Experimental and Numerical Investigationsof

Ultra-Cold Atoms

Magnus Rehn

Experimental and Numerical Investigations of Ultra-Cold AtomsMagnus Rehn

ISBN 978-91-7264-466-3

c! Magnus Rehn

Umea UniversityDepartment of PhysicsSE-901 87 UmeaSWEDEN

Printed by Print & media

Umea 2007

2

Abstract

I have been one of the main responsible for building a new laboratory for Bose-Einstein condensation with 87Rb. In particular, the experimental setup hasbeen designed for performing experiments with Bose-Einstein condensatesload into optical lattices of variable geometries.

All parts essential for Bose-Einstein condensation are in place. Atoms arecollected in a magneto-optical trap, transferred to another vacuum chamber,with better vacuum, and trapped in another magneto-optical trap. Atoms aresuccessfully transferred to a dark magnetic trap, and system for diagnosticswith absorption imaging has been realized. We have not yet been able toform a Bose-Einstein condensate, due to a range of technical di!culties.

Equipment for alignment of optical lattices with flexible geometry hasbeen designed, built, and tested. This tool has been proven to work asdesired, and there is a great potential for a range of unique experiments withBose-Einstein condensates in optical lattices of various geometries, includingsuperlattices and quasi-periodic lattices.

Numerical studies have been made on anisotropic optical lattices, andthe existence of a transition between a 2D superfluid phase and a 1D Mott-insulating phase has been confirmed. We have shown that the transition isof Berezinskii-Kosterlitz-Thouless type. In another numerical study it hasbeen shown that using stimulated Raman transitions is a practical methodfor transferring atoms between states in a double optical lattice. Thus, itwill be possible to transfer populations between the lattices, with furtherapplications in qubit read/write operations.

Acknowledgements

There are numbers of people that has been invaluable during my years as aPh.D. student and which I would like to thank.

My supervisor, Anders Kastberg, for tremendous support and infinitepatience.

Kristian Støchkel, Robert Saers and Martin Zelan, for all the great timewe have had together in lab and all the other fun we have had.

Peder Sjolund and Henning Hagman, for a lot of fun, sharing the struggleswith the labs and for fruitful collaborations.

Claude Dion, Svante Jonsell, Emil Lundh and Alberto Cetoli, for enlight-ment and great collaborations.

Sara Bergkvist and Anders Rosengren for collaborations.All master students.The rest of the department.

2

Comments to MyParticipation in the OverallWork and Summary of thePapers

I have been active on all parts of building and running the experiment, exceptthe recent rebuild of the vacuum system, where I have been involved in theplanning and in the discussions. I have played the major part in designingand building the dark magnetic trap, building electronics for switching thehigh currents in the magnetic trap, building electronics for fast switching ofthe shutters, and I have developed the scripting utilities for the timing andcontrol of the experiment.

Paper I

A set-up for flexible geometry optical lattices

Submitted to Eur. Phys. J. – Appl. Phys.

We have designed and built a tool for facilitating the alignment of opticallattices with flexible geometry. We have used this tool to setup up a threebeam 2D lattice, and to confirm, by measuring the expansion of the cloud ofatoms in two directions, that we have created the lattice that we expected.

I have been involved in all parts of the design, construction and testing ofthe lattice tool. I was also one of the main responsible for putting the papertogether.

3

Paper II

Transition from a Two-Dimensional Superfluid to aOne-Dimensional Mott Insulator

Phys. Rev. Lett., 99, 110401 (2007)

We have studied the transition between a 2D superfluid phase and a 1DMott insulator phase in a finite anisotropic optical lattice. This have beendone by using an operator loop scheme in a stochastic series expansion Monte-Carlo method. We have shown that this transition does exists and that it isof Berezinskii-Kosterlitz-Thouless type. We also calculated the experimentalsignature we expected from the above described phases. To find physicallyrealistic parameters for the tunneling and on site interaction used in theBose-Hubbard Hamiltonian, we had to calculate the Wannier function forthe atoms in the lattices. For non-cubic (or non-square) lattice, especiallyquasi-periodic ones, this can not be done in the conventional way.

The idea of investigating these properties of atoms in anisotropic opticallattices were born during preparations in the laboratory, and during manydiscussions in the group.

The actual calculations were made by Sara Bergkvist at KTH. I wasinvolved in discussions and in the preparation of the paper.

Paper III

One-dimensional phase transitions in a two-dimensionaloptical lattice

To be submitted.

This paper is a continuation and an extension of Paper II. We have calcu-lated the same exponents as above, but for a slightly di"erent tunneling inthe strongly coupled direction, and seen that those exponents behave as ex-pected. We also checked that the system behaves as expected for di"erentcommensurability for the weakly coupled, and the strongly coupled directionsrespectively. The ambition was furthermore to produce a platform for future,more detailed and diversified, studies of this line of physical problems. Thiswas also accomplished.

4

I prepared the theoretical investigations, wrote the program for the Monte-Carlo simulations and have been responsible for running the program andproducing the data for the article. I was partially responsible for writing thearticle.

Paper IV

Raman transitions in double optical lattices

To be submitted.

By numerically integrating the optical Bloch equations for a four-level system,we have investigated the possibility, and developed a scheme, for using Ramantransitions for transfer of atoms between di"erent hyperfine levels, and thusbetween di"erent lattices, in a double optical lattice. We have investigatedhow sensitive this process is to di"erent parameters, such as timing and errorsin the laser frequencies.

I took part in the discussions when formulating the problem theoretically,and I wrote the program used for solving the optical Bloch equations.

5

Contents

1 Introduction 1

2 Bose-Einstein Condensates 32.1 Some Past and Current Research on BEC . . . . . . . . . . . 6

3 Experimental Techniques 83.1 Atom-Light interaction . . . . . . . . . . . . . . . . . . . . . . 83.2 Magneto-Optical Cooling and Trapping . . . . . . . . . . . . 9

3.2.1 Optical Molasses . . . . . . . . . . . . . . . . . . . . . 93.2.2 Magneto-optical trap . . . . . . . . . . . . . . . . . . . 103.2.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 Optical Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 Magnetic traps and Evaporative Cooling . . . . . . . . . . . . 13

3.4.1 Evaporative Cooling . . . . . . . . . . . . . . . . . . . 133.4.2 Forced evaporation . . . . . . . . . . . . . . . . . . . . 143.4.3 Runaway evaporation . . . . . . . . . . . . . . . . . . 143.4.4 Radio frequency induced evaporation . . . . . . . . . . 15

4 Experiment 164.1 Path towards BEC . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 Atoms from heated getter . . . . . . . . . . . . . . . . . . . . 184.4 MOT I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.5 Transfer Between the MOT:s . . . . . . . . . . . . . . . . . . 194.6 The Dark Magnetic Trap . . . . . . . . . . . . . . . . . . . . 204.7 Transfer to the Magnetic Trap . . . . . . . . . . . . . . . . . 20

4.7.1 Optical Pumping . . . . . . . . . . . . . . . . . . . . . 224.7.2 Mode Matching . . . . . . . . . . . . . . . . . . . . . . 22

4.8 Evaporative Cooling . . . . . . . . . . . . . . . . . . . . . . . 23

6

4.9 Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.9.1 Optical Fibers . . . . . . . . . . . . . . . . . . . . . . 234.9.2 Shutters . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.10 Control and timing of the Experiment . . . . . . . . . . . . . 244.11 Imaging and Diagnostics . . . . . . . . . . . . . . . . . . . . . 254.12 Alignment of Optical Lattices . . . . . . . . . . . . . . . . . . 25

5 Status of the Experimental setup 28

6 Raman Transitions in Double Optical Lattices 316.1 Stimulated Raman transitions . . . . . . . . . . . . . . . . . . 316.2 Population Transfer in DOL . . . . . . . . . . . . . . . . . . . 32

7 Quantum Monte-Carlo Methods 367.1 Monte-Carlo methods . . . . . . . . . . . . . . . . . . . . . . 36

7.1.1 Detailed balance . . . . . . . . . . . . . . . . . . . . . 377.1.2 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . 377.1.3 The Metropolis algorithm . . . . . . . . . . . . . . . . 38

7.2 Stochastic Series Expansion . . . . . . . . . . . . . . . . . . . 397.3 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.4 Updating Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 40

8 Quantum Phase Transitions in Optical Lattices 418.1 The Bose-Hubbard Model . . . . . . . . . . . . . . . . . . . . 418.2 Phases in optical lattices . . . . . . . . . . . . . . . . . . . . . 42

8.2.1 Isotropic Lattices . . . . . . . . . . . . . . . . . . . . . 428.2.2 1D Lattices . . . . . . . . . . . . . . . . . . . . . . . . 428.2.3 Anisotropic 2D Lattices . . . . . . . . . . . . . . . . . 44

8.3 1D Phase Transitions in a 2D Optical Lattice . . . . . . . . . 45

9 Conclusions 47

7

Chapter 1

Introduction

1995, the first Bose-Einstein condensate was realised by the group of EricCornell and Carl Wieman at JILA, in Boulder, Colorado [1, 2, 3]. This was70 years after the prediction of this exotic state of matter, made by AlbertEinstein, building on work of Satyendra Nath Bose, [4, 5]. Bose had studiedthe statistical properties of photons. Einstein generalized Bose’s theory tomassive, what we now call bosonic, particles. Einstein realise that coolingatoms to very low temperature, the atoms would fall into the lowest possibleenergy state and condense into a new form of matter. This new form ofmatter would then form a giant (on the atomic scale) coherent matter wave,whose properties is still not fully understood.

One of the scientific advancements that made Bose-Einstein condensationpossible was the advancement of laser cooling [6, 7, 8]. With laser cooling,lasers are used to slow atoms down. During the last two decades, a hugevariety of techniques for controlling atoms with light have been developed.One of them is optical lattices, where standing waves of light produce periodicpotentials for the atoms [9, 10, 11].

The abilities to cool atoms to extremely low temperatures, as well as thenew ways to control atoms with light, have opened up a new angle of attackfor a lot of research; from fundamental studies of quantum mechanics of manybody systems to the prospect of building quantum computers.

The Ultra Cold Matter group at Umea University, has been working withlaser cooling and optical lattices for many years [12, 13, 14]. In recent years,we have also been working on an experimental setup for Bose-Einstein con-densation, with the intent to study coherent matter in optical lattices.

I have mostly been working on the experimental setup for Bose-Einsteincondensations. I have also been involved in some theoretical studies of phase

1

Introduction Chapter 1.

transitions of cold atoms in optical lattices, and in some calculations regard-ing Raman transitions in double optical lattices.

I will start this thesis with some of the theoretical backgrounds to Bose-Einstein condensation, laser cooling and optical lattices. That will be followedby a description of the experimental setup and of the current status of theexperiment. This part of the thesis also includes the introduction of a newtype of alignment tool for optical lattices. Finally, chapters 6-8 outlines workmade with numerical simulations and calculations for cold atoms in opticallattices, partially performed as as prestudies of planned experiment.

2

Chapter 2

Bose-Einstein Condensates

Quantum mechanical particles can be divided into two classes, with verydi"erent properties [15]. Fermions, have half integer spins and are describedby anti-symmetric wavefunctions. The anti-symmetry of the wavefunctionleads to the Pauli exclusion principle, meaning that no two fermions may bein the same state at the same time.

The bosons have integer spins and are described by a symmetric wave-function. They are thus very di"erent and there is no limit to the number ofbosons in the same state. It is even so that the more bosons there are in acertain state, the larger is the probability that another boson ends up in thesame state.

At high temperature, these properties do not make much di"erence. Themean occupation number for each state is then much less than one, so thatneither the Pauli exclusion principle nor the property that bosons tend tobunch together matter very much. Thus, at high temperature the distributionof states is described by the usual Maxwell-Boltzmann distribution [15]:

f(!) =1

e(!!µ)/", (2.1)

where f(!) is the number of particles per state with energy !. µ is thechemical potential and " = kBT , where kB is the Boltzmann constant.

At the very lowest temperatures, the situation changes. The fermions fillall the lowest states with exactly one particle in each state. The bosons, onthe other hand, end up with the vast majority of the particles in the loweststate, see fig. 2.1. A large collection of particles in exactly the same statewill form one large coherent wave-function.

3

Bose-Einstein Condensates Chapter 2.

a. b. c.

Figure 2.1: Illustration of di!erent distributions in an harmonic potential.a. “Hot” particles. b. Fermions. c. Bosons.

In the following, the conditions necessary for getting a significant part ofall atoms in the ground state will be derived, following [15].

The distribution of a collection of bosons are described by the Bose-Einstein distribution:

f(!) =1

e(!!µ)/" " 1. (2.2)

For a system with a given density of states, the chemical potential µ is de-termined by the total number of particles and the temperature. Setting theground state energy equal to zero, we have the ground state occupation:

f(0, ") =1

e!µ/" " 1. (2.3)

When the temperature approaches zero, all particles will be in the groundstate:

N =1

e!µ/" " 1# 1

1" (µ/")" 1= " "

µ. (2.4)

Thus, for low enough temperatures, the magnitude of

µ # " "

N(2.5)

4

Section 2.0

will be very small for any significant number of particles.For a particle in a box with volume V , the density of states is [15]:

D(!) =V

2#2

!2m

h2

"3/2

!1/2, (2.6)

where m is the particle mass.It is convenient to approximate the sum over the occupation numbers by

an integral, but this approximation breaks down close to the ground statesince D(!) = 0 for ! = 0, and for low temperatures the ground state can notbe neglected. In fact, the occupancy of the ground state might be very largefor low enough temperatures. We therefore count the number of particles inthe ground state N0(") and the excited states Ne(") separately to still beable to integrate over the excited states:

N =#

n

fn = N0(") + Ne(") = N0(") +$ "

0D(!)f(!, ")d!. (2.7)

The number of atoms in the excited states is then

Ne(") =$ "

0D(!)f(!, ")d! =

V

4#2

!2m

h2

"3/2 $ "

0

!1/2

e(!!µ)/" " 1d!, (2.8)

or with x $ !/"

Ne(") =V

4#2

!2m

h2

"3/2

"3/2

$ "

0

x1/2

e!µ/"+x " 1dx. (2.9)

The factor "3/2 gives the temperature dependence of the number of atoms inthe excited states.

Since µ is very small for low temperatures, see eq. 2.5, the approximatione!µ/" # 1 can be used. Using this we can evaluate the integral in 2.9:

$ "

0

x1/2

ex " 1= 1.306#1/2. (2.10)

Thus the number of atoms in the excited states is

Ne =1.306V

4

!2m"

#h2

"3/2

=2.612V

$3dB

(2.11)

where we have introduced the thermal de Broglie wavelength

$dB =

%2#h2

m". (2.12)

5

Bose-Einstein Condensates Chapter 2.

A transition to BEC is said to occur when there is a significant fractionof the atoms in the ground state, that is when

Ne =2.612V

$3dB

< N, (2.13)

which we write asN

V$3

dB > 2.612. (2.14)

Thus the condition for the phase transition to BEC is

n$3dB = 2.612, (2.15)

where n = N/V is the number density and

n$3dB (2.16)

is the phase space density.

2.1 Some Past and Current Research on BEC

One of the most pronounced things about the BEC, with all the atoms inthe same state, is that it is forming a giant coherent wave function. Oneof the first things that was done, as soon as anyone had managed to createcondensate with enough atoms, was to let two condensates interfere with eachother [16]. Letting two condensates fall and then observing the interferencefringes form when the condensates start to overlap.

When angular momentum is added to a BEC, either by varying the con-fining potential or stirring it with a laser beam, more of the very quantummechanical nature of a BEC is pronounced. As the angular momentum ofeach atom is quantized, the energetic most favourable configuration is notone big vertex of the whole condensate, but many small vortices. Thus, themore angular momentum that is added to the condensate, the higher numberof vertices. This is another important branch of research on BEC [17, 18].

Even though the above derivation of the conditions for Bose-Einsteincondensation is made for non-interacting particles, the interaction is veryimportant for the dynamics of the condensates. A lot of study has gone intothe dynamic e"ects of BEC, such as excitations of condensates and soundwaves in condensates [2, 3].

Studies has been done on condensates containing more than one speciesof atoms. Not all species of atoms have properties that make them suitablefor the cooling techniques used to create BEC:s. One way to get around this

6

Section 2.1 Some Past and Current Research on BEC

problem is to cool the species that is hard to cool with sympathetic cooling.Cooling a species that is hard to cool with one that is easy by letting themoverlap spatially and wait until they have attained the same temperature asa consequence of interspecies collisions [2, 3].

The scattering length of the atoms can be tuned by applying a strongexternal magnetic field, using Feshbach resonances. The interaction can thenbe tuned within a huge range. This has been used to study in detail how theinteraction influences the static and dynamic properties of the condensate.Also molecular condensates have been form with this technique [2, 3].

I would also like to mention the impressive work by the group of LeneHau. By manipulating the optical properties of BEC:s they have managedto slow and even temporarily stop light inside a BEC [19].

An Optical Lattice (OL) is a periodic potential suitable for cold atomsthat can be formed by interfering light beams. More about OL further onin this thesis. A lot of research have been centered around BEC:s in opticallattices [9]. Depending on the dimensionality of the lattice, BEC:s in one andtwo dimensions have been studied.

In one dimension, so called fermionization of strongly interacting bosonshave been studied. This research sprung from the observation done by Mar-vin Girardeau in 1960 [20] that there is a very simple mapping betweenstrongly repulsive bosons in one dimension and non interacting fermions inone dimension.

By varying the strength of the optical potential, transitions between dis-tinct phases have been observed. From a Mott-insulating phase, where eachatom is localised in a certain potential minima without any well defined phase,to a superfluid phase where the atoms are not localised but instead form onegiant wavefunction and having a well defined phase [21].

The unique control of coherent atoms made possible by OL:s have beensuggested to be used for quantum information manipulation, and ultimatelyas a way for quantum computations [22].

7

Chapter 3

Experimental Techniques

This chapter contains some of the theoretical background for the experimentaltechniques we use to trap and cool atoms, as well as a section about opticallattices.

3.1 Atom-Light interaction

The forces on an atom in a light field can be divided into two fundamentallydi"erent kinds [23]. There is a dissipative force that relies on the scatteringof photons. This force is strong when the frequency of the field is close to anatomic resonance, but quickly gets weaker when the detuning gets large. Italso saturates; increasing the irradiance over a certain level does not increasethe force significantly. We use the dissipative force for rapid acceleration ofan atomic cloud, sec. 4.5, but mostly to cool an ensemble of atoms, sec. 3.2.1and 4.4.

The other fundamental kind of force is the dipole force. It stems fromthe fact that the atomic energy levels shifts in the presence of a light field,resulting in a force that is proportional to the gradient of the irradiance[23]. This force is less dependent on the detuning, and it does not saturate.Since it is a conservative force, it can not be used for cooling on its own,but nevertheless plays important roles in di"erent cooling mechanisms, e.g.Sisyphus cooling [23]. We use it for optical lattices, created by standingwaves, see section 3.3.

8

Section 3.2 Magneto-Optical Cooling and Trapping

3.2 Magneto-Optical Cooling and Trapping

3.2.1 Optical Molasses

When an atom scatters a photon, a small part of the photon momentum istransferred to the atom: #p = h(k" k#), where k = 2#

$ and k# = 2#$! are the

angular wave numbers of the incoming and the scattered photon respectively.The scattering rate is [23]

dN

dt=

%

2s0

1 + s0 + (2&/%)2. (3.1)

% is the natural line width of the transition, & = 'L " 'A is the angulardetuning of the light from the transition resonance and s0 = I/Is the onresonant saturation parameter. Is = #hc%/3$3, where $ is the wavelength ofthe light, is the saturation irradiance.

For many scattering events, the momentum of the spontaneously emittedphotons average out, but if the incoming light has a certain direction, as ina plane wave, the scattered light will result in a net force. For many cycles,the time average of the force %F& = dN

dt hk, where dNdt is the scattering rate.

Thus, with expression 3.1, we can write

F =hk%

2s0

1 + s0 + (2&/%)2. (3.2)

In this expression, and from now on, F denotes the average force for manycycles.

An atom with velocity v will experience a Doppler shifted frequency,changing the e"ective detuning to:

&# = & " v · k. (3.3)

An atom moving towards the direction of a beam of light will experience lightcloser to resonance and thus a greater force. With two opposing lasers, withwave vectors k and "k the force will be

F =hk%

2s0

1 + s0 + (2(& " v · k)/%)2" hk%

2s0

1 + s0 + (2(& + v · k)/%)2. (3.4)

Expanding this for v around zero, the force becomes:

F =8hk2

%

&s0

(1 + s0 + 4&2/%2)2v + O(v3). (3.5)

9

Experimental Techniques Chapter 3.

With red detuned light (& < 0) this results in

F # "(v, (3.6)

where ( is equivalent to a friction coe!cient. Thus, this is a viscous force,slowing down atoms and thus cooling an ensemble [23].

3.2.2 Magneto-optical trap

In a magnetic field, B, the transition frequency will be shifted due to theZeeman e"ect. For simplicity, assume a two-level atom with ground stateJg = 0 and excited state Je = 1. The sub-levels in the excited state will beshifted by [23]

#E = BµBgJMJe , (3.7)

where µB = eh/2mec is the Bohr-magneton, gJ is Lande’s g-factor, MJe isthe quantum number for the projection of the magnetic moment onto thedirection of the magnetic field, and me is the electron mass.

In a gradient magnetic field, B(z) = B#z, the shifts of the magnetic sub-levels depend on the position of the atom, see fig. 3.1. Hence the e"ectivedetuning becomes

## = # + B#zµBgJMJe . (3.8)

)+-polarized light, propagating in the positive z direction, has a scatteringmaximum to the left (see fig. 3.1) of the center of the trap for the transitionbetween the ground state and the excited state Je = 1,MJe = 1. )! polarizedlight, propagating along "z, has a scattering maximum to the right of thecenter, for the transition between the ground state and the excited stateJe = 1,MJe = "1. The result is a net force that is everywhere directedtowards the center and that has a minimum there. Using expression 3.8 inexpression 3.4 and expanding for z we get:

F = " 8gJ hk##µms0B#

%&1 + s0 + 4!!2

%2

'z + O(z3) # "*z. (3.9)

This is a restoring force where * corresponds to a spring constant. Hence,the total force on the atoms in a MOT is [23]

F = "(v " *z, (3.10)

which is a damped harmonic oscillator. For realistic parameters, it is astrongly overdamped oscillator, and thus very e!cient for trapping and cool-ing.

10

Section 3.3 Magneto-Optical Cooling and Trapping

E

z

!-!+

MJe = 1

MJe = 0

MJe = -1

h!L

Jg = 0

Je = 1

MJg = 0

Figure 3.1: Scheme for the confining forces in a MOT. The magnetic sublevesare shifted due to a magnetic field gradient. !+-polarized light, propagatingin the positive z direction, has a scattering maximum to the left of the centerof the trap for the transition between the ground state and the excited stateJe = 1, MJe = 1. !! polarized light, propagating along !z, has a scatteringmaximum to the right of the center, for the transition between the groundstate and the excited state Je = 1, MJe = !1.

3.2.3 Limitations

Even though the scattered photons leave in random directions and aver-age out, the recoil energy will limit the minimum temperature, that can bereached in a MOT, to a few µK for Rb.

The denser the cloud is, the higher is the probability that a scattered pho-ton scatters again [23]. This will result in an outward pressure that limits theatomic density. The maximum number density, n, obtainable in a standardMOT is of the order of 1011 atoms/cm3. Thus, the space-phase density, eq.2.16, that can be reached in a MOT is of the order of 10!5. Comparing thisvalue with eq. 2.15, it is apparent that we need other techniques to reachBEC, such as evaporative cooling described in sec. 3.4.1.

11

Experimental Techniques Chapter 3.

3.3 Optical Lattices

For large detuning, the scattering rate, and thus the radiative pressure isnegligible. Instead, the dipole force is dominant, forming a potential whichis proportional to the irradiance1 [24]:

Udipole(r) =3#c2

2'0

%

&I(r). (3.11)

When the light is red-detuned, relative to the atomic transition frequency, theenergy shift is negative. When the light is blue-detuned the shift is positive.

By creating a standing wave of two or more laser beams, it is thus possibleto create a periodic potential of light, an Optical Lattice (OL). If we writethe field of a plane traveling wave as

Ei(r, t) = EiRe[+ie!i(&t!k·r!'i)], (3.12)

we can write the potential from several beams with angular frequency ',angular wave vectors ki, polarisations +i, and phases ,i as

U(R) = "3#c3!0

4'

%

&

(

)#

i

E2i +

#

i $=j

EiEj(+i · +j) exp i[(ki " kj) ·R + ,i " ,j ]

*

+ .

(3.13)Thus, the reciprocal lattice is given by the di"erences ki " kj between thedi"erent wave vectors [25].

The majority of the optical lattices in 2D and 3D have been constructedusing 2 and 3 orthogonal beam pairs respectively. These kinds of lattices arerelatively easy to align and the beams are more often compatible with theview ports in the vacuum chamber, containing the sample of atoms. Also, theresulting lattice will be simple cubic, or square in 2D, which has advantagesfor analysis and comparisons with theoretical models.

A disadvantage with creating e.g. a 3D lattice with three pairs of orthog-onal lattice beams is that that setup gives a system with 5 relative phases,but only three external degrees of freedom. This means that any fluctua-tions in the phases of the beams change the topography of the lattice [10].This can be solved without changing the beam path through the sample.One way is to interferometrically stabilize the phases of the beams [26]. Theother way is to have distinctly di"erent frequencies for the beam pairs [9].

1The dipole potential, as it is written here, is for a simple 2 level atoms. For a morerealistic multi-level atom, small corrections that depend on the polarization must be made[24].

12

Section 3.4 Magnetic traps and Evaporative Cooling

The latter method gives fast frequency beats, but as long as these are reallyfast, compared to the time scale of atomic motion, only the time average isrelevant.

One way to completely avoid the problems by phase fluctuations, in theorthogonal beam pair setups, is to use fewer beams and not align them or-thogonal to each other. For a periodic optical lattice, this means using 4beams for a 3D lattice and 3 beams for a 2D lattice [27, 25, 12, 13]. In thisway, the phase fluctuations only leads to global translations of the lattice andno change in topography. The translations is usually so slow (mechanic oracoustic) that it is irrelevant on the time scale of the studied processes.

Non-orthogonal lattice beams have more advantages than avoiding theproblem with phase fluctuations. Now, the angles can be chosen such thatany Bravais lattice can be created [25]. Furthermore, by introducing morebeams, more complex lattice potentials can be created, notably super-latticesand quasi-periodic ones [28].

3.4 Magnetic traps and Evaporative Cooling

An atom, in a particular hyper-fine structure state F , MF , and in a magneticfield, has the potential energy

E = gF µBMF B. (3.14)

MF is the projection of the angular momentum F along B. gF is the Landeg-factor that depends on the quantum numbers J , S, L, and F , see e.g. [23]for details. By producing a local minimum in the magnitude of a magneticfield, it is thus possible to make a magnetic trap for neutral atoms withoutlight. For details about the trap we are using, see section 4.6.

3.4.1 Evaporative Cooling

The magnetic trap is conservative, and often harmonic, depending on the con-figuration. To increase phase space density, we use evaporative cooling. Thismeans that the most energetic particles in a thermal distribution leave thesample, leaving the remaining particles with lower average energy. A close-at-hand example is hot co"ee that gets colder when the hottest moleculesleave. Evaporative cooling is reviewed in [29].

Particles in a thermal distribution with energy higher than the confiningpotential can escape and this leaves the reminder of the sample with a loweraverage energy. In a gas, the high energy tail will be continuously repopulatedby elastic collisions. The higher the potential, the more energy the escaping

13

Experimental Techniques Chapter 3.

particles carry from the distribution. In this way, the evaporation can bevery e!cient in the sense that a very small amount of particles take a verylarge amount of energy from the sample.

On the other hand, the higher the potential, the longer it will take forelastic collisions to redistribute the energy so that some particles have enoughenergy to escape. Thus, the evaporative process becomes very slow.

Other processes than evaporation lead to loss of particles. For 87Rb in adark magnetic trap, one major loss process is collisions with the hot back-ground that kicks out atoms. Under certain circumstances and for someatoms (notably Cs), inelastic collisions become an important loss process.These are two-body collisions, where spin relaxation and dipolar relaxationmake atoms leave the weak field seeking state, and three body recombinationprocesses where diatomic molecules can form [29]. These “bad” loss pro-cesses limit the lifetime in the trap and therefore limit the e!ciency of theevaporation.

3.4.2 Forced evaporation

In a potential, constant in time, the portion of the population that has energyhigher than the potential barrier decreases with temperature. This makes thespeed of the evaporation decrease exponentially with time [29]. Therefore,the potential should be lowered with temperature to keep a high evaporationrate. This is usually called forced evaporation.

3.4.3 Runaway evaporation

In a box potential, the sample becomes more and more dilute in the evapo-ration process. Therefore the evaporation will slow down, since it is propor-tional to the collision rate n)v, where n is the mean number density, ) is thecross section for elastic collisions and v is the mean velocity of the particles.

We define a parameter - in such a way that the volume of the samplescales with temperature as T ( . - = 3 for a 3D linear potential, - = 3/2for a 3D harmonic potential and - = 0 for a box potential, see [29]. In apower law potential in d dimensions, U(r) ' rd/( , the elastic collision rate isproportional to

n)v ' N1!)((!1/2), (3.15)

where N is the number of particles and

( =d(lnT )d(lnN)

(3.16)

14

Section 3.4 Magnetic traps and Evaporative Cooling

is a measure of the decrease in temperature per the number of particlesremaining. [29]. For

((- " 1/2) > 1 (3.17)

the elastic collision rate will increase with decreasing number of particles.Hence the evaporation process accelerate, which is known as runaway evap-oration.

3.4.4 Radio frequency induced evaporation

Lowering the potential by just weakening the magnetic field will also reducehow tightly the atoms will be confined in the trap, and thus the density.This makes it di!cult to maintain e!cient evaporation and to increase thephase-space density.

Atoms may be removed from the trap by inducing spin flips [2, 3]. This isdone with an oscillating electromagnetic field tuned to a hyperfine transition

'rf =B(r)µBgF

h. (3.18)

Since the trapping potential is

Utrap = MF µBgF B(r), (3.19)

only atoms with a total energy

E > h'rf |MF | (3.20)

will be evaporated. In this way it is possible to expel only atoms with highenergy and the evaporation can be finely controlled.

15

Chapter 4

Experiment

In this chapter, I will give an overview of the path towards a Bose-EinsteinCondensate (BEC) that we have followed. That will be followed by an de-scription of the experimental apparatus in use right now. One of the maingoals with the setup is to do experiments with cold atoms in optical latticeswith variable geometry. I will describe the equipment and procedures usedto do this.

4.1 Path towards BEC

Several steps is necessary to achieve the extremely low temperatures neededfor Bose-Einstein condensation. A hot vapor of rubidium atoms is generatedfrom a getter. We then trap the cold tail of the thermal distribution of theisotope 87Rb in a Magneto-Optical Trap (MOT), see sec. 3.2.2. 87Rb is theisotope that is used for the whole experiment. A partial energy level diagramis shown in fig. 4.1

The atoms need further cooling and compression than what is possible ina MOT (sec. 3.2.3), in order to reach BEC. That will be done with evapo-rative cooling in a Dark Magnetic Trap (DMT), sec. 3.4. The evaporativecooling process is slow. The hot vapor from which we have collected theatoms reduces the lifetime of the cloud of atoms in a magnetic trap. Thus,before we can start with evaporative cooling, we transfer the atoms from thechamber where we collected the atoms, to another vacuum chamber, wherethe pressure is about 2-3 orders of magnitude lower. The transfer is done bypushing the atoms, with a short pulse of a resonant beam, through a smalltube connecting the chambers. It is this small tube that make it possible to

16

Section 4.1 Path towards BEC

5p 2P3/2

5s 2S1/2

267 MHz

157 MHz

72 MHz

6.8 GHz

780.241 nm384.230 THz

!

Repumper

Probe and Push

MOT

F = 3

F = 2

F = 1F = 0

F = 2

F = 1

Figure 4.1: Transitions used for the cooling and trapping beams and for theprobe and push beams. The detuning " of the MOT-beams, is most of thetime 12.5 MHz—larger during a short cold MOT phase just before the transferto the magnetic trap.

have several orders of magnitude di"ering pressure between the chambers. Inthe second chamber, the atoms are again trapped in a MOT. Here the cloudis further cooled and compressed, before the atoms are transferred to a darkmagnetic trap. This transfer is performed by turning o" the MOT and thenramping up the magnetic fields that make the magnetic trap, in a controlledmanner. In the magnetic trap, the evaporative cooling will commence (sec.3.4.1). This is done by applying a radio frequency (RF) field, which is reso-nant with a transition to an untrapped state for atoms far from the center ofthe trap. The only atoms reaching this are the most energetic ones. In thisway, these are selectively removed from the trap. After thermalization, theremaining atomic cloud is cooler and denser than before. By slowly rampingdown the radio frequency, the atoms are cooled even further until the phasetransition to BEC is reached.

The setup for achieving a BEC is a complicated machine. Each part hasto finely tuned and each event has to be exactly timed. There is need for a

17

Experiment Chapter 4.

good vacuum system to achieve ultra high vacuum. We need a good source ofrubidium atoms; a source that gives rubidium but nothing else, which wouldinterfere with the vacuum. We need equipment for narrow band laser lightexactly tuned relative to the transition frequencies. We need stable magneticfields for the MOT:s and for the dark magnetic trap. We need a source for theRF-field for the evaporative cooling. A computer system is used to controlthis setup and to time each event. On top of that we need a system fordiagnostics and measurements.

4.2 Vacuum System

The background vapor of thermal atoms scatter atoms out from MOT:s andthe magnetic trap. This leads to a decreased lifetime of the sample in thetraps. The evaporative cooling is a slow process, and a too short a life-time makes it impossible to reach conditions for BEC. Thus we need a goodvacuum system for these experiments.

Our vacuum system consists of two chambers, fig. 4.2. The first, usedfor the initial loading of atoms from the hot Rb vapor, is made of stainlesssteel and has 9 viewports for the laser beams and diagnostics. This chamberis continuously pumped with an 8 l/s Varian ion pump. An vacuum gaugeis attached and also a valve used during the initial evacuation, which is donewith a turbomolecular pump.

The second chamber is completely made of quartz giving good opticalaccess. It is pumped by an 1000 l/s Non-Evaporative-Getter (NEG) pumpand a 20 l/s Star Cell ion pump. The chamber is not anti-reflection coated.

To reach the low pressures needed in these experiments, the system isheated (baked) during the initial pumping to evaporate residual gases fromthe walls of the chambers. Ideally, this should be done at 350 %C. Theexperiment chamber, made of quartz, risk cracking at this temperature, andis therefore heated only to about 90 %C. This chamber has very recently beenreplaced with a chamber that can be baked out at su!cient temperatures,see chapter 5.

4.3 Atoms from heated getter

As a source for Rb we use a getter from SAES. By sending a current of severalamperes through the getter it is heated to several hundred %C. By varyingthe current the amount of Rb released can be controlled.

The getter from SAES has very recently been replaced by vapour sourcesfrom Alvatec. The initial experience of these is very good.

18

Section 4.5 MOT I

Figure 5.4: The vacuum system, with the second chamber stripped from all opticsfor bake out. The ion pumps are not visible on the picture. The first chamber ionpump is hidden by NEG pump in the picture and the one from the second chamber ismounted on the top of the NEG pump.

atoms are collected from the partial pressure of rubidium, into the first MOT.A quartz cell with large optical access is used in the second chamber. Thesecond MOT and the magnetic trap center coincide with the center of this quartzcell, enabling a trap for the atomic sample with low background pressure, andtherefore long lifetime. The atomic sample is transferred from the first chamberto the second through a tube. Background atoms from the first chamber are nottransferred, in any large quantities, to the second chamber. This is realized byusing a narrow and long tube, keeping a presure di!erence between the chamberswhile providing a way to transfer atoms from one part of the system to another.

The vacuum system, without mounted optics is shown in figure 5.4. Thispicture is taken just after bakeout of the system when the set-up is not crampedwith optics and other hardware. Baking the vacuum system is done by heatingthe entire system to high temperatures, locally as high as 350!C, while pumpingwith external pumps.

5.4 Magneto Optic Traps - MOTs

Each MOT have six, pairwise counter-propagating laser beams with circularlypolarized light. The light to both MOTs come from the same source; the coolinglaser in section 5.2. The frequency is changed simultaneously in the two MOTsby tuning an AOM, and the light can be turned on and o! separately in theMOTs with shutters. Both MOTs work with retro reflected beams, and theirradiance and polarization of each retroreflected beam is controlled with waveplates and beam splitter cubes.

The cooling is performed with light, red detuned from the closed transition5s2S1/2 ! 5p2P3/2, F = 2 ! F = 3, see figure 5.3. Since the transition is

42

Figure 4.2: The vacuum system. The picture is taken before the magnetictrap and most of the optics were mounted. The ion pumps are either outsidethe picture or hidden by the first chamber. The picture is taken from [30].

4.4 MOT I

The atoms from the getter are trapped into MOT I (3.2.2). The purpose ofthis MOT is to collect enough atoms from the vapor produced by the getter.The MOT beams have diameters of approximately 1 cm. The power in eachMOT beam is approx. 20 mW, which is limited by the total available power.The magnetic gradient is 0.1 T/m, produced by a current of 0.3 A throughcoils in an anti-Helmholtz configuration with 700 windings each.

4.5 Transfer Between the MOT:s

To transfer the atoms between the chambers, we push the atoms with shortlaser pulses. The procedure for each push is as follows: the MOT I is loadedfrom the background for approximately 300 ms. By increasing the detuning,the atoms in MOT I are cooled even further for a few ms. Then, the MOT

19

Experiment Chapter 4.

I beams are shut o" and the push beam is turned on for 2 ms. This givesthe atoms a speed of approximately 25 m/s, which is slow enough for thesecond MOT to be able to catch the atoms, but fast enough for the atomsnot to have time to be accelerated too much by gravity, and the cloud not toexpand too much, before passing the thin tube between the chambers. Themagnetic field in MOT I is left on, as well as the repumper beam, see sec.4.9, during the push.

The procedure is repeated until equilibrium is reached between the lossesin the second MOT and the gain from atoms pushed from the first MOT.This takes about 120 pushes which leaves approximately half a million atomsin MOT II after a total loading time of approximately 40 seconds.

4.6 The Dark Magnetic Trap

The dark magnetic trap we have built is an Io"e-Pritchard trap [31] in clover-leaf configuration [32]. The clover-leaf configuration leaves very good opticalaccess in one plane.

The trap consists of 12 coils in total: 2 pinch coils for harmonic confine-ment in the z-direction, 2 coils to compensate for the o"set produced by thepinch coils and 8 coils to produce a magnetic gradient in the radial direction.The compensation coils double as MOT-coils for MOT II. Figure 4.3.

We must be able to turn o" the trap fast. Therefore it is designed withfew turns in order to minimize inductive load. To produce strong fields withfew turns we need high currents. The trap was designed for 400 A and 6 kW.

To dissipate the heat, the coils are made of copper tubes which we flushwater through to dissipate the heat. The water is lead in parallel througheach tube, which means that each coil is connected to 2 plastic pipes. Thus,a lot of pipes extends from the trap, fig. 4.4. We have more cooling capacitythan we need, and by leading the water in series through some of the coils,some of the bulky pipes could be removed.

4.7 Transfer to the Magnetic Trap

To transfer the atoms from MOT II to the dark magnetic trap in a con-trolled fashion, we turn o" the magnetic field and the light for the MOTbefore ramping up the magnetic field for the magnetic trap. There are somecomplications.

20

Section 4.7 Transfer to the Magnetic Trap

Ø162

6.2

Ø12.2

Ø25.4

Ø22

Ø60.1

Ø152

19

43.2

a)

b)

Figure 4.3: Scheme of the magnetic trap. All dimensions are in millimeters.The coils in the center are the curvature coils. Around those are the 2 times4 clover-leaf coils for the gradient and outermost is the anti-bias coils. In b),the quartz chamber is also shown.

21

Experiment Chapter 4.

Figure 4.4: The magnetic trap mounted around the new experiment cham-ber.

4.7.1 Optical Pumping

The dark magnetic trap only traps atoms in certain magnetic sub states (seesec. 3.4). But the atoms, shortly after turning o" the MOT, are not alignedrelative to any axis of quantization. In order not to not loose the majorityof the atoms during the transfer, we optically pump the atoms to the desiredsub state. That is, after the MOT is shut down, a short, low irradiance,resonant, )+ pulse is applied to pump the atoms to the magnetic sub stateMF = 2, see fig. 4.1 .

4.7.2 Mode Matching

To conserve phase space density, the potential in the magnetic trap shouldmatch the shape of the atomic cloud in the MOT. This is hard to achievewith the this setup. It takes 7 ms for the current in the magnetic trapto rise to the preset level. During that time, the cloud of atoms expandsconsiderable, and thus gains energy since they are far from the center ofthe potential. According to [33], the phase space density does not dependcritically on successful mode matching, and we have not been able to diagnosethe transfer well enough to know if this is critical in our setup.

22

Section 4.9 Evaporative Cooling

4.8 Evaporative Cooling

The radio-frequency field used for evaporative cooling, see sec. 3.4.1, is pro-duced with an antenna that consists of a circular coil with 5 cm diameter and3 turns. The high frequency current through the antenna is generated by ahigh frequency synthesizer and is then amplified. Preliminary tests indicatethat the exact position of the antenna or the amplitude of the signal is notcritical.

4.9 Light

Narrow band light, used for the MOT beams, for transfer between the MOT:sand for probing, is produced by a Toptica DLX high power diode laser system.Even though the MOT beams are locked near a closed transition, between thehyper fine levels F = 2 and F # = 3, fig. 4.1, there is still a finite possibilityfor the atoms to end up in F = 1. To avoid this problem, a repumper beamis used to pump the atoms back into the state F = 2. The repumper beamis produced by a home external built cavity diode laser [34].

A Ti:sapph laser (Coherent-MBR) is used for far detuned optical lattices.This laser is pumped by a frequency doubled Nd:Yag-laser (Coherent, Verdi-18) at 18 W, yielding a maximum output of 4 W.

Standard absorption spectroscopy is used to lock the lasers to the de-sired transitions. Acousto Optical Modulators (AOM:s) [35] are then used todetune the light to the desired wavelengths.

4.9.1 Optical Fibers

Optical fibers serves several purposes in the experiment. The Ti:Sapph laseris placed at another floor, than the rest of the experimental setup, and a fiberis used to guide the light to the experimental setup. The light out from anoptical fiber has a clean spatial mode, which is crucial for the imaging andmakes it easier to find an optimal alignment of the MOT:s. The fibers alsoserve to isolate the locking of the laser from the experiment. This makes itpossible to realign the absorption spectroscopy and the AOM:s while leavingthe alignment of the light in the other parts of the experiment intact.

4.9.2 Shutters

We have to be able to switch the light in the di"erent beams fast and at precisemoments in time. We use the AOM:s and mechanical shutters. The AOM:sare very fast, but always lets through a small part of the light. The atoms

23

Experiment Chapter 4.

in the dark magnetic trap is very sensitive to stray light. The mechanicalshutters completely shut out all light, but they are slow, with closing timesaround a half millisecond. Where the demands for speed is not very high,we use just the mechanical shutters. Where we have to shut of the light fastand precise, we use a combination of an AOM and a mechanical shutter.

The mechanical shutters are inexpensive camera shutters, manufacturedby Densitron. The shutters are closed by a current through an electromagnetand opened with a spring. Depending on the beam size, they shut o" thelight in about 0.5 ms, but there is always a delay of several ms before theyclose. Both the shut-o" time and the delay vary between di"erent shuttersand depend on the exact position of the shutter relative to the beam andthe size of the beam. Thus, where the demands are high and where we arenot able to use AOM:s, the delay and the shut-o" time must be measuredwith a fast photo detector. The timing in the control script is then adjustedaccordingly.

There are two problems with the shutters from Densitron. We have builtcircuits to deliver a high current pulse during the closing, but then a lowercurrent to keep the shutters in closed position. We have adjusted the lowcurrent to just keep the shutters closed. Even then, the closing coils heats upuntil the shutter get stuck if the shutter is closed for a longer period. We havesolved this by combining two shutters that takes turn where the shutters areclosed a long time in the experimental sequence.

High beam irradiance will eventually burn a hole in the shutter. Wherethis is a problem, we have glued aluminum foil to the shutter surface tominimize heat absorption. The excessive stray light must then be shielded.

4.10 Control and timing of the Experiment

Each cycle in the experiment involves a large number of events, such asswitching currents for magnetic fields and closing and opening shutters. Mostof these events have to be precisely timed. We need something that managestimescales from about 100 µs to hundreds of seconds.

For the hardware timing, we use a National Instruments NI-6229 digitaldata acquisition card. National Instruments provide their own programmingenvironment, LabView, for a graphical programming style: boxes symbolizingprocedures, functions or devices are connected by lines that provide the dataflow. I find LabView convenient for writing simple data acquisition programs,but awkward for writing more complex programs with more than the mostelementary structure.

To facilitate easy scripting of the experiment cycle, we have made a simple

24

Section 4.12 Imaging and Diagnostics

scripting language on top of GNU Octave1 [36, 37]. The scripting languageproduces an array with the state of the experiment at each moment in time.The array is then read in by a small LabView program that initialize theacquisition card and starts the experimental sequence.

4.11 Imaging and Diagnostics

Currently, all measurements are made with standard absorption imaging [33].The cloud of atoms is illuminated with a resonant laser beam. The atomsscatter light, thus attenuating the signal, and the shadow is registered by acamera. The attenuation follows Beer’s law

I = I0e!D (4.1)

where I0 is the irradiance of the incoming light, I the irradiance of the lightleaving the cloud and D the optical density. The column density at each pointin the image is D/), where ) is the absorption cross section for the relevantwavelength. Thus, by analyzing the image of the shadow, a 2-dimensionaldensity profile can be obtained.

We take three images to obtain one density profile: the first with theatoms to get the shadow image, the second with just the probe beam and noatoms to obtain I0 for each point in the image, and the third without anyprobe to be able to compensate for any stray light. We use an Alta U2000cooled CCD camera system for the absorption imaging.

By taking images at di"erent times, after releasing the atoms from theMOT or the magnetic trap, we can measure how the cloud expands, and fromthat calculate the temperature.

In addition to the absorption imaging, we use two cheap surveillancecameras to monitor the MOT:s during the experiment. These cameras arenot good for any quantitative measurements, but they are very useful foroptimizing the MOT:s, as well as for instantly letting us know if somethinghas gone wrong.

4.12 Alignment of Optical Lattices

One of the main goals of the experimental setup has been to study ultra-cold atoms in optical lattices, sec. 3.3. We have built an alignment-tool, tofacilitate the alignment of lattices with di"erent geometry.

1GNU Octave is a language primary intended for numerical computations. It is mostlycompatible with Matlab.

25

Experiment Chapter 4.

Top View

Atoms

Rails with carriers

Laser

Iris

Atoms

Front View

Iris

Figure 4.5: Schematic view of the alignment tool, shown from the top andfrom the front. The rails are mounted above the experimental chamber. Thetwo rails are connected by two carriers, so the top rail can be freely moved inthe horizontal plane. After measuring the exact position of the atomic cloud,relative to scales on the rails, the setup can be used to facilitate alignment oflaser beams with known angles relative to each other.

With the help of two rails with carriers, an iris is mounted so that it canbe moved freely in an horizontal plane above the experiment chamber, seefig. 4.5. The position of the MOT is determined relative to the horizontalcoordinate system defined by the scales on the rails, and the vertical distancebetween the plane and the MOT. The two points in space, the center of theiris and the position of the MOT defines the line for the lattice beam.

The lattice beams are far from resonance, which makes it hard to seeif they hit the atoms. Therefore, we use resonant beams during alignment.After ensuring that the iris is in the right position, the beam is alignedthrough the iris and made to hit the atomic cloud. With a resonant beam,it is immediately apparent if we hit the atoms, since then atoms are blownaway by the scattering force. After aligning a resonant beam, we overlap itwith a beam for the lattice. This can be done easily with a pair of irises andsome optics.

We have confirmed that the tool works as expected by measuring theexpansion of a cloud of cold atoms, with and without lattice beams. Somedata are presented in fig. 4.6. A detailed description of the alignment toolcan be found in Paper I.

26

Section 4.12 Alignment of Optical Lattices

Magnus Rehn et al.: A set-up for flexible geometry optical lattices 7

! " #! #"!

!$#

!$%

!$&

!$'

!$"

t()(*+

r()(**

Fig. 4. Root-mean-square radii of expanding atomic clouds,

acquired by Gaussian fits to images like the one in fig. 3. The

triangles show the case for free expansion, for the two direc-

tions, The lines are fits to a model for normal difussion. The

circles, filled and unfilled, shows data for the case where the

opotical lattice is present.

A natural way to developed the experiment further is

to trap the sample magnetically, and cool it by evapora-

tion. Then, the thermal energy of the atoms should be

reduced by orders of magnitude, and a Bose-Einstein con-

densate can be formed.

4 Conclusion

For complicated experiments with optical lattices, great

precision and flexibility for laser beam directions are needed.

A simple tool, built up by o!-the-shelf optomechanical

components has been developed. The tool has been cali-

brated, using reflections and geometry, and tested with a

far-detuned two-dimensional optical lattice. With the tool,

we can set beam angles in our experiment with a preci-

sion of 10 mrad. The tool and the procedures developed

for its use have proved to be functional. The set-up will

be used for a range of experiments, studying the trans-

port properties of Bose-Einstein condensates in periodic

and quasi-periodic optical lattices.

This work has been supported by Knut & Alice Wallenbergs

stiftelse, Carl Tryggers stiftelse, Kempestiftelserna and Veten-

skapsradet.

References

1. P.S. Jessen, I.D. Deutsch, Adv. At. Mol. Opt. Phys. 37,

95 (1996)

2. L. Guidoni, P. Verkerk, J. Opt. B: Semiclass. Opt. 1, R23

(1999)

3. G. Grynberg, C. Robilliard, Phys. Rep. 355, 335 (2001)

4. I. Bloch, Nature Physics 1, 23 (2005)

5. J.P. Barrat, C. Cohen-Tannoudji, J. Phys. Radium 22, 329

(1961)

6. C. Cohen-Tannoudji, Ann. Phys. (Paris) 7, 423 (1961)

7. C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Atom-

Photon Interaction (Wiley, New York, 1992)

8. S. Chu, Rev. Mod. Phys. 70, 685 (1998)

9. C. Cohen-Tannoudji, Rev. Mod. Phys. 70, 707 (1998)

10. W.D. Phillips, Rev. Mod. Phys. 70, 721 (1998)

11. A. Hemmerich, T. Hansch, Phys. Rev. Lett. 70, 410 (1993)

12. C. Monroe, Nature 416, 238 (2002)

13. P.B. Blakie, C.W. Clark, J. Phys. B: At. Mol. Opt. Phys.

37, 1391 (2004)

14. G. Grynberg, B. Lounis, P. Verkerk, J.Y. Courtois, C. Sa-

lomon, Phys. Rev. Lett. 70, 2249 (1993)

Figure 4.6: Root-mean-square radii of expanding atomic clouds, acquiredby Gaussian fits to absorption images. The triangles show the case for freeexpansion, for the two directions. The lines are fits to a model for normaldi!usion. The circles, filled and unfilled, shows data for the case where theoptical lattice is present. This shows that the desired confinement can berealized

27

Chapter 5

Status of the Experimentalsetup

We are able to trap atoms in a first MOT and transfer them to a second MOT,which resides in a chamber with better vacuum. Previously, we were able tocollect approximately 106 atoms in the second MOT. We were then able totransfer of the order of half of these atoms to a dark magnetic trap. Fromthe magnetic trap we are able to expel atoms with a radio-frequency field.We have a functioning system for absorption imaging for diagnostics. Untilrecently, we have had problems with too few atoms to make any meaningfuldiagnostics of the atoms in the magnetic trap. Thus we have not been able tostart optimising neither the transfer to the magnetic trap, nor the evaporativecooling.

Right now, it looks like these problems have been solved. We have recentlyrebuilt the vacuum system. The current experimental setup is shown in fig.5.1. Without having the time to do thorough measurements, we know thatthe vacuum has improved considerably and the number of atoms in MOT IIhas increased by orders of magnitude, see fig. 5.2 and 5.3. This means thatwe will be able to start measure what happens more in detail with the transferof atoms to the magnetic trap, as well as with the evaporative cooling. Thisgives very good hopes for the future.

28

Section 5.0

Figure 5.1: The current experimental setup. The setup is changed fromhorizontal to vertical transfer of atoms. On top is the first MOT-chamberwhere we collect atoms from a Rb-vapour. Those atoms are then transferredto the MOT II /Experimental chamber. There the cloud of atoms is furthercooled in a MOT and then transferred to the magnetic trap. In the foregroundone of the surveillance cameras is visible.

29

Status of the Experimental setup Chapter 5.

Figure 5.2: Recent absorption image of the atoms in a MOT, showing ap-proximately 4 million atoms

Figure 5.3: Recent density profile from the absorption image of the atomsin a MOT, acquired from the image in fig. 5.2.

30

Chapter 6

Raman Transitions in DoubleOptical Lattices

Optical lattices have been suggested as a platform for storing information forquantum computations and for performing quantum information processing[22]. Optical lattices have a number of advantages: It is a well controlledquantum system with long coherence times and they can be made large, thusscalability could be within reach.

To be able to realize quantum gates, some means of interaction betweenthe atoms in the optical lattice must be induced. This can be done by util-ising a double optical lattice (DOL) [12, 13, 14]. A DOL is two spatiallyoverlapping lattices with the same periodicity, trapping two distinct hyper-fine levels of the atoms. By translating the lattices relative to each other, itis possible to induce interactions between the atoms.

Thus, letting an atom with two hyperfine ground states being the funda-mental information unit in quantum computer, a qubit, and creating quan-tum gates by interacting the atoms in the distinct lattices, a scheme for aquantum computer could potentially be realized. Some means to set thequbits in the initial state of the processes is needed, i.e. some means totransfer the atoms between the di"erent hyperfine levels.

6.1 Stimulated Raman transitions

The Raman e"ect [38], is most commonly used and studied in terms of spon-taneous Raman scattering. Spontaneous Raman scattering is a weak e"ect,but it can be greatly enhanced in stimulated Raman scattering, where the

31

Raman Transitions in Double Optical Lattices Chapter 6.

201 MHz

151 MHz

5p 2P3/2

5s 2S1/2

251 MHz

9.19 GHz

852.347 nm351.726 THz

F’ = 5

F’ = 4

F’ = 3F’ = 2

F = 4

F = 3

Lattice 1

Raman CouplingLaser

Lattice 2

Figure 6.1: Cesium D2 transition, with hyperfine structure. Shown is alsothe light fields for the optical lattices as well as the laser that couples thehyperfine levels

e"ect is amplified many orders of magnitude by adding an extra beam tostimulate the desired transition. Here, also the possibility for coherent trans-fer increases.

Stimulated Raman transitions have been used in ion traps for quantuminformation [39]. In our case, we should couple the two hyperfine groundstates of Cs, see fig. 6.1, in a double optical lattice experiment.

6.2 Population Transfer in DOL

There is a limit to how far from resonance a double optical lattice can berealised [12, 13, 14]. In our case we are limited to about 1 GHz detuning,otherwise the lattices will no longer be independent. We have investigatedhow the coupling by stimulated Raman transitions transfers the populationsbetween the two optical lattices, and also the coherence of the transitions.

We formulate the optical Bloch equations [40] for the system in fig. 6.2,that corresponds to the relevant parts of fig. 6.1: EAC corresponds to Latticebeam 1, EBD corresponds to Lattice beam 2, and EBC corresponds to the

32

Section 6.2 Population Transfer in DOL

|D!

|C!

|B!

|A!

!BC

!AC

!BD

Figure 6.2: Four-level model of an atom in a DOL, made up of laser fieldsEAC and EBD, trapping respectively states |A" and |B". An extra laser EBC

enables a coupling between the two trapped states. Laser fields EAC, EBC,and EBD are detuned from resonance by "AC, "BC, and "EBD, respectively,illustrated here in the case of the resonant Raman transition "AC = "BC = ".

Raman coupling laser. |A& corresponds to F = 3,MF = +3, |B& correspondsto F = 4,MF = +4, |C& corresponds to F # = 4,MF = +4, and finally |D&corresponds to F # = 5,MF = +5.

The density matrix describing the atom, with elements

.)* =#

),*=A,B,C,D

|(& %*| , (6.1)

follows a time evolution governed by

ih. = [Ha + Hal, .] + Lspont.em.. (6.2)

Using the dipole approximation and a semi-classical representation of thelight-atom interaction, we arrive at the following system of equations (see

33

Raman Transitions in Double Optical Lattices Chapter 6.

further Paper IV)

.AA =ih

DACEAC (.CA " .AC) + $C&A.CC,

.BB =ih

[DBCEBC (.CB " .BC) + DBDEBD (.DB " .BD)] + $C&B.CC + $.DD,

.CC =ih

[DACEAC (.AC " .CA) + DBCEBC (.BC " .CB)]" $.CC,

.DD =ih

DBDEBD (.BD " .DB)" $.DD,

.AB = i ('B " 'A) .AB +ih

[DACEAC.CB "DCBECB.AC "DBDEBD.AD] ,

.AC = i ('C " 'A) .AC +ih

[DACEAC (.CC " .AA)"DBCEBC.AB]" $2

.AC,

.AD = i ('D " 'A) .AD +ih

[DACEAC.CD "DBDEBD.AB]" $2

.AD,

.BC = i ('B " 'C) .BC +ih

[DBCEBC (.CC " .BB) + DBDEBD.DC "DACEAC.BA]" $2

.BC,

.BD = i ('D " 'B) .BD +ih

[DBDEBD (.DD " .BB) + DBCEBC.CD]" $2

.BD,

.CD = i ('D " 'C) .CD +ih

[DACEAC.AD + DBCEBC.BD "DBDEBD.CB]" $.CD,

(6.3)along with equivalent equations for the other coherences, i.e., .*) = .')* .We solve this by numerically integrating the system of di"erential equationsafter some further simplifications.

Using these methods we have, in Paper IV, shown that it is possible toachieve high e!ciency in the transfer of population from one optical lattice toanother with the above described scheme for stimulated Raman transitions.In fig. 6.3, we see the time evolution for the four-level scheme for the case ofa far detuned lattice. The Raman coupling beam is turned o" at the peak ofthe population, i.e. after a pi-pulse [40]. Of interest, but hardly visible, isthe e"ect of spontaneous emission, which may turn out to constitute a severelimitation in coherence time.. We have also shown that the above schemecould be useful for Raman sideband cooling [12, 13, 14].

34

Section 6.2 Population Transfer in DOL

1.0

0.8

0.6

0.4

0.2

0.0

!

10x10-6

86420

[s]

!AA

!BB

!CC

!DD

Figure 6.3: Time evolution for the four-level scheme for a far-detuned case(# 1 GHz). The Raman-coupling beam is turned o! at the peak, after half aRabi period.

35

Chapter 7

Quantum Monte-CarloMethods for Soft CoreBosons in Optical Lattices

In this chapter I briefly describe Monte-Carlo methods in general and, inparticular, the specific methods we have used for the results presented inchapter 8.

7.1 Monte-Carlo methods

The average of any thermodynamic quantity a is given by

%a& =

,|)(%(|a exp("*H)|(&

Z, (7.1)

where the sum is over all states |(& of the system and H is the relevantHamiltonian. Z is the partition function and is the sum of the Boltzmannfactors %(| exp("*H)|(&. * is the inverse temperature.

For all but the smallest many body systems, the number of states ishuge, so a direct summation is not practical. When an analytic solution isn’tavailable, we can still randomly sample the distribution (the terms in thesum) to get estimates of the thermodynamic properties of the system.

A thermodynamic system in equilibrium is very sharply peaked aroundthe average. If we just sample the distribution randomly we will use too muchof our resources to sample parts of the distribution that contributes very little

36

Section 7.1 Monte-Carlo methods

to the averages. By using importance sampling, i.e. taking samples with aprobability that is proportional to the Boltzmann factor, we make sure thatour samples are taken where they are needed most.

A common technique is to generate a sequence of samples where each sam-ple is generated from the previous with some probability that only dependson the configuration of the previous sample.

7.1.1 Detailed balance

We define Pn(t) as the probability to find the configuration in state n aftercertain number t of updates of the configuration. t can be seen as a kindof time variable, since the development of the system depends on it, but ithas little to do with real time. We further define Wnm as the transitionprobability for a system in state n at any t to be in state m at t + 1.

We must have

Wnm ( 0 and#

m

Wnm = 1. (7.2)

That just expresses that the transition probabilities must be positive or 0,and that we always end up in some state. For a system in equilibrium, i.e.the probabilities to find the system in a certain state does not change, wemust also have #

n $=m

Pn(t)Wnm =#

n $=m

Pm(t)Wmn, (7.3)

for all t. This expresses that the probability to enter state m at all timesequals the probability to leave the same state. This gives the condition thatis known as detailed balance

Pn(t)Wnm = Pm(t)Wmn, (7.4)

i.e. the probability to enter state m from state n equals the probability togo from state n to state m.

7.1.2 Ergodicity

To successfully sample the system we must also fullfill the condition thatthere is a finite probability to enter any state in a finite number of steps.This is the principle of ergodicity.

37

Quantum Monte-Carlo Methods Chapter 7.

7.1.3 The Metropolis algorithm

One of the first algorithms for importance sampling, and one that is easy toexplain, at least for a classical system, is the Metropolis algorithm [41]. Theprobability for the n:th state to occur in a classical system is

Pn =exp("En*)

Z. (7.5)

Even if Z is not know in advance it is possible to generate a series of configu-rations since the di"erence in energy #E = En"Em between di"erent statesis known. Any transition rate that fulfills detailed balance is acceptable[41].The first transition rate used in statistical physics is the Metropolis form:

Wmn =-

exp("#E*), #E > 01, #E < 0 (7.6)

A simple recipe for the Metropolis algorithm is as follows [41]:

1. Choose an initial state.

2. Choose a local update of the system, e.g. a spin flip for a fermionsystem, or add or remove a boson from a lattice site.

3. Calculate the di"erence between the energy of the system before theupdate and the energy after the update.

4. Generate a random number r between 0 and 1

5. If r < exp("*#E), update the system.

6. Choose a new site and go to step 3.

After a number of local updates has been made, the properties are measuredand added to the averages.

For a quantum system, since the many-body function is unknown, notonly the distribution needs to be sampled. How this is solved is described inthe next section.

For both quantum and classical systems, as the temperature approachesthe phase transition, the correlation lengths of the system increases. Thismeans that any local update method becomes increasingly slower closer tothe critical point. This can be solved by using a non-local update scheme, asthe one described in [42, 43].

38

Section 7.3 Stochastic Series Expansion

7.2 Stochastic Series Expansion

To be able to evaluate the terms in the partition function

Z =#

)

%(| exp("*H)|(& (7.7)

we expand the exponential function:

exp("*H) = 1" *H +*H

2!" . . . =

#

n

("*)n

n!Hn. (7.8)

Thus,

Z =#

)

#

n

("*)n

n!%(|Hn|(& (7.9)

and%a& =

1Z

#

)

#

n

("*)n

n!%(|aHn|(&. (7.10)

Instead of sampling proportionally to the Boltzmann distribution, the sam-ples are taken from this sums with probabilities proportional to

(*)n

n!%(|Hn|(&. (7.11)

7.3 Lattices

For an optical lattice with bosons, we choose to represent the state as anumber state |(& = |n1n2 . . . nL& = |n1&|n2& . . . |nL& where ni is the numberof atoms at site i. This makes it possible to write the Hamiltonian as a sumof operators each operating on two lattice sites such as

%(|H|(& =#

b

%nb1nb2 |Hb|nb1nb2&. (7.12)

For a system with no long range interactions, b represents two neighboringlattice sites, e.g. b1 = i, b2 = i + 1.

Further, we separate the diagonal (dia) parts and the o"-diagonal (o")parts in the Hamiltonians and write

Hb =#

b

&Hdia

b + Ho"b

'(7.13)

39

Quantum Monte-Carlo Methods Chapter 7.

The n:th power of H can then be expressed as a sum over all permutationsof n diagonal- and o"-diagonal operators acting on all neighbor-pairs:

Hn =

.#

b

Hdiab +

#

b

Ho"b

/n

=#

{Sn}

0

i

Hai

bi (7.14)

{Sn} is the set of all permutations Sn of n operators. Each Sn is representedas a series of operator-neighbor pairs [a1, b1][a2, b2] . . . [an, bn] and the productis the product of the corresponding operators.

Defining |((p)& =1p

i=1 Haibi|((0)& for a sequence of operators defined as

above, we now write the partition function

Z =#

)

#

n

("*)n

n!

#

{Sn}

0

p

%((p)|Hap

bp |((p" 1)& (7.15)

A configuration will now be defined by a number state |((0)& and anoperator sequence Sn.

7.4 Updating Scheme

The description, outlined above, gives some background to the method usedto solve the problem of atoms in an anisotropic optical latice, see chapter8. During those calculations, we have used the directed loop algorithm,described in [42, 43]. The weights used for an interacting boson model istaken from [42].

40

Chapter 8

Numerical Investigations ofQuantum Phase Transitionsin Optical Lattices

The versatility of optical lattices makes it possible to explore novel periodic,superperiodic and quasi periodic systems. Partly as a preparatory studyto planned experiments, we have numerically investigated phase transitionsin anisotropic optical lattices. The first two sections in this chapter containssome background about phase transitions in optical lattices. The last containssome of the results of our investigations.

A simple cubic lattice can easily be created with three pairs of orthogonallaser beams, but also with other beam geometries. The potential barriersbetween the lattice sites can be controlled by changing the irradiance in abeam pair. By turning up the potential barrier in one direction, we cancreate, for all practical means, an array of two-dimensional lattices. Imaginea stack of pancakes, where each pancake is a 2D lattice. 2D lattices is themain concern in this chapter.

8.1 The Bose-Hubbard Model

Interacting bosons in an optical lattice can be described by the Bose-HubbardHamiltonian (BHH) [44, 45]

H = "tx#

<i,j>x

a†i aj " ty

#

<i,j>y

a†i aj +

U

2

#

i

ni(ni " 1)" µ#

i

ni. (8.1)

41

Quantum Phase Transitions in Optical Lattices Chapter 8.

The first two sums represent tunneling between lattice sites, with strengthstx and ty for the two orthogonal directions and the sums are over all neigh-bors in those directions. a and a† is the bosonic annihilation and creationoperators, ni = a†

i ai counts the number of bosons at site i, and the thirdterm represents the on-site interaction, with U corresponding to the strengthof the interaction. All long-range interactions have been neglected. Thefourth term represents the chemical potential, which is used to control themean number of atoms in the system. For this Hamiltonian to be valid, thetemperature must be low enough for all atoms to be in their lowest energystate.

The BHH written here is neither as general or as specific as it is commonlywritten. We have added terms to take into account di"erent tunneling indi"erent directions. It is trivial to add e.g. an o"set term for each site totake into account an extra smoothly varying potential from e.g. a magnetictrap, or to add terms for nearest site interactions between the atoms.

8.2 Phases in optical lattices

8.2.1 Isotropic Lattices

The zero temperature phase diagram for the BHH, eq. 8.1, has been describedby Fisher et al.. [44]. For a commensurable number of atoms, i.e. the densitybeing such as that the mean number of atoms per lattice site is an integer n,there are two distinct phases for an isotropic lattice (t $ tx = ty).

When the interaction is weak compared to the tunneling, t ) U , thesystem forms a Bose-Einstein condensed state of matter. Each atom is de-localized and the total system can be described by a giant coherent matterwave.

When the interaction is strong compared to the tunneling, t * U , thesystem enters a state of a Mott-insulator. The kinetic energy can not over-come the on site interaction. Thus, each atom is localized to a single latticesite. Therefore, this system can no longer be described by one giant matterwave. On the other hand, each site contains exactly n atoms. In figure 8.1,a sketch of the phase diagram for the Bose-Hubbard model for an isotropiclattice is shown.

8.2.2 1D Lattices

With the irradiance of two of the beam pairs turned up high, it is possi-ble to create essentially one-dimensional lattices. These consist of an two-dimensional array of tubes, where there is no interaction between atoms in

42

Section 8.2 Phases in optical lattices

SF

n = 1

n = 2

n = 3

t/U

µ/U

MI

MI

MI

Figure 8.1: Sketch of the phase diagram for an isotropic Bose-Hubbardmodel. It has Mott-insulating (MI) regions where the number of atoms equalsthe number of sites and the tunneling is low. It has a superfluid (SF) regionwhere the tunneling is high and the number of atoms is incommensurable withthe number of sites.

di"erent tubes. The beam pair in the last direction builds up the periodicpotential within each tube.

In [46, 47, 48], the 1D tubes were described using Tomonaga-Luttingerliquid (TLL) theory. A TLL is, loosely speaking, a 1D superfluid (SF). It ischaracterised by two parameters independent of statistics and the detailedproperties of the constituent particles. That is the sound velocity vs and theTLL parameter K, defined as

K =vF

vs. (8.2)

Here, vF = h#.1D/m, where .1D is the line density in 1D. The TLL param-eter K determines the behavior of the correlations, which obey the powerlaw

$(i, j) $ %a†i aj& ' |ri " rj |!1/(2K). (8.3)

43

Quantum Phase Transitions in Optical Lattices Chapter 8.

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

tx / U

ty / U

2D SF

1D MI

1D

MI

2D MI

Figure 8.2: The phase diagram for a commensurable number of atoms ina finite square lattice. For low tunneling, the system is a 2D Mott insulator(2D MI). Where both tunnelings are large, the system is a 2D superfluid (2DSF). When one tunneling is large, but the other is very small, the systemis superfluid in the direction with strong coupling, but a Mott insulator inthe direction of the weak coupling (1D MI). The areas for 1D MI are greatlyexaggerated.

8.2.3 Anisotropic 2D Lattices

In the following, we shall discuss the correlations along di"erent directions, sowe introduce the notation $x and Kx for the correlations within the stronglycoupled x direction, as well as $y and Ky for describing correlations alongthe weakly coupled y direction.

When we can tune the tunnelings tx and ty independently, the phasediagram looks like fig. 8.2 for a commensurate number of atoms. When tx orty are of the same order as, or greater than U , the system is in a superfluidphase. When tx and ty are small, we are in a Mott-insulating phase. Foran infinite system, those are the only phases that exists. If either tx or tyis large enough for superfluidity in respective direction, there will also besuperfluidity in the other direction.

However, as was noted by Ho et al. and Gangardt et al. [46, 47, 48],the situation is di"erent for a system which is finite in a strongly coupleddirection. In this case, it is possible for a state to be superfluid in onedirection and Mott-insulating in the other. This is the state called 1D Mott-insulator (1D MI) in fig. 8.2.

44

Section 8.3 1D Phase Transitions in a 2D Optical Lattice

10!3

10!2

10!1

100

10!8

10!6

10!4

10!2

100

ty / U

! s

Figure 8.3: Arbitrary selected curves showing the superfluid density as afunction of the ratio of the tunneling ty to the on site interaction U . Thedi!erent curves corresponds to di!erent lattice sizes and di!erent filling factors#.

8.3 1D Phase Transitions in a 2D Optical Lattice

We have numerically studied the transition between the 2D SF phase and the1D MI phase in finite systems, holding one of the tunneling parameters, tx,fixed at a quite high value, while varying the other, ty, around the transitionpoint.

In the numerical calculations, we used the stochastic series expansionmethod described in chapter 7. During the thermalization phase of thesimulation runs, µ was tuned until arriving at the desired densities. Theinverse fundamental temperature * = 1/(kBT ) was chosen to be at least* = 1000U!1 to make sure that we actually calculated the ground stateproperties of the system. The maximal number of atoms possible in a latticesite was chosen to be 6 to ensure that the finite maximum number of atomsper site did not influence the results.

We have run the simulations for a variety of system sizes and studiedhow the superfluid density and the correlations depend on ty and the size ofthe system. Figure 8.3 shows some of the results from the simulation runs.To extract relevant parameters, we had to go through an involved scaling

45

Quantum Phase Transitions in Optical Lattices Chapter 8.

procedure, described in paper III. Our calculations support the predictionsdone by the Tomonaga-Luttinger liquid theory [46, 47, 48]. The correlationsbetween the tubes determine the location of the transitions consistent withthose predictions. We have also showed that the transition occurs when thenumber of particles is commensurate with the number of tubes, and notnecessarily with the number of sites.

46

Chapter 9

Conclusions

I have been one of the main responsible for building a setup for Bose-Einsteincondensation with 87Rb. All parts essential for Bose-Einstein condensationare in place. We collect atoms in a MOT, transfer them to another vacuumchamber (with better vacuum) and trap them in another MOT. We cantransfer the atoms to a dark magnetic trap, but have not yet been able toform a BEC, due to a range of technical di!culties.

I have participated in the design of, the building, and the testing of equip-ment for alignment of optical lattices with flexible geometry. This tool hasbeen proven to work as desired, an there is a great potential for a range ofunique experiments with optical lattices of various geometries.

Numerical studies have been made on anisotropic optical lattices, andwe have confirmed the existence of a transition between a 2D superfluidphase and a 1D Mott-insulating phase. We have shown that the transitionis of Berezinskii-Kosterlitz-Thouless type. We have also, in another study,shown that using stimulated Raman transitions is a practical method fortransferring atoms between lattices in a double optical lattice. It will bepossible to transfer population between the lattices, with possible furtherapplications in qubit read/write operations.

47

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