiltonian is a direct sum of single-particle Hamiltonians
1p2+ mqt Ωq
(2.1)
H=
2m 2
where Ω is a 3 by 3 matrix whose eigenvalues are the trap frequencies along themain axes of the harmonic potential (that is the main axes of the equipotential
ellipsoids). According to the Bose-Einstein distribution function, the meanoccupation of state i of energy i isn̄i =1eβ(i −µ) − 1(2.2)

with β = 1/kB T and the chemical potential µ, which is determined
P from thecondition that the mean total atom number has to be N =i n̄i . Since
occupation numbers cannot become negative, the chemical potential has tostay below the ground state energy µ < 0 . As µ approaches 0 from below,
the occupation of the lowest energy state grows to larger and larger values
until eventually all available particles are in the ground state.
Let us thus separate the ground state occupation from the sum over allenergy levels before approximating the sum by an integral over the density ofstates D() assuming closely spaced energy levels and temperatures far above6the level spacings:N=1eβ(0 −µ) − 1

Z ∞
d
+
0

D()
.
eβ(−µ) − 1(2.3)

If the total atom number exceeds the number of available occupancies in thethermal part (integral term), the thermal gas saturates and the excess numberof particles have to populate the ground state of the system, i.e. it forms
the Bose-Einstein condensate. We can compute the fraction of atoms in thecondensate using the so called “Bose function” [54, 57], which is defined as
1
gp (z) =
Γ(p)

Z ∞
dx
0

∞
X
x1/2
zn
=
= Lip (z)z −1 ex − 1 n=1 np(2.4)

and which, more generally, is known as the polylogarithm function Lip (z). In
the special case z = 1 it reduces to the Riemann zeta function gp (1) = ζ(p).
The particle density in the thermal component in the semi-classical approximation [9] isβ(µ−V (r))
nT (r) = λ−3).
(2.5)
T g3/2 (eWhen the temperature drops below a critical value
Tc =

~ω̄
~ω̄ N 1/3
≈ 0.94 N 1/3 ,1/3
kB [ζ(3)]
kB

(2.6)

the ground state becomes macroscopically occupied and there is a phase transition between the normal phase and the Bose-Einstein condensate. Here,ω̄ = (ωx ωy ωz )1/3 is the mean trap frequency. For the ideal Bose gas in athree-dimensional harmonic trap, the condensate fraction behaves as
N0 /N = 1 − (T /Tc )3(2.7)

for T < Tc . Note the remarkable feature of BEC that the transition temperature at which the ground state starts to be macroscopically populated is far
above the energy distance to the first excited state for typical condensate atom
numbers, kB Tc  ~ω̄.
While the intuitive picture of Bose-Einstein condensation as describedabove captures the physical processes correctly, it fails in precisely defining
what a Bose-Einstein condensate is. This was achieved by Penrose and Onsager in 1956 [62] who studied the single-particle density matrix of the manybody system ρ(r1 , r2 ) = hΨ† (r1 )Ψ(r2 )i. A Bose-Einstein condensate exhibits
one macroscopically occupied eigenvalue of the density matrix, which measures

7

the condensate fraction, and off-diagonal long-rage order [63] throughout thewhole system. For Helium-4 at T = 0 the condensate fraction is only about
10%, whereas in dilute gases condensate fractions in excess of 95% can beachieved. The one-particle density matrix in a homogeneous system depends
only on the relative distance r = |r1 − r2 | between two locations. The zeromomentum component is macroscopically occupied if the one-particle density
matrix tends to a finite value for r → ∞, which is just the condensate fraction.

2.4

Taking into account interactions

The non-interacting Bose gas turns out to be quite an oversimplification evenin the case of trapped dilute ultracold gases, where in fact, despite their diluteness, the interaction energy dominates. At the temperatures we are interested
in here, only s-wave collisions contribute, since the atoms do not have enoughkinetic energy to overcome the centrifugal barrier that is present for non-zeropartial waves, i.e. collisions with angular momentum different from zero. Under the condition k  1/rc where k is the modulus of the relative collision
wave vector and rc is a characteristic length scale of the interatomic potential,
the potential can be safely replaced by a δ-function interaction potential,
V (r, r0 ) = g δ(r − r0 )(2.8)

Intuitively speaking, the wave packet of the atomic motion does not havesufficiently high Fourier components to sample the details of the potential.
This is often referred to as pseudo-potential approximation [54]. The strength
g of this pseudo-potential can be re-expressed in terms of another quantity
called the s-wave scattering length as , which can be interpreted in terms of
the phase shift the scattered wave undergoes in the collision. For two colliding
atoms of equal mass one obtains
g = 4π~2 as /m.

(2.9)

The problem of the interacting Bose gas can in general not be solved exactly any more. However, it has turned out that a simple mean-field solution
describes most relevant physical observations very well, except for strong interactions and in low-dimensional systems. Generally, the corrections to the
non-interacting model due to weak interactions are small. Finite-size effects
and interactions shift Tc only a few percent away from the non-interacting
value given above. While the transition temperature is not affected significantly by interactions, the shape of the condensate in the trap, its excitations
and its dynamics are altered considerably, as we will see below.
82.5

The Gross-Pitaevskii equation

The full Hamiltonian for a weakly-interacting Bose gas is given in second
quantization by
Z
H=



~2 2
d r Ψ (r) −∇ + Vext (r) Ψ(r)2m
ZZ
1
d 3 r d 3 r0 Ψ† (r)Ψ† (r0 )V (r − r0 )Ψ(r0 )Ψ(r) (2.10)
+
2
3

†

where the “field” operators Ψ(r) obey the canonical commutation relations forbosonsΨ(r), Ψ† (r0 ) = δ(r − r0 ),[Ψ(r), Ψ(r0 )] = 0,(2.11) †
Ψ (r), Ψ† (r0 ) = 0.
For a condensate, Ψ(r) assumes a non-zero expectation value ψ(r) = hΨ(r)i
which acts as a complex order parameter here. It can be regarded as a macroscopic wavefunction of the condensate. If one writes the field operator as its
expectation value plus a fluctuation operator Ψ(r) = ψ(r) + δΨ(r), the timedependent Schrödinger equation for the order parameter alone leads to theso-called Gross-Pitaevskii equation (GPE) for the condensate wavefunction [9]


~2 2
2
−
∇ + V(r) + g3D |ψ(r, t)| ψ(r, t) = i~ ∂t ψ(r, t)(2.12)
2m
where V(r) is the external potential, which includes the lattice as well as thetrapping potential, and g3D = 4π~2 as /m is the nonlinear coupling with thes-wave scattering length as . Here, the condensate wavefunction is normalized
to the total number of particles
ZZ
3
2d r |ψ(r, t)| = d3 r n(r, t) = N.
(2.13)
This is only possible in this form if the particle density falls off sufficiently
quickly for r → ∞, such that the integral exists, or correspondingly that thecondensate is confined in all three directions, which basically is the case in all
experimental realizations.

9

2.6

Density profile: Thomas-Fermi approximation

In experiments one is usually in a situation of a large interaction parameter
N as /aho because the atom number N is the range N ≈ 105 -106 . Under these
conditions, the kinetic energy term in the Gross-Pitaevskii equation may beneglected except for a small region at the surface of the cloud where the orderparameter changes more rapidly across space. To good approximation, theground state in the trap is then given by the Thomas-Fermi approximation,
which leads to a density distribution [9][µ − V(~r)] /g3D
V(~r) < µ2n(~r, t = 0) = N |Φ0 (~r)| =(2.14)
0V(~r) ≥ µ,with the chemical potential µ that is fixed by the atom number. The highest
particle density npeak occurs at the minimum of the trapping potential, which
we define as the zero point of the energy scale. Note the simple but useful
relation µ = npeak g3D . The density vanishes on and outside of the “ThomasFermi surface” given by V(~r) = µ.
We can also obtain a general result on the relation between the chemical
potential and the total number of particles. Since the density (2.14) is nonzero only inside the Thomas-Fermi surface, we only need to integrate it overthe “Thomas-Fermi volume” VTF to get the number of particles
R
Z
3
r)
V
TF VTFd r V(~R= VTF /g3D (µ − V ). (2.15)
N=
d3 r n(~r) = npeak VTF −
3
g3D
dr
VTF
VTF
Here V is the average potential inside the Thomas-Fermi volume. By definition
we get 0 < V < µ. Defining α = 1 − V /µ which depends on the actual trap
shape and lies between zero and one, we arrive at the general result
N = α µ/g3D VTF .

(2.16)

Note that VTF , V , and also α (unless V is linear in µ) depend on µ.
We generally deal with a harmonic trapping potential or a potential which
can be approximated as such. In normal coordinates, with the trap frequencies ωx,y,z the potential takes the form V(~r) = m/2 (ωx2 x2 +ωy2 y 2 +ωz2 z 2 ). Thus,
the Thomas-Fermi surface is an ellipsoid with half-axes Ri = (2µ/m)1/2 /ωi .

10The enclosed volume is 3/2
4π
4π 2µ
1
VTF =
Rx Ry Rz =.
33 m
ωx ωy ωz

(2.17)

The average potential becomes V = 3µ/5, and therefore α = 2/5 is a constant.
From eq. (2.16) we get N = 2/5 npeak VTF or in terms of µ
√
8π 8
1
5/2
N =µ
.
(2.18)
3/2
15 m g3D ωx ωy ωz

2.7

Elementary excitations and collective modes

The elementary excitations of the condensate ground state were investigatedby Bogoliubov using a theory of small fluctuations around the ground stateand linearizing the solutions for weak excitations. The Bogoliubov excitation
spectrum of a homogenous system reads
 2 2 2 2

~q
~q
2
(~ω) =+ 2gn .
(2.19)
2m
2m
This corresponds to phonon (collective) excitations ω ∝ k at low momenta andfree-particle like excitations ω ∝ k 2 at large momenta. The speed of sound forlow momenta isp(2.20)
c = gn/m.
For the harmonically trapped system, the wavelength of low-frequency excitations has a natural cutoff provided by the size of the harmonic trap. Hencethe lowest collective modes are quantized oscillation modes in the trapping
potential. Using the Thomas-Fermi density distribution


y2
z2
x2
(2.21)n(r, t) = npk (t) 1 − 2 − 2 − 2
Rx Ry Rz
small perturbations around the equilibrium lead to a simple scaling of thecloud radii according to
Ri (t) = Ri (0) bi (t)(2.22)and hydrodynamic equations, which can be derived from the time-dependent
Gross-Pitaevskii equation result in the equations of motion of the scaling parameters [64, 65]
2
ω0i
b̈i =
− ωi2 (t) bi .
(2.23)
bx by bz bi
11

Chapter 3
Fundamentals: Putting a handle
on atoms
The achievement of quantum control over positioning, motion, internal states,
and interactions of ultracold atoms involves a number of involved experimental techniques, most of which are built upon the interactions of atoms withexternal fields. In order to lay the foundations for the principal functioning of
the apparatus and the experiments discussed in the later chapters, I discuss
the fundamentals of the interaction of atoms with light and magnetic fields inthis chapter as well as the properties of atoms in periodic potentials.

3.1

Atom-Light Interaction

The techniques to cool atoms to Microkelvin temperatures, to manipulatethem in optical lattices, and to detect them are fundamentally based on theprinciples of light-matter interaction. Near-resonant laser light is used to cool
the atoms by the exchange of mechanical energy with photons and the dissipative force resulting from this interaction, and far-resonant light is used toprovide conservative potentials with a period of a few hundred nanometer instanding wave laser configurations, so called optical lattices. In this section, I
review the fundamentals of light-matter interaction.
Let us consider a strongly simplified atom with only two energy eigenstates|ai and |bi of energies Ea and Eb , respectively. In an external oscillating
electric field Eext (t) that we treat classically the two-level atom is describedby the Hamiltonian
H = Hatom + Hali .
(3.1)
Here, Hatom = Ea |aiha| + Eb |bihb| is the bare atomic Hamiltonian and Hali is12the interaction Hamiltonian
Hali = −d̂ · E(r, t)(3.2)

with the dipole operator d̂ = −e r̂, where we have used the electric dipole andlong-wavelength approximations, which usually are excellent approximations
for optical transitions. The matrix elements of the dipole operator d vanish
between states of the same parity. The wave functions of the states of ouratom therefore have to be of opposite parity, otherwise our example would berather uninteresting.
In the basis of eigenstates the dipole operator becomes
 0 dabha|d = |ai |bi
(3.3)
dba 0
hb|
with dba ≡ hb|d|ai. Note that the Hermiticity of the Hamiltonian necessitates
dba = d∗ba . This leads to the total HamiltonianEa−dab · Eext (t)
H=
.
(3.4)
−dba · Eext (t)Eb
For the electric field we assume a linearly polarized classical light field Eext (t) =ε̂E0 cos ωt with the polarization unit vector ε̂. Plugging that in givesEa~Ω0 cos ωt
H=(3.5)~Ω∗0 cos ωtEbwith the Rabi frequency ~Ω0 ≡ −(dab · ε̂)E0 . We can split up the cosine into
two exponentials and pull out ~ to obtain~ωaΩ0 (eiωt + e−iωt )
.
(3.6)
H=
ωb
2 Ω∗0 (eiωt + e−iωt )To diagonalize the Hamiltonian, we switch into a frame rotating at angularvelocity ω. In this frame, we obtain a time-independent term given by~ 0 Ω0,(3.7)
Hrot =
2 Ω∗0 −2δ
where δ ≡ ω − |ωb − ωa | is the detuning from resonance. The other term
proportional to e−2ωt rotates at 2ω and for |Ω0 |  ω it can safely be neglected(“rotating-wave approximation”), as its effects average out. The eigenvaluesp
are given by ±~Ω/2, with the generalized Rabi frequency Ω ≡Ω20 + δ 2 ,13

corresponding to ac Stark shifts of the ground and excited states. In the limitof large detuning, δ 2  Ω20 , the ac Stark shift of the ground state is given by∆U ' ~

Ω20
.
4δ

(3.8)

The corresponding eigenvectors (“dressed states”) are mixtures of the original two basis states. The populations in the two original states undergo Rabi
oscillations,
 
 2
Ωt
Ω0
2
,
(3.9)
Pb (t) =sinΩ
2
and Pa (t) = 1 − Pb (t). The state dynamics can also be visualized conveniently
on the Bloch sphere [66]. From the steady-state value of Pb it is possible tocalculate the spontaneous scattering rate Γsc = Pb Γ (where Γ is the naturalline width) in the far-detuned limit δ  Ω0 , Γ as
Γsc =

∆U
Ω20
Γ=Γ
.
2
4δ
~δ

(3.10)

One can derive the following important relation between the reduced dipole
matrix element D and the transition linewidth Γ (see for example Loudon [67])Γ=ω03 e2 D2.
3π0 ~c3(3.11)The saturation intensity Is of the atomic transition considered is defined as [50]2π 2 ~cΓ
~Γω03
Is ≡
=
,
3λ30
3 × 22 πc2(3.12)

where λ0 = 2πc/ω0 . With the cycle-averaged intensity I = c0 |E|2 /2 one
obtainsrI.
(3.13)Ω0 = Γ
2Is
In the above treatment we have neglected the counter-rotating term. If thedetuning becomes comparable to the atomic transition frequency, the counterrotating term has to be taken into account. In this case the ac Stark shift ofthe ground state, or in other words the optical potential, is given by


~Ω20
1
1
∆U = −+,(3.14)
4
ω0 − ω ω0 + ω

1415E h GHz

10|2,+2>

5

F=2

0
5

|1,+1>

10
15|2,-2>|1,-1>

F=1

0

2000

4000

B G

6000

8000

Figure 3.1: Energies of the ground state hyperfine levels of 87 Rb as a functionof an applied static magnetic field.

and the scattering rate byα3 ω 3 Γ2
Γsc =3 × 8ωRyd ω0



1
1
+
ω0 − ω ω0 + ω

2

I
Is

(3.15)

FOR
which is equal to ΓUdip/~δ when neglecting the “counter-rotating” term.

3.2

Static external magnetic field

An important part of the experimental sequence to produce and manipulate
ultracold atomic ensembles is the trapping in inhomogeneous magnetic fields.
In order to elucidate the underlying physics, I consider the general case of an
atom with total electronic angular momentum J = L + S (L is the orbital
angular momentum and S is the spin) and nuclear spin I in a homogeneous
magnetic field B. The Hamiltonian of this system is
H = −µ · B + A J · I,

(3.16)

with F = J + I, where A is the magnetic dipole hyperfine coupling constant,and µ = µB /~ (gJ J + gI I).
For the ground state of the experimentally relevant alkalis, which is 2 S1/2 ,we have L = 0 and S = 1/2, so that J can be replaced by S and gJ by
gS . In this case the Hamiltonian can be diagonalized analytically, yielding the15

so-called Breit-Rabi formula for the eigenenergies [68]
s
∆W
∆W
2mF
±
x + µB gI m F B
W =−1 + x2 +2(2I + 1)
2
I + 1/2(3.17)

with x = (gS − gI )µB B/∆W and ∆W = A(I + 1/2) is the zero-field hyperfine
splitting. The last term is small compared to the others and was neglected in
the original paper by Breit and Rabi.

State-Dependent Optical Potentials
mJ

P1/2

(a)
2

S 1/2

σ+

− 3 2 − 12 + 12 + 3 2

780 nm2

795 nm

P3/2

785 n
m

2

mJ

− 3 2 − 12 + 12 + 3 2

(b)

785 n
m

3.3

σ+

D1 D2

Figure 3.2: Illustration of the basic principle behind state-dependent opticalpotentials. For clarity, the hyperfine structure is neglected. (a) Atoms in themJ = −1/2 state are not affected by σ + light at a certain wavelength between
the D1 and D2 lines where the red and blue-detuned contributions to the lightshift cancel exactly. (b) Atoms in state mJ = +1/2, however, feel an attractive
optical potential, since only “red-detuned” intermediate states are accessible.
For σ − light, the roles of the two states are reversed, i.e. only atoms in statemJ = +1/2 feel the optical potential (not shown).

The ability to independently control the optical potentials experienced byatoms in different hyperfine states opens up many interesting experimental
possibilities. Here, I will focus on the implementation and the properties ofstate-dependent optical potentials.
The basic idea behind state-dependent optical potentials is to choose thewavelength λL and polarization of the trapping or lattice beam such that
there are red and blue-detuned contributions from intermediate excited states
which cancel for one state of but not for another. This concept is illustrated
in Fig. 3.2 for a simplified rubidium atom without hyperfine structure. For

16the alkali atoms, one chooses λL in between the D1 and D2 lines, such that
one is red-detuned with respect to the states in the 2 P3/2 manifold and bluedetuned with respect to 2 P1/2 . Now, from each of the two ground statesmJ = ±1/2 one kind of circularly polarized light (σ ∓ ) couples to both red andblue-detuned excited states, while the other kind (σ ± ) can couple only to themJ 0 = ±3/2 states in 2 P3/2 . By choosing the wavelength such that the red andblue-detuned contributions cancel for the transitions where both are present,
one can therefore obtain a situation in which σ ± light exclusively affects atoms
in the mJ = ±1/2 ground state [69–71].
In real life, the hyperfine structure splitting leads to additional complications compared to this picture, but the basic principle stays the same. A more
serious limitation of this scheme is due to the fact that the D1 and D2 lines in87
Rb are only 7 THz apart, such that one cannot arbitrarily reduce residual
spontaneous scattering, as the farthest off-resonance one can go is right in the
middle between the two lines.1 In practice, so far, we have not found heating due to spontaneous scattering to be a limiting factor in our experiments.
However, the possibility of such heating should be kept in mind.
In the following, I will outline the calculation of state-dependent optical potentials for multi-level atoms, along with scattering and heating rates. After ashort discussion of Kapitza-Dirac diffraction as a quick and reliable method forlattice depth calibration, I will present an example of measured lattice depths
as a function of polarization for the two hyperfine ground states considered inthis work.

3.3.1

Calculation of optical potentials

When an atom is put into a light field that spatially varies in polarization,
intensity, or both, the different atomic energy levels experience spatially dependent light shifts, i.e. the so-called optical dipole potential. If the light field
is far off-resonant from any atomic transitions (|∆|  Γ), the level shifts canbe calculated in very good approximation by treating the atom-light interaction as a perturbation to leading order [72]. Thus, for state |ii one obtains thelight shift
X |hl|d · E|ii|2 X |dli · E|2=(3.18)∆Ei =E
E
i − Eli − Ell6=il6=iwith the interaction Hint = −d · E in the electric dipole approximation, andwith the electric dipole matrix elements dli ≡ hl|d|ii = d∗il . The presence of a
1

For a quick comparison, the 1064 nm light used for the optical dipole trap is about
95 THz away from the closest resonance, i.e. from the D1 line. The scattering rate drops as
1/∆2 where ∆ is the detuning.

17homogeneous magnetic guiding field that defines a quantization axis and that
lifts the residual level degeneracies via the Zeeman shift is implied. As shownin detail in Appendix A, this eventually leads to the optical potential for statei ∈ 2 S1/2


2
2
li
li
2
cq
cq
ΓD2 X
3πc  ΓD1 X
 I(r), (3.19)+
2
Udip (r; i, q) =
3
3
2
ωD
ω
−
ω
ω
ω
−
ω
li
li
D
2
2
1
2
l∈ P1/2l∈ P3/2

where q denotes the polarization of the light (q = ±1 for σ ± polarization, q = 0
for π polarization), I(r) is the cycle-averaged intensity, ω is the frequency of thelight, and ~ωli is the energy difference between the atomic levels l and i. The
reduced dipole matrix elements have been expressed in terms of the naturallinewidths ΓD1/D2 and the transition frequencies ωD1/D2 , while the factors cliq
contain the geometrical dependence in terms of Clebsch-Gordan coefficients.
Equation (3.19) can be simplified further, since ωli ≈ ωD1 for all terms in
the first sum and similarly in the second sum. This is a good approximation if
the detuning |ω − ωli | is much larger than the hyperfine splitting, which is wellsatisfied for our purposes. The denominators can therefore be pulled out, thesums performed, and with the detunings ∆D1/D2 ≡ ω − ωD1/D2 one obtains [72]
πc2 ΓD2 1 − gF mF q 2 + gF mF q
Udip (r; F, mF , q) =+I(r)(3.20)
3
2ωD2
∆D1
∆D2
where mF is the magnetic quantum number and gF is the appropriate Landég-factor of state i (for 87 Rb we have gF = −1/2 for F = 1, and gF = +1/2
3
3.
≈ ΓD2 /ωD2
for F = 2). Here, we also made use of the fact that ΓD1 /ωD1
This simplified form allows for more qualitative insights into the dependenceof the potential on wavelength, polarization and internal state. For example,p ≡ gF mF q can only take five possible values for 87 Rb, namely p = 0, ±1/2, ±1.
Accordingly there are only five different equally-spaced potential curves versus
wavelength centered around the p = 0 curve (see the thin gray curves inFig. 3.3). For linearly polarized light all 2 S1/2 hyperfine states follow the p = 0
curve and experience identical light shifts.
For an optical lattice obtained through complete retro-reflection of an incoming beam of intensity Iin and wave vector kL = 2π/λL along the z direction
the resulting spatial intensity distribution is 4Iin cos2 (kL z). The lattice modulation depth is thus four times as large as the single-beam light shift. In
Fig. 3.3 (a), the wavelength dependence of the optical lattice depth is shownfor the two chosen hyperfine states |ai = |1, −1i and |bi = |2, −2i as calculatedusing (3.19). Note that on the scale of the plot the difference compared to the18lattice depth V0 (ER)

60
40

D2

0
20

Γsc (1/s)

6

D1

σ+
σ−

40

8

a
b

785.1 nm

20

60

pk

(a)

775

780

785

790

795

800

780

785

790

795

800

(b)

4
2
0

775

λL (nm)Figure 3.3: (a) Calculated lattice depths and (b) peak scattering rates Γpk
sc =
4Γin
for
the
two
hyperfinestates
|ai
=
|1,
−1i
and
|bi
=
|2,
−2i
(red
and
blue)
sc
for σ + and σ − polarized light (solid and dotted) as a function of wavelength(full retro-reflection of an incoming 30 mW beam with a 1/e2 -radius of 230 µm,
yielding Iin = 36.1 W/cm2 ). Negative depths indicate attraction to regions ofhigh intensity (red-detuned character), positive values indicate repulsion (bluedetuned character). At 785.1 nm the optical potential vanishes for |bi atoms
in a σ + lattice (blue square), while |ai atoms feel a deep lattice potential (redcircle, Va = 33.6ER ). The peak scattering rates are 2.35 s−1 for |ai, and1.32 s−1 for |bi at these settings. The thin gray curves are obtained from thesimplified formula showing all five possible values of gF mF q, as discussed in
the text. The dashed line at 787.5 nm represents the center frequency between
the D1 and D2 lines.

simplified eq. (3.20) (less than 1 %) is not noticeable.

3.3.2

Scattering rates and heating

The scattering rate for a multi-level atom can be obtained by summing overthe contributions from each possible excited state similar as in the expressions

19

for the dipole potential above. Recall that for a two-level atom in the far
off-resonant limit one has Γsc = ΓUdip /~∆. This can be applied to each termin (3.20) leading to the expression
πc2 ΓD2
1 − gF mF q
2 + gF mF q
Γsc (r; F, mF , q) =
ΓD1
+ ΓD2
I(r). (3.21)
3
2~ωD2
∆2D1
∆2D2
It is interesting to note that the same expression is obtained by a more rigorous
derivation using the Kramers-Heisenberg formula [67], which involves summingover all accessible final states. In Fig. 3.3 (b), the calculated scattering rate at
the intensity maxima of an optical lattice as a function of wavelength is shownfor the two states |ai and |bi. For a certain detuning very close to halfway
between the D1 and D2 lines (∆D1 ≈ −∆D2 ), the scattering rate becomes
independent of the hyperfine state and the light polarization.
These residual spontaneous emission processes kick the atoms into random directions and therefore result in a diffusion of the atomic momenta or,in other words, heating. Moreover, fluctuations of the dipole force itself also
contribute to the overall momentum diffusion [73–75]. The momentum diffusion constant Dp for a (motionless) two-level atom in the far off-resonant, low
saturation regime can be written as [75, 76]




~2
1 2
1 2
~2 3πc2 Γ Γ
22
2kL I(r) + ∇ I(r) =2kL + ∇ ΓTLA
Dp =
sc (r)
2 2~ω03 ∆2
2
2
2(3.22)
TLA
where Γsc (r) is just the local scattering rate for a two-level atom in a lightfield of intensity I(r). The first term proportional to the intensity is due tospontaneous scattering, while the second term proportional to the Laplaceianof the intensity represents the contribution from the fluctuations of the dipole
force. The kinetic energy of an atom of mass m increases at the rate Ė =Dp /m, also called the heating power [72]. By simply replacing the two-level
atom scattering rate in (3.22) by the one for the multi-level atom (3.21) we
obtain the heating power
1 2Ė = ER 2 + 2 ∇ Γsc (r)(3.23)
2kL
with the usual recoil energy ER = ~2 kL2 /2m. For a single beam (i.e. a travelingwave) of intensity Iin the heating power is Ė = 2ER Γin
sc , since the intensity
and thus the scattering rate are spatially homogenous. Each spontaneous
scattering event thus increases the energy by 2ER , where one recoil energy is
due to spontaneous emission and the other one is due to the prior absorption

20

that is not canceled by induced emission.2
Retro-reflection of the beam leads to a standing wave for the dipole po2
tential as well as for the scattering rate Γsc (r) = 4Γinsc cos (kL z). The peak
scattering rate at the intensity anti-nodes is thus four times as high as for the
single-beam case. If there was no contribution from the dipole force fluctuations, the spatially averaged heating power (assuming a homogeneous atom
distribution) would be twice as large as for a single beam. However, the Laplaceian term causes “cooling” at the anti-nodes and additional heating at thenodes of the intensity, where the dipole force is largest. In total the heatingmax
power becomes Ė = 4ER Γinwhich actually is still twice as much
sc = ER Γsc
as for a single beam, but surprisingly is independent of the position in thestanding wave [75–77].
It is also noteworthy in the context of state-dependent potentials that evenwhen the dipole potential for a certain state vanishes does the scattering rate
remain finite. Although one would not expect the spontaneous contribution
to the heating power to vanish, it is somewhat counter-intuitive at first, that
also the contribution due to the dipole force fluctuations in a standing wave
exists, even when the potential is flat. This flatness, however, comes fromthe cancellation of possibly deep red and blue-detuned lattices. The dipole
force fluctuations thus do not arise from the population fluctuations between
ground and excited states in this case, but from the population fluctuations
in between red and blue-detuned excited states.
Finally, the heating power due to momentum diffusion Ė can be translated
into an actual heating rate Ṫ [72]. Assuming thermal equilibrium, the mean
total energy per atom is a function of temperature. For example, in a threedimensional harmonic trap one has E = 3kB T , such that the heating rate
becomes Ṫ = Ė/3kB . Consequently, if we shine light onto an atomic cloudsitting in a harmonic trap, the cloud’s temperature increases at a rateṪ = 2ER Γsc /3kB = Γsc TR /3(3.24)

where the recoil temperature is TR ≡ 2ER /kB . For 87 Rb and 785 nm light, therecoil temperature is about 360 nK, such that a scattering rate of 1 s−1 leads
to a temperature increase of 12 nK in 100 ms. Once the thermal energy kB T
becomes comparable to the trap depth, more and more atoms are lost from thetrap due to evaporation and the temperature ceases to increase at a constantrate. Retro-reflection of the light to form a lattice would naively cause twice
the heating since the heating power is doubled. However, the relation between
energy and temperature would not correspond to a simple harmonic trap any
2

As an aside, the spontaneous force is completely negligible compared to the dipole force
in the regime considered here [72, 75].

21

lattice depth V0 (ER)λL = 785.1 nm

30
20

b
σ+

10
0

σ−

a

π

π
0º

30º

60º

90º

120º

150º

180º

quarter wave plate setting

Figure 3.4: Measured lattice depths in a simple retro-reflected beam configuration for the two states |ai ≡ |1, −1i and |bi ≡ |2, −2i as a function of thelattice beam polarization at a given intensity and at a wavelength of 785.1 nm.
The lattice depths were obtained separately for each state from Kapitza-Diracdiffraction patterns after a 10 µs lattice pulse. The solid lines are sinusoidal
fits [78].

more, at least along the direction of the beam. A more detailed treatment
thus seems necessary in this case.

3.3.3

Experimental Lattice Depth Determination

Experimentally determined lattice depths for the two hyperfine states |ai and|bi are shown in Fig. 3.4 as a function of the polarization of the lattice lightat constant intensity [78]. The polarization of the incoming beam can be
adjusted through a quarter wave plate. Since the lattice is formed by simple
retro-reflection of this beam, the polarizations of the two counter-propagating
beams are identical. As expected, the lattice depths for both states are thesame for linearly polarized light, while circular polarization leads to maximallydifferent lattice depths for the two states considered. In accordance with thetheoretical curves in Fig. 3.3 (a), at the chosen wavelength of 785.1 nm thepotential for state |bi approximately vanishes3 for σ + light while state |ai is at
its maximum lattice depth. Similarly, for a σ − lattice, |bi experiences a deep
lattice potential, while the lattice depth is minimal but non-zero for |ai, as
expected.
3

The non-zero lattice depth for |bi at σ + polarization in this set of measurements iscaused mainly by an imperfect alignment of the magnetic guiding field with respect to thelattice beam. Further details can be found in [78]. With careful alignment, the residual
lattice can be reduced down to virtually zero for all practical purposes.

22In order to experimentally determine the actual depth of the lattice potential V0 experienced by the atoms, we generally employ Kapitza-Dirac diffraction [79–81], which we have found to be quick and reliable in comparison toother possible methods [11]. The method consists of pulsing on the optical lattice onto a BEC for a short time τKD on the order of the recoil time τR ≡ h/ER .
The optical lattice acts as a grating from which the BEC is diffracted, verymuch in analogy to usual diffraction of light from a material grating, just thatthe roles of light and matter are interchanged. Here, the pulse duration corresponds to the thickness of the material grating in the optics analogy. Fromthe matter-wave diffraction pattern observed after time of flight, one can thendetermine the modulation depth of the optical lattice.
For typical experimental parameters considered here, the mean-field interaction can be neglected and it suffices to treat the condensate as a homogeneous non-interacting matter wave ψ(z, t) evolving according to the usual
Schrödinger equation

 2
~
2
∂zz + V0 sin (kL z) ψ
(3.25)
i~ ∂t ψ = −2m
for the duration of the lattice pulse.
If the pulse is sufficiently short, the spatial distribution of the atoms remains essentially unchanged and the gain in kinetic energy is small. However,
the optical lattice potential imprints a periodic phase variation onto the condensate. This thin-grating or√Raman-Nath regime is attained provided that
τKD  Tho , where Tho = τR /2 s (with s ≡ V0 /ER ) is the classical oscillation
period when approximating each lattice well by a harmonic potential. In thisregime, the kinetic energy term in (3.25) can be neglected. The time evolution
is then simply given
exp [−i (V0 τKD /2~) (1 − cos 2kL z)]. Use of the identity
P bynexp(i a cos φ) = n i Jn (a) exp(i nφ) leads to
V0 τKD
2
2(3.26)
P2n = |c2n | = Jn
2~
for the population distribution of the even diffraction orders 2n, where Jn
denotes Bessel functions of the first kind. The odd orders remain unpopulated.
More generally, the mode populations can be calculated numerically byintegrating the Schrödinger equation. We can expand ψ(z, t) in terms of planewaves propagating along z with multiples of the recoil velocity vR = ~kL /mψ(z, t) =

∞
X
n=−∞

23

cn (t) ei nkLz .

(3.27)

p / ħkL
−6

−4

−2

0

+2

+4

+6

Figure 3.5: Example of a Kapitza-Dirac (KD) diffraction pattern as seen inabsorption imaging after time of flight. The different diffraction orders areclearly visible. In the bottom the integrated optical density is shown. The
slight variation in the shapes of the different orders is caused by the mean-field
repulsion between the atoms that rapidly translates into kinetic energy after
release from the trap. (Here, the KD pulse is applied just before the trap isturned off.)

In this basis, (3.25) becomes a system of coupled differential equations for theamplitudes cn (t)i~ ċn = ER n2 cn + V0 21 cn − 14 (cn+2 + cn−2 ) .
(3.28)The initial conditions for Kapitza-Dirac diffraction of a motionless BEC arecn (t = 0) = δn0 , i.e. only the zero-momentum mode is occupied. According
to (3.28), the lattice potential only couples momentum modes that are 2~kLapart, and the equations are symmetric under interchange of positive andnegative momenta. With the above initial conditions, the mode populations
Pn (t) = |cn (t)|2 will thus be non-zero only for n even, and the distribution
will always be symmetric P−n (t) = Pn (t). These general considerations are ofcourse also valid in the Raman-Nath regime. For the determination of lattice
depths we do not rely on the Raman-Nath approximation, i.e. we fit numerically integrated diffraction patterns to the experimentally obtained ones. Moredetails on this method can be found in our publication Ref. [81].

24

3.4

Optical lattice band structure and wave
functions in terms of the Mathieu equation

When particles are confined to the periodic potential of a lattice, the usual freespace dispersion relation E = ~2 k 2 /2m is dramatically modified. As pointed
out by Felix Bloch and others in the early days of quantum mechanics, thedispersion relation becomes periodic in momentum space and energy gaps
open up in the energy spectrum. In this section, I will review the fundamental
properties of single particles in a periodic potential.
We assume that the lattice potential is formed by a one-dimensional opticalstanding wave, i.e. an optical lattice, which gives rise to the spatially periodic
potential (shown in Fig. 3.6)
V (x) = V0 sin2 (kL x),(3.29)

with kL = 2π/λ and λ the wavelength of the light. With this potential, we
can rewrite the Schrödinger equation as a Mathieu equation (in the notationof [82, 83])
y 00 (z) + [a − 2q cos(2z)] y(z) = 0.
(3.30)
Here the parameter q corresponds to the lattice depth, while a corresponds tothe energy as follows
q ≡ −s/4,a ≡ e − s/2 = e + 2q

(3.31)(3.32)

with s ≡ V /ER and e ≡ E/ER , i.e. measuring the lattice depth and the energy
in units of the recoil energy. Length is measured in units of 1/kL = λ/2π,
and the dimensionless momentum r is measured in units of ~kL . For the
Mathieu equation exact analytical solutions exist in terms of the Mathieufunctions [82, 83].

3.4.1

Energy bands

In the reduced zone scheme, the states are labeled by a pair (n, rred ) where n is
the band index and rred ∈ [−1, 1) is the quasi-momentum. In the expanded or
free scheme they are labeled, in principle, only by r ∈ R. However, for integer
values of r 6= 0 there are two states of different energy, such that a further

25

1.0

0.8

0.8

Eho sER VxsER 

1.0 (a)
0.6
0.4
0.2
0.0
1.0

0.6
0.4
0.2

(b)

0.0
0.5

0.0
xΛ

0.5

1.0

0

20

40

s

60

80

100

Figure 3.6: (a) Sinusoidal lattice potential V (x) as created by an opticalstanding wave. The dashed red curve shows the approximation by a harmonic
potential around the minimum of a single lattice well. Harmonic oscillator (ho)levels are shown for the case of s = 40. (b) Relative energy of ho levels fitting
into a single well, as the lattice depth s is increased. The second level gets
pulled in only at s = 9, the third at s = 25. The harmonic approximation isuseful only for ho levels lying sufficiently deep in the well (Eho  V0 = sER ).

distinction at these points is necessary in the expanded scheme4 . The energy
bands are thus given bye(r; s) = ar (−s/4) + s/2

r 6∈ Z(3.33)

everywhere except at the band edges.

3.4.2

Bloch wave functions

According to Bloch’s theorem, the wave functions of a particle in a general
periodic potential can be written in the formϕr (z) = ei rz ur (z)with ur (z + 2π) = ur (z).

(3.34)

Note that this does not mean that an arbitrary solution to the Schrödinger
equation is of this form, with counter-examples being the even and odd superpositions of two Bloch waves. For the a sinusodial potential, they are given
in terms of the even and odd solutions MathieuC[a, q, z] and MathieuS[a,q, z] provided by Mathematica as follows (where C and S basically take the4This distinction is provided by ar (q) and br (q), in Mathematica called
MathieuCharacteristicA and ...B, respectively. For given q, these functions coincide
everywhere except at r ∈ Z.

26

EER

15
10
5
0
4

0

2

2

4

kkL
Figure 3.7: Allowed energy bands E(k) of solutions propagating with wave
vector k, here at s = 5. Shown are the reduced zone scheme (first Brillouinzone only, black), the extended zone scheme (blue, relevant for adiabatic lattice
depth changes, cf. band map) and the repeated zone scheme (gray). The bandedges are marked by red dots in the extended scheme and orange dots in the
reduced scheme.

role of generalized cosine and sine functions):
ϕ±r (z) = MathieuC [ar (q), q, z] ± iMathieuS [ar (q), q, z] .

(3.35)In the notation of Ref. [82, 83] this is
ϕ±r (z) = cer (z, q) ± i ser (z, q).

(3.36)The sign determines in which direction the solution moves completely analogous to a plane wave, to which it reduces for q ∝ s = 0, i.e. cer (z, 0) = cos(rz).

3.4.3

Localized basis states – Wannier functions

Localized Wannier functions can be obtained as a superposition of the delocalized Bloch functions. For deep lattices the Wannier functions can be approximated by Gaussians, just as each optical lattice well can be approximatedby a harmonic potential of the form mω 2 z 2 /2 (see Fig. 3.9). The trapping
frequency (respectively the distance between the harmonic oscillator levels)
increases with the lattice depth as√~ω = 2ER s.
(3.37)

27

80

EER

60

40

20

0

0

20

40

s

60

80

100

Figure 3.8: The five lowest bands, i.e. the energy ranges where propagating
solutions exist, as a function of lattice depth s (shaded light blue). In the gaps
between these regions only localized and evanescent wave solutions exist. As
the lattice depth is increased, the bands flatten out and approach the harmonic
oscillator levels of the single well harmonic approximation shown as dashed red
curves. The dash-dotted blue line shows the modulation depth s of the lattice.
Naturally, the states above have predominantly free particle character, while
they have the character of bound states below.

The size of the wave function on a single site is then just the harmonic oscillator
lengthr
λ −1/4~σ==s.
(3.38)
mω
2π

28

4
3

|w (x)|2|ψ (x)|2

2
V(x)

1
0

0.4

0.0

0.2

x/λ

0.2

0.4

Figure 3.9: Ground state Wannier function |w0 (x)|2 localized at a single
site in a lattice potential V (x) of depth s = 50. For comparison the density
distribution of the delocalized Bloch function |ψ(x)|2 is shown as dashed line.

3.5

Strongly-correlated atoms and the BoseHubbard model

In second-quantized notation, the full Hamiltonian for a trapped Bose gas ina lattice potential interacting via an effective contact interaction is given by


Z
~2 2
3
†
∇ + Vlat (r) + Vtrap (r) Ψ(r)H = d r Ψ (r) −2m(3.39)
Z
1 4π~2 as
3††+
d r Ψ (r)Ψ (r)Ψ(r)Ψ(r).
2 m
The potentials of the opticalthe external trap
 are typically of 2 lattice and22
the forms Vlat (r) = V0 sin (kL x) + sin (kL y) + sin (kL z) for a simple-cubic
three-dimensional lattice, and Vtrap (r) = mω 2 r2 /2 for an isotropic harmonic
trap, respectively.
We now expand the field operators Ψ(r) that create or destroy an atom
at the point r in terms of the operators bni that create or destroy an atom of
the band n that is localized at lattice site i, described by the Wannier wave(n)function ψ(r) = w(n) (r − ri ) = wi (r). ExpandingXΨ(r) =
w(0) (r − ri ) b0i(3.40)
i29into the lowest band5 (in the following b0i ≡ bi ), we obtain the Hamiltonian in
the Wannier basis
XX
1 X(3.41)
H=−tij b†i bj +Vij b†i bj +Uijkl b†i bj b†k bl
2
i,j
i,j
i,j,k,lwith the tunneling matrix elements tij for tunneling from site i to site j, thematrix elements Vij from the trapping potential that leads to an on-site energy
offset, but may also induce directional tunneling (e.g. for a potential gradient),
and the general interaction matrix elements Uijkl involving four lattice sites.
The matrix elements introduced in (3.41) are given by


Z
~2 2
3
∗
∇ + Vlat (r) wj (r),(3.42)
tij = − d r wi (r) −
2m
Z
Vij =
d 3 r wi∗ (r) Vtrap (r) wj (r),(3.43)
Z
4π~2 asUijkl =
(3.44)
d 3 r wi∗ (r)wj∗ (r)wk (r)wl (r).
m
We note that the nth -nearest neighbor tunneling matrix elements ti,i+n canalternatively be obtained as the Fourier coefficients of the dispersion relationin the lattice. In figure 3.10 we show a comparison of the different ordertunneling terms for different lattice depths of the optical lattice. For large
lattice depths, higher order tunneling is small, such that a simple nearestneighbor-tunneling tight-binding model is well applicable.
As shown by Jaksch et al. in 1998 [14], the full Hamiltonian (3.41) can besignificantly simplified yielding the well-known Bose-Hubbard model (BHM)
under certain conditions that are easily realizable with ultracold atoms in
optical lattices. The Bose-Hubbard model had been used before to describe
Josephson junction arrays as realized by superfluid Helium-4 in porous media,
for example. Since porous media are irregular structures, early theoreticalattention had been focused on the more involved disordered case [84]. The
Bose-Hubbard model Hamiltonian with an external trapping potential reads
 1 X
X
X †
ni (ni − 1) +Vi ni
(3.45)
HBHM = −tbi bj + b†j bi + U
2
i
i
hiji

5

In order to leave the Hamiltonian unchanged,
we should use the full expansion includingP(i)higher Wannier levels or bands Ψ(r) =i,n w (r − ri ) bni . However, this would only
complicate the notation, and is not necessary for our purposes. One should keep in mind,
though, that this projection of H onto the lowest band technically is valid only at very low
energies.

30

0.25

0.10

nn

0.05
0

5

10
s

15

20

108

nnn
t3

104
106

nnn

(b)

t2

0.01

0.15

|tn t||t|ER

0.20

0.00

1

(a)

t4

0

5

10
s

15

20

Figure 3.10: (a) Size of the tunneling matrix elements as a function of s,
calculated as Fourier coefficients of the dispersion relation of the lowest band.
The higher-order matrix elements becoming small corresponds to the band
becoming approximately sinusoidal. Then, the nearest neighbor (nn) tunnelingt approaches 1/4 of the bandwidth, shown as dashed grey line. (b) Magnitudeof higher-order tunneling matrix elements |tn | compared to nearest neighbortunneling t. (nnn: next nearest neighbor).

where hiji denotes a sum over nearest neighbor links between sites i and j(here the link h12i is the same as the link h21i, these are undirected links, andthus each link appears only once in the sum; note that other conventions arealso used in the literature), ni ≡ b†i bi is the numberRoperator at site i, t is thenearest neighbor tunneling matrix element,RU = g d 3 r |w(r)|4 is the on-site
repulsion with g = 4π~2 as /m, and Vi = d 3 r |wi (r)|2 Vtrap (r) is the mean
external potential at site i. In order for ultracold atoms in optical lattices
to behave according to the BHM (3.45), the following conditions have to be
fulfilled: nearest-neighbor tunneling has to dominate, off-site interactions have to be negligible, only on-site interactions should
be sizeable (this and the previous condition require aho  λL /2),
 the external potential has to change sufficiently slowly on the length scaleof the lattice spacing, such that it does not induce tunneling betweensites,
 only the lowest band should play a role, excited states should not be
populated, the temperature therefore has to be sufficiently close to zero(we have already implied this above by projecting onto the lowest band).

Even without an external potential (Vi = 0) the BHM cannot be solved analytically for arbitrary parameter values. One therefore has to resort to analytical31

approximations or numerical methods such as exact diagonalization of small
systems using the Lanczos algorithm or the increasingly powerful quantum
Monte-Carlo (QMC) methods. However, in the two limiting cases of negligible tunneling and of negligible on-site repulsion, HBHM can be diagonalized
almost trivially, as discussed in the following. These solutions can then be
used as starting points for perturbation theory.

3.5.1

Exact solutions for special cases

Negligible tunneling (t  U )In this case (t = 0), the Hamiltonian reduces to the on-site repulsion part,and since the sites do not communicate with each other any more, it can be
written as a sum over single site Hamiltonians Hi
Ht=0 =X
X
1 X
Uni (ni − 1) +Vi ni =
Hi2i
ii

(3.46)

with the single-site Hamiltonians
Hi =1
U ni (ni − 1) + Vi ni .
2

(3.47)

The
are already diagonal in an occupation number basis |ni i = nHamiltoniansi†bi|0i with ni = 0, 1, 2, . . .. The Hamiltonian summed over all sites is
QM  † ni|0i.
diagonal in the basis |n0 , n1 , . . . , nM i = i=0 bi
Negligible on-site interaction (U  t)When the on-site interaction is turned off (U = 0), we have only the kinetic
energy term and the external potential
 X
X †
†
bi bj + bj b i +Vi ni .
(3.48)
HU =0 = −tihiji

For a vanishing external potential this is just a tight-binding Hamiltonian, thatcan be diagonalized by simply going over to momentum space via a Fourier
expansion
1 X
bk e(ik · ri ).
(3.49)
bi = √
N k

32

4

(a)
ntot, n0 , ndepleted

n4
ΜU

3
n3
2
n2
1
n1
0
0.00

0.05

(b)

4

0.10

0.15

z t U

0.20

3
2
1
0

0.25

0

1

2

3

4

ΜU

Figure 3.11: (a) Phase diagram of the Bose-Hubbard model at T = 0 according to the decoupling mean-field approach. The blue shaded regions are
Mott insulating with vanishing compressibility and with the atom number persite fixed to an integer. The red dots indicate the critical points at the tipsof the Mott lobes. The blue open square indicates the position of the n = 1
Mott lobe in 3D determined by quantum Monte-Carlo calculations [85]. (b)Atom number per site along the dashed line in (a). Blue: Total atom number.
The integer filling within the Mott lobes can clearly be seen. Red: Expectation
value of ψ 2 which gives the number of condensed atoms per site. Green: The
difference between total and condensed atom number per site, i.e. the numberof quantum depleted atoms per site. It coincides with the total atom number
only in the Mott insulating regions. Although the condensate fraction is small
in between the lobes, the whole system still shows a superfluid response there(similar to the situation in Helium-4 at T = 0, which has a superfluid fractionof 100%, while the condensate fraction is only about 10%).

The result is
HU =0 =

X

(k) nk(3.50)

kwith the dispersion relation (k) = −2t cos(ka).

3.5.2

Mean-field treatment and T = 0 phase diagram

Using a decoupling mean-field approach [86] for the tunneling term, one candetermine a phase diagram of the BHM. Essentially, we can write the creationand destruction operators in the kinetic energy term as a sum of a mean value
plus a fluctuation operator, A = hAi + ∆A. Neglecting terms of higher order33

than linear and reexpressing ∆A as A − hAi again, we obtain


†
†
†
†
†
bi bj ≈ hbi ibj + bi hbj i − hbi ihbj i = ψ bj + bi − ψ 2 .

(3.51)

Assuming a homogeneous system, the “mean-field” hbi i = ψ has the samevalue on each site, and we choose it to be real. It essentially represents thecondensate density. For given parameters, we now have to determine the valueof ψ that minimizes the system’s energy. In this approach, all lattice sites havebecome equal, such that it is enough to look at a single site with the effective
HamiltonianU
HMF = −zt b + b† ψ + ztψ 2 + n(n − 1) − µn,2(3.52)

where z is the number of nearest neighbors (z = 2d for a cubic lattice in ddimensions). This Hamiltonian can now be diagonalized numerically, yielding
the lowest energy for given parameters t, U, z, µ, ψ. Alternatively, one canstart from the eigenstates in the t = 0 limit and treat the first term in secondorder perturbation theory, which leads to analytical forms of the Mott lobe
boundaries which differ from the numerical values only by a few percent,
B(x) =(x − bxc) (dxe − x)1+x(3.53)

where x ≡ µ/U , B ≡ t/U , and bxc and dxe denote the floor and ceilingfunctions, respectively.
Figure 3.11 (a) shows the T = 0 phase diagram, while Fig. 3.11 (b) shows
the total number of atoms per site in a cut at constant zt = 0.08 as determinedusing exact diagonalization of the mean-field Hamiltonian, and then using
ntot = −∂hHMF (ψmin )i.
∂µ(3.54)

Also shown is the order parameter n0 = ψ 2 , which corresponds to the numberof condensed atoms per sites, and the resulting number of quantum-depleted
atoms per site ntot − n0 .
Note that the mean-field treatment is most accurate in infinite dimensions,so significant abberations from the exact phase boundaries are expected in one
and two dimensions, as verified by QMC calculations, showing that the lobes
get pointed tips extending much further out. The n = 1 Mott lobe in three
dimensions has also been extensively studied for finite temperatures using a
numerically exact QMC method [85] (with “numerically exact” meaning that34

4

MI (n=2)

3SF

MI (n=1) SF

ntot

2

μloc

1

ψ2

0
0.0

0.2

0.4

0.6

0.8

1.0

rRTF

Figure 3.12: Shell structure in an isotropic harmonic trap Vtrap (r) =mω 2 r2 /2. The dashed line is the local chemical potential µloc (r) = µ−Vtrap (r).
Here, zt = 0.08 and µ/U = 1.5 is chosen such that in the trap center we are in
the middle of the n = 2 Mott lobe (cf. Fig. 3.11). The n = 2 Mott plateau in
the center is surrounded by a superfluid shell, an n = 1 Mott-insulating shell,
and another superfluid shell at the edge.

no uncontrolled approximations have been made and the values obtained can,in principle, be made as close to the exact values as desired given enoughcomputing time). From this study, the tip of the Mott lobe is located at(U/t)c = 29.34(2) or (t/U )c = 0.03408(2), as indicated in the figure. To
date, much more elaborate versions of the mean-field approach such as dynamical mean-field theory which incorporates site-dependent potentials havebeen applied, for example, to the disordered BHM [29], and to two-component
mixtures [87].

35Chapter 4
Experimental Setup
In this chapter I will discuss our newly constructed apparatus for producing
magnetically and optically-trapped condensates in a moving-coil transporter
scheme [88]. A particularly attractive feature of our apparatus, that reduces
the overall complexity of the design, is that the movable quadrupole coils
are also used as an essential part of the final magnetic trap, the “McTOP”
trap, which belongs to the family of time-averaged orbital potential (TOP)traps [89]. As a stand-alone device, this trap reliably produces condensateswith minimal technical complexity. In experiments with optically-trapped
Bose-Einstein condensates, the quadrupole coils can be retracted before quantum degeneracy is reached, providing large optical access. The apparatus iswell-suited for experiments with optical lattices, as demonstrated by observing the superfluid-to-Mott insulator transition. This chapter is an extended
version of our publication Ref. [90].

4.1

Overview of our BEC machine

Our moving-coil transporter apparatus is illustrated in figure 4.1. A cylindricalglass vapor cell is connected to a UHV chamber with an attached small quartz
glass science cell (cf. also figure 4.5) through a differential pumping tube.
In the vapor cell, we typically collect 7 × 109 atoms in a standard six-beam87
Rb MOT, making use of light induced atom desorption (LIAD) [92–94] to
temporarily enhance the loading rate. After an 8 ms molasses phase thatlowers the temperature to about 25 µK, the atoms are optically pumped to
the |F = 1, mF = −1i hyperfine ground state, and subsequently caught in amagnetic quadrupole trap using the same coil pair as for the MOT at an axial
field gradient of Bz0 = 100 G/cm, yielding 2.2 × 109 trapped atoms at 150 µK.
The quadrupole coils are mounted on a mechanical translation stage assem-36

to pumps & Rb oven
UHV
chamber

vapor
cell

to
pumps
a

b

science cell
& bias coils
c

z
y
x

Figure 4.1: Transporter apparatus. The vapor cell is made from a 5.7 cmdiameter Pyrex tube and is connected through valves with an ion pump and arubidium reservoir (not shown), maintaining a rubidium background pressureof several 10−9 torr. A 28 cm-long differential pumping tube of 1 cm inner diameter connects the vapor cell to the UHV chamber and the attached science
cell, which are pumped down to less than 10−11 torr by an ion pump in combination with a titanium sublimation pump (not shown). The quadrupole coils(shown in MOT position ‘a’) are mounted on an aluminum holder (sliced toreduce eddy currents) that sits on two orthogonally stacked translation stages(Parker Daedal 404XR) (for details see [91]). The kink in the translation path
protects the science cell from fast ballistic atoms escaping from the vapor cell,and it enhances optical access to the condensate along the x-axis. A small gate
valve located midway in the pumping tube allows to completely disconnect thetwo vacuum regions for servicing.

bly [91], which is used to move the magnetically trapped cloud into the sciencecell. This is done as quickly as possible (a = 4.5 m/s2 , vmax = 0.94 m/s) inorder to minimize losses due to collisions with background gas atoms in thevapor cell. The quadrupole trap is simultaneously compressed to its final valueof Bz0 = 350 G/cm within 150 ms before the cloud reaches the pumping tube.
This adiabatically heats up the cloud to 450 µK. We have confirmed that nonadiabatic heating during the motion amounts to less than 10 µK. A detailed
study of the transfer in terms of atom number losses and heating can be foundin [95].
Once the cloud reaches the UHV chamber ‘b’, where the measured lifetime

37

exceeds 150 s, the transporter slows down and proceeds along a kinked path(a = 0.5 m/s2 , vmax = 0.18 m/s). After a total travel time of 3 s and a covered
distance of 66 cm, about 1.6 × 109 atoms, or 75% of the atoms initially caught(see also [95, 96]), arrive in the science cell ‘c’, where they are evaporatively
cooled as discussed in section 4.8.1. We attribute the loss of atoms during the
motion to background gas collisions while moving out of the vapor cell, andto a possible shaving off of hot atoms on the walls of the differential pumping
tube.
By performing the final evaporation in an optical trap, it is possible toeasily use the TOP trap for optical lattice experiments, thus avoiding theusual drawback of TOP traps: atomic micromotion [97] can lead to strong
heating due to an oscillatory motion of the atoms relative to the lattice at thefrequency of the rotating bias field [98].

4.2

Cooling and Imaging Laser System

Our laser setup, shown schematically in Fig. 4.4, is comparatively simple. All
the light used for laser-cooling and imaging is derived from two commercial
external cavity diode lasers (ECDLs). Both are locked to atomic transitions
using polarization (saturation) spectroscopy [99–101], which directly provides
an error signal that can be used for locking without the need for effect modulation or a Pound-Drever-Hall scheme. The cycling light along with the imaging
and depump light is generated by a Toptica DL pro ECDL with 60 mW at
780 nm. A tapered amplifier (Toptica BoosTA) provides the necessary power
for the MOT and molasses cycling light.1
A small fraction of the light is diverted for spectroscopy. The light forked off
to lock the lasers to atomic transitions is delivered to the spectroscopy setups
via PM fibers. The polarization spectroscopy cells are relatively sensitive totemperature changes, vibrations and magnetic field fluctuations, so that itproved advantageous for stability to place them not directly on the laser table,
but into light tight boxes away from the table.
Most beams are coupled into polarization-maintaining fibers which thendeliver the light to their destinations. Only the MOT repump and depump
1

As a historical aside, in the initial stages of the experiment we were struggling witha commercially available 1W tapered diode laser from a competitor. That laser proved
to have insufficient frequency stability for reliable absorption imaging (see [95]), and was
generally plagued by bad thermal behavior, drifting and several spontaneous laser diodedeaths. We were thus forced to image our first condensates on the repumping transition
which is not particularly effective, as it is not closed. The reliability of the machine increased
considerably after replacing this laser by a DL pro-BoosTA combination.

38

VaporMOT
Molassesscience cell
QPMT

vapor cell

Transport

RF Evap

TOP MT
TOP MT → XODT
condensation
MT BECXODT BEC

Figure 4.2: Schematic illustration of the steps to BEC in our apparatus.
In the room temperature vapor cell, a MOT captures and cools an atomic
cloud to µK temperatures. A short optical molasses phase further reduces thetemperature, before all laser-light is extinguished and the magnetic quadrupole
trap (QPMT) is rapidly turned on to catch the cloud of cold atoms. The
magnetic coils of the quadrupole trap along with the cold atoms trapped in it
are then quickly moved to the science cell where the ultrahigh vacuum allows fortrap lifetimes sufficiently long for evaporative cooling. As evaporative cooling ismore efficient in stiffer traps, the first part of the rf-induced forced evaporation
is done in the linear potential of the QPMT. Just before losses due to Majorana
spin flips at the field zero at the trap center set in, the trap is converted to
TOP trap by turning a the rotating bias field, yielding a harmonic potential.
Evaporative cooling continues in the TOP trap either until condensation setsin which provides us with a magnetically trapped BEC in the |1, −1i hyperfine
state. In the other path taken for our optical lattice experiments, evaporationin the TOP trap is stopped just before condensation sets in. The ultracold
thermal cloud is then transferred into a crossed optical dipole trap (XODT),
where the final evaporation down to an optically trapped BEC is performed bylowering the trap depth. Although approximately harmonic at the trap bottom
the potential in the XODT is anharmonic when moving away relatively little
from the trap bottom compared to the TOP trap, especially along the directionof gravity.

beams, which are relatively insensitive to alignment are delivered to the vapor
cell as free-space beams (in this case, more power is more important thanpointing stability).

39

For the imaging setup, we took care that the imaging beam paths aremechanically as rigid as possible, such that fringes in the illumination causedby dust and other artifacts do not move significantly between the absorption
image and the illumination reference image, greatly improving the quality of
our absorption images.
 Acousto-optical modulators are used to rapidly (on the time scale of afew microseconds) control the beam intensity and to shift the frequencyof the light.
 Cycler locked to less than 500 kHz using grating and diode current
feedback, 187 MHz below the F = 2 → F 0 = 3 cycling transition(cf. Fig. 4.3).
 Repumper locked (via grating feedback only) to 80 MHz below the F =1 → F 0 = 2 transition
 The depump beam (F = 2 → F 0 = 2) fulfills essentially the opposite
role of the repumper by removing all atoms from the cycling transition
and accumulating them in the F = 1 ground state. (It also can act
to partially close the 1 → 2 transition, e.g. for imaging) This is used to
optically pump the laser-cooled cloud into the F = 1 state before loading
it into the magnetic quadrupole trap.
 The power behind the MOT fiber is regulated via a feedback loop that
controls how much light the AOM diffracts, very similar to the lattice
beams.
 A double-pass AOM configurations for the cycling light which allows forreaching the depump transition with simple 80 MHz AOMs (3 × 80 =240 MHz) and allows for a larger accessible detuning range of the cycling
light (the single-pass AOM frequency change also affects the deflection
angle of the light, so in order to cancel this effect a back-reflected double
pass configuration is necessary).

4.3

Magnetic Traps and Transporter

4.3.1

Movable Quadrupole Coil Pair

Each of the water-cooled quadrupole coils consists of 33 turns of 1/4 inchdiameter coated hollow copper tubing. At a current of 425 A, the quadrupole
coils (L = 90 µH) produce an axial field gradient of Bz0 = 350 G/cm , which
can be switched off completely within less than 1 ms using an IGBT.
40

F=3

Γ ≈ 6 MHz∆c = −3.5 Γ

F=2267 MHz2

5 P1/2814 MHz×4·105×105

5 2P3/2

F=2

F=1

157 MHzF=1
72 MHz780.24 nm
384 230 484 MHz

377 107 463 MHz

Depump794.98 nm

D2

RepumpD1

Cycling

F=0

87

Rb

I = 3/2
S = 1/2

F=2×104

6 835 MHz

5 2S1/2

F=1

Figure 4.3: Relevant level-structure of rubidium-87 and transitions used forlaser-cooling and imaging (to scale, the fine-structure manifolds are magnified
by different factors to show the hyperfine structure).

4.3.2

Time-averaged Orbiting Potential Trap

Instead of producing the condensate in a separate Ioffe-Pritchard trap, as
generally found in transporter apparatus based on [88], we use the movable
quadrupole coils as an integral part of the final magnetic trap. This could bedone, in principle, by realizing either a TOP trap [89] or a QUIC trap [102].
For the latter, however, the trap bottom depends on the delicate cancelationof the much larger fields of the quadrupole coils and the Ioffe coil, making thetrap very sensitive to fluctuations of their relative positions.
For a TOP trap, the trap bottom is solely determined by the magnitude B0of the rotating bias field. It is therefore inherently insensitive to the relative
positioning of the coils provided that the bias field is sufficiently homogeneous.
With the bias field rotating in the xy-plane, the time-averaged magnetic potential at the center of the trap is given by11
V (ρ, z) = µB0 + m ω⊥2 ρ2 + m ωz2 z 222
41(ρ, z  ρ0 ).

(4.1)

Diagnostics−

Spectroscopy
Setup+80 MHz

Imaging
RepumpRb cell+80 MHz

PhotodiodePBS (Polarizing beam splitter)
PBS + half-wave plate (λ/2)
Quarter-wave plate (λ/4)
Shutter

MirrorMOT
Repump

780 nm+80 MHz

ECDL

Spectroscopy(F = 1 → F' = 2)

Toptica DL 100

Lens (plano-convex)−80 MHz

Fiber coupler

MOT
DepumpShutter+166 MHzAOM (Acousto-optic modulator)
Faraday isolator

MOT
Cycling

83 MHz+187 MHz93.5 MHz+187 MHz

Imaging

Spectroscopy(F = 2 → F' = 3)

780 nmECDL
Toptica DL pro

Toptica BoosTA

93.5 MHz

Diagnostics

Figure 4.4: Schematic laser setup for laser-cooling and imaging (folding mirrors and components for beam shaping such as fiber coupling lenses are omitted
for clarity). Also shown is a spectroscopy setup and an experimental polarization spectroscopy signal when scanning over the transitions involving the 87 Rb
F = 2 level. The largest amplitude line is the F = 2 → F 0 = 3 cyclingtransition which is the lock-point for the cycling laser (see also Fig. 4.3).

Here, m denotes the atomic mass, µ the magnetic moment, and ρ0 = B0 /B⊥0 is
the radius of the “circle-of-death” on which the field-zero is moving [89], whereB⊥0 = Bz0 /2 is the radial quadrupole field gradient.
√ The radial and axial trap0
1/2
frequencies are ω⊥ = B⊥ (µ/2mB0 ) and ωz = 8 ω⊥ , respectively. A further
advantage of a TOP trap is that magnetic field fluctuations on time scales
much slower than the trap frequencies do not affect the trap bottom, unlike
for dc magnetic traps. Our “moving-coil TOP trap” (McTOP) is formed by
combining the movable quadrupole coils with stationary bias-field coils at thescience cell, as illustrated in figure 4.5. We note that a similar strategy has
been used in [103] in conjunction with a magnetic waveguide.

42

z
x

y

Figure 4.5: Moving-coil TOP trap configuration (McTOP) around the science cell. The rectangular bias-field coils have dimensions of 56 mm×37 mm(inner coils) and 54 mm×54 mm (outer coils), and a center-to-center spacingof 25.5 mm and 29.5 mm, respectively. They are tightly sandwiched between
the movable quadrupole coils used for transport (6.0 cm vertical clearance),
which each have an inner (outer) diameter of 4.4 cm (11.5 cm) and a height of
4.0 cm. The science cell is a small quartz glass cell with inner dimensions of
10× 20×45 mm3 and 1.25 mm wall thickness, which is fused to a glass-to-metal
adapter (not shown). Also sketched are the laser beams forming the crossed
optical dipole trap.

Bias-field coils
The bias-field coils are designed to provide a maximally homogeneous bias field,
while minimally obstructing the optical access to the trap center for the given
science cell geometry, as shown in figure 4.5. The outer Bx (inner By ) coils eachconsist of 25 (20) turns of 24 AWG (0.5 mm-diameter) magnet wire and arewound onto a stiff fiberglass holder structure. Each coil pair produces a fieldof 7.5 G/A at the center of the trap with a simulated field inhomogeneity ofless than ±3 × 10−4 within a distance of 1 mm from the center. The air-cooledcoils can thermally withstand ac currents of 8 A amplitude for the durationof the evaporation, corresponding to a 60 G bias field and a circle-of-deathradius ρ0 = 3 mm at the maximum quadrupole field.

43

Vset
PI-control

Verr

Vact

RMS-to-DC

Vcontrol
FG

VCA

ADG419AD633

10 kHzVsensePAQSC Audio
RMX 850

On/Off

90˚

AD737

same
for y-coils

Cmatch

CS

Rcoil
Lcoil

F.W. Bell
CLSM-25

IDC
Cbias

Lbias

Figure 4.6: Closed-loop bias coil driving circuit (shown for one coil pair). An
audio power amplifier (PA) resonantly drives the coils at 10 kHz. The coil current is sensed by a closed-loop Hall-effect current sensor (CS) whose output isconverted into a dc voltage Vact , corresponding to the amplitude of the ac current, by an rms-to-dc converter which determines the regulation bandwidth ofabout 200 Hz. The error signal Verr = Vset − Vact is fed into an op-amp integrator that acts as a PI-controller and adjusts the gain of the voltage-controlled
amplifier (VCA) to counteract any deviations from the desired amplitude Vset .
The ac current can be switched off rapidly with an analog switch. A bias-T
allows dc currents to be run through the coils independent of the ac operation,e.g. for earth-field compensation.

AC bias-field coil drivers
To drive the bias-field coils, we use an 800 W audio power amplifier (PA). For
each coil pair, the impedance at the 10 kHz driving frequency is minimized
down to the ohmic resistance by canceling the inductance with a matchingcapacitor. The measured inductance of the outer (inner) coil pair is 190 µH(110 µH). The PA can easily supply the ac currents for a 60 G bias field directly
into the resulting loads of ∼ 1 Ω, without the need for step-up transformers.
The amplitude of the ac current through each coil pair is actively stabilized
to within ∼ 10−4 using the circuit outlined in figure 4.6. The regulation
bandwidth of about 200 Hz is more than sufficient for compensating thermal
drifts in the coil resistance 2 . The current can be switched off within less than
1 ms limited by the quality factor of the matched coil pair.

2

An alternative, fast feedback scheme has been described in [104], where the focus ison reducing current noise to improve the coherence time for condensate interferometry in a
special TOP waveguide [103].

44

B

ωX-ODT ≈ 2π × 50 Hzinsciencecell

X-ODT &┴-lattices
1064 nmretroz(w0 = 140 µm)

x'z-lattice785 nmy'(and1064 nm)state-dependent(w0 = 230 µm)

Figure 4.7: Illustration of the dipole trap and lattice beam configuration withrespect to the science cell. The two linearly polarized, orthogonal 1064 nm
beams in the xy-plane (red) form a crossed optical dipole trap. These two
beams can be continuously converted into lattice beams by turning on a retroreflected beam using a double AOM setup shown in Fig. 4.13. Along the verticalz-axis there is a third state-independent 1064 nm lattice beam, as well as a circularly polarized 785 nm lattice beam that creates a state-dependent opticallattice (orange). The vertical lattices are obtained by simple retro-reflection
with a mirror. When using the state-dependent lattice, a small magnetic guiding field is present along the vertical direction. Absorption imaging of theatoms can be performed along both horizontal beam axes.

4.4

Optical dipole trap

The focus of a red-detuned Gaussian beam constitutes the simplest optical
trap. The optical dipole potential is essentially proportional to the intensity,

45

which for a Gaussian beam is given byI(ρ, z) =2P22e−2ρ /w (z)2πw (z)(4.2)

where P is the beam power, and w(z) = w0 (1 + z 2 /zR2 )1/2 is the 1/e2 -radiusof the beam, w0 is the beam waist, i.e. the 1/e2 -radius at the focus, andzR = πw02 /λ is the Rayleigh range.
The potential can be expanded around the minimum which yields the harmonic trap frequencies ωz and ωρ . Note that gravity displaces the trap minimum below the point of maximum intensity, and reduces the trap frequency
along the vertical direction. Further, there is a critical potential depth below
which the trap cannot support the atoms against gravity. The confinement
along the beam axis is typically very weak. This can be overcome by adding
a second beam perpendicular to the first, thus forming a crossed-beam optical
dipole trap.
The beams of our crossed optical dipole trap (cf. figure 4.7) have a 1/e2
radius of ∼ 135 µm and a combined power of 3 W. They are derived from asingle-frequency 1064 nm ytterbium fiber laser (IPG YLR LP-SF series) witha relative frequency offset of 20 MHz to average out interference effects. The
maximum depth of the optical trap, including gravity, is 6 µK in the horizontal
and 1 µK in the vertical direction, which allows for efficient gravity-assisted
evaporation of atomic clouds.

4.5State-dependent lattice setup

The 785 nm light for the state-dependent optical lattice potential is derived
from a Coherent 899 Titanium-Sapphire ring laser that is pumped by a 10 W
diode-pumped solid-state laser Coherent Verdi V10 at 532 nm. More details
about this laser setup, e.g. a measurement of its line-width by heterodyning
it with our cycling laser can be found in [78]. After passing trough an AOM
for fast intensity control and a shutter, the light is delivered to the experiment
via a polarization-maintaining optical fiber. Behind the fiber, which is also
used to provide a clean spatial mode, the polarization is cleaned up with apolarizing beam splitter cube, part of the light is picked up onto a photodiodefor intensity stabilization via a PID-controller loop, and the beam is shapedand delivered to the science cell. The polarization can be adjusted manually
via a quarter-wave plate situated directly before the beam enters the science
cell.

46

4.6

RF Setup

The setup for the generation of radio-frequency radiation at the position of
the atoms is described in detail in [96]. It is used for the forced evaporation in
the magnetic quadrupole and TOP traps. Once in optical trap, it is also used
for hyperfine state manipulation within the hyperfine state manifolds and for
combined microwave and rf two-photon transitions.

4.7

Microwave Setup for Hyperfine State Manipulation

4.7.1

Microwave Generation

The experimental setup for the generation and delivery of microwave radiation
to the atoms in the science cell is shown schematically in Fig. 4.8. In thefollowing, I will briefly describe the functions and properties of the individualcomponents.
Microwave synthesizer
The microwave carrier signal is produced by an ultra-low-noise frequency synthesizer module (Microsource SNP-0608-520-02) that is locked to an external
rubidium frequency standard (SRS SIM 940). The phase noise performance ofthe synthesizer module is comparable to high-end microwave generators (e.g.
-104 dBc/Hz at 1 kHz away from carrier), but comes at a fraction of the price.
The carrier frequency fc can be programmed via a serial port to anywhere between 6 and 8 GHz with a resolution of 1 Hz. For the experiments presented
in this thesis, the carrier was typically set to fc = 6827 MHz.
Mixer and IF modulation signal
In order to achieve fast frequency and amplitude control of the microwave
signal, we use a quadrature double-balanced diode mixer (Marki IQ-4509L)
to modulate a so-called intermediate frequency (IF) signal fmod onto the microwave carrier. This quadrature (or IQ) mixer in principle allows for extensive
control of the amplitude, frequency and phase of the microwave signal with an
IF signal bandwidth of up to 500 MHz, if both the in-phase (I) and quadrature (Q) ports are driven separately from a two-channel arbitrary waveformgenerator. So far, however, we use a simple single-sideband scheme, by feeding
the two ports with the same modulation signal, but phase-shifted by 90◦ for47

Frequency,
Power

PULSE ONTTL

GPIB

Function
Generator

B1(t)

fmod

−20 dB

Tektronix
AFG 3021B

Mini-Circuits
ZX80-DR230

90˚ Splitter

Frequency

SRS
SIM 940

10 MHzMicrowave
Synthesizer
6 - 8 GHz
Microsource
SNP-0608-520-02

fc

0˚

90˚

I

Q

IQ-Mixer
Marki
IQ-4509L

Mini-Circuits
PSCQ-2-8

−2 d

BEC

x

z
y

B

AMP ONTTL

RS-232Rubidium
Frequency
Standardm.e.c
H3HN-506

RF Switch

fc+fmod+8 dB+39 dB

Mini-Circuits Stealth Microwave
ZX60-8008E
SM6471-37HS

Figure 4.8: Schematic illustration of the experimental setup for the generationand delivery of microwave radiation to the atoms in the science cell.

the Q-port. This is done with a 90◦ power splitter (Mini-Circuits PSCQ-2-8),
which limits the modulation signal frequency to a range from 2 to 8 MHz.
The microwave signal thus can be quickly adjusted within the frequency range
fc + 5 ± 3 MHz. Despite being rather narrow, this “instantaneous bandwidth”
of 6 MHz has proven sufficient for the small magnetic fields at which we havebeen working so far, and at which the Zeeman splitting is less than a few MHz.
Also, to drive transitions at higher magnetic fields, the carrier frequency canbe reprogrammed easily within about 25 ms to center the 6 MHz band aroundthe desired frequency.
With this configuration of the IQ mixer, the IF signal amplitude controls
the upper side band (USB) fc + fmod , while the lower side band fc − fmod isrejected by about 33 dB in comparison to the USB. The carrier feed-through isconstant and suppressed by about 30 dB. At high modulation signal power, thenonlinear characteristics of the diode mixer become apparent with the powerin higher and lower order sidebands growing at the cost of the USB.
The modulation signal fmod is generated by an ordinary function generator(usuallly a Tektronix AFG 3021B) that can be programmed via GPIB. For
simple pulse shaping we employ an RF switch (Mini-Circuits ZX80-DR230)
providing about 90 dB isolation when off. The rise and fall times are 1 µs and1.6 µs, respectively.
With the current, low IF frequencies the use of a narrow-band filter after48

B′
gradient
pulse


g

B'

TOF
trap
absorption
releaseimaging

1,−1

2,−2

Figure 4.9: Superposition of absorption images taken at different times offlight (TOFs) 2 ms apart after a BEC mixture of the |1, −1i and |2, −2i hyperfine states is released from the trap. The yellow shading indicates the time
during which a magnetic field gradient is applied along the horizontal axis.
The gradient pulse accelerates the two different components in opposite directions due to their different magnetic moments, which leads to spatial separation
during TOF.

the mixer to remove the residual carrier and undesired higher order sideband
signals is not possible. They are thus also getting amplified in the following amplifier stages, which reduces the power available for the desired signal,and creates more intermodulation frequency components. These are, however,usually much smaller than the signal unless the mixer saturates.
Amplification
The modulated microwave signal is then sent through a pre-amplifier (MiniCircuits ZX60-8008E) into a narrow-band, solid-state 5 W linear power amplifier (6.4 to 7.1 GHz, Stealth Microwave SM6471-37HS). The power amplifier
can be turned off via a TTL signal (more than 100 dB suppression whenoff). This is a very useful feature, since even when the modulation signal isswitched off, the amplified carrier signal leaking through the mixer is enoughto adversely affect the evaporative cooling in the magnetic trap due to parasitic microwave evaporation. The amplifier can be turned on and off withinroughly 5 µs and 300 µs, respectively.
Microwave antenna
A 6-feet cable (-2 dB loss) directly connects the power amplifier to the microwave antenna (Microwave Engineering Corp. H3HN-506). The antenna is
49

mF

−2

0

−1

+1 + 2

F=2
2

S 1/2

F=1

2,−2

1,−1
0

25

50

75

100
τ (µs)

125

150

175

200

Figure 4.10: Example of Rabi oscillations between the hyperfine states |1, −1i(marked red) and |2, −2i (marked blue) as the microwave pulse duration τ isvaried. Shown are absorption images after Stern-Gerlach separation and timeof flight. Here, the microwave power was adjusted to yield an on-resonanceRabi frequency Ω0 of about 2π × 10 kHz.

essentially a matched sawed-off rectangular waveguide pointing towards thescience cell. At the position of the atoms it creates a horizontally polarized
oscillating magnetic field. The opening of the waveguide is located about 8 cm
away from the position of the atoms. Although a closer position would bepossible mechanically, we found that, when the antenna is placed too close,
eddy currents around the opening of the waveguide affect the turn-off of themagnetic quadrupole trap and slightly kick the atoms away from the waveguide.

4.7.2

Hyperfine State Manipulation and Detection

The purpose of the microwave radiation produced by the setup just describedis to drive magnetic dipole transitions between the two ground state hyperfine
manifolds F = 1 and F = 2. At a vanishing external magnetic field B0 = 0,
they are separated by 6834 MHz and are (2F +1)-fold degenerate. As discussed
in section 3.2, a non-zero magnetic field lifts the degeneracy of the hyperfine
levels and for small fields the energy difference between adjacent magnetic sublevels grows at about 0.7 MHz/G, as illustrated in Fig. 3.1. In order to detect
the population of the different internal states we map the internal states todifferent motional states via Stern-Gerlach separation in a pulsed on magnetic
field gradient during time of flight (see Fig. 4.9). The transition frequencies
between the different magnetic sub-levels can be calculated analytically with50

14.8 MHz14.1 MHzz

optical density

14.5 MHzN =3.3 ×10

6

T=280 nK

N =1.4×106

T=175 nK

N =0.6×105(T=65 nK)

T
4
N0 =0
3
2
1
0

T
4
N0 =2.8×105
3
2
1
0
T
4
5
N0 =4.7×10
32
1
0-200 -1000z (µm)

100 200

Figure 4.11: Condensate production in the McTOP trap: Phase transition
as seen in near-resonant absorption images (F = 2 → F 0 = 3) after 16 ms timeof flight along with vertical cuts through the density profiles. The data are
fitted with the sum of two-dimensional Bose-enhanced Gaussian (thin line) andThomas-Fermi distributions [8]. Condensation sets in at a critical temperatureof (223 ± 12) nK at which the cloud contains (2.7 ± 0.4) × 106 atoms. For thequasi-pure condensate shown, the temperature in brackets is an estimate basedon the condensate fraction assuming an ideal Bose gas.

the Breit-Rabi formula (3.17), and close to a particular transition the system
can be described as a two-level system, undergoing coherent Rabi oscillations
as shown in Fig. 4.10. On resonance, full population transfer occurs and the
Rabi frequency becomes minimal at fixed power. From the observed resonanceposition one can therefore calibrate the magnitude of the magnetic field B0 .

4.8

Condensate production

4.8.1

Condensate production in the McTOP trap

After transport into the science cell, the atom cloud typically contains 1.6 ×
109 atoms in the |1, −1i state at a phase-space density of 5 × 10−7 . Forced

51

radio-frequency (rf) evaporative cooling is initially performed in the stiff linear
potential of the fully-compressed quadrupole trap (350 G/cm axial gradient),
where it is more efficient until Majorana losses outweigh the advantage of alinear potential [105]. After a 14 s-long linear rf evaporation ramp down toa temperature of 75 µK and an atom number of 7 × 107 , the phase-spacedensity has increased to 1 × 10−4 and the lifetime in the quadrupole trap
due to Majorana losses has decreased to 35 s. At this point, the trap isconverted into a TOP trap by switching on an 18 G rotating bias field, which
preserves the atom number to within 15% and the phase-space density towithin a factor of two [96]. The resulting harmonic trapping potential has
measured trap frequencies of 70.4 Hz in the axial and 25.0 Hz in the radial
direction. The trap parameters are held constant for the remaining 30 s ofthe evaporation sequence, during which another piecewise-linear rf ramp takes
the cloud to quantum degeneracy, cf. figure 4.11(a). A detailed description ofvarious evaporation scenarios in the magnetic trap including the one currently
used can be found in [96].
Position and Atom Number Reproducibility
We have found the shot-to-shot position reproducibility of the condensate
to be consistent with the specified positioning uncertainty of the translation
stages (3σ = 3µm) once the system is warmed up, as shown in figure 4.11(b).
For this measurement, the condensate was imaged simultaneously along two
orthogonal axes in the horizontal xy-plane with resonant absorption imaging
on the repump transition 5S1/2 (F = 1) → 5P3/2 (F 0 = 2), immediately after
the magnetic trap had turned off. We have not observed any systematic shiftsof the condensate position caused by the “dual” imaging itself for the beam
intensities used.
During the first 63 of a total of 173 consecutive runs after a cold start, thecondensate position drifts by roughly 15 µm, as seen in figure 4.11(b). In thez-direction we find a similar drift of about 7 µm. This initial drift is directly
correlated with the slow temperature increase and subsequent stabilization
at ∼ 75◦ C of the supply cables (4/0 AWG, i.e. 11.6 mm core diameter) of
the quadrupole coils. As the cables warm up, their thick rubber insulation
becomes much less rigid, thus changing the mechanical torque exerted on thecoil holder. This potential problem can most easily be avoided by pre-warming
the cables at high quadrupole coil currents.
We have found the condensate atom number to be stable to within 510% depending on the performance of the MOT. For optimized conditions,condensates containing up to 1×106 atoms have been observed. No correlations
were found between the atom number and the position jitter of the cloud in52

5

imagingaxes

2σ

y (µm)

0

warm-up
drift

2σ

-5

3σ

positioner
repeatability(nominal)

-10

-15

-10

-5

x (µm)

0

5

Figure 4.12: Condensate production in the McTOP trap: Reproducibilityof the condensate position in the horizontal plane. The filled circles show thein-situ positions of the condensate (as determined from simple Gaussian fits)
for the last 110 of a series of 173 runs. The 2σ ellipsoid (shaded) with 95% of
the runs has half-lengths of 3.3 µm and 1.8 µm, comparable to the specified
repeatability of the translation stages. The open circles represent a warm-up
drift during the first 63 runs (see text). The arrows indicate the two imaging
directions used, and the error bars indicate the maximum uncertainty in the fits
used to determine the condensate position. The radial Thomas-Fermi diameterof the trapped condensate is ∼ 40 µm.

the science cell.

4.8.2

Condensate production in the optical trap

For condensate production in an optical trap, the evaporation in the McTOP
trap can also be used as “phase-space funnel” to load an optical dipole trap
after which the quadrupole coils are moved out of the way. The stationarybias-field coils can then still be used to control the spin quantization axis, forexample.
The experimental procedure is as follows. After RF evaporation in themagnetic potential, the atoms are loaded adiabatically into a crossed-beam
optical dipole trap formed by two orthogonally intersecting laser beams. Thisis done by smoothly ramping up the optical potential over 400 ms and thensmoothly ramping down the magnetic confinement over another 400 ms. After the transfer, the quadrupole coils are moved back to the intermediate53

80 MHz

(a)

BEC
+1

−1

(b)

18 E r0

13 E r
80 ms

13 E r
10

0

80 ms

Figure 4.13: All-optical trapping and manipulation of a condensate after
transfer from the McTOP trap. (a) Conversion of each of the two beams ofthe crossed dipole trap (cf. figure 4.7) into an optical lattice beam by partial retro-reflection, using a double-pass AOM configuration [106] with zeronet frequency shift and rapidly adjustable reflectivity of 10−6 -10−1 . To realize a three-dimensional lattice, a third beam pair (with full retro-reflection)is added along z. (b) Superfluid-to-Mott insulator transition and back in athree-dimensional state-independent optical lattice (1064 nm)3 as observed inabsorption images after 18 ms time of flight. The lattice depth follows exponential ramps of 80 ms duration, separated by a 10 ms hold time at a depth of
18 recoil energies ER (solid line).

position ‘b’ indicated in figure 4.1.
We typically load the optical trap with clouds at ∼ 250 nK and then ramp
down the trap depth to 5 µK in the horizontal and 200 nK in the verticaldirection where condensation sets in. At this point, the trap is nearly isotropic
with measured frequencies between 50 and 60 Hz. The loading procedure
results in nearly pure condensates with atom numbers that are within 90% of
those reached in the McTOP trap.

4.9

Demonstrating the Superfluid-to-Mott insulator Transition with our Apparatus

By performing the final evaporation in an optical trap, it is possible to easily
use the TOP trap for optical lattice experiments, thus avoiding the usual
drawback of TOP traps: atomic micromotion [97] can lead to strong heating54

due to an oscillatory motion of the atoms relative to the lattice at the frequencyof the rotating bias field [98]. To demonstrate the suitability of our apparatus
for optical lattice experiments, we have observed a reversible superfluid-toMott insulator transition in a three-dimensional (state-independent) opticallattice [13, 14], as shown in Fig. 4.13(b). Since this initial data has been taken,
we have improved the quality of our lattice beams and the lattice ramps, suchthat much cleaner single-component SF-MI transitions can be realized at thetime of writing this thesis.

55

Chapter 5Interacting Bosonic Mixtures in
Optical Lattices
In this chapter, I report the observation of many-body interaction effects for ahomonuclear bosonic mixture in a three-dimensional optical lattice with variable state dependence along one axis. Near the superfluid-to-Mott insulatortransition for one component, we find that the presence of a second component can reduce the apparent superfluid coherence, most significantly when
the second component either experiences a strongly localizing lattice potential
or none at all. We examine this effect by varying the relative populations andlattice depths, and discuss the observed behavior in view of recent proposals
for atomic-disorder and polaron-induced localization. This chapter is basedon our publication Ref. [107].

5.1

Introduction

Bosonic mixtures in optical lattices allow for the study of many interesting
topics, such as the two-component Bose–Hubbard model [15–18], as illustrated
in Fig. 5.1, with its connection to quantum magnetism [15, 19] (see Fig. 5.2)
and models for decoherence mechanisms [20, 21]. In regard to condensed
matter physics simulations, the introduction of a second component allows forthe investigation of phenomena such as polaron physics [22–24] and phononmediated long-range interactions [18, 23], as well as effects of impurities anddisorder [25–29].
Recent experiments have addressed heteronuclear mixtures of atoms in optical lattices, both for the boson-boson [108] and boson-fermion [109–111]
cases. The superfluid coherence of the heavier bosonic component was universally found to decrease in the presence of the lighter second species, indepen-

56

ta

tb

Uab

Ub

Ua

a
b
Figure 5.1: Illustration of the two-component Bose-Hubbard model. Here,the two components are realized by two different internal states |ai and |bi.
Each component has its own tunneling rates ta and tb between nearest-neighborsites, and intra-component on-site repulsions Ua and Ub . Additionally, the twocomponents interact via the inter-component on-site repulsion Uab .

dent of the sign of the interaction, even for small overlap between the components [108]. There exists a number of explanations for the observed behavior,ranging from localization due to impurities [110], self-trapping [111], the formation of composite particles [109], incoherent scattering of phonons [108], tothermalization effects [109] and changes in the chemical potential [112]. Compared to the single-component case, the description of mixtures presents richer
physics, but also depends on additional parameters, such as the ratio of thetunneling rates, the interspecies interaction, and the relative atom numbers.
In this chapter we study how the superfluid coherence of bosons in a lattice is affected by a variable bosonic “background” medium. Through a statedependent optical lattice we can vary the localization of the medium’s constituent atoms, and we find a non-monotonic dependence of the coherence
properties of the primary species (“foreground atoms”) on this localization.
For a binary mixture of 87 Rb hyperfine states, we demonstrate a reversible
state-dependent transition from the superfluid into the Mott regime, and wesystematically examine interaction effects by varying both the relative populations and the respective lattice depths. Remarkably, the superfluid coherence
decreases not only when the bosons of the medium are strongly localized, butalso when they are very delocalized.

57

0.8

↑=
1

Uab / U

0.6

0.4

2

↓=

(↑ + ↓) =

“

“

z-FM
xy-FM (SCF)

0.2
0.0

(U=Ua=Ub , µa=µb )

z-AF
0.2

0.4

ta /tb

0.6

0.8

Figure 5.2: Ordered phases of the two-component Bose-Hubbard model inthe deep Mott-insulator limit at unit filling (na + nb = 1) at T = 0 obtained
via a simple mean-field ansatz following Ref. [17] (details of the calculation canalso be found in [78]). In this limit, the model can be mapped to the spin1/2 XXZ anisotropic Heisenberg model, and the two types of bosons can beviewed as the eigenstates of a pseudo-spin 1/2. Depending on the tunnelingratio ta /tb and the strength of the inter-species interaction Uab , the system favors different “magnetically” ordered phases, such as an easy-axis ferromagnet(z-FM), which corresponds to phase-separation in the particle picture, an Ising
antiferromagnet (z-AF) or checkerboard-ordered state, and an easy-plane ferromagnet (xy-FM). In the particle point of view, this last phase is a so-called
counterflow superfluid [15], and represents the most interesting of the threemagnetic phases shown here.

5.2

Experimental Procedure

Our experimental setup has been described in detail in chapter 4. In brief, we
produce a 87 Rb Bose–Einstein condensate in the |F, mF i = |1, −1i hyperfine
ground state containing 3 − 5 × 105 atoms in a crossed-beam optical dipole
trap (ODT) at λ⊥ = 1064 nm, with a mean trap frequency of about 50 Hz.
We prepare mixtures of the |1, −1i ≡ |ai and |2, −2i ≡ |bi hyperfine groundstates, with variable fractional populations fa and fb = 1 − fa , via microwave
Landau–Zener sweeps [113]. These mixtures are subsequently loaded into an
adiabatically ramped up 3D optical lattice. The state-independent transverse
lattice potentials along x and y are generated by partial retroreflection of the
ODT beams [90], maintaining constant gravitational sag through coordinationof the forward beam intensities with the amount of retroreflection. The statedependent lattice along z is generated by full retroreflection of a beam at58

λz = 785.1 nm (1/e2 radius: 230 µm). At this wavelength (between the D1
and D2 lines of 87 Rb), the individual depths Va and Vb of the z-lattice, for |ai
and |bi, are strongly polarization dependent [69]. A 0.4 G magnetic bias fieldalong z defines the quantization axis. By varying the polarization of the beamfrom σ + to σ − , we can adjust the ratio Vb /Va from approximately 0 to 3.5.
We characterize the many-body state in the lattice by releasing the atoms andthen absorptively imaging both components on the F = 2 → F 0 = 3 cyclingtransition after 18 ms time-of-flight (TOF), concurrent with optical pumping
from F = 1 → F 0 = 2. Additionally, we use a magnetic Stern–Gerlach pulse
to separate the states in TOF.
The use of a homonuclear mixture avoids differential gravitational sag [108]in our far-detuned ODT, and the similarity of all relevant scattering lengths(difference < 2% [114]) precludes macroscopic phase separation. We characterize component overlap in the ODT using collinear two-component four-wave
mixing as a sensitive probe [115]. Residual magnetic field gradients were carefully canceled by maximizing component overlap and inferred to be less than
40 µG/cm from hyperfine Ramsey measurements.
We first demonstrate a state-dependent transition from the superfluid tothe Mott regime, keeping the final transverse lattice depth constant at 12 ER⊥
2(ER⊥ = ~2 k⊥/2m, k⊥ = 2π/λ⊥ ) and ramping up and down the z-lattice depth1 .
The transverse lattice depth is slightly below the superfluid-to-Mott insulatortransition in an isotropic 3D cubic lattice (at a tunneling-to-interaction ratiot/U ≈ 1/36 [13]), and sufficiently large for undesired four-wave mixing effects
in TOF [115] as discussed in section 6.6 to be negligible. Fig. 5.3 shows TOF
images for the case Va = 4Vb and a ramp of Va up to 15 ER (ER = ~2 kz2 /2m,kz = 2π/λz ). The |ai atoms undergo a reversible transition into the Mott
regime, while the |bi component remains superfluid.

5.3

Visibility and diffraction peak width as coherence measures

To characterize the many-body states of the two components, we analyze thevisibility [116] and peak width [117] of their TOF diffraction patterns. As
our optical lattice is anisotropic (transverse and vertical axes) in both lattice
constant and depth, we employ a single-axis visibility measure along the state1

Our initial attempts with a slightly different lattice loading sequence are described indetail in Ref. [78]. There, however, we were not able to observe any interspecies interaction
effects, simply because the two components got vertically separated due to a change in
gravitational sag during the lattice ramp up, which we had not considered at that time.

59

(C)

(A)

a
z
y
x

b

(B)
lz
lx

15 ER

Va

Vb = Va/4
80 ms

7.5 ER
2 ms

30 ms

Figure 5.3: State-dependent transition from the superfluid to the Mottinsulating regime. (A) Surfaces of equal probability density (1/e2 ) for noninteracting atoms of type |ai (red) and |bi (blue) in a three-dimensional lattice⊥ in the transverse directions (x, y), and 15 (3.8) E alongwith depths of 12 ERRz for |ai (|bi) atoms. (B) Time-of-flight (TOF) absorption image after releaseof a balanced mixture (fa ≈ fb ). The aspect ratio lz /lx is given by kz /k⊥ . (C)TOF images after Stern–Gerlach separation. The |ai component (red) enters
the Mott regime (with ta /Uaa ≈ 1/39, and t⊥ /Uaa ≈ 1/38), whereas the |bicomponent (blue) remains superfluid. The bottom graph illustrates the statedependent z-lattice ramp. The transverse lattice ramp (not shown) follows a⊥ as thesigmoid curve of 115 ms duration and reaches its full depth of 12 ER
z-lattice goes through half of its maximum depth.

dependent vertical axis. A 600 µs gravitational π-phase shift [13, 118] between
adjacent vertical lattice sites produces a symmetric diffraction pattern withtwo vertical peaks separated by 2~kz (in contrast to the patterns of Fig. 5.3
without a shift). This is accomplished by turning off the ODT while keeping on
the z-lattice. The visibility is then defined as γ = N− /N+ , where N± = N+1 +N−1 ± 2N0 are the sum and difference of the atom numbers in the diffraction
peaks (N+1 , N−1 ) and intermediate region (N0 ), as shown in Fig. 5.4 (I-A).
To determine the peak width σ (1/e half-width), we sum over a vertical strip
and fit the projection with two Gaussian peaks on top of a broad Gaussian
background.
60

In the following, we examine how the coherence properties of a given superfluid component depend on the presence of a background of either a lighter(in terms of the band mass m∗z ) superfluid, or heavier “impurity” atoms. Thisis implemented by a halving (I) and doubling (II) of the maximum z-lattice
depths as compared to Fig. 5.3. In the first case (I) the background medium isformed by |bi atoms in a 2 ER deep potential, and in the second case (II) by |ai
atoms in a potential with 31 ER depth. In both cases, the foreground component (|ai atoms in (I), |bi atoms in (II)) experiences an 8 ER deep lattice, forwhich the visibility displays a strong differential dependence on lattice depth
and tunneling (at the chosen transverse lattice depth of 12 ER⊥ ). In both cases,
we vary the relative populations fa and fb = 1 − fa of the two components
while keeping the total atom number constant, which allows for separating out
effects of interspecies coupling from simple overall density effects.

5.4

Delocalized background medium

In case (I) the background density is only weakly modulated by the 2 Er deep
z-lattice, while the foreground atoms are concentrated on lattice sites to agood approximation, but can still tunnel appreciably. The collisional interaction will thus tend to repel background atoms away from sites populatedby the foreground atoms, as shown schematically in Fig. 5.5 (A) . The resulting dips in the background density (within a characteristic range givenby the mean-field healing length ξ) in turn increase the foreground atoms’
localization. This mechanism, similar to self-trapping of impurities in a condensate [22], corresponds to the formation of polarons [23] – composite quasiparticles consisting of individual foreground atoms surrounded by a coherent
phonon cloud comprising the background’s local mean-field depression. The
immersed foreground atoms should thus have an increased effective mass anda lower tunneling than bare foreground atoms, leading to a degradation of theforeground’s observed coherence.
Data for case (I) are shown in Fig. 5.4 (I). As the superfluid backgroundof |bi atoms gets more and more populated at the expense of the foreground|ai component, the |ai visibility drops continuously and the diffraction peaks
get broader. A reference without background, in which the |ai atom number
is correspondingly varied, in contrast reveals a slight increase in visibility witha reduction in |ai atom number, consistent with the naive expectation for adecrease in on-site density. We note that atom transfer from the |ai component
to the less localized |bi component lowers the on-site density as well, which
naively should also lead to an increase in visibility. The observations thus are
not consistent with a simple overall density effect. Additionally, the visibility

61

I
N+1

(A)

II

2.1 ER

8.3 ER

31 ER

N0
b N-1

a

a

“Background Medium” (b)

peak width - σ (ħk)

0.75

γa

0

-0.25

σb

0.1

b
“Background Medium” (a)

1
0.8
0.2

γb

Visibility - γ

1.00

(B)

7.8 ER

0

0

1

fb

1

fa

0.5
0.25
0.45

a onlyb onlya with bb with a

0.35
0.25
0.15
0

0.2

0.4

0.6

0.8

background fraction (fb)

1

0

0.2

0.4

0.6

0.8

1

background fraction (fa)Figure 5.4: Effects of a background medium on the superfluid coherence.
(I;left column) |ai atoms (Va = 8.3 ER ) in contact with superfluid |bi atoms(Vb = 2.1 ER ) and (II;right column) |bi atoms (Vb = 7.8 ER ) in contact withlocalized |ai atoms (Vb = 31 ER ). (A) Stern–Gerlach separated TOF images(for fa ≈ fb ) after a gravitational phase shift. The circles in (I) denote apertures
used to determine visibility. (B) Dependence of the visibility and the peakwidth on the relative population of the background medium, with constanttotal atom number 1.7(1) × 105 ; the insets show corresponding data for the
background medium. The open circles denote reference measurements withoutbackground medium, in which the atom numbers are varied correspondingly.
⊥.
All ramp shapes are as in Fig. 5.3, with V⊥ = 12 ER

could in principle be affected by incoherent intra- and interspecies collisions
after release from the trap. However, both would similarly degrade the |bi
visibility which, as seen in the inset of Fig. 5.4 (I-B), remains at a consistently
high value throughout.

62

The observed reduction of the |ai visibility from γa ≈ 0.5 (no atoms inbackground) to γa ≈ 0.3 (95% in background), and the associated increase inpeak width, would correspond in the single-component case to a reduction oftunneling by about 40%, as estimated from an independent single-component
reference measurement in which the lattice depth was varied. For a single immersed |ai atom, the surrounding dip of the |bi component density leads to an
on-site polaronic energy shift Vp = gab,1D /ξ ∼ 0.2 ER [23] (where ξ ∼ 200 nm,
gab,1D = 2aab ~ω⊥ and aab ≈ abb ≈ 100a0 is the interspecies scattering length),and one can roughly estimate that for the modified on-site potential depth∼ Va +Vp there is an approximately 5% reduction in tunneling. Therefore, thissingle-polaron effect alone seems insufficient to explain the observed change invisibility. However, given the large number of |ai atoms even at fb = 0.95, additional localization can be expected due to polaron clustering [23], illustrated
in Fig. 5.5 (A), which results from attractive off-site interactions mediatedby the superfluid background [18]. The mutual exponential localization of polarons in clusters has been predicted to lead to a broadening of the momentum
distribution [23], similar to our observations.
With regard to possible temperature effects, adiabatic loading of bosons
into the lowest band of a 3D optical lattice should lead to lower temperaturesdue to the reduction in band width [119]. Given the unequal z-lattice depths,
one thus would expect unequal final temperatures for the two components
if loaded separately, leading to thermalization in a mixture [77]. However,thermalizing elastic collisions should be largely suppressed due to the mismatchin band structure [77, 115]. Moreover, the expected temperatures in the lattice
are incompatible with the magnitude of the observed effects, even assuming
thermalization. Reference measurements with only |ai atoms showed thatvisibilities below 0.3 require initial loading temperatures exceeding 75 nK,
which can be estimated 2 to yield final temperatures above 14 nK. However,for the mixture we start with much lower initial loading temperatures ∼15 nK
that should accordingly be reduced to below 4 nK even in the weaker |bi lattice(also, these temperatures  Vp /kB are sufficiently low to suppress effectsof thermal phonons on polaron tunneling [23]). We can also exclude trivial
heating effects due to spontaneous photon scattering from the state-dependentlattice beam, since the |bi atoms are not only less confined to intensity maxima,
but also have a slightly lower scattering rate, so that such heating should belower for higher values of fb .

2

pUsing Tf /Ti ∼ m/m∗ [119], with m∗ = 3 m∗x m∗y m∗z for our anisotropic lattice [120]

63

(A)

(B)

Figure 5.5: (A) The delocalized background component (blue) forms a superfluid bath for the foreground atoms (red). Because of the inter-species repulsion
a foreground atom creates a region of reduced background density around it,
and effectively gets “dressed” by the phonon excitations of the bath, forminga polaron [23]. The tunneling of this polaron is reduced compared to a bareforeground atom since, pictorially speaking, it has to carry the dressing cloudalong as it moves through the bath. With the density dip extending over aslightly larger range than the lattice spacing, the foreground atoms should further feel off-site attractive interactions mediated through the background component, which can lead to clustering and localization. Both, reduced tunnelingand clustering will reduce the superfluid coherence of the foreground component compared to the single-component superfluid-to-Mott insulator transition.
(B) The opposite case of a strongly localized background medium, now formedby the red atoms, which act as static local impurities for the blue foregroundcomponent. Here, as for the first case, the red curve indicates the optical lattice potential felt by the red atoms, while the blue curve shows the potential
seen by the blue atoms through the interaction with the red ones. The random atomic disorder potential can lead to Anderson localization and Bose-glass
physics [25], in effect reducing the superfluid coherence of the foreground component, as in the first case. One could think that the red atoms should forma regular, Mott insulating phase instead of a disordered assembly. However,when co-loading two interacting components into a state-dependent opticallattice, in a certain parameter range this can result in a disordered mixtureof small droplets of the two components, a so-called “quantum emulsion” [27].
Another interpretation is through a percolation phenomenon [28, 29], i.e. atsome point when the impurity density has become large enough, there exists
no percolating path connecting the different condensate droplets any more, sothat the overall superfluid coherence suddenly disappears.

64

5.5

Strongly-localized background medium

Observations
We now turn to the case (II) of a superfluid coupled to a background mediumthat is pinned to the optical lattice. We ramp the z-lattice to Vb(a) ≈ 8(31) ER ,for which the |ai component, now playing the role of the background, is deepin the Mott regime. With an increasing background fraction, we again observea drop in visibility and an increase in peak width of the foreground |bi atoms,
as is shown in Fig. 5.4 (II).
Possible causes and interpretations
The observed changes for the |bi component could be caused by a simple increase of on-site interactions due to localized |ai atoms. However, the changes
to both visibility and peak width are quite abrupt, which may be indicative of a disorder effect. The simultaneous loading of multi-component gaseswith differing lattice parameters is predicted to result in a “quantum emulsion
state” [27, 112] similar to a Bose-glass, with the less mobile species acting as
quasi-static impurities [25], as illustrated in Fig. 5.5 (B). The observed sudden drop could thus be caused by the concentration of localized |ai atoms
increasing beyond a percolation threshold, leading to localization of the |bi
atoms [28, 29]. Another possible contribution, heating due to spontaneous photon scattering by |ai(|bi) atoms, is measured to amount to less than 14(8) nK
for symmetric ramps to 31(8) ER and back down. However, thermalization(which by the earlier argument should be largely suppressed due to disparate
band structures) would not appear to be consistent with the suddenness of thechange in |bi visibility and peak width at fa ∼ 0.8 for a continuous increase of
background |ai population.

5.6

Varying the localization of the backgroundmediumFinally, having studied these two extreme cases, we examine more closely
how the interspecies effects depend on the background medium’s degree of
localization at a fixed background fraction. Using the polarization dependenceof our lattice, we keep Va fixed at 12 ER while tuning Vb over a wide range,changing the character of the |bi background component from superfluid tolocalized. Shown in Fig. 5.6 are results for a mixture with fb ≈ 3/4. The
addition of |bi atoms leads to a reduction of the |ai visibility, most prominently65σa (ħk)
γa

0.4
0.3
0.2
0.6
0.4
0.2

Vb = Va0

1020

Vb (E R )30

40

Figure 5.6: Dependence of superfluid coherence on the localization of thesecond component. Visibility (γa ) and peak width (σa ) (filled circles) of the⊥,|ai component, with fixed lattice parameters of Va = 12 ER and V⊥ = 12 ER
5in the presence of |bi atoms (fb ≈ 3/4, Na + Nb = 3.7(2) × 10 ) as Vb isincreased. In references (open circles) taken without |bi atoms and with Na =1.0(1) × 105 , the visibility and peak width are roughly constant at 0.52(3) and0.26(2) (dashed lines). Data points are averaged over 3-5 runs, with statisticalerror bars shown. Uncertainties in lattice depth are ∼5% (not shown).

when the |bi component is either much lighter or much heavier, in terms ofband mass. For Vb = Va , the visibility is lower than in a reference measurementin which only |ai atoms were present, which is most likely caused by thenear doubling of the chemical potential due to the increase in overall atom
number. We note that the strongest interspecies effects are observed when
the background |bi atoms are highly localized, while no effect is observed for asimilar background fraction of localized atoms in Fig. 5.4 (II). However, here wehave a much greater total population (more than twice that of Fig. 5.4 (II)),so that the concentration of sites occupied by localized background atoms
will be greater for a given background fraction, which should enhance thedisorder-induced localization of the foreground component as discussed above.
In general, the foreground visibility at a given lattice depth should depend on
the modified tunneling and on-site interactions due to background atoms, aswell as non-trivial effects of clustering and disorder as discussed above. Thus,a determination of where the visibility maximum is located would require aquantitative model for the detailed interplay between these effects.

66

5.7

Conclusion

In summary, we have studied interspecies effects for a binary bosonic mixture,
in a three-dimensional optical lattice with tunable state dependence along one
axis. We observe a reduction of apparent superfluid coherence, most strongly
for large population imbalances and tunneling rate asymmetries of the twocomponents. The observed reduction in coherence for the addition of both adelocalized and also a localized second species suggest polaron-related effects
and atomic-disorder, respectively. Our system should be of interest for future
investigations of phonon-mediated interactions in polaron systems and in the
spin-boson model.

67

Chapter 6
Collinear Four-Wave Mixing of
Two-Component Matter Waves
We demonstrate atomic four-wave mixing of two-component matter waves ina collinear geometry. Starting from a single-species Bose–Einstein condensate, seed and pump modes are prepared through microwave state transfer
and state-selective Kapitza–Dirac diffraction. Four-wave mixing then populates the initially empty output modes. Simulations based on a coupled-mode
expansion of the Gross–Pitaevskii equation are in very good agreement withthe experimental data. We show that four-wave mixing can play an importantrole in studies of bosonic mixtures in optical lattices. Our system should also
be of interest in the context of quantum atom optics [47]. This chapter isbased on our publication Ref. [115].

6.1

Introduction

Four-wave mixing is a fundamental, well-studied concept in nonlinear optics
and spectroscopy [30]. Its matter-wave analogue, based on binary collisions
in ultracold atomic gases, was first demonstrated experimentally a decadeago [31–33], establishing the field of nonlinear atom optics [34]. In four-wave
mixing, two waves form a grating from which a third wave diffracts, thus generating a fourth wave. This process has been used for coherent matter-wave
amplification [35, 36], and for the generation of correlated atom pairs [34–38].
Energy and momentum conservation require the magnitudes of all atomic momenta in the center-of-mass frame to be equal which, for atoms in a single
internal state, necessitates a two-dimensional geometry [32, 33]. By modifying the dispersion relation with an optical lattice, non-degenerate four-wave
mixing of a single species becomes possible also in one dimension [39–41].

68

Despite considerable theoretical work on atomic four-wave mixing (FWM)with more than one internal state [42–45], experiments have only very recently
started to explore possible mechanisms for such FWM [38, 46]. The additional internal degree of freedom allows for degenerate FWM to occur in one
dimension, with pairs of waves in different internal states sharing the same
momentum mode, opening up possibilities to generate non-classical matterwave states, e.g. with macroscopic spin entanglement [43, 44]. In this chapter,we demonstrate free-space collinear atomic FWM involving two internal stateswith distinct, macroscopically populated momentum modes.
Apart from the relevance for quantum atom optics, another important context arises in experimental studies of bosonic mixtures in optical lattices [77,108, 121]. These systems are of high interest not only in connection with applications to quantum magnetism [15–17], but also for studies of decoherence
mechanisms [20, 122], and for lattice thermometry [77, 121]. Most experiments with ultracold atoms in optical lattices to date rely on time-of-flightinformation. In particular, a sudden release from the lattice projects the band
populations onto plane-wave states [123]. We find that for a homonuclear
mixture of interacting superfluids, four-wave mixing processes can alter the
expected momentum-space distributions, masking or even mimicking in-situ
interaction effects.

6.2

Experimental Procedure

In order to induce collinear two-component four-wave mixing, we apply astate-selective optical lattice pulse to a Bose–Einstein condensate containing
atoms in two internal states |ai and |bi, as illustrated in Fig. 6.1 (A). The
pulse induces Kapitza-Dirac (KD) diffraction [79–81] producing recoiling |ai
atoms in both positive and negative momentum modes |±2i ≡ |±2~kL i wherekL = 2π/λL , while the |bi atoms remain unaffected. Subsequently, as illustrated in Fig. 6.1 (B), the |bi atoms Bragg diffract from the density modulation
formed by the interference of the recoiling |ai atoms |a, +2i ≡ |ai ⊗ |+2i withthose at rest, |a, 0i. Due to momentum exchange collisions, the recoiling |ai
atoms are coherently transferred back into |0i, as recoiling atoms |b, +2i areproduced. This process is formally not distinguishable from coherent (pseudo-)
spin exchange. Our system might thus pose an interesting alternative to spinor
condensates for the creation of non-classical states [43, 44, 46]. We note that
due to the symmetry of the KD pulse, another, independent “copy” of theFWM process occurs on the negative momentum side. For quantum atom
optics purposes, this can easily be avoided by using a state-selective Bragg
pulse instead, which also allows for extended control of the initial mode pop-

69

(A)
−ħkL

τKD

B

BEC

0

+ħkL

σ+

−2

σ

+

(B) a

b

p/ħkL

2

Figure 6.1: Experimental scheme. (A) State-selective Kapitza–Dirac diffraction of a two-component Bose–Einstein condensate in hyperfine states |ai and|bi, which populates the recoiling modes |a, ±2~kL i while the |bi component
remains unaffected. (B) Four-wave mixing (solid arrows) with pump modes|b, 0i, |a, 2i and seed mode |a, 0i transfers |bi atoms to the output mode |b, 2i.
Due to the symmetry of the problem, the process also occurs for the modes|a, −2i and |b, −2i (dashed arrows).

ulations. In the present work, however, we are content with applying a KD
pulse, mainly out of technical convenience.
Our experimental setup has been described in detail in Ref. [90]. In acrossed-beam optical dipole trap at 1064 nm wavelength, we produce nearlypure 87 Rb Bose–Einstein condensates in the |ai ≡ |F = 1, mF = −1i hyperfine state typically containing about 1.6 × 105 atoms. The trap is approximately isotropic with a mean trap frequency around 50 Hz and an alignmentdependent vertical frequency ωz /2π between 40 and 50 Hz. We then apply amicrowave Landau–Zener sweep to transfer a variable fraction of the condensate into the |bi ≡ |2, −2i hyperfine state [113]. Immediately afterwards, astate-selective lattice beam [69, 70] of wavelength λL = 785.1 nm is pulsed onalong the vertical z-direction (1/e2 radius 230 µm) for a time τKD . The polarization of the light (σ + ) is chosen such that only the |ai atoms feel the opticallattice potential formed through retro-reflection of the beam. A small magnetic field of ∼0.4 G along the beam axis defines the quantization axis. Afterrelease from the trap and a few milliseconds of free evolution, during which
the FWM occurs, a magnetic field gradient (Stern–Gerlach pulse) spatiallyseparates the two hyperfine states along the horizontal x-axis for detection.
Finally, the atoms are imaged along the y-axis after a total time of flight of
15 ms by a 100 µs long pulse of near-resonant F = 2 → F 0 = 3 imaging
light, combined with F = 1 → F 0 = 2 repumping light, which ensures equal
detection efficiencies for both hyperfine states.

70

(A)

(B)

(C)
2
0
−2

z

a

x

b

Figure 6.2: Typical absorption images taken after application of the stateselective Kapitza–Dirac pulse (τKD = 25 µs, Va = 6 ER ), 15 ms time of flightand Stern–Gerlach separation along x, for the case of (A) only |ai atoms
present (fa = 1), (B) only |bi atoms (fa = 0), and (C) equal populations ofboth components (fa = 0.5).

6.3

Results

In Fig. 6.2, typical absorption images are shown for three different fractions of|ai atoms fa ≡ Na /N . The KD pulse duration (25 µs) and the lattice depth
Va for atoms of type |ai (6 ER , where ER = ~2 kL2 /2m is the recoil energy),
are chosen such that half of the population of |ai atoms is diffracted into|±2i, while higher orders remain largely unpopulated. By analyzing singlecomponent diffraction patterns [81], we have determined the lattice depths foreach component, confirming that atoms of type |bi experience < 5% of thelattice depth seen by the |ai atoms. On their own, the |bi atoms therefore are
not affected by the lattice pulse, as shown in Fig. 6.2 (B). However, when bothcomponents are present, a significant fraction of |bi atoms is transferred into
the |±2i momentum modes (Fig. 6.2 (C)).
We have measured the amount of diffracted atoms in each state as afunction of fa . As shown in Fig. 6.3, the fraction of diffracted |bi atoms(Nb,+2 + Nb,−2 )/Nb monotonically increases from zero towards a maximum as
fa is increased, consistent with the picture that the grating formed by interference of the |a, 0i and |a, ±2i modes, from which the atoms in |b, 0i diffract, gets
deeper as the number of |ai atoms grows. The relative number of diffracted|ai atoms has a pronounced minimum near fa = 0.5, which can be interpreted
as a “back-action” of the |bi atoms onto the |ai grating.

71fraction in ±2

0.8
0.6

a

0.4
0.2
0.0

b
0.0 0.2 0.4 0.6 0.8 1.0
fraction of a atoms

Figure 6.3: Fraction of atoms with momenta ±2~kL in state |ai (green circles)
and |bi (orange squares) after four-wave mixing. The solid curves are obtained
from the theoretical model described in the text using the same parameters as
in Fig. 6.4.

6.4

Theoretical Model

To obtain a more quantitative understanding, we theoretically model our system starting from the coupled Gross-Pitaevskii equations (GPE) for the orderparameters Φα (r, t) of the two components α ∈ {a, b}


2
X
~
i~ ∂t Φα = −
∇2 + Vαtot +gαβ |Φβ |2  Φα ,(6.1)
2m
β∈{a,b}where gαβ = 4π~2 aαβ /m, m is the atomic mass, and the intra and inter-speciess-wave scattering lengths aaa , abb , and aab in units of a0 are 100.4, 99.0, and99.0, respectively [114, 124]. The trapping and lattice potentials are given
by Vαtot = Vtrap (r, t) + Vα (t) sin2 (kL z). Similar to the slowly-varying envelope
approximation (SVEA) used in Refs. [32, 33, 45], we approximate the solutionof Eq. (6.1) as an expansion in terms of momentum modes, or wave packets,
moving along z with multiples of the recoil velocity vR = ~kL /mΦα (r, t) =∞X

cnα(t) ei nkLz Φ0 (r − ẑ nvR t, t) .

(6.2)

n=−∞

However, since we are mainly interested in the mode populations, we further
assume that the wave packets Φ0 are of Thomas–Fermi form and expand hydrodynamically after release from the trap [9], which leads to a significant
72simplification compared to a full SVEA simulation. On the time scales ofinterest, phase-separation [125–127] can be neglected, and we have Φα ∝ Φ0for both components immediately following the microwave transfer [125, 126].
Since the momentum-spread of Φ0 is much less than ~kL , the modes in the
expansion are quasi-orthogonal. After inserting the ansatz (6.2) into Eq. (6.1),
we arrive at a system of coupled equations for the amplitudes an (t) ≡ cna (t)i~ ∂t an = ER n2 an + Va (t) 12 an − 41 (an+2 + an−2 )X+(gaa a∗m am0 + gab b∗m bm0 )an0 hnmm0 n0 (t)mm0 n0and similarly for the other component bn (t) ≡ cnb (t). Here, hnmm0 n0 (t) ∝δ(n + m − m0 − n0 ) denotes overlap integrals that include the effective temporal
decay of the nonlinear interaction, as the different wave packets separate, andas the density decreases during the expansion. The terms responsible for FWM
are of the form b∗m bn am (and a∗m an bm for the |bi component) with m 6= n.
After adjacent modes (|m − n| = 2), for which the overlap decays the slowest,
have completely separated, the populations remain frozen, since only equalmomentum self and cross-phase modulation terms of the forms |an |2 an and|bn |2 an survive. With a typical Thomas–Fermi radius Rz ∼ 10 µm, we obtaina typical separation time tsep ≈ 2Rz /2vR of 1.7 ms.

6.5

Discussion

The full set of observed populations |an |2 and |bn |2 after FWM is plottedin Fig. 6.4, along with predictions of our model obtained with parameters
according to the experimental ones, leaving only the total atom number N asa fit parameter. The overall agreement between data and theory is remarkable.
The maximum FWM yield occurs near fa = 2/3 where the initial populationsof the pump and seed modes are equal, maximizing the FWM term a∗m an bm
at t = 0 [32, 33]. The data also clearly show the correlated growth of |b±2 |2
and |a0 |2 , along with a corresponding depletion of the pump modes |b0 |2 and|a±2 |2 , as detailed in the inset.
We note that since the FWM yield isto the inter-species scatR proportional
32
2
tering length as well as to the overlap d r |Φa | |Φb | of the two components,
it can serve as a sensitive probe for both quantities. As a practical example, we
use two-component FWM as a clear “single-shot” diagnostic for the optimization of component overlap. By carefully canceling magnetic field gradients weare able to sustain overlap, i.e. FWM yield, for up to 2 s after the microwave
transfer.

73

1.0

0.15
ΔNb,±2 /N

0.10

fraction of total atom number

0.8

0.05
0.00

0.6

a

0.00

b
2

0.05 0.10
ΔNa0 /N

0.15

(1)

0
−2

0.4

(2)
0.2

(3)

0.0
0.0

0.2

0.6
0.4
fraction of a atoms

0.8

1.0

Figure 6.4: Mode populations after four-wave mixing. Populations |cnα |2 =Nnα /N of the modes |α, ni, as indicated in the plaquette. The dashed lines(1), (2), and (3) indicate the initial conditions before FWM, |b0 (0)|2 , |a0 (0)|2 =|a±2 (0)|2 , and |b±2 (0)|2 (where ±2 indicates the combined populations). The
arrows indicate the temporal evolution of the populations. The solid lines
represent the predictions of the model (Va = 5.6 ER , τKD = 25 µs, ωz =2π × 51 Hz, N = 1.4 × 105 ). The inset shows the transferred |bi population
vs. the transferred |ai population, where the dashed line represents a slope ofunity.

To further confirm the coherence of the observed two-component FWM as
implied by our model, we directly map out the time evolution of the output
mode population |b±2 |2 by interrupting the FWM process after a variable timethrough a selective removal of |ai atoms. This is achieved by a 50 µs long“blast” pulse of light resonant with the F = 1 → F 0 = 2 repumping transition.
As shown in Fig. 6.5, the atom number in the |b, ±2i modes smoothly grows
from zero to a maximum value reached around the expected separation time.
The nonlinear, initially quadratic growth is indicative of a coherent process [33,74Nb,±2 (arb. units)

80
60 ×1/2
40
Vb = 020
0Vb = Va

0.0 0.5 1.0 1.5 2.0 2.5
τ (ms)

Figure 6.5: Growth of the population in the output mode |b, ±2i following the
Kapitza–Dirac pulse for Vb = 0 (orange squares) and for Vb = Va (blue triangles,×1/2). The FWM was interrupted after a variable time τ by blasting away the|ai atoms. Each data point is averaged over 2–6 runs (here ωz = 2π × 41 Hz,
fa = 0.5). The dashed lines are the predictions of the uncorrected model(including higher order FWM terms), whereas the solid lines take into accountthe loss of atoms during the blasting process (see text). The blast-loss model
was calibrated by fitting to the Vb = Va data.

35] (other signs would be an overshoot and oscillations, which however would
require higher densities or longer overlap). To exclude the possibility that theobserved growth of the output mode population is merely an artifact caused bydensity-dependent losses of |bi atoms accompanying the blast (due to collisions
with |ai atoms), we repeat the experiment with the polarization of the latticebeam chosen such that both components experience the same lattice depthof about 6 ER . In this case, we expect the |b, ±2i modes to be populated
immediately after the KD pulse, as indeed can be seen in Fig. 6.5. Further,no FWM is expected to occur for Vb = Va , as the internal and external state
dynamics are decoupled. By comparing the observed time evolution for thisreference case with the expected one, we can calibrate our model for the blastinduced losses, which assumes a relative loss of |bi atoms proportional to thedensity of |ai atoms in the overlap region. With this correction, the theoretical

75

30

a

b

Va(t)

τ

Nb,±2 (arb. units)

(A)

F=1
blast

(B)

20
10
0

0.0

0.5

1.0

1.5

τ (ms)

2.0

2.5

Figure 6.6: Four-wave mixing effects for an adiabatically ramped-up opticallattice. (A) An |ai-selective lattice is ramped up to a depth of 6.0 ER within
100 ms onto an equal mixture (fa = 0.5). After release and 17 ms time of flight,|bi atoms appear in |±2i. (B) Growth of population in |b, ±2i as determinedby blasting away the |ai atoms after an evolution time τ . Each data point isaveraged over 6 runs.

time evolution for Vb = 0 matches the experimental data very well.

6.6

FWM after release from a slowly-loaded
lattice

So far, we have discussed controlled FWM after application of a short opticalpulse to induce diffraction. Now, we turn to the question whether FWM is also
relevant in the case of adiabatically ramped-up, state-selective optical lattices.
For such a system, inter-species interactions can be expected to give rise todiffraction effects qualitatively similar to those due to FWM. The density profile of the |ai component gets spatially modulated by the optical lattice, thus
forming an “atomic lattice” that, in turn, should modulate the density of the|bi component. The momentum distribution after release should then exhibit
diffraction peaks at ±2~kL , which thus could be interpreted as a signature ofsuch a density modulation. However, we find that, at least as long as bothcomponents are in the superfluid state, FWM is by far the dominant mechanism for the emergence of recoiling |bi atoms, caused by the projection of the|ai component into plane-wave momentum modes after release. We note thatthe mismatch between the dispersion relations for |ai and |bi atoms suppresses
FWM while the lattice is on. For fa = 0.5 and Va = 6 ER (Vb = 0), as shownin Fig. 6.6 (A), we measure a relative population of up to 1.5 % in each of76

the |b, ±2i states. Assuming this to be caused by a density modulation would
require an atomic lattice modulation depth of 2 ER , more than the chemical
potential of the condensate in the lattice. A blast measurement as discussed
above shows that the population in the observed peaks slowly grows only after
release from the lattice (Fig. 6.6 (B)), indicating that the peaks are indeed
caused by FWM.
We have performed analogous experiments for different final lattice depths
Va , and with additional, state-independent lattices along the x and y directions. In brief, we observe similar FWM effects along the state-dependent axis
(cf. also [77], Fig. 8); however, we find that no FWM peaks are produced whenthe |ai atoms are in the Mott regime. This is consistent with the notion that
FWM as described relies on the existence of a well-defined macroscopic phaseand thus bears the potential to be used as a sensitive probe of phase coherence.

6.7

Conclusion

To summarize, we have demonstrated collinear four-wave mixing in a twocomponent mixture of bosonic atoms, and find excellent agreement with asimple theoretical model. Our work is of relevance both in the context ofquantum atom optics [47], and for experimental studies of bosonic mixturesin optical lattices.

77

Chapter 7
Conclusion
In this thesis, I have described a versatile and simple moving-coil transporter
apparatus for the production and study of magnetically and optically-trapped
Bose-Einstein condensates. Further, I have reported on our first experiments
using hyperfine-state mixtures of ultracold atoms in a state-selective opticallattice potential, which span from the strongly correlated regime to the fieldof nonlinear atom optics.
Using our BEC apparatus, in which the movable quadrupole coils are also
used as an essential part of the final magnetic TOP trap, we have obtained
the first BEC in downstate New York. In experiments with optically-trapped
Bose-Einstein condensates, the quadrupole coils can be retracted before quantum degeneracy is reached, providing large optical access. Our apparatus
has proven to be well-suited for experiments with optical lattices, as we havedemonstrated by observing the superfluid-to-Mott insulator transition [90].
Further, we have observed many-body interaction effects in a homonuclear
bosonic mixture in a three-dimensional optical lattice with variable state dependence along one axis [107]. Near the superfluid-to-Mott insulator transition
for one component, we have found that the presence of a second component can
reduce the apparent superfluid coherence, most significantly when the second
component either experiences a strongly localizing lattice potential or almost
none at all. We have examined this effect by varying the relative populations
and lattice depths, and have discussed the observed behavior in view of recent
proposals for atomic disorder and polaron-induced localization.
We have demonstrated atomic four-wave mixing of two-component matter waves in a collinear geometry with macroscopic mode populations [115].
Starting from a single-species Bose–Einstein condensate, we have prepared
seed and pump modes through microwave state transfer and state-selective
Kapitza–Dirac diffraction. We have observed how four-wave mixing subsequently populates the initially empty output modes, in very good agreement78with simulations based on a coupled-mode expansion of the Gross–Pitaevskii
equation. We have shown that four-wave mixing can play a role in studies ofhomonuclear bosonic mixtures in optical lattices. Moreover, our system of internal and momentum modes coupled via four-wave mixing might have futureapplications in sub-shot noise atom interferometry [47].
In conclusion, the experimental system that we have realized over thecourse of the last few years has allowed us to perform a wide range of experiments involving two-component bosonic mixtures in state-dependent latticepotentials. Beyond the experiments I have described in this thesis, we havebeen able to demonstrate Bragg diffraction of a matter wave from a diluteatomic crystal [128], which can be used as a sensitive probe of the structure
and long-range order of the atomic crystal. Further, we have by now more extensively explored the case of a strongly localized background component [129],
which acts as a fine-grained atomic disorder potential, and have studied how
this type of disorder affects the transport and localization behavior of the foreground atoms, in comparison to the pseudo-random potential created by anincommensurate bichromatic optical lattice.
Apart from the experimental investigation of the phases of the twocomponent Bose-Hubbard model such as the counterflow superfluid [15], natural future areas of study may include disordered systems [25, 26], polaronsystems [23] and phonon-mediated interactions [130], as well as decoherence
mechanisms and quantum dissipative systems [20, 21].

79

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91Appendix A
Dipole Matrix Elements forMulti-level Atoms
In this this appendix we explicitly calculate the dipole matrix elements for a
multi-level atom interacting with a classical light-field.

A.1

Atom-light interaction

The atom-light interaction Hamiltonian assuming a classical light field E isHali = −d̂ · E(r, t)(A.1)

with the dipole operator d̂ = −e r̂. In terms of the polarizability it is1
Hali = − E† α̂ E.
2

(A.2)

Writing the interaction Hamiltonian in the basis of the atomic eigenstates|ii we obtain the matrix elements
Z(A.3)
hj|Hali |ii = e hj|r̂ · E|ii = e d 3 r ψj∗ (r)ψi (r) r · E(r, t),where ψi (r) = hr|ii. Since the spatial variation of the electric field is usually
very small over the size of the atomic wave functions, the electric field can be
regarded as uniform over the size of the atom, and can be pulled out of theintegral. This leads toZ3∗
hj|Hali |ii = e hj|r̂|ii · E = e
d r ψj (r)ψi (r) r · E ≡ e dji · E.
(A.4)

92

(a)

(b)

Figure A.1: The two possible second order perturbation theory diagrams tocalculate the light shift. The dotted line represents the atom, the wavy lines
represent quanta of the radiation field. Time is in the horizontal direction. (a)The atom absorbs light and gets into an intermediate higher internal state, thenreemits the light, going back to the original state. (b) The opposite process isalso possible, i.e. the atom emits light and goes into a lower internal state, andthen reabsorbs a quantum from the radiation field. This process corresponds
to the counter-rotating term that is usually neglected in the rotating wave
approximation.

The here defined dipole matrix elements dji accordingly have the propertyd∗ji = dij ,(A.5)

which simply means that the dipole matrix (actually there are three) is Hermitian.

A.2

Optical Dipole Potential

The optical dipole potential corresponds to the spatially varying light shiftof the atomic energy levels that they experience in an light field of varying
intensity. The light shift for a multi-level atom can be calculated to very good
approximation by treating the atom-light interaction −d · E as a perturbation.
To leading order one obtains∆Ei =X |hl|d · E|ii|2
l6=i

Ei − El=X |dli · E|2
l6=i

Ei − El=

X (E · dil ) (dli · E)
l6=i

Ei − El(A.6)

with dli ≡ hl|d|ii = d∗il . There is no first-order correction, since the dipole
matrix element between same states vanishes due to parity.
It should be noted that the energy of the intermediate state El is not simply
the eigenenergy of the atomic level, but it includes the state of the radiation

93

field, which has lost an energy quantum to the atom in the intermediate state,such that El = El − ~ω. Although much less probable, another possible process is that the atom first emits and then reabsorbs a photon. In this case,illustrated in Fig. A.1(b), the energy of the intermediate state is El = El + ~ω.
This case corresponds to the counter-rotating term usually neglected in therotating-wave approximation. So the denominator in (A.6) becomes


1
1
1
1
1
1
=
+
=−
+
(A.7)
Ei − ElEi − El + ~ω Ei − El − ~ω~ ωl − ω ωl + ω
where ~ωl ≡ El − Ei is the energy difference between the fixed initial state andthe intermediate state l. For a laser wavelength in between the D1 and D2lines of rubidium, the contribution of the “counter-rotating” term is less than
1 % compared to the first term and may be neglected. For 1064 nm light,
however, both terms should be retained, as the second term is already about
15 % of the first.
Here the electric field can be treated as a classical external field, since thelight produced by a laser is essentially in a coherent quantum state. We canwrite the electric field at the position of the atom as
E = Re E(+) e−i ωt =


1 (+) −i ωt
E e+ E(−) ei ωt
2

(A.8)

where the complex vector
E(+) is the “positive” frequency component of the∗field, and E(−) = E(+) [67, 75, 131]. In a quantum treatment E(+) would
be a destruction operator for the electric field at a certain position [67].
Plugging this into the numerator of (A.6) and performing a cycle averageover the fast optical frequency yields
1  (+)
E · dil dli · E(−) + E(−) · dil dli · E(+) .
4(A.9)
These two terms can be identified with the two possible processes mentioned
above, so that we arrive at
"

#


E(+) · dil dli · E(−)
1 X E(−) · dil dli · E(+)+.
h∆Ei iT = −4~ l6=iωl − ω
ωl + ω
h(E · dil ) (dli · E)iT =(A.10)
Note that the numerators are not necessary equal, since total angular momentum has to be conserved. In other words, depending on the magnetic quantum
number of the initial state, one of the two processes might not be possible forcircularly polarized light due to the lack of appropriate intermediate states.
94

Now, the dipole matrix elements dli can be factored into a universal angularpart expressed in terms of Clebsch-Gordon coefficients and a residual reduced
dipole matrix element. For this we need to change into a spherical basis.
√ The
orthonormal spherical basis vectors are defined as ê±1 ≡ ∓(êx ± i êy )/ 2 andPP(+)
ê0 ≡ êz . In this basis we have E(+) = q êq Eq and dli = q êq dliq where theindex q ∈ {−1, 0, +1}. This finally leads to
P li (+) 2
P il (+) 2
q dq Eq
1 X  q dq E qh∆Ei iT = −+(A.11)
.
4~ l6=iωl − ω
ωl + ω
In the following, we will extract the angular dependence of the dipole matrix
elements.

A.3

Factoring out the angular dependence ofthe dipole matrix elements

Since the atomic energy eigenstates are eigenstates of the total angular momentum as well, we can use the Wigner-Eckart theorem [50, 132, 133] to separate
the matrix element into a universal angular part and an atom specific radial
part1
dq,i0 ←i = hα0 F 0 m0 |rq |αF mi = hα0 F 0 krkαF i

hF 1mq|F 0 m0 i√
2F 0 + 1(A.12)

with the Clebsch-Gordan coefficients hF 1mq|F 0 m0 i. These can be expressed interms of the Wigner 3-jm symbol (which make the symmetries more apparent,
i.e. are more handy for calculations)
hF 1mq|F 0 m0 i
1 F F0F 0 −m0√= (−1)(A.13)
q m −m0
2F 0 + 1

1

We use the usual definition for the reduced matrix element as in Refs. [50,√56, 132].
Note that Steck [134] uses a slightly different convention by including the factor 1/ 2F 0 + 1
into the reduced matrix element.

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With F = J + I and r only acting on J, we can reduce the double-bar matrix
element further [132]0hα0 J 0 IF 0 krkαJIF i = hα0 J 0 krkαJi (−1)J +I+F +1
 0

pJ F0 I
0× (2F + 1)(2F + 1)
(A.14)F J 1
where the curly brackets denote the Wigner 6-j symbol. Also note that thenuclear spin I is unaffected by the position operator that only acts on J,
and therefore is constant throughout, just as the electronic spin S. The new
reduced matrix element can be reduced once more due to J = L + S in thesame manner
hα0 L0 SJ 0 krkαLSJi = hα0 L0 krkαLi × 0p
L J0 S
L0 +S+J+1
0(2J + 1)(2J + 1)
. (A.15)× (−1)J L 1
After plugging everything in we finally obtain
0

0

0

0

0 0
L +J+J +F +F −mhα0 L0 J 0 F 0 m0 |r(1)q |αLJF mi = hα L krkαLi (−1)
p(2J 0 + 1)(2J + 1)(2F 0 + 1)(2F + 1)
 0
 0


L J 0 1/2
J F 0 3/2
1 F F0×(A.16)J L 1
F J1
q m −m0where we already assumed the relevant values S = 1/2 and I = 3/2 in thecase of 87 Rb.

96