Quantum Gases in State-Dependent
Optical Potentials

A Thesis PresentedbyRené Harald Reimann
to
The Graduate Schoolin Partial Fulfillment of the Requirements
for the Degree ofMaster of ArtsinPhysics

Stony Brook University
August 2009

Stony Brook University
The Graduate School

René Harald Reimann
We, the thesis committee for the above candidate for the Master of Arts degree,
hereby recommend acceptance of this thesis.

Dominik Schneble – Thesis Advisor
Assistant Professor, Department of Physics and Astronomy
Stony Brook University

Harold Metcalf – Chairperson of Defense
Distinguished Teaching Professor, Department of Physics and Astronomy
Stony Brook University

Thomas Bergeman
Adjunct Professor, Department of Physics and Astronomy
Stony Brook University

This thesis is accepted by the Graduate School.

Lawrence Martin
Dean of the Graduate Schoolii

Abstract of the Thesis

Quantum Gases in State-Dependent Optical
Potentials
byRené Harald Reimann
Master of ArtsinPhysics
Stony Brook University
2009This thesis describes preliminary results for the first experimental
implementation of a state-dependent two-component superfluidMott insulator transition for two inter-convertible atomic species.
The system is realized in a 87 Rb moving coil transporter BoseEinstein condensate machine.
The implementation of a crossed optical dipole trap and optical lattices is described. By experimentally performing a one-component
superfluid-Mott insulator transition the reliability of the BoseEinstein condensate machine is shown.
State-dependent optical potentials and scattering rates are treatedtheoretically as well as experimentally in good agreement with eachother.
Finally the first state-dependent two-component superfluid-Mott
insulator transition is presented and a comparison of the systemwith theory is discussed, as well as issues of interaction between
the two atomic components.

iii

To Christina

Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . .

x

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 Basic Aspects of Bose-Einstein Condensates and Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 BEC Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Non-Interacting Bosons in a Harmonic Trap . . . . . .
2.1.2 Effects of Interaction . . . . . . . . . . . . . . . . . . .
2.2 Time-of-Flight Measurement and Absorptive Imaging . . . . .
2.3 Rabi Pulses, Stern-Gerlach Method and Visibility Measure . .
2.3.1 Rabi Pulses . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Stern-Gerlach Method . . . . . . . . . . . . . . . . . .
2.3.3 Visibility Measurement . . . . . . . . . . . . . . . . . .

4
4
5
6
7
9
9
10
10

3 Experimental Basis . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Vacuum System, Quadrupole Coils and Transportation . . . .
3.3 Production of Bose-Einstein Condensates . . . . . . . . . . . .

13
13
16
17

4 Trapping and Manipulating Quantum Gases with Laser Beams 19
4.1 The Dipole and the Radiation Force . . . . . . . . . . . . . . .
19
4.2 Red-Detuned Crossed Dipole Trap . . . . . . . . . . . . . . . .
22
4.2.1 Theoretical Description of Crossed Optical Dipole Trap
22
4.2.2 Measurement of XODT Trap Frequencies . . . . . . . .
25
4.3 Optical Lattices . . . . . . . . . . . . . . . . . . . . . . . . . .
28
4.3.1 Lattice Envelope and Lattice Ramp . . . . . . . . . . .
31v

4.3.2 Lattice Sites . . . . . . . . . . . . . . . . . . . . . . . .
Kapitza-Dirac Diffraction . . . . . . . . . . . . . . . . . . . . .
State-Dependent Optical Lattice . . . . . . . . . . . . . . . . .
4.5.1 State-Dependent Potential . . . . . . . . . . . . . . . .
4.5.2 State-Dependent Scattering Rate . . . . . . . . . . . .

36
36
40
40
47

5 Optical Lattice Bandstructure and Bose-Hubbard model . .
5.1 Solid State Description . . . . . . . . . . . . . . . . . . . . . .
5.2 The Bose-Hubbard Model . . . . . . . . . . . . . . . . . . . .
5.2.1 The One-Component Bose-Hubbard Model . . . . . . .
5.2.2 The Two-Component Bose-Hubbard Model . . . . . . .

51
51
55
56
63

6 State-dependent two-component Superfluid - Mott insulatortransition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 Observing the phase transition . . . . . . . . . . . . . . . . . .
6.2 Separation of Species . . . . . . . . . . . . . . . . . . . . . . .
6.3 Simulation of misaligned 785 nm beam . . . . . . . . . . . . .
6.4 Comparison with Theory and Outlook . . . . . . . . . . . . .

70
70
74
76
79

7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

A Linewidth of Coherent 899-21 Ring Laser . . . . . . . . . . .

92

B List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . .

96

4.4
4.5

vi

List of Figures
2.1
2.2

Stern-Gerlach pulse . . . . . . . . . . . . . . . . . . . . . . . .
Definition of visibility . . . . . . . . . . . . . . . . . . . . . . .

11
12

3.1
3.2
3.3

87

Rb level scheme . . . . . . . . . . . . . . . . . . . . . . . . .
Laser setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transporter apparatus . . . . . . . . . . . . . . . . . . . . . .

14
15
16

Approximation of dipole potential . . . . . . . . . . . . . . . .
Gaussian dipole potential . . . . . . . . . . . . . . . . . . . . .
Dipole potential with gravitation . . . . . . . . . . . . . . . .
Measurement: XODT trap frequencies . . . . . . . . . . . . .
Setup of crossed dipole trap and optical lattice . . . . . . . . .
Structure of standing wave potential . . . . . . . . . . . . . .
Change of ωenv when ramping up lattice . . . . . . . . . . . .
Measurement: Change in gravitational sag when ramping up
the lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Overview of lattice parameters . . . . . . . . . . . . . . . . . .
4.10 Typical Kapitza-Dirac diffraction pattern . . . . . . . . . . . .
4.11 Hyperfine structure of 87 Rb . . . . . . . . . . . . . . . . . . .
4.12 State-dependent potentials . . . . . . . . . . . . . . . . . . . .
4.13 z-lattice setup . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.14 Measurement: Polarization dependent KD patterns for |F =2, mF = −2i state . . . . . . . . . . . . . . . . . . . . . . . . .
4.15 Measurement: Polarization dependent lattice depth for the |F =1, mF = −1i and the |F = 2, mF = −2i state . . . . . . . . . .
4.16 Measurement: State-dependent scattering rate for |F = 1, mF =−1i atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.17 State-dependent scattering rate for |F = 1, mF = −1i and |F =2, mF = −2i atoms . . . . . . . . . . . . . . . . . . . . . . . .

21
23
24
26
29
31
33
35
37
39
41
42
44

5.1
5.2

53
54

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8

Band structure for a 1D sinusoidal lattice . . . . . . . . . . . .
1D Wannier functions . . . . . . . . . . . . . . . . . . . . . . .

vii

44
45
48
49

5.3
5.4
5.5
5.6

Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . .
Measurement: one-component phase transition . . . . . . . . .
Cartoon of the two-component BHM . . . . . . . . . . . . . .
Two-component BHM phase diagram deep in the MI phase . .

60
62
65
68

6.1
6.2
6.3
6.4

Measurement: State-dependent SF-MI transition . . . . . . . .
Decoherence of coherent superposition state . . . . . . . . . .
Measurement: visibility - effect of |F = 2, mF = −2i state . .
Simulation of misaligned z-beam . . . . . . . . . . . . . . . . .

72
73
75
77

A.1 Simplified setup for the linewidth measurement . . . . . . . .
A.2 Photo of beat signal . . . . . . . . . . . . . . . . . . . . . . .

93
94

viii

List of Tables
4.1
4.2
4.3

Measurement: XODT trap frequencies and decay constants . .
Fit parameters for polarization dependent lattice measurement
Measurement: Scattering Rate . . . . . . . . . . . . . . . . . .

ix

27
46
47

Acknowledgments
Even though my girlfriend Christina might not like to hear these words,
I have to say that my year in Stony Brook passed by pretty fast. I mostly
contribute this to the “cool” atmosphere in the BEC lab, where work is fun.
Becoming serious now I would like to thank my advisor Dominik Schneble who
accepted me with open arms as I arrived in August 2008. Even if he would not
pay me right now for working in his lab, I certainly would note that Dominik
is the best kind of advisor a student can have.
I thank Dominik for his cheerfulness that allows me to write my acknowledgments the way I do. Of course he is a great guy. It has never happened thathe left me alone with a question I asked him (“Dominik, how is it with....”- “GREAT!!! Ja, what’s up?”). He is very often in the lab and motivates
through his advise and action. On the other hand I never felt “negative” pressure from his side which leads to the fact that I love spending my time in the
BEC lab.
The lab atmosphere was completed in a perfect manner by Daniel Pertot,
Bryce Gadway, Bartosz Bogucki and Marty Cohen.
Daniel is very well educated in both, theory and experiment and acts andthinks in a very accurate way. Thank you Daniel for answering many of myquestions and for driving me home in rainy nights.
Bryce is pretty smart, too. His talent in physics is amazing. Furthermore it
appears that he is experimentally immovable. No experimental disaster seems
to be able to make him desperate.
Bryce and Daniel have made a huge experimental contribution to the work
presented in this thesis.
Unfortunately I never had a closer interaction with Bart, but he certainly isa very nice guy and can build electronic devices that do not even have thepossibility of being, in my mind.
I am also grateful to Marty who often helped when specific experimental questions arose, and proofread this thesis with Bryce and Daniel. Thank you allfor your helpful comments and corrections.
Harold Metcalf, Thomas Bergeman and Dominik are in my thesis committee

and gave me very constructive feedback - thank you very much for that.
Of course I do not want to miss the opportunity to express my gratefulness
to Fakher Assaad, Laszlo Mihaly and Peter Koch who made the exchange between Würzburg and Stony Brook possible. Furthermore I would like to thank
Harold Metcalf who was there at the very beginning to answer my questions
and helped me out by lending me a bike.
With switching from work to weekend, I would like to mention my close friend
Ali. Having dinner with you on Saturdays and discussing cultural, political,
philosophical and personal issues with you enlarged my horizon immensely.
People like you are rare.
Finally I want to thank my girlfriend Christina. Even though you sometimes
might have the feeling that physics is more important to me than you, I certainly know that this is not true. Thank you for staying by my side for thisyear. I really look forward to have you again. This thesis is dedicated to you.
I acknowledge support of DAAD, the German government and the Research
Foundation of the State University of New York.

Chapter 1
Introduction
The theoretical foundation for Bose-Einstein condensation (BEC) was formulated in 1924 (for a theoretical review on BEC in dilute trapped gases see[1]). In short, condensation happens at low enough temperatures when bosons
macroscopically occupy the energetic ground state of the system. For theachievement of BEC in dilute atomic gases (for an experimental review see[2]) the invention of lasers [3] in 1958 and laser cooling in the 1980s [4] werenecessary. After a rather short time the first condensates with alkali metals were achieved in 1995 [5–7]. To date many interesting experiments with
BECs have been performed and studied, for instance, interference between two
BECs, collective excitations, atom lasers or vortices in BECs (for reviews see[1, 2, 8, 9]).
Here we will concentrate on the scenario of ultracold atoms in an opticallattice. When two laser beams of the same wavelength counter-propagate,
they form a 1D optical lattice due to interference. Using the freedom of polarization and more laser beams optical lattices of higher dimensionality andcomplexity can be formed [10]. Atoms interact with these lattices via thedipole force [11]. For now we restrict ourselves to far detuned optical lattices
where scattering can be neglected and the internal structure of the atoms is
not important. When studying cold atoms in an optical lattice, one has todistinguish between laser cooled atoms (T ∼ 100 µK, n ∼ 1011 cm−3 ) or a
BEC (T ∼ 100 nK, n ∼ 1014 cm−3 ) as starting point. The former scenario
is in the weakly-interacting regime because the density of the atoms in the“light crystal” is so low that the atoms practically never see each other [8].
The latter case has the potential to explore the strongly-interacting regime
where the filling factor (the number of atoms per lattice site) is on the orderof one and atom-atom interactions are important. Both cases, however, havein common that the systems are well described by basic solid state theory,
which, however, cannot always be probed in a solid. The main advantages of1

a “light crystal” over an actual one, are the missing defects of the “light crystal” and the possibility to easily vary the lattice properties by changing theintensities, polarizations and wavelengths of the lattice light nearly without
any experimental restrictions. Furthermore, lattice experiments with different
dimensionality can be performed and the lattice can be turned on and off arbitrarily. These facts enable the system to probe solid state theories in regions
where no solid exists.
In the 1990s the weakly-interacting regime has been studied a lot. At the
beginning of the decade the localization of laser cooled atoms to less than an
optical wavelength in a 3D lattice was reported [12]. A few years later thequantized motion of cold atoms [13] (in three dimensions), [14] (in two dimensions), [15] (in two and three dimensions) was observed. The first experimental
connections between typical solid state scenarios and atom-optical lattice systems where found by the groups of C. Salomon [16] and M. G. Raizen [17, 18].
The famous Bloch oscillations and Wannier-Stark ladders, both predicted forsolid state systems, were observed in an artificial light crystal. The static electric field that accelerates the electrons in the solid state case was modeled byaccelerating the optical lattice via detuning the frequencies of the two beams.
The advantage of the new atom-light system indeed becomes clear in the caseof Bloch oscillations. These oscillations have already been observed in 1992
in semiconductor superlattices [19]. In this case the oscillations were damped
because the typical timescale of the oscillations was on the same order of magnitude as relaxation processes (scattering from lattice defects, phonons,...).
The damping in the atom-light experiments, however can be adjusted by the
detuning of the lattice lasers1 [20] and Feshbach resonances [21]. Therefore
new regimes can be entered.
In 2001 the group of M. A. Kasevich explored the strongly-correlated regime
where atom-atom interactions become important, in a one dimensional experiment [22]. One year later the group of T. W. Hänsch opened up a new branchof BEC experiments [23]. Using a 3D optical lattice for 87 Rb atoms they
were able to leave the weakly-interacting BEC regime and entered a strongly
correlated regime that is known as the Bose-Hubbard regime [24]. A phasetransition from the superfluid to the Mott insulating phase was observed. In
the former phase the atoms can move freely from one lattice site to the next
and a global phase is established (like in the BEC case without lattice) while
the latter phase is characterized by a fixed atom number at each site and no
global phase coherence. Since then quantum transitions in different dimensions
and dynamics of the boson-lattice systems have been studied. Furthermore,
1

Far-detuned from atomic resonance → no damping. Detuning close to resonance →strong damping.

2

fermions have been introduced (for a review see [8]). In 2003 T. W. Hänsch’sgroup used the idea of spin-selective potentials [25, 26] to transport and collide
atoms of different hyperfine states [27, 28] in a controlled way. A spin-selective
or state-dependent potential could, for instance, be chosen in a way such that
one hyperfine state of an atom conservatively interacts with the light that
forms the potential but another hyperfine state of the same atom does not atall.
In this thesis experiments which combine the SF-MI approach with spinselective potentials to a state-dependent 3D superfluid-Mott insulator (SF-MI)
transition are discussed.
The thesis is divided into five parts. First a general BEC and optical lattice
toolbox is introduced. An explanation of experimentally useful techniques liketime-of-flight imaging or visibility measurements follows a short theoretical
BEC discussion. The general tools provided in this chapter will be used during the entire thesis.
In the third chapter we partially recall the work that has been done before in
our group as an experimental basis. The machine with its moving coil transporter is discussed as well as the laser and vacuum systems. Finally the BEC
production is reviewed.
The fourth chapter starts with a theoretical description of the interaction between atoms and light. This description is used to introduce the crossed optical
dipole trap (XODT). A measurement of the XODT frequencies follows. In addition optical lattices and state-dependent potentials and scattering rates arecovered, both theoretically and experimentally. A reliable method for lattice
depth calibration via the Kapitza-Dirac effect is established.
In the fifth chapter the well-known one-component SF-MI transition is repeated. The Bose-Hubbard model as well as a useful solid state description of
the atom-light system is given.
Finally the first experimentally achieved state-dependent two-component SFMI transition for two inter-convertible species is shown. Comparisons withtheory and experimental challenges are discussed and analyzed.

3Chapter 2
Basic Aspects of Bose-Einstein
Condensates and Experimental
Methods
In this chapter we introduce general theoretical descriptions which will
provide us with experimentally useful statements and tools. This chapter iswritten to give a fundamental basis which is needed in many parts of the thesisor to explain general concepts of the BEC theory or related topics. Discussions
which are more specific to this work will appear in later chapters.

2.1

BEC Theory

In a simple model one can think of bosonic atoms as point particles with amass m but no further intrinsic properties. In a quantum mechanical formalism
these atoms can be described as “wave packets” with a typical
p extension scalethat is given by their thermal de Broglie wavelength λT = (2π~2 )/(mkB T )with T being the temperature and kB being Boltzmann’s constant. A quantum
mechanical description becomes inevitable when this wavelength extends toscales which are comparable to intrinsic scales of the considered system. Herethis intrinsic scale is the inter particle spacing d. Assuming that each particle
has a small cubic box with volume V = d3 it can call its own, the particle
density n = N/V connects to particle spacing by nd3 = 1. In our simple
picture Bose-Einstein condensation appears when the atomic “wave packets”
start to overlap which means λT ∼ d. Therefore condensation starts to occur
fornλ3Tc ∼ 1
4

(2.1)

where every particle starts to be at “any place” in the system because of
the fundamental principle of indistinguishability in quantum mechanics. Tc
is called the critical temperature for which the transition starts and a macroscopic number of particles occupy the ground state. For T=0 condensationis complete and all particles are in the lowest energetic state. The system issometimes said to be a “giant matter wave” [2] because it can be describedby a macroscopic wave function. The quantity nλ3T of eq. 2.1 is called thephase space density. Ref. [2] gives a very nice introduction to BECs from anexperimental point of view. Performing a proper statistical calculation for ahomogeneous bosonic gas in 3D [29], one gets nλ3Tc = ζ3/2 [1] ≈ 2.6, where ζn [x]is the Riemann zeta function.

2.1.1

Non-Interacting Bosons in a Harmonic Trap

In all the experiments we will discuss later, the external trap confinement
for the atoms can be approximated harmonically which gives a general potential of the formm X 22
ω i with
2 i=x,y,z iX~ωi (ni + 1/2)
eigenenergies nx ny nz =Vext [r] =(2.2)

i=x,y,zThe eigenenergies belong to the single particle Hamiltonian of the non-interacting
system and the ni are non negative integers. The quantum-statistical calculation starts with the Bose-Einstein distribution which gives the average particle
number in the quantum state k with energy k
f [k ] =1with β = (kB T )−1
exp[β(k − µ)] − 1(2.3)

From eq. 2.2 and eq. 2.3 all needed relations can be derived with the help of
the formalism of statistical mechanics (cf. [1, 29]) in the frame of the grandcanonical ensemble. Doing so one can derive the critical temperature1 Tc atwhich the phase transition to the BEC happens, by setting µ = 000 .


~ω̃
N
Tc =
with ω̃ = (ωx ωy ωz )1/3
(2.4)
kB ζ3 [1]1

In a lattice, the critical temperature is reduced compared to the described case. This
happens due to an increase of the effective mass [8].

5As stated above, only at T = 0 are all N atoms in the condensate. For
0 < T < Tc some of the bosons are not in the ground state and contribute toa thermal background. The condensate fraction is then given by
 3
T
N0
=1−
(2.5)
N
Tc
where N0 gives the number of atoms being in the ground state. It is impressivethat the BEC phase transition happens only because of quantum statistics andnot because of interactions between the particles.

2.1.2

Effects of Interaction

Even though the non-interacting gas model gives a qualitatively correct description details like the shape of the condensate cannot be explained. Therefore interactions have to be introduced.
Gross-Pitaevskii Equation
In 1947 Bogoliubov treated the superfluid problem with a mean field approach [1]. Without going into details, for N0  1 he wrote the bosonic field
operator as sum of a complex function Ψ[r, t] and a small perturbative field
operator. Ψ[r, t] is called the condensate wavefunction and relates to the condensate number density by n0 [r, t] = |Ψ[r, t]|2 .
Starting with this idea Gross and Piteavskii derived in 1961 their well-known
equation which reads in the time independent form~2
2
∆ + Vext [r] + g|Ψ[r]| Ψ[r] = µΨ[r]
(2.6)
−
2m
A derivation can be found in [30]. The interaction potential for eq. 2.6 has
been approximated by a pseudo contact potential which is characterized by
4π~2 as(2.7)
mThis can be done because of the diluteness (d  as ) of the gas which leads toa simplification of the complex many body potential. Interactions are basedon elastic two particle collisions which are described by the s-wave scattering
length as and a mean-field energy. as will be positive in all discussed cases
herein. The interaction potential is therefore repulsive. Eq. 2.6 is only strictly
valid for T=0 when all the atoms are in the condensate.
g=6

Thomas-Fermi Approximation
If one neglects the kinetic term in eq. 2.6, a simple equation for the density
profile is achieved
n0 [r] = (µ − Vext [r])/g(2.8)

for Vext [r] < µ and n0 = 0 elsewhere. This approximation is referred to theThomas-Fermi approximation and is valid forp
the limit Eint /Ekin ∼ N0 as /aho 
1 with the typical trap length scale aho = ~/(mω̃). For our experimental
situation this translates to 10−3 N0  1 which is fulfilled in all of the experiments that will be discussed. The density profile of the cloud in eq. 2.8 is aninverted paraboloid. The approximation is very good except for the outer shellof the condensate were the quantum pressure (∆Ψ[r]-term) is not completely
insignificant. In discussing the size of a condensate, the so-called ThomasFermi radius is often considered. This radius measures from the center of thecondensate to the point where the density in eq. 2.8 becomes 0. For a spherical
trapping potential, it is [1]
RTF = aho

15N0 as
aho

1/5
(2.9)

This concludes the short theoretical BEC description. Refs. [1, 2, 8, 30]
provide a good introduction on a much broader basis than given here.

2.2

Time-of-Flight Measurement and Absorptive Imaging

With a BEC confined in a harmonic trap, one can generally image thetrapped cloud (in-situ), or switch the trap off and image the atoms after a certain time-of-flight (TOF). The TOF method has the advantage that it reveals
the momentum distribution of the BEC after the ballistic expansion. This happens due to the quantum pressure of the BEC in the trap (see eq. 2.6). For
an anisotropic trap this leads to a bigger momentum along the more confined
directions and therefore to an anisotropic expansion of the BEC cloud with
TOF [31]. Assuming thermal equilibrium between the thermal backgroundand the BEC the temperature of the system can be calculated by fitting the
BEC with a Thomas-Fermi parabola and the thermal background with a thermal Gaussian [2].
Ref. [2] also discusses different techniques for imaging the atom cloud. In
our case, we exclusively use absorptive imaging [32, 33]. The principle of this7

technique is very simple. The atoms absorb photons of a laser beam that
illuminates them for imaging. The emission of the absorbed photons follows
also spontaneously in a random direction. This means that the atom cloud
can be seen as a shadow in the imaging beam which shines on a CCD chip ofa camera. To cancel inhomogeneous illumination and background noise of thedata, three pictures are taken. One with atoms and imaging light which leads
to an intensity Iatoms [x, y] in the xy-plane of the chip. One without atoms butwith imaging light (Iref [x, y]) and one without light (Ibkg [x, y]). The laser beam
propagates along the z-direction where x,y and z from a orthogonal coordinate
system. The theoretical intensity on the chip with beam and atoms is givenbyI[x, y] = I0 [x, y] exp[−OD[x, y]] with OD[x, y] = σ̃ · n2D [x, y](2.10)

I0 [x, y] is the intensity without atoms but only with the imaging beam. The
scattering cross section σ̃ depends on the atomic resonance frequency,
the lightZ∞

frequency (and linewidth) and on the intensity2 [34]. n2D [x, y] =n[x, y, z]dz−∞

is the two dimensional column density, where n[x, y, z] is the real atomic density. OD is called optical density and is experimentally given by
Iatoms [x, y] − Ibkg [x, y](2.11)
OD[x, y] = − ln
Iref [x, y] − Ibkg [x, y]
The number of atoms imaged is now
Z ∞Z ∞
OD[x, y]dxdy
N = const−∞(2.12)−∞

with const being a constant (if effects of saturation can be neglected) which
depends on σ̃ and the length calibration of the camera [33].
In conclusion the integrated optical density OD gives a measure for the imaged
atom number. Our only way in extracting information from the system isprocessing the optical density. Since the used imaging method is destructive
a new condensate has to be prepared for each image.

2

If the laser power stays well below the saturation intensity, the intensity dependence
can be neglected.

8

2.3

Rabi Pulses, Stern-Gerlach Method and
Visibility Measure

The tools provided in this section are a little more specific to our experiments. The transfer of atoms in one atomic state to another one is treated.
Then the question arises of how to distinguish these two states on the recorded
image after TOF. Finally the visibility measurement is discussed which is a
specific technique for lattice experiments.

2.3.1

Rabi Pulses

Experimentally we are at the point where we want to have two atomic
Rb species in our BEC. Therefore we want to transfer a certain fractionof the species (= hyperfine state) we start with (|1i), to another atomic state
which is then labeled as the second species (|2i). This can be done by applying
a microwave Rabi pulse of certain length and strength to the atoms of the first
species.
The theory behind this problem was introduced by Rabi [35] and is included
in many textbooks (e.g. [4]). Having a driving electro-magnetic field withfrequency ωl which has a much narrower linewidth than the atomic transition(which is characterized by ωt ) that it drives, it is possible to consider thesystem as a two level one. This can be done as long as the detuning ∆ =ωl − ωt is small enough that coupling to other existing atomic levels does not
become important. The Hamiltonian of the system can then be truncated.
After some algebra and approximations, two coupled differential equations
remain [4]. Solving these equations leads to the time dependent populations
P1 (population of 1st species) and P2 (population of 2nd species)
 0 
 2
Ω
2 Ω
sin
t and P1 = 1 − P2
(2.13)
P2 =
Ω0
2
87

Ω = E0 h1|er|2i/~ is called the Rabi frequency, where E0 is the electric field
amplitude of the driving field. The dipole matrix element that is an importantpart of the Rabi frequency
is taken between the atomic state of the 1st and√022
2nd species. Ω = Ω + ∆ is called the generalized Rabi frequency. eq. 2.13
shows that by applying a driving pulse which is close enough to the frequency ofthe desired transition to stay within the limits of the approximations used, one
can get arbitrary mixtures between the species. The pulse just has to be turned
off at the right time. It has to be pointed out that directly after applying

9the pulse, each of the atoms is in a coherent superposition3 between |1i and|2i. Imaging the atoms however projects the states out and a population
distribution in agreement with eq. 2.13 can be measured.
This technique can now be applied to transfer atoms from one to anotheratomic state. However it can happen that the transition frequency between
the states depends crucially on magnetic fields (Zeeman effect). A changeof the magnetic field or unwanted gradients therefore lead to experimental
issues. A solution to this is to apply a so called Landau-Zener sweep [36–38].
This technique is much less sensitive to small changes in the magnetic field or
inhomogeneities over the extension of the atom cloud because the frequency ofthe driving field is swept over the resonance with a certain constant speed. By
varying the sweep speed and the power of the driving frequency, the populationtransfer can be adjusted.

2.3.2

Stern-Gerlach Method

The Zeeman effect can now be used to separate atoms in atomic stateswith different magnetic moments. For weak4 magnetic fields, the energy offset
of an atom in a state with the magnetic quantum number mF and the g-factorgF (values for 87 Rb see Fig 3.1) is [39]E = mF gF µB B0(2.14)

where µB is the Bohr magneton and B0 the amplitude of the magnetic field.
Applying an inhomogeneous magnetic field, atoms with different magnetic moments will separate. This is the idea of a Stern-Gerlach pulse where a magnetic
gradient field is pulsed on for a short time to have separated species when theimage is taken after a certain TOF [40, 41]. Fig. 2.1 shows the separation of
the species after applying a magnetic gradient pulse. Note that the magnetic
moment of |F = 2, mF = −2i is twice as big as the one of |F = 1, mF = −1i.
This can be seen by the seperation in Fig. 2.1.

2.3.3

Visibility Measurement

In the context of quantum gases in optical lattices, interference effects will
be measured (see Fig. 5.4). In order to quantify the degree of coherence of thesystem that leads to the interference, a measure has to be defined. Fig. 2.2
shows two absorbtion images which illustrates our definition of visibility. The
3

For an integer π-pulse pure states are obtained.
Weak means that the splitting of the states due to the field is small compared to thefine structure spitting.
410

|F=1,mF=-1〉g

B‘0|F=2,mF=-2〉

Figure 2.1: Stern-Gerlach pulse applied to 87 Rb atoms in the ground states|F = 1, mF = −1i and |F = 2, mF = −2i (compare to Fig. 3.1). The images
are shown in a stroboscopic manner from TOF= 0 ms (left) to TOF= 14 ms(right). The direction of the strongest B-field gradient is indicated. The lengthof the Stern-Gerlach pulse was 4.5 ms.
left part of the figure shows a coherent scenario. Most of the atoms are in theinterference peaks. In this case the visibility is close to 1 because the peaks
can be seen and distinguished perfectly. The right part shows a case where
the interference peaks cannot be seen in a very clear way. Here the visibility
is closer to zero than to one.
For the definition of the visibility V, we need the optical density OD from eq.
˜ peak as the integrated OD that is enclosed by the four green
2.11. We define OD
˜ bkg as the one that is enclosed by the four redsolid circles in Fig. 2.2 and OD
dotted circles. All circles have the same area.
The visibility is defined as [42]V=˜ peak − OD˜ bkg
OD
˜ peak + OD
˜ bkg
OD

(2.15)

This definition fulfills the requirements given above and is a measure of coherence of the system5 . The radii of the circles should be chosen such that
all interference peaks of the most coherent image are a little bigger than thecircles themselves (in the figure, the circles are therefore drawn to big). Of
course once chosen, the radii remain constant.

5

More about this in 5.2.

11

Figure 2.2: Definition of visibility. The left side shows a coherent case, while
the right side shows a image that is less coherent and has smeared out interference peaks. The ellipse that connects the strongest peaks as well as thedotted gray lines are guides to the eye. The green solid circles are centered on
the peaks while the red dotted ones lie on the ellipse and the diagonals.

12Chapter 3
Experimental BasisThis chapter partially summarizes the work that has been done in earlier
times in our group but is important as a basis for this thesis. Therefore
the fundamental setup of the machine and how it produces a BEC, will be
discussed. Detailed information can be found in [32, 33, 43, 44].

3.1

Laser System

Fig. 3.1 shows the 87 Rb energy level diagram with the relevant cooling andimaging structure.
The cycling transition, which is used for cooling, is operated by a DL pro 780
that is amplified by a BoosTA (see Fig. 3.2). The DL pro 780 and the DL
1001 , that is used for repumping are both external cavity diode lasers from
Toptica. The BoosTA is a tapered amplifier that amplifies 65 mW from the
DL pro to 700 mW.
For laser cooling generally, a closed transition is needed which means thatthe atoms cycle between the two chosen levels (here F=2 and F=3) but do
not escape to other levels. However there is a small probability that an atom
escapes and ends up in the F=1 ground state. To get these atoms back into
the cooling cycle, the repump beam is used. For a detailed discussion of laser
cooling see [4]. The purpose of the depump beam is to transfer atoms fromthe F=2 to the F=1 ground state. For absorptive imaging a closed transition
is preferred, too. Therefore the cycling light is used for imaging (∼ 100 µs)
while the repumper pumps F=1 atoms to the F=2 manifold at the same time.
Both diode lasers are frequency stabilized by a PID loop that uses a signal
from 87 Rb polarization saturation spectroscopy as reference [33, 46]. The
desired frequency is generated with acousto-optic modulators (AOM [47]).
1

not shown in Fig. 3.2 but in [33]13F = 3 gF = 2/3
266.6500(9) MHz

5 2P3/2

72.9112(32) MHz
156.947(7) MHz
72.218(4) MHz

Cycling

5 2P1/2
794.978 851 156(23) nm
377.107 463 380(11)THz
1.559 591 016(38) eVD1

780.241 209 686(13) nm
384.230 484 468 5(62)THz
1.589 049 462(38) eVD2

F = 2 gF = 2/3

F = 1 gF = 2/3
F=0RepumpDepumpF=2

gF = 1/2

2.563 006 GHz

5 2S1/2
4.271 677 GHz

F = 1 gF = – 1/2

Figure 3.1: 87 Rb level scheme with experimentally relevant fine and hyperfine
splitting. The data are taken from [45].
These devices are also used for modulating the intensity of the beams andas shutters (see Fig. 4.5).
Fig. 3.2 furthermore shows the laser system that is used for creating the statedependent optical lattice along the z-direction (see Fig. 4.13). This system
consists of a 899-21 Titanium:Sapphire Ring Laser which is pumped by a Verdi
V-10. The Verdi V-10 is a diode-pumped, frequency-doubled solid state laser.
The wavelength of the ring laser can be adjusted between 700 - 825 nm [48].
The properties of this system are discussed in [49]. Appendix A shows that the
linewidth of the ring laser, at the used wavelength of 785 nm, is good enoughfor a lattice experiment. For this system, the use of an AOM as an intensity
modulator can be seen in Fig 3.2.
The same principle is used to control the intensity of the crossed optical dipole
trap and the state independent 1064 nm lattice (see 4.2 and 4.3). The 1064
laser light is produced by an IPG fiber laser (IPG YRL LP-SF series) withmaximum output of 20 W and a linewidth of 50 kHz (not shown in Fig. 3.2).

14

diagnostic

AOM

Verdi V10

to z-lattice

Ring 899
Faraday Isolator:

Mirror:

PBS:

Photo diode:Prism:

Lens:

Pickoff:

Optical Fiber:AOM:λ/4, λ/2 -plate:diagnositc

AOM

Shutter:

250 mW

cycling MOT

depumpAOMAOM

F=2 imaging,

AOM

DL Pro 780

to spectroscopy

AOM

BOOSTA

Figure 3.2: Setup of new cycling and lattice laser system. On the bottom of
the figure the laser system for cycling and depumping is shown. The system
is isolated by a box (gray in the picture). On the top of the figure, the laser
system for the state-dependent lattice is shown.

15

3.2

Vacuum System, Quadrupole Coils and Transportation
to pumps & Rb oven
UHV
chamber

vapor
cell

to
pumps
a

b

science cell
& bias coils
c

z
y
x

Figure 3.3: Transporter apparatus [44]. The figure shows the transporter
apparatus and its path (red arrows) from the vapor to the science cell. The
quadrupole coils are shown in the MOT (magneto-optical trap) position. Moving these coils, the cooled atom cloud can be transfered to the science cell.
Around the science cell the bias coils can be seen. More coils (not shown here)
exist that can be used to vary magnetic field amplitudes and gradients along
different directions. The lab coordinate system is shown. Figure included withkind permission of D. Pertot.
The vacuum system is divided into a low and a high pressure region. Fig.
3.3 shows both regions.
In the vapor cell or MOT region the Rb background pressure is ∼ 10−9 torr
which is necessary for loading the MOT. The atoms get into the vapor cellfrom a rubidium oven that can be opened and closed manually (not shown infigure). In this high pressure region, the atoms are confined by the magnetic
potential (physical principle: eq. 2.14) of quadrupole coils in anti-Helmholtz
configuration. About 1010 atoms are cooled by a standard 87 Rb MOT. The
principle of laser cooling is based on a dissipative interaction between light16

and atoms which means that photons scatter (see 4.1). This causes a net
momentum transfer to the atoms which cools them down to, in our case, 25µK. In the end the atoms are optically pumped into the |F = 1, mF = −1iground state and the field gradient along the z-direction is increased to stiffen
the trap prior to transport.
The atom cloud is then quickly moved to the ultra-high vacuum region in thescience cell. This is done by the quadrupole coils which are mounted to themechanical transporter. The pressure in the science cell is < 10−11 torr andachieved by an ion pump combined with a Ti sublimation pump. The twopressure regions are connected by a differential pumping tube. The transport
from the MOT region to the science cell happens in 3 s. It must be this fast
because the high pressure in the MOT region strongly decreases the lifetimeof the atom cloud. During the transport, the cloud is compressed by a further
increase of the magnetic field gradient along the z-direction to 350 G/cm.

3.3

Production of Bose-Einstein Condensates

The stiff linear potential of the quadrupole trap only contains atoms in thehyperfine ground state |F = 1, mF = −1i. Atoms with a non-negative magnetic quantum number are not trapped. The atoms are now cooled further
by applying a 14 s long radio frequency (RF) ramp, linearly sweeping fromhigher to lower frequencies. The principle here is called evaporative cooling:
the atoms with high temperatures can travel to regions of high magnetic fields(see eq. 2.14). In these regions the RF drives transitions to magnetically nontrapped states and the atoms with high thermal energies are removed from thetrap. Since the RF sweep is slow enough to allow for thermalization (elastic
two body collisions) the cloud becomes colder. At cold temperatures (here 75µK) a problem occurs [50]. The atoms amplified (compared to higher temperatures) undergo Majorana losses which means that their magnetic quantum
number flips near the center of the trap where the magnetic field is zero. This
problem is solved by introducing homogeneous bias field coils (see Fig. 3.3).
These coils produce a rotating bias field in the xy-plane which is added to thequadrupole field of the moving coils. The result is that the atoms see a timeaveraged harmonic potential without a field zero. This kind of trap is called atime-averaged orbiting potential trap (TOP trap). Therefore Majorana losses
do not appear. With this configuration, we can go on with RF evaporation
untill the cloud has a temperature of ∼ 250 nK.
At this temperature we transfer the atoms into the crossed optical dipole trap
(XODT) (see Fig. 4.5) by smoothly ramping up the XODT potential andthen smoothly ramping down the magnetic confinement which together takes17∼ 1s. Then the XODT potential is lowered to force evaporative cooling until
a nearly pure 87 Rb BEC with ∼ 105 atoms in the |F = 1, mF = −1i groundstate is produced. Whenever we refer to the XODT in the following text, wemean this all-optical potential which contains the BEC.
One advantage of our machine is the possibility to move the quadrupole coils
away from the science cell after loading the XODT. This guarantees good optical access. The production time for on BEC is about one minute. Bothreproducibility in atom number and in position are discussed in detail in [44].

18Chapter 4
Trapping and Manipulating
Quantum Gases with Laser
Beams
Laser light exerts two fundamental forces on atoms. One has a dispersive
nature and is called the optical dipole force which can form a conservative trap
for the atoms. The second part of the atom-light interaction is of absorptive
nature and is dissipative. This radiation force leads to a scattering of photons.
For trapping quantum gases the dipole force is widely used, whereas the radiation force is to be avoided, since it causes heating and minimizes the lifetimeof the trapped atoms.
The use of the dipole force for trapping atoms in a 1-D standing light wave
was suggested by Letokhov in 1968 [51]. In 1986 Chu et al. [52] reported
the first experimental success in optically trapping atoms, where the sodium
atoms were not confined in a standing light wave but by a strongly focused
Gaussian ”one-way” beam.
This chapter introduces the theoretical basis for understanding the radiationand dipole force. In addition we describe our optical trap, optical lattices ingeneral, and state-dependent optical lattices in particular. This discussions
forms the main part of the chapter.

4.1

The Dipole and the Radiation ForceThe semi-classical Lorentz model [53] describes the electron as an oscillator
in the electric light field. This light field induces an electric dipole moment at
the atom19

p = αE(4.1)

where E is the real and time dependent electric field at the place of the atomand α is the complex polarizability. The induced dipole moment leads directly
to the dipole potential
1
1
Udipole = − hRe(p)Ei = −Re(α)I
(4.2)
2
20 c
The brackets in eq. 4.2 indicate the time average over the fast oscillating
electric field. Furthermore the relation between the electric field amplitude
and the intensity I = 21 0 cE0 2 has been used. From the equation one can seethat the only spatial part that could be used for trapping the atoms inside
Udipole is induced by the intensity. Therefore the gradient of the light intensity
is directly connected to the attracting or repelling force that acts on the atoms.
To gain deeper insight into the dipole potential, the polarizability α needs tobe calculated. This can be done classically by solving [34] the damped equationof motion of the vibrating electron that is driven by the light field:
α = 6π0 c3

Γt /ωt 2
ωt 2 − ωl 2 − iΓt ωl 3 /ωt 2(4.3)

with ωl being the frequency of the light and ωt as the transition frequency of
the atom that is considered. eq. 4.3 holds for large detuning (no saturation).
To obtain a proper quantum mechanical description, the damping rate (=spontaneous decay rate) has to be described as [11]
Γt =

ωt 3
2
3 |hnt |er|mi|3π0 ~c(4.4)

where the dipole matrix element is taken between the ground state |mi andthe excited state |nt i with respect to the specific atomic transition. For not
too small detunings eq. 4.2 and 4.3 lead in a very good approximation to


3πc2
Γt
Γt
Udipole,t =
−
I
(4.5)
2ωt 3 ωl − ωt ωl + ωt
Fig. 4.1 shows that the error introduced by using the approximation that
leads to eq. 4.5 is insignificant, as long as the laser is not operated very close
to an atomic resonance. At resonance, or for too high intensities, saturation
occurs which makes the above equations invalid. These cases however do not
eventuate for our experiments.
The polarizability also relates to the scattering rate that gives the number ofscattered photons per unit time. An expression for this rate can be derived by
20

( U - Uexact ) / Uexact

0.001
10- 5
10- 7
10- 9
10 - 11
10- 13

775

U

780

785

λ [nm]

790

795

800

−U

exact
with respect to the laser wavelength λ. Uexact
Figure 4.1: Plot of dipole,tUexact87
is the potential for Rb that can be calculated from eq. 4.2 and 4.3 without
any approximation. The solid curve corresponds to the D2 , the dashed curveto the D1 transition. The hyperfine splitting is not taken into account (2x twolevel atom). Data for calculation are taken from [45].

dividing the power the dipole absorbs from the light field (and emits again)by the energy of the corresponding photon of the light field [11]ωl Im(α)I/(0 c)
1P==
Im(α)I
(4.6)
~ωl
~ωl
0 c~
Now again eq. 4.3 is used for the imaginary part of α. Performing a similar
approximation as used for eq. 4.52
3πc2 ωl 3ΓtΓt
Γscatt,t =
−
I
(4.7)
2~ ωt 6 ωl − ωt ωl + ωtΓscatt =is achieved. The area in which Γscatt,t is valid follows the same restrictions as
those for the dipole potential. The force that is connected to the scattering
rate is caused by the photon recoil momentum ~k of the laser light field.
A deeper discussion on both the dispersive and the absorptive force can befound in [11]. The dipole potential can also be derived using the dressed atom
approach which gives an intuitive understanding due to energy shifts of theeigenstates of the atom-light Hamiltonian (dressed state picture)[54].

21

4.2

Red-Detuned Crossed Dipole Trap

To get a qualitative impression that often is important for understanding
experiments, eq. 4.5 and 4.7 can be considered for a simple 2-state atom.
Furthermore the so called rotating wave approximation (RWA) that sets ωl /ωt
to 1 (only for the fraction in eq. 4.7) and neglects the terms containing ωl + ωt
can be applied. This approximation is valid for rather small detunings but
even for larger detunings - in our case resonances for 87 Rb at about 795 and780 nm, laser at 1064 nm - the discrepancy is only on the order of a few
percent. eq. 4.5 and 4.7 become
3πc2
Udipole,RWA =
2ωt 3



Γt
∆



3πc2
I and Γscatt,RWA =2~ωt 3



Γt
∆

2
I

(4.8)

with ∆ = ωl −ωt . Equ 4.8 shows two important physical features of dipole traps
and optical lattices. First the dipole potential is attractive for red-detuning(∆ < 0) and repulsive for blue one (∆ > 0). Second large detuning is needed
to minimize the scattering rate which destroys the coherence. But since ∆goes linearly into the potential and quadratically into the scattering rate, theformer one is less affected by a growing detuning than the latter. Therefore
high intensities in combination with large detunings can be used to minimize
the scattering rate without significantly affecting the potential.
Eq. 4.8 suggests trapping the ultra cold atoms in the intensity maximumof a far red detuned laser beam. For the sake of completeness it should be
mentioned that blue-detuned traps can be used, too. Here the atoms aretrapped in the dark by a repulsive force. The advantage of the blue-detuned
dipole trap is the further reduction of the scattering rate. For a review on blue
dipole traps, see for example [55].

4.2.1

Theoretical Description of Crossed Optical Dipole
Trap

In our case however a red detuned trap is implemented. It will turn outthat a TOP trap as described in 3.3 is not usable for our experiments because
its potential strongly depends on the hyperfine state of the atoms (see eq.
2.14). Furthermore micromotion of the atoms in the TOP trap might cause
problems [56].
The crossed optical dipole trap (XODT) does not have these drawbacks. It

22

is formed by two crossed1 Gaussian laser beams. Each beam forms an optical
dipole trap (ODT) and is labeled with 1 or 2, respectively (see Fig. 4.5). The
intensity of such one beam is described by


−2r2
2P
exp
I[r, z] =πw2 [z]
w2 [z]

withw[z] = w0p

1 + (z/zR )2(4.9)

The laser beam propagates along the z-direction and shows a radial symmetry.
w[z] is the 1/e2 -radius of the beam and has its minimum w0 at z = 0. w0
is commonly called the (smallest) beam waist. zR = πw0 2 /λ is named theRayleigh range. It describes how fast the beam diverges. The power of thebeam P is related to the peak intensity I0 ≡ I[0, 0] by
2P(4.10)
πw0 2
A Gaussian laser beam is therefore fully characterized by its power, wavelength, propagation direction, waist minimum w0 and location of w0 .
I0 =

z

r

Figure 4.2: Plot of Dipole potential for a 1064 nm beam with a smallest waist
w0 =140 µm. The radial and the axial scale is the same. The ultra cold87
Rb atoms are trapped at the minimum of the potential. The gravitational
potential is not considered.
Fig. 4.2 shows a typical (far red-detuned) optical dipole trap as it is usedin the experiment. Since the spatial axes of the plot have the same scale it
becomes clear that the axial confinement is much weaker than the radial one.
This is one of the reasons why a crossed optical dipole trap is used. The
beams of the XODT cross at their smallest waists. For smallest waists of the
Crossing angle = 90◦ ; the normal vector of the plane formed by the crossing beams isparallel to the gravitational force, see Fig. 4.5.
1

23

same size and the right power (to compensate the gravitational potential) a
confinement with approximately equal trap frequencies in all three directions
can be achieved.

+O
D

100

ODT2

n
tio
a
t
i
rav
g
+
T2

OD
T1

potential [Erec]ωz, ODT1,ODT2~ 2π 50 Hz
200 depthz~ 0.2 µK
depthODT1,ODT2~ 5 µK

0

ODT1

100
200

gravitational sag
200

0

100

100

200

z [µm]
Figure 4.3: Plot of 1064 nm XODT dipole potential for 87 Rb with gravitation(solid line) with smallest waists w0 = 135, 141 µm and power P = 1.27,
1.08 W for ODT1, ODT2-beam. The dashed curves show the ODT1 andODT2 potentials without gravitation. The horizontal axis points along thegravitational force. The gray dashed lines indicate the minimum of the dipole
potential with and without gravitation, respectively. The energy is measured
in units of the recoil energy Erec = (~k)2 /(2m). k is the absolute value of the
wave vector of the light used and m is the mass of one atom.
Fig. 4.3 shows the dipole potential of the XODT with parameters typical forour experiment. Comparing the dotted vertical lines in the plot the effectof gravitation can be seen: it lowers the equilibrium position of the trap.
The potential is calculated for 87 Rb atoms in the hyperfine ground state|Fthe sum
P = 1, mF = −1i. This is done by using eq. 4.5 and performing
87
Rb. The dipolet Udipole,t by summing over all excited D1 and D2 states ofmatrix element of eq. 4.4 is evaluated by calculating the particular ClebschGordan coefficients as suggested in [45]. These matrix elements depend on thepolarization of the laser light. The plot shows the potentials for π-polarized
light which are only slightly different from σ + or σ − light potentials. Further24

more the trap is not state-dependent and therefore traps any hyperfine groundstate without significant discrepancy. This is because the hyperfine structure
is not resolved, due to the large detuning of the XODT. Therefore the exact
summation which has been performed for the XODT could be reduced to the
two-level case without a significant error2 .
Finding the minimum of the three-dimensional XODT potential and expanding the potential around this minimum to the second polynomial order, thetrap frequencies in the directions of ODT1, ODT2 and z can be found. Itturns out that the angular trap frequencies are on the order of 2π×50 Hz for
all three directions. The depth of the trap is a few µK in the ODT1 andODT2 directions and some hundreds of nK in the z direction. A more accurate specification on the basis of a calculation cannot be given, since all theabove parameters depend sensitively on the smallest waists and the power ofthe beams. The smallest waists especially cannot be measured to a higher
accuracy at the required position. Furthermore a misalignment of the beams
may change the parameters of the trap.
P
Using eq. 4.7 and performing the sum t Γscatt,t in the same way as described
for the XODT potential, the scattering rate can be obtained. Since the BEC
is located at the trap minimum the potential with gravitation (solid line in
Fig. 4.3) has to be considered for a reference point. The calculation showsthat the scattering rate at this point is on the order of 10−2 s−1 . This value
includes a reduction of scattering by 1/3 compared to the XODT potential
without gravitation, since the atoms are not sitting at the intensity maximumof the Gaussian ODT1 and ODT2 beams when gravitation is considered. The
argument for the 1064 nm scattering rate is the same as for the 1064 nm potential. Hence it is neither sensitive to the polarization of the trap light nor
to the trapped atomic state. A scattering rate of ∼ 10−2 s−1 is very low andtherefore sets the lifetime of the ultra cold atoms in the trap to a high value,
which is experimentally favorable.

4.2.2

Measurement of XODT Trap Frequencies

The values of the trap frequencies need to be known to determine the equilibrium properties of the BEC such as the Thomas-Fermi radius (see 2.1.2).
Experimentally the XODT trap frequencies can be measured by kicking thecondensate, letting it evolve freely in the trap, switching off the trap after acertain but variable holdtime and taking a picture of the atom cloud after afixed TOF. The displacement of the BEC from its equilibrium position (without kick) gives the velocity of the BEC at the time when the trap was switched
2

This cannot be done for the state-dependent potentials which will be discussed later.

25

data taken 2009.06.02
ODT1 imaging - ωz4displacement [a. u.]displacement [a. u.]

ODT1 imaging - ωODT1
2
0
2
4
0

50

100

150

50

100

150

displacement [a. u.]

0
2
4
0

kinetic energy [a.u.]
0

2

200

ODT2 imaging - ωODT2
6
4
2
0
2
4
6

4

200

8
7
6
5
4
3
2
1
0

50

100

150

200

ODT1,2 imaging - Εkin(cm)20

holdtime [ms]

40

60

8
7
6
5
4
3
2
1
0

80 100 120 140-0.0123
pm 0.0027
andof5.41
Figure 4.4: XODT trap frequencies. Thelinear
plots fit:show
the time
evolution
the pm 0.24
kicked BEC in XODT trap. The data of ODT1-imaging ωODT1 and ODT2imaging are fitted by a function A exp[γt] sin[ωt + φ] with A, γ, ω and φ as
free parameters. All shown fits are best fits with respect to the data whichlie in the time interval of the extension of each plotted fit function. All thedata are taken after a TOF of 12 ms. The plot in the lower right corner shows(cm)
the center of mass kinetic energy Ekin of the BEC cloud. It can be seen that(cm)
Ekin decreases with time as indicated by the gray linear fit.

26

20

40

60

off. Assuming a harmonic trap, one can easily find the trap frequencies by taking pictures of the condensate for different holdtimes. In our case the BEC iskicked by applying a short magnetic pulse with the quadrupole coils (see Fig.
3.3) which are ∼10 cm away from the science cell. This leads to a force mainly
along the x-direction (compare Fig. 4.5). Fig. 4.4 shows the displacement of
the cloud as a function of the holdtime. The position of the cloud is foundby applying a Gauss fit to the data. Table 4.1 shows the frequencies and thedamping constants of the XODT. Fig. 4.4 reveals two qualitative behaviors.
First a general damping can be seen. All the oscillations tend to zero for very
long times. This could be explained by a disorder of the dipole potential that
forms the XODT [57]. This disorder (small kinks in the transverse Gaussian
distributions) could be caused by dirt on the optics that are used to form the
XODT. An explanation which is more likely is that atoms with high energies
fall out of the trap because a clear decrease in atom number for longer holdtimes can be seen by summing over the optical densities (see 2.2) of the TOF
images. On the other hand this decrease is possibly caused by the normal decrease of the atom number over time. Another option that has been checked isenergy transfer to shape oscillations of the cloud. These oscillations can clearly
be seen but they also decrease with time without any interesting behavior like
coupling between different directions or revivals. In the end, it is not clear
where the energy that disappears in the center of mass frame, occurs again.
A mixture between the suggested mechanisms is likely.

ω/(2π) [Hz]γ [s−1 ]ODT159.2 ± 1.0−11.8 ± 1.0

ODT246.9 ± 2.0−24.6 ± 3.7

z51.1 ± 1.5
21.5 ± 3.0

Table 4.1: XODT trap frequencies and decay constants. The errors of the trap
frequencies are estimated for different possible best fits in the time intervall
from 0 to 60 ms (see Fig. 4.4). The decay constants belong to the fits shownin Fig. 4.4 and are given with the corresponding best fit errors. ωODT1 is thefrequency of the oscillation along the ODT2 direction, and vice versa. ωz is
the frequency of the oscillation along the z-direction.
The second interesting observation that can be made is that there seems to bea strong coupling between the z-oscillation and the ODT2-oscillation. Considering these two signals for times smaller than 50 ms one sees a clear increasein the amplitude of the ωz -signal whereas the ωODT2 -signal decays. The ODT1and ODT2 imaging3 data are taken at the same time by dual imaging. The
decay constants of the ωODT1 and ωz -signal are well agreeable (see Table 4.1)
3

The imaging along ODT1 and ODT2 works with the same principle as shown in Fig.

27

which suggests that the main energy transfer happens from the ωODT2 to theωz -signal. Furthermore it can be seen that the ωODT2 -signal peaks around 100
ms wheres the ωz -signal has a local minimum at this time. This strong coupling indicates an anharmonicity of the trap.
It has to be mentioned that the trap frequencies of the XODT depend on thepolarization of the beams only because the polarization changes the transmitted power through the glass cell. This measurement was taken after changing
the polarization of the ODT2 beam and realigning the trap (only ODT2 beam).
This was done in order to avoid interferences of the ODT2 beam with the science cell that could change the beam profile. Now the polarizations of ODT1and ODT2 are parallel. Cross interferences between the beams, however, do
not happen since the two beams are detuned by 20 MHz with respect to eachother.
In conclusion the measured trap frequencies match up well with the calculated
ones (see Fig. 4.3).

4.3

Optical Lattices

The physical principle that applies to an optical lattice (OL) is rather simple. A laser beam with wavelength λ is reflected back on itself by a fixedmirror, forming a standing light wave with a periodicity of λ/2 and a node at
the surface of the mirror. Ultra cold atoms can now be trapped in the nodes(blue-detuned OL) or antinodes (red-detuned OL) due to the dipole force (seeeq. 4.8). To insure that no phase shifts destroys the homogeneity of the 1-D
lattice the coherence length of the light has to be larger than twice the distance of a point in the lattice to the surface of the mirror. This can easily beachieved since the linewiths of the lasers used in the experiment are all below
1 MHz which corresponds to a coherence length of 300 m. By superimposing
additional 1-D lattices, one can create higher dimensional OL.
Standing light waves have been used to diffract atoms at an optical grating.
Hereby the Kapitza-Dirac effect [58] and the well-known Bragg effect [59] havebeen probed. For an introduction to this topic see [60].
Fig. 4.5 shows the setup of the XODT and the OL in the xy-plane. The ODT1
and 2 beams are focused such that their smallest waists (135 µm for ODT1,
141 µm for ODT2) are located at the BEC position. Since the Rayleigh rangeof the beams is on the order of 5 cm the position of the mirrors does not haveto be very accurate in order to assure a good overlap of the incoming and thereflected beams at BEC position. The switching between ODT and OL in the
4.13. The XODT is built with dichroic mirrors that transmit the imaging light. Therefore
imaging along the XODT directions is possible (not shown in simplified setup of Fig. 4.5).

28

xy-plane is done using two AOMs for each beam. The first AOM puts the lightin its first diffraction order and the second one in its minus first diffraction
order. Going twice through this shutter, the frequency and propagation direction of the diffracted light is not changed. The advantage of this shutter is that
it is fast (∼ µs) and has negligible leakage if closed. Furthermore the intensityof the reflected light can be varied by changing the power of the RF applied to
the AOMs. This way it is possible, e.g., to ramp up the lattice adiabaticallywith respect to the trap frequencies. The two-way intensity efficiency of the
AOM shutters is around 10%.

RF Source

OD

T

2

M

AOM

RF Source

Science Cell

L

x’

+1

T

OD

z’

1

-1

y

y’
x

z

Figure 4.5: Setup of XODT (simplified) and OL. The two beams which act
as dipole trap are retro reflected with mirrors (M) when the AOM-shutter isopen. ODT1 and ODT2 then become OL1 and OL2. A third standing lightwave along the z-direction (see Fig. 4.13) can be added. The wavelength of
the beams is 1064 nm. All beams are focused on the BEC with lenses (L). The
xyz-coordinate system is the lab coordinate system (compare to Fig 3.3). The
gravitational force points along the negative z-direction. The x0 y0 z0 -coordinate
system is created by rotating the xyz-system by (3/4)π where z0 =z. The
primed coordinate system is a natural choice for the XODT.

29

The two linearly polarized standing light waves are detuned from each otherby 20 MHz4 . This is done to avoid cross interferences between the beams. The
two-way detuning minimizes the effect of cross interferences since the atoms
cannot follow the rapid oscillations of the unwanted potential. To understandthe potential that is formed by the standing light fields with wavelength λ, it
is useful to consider the effect of one standing wave. For one plane standing
wave the intensity which, is proportional to the potential (eq. 4.5), varies sinusoidally in the direction of propagation but is independent of the transverse
directions. Therefore it forms a sheet of constant potential for every plane perpendicular to the propagation direction. Leaving out the symmetry breaking
effect of gravitation the atoms would be confined in light sheets with a spacingof λ/2. Adding now a second standing wave with perpendicular propagation,
the equipotential planes of the two fields cross and form 1D tubes of confinement that are parallel and in the order of a 2D square lattice. The potential
sheets of a third beam (again perpendicular to the first and second one) would
now cut the tubes and form a 3D simple cubic lattice with a lattice spacingof λ/2. The atoms are trapped at the points of the minimum potential which
are the lattice sites. The potential formed by three standing plane waves is
described asX
Vpw (x, y, z) = V0
sin2 [ki x](4.11)
i=x,y,zIn our case however the Gaussian beams are an overlying potential according
to eq. 4.9 [61]y2 + z2
sin2 [kx x] +VG (x, y, z) = Vx exp −2 2
w [x]


x2 + z 2
sin2 [ky y] ++Vy exp −2 2
w [y]


x2 + y 2
sin2 [kz z]+Vz exp −2 2w [z]

(4.12)

with ki = 2π/λi . Here the possibility of different lattice periodicity along
different axes is taken into account. This Gaussian confinement can be approximated harmonically at the position of the BEC (trap center). For deep
lattices the potential of one lattice well at the trap center is well approximatedby4

If a third 1064 nm beam along the z-direction is added, the beam is detuned by 10
MHz with respect to both XODT beams.

30

r
ωi =

4.3.1

k2~
2Vi k 2
=
m
m

r

Vi
Erec

(4.13)

Lattice Envelope and Lattice Ramp

In Fig. 4.3 the dipole potential for the XODT is shown. As discussed
above the trap frequencies of this XODT are ∼ 2π·50 Hz for a laser powerof ∼ 1 W and the known beam parameters. When a certain fraction of thebeams is reflected back to form a standing light wave with the incoming beams
the situation changes. Now there are small wells which are formed due to thestanding wave and are equally spaced at the trap center. These small wells
are enveloped by the Gaussian potentials described above.

x’y’

Figure 4.6: Standing wave potential of lattice beams along XODT directions
and corresponding envelope for a constant z value. The beam waist is multiplied by a factor of 100 and the plots are flipped, both to make the structureof the potential clear.
Fig. 4.6 shows the potential of the OL1 and OL2 standing waves. If thepotential were drawn to scale it would become clear that the envelope changes
only slightly over the region in which the BEC is localized and that around502 lattice sites extend over this region. To describe the situation adequately,
the parameters of the envelope and the lattice sites (see eq. 4.12) have to beconsidered separately. First I will discuss the envelope of the OL1-OL2 2D
lattice. The intensity distribution of the envelope is given bypp(4.14)
Ienv [x, y, z] = Ix (1 + Rx + 2 Rx ) + Iy (1 + Ry + 2 Ry )where Ix,y describes the intensity of a Gaussian beam traveling in the x0 (Ol1)or y0 (OL2)-direction respectively, and is given by eq. 4.9 with r replaced byp{y 2 , x2 } + z 2 . The factors containing Rx,y occur due to interference of the31incoming and reflected beams in the x- and y-direction. As described above,no interference between the x- and y-beams must be taken in account. Rx,y isthe intensity fraction which is reflected back to form the standing waves along
the x- and y-axes. eq. 4.14 indicates that the envelope potential and therefore
the envelope trap frequencies ωenv change as a function of Rx,y . This can cause
experimental problems when ramping up the lattice which means increasing
Rx,y .
The 1D ramp up of a lattice has been treated experimentally and theoretically[62, 63]. Increasing the lattice potential V0 from 0 to 1 Erec has to happen
in a time bigger than the recoil time mRb λ2 /(~π) [64] which is ∼ 490 µs for1064 nm light. The reason for this lies in the band structure (see Fig. 5.1).
For a very flat potential, the energy gap between the first and higher bands isvery small. Ramping up the lattice in a timescale faster than the recoil time
would therefore excite the atoms into a higher band. On the other hand theramps also have to be slow for a deep lattice to stay in the adiabatic regime.
Here the atoms are confined to lattice wells and can not move freely as in thepure XODT case but go from one lattice site to the next by hopping which
is described by the rate t/~ (with t being the hopping amplitude, see equs.
5.7, 5.10). The deeper the lattice, the smaller the hopping rate. To avoid
excitations of the BEC, now a change of the envelope trap frequencies has tobe slow enough that the atoms can follow by hopping. This sets an upper limit
for the lattice ramp up speed which may lead to an increase of decoherences
due to a longer ramp time.
To analyze the lattice ramp problem it is helpful to calculate the detailed
change of the envelope frequencies when ramping up the lattice equally in
ODT1- and ODT2-direction. Fig 4.7 shows the result of such a calculation on
the basis of eq. 4.14. The envelope trap frequencies are evaluated following
the description in 4.2.
The plot shows two main points. First the the envelope frequencies change
significantly when ramping up the lattice and second this change leads to ashift of the BEC position in z-direction (gravitational sag). The reason forthis connection becomes clear when one remembers that the dipole force in a
red-detuned trap pulls the atoms to the places of high intensity that are thecenters of the Gaussian beams. If the intensity of the beams is increased due
to reflecting back parts of the light (cf. eq. 4.14) the dipole force becomesstronger with respect to the gravitational force which leads to a “pull-up”
effect for the BEC. At the new position the envelope frequencies differ fromthe old ones since the trap minimum now is closer to the beam centers. As aresult the trap frequencies are directly connected to the gravitational sag. To
avoid the change of the envelope frequencies, the power of the incoming beams

32

120

ωz

ωenv/(2π) [Hz]

100

ωODT2

80
60

ωODT1
no lattice

40
0

2

4

6

8

10

12

14

10

12

14

grav. sag [µm]

50
45
40
35

max lattice depth

30
25
20

0

2

4

6

8

back reflected intensity R [%]Figure 4.7: Change of ωenv due to ramping up the OL in the xy-plane of thelab coordinate system. The parameters of the beams are the same as for Fig.
4.3. It is assumed that the smallest waists of the reflected beams have thesame sizes and positions as for the incoming beams. The parameter R on theabscissa equals both, Rx and Ry . The lower graph shows the change of thegravitational sag (compare Fig. 4.3) as a function of R.

33

can be decreased proportionally when Rx,y is increased.
Experimentally this can be done by looking at the change of the position of
the BEC in z-direction. Fig. 4.8 shows data from such an experiment. In
detail the BEC was loaded into the XODT and the lattice in OL1-direction
was ramped up exponentially while the power in the ODT2 beam was held
constant. At a certain point of the lattice ramp an in situ image of the BEC
was taken. The z-position of the BEC of this image is now compared witha reference image of the BEC held in the full power XODT with no lattice
beams (point without error bars in Fig. 4.8). If the lattice in OL1-direction
is ramped up with ODT1 and ODT2 beams at full power a clear shift (up to∼ 15 µm for max. lattice depth) in z-direction (with respect to the z-positionof the reference file) against gravity can be seen. The data points in Fig.
4.8 are obtained by looking at this shift for a certain back reflected intensity
along Ol1-direction (which corresponds to a certain lattice depth along thisdirection) and resetting it (the shift) with respect to the reference by taking
power out of the ODT1 beam. The black line in the figure is a best fit withthe model developed for Fig. 4.7. The theory contains the powers, the waists
and the percentage of back reflected light (to form the lattice) of the beams
as variables. For the data in Fig. 4.8 the values of the ODT2 beam do notmatter since the BEC is always sitting at the same position in this beam andthe ODT2 parameters do not change. Due to the normalization in Fig. 4.8
the power and the waist of ODT1 can be changed without affecting the fit.
Therefore the maximum value for Rx (see eq. 4.14) is the only free parameter.
(max)
It is evaluated to Rx= (1.22 ± 0.20) %. This value corresponds to the(max)max. lattice depth in Fig. 4.7. In this figure however Rx
is shown to
be 10 %. One has to remember now that Fig. 4.7 is based on a calculation
where the incoming and reflected beams have the same waist properties at the(max)
BEC-position. Since a beam pick-off was used to measure Rx,y between thescience cell and the AOM-shutters (compare Fig. 4.5) and 10 % were found it
is very likely that the waists of the back reflected beams are bigger than theones of the incoming beams. eq. 4.9 and eq. 4.2 tell that the dipole potential
the atoms see is proportional to 1/w2 . This means that the waist of the backreflected beam at BEC position should be ∼3 times bigger than the ODT1
waist.
For the ODT2 direction the same data as shown in Fig. 4.8 has been taken.
(max)
It comes out that Ry= (2.10 ± 0.20) %. Therfore an equal argumentation
holds for the waists of the OL2 beams as for the OL1 ones.
With the data of Fig. 4.8 and the corresponding data set for the ODT2 direction the lattice in ODT1- and ODT2-direction can now be ramped withoutchanging the position of the BEC and without changing the envelope trap

34

ODT 1 power [a.u.]

1.00
0.95
0.90
0.85
0.80
0.0

0.2

0.4

0.6

0.8

1.0

back reflected intensity [a. u.]
Figure 4.8: Canceling the shift of the BEC position (found by simple Gaussfit) against gravity when ramping up the lattice. The ODT2 beam stays atconstant power whereas the ODT1 power (vertical axis) is reduced when thelattice in OL1 direction (horizontal axis) is ramped up. For each data point
the power in the ODT1 beam is chosen such that the z-position of the BEC isalways the same. The black line is numerically found with the theory for Fig.
4.7 by requiring no change in the gravitational sag (cf. Fig. 4.3). The only
free parameter in the fit (again assumption: waist of incoming and reflectedbeam have same size and sit at BEC position) is the maximum reflectivity Rx(see eq. 4.14) which corresponds to the very right data point in the plot. The(max)
fit suggests Rx= (1.22 ± 0.20) %.
The errors of the data arise from the calibration of the AOM shutter (±3%horizontal) which regulates the lattice depth (see Fig. 4.5), and from thevariation of the z-position for 3 different runs (vertical error) for fixed hoizontal
and vertical values in the plot.

35

frequencies (compare to Fig. 4.7). This has the two advantages that first theatoms do not have to redistribute due to a change of the external confinement
and second the BEC is not pulled through the lattice which can be appliedalong the z-direction. Both advantages reduce heating of the atoms whenloading them into the lattice.

4.3.2

Lattice Sites

The second important part of the lattice (besides the envelope) of course
is formed by the lattice sites themselves. For an expansion of the BEC over ∼50 lattice sites per direction the envelope only changes slightly from the center
to the last occupied site. Therefore it is a good approximation to describe
the lattice sites by the parameters of a site at the trap center. For our setup
the light field at the trap center can be approximated as plane waves with aGaussian envelope. A simple calculation on the basis of the dipole force of aplane wave with a Gaussian envelope gives the site frequencies and the lattice
depth as a function of the back reflected power. Fig. 4.9 shows the justification(part a)) for a description of all 50 occupied sites along one direction by thecenter lattice site. From the plots it can be seen that high lattice depth canbe achieved assuming that the smallest waist of the back reflected beams havethe same size and position as for the incoming beams. Part d) shows the effectof lowering the incoming power on the site frequencies. This is of interest if
one wants to maintain the envelope frequencies (see above). The depth ofthe lattice sites decreases proportionally with the incoming power because theincoming intensity is just a factor in the formula describing the lattice depth(compare to eq. 4.14).

4.4

Kapitza-Dirac Diffraction

To be able to make quantitative statements it is important to know thelattice depth of the applied OL. Since the lattices along different directions aredecoupled in our case it is sufficient to concentrate on the 1D case. In 1933,
Kapitza and Dirac proposed that electrons should be diffracted by a standing
light wave [65]. This effect also happens for atoms [58] instead of electrons
and is even stronger pronounced for equal light intensities.
For the purpose of lattice calibration the BEC is held in the XODT and a lattice along a certain direction is pulsed on (=Kapitza-Dirac pulse =KD pulse).
The system can now be described by the Hamiltonian:22b = − ~ ∂ + V0 cos2 [kx]H
2m ∂x236(4.15)

E [Erec]

160
180

OL1, R=10%

200
10

0

5

5

depthsite [Erec]

a)

140

100

OL1

60
40

OL2

20
0

10

b)

80

0

2

c)

ωsite/(2π) [Hz]ωsite/(2π) [Hz]

40000

OL1

30000

OL2

20000
10000
0

0

2

4

6

8

4

6

8

10

12

14

back reflected intensity R [%]x [µm]

10

12

35000

trap opens

30000
25000

OL2

R=10%

20000
50

14

OL1

d)

60

70

80

90

100

fraction of max. incoming power [%]back reflected intensity R [%]Figure 4.9: Overview of lattice parameters. Part a) shows the red latticesites (around 50 here) for R=10% along the OL1 direction. The black line
at the bottom of the plot is the envelope function which occurs due to thetransverse Gaussian distribution of the OL2 beams. All the plots are basedon the same beam parameters as in Fig. 4.7 and Fig. 4.3 as long as nothingelse is mentioned especially. Part b) shows the depth and part c) the site
frequencies along the OL1- and OL2-axis. Part d) shows the change of thesite frequencies for a back reflected intensity of 10% when one lowers thepower of both incoming beams equally. The end of the plot between 50 and
60% indicates that the power of the beams becomes to low to hold the atoms
against gravity.

37

where m is the mass of the 87 Rb atoms, V0 the lattice depth and k the wave
vector of thef laser light that forms the lattice. Inserting the Ansatz
XΨ=
cn [t] exp[i2nkx](4.16)nP
with n being an integer, cn [t = 0] = δ0,n and n |cn [t]|2 = 1 into the timedependent Schödinger equation with the Hamiltonian from eq. 4.15 leads to2~n2 k 2V0
cn [t] + (cn−1 [t] + 2cn [t] + cn+1 [t])(4.17)
m
4~
The coefficients cn tell how much the nth diffraction order is populated by the
2 2condensate wave function. The term 2~nm k (kinetic energy) becomes more
important for longer interaction times (pulse lengths). Note that a change ofn by 1 corresponds q
to a change of 2 in terms of scattered photons.
1
For τ  1/ωosc ∼ k Vm0 the first term on the right hand side (kinetic energy)
iċn [t] =of eq. 4.15 can be neglected and the coupled differential equations 4.17 havethe analytic solutions cn [t] = in exp[−i V2~0 t]Jn [− V2~0 t] with Jn being the Bessel
functions of the first kind. This approximation is called the Raman-Nathapproximation and can be made when the length τ of the applied lattice pulse
is much smaller than the oscillation time of a particle in a lattice well. Now
the population in the nth diffraction order due to atom photon scattering fromthe KD pulse is given by the absolute value of cn [t] squared


2 V0
τ
(4.18)
Pn = Jn
2~
For sufficiently short pulses (note: short is here always dependent on the lattice
depth V0 ) equation 4.18 can be used to calibrate the lattice depth.
Fig. 4.10 shows a typical KD pattern. Wanting to know the the lattice depth
one could use eq. 4.18 as a fit function. However especially for deep lattices5
the above condition, for eq. 4.18 to be valid, is easily broken (unless pulse
durations are very short which gives a technical limit) and the suggested fit
introduces a uncertainty. To overcome this drawback but not wanting to loose
an analytical fit function, different simple extensions to the model are possible[66]:
0
The maximum occupied diffraction order nmax is given by n2max = 4EVrec
wherethe lattice energy is fully converted into kinetic energy [67]. For not to high5

The lattice for Fig. 4.10 turns out to be ∼ 20 Erec deep. A fit with eq. 4.18 howeverrequires V0  10 Erec for a 10µs pulse. And even for shallow lattices corrections have to be
added here [66].

38

P-3

P-2

P-1

P0

P1

P2

P3

2hk
Figure 4.10: Typical diffraction pattern after 12 ms of free expansion from
XODT. The length of the KD square pulse equals 10 µs. The pulse is generated
with the 785 nm lattice along the z-direction (cf. Fig. 4.5 and Fig. 4.13)
and controlled with an AOM. The peaks are separated by 2~k which canbe understood with eq. 4.16 and 4.18. For an analysis in terms of latticecalibration one has to evaluate the relative atom number in each diffracted
peak by summing over the optical density (see 2.2).
lattice depth (V0 . 4Erec ) only the 0th and the 1st diffraction patterns are
populated. This means eq. 4.18 can be truncated for |n| > 1. The remaining
three coupled equations can be solved analytically and give 
1  χ 2 2 Ωτand P0 = 1 − 2P±1
sin
(4.19)
P±1 =
2 Ω
2
p
with a generalizedRabifrequencyΩ=χ2 + ∆2 , a resonant coupling term√χ = V0 /( 2~) and a detuning of ∆ = 4Erec /~. The analogy in terms ofstructure to a Rabi frequency that drives a two level atom can be drawn because of the symmetry of the problem which generally has its manifestation in
Pn = P−n .
Thinking in terms of the Rabi formalism (see 2.3.1) helps to motivate an advancement to eq. 4.18. A scattering event that puts an atom from the |0~ki tothe |±2~ki state needs to conserve energy. Energy conservation can be fulfilled
by a detuning of ∆ of the scattered photon with respect to its standing lightwave. Now it is well known that there is a connection between the temporal
width and the frequency width of a light pulse. The connection is formed by a
Fourier transformation where the two widths behave reciprocally with respect
to each other: the longer the pulse in the time domain the narrower is its
frequency width and vice versa. In our case this translates to a bigger ∆ andtherefore an occupation of higher momentum states for a smaller pulse width.
The used pulses are rectangular in the time domain. The Fourier transform ofsuch a rectangular function is a sinc function and well known from diffraction
at a slit. Here this motivates to extend the Bessel function in eq. 4.18 to

39

h
αi
Pn = Jn2 β sinc(4.20)
2
The two dimensionless parameters α = (4Erec /~)τ and (V0 /2~)τ are introduced
for convenience. Indeed the two formulas eq. 4.20 and eq. 4.19 agree well up
to O(α2 β 2 ). These two analytical formulas fit much better with experimental
observations than the simple Bessel function from eq. 4.18 [66].
The possibility of using these functions however ends with stronger latticepulses when the kinetic energy term in eq. 4.17 cannot be neglected or higher
momentum orders are populated by the atomic wave function. In this case(observed)
eq. 4.17 can be truncated for n  nmax
and numerically integrated. Thismethod is reliable without any restrictions (except of the disregard of a meanfield interaction in eq. 4.15) and is always used within the frame of this thesis
when lattice depths are named.

4.5

State-Dependent Optical Lattice

So far optical lattices in general have been discussed. This discussion entirely ignored the atomic hyperfine the 87 Rb atoms (although it was includedin the calculations). This has been valid since the used lattice wavelength of1064 nm is so far detuned from the atomic transitions that the finer structure
is not resolved. Even the very rough formula 4.8 gives a very good approximation of the potential the atoms see independently of their internal state.
This means that this OL is state independent and traps all atomic statesequally. For a lattice wavelength close to the atomic transitions of 87 Rb (D1and D2-line) the situation becomes much richer [25, 26]. Fig. 4.11 shows thehyperfine structure of 87 Rb. The two ground states |F = 1, mF = −1i and|F = 2, mF = −2i are marked in the scheme. These two levels are chosen to
be the quantum states of interest for the 2 component Bose-Hubbard model
which will be discussed later.

4.5.1

State-Dependent Potential

In order to calculate the potential that acts on the atoms eq. 4.5 has to be
consulted. The dipole potential that is described in this equation is a functionof the dipole matrix element given with eq. 4.4. To derive an equation
P for thepotential an atom in one particular hyperfine state feels, the sum t Udipole,tis evaluated. As described above, this means summing over all levels that cancouple to the fixed hyperfine state of interest. Doing so the only problem thathas to be solved lies in the calculation of the dipole matrix elements in eq. 4.4.

40

σ+
π
σ−

{

5 2P1/2

{

F=2
F=1

5 2S1/2

{

F=2
F=1

{
{

F=3

D1= 794.98 nm

-3

-2

-1

0

1

2

{

mF =

D2= 780.24 nm

5 2P3/2

F=2
F=1
F=0

3

Figure 4.11: The hyperfine structure of 87 Rb with its magnetic sublevels (noton scale). The states |F = 1, mF = −1i and |F = 2, mF = −2i are marked
since the atoms are only prepared to be in these states in the experiment. Somepossible dipole transitions regarding these states of interest are indicated. The
D1- and the D2-line are shown to give a scale of the transitions in terms ofwavelength.

41

It turns out that these matrix elements can be reduced to a constant for eachline (D1, D2) which is multiplied by Clebsch-Gordan coefficients that dependon the polarization of the light that forms the dipole potential (see [45]). As
an example the state |F = 2, mF = −2i could be considered. For this state
most of the dipole matrix elements are 0 which can be seen by calculatingthem or just by looking at Fig. 4.11. It is clear that none of the S-states
can give a contribution since the dipole selection rule forbids transitions with∆l = 0. Due to conservation of angular momentum all matrix elements withmF ≥ 0 are 0 because none of the existing polarizations of light can induce
single photon transitions with |∆m| > 1. The remaining possible transitions
that contribute to the potential with the corresponding dipole matrix element
now depend on the chosen polarization of the light. Fig. 4.11 shows somepossible transitions. In order to follow the chosen example, one could choose
σ + -polarized light. This means that all the matrix elements that connect to
the P-states with mF = −1 can give non-zero
P numbers. Therefore 5 matrix
elements have to be calculated to be put in t Udipole,t where the ground state
is |F = 2, mF = −2i and the light is σ + -polarized. This can be done for otherpolarizations and for the |F = 1, mF = −1i state.

|2,-2

>

50

0

E [Erec]

E [Erec]

50

-50

|1,-1

>

0

- 50

-100
782

784

786

788

λ [nm]

790

792

794

782

784

786

788

λ [nm]

790

792

794

Figure 4.12: State-dependent potentials for the states |F = 2, mF = −2i and|F = 1, mF = −1i. The solid lines correspond to π, the thick dotted lines toσ + and the dashed lines to σ − polarized light. The vertical dashed dotted lines
indicate the zero of potential for the |F = 2, mF = −2i state and σ + polarized
light which happens at λ= 785.12 nm. The potential is calculated for 87 Rb
atoms sitting at the smallest waist of a 230 µm laser beam (no standing wave)
with 125 mW power.
Fig. 4.12 shows the result of the calculation. In order to define a quantization axis for the polarization a small magnetic field which points along the
propagation direction of the light has to be applied. The small effect of thisfield (Zeeman shift) can be neglected in the calculations. The direction of the42magnetic field however is crucial since an angle between the laser beam andthe magnetic field lines breaks the rotational symmetry of the potential andleads to coherences between the magnetic sublevels. This can lead to off diagonal terms in the Hamiltonian of the problem and changes the structure of
the potential6 . Here this is not done but the magnetic field is aligned parallelto the laser beam and has a magnitude of ∼ 300 mG.
We now want to concentrate on σ + -light (thick dotted line in Fig. 4.12). As
indicated by the dashed dotted line the potential for the |F = 2, mF = −2istate vanishes (of course independently of the intensity since the intensity
is just a factor in eq. 4.5) for a laser wavelength of about 785 nm. The|F = 1, mF = −1i state however feels an attractive force at this wavelength.
The possibility of a laser wavelength with vanishing dipole potential could
have been guessed with the simplified formula eq. 4.8. For red-detuning withrespect to a transition this potential is attractive for blue-detuning however repulsive. Furthermore the strength of the potential becomes bigger the smaller
the detuning is chosen. Having now two transitions (here D1 and D2-line) it
seems to be natural that there is a wavelength in between the lines where theattractive (D2-transition) and the repulsive (D1-transition) forces cancel out.
This is the main idea for a state-dependent optical lattice (SDOL), where thebeam of a wavelength and polarization at which one species sees a lattice but
the other one does not, is reflected back to itself and forms a 1D OL for one
species but not for the other one.
In the experiment the SDOL beam is set up along the z-direction in the lab
coordinate system as shown in Fig. 4.13.
To test the described idea of a SDOL experimentally the BEC is loaded
into the XODT. Then a 10 µs long 785 nm lattice pulse with constant depth
is applied along the z-direction, the atom cloud is dropped by switching of the
XODT and an absorptive Image is taken after a TOF of 12 ms. Fig. 4.10 shows
an image that led to a data point for the measurement. The lattice depth isthen found with the procedure described in 4.4. To introduce the effect of
the state-dependence the measurement is once done with the atoms in the|F = 1, mF = −1i and once with the atoms in the |F = 2, mF = −2i state.
In the second case the nearly full population transfer from |F = 1, mF = −1i
to |F = 2, mF = −2i is driven by a Landau-Zener sweep (see 2.3.1) with theatoms sitting in the XODT. For both of the named atomic states the angle ofa λ/4-plate that changes the polarization of the applied 785 lattice is scanned
through.
Fig. 4.14 shows the obtained KD patterns for the |F = 2, mF = −2i state. One
6

Therefore it is possible to create double well potentials by choosing the right angle
between the magnetic field and the direction the beam propates along (cf. [25]).

43

z-direction
gravitation

science cell

λ/4

mirror for 1064 nm
transparent for 785 nm

mirror for 785
and 1064 nm

180

170

160

150

140

130

120

110

100

90

80

70

60

50

40

30

20

0



10

Figure 4.13: Simplified setup of z-lattice. Along the z-direction the state
independent 1064 nm beam can be exchanged by a state-dependent 785 nm
beam. If we are talking about a SDOL it means that there is no 1064 but only
the 785 beam along the z-direction. Along the x0 and y0 -direction (compare to
Fig. 4.5) no SDOL exists. For switching between the 1064 and the 785 lattice
the setup does not have to be changed. The trick is to use a dichroic mirror
for 1064 light that is (nearly) transparent for 785 light. For the SDOL a 300
mG magnetic field is applied along the z-direction.

,

Figure 4.14: The picture shows KD-patterns for a 10 µs 785nm lattice pulseof constant intensity applied to atoms in the |F = 2, mF = −2i state. The
numbers above the images give the the angle (in degree) of the λ/4-plate (seeFig. 4.13) that changes the polarization of the lattice beam44

sees that the effect of the KD pulse becomes minimal at an angle of ∼ 160◦ .
Evaluating the lattice depth for both states as a function of the angle of the
λ/4-plate leads to Fig. 4.15.

lattice depth [Erec]

35
30

π

25
20

σ-

15

σ+

10
5
0

0

50

100

150

angle of λ/4 - plate [deg]Figure 4.15: Polarization dependent lattice depth for the |F = 1, mF = −1i
and the |F = 2, mF = −2i state. The lattice depth is plotted as a function of
the polarization of the 785 nm light that forms the OL. The locations of typicalpolarizations are indicated by transparent gray vertical bars. The error of the
wave plate angle is estimated with ±2◦ . The error of the lattice depth rises
because of a ±5% variation of the monitored back reflected light intensity3 .
An error of the lattice depth fit which tends to overestimate the lattice depth
for shallow lattices is not included. The blue and the red curve are best fits(see text).
π
x + φ + of f . Table
The data of Fig. 4.15 is fit with the fit function A sin 90
4.2 shows the values of the fit parameters.
Comparing the theoretical result form Fig. 4.12 with the obtained experimental result from Fig. 4.15 good agreement can be seen. First Table 4.2 gives thephases φ of the sinusoidal fit function which are off by exactly π with respect
3

The back reflected 785 beam that forms the lattice with in this case constant ingoing
beam, can be monitored with a photo diode that measures a small part of the back reflected
intensity which is deflected by a beam pickoff.

45

A [Erec ]7.78 ± 0.20
14.57 ± 0.17

of f [Erec ]15.71 ± 0.14
15.98 ± 0.12φ
|F = 1, mF = −1i2.156 ± 0.026|F = 2, mF = −2i−1.013 ± 0.012π
Table 4.2: Best fit parameters to data shown in Fig. 4.15 with A sin 90
x+φ +of f as fit function. The errors are the standard errors from the fit.
to each other. This is in nice agreement with the theory since the σ + andthe σ − -potential lines in Fig. 4.12 are flipped for the |F = 1, mF = −1i andthe |F = 2, mF = −2i state. The offset parameter of f is about the same forboth states. This parameter tells the height of the point symmetry center of
the two data sets. This center lies right in between σ + and σ − polarized lightand is realized by the data when the λ/4-wave plate does not affect the linear polarization of the light. Therefore the light remains π-polarized and theparameter of f gives the corresponding lattice depth. Comparing the resultthat both states should see nearly the same lattice for 785 nm π polarized
light with the theory leads again to a nice agreement. In the end the relative
depth of the potentials can be compared with theory. This can be done bycalculating the ratio of the measured curves for σ − -light and comparing thiswith the theoretical values for 785 nm light of the same polarization. This
leads to−785 nm, σEmeasured[|F=1,mF =−1i]−785 nm, σEmeasured[|F=2,mF =−2i]f [|F=1,mF =−1i]−A[|F=1,mF =−1i]= of= 0.260 ± 0.08
of f [|F=2,mF =−2i]+A[|F=2,mF =−2i]−785 nm, σEtheory[|F=1,mF =−1i]−785 nm, σEtheory[|F=2,mF =−2i]= 0.257

Again theory and experiment agree within the margin of error that is given
by simple error propagation. For π-light experiment and theory are in good
agreement, too.
However looking at σ + -light one would expect no potential for the |F = 2, mF =−2i state, but there clearly is a residual lattice in Fig 4.15 for this state. One
explanation for this is that the numerical lattice depth fit based on eq. 4.17
tends to overestimate the lattice depth for very weak lattices. On the other
hand one can still see very dilute diffraction peaks in Fig. 4.14 for an angleof 160◦ (that is the lowest fitted lattice depth for the |2, −2i atoms). Thiscan either be explained by weak residual magnetic fields which do not point
along the 785 lattice beam direction or by the fact that the laser wavelength
does not exactly correspond to the calculated magic wavelength of 785.12 nm.

46

Both effects lead to a non-vanishing potential for the mentioned atomic state.
To draw a conclusion the experimental results are in good agreement with thetheoretical calculation.

4.5.2

State-Dependent Scattering Rate

The availability of a state-dependent optical potential has the great advantage to give one the possibility to explore new and rich physics in the ultra cold
regime. However a potential with a wavelength between the D1- and D2-lineof the atom under study results in a much higher scattering rate than for the1064 nm potential of the XODT.
Fig. 4.16 shows the measurement of the scattering rate for 780.8 nm lightwith respect to 87 Rb atoms in the |F = 1, mF = −1i state. This measurement
is repeated for 785.3 and 779.0 nm. Table 4.3 gives the measured scattering
rates for these wavelengths.
λ [nm]
Γscatt [1/s]P [mW]

779.0 ± 0.3
0.62 ± 0.14
11.75 ± 0.59

780.8 ± 0.3
6.1 ± 1.3
1.48 ± 0.07

785.3 ± 0.3
0.221 ± 0.051
22.1 ± 1.1

Table 4.3: Values of measured scattering rate (σ + -light) for a beam powerof 10 mW (scaled!) and waist of 230 µm (experimental waist) at the BEC
position. The errors result from a error analysis of the errors of the measured
power (5%) and waist of the beam (10%) and of the fit error (compare Fig
4.16). The wavelength error is estimated (calibration issue with wave meter).
The last line gives the experimental power of the beam at BEC position.
In order to compare the experimental results with theory eq. 4.4 and 4.7 haveto be consulted. The scattering rate is then evaluated in exactly the same
way as described in the previous part of this chapter. Fig 4.17 shows thecalculated scattering rates for different polarizations of the light. Part c) gives
the comparison between the measured data and the theoretical curve with nofree parameters.
All points except the measured point at around 785 nm agree well with thetheory. The measured scattering rate at ∼ 785 nm however is slightly higher
than the theory value. One systematic error in the measurement could be thata scattered photon removes more than one atom from the BEC by a avalanchelike effect that comes in by atom-atom interaction. Furthermore the theory
does not implement saturation effects. This, however, is insignificant in ourcase because the used intensity stays far below the saturation intensity. In
conclusion the theory fits well with the experimental data and the scattering47

rel. XODT atom # [a.u.]

1.0
0.8
time=0.8 s

0.6time= 2.4 s

0.4
0.2
0.0

0

1

2

3

time [s]4

5

Figure 4.16: State-dependent scattering rate measured for |F = 1, mF = −1iatoms and σ + -light at λ= (780.8±0.3) nm with a power of the 780.8 nm beamin the science cell of (1.48±0.08) mW. The beam has a smallest waist at the
BEC position of (230±23) µm. The time on the abscissa of the plot gives
the exposure time of the BEC to the 780.8 nm light. The BEC is trapped in
the 1064 nm XODT. The y-axis shows the remaining atom # after a certain
exposure time to the 780.8 nm light divided by a reference atom # that wasmeasured without the 780.8 nm light. The holdtime of the reference run in
the XODT equals the holdtime for the corresponding 780.8 nm run that was
taken directly before the reference run to exclude the effect of long time drifts
in the atom #. The plot shows the best fit of the data with the fit function
A · exp[−Γscatt t] with A and Γscatt as free parameters. The two insets show
example images of the BEC after a certain exposure time to the 780.8 nm
light.

48

a)

8

b)

6

|2,-2〉|1,-1〉

64
4
2

Γscatt [1/s]

2

0

782

784

786

788

790

792

794

0

782

784

786

788

790

792

794

100

c)

80

|1,-1〉

60

40

20

0

778

780

782

λ [nm]

784

786

Figure 4.17: State-dependent scattering rate. a) and b) show the scattering
rate for the two 87 Rb states of interest. Again the solid lines correspond to π,
the thick dotted lines to σ + and the dashed lines to σ − polarized light. Part c)shows the data from table 4.3 with the corresponding theoretical curve. The
dashed dotted line is again drawn at the “zero-lattice” wavelength (see Fig.
4.12). All plots are shown for atoms which sit at the intensity maximum of a
Gaussian beam with a waist of 230 µm and a power of 125 mW.

49

rate at ∼ 785 nm is low enough to give a sufficiently long time scale for
performing experiments with a SDOL without leaving the quantum gas regime.

50

Chapter 5
Optical Lattice Bandstructure
and Bose-Hubbard model
Here we want to give the theoretical and experimental basis for a statedependent two-component lattice experiment. Therefore we first discuss theoretical models of bosons in a lattice and then check the functionality of oursystem with a one-component SF-MI transition. The two-component case willbe treated theoretically at the end of this chapter.

5.1

Solid State Description

The system of atoms in an OL shows strong parallels to the typical solidstate system of electrons moving in the potential of atomic ions. Both systems
exhibit the periodic structure of the potentials and in both the particles whichmove in the potentials can (in a first step) be considered without mutualinteractions. Therefore the atom-OL system can be treated with well known
formalisms from solid state physics in a time independent way. For a latticepotential of infinite extent the eigenenergies of the system lie in continuous
bands which are separated by gaps. The wave function of a particle in such a
1D lattice can be written as a plane wave multiplied by a function with thelattice periodicity (Bloch theorem) [68, 69]:
h qx iuq(n) [x](5.1)
Φq(n) [x] = exp i~where q is called the quasimomentum, is continuous and can be reduced to the
1st Brillouin zone due to the periodicity of the problem whereas n is a positive
integer and labels the energy bands. Inserting this Ansatz into the standard
time-independent Schrödinger equation of a particle in a periodic potential51

V [x] leads to:b = 1 (bp + q)2 + V [x](5.2)
with H2m
The standing light wave potential (neglecting the envelope) may now be written as
V0
V [x] = V0 sin2 [kx] = − (exp[2ikx] + exp[−2ikx] − 2)
(5.3)
4
The above form of the potential suggests to write the lattice part of the Blochfunction as a Fourier sum:X
exp[2iskx]
(5.4)
[x]
=
c(n,q)u(n)sqb (n) [x] = E (n) u(n) [x]
Hu
q
q
q

s

Inserting this sum into eq. 5.2, writing pb as −i~∂x and remembering that noneof the exponential functions in the sum can be replaced by a linear combinationof other exponential functions of the same kind leads to
q 2 V0 (n,q) V0 (n,q) V0 (n,q)
Erec (2s +) +cs − cs+1 − cs−1 = Eq(n) c(n,q)
s
~k
2
4
4

(5.5)
(n)

c0 c = Eq c, withIn terms of linear algebra this equation can be written as H
c0 has one diagonalc being a vector of infinite extension. The Hamiltonian H
and two off-diagonal terms per row. The other entries are zero. Of course the
Hamiltonian is of infinite extension, too. However it happens that the components of the vector decay rather quickly for big |s|. Therefore it is possible totruncate the Hamiltonian and only solve it for |s| ≤ 5 as long as higher bands
are not of interest [61]. Following this path an 11×11 Hamiltonian with real
entries remains. The eigenvalues of this Hamiltonian depend on q and n, givethe eigenenergies of the system and can be calculated by numerically solving
its 11th order polynomial determinant equations. Fig. 5.1 shows the band
structure of the described system for different lattice depths. For an increasing depth V0 the band gap between the 1st and the 2nd band increases whereas
the width of the 1st band decreases. The strict theory applies to a lattice of
infinite size. It can be shown, however, that for a finite lattice the general
results remain valid, but the continuous bands become quasi continuous [69].
(n,q)
The vector components cs
can now (with the diagonal matrix equation that
follows from the described procedure from eq. 5.5) numerically be obtainedfor certain n and q. This leads with eq. 5.4 and eq. 5.1 to the Bloch func(n)tions Φq [x]. It is well known that the Bloch functions spread out over thewhole lattice. In our case we are interested in processes in the lattice such as
52

20

20

20

15

15

20

V0= 0 Erec

E [Erec]15

V0= 10 Erec15

V0= 20 Erec

V0= 5 Erec
10

10

10

10

5

5

5

5

0
1.0 0.5 0.0 0.5 1.00
1.0 0.5 0.0 0.5 1.00
1.0 0.5 0.0 0.5 1.00
1.0 0.5 0.0 0.5 1.0

q [hk/(2π)]
Figure 5.1: 1D band structure of a sinusoidal lattice potential. The plot shows
the first energy bands (different bands in different colors). It can be seenclearly that a band gap between the first (black) and the second (red) opens
up when a lattice is applied (V0 = 5Erec ) and that this gap becomes larger for
deeper potentials. Furthermore the width of the first band becomes smallerwith higher lattice depths.
tunneling or hopping of particles to neighboring sites. Those processes can be
described best with a set of localized wave functions that can be addressed
to one specific lattice site. The so-called Wannier functions which are widelyused in solid state physics represent such a basis set. Therefore we change ourbasis from Bloch to Wannier functions. The connection between the two basis
sets is given by [69]
iqxj
1 X
exp[
] · Φ(n)wn [x − xj ] = √
q [x]
~
N q

(5.6)

where N is a normalization constant and xj labels the lattice site at which the
Wannier function is localized. For us the Wannier function for different bands
and lattice depth can be found with eq. 5.6 by numerically integrating over the
known Bloch functions. Fig. 5.2 shows the real Wannier functions of the firstband for two lattices of different depth. Furthermore the Gaussian groundwave function of a corresponding harmonic potential is shown. It becomes
clear that a particle in the 1st band of a deeper lattice can approximately
be treated like a particle in a harmonic potential. This means also that theWannier function becomes more localized the deeper the lattice is.
A particle in the lattice at the lattice site j in the lowest (=1st) band can

53lattice

Gauss

Wannier3

2.5

V0= 3 Erec

2.0

V0= 10 Erec
2

1.5
1

1.0
0.5

0

0.0
−λ

−λ/2

0

λ/2

λ/2

1
−λ

−λ/2

0

λ/2

λ/2

x [a. u.]Figure 5.2: Wannier functions of the 1st band (black) for different lattice
depth and ground function of harmonic potential (blue, dashed) that taylors
the corresponding lattice potential at its minimum. It can be seen that thepotential becomes more harmonic for deeper lattices.

54now be described by w1 [x − xj ]. In the Bose-Hubbard model (see below) the
hopping or tunneling matrix element t of a particle to a neighboring latticesite is given by [70]:
(1)

(1)

max(Eq ) − min(Eq )(5.7)
4which is the width of the 1st band divided by 4. This is not surprising since thewidth of the 1st band decreases with higher lattice depth (see Fig. 5.1) and one
intuitively expects less tunneling for deeper lattice wells. eq. 5.7 agrees betterwith the exact value of t the deeper the lattice becomes [8]. For a lattice depthof 3 Erec the deviation is 1.8% whereas it reduces to 0.022% for 10 Erec lattice
depth. Approximation eq. 5.7 always overestimates t and stays below 10%
deviation with respect to the accurate value. The other important parameterof the one species Bose-Hubbard model is given by the onsite interaction U
with
Z
4π~2
as |w1 [x]|4 d3 x
(5.8)
U=
m
This term that gives the energy difference between one and two
Q particles persite becomes bigger when the 3D-Wannier function w1 [x] = i=x,y,z w1 [i] ismore localized, meaning that the nearest neighbor tails of the Wannier function
which control tunneling are vanishing more and more. The factor 4π~2 as /mthat contains the s-wave scattering length as can be positive (repulsive interaction as in our case) or negative (attractive interaction) and adjusts the onsite
interaction strength.
The above discussion has been done for the 1D case. For the 3D-case it can beshown however that the problem decouples1 as indicated by eq. 5.8 as long as
the 3D potential can be written as sum of the 1D potentials [8]. This is always
true for orthogonal optical 1D lattices with no effective mutual interferences.
t≈

5.2

The Bose-Hubbard Model

In solid state physics electrons with small energy bands in a crystal canbe described by the Hubbard model [71]. This tight-binding model can for
example be applied to explore physics of band magnetism, high-temperature
superconductors or metal-insulator transitions (Mott-transitions). The extension of this fermionic model to bosons is called Bose-Hubbard model. Fisher
1This means that mathematically now new term or step shows up in the calculation.
Therefore the 3D solution is known if the 1D problem is solved.

55

et al. [24] did this expansion in 1989 and proposed a bosonic lattice systemwith a superfluid and a Mott insulating phase. For lattices with disorder athird phase, the Bose glass phase, was predicted. In 1998 Jacksch et al. [26]recognized that a 3D optical lattice filled with bosons is a system that realizes
the Bose-Hubbard model (BHM) and exhibits a tunability that allows one to
explore the predicted phase diagram.
This chapter introduces the standard one-component BHM and the more complex and much more interesting two-component BHM.

5.2.1

The One-Component Bose-Hubbard Model

The system under study consists of ultra cold bosonic atoms in a periodic
lattice with an harmonic overlying slowly varying trapping potential VT [x]from the Gaussian confinement of the laser beams. In second quantization the
Hamiltonian reads [26]


~2
†
b +
b
∆ + V [x] + VT [x] ψ[x]d x ψ [x] −2mZ
1 4πas ~2b ψ[x]bb † ψb† [x]ψ[x]+d3 x ψ[x]
2 m
Z
b =
H

3

(5.9)

b a bosonic field operator, as the s-wave scattering length of the atoms,
with ψ[x]P
and V [x] = V0 3m=1 sin2 [km xm ] the periodic potential (cf. eq. 5.3). The firstterm of eq. 5.9 contains the kinetic energy and the energies of the potentials
one atom sees. The second term takes interactions between the bosons by thepseudo potential (see 2.1.2) into account. m is the mass of one atom. The
bosonic field operator is now expanded in terms of Wannier functions (see eq.
b =P b5.6) ψ[x]i ai w1 [x − xi ] with the assumption that all the atoms are andstay in the lowest state or band. With this and eq. 5.9 the one-component
Bose-Hubbard Hamiltonian (BHH) follows [26, 72]:

bBH,1 = −t
H

X
hiji

(b
a†i b
aj + ba†j bai ) +X
1 X(i − µ) n
bi + Unbi (bni − 1)2ii(5.10)

Generally the Hamiltonian describes no long range interactions between thebosons. These interactions are small compared to the nearest neighbor energies. Even the next nearest neighbor interactions are 2 orders of magnitude
smaller than the nearest neighbor ones [26]. The terms of the one-component
BHH can be understood intuitively [61].

561. Tunneling or Hopping - The first term of the Hamiltonian describes
hopping of atoms to neighboring sites. The sum is taken over all possible
nearest neighbor combinations. The scalar t gives the hopping amplitude
and is evaluated by eq. 5.7q
or can be approximated (for deep lattices) by
0
0)3/4 exp[−2 EVrec] [8]. In second quantization the hoppingt ≈ √4π Erec ( EVrec
picture translates to annihilation of an atom at site i and creation of an
atom at site j. This process is expressed by the annihilation and creation operators b
ai and ba†j , respectively. The operators obey the canonicalcommutation relations: [bai , ba†j ]− = δi,j , [bai , baj ]− = 0 and [ba†i , ba†j ]− = 0.

2. External confinement and chemical potential - The second termof the BHH contains the energy offset of Rthe ith lattice site, which comesfrom the external confinement [26]: i = d3 xVT [x]|w1 [x−xi ]|2 ≈ VT [xi ].
The last approximation is valid since the external potential VT is only
slowly varying.
The chemical potential µ is a Lagrange multiplier that guarantees for a
fixed mean particle number in a grand canonical calculation.
3. Onsite interaction - The last term of the Hamiltonian describes theinteraction of one particle at lattice site i with all the other particles(n − 1). In the derivation of this term the sum is taken over several
different lattice sites, but can be truncated to one because the interaction(localized Wannier functions) between the atoms is of very short range.
In the tight binding regime
is given by 5.8 canpthe parameter U which
3/4
be approximated by U ≈ 8/πkas Erec (V0 /Erec ) [8]. This is done by
expanding the lattice sites into harmonic potentials and correspondingly
using eq. 5.8 for a Gaussian function. The onsite interaction term isthe complement to t: it localizes one atom at a specific site. The ratio
between U and t plays a crucial role in the phase diagram of the systemthat correspondingly shows two extrema. In one case the atoms are
pinned at the lattice sites (Mott insulator) and in the other case they
are delocalized across the whole system (superfluid phase) [73].
In the experiment the system is realized by adiabatically loading ultra cold87
Rb atoms into a far red detuned optical lattice (see Chapter 4.3). The
parameters U and t can easily be manipulated by changing the intensity ofthe back reflected light and therefore the lattice depth of the system. The
deeper the lattice becomes the more the atoms are pinned at their specificlattice sites. This means that U increases and t decreases with an increasing
lattice depth. It shall be pointed out again that the whole description of thesystem is only valid for the ground state.
57

The Superfluid (SF) Phase
If the onsite repulsion U goes to 0 (very shallow lattice) and the system
is considered to be in its homogeneous limit (i → 0) the ground-state wave
function of the BHH can be approximated as [70]
M
X
b
a†i

|ΨSF it/U →∞ ∝!A
|0i

(5.11)

i=1

A is the number of atoms in the lattice and M is the number of lattice sites.
To make clear what the superfluid wave function looks like it is helpful to write
it explicitly out for freely chosen numbers A and M . For the sake of simplicity
I choose A=4 and M=2. The wave function than reads:|ΨSF (A = 4, M = 2)i ∝ |4, 0i + |0, 4i + 4|3, 1i + 4|1, 3i + 6|2, 2i
From this example it becomes clear that every atom has the same probability to be in any lattice site. Due to the fundamental indistinguishabilityof the atoms, states with symmetric populations (here |2, 2i) are more likely.
Effectively one atom is not pinned at one lattice site but can move freely overthe entire system. Therefore the atom number at one site is completely undefined. Within a measurement one would find a random number of atoms at
a lattice site with a Poisson distribution for repeating this measurement. On
the other hand the phase of the state that can be described as a macroscopic
wave function is well defined which can be seen by looking at the non zero order parameter hb
ai i = hφi |bai |φi i [70] with φi being the wave function at latticesite i. In our example it can be seen that hb
a1 i = hb
a2 i =
6 0. Neglecting U the
superfluid state can be written as a coherent state [61] which is an eigenstateof the annihilation operator and intrinsically exhibits a Poisson distribution.
The Mott Insulator (MI) Phase
In the other limiting case of the homogeneous system an infinitely deep
lattice has to be considered. Now the ground state can be approximated as:|ΨM I it/U →0 ∝

M  
Y
N
|0i
b
a†i

(5.12)

i=1

This state holds as ground function for so called commensurate filling whichmeans N = A/M ∈ integer. For N ∈/ integer a Mott insulating core according
to eq. 5.12 with a new N being the biggest possible integer below the old N
is formed. The rest of the atoms go into the SF state (eq. 5.11). To illustrate58

the MI state I give the same example as for the SF one:
ΨM I (A = 4, M = 2)i ∝ |2, 2iThis is a Fock state with 2 atoms per lattice site. In contrast to the SFcase the atoms are pinned at their sites. In a measurement one always would
get the same atom number per site. In general the MI state has exactly N
atoms per site. The fixed atom number minimizes any phase correlation between the atoms. There are no coherences left. Again this can be seen by theorder parameter hb
ai i which becomes zero. In our example hb
a1 i = hb
a2 i = 0.
A further difference between the SF and the MI phase is that the MI phasehas a finite gap in its excitation spectrum and is incompressible (∂n/∂µ = 0).
Deep in the MI regime the gap has the size of U .
The BH Phase Diagram
From the two previous subsections it has become clear that two extreme
cases exist. In the SF phase the atoms move freely in the lattice potential andthe system can be described by a macroscopic wave function. In the opposite
case the atoms are confined at specific lattice sites and coherences are suppressed. With the description below eq. 5.10 it is clear that the parameters t
and U that drive the transitions from the SF to the MI regime and vice versa
can be adjusted by changing the depth of the optical lattice.
To get a qualitative answer to how a variation of t and U changes the groundstate the phase diagram of the system has to be calculated. This can be donein a mean field approach following [72, 74, 75]. After finding the effective onsite Hamiltonian with a decoupling approximation, second order perturbation
theory is applied to find the ground state energy of the system. After thisthe standard Landau procedure is used which yields the borders of the phases,
dependent on the number of nearest neighbors in the lattice.
Fig. 5.3 shows the result of the calculation for a simple cubic lattice. The
gray loops show the regions where the system is in the MI state. The phase
diagram shows what one expects intuitively: in order to drive the system tothe MI state a sufficiently deep lattice has to be used (t/U small enough).
Furthermore it can be seen that for a high enough local chemical potential µi
at the center of the system (yellow dot), Mott phases with different integer
fillings per site can exist. The more atoms are loaded into the lattice the higher
the local chemical potential at the trap center will be and consequently thehigher the MI filling (labeled by N in the figure) in the center can be.

59

5

N=5
4

ith site

µi

µ /U

N=4
3

N=3

εi

2

N=2

SF

1

N=1
0
0.000

0.005

MI
0.010

0.015

0.020

t /U

0.025

0.030

0.035

Figure 5.3: Phase diagram of one-component BHM for the three dimensionalcase of a simple cubic lattice (6 nearest neighbors). The drawing on the upper
right shows a cartoon of the BEC in the lattice with a Gaussian envelope. The
local chemical potential µi which leads to an onion structure (for sphericalsymmetry) of the system, gives a constant energy if added to the energy offset
i (compare to eq. 5.10). The two arrows (one in the phase diagram, the otherone in the cartoon) show one possible way from the core to the surface of thecondensate.

60Measuring the Phase Transition
In chapter 4 I described theoretically and experimentally how to trap andmanipulate a system of ultra cold atoms in an optical lattice. In this chapter
so far I described the theoretical physics of a system of ultra cold atoms in
an OL in terms of the band model and the BHM. Now these two chapters aregoing to be combined by performing an experiment in analogy to [23] whichuses the setup described in chapter 4 within the treated theoretical frame.
The setup shown in Fig. 4.5 is used with laser light of a wavelength of 1064
nm along all three beam directions. Therefore the potentials are non statedependent (see chapter 4.2.1). The BEC is held in the XODT in the xy-plane.
Now the lattices along all three beam directions are ramped up such that thelattice depth along each direction is the same at every time. The lattice ramp
is sketched in Fig. 5.4 and is exponential which takes care of the adiabaticity
problem (see discussion in 4.3).
Furthermore Fig. 5.4 shows the atom cloud 18 ms after falling freely from a
3D lattice of a certain depth. The first image shows the BEC in the XODTwith no optical lattice. The next image shows the atoms after being confinedin a 3D OL with a depth of 3 Erec . Clear diffraction peaks can be seen whichindicate the the SF phase where the atom cloud can be described with amacroscopic phase that leads to coherences. The shape of the pattern can beunderstood by using the solid state description from 5.1. First one notices
that our lattice is a simple cubic lattice. Performing TOF measurement witha long enough flight time corresponds to a Fourier transform that reveals themomentum distribution of the atoms [76]. For the solid state system understudy this corresponds to a transformation from the real to the reciprocal
space. Since the reciprocal lattice of a simple cubic lattice is again simple
cubic, one expects a highly symmetric interference pattern. This pattern canbe seen in the images and looks always the same no matter along which beam
direction the absorptive image is taken. Therefore the spacing between two
peaks equals 2~k as indicated in one image, where k is the absolute value ofthe wave vector of the used lattice light. In the atom-light interaction picture
this means that atoms in a certain peak always scattered with 2n photons
where n={0,1,2,...}.
For deeper and deeper lattice depth however the interference peaks become
more and more washed out and vanish for sufficiently deep lattices. This
indicates the transition of the system from the SF to the MI state. Fromthe calculation that was used for the plot in Fig. 5.3 and the approximative
formulas for t and U which are given below eq. 5.10 one can estimate the valueof the lattice depth where the system starts to enter the N = 1 MI phase. This
happens theoretically at ∼ 12Erec (see also [8]). Indeed it becomes clear by

61

↓

3

↓

8

13

↓

18

13 ↓

8↓

3↓

0↓

2 hk

↓

0

0 Erec18 Erec

80 ms

10 ms

80 ms

Figure 5.4: Quantum phase transition of |F = 1, mF = −1i atoms. The
lattice is ramped exponentially to a certain value at which all lattice andtrapping beams are switched off and the atoms fall freely for 18 ms where aabsorptive image is taken. The small arrows in the images that are next to thenumbers which give the lattice depth (directly before dropping the cloud) in
Erec indicate if the lattice is ramped up or down. Where ↑ stands for up and↓ for down. The ramp is symmetric, 170 ms long and sketched at the bottomor the figure.

62

looking at Fig. 5.4 that the interference peaks in the 13 Erec ↑ image are lesspronounced than the peaks in the 8 Erec ↑ image. This can be affirmed byan evaluation of the visibility (see eq. 2.15) which is bigger for the later case.
Going to an even higher lattice depth of 18 Erec the peaks vanish and thesystem is clearly in the MI state.
The images which are taken on the the way down (↓) show that the phasetransition happened in a good approximation adiabatically because the SF
phase is reentered for lower lattice depth. This proves that the image at 18
Erec does not show just a thermal cloud due to heating.

5.2.2

The Two-Component Bose-Hubbard ModelIn the foregoing section the one-component BHM was described theoretically and it was shown that the experimental reality agrees well with what
follows from the theory. The next step is now to enrich the system by having
two different hyperfine states in the frame of an OL. Interesting physics canbe explored if the parameters that drive the dynamics and transitions of thesystem can be varied differently for each atomic hyperfine state. It is clearthat the SDOL which was described in 4.5 will play an important role in doingso.
The Hamiltonian
With similar approximations as introduced for the one-component BHM
one can derive a low-energy Hamiltonian in second quantization [77, 78]
bBH,2 = −
H

X

tα (b
a†i,αbaj,α + ba†j,αbai,α ) +(5.13)

hiji,αX
1X
Uα nbi,α (bni,α − 1) + U1,2
nbi,1 n
bi,2
+
2 i,α
i
bBH,2 with H
bBH,1 of eq. 5.10 it first has to be pointed out that
Comparing H
the same approximations (only nearest neighbor hopping, atoms in lowest
Bloch band) have been made for the derivation eq. 5.13. The term that tookbBH,1 is now left out for thecare of the external confinement in the case of H
sake of simplicity. The main and most interesting difference between the twomodels however is a new term in the two-component BHM that takes careof interactions between the two different atomic species. Even though thebBH,2 do not change compared toprincipal meanings of the different terms of H

63

bBH,1 it makes sense to quickly recall their meanings and point out changes.
H
1. Tunneling or Hopping - This first term is equal to the hopping term in
the one-component case with the enhancement that the index α denotes
a summation over the different species 1 and 2. The brackets aroundthe letters i and j indicate the summation over adjacent lattice sites.
The hopping amplitude is now part of the atomic summation and in aharmonic approximation givenq by(αtα ≈ √4π Erec ( VErec) )3/4 exp[−2V (α)] where the lattice depth V (α) equals
Erec[bai,α , ba†j,α ]− = δi,j , [bai,α , baj,α ]− = 0 and

V0 for the atomic species α.
[ba†i,α , ba†j,α ]− = 0 hold. Operators of different atomic species always commute.

2. Onsite intra-species interaction - The second term describes theenergy that has to be used when an atom of species α is put into thesame lattice well with another atom of species α. This term (and the lastone) plays the complement to the delocalizing tunneling term describedby tα if all involved scattering
length are positive. The onsite interactionp(α)(α) (α) (α)
energy is given by Uα ≈ 8/πkas Erec (V1 V2 V3 /Erec 3 )1/4 . Here the(α)
model is slightly extended because different potentials Vκ are possible(α)
along the three axes of the simple cubic optical lattice2 . as gives the swave scattering length that describes the interaction between two atomsof the same kind. Since this onsite term describes an interaction between
equal atoms it is referred to as intra-species interaction. The number
operator nbi,α is given by ba†i,αbai,α .
3. Onsite intra-species interaction - The third term is the new andinteresting one which does not exist in the one-component case. It describes the case when an atom of species 1 happens to sit in the samelattice well as an atom of species 2. Comparing the operator terms between intra and inter-species interaction one sees that in the first case
an atom interacts with all the atoms of its species where self interaction is excluded (that is how the -1 comes in). In the second casethe atom of species 1 interacts with all atoms of species 2. An exclusion of a self interaction is therefore not needed (no -1). The energy of the inter-species interaction in harmonic approximation is givenp(1,2) (1,2) (1,2)(1,2)
by [77] U1,2 =
8/πkas Erec (V 1 V 2 V 3 /Erec 3 )1/4 . The scat(1,2)
tering length as
describes a collision between two atoms with two2

Of course also three different hopping terms for the different directions exist. In the(α)
equation for the hopping term V (α) just has to be replaced by Vκ

64

a)

b)
t1

t2

U2

c)

U1,2

U1

Figure 5.5: Cartoon for two-component BHM. The two different atomic hyperfine states of the atoms are shown in different colors (red and blue). a)shows the hopping of the blue and red atoms to neighboring lattice sites. Of
course the atoms can not only hop to empty sites. t1 and t2 are the hopping
amplitudes (see text). b) shows the intra-species onsite interactions of theatoms which are quantified by U1 and U2 . c) sketches the inter-species onsite
interaction. The magnitude here is given by U1,2 . This interaction between
the two different species gives rise to new and interesting physics.
different internal states.
overlinedq The q
2 lattice potentials are given by(1,2)(1) (2)(1)(2)
V κ = 4Vκ Vκ /Vκ + Vκ.
Fig. 5.5 shows a cartoon of the situation of two different atoms in an OL
potential within the framework of the two-component BHM. In an experiment
the first state could be the 87 Rb hyperfine ground state |F = 1, mF = −1i(red) and the second state |F = 2, mF = −2i (blue). These two states arechosen later for an experiment with two different types of bosons in a SDOL.
Discussions of the two-component BHM and theoretical proposals for different
scenarios can be found in [77–81]. It turns out that there is still (like for the
one-component BHM) a transition between the SF and the MI regime but
different new sub phases appear and the transition points are different.
Example: Phase Diagram Deep in the Mott Phase.
Now we will look at one very specific case which is derived in [80]. The
reason for picking this special case does not lie in its theoretical importance but
in its interesting results which can be achieved with a minimum of numericalcomputation. Therefore this section shall give a motivation why to study
the two-component BHM more closely in a laboratory. Of course many morescenarios in terms of phase diagrams are (theoretically) available.
The parameters for our case are set as follows. The intra-species interactionenergies U1 and U2 of the two different atomic species are equal and called U .
65

We are not in the region where the inter-species term is comparable with theintra-species one but we stay well below (U1,2 < U ), nor can this theory be
applied in the region where U1,2 goes to zero. Furthermore this theory is valid
deep in the MI regime (compare to 5.2.1) where the atoms are pinned down
at one site. Mathematically this is described by U, U1,2 , |U − U1,2 |  t1 , t2 .
For convenience t2 ≤ t1 is taken. The calculation in [80] only holds for an odd
number of atoms at each lattice site. Here we will take one atom for each site
in the MI. To allow for possible checkerboard phases the lattice is divided into
sublattices A and B. These two sublattices are defined that way that theybuild a well known NaCl structure. Now the Ansatz
|ψi =

YY

|ϕA ii |ϕB ij

i∈A j∈B

with|ϕA,B ik = cos [θA,B /2] |1, 0ik + exp [iφA,B ] sin [θA,B /2] |0, 1ik(5.14)

is used. |n1 , n2 ik means n1 and n2 atoms of species 1 and 2 are at site k.
eq. 5.14 tells that at any site there is always one atom but it can be in any
coherent superposition of species 1 and 2.
A perturbative calculation leads now to an energy correction of the groundstate of the MI system. This correction turns out to be [80]

Ecorr

(
N zt21(1 + (t2 /t1 )2 )(1 + cos[θA ] cos[θB ]) += −2U

(5.15)+(1 − (t2 /t1 )2 )(cos[θA ] + cos[θB ]) ++(1 + (t2 /t1 )2 )(1 − cos[θA ] cos[θB ])+ sin[θA ] sin[θB ] cos[φA − φB ]1
2U1,2 /U)+

t2 /t1
U1,2 /U

N is the number of lattice sites and z the number of nearest neighbors. For acertain set of parameters always the state of the system with minimal energy
is occupied. Ecorr has t2 /t1 and U1,2 /U as parameters. Choosing numbers forthese parameters within the allowed boundaries and numerically minimizing
Ecorr with respect to θA,B and φA,B , yields these angles for the chosen parameters where especially θA,B is of interest. φA,B is just a phase (see eq. 5.14).
Doing so one finds that three different physical cases as functions of the parameters exist.

66

1. Ferromagnetic phase (FM) - Here θA = θB = 0. According to eq.
5.14 all atoms are in the state |1, 0i. This reminds one of a ferromagnet
where the magnetic spins are all well aligned in their Weiss domain. The
breaking of symmetry (why not |0, 1i?) happens because t2 ≤ t1 has
been chosen above.
2. Antiferromagnetic phase (AF) - This phase is described by θA(B) =0 ∧ θB(A) = π. This phase shows that the sublattices A and B were notintroduced without any reasoning. One sublattice is filled with atoms of
the first species the other one with atoms of the second kind. Therefore
the system looks like a three dimensional checkerboard.
3. XY phases - Here the plural is used because the angles do not have as
clear restrictions: θA = θB 6= 0. This means the atoms in this phase arein coherent superpositions of |1, 0i and |0, 1i with varying parts of bothas a function of t2 /t1 and U1,2 /U in the XY region.
Fig. 5.6 shows the calculated phase diagram of the described situation. It canbe calculated as described with eq. 5.15. The phase transitions between thephases are found with a simple bisection algorithm.
The discussed phase diagram covers only the deep MI regime. In [80] however
diagrams showing transitions from different SF to the discussed MI phases arecalculated. Interesting scenarios like one species being in the MI state while
the other one still remains in the SF state can exist. Looking at Fig. 5.6 three
general questions arise:
1. The first one regards the experimental realization. From an experimental point of view it has to be pointed out that realizing a system under the above conditions is challenging. Since the hopping rate
(in recoil energy units) of one species does not depend on the latticedirection, the lattice depths V (1) and V (2) can be chosen differently
from each other but have to be the same along every lattice direction, respectively. The onsite energies given in the discussion below
3/4
3/4
(1)
(2)
eq. 5.13 therefore reduce to U1 ∝ as (V (1) ) , U2 ∝ as (V (2) )and
2 3/4
√√(1,2). To explore the calcuV (1) + V (2)U1,2 ∝ as
4V (1) V (2) /lated phase diagram the ratio between U1,2 and U = U1 = U2 and theratio between t2 and t1 have to be scanned. To scan the ratio between
the hopping rates the lattices depths have to be changed. This could
be done by changing the polarization of the lattice light (see Fig. 4.12).
However, to guarantee that U1 = U2 stays valid, the scattering lengths
67

0.8
0.7

FM

U1,2 / U

0.6
0.5
0.4

XY

0.3
0.2
0.1
0.0

AF
0.2

0.4

0.6

0.8

t2 / t1

Figure 5.6: Two-component BHM phase diagram deep in the MI phase withexactly one atom per lattice site (more details: see text). The different phases
are illustrated with small lattice cartoons. The point where all three phases
meet sits at t2 /t1 = 0 and U1,2 /U = 0.25.

68

(1)

(2)

as and as must be changed correspondingly. This can be done by Feshbach resonances (see 6.4). However it is not clear how one can drive twodifferent Feshbach resonances at one time. When changing the lattice
depths, U1,2 is also modified. Another option might be to shift the lattices of species one and two with respect to each other [27]. Therefore allphases of the diagram could be explored. For a more general discussionof an experimental realization of theories in our system see 6.4.
2. The second question is how one would measure these phases if the system
is realized experimentally? The answer is simple [80]: A TOF measurement in combination with a Stern-Gerlach pulse is used (see 2.2 and 2.3).
Having two hyperfine states with different magnetic moments, the two
species would separate. In our case the FM phase would be revealed byonly having one cloud after applying the described method. In the AFcase two clouds with equal atom numbers would exist whereas the ratioof the two clouds in the XY case would be arbitrary. These statements
are only valid as long as the system is not inhomogeneous, such thatdifferent phases cannot exist for one set of parameters. This gives riseto the third question.
3. The third question is more fundamental and difficult. It is well known
that the total spin number Sz of a closed system has to be conserved.
The two-component BHM is equivalently described by labeling the two
atomic species as spins (e.g. ↑ and ↓). If one now starts with a FM
(N↑ 6= N↓ = 0 → SzFM = N↑ ) how could one for instance drive atransition down to the AF phase (compare Fig. 5.6) by varying thelattice parameters, although the total spin is not conserved(N↑ = N↓ → SzAF = 0)? One possible answer could be that the system
has an inhomogeneity that is big enough that different parameters of
the phase diagram at different places are realized. But still the question
remains what happens in a rather homogeneous experimental case? And
could the spin conservation problem suppress a phase transition or doother effects (e.g. Einstein-de Haas effect which describes a transfer toorbital angular momentum [82]) happen?
These are very interesting questions which complete the motivation why thetwo-component BHM should be studied experimentally. The following chapter
will describe our first steps towards the experimental realization of such asystem.

69

Chapter 6
State-dependent two-component
Superfluid - Mott insulatortransition
In the last chapter the functionality of our machine has been shown by
repeating the one-component SF-MI transition [23]. Furthermore theoreticalmodels like the band model and the one and two-component BHM have been
discussed. It has been pointed out that theoretical predictions promise interesting new physics in the case of the two-component BHM. In 4.2, 4.3, 4.4 and4.5 the specific experimental features like the XODT and the lattices (stateindependent and state-dependent) have been probed, both theoretically andexperimentally. This means the needed experimental conditions for a latticeexperiment with two bosonic species in an OL are available.

6.1

Observing the phase transition

A BEC in the |F = 1, mF = −1i state is prepeared in the XODT. Now
three different cases are probed. In the first one, the BEC stays in its originalstate. In the second, the BEC is put into a coherent superposition between|F = 1, mF = −1i (red state) and |F = 2, mF = −2i (blue state) by applying a
microwave pulse which drives a resonant Rabi coupling between the two states(see 2.3.1). In the third case, the atoms are transfered completely into theblue state by a Rabi pulse.
The following procedure is the same for every case: 4 ms after applying the
Rabi pulse the 3D lattice is exponentially ramped to a certain value. Now the3D lattice is switched off and the atoms are imaged after a TOF of 18 ms.
Directly after switching the lattice off, a magnetic Stern-Gerlach pulse that70

separates blue and red atoms is applied. The 3D lattice consists of two 1064
nm lattices in the xy-plane along the ODT1 and ODT2-directions (compare
Fig 4.5) and a state-dependent lattice at 785 nm (σ + -polarized) along the zdirection (see Fig. 4.13). The lattice depth at any point in time is the samefor all directions1 . Fig. 6.1 shows the data. Looking at the left column it canbe seen that the red state undergoes a phase transition from SF to MI just likein Fig. 5.4. This is not surprising and there is no new physics compared toFig. 5.4. Again the separation of the interference peaks equals 2~k. Since theimaging direction is along OL1 one can see that the vertical peak separationis bigger than the horizontal one, with a ratio of 1064/785.
The case of having only blue atoms (right column) is perhaps more interesting.
In good agreement with Fig. 4.15 the |F = 2, mF = −2i state does nearly not
see the 785 lattice. The BEC therefore stays in the 2D SF state which isformed by the 1064 lattice. Comparing the red and the blue BEC in the left
and right column one sees that the blue BEC on the end of the ramp (a↓)) ismuch clearer than the red one. This could be explained by the state-dependent
scattering rates (see Fig. 4.17). For σ + -light at 785 nm the scattering rate of
the red atoms equals 2.7 times the scattering rate of the blue ones. Therefore
more heating in the red case is expected.
The most interesting case is shown in the middle column. Here 50% of theatoms are in the red and 50% are in the blue state. At point c) in Fig. 6.1 thered state sees the full lattice in all directions while the blue state only sees a
non-neglegible lattice in the xy-plane. Consequently the red state goes to theMI phase while the blue one stays in the SF phase at the same time. Therefore
this is the first state-dependent SF-MI transition of 2 inter-convertible speciesof bosons in one lattice where one species stays in the SF phase while the otherone goes into a MI. This proves that the high scattering rate of the 785 light(see 4.5.2) is not a limiting factor for such an experiment.
Looking at the top of Fig. 6.1 one recognizes that the atoms in the middle
column initially have been driven to a coherent superposition between the|F = 2, mF = −2i and the |F = 1, mF = −1i state. However in the text above
a 50:50 mixture which is indeed statistical was mentioned. This mismatch is
explained with Fig. 6.2. It can be seen that atomic spins decohere withina timescale of a few ms. With a spin echo pulse the coherence time can beincreased up to a few 10 ms, but this is still on a smaller timescale than theapplied lattice ramp.
Fig. 6.2 shows that the fraction of |F = 1, mF = −1i atoms does not equilibrate
at 0.5 but at a lower value. This could be because the microwave frequency
1This means the depth measured in Erec is the same in every direction. The recoil energy
depends on the light used in the considered direction.

71

]

1
2

]

+

Lattice ramps - Stern - Gerlach - TOFa )U  0 (OL1, OL2)
=(z)
Er 0,0

b )

U 8
=
Er 9, 1c)U  16=Er 18, 2

U

8

r



U

0

r



b ) E = 9, 1a ) E = 0, 0

a )

c)

b )

54 ms
64 ms

5 ms

b )

a )

54 ms
64 ms

Figure 6.1: State-dependent SF-MI transition. The first column shows only
|F = 1, mF = −1i (red) and the third column only |F = 2, mF = −2i (blue).
The second column shows a 50:50 mixture between red and blue. Note thateach of the 15 pairs of images is a single shot. The first value behind the curly
braces gives the state-independent 1064 nm lattice depth in OL1 and OL2direction. The second and third value behind the curly braces give the 785
nm lattice depths for the red state and the blue state, respectively. Again allenergies are given in units of Erec with respect to the considered wavelength.
At the bottom the symmetric lattice ramp with the drop points of the data(for instance a↓) is shown. All images are taken along the OL1 direction.

72

Fraction of |1,-1〉 atoms

1.0

1.0

different T’s

0.8

different τ’s

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0

1

2

3

4

5

guide to the eye

0.8

6

7

0.0

0

10

20

30

t [ms]

t [ms]

t

t
τπ/2

T

40

τ

τπ/2

T/2

τπ

T/2

τ

Figure √6.2: Decoherence of coherent superposition state. The atoms are put
into 1/ 2(|F = 2, mF = −2i + |F = 1, mF = −1i) by a π/2-pulse. After a holdtime t they are sampled by different short Rabi pulses of lenght τ that covera full Rabi oscillation. The difference between the right and the left data set
is that a spin echo sequence [83] was implemented into the right one.

73

might not exactly hit the atomic resonance (see eq. 2.13). Besides timedependent magnetic fields, the exact length of the first applied Rabi pulse andtechnical issues (imaging) play a role, too.
Therefore Fig. 6.1 shows a statistical mixture of red and blue bosons in the
middle column because the ramp time is much bigger than the decoherence
time.

6.2

Separation of Species

In the recent time there have been discussions and experiments on how
Bose-Fermi mixtures in optical lattices behave [84–87]. For instance there aretheoretical proposals how the presence of fermions would influence the SF-MI
transition [86]. Such an effect is also expected for our two boson mixture
experiment (Fig. 6.1). Here a suppression of the MI is expected [88]. Thiscan be understood if one thinks about the role of the blue atoms that do
not undergo a phase transition into the MI state but stay superfluid. These
atoms interact with the red atoms which are more and more confined whenramping up the state-dependent 785 nm lattice. But there still will be a higher2
effective hopping term of the red atoms because they can use the blue atoms
as “medium”. This would lead to an enhancement of the SF phase.
Experimentally, we try to explore this effect by comparing the scenario shownin the left column of Fig. 6.1 where only red atoms exist with the scenario
in the middle column where red and blue atoms (ratio 1:1) are trapped. The
visibility (compare 2.3.3) of the red atoms is evaluated for both cases. The
lattice ramp is exponential and switched off at certain depths3 (see Fig. 6.3)
on the way up. The ramp time for maximum lattice depth is 100 ms. Since
the visibility is a measure of coherence it tells when the SF phase is left behindand the MI phase is entered (visibility goes from 1 to 0). Fig. 6.3 shows thedata.
It is guaranteed that atom number of the red state does not affect the measurement. Therefor in any case the atoms are put into a 50:50 mixture by amicrowave pulse (compare 2.3.1) while they are confined in the pure XODT.
The green data (joined by dashed green line) is now measured by performing
the above described experiment.
For the red data (joined by red solid line) the |F = 2, mF = −2i atoms (bluestate) are blown away by a 100 µs cycling light pulse (see Fig. 3.1). Thisis done with a small magnetic gradient that separates the species such that
2Compared to the one-component case.
The lattice depth is for all three directions about the same in corresponding recoil
energy units.
374

visibility of |1,-1〉

1.0
0.8

|1,-1〉 withpresence of |2,-2〉

0.6
0.4

only |1,-1〉

0.2
0.0

5

10

15

lattice depth [a.u.]
Figure 6.3: Measurement of visibility in a two-component scenario. The redpoints show the visibility as a function of the lattice depth the red atoms see,for the case of a normal one-component |F = 1, mF = −1i BEC (red state,dashed line) as starting point. The lattice depth is given in arbitrary units
because of calibration issues. The green data points show the visibility of the
red state in presence of the blue state. Each data point is measured threetimes. The error of the visibility arises from the standard deviation of thesethree measurements. The error of the lattice depth is estimated with 3%. Thelines are guides to the eye.

75heating of the red state is minimized.
In both cases the data are taken after a Stern-Gerlach pulse to separate thetwo-components for imaging and a TOF of 18 ms. Fig. 6.3 shows that there
seems to be no fundamental difference between only red atoms in the systemthat undergo the phase transition and red atoms that undergo the phase transition in the presence of a superfluid blue background.
One reason for this could lie in residual magnetic gradients that pull the blue
and the red state apart (∼ ms). Therefore the system would not really be a
two-component system but the red state would sit in one part of the lattice
and the blue one in the other part.
However, this spatial separation due to different magnetic moments of thestates is minimized by nulling the magnetic field gradient at the BEC position
with gradient coils. Doing so, partial spatial overlaps of the two states up to
2 s have been achieved. The shown visibility data was measured under suchconditions.
Another problem might occur if the 785 nm beam is slightly misaligned, hasa non Gaussian internal structure, or an internal polarization profile. In thiscase the beam which is only talking to the red atoms4 will separate the states,
too. In the following section such a beam misalignment is simulated.

6.3

Simulation of misaligned 785 nm beam

As described in the preceding section the experimental problem of a spatial
separation of the atomic species in the XODT was found. Besides residual
magnetic field gradients at the BEC position, a misalignment of the 785 beam
also leads to a separation of the species. Fig. 6.4 shows the result of a simple
classical calculation. It is assumed that the 785 beam is misaligned but still
propagates along the z0 -direction (see Fig. 4.5). This beam with a smallest
waist of 230 µm at x0 = y0 = ∆z0 = 0 is ramped up exponentially to 200 mW
in 50 ms and then held at constant power. The atoms are assumed to be in the
red state. Before the 785 beam is ramped up, they are sitting at the 1064 nm
XODT minimum with the fixed parameters of Fig. 4.3. The calculation is done
under the simplification that there is no interaction between the atoms. The
calculation is a classical one and based on a numerical solution of Newton’s
equations for the center of mass of the BEC in the given beam potentials(time-dependent for the z- beam).
This means that in none of the three directions a lattice is implemented. For
4

An internal polarization profile leads to a non-zero potential seen by the blue state.

76

0

a)

10

t= 0 ms

5

4
6
8

tend =143 ms
0

b)

2

∆z’ [µm]y’ [µm]

15

10
0

5

10

0

15

20

40

60

80

t [ms]

100

120

140

25

c)

20

e)

x’

OD

T

2

x’ [µm]785 z-beam

15

z’
y’

10

0

40

60

80

100

120

140

25

1

20

T

0

OD

r’ [µm]5t=50 ms, Pz=0.2 W

20
15

f)

d)

10

t=43 ms, Pz=0.1 W

5
0

∞

0

200

400

600

800

1000

t=0, Pz=0t [ms]

Figure 6.4: Simulation of misaligned z-beam. The misaligned 785 z-beam isramped up on the atoms in the |F = 1, mF = −1i state, which are sittingin the XODT. Part a) shows the displacement of the atoms in the XODTplane. b) shows the time resolved displacement in the z0 -direction. Part c)and d)
p illustrate the radial displacement for short and long timescales, wherer0 = (x0 )2 + (y0 )2 + (∆z0 )2 . ∆z0 = 0 marks the BEC equilibrium z0 -position
if there is no 785 z-beam, and grows in the negative direction if the BEC falls.
P 0.2;offx  30 106;offy  30 106;zo PotAllD[Econst,1064,1.27,1.08,135,141, 9.81,0,0,0]e) is a cartoon of the XODT and the misaligned z-beam and f) shows the used
exponential ramp. For more information see text.

77the implementation of a lattice one would have to consider the band model (see5.1) in a dynamical way which would complicate the calculations immensely.
For the shown figure the displacement of the 785 beam was chosen to be 30µm away from both ODT axes.
Fig. 6.4 shows the separation between the red and the blue species because
the blue atoms do not see the 785 light. The position of the blue species isx0 = y0 = ∆z0 = 0. The oscillations of the plots in the figure never decay. Thisis because no damping was inserted into the equations. In the real experiment
the spacial oscillations will damp out because of interactions between the atoms
which finally leads to a heating of the BEC. For the shown calculations thiswould lead to a heating of the BEC of 10 nK, with the assumption that there
is no energy transfer to the blue atoms. This also leads to an atom loss
mainly along the z0 -direction. Part a) and b) show that the amplitudes of theoscillations along the x0 and the y0 -direction are by a factor of ∼ 2 smaller than
the amplitude of the z0 -signal. The reason for this is the confinement of the trap
along the z0 -direction (see Fig. 4.3) which is weak compared to the confinementin the x0 y0 -plane. Furthermore by looking at b) and c) it can be seen that theoscillation along all directions start around the same time. This and the nonharmonic oscillation shapes of the curves indicate the anharmonicity of thepotential. It turns out that there is a strong coupling between the x0 andthe z0 , and the y0 and the z0 -direction but a weak one between the x0 andthe y0 -direction. A similar effect has been observed for XODT trap frequency
measurement shown in Fig. 4.4. All oscillation periods differ from the ones
which have been calculated for Fig. 4.3. There are two reasons for this. First,
the simulated oscillation is not harmonic (as discussed above), and second
the trap point of equilibrium has changed due to the misaligned beam. This
also changes the oscillation frequency along the z0 -direction. A change of theconfinement along this direction because of the z-beam itself can be neglected
since the Rayleigh range of the beam is ∼ 20 cm. All statements made in thisclause are qualitatively true, independent of specific simulation parameters.
In conclusion the simulation has shown that under simplified conditions theseparation of the species always happens on the timescale of the 785 ramp (seeparts c) an f)). This however will not hold for the case of an added 3D latticeramp. Here the separation will occur on a slower timescale, especially for deep
lattices and will stop if the MI phase is reached before the BEC went to its
above calculated equilibrium position. If this is not the case separation of
the species will be the same as the calculated one if one applies the suggested
technique (see Fig. 4.8) in order to keep the external confinement constant.
Another 785 beam effect that leads to a separation of the species is an internal
non Gaussian structure (fringes, etc.) of the beam. There are indications that78

this could be the case in our setup.
Therefore the challenge in the near future will be to increase the spatial overlapof the two species by upgrading the properties and the alignment of the 785
beam. With spatial overlap of the two states in the lattice, a clear effect on
the visibility of the red state is expected.

6.4

Comparison with Theory and Outlook

In general comparison with sophisticated models [77–81] is not straightforward. There are several reasons for this. Some theoretical proposals do
simply not fit to our experimental possibilities. For instance [89] requires heteronuclear bosons. But even for porposals that seem to fit our system quite
well [77, 80] challenges occur. To the best of my knowledge, to date there isno paper that discusses the case of equal lattice spacing along two directions(here 1064 nm lattice) but a different spacing along the third direction (785
nm lattice). The extension of the existing theories to this case seems to be
non-trivial, since the symmetry of the system is broken which for instance has
its manifestation in different hopping amplitudes of one species for different
directions. However equal hopping rates along all directions in our setup canexperimentally be achieved via adjusting the 785 and 1064 lattice depths.
Effects of the experimentally indispensable external confinement are often ignored in theoretical treatments. This confinement however might lead to different phases for one specific system with fixed parameters and raises thequestion how one experimentally can distinguish different cases. One experimental attempt could be to try to reduce the external confinement. This
procedure however has boundaries which will be reached quickly in our setup.
Other theoretical restrictions like a fixed chemical potential for both species
or certain filling factors in principle hinder direct comparisons.
On the other hand, it is likely that fundamental properties like new (in comparison to the one-component case) phases still can be seen with our system.
But even if a direct comparison to theory might not be our aim challenges still
exist. To explore different phases in a phase diagram it is necessary to haveaccess to the parameters that drive the transitions. In 5.2.2 it is shown thatall parameters depend on V0 . Therefore the lattice depth that can simply bevaried by changing the lattice laser intensity as an important parameter forscanning through the phase diagram. Shifting the lattices for the two different
species with respect to each other [27] can lead to interesting scenarios, too.
All onsite interaction terms in eq. 5.13 depend on scattering lengths. Looking
at Fig. 5.6 one might want to change the ratio of the inter-species and intraspecies interaction in order to try to drive a phase transition from the FM to79

the AF phase. The scattering lengths could be changed via Feshbach resonances [30, 90, 91]. For 87 Rb these resonances are experimantally not easily
accessible [92, 93]. But the ratio of the interaction terms is experimentally wellaccessible by tuning the lattice depths and the relative position of the atoms(by shifting the SDOL [27]). New phases are expected to be seen clearly forunequal scattering lengths. This is intuitively clear because one importantdifference between the one and the two-component BHM is the inter-species
interaction which significantly changes the scenario if it is preferably unequal
to the intra-species interaction energy. The phase diagram shown in Fig. 5.6
for instance is not valid for U ≈ U1,2 . The scattering lengths of the 87 Rb statesof interest (|F = 2, mF = −2i and |F = 1, mF = −1i) are all ∼ 100 a0 [94].
This could impede the research on new phases. One advantage of our systemover a system with two different bosonic [95] atoms, however, is that we caninter-convert the atomic states.
In conclusion, challenging future experiments have the potential to reveal new
interesting physics.

80

Chapter 7
Conclusion
We have done the first two-component state-dependent SF-MI transition
with dilute 87 Rb quantum gas and thereby proved that the high scattering
rate of light that forms state-dependent potentials, is in principle not a limiting factor for such an experiment.
Furthermore we have discussed the properties of the XODT alone and in connection with back reflected light that leads to the OL1 and OL2 potentials,
and can affect the envelope shape of the dipole trap. A reliable method forlattice calibration has been introduced.
We have performed measurements on the basis of state-dependent potentials
and scattering rates in good agreement with theory and have shown the reliability of the BEC machine by driving the system to a one-component SF-MI
transition.
The next steps will include a search for interactions between the two usedhyperfine species in the two-component case. This will be done by increasing
the spatial overlap between the two species.

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Appendix A
Linewidth of Coherent 899-21
Ring Laser
The linewidth1 of the 899 Ring Laser needs to be known to make sure that
it is narrow enough to use the laser for optical lattice experiments. A too
broad linewidth leads to heating in the atom-lattice system and the desired
circumstances of the BHM are not given any more (see 5.2).
Homodyning2 is a well-known technique for measuring the linewidth of a laser.
Using this technique the laser output is combined with a delayed and frequency
shifted version of itself. The combined signal can then be monitored with aspectrum analyzer [96]. The delay has to be on the order of the coherence
length of the laser light to ensure that in the end the signal consists of two
different wave packets. To achieve this delay, generally an optical fiber isused. In our experiment the linewidth of the 899 Ring laser is expected to bearound 0.5 MHz, requiring a fiber with a length of several hundreds of meters
for a sufficient resolution. Even though there have been improvements on the
resolution of the method [97] still a fiber of the mentioned order of length
would be needed.
To overcome the lack of such a fiber a second laser with an adequate small
linewidth can be used as a reference signal. In a simple picture one can assume
an infinitely small linewidth for two lasers with different frequencies ω1 and ω2 .
Beating this two lasers against each other, one would anticipate interferenceterms at ω1 + ω2 and at ∆ = |ω1 − ω2 |. Since lasers emit light in the range ofhundreds of nm the first term would be in the region of 1015 Hz and therefore
too fast oscillating to be measured electronically. However it is possible tomeasure the-so called beat signal of the two lasers at ω = ∆, as long as the12

linewidth=FWHM
also called self-heterodyning92

spacing is within the frequency range of an oscilloscope or spectrum analyzer(typical value ∼ GHz).
In our case we want to measure the linewidth of the 899 Ring Laser Γ899 byoverlapping its beam with the beam of the DL Pro 780 Laser. The linewidth of
the DL Pro 780 Laser Γ780 is specified to be smaller than 200 kHz and therefore
sharp enough to act as a reference signal. The setup for the measurement isshown in Fig. A.1. The beams of the two lasers with parallel polarizations
are coupled into a fiber to guarantee spatial overlap. Both laser outputs aremonitored with a Fabry-Perót interferometer to ensure single mode operation.

DL Pro 780

PBS
Ring 899

Spectrum Analyzer

λ/2

50:50

L

PD

Figure A.1: Simplified setup for the linewidth measurement of the Ring Laser
899. The polarization of the two laser beams is cleaned up before combining
them on a non-polarizing 50-50 beam splitter. The beams with equal polarization are now coupled into a fiber to ensure perfect overlap. Since the fiber is
not polarization maintaining a third PBS is used before measuring the signal
with a spectrum analyzer via a photodiode. The graphic does not show the
setups used for the polarization spectroscopy, Fabry-Perót interferometer andthe mirrors for aligning the beams.
To find the beat note, both lasers scan at maximum scan width and the signal
at the photodiode3 is monitored with an oscilloscope. By changing the wavelength of the 899 Ring Laser slightly the beat signal at the oscilloscope can be
3

Thorlabs DET 110 with 50 Ω terminating resistor, bandwidth ∼ 160 MHz

93

observed. In order to lock the two lasers at nearly the same frequency, the scanwidth of the lasers has to be lowered by small amounts while the beat notemust not disappear from the oscilloscope. The lasers have to be locked closely
to each other to stay within the bandwidth of the photodiode. The DL Pro
780 is locked to its spectroscopy signal while the 899 Ring Laser is locked toits external cavity. Now one can monitor the signal with a spectrum analyzer4
to view the combined linewidth of both lasers Γc . With a scan time of ca. 2 sfor the whole line, the combined linewidth is measured as Γc = (800 ± 50) kHz,
where the height at half width full maximum equals -6 dBm compared to the
maximum of the line. Fig. A.2 shows a screenshot of the spectrum analyzer
with the described line.

Figure A.2: Photo of spectrum analyzer with beat signal of DL Pro 780 and
899 Ring Laser. The amplitude scale is linear. The span of the whole screen
is 15.2 MHz.
To check the measurement for consistency the 899 Ring Laser is exchanged forthe locked DL 100 Laser which has a specified linewidth of ∼ 500 kHz. In thiscase the combined linewith is (500 ± 50) kHz. Therefore the DL Pro 780 Laser
appears indeed to be a sharp reference signal. In conclusion the linewidth of
the 899 Ring Laser can be estimated to be:Γ899 . 800 kHz
Since a slow drift of the line Γc on the order of 100 kHz·s−1 is observed thegiven value can be considered as an upper bound. For the beat signal of the4

HP 8566A Spectrum Analyzer

94

DL Pro 800 and the DL 100 the combined line was much more stable in time(slow jitter which remains within 50 kHz around the center of the line). Thismeans that the slow drift of Γc is induced by the 899 Ring Laser.
The 899 Ring Laser is well suited for performing an OL experiment.

95

Appendix B
List of Abbreviations
AF

Antiferromagnetic (phase). This phase of the two-component
BHM has a checkerboard structure where atoms of different
species alternate.

AOM

Acousto-optic modulator. This device is used to diffract lightby a sound wave. Frequency shifts of the light in the MHz
range are the result.

BEC

Bose-Einstein condensate. For very low temperatures bosonicatoms macroscopically occupy the lowest energetic state of
their system which then can be described by a single wave
function.

BHH

Bose-Hubbard Hamiltonian. The quantum mechanical Hamiltonian of the BHM.

BHM Bose-Hubbard model. This model describes bosons in a lattice. The dynamic is driven by tunneling rates to neighboringsites and onsite interaction energies.
CCD

Charge-coupled device. The principle is to convert photonic
energy to electrical one. CCD-chips are used in most modern
cameras.

FM

Ferromagnetic (phase). In this phase of the two-component
BHM all lattice sites are occupied by one atomic species.

96

MI

Mott insulator. In this phase of the BHM the atoms are
pinned to specific lattice sites. There is no macroscopic phasewhich would lead to interferences as for the SF phase.

MOT

Magneto-optical trap. Combining red detuned laser beams
and magnetic field gradients, the MOT traps and cools atom
down a few tens of µK

OL

Optical lattice. The simple principle of a standing light wave
and the dipole force (see. 4.1) are here used to trap quantum
gases in a 3D OL.

PID

Proportional - integral - derivative (controller). The controller combines the advantages of an proportional, integral
and derivative controller. The device is used to stabilize asystem at a wanted point by correcting the error with respect
to the wanted point.

RF

Radio frequency. Electromagnetic radiation in the ∼MHZ
range.

SDOL State-dependent optical lattice. An OL that has significantly
different potentials for different internal atomic states. The
state dependence is achieved by the choice of a certain wavelength and polarization of the lattice light.
SF

Superfluid (phase). In this phase of the BHM the atoms movefreely in the OL. There is a macroscopic phase which leads tointerferences in contrast to the MI phase.

TOF

Time-of-flight. The time of flight technique is widely usedin the BEC community. The atoms fall under the effect ofgravitation after switching of the trap. The quantum pressureof the cloud reveals the BEC momentum distribution for longenough TOF.

TOP

Time-averaged orbiting potential (trap). To avoid that ultra
cold atoms undergo Majorana flips in the field zero of thetime independent magnetic trap, a magnetic time orbiting
potential is added which results in a harmonic confinement
for the atoms with no field zero.
97

XODT Crossed optical dipole trap. The XODT is used to trap theatoms independently of their internal state. The atom lightinteraction comes in due to the conservative dipole potential(see. 4.1).
XY

XY (phases). These phases of the two-component BHM consist of coherent superpositions of atoms in state one and two
which are not described by the FM or AF phase.

98