96

CHAPTER 4

NUMERICAL ANALYSIS OF THE GPE WITH

A LINEAR POTENTIAL

4.1 INTRODUCTION

After a successful experimental detection of Bose-Einstein

condensates (BEC) of dilute trapped bosonic alkali-metal atoms Li7, Na

23, and

Rb87

, at ultra-low temperatures, there have been intense theoretical activities in

studying properties of the condensate using the time-dependent mean-field

Gross-Pitaevskii (GP) equation under different trap symmetries.

Among many possibilities, the following traps have been used in

various numerical studies: three-dimensional (3D) spherically-symmetric,

axially-symmetric and anisotropic harmonic traps, two-dimensional (2D)

circularly-symmetric and anisotropic harmonic traps, and one-dimensional

(1D) harmonic trap.

Dalfovo et al (1999), deals in detail about the various aspects of

symmetric traps. According to him, the symmetry of the Bose-Einstein

condensate is fixed by the shape of the confining field. Though there are two

types of symmetry, namely the spherical and axially symmetric traps, the first

experiments on rubidium and sodium were carried out with axial symmetry.

The ratio between the axial and radial frequency fixes the asymmetry of the

trap. If the ratio of the traps less than one, the trap is cigar shaped and if the

ratio of the trap frequencies is greater than one, the trap is disk shaped.

97

The inter-atomic interaction leads to a nonlinear term in the GP

equation, which complicates its accurate numerical solution, especially for a

large nonlinearity. The nonlinearity is large for a fixed harmonic trap when

both the number of atoms in the condensate or the atomic scattering length is

large and this is indeed so under many experimental conditions.

In particular, the experimental realization of BECs in dilute quantum

gases have opened the floodgate in the field of atom optics and condensed

matter physics. At the same time, the collective excitation of matter waves in

BECs has also drawn a great deal of interest to explore the dynamics of BECs

deeply from both experimental and theoretical perspectives, such as matter

wave solitons, periodic waves, shock waves, vortex and necklaces.

According to Parker et al (2004), the fundamental zero-temperature

excitations in a BEC are divided into collective and topological excitations as

discussed by Inguscio et al (1999). Collective excitations are density

perturbations, with two regimes depending on the wavelength of the excitation

relative to the size of the condensate R. For < R, the excitations are sound

waves (phonons), while for >R, the excitations represent large-scale

oscillations of the fluid like breathing and quadrupole modes suggested by

Durfee et al (2000)

Bao and Tang (2003) presented a general method to compute the

ground state solution by directly minimizing the energy functional and used it

to compute the ground state of the GPE in different cases. Edwards and

Burnett (1995) introduced a Runge-Kutta type method and employed it to

solve the spherically symmetric time independent GPE. Adhikari (2000) used

this approach to obtain the ground state solution of the GPE in two dimensions

with radial symmetry. Other approaches include a finite difference method

proposed by Chiofalo et al (2000) and Schneider et al (1999) and a simple

analytical method proposed by Dodd (1996). According to Bao (2003), for the

98

numerical solution of the time-dependent GPE only a few methods are

available, a particle-inspired scheme proposed by Cerimele et al (2000) and a

finite difference method used by Ruprecht et al (1995) and Ensher et al (1996).

Study of the dynamical properties of a BEC has been a very active field

of research since the experimental realization of BECs in dilute atomic gases.

Atomic interactions play a crucial role in the prediction and explanation of a

wide range of observable phenomena including free condensate expansion,

collective excitations, nonlinear atom optics, solitons, vortices and tunneling.

4.2 SPLIT-STEP FOURIER METHOD

The Split-Step Fourier method was first propounded by Ghafouri et

al (1998) and was applied to the analysis of soliton propagation in optical

fibers. It is useful to run through the various steps involved in this method, as

it is the basis on which our Split-Step Crank-Nicolson method is built. The

Split-Step Fourier Method basically consists of following steps:-

1. Solve the nonlinear part in the time domain analytically.

2. Solve the linear part in the frequency by means of Fourier

transform.

3. To apply the SSFM, the nonlinear partial differential equation

should first be replaced by two equations which are integrated

alternately by the same step length x

4. After splitting the nonlinear equation into real and imaginary

parts, analytic solutions can be obtained in the time domain for

the nonlinear part and in the frequency domain for the linear

part.

99

5. An initial condition is necessary to start the calculation and the

fast Fourier transform (FFT), is employed for the

transformation of the time signal to the frequency domain.

6. After being propagated and distance of x in both the linear

and nonlinear parts, results can be obtained, and these results

will be used as the initial condition for another propagation

distance of x .

7. The above processes should be repeated until the required

propagation distance is achieved

The above steps of the algorithm are represented in the form of a

flow chart in Figure (4.1).

Properties of the Split-Step Fourier Method

1. For the SSFM, the computational load of the FFT to transform

sampling points between the time and frequency domains is

very heavy, and this is the reason more computational time is

required.

2. Using SSFM, we may need to employ other numerical methods

to solve the higher order nonlinear terms (e.g. finite difference

methods) if one or both split equations cannot be solved

analytically. Thus the additional numerical methods will

introduce extra errors, more CPU time and greater difficulties

in the control of the accuracy.

100

Figure (4.1): Algorithm of the Split-Step Fourier Method

NLPDE to

be solved

Nonlinear part

solvable in time

domain

Linear part

solvable in

frequency domain

Analytic solution

in time domain

Analytic solution in

frequency domain

Time Domain Frequency Domain

Is the required

propagation

distance achieved?

Final solution after propagating

distance of x

Apply initial condition

Solution after propagating

distance x along fibre

Apply IFFT

Apply initial condition

Apply FFT

Solution after propagating

distance x along fibre

Yes

No

101

4.3 SOLVING GPE WITH LINEAR POTENTIAL USING CNFD

METHOD

According to Adhikari and Muruganandham (2010), there are

several methods for the numerical solution of the time dependent Gross-

Pitaevskii equation. Most of them adopt a time-iteration procedure in

execution, although the details could be different from method to method. The

time dependent GPE is first discretized in space and time and then solved

iteratively with an input solution.

In the spherically symmetric case the GPE is solved by

discretization with the Crank-Nicolson scheme for the time propagation is

complimented by the known boundary condition. However, in the axially

symmetric case a two-step procedure is first used to separate the radial and

axial parts of the Hamiltonian before applying the explicit Crank-Nicolson

scheme for the time propagation in each step.

Properties of the Crank-Nicolson Finite Difference Method:

1. In all these approaches the nonlinear term is treated together

with the explicit Crank-Nicolson scheme. Such a Crank-

Nicolson method may suffer from a possible numerical

amplification of random numerical noise, due to nonlinearity,

leading to spurious input of kinetic energy and limiting the

stability of the algorithm to a short time.

2. Crank-Nicolson finite difference method limits the

applicability only to small values of the nonlinearity.

102

In order to check the truth of the above properties, we try to solve

the GP equation using the Crank Nicholson finite difference method. The

Gross-Pitaevskii equation (GPE) determines the quantum wave function of a

gas of bosons in the limit in which thermal and quantum fluctuations are

negligibly small. Solving the GPE lets us describe a dilute BEC’s ground state

as well as its linear and nonlinear excitations and transport properties at

ultralow temperatures. Mathematically, the time-dependent GPE in three

dimensions takes the form of a cubic Schrödinger equation,

),(),(),()(),(2

),( 20

22

trtrUtrrVtrmt

tri (4.1)

Where 0U is the effective two-particle interaction, which may be expressed in

terms of the scattering length a according to m

aU

2

04

, where m is the mass

of an atom. The confining potential is denoted by V(r). Reducing the three

dimensional GPE to a one dimensional PDE with a linear potential and

rescaling the length and time to and with the transformation

3/12

2mFand

22m

and the wavefunction to

20

8

1

a

where F is the force constant arising due to linearization of

the harmonic oscillator potential, we obtain the dimensionless equation

),(),(),( 22

2

2

txtxpxxt

txi (4.2)

Where 12p for repulsive interactions and 12p for attractive

interactions.

103

In this finite difference method, the GP Equation (4.2) is approximated by

replacing time with the backward difference approximation and space with the

central difference approximation. Here we split the wave function into two

parts of real and imaginary values.

),(),(),( txivtxutx (4.3)

where u(x,t) and v(x,t) are real functions

Substituting Equation (4.3) in (4.2) and equating real and imaginary

parts on either side we get two symplectic equations

uvuxux

u

t

v)(

22

2

2

(4.4)

vvuxvx

v

t

u)(

22

2

2

(4.5)

Writing the initial condition for the two equations

)2cos()2(sec8.2)0,( xxhtxu

)2sin()2(sec8.2)0,( xxhtxv (4.6)

The boundary conditions for the equations are

0),2(),2( tutu

0),2(),2( tvtv (4.7)

To approximate the Equations (4.4) and (4.5) by finite differences,

we divide the closed domain [-2,2] X [0,tF] by a set of lines parallel to x- and

104

t-axes to form a grid or mesh where tF can be as large as we choose. We shall

write x and t for the line spacing. The crossing points

,( xjx j )tntn , Jj ,...2,1,0 and ,...1,0n (4.8)

Jhx

4and kt (4.9)

are called the grid points or mesh points. Approximations of the solution of

these mesh points are sought and are denoted by

),( njjn

txuU (4.10)

Writing the finite difference formulas for Equations (4.4) and (4.5)

using the Crank Nicholson scheme, we have

njnjnjnj

njnjnj

njnjnj

njnj

uvuxu

h

uuu

uuu

k

vv

,2

,2

,,

2

1,11,1,1

,1,,1

,1,

)(

2

2

2

1

(4.11)

njnjnjnj

njnjnj

njnjnj

njnj

vvuxv

h

vvv

vvv

k

uu

,2

,2

,,

2

1,11,1,1

,1,,1

,1,

)(

2

2

2

1

(4.12)

With2h

k rewriting Equations (4.11) and (4.12)

105

njnjnjnjnjnj

njnjnjnj

uukjhkvuvu

uuvu

,1,2

,2

,,,1

1,11,1,1,1

))2)((2(

)22(

(4.13)

njnjnjnjnjnj

njnjnjnj

vvkjhkvuuv

vvuv

,1,2

,2

,,,1

1,11,1,1,1

))2)((2(

)22(

(4.14)

When2

1

2

1

2h

k If 1h then 5.0k . Equations (4.13) and

(4.14) then become

njnjnjnjnjnj

njnjnjnj

njnjnjnjnjnj

njnjnjnj

vvjvuuv

vvuv

uujvuvu

uuvu

,1,2

,2

,,,1

1,11,1,1,1

,1,2

,2

,,,1

1,11,1,1,1

))2)((4(

)24(

))2)((4(

)24(

(4.15)

Equation (4.15) forms a coupled set of algebraic equations which can be

solved using matrix inverse method.

4.4 STABILITY CONSIDERATION

Stability considerations are very important in getting the numerical

solution of a partial differential equation, using finite difference methods. The

solutions of the finite difference equation are stable, if the error does not grow

exponentially as we progress from one time step to another. Matrix method

and Von-Neumann Fourier series expansion methods are the two standard

methods for investigating growth of errors, in solving finite difference

equations. Stability simply means that the finite difference scheme does not

amplify errors. Obviously this is very important, since errors are impossible to

avoid in any numerical calculation. Using the Von Neumann stability analysis

for PDEs, we write the real functions u and v varying in space and time

coordinates as

106

))(exp( xjiUu nnj ))(exp( xjiVv nn

j (4.16)

Substituting Equations (4.16) in (4.15) and simplifying we get two

equations

UeUjVUVUe

UeUVUe

xinxi

xixi

)]2)([4(

)24(

222 (4.17)

VeVjVUUVe

VeVUVe

xinxi

xixi

)]2)([4(

)24(

222 (4.18)

Multiplying Equation (4.17) by V and Equation (4.18) by U and

adding them we get

22

22

UV

UV (4.19)

For stability the condition is 1122

22

UV

UV for all values

of . Thus we see that the CN method is unconditionally stable for all values

of .

4.5 SOLVING GPE WITH LINEAR POTENTIAL USING SSCN

METHOD

The Split-Step Crank-Nicolson method is the new method suggested by

Adhikari and Muruganandham (2010) for the solution of the time-dependent

GPE. In this method, the full Hamiltonian is split into three parts. The

nonlinear as well as the different linear (non-derivative) terms (excluding

spatial derivatives) are treated in one step. In the spherical case the spatial

derivative is treated in another step. In the axially symmetric case the radial

107

and axial derivatives are dealt with in two separate steps. In the treatment of

the non-derivative terms, the numerical error is kept to a minimum. In dealing

with the derivative kinetic energy terms the GPE is solved by discretization

using the Crank-Nicolson scheme for time propagation complimented by the

known boundary conditions.

The GPE can be written in the operator form as

Ht

i (4.20)

Where the Hamiltonian H contains the different nonlinear and linear terms

including the spatial derivatives. In the split step Crank Nicholson Method, the

iteration is done in several steps by breaking up the full Hamiltonian into

different derivative and non-derivative parts. So 321 HHHH where

21 xH (4.21)

2

2

2x

H (4.22)

13 HH (4.23)

The time variable is discretized as ntn where is the time step.

The solution is advanced first over the time step at time nt by solving the

GPE (4.20) with 1HH to produce the first intermediate solution and from

this we generate the second intermediate solution by following semi-implicit

Crank-Nicholson scheme and then obtain the final solution.

The advantage of the above split-step method with small time step is

due to the following

108

1. All iterations conserve normalization of the wave function.

2. The error involved in splitting the Hamiltonian is proportional to two and

can be neglected and the method preserves the symplectic structure of the

Hamiltonian formulation.

3. Finally, as a major part of the Hamiltonian including the nonlinear term is

treated fairly accurately without mixing with the delicate Crank-Nicolson

propagation, the method can deal with an arbitrarily large nonlinear term and

lead to stable and accurate converged result.

4. In the SSCN method the nonlinear and linear non-derivative terms can be

treated very precisely and this improves the accuracy and the stability of the

method compared to the other methods

5. We could easily find accurate solution of the GPE for very large

nonlinearity, which remains stable even after millions of iteration.

6. The SSCN method is advantageous for the study of time evolution problems

during a large interval of time.

A program to perform the various operations was written in matlab.

The algorithm of the SSCN method is a combination of the split step Fourier

method (SSFM) found in the literature and the Crank-Nicolson finite

difference method.

Splitting Equation (4.2) which is the equation to be solved

numerically into a linear part and a nonlinear part, we have the linear part as

),(),(

2

2

txxxt

txi (4.24)

109

And the nonlinear part as

),(),(),( 22 txtxp

t

txi (4.25)

Solving the linear part using Fourier Transform in the frequency

domain we have a solution for Equation (4.24) in the form

),(exp),(2

2

txxx

tittx (4.26)

Using the expansion for the exponential term as:-

0 !n

n

ne (4.27)

Rewriting Equation (4.26) using Equation (4.27) we have

),(2

exp2!

1

2exp),(

2

2

txtxi

x

ti

n

txittx

n

(4.28)

We should deal with an object of the form

02

2 )(

!nn

nn

x

xf

n

a (4.29)

where

2

tia and ),(

2exp)( tx

txixf (4.30)

Applying Fourier Integral theorem and integrating by parts we have

110

),(2

exp2

exp2

exp),(2

1 txtxi

Ftki

Ftxi

ttx (4.31)

The codes in matlab were developed after slight modification of the

already available codes for the split step Fourier method. The second order

space derivative was replaced by the finite difference formula of Crank-

Nicolson. We found that the program was much more efficient as the time

taken for obtaining a Fourier transform and then once again the inverse Fourier

transform for the SSFM was reduced in the SSCN method.

4.6 RESULTS AND DISCUSSION

Our CNFD scheme is used to solve Equations (4.4) and (4.5). The

resolution used is (h,k) = (1,0.5). After analyzing the analytical solution of

Equation (4.2) provided by Khawaja (2007) using Darboux transformation, we

take the exact solution to be

)]23

(2cos[)4(sec8.23

22 tt

xttxttxhu (4.32)

)]23

(2sin[)4(sec8.23

22 tt

xttxttxhv (4.33)

In Table (4.1) and Table (4.2) we list the numerical results by the

exact results as well as the absolute error at 5 evenly distributed grid points

across [0,5] at t =1 for the values of u and v respectively. Note that the

truncation error is of the order 5.01

5.0

22h

k. The absolute error in our CNFD

method appears to be very high. The reason is because of the splitting of the

wave function into two different u and v parts. The one soliton and two

soliton solution plots by the SSCN method is given in Figure (4.2) and Figure

(4.3).

111

Table 4.1 Solutions to Equation (4.4) at t =1

x u (CNFD) Exact solution Absolute error

-2 0.0000 0.0000 0.0000

-1 0.6223 0.1024 0.5199

0 3.7025 0.0375 3.6650

1 0.4840 0.0136 0.4704

2 0.0000 0.0000 0.0000

Table 4.2 Solutions to Equation (4.5) at t =1

x v (CNFD) Exact solution Absolute error

-2 0.0000 0.0000 0.0000

-1 1.2055 0.0048 1.2007

0 2.8852 0.0044 2.8852

1 0.9548 0.0016 0.9532

2 0.0000 0.0000 0.0000

112

-10

-5

0

5

10

0

0.005

0.010

0.5

1

1.5

2

2.5

3

distance

Numerically solving GPE of a BEC with linear potential

time

inte

ns

ity

Figure 4.2 Matlab plot of one soliton solution using SSCN method

-10

-5

0

5

10

0

0.005

0.010

0.5

1

1.5

2

2.5

3

distance

Numerically solving GPE of a BEC with linear potential

time

inte

nsity

Figure 4.3 Matlab plot of two soliton solution using SSCN method

113

4.7 CONCLUSION – COMPARISON BETWEEN THE TWO

NUMERICAL METHODS

The CNFD method was easy to perform manually, theoretically

stable but in practice produced huge errors due to the splitting of the wave

function, which was done to handle the imaginery value i in the GPE. Writing

matlab codes for the CNFD method was difficult so automation and better

resolution could not be achieved. On the other hand, the SSCN method was

completely automated, so much faster and much more stable. The SSCN

method was similar to the split step Fourier method (SSFM) but different in

the fact that there was no change to the Fourier space and back to co-ordinate

space. So the SSCN method was much more efficient than the SSFM method.

The SSCN method is thus the best numerical method for solving the given

nonlinear partial differential equation as it took features from the SSFM

method and prevented the disadvantage of the SSFM by combining an

efficient finite difference method like Crank-Nicolson method.