Jan Max Walter Kruger- Dynamical Behaviour of Radio Frequency Output Couplers for a Bose-Einstein Condensate

8/3/2019 Jan Max Walter Kruger- Dynamical Behaviour of Radio Frequency Output Couplers for a Bose-Einstein Condensate

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Dynamical Behaviour of Radio

Frequency Output Couplers for a

Bose-Einstein Condensate

Jan Max Walter Kr¨ uger

March 2000

A thesis submitted for the degree of

Master of Science in Physics

of the University of Otago, Dunedin

New Zealand



Abstract

In the ultra-low temperature regime, Bose-Einstein condensates are welldescribed by the Gross-Pitaevskii (GP) equation for the single particle wave-function Ψ( r , t ). In this thesis we use the GP equation to study the behaviourof two-component Bose-Einstein condensates in two spatial dimensions, cou-pled by an rf eld. These coupled equations represent a model for an atomlaser output coupler, and we study aspects of the coupler behaviour in twodistinct scenarios.

In the rst, we simulate continuous-wave output in a harmonic trap underthe inuence of gravity. We study the effects of spatially selective coupling,and show how it depends on the usual mechanisms of bandwidth for couplingof two quantum states, including eld strength, detuning and pulse length.We investigate the effects of noisy coupling elds on the coherence of theoutcoupled populations.

In the second scenario, we simulate the full dynamical behaviour of theTime-Orbiting Potential (TOP) trap in regimes of strong and weak couplinginteraction. We examine the evolution in the two condensate states under theinuence of the rotating cw coupling eld, and we discover rich vortex dynam-ics. We present a simple model of vortex formation and give an explanationfor vortex formation and temporal evolution in the TOP trap. Furthermore,we identify two distinct regimes in which we can interpret the mechanisms of

vortex formation.



i

Acknowledgements

First of all, I would like to thank my excellent supervisor Robert J. Ballagh.He was always very helpful and encouraging and despite all his other commit-ments, he always had enough time for me. Thanks also to my lovely girlfriendKatharine J. Webb, who kept me sane by dragging me out into the outbackmore than once. She believes that physicists are the only people in the worldwho say, “Oh, nooo! It’s Friday again !”.

I am also very much indebted to all other members of the theory group whohad to put up with my questions. Blair Blakie, Ben Caradoc-Davies, AdamNorrie, Andreas Penckwitt and Terrence Scott. They all were very helpful andenlightened me on many things. Furthermore, they shared my fascination foreverything GNU and Linux, thanks to which no Micros˜1 software is neededat all. Credits also go to Malcolm Fraser, our group’s cluefull system adminis-trator and my later atmate. Thanks especially to Andreas for challenging myGerman and for being a brother in arms, struggling to apply fading Germanlanguage abilities to physics. Often, we remained speechless and were forcedto fall back to English.

I wish to thank the experimentalists for helpful input and ideas, especiallyAndrew Wilson, Jos Martin and Callum McKenzie. Thanks also to the rest of the BEC gang, Reece Geursen, Nick Thomas and Nicola van Leuwen.

Thanks go to my New Zealand friends and atmates inside and outside thephysics department. Sadly, many of them left Dunedin at some stage duringthe two years of my stay, after completing their degrees.

I gratefully acknowledge nancial support by the University of Otago 1999Study Award, which paid for my tuition fees and living expenses, and I con-clude the credits with a big “Thank You!” to my brother Hannes, sister Anna

and my parents and relatives who supported me and had to put up with megoing so far away for such a long time. However, thanks to the internet, I wasnever a lot further away than the next keyboard. My apologies to numerouspeople who I forgot to mention here.



Contents

1 Introduction 1

1.1 Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . 11.2 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Background 5

2.1 Mean Field theory and the GP equation . . . . . . . . . . . . . 52.1.1 Thomas-Fermi Approximation . . . . . . . . . . . . . . . 102.1.2 Condensate Energy . . . . . . . . . . . . . . . . . . . . . 112.1.3 Energy conservation . . . . . . . . . . . . . . . . . . . . 12

2.2 The two-state Atom . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Pseudospin vector description of a two-state Atom . . . . 132.2.2 The Rotating Wave Approximation . . . . . . . . . . . . 152.2.3 Rabi-solution . . . . . . . . . . . . . . . . . . . . . . . . 162.2.4 Coupling Pulses . . . . . . . . . . . . . . . . . . . . . . . 18

3 Phase Evolution 20

3.1 Dressed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Shift of Energy Levels due to Coupling . . . . . . . . . . . . . . 22

3.2.1 Using detuned lasers to shape trapping potentials . . . . 253.3 Phase Evolution in a driven Two State System . . . . . . . . . . 26

3.3.1 Phase evolution for arbitrary initial inversions . . . . . . 29

4 Numerical Implementation 30

4.1 Dimensionless Units . . . . . . . . . . . . . . . . . . . . . . . . . 304.2 Interaction Picture . . . . . . . . . . . . . . . . . . . . . . . . . 314.3 Fourier Transform Method . . . . . . . . . . . . . . . . . . . . . 324.4 Runge-Kutta Method . . . . . . . . . . . . . . . . . . . . . . . . 33

ii



CONTENTS iii

4.5 Numerical Limits . . . . . . . . . . . . . . . . . . . . . . . . . . 344.5.1 Limitations imposed by grid resolution . . . . . . . . . . 344.5.2 Limited simulation stepsize . . . . . . . . . . . . . . . . 36

5 RF Output Coupler in 2D 38

5.1 Output Couplers . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2 Spatial Effects in Coupling . . . . . . . . . . . . . . . . . . . . . 395.3 Inuence of gravity on trapped BEC . . . . . . . . . . . . . . . 435.4 Localized Output Coupling . . . . . . . . . . . . . . . . . . . . . 435.5 Coupling Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . 49

5.5.1 Power Broadening . . . . . . . . . . . . . . . . . . . . . . 495.5.2 Broadening through nite Pulse Times . . . . . . . . . . 515.5.3 Broadening using Noisy Signals . . . . . . . . . . . . . . 55

5.6 Limits of the Simulation . . . . . . . . . . . . . . . . . . . . . . 595.7 Absorber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.7.1 Spatially ltering the Wavefunction . . . . . . . . . . . . 615.7.2 Complex Potential . . . . . . . . . . . . . . . . . . . . . 61

5.8 Validation of simulations . . . . . . . . . . . . . . . . . . . . . . 63

6 TOP Trap Simulations 686.1 The Quadrupole Trap . . . . . . . . . . . . . . . . . . . . . . . 696.2 Time-Orbiting Potential (TOP) Magnetic Trap . . . . . . . . . 70

6.2.1 Calculating the Time Average . . . . . . . . . . . . . . . 716.3 Numerical Model of a TOP Trap . . . . . . . . . . . . . . . . . 736.4 Spatial coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.4.1 Coupling Bandwidth . . . . . . . . . . . . . . . . . . . . 766.4.2 Localized evolution of population and phase . . . . . . . 77

6.4.3 Force on a condensate in a TOP trap . . . . . . . . . . . 806.5 Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.6 Vortex dynamics in a TOP trap . . . . . . . . . . . . . . . . . . 83

6.6.1 Vortices in the strong coupling regime . . . . . . . . . . 846.6.2 Model of vortex formation . . . . . . . . . . . . . . . . . 916.6.3 Vortices in the weak coupling regime . . . . . . . . . . . 966.6.4 Angular momentum . . . . . . . . . . . . . . . . . . . . 102

6.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104



CONTENTS iv

7 Conclusion 105

A Adiabatic Elimination 108A.1 Two-State System in Rotating Wave Approximation . . . . . . . 108A.2 Adiabatic Elimination . . . . . . . . . . . . . . . . . . . . . . . 110A.3 Accuracy of the approximation . . . . . . . . . . . . . . . . . . 111

B Simulation and Experiment 113



Chapter 1

Introduction

1.1 Bose-Einstein Condensation

Quantum Mechanics classies all particles either as fermions or as bosons.Systems of fermions are described by antisymmetric wavefunctions and havehalf-integral spin, while systems of bosons are described by symmetric wave-functions and have integral spin. Although the difference between a fermionicand a bosonic gas is small at high temperatures, both families of particles follow

different quantum statistics with striking differences in the low temperatureregime.

The Pauli exclusion principle prohibits two fermions to occupy the samesingle-particle quantum state of a system, while bosons have no such funda-mental restriction. In fact, a single-particle quantum state of a bosonic systemcan be occupied by a macroscopic number of particles. At very low tempera-tures, more and more particles in a bosonic system will transfer into the single-particle state with the lowest energy, leading to the build-up of a macroscopic

population. Macroscopic population further increases scattering into the samestate, and one can refer to this condensation phenomenon as a phase transitioninto a fth state of matter. In such a Bose-Einstein condensate (BEC), thedeBroglie wavelength of the cold bosons reaches the mean particle separation.The bosons lose their individual identity and the whole assembly starts actinglike, and can be described by, a single coherent macroscopic wavefunction.

Einstein predicted BEC as a consequence of Bose statistics in 1924. He gen-eralized the statistical methods used by Bose to describe photons to the case

1



CHAPTER 1. INTRODUCTION 2

of arbitrary indistinguishable particles. In 1938, London attributed the phe-nomenon of superuidity in liquid helium 4He to Bose-Einstein condensation.It has subsequently been established that because of the strong interactionsbetween particles in a liquid, only 10% of liquid 4He are condensed while 90%are depleted into quantum states of higher energies. Experimental realizationof BEC in weakly interacting atomic systems requires advanced techniques incooling dilute atomic clouds to ultra-low temperatures. Diluteness is an impor-tant requirement, because the negligible probability of three-particle collisionsprevents condensate depletion by classical condensation into liquids or solids.

A theoretical description of BEC was formulated in 1961, when the Gross-

Pitaevskii (GP) equation was derived. This equation describes the single par-ticle state for a weakly interacting BEC, and incorporates the interparticleinteraction in an approximation using the condensate’s mean eld. The fullsystem state for N particles is then assumed to be the simple product of N suchsingle particle wavefunctions, neglecting particle correlations, depletion and -nite temperature effects. The GP equation could not be applied to describeBEC in liquid 4He because of the large interaction and depletion.

In the early 1990s, advances in technologies for magnetic trapping, and for

laser cooling atoms, paved the road to achieving BEC experimentally, and in1995, Anderson et al. [2] achieved BEC in a weakly-interacting dilute gas of magnetically trapped 87Rb atoms. At the present time, a number of groupshave experimentally created BEC in dilute vapours of alkali metals and inhydrogen, including one at Otago University, using 87Rb.

This thesis presents a study of solutions of the GP equation, a nonlinearpartial differential equation, for a variety of scenarios related to current ex-perimental concerns. Numerical methods are used to solve the two componentGP equations, representing a condensate with two hyperne internal states,coupled by classical radio frequency elds. This system can be used to modelan output coupler [30], and this is the major focus of our work. Output cou-pling of BEC led to experiments trying to realize an “atom laser” in [8], thenotion of which has been strictly dened in [39]. Atom lasers are promising torevolutionize atom optics and are subject of huge interest in current research.

The rst theoretical work in this area was done in one spatial dimension byBallagh et. al. in [3]. Subsequently, the two-component equations have also




been solved in two spatial dimensions. In [14], Eschmann et. al. investigatedthe formation of Ramsey fringes in pulsed two-state coupling in a two dimen-sional condensate system. The two dimensional treatment allowed the effectsof gravity to be modelled by Zhang in [40], Wallis et. al. [35] and by Jacksonin [22], where the behaviour of out-coupled pulses, and also the ow througha potential constriction, have been studied.

The two-component 2D version of the GP equation also allows us to modelsome detailed dynamics of the TOP trap. This implementation of magnetictrapping is an important current tool in experimental BEC, and in most casesit is assumed to provide a static harmonic magnetic trapping eld. However,

in fact only its time-averaged eld is harmonic, and we show in chapter 6 thatthe dynamical behaviour of the trapping eld, in conjunction with the radiofrequency coupling eld, can lead to some interesting dynamical behaviour of the condensate, including the formation of vortices.

1.2 Thesis Overview

Chapter 2 introduces the reader to the general background necessary to un-

derstand the simulations which were carried out and described in this work.It outlines the description of a Bose-Einstein condensate (BEC) as a quantummechanical single particle state wavefunction obeying the GP equation, and itintroduces the physics of coupled two-state systems. Couplers permit outputcoupling of population density from a trapped condensate into a ne coherentatom beam. This conguration represents a basic element of the so calledatom lasers [21] and is thus of considerable importance.

In Chapter 3, we review the mechanism of phase encoding of an atomicwavefunction by potentials and coupling interactions, and give detailed andto our knowledge novel interpretation of the mechanisms. Manipulation of condensate phase is an essential element for the manipulation and control of coherent atomic beams. In normal optics, this phase manipulation is achievedwith well understood devices such as lenses, but a new method is required foratom optics.

In chapter 4, we outline and discuss the numerical implementation of thephysical problem, including the methods used to process and simulate the phys-




ical system in a computer. We discuss issues of accuracy and computationallimitations of the simulated condensate systems.

In chapter 5, the results of output coupling simulations under gravity arepresented. Spatial coupling effects are used to create shaped output pulses andbeams. We discuss bandwidth effects of radio frequency coupling, investigatethe inuence of a noisy rf coupling eld on spatial coherence of an outcou-pled “beam”, and we propose and investigate absorber methods that alleviateproblems arising from condensate population, which reaches the edges of thecomputational grid.

Chapter 6 presents the model used to describe and simulate BEC in a

dynamical TOP trap in two spatial dimensions. TOP traps are based on thesimple quadrupole trap with a magnetic eld zero in its centre and a linearincrease in all radial directions. By means of a rotating bias eld, the point of the magnetic eld zero is rotated in the xy plane, yielding a dynamic magnetictrapping eld that time-averages to a harmonic eld. We present results of two-state coupling simulations and show that under certain conditions vorticesform. We also present a simple model explaining the formation of vortices intwo-state coupling in TOP traps, which we have developed, and we identify

two simple parameter regimes in which the mechanics of vortex formation canbe analyzed.In chapter 7 we present the conclusions from this thesis.

This thesis is also available electronically from the author’s internet pages athttp://hubble.physik.uni-konstanz.de/jkrueger/ orhttp://max.yi.org/jkrueger/ The author’s email address [email protected](July 2000.)



Chapter 2

Background

In this chapter we outline the theoretical framework the work in this thesis isbased upon. We outline the mathematical description of Bose-Einstein con-densates as macroscopic single particle wavefunctions, following a mean eldapproach. More comprehensive discussions of the subject are in [18][31]. In thesecond part of this chapter we introduce the concept of two-state coupling andoutline the model of the two-state atom [1] which is the basis of our simulationsof two-component BEC.

2.1 Mean Field theory and the GP equation

In an ultra-cold atomic cloud undergoing Bose-Einstein condensation, the ma- jority of the atoms condense into the same single particle quantum state andlose their individuality. Since any given atom is not aware of the individualbehaviour of its neighbouring atoms in the condensate, the interaction of thecloud with any single atom can be approximated by the cloud’s mean eld, and

the whole ensemble can be described by the same single particle wavefunction.Simple mean eld descriptions have widely been used to theoretically describeand simulate BEC. Although effects like depletion of the condensate by higherorder interactions between the condensate atoms, and the existence and in-teraction with a thermal fraction, are neglected in these descriptions, theygive satisfactory results in the ultra-low temperature limit. In the following, aderivation of the Gross-Pitaevskii (GP) equation will be outlined.

The hamiltonian of a gas consisting of N bosons trapped by an external eld

5



CHAPTER 2. BACKGROUND 6

V trap can be described quantum mechanically using the boson eld operatorΨ(r , t ). Assuming solely pairwise collisional interaction in the very dilute gas,the hamiltonian operator H is

H = d3r Ψ†(r , t ) −h2 2

r

2m+ V trap (r , t ) Ψ(r , t )

+12 d3r d3r Ψ†(r , t )Ψ†(r , t)V (r −r )Ψ(r , t )Ψ(r , t ) (2.1)

V (r − r ) represents an interatomic potential characteristic for species andinternal states of the atoms. This full potential is commonly approximatedheuristically by a simplied binary collision pseudo-potential

V (r −r ) = U 0δ(r −r ), (2.2)

treating binary collisions as hard-sphere collisions. The effective interactionstrength U 0 is related to the s-wave scattering length a by

U 0 =4πh2a

m, (2.3)

where m is the atomic mass. In this thesis we only consider the dilute near-zero temperature regime with a λdB , i.e. scattering length a much lessthan the deBroglie wavelength λdB = 2π h 2

mkT , excluding the treatment of length

scales shorter than the scattering length. There is a considerable body of literature about this. A very comprehensive discussion of interaction effectscan be found in [31]. Morgan showed how divergences of the above pseudo-potential approximation in the nite temperature regime can be mitigated bymaking the approximation on higher order terms of the scattering T matrixseries, which V is the rst term of.

There are a number of different ways of deriving the GP equation, perhapsthe most conceptually consistent of which can be found in [18]. Other dis-cussions are also in [31] [34]. In the following, we will give a somewhat naivetreatment which nevertheless has the virtue of indicating in a simple way howthe nonlinear term in the GP equation arises.

The boson eld operators Ψ(r , t ) satisfy the following commutation rela-tions:

Ψ(r , t ), Ψ(r , t ) = Ψ†(r , t ), Ψ†(r , t ) = 0 (2.4)

Ψ(r , t ), Ψ†(r , t ) = δ(r −r ) (2.5)




From these relations, the Heisenberg equation of motion for Ψ(r , t ) can becalculated and one obtains

ih∂ Ψ(r , t )

∂t= Ψ(r , t ), H (2.6)

= −h2 2

r

2m+ V trap (r , t ) + U 0Ψ†(r , t )Ψ(r , t ) Ψ(r , t )

The eld operators Ψ(r , t ) can in general be written as a sum over all par-ticipating single-particle wave functions Ψ i(r , t ) and the corresponding bosoncreation and annihilation operators.

Ψ(r , t) =i

Ψi(r , t )a i (2.7)

The boson creation and annihilation operators obey the commutation rules

a i , a† j = δij

[a i , a j ] = a†i , a† j = 0 (2.8)

and their operation on the system state in Fock space (tensor product of allsingle state Hilbert spaces) is dened as

a†i |n0, n 1, . . . , n i , . . . = √n i + 1 |n0, n 1, . . . , n i + 1 , . . .

a i |n0, n 1, . . . , n i , . . . = √n i |n0, n 1, . . . , n i −1, . . . . (2.9)

where the n i denote the bosonic populations of the particle states.Since the main characteristic of BEC is that most participating particles

condense into the lowest single particle quantum state, it is possible to sepa-rate out the condensate part Ψ 0(r , t )a0 of the generalised mean eld operator.

With a total number of particles N , the population n 0 of the lowest state ismacroscopic such that N −n0 N . In this case (with N 0 ≡ n0), there isno signicant physical difference between states with N 0 and N 0 + 1 so thatoperators a0 and a†0 (dened above) in the nonlinear mean eld part of thegeneralised version of equation (2.1) can be replaced by √N 0. This is wellknown as the Bogoliubov approximation . Using the Bogoliubov approxima-tion, the eld operator Ψ(r , t ) is written as a sum of its expectation valueΨ(r , t ) = Ψ(r , t ) = √N 0 Ψ0(r , t ) and an operator φ(r , t ) representing the




remaining populations in thermal states, which can be considered vanishinglysmall in the zero temperature BEC regime.

Ψ(r , t ) = Ψ( r , t ) + φ(r , t ) (2.10)

The decomposition (2.10) leads to the following expression for the Ψ†ΨΨ termin (2.1) [34]

Ψ†ΨΨ = |Ψ|2Ψ + 2 |Ψ|2φ + 2Ψ φ†φ + Ψ 2φ†+ Ψ ∗φφ + φ†φφ (2.11)

Only the rst term of this result is used in the GP equation. The next fourterms represent interactions between the condensate component and excitedstates, which are neglected. The last term represents more complex interac-tions between particles, which are also neglected. Thus, it should be kept inmind that the GP equation we are about to obtain is an approximation to amore complete quantum eld theoretical description, with validity restrictedto applications of dilute gases without signicant depletion of the condensedcomponent at temperatures of about T ≈ 0. A conceptual difficulty worthnoting, and arising from this simple form of the mean eld approach, is the in-consistency of a non-zero average of the eld operator a0 with the conservationof particle number. This is sometimes referred to as the issue of broken sym-

metry. Again, the interested reader is referred to a comprehensive resolutionof these difficulties by Gardiner [18].

By substituting the decomposition (2.10), within the approximation of (2.11) as outlined above, into equation (2.1), and normalising the condensatewavefunction Ψ( r , t ) to dr |Ψ(r , t )|2 = N 0, one obtains the Gross-Pitaevskii equation

ih∂ Ψ(r , t )

∂t= −

h2 2r

2m+ V trap (r , t ) + U 0|Ψ(r , t )|2 Ψ(r , t ) (2.12)

As indicated above, all terms containing the perturbation operator φ(r , t ) havebeen neglected in (2.12). Strictly speaking, it is therefore only valid in the zerotemperature case when all atoms are in the condensate component. However,even in that case, the GPE is an approximation, since even at T = 0 , acertain fraction of the atoms is still depleted from the ground state due tointeractions. In 4He, the depletion of the condensed component is about 90%,while in condensed alkali gases such as Rb and Cs, the depletion is in the orderof a mere 0.5%, justifying application of the GP equation in the latter case.




Coupled two-component GP equations

In this work we consider coupled two-component BEC, as have previously beeninvestigated computationally in one dimension in [3] and [14]. We are going toanalyse and describe properties of such systems in two dimensions in detail ata later time in this thesis. However, in context with outlining the derivationof the single component GP equation, we want to state at this point how theextension to two components is done.

The GP equations for a coupled two state system under the inuence of gravity, as used in chapter 5, introduce the concept of two coupled states (seesection 2.2) and include the potential of gravity in addition to the harmonictrapping potential. The coupled equations for the two components, which aredenoted by the appropriate subscripts, are:

∂ψ1

∂t= i 2ψ1−i

r 2

4ψ1 + i

Ω2

ψ2−iC |ψ1|2 + w|ψ2|2 ψ1−iGyψ1 (2.13)

∂ψ2

∂t= i 2ψ2−ik

r 2

4ψ1 + i

Ω2

ψ1−i∆ ψ2−iC |ψ2|2 + w|ψ1|2 ψ2−iGyψ2 (2.14)

The units are given in a dimensionless computational basis, which is describedin detail in section 4.1. We assume a harmonic trapping potential with an un-trapped, equally trapped or stronger conned second state (adjustable throughk = [0, 1, 2]), and Ω is the Rabi frequency providing the coupling interactionbetween the two states. ∆ is the detuning in the coupling process, and w isthe ratio of intra– and intercomponent scattering lengths. It is considered tobe w = 1, as we will see. C represents the coefficient of the nonlinear term,and G accomodates the potential of gravity, which acts along the positive yaxis in the representation we have chosen in our simulations.

The time-independent GP equation

In certain cases, i.e. for eigenstates of a harmonic trap, the wavefunctionΨ(r , t ) can be separated into parts of spatial and time dependence

Ψ(r , t ) = Ψ( r )e−i µth (2.15)

with eigenvalue µ representing the chemical potential of the system at zerotemperature. Substitution of 2.15 into the time-dependent GP equation leads




to the time independent GP equation

µΨ(r ) = −h2 2

r2m + V trap (r ) + U 0|Ψ(r )|2 Ψ(r ). (2.16)

While this thesis concentrates on the time-dependent GP equation, the steadystate case of equation (2.16) is needed for analysis purposes and for com-putation of initial (eigen-)states. A comprehensive discussion of the time-independent GP equation and the generation of eigenstates in harmonic trapscan be found in [7].

2.1.1 Thomas-Fermi ApproximationThe time independent GP equation 2.16, with nonlinearity C , and for a har-monic trapping potential

V trap =r 2

4, (2.17)

can be simplied in the so-called “Thomas-Fermi Approximation”: For largenonlinearities C , and thus for high particle numbers and high self-energy as inreal experimental BEC, the kinetic energy term 2Ψ becomes small comparedto the high self-energy and can be neglected. We then get

µ =r 2

4+ C |Ψ|2, (2.18)

which gives a simple analytic solution for the wavefunction. The population

| Ψ | 2

x=2 µ

-30 -20 -10 0 10 20 30x

Harmonic 1D Eigenstate in TF Limit, C=1000

Figure 2.1: Shape of a harmonic eigenstate in 1D within the TF limit. C=1000.




density |Ψ|2 for this case of a harmonic trap is plotted in gure 2.1 and goesto zero at x = 2√µ. Consequently the normalisation condition in 1D is

2√µ

−2√µ

|Ψ|2dx ≡1, (2.19)

which leads to the result for the chemical potential µ

µ =3C 8

23

. (2.20)

In two dimensions, the normalisation condition is2π

0

2√µ

0|Ψ|2 rdrdφ ≡1, (2.21)

which results in

µ =C 2π

12

, (2.22)

and for completeness, in 3D we have normalisation condition

π

0

2π

0

2√µ

0|Ψ|2r 2sin (θ) drdθdφ ≡1, (2.23)

resulting in

µ =15C 64π

25

. (2.24)

It can be seen that the C values for given chemical potentials increase withdimensionality.

2.1.2 Condensate Energy

The total energy of a condensate is the sum of three components:

• kinetic energy E kin = − V Ψ∗2ΨdV

• potential energy E pot = V Ψ∗V trap ΨdV

• interaction energy E self = 12 C

V |Ψ|4dV




The kinetic energy arises from the Laplacian term in the Hamiltonian (2.1), andgives rise to the spatial movement of the condensate atoms. In the Thomas-Fermi approximation this term is zero. Potential energy originates from theconning trap and, in some of the cases we consider, also from gravity. Theinteraction energy, also called the selfenergy, originates from collisional forcesbetween the atoms, and in multicomponent cases can be further subdividedinto intra- and intercomponent interaction energies. In the cases we consider,the interaction energies are always positive. The condensate energy expecta-tion value for a single component BEC is

E = Ψ|H |Ψ = V −Ψ∗2

Ψ + Ψ∗V Ψ +12C |Ψ|

4

dV. (2.25)

It is worth remarking on the difference in the factor of 12 that accompanies

the self energy term in the Hamiltonian (2.1) and in the GPE (2.12). Whilethe chemical potential µ in the GP equation (2.16) represents the energy oneadds to the system by adding one more atom, the Hamiltonian represents theaverage energy of an atom within the condensate.

In the next section it will be shown that total energy is conserved in systemswithout interaction of an external coupling eld. This is also given for two-state systems. Section 5.8 outlines, how energy components are calculated innumerical simulations.

2.1.3 Energy conservation

The GP equation is unitary and preserves population. It also conserves energyas we now show. The time derivative of equation (2.25) is

dE

dt=

V −∂ Ψ∗

∂t2Ψ

−Ψ∗ 2 ∂ Ψ

∂t+

∂ Ψ∗

∂tV Ψ + Ψ∗V

∂ Ψ

∂t

+ C ∂ Ψ∗∂t

Ψ∗Ψ2 + C ∂ Ψ∂t

ΨΨ∗2 dV.(2.26)

For energy conservation this expression must vanish. To simplify the above,Green’s second identity [9] can be used.

V u 2v −v 2u dV = S

(u v −v u) ·da (2.27)




When u and v are wavefunctions with zero density for r → ∞, the surfaceintegrals on the RHS of Green’s second identity vanish, leading to the followingequality:

V u 2v dV = V

v 2u dV. (2.28)

Using the above equation allows us to rearrange the order of the Ψ ∗ 2 ∂ Ψ∂t term

in (2.26) before substituting the time dependent GP equation and its complexconjugate into it. With this, it is straightforward to show that equation (2.26)becomes

dE dt

=ddt

Ψ|H |Ψ = 0 . (2.29)

Thus energy is conserved in single component simulations.Because the classical coupling elds we use to model output coupling in

chapters 5 and 6 are not conserving energy, we do not expect energy conser-vation in our coupled two-state simulations.

2.2 The two-state Atom

In this section, a review of the physical description of coupled two-state systems

will be given. The signicance of the model becomes evident when we takeaway kinetic energy and nonlinear self-interaction terms from the coupled two-component GP equations, as we have done e.g. for localized analysis and fordebugging purposes. In this case, the behaviour of each microscopic pointin a condensate accurately follows the coupled two-state model, and thus itmust be considered a fundamental principle of our two-component couplingsimulations.

2.2.1 Pseudospin vector description of a two-state AtomBloch’s spin vector formalism [1] provides a convenient way of visualizing theeffects of coupling between the two states of a spin 1

2 system (or a generaltwo-state system) such as an atom in an electromagnetic eld.

If the system state is a linear combination of two states as follows,

|Ψ(t) = a(t)|+ + b(t)|− , (2.30)




it proves useful to dene the following quantities

u = ab∗+ a∗bv = −i(ab∗−a∗b) (2.31)

w = a∗a −b∗b.

For a spin 12 system these can be immediately recognised as the spin com-

ponents sx , sy and sz . More generally, for an arbitrary two-state system, weinterpret these as the in-phase and out of phase components of the magneticdipole, and the inversion. From population conservation in the two states, i.e.

|a

|2 +

|b

|2 = 1, it can be shown that probabilities are conserved over time, and

that vector s(t) = ( u,v,w ) traces out curves on the Bloch unity sphere as itevolves.

u2(t) + v2(t) + w2(t) = ( |a(t)|2 + |b(t)|2)2 = 1 . (2.32)

The system state |Ψ (2.30) evolves as

ih∂ ∂t |Ψ(t) = H |Ψ(t) (2.33)

with Hamiltonian

H = h 0m

·B

h cos(ωt)m ·Bh cos(ωt) ω0. (2.34)

The off-diagonal coupling terms represent the magnetic dipole interaction of dipole moment m of the atomic state and the oscillating magnetic componentB (t) = B 0 cos(ωt) of the coupling eld, while hω0 is the energy differencebetween the two states. (A discussion purely in terms of the two states can befound in appendix A.1). With the above hamiltonian, the components of s(t)become

u(t) = −ω0v(t),

v(t) = ω0u(t) + 2mB (t)h w(t), (2.35)

w(t) = −2mB (t)h v(t).

Feynman, Vernon and Hellwarth [15] showed that this has a geometricalexplanation. Dening a vector

Ω = −2m ·B

h, 0, ω0 , (2.36)




the equations for components u, v and w of s(t) obeyddt s = Ω ×s , (2.37)

which is exactly the equation for a magnetic dipole ( s) precessing in a torque(Ω ).

2.2.2 The Rotating Wave Approximation

For all practical purposes the transition energy ¯ hω0 is orders of magnitudelarger than the magnetic dipole interaction energy m · B , even for strongcoupling elds. Thus, Ω has its largest component along the z axis, with asmall component along the x axis, oscillating very fast with the frequency ω0

of the interacting eld B . It is convenient to decompose the oscillation alongx into two circular components, so that Ω can be written as

Ω = Ω + (t) + Ω−(t) + Ω 0 (2.38)

where

Ω 0 = (0 , 0, ω0) (2.39)

Ω + = (

−2mB

h cos ωt,

−2mB

h sin ωt, 0) (2.40)

Ω− = ( −2mBh cos ωt, 2mB

h sin ωt, 0) (2.41)

Ω + represents a vector, rotating clockwise at the driving frequency ω whereasΩ − rotates counter-clockwise. In the absence of coupling, the torque vectorcauses precession of s about the z axis at a rate ω0. We can simplify thedescription by moving into a frame of reference which is co-rotating with theprecession. Assuming conditions near resonance, i.e. ω ≈ω0, Ω + rotates with(small) frequency ω−ω0 in this frame and accounts for a steady and cumulativeinuence on the vector of the system state. Vector Ω − is counter-rotating withω+ ω0, which is a comparatively large frequency, and thus it does not exhibit acumulative effect on the system state. It reverses itself far too quickly and hasa negligible time-average. The Rotating Wave Approximation (RWA) neglectsΩ − .

After dropping the Ω − term, the torque vector Ω becomes Ω within therotating wave approximation, and has the components

Ω = −2m ·B

h, 0, ω0 −ω = ( −Ω0, 0, ∆) . (2.42)




Now vector equationd

dts = Ω

×s (2.43)

becomes

ddt

uvw

=0 −∆ 0∆ 0 Ω0

0 −Ω0 0

uvw

(2.44)

with Ω0 = 2mBh and after transforming s into its rotating frame equivalent

s = ( u , v , w ) using transformation

u

vw

=

cos ωt sin ωt 0

−sin ωt cos ωt 00 0 1

u

vw

. (2.45)

Above, we specied the Rabi-frequency Ω in terms of the interaction be-tween magnetic moment m and the coupling magnetic eld B . The Rabi-frequency Ω is the rate which coherent transitions between the two levels of the system are induced at, when the system is on resonance, i.e. ∆ = 0. Wewill call this Ω0 from now on. For non-zero detunings ∆, the Rabi frequency asthe oscillation frequency between the two states becomes detuning dependent,

and we getΩ(∆) = Ω2

0 + ∆ 2. (2.46)

In the rotating wave approximation we made the assumption that the mag-nitude of the magnetic dipole interaction is small compared with the transitionenergy. In the case of optical transitions and electric dipole interaction, cautionis indicated only for the strongest interacting laser elds [38]. For magneticdipole interactions, transition energy and dipole interaction both are orders of magnitude smaller. Thus the rotating wave approximation can be considered

to be valid for all magnetic interaction elds that can practically be generatedin a laboratory.

2.2.3 Rabi-solution

Some well known solutions exist for the steady state of (2.44), for the case of constant ∆ and Ω 0. The “Rabi solution” can be obtained by two consecutiverotations of the “torque” vector Ω . The rst rotation is about the v axis to




align Ω with the u axis. (In the general case with detuning, ∆ is in the u −vplane.)

u

v

w

∆

χ

Ω

Ω0

Figure 2.2: Torque vector Ω is constant in the rotating frame and lies in the u-w plane.

The angle Ω is rotated by is tan χ = ∆Ω0

as is illustrated in gure 2.2. Thisrotation is decribed by

uvw

= cos θ 0 sin θ0 1 0

−sin θ 0 cosθ

uvw

. (2.47)

After this rotation, the time evolution of the Bloch vector (2.44) simplies to

ddt

uvw

=0 0 00 0 Ω(∆)0 −Ω(∆) 0

uvw

. (2.48)

And vector ρ is now counter-rotated by an angle of

−Ω(∆) t, thus moving into

a coordinate frame where s = ( u , v , w ) is stationary. The relation betweenthe two rotated vectors s and s is:

uvw

=1 0 00 cos(Ω(∆) t) −sin(Ω(∆) t)0 sin(Ω(∆) t) cos(Ω(∆) t)

uvw

(2.49)

Now, knowing the initial value of the Bloch vector s = ( u0, v0, w0), an an-alytic solution for all times can be obtained by applying all three rotations




consequently in the order discussed above: Rotate s until Ω is parallel withthe u axis, rotate depending on the length of the application of the “torque”and reverse the rst rotation. The resulting matrix multiplication leads to ananalytical expression for the temporal evolution of the inversion:

w(t, ∆) = −u0∆ ·Ω0

Ω(∆) 2 (1 −cos(Ω(∆) t))

−v0Ω0

Ω(∆)sin(Ω(∆) t) (2.50)

+ w0∆ 2 + Ω2

0 cos(Ω(∆) t)Ω(∆) 2

An interesting special case of this general solution is that of an initial conditionu0 = v0 = 0 and w0 = −1 (ground state):

w0(t, ∆) = −1 +2Ω2

0

Ω(∆) 2 sin2(Ω(∆)t2

) (2.51)

From an initial inversion w0 = −1, the inversion oscillates with a frequencydepending on the effective Rabi frequency Ω(∆), with a maximum inversionof 2 Ω2

0

Ω20 +∆ 2 −1 , which also depends on the detuning.

This is illustrated for four different detunings in gure 2.3. The illustrationalso shows the effect of gradual “dephasing” of Rabi oscillations with differentdetunings, with an initial simultaneous population transfer across a wide rangeof detunings during an initial time t ≈ π/ 2. In the case of zero detuning,the previous expression simplies to w(t;0) = −cos(Ω0t), describing simpleresonant Rabi-cycling with complete population transfer.

2.2.4 Coupling Pulses

As has been shown in the previous section, that if one starts with a systemin its ground state, so that w0 = −1 and u0 = v0 = 0, complete populationinversion can be achieved with a so called “π-pulse”. After a time δt such that

Ωδt = π, (2.52)

where Ω is the effective Rabi frequency. The solutions of equation (2.51)show that, for Ω = Ω 0, w = 1 after a coupling pulse like this. For non-zerodetunings, the inversion after a “ π-pulse” is w = 2 Ω2

0

Ω20 +∆ 2 −1. For other times




2π 4π 6π-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

I n v e r s

i o n

t

Figure 2.3: Rabi solutions for Ω0=1 showing the effects of different detunings ∆ . The highest curve represents resonance. The curves with lower peaks are detuned from resonance by 0.2, 1 and 2 times the on-resonance Rabi frequency,respectively.

δt the quantity Ω 0(t2 −t1) gives the overall angle the pseudospin vector s wasrotated by due to the pulse, to calculate the corresponding inversion.

In the more general case of time dependent Rabi frequencies due to timedependent elds, the relevant property of the pulse is its envelope area

A(t) =t

−∞

Ω(t )dt = θ(t) (2.53)

Pulses of fπ , where f is some fraction between 0 and 1, have been used ina huge variety of applications, such as atomic clocks, Ramsey fringes [14], andmore recently in measuring phase coherence of BECs [20].



Chapter 3

Phase Evolution

Control over phase is the key to all wave optics. In classical optics, control-ling the phase of waves permits engineering beam shapes. Lenses of mediawith different phase velocities are used to impose specic phase signatures onbeams as they pass through to make them converge or diverge. In atom optics,phase has the same importance, but also has a specic interpretation: Phasegradients correspond to velocities. By imposing specic phase proles on mat-ter waves, they can be made evolve in similar ways as their classical optical

analogs, exhibiting effects like interference and diffraction.Because of the central importance for atom optics, in this chapter we will

describe the evolution of phase in coupled two-state systems in detail.

3.1 Dressed States

In Atom Optics, coupled systems may be described in the familiar “dressedstate” picture [12]. It is not only the energy levels of the atomic system that

undergo a quantized treatment, but also the interacting radiation eld. Thus,atoms and eld are included in the system hamiltonian and diagonalisationyields the dressed states.

Looking at a two state system interacting with a strong radiation eld likethat of a laser, the description will be in terms of states that describe theatomic states of |u (“upper”) and |l (“lower”) at the same time as the Fockstates of the radiation eld. While there is an arbitrary number of general

20



CHAPTER 3. PHASE EVOLUTION 21

atom-eld states for all numbers N

|u |N , ∀N (3.1)

|l |N , ∀N,

the following atom-eld states are chosen as a two state basis that constitutea closed system, within some well dened approximation.

|I = |u |N −1 (3.2)

|II = |l |N

In state |II the system is in the upper atomic state |u after having absorbedone photon from the N photon radiation eld. The unperturbed energies of these two atom-eld states are:

H 0|I = h(ω0 −ωL ) = −h∆ (3.3)

H 0|II = 0 .

Here, we introduced the unperturbed hamiltonian H 0 of the atoms plus radi-ation, and the detuning ∆ as the difference between the transition frequency

and the laser frequency coupling the two states. The energy is arbitrarily setto zero for state |II and H 0 is diagonal in the |I , |II basis. When thephoton energy is higher than the transition energy, ωL > ω0 (positive detun-ing), state |II is energetically higher than state |I . For negative detuningsthe situation is reversed.

Due to the coupling radiation eld of frequency ωL , a perturbation is intro-duced into the system and the states of equation (3.2) are no longer eigenstates.Under the rotating wave approximation (compare section 2.2.2), diagonalisa-

tion of the perturbed hamiltonian (3.9) connecting states |I and |II yieldsthe eigenstates |N + and |N − . These so-called “dressed states” [11] are linearcombinations of the atom-eld states (3.2).

|N + = α|I + β |II (3.4)

|N − = β |I −α|II

Eigenenergies of the dressed states are E + and E −, also found by diagonalisingthe perturbed hamiltonian.




3.2 Shift of Energy Levels due to Coupling

Without coupling, the eigenenergies of the states |I and |II are in general

H 0|I = E 1|I (3.5)

H 0|II = E 2|II .

Coupling between the two states introduces a perturbation W so that thehamiltonian of the coupled system becomes

H = H 0 + W. (3.6)

This perturbation shifts the energy levels of the original states and the eigenequa-tions become

H |N + = E + |N + (3.7)

H |N − = E −|N − .

Assuming a purely non-diagonal coupling matrix (W), with Rabi frequency Ω 0

characterising the coupling strength, where

W 12 = W ∗21 =hΩ0

2, (3.8)

the matrix representing H in the |I , |II basis is written

(H ) =E 1 W 12

W 21 E 2(3.9)

and one nds the eigenenergies

E + =12

(E 1 + E 2) +12 (E 1 −E 2)2 + 4 |W 12|2

E − =12(E 1 + E 2) −

12 (E 1 −E 2)

2

+ 4 |W 12|2

. (3.10)The relations between the |I , |II basis and the new eigenvector basis

|N + , |N − are

|N + = α|I + β |II (3.11)

|N − = β |I −α|II

|I = α|N + + β |N − (3.12)

|II = β |N + −α|N −




where α and β are:

α = Ω + ∆

2Ω , β = Ω

−∆

2Ω (3.13)

Here, the generalized Rabi frequency Ω , which depends on the detuning h∆ =E 1 −E 2, is dened as

Ω = ∆ 2 + Ω20 =

1h (h∆) 2 + (2 |W 12|)2. (3.14)

E

+E

E

E

o

E m

-E

Ω /2

∆

’Ω/21

2

Figure 3.1: Variation of the eigenenergies E + and E − of the coupled systemwith respect to the detuning ∆ . In the absence of coupling interaction, the en-ergies E 1 and E 2 cross at the origin. Under the effect of non-diagonal coupling,the energy levels repel each other, leading to the “avoided crossing”.

Using E m = 12 (E 1 + E 2) and the above expressions, the eigenenergies be-

come

E + = E m +hΩ

2(3.15)

E − = E m −hΩ

2.

An illustration visualising the effect of detuning in a coupled system isfound in gure 3.1. Without the coupling W , the eigenenergies cross at the




origin for zero detuning ∆ = 0. The coupling interaction between the twostates separates the energy levels for all detunings, leading to the “avoidedcrossing” in the illustration. For ∆ = 0, the shift is ±|W 12| for the respectivestates, separating the levels by the energy ¯hΩ0.

The asymptotic nature of this level shift at large detuning is easily foundfrom a power series expansion [11] of Eq. (3.10) in terms of |W 12

h∆ |:

E ± = E m ±h∆2

1 + 2W 12

h∆

2

−. . . . (3.16)

Figure 3.2 illustrates the situation of the level shifts for a xed detuning ∆.

The “bare” atom-eld states are separated by the energy ¯ h∆, the dressed states

(Ω−∆)/2

Unperturbed levels Levels perturbed by coupling

|l>|N>

|u>|N-1>

,

,

∆h Ω,

h

h

(Ω−∆)/2h-

Figure 3.2: Shift of dressed state energy levels in the presence of a detunedcoupling eld. (Diagram drawn for the case ∆ > 0).

are separated by hΩ . Since equations (3.10) describe a symmetric level shift,the magnitude of the shifts of the respective levels is h Ω −∆

2for all values of

∆.Notice that for the bare atom-eld states, state |l |N has higher energy

than |u |N −1 for positive detuning ∆, while for negative detunings the orderis reversed.




Level shift of dressed state for detuningCorresponding atom-eld state ∆ > 0 ∆ < 0

|u |N −1 −h Ω −|∆ |2 −h Ω + |∆ |2

|l |N + h Ω −|∆ |2 + h Ω + |∆ |2

Coupling always leads to new eigenstates which are energetically separatedfurther than the bare atom-eld states.

3.2.1 Using detuned lasers to shape trapping potentials

It is worth noting that the potentials created by far detuned laser beamsused for creating spatially controlled perturbations of the potentials conningcondensate clouds [10] [26], are based on exactly the same physical principleof the level shifts described above, expressed in Eq. (3.16). A far blue detunedbeam, for example, causes a positive energy shift Ω2

4∆ of the ground state (seeappendix A) and thus behaves as a repulsive potential. A red detuned beambehaves as an attractive potential.

Fundamentally, the potential, or “force”, arises by the mechanism of co-

herent scattering of photons from one mode (or k vector) to another, and onecan see that for a net exchange of momentum between the atom and the radi-ation eld, a spatially shaped, or non zero intensity gradient is required. Thisso-called dipole force is discussed in detail in [12].




3.3 Phase Evolution in a driven Two State

SystemThe temporal evolution of a system state can be easily expressed in terms of the eigenstates and eigenenergies. For the case of the initial population beingfully in state |I , temporal evolution can be expressed as follows:

e− iHt

h |I = e− iHt

h [α|N + + β |N − ] (3.17)

= αe− iE + t

h |N + + βe− iE − t

h |N −

Now expressing the eigenstates |N + and |N − in terms of the bare atom-eldstates, equation (3.17) becomes

= ei ∆ t

2 cos Ω t2 − i∆

Ω sin Ω t2 |I −Ω

Ω i sin Ω t2 |II (3.18)

≡ a(t)|I + b(t)|II

The magnitudes of the complex coefficients a(t) and b(t) relate to the pop-ulations in the respective states |I and |II , while the phase of the statesis evaluated from the complex arguments of the coefficients. Conservation of

total population is expressed as

|a(t)|2 + |b(t)|2 = 1 . (3.19)

Now if we writea(t) = |a(t)|eiΦ| I (t ) (3.20)

andb(t) = |b(t)|eiΦ| II (t ) , (3.21)

it is easy to see that Φ|II

evolves as ∆

2t, but with a π phase jump every period

T = 2πΩ , as illustrated in gures 3.3 and 3.4. The phase jumps occur when

the upper state population goes to zero. Averaging over an integral numberof periods, we nd

Φ|II (t) =Ω + ∆

2t. (3.22)

For state |I , it is easy to show that

Φ|I (t) =∆ t2 −tan −1 ∆

Ωtan

Ω t2

. (3.23)




This solution is smooth (e.g. see gures 3.3 and 3.4). Over an integer numberof periods it averages to

Φ|I (t) = −Ω −∆2

t. (3.24)

a)

-15

-10

-5

0

5

10

15

20

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

P h a s e a n g

l e [ r a

d ]

b)

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

R e

l a t i v e p o p u

l a t i o n s

Time [t/T]

Figure 3.3: a) phase evolution b) population evolution for states |I (solid lines)and |II (dashed lines), driven by a cw eld for the case Ω0 = 1 , ∆ = 0 .1,

initial inversion w = −1. Time scales in terms of 2π

Ω0 .For large detunings ∆, i.e. ∆ 2 Ω2, this phase slope is approximated by

ΦI (t) = −Ω2

4∆ t. This result is in agreement with the phase encoding that can becalculated directly for the far detuned regime, when the two-state system canbe approximated as a single-state (ground state) system using the so-called“adiabatic elimination”. (We have outlined this in Appendix A.)

Equations (3.22) and (3.24) describe the averaged phase evolution accu-rately for ∆ > 0 as well as for ∆ < 0. The equivalence of the gradients of




a)

-5

0

5

10

15

20

25

30

0 1 2 3 4 5

P h a s e a n g

l e [ r a

d ]

b)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

R e

l a t i v e p o p u

l a t i o n s

Time [t/T]

Figure 3.4: a) phase evolution b) population evolution for states |I (solid lines)and |II (dashed lines), driven by a cw eld for the case Ω0 = 30 , ∆ = 100 ,initial inversion w = −1. Time scales in terms of 2π

Ω0.

phase evolution with the shift of energy levels previously discussed in section3.2 will be interpreted in the following section.

Note that we have chosen the energy of atom-eld state

|I to be E 1 = 0,

which sets the origin of the energy scale. All level shifts and, as we will see,all phase gradients are measured relative to this energy level.





Chapter 4

Numerical Implementation

This section describes the implementation of the method that was used tosolve numerically the nonlinear GP equations. The Fourier Transform method,which we outline below, was chosen because it is a well developed algorithmthat was available within the group [16].

4.1 Dimensionless Units

To overcome problems of machine precision and numerical errors, which arisethrough truncations and roundings, it is worthwhile in a numerical implemen-tation of a physical problem to transform the equations into “computationalunits”. For reasons of simplicity we will discuss the single component GPequation only, since it is trivial to extend this description to the coupled twocomponent case. The GP equation, for a trapped BEC in two spatial dimen-sions, and in a gravitational eld is

ih∂ Ψ∂ t = −

h2

2m2 ¯Ψ +

12m(ω

2x x

2

+ ω2y y

2

)¯Ψ −NU 0|

¯Ψ|

2 ¯Ψ −mgy

¯Ψ (4.1)

Equation (4.1) is written in SI units (marked with bars on the symbols) andallows for nonisotropic traps by introducing different trap frequencies in x andy directions, such that ωy = λωx .

We transform t, x, y, Ψ into dimensionless (i.e. computational) quantities

30



CHAPTER 4. NUMERICAL IMPLEMENTATION 31

by the transformations

x = x0xy = x0y

t = t0t (4.2)

Ψ = Ψ 0Ψ

in which t0, x0, y0 and Ψ0 carry dimensions. Substituting equations (4.2) intoequation (4.1), we nd that by setting t0 ≡ 1

ωx, equation (4.1) reduces to

∂ Ψ

∂t

= i 2Ψ

−

i

4

(x2 + λy2)Ψ + iC

|Ψ

|2Ψ + iGyΨ (4.3)

where

x0 ≡ h2mω x

C ≡ NU 0hω x x2

0(4.4)

G ≡ mghω x

x0.

Notice that Ψ 0 has the dimension 1x0

, as can be inferred from its normalisationequation

∞

−∞|Ψ|2 dxdy ≡1. (4.5)

In the above denition of C , U 0 has the dimensions of an area, so that Cis nondimensional. The two-dimensional nonlinearity used in our simulationscan be derived from the three-dimensional value using an integration methodas used in [41], or by matching density peaks of corresponding 3D- and 2D-wavefunctions, as outlined in [29].

4.2 Interaction PictureIt is convenient to separate the right hand side of equation (4.3) into the partinvolving spatial derivatives, and the remainder. We write

∂ Ψ∂t

= DΨ + N Ψ (4.6)

whereD = i 2, N = −

i4

(x2 + λy2) + iC |Ψ|2 + iGy (4.7)




This allows the Laplacian operator to be handled separately from effects of thenonlinear term in the GP equation. Since it is easy to account for the kineticenergy term ( i 2) alone, we transform into an interaction picture as follows

ΨI (τ,x,y ) = e−D (τ −t )Ψ(τ,x,y ), (4.8)

where τ is the time at which the interaction picture separates from the normalpicture. The GP equation (4.6) now becomes

∂ ΨI (τ ,x ,y )∂t

= N I (τ,x,y )Ψ I (τ,x,y ) = e−D (τ −t )N Ψ(τ,x,y ), (4.9)

which is formally similar to an ordinary differential equation, and thus can be

solved by Runge Kutta methods. The only remaining problem is the evaluationof the right hand side of equation (4.9), which is described in the followingsection.

4.3 Fourier Transform Method

For the sake of simplicity, the following description applies to the one-dimensionalcase. The extension to two spatial dimensions is relatively straightforward con-

ceptually, but computationally intensive.The use of Fourier transforms (FTs) simplies the evaluation of the right

hand side of equation (4.9), since the operator D is diagonalised in this repre-sentation.

Dening the Fourier transform of N (τ, x)Ψ(τ, x) (as in [33])

θ(τ, f ) =∞

−∞

N (τ, x )Ψ(τ, x)ei2πf x dx, (4.10)

then the inverse transform will be

N (τ, x )Ψ(τ, x) =∞

−∞

θ(τ, f )e−i2πf x df. (4.11)

It is now easy to calculate the right hand side of equation (4.9) as follows:

e−D (τ −t )N (τ, x)Ψ(τ, x) = ∞

−∞θ(τ, f )e−i(τ −t ) 2

e−i2πf x df (4.12)

= ∞

−∞

θ(τ, f )e−i(τ −t )(−i2πf )2e−i2πf x df




To summarize the whole calculation, if one denes

˜θ(τ, f ) = θ(τ, f )e−

i(τ

−t )(

−i2πf )2

, (4.13)the calculation of (4.9) can be described as follows:

• Calculate θ(τ, f ) by simple FT.

• Multiply each coefficient with e−i(τ −t )(−i2πf )2to obtain θ.

• Carry out inverse FT on θ to obtain e−D (τ −t )N Ψ.

4.4 Runge-Kutta MethodThe Fourier method described in the previous section allowed us to conve-niently evaluate the function f (Ψ I , t ) representing the derivative in equation(4.9).

∂ ∂t

ΨI (t) = f (Ψ I , t ) (4.14)

The Runge-Kutta (RK) method provides a well tested “workhorse” for in-tegrating differential equations, and a 4th order implementation to equation

(4.9) is convenient.For each timestep the solution is advanced by the RK4 algorithm, thederivatives are evaluated four times in different places; once at the initial point,twice at the center between the step points and once at a trial endpoint. The“true” endpoint, the solution of the timestep, is then calculated as a weightedsum of the four derivative evaluations as follows. Dening [33]

k1 = f (Ψ I , tn )

k2 = f (Ψ I + k1∆ t2 , tn + ∆ t

2 )

k3 = f (Ψ I + k2 ∆ t2 , tn + ∆ t

2 ) (4.15)

k4 = f (ΨI + k3∆ t, t n + ∆ t),

the weighted sum is

ΨI (tn +1 ) = Ψ I (tn + ∆ t) = Ψ I (tn ) +k1

6+

k2

3+

k3

3+

k4

6∆ t. (4.16)

An advantage of using the interaction picture is that by chosing the originof the picture ( τ ) at each time step to be at the centre of the step, the factor




of e−D (τ −t ) disappears from the derivative, so that FFTs are saved in two of the four evaluations of the differential necessary in each timestep.

4.5 Numerical Limits

Our computer simulations are limited, as usual, by the available resources of CPU power, memory and data storage. The main impact on our work is on thechoice of grid- and stepsize. We discuss the implications of these limitationsin the following sections.

4.5.1 Limitations imposed by grid resolution

The velocity of a coherent BEC superuid is a linear function of its phasegradient (see section 6.5). In our computational units, the following relationholds:

v = 2 S (4.17)

where S is the phase of a complex wavefunction of the form |Ψ|eiS . To simulate

v

S ∆

-40

-30

-20

-10

0

10

20

30

40

0 5 10 15 20 25 30 35 40

v e

l o c

i t y

Phase Gradient

Figure 4.1: Centre of mass condensate velocity vs. phase gradient. Aliasing occurs at the critical velocity limit v=19.95. Solid line for a Gaussian eigenstate and dotted line for a plane wave with zero bandwidth. In the plane wave case the velocity changes sign exactly at the limit.




a BEC cloud moving through computational space, we apply a phase gradientS across a BEC eigenstate and propagate it in time. Now, if 20 spatial

units of computational space are stretched out over 128 gridpoints, the spatialresolution is ∆ x = 20/ 127 = 0.1575 . . . , and the phase change ∆ S from onegridpoint to the next must be less than π for a positive velocity, since a phasegreater than exactly π can be ambiguously interpreted. Additionally, one mustconsider the problem of aliasing. The Fourier spectrum associated with the grid(kmax = 1

∆ x ) must be large enough to accommodate the Fourier componentsof the BEC.

Figure 4.1 the centre of mass velocity of the simulated BEC cloud is plot-

ted vs. the applied phase gradient (solid line). Since the BEC cloud has anear-Gaussian Fourier spectrum (Gaussian for a C =0 eigenstate), aliasing oc-curs in a region of a phase gradient between 18 and 22, while a plane wavewith vanishing bandwidth exhibits a sharp aliasing “transition” at a phasegradient of exactly π/ 0.1575· · · ≈19.95 (dotted line). This demonstrates thatsimulations of dynamic processes, i.e. output coupling with acceleration undergravity, require minimum grid resolutions, depending on the expected max-imum velocities, in order to yield physical results. To allow high velocities

in output coupling simulations, i.e. velocities of up to v=80, we used spatialresolutions of ∆ x = 20/ 255 = 0.078 . . .




4.5.2 Limited simulation stepsize

It is desirable to use simulation timesteps as large as possible, to limit com-putational time. The RK based algorithm described in sections 4.3 and 4.4has been shown to be accurate to 4th order in step size [16], and thus ensuresaccurate solutions with larger step size than in a standard split step method.Nevertheless, the step size cannot be chosen too large.

The temporal phase gradient of an eigenstate (compare equation 2.15), isthe chemical potential µ as dened in equation (2.16). For two-dimensionalwavefunctions, the chemical potential is evaluated as µ = C

2π , as shown insection 2.1.1. Thus, the aliasing limit for a C =200 eigenstate in 2D is reachedfor stepsizes greater than ∆ t = π/µ = 0 .54.

In gure 4.2, the centre of mass (c.o.m.) velocity of a BEC eigenstate witha constant phase gradient of 10 and a corresponding velocity of vx = 20 isplotted vs. temporal step size; the initial state was propagated for a singletime step, after which the c.o.m. velocity was calculated. With increasingstepsize, the aliasing limit is reached at ∆ t = 0 .54 and the measured c.o.m.velocity is reversed, i.e. phase rotations larger than π within one timestep areinterpreted as negative phase rotations. With further increasing stepsize, mea-

sured velocity reverses several times while tiny differences in the condensate’sphase gradient (due to adiabatic expansion in absence of a trapping potential)become increasingly important, leading to a “washing out”, and causing theaverage c.o.m. velocity eventually to go to v = 0.

When practically determining the step size for dynamic numerical simu-lations, we must also consider effects of changing wavefunction amplitudes,which can cause large temporal phase gradients. Furthermore, although theRK method works very well, it cannot accurately follow large gradients close

to the aliasing limit. Thus, for all practical purposes, the step sizes we chosewere many orders of magnitude smaller than the step size at the aliasing limitin the example described above. For example, while the aliasing limit for aC =200 2D eigenstate is at 12 steps per trap cycle T trap = 2 π, we typically usedstep sizes as small as 1000 or 2000 steps per T trap .




10−1

100

101

102

−10

−5

0

5

10

15

20

temporal step size

c l o u d c

. o . m .

v e l o c i t y

Figure 4.2: Illustration of simulation failure for too large stepsizes. The temporal phase gradient µ (chem. potential) of a C =200 condensate 2D eigenstate reaches the aliasing limit for a step size of ∆ t=0.54.



Chapter 5

RF Output Coupler in 2D

This chapter describes simulations of continuous wave (cw) radio frequency (rf)output coupling in a two-state system in two dimensions. While cw outputcoupling has been simulated in one spatial dimension [3], a key difference inthe present work is the addition of gravity. The setup of a cw output couplerfor BEC has been called an “atom laser” and has been realized experimentally[8]. While the term “atom laser” appears to be an appropriate descriptionfor a “beam” of matter with properties similar to optical laser beams, most

importantly with spatial coherence, the notion is not universally agreed uponand still a matter of controversy. A strict denition has been proposed byWiseman [39].

The effects of gravity open up a range of new behaviour, but also give riseto certain computational problems. For example, gravity provides an easy wayto get rid of outcoupled untrapped populations by just letting it fall out of thetrap, thus forming a “beam”. However, the output “beam” will now inevitablyreach the edge of the computational grid in the consequent acceleration. Fur-thermore, increased velocities in the simulation demand higher computationalgrid resolutions and smaller temporal stepsizes.

5.1 Output Couplers

To obtain Bose-Einstein condensation, atoms are trapped magnetically, uti-lizing the Zeeman shift of their hyperne states. Different spin states, i.e.M F = 1 , 0, −1, have different trap properties since the force, which the trap-

38



CHAPTER 5. RF OUTPUT COUPLER IN 2D 39

ping eld exercises on the atoms, is

F ∝M F ∂B (r )

∂r , (5.1)

where B is the magnetic eld and r the distance from the eld zero, i.e. thetrap centre. While, for example, atoms with M F = 1 are trapped, atoms withM F = 0 are indifferent to a magnetic eld, i.e. they are untrapped, and atomswith M F = −1 are “anti-trapped” and repelled from the trap.

The energy difference between trapped and untrapped states (we will useM F = 1 and M F = 0 for this purpose from now on) is in the radio frequency(rf) regime, so that two basic strategies for an output coupler are viable: Pop-ulation can be transferred and deposited into an untrapped state by simpleradio frequency coupling, and left there to evolve from the same spatial re-gion (rf coupler), or it can be transferred and actively ejected from the spatialregion of coupling (Raman couplers). We will only discuss in this work thesimple rf coupler case, as has been experientally realised in [30] and [8].

5.2 Spatial Effects in Coupling

A coupled two-state system is described by the following set of GP equations

∂ψ1

∂t= i 2ψ1 −i

r 2

4ψ1 + i

Ω2

ψ2 −iC |ψ1|2 + w|ψ2|2 ψ1 (5.2)

∂ψ2

∂t= i 2ψ2 −ik

r 2

4ψ1 + i

Ω2

ψ1 −i∆ ψ2 −iC |ψ2|2 + w|ψ1|2 ψ2. (5.3)

Ω is the Rabi frequency, i.e. the coupling strength, ∆ is the detuning of thecoupling eld for the centre of the trap, and k is the relative trap parame-

ter of the second component. For an untrapped second state, k is equal tozero. Other physical systems with different magnetic momenta, e.g. in theM F = [2, . . . , −2] manifold, can be simulated by adjusting this parameter ap-propriately. w is the relative inter-state collision strength, the ratio betweenthe inter-component interaction and the self interaction. The magnitude of whas been a matter of discussion and was incorrectly set at w = 2 in reference[3]. Recent experimental results in 87Rb vapour [19] show scattering lengthratios of a1 : a12 : a2 :: 1.03 : 1 : 0.97, where a1 and a2 are the scattering




lengths within the rst and second component, and a12 is the intercomponentscattering length. Since all three values are very close to each other and ap-proximately equal to one, using a single value of w = 1 for all three scatteringlengths for the numerical simulations in this work seems to be justied.

In our simulations of output coupling, only the rst state is trapped by aharmonic potential, while the second state is untrapped. Thus the detuningbetween the two components is not uniform at all spatial positions in the cloud.In the absence of the 2 term, i.e. within the TF approximation, we can denean effective detuning ∆ eff 5.1):

∆eff

(r ) =

−C (w

−1)

|Ψ

2(r , t )

|2

− |Ψ

1(r , t )

|2 + ∆

−1

4r 2. (5.4)

which is obtained by taking the difference of the diagonal terms of (5.2) and

r

14

r 2

Trapping potentialfirst state

Untrappedsecond state

E

Figure 5.1: Spatial dependence of energy difference between trapped and un-trapped states. ( w=1)

(5.3). ∆ eff reduces to ∆ at the centre of the trap, when w=1, and increasesradially, as we illustrate in gure 5.1.

As we will show in detail in section 5.5, the coupling strength depends onthe detuning. We dene a spatially dependent Rabi frequency:

Ω(r ) = ∆ 2eff (r ) + Ω 2

0. (5.5)

The spatial dependence of the effective detuning (and hence the generalizedRabi frequency) lead to “spatial effects”: Coupling is no longer homogeneous




throughout the cloud and is more effective at certain spatial locations. Asshown in [3], for spatial effects to be negligible, i.e. for homogeneous populationtransfer, we require that Ω( r ) is approximately constant over the whole cloudand thus approximately equal to the trap centre cycling frequency Ω 0. Thisrequirement can be expressed as

Ω(r )Ω0 ≈1. (5.6)

Using the Thomas-Fermi expression (see section 2.1.1)

C

|Ψ

|2 = µ

−1

4r 2, (5.7)

we see that for w = 2 as in [3], ∆ eff is reduced to an initial value (for vanishingpopulation in the second state) of

∆ eff (r ) ≈µ + ∆ −12

r 2. (5.8)

For w = 1, we have no net contribution of self energies, and we get

∆ eff (r ) ≈∆ −14

r 2. (5.9)

Evaluating (5.6) in the range −2√µ < r < 2√µ, where the wavefunction isnonzero within the Thomas-Fermi limit, the condition for spatially indepen-dent Rabi cycling (i.e. no spatial effects of coupling) for both values of w(w=1,2) becomes

∆ 2 + Ω20 µ(µ + 2 |∆ |). (5.10)

For the remainder of this thesis, we will always assume a value of w = 1.Since inequality 5.10 is not easy to interpret, we illustrate it for a 2D

C =200 eigenstate in gure 5.2. Curves are plotted for an inequality factor of 10, and also for a factor 1. This provides a rough impression of the shape of the parameter regime, in which spatially dependent coupling can be expected,but it should be interpreted with appropriate care. Spatial effects will becomemore signicant with smaller ∆ and Ω.

Spatially homogeneous coupling can occur for two distinct reasons: Ei-ther large bare Rabi frequencies or large detuning ∆, dominating the spatialvariation in the effective detuning ∆ eff .




C o u p

l i n g

Detuning ∆

Ω 0

0

10

20

30

40

50

60

-150 -100 -50 0 50 100 150

Figure 5.2: Illustration of regimes in which spatial effects are expected. Pa-rameter µ=5.64 represents a 2D C =200 eigenstate. Plotted are functions ∆ 2 + Ω2

0 = µ2 + 2 µ|∆ | (solid line) and ∆ 2 + Ω20 = 10( µ2 + 2 µ|∆ |) (dashed

line). Equation (5.10) is valid for parameters under the dashed curve.

−10

−5

0

5

10

−10

−5

0

5

10

0

0.5

1

1.5

2

2.5

3

x 10−3

a)

| Ψ | 2

x

y−8 −6 −4 −2 0 2 4 6 80

0.5

1

1.5

2

2.5

3x 10

−3

x

| Ψ | 2

b)

Figure 5.3: Two dimensional normalised C =200 eigenstate. a) Spatial prob-ability density |Ψ|2. b) Prole along y=0 (solid line) and Thomas-Fermi approximation (dashed line).




5.3 Inuence of gravity on trapped BEC

Gravity adds another potential to the coupled GP equations. This potentialis linear and vertically increasing, and it leads to an offset of the condensate’sequilibrium position from the center of the magnetic trapping potential. Thisgives rise to new behaviour of output couplers.

The coupled nonlinear Gross-Pitaevskii equations for a two-level systemunder the inuence of gravity are (in computational units):

∂ψ1

∂t= i 2ψ1

−iV trap ψ1 + i

Ω

2ψ2

−iC

|ψ1

|2 + w

|ψ2

|2 ψ1

−iGyψ1 (5.11)

∂ψ2

∂t= i 2ψ2−ikV trap ψ2 + i

Ω2

ψ1−i∆ ψ2−iC |ψ2|2 + w|ψ1|2 ψ2−iGyψ2 (5.12)

G is the acceleration of gravity scaled into dimensionless units in equation(4.4). All other symbols have been discussed in the previous section.

The equilibrium offset of the ground state condensate centre of mass, rel-ative to the center of the magnetic trapping potential is easily calculated. Wehave the vertical component of the trapping potential V trap,v = −1

4 y2 and thelinear vertical potential of gravity V grav =

−Gy. The condensate equilibrium

position is where the joint potential is at a minimum, i.e. where the forcesresulting from the harmonic trapping potential and gravity are equal. This iseasily found to be at y = 2 G.

In our simulations of output coupling, we use nonlinear two-dimensionalinitial states with values of C =200, unless stated differently. We plot in gure5.3a a true 2D wave function, and in gure 5.3b, we compare a prole of thiseigenstate with the corresponding TF approximation. We can see that thelatter is a very accurate approximation, except at its edges.

5.4 Localized Output Coupling

While gravity shifts the equilibrium position of the condensate, the spatiallocation of the coupling is still determined by the effective detuning. (No-tice that gravity effects both components equally and thus does not alter theexpressions for ∆ eff of equation 5.4.) Output coupling occurs along the sur-face of a sphere (a circle in the 2D representation) centred at the trap origin,




i.e. along a magnetic equipotential surface, and this is no longer symmetricabout the condensate due to its offset of y = 2 G. This is illustrated in gure5.4. Population coupled into the untrapped second state is released from this“cut” along the magnetic equipotential line to fall down under the inuenceof gravity.

¡ ¡ ¡ ¡

¡ ¡ ¡ ¡

¡ ¡ ¡ ¡

¡ ¡ ¡ ¡

¢¡ ¢¡ ¢¡ ¢

¢¡ ¢¡ ¢¡ ¢

¢¡ ¢¡ ¢¡ ¢

¢¡ ¢¡ ¢¡ ¢

y

x

g

of trapping fieldmagnetic equipotential lines

condensate cloud

Figure 5.4: Equlibrium position of condensate cloud in a harmonic trapping potential under the inuence of gravity.

The position of the “cutting” magnetic equipotential line can be determinedby setting the effective detuning in equation (5.4) to ∆ eff = 0. For w = 1 thisbecomes independent of populations and nonlinearity, and the solution is

r = 2√∆ . (5.13)

For arbitrary values of w, the cutting position depends on the componentpopulation densities according to

r = 2 ∆ −C (w −1)[|ψ2|2 − |ψ1|2]. (5.14)

In gure 5.5, the relationship between spatial position of coupling anddetuning is illustrated. (Data acquired in simulations in the TF regime, i.e.




∆

0

2

4

6

8

10

12

14

16 0 5 10 15 20 25 30 35 40 45 50

y

detuning

Figure 5.5: Spatial region of maximum coupling in a prole along the x=0axis. Function y = 2√∆ (solid line). Simulation data points for w=1 (crosses)and data points for w=2 (stars). Condensate equilibrium position at y=10 forG=-5.

2 term ignored.) For w=1, all data points obtained from simulation-runs lieon the theoretical curve corresponding to Eq. (5.13). For w=2 the situationis slightly different. Since coupling is relatively weak, population in state ψ2

remains small and equation 5.14 can be approximated by

r = 2 ∆ + C |ψ1|2. (5.15)

Higher self-energy increases the effective detuning between the two compo-nents at a certain spatial position (5.4). Since coupling occurs along a lineof constant detuning ∆, this moves the coupling region to a spatial positionwhere an increased trap potential compensates the shift, and coupling occursat a different spatial position, i.e. at increased y. This shift is dependent onpopulation densities, and thus, the w=2 data points edge closer to the line,which represents w=1, with increasing distance y from the condensate centreat density maximum at y=10 in gure 5.5.

Figure 5.6 shows the temporal evolution of an output coupled populationstream in the weak coupling limit. The initial state (a) is left almost unchangedduring the time of the simulation, while a region of the cloud is subject to Rabi




y

t = 0.42412

a)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

t = 0.42412

b)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

10−2

10−3

10−4

10−5

10 −6

10−7

10−8

x

y

t = 1.3666

c)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

x

t = 2.5447

d)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

10−2

10−3

10−4

10 −5

10−6

10−7

10 −8

Figure 5.6: Population densities in condensate states |1 and |2 for cw outputcoupling in a gravitational eld. (a) state |1 , (b) state |2 at the same time t as in (a). A temporal sequence of the evolution of state |2 is shown in (b),(c) and (d), plotted in a logarithmic density scale. (Simulation parameters:Ω0=0.2, ∆ =40, G = −5, C = 200, 256 by 512 grid.)




y

t = 0.42412

a)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

t = 0.42412

b)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

π

π /2

0

− π /2

− π

x

y

t = 1.3666

c)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

x

t = 2.5447

d)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

π

π /2

0

− π /2

− π

Figure 5.7: Phase of condensate states |1 and |2 for the same simulation as in gure 5.6. Phase plotted for population densities larger than 10−8.




cycling with the second state, shown in (b) at the same time t as (a), and in(c) and (d) at later times. (At a Rabi fequency of Ω=0.1, a full Rabi cycletakes ∆t=20 π, thus the small effect on the rst state.)

In gure 5.7 we plot the phase of the two wavefunctions for the same caseas in gure 5.6. Gravity accelerates the outcoupled population downwards,leading to an increasing phase gradient. At the same time, self energy andinteraction energy (mutual repulsion between the states) cause the untrappedcloud to expand and accelerate radially. Overlaying these two gradients leadsto the “interference” fringes most clearly seen in (d). A magnication of theinteresting region is shown in gure (5.8). It has been veried that this is

not an artefact of printer resolution by taking the population in question andevolving it after subtracting the phase gradient caused by gravity. Note thatin gures (5.7) and (5.8) we have plotted phase only for population densitiesgreater than 10 −8 in front of a uniform background.

x

y

−15 −10 −5 0 5 10 15

25

30

35

40

45

50

55

π

π /2

0

− π /2

− π

Figure 5.8: Magnication of gure 5.7d. Population falling under gravity (strong vertical phase gradient) while undergoing free expansion (small radial phase gradient) leads to an “interference like” phase plot.




5.5 Coupling Bandwidth

In the previous sections we have shown how localized spatial coupling can beaccomplished by controlling the coupling eld strength and detuning underappropriate trapping conditions. We have also seen that the coupling band-width plays an important role in output coupling simulations by determiningthe spatial width of the coupling region, i.e. very high bandwidths lead tohomogeneous coupling through power broadening. For a precise engineering of output coupled “beams”, controlled small coupling bandwidths are essential.

5.5.1 Power BroadeningThe natural linewidth of our transitions depends on upper state lifetime τ ,leading to a Lorentzian shaped spectrum of

|E (ω)|2∝1

∆ 2 + γ 24

(5.16)

for detunings ∆ = ω −ω0 and with γ = 1 /τ . Compared with optical dipoletransitions, the spontaneous decay rate for magnetic dipole transitions betweendifferent M F states is negligibly small, even on much larger timescales thanthat of our simulation. Furthermore, collisional coupling at low densities isalso negligible, and thus power broadening is the main broadening mechanism.

It is well known from the Rabi solutions (e.g. see chapter 2.2.3 or [25])that the maximum upper state population in a coupled system undergoingRabi cycling with a detuning ∆ is

pmax =Ω2

0

∆ 2 + Ω20

=Ω2

0

Ω 2 (5.17)

This leads to a Lorentzian shaped curve of maximum population vs. detuning,with a HWHM value ∆ ωH given by the Rabi frequency

∆ ωH = Ω0. (5.18)

In the output coupler described in the previous sections, the detuning ∆ in(5.17) is replaced by the spatially dependent effective detuning ∆ eff (eq. 5.4).The position of the centre of coupling was found to be r = 2√∆. Using theHWHM condition

∆ eff = ∆ ωH = Ω0, (5.19)




0

0.5

1

1.5

2

2.5

3

3.5

4

0 2 4 6 8 10 12 14

S p a t i a l F W H M o f c o u p l i n g r e g i o n

Distance of centre of coupling region from trapping field zero

Figure 5.9: Width of coupling region plotted vs. distance of the resonantcoupling region from the trapping magnetic eld zero ( r =2 √∆ ). Widths cal-culated from equation (5.20) for a purely power broadened transition. Curves are plotted for Ω=0.2, 0.4, 1, 2 (solid, dot-dashed, dashed and dotted lines respectively).

we calculate the full spatial width of the (FWHM) coupling region ∆ y as:

∆ y = 2 ∆+Ω 0 − ∆ −Ω0 for ∆ > Ω0, (5.20)

∆ y = 2 ∆+Ω 0 for −Ω0 < ∆ < Ω0,

and ∆ y = 0 for ∆ ≤ −Ω0.

Figure 5.9 illustrates how the power broadening converts into a region of

spatial coupling for various choices of the centre of the coupling.




5.5.2 Broadening through nite Pulse Times

x

y

t = 2.4976

a)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

x

t = 2.4976

b)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

10−2

10−3

10−4

10 −5

10−6

10−7

10 −8

Figure 5.10: Population density in condensate states |1 (a) and |2 (b) on a logarithmic scale, illustrating coupling linewidth (gradually approaching the power broadened limit) for long pulse-time at t = 2 .4976. Simulation in the TF limit. Ω=0.2, ∆ =40, G=-5, C = 200.

In this and the previous sections, we develop a concept for interpretingresults of dynamic coupling bandwidths, which we nd in our simulations.Note that these are not models on which our simulation code is based, sincethe GP equation takes care of all this. We are merely interpreting the results,solutions of the GP equation.

The most important contribution to coupling linewidths in our simulations

originates from the pulse-like nature of our coupling eld. It has a nite pulse-time τ pulse starting at t=0, leading to a spectral FWHM width of

Γ pulse =2π

τ pulse. (5.21)

Neglecting the insignicantly small natural linewidth, power broadening andpulse spectral width combine to a linewidth Γ which is calculated from aconvolution of the spectra of the two separate broadening mechanisms. For all




practical purposes in this work, it can be approximated by

Γ ≈Γ power + Γ pulse . (5.22)(Note that the GP equation provides an accurate description of the broadeningphenomena without approximations in any case.)

Pulse spectral width is the reason why broad coupling over the whole cloudis observed at the early times of a coupling simulation. As the coupling (pulse-)time increases, bandwidth decreases signicantly, ultimately approaching thelimit of a purely power broadened signal.

Figure 5.10 illustrates a very narrow bandwidth, coupling an arc of high

population density from the rst into the second component. Pulse time inthis simulation is approaching half a trap period (which is T trap /2= π) andthe simulation is performed in the TF regime to prevent the second statecomponent to fall down. In gure 5.11 we illustrate the temporal evolution of the measured spatial bandwidth. It is evident that nite pulse time constitutesthe main broadening mechanism on a timescale of full trap cycles.

The effect of the bandwidth due to transient effects is also dramatically il-lustrated in the case of negative detuning. In this case, ∆ eff is not resonant atany spatial point, and thus we do not expect long term coupling. Figure 5.12shows a temporal sequence of the evolution in the second (untrapped) state:Initially, after turning on the coupling cw eld, coupling occurs throughout thewhole cloud despite the negative effective detuning. With increasing interac-tion time, the bandwidth of the coupling eld decreases signicantly, leadingto a very narrow energy range of coupling, which is not resonant at any posi-tion in the cloud. The system eventually settles into a state of constant lowcoupling bandwidth where rapid, small amplitude Rabi cycling occurs betweenthe two states without signicant output coupling. The part of the conden-sate that was output coupled in the initial high bandwidth period falls downunder the inuence of gravity forming a low density crescent shaped pulse asits selfenergy causes it to expand (reminiscent of the original MIT output cou-pler [30]). Repetitive crescent shaped pulses like the above are obtained whena resonant coupling eld is operated in pulse mode [22]. Due to inter-staterepulsion with the falling outcoupled population, the parent cloud, i.e. state

|1 component, experiences a small “recoil” directed upwards. Because of this,the parent cloud undergoes small amplitude oscillation in its trap. However,




0.5 1 1.5 2 2.5

0

5

10

15

20

25

t

∆ y

Figure 5.11: Temporal evolution of the spatial FWHM coupling bandwidth∆ y of the simulation in gure 5.10 (solid line). The dashed line represents the bandwidth component caused by nite pulse time (5.21) and the dot-dashed line at ∆ y=0.063 represents the small bandwidth contribution of power broadening at a coupling strength of Ω=0.2 at a detuning of ∆ =40.

in the low coupling strength regimes of our simulation (i.e. low Rabi frequen-cies and small outcoupled densities), this effect, which is described in [22], wastoo small to measure. In regimes of stronger coupling, we did not obtain anyconclusive results.

Increasing energy selectivity of the coupling eld (i.e. bandwidth narrows),as time progresses is also evident in Fig. 5.6 on page 46, where it leads to fringesof shrinking width as population undergoes quick Rabi oscillations between the

two states, with a spatially dependent Ω( r ), Eq. (5.5), in (b) and (c). Thiseffect combined with vertical acceleration of outcoupled population also leadsto the density “pulse”-like uctuations in the output coupled stream, whichare visible in (d). These “pulses” are an effect of the transient due to turningon the coupling eld, (i.e. high coupling bandwidth, or low energy selectivity)and subside as time progresses.




y

t = 0.1885

a)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

t = 0.6597

b)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

10 −5

10 −6

10−7

10−8

x

y

t = 1.3666

c)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60x

t = 2.3091

d)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

10−5

10−6

10−7

10 −8

Figure 5.12: Output coupling with negative detuning. Pulse output is due totemporal broadening at early times. Temporal evolution sequence of probabil-ity density of component |2 in gures (a)-(d). (Parameters: Ω=0.2, ∆ =-20,C = 200, 256 by 512 grid)




5.5.3 Broadening using Noisy Signals

We have considered the effect of noisy cw coupling elds on the output couplingprocess using a simple stochastic model. The noise was modelled as small timedependent uctuations of the driving frequency, implemented as random phase jumps between temporal simulation steps.

We use a complex amplitude

z(t) = z0eiω(t )t , (5.23)

where the frequency exhibits small time dependent uctuations α(t) around

the mean ω0. ω(t) = ω0 + α(t) (5.24)

Formally, this can be represented by the Stratonovich equation [17]

dz = i ω0dt + 2γ dW (t) z (5.25)

where γ is a measure of the strength of the noise on our complex amplitudeand W (t) is the Wiener function. The solution to (5.25) is

z(t) = z0ei[ω0 t+ √2γW (t)]

. (5.26)The time dependent frequency uctuation α(t) becomes

α(t) =√2γ

tW (t) (5.27)

and (5.26) gives a frequency spectrum

S (∆) =I π

γ ∆ 2 + γ 2

, (5.28)

with a HWHM bandwidth of ∆ ω = γ .Randomness in the simulation was represented as a Wiener function W (t)

with expectation value W (t) = 0 and variance W (t)2 = t. Numerical sim-ulation of this process was implemented using the MATLAB function randn() ,which is a timer seeded Gaussian pseudo random number generator (PRNG),with variance v = σ2 = 1. By multiplying the output numbers from randn()with √∆ t, where ∆ t is the simulation timestep, we obtain a Wiener randomprocess function with variance W (t)2 = t.




y

t = 0.7697

a)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

t = 1.2881

b)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

10−4

10−5

10−6

10 −7

10 −8

x

y

t = 1.8064

c)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

x

t = 2.3405

d)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

10−4

10−5

10−6

10−7

10−8

Figure 5.13: Output coupling with a noisy cw eld. (a)-(d) gives a temporal sequence of the probability density of untrapped component |2 . (Parameters:Ω=0.2, ∆ =10, C = 200, γ =10, 256 by 512 grid)




In output coupling simulations, noise destroys the coherence of the outcou-pled beam. Noise leads to density uctuations and stochastic phase evolutionalong the output “beam” (see Fig. 5.13) and thus causes a loss of the propertiesthat are characteristic for “atom lasers” [39].

To demonstrate this in more detail, we compared the self correlation func-tions

Γ(y) = Ψ∗2(y )Ψ2(y) (5.29)

of the outcoupled state at time t = 2 .3405 of a simulation without noise, tovarious realizations of a simulation at the same noise level (e.g. one of therealizations is in Fig. 5.13d). In each case we selected a prole at x = 0.

Figure 5.14a shows the function Γ( y) for a simulation with noiseless rf coupling eld, with the xed value y = 16 .6 (vertical mark). We note that ywas chosen somewhat arbitrarily, but any y within the outcoupled populationyields similar results, a smooth almost uniform correlation along the beam.

The situation is different for a simulation with a noisy coupling eld, asshown in gure 5.14b. The magnitude of the correlation function exhibits largeuctuations which are also apparent in a density prole (e.g. see Fig. 5.13d).The loss of coherence, which is not illustrated very well by the magnitude

function, becomes evident by averaging Γ( y) over a number of simulationswith the same noise level. All realisations have different random dephasingswhich eventually average to zero when enough individual runs are included.Figure 5.14c shows an average of 25 realisations of the same simulation. Thewidth of the peak at y can be regarded as the coherence length of the “beam”as we will see shortly. Figure 5.14d shows the same data but using y = 33 .855.The correlation peak decreases as the population density decreases and widensbecause of the spread due to gravitational acceleration.

A rough estimate of the coherence lengths can be made as follows: A noiselevel of γ = 10 has a bandwidth of ∆ ω = 10, as found in equation (5.28). Thisbandwidth relates to a spectral time τ pulse = 2 π/γ = 0 .628. Now, using simpleNewtonian mechanics, we can calculate the time t taken by the outcoupledpopulation to fall from the coupling region to position y . We expect thebeam to be coherent between times t + τ pulse / 2 and t −τ pulse / 2, which is easilyconverted to a corresponding spatial distance along the beam.




0 10 20 30 40 50 600

1

2x 10

−5

0 10 20 30 40 50 600

0.5

1x 10

−5

0 10 20 30 40 50 600

0.5

1x 10

−5

0 10 20 30 40 50 600

0.5

1x 10

−5

y

c)

| Γ |

a)

b)

d)

| Γ |

| Γ |

| Γ |

Figure 5.14: Spatial coherence analysis of outcoupled “beams”; (a): No noise present. (b): Single realization of simulation at noise level γ = 10 . (c):Average over 25 different realisations of (b). (d): same as (c) but with differenty . The positions of the y values used are marked by vertical solid lines.Proles at x = 0 of simulations with Ω0 = 0 .2, ∆ = 10 , C = 200, γ = 10 in a 256 by 512 grid at t = 2 .3405.




For gure 5.14c, the calculation yields ∆ y ≈ 6.3 and for gure 5.14d weget ∆ y

≈8.4, which agrees with the graphs.

5.6 Limits of the Simulation

There are signicant difficulties in simulating output couplers compared tosimulating BEC in a trap. First, the output coupled beam, falling undergravity, can reach the critical speed where reections due to numerical aliasingoccur, as described in section 4.5. This can be resolved by increasing the gridresolution, and typically we used grids of 256 (horizontal) by 512 (vertical).

The ultimate restriction on the size of grid that can be used is set by theavailable physical memory of the machine. The grid matrix must t intomemory, since many matrix operations and Fourier transforms are performedon it in every timestep of the simulation. A typical (i.e. 256 by 512 grid)simulation running in MATLAB on a Pentium Pro 200 CPU with 128MBRAM took approximately eight hours to propagate through a full trap cycleT trap .

We illustrate in gure 5.15 how aliasing in a coarse grid leads to the reversal

of the outcoupled “beam”. The critical velocity of v=40 is reached at aroundy=35. Consequently, the reversed beam interferes with the initial beam.

The other, more difficult problem was that of output coupled material hit-ting the edges of our simulation grid and reecting back– clearly an unphysicalscenario. In an ideal case, material “falling” out of the grid should just disap-pear, without causing any reections or refractions that can interact with theinitial populations.

We have mitigated the effects of these reections by two separate strategies.The simplest was to monitor population density in the grid border regions asthe simulation progressed, stopping the simulation run when a certain limitwas reached. Typically, we stopped the simulations when the combined relativedensity of both states in a 5% rim around the grid exceeded 0.001.

The other strategy was to add a thin absorbing layer at the edge of thegrid, as we discuss in the next section.




y

t = 1.3666

a)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

t = 1.3666

b)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

10−2

10−3

10−4

10−5

10 −6

10 −7

10−8

x

y

t = 2.0735

c)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

x

t = 2.7803

d)

−20 −10 0 10 20

−20

−10

0

10

20

30

40

50

60

10−2

10 −3

10−4

10−5

10−6

10−7

10 −8

Figure 5.15: Reection due to velocity aliasing on spatial resolution limit.Figure a) shows the rst component, mostly unchanged in the coupling process.Figures b), c) and d) show the evolution of the outcoupled population in the second component, reecting at the Nyquist limit and interfering with the initial beam. Figures a) and b) show the two states at the same time t.(Parameters: Ω=0.2, ∆ =40, C = 200, 128 by 256 grid)




5.7 Absorber

Our simulation should model as closely as possible the real physical situation,where the outcoupled beam rapidly leaves the region of interaction. We haveattempted to achieve this by using a thin absorbing shell at the edge of thecomputational grid, which ideally should remove all condensate populationthat enters the shell, and cause no reections and no interactions with theincoming beam.

5.7.1 Spatially ltering the Wavefunction

The most straight forward implementation of an absorber just spatially ltersthe wavefunction in every simulation timestep, multiplying it with what es-sentially is a unity matrix that smoothly goes to zero at the borders. Thisis computationally very inexpensive and exible. However, it is not really anabsorber, but an ongoing reinitialization of the simulation. A smooth functionthat has proven to be very well suited is a sin 1/ 8 (rise) and a cos1/ 8 (falloff) ina shell at the edge of the grid. An illustration for a 128 point grid and a shellwidth of 20% of the grid can be found in gure 5.16a. For larger grid sizes,

the relative width of the shell can be reduced to 10% or even 5%, as long asabout 20 to 30 grid points are used for the edge functions.

5.7.2 Complex Potential

An alternative approach to the absorber problem is to use a complex compo-nent added to the trapping potential in the edge regions, where absorption isrequired. The same considerations as above, concerning the width of the shellin the edge regions, apply in this situation. Considering only the trap part of

the time dependent GP equation

i∂ψ∂t

= . . . V ψ . . . , (5.30)

we see that for a complex potential V = −ik, equation (5.30) becomes

∂ψ∂t

= · · ·−k ψ . . . (5.31)

which has the solutionψ(t) = ψ0e−kt . (5.32)




This describes amplitude, damping with absorption coefficient k. By addinga complex component to the potential in the grid edge regions, we design aspatially dependent absorption coefficient. To avoid reections, the complex

−10 −8 −6 −4 −2 0 2 4 6 8 10−120

−100

−80

−60

−40

−20

0

grid position

c o m p l e x p o t e n t i a l c m p t .

−10 −8 −6 −4 −2 0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

f a c t o r a)

b)

Figure 5.16: Proles of matrices used in the two different absorber approaches.

Figure a): shaping method with a sin1/ 8

rise at the left edge and a cos1/ 8

falloff at the right edge of the grid. Figure b): complex component of the trapping potential with a ( −sin1/ 8 ·exp) falloff to the right edge. Opposite edges are mirror images, and the functions in both approaches affect 20% of the grid oneach edge in this illustration.

potential component should increase as smoothly as possible from zero at theinner border of our absorbing “shell” and reach high magnitudes at the gridedge to prevent tunneling “transmissions” due to periodic boundary effects.We used an exponential function to obtain a steep increase and multiplied itwith −(1−cos)1/ 8 (and −sin1/ 8 at the opposite edge) to smooth the transitionto zero at the inner edges. (Sine and cosine periods were adjusted to obtainan increase/decrease between 0 and 1 over the appropriate intervals.) Thismethod is mathematically more elegant and justies the notion of an absorberbetter than the previous approach.

We carried out a comparison of the performances of the two different meth-ods. An isotropic two dimensional condensate wavefunction of Gaussian ra-




dial prole, i.e. C = 0, was placed in the centre of a 1282 computationalgrid (spatial extents of the grid x, y = [

−10, 10]), with an absorbing shell of

20% of the grid, and was given a centre of mass velocity by imposing a linearphase gradient on the wavefunction. The critical velocity, where grid aliasingreverses the cloud’s motion, is v = 20 at this grid size. On the velocity in-terval v = [6, 14], well under the aliasing limit, yet with reasonably fast cloudmovement, we measured the residual overall population after the cloud hascompletely passed through the absorbing region. The results are plotted ingure 5.17.

We found that the complex potential method performed better for higher

velocities while the shaping method performs well for low velocities. For lowvelocities, i.e less than v = 10, mostly reections lead to residual populations.For cloud velocities higher than v = 10, the main cause of residual populationwas tunneling “transmission” to the opposite side of the grid (due to periodicboundary effects).

Note that the performance analysis discussed above represents a very spe-cic case and that adjustments to the width of the absorbing shell, to thegridsize and to the functions used to model the absorption coefficients leave

a wide margin for customizations. In general, however, the complex potentialmethod appears to be superior to the shaping method for higher velocities,while the reverse is true for lower velocities. Since the outcoupled streams inour cw rf coupling simulations accelerated to near critical grid velocities whilefalling under gravity towards the edge of the grid, we have chosen to use thecomplex potential method in our simulations.

5.8 Validation of simulations

Since coding errors and conceptual mistakes are easily made, simulation codemust be tested to make sure that simulations are physically valid. We can testsimulation code on simple scenarios with predictable outcome. A very simpleand predictable simulation is, for example, that of an atomic cloud undergoingtrap oscillations. The cloud oscillates with the the trap frequency around thetrap equilibrium and, because of the absence of external interactions, the totalenergy and the normalisation, the particle number, are conserved [32].




6 7 8 9 10 11 12 13 1410

−8

10−7

10−6

10−5

10−4

10−3

velocity

r e s i d

u a l p

u p o l u a t i o n

Figure 5.17: Residual relative population after a moving condensate cloud has completely passed through the absorbing or spatially ltering region at the grid edge. The solid line is the result from the shaping method and the dashedline is from the complex potential method.

The diagrams in gure 5.18 illustrate the simulation of a cloud at initialposition y=0, oscillating about the equilibrium position at y=10 (vertical offsetdue to gravity). Using Ψ for Ψ( x,y,t ), the values monitored are the verticalcentre of mass position

y(t) =∞

−∞

∞

−∞

|Ψ|2ydxdy, (5.33)

the kinetic energy,

E kin (t) = −∞

−∞

∞

−∞

Ψ∗ 2Ψ dxdy, (5.34)

and the potential energy

E pot(t) =∞

−∞

∞

−∞

14

r 2|Ψ|2 dxdy. (5.35)




The internal energy, or self energy, of the cloud is calculated as

E self =∞

−∞

∞

−∞

12

C |Ψ|4 dxdy. (5.36)

Self energy does not change over time in the simulation described above, sincethe cloud retains its shape undergoing oscillations in the trap.

equilibriumtrap

a) 0

5

10

15

200 2 4 6 8 10 12 14

c e n t r e o f m a s s p o s i t i o n y

-1

0

1b)

0 2 4 6 8 10 12 14

e n e r g y

[ a r

b i t

r .

u n

i t s

]

t

Figure 5.18: (a) Single component condensate cloud in vertical oscillating around trap equilibrium at y=10. While the cloud oscillates with ωtrap =1,(b) kinetic energy E kin (solid line) and potential energy E pot (dotted line) os-cillate with ω=2. Total energy E tot (dashed line) remains constant. Self energy E self is also constant (but not plotted).

In general, conserved quantities are calculated and monitored as a rou-tine operation since these are very sensitive indicators of simulation validity.Normalisation of the wavefunction, for example, is the single most importantindicator for code errors and miscongurations. In coupled two-state 2D simu-lations, normalisation over the two present components is dened as the spatial




integral over the sum of the two states

∞

−∞

∞

−∞|Ψ1|2 + |Ψ2|2 dxdy ≡1, (5.37)

representing bosonic particle conservation over all participating components.Generally, normalisation uctuates slightly during the course of a simulationbecause of rounding errors. In valid simulations of 256 by 512 point grids,these uctuations were within the order of 10 −5 or 10−4. For lower grid reso-lutions, these uctuations were higher, starting to become intolerable for 64 2

grids. Although we used large grids for “production” simulations, quick runson 1282 grids could be used as rough indicators or for rapid probing of pa-rameter regimes. This posed another validity test: Changing the grid size,or, as another option, decreasing the time step size, should not change thefundamental physical outcome of any simulation run.

To evaluate the magnitude of numerical errors, we can stop our simulationsafter a certain time and then reverse the time steps. Running the simulationbackwards to t = 0 should nally reconstitute the initial state, since the GPequation is unitary. All deviations are then due to numerical errors. Output

coupling and TOP simulations (presented in chapter 6) reconstituted the initialstate with a difference between initial state Ψ i and nal state Ψ f

∞

−∞

∞

−∞

|Ψf |2 − |Ψi|2 dxdy (5.38)

of less than 10−4 or 10−3, depending on how far forward it was run initiallyand what grid and time step size was used. Time reversal of simulations whichincluded damping at grid edges, vastly amplied numerical errors and can thusnot be considered representative for the precision of the simulation code.

For debugging purposes and in order to illustrate specic interpretations,it was very convenient to disable spatial diffusion by omitting the ( 2Ψ)-terms in the coupled GP equations, thus effectively making the Thomas-Fermiapproximation which has been described earlier. For example, gure 5.10on page 51, which illustrates the effect of nite pulse time broadening, wasgenerated with diffusion disabled. With population densities staying in placeunder this provision, spatial effects in coupling can be investigated in greater




detail and evolution at individual spatial points can be checked against analyticsolutions.



Chapter 6

TOP Trap Simulations

One of the most commonly used magnetic traps in BEC is the Time-averagedOrbiting Potential (TOP) trap, and in particular it is the trap used in theOtago BEC experiment. This ingenious conguration is based on the simplequadrupole trap, consisting of two Helmholtz coils on the z axis, generatingmagnetic elds of opposite polarity with a magnetic eld zero in the trap centrebetween the coils. The magnetic eld increases linearly in all spatial directionsand thus atoms in low eld seeking atomic states are trapped. However, atoms

undergo spin ips into untrapped states in the zero-eld region and can thusescape from the simple quadrupole trap.

Cornell et. al. proposed to introduce a magnetic bias eld of constantmagnitude in the xy plane. This bias eld moves the magnetic eld zero awayfrom the trap centre. By rotating the bias eld, which is generated by twosets of Helmholtz coils on the x and y axis respectively, the magnetic eld zerocan be moved on a circle in the xy plane around the centre of the trap. Thisrotation can be done so quickly that the atoms in the trap centre do not havetime to re-centre at the shifted magnetic eld zero and thus never enter regionsof near-zero magnetic eld magnitudes, where trap loss can occur.

The physics and the numerical implementation of quadrupole and TOPtraps will be described in detail in the following sections. We simulate two-statecoupling in the TOP trap in two different parameter regimes. We describe andanalyse the dynamic vortex behaviour which we discovered, and we propose amodel describing the spatial coupling effects and the formation of vortices.

68



CHAPTER 6. TOP TRAP SIMULATIONS 69

6.1 The Quadrupole Trap

¡ ¡ ¡ ¡

¡ ¡ ¡ ¡

¡ ¡ ¡ ¡

¢¡ ¢¡ ¢¡ ¢¡ ¢

¢¡ ¢¡ ¢¡ ¢¡ ¢

¢¡ ¢¡ ¢¡ ¢¡ ¢ x

y

z

1

2

B

B

magnetic field zero pointCloud of trapped atoms around

Helmholtz coilsPair of anti-

Figure 6.1: Schematic illustration of a simple quadrupole trap. The Helmholtz coils positioned on the z axis carry anti-polar currents and create anti-polar

elds B1 and B2. The cloud of trapped atoms rests near the magnetic eldzero point at the origin.

A quadrupole trap consists of two Helmholtz coils centred on the z axiswith equal currents in opposite directions (see gure 6.1), creating a magneticeld zero in the centre between the coils and a linearly increasing magnetic eldamplitude in all spatial directions. For small displacements from the centreof the trap, i.e. less than the radius of the Helmholz coils and less than theseparation of the two, the magnetic eld can be described as

B = Bx x + By y + Bzz = ( Bqx)x + ( Bqy)y + (2 Bqz)z (6.1)

where Bq, the gradient of the magnetic eld in radial or ( x,y) directions, isconstant, and x, y and z are displacements from the origin. Note that theeld gradient in the axial direction is twice as large as in the radial, leadingto a stronger trap connement along the z axis (and ultimately to “pancakeshaped” condensates).




The potential energy of a magnetic dipole µ in a magnetic eld B isU =

−µ

·B . Under the assumption that the atoms’ magnetic moments al-

ways align with the magnetic eld lines and follow any small changes in B

adiabatically, this simplies to U = −µB . The trapping force is proportionalto the local eld gradient, and given by

F = − U (6.2)

Thus the magnitude of the trapping force e.g. in the z direction is

F =∂U ∂z

= µ∂B∂z

= gF M F µB∂B∂z

. (6.3)

87

Rb F =2 states, as used in the Otago experiment, have a Lande-factor of gF = 1

2 , so that they are low eld seeking for magnetic quantizations M F =[1,2],magnetically untrapped with M F =0 and high eld seeking, and thus expelledfrom the trap, for magnetic quantizations of M F =[-2,-1]. Trapping low eldseeking states works as long as the Larmor frequency

ωLarmor = gF µB

hM F B (6.4)

of the atoms is larger than the rate of change of the magnetic eld due tothe atom’s movement in the inhomogeneous magnetic trapping eld. This isrequired so that the rapid precession of the atom’s magnetic moment aroundthe magnetic eld lines allows the atom’s mean magnetic moment to followadiabatically the changes in local eld direction. At the trap centre, the mag-netic eld B and the Larmor frequency go to zero, allowing trapped atoms toundergo Majorana spin-ip transitions into untrapped states [24]. This pro-cess constitutes the trap “hole” mentioned before, the major drawback of thequadrupole trap.

6.2 Time-Orbiting Potential (TOP) MagneticTrap

The “hole” in the bottom of the quadrupole trap can be plugged by rotatingthe homogeneous bias eld of two pairs of Helmholtz coils in the x-y plane.This rotating eld,

B top = Bbias [cos(ωtop t)x + sin( ωtop t)y ] , (6.5)




constitutes the so called “TOP” eld. The bias eld moves the magnetic eldzero point away from the origin. Because of the geometry of the bias eld coilscentred on x and y axes, the resultant eld is simply the original eld displacedsideways by an amount Bbias in the xy plane. The rotation of the bias eldBbias causes the displaced quadrupole, and in particular the point of zero eld,to rotate around the origin with an angular velocity ωT OP at a radius

r 0 = x2 + y2 =Bbias

Bq. (6.6)

The B eld zero traces out a trajectory which is commonly known as “thecircle of death”. Actually, there is a small spatial region about the zero eldpoint with relatively high probability of population loss into untrapped states,and this is more accurately called the “Doughnut of Doom” [28]. Neglectinginuences of gravity, the trapped atomic cloud resides at the origin, well withinradius r 0, as we will demonstrate in the following section.

Adding the quadrupole eld and the rotating bias, we get the following timedependent magnetic eld, which we will occasionally refer to as the “rotatingTOP eld” [4]:

B = ( Bqx + Bbias cos(ωtop t)) x + ( Bqy + Bbias sin(ωtop t)) y + (2 Bqz)z (6.7)

A schematic illustration of the TOP eld is shown in gure 6.2. We showthe trajectory of the magnetic eld minimum around the origin, where thetrapped cloud rests, and an instantaneous projection of the magnetic eldmagnitude in the xy plane. The height of the cone, which increases linearly inx and y directions from the minimum point, over the xy plane characterisesthe eld magnitude. Note that this simplied gure is only illustrating theeld magnitude in the xy plane, the projection we chose for our numericalsimulations.

6.2.1 Calculating the Time Average

The reason why TOP traps are extremely useful and in widespread use in BECis that the orbiting potential time-averages to a harmonic potential, as we nowshow. Starting with equation (6.7) and integrating over one TOP rotation




¡ ¡

¡ ¡

¡ ¡

¡ ¡

¢¡ ¢¡ ¢

¢¡ ¢¡ ¢

¢¡ ¢¡ ¢

¢¡ ¢¡ ¢

x

y

z

Atom cloud in trap centre

path of magnetic field B=0 region

quadB

"circle of death" in x-y plane

Figure 6.2: Schematic illustration of the eld in the xy plane of a TOP trap.Instantaneous view of the displaced quadrupole eld. (Topographical view of magnetic eld magnitude in the xy plane in a 2D projection along the z-axis.)

θ = [0, 2π], with r = x2 + y2, we obtain

Baverage =1

2π 2π

0dθ r 2B 2

q + B 2bias + 2 BqBbias (x cos θ + y sin θ) + 4 z2B 2

q

(6.8)This can be rearranged in terms of the radius of the circle of death (c.o.d.)r 0 = Bbias /B q, and we get

Baverage =Bq

2π

r 2

0 + r 2 + 4 z2

2π

0dθ 1 +

2r 0(x cos θ + y sin θ)r 2

0 + r 2 . (6.9)

We can make a binomial expansion of Eq. (6.8), which is valid for regions wellwithin the c.o.d., which is where most atoms reside. The zeroth order yields aconstant term and rst order integrates to zero, and thus we need to includethe second order term to get a result that varies with position. We obtain

Baverage ≈ Bqr 0 1 +r 2 + 4 z2

2r 20

1 −r 2

4r 20

= Bq r 0 +r 2 + 8 z2

4r 0 −r 4 + 4 z2r 2

8r 30

+ . . . (6.10)




When terms in r −30 and smaller are neglected, we are left with a time-averaged

harmonic potential in all three dimensions, where the trap “spring” constantin z direction is eight times larger than that in the xy plane. This leads to“pancake” shaped, or elliptic conning potentials, and condensates that areattened in the z direction because of the stronger connement, and circularin their xy plane projection. The constant offset eld B qr 0 does not affect thetrapping properties (since the trapping force is proportional to the eld gradi-ent), but remedies the zero eld hole of the simple quadrupole arrangement.

The usual form taken for the time averaged potential is (from 6.10)

V = µBaverage ≈µBbias + µB 2

q

Bbias

14(r

2+ 8 z

2). (6.11)

Here, we can see that the trap’s “spring constant” is given by the ratio of ( B q)2

to Bbias . Thus, by increasing the magnetic eld gradient Bq or by decreasingthe bias eld Bbias , we can increase trap stiffness. In dimensionless units, weget

V =V

hωr ≈µBbias

hωr+

µx20

hωr

B 2q

Bbias

14

(r 2 + 8 z2). (6.12)

A very important parameter is the angular velocity ωT OP of the rotating

TOP eld. ωT OP must be chosen large enough so that the trapped atoms donot have enough time to recentre in the shifted eld and small enough so thatthe atomic magnetic momenta will remain adiabatically aligned with the elddirection.

ωLarmor ωT OP ωtrap (6.13)

6.3 Numerical Model of a TOP Trap

Our interest in this chapter is to dynamically simulate condensate evolutionin the presence of the rotating eld in a TOP trap. We explicitly take intoaccount the time evolution of the magnetic eld and determine its effect on thecondensate dynamics. However, we are limited by computational resources toconsider only a two dimensional model of the trap. We have chosen to considerthe behaviour in the z = 0 plane.

In SI units, the nonlinear GP equations for a coupled two component con-densate with trapped rst and untrapped second state in a “dynamic” TOP




trap are (symbols with bars are in SI units, and we use ωT ≡ωT OP for clarity)

ih∂ Ψ1

∂ t = −h2

2m2

Ψ1 + µBq

(x −r 0 cos(ωT t ))2

+ ( y −r 0 sin(ωT t ))2

Ψ1 (6.14)

+ ihΩ2

Ψ2 −NU 0 |Ψ1|2 + w|Ψ2|2 Ψ1

ih∂ Ψ2

∂ t= −

h2

2m2Ψ2 + i

hΩ2

Ψ1 −ih∆ Ψ2 −NU 0 |Ψ2|2 + w|Ψ1|2 Ψ2 (6.15)

In our dimensionless units, the coupled GP equations are∂ Ψ1

∂t= i 2Ψ1 −iκ (x −r 0 cos(ωT t)) 2 + ( y −r 0 sin(ωT t))2 Ψ1 (6.16)

+ iΩ2 Ψ2 + iC |Ψ1|

2

+ w|Ψ2|2

Ψ1

∂ Ψ2

∂t= i 2Ψ2 + i

Ω2

Ψ1 −i∆Ψ 2 + iC |Ψ1|2 + w|Ψ2|2 Ψ2 (6.17)

where we dene the dimensionless parameter

κ ≡µBqx0

hωr=

µBq

hωr h2mωr, (6.18)

and where ωT ≡ωT OP has been scaled with the trap frequency ωr , and ωT =ωT ωr

.

The rotating TOP eld averages to a harmonic eld (Eq. 6.12) with acoefficient of µx 2

0 B 2q

4hω r B bias, so that we need to set

µx20

hωr

B 2q

Bbias= 1 , (6.19)

in order to obtain a coefficient of 14 for the time-average of this trap. We are

motivated to obtain a time-averaged trapping potential of

U =14

r 2, (6.20)

by the fact that we would be able to re-use in our TOP simulations the initialeigenstates previously generated for our harmonic simulations in the previouschapters. We can calculate Bq from (6.19) and then insert it into κ (6.18).Using (6.6), this yields that κ needs to be equal to r , the dimensionless radiusof the “circle of death”,

κ =r 0

x0= r. (6.21)

With this denition, Eq. (6.11) describes an offset harmonic potential withthe desired coefficient 1

4 .




6.4 Spatial coupling

In a TOP trap, the dynamically evolving trapping eld leads to a behaviour inthe presence of coupling elds that is qualitatively different from the behaviourexpected in a harmonic trap.

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of the cloudresonance at the centre

Region of strongest

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Figure 6.3: Motion of the instantaneous coupling surface during the TOP rotation. (Illustration not to scale.)

For Rabi frequencies Ω larger than the frequency ωT OP , the coupling inter-action occurs on a smaller timescale than the TOP eld rotation period T T OP ,and thus the resonant instantaneous coupling surface is a surface of constantTOP eld magnitude and not a surface of constant time averaged (harmonic)

eld magnitude, as would be the case in a genuine harmonic trap. Thus, anapproximate picture for the coupling interaction is a “static” B eld, centredon the instantaneous eld zero, as illustrated in gure 6.3. For TOP rota-tion frequencies close to, or larger than the Rabi frequency, this interpretationbecomes problematic, and in fact, the TOP trap becomes more similar to asimple harmonic trap.

In all of our TOP trap simulations involving coupling elds, the centre of the condensate is tuned to resonance, i.e. the effective detuning at the cloud






6.4.2 Localized evolution of population and phase

The TF approximation is valid for times t T trap , i.e. for timescales of afew TOP eld rotations. (Where T trap is the harmonic trap oscillation period.)Within this timescale, diffusion caused by non-zero kinetic energy is negligibleand grid points can be treated as if they were independent of each other. Inother words, as long as we only look at a few TOP rotations, simulations inthe TF limit do not differ signicantly from simulations including the kineticenergy term. Under the Thomas-Fermi (TF) approximation of zero kineticenergy, the set of coupled GP equations simplies to a set of linear differentialequations, and the evolution of population and phase at any position in ourtwo-dimensional computational grid can be described independently using thesimple two-state model outlined in section 2.2. This enables a signicant sim-plication of the full equations, for the purposes of interpreting the mechanismsat work.

An important process in systems with time-dependent detunings is thatof adiabatic inversion of population, which we will outline in the followingparagraphs.

Adiabatic inversion and adiabatic following

In the geometric representation of the system state within the rotating waveapproximation, as outlined in section 2.2, the state vector s = ( u,v,w ) (Eq.2.31) precesses around the coupling “torque” vector Ω= ( −Ω0, 0, ∆) (Eq. 2.37).By using a time dependent detuning ∆( t) = −µt, it is possible to adiabaticallymove, and ultimately invert, the state vector from population component w =1 to w = −1 [37]. Starting at t0 = −∞, we reach inversion w = 0 at timet = 0. This is illustrated in gure 6.4. Ω starts out almost parallel to the 3

axis, moves through ( −Ω0, 0, 0) at t = 0, and ends up parallel to the negative3 axis, and the state vector follows it along, as it precesses around it. For anadiabatic following description to be valid, we require

∂ ∆( t)∂t

Ω, (6.23)

i.e., the precession frequency must be larger than the chirp rate of the detuningsweep. As long as this requirement is met, the inversion process is qualitativelyindependent of the strength of the coupling interaction Ω 0.




2

1

t−µ

−Ω0

Ω

s

3

Figure 6.4: Adiabatic population inversion by means of a detuning sweep from∆ = ∞ to ∆ = −∞.

While requirement (6.23) must be met for a truly adiabatic inversion pro-cess, inversions with residual population in the originating state can still be

achieved when the chirp rate is larger than the precession rate by factors of up to around 20 (as illustrated in gure 6.5b). The magnitude of residualpopulations in these cases can depend on the specic phase relation betweenpopulation and coupling interaction during the sweep, and the notion of “in-version” may not justied in all cases. If the chirp rate is even greater thanthat, the population does not pass the equilibrium point and may even partlycouple back into the originating state in the second half of the chirp. Figure6.5c illustrates a case where the sweep rate is 40 times as large as the precession

frequency, well out of the adiabaticity condition. The population equilibriumpoint between the two states is reached only after resonance at t = 0 .5, and wenote that for an oscillating detuning in a TOP trap, when sweep rates in thisorder of magnitude are reached, population may not even cross the equlibriumpoint before it “swings” back due to a reversed detuning sweep.

When we apply this model to the situation in the TOP trap, we also need tounderstand that in most positions close to the cloud centre we are not lookingat a complete inversion process since the detuning only oscillates around values




0 1 2 3 4 5 6 7 8 9 100

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

t

a)

b)

c)

Figure 6.5: Adiabatic population inversion for Ω0 = 40 and sweep rates of (a)µ =

−20, (b) µ =

−800 and (c) µ =

−1600. Resonance ( ∆ = 0 ) is reached at

t = 5 in (a) and at t = 0 .5 in (b) and (c).

close to zero. Thus, the behaviour in this regime is dominantly the usualRabi oscillations with a small modulation of the peaks due to the oscillatingdetuning.

For points off the centre of the condensate at r = ( x, y), the detuning os-cillates with the TOP eld rotation and the time-dependent effective detuning

is

∆ eff (t) = Bq r 0 − (x −r 0 cos(ωT OP t))2 + ( y −r 0 sin(ωT OP t))2 , (6.24)

where r 0 is the radius of the c.o.d. For displacements r from the condensatecentre, small compared to r 0, i.e. for all positions within the condensate, theeffective detuning takes the simple form

∆ eff (t) = −r 0r cos(ωT OP t) (6.25)




where Bq = r 0 is the magnetic eld gradient (Eq. 6.21). This yields anapproximate maximum rate of change of the detuning of

∂ ∂t

∆ eff (t) = r 0rωT OP . (6.26)

For ωT OP = 60 and r 0 = 20, we get a maximum rate of change in thedetuning at the Thomas-Fermi “cloud border” (i.e. at r ≈7) of ∂ t ∆ eff,max =8400. Compared with the coupling frequency we used in the strong couplingcase (where Ω0 ωT OP ), Ω0 = 350, there is a factor of 24 difference, placingthe strong coupling regime into the region where “pseudo-adiabatic” inversion,or pseudo-adiabatic following, can be expected (gure 6.5b). In the weakcoupling regime, i.e. where Ω ≈ ωT OP , this ratio is an order of magnitudelarger, i.e. 140, so that only regions close to the condensate centre exhibit abehaviour, qualitatively comparable to the strong coupling regime with pseudo-adiabatic following. At radii of larger than r ≈3, the coupling process cannotbe as easily interpreted as in the strong coupling case, where pseudo-adiabaticfollowing and Rabi cycling are dominant. We will return to the case, wherepseudo-adiabatic following fails, in section 6.6.3, and we dene the pseudo-adiabatic following regime to be where

1100

∂ ∂t

∆ eff (t) Ω0. (6.27)

Specically for positions r in a TOP trap, with Eq. (6.26) this becomes

1100

rr 0ωT OP Ω0. (6.28)

Note that in the full GP equation treatment, the changing effective detun-ing is fully accounted for in form of a complete dynamic description of the

TOP potential. The above discussions only concerns the interpretation of ob-servations in our simulations, and the numerical solutions of the GP equationsare accurate in all cases.

6.4.3 Force on a condensate in a TOP trap

To understand the coupling effects in a TOP trap, it is important to look at theevolution of phase and population on a small time scale, i.e. on the time scalecharacteristic of the coupling, which is less than one TOP eld rotation. When




integrated over one TOP period T T OP , the rotating linear eld averages to astationary harmonic eld. The situation is more complicated on shorter timescales. For an eigenstate in a harmonic trap, phase remains at and uniform,evolving slowly depending on the eigenvalue of the condensate. In a TOP trap,however, the cloud keeps developing a phase gradient towards the eld zeroand develops a velocity in response to the force, and a phase gradient. Sincethe eld zero moves, this gradient also rotates. As we will show, it rotateswith half of the TOP eld rotation frequency and reaches its maximum afterhalf a TOP eld rotation.

Since condensate velocity is proportional to the phase gradient (equation

4.17), the trap eld rotation causes the condensate, which is starting out froma harmonic eigenstate at the trap centre, to gradually start “sloshing around”in our simulated trap as its centre of mass moves in the direction y of the timeaverage of the phase gradient.

In order to understand this behaviour, we consider the following simpletreatment in terms of classical mechanics. The force F towards the magneticeld minimum of the rotating TOP eld in the condensate in the trapped statecan be expressed as

F∝cos(ωT OP t) x + sin ( ωT OP t) y . (6.29)

The condensate velocity due to this force is proportional (and in dimensionlessunits equal to)

v∝

t

0

F dt =1

ωT OP sin(ωT OP t) x +

2ωT OP

sin2 ωT OP

2t y . (6.30)

This describes a simple oscillation along the x axis, and an oscillation with

half the TOP frequency in the positive y direction. From (6.30), the evolutionof the condensate displacement ∆ r (t) can be calculated. We get

∆ r =t

0

v dt =1

ω2T OP

(1 −cos(ωT OP t)) x +1

ω2T OP

(ωT OP t −sin(ωT OP t)) y

(6.31)This calculation shows that the condensate centre of mass oscillates with theTOP frequency around a point slightly offset in positive x . The magnitude and




offset of this oscillation decreases with the inverse square of the TOP frequency.A small drift occurs in positive y direction. The y offset increases with t/ω T OP

due to this drift. This becomes more evident for longer simulation times, butit is negligible for simulations of only a few (i.e. 10) TOP eld rotations.

6.5 Vortices

In order to understand the nature of superuids like BEC, and the occurrenceof vortices, we consider the change of population density in a volume V andderive the probability current density vector

∂ ∂t V |Ψ|2dr = V

Ψ∗∂ Ψ∂t

+∂ Ψ∗∂t

Ψ dr

=ih2m V ·[Ψ∗( Ψ)−( Ψ∗)Ψ]dr = − V · jdr , (6.32)

where the second line is obtained by using the time-dependent GP equation

ih∂ ∂t

Ψ = −h2

2m2 + V + U 0|Ψ|2 Ψ (6.33)

and its complex conjugate. For notational convenience we use Ψ for Ψ( r , t ).Equation (6.32) allows us to interpret j as

j(r , t ) =h

2miΨ∗( Ψ)−( Ψ∗)Ψ . (6.34)

It is possible to write a condensate wavefunction Ψ = |Ψ0|eiS where S is a realquantity, and can be interpreted as the phase, which leads to the followingexpression for the velocity [13]:

v (r , t ) = |Ψ0|−2 j(r , t ) =hm S (r , t ) (6.35)

From this result follows

×v = 0 . (6.36)

This means that BECs are irrotational superuids, since the curl of the con-densate velocity vanishes everywhere where v is dened. The only way thesuperuid can exhibit rotational ow is by having points where phase, andthus velocity, are undened, i.e. a zero of the wavefunction. A structure in




which the phase undergoes a positive or negative 2 π winding is called a vortex(and in three spatial dimensions it becomes a vortex line).

In stirred homogeneous superuids like liquid helium, the ow is quantized,and one vortex forms for every unit of angular momentum [36]. Clouds of magnetically trapped alkali BECs are not of homogeneous density and thusthere is no strict quantization of the superuid ow. While a single centrevortex represents a unity ow, fractional Lz values occur while a single vortextravels from the low population density cloud border to the high density centre[10].

6.6 Vortex dynamics in a TOP trap

The external coupling eld connects trapped condensate state |1 and un-trapped condensate state |2 . This creates a situation where population of state |1 with a specic phase gradient, which has developed due to the trap-ping potential, is temporarily transferred into and “stored” in state |2 . In thisstate, in the absence of a trapping eld, which causes a radial phase gradientto develop on the time scale of T trap in the trapped state, the phase prole

remains largely unchanged, evolving only slowly due to the slight energy shiftprovided by the coupling interaction as outlined in chapter 3. Half a Rabi cy-cle later, this “stored” population is transferred back from |2 and superposedonto the residual population in initial state |1 , which, being “exposed” tothe rotating magnetic TOP eld, has in the meantime evolved into a slightlydifferent, rotated, phase prole. This is a situation in which phase singulari-ties, “vortices”, can occur, as we will describe in detail in our model of vortexformation in section 6.6.2.

In our simulations we can destinguish between two regimes with differentvortex dynamics

• Strong coupling, i.e. Ω ωT OP and fully in the pseudo-adiabatic fol-lowing regime.

• Weak coupling, i.e. Ω ≈ωT OP and only partially (condensate centre) inthe pseudo-adiabatic following regime.




Note that these are the same two regimes that we distinguished before, insection 6.4.2.

Since the medium of choice for the presentation of our simulations is acomputer generated and played movie, it is difficult to illustrate some of thecomplex behaviour, which we see in many simulations, by means of limitedsequences of static images. With this in mind, we hope the reader can excuseour somewhat excessive use of plots in the following sections.

6.6.1 Vortices in the strong coupling regime

In the strong coupling case, large Rabi frequencies and pseudo-adiabatic follow-ing within the entire condensate cloud lead to signicant population transferbetween the two states at all spatial positions on time scales much smaller thanthe TOP eld rotation frequency. In this case, the picture of spatially depen-dent coupling, which we developed in section 6.4, is applicable and the TOPeld rotates through only a small angle during one complete Rabi cycle. Also,the coupling bandwidth is large due to power broadening, and coupling occurson a wide band across the whole condensate cloud. This band, however, has aLorentzian prole across the cloud, so that the spatial coupling prole changesas the TOP eld rotates around. This also leads to a slightly non-uniformdensity distribution of |1 compared with |2 , because of a changed couplingprole between two maxima of the Rabi cycle. We keep this in mind, becauseit is important for vortex formation by two-state coupling, as we will elaboratelater. In this regime, strong coupling leads to effects that occur periodicallywith the Rabi frequency, making it the simplest to describe and interpret.

In gure 6.6, we present a temporal sequence of plots of density and phaseevolution of coupled condensates showing vortex formation. The left hand

column shows the population density on a logarithmic scale, and the right handcolumn shows the condensate phase. The number N on the images indicatesthe number of TOP eld rotations that have occured since the beginning of the simulation. Fig 6.7 and on page 89 and Fig. 6.9 show the evolution of total population in each state during this process. Two solid vertical lines in6.7 mark the position of the beginning and the end of the sequence shownin 6.6, i.e. the beginning and end of the period between vortex creation andannihilation.




The sequence in Fig. 6.6 displays a short fraction of a Rabi cycle, severalRabi cycles into the simulation. We see from Fig. 6.7, that at the rst imageof 6.6, the total population density in state |1 is decreasing towards zero, dueto coupling into component |2 . At this time, coupling starts to transfer popu-lation back from |2 into state |1 , starting at the low density edge region of thecondensate on the axis, on which we will see vortices travelling in a short timelater. This coupling- in appears to start at the condensate cloud edge beforethe total population density minimum in the concerning state is reached. Anexplanation for the observed “inward sweep” of population minima and on-setting in-coupling is the delay between the start of in-coupling and the time

when the in-coupling wavefunction actually starts to increase the populationdensity. Initially, the in-coupling wavefunction may decrease the total ampli-tude at certain locations (like the diameter along which the vortices travel)in a complex addition to the residual wavefunction. This causes a transientdecrease in local population density, which depends on the relative phases, andalso the wavefunction amplitude proles, of the two coupled components.

Population from untrapped state |2 is now (in Fig.6.6b, c) coupled back,in the presence of a magnetic trapping prole that differs slightly from the one

it was coupled out by, because the coupling region has rotated by a small angledue to the TOP eld rotation. Thus, the phase prole of state |2 “lags” (hasa gradient of a slightly different direction and magnitude) behind the one of the residual population in |1 , onto which it is now being added in the shapeof the new (slightly rotated) spatial coupling region.

Two vortices, minima in the density prole, appear at the positions alonga cloud diameter, where the addition of complex wavefunctions yields a zeroresult. The two vortices are of opposite polarity and they travel inwards duringonly small fractions of a TOP cycle. In the condensate centre, they meet andannihilate shortly after the total population density goes through its minimum(in Fig. 6.7) at approximately N = 0 .77.

Summarizing, we can say that what we are observing is a radial inwardsweep of superimposing one complex wavefunction prole onto another with adifferent phase and amplitude prole. This will be explained in more detail inour simplied model of vortex formation in section 6.6.2.

In gure 6.7, the rst vertical mark also represents the minimum in local




population density at the cloud border, on the track, which the vortex travelsalong, and the minimum of the curve of local Rabi cycling Ω eff . It is importantto note, that the vortices appear (with a short delay depending on the residualwavefunction amplitude) when peripheral coupling starts to couple populationinto the concerned condensate component. This happens alternatingly in bothstates |1 and |2 , with periodicity of the Rabi frequency Ω 0 at the condensatecentre, with half a Rabi cycle difference between the two states. At a Rabifrequency of Ω = 350 and a TOP rotation frequency of ωT OP = 60, thisvortex scenario occurs almost 6 times in both states during one TOP rotationcycle. Vortices occur for the rst time during the rst minimum in state |2(initially, at N = 0, the full population is in |1 ). Surprisingly, no vorticesappear during the rst minimum of state |1 . This may be due to high initialbandwidth with very homogeneous coupling preventing a spatially differentdensity prole to develop between the two coupled states immediately afterturning on the coupling interaction at N = 0.

The diameter, along which the vortices travel inwards at certain fractionsof the Rabi cycle, rotates with the direction of the phase gradient of state |1 ,i.e. with ωT OP / 2, and there appears to be a 90 angle difference between the

orientation of these diameters in states |1 and |2 .For simulations with nite C values (shown graphs represent simulationswith C = 0 for reasons of simplicity), the same effects can be observed. How-ever, the condensate clouds are slightly bigger. Also, the presence or absenceof the kinetic energy term in the GP equation does not qualitatively changethe simulations. This can be understood easily in terms of separated timescales. Diffusion occurs only on a much larger time scale than the coupling inthe strong coupling regime.

A situation which appears qualitatively different to the one described aboveoccurs when the phase proles of |1 and |2 , which are added in the couplingprocess, are nearly “at”, i.e. do not have a gradient. A situation like thisoccurs during completion of (or in the early parts of) a full TOP rotation,i.e. around N = 1. This case is illustrated for state |2 in gures 6.8, us-ing the same representation as previously in gures 6.6. Evolution of totalpopulation density is plotted in gure 6.9. The sequence shows that for tinyphase gradients across the condensate at the time of a specic Rabi cycle, the




two vortices of opposite polarity “spread out” into oval or stretched densityminima, connecting to form a complete “vortex-ring” within the condensatecloud.

This “ring” contracts as the vortex pair travels inwards. The vortex pairannihilates in the same way as described before for the case of Fig. 6.6 andthe total population density in Fig. 6.9 evolves as expected and similar to Fig.6.7.

The vortex ring travels into the condensate centre at the same velocity asany other typical vortex pair in the same coupling scenario. We conclude thatthe velocity, which the vortices travel at, depends on the Rabi frequency and

not on the phase gradient across the condensate cloud, because the occuranceof vortices is a coupling phenomenon as we will show in detail in the nextsection.




10−2

10 −3

10−4

10 −5

10 −6

10−7

10 −8

t = 0.077493 N = 0.74a1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.077493 N = 0.74a2)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

π

π /2

0

− π /2

− π

10−2

10 −3

10−4

10 −5

10 −6

10−7

10 −8

t = 0.079063 N = 0.755b1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.079063 N = 0.755b2)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

π

π /2

0

− π /2

− π

10−2

10 −3

10 −4

10 −5

10 −6

10−7

10 −8

t = 0.080634 N = 0.77c1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.080634 N = 0.77c2)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

π

π /2

0

− π /2

− π

10−2

10 −3

10−4

10 −5

10 −6

10−7

10 −8

t = 0.081158 N = 0.775d1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.081158 N = 0.775d2)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

π

π /2

0

− π /2

− π

Figure 6.6: Temporal evolution of condensate state |1 in the large coupling regime, during a small fraction of a Rabi cycle, after N = 0 .74 TOP eldrotations. Left hand column, population probability density. Right hand col-umn, phase. The bar on the left gives the density scale and that on the rightgives the phase scale. (Parameters: 2562 grid, Ω0 = 350, ωT OP = 60 , r 0 = 20 ,C = 0 , coupling on resonance at cloud centre.)




0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.860

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

N=fTOP

t

| Ψ | 2

Figure 6.7: Temporal evolution of the total population densities for gures 6.6.Solid lines represent state |1 and dashed lines represent state |2 . Vortices appear before the minimum in total density is reached (rst solid vertical mark), and annihilate shortly after the minimum (second solid vertical mark).




10−2

10 −3

10−4

10 −5

10 −6

10−7

10 −8

t = 0.10472 N = 1a1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.10472 N = 1a2)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

π

π /2

0

− π /2

− π

10−2

10 −3

10−4

10 −5

10 −6

10−7

10 −8

t = 0.10577 N = 1.01b1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.10577 N = 1.01b2)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

π

π /2

0

− π /2

− π

10−2

10 −3

10 −4

10 −5

10 −6

10−7

10 −8

t = 0.10734 N = 1.025c1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.10734 N = 1.025c2)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

π

π /2

0

− π /2

− π

10−2

10 −3

10−4

10 −5

10 −6

10−7

10 −8

t = 0.10838 N = 1.035d1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.10838 N = 1.035d2)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

π

π /2

0

− π /2

− π

Figure 6.8: Temporal sequence of condensate state |2 in the large coupling regime, during a small fraction of a Rabi cycle after N = 1 TOP rotation. Lefthand column, population probability density. Right hand column, phase. The bar on the left gives the density scale and that on the right gives the phase scale. (Parameters: 2562 grid, Ω0 = 350, ωT OP = 60 , r 0 = 20 , C = 0 , coupling on resonance at cloud centre.)




0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 1.120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

N=fTOP

t

Figure 6.9: Temporal evolution of the total population densities for gures 6.8.Solid lines represent state |1 and dashed lines represent state |2 . Vortices appear before the minimum in total density is reached (rst solid vertical mark), and annihilate shortly after the minimum (second solid vertical mark).

6.6.2 Model of vortex formation

In this section we explain the mechanism of vortex formation that was observedin our TOP simulations of the previous section. From the previous section wesee that the basic condition for vortex formation is the following: We havean existing population in, for example, state |1 , with a given (near uniform)non-zero phase gradient. A second component in state |2 , with a different

spatial density prole and a (near uniform) phase gradient, which points in adifferent direction than that of the existing population in |1 , is added to theexisting population in state |1 . We can understand how this leads to vorticesby means of the following simple model.

Let us consider adding a wavefunction Ψ 1 with a simple density hump (aninverted parabola) to a homogeneous (i.e. constant) wavefunction Ψ 2, wherethe two wavefunctions have some relative velocity. Without loss of generalitywe can assume that Ψ 1 has a constant phase gradient along the x axis (g




−5

0

5

−5

0

50

5

10

15

20

xy

| Ψ |

a) |Ψ1 |

−5

0

5

−5

0

50

5

10

15

20

xy

| Ψ |

b) |Ψ2 |

Figure 6.10: Simplied model of vortex formation. Plotted are probability amplitudes |Ψ| of (a) residual population with linear phase gradient along xaxis, and (b) homogeneous probability amplitude of “in-coupled” populationwith “at” phase.

6.10a) and the homogeneous wavefunction Ψ 2 (g 6.10b) has a constant phase(i.e. a zero gradient).

The superposition of Ψ s = Ψ 1 + Ψ 2 at any spatial point is obtained byadding the complex values of the elds at that point. In order to obtain a zeroresult, it is necessary that the magnitudes of the two added wavefunctions areequal, i.e. |Ψ1| = |Ψ2|, and in our model this occurs along a circle centred inthe origin. For a zero result, furthermore it is also necessary that the phase of Ψ1 relative to Ψ 2 be π, and we can easily see that due to its phase gradient, Ψ 1

has the appropriate π phase difference along lines of constant x, of which theremay be zero, one, two, three etc., intersecting the ring which is dening equaldensities. The simplest case, which is adequate for our discussion, is whenthere is only one line of this appropriate phase difference intersecting the ring,

and we see immediately that there will be two zeros of net amplitude, one ateach end of this line.

Furthermore, the superposition of Ψ 1 and Ψ2 in the region about each zeroleads to a 2π phase circulation as we now demonstrate. (Example with threevortex pairs shown in g. 6.11.) We can visualize the situation with the helpof Fig. 6.12, where we represent the superposition of the complex elds at anumber of points on a small circle about the zero net amplitude of Ψ 1 andΨ2. In each of Fig. 6.12 a, b, c, the solid arc represents (part of) the circle




in the xy plane where |Ψ1| = |Ψ2|. The cross ‘x’ marks the point on the arcwhere the phases differ by π, and hence result in zero net eld Ψ s . On thedashed circle, we have shown at a number of points the complex value of of the eld, by means of a vector with length representing amplitude and anglerepresenting phase.

Fig. 6.12a gives the eld Ψ1. We see that the length of the vectors increasesas we move vertically up the page, i.e. as we move closer to the centre of theinverted parabola (at the origin of the xy coordinate system). The phase isconstant along lines of constant x.

In Fig. 6.12b, we represent the amplitude of the eld Ψ 2. This is homo-

geneous, and all the vectors are of the same length and point in the samedirection.

Finally, in Fig. 6.12c, we represent the sum Ψ s of Ψ1 and Ψ2. It is clearthat as we move around the small circle, the phase of Ψ s goes through 2π, i.e.we have a 2π phase circulation and thus the structure is a vortex.

For a case where wavefunction Ψ 2 increases magnitude over time, i.e. dueto population transfer by coupling, the circle |Ψ1| = |Ψ2| shrinks, i.e. movesinwards, as “higher” regions of the inverted parabola Ψ 1 intersect with rising

Ψ2. This causes the vortices in Ψ s to move inwards and ultimately to disappear(at the latest, in case the π phase difference occurs along x = 0) when Ψ 1

reaches the level of Ψ2 at the top of the inverted parabola.




10 −2

10 −3

10 −4

10 −5

10 −6

10−7

10 −8

x

y

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

a)

x

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

b)π

π /2

0

− π /2

− π

−50

5

−5

0

51

2

3

4

5

6

7

xy

l o g

( | Ψ | 2 )

c)

Figure 6.11: Superposition of the two different wavefunctions of gure 6.10.

Three pairs of vortices of opposite signs appear. (a) shows the populationdensity on a logarithmic scale, (b) shows the phase. The solid arrow points tothe vortex studied closer in gure 6.12. In (c) a meshplot of the populationdensity on a logarithmic scale is shown. The six vortices are the singularity points ( → −∞) on the logarithmic density scale.




a)

To populationmaximum atparabolacentre

−β β

α

−α α

−α

x

y|Ψ |=|Ψ |1 2

Ψ1

b)

Ψ2

c)

Ψs

Figure 6.12: Vortex formation in Ψs (c) by superposition of complex elds (a) Ψ1, an inverted parabolic amplitude with constant phase gradient along x,and (b) Ψ2, homogeneous population with zero phase, in a xy plane projection.Length of solid arrows symbolizes complex wavefunction amplitude, directionsymbolizes relative phase angle. The vortex point is located in the centre of the dashed circle. (c) Ψs = Ψ 1 + Ψ 2 exhibits a 2π phase rotation around the vortex point.




6.6.3 Vortices in the weak coupling regime

In the case of weak coupling interaction between the two states, i.e. withRabi frequencies in the order of Ω ≈ωT OP , we are in a regime, where pseudo-adiabatic following (p.a.f.) does not occur through the entire condensate cloud.In this case, where p.a.f. is restricted to centre regions of the condensate, wehave observed a more complex dynamical behaviour. In this regime, powerbroadening is not the main broadening mechanism of interest. Finite pulse timeleads to signicant initial broadening, and the width of the interacting regionexhibits a signicant and ongoing temporal decrease during a few TOP eldrotations. This has previously been illustrated in gure 5.11. Furthermore,within time scales of full TOP rotations, the population in state |2 remainsuntrapped for longer than in the previous case, and thus the simple spatialcoupling model of gure 6.3 can not be applied, since the time scales are noteasily separated. For the sake of clarity, we have stopped our simulations aftertwo TOP eld rotations. All important effects can be illustrated within thisperiod.

In contrast to the strong coupling regime (where p.a.f. does occur throughthe entire condensate), simulations in the weak coupling regime show vortices

and features of spatial coupling effects in both condensate components at thesame time, in different spatial regions. We observe the creation of one vortex-pair in each state per Rabi cycle. The vortex pair does not travel to thecloud centre, however, but rotates around the centre. One pair of vortices of opposite polarity orbits the cloud centre in each condensate state, positionedon rotating cloud diameters with a π/ 2 angle difference. Figures 6.13 (state

|1 ) and 6.14 (state |2 ) show the dynamics of each of the states during therst two TOP eld rotations at rotation numbers N = (a) 0 .325, (b) 0.65, (c)

1, and (d) 2. To illustrate that vortices occur in both states at the same timein different spatial positions, we repeat the density proles of gures 6.13 and6.14 side by side in Fig. 6.15.

To rule out effects of expansion in the untrapped state, we simulated aC = 0 state in the TF limit, i.e. simulation without the diffusion term in theGP equation. While the situation for states with nite C values is qualitativelythe same as in the gures shown, the diffusion term adds a signicant expansion(and thus a radial phase gradient) to state |2 . This gradient couples over into




state |1 , so that eventually both states expand.In frames (a) of gures 6.13 and 6.14, we can see the rst vortex pair (of

opposite polarity) forming in |2 as population with a quickly increasing androtating phase gradient is transferred into this state. This is essentially thesame mechanism of vortex formation, by superposition of two different densityproles of different spatial phase, that leads to vortices in the large couplingregime. At N = 0 .5, a situation between frames (a) and (b) which is notshown, the vortex pair in |2 lies on a vertical condensate diameter parallelto x = 0 and coupling has carved out a horizontal line of density minimumin state |1 while the TOP eld has travelled half way around. This density

minimum in |1 evolves into a vortex pair of opposite polarity, due to the samemechanism as before as population is transferred back into this state after half a TOP eld rotation. After a full TOP eld rotation and one complete Rabicycle (at resonance in the cloud centre), instead of a full reversion resultingin complete population transfer back into state |1 , as one would expect in asimple harmonic trap, state |1 is left with a vortex pair on a vertical axis whilea ring and a horizontal line of density minimum at the end of the completedRabi cycle appear in state |2 ; shown in frames (c).

A second full TOP eld rotation superposes a repetition of the above de-scription with a simple evolution of the state as found at N = 1 (after one com-plete TOP rotation). New vortex pairs, created in the second TOP rotationperiod, appear closer to the cloud centre. In further TOP eld rotations, thespatial separation of the vortex pairs, and thus the radius of vortex “rings”, de-creases. The simulation state after two complete TOP eld rotations is shownin frames (d).

The dynamics in the weak coupling regime can be interpreted using thesame approach as to the strong coupling regime, using the model of vortexformation for the strong coupling regime as presented in the previous section,and keeping in mind two key differences.

• Coupling width decreases signicantly during a few TOP eld rotations,and the power broadened bandwidth is small compared with the strongcoupling case, so that coupling the spatial coupling width with signicantpopulation transfer rapidly decreases to a slim rotating band across thecondensate centre.




• The simple spatial model in gure 6.3 and the pseudo-adiabatic followingmodel (in section 6.4.2) fail for all but the immediate condensate centreregions.

Hence, we only get complete population transfer by Rabi cycling in a smallcentre region of the condensate, the radius of which decreases with increasinginteraction time. This explains the decreasing radius of the vortex occurancesand ring features, which enclose the region where the simple models are stillvalid. The validity is limited to the centre regions of the condensate, wherethe temporal oscillation of the detuning is small enough (Eq. 6.28) comparedwith the Rabi frequency.

In respect to the strong coupling regime, this TOP trap scenario can beconsidered a transition to a regime where ωT OP Ω. For increasingly largeTOP rotation frequencies ωT OP , the behaviour of the TOP trap becomes moreand more similar to that of a simple harmonic trap.




10−2

10 −3

10−4

10 −5

10 −6

10−7

10 −8

t = 0.051051 N = 0.325a1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.051051 N = 0.325a2)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

π

π /2

0

− π /2

− π

10−2

10 −3

10−4

10 −5

10 −6

10−7

10 −8

t = 0 .1 02 1 N = 0 .6 5b1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.1021 N = 0.65b2)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

π

π /2

0

− π /2

− π

10−2

10 −3

10 −4

10 −5

10 −6

10−7

10 −8

t = 0 .15708 N = 1c1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.15708 N = 1c2)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

π

π /2

0

− π /2

− π

10−2

10 −3

10−4

10 −5

10 −6

10−7

10 −8

t = 0 .3 14 16 N = 2d1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.31416 N = 2d2)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

π

π /2

0

− π /2

− π

Figure 6.13: Temporal sequence of dynamics in condensate state |1 in the weak coupling regime during two complete TOP eld rotations. Left handcolumn, population probability density. Right hand column, phase. The bar on the left gives the density scale and that on the right gives the phase scale.(Parameters: Ω0 = 40 , ωT OP = 40 , r 0 = 20 , C = 0 , 2562 grid.)




10−2

10 −3

10−4

10 −5

10 −6

10−7

10 −8

t = 0.051051 N = 0.325a1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.051051 N = 0.325a2)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

π

π /2

0

− π /2

− π

10−2

10 −3

10−4

10 −5

10 −6

10−7

10 −8

t = 0.1021 N = 0.65b1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.1021 N = 0.65b2)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

π

π /2

0

− π /2

− π

10−2

10 −3

10 −4

10 −5

10 −6

10−7

10 −8

t = 0 .1 57 08 N = 1c1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.15708 N = 1c2)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

π

π /2

0

− π /2

− π

10−2

10 −3

10−4

10 −5

10 −6

10−7

10 −8

t = 0.31416 N = 2d1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.31416 N = 2d2)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

π

π /2

0

− π /2

− π

Figure 6.14: Temporal sequence of dynamics in condensate state |2 in the weak coupling regime during two complete TOP eld rotations. Left handcolumn, population probability density. Right hand column, phase. The bar on the left gives the density scale and that on the right gives the phase scale.(Parameters: Ω0 = 40 , ωT OP = 40 , r 0 = 20 , C = 0 , 2562 grid.)




t = 0.051051 N = 0.325a1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.051051 N = 0.325a1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

10−2

10 −3

10−4

10 −5

10 −6

10−7

10 −8

t = 0 .1 02 1 N = 0. 65b1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.1021 N = 0.65b1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

10−2

10 −3

10−4

10 −5

10 −6

10−7

10 −8

t = 0 .15708 N = 1c1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0 .1 57 08 N = 1c1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

10−2

10 −3

10 −4

10 −5

10 −6

10−7

10 −8

t = 0 .3 14 16 N = 2d1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

t = 0.31416 N = 2d1)

−10 −5 0 5 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

10−2

10 −3

10−4

10 −5

10 −6

10−7

10 −8

Figure 6.15: Rearrangement of gures 6.13 and 6.14 for a more convenientcomparison. A temporal sequence of dynamics in states |1 (left column)and

|2 (right column) in the weak coupling regime. Plotted are population

densities on a logarithmic scale.




6.6.4 Angular momentum

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.06

−0.04

−0.02

0

0.02

0.04

0.06

L z

a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

b)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x 10−4

N

L z

c)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2

−1

0

1

2

3

4

x 10−4

N

d)

Figure 6.16: Temporal evolution of angular momenta in simulation of gures 6.13 and 6.14. For gure 6.13: (a) state |1 , (b) state |2 . For gure 6.6: (c)state |1 and (d) state |2 . The solid lines represent total angular momentumLz , dashed lines represent angular momentum of the cloud centre of mass

around the coordinate origin Lz,origin , and dot-dashed lines represent the difference, “residual” angular momentum about cloud centre of mass Lz,com .

A mirror-symmetry, in all of the TOP trap simulations shown, about an axisthrough the condensate components can be found in all images of the temporalsequences. We have symmetry along the x axis at integer number of TOPeld rotations, and vortices only appear in form of pairs of opposite polarity.Looking at the development of angular momentum of the two componentsundergoing coupling, we nd, however, that this symmetry is not perfect. Wecalculated global angular momentum as

Lz = x ˆ py −yˆ px (6.37)




and the angular momentum arising from centre of mass condensate motionaround the trap origin as

Lz,origin = x ˆ py −y ˆ px . (6.38)

While total angular momentum and angular momentum around the coordinateorigin were expected to oscillate, because of the coupling process and the slight“sloshing” of the condensate in the TOP trap as described in section 6.4.3, thecompensated centre of mass angular momentum of the cloud,

Lz,com = Lz −Lz,origin , (6.39)

was expected to vanish because of said symmetry. This was not the case,however, and a typical result for the temporal evolution of the angular mo-menta is shown in Fig. 6.16. These results can be explained by the asymme-try of the coupling region, illustrated in Fig. 6.3. This asymmetry, and thetime-dependent width of the coupling region, account for the residual angularmomenta about the cloud centre of mass.

Figures 6.16 illustrate the differences between the two coupling regimes,which we have discussed. In the strong coupling case, i.e. in frames (c) and(d), the rapid periodicity with Ω 0 almost entirely reverses population transfersin every Rabi cycle, while the spatial coupling width decreases on time scales of ωT OP , due to increasing interaction time. This explains the fact that all angularmomentum components vanish after complete TOP rotations at N = 1 andN = 2.

In the weak coupling regime, i.e. Fig. 6.16, frames (a) and (b), popu-lation transfers occur on time scales of the TOP eld rotation, so that thespatial bandwidth and the coupling region change signicantly before transfers

can be reversed by Rabi cycling. This leaves behind asymmetric populationdensity proles, which account for the angular momenta. Note that the angu-lar momenta in the strong and weak coupling regimes differ by two orders of magnitude.




6.7 Discussion

Our models for vortex formation in a spatially dependent coupling has enabledus to explain the formation of vortices, which we have seen in our TOP trapsimulations in the strong coupling regime. This regime is also used by theexperimental BEC group, as discussed in Appendix B.

In the weak coupling regime, we have also seen vortex formation, but thisis more difficult to interpret and it remains a problem of interest. However,we understand the ways in which our models fail, and we can interpret theweak coupling regime as a transition from the strong coupling regime to abehaviour more similar to that of a static and purely harmonic trap, as weapproach Ω0 ωT OP .

Investigation of TOP trap dynamics showed that the TOP trap is far morecomplex than expected and not very well understood, although it is widely usedby experimental groups. Problems with TOP traps, like unexpected heatingas described in [27], may be attributable to false assumptions of similarity withharmonic traps. Within the TOP trap condition (6.13)

ωLarmor ωT OP ωtrap ,

these problems can possibly be avoided or mitigated by moving the experi-mental setup to smaller coupling interactions and larger TOP frequencies.



Chapter 7

Conclusion

The Gross-Pitaevskii (GP) equation provides a relatively simple but surpris-ingly accurate description of dilute Bose-Einstein condensates in the ultra-lowtemperature regime. In this thesis we have applied the GP equation in nu-merical simulations in two spatial dimensions to two qualitatively differentdynamic scenarios. In both situations we simulate a two-component conden-sate system, corresponding to two different hyperne states for the condensateatoms, which are coupled by a radio frequency eld.

In the rst scenario, we have simulated the dynamic behaviour of a simplerf output coupler for a BEC in a harmonic trap. The two different condensatestates are assumed to possess different trapping properties, with one hypernestate being trapped by a harmonic magnetic potential, while the second feelsno trapping potential and escapes from the trapping region, accelerating underthe inuence of gravity.

We have studied output coupling for different detunings and different cou-pling strengths, and we have shown that the combined effect of the static mag-netic eld and the rf coupling eld leads to spatially localized output coupling.We have also shown that a regime exists, in which a more or less continuousstream of matter is released from a trap, and we have looked at the effectsof bandwidth. We have modelled noisy coupling elds, implemented usingstochastic phase jumps of the coupling eld, and we showed that this destroysthe coherence properties of outcoupled beams.

The introduction of gravity into the system of coupled GP equations gaverise to certain technical problems in the numerical implementation of our sim-

105



CHAPTER 7. CONCLUSION 106

ulations, including the limitation on length of simulations due to populationreaching the grid edge, falling under the inuence of gravity. To overcome thislimitation, we investigated two different methods of “absorbing” and remov-ing population density from the system at the grid edges without interferingwith the simulation core. We found that both of the two approaches haveadvantages in different parameter regimes.

The second major part of this work was the implementation of a dynam-ical model of the time-orbiting potential (TOP) magnetic trap. Since com-putational resources limited us to two spatial dimensions, we modelled thisin the plane of the TOP eld orbit, which is perpendicular to gravitational

acceleration.In both the weak and, in particular, in the strong coupling regime, which

is currently typical for the BEC experiment at Otago, surprisingly, we dis-covered a rich dynamic vortex behaviour, when a trapped and an untrappedatomic hyperne state are rf-coupled in a TOP trap. We developed a model,explaining vortex formation in TOP trap two-component coupling in terms of two superposed density proles of different density and phase. In the strongcoupling regime, where pseudo-adiabatical following (as described in section

6.4.2) occurs through the entire condensate cloud, the interpretation of vor-tex formation was possible applying our model. Our exact model fails inthe weak coupling regime, where pseudo-adiabatic following does not occurthrough the entire cloud. A qualitative interpretation for the observed spatialfeatures within this regime could be given nevertheless, and the weak couplingregime can be interpreted as a transition from the distinct spatial couplingbehaviour, which is characteristic for the TOP trap in the strong coupling/pseudo-adiabatic following regime, to the weak coupling regime, where theTOP trap (for increasing TOP rotation frequency ωT OP Ω0) becomes moreand more like a simple harmonic trap.

Future work

In TOP simulations within the strong coupling regime, which can be inter-preted using our model of vortex formation, we found short lived vortex pairs,spread out to rings of near-zero density in situations, when the phase of thetwo coupled condensate components happened to be near-homogeneous, i.e.



CHAPTER 7. CONCLUSION 107

“at”. This happens after integer numbers of TOP eld rotations as outlinedin section 6.4.3. It has been suggested that these rings are solitonic in nature.However, we did not have sufficient time to pursue a profound investigation of this matter. At this time, we leave the question unanswered, and we encouragefuture work in this direction.

Future work can also go into further investigation of TOP trap dynamics,and in particular, into a more complete model of vortex formation includingthe weak coupling regime. A matter of interest is the extension of this workto three spatial dimensions, where vortex points of our xy plane projectionbecome vortex lines. A two dimensional treatment of the situation appears to

be justied, since the third spatial dimension may be thought of as “frozen”out by a strong connement along the z axis of the TOP trap, but a threedimensional treatment is of great interest and will likely yield further insightinto the problem.

Furthermore, the immediate effects of vortex formation and dissipation onthe condensate states could be investigated within a nite temperature modelin order to understand “heating” effects, which have been described in [27].Dissipation of vortices created and annihilated by the mechanisms described

in this work may give an explanation for these observations of heating, sinceheating and vortex annihilation seem to be closely related [23].

This thesis is also available electronically from the author’s internet pages athttp://hubble.physik.uni-konstanz.de/jkrueger/ orhttp://max.yi.org/jkrueger/ The author’s email address [email protected](July 2000.)



Appendix A

Adiabatic Elimination

Many dynamic physical systems depend on a number of variables that evolvedifferently in time. In some situations, and given suitable parameter regimes,it is possible and justied to make simplications by approximating their be-haviour. For example, a system may respond very slowly to interacting elds,so that a rapidly revolving eld has little or no effect compared to a slowlyvarying or constant interaction.

In the following paragraphs, application of the so called Adiabatic Elimi-

nation will be demonstrated on the example of phase evolution in a coupledtwo-state system [5][6].

A.1 Two-State System in Rotating Wave Ap-proximation

The general state of a two-state system is a superposition of its eigenstates

|0 ,

|1

:

|Ψ(t) = a(t)|0 + b(t)|1 (A.1)

Assume a classical magnetic eld

B (t) = B 0 cos(ωt) (A.2)

causing a magnetic dipole interaction

hΩ = m ·B 0. (A.3)

108



APPENDIX A. ADIABATIC ELIMINATION 109

In a matrix representation

|0 = 10 |1 = 0

1(A.4)

(A.1) satises Schr odinger equation

ih∂ ∂t |Ψ(t) = H |Ψ(t) (A.5)

with a Hamiltonian matrix

H = h0 Ω cos(ωt)

Ω∗cos(ωt) ω1. (A.6)

Now this is simplied by moving into a frame, rotating with the frequency of the interacting eld. Rotating

|Ψ (t) = Γ |Ψ(t) (A.7)

leads to the rotated Schr¨odinger equation

ih∂

∂t |Ψ (t) = ih

∂ Γ

∂t+ Γ H Γ−1

|Ψ (t) = H 0

|Ψ (t) (A.8)

with

H 0 = h0 Ω cos(ωt)e−iωt

Ω∗cos(ωt)eiωt −∆, (A.9)

where ∆ = ω1 −ω. The time varying terms can now be expanded

Ωcos(ωt)e−iωt =Ω2

1 + e−2iωt , (A.10)

yielding a constant and a rapidly oscillating term. In the Rotating Wave

Approximation this term is ignored since its average value vanishes. Thus, inthe rotating frame, the evolution equation simplies to

ih∂ ∂t |Ψ (t) = h

0 Ω2

Ω∗2 −∆ |Ψ (t) = H 0|Ψ (t) . (A.11)




A.2 Adiabatic Elimination

For high detunings |∆ | |Ω| and for a situation where the entire populationis initially in the ground state, a(0) = 1, b(0) = 0, we can now adiabaticallyeliminate terms from the coupled Sch¨odinger equations (they represent thesystem in the rotating frame under the rotating wave approximation)

ih∂a (t)

∂t= h

Ω2

b(t) (A.12)

ih∂b(t)

∂t= h

Ω∗2

a(t) −∆ b(t) . (A.13)

Now b(t) can be formally integrated:

∂b(t)∂t

= i∆ b(t) −iΩ∗2

a(t) (A.14)

∂ ∂t

(b(t)e−i∆ t ) = −iΩ∗2

a(t)e−i∆ t (A.15)

b(t) = −iΩ∗2

ei∆ t t

0a(t)ei∆ s ds (A.16)

Equation A.16 can be integrated by parts and one obtains

b(t) = iΩ

2ei∆ t a(s)

e−i∆ s

i∆

t

0

+

t

0

∂a (s)

ds

e−i∆ s

i∆ds (A.17)

The second term in the equation above can be neglected since the rate of changein the lower state population almost vanishes under our set of assumptions.One retains the following expression:

b(t) =Ω

2∆[a(t) −a(0)ei∆ t ] ≈

Ω2∆

a(t) (A.18)

The last step is possible in regimes with high detunings, since the exponentialterm represents another fast oscillation that averages to zero. Now, puttingthis result back into A.12, one obtains

∂a (t)∂t

= −iΩ2

b(t) = −iΩ2a(t)

4∆(A.19)

This represents the phase evolution of

Φ(t) = −Ω2

4∆t (A.20)

for the lower state in the rotating frame. Inuences of the upper state havebeen successfully eliminated from the evolution equation for the lower state andthus the two-state problem has been approximated by a single state problem.




A.3 Accuracy of the approximation

In order to understand how precise the approximation made in the previoussection is, the magnitude of next order correction terms is required.

Equation A.11 has eigenvectors (dressed states, denoted as |−, |+ ) andeigenvalues as follows:

H 0|± = hω±|± (A.21)

eigenvalues are given by

ω± =∆2 ±

12 ∆ 2 + |Ω|2 (A.22)

with eigenvectors e + = cos( θ2 )sin( θ

2 )and e− = −sin( θ2 )

cos( θ2 )

in the |0 , |1 basis. Here, tan( θ) = |Ω|∆ represents the angle of the diagonalising matrixrotation. We have

|Ψ(t) = a(t)|0 + b(t)|1 = a (t)|− + b (t)|+ (A.23)

Vectorising the state coefficients in the original base as a and in the eigenvectorbase as a , the basetransformation a = La is achieved by the rotation matrix

L = cos( θ2 ) sin( θ2 )−sin( θ

2 ) cos( θ2 )

. (A.24)

To obtain an expression for the evolution of the original basis, one transformsinto the diagonalised basis, where evolution is very simple, decribed by matrix

U (t) =eiω− t 0

0 e−iω+ t , (A.25)

and after that, one transforms back into the original basis. So

a(t) = LT

U (t)La(0). (A.26)needs to be calculated within the parameters applied before (for high de-tunings). Using the approximations cos( θ

2 ) ≈ 1, sin( θ2 ) ≈ |Ω|2∆ and ω± ≈

∆2 ± ∆

2 1 + |Ω|22∆ 2 the rst order approximation obtained is

a(t) =e−i | Ω | 2

4∆ t + |Ω|24∆ 2 ei(∆+ | Ω | 2

4∆ )t

Ω2∆ ei(∆+ | Ω | 2

4∆ )t −e−i | Ω | 2

4∆ t=

e−i | Ω | 2

4∆ t

−Ω2∆ e−i | Ω | 2

4∆ t+

|Ω|24∆ 2 ei(∆+ | Ω | 2

4∆ )t

Ω2∆ ei(∆+ | Ω | 2

4∆ )t

(A.27)




Ignoring the fast rotating parts that average to zero, one gets the same resultsas before and it can be seen that the correction term for the ground statecoefficient a(t) is second order in Ω

∆ , truely small.





APPENDIX B. SIMULATION AND EXPERIMENT 114

is orders of magnitude larger than in our simulations, where we chose C = 200(and C = 0 for analytical purposes in some simulations). Higher C values, dueto large atom numbers and three spatial dimensions, lead to larger condensateclouds and a magnetic eld gradient across the condensate, which is muchlarger compared with the cloud diameter than in the simulation. This causesspatially more selective coupling.

Note that the differences between experiment and simulation describedabove do not change the fundamental processes in the TOP trap, so that ourresults should be applicable to the experimental situation.

Observation of the vortices created by the coupling process may be very

difficult, however, for two reasons. First, vortices occur on time scales shortcompared with the TOP eld rotation frequency, so that an excellent control of the elds is necessary in order to observe “static” snapshot pictures of the dy-namic process in time-of-ight imaging methods. This also applies, to a lesserextent, to experiments in the weak coupling regime. Second, vortices occur inthe weakly conned xy plane (presumeably in shape of vortex lines in threespatial dimensions), which is the plane with the smallest expansion rate dur-ing time-of-ight. These unfavourable circumstances will make experimental

observations of the effects described in this work difficult.



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Documents

Jan Max Walter Kruger- Dynamical Behaviour of Radio Frequency Output Couplers for a Bose-Einstein Condensate