phase dislocations and solitons in bose-einstein condensates

Phase Dislocations and Solitons in

Bose-Einstein Condensates

Theoretical Physics Honours Thesis

Shekhar Suresh Chandra1

November 4, 2005

1Monash University. Email: [email protected]

mailto:[email protected]

Contents

Acknowledgments vii

Abstract viii

Conventions ix

1 Introduction: Bose-Einstein Condensation 1

1.1 Bosons - The Key Ingredient . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Bose-Einstein Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Bose-Einstein Condensates (BECs) . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Experimental Processes for Forming a BEC . . . . . . . . . . . . . . . . . . 4

1.4.1 Cooling Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.2 Laser Pre-Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.3 Magnetic/Laser Trapping . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.4 Evapourative Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Imaging of a BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Gross-Pitaevskii-Bogoliubov Theory . . . . . . . . . . . . . . . . . . . . . . 7

2 Vorticity 11

2.1 Vortices in a Bose-Einstein Condensate . . . . . . . . . . . . . . . . . . . . . 12

2.2 Experimental Realisation of Vortices in a BEC . . . . . . . . . . . . . . . . 13

2.3 Numerical Simulations of Vortices . . . . . . . . . . . . . . . . . . . . . . . 14

i

CONTENTS ii

3 The Aharonov-Bohm (AB) Effect 16

3.1 The Aharonov-Bohm Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 The Aharonov-Bohm Phase-Shift . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 A Classical Analogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 The AB Effect in a BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Phase Retrieval 23

4.1 Generalised Gerchberg-Saxton (GGS) Algorithm . . . . . . . . . . . . . . . 23

4.2 Measure of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Solitons in Two-Component BECs 29

5.1 Theory of Two-Component BECs . . . . . . . . . . . . . . . . . . . . . . . . 29

5.2 Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.3 Dark Soliton Rings in a Two-Component BEC . . . . . . . . . . . . . . . . 30

6 Conclusions 32

A Numerical Analysis 34

A.1 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

A.2 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 34

A.3 Derivation of the Numerical Equations . . . . . . . . . . . . . . . . . . . . . 35

A.3.1 Time-Independent Solutions . . . . . . . . . . . . . . . . . . . . . . . 35

A.3.2 Time-Dependent Solutions . . . . . . . . . . . . . . . . . . . . . . . . 38

A.4 Test/Stress Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

A.4.1 Test Case 1 - Vortex Nucleation . . . . . . . . . . . . . . . . . . . . 42

A.4.2 Test Case 2 - Closed Loop Evolution . . . . . . . . . . . . . . . . . . 43

A.5 Listing of Numerical Parameters . . . . . . . . . . . . . . . . . . . . . . . . 44

B The qC++ Toolkit and Pseudo-code 46

B.1 Quantum Construct (qC++) Toolkit . . . . . . . . . . . . . . . . . . . . . . 46

CONTENTS iii

B.2 Pseudo-code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

B.2.1 Generalised Gerchberg-Saxton (GGS) Algorithm . . . . . . . . . . . 46

B.2.2 Fast Semi-Implicit (FSI) Algorithm . . . . . . . . . . . . . . . . . . . 47

C Dirac’s Magnetic Phase Factor Prescription 49

List of Figures

1 Plot Legends (a) The legend of the Probability Density plots. (b) The

legend of the Phase plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1.1 The steady state radial solution of the ground state of a BEC (g = 1). A

Gaussian function is shown which corresponds to a g = 0 solution. The

y-axis represents probability density. Notice that the maximum probability

density occurs at r = 0 because of the harmonic confining potential. . . . . 10

2.1 The helical phase pattern of the phase winding of a vortex in a BEC. (a)

The phase winding of the vortex located at the centre (plan view). (b) The

same vortex phase winding (isometric view). . . . . . . . . . . . . . . . . . . 13

2.2 The radial solution of the BEC with a vortex . . . . . . . . . . . . . . . . . 14

3.1 Aharonov-Bohm effect in which the two paths (ABF and ACF) undergo

different phase shifts due to the non-zero magnetic vector potential outside

the solenoid. Figure taken from (Aharonov and Bohm, 1959). . . . . . . . . 17

3.2 Aharonov-Bohm effect for water waves. Water waves pass a vortex, giving

rise to phase shifts known as wave dislocations. Propagation is from right

to left and the vortex is circulating clockwise. Figure taken from (Berry

et al., 1980). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

iv

LIST OF FIGURES v

3.3 The initial phases of the wavefunction for the AB effect in a BEC. The

vortex phase winding is superimposed with a step function having a finite

gradient. (a) The phase of the BEC wavefunction (plan view) and (b) The

phase of the BEC wavefunction (isometric view). . . . . . . . . . . . . . . . 20

3.4 The phase dislocations in a Bose-Einstein condensate. A phase disturbance

propagates from right to left and forms phase dislocations past the vortex

situated at the centre. (a) Probability density of the BEC and (b) The

phase of the BEC. (c) Probability density of the BEC in a rotating trap,

(d) The phase of the BEC in a rotating trap. . . . . . . . . . . . . . . . . . 21

3.5 The effect of non-linearity g on the phase dislocations in a BEC. (a) Prob-

ability density of the BEC with g = 1 and (b) Probability density of the

BEC with g = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1 The results of the GGS Phase Retrieval Technique of Tan et al. (2003)

applied to nucleated vortices. Phase retrieval is valid only for the inner

region for the above image. (a) The actual phase distribution, and (b) The

retrieved phase from the method. . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 The convergence of the GGS algorithm for nucleated vortices of the image

in Figure 4.1. The smallest RMS value corresponds to about 1.4% deviation

from the actual image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.3 The results of the GGS algorithm applied to the wavefront dislocations in a

rotating trap. Phase retrieval is valid only for the inner region for the above

image. (a) The actual phase distribution, and (b) The retrieved phase. The

phase retrieval for the winding of the vortex was unsuccessful (the “hole”

in the central region represents where the vortex winding should be). . . . . 28

5.1 The dark ring soliton in one of the components of a numerically simulated

two-component BEC. The result is identical for the other component. . . . 31

LIST OF FIGURES vi

A.1 The probability density of the nucleated vorticies (just off center towards the

top left-hand corner). Two vortices with opposite circulation were nucleated

in this case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

A.2 The probability density for the result of the closed loop evolution. The

disturbance has propagated forward to form dislocations and back to its

initial position (as shown in section 3.4). . . . . . . . . . . . . . . . . . . . . 44

A.3 The phase map for the result of the closed loop evolution. The disturbance

has propagated forward to form dislocations and back to its initial position

(as shown in section 3.4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Acknowledgments

I wish to thank Dr. Rotha Yu for his experience and information pertaining to the

numerical work of the project, and honours colleague, Gary Ruben for valuable information

and ideas regarding the development of this thesis. Last and not least, my supervisor

Associate Professor Michael Morgan for his insightful vision, critical input and guidance

on the project. Without him, this project would not be possible.

vii

Abstract

The dynamics of single and two-component Bose-Einstein condensates (BECs) are stud-

ied in order to understand how vortices and solitons arise in these systems. In particular

we examine wavefunction dislocations in the complex-valued order parameter of a BEC,

which arise when a disturbance propagates past a vortex. These phase dislocations are

analogous to a hydro-dynamical system studied by Berry et al. (1980), and constitute

an example of the Aharonov-Bohm (AB) effect in a BEC. The time dependent Gross-

Pitaevskii equation was numerically solved (based on a fast semi-implicit fractional time

step algorithm) to simulate vortices and solitons. A generalised Gerchberg-Saxton algo-

rithm is applied to retrieve the full complex wavefunction of the condensate and to provide

an experimental means to measure the non-linear interference fringes observed.

viii

Conventions

• Bold face represents Vectors, except ` which is also a vector.

• The colour conventions of the plots are shown in figure 1.

Figure 1: Plot Legends (a) The legend of the Probability Density plots. (b) The legend

of the Phase plots.

ix

Chapter 1

Introduction: Bose-Einstein

Condensation

In this thesis we investigate the dynamics of Bose-Einstein condensation. In particu-

lar, we examine vorticity in both single and two-component Bose-Einstein Condensates

(BECs), and use this phenomenon as a vehicle to explore the Aharonov-Bohm effect.

Moreover, to provide a full picture of the complex-valued wavefunction (the order param-

eter) that describes a BEC, a method of iterative phase retrieval developed by Tan et al.

(2003) was implemented for a single component BEC1.

Berry et al. (1980) demonstrated a hydro-dynamical analogue of the Aharonov-Bohm

effect2. Here we setup a quantum mechanical analogue to this experiment using a BEC. A

phase retrieval method, the Generalised Gerchberg-Saxton (GGS) algorithm (Tan et al.,

2003), was used to provide a means of studying the observations experimentally.

We begin with a discussion of how Bose-Einstein Condensates are formed. This closes

with the derivation of the theory that describes BECs - the Gross-Pitaevskii-Bogoluibov

Theory. Chapter 2 covers the theory of vortices including the classical and quantum1As the phase is not experimentally measurable.2Which is a quantum mechanical effect of the magnetic vector potential discussed in detail in Chapter 3.

1

CHAPTER 1. INTRODUCTION: BOSE-EINSTEIN CONDENSATION 2

manifestations of vorticity, as well as viewing the Gross-Pitaevskii-Bogoliubov theory in

light of vortices. The Aharonov-Bohm Effect is described in Chapter 3, including a classical

analogue of Berry et al. (1980), and its quantum realisation in a BEC. Chapter 4 covers

an iterative phase retrieval - the Generalised Gerchberg-Saxton (GGS) algorithm, and its

application to BECs (Tan et al., 2003). Finally, the thesis concludes with a discussion

of solitonic solutions of two-component BECs. Additional details of the numerical work

reported in this thesis is provided in the accompanying Appendices A and B.

We begin with a description of the phenomenon of Bose-Einstein condensation. The

fundamental particles involved are the bosons, obeying Bose-Einstein Statistics.

1.1 Bosons - The Key Ingredient

All elementary particles can be classed into two groups, namely bosons or fermions.

When a particle has integral spin (in terms of h) then that particle is classed as a boson3.

Examples of bosons include photons, W & Z intermediate vector bosons and atoms with net

integral spin, such as Helium-4 (4He), Rubidium (87Rb) and Sodium-23 (23Na) (Ketterle,

2002). These atoms are bosons because the number of fermionic constituents is even (i.e.,

the total number of protons, neutrons and electrons is even) (Ketterle et al., 1999)4. An

important consequence of Bose-Einstein statistics (reviewed section 1.2) is that bosons can

occupy the same quantum state. This is entirely different for fermions, where the Pauli

Exclusion Principle5 applies.3Since the wavefunction (in 3-dimensions) is either symmetric or anti-symmetric under particle inter-

change we have Ψ(1, 2) = ±Ψ(2, 1). The positive sign (symmetric interchange) corresponds to bosons,

whereas the negative sign (anti-symmetric interchange) corresponds to fermions.4For the duration of the thesis, the term “boson” shall be replaced with “atoms” for sections involving

Bose-Einstein condensation because all condensates are formed from bosonic atoms5No two fermions can occupy the same quantum state.


1.2 Bose-Einstein Statistics

As noted in section 1.1, bosons have the ability to occupy the same quantum state.

In 1924, Indian physicist Satyendra Nath Bose provided an alternative derivation of the

Planck blackbody radiation law, in which he applied statistical mechanics and treated

photons as indistinguishable particles (Bose, 1924). Since bosons are indistinguishable

particles described by a symmetric wavefunction, they are not subject to the Pauli Ex-

clusion Principle. The theory was then further developed by Albert Einstein to form the

quantum statistics describing bosons, called the Bose-Einstein distribution fBE , which has

the form

fBE(E) =1

BeE/kBT − 1, (1.1)

where E is the energy of the bosons and kB is Boltzmann’s constant. The most important

feature of the distribution (1.1) pertinent to condensation is its behaviour as E → 0; in

this case fBE(E) → ∞. This is interpreted as the occupancy of energy states tending

to infinity and, since the ground state is the lowest energy state, the number of bosons

occupying the ground state will be infinite. So, as the energy of the bosons are reduced,

bosons will tend to occupy the ground state more and more, until all the bosons in the

sample are in the ground state. Therefore, the distribution predicts the existence of a

phase transition to a condensate ground state. In the next section we describe how a

condensate can form.

1.3 Bose-Einstein Condensates (BECs)

As noted in the previous section, Bose-Einstein statistics predicts the existence of a

condensate at a very low energies. However, it was not until 1995 when a BEC was

realised experimentally (Anderson et al., 1995; Davis et al., 1995; Bradley et al., 1997).

In the wave nature of particles, the spatial extent of the atom is of the order of the de

Broglie wavelength λdB, which is given by

λdB =h

p, (1.2)


where, in the case of an atom, the linear momentum p is thermally driven and so has the

form p =√

2πmkBT . Here h is Planck’s constant, m is the mass of the atom, kB is the

Boltzmann’s constant and T is the absolute temperature of the atomic vapour. Lowering

the temperature causes an increase in the spatial extent of the wavefunction. This process

can continue until the de Broglie wavelength is comparable to the inter-atomic separation

of the atoms and the atomic wave packets “overlap”. When overlapping occurs, the

atoms have a finite probability of occupying the same quantum state (Ketterle, 2002;

Anglin and Ketterle, 2002; Ketterle et al., 1999). At a critical temperature TC , a phase

transition occurs, causing a cloud of coherent atoms to form, where all the atoms occupy

the same quantum state. In this phase transition the atoms lose their individuality, i.e. the

individual atoms cannot be distinguished from the entire collection of atoms and become

a single macroscopic quantum object - the Bose-Einstein condensate.

The loss of individuality has two important ramifications. First, because the atoms

are now strongly coupled to each other (a consequence of the overlapping of their wave

packets), the quantum theory required is no longer linear6. Secondly, because of this non-

linearity and the resulting quantum mechanical macroscopic extension of the condensate,

a single wavefunction called the complex order parameter Ψ, can be used to describe the

system of atoms. The order parameter satisfies a non-linear Schrodinger equation - the

Gross-Pitaevskii equation, which will be examined in section 1.6. We turn our attention

to the processes involved in cooling and maintaining a BEC.

1.4 Experimental Processes for Forming a BEC

In order to experimentally realise a BEC, it is necessary to cool a dilute gas of alkali

atoms to very low temperatures (< 200nK) and confine the resulting condensate in a trap

potential.6The non-linear Schrodinger equation does not obey the superposition. Consequently, the concept of

interference becomes more complicated


1.4.1 Cooling Conditions

Before cooling is carried out, the sample must be prepared so that conventional con-

densation does not occur; or in the words of Ketterle (2002) it is necessary to avoid “pre-

emption of the Bose-Einstein condensation by the more familiar liquid or solid state”.

This pre-emption can be avoided by using extremely low densities. In this dilute envi-

ronment, the clusters formed by three-body (or indeed many-body) interactions are very

rare compared to the lifetime of the condensate. This means that the two-body (binary)

interactions are more common and that the formation of a conventional liquid condensate

does not during the cooling process. The first cooling step is laser pre-cooling, followed

by evaporative cooling to achieve the condensate.

1.4.2 Laser Pre-Cooling

Laser pre-cooling, as noted by Vuletic and Chu (2000), and also recognised by Ketterle

(2002), is carried out to allow the bosonic atoms to be contained in a wall free containment

system such as magnetic or laser traps. The laser pre-cooling predominantly utilizes

Doppler cooling, where the atoms are illuminated by a laser of specific wavelengths and

the atoms scatter the photons. The scattered photons are on average blue-shifted with

respect to the incident radiation and therefore carry away more energy than is absorbed

by the atoms. This process ceases to be effective below a certain temperature, which

unfortunately is well above that of the critical temperature. This is a consequence of the

interaction process in which an atom interacts with the photons, after which the atom is

in a different internal state and undergoes no further interactions with the photons. So

laser pre-cooling cannot be used to reach the phase transition temperature. However, once

cooled enough, atoms are transferred to a trap for further cooling.

1.4.3 Magnetic/Laser Trapping

A trap is required for two main reasons. First, it facilitates the second stage of cooling,

called evaporative cooling, which is required to reach the critical temperature for Bose-

Einstein condensation. Secondly, the cooled atoms are not allowed to interact with the


walls of a containment system. This would result in the atoms sticking to the walls

facilitating many-body interactions and leading to a conventional liquid condensate. This

trap is normally of the form where it can be modelled by a harmonic potential7. This

harmonic potential is a paraboloid with zero potential energy at the origin, making it

energetically favourable for the condensate to be confined to this region. The traps are

either magnetic or laser based. Laser traps, as demonstrated by Barrett et al. (2001),

are more recent and have the advantage in that it has faster achievement of the phase

transition than magnetic traps. However, magnetic traps have been favoured in the past

due to their simplicity.

1.4.4 Evapourative Cooling

Once confined to the trap, the second and final stage of cooling begins. This involves

the reduction of the trap depth8 to allow the energetic atoms to escape. The reduction

in the height of the potential barrier means the particles with energy greater than this

height escape the trap. In doing so, the particles take away some of the energy present

within the sample. As Masuhara et al. (1988) showed, the result is that the rest of the

sample re-thermalizes at a lower temperature. The cooling is done until the condensate

is achieved. This whole cooling process lasts in the order of seconds to minutes, and the

resultant condensate has a density between 1014cm−3 and 1015cm−3. Once the condensate

has been formed and contained, the experiment is conducted and images of the BEC are

taken. We discuss the imaging technique in the following section.

1.5 Imaging of a BEC

In the imaging of BECs, there are two methods predominantly used, both of which are

based on laser probing. The first and most common method is that of absorption imaging.

Here the BEC is illuminated by a laser set to the resonate frequency of the condensate.7The harmonic potential has the form 1

2mω2r2 where r is the radial coordinate, m is the mass of the

condensate atom and ω is the angular frequency of the trap.8By trap depth we mean the overall potential height.


The trap is turned off and the condensate allowed to expand before an image is taken9.

The images produced are darkest where the density is the highest. The problem with

this technique is that the BEC is destroyed by the light and the expansion. The second

method is dispersive imaging. Here the laser is detuned far from the resonant frequency

of the condensate and the elastically scattered photons are collected to form an image.

This latter method is not destructive, but lacks the resolution of the absorption method10

(Ketterle et al., 1999). We now turn to the quantum theory that describes BECs - the

Gross-Pitaevskii-Bogoluibov theory.

1.6 Gross-Pitaevskii-Bogoliubov Theory

We have already noted that the diluteness of the bosonic gas is important to achieve

Bose-Einstein condensation because of the requirement of binary interactions. The the-

oretical description of Bose-Einstein condensates requires a non-linear model because of

the interactions of the constituent atoms. We will now introduce the mean field theory

known as the Gross-Pitaevskii-Bogoluibov theory of dilute boson gases.

The idea of a mean field description of a dilute boson gas was first formulated by

Bogoliubov (1947), where the “first order” theory, in which atoms scatter into the excited

states from the condensate state, was developed. The “zeroth order” theory, which is the

theory used for Bose-Einstein condensation in this thesis, was independently developed by

Gross (1961) and Pitaevskii (1961). Here all the atoms occupy the same condensate state.

This zeroth order theory is encompassed in the time-dependent Gross-Pitaevskii (TDGP)

equation, which has the form of a non-linear Schrodinger equation:

ih∂Ψ∂t

=

[− h2

2m∇2 +

12mω2|r|2 + g|Ψ|2

]Ψ, (1.3)

where Ψ is the complex order parameter (which is treated as a complex-valued scalar

field), r is the radial coordinate, m is the mass of the condensate atom, ω is the angular9The optical density of a BEC is very high, making the transmission coefficient small

10The density reduction applied magnifies the condensate, making it easier to see features of the con-

densate.


frequency of the trap and ∇2 is the Laplacian operator. We will now derive this equation

using a second quantisation formalism. Let us begin by taking the simplified case of no

interactions between the atoms, then our Hamiltonian in second quantised form is

H =∫d3rΨ†(r)

[h2

2m∇2 + Vtrap(r)

]Ψ(r), (1.4)

where the boson field operators have the form

Ψ†(r) =∑α

Ψαa†α, (1.5)

Ψ(r) =∑α

Ψαaα, (1.6)

where aα and a†α are the bosonic annihilation and creation operators, respectively and Ψα

are the single particle wavefunctions. These operators obey the commutation relations

[aα, a†β ] = δαβ , (1.7)

[aα, aβ ] = 0, (1.8)

[a†α, a†β ] = 0. (1.9)

The above commutation relations have Bose-Einstein statistics built into them. The so-

lution to equation (1.4) is a Gaussian function when the trap potential is harmonic. If we

now include interactions between the atoms in the gas (dominated by two-body interac-

tion because of the dilute nature of the gas), our Hamiltonian in equation (1.4) becomes

a many-body Hamiltonian of the form

H =∫d3rΨ†(r)

[− h2

2m∇2 + Vtrap(r)

]Ψ(r)

+12

∫d3r

∫d3r′Ψ†(r)Ψ†(r′)U(r− r′)Ψ(r′)Ψ(r), (1.10)

where the last term is the Hamiltonian of the two-body inter-atomic interaction (Dalfovo

et al., 1999). Due to the low temperature and diluteness, the interactions will be dominated

by s-wave scattering (characterised by the scattering length as). We can approximate the

interactions by a hard sphere potential of the form

U(r− r′) = gδ(r− r′), (1.11)


where the interaction coefficient g is given by

g =4πh2as

m. (1.12)

The many-body description above is very complicated due to the large number of particles

involved. To make progress we reduce equation (1.10) to a mean field form. The time

evolution equation of the field operator can be written using the Heisenberg equation

ih∂

∂tΨ(r, t) =

[Ψ, H

](1.13)

=

[− h2

2m∇2 + Vtrap(r)

]Ψ(r, t) + gΨ†(r, t)Ψ(r, t)Ψ(r, t). (1.14)

In the mean field description we can decompose the field operator as

Ψ(r, t) = Ψ(r, t) + Ψ(r, t), (1.15)

where Ψ(r, t) is the expectation value of the field operator <Ψ(r, t)> and Ψ(r, t) describes

the fluctuations around the mean value. Thus Ψ(r, t) is a classical field having the mean-

ing of an order parameter and is often called the “the wavefunction of the condensate”

(Dalfovo et al., 1999). At zero temperature, where a pure condensate is formed, there

are no fluctuations and Ψ(r, t) is negligible, and so we can replace Ψ(r, t) with Ψ(r, t)

in equation (1.14). This gives us the TDGP equation of (1.3). The ground state radial

solution of this has the form shown in equation (A.13) (see Appendix A.3) and is shown in

figure 1.1. In the absence of interactions, the solution would be a Gaussian wavefunction.

The above solution was found using the Fixed-Point iterative method, which is discussed

in detail in Appendix A.

In summary, we have seen how the atomic gas has to be dilute and confined in a

magnetic trap in order to achieve Bose-Einstein condensation. Theory is non-linear due

to the coupling of the atoms and is decribed by the TDGP equation. In the next chapter

we look at vortices and the effect of vorticity in BECs.


Figure 1.1: The steady state radial solution of the ground state of a BEC (g = 1). A

Gaussian function is shown which corresponds to a g = 0 solution. The y-axis represents

probability density. Notice that the maximum probability density occurs at r = 0 because

of the harmonic confining potential.

Chapter 2

Vorticity

Vortices have been studied extensively in classical fluid dynamics and indeed in many

other classical systems. Recently their quantum ‘cousins’ have sparked interest especially

in BECs. In fact, the initial theoretical work describing dilute bosonic gases were to

determine vortex states (Gross, 1961; Pitaevskii, 1961).

Mathematically, vorticity is the non-zero circulation Γ of a vector field (say u) defined

by

Γ =∮

Cu · d` =

∫S∇× u · ds, (2.1)

where Stokes’ Theorem has been used to transform the path integral to a surface integral

(Kreyszig, 1999). In classical systems, the vector field in question is normally a velocity

field of physical system represented classically (e.g., the velocity field of a classical fluid

such as water). In quantum systems, the vector field may not be necessarily be a velocity

field directly and vortices in these systems do not interact in the same way as classical

vortices do. In a BEC, the vector field corresponds to the gradient of the phase of the

condensate wavefunction as we shall see in the next section.

11

CHAPTER 2. VORTICITY 12

2.1 Vortices in a Bose-Einstein Condensate

To understand vorticity in a BEC, let us look at the complex value of the order param-

eter. This will have the form

Ψ = |Ψ|eiθ, (2.2)

where θ is a real number representing phase. Now the TDGP equation (1.3) contains a

Laplacian term. This is essentially a squared velocity term1. Since vorticity is related to

the velocity field, a vortex in a quantum mechanical system requires that the phase be

spatially dependent in such a way that there is a gradient in the phase directed around

the vortex. This phase winding is also quantised and according to∮C∇θ · d` = 2πκ, (2.3)

where κ represents the circulation/winding number of the vortex. In practice a vortex

is created in a BEC by applying angular momentum to the condensate. This is also

illustrated in figure 2.1, which shows the phase imprinted on the condensate to achieve a

vortex. It is also useful to point out the topological nature of the vortex in figure 2.1. Note

that if one were to draw a circle around the vortex, the phase is different at each point on

that circle; as one shrinks the circle to the point containing the vortex, there is a phase

ambiguity because the phase values cannot be reconciled. Therefore the modulus of the

order parameter has to vanish at the centre of the vortex. This is called the vortex core

and this ambiguity does not depend on the shape of the path chosen. We shall refer to

this as the topological phase of the system. Although numerically imprinting a topological

phase in a BEC is simple, the experimental realisation is not trivial. This hurdle is due

to the small size of most BEC samples. Despite this, it was experimentally realised by

Matthews et al. (1999) after the prediction of vortices by Williams and Holland (1999).

In what follows, we discuss the experimental methods for producing vortices.1This is because of canonical quantisation where the momentum operator has the form p = −ih∇. We

shall make use of this point again when describing propagating disturbances in chapter 3.


Figure 2.1: The helical phase pattern of the phase winding of a vortex in a BEC. (a) The

phase winding of the vortex located at the centre (plan view). (b) The same vortex phase

winding (isometric view).

2.2 Experimental Realisation of Vortices in a BEC

The vortex achieved by Matthews et al. (1999) was through the use of laser and radio

frequency fields to imprint a phase pattern (similar to figure 2.1) in a two-component

BEC2. The phase pattern imprinted has a phase winding of 2π around the vortex core.

Another technique for forming vortices was used by Madison et al. (2000) at ENS3 in

Paris, a few months later. They used a rotating laser beam to spin up a condensate (a

process called “stirring”), which resulted in the formation of vortices near the edge of the

BEC; the vortices eventually settled into a lattice configuration. These lattices have been

studied quite extensively by Fetter and Svidzinsky (2001) and can be explained in terms

of an analogy to a rigid body (whose vorticity is constant), in which the BEC attempts

to distribute the vorticity as uniformly as possible (Abo-Shaeer et al., 2001). In the2The two component BEC consists of a bosonic atom with two possible internal states as we shall see

in chapter 5.3Ecole Normale Superieure.


numerical work reported in this thesis, the steady state vortex solutions were determined

and is discussed in the following section.

2.3 Numerical Simulations of Vortices

Vortices are numerically simulated by integrating the TDGP equation (1.3) with an

extra term involving angular momentum. This angular momentum term involves the

angular momentum operator Lz, i.e.,

ih∂Ψ∂t

=

[− h2

2m∇2⊥ +

12mω2|r|2 − LzΩ + g|Ψ|2

]Ψ, (2.4)

where Ω is the rotational frequency of the condensate. However, before numerically solving

equation (2.4) it is convenient to transform to cylindrical coordinates and apply separation

of variables (refer to appendix A.3.1 for details). In setting up a stable vortex for a two

dimensional BEC, the radial solution of the BEC with a vortex (illustrated in figure 2.2)

was used for the magnitude of the order parameter and phase winding of figure 2.1 was

Figure 2.2: The radial solution of the BEC with a vortex


used for the phase of the order parameter. Nucleation of vortices was also done, but only

as a test for the validity of the numerical code (refer to Appendix A.4.1).

Chapter 3

The Aharonov-Bohm (AB) Effect

In an important paper, Aharonov and Bohm (1959) showed that the magnetic vector

potential A phase shifted charged particles and consequently had a physical meaning as

opposed to just a mathematical one in classical physics. This has been verified recently

by Tonomura et al. (1986) and we discuss it in the following section.

3.1 The Aharonov-Bohm Setup

A schematic of the Aharonov-Bohm effect is shown in figure 3.1, where an electron beam

is split coherently into two parts and travel around a solenoid. The electron wavefunc-

tions are recombined past the solenoid to produce an interference pattern. Aharonov and

Bohm (1959) supposed that neither the electron, nor the field could penetrate the solenoid

boundary. The magnetic flux, ΦB is evaluated via Stokes’ theorem

ΦB =∫

SB · ds =

∫S∇×A · ds =

∮C

A · d`, (3.1)

where C is a path that encirlces the solenoid The magnetic vector potential is non-zero

outside the solenoid. This led Aharonov and Bohm (1959) to conclude that the magnetic

vector gives rise to observable phase shifts. Note that in the case of vorticity we mentioned

in the chapter 2 that Stokes’ theorem related vorticity to the circulation Γ. Here the phys-

ical phenomenon being quantified is not the circulation but the magnetic flux ΦB. They

16

CHAPTER 3. THE AHARONOV-BOHM (AB) EFFECT 17

are mathematically equivalent, but not necessarily physically equivalent. This conclusion

gave the magnetic vector potential physical meaning and in what follows, we determine

the form of the phase shifts.

3.2 The Aharonov-Bohm Phase-Shift

Consider a B field confined within a solenoid aligned with the z-axis and a containing

flux ΦB. Let us now consider charged particles with energy E and mass m incident as

shown in figure 3.1. In this setup, the vector potential A is non-zero in all regions around

and inside the solenoid, but the B field is zero outside and non-zero inside the solenoid.

The incident charged particle undergoes a phase change as it traverses different paths

around the solenoid, although the region inside the solenoid containing the B field is

inaccessible to the particles (Berry, 1980). More precisely, if the potential is given by the

simplest case, namely

A(r) =Φ

2πrθ, (3.2)

Figure 3.1: Aharonov-Bohm effect in which the two paths (ABF and ACF) undergo differ-

ent phase shifts due to the non-zero magnetic vector potential outside the solenoid. Figure

taken from (Aharonov and Bohm, 1959).


where θ is the azimuthal unit vector. We assume a simple incident wavefunction (that of

a plane wave) of the form

ψ0(r) = exp (ik · r) = exp (i|k||r| cos θ) , (3.3)

where the wavenumber |k| =√

2mE/h and equation (3.3) applies for the idealised case of

a thin and infinitely long solenoid. Using the boundary conditions of inaccessibility to the

inside of the solenoid (i.e., ψ0(0) = 0), the magnetically shifted wavefunction becomes

ψD(r) = ψ0(r) exp

(iα

∫ θ

0

1rθ′ · dr′

)= ψ0(r) exp (iαθ)

= exp (ik · r + iαθ) (3.4)

where α = eΦ/h, e is the electron’s charge and we have used Dirac’s magnetic phase factor

prescription (see appendix C for details). Equation (3.4) shows that the wavefunction

acquires a phase shift. Figure 3.1 illustrates how two paths will undergo different phase

changes and therefore interfere with each other. It is possible to setup a classical analogue

of the above dislocations in the form of water-waves. We study this analogy in the next

section.

3.3 A Classical Analogue

Berry et al. (1980) envisaged the changes in phase of the Aharonov-Bohm effect as anal-

ogous to phase dislocations of wavefronts (phase contours) traversing through a classical

fluid containing a vortex. They constructed a water-wave experiment to illustrate this,

in which a hydro-dynamical vortex was formed and waves propagated through the fluid.

The theory for the Aharonov-Bohm effect was applied to the water-wave scenario, and the

results obtained were found to be in accordance with this theory. A typical experimental

result is shown figure 3.2. Berry et al. (1980) described the shifts in the wavefronts as

dislocations by analogy to dislocations of atomic planes in crystals. Their results also in-

dicated (as their theory predicted) that the dislocations are dependent on the circulation

of the water through 2.1. In the spirit of this analogy, we construct a quantum mechanical


Figure 3.2: Aharonov-Bohm effect for water waves. Water waves pass a vortex, giving

rise to phase shifts known as wave dislocations. Propagation is from right to left and the

vortex is circulating clockwise. Figure taken from (Berry et al., 1980).

analogue of the Aharonov-Bohm effect in a BEC. We also provide a possible application

of the effect in measuring the self-interaction of a BEC.

3.4 The AB Effect in a BEC

Having discussed the original AB effect as well as the classical analogue, we now “close

the loop” by investigating the AB Effect in a BEC. The AB effect in a BEC was inves-

tigated by forming a steady state vortex to the TDGP equation (see section 2.3). Then

a disturbance in the condensate wavefunction was propagated through the BEC. In set-

ting up a disturbance to propagate through the BEC, it is required that the disturbance

have a phase gradient. Therefore, the form of the disturbance used was a one-sided step


function of finite gradient shown in figure 3.3. The phase dislocations in the condensate

Figure 3.3: The initial phases of the wavefunction for the AB effect in a BEC. The vortex

phase winding is superimposed with a step function having a finite gradient. (a) The

phase of the BEC wavefunction (plan view) and (b) The phase of the BEC wavefunction

(isometric view).

wavefunction is shown in figure 3.4 with the effect of non-linearity shown in figure 3.5.

Both figures clearly show the interference pattern of the phase dislocations. The loop is

indeed complete and the AB effect may be a common phenomenon in vector fields.

The results show that an increase of the non-linear co-efficient g, which quantifies in-

creased repulsion; the condensate spreads out and consequently so does the disturbance

and dislocations. The decrease in the separation of the fringes is also observed and this

is most likely due to the increase in the non-linear “super-position” of the dislocations to

form the fringes. This property of the fringes may provide a method of measuring the

self-interaction of BECs, though such precise measurements may be problematic. The

results of figure 3.5 suggest that the distance between fringes d and the self-interaction

parameter g may be related as

g ∝ 1d. (3.5)


Figure 3.4: The phase dislocations in a Bose-Einstein condensate. A phase disturbance

propagates from right to left and forms phase dislocations past the vortex situated at the

centre. (a) Probability density of the BEC and (b) The phase of the BEC. (c) Probability

density of the BEC in a rotating trap, (d) The phase of the BEC in a rotating trap.

The strength of the dislocations may also provide a quantitative measure of the vorticity

in BECs. Also note that these dislocations are not as disconnected as those illustrated in

figure 3.2. This is because the circulation of the two systems is not the same and the fact

the BEC is a highy non-linear system. It is plausible that a BEC can be setup to produce a

form of the dislocations as in figure 3.2, but instead of a study of the dislocation strengths1,1Studied in detail by Berry et al. (1980).


Figure 3.5: The effect of non-linearity g on the phase dislocations in a BEC. (a) Probability

density of the BEC with g = 1 and (b) Probability density of the BEC with g = 20.

the project focused on a means to recover the phase information on these dislocations in

BECs so they can be experimentally verified through both the phase and probability

density and without interferometry2. This is discussed in the following chapter.

2Interferometry applies in a linear theory, however the TDGP equation describing the BEC is non-linear.

Chapter 4

Phase Retrieval

In the previous chapter, we discussed how dislocations arise when a wavefunction dis-

turbance propagates past a vortex. The resulting fringe pattern is dependent on the self-

interaction (or non-linearity) of the condensate. Although a fringe pattern is observed,

the notion of interferometry is complicated by the non-linear nature of the TDGP equa-

tion. Here we discuss a phase retrieval technique which is not limited by considerations of

linearity. This method is based on an iterative phase retrieval technique which we discuss

in the next section.

4.1 Generalised Gerchberg-Saxton (GGS) Algorithm

The original Gerchberg-Saxton algorithm (Gerchberg and Saxton, 1972) used modulus

data at the image and diffraction planes to reconstruct the phase distribution (based on

Fourier transforms) in optics (e.g., in microscopy). Tan et al. (2003) provided a general-

isation of this algorithm to multiple sets of modulus data (what we shall now refer to as

images) and a non-linear evolution equation. The exact algorithm is as follows for a set

of images ordered in time:

1. Construct the trial function for the first image as

ΨT = |ΨA|eiθT , (4.1)

23

CHAPTER 4. PHASE RETRIEVAL 24

where T denotes trial/guess values and A the actual values.

2. Evolve the trial function to the next image using the TDGP equation and update

the modulus of the trial wavefunction to that of the actual modulus.

3. Repeat step 2 until the last image is reached. Then repeat the procedure backwards

through the images (i.e., evolve backwards in time until the first image is reached).

This represents a single iteration of the GGS algorithm1.

4. Loop the steps 2 and 3 until the desired convergence criterion has been satisfied.

Essentially, the system is allowed to evolve using the guessed phase, but the updated

modulus at every image forces the phase to iteratively converge to the actual phase of

the images. It should be pointed out that the convergence of the method is dependent on

three important aspects.

1. The difference in the moduli of the images. The moduli of the images must be suffi-

ciently different so as to cause the phases to adjust through evolution. However, one

must balance the difference between the moduli of the images with the time differ-

ence between them since the images must contain sufficient evolutionary information

to facilitate the phase retrieval.

2. The phase reconstruction only applies to the central region of the images where there

are no edge/boundary effects from numerical simulations. This is because the freely

propagating solutions reach the artificial boundaries and become invalid solutions,

this makes the phase in these regions meaningless.

3. The algorithm is not well suited in regions where the modulus is very close to zero.

In these regions, the modulus will not change very much and so the phase will not

either.

As we reconstruct the phase, a measure of how well the reconstruction is progressing is

required. We discuss this convergence issue in the next section.1This is slightly different to that of Tan et al. (2003), where one iteration is defined after having

completed step one.


4.2 Measure of Convergence

There are a number of ways to quantify convergence and the error between the results

and the actual images. A convenient measure is the RMS error σ, which is defined as

follows

σ =

√√√√∑Njk (|ΨA(j, k)| − |ΨR(j, k)|)2∑N

jk |ΨA(j, k)|2, (4.2)

where A represents the actual image values, R represents the resultant image values, j

and k are grid points and N2 is the total number of grid points. This measure was used

to explore the convergence of the numerical results.

4.3 Numerical Results

In our first attempt in developing the GGS algorithm, vortices were nucleated and

the phase winding of the vortices retrieved through the GGS algorithm. Seven images

were used for the phase retrieval of the nucleated vortices of appendix A.4.1. The initial

condition consisted of a ground state solution with random phases in each quadrant and

an image at every 300 timesteps2. The result of the phase retrieval is shown in figure 4.1

and the convergence of the results are shown in figure 4.2. The main issue with the

phase retrieval of the vortices was the time it took to get the results. Running at a low

spatial resolution and reasonable time resolution, the process took several hours to run3.

This combined with the trial and error of determining the optimal number of images and

differences between images, made the implementation of the phase retrieval method time

consuming. Despite this, the resultant deviation was within a couple of percent of the

actual image.

We next attempted the phase retrieval of the wavefront dislocations discussed in sec-

tion 3.4. The result of the phase retrieval is shown in figure 4.3. This proved a challenge

for the phase retrieval method as the first criteria mentioned in section 4.1 was not fully2Other simulation parameters are noted in section A.5.3Simulations were done on a 2.4 GHz PC with 1GB RAM.


Figure 4.1: The results of the GGS Phase Retrieval Technique of Tan et al. (2003) applied

to nucleated vortices. Phase retrieval is valid only for the inner region for the above image.

(a) The actual phase distribution, and (b) The retrieved phase from the method.

met; consequently the phase of the vortex was not fully recovered. As can be seen from

the results, the phase of the dislocations and interference pattern has been reconstructed

although they are barely visible. Closer examination with qC++’s4 3D viewing abilities

reveal these dislocations more clearly than can be shown here5. However, the phase wind-

ing associated with the vortex solution (shown in figure 2.1) was not retrieved successful.

We mentioned in section 4.1 that the images have to be sufficiently different to get con-

vergence. Here the vortex was in steady state. Consequently, the convergence to the

vortex phase winding was not reached. It is plausible however, that a nucleated vortex

with dislocations (i.e., using images of the nucleation and the dislocations together for the

phase retrieval) will overcome this problem. Unfortunately, it was not possible to conduct

further numerical simulations because of the time constraints involved.

4Refer to Appendix B for details on qC++.5The results shown in figure 3.4(d) are at a higher resolution than in this simulation due to the amount

of time it takes to run the GGS algorithm.


Figure 4.2: The convergence of the GGS algorithm for nucleated vortices of the image in

Figure 4.1. The smallest RMS value corresponds to about 1.4% deviation from the actual

image.


Figure 4.3: The results of the GGS algorithm applied to the wavefront dislocations in a

rotating trap. Phase retrieval is valid only for the inner region for the above image. (a)

The actual phase distribution, and (b) The retrieved phase. The phase retrieval for the

winding of the vortex was unsuccessful (the “hole” in the central region represents where

the vortex winding should be).

Chapter 5

Solitons in Two-Component BECs

A BEC is described by a non-linear evolution equation (the TDGP equation) charac-

terised by a self-interaction parameter g (see section 1.6). In this chapter, we examine a

BEC that has two internal states, both of which are described an interaction parameter.

This system constitutes a two-component BEC.

5.1 Theory of Two-Component BECs

A two-component BEC has two internal states, each of which interacts with itself and

with the other internal state. These systems are described by a Gross-Pitaevskii equation

for each of the two internal states, i.e.,

ih∂Ψ1

∂t=

[− h2

2m∇2⊥ +

12mω2|r|2 − LzΩ + g11|Ψ1|2 + g12|Ψ2|2

]Ψ1, (5.1)

ih∂Ψ2

∂t=

[− h2

2m∇2⊥ +

12mω2|r|2 − LzΩ + g21|Ψ1|2 + g22|Ψ2|2

]Ψ2, (5.2)

where Ψ1 represents the order parameter of the first component, Ψ2 represents the order

parameter of the second component and g`m(`,m = 1, 2) represents the ‘self-interaction’

coefficients of the internal states. Possible internal states include a mixture of two different

bosonic atoms, such as 41K & 87Rb studied by Modugno et al. (2001), as well as bosonic

atoms with different internal spin states studied by Myatt et al. (1997). Basically, each

of the internal states sets up a potential for the other, as well as apart a potential for

29

CHAPTER 5. SOLITONS IN TWO-COMPONENT BECS 30

itself. The resulting non-linear coupled system allows the formation of an interesting

phenomenon, called solitary waves or solitons, which we discuss in the next section.

5.2 Solitons

In 1834, a young Scottish engineer by the name of John Scott Russell noticed an unsual

wave phenomenon while observing a boat being drawn through a canal by a pair of horses.

He noted in his report that

“... it accumulated round the prow of the vessel in a state of violent agitation,

then suddenly leaving it behind, rolled forward with great velocity, assuming

the form of a large solitary elevation, a rounded, smooth and well-defined heap

of water, which continued its course along the channel apparently without

change of form or diminution of speed.”

Although he named the phenomenon “Wave of Translation” (Russell, 1844), it would be

later given the name Solitary Wave or soliton. A modern description of a soliton is that

it is a disturbance that is non-dispersive, moving with constant velocity and associated

with non-linear wave equations (Berloff, 2005). Essentially, the phenomenon is due to the

non-linearity of the wave offseting the dispersion of the components in the wave. It is no

surprise then that solitonic solutions are possible in a BEC due to the non-linear nature of

the condensate. A solitonic solution obtained in this project is discussed in the following

section.

5.3 Dark Soliton Rings in a Two-Component BEC

A “dark” soliton ring is a ring-shaped soliton which has less density than the surrounding

density, giving it a dark appearance. A recent study by Xue (2005) has demonstrated the

dynamics of such soltions in two-component BECs. In what was hoped to be a further

application of the phase retrieval method metioned in Chapter 4, the numerical simulation

of a two-component BEC was conducted. The simulation with a steady state vortex in the

CHAPTER 5. SOLITONS IN TWO-COMPONENT BECS 31

centre of the grid for both components (with opposite circulation to each other) was setup.

The system was then evolved and the resulting dark soliton ring is shown in figure 5.1.

It was hoped that a further study of the types of solutions and topological defects (e.g.,

Figure 5.1: The dark ring soliton in one of the components of a numerically simulated

two-component BEC. The result is identical for the other component.

vortices) could be studied using this model. However, time did not permit this, and

therefore will constitute further work.

Chapter 6

Conclusions

This thesis investigated Bose-Einstein condensation, with the intention of studying the

Aharonov-Bohm effect and the phase dislocations associated with the effect in BECs. The

work of Berry et al. (1980) demonstrated that this effect can be observed in a water-wave

experiment. We showed that a similar phenomenon can be realised in a BEC. An iterative

phase retrieval method was also applied to reconstruct the complex wavefunction of the

condensate.

The simulation for the Aharonov-Bohm effect in a BEC was conducted by setting up

a steady state vortex in the condensate and propagating a disturbance in the complex

wavefunction (order parameter) past the vortex. The form of the disturbance was that

of a one-sided step function with a finite gradient. The simulation clearly showed the

formation of an interference pattern past the vortex, indicative of wavefront dislocations

as by Berry et al. (1980) for the Aharonov-Bohm effect. This interference pattern seemed

to be dependent on the degree of non-linearity of the condensate, and the fringes observed

were a direct consequence of phase shifts associated with the vortex in the BEC. As such,

standard interferometric measurements cannot be used to determine the phase shifts due

to the non-linear nature of the BEC. A method of iterative phase retrieval (Tan et al.,

2003) was implemented and demonstrated that it is possible to reconstruct the complex

wavefunction (modulus and phase) for a BEC. However, problems were encountered with

32

CHAPTER 6. CONCLUSIONS 33

the phase winding of a stationary vortex due to the absence of changes in the images

between time steps. As a further application of the phase retrieval method, solitonic

solutions to a two-component BEC was also simulated. This yielded dark ringed solitons

consistent with the work of Xue (2005).

Future work will be directed to using the fringes (wavefront dislocations) to quantifying

the magnitude of self-interaction parameter g. There are also many interesting possibilities

associated with topological defects and solitons in multi-component BECs. The GGS

algorithm may provide a useful tool for studying the phase structures in these systems.

Also work on the Aharonov-Bohm effect in a multi-component, 3-dimensional BEC can

be done with further applications of the phase retrieval method to this system.

Appendix A

Numerical Analysis

A.1 Numerical Methods

There are special requirements for numerical methods involving quantum mechanics.

Primarily, the method must be unitary (i.e., it conserves probability) as well as the stan-

dard requirements of stability and accuracy. There are two main approaches in developing

numerical methods for quantum mechanical systems. The first method is the Runge-Kutta

forth order method, which evolves the system in imaginary time (τ = it). With sufficiently

small time steps, the method satisfies the above criteria. Another approach is the method

employed by Winiecki and Adams (2002), called the Fast Semi-Implicit Method. Here the

semi-implicit refers to the semi-implicit nature of the right hand side of the equation. In

this method the system is evolved using the Crank-Nicolson Implicit Method incoporat-

ing the Method of Approximate Factorization. This approximate factorisation essentially

reduces the system to solving a tridiagonal system for each dimension. It is this second

approach that is used for the simulations in this thesis.

A.2 Initial and Boundary Conditions

In all simulations reported in this thesis, the order parameter was set to vanish at the

boundaries. The initial conditions used for the simulations were the steady state solutions

34

APPENDIX A. NUMERICAL ANALYSIS 35

of the TDGP equation. The exact initial condition for each simulation is noted in their

relevant sections.

A.3 Derivation of the Numerical Equations

The following is a ‘walk-through’ of the equations used for the simulations, including

derivations and numerical considerations.

A.3.1 Time-Independent Solutions

Separating the Variables

As we have seen in section 1.6, the Gross-Pitaevskii equation describing a BEC, in a

harmonic trap, with applied angular momentum (discussed in section 2.3), in one time

and 2-spatial dimensions is given by

ih∂Ψ∂t

=

[− h2

2m∇2⊥ +

12mω2|q|2 − LzΩ + g|Ψ|2

]Ψ, (A.1)

where q is a generalised coordinate and

∇2⊥ =

∂2

∂x2+

∂2

∂y2(A.2)

=∂2

∂r2+

1r

∂

∂r+

1r2

∂2

∂φ2(A.3)

=1r

∂

∂r

(r∂

∂r

)+

1r2

∂2

∂φ2. (A.4)

Lz = −ih(x∂

∂y− y

∂

∂x

)(A.5)

= −ih ∂

∂φ. (A.6)

Let us consider that the above is Variable Separable of the form

Ψ(x, y, t) = Q(q)T (t), (A.7)

where the T (t) is the time part and Q(q) the spatial part. Inserting (A.7) into equa-

tion (A.1), we get

ih∂

∂tQ(q)T (t) = − h2

2m∇2⊥Q(q)T (t)+

12mω2|q|2Q(q)T (t)−LzΩQ(q)T (t)+g|Q(q)T (t)|2Q(q)T (t).


The partial derivatives now become ordinary derivatives, and hence

ihQ(q)T (t) = − h2

2mQ′′(q)T (t) +

12mω2|q|2Q(q)T (t)−ΩQ′(q)T (t) + g|Q(q)T (t)|2Q(q)T (t).

Dividing through by Q(q)T (t) and noting that |T (t)|2 = 1 because of the unitary require-

ment of time evolution, one gets

ih1

T (t)T (t) = − h2

2m1

Q(q)Q′′(q) +

12mω2|q|2 − Ω

1Q(q)

Q′(q) + g|Q(q)|2. (A.8)

Note that the left hand side is only a function of time and time derivatives and that

the right hand side is only a function of space and spatial derivatives. In order for the

time derivatives to equal spatial derivatives, both must equal a constant, the constant of

separation, which we shall call µ. This constant allows us to separate the time and spatial

parts of the above into

ih1

T (t)dT (t)dt

= µ (A.9)

− h2

2m1

Q(q)Q′′(q) +

12mω2|q|2 − Ω

1Q(q)

Q′(q) + g|Q(q)|2 = µ. (A.10)

Let us proceed to determine the steady state or time independent equation of the system

by writing the spatial equation above in cylinderical co-ordinates (i.e. Q(q) = R(r)Φ(φ))

− h2

2m∇2⊥R(r)Φ(φ)+

12mω2r2R(r)Φ(φ)−LzΩR(r)Φ(φ)+g|R(r)Φ(φ)|2R(r)Φ(φ) = µR(r)Φ(φ).

(A.11)

We use equations (A.3) and (A.6) to write (A.11) as

− h2

2m

[∂2R(r)∂r2

+1r

∂R(r)∂r

+1r2∂2φ(φ)∂φ2

]+

12mω2r2R(r)Φ(φ)

+ ihΩR(r)∂Φ(φ)∂φ

+ g|Ψ|2R(r)Φ(φ) = µR(r)Φ(φ). (A.12)

From here we can form two steady state solutions. For the case of the ground state

solution of a BEC, we assume there is no angular dependence and so our equation can be

writen

− h2

2m

[∂2R(r)∂r2

+1r

∂R(r)∂r

]+

12mω2r2R(r) + g|Ψ|2R(r) = µR(r). (A.13)


Including angular momentum leads to the vortex states. We assume an ansatz of the form

Φ(φ) = e−inφ, (A.14)

where n is the winding number (charge) of the vortex1 and φ is the phase angle. Substi-

tuting this into equation (A.12) and multiplying through by Φ(φ), we obtain

− h2

2m

(d2R(r)dr2

+1r

dR(r)dr

− n2

r2

)+

12mω2r2R(r) + nhΩR(r) + g|R(r)|2R(r) = µR(r).

(A.15)

Discretization

In order to solve these non-linear differential equations, one must discretise the equations.

First, we discretize the time independent (or steady state form) of the Gross-Pitaevskii

equation, equation (A.15). Equation (A.15) needs to be discretized using a stable and

reasonably accurate numerical method. For accuracy, we shall keep it to the order of the

derivatives involved, in this case, second order. The second derivative to second order

accuracy is given byd2ψ

dr2=(ψj+1 − 2ψj + ψj−1

∆r2

), (A.16)

where now the j represents the grid position. The above is a centered difference method.

We shall also discretize the first derivative to second order

dψ

dr=(ψj+1 − ψj−1

2∆r

). (A.17)

Our finite difference form of the steady state equation is

− h2

2m

[(ψj+1 − 2ψj + ψj−1

∆r2

)+

1r

(ψj+1 − ψj−1

2∆r

)− n2

r2

]

+12mω2r2ψj + nhΩψj + g|ψj |2ψj = µψr. (A.18)

This can be solved using the Fixed Point Method to acquire the steady state solution.1In a BEC the vorticity is quantised according to the winding number n


A.3.2 Time-Dependent Solutions

In order to evolve the system in time, one needs to solve equation (A.9). The full wave-

function is

Ψ(q, t) = Ψ(q)e−iµt/h. (A.19)

Here the evolution of the system is governed by the time evolution equation

|Ψ(t)> = e−iHt/h|Ψ(0)> . (A.20)

Because the evolution must be unitary for the conservation of probability, the method

of integration must be unitary as well as stable in the complex domain and reasonably

accurate. Such a method is the Crank-Nicolson Method. We begin with the Cartesian

form of the Gross-Pitaevskii Equation

ih∂Ψ∂t

=

[− h2

2m

(∂2

∂x2+

∂2

∂y2

)+

12mω2(x2 + y2) + ihΩ

(x∂

∂y− y

∂

∂x

)+ g|Ψ|2

]Ψ.(A.21)

Using n to denote the nth timestep, k and ` for the x and y spatial points respectively,

we write the forward difference approximation of the left-hand side of equation (A.21) as

ih∂Ψ∂t

= ih

(Ψn+1

k,` −Ψnk,`

∆t

). (A.22)

The explicit centred difference approximation for the x part of the spatial derivatives

becomes

− h2

2m∂2

∂x2Ψ = − h2

2m

(Ψn

k+1,` − 2Ψnk,` + Ψn

k−1,`

∆x2

)(A.23)

and

− h2

2m∂2

∂y2Ψ = − h2

2m

(Ψn

k,`+1 − 2Ψnk,` + Ψn

k,`−1

∆y2

). (A.24)

Finally, the forward difference approximation to the angular momentum term becomes

x∂Ψ∂y

= x

(Ψn

k,`+1 −Ψnk,`

∆y

)(A.25)

y∂Ψ∂x

= y

(Ψn

k+1,` −Ψnk,`

∆x

). (A.26)


The implicit versions of the spatial derivatives above involve n = n + 1. For the Crank-

Nicolson method we use the average of the explicit and implicit forms. This gives

− h2

2m∂2

∂x2Ψ =

−h2

4m∆x2

[Ψn

k+1,` − 2Ψnk,` + Ψn

k−1,` + Ψn+1k+1,` − 2Ψn+1

k,` + Ψn+1k−1,`

](A.27)

− h2

2m∂2

∂y2Ψ =

−h2

4m∆y2

[Ψn

k,`+1 − 2Ψnk,` + Ψn

k,`−1 + Ψn+1k,`+1 − 2Ψn+1

k,` + Ψn+1k,`−1

](A.28)

x∂Ψ∂y

=x

2∆y

[Ψn

k,`+1 −Ψnk,` + Ψn+1

k,`+1 −Ψn+1k,`

](A.29)

y∂Ψ∂x

=y

2∆x

[Ψn

k+1,` −Ψnk,` + Ψn+1

k+1,` −Ψn+1k,`

]. (A.30)

Likewise we take the average of the other functions of Ψn

12mω2(x2 + y2)Ψ =

12mω2(x2 + y2)

(Ψn

k,` + Ψn+1k,`

)(A.31)

g|Ψ|2Ψ =12g(|Ψn

k,`|2Ψnk,` + |Ψn+1

k,` |2Ψn+1

k,`

). (A.32)

Combining equations (A.22), (A.27), (A.28), (A.29), (A.30), (A.31) and (A.32), we get the

Gross-Pitaevskii equation in implicit finite difference form

ih

(Ψn+1

k,` −Ψnk,`

∆t

)= −1

2h2

2m∆x2

[(Ψn

k+1,` − 2Ψnk,` + Ψn

k−1,`) + (Ψn+1k+1,` − 2Ψn+1

k,` + Ψn+1k−1,`)

]− 1

2h2

2m∆y2

[(Ψn

k,`+1 − 2Ψnk,` + Ψn

k,`−1) + (Ψn+1k,`+1 − 2Ψn+1

k,` + Ψn+1k,`−1)

]+

xihΩ2∆y

[Ψn

k,`+1 −Ψnk,` + Ψn+1

k,`+1 −Ψn+1k,`

]− yihΩ

2∆x

[Ψn

k+1,` −Ψnk,` + Ψn+1

k+1,` −Ψn+1k,`

]+

12mω2(x2 + y2)

(Ψn

k,` + Ψn+1k,`

)+

12g(|Ψn

k,`|2Ψnk,` + |Ψn+1

k,` |2Ψn+1

k,`

). (A.33)

We need to separate the known values (those at timestep n) and those that are unknown.

We do this first by breaking up the finite time difference

Ψn+1k,` = Ψn

k,` +ih∆t

4m∆x2

[(Ψn

k+1,` − 2Ψnk,` + Ψn

k−1,`) + (Ψn+1k+1,` − 2Ψn+1

k,` + Ψn+1k−1,`)

]+

ih∆t4m∆y2

[(Ψn

k,`+1 − 2Ψnk,` + Ψn

k,`−1) + (Ψn+1k,`+1 − 2Ψn+1

k,` + Ψn+1k,`−1)

]+

xΩ2∆y

[Ψn

k,`+1 −Ψnk,` + Ψn+1

k,`+1 −Ψn+1k,`

]− yΩ

2∆x

[Ψn

k+1,` −Ψnk,` + Ψn+1

k+1,` −Ψn+1k,`

]+

∆t2ih

mω2(x2 + y2)(Ψn

k,` + Ψn+1k,`

)+

∆t2ih

g(|Ψn

k,`|2Ψnk,` + |Ψn+1

k,` |2Ψn+1

k,`

). (A.34)


For convenience, we can write

δ2xΨn+1 =(Ψn+1

k+1,` − 2Ψn+1k,` + Ψn+1

k−1,`

)(A.35)

δ2xΨn =(Ψn

k+1,` − 2Ψnk,` + Ψn

k−1,`

)(A.36)

δxΨn+1 =(Ψn+1

k+1,` −Ψn+1k,`

)(A.37)

δxΨn =(Ψn

k+1,` −Ψnk,`

). (A.38)

The same notation will be adopted for the y component. Then equation (A.34) can be

more conveniently rewriten as

Ψn+1k,` = Ψn

k,` +ih∆t

4m∆x2

[δ2xΨn + δ2xΨn+1

]+

ih∆t4m∆y2

[δ2yΨ

n + δ2yΨn+1

]+

xΩ2∆y

[δyΨn + δyΨn+1

]− yΩ

2∆x

[δxΨn + δxΨn+1

]+

∆t2ih

mω2(x2 + y2)(Ψn

k,` + Ψn+1k,`

)+

∆t2ih

g(|Ψn

k,`|2Ψnk,` + |Ψn+1

k,` |2Ψn+1

k,`

). (A.39)

We now take separate the implicit part from the explicit part by taking the implicits terms

to the left-hand side of the equation A.39. This left-hand side now becomes

Ψn+1k,` − ih∆t

4m

[δ2xΨn+1

∆x2+δ2yΨ

n+1

∆y2

]− Ω

2

[xδyΨn+1

∆y− y

δxΨn+1

∆x

]

−∆t2ih

mω2(x2 + y2)Ψn+1k,` − ∆t

2ihg|Ψn+1

k,` |2Ψn+1

k,` , (A.40)

and the right-hand side is

Ψnk,` +

ih∆t4m

[δ2xΨn

∆x2+δ2yΨ

n

∆y2

]+

Ω2

[xδyΨn

∆y− y

δxΨn

∆x

]+

∆t2ih

mω2(x2 + y2)Ψnk,` +

∆t2ih

g|Ψnk,`|2Ψn

k,`. (A.41)

The right side of the equation can be set equal to a single value d, since all values are known

and can be evaluated explicitly. Therefore, the Gross-Pitaevskii equation in semi-implicit

form can be written as

Ψn+1k,` − ih∆t

4m

[δ2xΨn+1

∆x2+δ2yΨ

n+1

∆y2

]− Ω

2

[xδyΨn+1

∆y− y

δxΨn+1

∆x

]

−∆t2ih

mω2(x2 + y2)Ψn+1k,` − ∆t

2ihg|Ψn+1

k,` |2Ψn+1

k,` = d. (A.42)


The above equation has to be put into tridiagonal form for each spatial dimension to be

solved conveniently.

Method of Approximate Factorization

Using the Gross-Pitaevskii Equation in the form[1− ih∆t

4mδ2x

∆x2− ih∆t

4mδ2y

∆y2

]Ψn+1 =

[1 +

ih∆t4m

δ2x∆x2

+ih∆t4m

δ2y∆y2

]Ψn+

∆t2

[fn+1 + fn

],

(A.43)

where

fn+1 =∆t2ih

mω2(x2 + y2)Ψn+1k,` +

∆t2ih

g|Ψn+1k,` |

2Ψn+1k,` +

Ω2

[xδyΨn+1

∆y− y

δxΨn+1

∆x

]. (A.44)

We approximately factorise the terms containing δ2 in the following way[1− ih∆t

4m∆x2δ2x

] [1− ih∆t

4m∆y2δ2y

]Ψn+1 =

[1 +

ih∆t4m∆x2

δ2x

] [1 +

ih∆t4m∆y2

δ2y

]Ψn

+∆t2

[fn+1 + fn

]. (A.45)

Notice how the left-hand is semi-implicit, this helps to stablize the method (Ames, 1992).

We let

αx =ih∆t

4m∆x2(A.46)

αy =ih∆t

4m∆y2. (A.47)

Each of the factors on the left is of the form of a tridiagonal system, i.e.

Ax = Ψn+1k,` − αx

(Ψn+1

k+1,` − 2Ψn+1k,` + Ψn+1

k−1,`

)(A.48)

= −αxΨn+1k+1,` + (1 + 2αx) Ψn+1

k,` − αxΨn+1k−1,`, (A.49)

where the tridiagonal system is of the form

Ai ·Ψ = d (A.50)

bi ci 0 . . .

ai bi ci . . .

0 ai bi . . ....

......

. . .

Ψ0

Ψ1

...

ΨN

=

d0

d1

...

dN

. (A.51)


In the case of equation (A.49), the elements for Ax are

ax = −αx, (A.52)

bx = (1 + 2αx) , (A.53)

cx = −αx. (A.54)

Therefore equation (A.45) can be rewriten as

AxAyΨn+1 = BxByΨn +∆t2

[fn+1 + fn

], (A.55)

where each of the A’s is a tridiagonal form. To complete the method, the tridiagonal form

must be solved using fractional stepping, i.e.

AxΨn+1/2 = BxByΨn +∆t2

[fn+1 + fn

], (A.56)

AyΨn+1 = Ψn+1/2. (A.57)

This method can be applied to multiple dimensions. In order to evaluate the fn+1 term, a

number of iterations has to be carried out per timestep because of the non-linear terms in

the equation. At the first iteration, the value fn+1 = fn can be used, and updated values

employed in further iterations. Winiecki and Adams (2002) showed that it converges when

following this proceedure and that it has an optimum number of iterations of three per

timestep.

A.4 Test/Stress Cases

Various test cases were implemented2 to ensure the evolution code worked correctly.

These are illustrated in the following sections.

A.4.1 Test Case 1 - Vortex Nucleation

To test the validity of the evolution code. We nucleated vortices in the BEC using the

Kibble-Zurek Mechanism3. Random phases were assigned to all points in the simulation2Simulations were done on a AMD 2700XP+ with 1 GB RAM System.3Essentially the phase of the order parameter is random at all points in space


grid and the simulation initiated. However, this produced too many vortices, which we

were unable to resolve. So a simpler nucleation system was adopted in which a random

phase was assigned to each quadrant of the grid. This produced the results shown in

figure A.1. This show the validity of the code and next we checked the stability and

correctness of the code.

Figure A.1: The probability density of the nucleated vorticies (just off center towards the

top left-hand corner). Two vortices with opposite circulation were nucleated in this case.

A.4.2 Test Case 2 - Closed Loop Evolution

The next test was to check the correctness and stability of the evolution code. This was

done by evolving the system forward in time by a certain number of timesteps and then

evolving the system backwards in time by the same number of timesteps. This forms a

closed loop in time. If the code worked correctly, then the system would revert back to

the initial state at the start of the evolution. The results are shown in figures A.2 and A.3

and demonstrated that the code worked correctly.


Figure A.2: The probability density for the result of the closed loop evolution. The

disturbance has propagated forward to form dislocations and back to its initial position

(as shown in section 3.4).

A.5 Listing of Numerical Parameters

The following parameters were used to obtain the numerical results in this thesis. Here

ht, hx and hy are the time, x and y steps respectively, that were used.

Grid Size ht hx hy Total Timesteps

Nucleation 100x100 1× 10−3 0.4 0.4 2200

Closed Loop 400x400 1× 10−4 0.2 0.2 10000


Figure A.3: The phase map for the result of the closed loop evolution. The disturbance

has propagated forward to form dislocations and back to its initial position (as shown in

section 3.4).

Appendix B

The qC++ Toolkit and

Pseudo-code

B.1 Quantum Construct (qC++) Toolkit

The qC + + Library is a toolkit designed and implemented in C++ to allow the rapid

development of quantum mechanical simulations with 3D plots and graphical user in-

terfaces (GUIs). It was constructed from scratch and explicitly for this thesis and is

intended to be used to investigate other quantum numerical systems and numerically

evolve them. The qC + + toolkit is fully open source and is supported at Source-

Forge.net and the toolkit as well as all the documentation can be found at the homepage

(http://qcplusplus.sourceforge.net/index.html). All numerical simulations were done on a

AMD 2700XP+, 1 GB RAM System or on a AMD 2400XP+, 0.5 GB RAM System.

B.2 Pseudo-code

B.2.1 Generalised Gerchberg-Saxton (GGS) Algorithm

BEGIN GGS

Initialize Grids

Set Phase Guesses to Random

46

https://sourceforge.net/projects/qcplusplus/

https://sourceforge.net/projects/qcplusplus/

http://qcplusplus.sourceforge.net/index.html

APPENDIX B. THE QC++ TOOLKIT AND PSEUDO-CODE 47

LOOP For Number of Iterations

Set to Evolve Forwards

LOOP For All Images

Update Magnitude

Save Current Phase Found for the Image

Evolve Guess to Next Image

END LOOP

Set to Evolve Backwards

LOOP For All Images

Evolve Guess to Next Image

Update Magnitude

Save Current Phase Found for the Image

END LOOP

END LOOP

Output Result

END GGS

B.2.2 Fast Semi-Implicit (FSI) Algorithm

BEGIN FSI

Initialize Arrays

LOOP For 3 Iterations

Solve x-dimension()

Solve y-dimension()

Set Boundary Conditions (set normally to vanish)

END LOOP

END FSI

BEGIN Solve x-dimension()

LOOP For All Grid Points

APPENDIX B. THE QC++ TOOLKIT AND PSEUDO-CODE 48

Set the Tri-diagonal Arrays a,b,c,d

Solve Tridiagonal

Place solution into grid

END LOOP

END Solve x-dimension()

BEGIN Solve y-dimension()

LOOP For All Grid Points

Set the Tri-diagonal Arrays a,b,c,d

Solve Tridiagonal

Place solution into grid

END LOOP

END Solve y-dimension()

Appendix C

Dirac’s Magnetic Phase Factor

Prescription

The quantum mechanics of a particle in a magnetic field is governed by the Hamiltonian

of the form

H(~r, ~p) = H0(~r, ~p− e ~A), (C.1)

where e is the charge of the particle, H0 is the Hamiltonian without the magnetic field, ~p

is the momentum operator −ih∇ and r is the radial coordinate. Dirac showed however,

that if one takes the Hamiltonian which corresponds to particles of fixed energy E with

out a magnetic field, namely

H0(~r, ~p)ψ0 = Eψ0. (C.2)

One can construct a solution to equation (C.2) in terms of the wavefunction in the absence

of the magnetic field multiplied by a magnetic phase factor involving the vector potential

~A (Berry, 1980), as in the following

ψD(r) = ψ0(r) exp(ie

h

∫ r

r0

~A(r′) · dr′), (C.3)

where r′ is the dummy integration variable. Equation (C.3) has its limitations however

(outside the scope of this review)1, but has been used in an inexact way to make predictions1See Berry (1980) for the exact treatment.

49

APPENDIX C. DIRAC’S MAGNETIC PHASE FACTOR PRESCRIPTION 50

which have been verified experimentally.

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