mathematical analysis and numerical simulation for bose-einstein condensation weizhu bao department...

Mathematical Analysis and Numerical Simulation for Bose-Einstein Condensation

Weizhu Bao

Department of Mathematics& Center of Computational Science and Engineering

National University of SingaporeEmail: [email protected]

URL: http://www.math.nus.edu.sg/~bao

Collaborators

External – P.A. Markowich, Institute of Mathematics, University of Vienna, Austria – D. Jaksch, Department of Physics, Oxford University, UK– Q. Du, Department of Mathematics, Penn State University, USA– J. Shen, Department of Mathematics, Purdue University, USA– L. Pareschi, Department of Mathematics, University of Ferarra, Italy– I-Liang Chern, Department of Mathematics, National Taiwan University, Taiwan– C. Schmeiser & R.M. Weishaeupl, University of Vienna, Austria – W. Tang & L. Fu, Beijing Institute of Appl. Phys. & Comput. Math., China

Internal– Yanzhi Zhang, Hanquan Wang, Fong Ying Lim, Ming Huang Chai– Yunyi Ge, Fangfang Sun, etc.

Outline

Part I: Predication & Mathematical modeling– Theoretical predication– Physical experiments and results– Applications– Gross-Pitaevskii equation

Part II: Analysis & Computation for Ground states – Existence & uniqueness – Energy asymptotics & asymptotic approximation– Numerical methods– Numerical results

Outline

Part III: Analysis & Computation for Dynamics in BEC– Dynamical laws– Numerical methods– Vortex stability & interaction

Part IV: Rotating BEC & multi-component BEC– BEC in a rotational frame– Two-component BEC– Spinor BEC– BEC at finite temperature– Conclusions & Future challenges

Part I

Predication &

Mathematical modeling

Theoretical predication

Particles be divided into two big classes– Bosons: photons, phonons, etc

• Integer spin• Like in same state & many can occupy one obit• Sociable & gregarious

– Fermions: electrons, neutrons, protons etc• Half-integer spin & each occupies a single obit• Loners due to Pauli exclusion principle


For atoms, e.g. bosons– Get colder:

• Behave more like waves & less like particles– Very cold:

• Overlap with their neighbors

– Extremely cold: • Most atoms behavior in the same way, i.e gregarious • quantum mechanical ground state, • `super-atom’ & new matter of wave & fifth state


S.N. Bose: Z. Phys. 26 (1924) – Study black body radiation: object very hot– Two photons be counted up as either identical or different – Bose statistics or Bose-Einstein statistics

A. Einstein: Sitz. Ber. Kgl. Preuss. Adad. Wiss. 22 (1924) – Apply the rules to atoms in cold temperatures– Obtain Bose-Einstein distribution in a gas

1

1)(

/ Tki Bie

f

Experimental results

JILA (95’, Rb, 5,000): Science 269 (1995)

–Anderson et al., Science, 269 (1995),

198: JILA Group; Rb

–Davis et al., Phys. Rev. Lett., 75 (1995),

3969: MIT Group; Rb

–Bradly et al., Phys. Rev. Lett., 75 (1995),

1687, Rice Group; Li

Experimental results

Experimental implementation– JILA (95’): First experimental realization of BEC in a gas – NIST (98’): Improved experiments– MIT, ENS, Rice, – ETH, Oxford, – Peking U., …

2001 Nobel prize in physics:– C. Wiemann: U. Colorado– E. Cornell: NIST– W. Ketterle: MIT ETH (02’, Rb, 300,000)

Experimental difficulties

Low temperatures absolutely zero (nK)Low density in a gas

Experimental techniques

Laser coolingMagnetic trappingEvaporative Cooling

($100k—300k)

Possible applications

Quantized vortex for studying superfluidity

Test quantum mechanics theoryBright atom laser: multi-componentQuantum computingAtom tunneling in optical lattice trapping, …..

Square Vortex lattices in spinor BECs

Giant vortices

Vortex latticedynamics

Mathematical modeling

N-body problem– (3N+1)-dim linear Schroedinger equation

Mean field theory: – Gross-Pitaevskii equation (GPE): – (3+1)-dim nonlinear Schroedinger equation (NLSE)

Quantum kinetic theory – High temperature: QBME (3+3+1)-dim – Around critical temperature: QBME+GPE– Below critical temperature: GPE

(nK)cT T O

Gross-Pitaevskii equation (GPE)

Physical assumptions– At zero temperature– N atoms at the same hyperfine species (Hartree ansatz)

– The density of the trapped gas is small

– Interatomic interaction is two-body elastic and in Fermi form

t),ψ(t),,,,(N

iiNN xxxxΨ

1

21

.1,1|| ||

3 aa

sas

Second Quantization model

The second quantized Hamiltonian:– A gas of bosons are condensed into the same single-particle state – Interacting by binary collisions – Contained by an external trapping potential

† † †0 int

1( ) ( ') ( ') ' ( ') ( ) ( ', ) ( ') ( ) '

2H x H x dx x x V x x x x dx dx

2

0 ext

†

( ) : ( , ) : Bose field operator

( ') : single particle Hamiltonian2

( , , ) : moentum operator

( ) & ( ) : creation & annihilation of a particle at position

Tx y z

x x t

PH V x

m

P i p p p

x x x

Second quantization model

– Crucial Bose commutation rules:

– Atomic interactions are low-energy two-body s-wave collisions, i.e. essentially elastic & hard-sphere collisions

– The second quantized Hamiltonian

† † †( '), ( ) ( ' ), ( '), ( ) ( '), ( ) 0x x x x x x x x

2int 0 0( ', ) ( ') with 4 /sV x x U x x U a m

† † †00( ) ( ') ( ') ' ( ') ( ') ( ') ( ') '

2

UH x H x dx x x x x dx

Second quantization model

The Heisenberg equation for motion:

For a single-particle state with macroscopic occupation

– Plugging, taking only the leading order term – neglecting the fluctuation terms (i.e., thermal and quantum depletion of the

condensate) – Valid only when the condensate is weakly-interacting & low tempertures

( ) ( , ) ( , )

( , ) : marcoscopic wave function (= ( ) / : expectation value of ( ))

( , ) : fluctation operator satisfies ( , ) 0

x N x t x t

x t x N x

x t x t

†0 0( ) ( ), ( ) : ( ) ( ) ( )i x x H H U x x x

t

Gross-Pitaevskii equation

The Schrodinger equation (Gross, Nuovo. Cimento., 61; Pitaevskii, JETP,61 )

– The Hamiltonian:

– The interaction potential is taken as in Fermi form

22

( ) *( , ) [ ( , )] ( , )2

1*( , ) *( , ) ( ) ( , ) ( , )

2

H x t V x t x t dxm

x t x t x x x t x t dx dx

( ), ( , , )

*( , )

ψ(x, t) Hi x x y z

t x t

.4),()1()(2

00 mxNx aUU s


The 3d Gross-Pitaevskii equation ( )

– V is a harmonic trap potential

– Normalization condition

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

),( 20

2

2

txtxUNtxxVtxm

txt

i

)(2

)( 222222zyx

mxV

zyx

.1|),(| 2

R3 xdtx

),,( zyxx


Scaling (w.l.o.g. )– Dimensionless variables

– Dimensionless Gross-Pitaevskii equation

– With

3/ 2

0 00

, , ( , ) ( , ),x

x

xt t x x t x t

a ma a

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

1),( 22 txtxtxxVtxtx

ti

0

22222 4,,),(

2

1)(

a

aNzyxxV s

x

z

zx

y

yzy

x y z


Typical parameters ( )– Used in JILA

– Used in MIT

]Js[1005.1 34

Rb87

NN

m

m

aaaa s

xs

xzyx

01881.04

],m[103407.0],nm[1.5

8],s/1[210],kg[1044.1

0

5

0

25

Na23

NN

m

m

aaaa s

zs

zyx

003083.04

],m[101209.1],nm[75.2

s]/1[25.3],s/1[2360],kg[108.3

0

5

0

26


Reduction to 2d (disk-shaped condensation)– Experimental setup– Assumption: No excitations along z-axis due to large energy

2d Gross-Pitaevskii equation ( )

1,1, zyxzyx

2

1

12 3

1/ 4/ 22 1/ 2

3 hoR

( , , , ) ( , , ) ( ) with

( ) ( | ( , , ) | ) ( ) z zzg

x y z t x y t z

z x y z dxdy z e

2 2 22

2

4 42 3 ho 2

1( , ) | | ,

2 2

( ) ( ) :2

y

az

x yi x t

t

z dz z dz

12),,( yxx

Numerical Verification

Numerical Results

Bao, Y. Ge, P. Markowich & R. Weishaupl, 06’


General form of GPE ( )

with

Normalization condition

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

1),( 22 txtxtxxVtxtx

ti dd

dx R

3

2

1

),(2

1

),(2

1

,2

1

)(

,

,2

)(

,2

),(

22222

222

2

4

3

R

4

232

d

d

d

zyx

yx

x

xVdzz

dydzzy

zy

ydz

zy

d

.1|),(| 2

R xdtx

d


Two kinds of interaction– Repulsive (defocusing) interaction

– Attractive (focusing) interaction

Four typical interaction regimes:– Linear regime: one atom in the condensation

– Weakly interacting condensation1|| d

00 dsa

00 dsa

0d


– Strongly repulsive interacting condensation

– Strongly attractive interaction in 1D

Other potentials– Box potential– Double-well potential– Optical lattice potential– On a ring or torus

1d

1 10 & | | 1


Conserved quantities– Normalization of the wave function

– Energy

Chemical potential

d

NxdtxtNR

2 1))0((|),(|))((

2 2 4

R

1( ( )) [ | ( , ) | ( )| ( , ) | | ( , ) | ]

2 2( (0))

d

ddt x t V x x t x t dxE

E

xdtxtxxVtxt ddd

]|),(||),(|)(|),(|

2

1[))(( 422

R

Semiclassical scaling

When , re-scaling

With

Leading asymptotics (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci., 05’)

1d 1/ 2 / 4 2 /( 2)1/d ddx x

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

),( 222

txtxtxxVtxtxt

i d

)1(]||2

1||)(||

2[)( 422

R

2

OxdxVE dd

1 1 2/( 2)

1 1 2/( 2)

( ) ( )

( ) ( )

dd

dd

E E O O

O O

Comparison of two scaling

1/ 2 1/( 2) 1/( 2)0 0 0 0

0

3/ 2 3/ 2 / 40 0

21

2

2/( 2)

2 /( 2)

Quanties Thomas-Fermi scaling Semiclassical scaling

1/ 1/

4/ ( )

:

Energy (1)

Chemical potential (1

s x x

d dss x d

ds

ss x x

s

dd

dd

t

Nax a m a a a

a

a a

mxE

t

E O O

O O

1/( 2)

/ 2( 2)

)

length of wave function ( 2 ) (1)

height of wave function ( / ) (1)

dd

d dd d

O O O

O O O

Quantum Hydrodynamics

Set

Geometrical Optics: (Transport + Hamilton-Jacobi)

Quantum Hydrodynamics (QHD): (Euler +3rd dispersion)

)2/(2/ /1,,, dd

iS vJSve

22

t

( ) 0,

1 1( )

2 2

t

d

S

S S V x

2

2

( ) 0

( ) ( ) ( ) ( ln )4

( ) / 2

t

t d

v

J JJ P V

P

Part II

Analysis & Computation

for

Ground states

Stationary states

Stationary solutions of GPE

Nonlinear eigenvalue problem with a constraint

Relation between eigenvalue and eigenfunction

)(),( xtx eti

1|)x(|

R),(|)(|)()()(2

1)(

2

R

22

xd

xxxxxVxx

d

ddd

xdEd

d 4

R|)x(|

2)()(

Stationary states

Equivalent statements:– Critical points of over the unit sphere – Eigenfunctions of the nonlinear eigenvalue problem– Steady states of the normalized gradient flow:(Bao & Q. Du, SIAM J. Sci. Compu., 03’)

Minimizer/saddle points over the unit sphere : – For linear case (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci., 05’)

• Global minimizer vs saddle points– For nonlinear case

• Global minimizer, local minimizer (?) vs saddle points

( )E

2 22

0 0

( )1( , ) [ ( ) | | ] ,

2 || ||

( ,0) ( ) with || ( ) || 1.

t x t V x

x x x

0d

0d

2| 1, ( )L

S E

Ground state

Ground state:

Existence and uniqueness of positive solution :– Lieb et. al., Phys. Rev. A, 00’

Uniqueness up to a unit factor

Boundary layer width & matched asymptotic expansion– Bao, F. Lim & Y. Zhang, Bull. Institute of Math., Acad. Scinica , 07’

4gR|| || 1

( ) min ( ), ( ) ( ) | ( ) |2 d

dg g g gE E E x dx

0d

00with any constant i

g g e

Excited & central vortex states

Excited states:Central vortex states:

Central vortex line states in 3D:Open question: (Bao & W. Tang, JCP, 03’; Bao, F. Lim & Y. Zhang, TTSP, 06’)

,,, 321

???????)()()(

)()()(

,,,

21

21

21

g

g

g

EEE

immm eryxtyx m

ti

m

ti

ee )(),(),,(

2 22

22

2

0

( )1( ) ( ) | | , 0 r

2 2 2

2 ( ) 1, (0) 0

mm m m m m

m m

d rd m rr r r

r dr dr r

r r dr

Approximate ground states

Three interacting regimes– No interaction, i.e. linear case– Weakly interacting regime– Strongly repulsive interacting regime

Three different potential– Box potential– Harmonic oscillator potential– BEC on a ring or torus

Energies revisited

Total energy:

– Kinetic energy:– Potential energy:– Interaction energy:

Chemical potential

2 2 4

kin pot intR

1( ) [ | ( ) | ( )| ( ) | | ( ) | ] : ( ) ( ) ( )

2 2d

ddx V x x x dxE E E E

2

kin R

1( ) | ( ) |

2 dx dxE

2

pot R( ) ( )| ( ) |

d dV x x dxE

4

int R( ) | ( ) |

2 d

d x dxE

2 2 4

R

int kin pot int

1( ) [ | ( ) | ( )| ( ) | | ( ) | ]

2 = ( ) ( ) ( ) ( ) 2 ( )

d d dx V x x x dx

E E E E E

Box Potential in 1D

The potential:The nonlinear eigenvalue problem

Case I: no interaction, i.e. – A complete set of orthonormal eigenfunctions

0, 0 1,( )

, otherwise.

xV x

2

12

0

1( ) ( ) | ( ) | ( ), 0 1,

2

(0) (1) 0 with | ( ) | 1

x x x x x

x dx

0

2 21( ) 2 sin( ), , 1, 2,3,

2l lx l x l l

Box Potential in 1D

– Ground state & its energy:

– j-th-excited state & its energy

Case II: weakly interacting regime, i.e.– Ground state & its energy:


20 0 0

0 0( ) ( ) 2 sin( ), : ( ) : ( )2g g g g g gx x x E E

2 20 0 0

0 0

( 1)( ) ( ) 2 sin(( 1) ), : ( ) : ( )

2j j j j j j

jx x j x E E

| | (1)o

2 20 0 03

( ) ( ) 2 sin( ), : ( ) ( ) , : ( ) ( ) 32 2 2g g g g g g g gx x x E E E

2 20 0

2 20

( 1) 3( ) ( ) 2 sin(( 1) ), : ( ) ( ) ,

2 2

( 1): ( ) ( ) 3

2

j j j j j

j j j

jx x j x E E E

j

Box Potential in 1D

Case III: Strongly interacting regime, i.e.– Thomas-Fermi approximation, i.e. drop the diffusion term

• Boundary condition is NOT satisfied, i.e. • Boundary layer near the boundary

1

TFTF TF TF 2 TF TF

1TF 2

0

TF TF TFg g g

( ) | ( ) | ( ), 0 1, ( )

| ( ) | 1

(x) ( ) 1, E E , , 2

gg g g g g

g

g g g

x x x x x

x dx

x

TF TF(0) (1) 1 0g g

Box Potential in 1D

– Matched asymptotic approximation• Consider near x=0, rescale• We get

• The inner solution

• Matched asymptotic approximation for ground state

1, ( ) ( )g

g

x X x x

31( ) ( ) ( ), 0 ; (0) 0, lim ( ) 1

2 XX X X X X

( ) tanh( ), 0 ( ) tanh( ), 0 (1)gg gX X X x x x o

MAMA MA MA MA

1MA 2 MA TF

0

( ) ( ) tanh( ) tanh( (1 )) tanh( ) , 0 1

1 | ( ) | 2 1 2 2 1 2, 1.

gg g g g g

g g g g

x x x x x

x dx

Box Potential in 1D

• Approximate energy

• Asymptotic ratios:

• Width of the boundary layer:

MA MAint, int,

MAkin, kin,

4 21 2, 1,

2 3 2 32

1 23

g g g g

g g

E E E E

E E

(1/ )O

int, kin,1lim , lim 1, lim 0,

2g g g

g g g

E E E

E E

Numerical observations:

2

3 / 2 3 / 2 3 / 2MA MA MA

MA MA MAkin, kin, int, int,

( ), ( ), ( )

(1/ ), (1/ ), (1/ )

g g g g g gL L

g g g g g g

O e O e O e

E E O E E O E E O

Box Potential in 1D

• Matched asymptotic approximation for excited states

• Approximate chemical potential & energy

MA [( 1) / 2]MA MA

0

[ / 2]MA MA

0

2( ) ( ) [ tanh( ( ))

1

2 1tanh( ( )) tanh( )]

1

jj

j j gl

j

g j gl

lx x x

j

lx C

j

MA 2 2

MA 2

MA 2int, int,

MA 2 2kin, kin,

2( 1) ( 1) 2( 1) ,

4( 1) ( 1) ,

2 32

( 1) ( 1) ,2 32

( 1) ( 1) 2( 1)3

j j

j j

j j

j j

j j j

E E j j

E E j j

E E j j j

Fifth excited states

Energy & Chemical potential

Box potential in 1D

• Boundary layers & interior layers with width

• Observations: energy & chemical potential are in the same order

• Asymptotic ratios:

• Extension to high dimensions

1 2 1 2( ) ( ) ( ) ( ) ( ) ( )g gE E E

int, kin,1lim , lim 1, lim 0,

2

lim 1, lim 1, lim 0,

j j j

j j j

j j j

g g g

E E E

E E

E E

E

(1/ )O

Harmonic Oscillator Potential in 1D


Case I: no interaction, i.e. – A complete set of orthonormal eigenfunctions

2

( )2

xV x

2 21( ) ( ) ( ) ( ) | ( ) | ( ), with | ( ) | 1

2x x V x x x x x dx

0

2

2

2

1/ 2 / 21/ 4

20 1 2

1 1( ) (2 !) ( ), , 0,1,2,3,

2

( ) ( 1) : Hermite polynomials with

( ) 1, ( ) 2 , ( ) 4 2,

l xl l l

l xl x

l l

lx l e H x l

d eH x e

dx

H x H x x H x x




Case II: weakly interacting regime, i.e.– Ground state & its energy:


20 / 2 0 00 01/ 4

1 1( ) ( ) , : ( ) : ( )

2x

g g g g g gx x e E E

20 1/ 2 / 2 0 00 01/ 4

1 ( 1)( ) ( ) (2 !) ( ), : ( ) : ( )

2j x

j j j j j j j

jx x j e H x E E

| | (1)o

20 / 2 0 00 01/ 4

1 1 1( ) ( ) , : ( ) ( ) , : ( ) ( )

2 2 2x

g g g g g g g gx x e E E E C C

0 0

0 0 4j j

-

( 1)( ) ( ), : ( ) ( ) ,

2 2

( 1): ( ) ( ) with C = | ( ) |

2

j j j j j j

j j j j

jx x E E E C

jC x dx


Case III: Strongly interacting regime, i.e.– Thomas-Fermi approximation, i.e. drop the diffusion term

– Characteristic length:– It is NOT differentiable at– The energy is infinite by direct definition:

1

TF 2 TFTF TF TF TF 2 TF TF

TF 3/ 2TF 2 TF 2/3

-

( / 2) / , | | 2( ) ( ) ( ) | ( ) | ( ) ( )

0, otherwise

2(2 ) 1 3 1 | ( ) | ( )

3 2 2

g gg g g g g g

gg g g

x xx V x x x x x

x dx

1/3( )O TF2 gx

TF TFkin( ) , ( )g gE E


– A new way to define the energy

– Asymptotic ratios

TF TF 2/3int, int, int

TF TF 2/3pot, pot, pot

TF TF 2/3 TFint, int,

TF TF TFkin, int, pot,

1 3( ) ( ) ,

5 21 3

( ) ( )10 2

3 3( ) : ,

10 2

0

g g g

g g g

g g g g g g

g g g g

E E E

E E E

E E E E

E E E E

int, pot , kin,3 2 1lim , lim , lim , lim 0,

5 3 3g g g g

g g g g

E E E E

E E E

Numerical observations:

2

TF MA MA2/5 2/5 2 /3

MA MA MAkin, kin, int, int,2 /3 2 /3 2 /3

ln ln ln( ), ( ), ( )

ln ln ln( ), ( ), ( )

g g g g g gL L

g g g g g g

O O O

E E O E E O E E O


– Thomas-Fermi approximation for first excited state

• Jump at x=0!• Interior layer at x=0

TF TF TF TF 2 TF1 1 1 1 1

TF 2 TFTF 1 1

1

TF 3/ 2TF 2 TF 2/31

1 1 1

-

( ) ( ) ( ) | ( ) | ( )

sign( ) ( / 2) / , 0 | | 2( )

0, otherwise

2(2 ) 1 3 1 | ( ) | ( )

3 2 2

x V x x x x

x x xx

x dx


– Matched asymptotic approximation

– Width of interior layer:

– Ordering:

MA MA 2 MAMA MA1 1 1

1 1

sign( ) ( / 2) / , 0 | | 2 ( ) [tanh( ) sign( )]

0, otherwise

x x xx x x

MA 1/3 MA 2/31 1(1/ ) (1/ ) ( )O O O

1 1( ) ( ) ( ) ( )g gE E

Harmonic Oscillator Potential

Extension to high dimensionsIdentity of energies for stationary states in d-dim.

– Scaling transformation

– Energy variation vanishes at first order in

kin pot int2 2 0E E d E

/ 20 0( ) (1 ) ((1 ) ) with ( ): a stationary statedx x x

2 2kin 0 pot 0 int 0

0

( ( )) (1 ) ( ) (1 ) ( ) (1 ) ( )

( ( )) | 0

dE x E E E

dE x

d

BEC on a ring


For linear case, i.e.– A complete set of orthonormal eigenfunctions

( ) 0 on an interalV x

2

22

0

1( ) ( ) | ( ) | ( ), 0 2 ,

2

( 2 ) ( ), with | ( ) | 1d

0

0 2 2 1

2

0 2 2 1

1 1 1( ) , ( ) cos( ), ( ) sin( );

2

0, , 1, 2,3,2

l l

l l

l l

ll

BEC on a ring



Some properties– Ground state & its energy – With a shift:

– Interior layer can be happened at any point in excited states

0 0 00 0

1( ) ( ) , : ( ) 0 : ( )

2g g g g g gx x E E

20 0 0

0 0

1( ) ( ) cos( ), : ( ) : ( )

2j j j j j j

jx x l E E

| | (1)o

0 0 01( ) ( ) , : ( ) , : ( )

8 42g g g g g gx x E E

0( ) is a solution ( ) is also a solution

Numerical methods for ground states

Runge-Kutta method: (M. Edwards and K. Burnett, Phys. Rev. A, 95’)

Analytical expansion: (R. Dodd, J. Res. Natl. Inst. Stan., 96’)

Explicit imaginary time method: (S. Succi, M.P. Tosi et. al., PRE, 00’)

Minimizing by FEM: (Bao & W. Tang, JCP, 02’)

Normalized gradient flow: (Bao & Q. Du, SIAM Sci. Comput., 03’)

– Backward-Euler + finite difference (BEFD)– Time-splitting spectral method (TSSP)

Gauss-Seidel iteration method: (W.W. Lin et al., JCP, 05’) Spectral method + stabilization: (Bao, I. Chern & F. Lim, JCP, 06’)

( )E

Imaginary time method

Idea: Steepest decent method + Projection

Physical institutive in linear case– Solution of GPE: – Imaginary time dynamics:

.1||)(|| with )x()0,(

,2,1,0,||),(||

),(),(

,||)(2

1)(

2

1),(

00

1

11

122

xx

nx

xtx

tttxVE

tx

ttn

nn

nnt

0

1

2 1̂

??)()(

)()ˆ(

)()ˆ(

01

11

01

EE

EE

EE

g

00 0

( , ) ( ) with ( ,0) ( ) ( )ji t

j j j jj j

x t a e x x x a x

i t 0 1

0 00 0

( , )( , ) ( , ) ( ) ( ) : grond state j

j jj

xx x t a e x x

a e

Mathematical justification

For gradient flow (Bao & Q. Du, SIAM Sci. Comput., 03’)

For linear case: (Bao & Q. Du, SIAM Sci. Comput., 03’)

For nonlinear case: ???))0(.,())(.,())(.,( 0010 EtEtE nn

Mathematical justification

Normalized gradient glow

Idea: (Bao & Q. Du, SIAM Sci. Comput., 03’)

– The projection step is equivalent to solve an ODE

– Gradient flow with discontinuous coefficients:

– Letting time step go to 0

– Mass conservation & Energy diminishing

2 22

0 0

( (., ))1( , ) ( ) | | , 0,

2 || (., ) ||

( ,0) ( ) with || ( ) || 1.

t

tx t V x t

t

x x x

0,0))(.,(,1||||||)(.,|| 0 ttEtd

dt

2

1 1 1

1( , ) ( , ) ( , ), with ( , ) ln ( , ) & ( , ) ( , )

2t n n n n n nn

x t t t x t t t t t k x t x t x tt

2 21( , ) ( ) | | ( , ) , 0,

2t nx t V x t t t

Fully discretization

Consider in 1D:

Different Numerical Discretizations– Physics literatures: Crank-Nicolson FD or Forward Euler FD – BEFD: Energy diminishing & monotone (Bao & Q. Du, SIAM Sci. Comput., 03’)

– TSSP: Spectral accurate with splitting error (Bao & Q. Du, SIAM Sci. Comput., 03’)

– BESP: Spectral accuracy in space & stable (Bao, I. Chern & F. Lim, JCP, 06’)

– Crank-Ncolson FD for normalized gradient flow

21

11 0 0

1

1( , ) ( ) | | , ( , ), , ( , ) ( , ) 0

2

( , )( , ) , ( ,0) (x) with || ( ) || 1.

|| ( , ) ||

t xx n n

nn

n

x t V x x a b t t t a t b t

xx t x x

xtt

Backward Euler Finite Difference

Mesh and time steps:

BEFD discretization

2nd order in space; unconditional stable; at each step, only a linear system with sparse matrix to be solved!

; 0;b a

h x k tM

j n , 0,1, , ; , k=0,1,2, ; (x ,t ) nj n jx a j h j M t n k

Backward Euler Spectral method

Discretization

Spectral order in space; efficient & accurate

Ground states

Numerical results (Bao&W. Tang, JCP, 03’; Bao, F. Lim & Y. Zhang, TTSP, 06’)

– In 1d• Box potential:

– Ground state; excited states: first fifth

• Harmonic oscillator potential: – ground & first excited & Energy and chemical potential

• Double well potential :– Ground & first excited state

• Optical lattice potential:– Ground & first excited state with potential next

otherwise;100)( xxV

2/xV(x) 2

2/)4()( 22xxV

)4(sin122/)( 22 xxxV

1)(

)(lim1

)(

)(lim

fixedany for ,)()()(,)()()(

nsObservatio

419.32691.31971.30259.30383.20629.19891.18171.1871.313

252.21512.20784.19070.19719.13944.12191.12464.11855.156

8349.80432.82802.75527.64043.65573.57438.49810.3371.31

1919.42590.33578.25266.18505.38865.29414.10441.11371.3

500.35000.25000.15000.0500.35000.25000.15000.00

)()()()()()()()(

11

12121

3213211

g

j

g

j

gg

gg

E

E

EEE

EEEE

back

Ground states

Numerical results (Bao&W. Tang, JCP, 03’; Bao, F. Lim & Y. Zhang, BIM, 07’)

– In 2d• Harmonic oscillator potentials:

– ground

• Optical lattice potential:– Ground & excited states

– In 3D • Optical lattice potential: ground excited states next

Part III

Analysis & Computation

for

Dynamics in BEC

Dynamics of BEC

Time-dependent Gross-Pitaevskii equation

Dynamical laws – Time reversible & time transverse invariant– Mass & energy conservation– Angular momentum expectation– Condensate width– Dynamics of a stationary state with its center shifted

)()0,(

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

1),(

0

22

xx

txtxtxxVtxtxt

i dd

Angular momentum expectation

Definition:

Lemma Dynamical laws (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci, 05’)

For any initial data, with symmetric trap, i.e. , we have

Numerical test next

0,)(**:)( txdxyixdLtLdd R

yx

R

zz

0,|),(|)()( 222 txdtxxy

dt

tLd

dR

yxz

yx

,0 ,0 0( ) (0), ( ) ( ), 0.z zL t L E E t

back


Energy

Dynamics of condensate width

Definition:

Dynamic laws (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci, 05’)

– When for any initial data: – When with initial data Numerical Test– For any other cases:

xdtxtxdtxyxtdd RR

r

22222 |,(|)(,|,(|)()(

22

02

( )4 ( ) 4 ( ), 0r

x r

d tE t t

dt

yxd &2

yxd &2 imerfyx )(),(0

0),(2

1)()( tttt ryx

22

02

( )4 ( ) 4 ( ) ( ), 0

d tE t f t t

dt

next

back

Symmetric trap Anisotropic trap

Dynamics of Stationary state with a shift

Choose initial data as: The analytical solutions is: (Garcia-Ripoll el al., Phys. Rev. E, 01’)

– In 2D:

– In 3D, another ODE is added

)()( 00 xxx s

( , )0( , ) ( ( )) , ( , ) 0, (0)si t iw x t

sx t e x x t e w x t x x

2

2

0 0

( ) ( ) 0,

( ) ( ) 0,

(0) , (0) , (0) 0, (0) 0

x

y

x t x t

y t y t

x x y y x y

20( ) ( ) 0, (0) , (0) 0zz t z t z z z

Solution of the center of mass

Center of mass: Bao & Y. Zhang, Appl. Numer. Math., 2006

In a non-rotating BEC:

– Trajectory of the center Motion of the solution– Pattern Classification:

• Each component of the center is a periodic function • In a symmetric trap, the trajectory is a straight segment• If is a rational #, the center moves periodically with period • If is an irrational #, the center moves chaotically, envelope is a rectangle

2 2( ) : | ( , ) | | ( ( )) | ( )d d

sx t x x t dx x x x t dx x t

0 0( ) cos( ), ( ) cos( ), 0x yx t x t y t y t t

/y x /y x

2p

next

Numerical methods for dynamics

Lattice Boltzmann Method (Succi, Phys. Rev. E, 96’; Int. J. Mod. Phys., 98’)

Explicit FDM (Edwards & Burnett et al., Phys. Rev. Lett., 96’)

Particle-inspired scheme (Succi et al., Comput. Phys. Comm., 00’) Leap-frog FDM (Succi & Tosi et al., Phys. Rev. E, 00’)

Crank-Nicolson FDM (Adhikari, Phys. Rev. E 00’) Time-splitting spectral method (Bao, Jaksch&Markowich, JCP, 03’)

Runge-Kutta spectral method (Adhikari et al., J. Phys. B, 03’)

Symplectic FDM (M. Qin et al., Comput. Phys. Comm., 04’)

Time-splitting spectral method (TSSP)

Time-splitting:

For non-rotating BEC – Trigonometric functions (Bao, D. Jaksck & P. Markowich, J. Comput. Phys., 03’)

– Laguerre-Hermite functions (Bao & J. Shen, SIAM Sci. Comp., 05’)

2

2

2

( ( ) | ( , )| )1

1 Step 1: ( , ) ,

2

Step 2: ( , ) ( ) ( , ) | ( , ) | ( , )

| ( , ) | | ( , ) |

( , ) ( , )d d n

t

t d d

n

i V x x t tn n

i x t

i x t V x x t x t x t

x t x t

x t e x t

Time-splitting spectral method

Properties of TSSP

– Explicit, time reversible & unconditionally stable– Easy to extend to 2d & 3d from 1d; efficient due to FFT– Conserves the normalization– Spectral order of accuracy in space – 2nd, 4th or higher order accuracy in time– Time transverse invariant

– ‘Optimal’ resolution in semicalssical regime

unchanged|),(|)()( 2txxVxV dd

)2/(2/1,, ddOkOh

Dynamics of Ground states

1d dynamics: 2d dynamics of BEC (Bao, D. Jaksch & P. Markowich, J. Comput. Phys., 03’)

– Defocusing: – Focusing (blowup):

3d collapse and explosion of BEC (Bao, Jaksch & Markowich,J. Phys B, 04’)

– Experiment setup leads to three body recombination loss

– Numerical results: • Number of atoms , central density & Movie

xx1 40,at t 100

yy 2,2 0at t,20 xx2

5040 0At t 2

next

420

22 ||||)(2

1),( ixVtx

ti

Collapse and Explosion of BEC

back

Number of atoms in condensate

back

Central density

back

Central quantized vortices

Central vortex states in 2D:

with

Vortex Dynamics – Dynamical stability – Interaction

• Pattern I• Pattern II

immm eryxtyx m

ti

m

ti

ee )(),(),,(

1)(2

r0,||)(22

)(

2

1)(

0

2

22

2

2

2

drrr

rr

r

m

dr

rdr

dr

d

rr

m

mmmm

mm

01

( , ) ( , ) / || ||j

N

n j jj

x y x x y y

),()()( txWxVxV

01

( , ) ( , ) / || ||j

N

n j jj

x y x x y y

Central Vortex states

Vortex stability & interaction

Dynamical stability (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci., 05’)

– m=1: stable velocity– m=2: unstable velocity

Interaction (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci., 05’)

– N=2: Pair: velocity trajectory phase phase2 Anti-pair: phase trajectory angular trajectory2 – N=3: velocity trajectory – Pattern II: Linear nonlinear

Interaction laws: – On-going with Prof. L. Fu & Miss Y. Zhang next

back

Linear case

back

Noninear case: BEC

back

Linear case

Some Open Questions

Dynamical laws for vortex interaction

With a quintic damping, mass goes to constant

Convergence & error estimate of the TSSP?Energy diminishing of the gradient flow in nonlinear case & error estimate ?

( )?????jdx t

dt

20( ) | ( , ) | 0 ????

d

t

N t x t d x C

Part IV

Rotating BEC

&

multi-component BEC

Rotating BEC

The Schrodinger equation ( )

– The Hamiltonian:

– The interaction potential is taken as in Fermi form

xdxdtxtxxxtxtx

xdtxLxVm

txH z

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

1

),(])(2

[),(*)(22

*

)(

H

t

t),xψ(i

.4),()1()(2

00 mxNx aUU s

),,( zyxx

Rotating BEC

The 3D Gross-Pitaevskii equation ( )

– Angular momentum rotation

– V is a harmonic trap potential

– Normalization condition

]||)(2

[),( 20

2

2

UNLxVm

txt

i z

)(2

)( 222222zyx

mxV

zyx

1|),(| 2

R3 xdtx

),,( zyxx

iPPxLiyxiypxpL xyxyz ,,)(:

Rotating BEC

General form of GPE ( )

with

Normalization condition

),(]||)(2

1[),( 22 txLxVtx

ti dzd

dx R

3),(

2

1

2),(2

1

)(

,

,2

)(22222

2224

3

dzyx

dyxxV

dzz

zy

y

d

z

d

.1|),(| 2

R xdtx

d

iyxiL xyz )(:

Rotating BEC


– Energy

Chemical potential

d

NxdtxtNR

2 1))0((|),(|))((

))0((

]||2

*||)(||2

1[))((

,

422

R,

E

E xdLxVt dzdd

xdLxVt dzdd

]||*||)(||

2

1[))(( 422

R,


When , re-scaling

With

Leading asymptotics

1d)2/(24/2/1 /1 d

ddxx

]||)(2

[),( 222

zd LxVtxt

i

)1(

]||2

1*)(||)(||

2[)( 422

R

2

,

O

xdLxVE zdd

)2/(2,

)2/(21,

1, )(,)()(

dd

dd OOOEE

Quantum Hydrodynamics

Set

Geometrical Optics: (Transport + Hamilton-Jacobi)

Quantum Hydrodynamics (QHD): (Euler +3rd dispersion)

)2/(2/ /1,,, dd

iS vJSve

1

2ˆ)(

2

1

)(:ˆ,0ˆ)(2

2

t SLxVSS

yxLLS

zd

xyzzt

01

10A ,J , 2/)(

)ln(4

ˆ)()()(

0ˆ)(

2

2

vP

JAJLVPJJ

J

Lv

zdt

zt

Stationary states

Stationary solutions of GPE

Nonlinear eigenvalue problem with a constraint

Relation between eigenvalue and eigenfunction

)(),( xtx eti

1|)x(|

),(]|)(|)(2

1[)(

2

R

22

xd

xxLxVx

d

dzd

xdEd

d 4

R,, |)x(|2

)()(

Stationary states

Equivalent statements:– Critical points of over the unit sphere– Eigenfunctions of the nonlinear eigenvalue problem– Steady states of the normalized gradient flow:

Minimizer/saddle points over the unit sphere : – For linear case

• Global minimizer vs saddle points– For nonlinear case

• Global minimizer, local minimizer (?) vs saddle points

)(, E

.1||)(||with)()0,(

,]||||

)(||)(

2

1[),(

00

2

,22

xxx

LxVtx zt

1&0 d

1&0 d

Ground state

Ground state:

Existence:– Seiringer (CMP, 02’)

Uniqueness of positive solution:– Lieb et al. (PRA, 00’)

Energy bifurcation:– Aftalion & Du (PRA, 01’); B., Markowich & Wang 04’

xdxEEEd

dgggg

4

R g,,,1||||

, |)(|2

)()(),(min)(

0&1|| d

0&0 d

10 c

Numerical results

– Ground states: in 2D in 3D isosurface – Quantized vortex generation in 2D

• surface contour

– Vortex lattice• Symmetric trapping anisotropic trapping

– Giant vortex generation in 2D• surface contour

– Giant vortex• In 2D In 3D

next

Numerical & Asymptotical results

Critical angular frequency: symmetric state vs quantized vortex state

Asymptotics of the energy:

Ratios between energies of different states

Rank according to energy and chemical potential– Stationary states are ranked according to their energy, then their chemical

potential are in the same order. Next

)()(,10 00,10, EEcc

),()(;1),1( )2/(2

,1,,d

sgg OEOEE

1||0,constlim.,constlim;0,1lim,1lim,

,

,

,

,

,

,

,

g

s

g

s

g

s

g

s

dddd E

E

E

E

Dynamical laws of rotating BEC


Dynamical laws – Time reversible & time transverse invariant– Conservation laws– Angular momentum expectation– Condensate width– Dynamics of a stationary state with its center shifted

iyxiLxx

LxVtxt

i

xyz

dzd

)(:),()0,(

]||)(2

1[),(

0

22

Conservation laws


– Energy

Chemical potential

d

NxdtxtNR

2 1))0((|),(|))((

))0((

]||2

*||)(||2

1[))((

,

422

R,

E

E xdLxVt dzdd

xdLxVt dzdd

]||*||)(||

2

1[))(( 422

R,


Definition:Lemma The dynamics of satisfies

For any initial data, with symmetric trap, i.e. , we have

Numerical test next

Bao, Du & Zhang, SIAM J. Appl. Math., 66 (2006), 758

0,)(**:)( txdxyixdLtLdd R

yx

R

zz

0,|),(|)()( 222 txdtxxy

dt

tLd

dR

yxz

yx

( )zL t

,0 ,0 0( ) (0), ( ) ( ), 0.z zL t L E E t

back


Energy

Dynamics of condensate width

Definition: Bao, Du & Zhang, SIAM J. Appl. Math., 66 (2006), 758

Dynamic laws– When for any initial data: – When with initial data Numerical Test– For any other cases:

xdtxtxdtxyxtdd RR

r

22222 |,(|)(,|,(|)()(

0),(4)(4)( 2

0,2

2

ttEdt

tdrx

r

yxd &2

yxd &2 imerfyx )(),(0

0),(2

1)()( tttt ryx

0),()(4)(4)( 2

0,2

2

ttftEdt

td

next

back

Symmetric trap Anisotropic trap

Dynamics of Stationary state with a shift

Choose initial data as: The analytical solutions is: Bao, Du & Zhang, SIAM J. Appl.Math., 2006

– In 2D:

– In 3D, another ODE is added

)()( 00 xxx s

( , )0( , ) ( ( )) , ( , ) 0, (0)si t iw x t

sx t e x x t e w x t x x

2 2

2 2

0 0 0 0

( ) 2 ( ) ( ) ( ) 0,

( ) 2 ( ) ( ) ( ) 0,

(0) , (0) , (0) , (0)

x

y

x t y t x t

y t x t y t

x x y y x y y x

20( ) ( ) 0, (0) , (0) 0zz t z t z z z


Center of mass: Bao & Zhang, Appl. Numer. Math., 2006

In a non-rotating BEC:

– Pattern Classification: • Each component of the center is a periodic function • In a symmetric trap, the trajectory is a straight segment• If is a rational #, the center moves periodically with period • If is an irrational #, the center moves chaotically, envelope is a rectangle

2 2( ) : | ( , ) | | ( ( )) | ( )d d

sx t x x t dx x x x t dx x t

00 0( ) cos( ), ( ) cos( ), 0x yx t x t y t y t t

/y x /y x

2p


In a rotating BEC with a symmetric trap:

– Trajectory of the center – Distance between the center and trapping center – Motion of the solution: 0.5 1 2 4– Pattern Classification:

0 0

0 0

2 2 2 20 0

| |( ) cos( ) cos( ) sin( ) sin( ) , | |

2 2| |

( ) cos( ) cos( ) sin( ) sin( ) , | |2 2

| ( ) |: ( ) ( ) | cos( ) |

x

x

x

x yx t a t b t a t b t a

y xy t a t b t a t b t b

x t x t y t x y t

next

back

1/5, 4/5, 1

3/2, 6, Pi

Pattern Classification

Pattern Classification: Bao & Zhang, Appl. Numer. Math., 2006

– The distance between the center and trap center is periodic function– When is a rational #

• The center moves periodically• The graph of the trajectory is unchanged under a rotation

– When is an irrational #, • The center moves chaotically• The envelope of the trajectory is a circle

– The solution of GPE agrees very well with those from the ODE system

/ qp

/ qp

back


In a rotating BEC with an anisotropic trap– When results

• The trajectory is a spiral coil to infinity • The trajectory is an ellipse

– Otherwise result1 result2• The center moves chaotically & graph is a bounded set

• The center moves along a straight line to infinity

||or xy

next

Total density with dissipation


Lemma The dynamics of total density satisfies

– The total density decreases when density function energy next

2 2

0

1( , ) [ ( ) ( , ) | | ]

2

( ,0) ( ), : ( )

d z d

z y x

i x t V x W x t Lt

x x L i x y i

2, 2

2()()|(,)|()0,0

1 d

dNtxtdxt

dt

0& | | min{ , }x y

Numerical Methods

Time-splitting pseudo-spectral method (TSSP)

– Use polar coordinates (B., Q. Du & Y. Zhang, SIAP 06’)– Time-splitting + ADI technique (B. & H. Wang, JCP, 06’)– Generalized Laguerre-Hermite functions (B., J. Shen & H. Wang, 06’)

|),(||),(|),(|),(|),()(),( :2 Step

)(:,2

1),( :1 Step

2

2

nddt

xyzzt

txtxtxtxtxxVtxi

iyxiLLtxi

Numerical methods for rotating BEC

Numerical Method one: (Bao, Q. Du & Y. Zhang, SIAM, Appl. Math. 06’)

– Ideas• Time-splitting • Use polar coordinates: angular momentum becomes constant coefficient • Fourier spectral method in transverse direction + FD or FE in radial direction• Crank-Nicolson in time

– Features• Time reversible• Time transverse invariant• Mass Conservation in discretized level• Implicit in 1D & efficient to solve• Accurate & unconditionally stable

Numerical methods for rotating BEC

Numerical Method two: (Bao & H. Wang, J. Comput. Phys. 06’)

– Ideas• Time-splitting • ADI technique: Equation in each direction become constant coefficient • Fourier spectral method

– Features• Time reversible• Time transverse invariant• Mass Conservation in discretized level• Explicit & unconditionally stable• Spectrally accurate in space

Dynamics of ground state

Choose initial data as: : ground state

Change the frequency in the external potential:– Case 1: symmetric: surface contour– Case 2: non-symmetric: surface contour– Case 3: dynamics of a vortex lattice with 45 vortices: image contour next

)()(0 xx g

21:&21: yx

2.21:&8.11: yx

1,8.0,100 zy

canisotropi :),(,9.0,1000 txV

Interaction of two vortices in linear

0

1/ 3


1/ 2

1


1/

Interaction of vortices in nonlinear

1/


0

1/ 2


1

4


1/

Some Open Questions

Dynamical laws for vortex interaction

With a quintic damping, mass goes to constant

Semiclassical limit when initial data has vortices??? Vortex line interaction laws, topological change?What is a giant vortex?

( )?????jdx t

dt

20( ) | ( , ) | 0 ????

d

t

N t x t d x C

Two-component BEC

The 3D coupled Gross-Pitaevskii equations

Normalization conditions

Intro- & inter-atom Interactions

2

2 2 21 11 1 12 2 1 2

2

2 2 22 21 1 22 2 2 1

( , ) [ ( ) | | | | ]2

( , ) [ ( ) | | | | ]2

z

z

i x t V x L U Ut m

i x t V x L U Ut m

3 3

22 0 0 0 2

1 21

( ) | ( , ) | : with | ( ,0) | ,j j jj

N t x t dx N N N N x dx

2

12 21

4 with jl

jl

aU a a

m

Two-component BEC

Nondimensionalization

Normalization conditions– There is external driven field

– No external driven field

2 2 21 11 1 12 2 1 2

2 2 22 21 1 22 2 2 1

1( , ) [ ( ) | | | | ]

2

1( , ) [ ( ) | | | | ]

2

z

z

i x t V x Lt

i x t V x Lt

3 3

2 21 2( ) | ( , ) | | ( , ) | 1N t x t dx x t dx

3 3

0 02 21 2

1 2| ( , ) | , | ( , ) |N N

x t dx x t dxN N

0

0

Two-component BEC

Energy

Reduction to one-component:

2 22 2 2 2 *

1 2Rj=1 1

1( ) [ ( | | ( )| | * | | | | ) 2 Re( )]

2 2d

jlj j j z j j l

l

E V x L dx

0 0 01 2 10, , ( )N N N O N

3 3

0 02 22 1

2 2 1 1( ) | ( , ) | : 1, ( ) | ( , ) | : 1 1N N

N t x t dx N t x t dxN N

2 2

0 01 1 1 11

1( , ) [ ( ) | | ] ( , ),

2

| ( ) ( ) |( , ) / ( , ) & = / ( )

( )

z

s

s

i x t V x L x tt

E Ex t N N x t N N O

E

Two-Component BEC


Semiclassical limit– No external field:

• WKB expansion, two-fluid model– With external field:

• WKB expansion doesn’t work, Winger transform

22 2 2

1 11 1 12 2 1 2

22 2 2

2 21 1 22 2 2 1

( , ) [ ( ) | | | | ]2

( , ) [ ( ) | | | | ]2

z

z

i x t V x Lt

i x t V x Lt

0

0

Ground state

No external field:

Nonlinear eigenvalue problem

Existence & uniqueness of positive solutionNumerical methods can be extended

0

1 21 2|| || ,|| ||

min ( , ) with 1E

2 2 21 1 11 1 12 2 1

2 2 22 2 21 1 22 2 2

1( ) [ ( ) | | | | ]

21

( ) [ ( ) | | | | ]2

z

z

x V x L

x V x L

Ground states

crater

Ground state

With external field:

Nonlinear eigenvalue problem

Existence & uniqueness of positive solution ???Numerical methods can be extended????

0

2 21 2

1 2|| || || || 1

min ( , )E

2 2 21 11 1 12 2 1 2

2 2 22 21 1 22 2 2 1

1( ) [ ( ) | | | | ]

21

( ) [ ( ) | | | | ]2

z

z

x V x L

x V x L

Dynamics

Dynamical laws:– Conservation of Angular momentum expectation– Dynamics of condensate width– Dynamics of a stationary state with a shift– Dynamics of mass of each component, they are periodic

function when – Vortex can be interchanged!

Numerical methods– Time-splitting spectral method

11 12 22

Dynamics

Spinor BEC

Spinor F=1 BEC

With

2

2 * 21 1 1 0 1 1 1 0

2

2 *0 0 1 1 1 1 1 0

2

2 * 21 1 1 0 1 1 1 0

[ ( ) ] ( )2

[ ( ) ] ( ) 22

[ ( ) ] ( )2

z n s s

z n s s

z n s s

i V x L g g gt m

i V x L g g gt m

i V x L g g gt m

2 22 0 2 2 0

1 0 1

0 2

24 4, | | , ,

3 3, : s-wave scattering length with the total spin 0 and 2 channels

j j n s

a a a ag g

m ma a

Spinor BEC

Total mass conservation

Total magnetization conservation

Energy conservation

3 3

12 0 0 0 0 2

1 0 11

( ) | ( , ) | : with | ( ,0) | ,j j jj

N t x t dx N N N N N x dx

3 3

2 2 0 01 1 1 1( ) | ( , ) | | ( , ) | :M t x t dx x t dx N N M

212 2 2

Rj=-1

2 2 * 2 * * 21 1 1 0 1 0 1 1 1 0 1 1 0 1

( ) [ ( | | ( )| | * )2m 2

( 2 2 2 ) ( ( ) )]2

d

nj j j z j

ss

gE V x L

gg dx

Spinor BEC

Dimension reductionGround state– Existence & uniqueness of positive solution??– Numerical methods ???

Dynamics – Dynamical laws – Numerical methods: TSSP

Semiclassical limit & hydrodynamics equation??

BEC at Finite Temperature

Condensate coexists with non-condensed thermal cloud Coupled equations of motion for condensate and thermal cloudMean-field theory in collisionless regimeZGN theory in collision dominated regime

Mean-field Theory

Evolution of quantum field operator

where is the annihilation field operatorand is the creation field operatorMean-field description

Condensate wavefunction

ˆˆˆˆ

2

ˆ†2

2

gVmt

i ext

0),(~

),(),(

),(~

ˆ),(),( 0

tx

txtx

txatxtx

mag s /4 2

~~~~2

2†*2

2

gmggngnVmt

i Tcext

),(ˆ tx

),(ˆ † tx

Mean-field Theory

Generalized GPE for condensate wavefunction

Temperature-dependent fluctuation field for non-condensate

functionn correlatio field-hree t ~~~

density condensate-non diagonal-off ~~

),(~

density condensate-non ~~

),(

density condensate ),(

~~~~22

†

†

2

†*22

txm

txn

txn

gmggngnVmt

i

T

c

Tcext

~~~~~

22

~††2

2

ggmgnVmt

i ext

†††

††

~~~~~~2

~~~

~~~~

~~~~

),(~),(),(

),(),(),(2 txmtxtxm

txntxntxn Tc

Hartree-Fock Bogoliubov Theory

Ignore the three-field correlation function

Bogoliubov transformation

where creates (annihilates) a Bogoliubov quasiparticle of energy εj

The quasiparticles are non-interacting

*22

~22

mgnngVmt

i Tcext

~~~ †

jjjj

jjjj

txvtxutx

txvtxutx

ˆ),(ˆ),(),(~

ˆ),(ˆ),(),(~

†j

*†

†j

*

†22 ~~

22

~

gmgnVmt

i ext

)ˆ(ˆ †j j

Hartree-Fock Bogoliubov Theory

Bogoliubov equations for non-condensate

where

jjextj

jjextj

ugmvgnVmt

vi

gmvugnVmt

ui

*22

22

22

22

1)/exp(

1ˆˆ

)21(),(

)1(),(

),(

†

*2

22

2

kTN

Nvutxm

NvNutxn

txn

jjjj

jjjj

jjjjjT

c

Time-independent Hartree-Fock Bogoliubov Theory

Stationary states

Time-independent generalized GPE and Bogoliubov equations

//

//

/

)(),(

)(),(

)(),(

titijj

titijj

ti

eexvtxv

eexutxu

extx

j

j

*2

2~2

2mgnngV

m Tcext

jjjjext

jjjjext

vugmvgnVm

ugmvugnVm

*22

22

22

22

HFB-Popov Approximation

HFB produces an energy gap in the excitation spectrumSolution: leave outGeneralized GPE and Bogoliubov equations within Popov approximation (gapless spectrum)

*22

22

Tcext nngVmt

i

jjextj

jjextj

ugvgnVmt

vi

vgugnVmt

ui

2*22

222

22

22

m~

Hartree-Fock Approximation

Approximate Bogoliubov excitations with single-particle excitations, i.e. let

Restricted to finite temperature close to Tc, where the non-condensed particles have higher energies

0jv

*22

22

Tcext nngVmt

i

jextj gnV

mti

22

22

jjjT

c

Ntxn

txn2

2

),(

),(

ZGN Theory

Mean-field theory deals with BEC in collisionless region (low density thermal cloud):

l >> l is the collisonal mean-free-path of excited particlesis the wavelength of excitations

In collision-dominated region l << (higher density thermal cloud)the problem becomes hydrodynamic in nature

ZGN theory (E. Zaremba, A. Griffin, T. Nikuni, 1999) describes finite-T BEC with interparticle collisions in the semi-classical limit

kBT >> ħ: trap frequency) kBT >> gn

ZGN Theory

Apply Popov approximation (ignore ) but include the three-field correlation functionGPE for condensate wavefunction

Quantum Boltzmann equation for phase-space distribution function of non-condensate

~~~

22

†22

gnngVmt

i Tcext

][2

22 12

22

fn

inngV

mti

cTcext

][][),,(

),,(2 2212 fCfCt

tpxftpxfgnV

m

p

t collisionpextx

][)2(

~~~Im

2][ termsource 123

†*12 fC

pdgf

~~~ †

m~

ZGN Theory

Thermal cloud density

Collision between condensate and non-condensate-- transfer atoms from/to the condensate

Collision between non-condensate particles

),,(2

),( 3 tpxfpd

txnT

)1)(1()1(

)()()()(

)()2(

2][

321321

321321

32132142

2

12

ffffff

pppppp

pppvmpdpdpdng

fC

pppc

sc

)1)(1()1)(1()(

)()2(

2][

321321321

32132175

2

22

ffffffff

pppppdpdpdg

fC

pppp

ZGN Theory

Energy of condensate atoms

Local chemical potential

Superfluid velocity

Energy of non-condensate atoms – Hartree-Fock energy

Limited to high temperature (close to Tc)For lower temperature, the spectrum of excited atoms should be described by Bogoliubov approximation

gnVm

ptx extp 2

2),(

2

),(2

1),(),( 2 txmvtxtx scc

Tcext

c

cc gngnV

n

n

mtx 2

2),(

22

),(),( txm

txvs

),(exp),(),( txitxntx c

Open questions

Mathematical theory– Quantum Boltzmann Master equation (QBE)– GPE with damping term– Coupling QBE +GPE

Numerical methods– For QBE: P. Markowich & L. Pareschi (Numer. Math., 05’)

– For QBE+GPE – Comparison with experiments– Rotational frame

Conclusions

– Review of BEC– Experiment progress– Mathematical modeling– Efficient methods for computing ground & excited states– Efficient methods for dynamics of GPE– Comparison with experimental results– Vortex dynamics– Quantized vortex stability & interaction

Future Challenges

– Multi-component BEC for bright laser– Applications of BEC in science and engineering– Precise measurement– Fermions condensation, BEC in solids & waveguide– Dynamics in optical lattice, atom tunneling– Superfluidity & dissipation, quantized vortex lattice – Coupling GPE & QBE for BEC at finite temperature– Mathematical theory for BEC– Interdisciplinary research: experiment,physics, mathematics, computation, ….

References

[1] M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman and E.A. Cornell, Science 269 (1995) 198-201.

[2] W. Bao, J. Shi and P.A. Markowich, J. Comput. Phys. , Vol. 175, pp. 487-524, 2002. [3] W. Bao and W.J. Tang, J. Comput. Phys., Vol. 187, No. 1, pp. 230 - 254, 2003. [4] W. Bao, D. Jaksch and P.A. Markowich, J. Comput. Phys., Vol. 187, No. 1, pp. 318 - 342,

2003.[5] W. Bao, S. Jin and P.A. Markowich, SIAM J. Sci. Comput., Vol. 25, No. 1. pp. 27-64, 2003. [6] W. Bao and D. Jaksch, SIAM J. Numer. Anal., Vol. 41, No. 4. pp. 1406-1426, 2003. [7] W. Bao, D. Jaksch and P.A. Markowich, J. Phys. B: At. Mol. Opt. Phys., Vol. 37, No. 2, pp.

329-343, 2004. [8] W. Bao, Multiscale Modeling and Simulation: a SIAM Interdisciplinary Journal, Vol. 2, No. 2.

pp. 210-236, 2004. [9] W. Bao and Q. Du, SIAM J. Sci. Comput. , Vol. 25, No. 5. pp. 1674-1697, 2004.

References

[9] W. Bao and Q. Du, SIAM J. Sci. Comput. , Vol. 25, No. 5. pp. 1674-1697, 2004.[10] W. Bao, H.Q. Wang and P.A. Markowich, Comm. Math. Sci. , Vol. 3, No. 1, pp. 57-88,

2005. [11] W. Bao, P.A. Markowich, C. Schmeiser and R. M. Weishaupl, Math. Mod. Meth. Appl.

Sci. , Vol. 15, No. 5, pp. 767-782, 2005. [12] W. Bao and J. Shen, SIAM J. Sci. Comput. , Vol. 26 , No. 6, pp. 2010-2028, 2005. [13] W. Bao and Y.Z. Zhang, Math. Mod. Meth. Appl. Sci. , Vol. 15 , No. 12, pp. 1863-1896,

2005. [14] W. Bao, Qiang Du and Yanzhi Zhang, SIAM J. Appl. Math., Vol. 66 , No. 3, pp. 758-

786, 2006. [15] W. Bao and H. Wang, J. Comput. Phys., Vol. 217, No. 2, pp. 612-626, 2006.[16] W. Bao, I-L. Chern and F. Y. Lim, J. Comput. Phys., Vol. 219, No. 2, pp. 836-854, 2006[17] W. Bao and Y. Zhang, Appl. Numer. Math., Vol. 57, No. 5-7, pp. 697-709, 2007.

References

[18] W. Bao, F. Y. Lim and Y. Zhang, Bulletin of the Institute of Mathematics, Academia Sinica, Vol. 2, No. 2, pp. 495-532, 2007.

[19] W. Bao, H.L. Li and Y. Zhang, Physica D: Nonlinear Phenomena, Vol. 234, pp. 49-69, 2007.[20] W. Bao, Y. Ge, D. Jaksch, P. A. Markowich and R. M. Weishaeupl, Comput. Phys. Comm.,

Vol. 177, No. 11, pp. 832-850, 2007. [21] W. Bao and H. Wang, A mass and magnetication conservative and energy diminishing

numerical method for computing ground state of spin-1 Bose-Einstein condensates, SIAM J. Numer. Anal., Vol. 45, No. 5, pp. 2177-2200, 2007.

[22]. A. Klein, D. Jaksch, Y. Zhang and W. Bao, Dynamics of vortices in weakly interacting Bose-Einstein condensates, Phys. Rev. A, Vol. 76, article 043602, 2007.

[23]. W. Bao and M.-H. Chai, A uniformly convergent numerical method for singularly perturbed nonlinear eigenvalue problems, Commun. Comput. Phys., to appear.

[24]. W. Bao and F. Y. Lim, Computing Ground States of Spin-1 Bose-Einstein Condensates by the Normalized Gradient Flow, arXiv: 0711.0568.

References

[25] Bradly et al., Phys. Rev. Lett., 75 (1995), 1687.[26] Davis et al., Phys. Rev. Lett., 75 (1995), 3969.[27] A.L. Fetter and A. A. Svidzinsky, Vortices in a trapped dilute Bose-Einstein condensate

(topical review), J. Phys.: Condens. Matter 13 (2001), 135-194. [28] A.J. Leggett, Bose-Einstein condensation in the alkali gases: some fundamental concepts,

Rev. Modern Phys., 73 (2001), 307-356.[29] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford University Press, 2003.[30]E.H. Lieb, R. Seiringer, J.P. Solovej and J. Yngvason, The Mathematics of the Bose Gas and

its Condensation, Birkhauser, 2000.[31] A. Aftalion, Vortices in Bose-Einstein Condensates, Birkhauser, 2006.[32] F. Dalfovo, S. Giorgini, L.P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation

in trapped gases, Rev. Modern. Phys., 71 (1999), 463-512.[33] C.J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge

University Press, 2002.

mathematical analysis and numerical simulation for bose-einstein condensation weizhu bao department...

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