Numerical studies of rotating bose einstein condensates via rotating lagrangian coordinates

And we study the conservation and dynamical laws of angular tum expectation and condensate width.. 64 3.2 Dynamics of angular momentum expectation hLzi t solid line, 3.5 Time evolution o

Bose-Einstein Condensates via Rotating Lagrangian Coordinates

CAO XIAOMENG(B.Sc.(Hons.), NUS,Diplˆome de l’Ecole Polytechnique)

A THESIS SUBMITTED FOR THE DEGREE OF

MASTER OF SCIENCE

DEPARTMENT OF MATHEMATICS

NATIONAL UNIVERSITY OF SINGAPORE

2012

I would like to extend my heartfelt gratitude to my supervisor: ProfBao Weizhu, for his guidance and advice throughout the researchprocess He has provided me great ideas and strong encouragement.

In addition, I’d like to express special thanks to Mr Tang Qinglinfor all his help and guidance

Moreover, I’d like to thank the professors in National University

of Singapore and Ecole Polytechnique in France, for their guidancethrough out my two years studies in Singapore and two years studies

in France under the French-Grandes Ecoles Double Degree Program(FDDP) I’d like to thank Assoc Prof Wong Yan Loi for his coor-dination and help in FDDP during the years

And finally, I wish to thank my parents for their understanding,unconditional support and sacrifice over the years I would neverarrive at this stage without them

1.1 Bose-Einstein condensates 1

1.2 The Gross-Pitaevskii equation 3

1.3 Existing numerical methods 4

1.4 Purpose of the study and structure of the thesis 5

2 Methods and analysis for rotating BEC 7 2.1 Dynamical laws in the Cartesian coordinate 7

2.2 GPE under a rotating Lagrangian coordinate 10

2.3 Dynamical laws in the Lagrangian coordinate 11

2.4 Numerical methods 21

2.4.1 Time splitting method 21

2.4.2 Discretization in 2D 23

2.4.3 Discretization in 3D 24

2.5 Numerical results 24

2.5.1 Accuracy test 25

2.5.2 Dynamical results in 2D 26

2.5.3 Quantized vortex interaction in 2D 33

3 Extention to rotating two-component BEC 45

3.1 Introduction 45

3.2 Coupled Gross-Pitaevskii equations 46

3.3 Dynamical laws in the Cartesian coordinate 47

3.4 The Lagragian transformation 50

3.5 Dynamical laws in the Lagrangian coordinate 51

3.6 Numerical methods 62

3.7 Numerical results 64

3.7.1 Dynamics of energy and density 64

3.7.2 Conservation of angular momentum expectation 65

3.7.3 Center of mass 65

3.7.4 Condensate width 67

3.7.5 Dynamics of vortex lattices 68

Since the realization of Bose-Einstein Condensation (BEC) in lute bosonic atomic gases [54, 5, 16], significant experimental andtheoretical advances have been developed in the field of research[56, 64, 4, 3, 13, 35, 33] which permits an intriguing glimpse intothe macroscopic quantum world Quantized vortices in rotating BEChave been observed by several groups experimentally, e.g the JILAgroup [56], the ENS group [54], and the MIT group [64] There arevarious methods to generate quantized vortices, including imposing

di-a rotdi-ating ldi-aser bedi-am with di-anguldi-ar velocity on the mdi-agnetic trdi-ap [21]and adding a narrow, moving Gaussian potential to the stationarymagnetic trap [45] These observations have spurred great excite-ment in studying superfluidity

In this thesis, the dynamics of rotating BEC is studied analyticallyand numerically based on introducing a rotating Lagrangian coordi-nate Based on the mean field theory, the rotating one-componentBEC is described by a single Gross-Pitaevskii equation (GPE) in

a rotating frame By introducing a rotating Lagrangian coordinatetransform, the angular momentum term has been removed from theoriginal GPE and is replaced by a time-dependent potential Wefind the formulation for energy and proved its conservation And

we study the conservation and dynamical laws of angular tum expectation and condensate width We investigate the center

momen-of mass with initial ground state with a shift A numerical method,which is explicit, stable, spectral accurate is presented Extensive nu-merical results are presented to demonstrate the dynamical results

The first chapter of this thesis will focus on the background of BECand existing numerical methods The work in this thesis will be in-troduced as well.

Chapter 2 will focus on the single-component BEC We apply thecoordinate transformation methodology The dynamical laws of therotating BEC under the new coordinate system will be discussedand presented in details We approximate the rotating BEC usingtime splitting method for temporal direction and spectral discretiza-tion method for spatial direction Numerical results will also be pre-sented

Our investigation is extended to two-component rotating BEC inChapter 3 We apply a similar approach to the coupled GPE wherethe dynamics is studied both analytically and numerically

List of Tables

2.1 Spatial error analysis: Error ||φe(t) − φh,k(t)||l 2 at t = 2.0 with

k = 1E − 4 252.2 Temporal error analysis: Error ||φe(t) − φh,k(t)||l 2 at t = 2.0 with

h = 1/32 26

List of Figures

1.1 Velocity-distribution data of a gas of Rubidium (Rb) atoms, firming the discovery of a new phase of matter, the Bose-Einsteincondensate Left: just before the appearance of a Bose-Einsteincondensate Center: just after the appearance of the condensate.Right: after further evaporation, leaving a sample of nearly purecondensate 22.1 Dynamics of mass and energies under Ω = 0, γx = γy = 1 282.2 Dynamics of mass and energies under Ω = 0, γx = 1, γy = 8 282.3 Dynamics of mass and energies under Ω = 1, γx = γy = 1 292.4 Dynamics of mass and energies under Ω = 4, γx = 1, γy = 2 292.5 Dynamics of condensate width and angular momentum under Ω =

con-1, γx = γy = 1, x0 = 1, y0= 1 302.6 Dynamics of condensate width and angular momentum under Ω =

1, γx = 1, γy = 2, x0 = 0, y0 = 1 302.7 Dynamics of condensate width and angular momentum under Ω =

0, γx = 1, γy = 2, x0 = 0, y0 = 1 312.8 Dynamics of condensate width and angular momentum under Ω =

0, γx = 1, γy = 2, x0 = 1, y0 = 1 312.9 Trajectory of center of mass under original and transformed framewhen γx = γy 352.10 Trajectory of center of mass under original and transformed framewhen Ω = 0, γx = 1, γy = 8 35

2.11 Trajectory of center of mass under original and transformed framewhen Ω = 0, γx = 1, γy = 2π 352.12 Trajectory of center of mass under original frame when Ω =1/5, γx = γy = 1 362.13 Trajectory of center of mass under original frame when Ω =4/5, γx = γy = 1 362.14 Trajectory of center of mass under original frame when Ω =

1, γx = γy = 1 362.15 Trajectory of center of mass under original frame when Ω =3/2, γx = γy = 1 372.16 Trajectory of center of mass under original frame when Ω =

6, γx = γy = 1 372.17 Trajectory of center of mass under original frame when Ω =

π, γx= γy = 1 372.18 Trajectory of center of mass under transformed frame when Ω =

1, γx = 1, γy = 2, (x0, y0) = (1, 1) 382.19 Trajectory of center of mass under original frame when Ω =

1, γx = 1, γy = 2, (x0, y0) = (1, 1) 382.20 Trajectory of center of mass under transformed frame when Ω =1/2, γx = 1, γy = 2 392.21 Trajectory of center of mass under original frame when Ω =1/2, γx = 1, γy = 2 392.22 Trajectory of center of mass under transformed frame when Ω =

4, γx = 1, γy = 2 402.23 Trajectory of center of mass under original frame when Ω =

4, γx = 1, γy = 2 402.24 Trajectory of center of mass under transformed frame when Ω =1/2, γx = 1, γy = π 402.25 Trajectory of center of mass under original frame when Ω =1/2, γx = 1, γy = π 41

2.30 Case III density contour plot, x01= (0.5, 0), x02 = (−0.5, 0), (m1, m2) =

(1, −1) 43

2.31 Case IV density contour plot, x01 = (0.5, 0), x02= (0, 0), (m1, m2) =

(1, −1) 43

3.1 Dynamics of total density and density of each component for case

i (left) and case ii (right) 64

3.2 Dynamics of angular momentum expectation hLzi (t) (solid line),

3.5 Time evolution of density surfaces for component one (left) and

component two (right) at different times for case I From top to

bottom: t = 0, 5, 10, 15 69

3.6 Time evolution of density surfaces for component one (left) and

component two (right) at different times for case II From top to

bottom: t = 0, 2.5, 5, 7.5 70

3.7 Time evolution of density surfaces for component one (left) andcomponent two (right) at different times for case III From top tobottom: t = 0, 2.5, 5, 7.5 713.8 Dynamics of center of mass Left: trajectory of total center ofmass Right: the time evolution of center of mass of componentone (top), time evolution of center of mass of component two(bottom) 723.9 Dynamics of center of mass Left: trajectory of total center ofmass Right: the time evolution of center of mass of componentone (top), time evolution of center of mass of component two(bottom) 723.10 Dynamics of condensate widths σx(t), σy(t) and σr(t) when λ = 0and V1(x) = V2(x) 733.11 Dynamics of condensate widths σx(t), σy(t) and σr(t) when λ 6= 0and V1(x) = V2(x) 733.12 Dynamics of condensate widths σx(t), σy(t) and σr(t) when λ = 0and V1(x) 6= V2(x) 733.13 Dynamics of vortex lattices when N = 4 for component one (left)and component two (right); From top to bottom, t = 0, 0.7, π/2, 2.3, π 743.14 Dynamics of vortex lattices when N = 9 for component one (left)

and component two (right); From top to bottom, t = 0, 0.7, π/2, 2.3, π 75

The idea of BEC was first predicted by Albert Einstein in 1924 He has dicted the existence of a singular quantum state produced by the slowing ofatoms using cooling apparatus [31] He reviewed and generalized the work ofSatyendra Nath Bose [14] on the statistical mechanics of photons The result

pre-of the combined efforts pre-of Bose and Einstein forms the concept pre-of a Bose gas,governed by Bose-Einstein statistics, which describes the statistical distribution

of identical particles with integer spin, known as bosons In 1938, Fritz Londonproposed BEC as a mechanism for superfluidity in liquid helium and supercon-ductivity [15,53] Superfluid helium has many exceptional properties, includingzero viscosity and the existence of quantized vortices It was later discoveredthat these properties also appear in the gaseous BEC, after the first experimen-tal realization of BEC, by Eric Cornell, Carl Wieman and co-workers at JILA

on June 5, 1995 in vapours of 87Rb (cf Fig 1.1) [5] About four months later,

an independent effort led by Wolfgang Ketterle at MIT created a condensate

made of 23Na [29] The condensate had about a hundred more atoms, allowinghim to obtain several important experimental results, such as the observation

of quantum mechanical interference between two different condensates Cornell,Wieman and Ketterle won the 2011 Nobel Prize in Physics for their achieve-ments One month after the JILA work, a group led by Randall Hulet at RiceUniversity announced the create of a condensate of 7Li atoms [16] Later, itwas achieved in many other alkali gases, including85Rb [27],41K [57],133Cs [73],spin-polarized hydrogen [36] and metastable triplet4He [65,67] These systemshave become a subject of explosion of research

The most striking feature of BEC is that due to the condensation of a large

Figure 1.1: Velocity-distribution data of a gas of Rubidium (Rb) atoms, ing the discovery of a new phase of matter, the Bose-Einstein condensate Left:just before the appearance of a Bose-Einstein condensate Center: just afterthe appearance of the condensate Right: after further evaporation, leaving asample of nearly pure condensate

confirm-fraction of identical atoms into the same quantum state, the wave-like behaviour

is exhibited on a macroscopic scale, which is distinguishable to the behaviours ofparticles following classical Newton’s second law Another intriguing property

is the unrestricted flow of particles in the sample, such as the flow of currentswithout observable viscosity and the flow of electric currents without observable

resistance [75] These properties can be explained by the macroscopic occupation

of a quantized mode which provides a stabilized mechanism Many experimentshave been carried out to study the superfluid properties of BEC, which has a par-ticularly interesting signature of supporting quantized vortex states [35,48,72]

In 1999, a vortex was first created experimentally at JILA using87Rb containingtwo different hyperfine components [56] Soon after, the ENS group has createdvortex in elongated rotating cigar-shaped one component condensate, with smallvortex arrays of up to 11 vortices being observed [24,26,54,55] Recently, MITgroup has created larger rotating condensates with up to 130 vortices being ob-served [1] More recently Leanhardt et al have created a coreless vortex in aspinor F = 1 BEC using “topological phase imprinting” [51] It is hence of greatimportance to study the quantized vortex state to better understand the aboveobservations as well as superfluidity [35, 48, 68, 17] Quantized vortex statescan be detected in the experiments of rotating single-component BEC, rotatingtwo-component and spin F = 1 BEC Mean-field theory is widely applied to ap-proximate the BEC The main idea of the theory is to replace interactions of allparticles in the system to any one body with an average or effective interaction,sometimes called a molecular field [23] The multi-body problem can be reducedinto a one-body problem In this case, the interactions between particles in adilute atomic gas are very weak and the system can be regarded as being dom-inated by the wave-like condensate One can hence apply the main-field theoryand sum the interaction of all of the particles to get an effective one-body prob-lem, which can be approximated using Gross-Pitaevskii equation (GPE) [28,30]

The Gross-Pitaevskii equation was first derived in the early 1960s and namedafter Eugene P Gross [42] and Lev Petrovich Pitaevskii [62] According to theory,the rotating one-component condensate can be described by a single GPE in arotating frame [4, 12, 21,35,39,37] At temperature T which is much smaller

than the critical temperature Tc, a BEC could be described by the macroscopicwave function ψ := ψ(˜x, t), whose evolution is governed by a self-consistent,mean field nonlinear Schr¨odinger equation (NLSE) in a rotational frame, alsoknown as the GPE with the angular momentum rotation term [21,8,34]:

To study the dynamics of BEC, it is essential to have an efficient and rate numerical method to analyse the time-dependent GPE In literature, manynumerical methods have been proposed to study the dynamics of non-rotatingsingle component BEC, which can be grouped into two types One is finite dif-ference method, such as explicit finite difference method [22], Crank-Nicolsonfinite difference method [66] and alternating direction method [71] Gener-ally, the accuracy can be second or fourth order in space The other method

accu-is pseudo-spectral method, for example, Bao [13] has proposed a fourth-ordertime-splitting Fourier pseudo-spectral method (TSSP) and a fourth-order time-

splitting Laguerre-Hermite pseudo-spectral method (TSLH) [11], Adhikari et

al have proposed Runge-Kutta pseudospectral method [2,59] Researches havedemonstrated that pseudospectral method is more accurate and stable than finitedifference method However, for a rotating BEC, due to the appearance of theangular rotating term, the above methods can no longer be applied directly Lim-ited numerical methods have been proposed to study the dynamics of rotatingBEC, but they are usually low-order finite difference methods [4,20,19,75,18].Some better performed methods were designed, for example, Bao et al [8] hasproposed a numerical method by decoupling the nonlinearity in the GPE andadopting the polar coordinates or cylindrical coordinates to make the angularrotating term constant It is of spectral accuracy in transverse direction but ofsecond or fourth-order accuracy in radial direction Another leap-frog spectralmethod is proposed, which is of spectral accuracy in space and second-orderaccuracy in time [76] However, it has a stability time constraint for time step[76]

For coupled-GPEs, there have also been quite a few existing numerical methods,such as finite difference method and pseudospectral method [6, 38, 25] Butfor rotating coupled-GPEs, due to the rotational term, difficulties have beenintroduced as the case for single-component BEC

1.4 Purpose of the study and structure of the thesis

Hence, it is of a strong interest to develop an accurate, stable and efficient merical method In this paper, we have proposed such a numerical method andstudied the dynamics of the rotating BEC by using it The key feature of themethod is: By taking an orthogonal time-dependent Lagrangian transformation,the rotational term in GPE can be eliminated under the new rotating Lagrangiancoordinate We can therefore apply previous numerical methods proposed fornon-rotating BEC on the transformed GPE In this paper, we have studied therotating single component BEC and rotating two-component BEC We have ap-plied a second-order time splitting method and spectral method in space, which

nu-is very efficient and accurate New forms of energy, angular momentum as well

as center of mass for the transformed rotating BEC are defined and presentedboth analytically and numerically

The paper is organized as follows: In Chapter 2, we analyse the rotating singlecomponent BEC And in Chapter 3, we extend the numerical methods to two-component BEC The two chapters follow the same structure We first begin bypresenting the dynamical laws of the BEC We proceed to apply the orthogonaltime dependent matrix transformation method and study the dynamics of the ro-tating BEC under the new Lagrangian coordinates, we redefined and studied theconservation of density and energy, as well as angular momentum conservationunder certain conditions Dynamical laws for condensate width and analyticalsolutions for center of mass are presented as well We follow by presenting anefficient and accurate numerical method for the simulation of transformed rotat-ing BEC Numerical results after applying the numerical method are discussed

in section 4, which include accuracy test, dynamical results and quantized vortexinteraction Finally, some conclusion and further study directions are drawn

There have been many researches done to study the dynamics of rotating BEC[76,8], we present a brief review of the dynamical laws of rotating BEC in theCartesian coordinate.

I) Energy and density

There are two important invariants: density and energy and they are defined as

N (ψ) =

Z

R d|ψ(˜x, t)|2d˜x, t ≥ 0, (2.1)E(ψ) =

where ψ∗ denotes the complex conjugate of ψ

II) Angular momentum expectation

Angular momentum expectation is defined as:

III) Condensate width

We define the condensate width as follows along the α-axis (α = x, y, z for 3D),

to quantify the dynamics of the problem (2.13):

hδαi (t) =pδα(t), with δα= 2 (t) =

Z

R d

α2|ψ|2d˜x, α = ˜x, ˜y, ˜z (2.6)

We have the following dynamical law for the condensate width:

Theorem 2.1.2 i) Generally, for d=2,3, with any potential and initial data,

the condensate width satisfies:

x [1 − cos(2γxt)]+δx(0)˜ cos(2γxt)+δ

(1)

˜ x

2γx sin(2γxt).(2.8)

iii) For all other cases, we have , for t ≥ 0,

δα(t) = E(ψ0) + Ω hLzi (0)

γ2

α [1 − cos(2γαt)]+δ(0)α cos(2γαt)+δ

(1) α

IV) Center of mass.

The center of mass is defined as follows:

h˜xi (t) =

Z

R d

˜x|ψ|2d˜x =: (˜xc(t), ˜yc(t), ˜zc(t))T (2.10)

By [8], the center of mass satisfies a 2nd order ODE and can be solved cally

Since the rotational term Lz is the key ‘bottle-neck’ when one derives a ical method, we now take an orthogonal rotational transformation for (1.1) inspatial space to deduce this rotational term and waive the difficulty Denote theorthogonal rotational matrix as follows:

numer-A(t) :=





cos(Ωt) − sin(Ωt)sin(Ωt) cos(Ωt)

we can therefore cancel the rotational term in (1.1) and instead solve the ing problem:

follow-





i∂tφ = −12∇2φ + V (x, t)φ + β|φ|2φ,φ(x, 0) = φ0(x), x ∈ Rd, d = 2, 3,

In this section, we provide some analytical results on the definition and thedynamical laws of the following quantities for the inhomogeneous GPE (2.13):energy, density, angular momentum expectation, condensate width and the cen-ter of mass

I) Energy and density

We introduce two important invariants of (2.13), which are the normalization ofthe wave function:

We have the following theorem for the conservation of energy and density:

Theorem 2.3.1 Conservation law for the energy and density:

Conservation of N (φ) follows directly from the conservation of N (ψ) We begin

with the equation (2.13) to show the energy conservation Starting from

R dβ|φ|4dx

= ∂t 1

2Z

R d|∇φ|2dx + β

2Z

R d|φ|4dx

+ ∂tZ

∂sV (x, s)|φ(x, s)|2ds

dx

dt .Hence we have the energy conservation law as stated above

II) Angular momentum expectation

Angular momentum is another important quantity to study the dynamics of

rotating BEC It is a measure of the vortex flux and is defined as follows:

parts, and taking into account that φ decreases to 0 when |x| → ∞, we have

d hLzi (t)

Z

R d(φ∗tLzφ + φ∗Lzφt) dx

=Z

R d[−(iφt)∗Jzφ + φ∗Jz(iφt)] dx

= −Z

2Z

We define the condensate width as follows along the α-axis (α = x, y, z for 3D),

to quantify the dynamics of the problem (2.13):

hδαi (t) =pδα(t), with δα = 2 (t) =

Z

R d

α2|φ|2dx, α = x, y, z, (2.28)

we have the following dynamical law for the condensate width:

Theorem 2.3.3 i) Generally, for d=2,3, and any potential and initial data,

the condensate width satisfies:

x [1 − cos(2γxt)]+δ(0)x cos(2γxt)+δ

(1) x

2γx sin(2γxt).(2.31)

iii) For all other cases, we have , for t ≥ 0,

δα(t) = E(φ0) + Ω hLzi (0)

γ2

α [1 − cos(2γαt)]+δα(0)cos(2γαt)+δ

(1) α

R d

α2[φ(i∂tφ)∗− φ∗(i∂tφ)] dx

= i2Z

R d

α2[φ(i∂tφ)∗− φ∗(i∂tφ)] dx

= i2Z

R d∇φ∗(α2∇φ + φ∇α2) − ∇φ(α2∇φ∗+ φ∗∇α2) dx

= iZ

= iZ

R dα(φtφ∗α− φ∗tφα) dx − i

Z

R d[φ∗t(αφα+ φ) − φt(αφ∗α+ φ∗)] dx

=Z

−4Z

R dV (x, t)|φ|2dx + 4Ω hLzi (t) − 2

Z

R d(x∂xVrot+ y∂yVrot)|φ|2dx

Noticing that when γx= γy = γr, we have:

Z 2π 0

r2cos2θ|f (r, t)|2rdθdr

= π

Z ∞ 0

r2|f (r, t)|2rdr =

Z ∞ 0

Z 2π 0

r2sin2θ|f (r, t)|2rdθdr

=Z

R d

y2|φ|2dx = δy(t) = 1

2δr(t).

Thus we could show the result above in (2.31)

(iii) In general, we take a similar approach as what we have done in (2.35) and

(2.36), by combining energy expression as in (2.27), we have:

IV) Center of mass

In this section, we would like to study the analytical solutions for the center ofmass Denote φ as the solution of GPE, the center of mass is defined as:

˙hxi(0) = 0,

2





cos(2Ωt) sin(2Ωt)sin(2Ωt) − cos(2Ωt)





, (2.42)

R d(φt∇φ∗− φ∗t∇φ) dx

=Z

Case I: γx= γy = γror Ω = 0, we have B(t) = diag(γ2

x, γ2, γ2), and the solution

should be of the form:

d2th˜xi (t) + AΛ h˜xi (t) = 0, (2.48)

by noticing that dd22tA(t) = −Ω2I˜2A(t) with

By multipling (2.48) to the orthogonal matrix A(t), we have:

In this section, we present an accurate and efficient numerical method whichsolves the transformed rotating GPE under a rotating Lagrangian coordinate asshown in (2.13) Without loss of generality, we take d = 2

Different from other studies to solve the rotating BEC, by introducing an onal transformation, we have reduced the rotational term in the GPE We finallyhave a standard GPE with inhomogeneous potential, which could be solved bystandard numerical methods in a more stable way, compared to previous re-searches done in this area

orthog-We begin by applying the time splitting method, and then proceed with Fourierspectral method in x and y direction

2.4.1 Time splitting method

We take ∆t > 0 as a time step For n = 0, 1, 2, , N from time t = tn= n∆t to

t = tn+1= tn+ ∆t, we could solve the transformed GPE (2.13) in the followingtwo steps:

(2.49)

These two steps are solved for the same time step of length ∆t A point need to

be noted that compared to the non-rotating BEC, the time step has not beenmuch affected with a small Ω For a very big Ω, a smaller time step is required

to well capture the rotation For step I (2.49), we will discuss in details in thefollowing two subsections Step II (2.50) can be solved analytically We firstdemonstrate that the ODE is linear by showing:

Ω(cos 2Ωt − cos 2Ωtn)

#.(2.54)

We can also apply numerical quadrature method, e.g Simposon rule to imate Rt n+1

φlk(t) is the Fourier coefficient for the lth mode in x and kth mode in y

Differentiate (2.55) with respect to t, and noticing the orthogonality of theFourier functions, we obtain:

φlk(tn+1) We take an Inverse Fourier Transform to get ˆφlk(tn+1)

In practice, we often apply the second order Strang splitting [70,76].

2.4.3 Discretization in 3D

When d = 3, for a defined domain [a, b] × [c, d] × [e, f ], we use a similar approach

as what we have discussed above in (2.55) We take:

Without loss of generality, we have taken d = 2 for numerical computations The3D case is quite similar In this section, we first test the numerical accuracy ofthe method proposed in section 2 Then we proceed to study the dynamics of thequantities discussed above, by choosing a gaussian initial data φ0(x) which is astationary state with its center shifted We will look at the conservation of energyand density, as well as the dynamical laws of angular momentum expectation,condensate width For center of mass, we will compare the numerical solutionswith the exact analytical solutions that we obtained by solving related ODE

We will also discuss the interaction between a few central vortices by looking attheir trajectories

2.5.1 Accuracy test

In this subsection, we will present the numerical results obtained to show aspectral accuracy in space and second order accuracy in time To this end, wetake the initial data as:

h and time step k

First, we test the spectral accuracy of TSSP2 in space We have three different

β, and for each one, we solve the numerical solution with a very small time step

k = 0.0001 and different mesh sizes h, as shown in Table 2.1 Since the timestep is chosen as small as our ‘exact’ solution, we could neglige the truncationerror resulted from time discretization compared to space discretization

Then we use a similar approach to test the time accuracy, as in Table2.2 In

Table 2.1: Spatial error analysis: Error ||φe(t) − φh,k(t)||l 2 at t = 2.0 with

k = 1E − 4

β = 10 1.114E-2 9.932E-7 9.6613E-13 <E-13

β = 20√

a strong repulsive interaction regime or semi-classical regime, where β ≫ 1, weare interested to find how to choose the “correct” mesh size h and time step △t.After a rescaling of equation (2.13) under normalization, we get:

iǫ∂tφ(x, t) = −ǫ

2

2∇2φ + Vd(x, t)φ + |φ|2φ, x ∈ Rd, (2.58)

Table 2.2: Temporal error analysis: Error ||φe(t) − φh,k(t)||l 2 at t = 2.0 with

h = 1/32

β = 10 1.002E-3 2.582E-4 5.661E-5 1.153E-5 3.528E-6

β = 20√

2 4.132E-3 1.341E-3 3.513E-4 9.382E-5 2.496E-5

β = 80 2.154E-2 5.381E-3 1.463E-3 3.652E-4 9.563E-5

with x → ǫ−1/2x, φ → ǫd/4.φ, ǫ = βd−2/(d+2)

As also demonstrated in [8, 9, 10], the suitable meshing strategy which bestapproximates the “correct” solution should be:

h = O(ǫ), k = O(ǫ)

Thus for a strong repulsive interaction, we take:

h = O(ǫ) = O(1/βd2/(d+2)), k = O(ǫ) = O(1/βd2/(d+2)), with d = 2, 3

2.5.2 Dynamical results in 2D

To verify the analytical solutions obtained in section II like the density andmass conservation, and to study the dynamical laws of a rotating BEC under aLagrangian coordinate, we take a Gaussian initial condition as stated in (2.57).(I) Dynamics of density and energy

As defined in (2.16), the energy is expressed as:

∂sV (x, s)|φ(x, s)|2ds

#dx:= Ek(φ) + Eint(φ) + Ep(φ) + Erot(φ), (2.60)

where Ek(φ) is the kinetic energy, Eint(φ) stands for interaction energy, Ep(φ)

is the potential energy and Erot(φ) is the rotating energy

We have energy and density conservation for any given initial state as discussed

above We take β = 10, T = 10, x0 = y0 = 1 We have listed four cases asfollows with different Ω and γx, γy: As we could see, in all the four cases, andalso the other examples that we have not listed here, the energy and density arewell conserved.

And when Ω = 0 or γx= γy, as shown in Figure 2.1, Figure 2.2and Figure2.3,the rotating energy equals zero This could be explained by our analytical results

in (2.53), (2.54) and (2.16) By comparing Figure 2.1 and Figure2.2 which aredifferent by the value of γy, we could see that the period becomes smaller with

an increasing γy The same follows when we change γx

(II) Dynamics of condensate width and angular momentum

We solve the problem on the domain [−10, 10] × [−10, 10] under a mesh size

h = 1/8 and time step 0.001, with homogeneous Dirichlet boundary conditionand initial condition defined in (2.57) We take β = 100 and T = 10, set differentvalues for Ω, γx, γy and the starting point x0, y0 to show the dynamical laws ofcondensate width and angular momentum

As discussed in Theorem 2.3.2, under the rotating Lagrangian coordinate, wehave the angular momentum conservation law in two cases:

(i) γx= γy, for any initial data and Ω given This is shown in Figure2.5, where

Ω = 1, γx = γy = 1, x0= 1, y0 = 1

(ii) Ω = 0 and the initial data φ0(t) is symmetric in either x or y direction

We could find that in Figure 2.7, where Ω = 0, γx = 1, γy = 2, x0 = 0, y0 = 1,initial data is symmetric in x direction, the angular momentum is conserved Tocompare, Figure2.8 has almost the same quantities except that x0 = 1, y0 = 1,where the initial data does not have any symmetric property, and the angularmomentum is not conserved in this case And in Figure 2.6, where Ω = 1, γx =

1, γy = 2, x0 = 0, y0 = 1, we have a symmetric initial data but with a nonzero

Ω, the angular momentum is not conserved

For condensate width, as we have discussed in Theorem2.3.3, when γx= γy = γr,

δx and δy are periodic with period T = 2πδ

r This is shown in Figure 2.5 Forother Figures, although the condensate width are not periodic, due to γy = 2γx,the oscillation frequency for δy is roughly double that of δx, and the amplitudes