Numerical Simulations for Bose-Einstein Condensation (BEC)bao/teach/ma5232/Lecture-BEC.pdf ·...

Mathematical Models & Numerical Simulation for Bose-Einstein Condensation

Weizhu Bao

Department of Mathematics National University of SingaporeEmail: [email protected]

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

BEC@JILA, 95’

Vortex@ENS

mailto:[email protected]

http://www.math.nus.edu.sg/%7Ebao

http://jilawww.colorado.edu/bec/hi_res_pic_album_macromedia/images/BEC_peaks_jpg.jpg


Outline

Part I: Predication, Experiment & Mathematical Models– Theoretical predication– Physical experiments and results– Applications– Gross-Pitaevskii equation (GPE)

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

BEC@ETH, 98’

Outline

Part III: Analysis & Computation for Dynamics of BEC– Dynamical laws– Numerical methods– Numerical results

Part IV: Rotating BEC & Multi-component BEC– BEC in a rotational frame– BEC with nonlocal dipole-dipole interaction– Two-component BEC– Spinor BEC– Conclusions & Future challenges BEC@MIT, 00’

Part I

Predication, Experiment &

Mathematical models

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

Theoretical predication

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

Theoretical predication of BEC

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 cold gas

( ) 0 1/

1 : ( ) , 0,1,2, ,1i Bi ik Tn f j

e ε µ ε µ ε ε−= = = < < <−

Einstein’s prediction on BEC: At zero or `low’ temperatures, most particles behavior in the same way (at quantum mechanical ground state)! –gregarious behavior----- quantum phase transition happens at extremely low temperatures --- degenerate quantum gas -- `super atom’ –--fifth matter of state.

Experimental difficulties

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

Experimental techniques

Laser coolingMagnetic trappingEvaporative Cooling

($100k—300k)

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., NUS, …

2001 Nobel prize in physics:– C. Wiemann: U. Colorado– E. Cornell: NIST– W. Ketterle: MIT

ETH (02’, Rb, 300,000)

Properties of BEC in experiments

87 7 23 52 4

11 15 3

Kinds of atom Rb, Li, Na, Cr, He, Temperatures 50nK 2 K

Density 10 10 cm µ

−

− − −− − −

7 6 23 9 4# of atoms 100( Li) 10 ( Na) 10 ( He) Sphere at diameter 10--15 m

Spatial sizecigar-shaped at length 300 m & diameter 15μm

Life span A few seconds to several minutes

µµ

− − − − −

Brief BEC research history

Milestones– 1924 ----- Prediction of BEC by A. Einstein

– 1938-----London & Tisza linked BEC & superfluidity in liquid helium 4

– 1995 --- Experiments of BEC in ultracold gas of alkali atoms

– Since then--- BEC begins a new era in atomic, molecular & optical (AMO) physics & quantum optics; attracts interests of computational, applied & pure mathematicians.

Some famous physicists in BEC research – Nobel Laureates: A. Einstein (1921), L. D. Landau (1961), L. Onsager (1968), R. Feynmann

(1965),T.-D. Lee&C.N. Yang (1957), A. Leggett (2003), E. Cornell, C. Weimann&W. Ketterle (2001),

– Others: S. Bose, F. London, L. Tisza, N. Bogoliubov, R. Penrose, K. Huang, D. Beliaev, L. Pitaevskii, E. Gorss, A. Fetter, P. Zoller, D. Jaksch, I. Bloch, R. Hulet, J. Yngvason, S. Stringari, E.H.. Lieb, …..

Mathematical models

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

Mean field theory--zeor or `extremely’ low temperatures

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

Quantum kinetic theory-- high temperatures

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

(nK)cT T O< =

Model for a BEC at zero temperature– with N identical bosons

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

Hartree ansatzFermi interactionDilute quantum gas -- two-body elastic interaction

1 2 1 2

22

int1 1

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

( ) ( )2

t N N N N N

N

N j j j kj j k N

i x x x t H x x x t

H V x V x xm= ≤ < ≤

∂ Ψ = Ψ

= − ∇ + + −

∑ ∑

31 2

1( , , , , ) ( , ),

N

N N j jj

x x x t x t xψ=

Ψ = ∈∏

2

int4( ) ( ) with s

j k j kaV x x g x x g

mπδ− = − =

31( ) : ( )--energy per particle

NN N N N N NE H d x d x N E ψΨ = Ψ Ψ ≈∫

Model for a BEC – with N identical bosons

Energy per particle – mean field approximation (Lieb et al, 00’)

Dynamics (Gross, Pitaevskii 1961’; Erdos, Schlein & Yau, Ann. Math. 2010’)

Proper non-dimensionalization & dimension reduction– GPE/NLSE

3

22 2 4( 1)( ) ( ) with : ( , )

2 2N gE V x dx x t

mψ ψ ψ ψ ψ ψ

−= ∇ + + =

∫

222 3( )( , ) ( ) ( 1) ,

2tEi x t V x N g x

mδ ψψ ψ ψδψ

∂ = = − ∇ + + − ∈

2 21( , ) ( ) | | , 2

dti x t V x xψ ψ ψ β ψ ψ∂ = − ∇ + + ∈

4 ( 1) 4 s s

s s

N a Nax x

π πβ −= ≈

Gross-Pitaevskii equation (GPE)

The 3D Gross-Pitaevskii equation ( )

– V is a harmonic trap potential

– Normalization condition

22 2( , ) ( , ) ( ) ( , ) ( 1) | ( , ) | ( , )

2i x t x t V x x t N g x t x t

t mψ ψ ψ ψ ψ∂

= − ∇ + + −∂

)(2

)( 222222zyxmxV zyx ωωω ++=

3

2| ( , ) | 1.x t dxψ =∫

),,( zyxx =

Gross-Pitaevskii equation

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

– Dimensionless Gross-Pitaevskii equation

– With

3/2, , ( , ) ( , ),x s ss x

xt t x x t x tx mx xψ ψω ω

= = = =

),(|),(|),()(),(21),( 22 txtxtxxVtxtx

ti ψψβψψψ ++∇−=∂∂

2 22 2 21 4 ( 1) 4( ) ( ), , ,2

y s szy z y z

s sx x

N a N aV x x y zx x

π πβω ωγ γ γ γω ω−

= + + = = = ≈

x y zω ω ω≤ ≤


Typical parameters ( )– Used in JILA

– Used in MIT

]Js[1005.1 34−×=

Rb87

25

5

1.44 10 [kg], 10 2 [1 / s], 8

45.1[nm], 0.3407 10 [m], 0.01881

x y z x

ss s

x s

m

NN

maa x x

π

πβ

ω ω ω ω

ω

−

−

= × = = × =

= = = × = =

Na23

26

5

3.8 10 [kg], 360 2 [1 / s], 3.5 2 [1 / s]

42.75 [nm], 1.1209 10 [m], 0.003083

x y z

ss s

z s

m

NN

maa x x

π π

πβ

ω ω ω

ω

−

−

= × = = × = ×

= = = × = =


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 γγωωωω

2

2

12 31/4

/22 1/23 ho

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

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

x y z t x y t z

z x y z dxdy z e γ

ψ ψ φ

γφ φ φπ

−

=

= ≈ = ∫

2 2 22

2

4 42 3 ho 2

1( , ) | | ,2 2

( ) ( ) :2

y

az

x yi x t

t

z dz z dz

γψ ψ ψ β ψ ψ

γβ φ φ β βπ

∞ ∞

−∞ −∞

+∂= − ∆ + +

∂

= ≈ = =∫ ∫

12),,( ψψ == yxx

Mathematical model for BEC—mean field theory

The Gross-Pitaevskii equation (GPE/NLSE)

– t : time & : spatial coordinate (d=1,2,3)– : complex-valued wave function– : real-valued external potential– : dimensionless interaction constant

• =0: linear; >0(<0): repulsive (attractive) interaction– :Schrodinger equation (E. Schrodinger 1925’)

– :GPE (E.P. Gross 1961’; L.P. Pitaevskii 1961’)

2 21( , ) ( ) | |2ti x t V xψ ψ ψ β ψ ψ∂ = − ∇ + +

( )dx ∈

( , )x tψ

( )V x

β

0β =0β ≠


Two kinds of interaction– Repulsive (defocusing) interaction

– Attractive (focusing) interaction

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

– Weakly interacting condensation| | 1β <<

0 0sa β> ⇒ ≥

0 0sa β< ⇒ ≤

0β =


– Strongly repulsive interacting condensation

– Strongly attractive interaction

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

1β >>

0 & | | 1β β< >>

Other applications of GPE/NLSE

For laser beam propagation In plasma physics: wave interaction between electrons and ions

– Zakharov system---NLSE +wave equation, …..

In quantum chemistry: chemical interaction based on the first principle– Schrodinger-Poisson system

In materials science: – First principle computation, DFT, …– Semiconductor industry

In nonlinear (quantum) opticsIn biology – protein foldingIn superfluids – flow without friction, liquid

2 2 22 2H O H O+ →

4 He

http://en.wikipedia.org/wiki/File:RGB_laser.jpg

http://en.wikipedia.org/wiki/File:RGB_laser.jpg

Part II

Analysis & Computation

for

Ground states

Conservation laws

DispersiveTime symmetric: Time transverse (gauge) invariant

Mass conservation

Energy conservation

2 21( , ) ( ) | |2ti x t V xψ ψ ψ β ψ ψ∂ = − ∇ + +

& take conjugate unchanged!!t t→ − ⇒

2( ) ( ) --unchanged!!i tV x V x e αα ψ ψ ρ ψ−→ + ⇒ → ⇒ =

2 2( ) : ( ( , )) ( , ) ( ,0) 1, 0

d d

N t N t x t d x x d x tψ ψ ψ= = ≡ = ≥∫ ∫

2 2 41( ) : ( ( , )) ( ) (0), 02 2d

E t E t V x d x E tβψ ψ ψ ψ = = ∇ + + ≡ ≥ ∫

Stationary states

Stationary states (ground & excited states)

Nonlinear eigenvalue problems: Find

Time-independent NLSE or GPE:Eigenfunctions are– Orthogonal in linear case & Superposition is valid for dynamics!!– Not orthogonal in nonlinear case !!!! No superposition for dynamics!!!

2 2

2 2

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

with : | (x) | 1d

dx x V x x x x x

d x

µ φ φ φ β φ φ

φ φ

= − ∇ + + ∈

= =∫

( , ) s.t.µ φ( , ) ( ) i tx t x e µψ φ −=

2 21( , ) ( ) | |2ti x t V xψ ψ ψ β ψ ψ∂ = − ∇ + +

Ground states

The eigenvalue is also called as chemical potential

– With energy

Ground states -- nonconvex minimization problem

– Euler-Lagrange equation nonlinear eigenvalue problem

4: ( ) ( ) | (x) |2 d

E d xβµ µ φ φ φ= = + ∫

2 2 41( ) [ | ( ) | ( ) | ( ) | | ( ) | ]2 2d

E x V x x x d xβφ φ φ φ= ∇ + +∫

( ) min ( ) | 1, ( )g SE E S E

φφ φ φ φ φ

∈= = = < ∞

Existence & uniqueness

Theorem (Lieb, etc, PRA, 02’; Bao & Cai, KRM, 13’) If potential is confining

– There exists a ground state if one of the following holds

– The ground state can be chosen as nonnegative – Nonnegative ground state is unique if– The nonnegative ground state is strictly positive if– There is no ground stats if one of the following holds

| |( ) 0 for & lim ( )d

xV x x V x

→∞≥ ∈ = ∞

( ) 3& 0; ( ) 2 & ; ( ) 1&bi d ii d C iii dβ β β= ≥ = > − = ∈0, . . i

g g gi e e θφ φ φ=

0β ≥2loc( )V x L∈

( ) ' 3& 0; ( ) ' 2 & bi d ii d Cβ β= < = ≤ −

2 2 2 2

1 24 2

2 2

( ) ( )40 ( )

( )

inf L Lb f H

L

f fC

f≠ ∈

∇=

Key Techniques in Proof

Positivity & semi-lower continuous

The energy is bounded below if conditons (i) or (ii) or (iii) and strictly convex if Confinement potential implies decay at far field The set is convex inUsing convex minimization theorem Non-existence result

2( ) (| |) ( ), with | |E E E Sφ φ ρ φ ρ φ≥ = ∀ ∈ =

( ) : ( )E Eρ ρ=

2

/41 | |( ) exp , with 0

(2 ) 2d

dxx xεφ ε

πε ε

= − ∈ →

| ( ) =1 & ( )d

S x d x Eρ ρ ρ = < ∞

∫

ρ

0β ≥

0( )E

ε

εφ→

→−∞

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’)

Fundamental gaps

,,, 321 φφφ

1 2

1 2 1 2

, , ,( ) ( ) ( ) ( ) ( ) ( ) ???????

g

g gE E Eφ φ φ

φ φ φ µ φ µ φ µ φ< < < ⇒ < < <

θφφψ µµ immm eryxtyx mti

mti ee )(),(),,( −− ==

2 22

22

2

0

( )1( ) ( ) | | , 0 r2 2 2

2 ( ) 1, (0) 0

mm m m m m

m m

d rd m rr r rr dr dr r

r r dr

φµ φ φ β φ φ

π φ φ∞

= − + + + < < ∞

= =∫

1 1( ) : ( ) ( ) 0, ( ) : ( ) ( ) 0, 0E g gE E C C C Cβ µ βδ β φ φ δ β µ φ µ φ β= − > ≥ > = − > ≥ > ≥

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 4kin pot int

1( ) [ | ( ) | ( ) | ( ) | | ( ) | ] : ( ) ( ) ( )2 2d

E x V x x x dx E E Eβφ φ φ φ φ φ φ= ∇ + + = + +∫

2kin

1( ) | ( ) |2 d

x dxE φ φ= ∇∫

2pot( ) ( ) | ( ) |

dV x x dxE φ φ= ∫

4int( ) | ( ) |

2 dx dxE βφ φ= ∫

2 2 4

int kin pot int

1( ) [ | ( ) | ( ) | ( ) | | ( ) | ]2

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

dx V x x x dx

E E E E E

µ φ φ φ β φ

φ φ φ φ φ

= ∇ + +

+ = + +

∫

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.x

V x≤ ≤

= ∞

2

12

0

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

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

x x x x x

x dx

µ φ φ β φ φ

φ φ φ

′′= − + < <

= = =∫0β =

2 21( ) 2 sin( ), , 1, 2,3,2l lx l x l lφ π µ π= = =

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

2g g g g g gx x x E E πφ φ π φ µ µ φ= = = = = =

2 20 0 0( 1)( ) ( ) 2 sin(( 1) ), : ( ) : ( )

2j j j j j jjx x j x E E πφ φ π φ µ µ φ+

= = + = = = =

| | (1)oβ =

2 20 0 03( ) ( ) 2 sin( ), : ( ) ( ) , : ( ) ( ) 3

2 2 2g g g g g g g gx x x E E E π β πφ φ π φ φ µ µ φ µ φ β≈ = = ≈ = + = ≈ = +

2 20 0

2 20

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

( 1): ( ) ( ) 32

j j j j j

j j j

jx x j x E E E

j

π βφ φ π φ φ

πµ µ φ µ φ β

+≈ = + = ≈ = +

+= ≈ = +

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

1β

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

µµ φ β φ φ φ

β

φ

βφ φ µ µ β

= < < ⇒ =

⇓ =

≈ = ≈ = ≈ =

∫

TF TF(0) (1) 1 0g gφ φ= = ≠

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 xµ

φβµ

= = Φ

31( ) ( ) ( ), 0 ; (0) 0, lim ( ) 12 X

X X X X X→∞

′′Φ = − Φ +Φ ≤ < ∞ Φ = Φ =

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

µφ µ

βΦ = ≤ < ∞ ⇒ ≈ ≤ =

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

µφ φ µ µ µ

β

φ µ µ β β µ β β

≈ = + − − ≤ ≤

= ⇒ ≈ = + + + = + + +∫

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 3

2 1 23

g g g g

g g

E E E E

E E

β ββ β

β

≈ = + + + ≈ = + +

≈ = + +

(1/ )O β

int, kin,1lim , lim 1, lim 0,2

g g g

g g g

E E EE Eβ β βµ→∞ →∞ →∞

= = =

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

β β βφ φ φ φ µ µ

β β β

∞

− − −− = − = = +

= + = + = +

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 xj

l x Cj

µφ φ µ

β

µ µ

+

=

=

≈ = − ++

+− −

+

∑

∑

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

µ µ β β

β β

β β

β

≈ = + + + + + +

≈ = + + + +

≈ = + + + +

≈ = + + + + +

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 Eφ φ φ µ φ µ φ µ φ< < < ⇒ < < <

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 EE E

E EE

β β β

β β β

µ

µµ µ

→∞ →∞ →∞

→∞ →∞ →∞

= = =

= = =

(1/ )O β

Harmonic Oscillator Potential in 1D


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

2

( )2xV x =

2 21( ) ( ) ( ) ( ) | ( ) | ( ), with | ( ) | 12

x x V x x x x x dxµ φ φ φ β φ φ φ∞

−∞

′′= − + + =∫0β =

2

22

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

x l e H x l l

d eH x edx

H x H x x H x x

φ µπ

− −

−

= = + =

= −

= = = −




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


20 /2 0 01/41 1( ) ( ) , : ( ) : ( )

2x

g g g g g gx x e E Eφ φ φ µ µ φπ

−= = = = = =

20 1/2 /2 0 01/41 ( 1)( ) ( ) (2 !) ( ), : ( ) : ( )

2j x

j j j j j j jjx x j e H x E Eφ φ φ µ µ φ

π− − +

= = = = = =

| | (1)oβ =

20 /2 0 00 01/4

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

xg g g g g g g gx x e E E E C Cβφ φ φ φ µ µ φ µ φ β

π−≈ = = ≈ = + = ≈ = +

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

j C x dx

βφ φ φ φ

µ µ φ µ φ β φ∞

∞

+≈ = ≈ = +

+= ≈ = + ∫


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:

1β

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

µ β µµ φ φ β φ φ φ

µ βφ µ µβ

∞

∞

− ≤= + ⇒ =

= = ⇒ ≈ =∫

1/3( )O βTF2 gx µ= ±

TF TFkin( ) , ( )g gE Eφ φ= ∞ = ∞


– 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 2

1 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

βφ

βφ

βµ µ

≈ = =

≈ = =

= − ≈ − = =

≈ − − =

int, pot, kin,3 2 1lim , lim , lim , lim 0,5 3 3

g g g g

g g g g

E E E EE E Eβ β β βµ→∞ →∞ →∞ →∞

= = = =

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

β β βφ φ φ φ µ µβ β ββ β β

β β β

∞− = − = = +

= + = + = +


– 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

µ φ φ β φ φ

µ β µφ

µ βφ µ µβ

∞

∞

= +

− < ≤=

= = ⇒ ≈ =∫


– Matched asymptotic approximation

– Width of interior layer:

– Ordering:

MA MA 2 MAMA MA1 1 1

1 1sign( ) ( / 2) / , 0 | | 2 ( ) [tanh( ) sign( )]

0, otherwisex x xx x xµ µ β µφ µ

β − < ≤= − +

MA 1/3 MA 2/31 1(1/ ) (1/ ) ( )O O Oµ β µ β= ⇐ =

1 1( ) ( ) ( ) ( )g gE Eφ φ µ φ µ φ< ⇒ <

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 xψ ν ψ ν ψ= + +

2 2kin 0 pot 0 int 0

0

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

( ( )) | 0

dE x E E Ed E x

d ν

ψ ν ψ ν ψ ν ψ

ψν

−

=

= + + + + +

=

ν

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 | ( ) | 1dπ

µ φ θ φ θ β φ θ φ θ θ π

φ θ π φ θ φ θ θ

′′= − + ≤ ≤

+ = =∫0β =

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

l l

φ θ φ θ θ φ θ θπ π π

µ µ µ

+

+

= = =

= = = =

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 01( ) ( ) , : ( ) 0 : ( )2g g g g g gx x E Eφ φ φ µ µ φπ

= = = = = =

20 0 01( ) ( ) cos( ), : ( ) : ( )

2j j j j j jjx x l E Eφ φ θ φ µ µ φ

π= = = = = =

| | (1)oβ =

0 0 01( ) ( ) , : ( ) , : ( )8 42g g g g g gx x E E β βφ φ φ µ µ φπ ππ

= = = = = =

0( ) is a solution ( ) is also a solutionφ θ φ θ θ⇒ +

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)– Backward Euler + spectral method (BESP): (Bao, Chern & Lim, JCP, 06’)

Gauss-Seidel iteration method: (W.W. Lin et al., JCP, 05’)

Spectral method + stabilization: (Bao, I. Chern & F. Lim, JCP, 06’)

Reguarized Newton method: (Bao, Wen & Xu, 15’)

( )E φ

Imaginary time method

Idea: Steepest decent method + Projection

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

2 21

11

1

0 0

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

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

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

t n n

nn

n

Ex t V x t t t

xx t n

xx x

tt

δ φφ φ φ β φ φδ φ

φφ

φ

φ φ φ

+

−

++ −

+

= − = ∇ − − ≤ <

= =

= =

0φ

1φ

2φ 1φ

??)()()()ˆ(

)()ˆ(

01

11

01

φφφφ

φφ

EEEE

EE

<<

<

gφ

00 0

( , ) ( ) with ( ,0) ( ) ( )ji tj j j j

j jx t a e x x x a xµψ φ ψ ψ φ

∞ ∞−

= =

= = =∑ ∑

i tτ =0 1

0 00 0

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

j

xx x t a e x xa e

µ µ τµ τµ τ

φ τφ τ ψ φ φ∞ < < →∞

−−

=

= = ⇒ →∑

Normalized gradient flow

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.

ttx t V x t

tx x x

µ φφ φ φ β φ φ φφ

φ φ φ

= ∇ − − + ≥

= =

0|| (., ) || || || 1, ( (., )) 0, 0dt E t td t

φ φ φ= = ≤ ≥

2

1 1 11( , ) ( , ) ( , ), 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 ttφ φφ µ φ µ φ φ φ− + −

+ + += ∆ < < = − =∆

2 21( , ) ( ) | | ( , ) , 0,2t nx t V x t t tφφ φ φ β φ φ µ φ= ∇ − − + ∆ ≥

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’)

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

– Crank-Ncolson FD for continuous normalized gradient flow

21

11 0 0

1

1( , ) ( ) | | , ( , ), , ( , ) ( , ) 02

( , )( , ) , ( ,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

φ φ φ β φ φ φ φ

φφ φ φ φ

φ

+

−

++ −

+

= − − ∈Ω = ≤ < = =

= = =

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 ah x k tM−

= ∆ = = ∆ >

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

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 & asymptotic excited state• Optical lattice potential:

– Ground & asymptotic excited state with potential next

otherwise;100)( ∞≤≤= xxV

2/xV(x) 2=

2/)4()( 22xxV −=

)4(sin122/)( 22 xxxV +=

1 2 3 1 2 3( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 0.5000 1.5000 2.5000 3.500 0.5000 1.5000 2.5000 3.500

3.1371 1.0441 1.9414 2.8865 3.8505 1.5266 2.3578 3.2590 4.191931.371 3.9810 4.7438 5.5573 6.4043 6.5527 7.2802 8.0432 8.8349

156.

g gE E E Eβ φ φ φ φ µ φ µ φ µ φ µ φ

1 2 1 2

855 11.464 12.191 12.944 13.719 19.070 19.784 20.512 21.252313.71 18.171 18.891 19.629 20.383 30.259 30.971 31.691 32.419Observations

( ) ( ) ( ) ( ) ( ) ( ) , for any fixed ( )

lim 1 lim( )

g g

j

g

E E EEEβ β

φ φ φ µ φ µ φ µ φ β

φ µφ→+∞ →+∞

< < < ⇒ < < <

=

( )1

( )j

g

φµ φ

=

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 & asymptotic excited states

– In 3D • Optical lattice potential:

ground asymptotic excited states next

Part III

Analysis & Computation

for

Dynamics in BEC

Dynamics of a BEC

Time-dependent NLSE / GPE

Well-posedness & dynamical laws – Well-posedness & finite time blow-up– Dynamical laws

• Soliton solutions• Center-of-mass • An exact solution under special initial data

– Numerical methods and applications

2 2

0

1( , ) ( ) | | , , 02

( ,0) ( )

dti x t V x x t

x x

ψ ψ ψ β ψ ψ

ψ ψ

∂ = − ∇ + + ∈ >

=

Dynamics with no potential

Momentum conservationDispersion relationBright soliton in 1D:

( ) 0, dV x x≡ ∈

( ) : Im (0) 0d

J t d x J tψ ψ= ∇ ≡ ≥∫

2( ) 21( , )2

i k x tx t Ae k Aωψ ω β−= ⇒ = +

2 20

1( ( ) )2

0( , ) sech( ( ))i vx v a tax t a x vt x e

θψ

β− − +

= − −−

Dynamics with harmonic potential

Harmonic potential

Center-of-mass:

An analytical solution if

2 2

2 2 2 2

2 2 2 2 2 2

11( ) 22

3

x

x y

x y z

x dV x x y d

x y z d

γγ γ

γ γ γ

== + = + + =

2( ) ( , )

dcx t x x t d xψ= ∫

2 2 2( ) diag( , , ) ( ) 0, 0 each component is periodic!!c x y z cx t x t tγ γ γ+ = > ⇒

0 0( ) ( )sx x xψ φ= −

( , )0

2 2

( , ) ( ( )) , (0) & ( , ) 0( , ) : | ( , ) | | ( ( )) | -- moves like a particle!!

si t iw x ts c c

s c

x t e x x t e x x w x tx t x t x x t

µψ φ

ρ ψ φ

−= − = ∆ =

⇒ = = −

2 21( ) ( ) | |2s s s s s sx V xµ φ φ φ β φ φ= − ∇ + +

Well-posedenss

Theorem (T. Cazenave, 03’; C. Sulem & P.L. Sulem, 99’’) Assumptions

– Local existence, i.e.

– Global existence, i.e. if

2 2

2 2 210

(i) ( ) ( ), ( ) 0, & ( ) ( ) | | 2

(ii) ( ) | ( ) ( )d

d d d

dX L L

V x C V x x D V x L

X u H u u u V x u x d x

α α

ψ

∞ ∞∈ ≥ ∀ ∈ ∈ ≥

∈ = ∈ = + ∇ + < ∞

∫

2

20 ( )

1 or 2 with / or 3& 0db Ld d C dβ ψ β= = ≥ − = ≥

max max(0, ], s. t. the problem has a unique solution ([0, ), )T C T Xψ∃ ∈ ∞ ∈

maxT = +∞

Finite time blowup

Theorem (T. Cazenave, 03’; C. Sulem & P.L. Sulem, 99’) Assumptions

– There exists finite time blowup, i.e. if one of the following holds

– Proof:

2 20 V 0

<0 & ( ) ( ) 0, with 2,3

with finite variance (0):= | ( ) |d

dV x d x V x x d

X x x d x

β

ψ δ ψ

+ ⋅∇ ≥ ∀ ∈ =

∈ < ∞∫

maxT < +∞

2

0

0 0 0

0 0 0 0 0

(i) ( ) 0

(ii) ( ) 0 & Im ( ) ( ( )) 0

(iii) ( ) 0 & Im ( ) ( ( )) ( )

d

dL

E

E x x x d x

E x x x d x E d x

ψ

ψ ψ ψ

ψ ψ ψ ψ ψ

<

= ⋅∇ <

> ⋅∇ < −

∫

∫

2 20

2 '0

( ) : | ( , )| ( ) 2 ( ), 0, 2,3

( ) ( ) (0) (0) 0 * & ( *) 0!!d

V V

V V V V

t x x t d x t d E t d

t d E t t t t

δ ψ δ ψ

δ ψ δ δ δ

= ⇒ ≤ ≥ =

⇒ ≤ + + ⇒ ∃ < < ∞ =

∫

Numerical difficulties for dynamics

Dispersive & nonlinearSolution and/or potential are smooth but may oscillate wildlyKeep the properties of NLS on the discretized level– Time reversible & time transverse invariant – Mass & energy conservation– Dispersion relation

In high dimensions: many-body problems

Design efficient & accurate numerical algorithms– Explicit vs implicit (or computation cost)– Spatial/temporal accuracy, Stability– Resolution in strong interaction regime: 1β

2 2

0

1( , ) ( ) | | , , 02

with ( ,0) ( )

dti x t V x x t

x x

ψ ψ ψ β ψ ψ

ψ ψ

∂ = − ∇ + + ∈ >

=

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: ( , ) ( ) ( , ) | ( , ) | ( , )

| ( , ) | | ( , ) |

( , ) ( , )n

t

t

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 tβψ

ψ ψ

ψ ψ β ψ ψ

ψ ψ

ψ ψ− + ∆+

= − ∇

= +

⇓ =

=

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

2( ) ( ) | ( , ) | unchangedV x V x x tα ψ→ + ⇒

( ) ( ) 2/(2 ), , 1 / dh O k Oε ε ε β += = =

Crank-Nicolson finite difference (CNFD) method

Crank-Nicolson finite difference (CNFD)method

– Implicit: need solve a fully nonlinear system per time step via iterations!!!– Time reversible: Yes– Time transverse invariant: No– Mass conservation: Yes

CNFD

– Stability: Yes– Energy conservation: Yes– Dispersion relation without potential: No– Accuracy

• Spatial: 2nd order• Temporal: 2nd order

– Resolution in semiclassical regime (Markowich, Poala & Mauser, SINUM, 02’)

– Error estimate: Yes

2 2( ) & ( ) ( ) & ( )h o o h O Oε τ ε ε τ ε= = ⇐ = =

Error estimates for CNFD

Assume

Theorem: Assume , there exist sufficiently small, when , we have the following error estimate

In addition, if either , we have

3 1, 2 3, 0 5, 1 10([0, ]; ) ([0, ]; ) ([0, ]; ) &C T W C T W C T W H V Cψ ∞ ∞ ∞∈ ∩ ∩ ∩ ∈

( , )n nj j n je x tψ ψ= −

0C hτ ≤ 0 00 & 0h τ> >

0 00 & 0h h τ τ< ≤ < <

2 2 3/2 3/2[ ] & [ ], 0 /n ne C h e C h n Tτ δ τ τ+≤ + ≤ + ≤ ≤

2 2[ ], 0 /n ne e C h n Tδ τ τ++ ≤ + ≤ ≤

0 20( ) | 0 or ([0, ]; )nV x C T Hψ∂Ω∂ = ∈

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 Jets formation & Movie

x x100 at t 0, 4β ω ω= = →

x x20, at t 0 2 , 2y yβ ω ω ω ω= = → →

At t 0 40 50β= = → −

next

ψψβδψψβψψψ 420

22 ||||)(21),( ixVtx

ti −++∇−=∂∂

Collapse and Explosion of BEC

back

Number of atoms in condensate

back

Central density

back

Numerical results (Bao et., J Phys. B, 04)

Jet formation back

Semiclassical scaling

When , re-scaling

With

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

1β >> 1/2 /4 2/( 2)1 /d dx x εε ψ ψ ε ε β− +→ = =

22 2( , ) ( , ) ( ) ( , ) | ( , ) | ( , )

2i x t x t V x x t x t x t

tε ε ε ε εεε ψ ψ ψ ψ ψ∂

= − ∇ + +∂

22 2 41( ) [ | | ( ) | | | | ] (1)

2 2dE V x dx Oε ε ε ε εεψ ψ ψ ψ= ∇ + + =∫

( ) ( )( ) ( )

1 1 2/( 2)

1 1 2/( 2)

( ) ( )

( ) ( )

d

d

E E O O

O O

ε ε

ε ε

ψ ε ψ ε β

µ ψ ε µ ψ ε β

− − +

− − +

= = =

= = =

Comparison of two scaling

( )( )

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

3/2 3/2 /4

21

2

2/( 2)

2/( 2)

Quanties Thomas-Fermi scaling Semiclassical scaling1 / 1 /

4/ ( )

:

Energy (1)

Chemical potential (1)

le

s x x

d dss s x s s s

sd

s s s

ss x x

s

d

d

tNax x m x x xx

x xmxEt

E O O

O O

ω ωπω ε β

ψ ε

ω ω ε

β

µ β

− + +

− −

−

+

+

= = =

=

( )( )

1/( 2)

/2( 2)

ngth of wave function ( 2 ) (1)

height of wave function ( / ) (1)

d

d d

O O O

O O O

β µ

β µ β

+

− +

=

=

Semiclassical limits

WKB analysis -- Gregor Wentzel, Hans Kramers & Leon Brillouin, 1926

– Formally assume

– Geometrical Optics: Transport + Hamilton-Jacobi

0

22 2

( ) /0 0

( , ) ( ) | |2

( ,0) : ( ) ( )

t

i S x

i x t V x

x x x eε

ε ε ε ε ε

εε ε ε

εε ψ ψ ψ β ψ ψ

ψ ψ ρ

∂ = − ∇ + +

= =

/ , ,iSe v S J vεε ε ε ε ε ε ε εψ ρ ρ= = ∇ =

22

t

( ) 0,1 1( )2 2

t

d

S

S S V x

ε ε ε

ε ε ε ε

ε

ρ ρ

ερ ρρ

∂ +∇• ∇ =

∂ + ∇ + + = ∆

0 1: ε

ε

ψ ψ

<

=

From QM to fluid dynamics

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

– Formal Limits --- Euler equations for fluids

Mathematical justification: G. B. Whitman, E. Madelung, E. Wigner, P.L. Lious, P. A. Markowich, F.-H. Lin, P. Degond, C. D. Levermore, D. W. McLaughlin, E. Grenier, F. Poupaud, C. Ringhofer, N. J. Mauser, P. Gerand, R. Carles, P. Zhang, P. Marcati, J. Jungel, C. Gardner, S. Kerranni, H.L. Li, C.-K. Lin, C. Sparber, …..

– Linear case – NLSE before caustics

2

2

( ) 0 ( ) / 2

( ) ( ) ( ) ( ln ) 4

t

t

v P

J JJ P V

ε ε ε

ε εε ε ε ε ε

ε

ρ ρ ρ β ρ

ερ ρ ρ ρρ

∂ +∇• = =

⊗∂ +∇• +∇ + ∇ = ∇ ∆

0 0 0 2

0 00 0 0

0

( ) 0 ( ) / 2

( ) ( ) ( ) 0

t

t

v P

J JJ P V

ρ ρ ρ β ρ

ρ ρρ

∂ +∇• = =

⊗∂ +∇• +∇ + ∇ =

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

′′′−Φ′+

Ω−+−=

∫

∇∫

),(),()(),(*),(*21

),(])(2

[),(*)(22

ψψψψ

ψψψ

*)(

ψδψδ H

tt),xψ(i =

∂∂

.4),()1()( 200 mxNx aUU s

πδ =−=Φ

),,( zyxx =

Rotating BEC

The 3D Gross-Pitaevskii equation ( )

– Angular momentum rotation

– V is a harmonic trap potential

– Normalization condition

ψψψ ]||)(2

[),( 20

22

UNLxVm

txt

i z +Ω−+∇−=∂∂

)(2

)( 222222zyxmxV zyx ωωω ++=

3

2| ( , ) | 1x t dxψ =∫

),,( zyxx =

∇−=×=∂−≡∂−∂−=−=

iPPxLiyxiypxpL xyxyz ,,)(: θ

Rotating BEC

General form of GPE ( )

with

Normalization condition

2 21( , ) [ ( ) | | ] ( , )2 zi x t V x L x t

tψ β ψ ψ∂

= − ∇ + −Ω +∂

dx R∈

22 24

32 22 2 2

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

z y

y z

x y dz dzV x

x y z d

γβ ββ π

β

γϕγ γ

∞

−∞

+ =≈ = = + + =

∫

2| ( , ) | 1.d

x t dxψ =∫

θ∂−≡∂−∂−= iyxiL xyz )(:

Rotating BEC

Conserved quantities– Normalization of the wave function

– Energy

Chemical potential

∫ ===d

NxdtxtNR

2 1))0((|),(|))(( ψψψ

2 2 41( ( )) [ | | ( ) | | * | | ]2 2

( (0))

d zt V x L dxEE

βψ ψ ψ ψ ψ ψ

ψ

Ω

Ω

= ∇ + −Ω +

=

∫

2 2 41( ( )) [ | | ( ) | | * | | ]2d zt V x L dxψ ψ ψ ψ ψ β ψµ

Ω= ∇ + −Ω +∫

Stationary states

Stationary solutions of GPE

Nonlinear eigenvalue problem with a constraint

Relation between eigenvalue and eigenfunction

)(),( xtx e ti φψ µ−=

2 2

2

1( ) [ ( ) | ( ) | ] ( ),2

| (x) | 1d

zx V x L x x

dx

µ φ β φ φ

φ

= − ∇ + −Ω +

=∫

4( ) ( ) | (x) |2 d

E dxβµ µ φ φ φΩ Ω= = + ∫

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’

4g|| || 1

( ) min ( ), ( ) ( ) | ( ) |2 dg g g gE E E x dx

φ

βφ φ µ µ φ φ φΩ Ω Ω Ω== = = + ∫

0&1|| ≥<Ω dβ

0&0 ≥=Ω dβ

10 ≤Ω< cβ

Numerical results

– Ground states: in 2D in 3D isosurface

– Vortex lattice• Symmetric trapping anisotropic trapping

– Giant vortex• In 2D In 3D

next

Ground states of rapid rotation

BEC@MIT

back

http://cua.mit.edu/ketterle_group/Projects_2001/Vortex_lattice/GrayLattice.jpg

http://cua.mit.edu/ketterle_group/Projects_2001/Vortex_lattice/GrayLattice.jpg

Dynamical laws of rotating BEC

Time-dependent Gross-Pitaevskii equation

Dynamical laws – Time reversible & time transverse invariant– Conservation laws– Well-posedness & finite-time – Dynamics of a stationary state with its center shifted

Numerical methods & results

2 2

0

1( , ) [ ( ) | | ]2

( ,0) ( ), : ( )

z

z y x

i x t V x Ltx x L i x y i θ

ψ β ψ ψ

ψ ψ

∂= − ∇ + −Ω +

∂= = − ∂ − ∂ = − ∂

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., H. Li &J . Shen, SISC 09’)

A method via rotating Lagrange coordinate (B., et al. SISC, 13’)

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

)(:,21),( :1 Step

2

2

nddt

xyzzt

txtxtxtxtxxVtxi

iyxiLLtxi

ψψψψβψψ

ψψψ θ

=⇒+=

∂−=∂−∂−=Ω−∇−=

TSFP 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

TSFP 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

TSLHP for rotating BEC

Numerical Method three: (Bao, LI & Shen, SISC 09’)

– Ideas• Time-splitting • Polar/cylindrical coordinates in 2D/3D, respectively• Lagurre + Fourier basis in 2D• Lagurre + Fourier +Hermite in 3D

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

Numerical methods A new formulation – A rotating Lagrange coordinate:

– GPE in rotating Lagrange coordinates

– Analysis & numerical methods -- Bao & Wang, JCP6’; Bao, Li & Shen, SISC, 09’; Bao, Marahrens, Tang & Zhang, 13’, …….

cos( ) sin( ) 0cos( ) sin( )

( ) for 2; ( ) sin( ) cos( ) 0 for 3sin( ) cos( )

0 0 1

t tt t

A t d A t t t dt t

Ω Ω Ω Ω = = = − Ω Ω = − Ω Ω

1( ) &

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

x A t x

x t x t A t x tφ ψ ψ

−=

= =

2 21( , ) [ ( ( ) ) | | ] , , 02

dti x t V A t x x tφ β φ φ∂ = − ∇ + + ∈ >

A Method via Rotating Lagrange Coordinate

Numerical methods A new formulation – A rotating Lagrange coordinate:

– GPE in rotating Lagrange coordinates

– Analysis & numerical methods -- Bao & Wang, JCP6’; Bao, Li & Shen, SISC, 09’; Bao, Marahrens, Tang & Zhang, 13’, …….

cos( ) sin( ) 0cos( ) sin( )

( ) for 2; ( ) sin( ) cos( ) 0 for 3sin( ) cos( )

0 0 1

t tt t

A t d A t t t dt t

Ω Ω Ω Ω = = = − Ω Ω = − Ω Ω

1( ) &

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

x A t x

x t x t A t x tφ ψ ψ

−=

= =

2 21( , ) [ ( ( ) ) | | ] , , 02

dti x t V A t x x tφ β φ φ∂ = − ∇ + + ∈ >

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 φψ =

21:&21: →→ yx γγ

2.21:&8.11: →→ yx γγ

1,8.0,100 ===Ω= zy γγβ

canisotropi :),(,9.0,1000 txV

=Ω=β


When , re-scaling

With

Leading asymptotics

1β >> 1/2 /4 2/( 2)1 /d dx x εε ψ ψ ε ε β− +→ = =

22 2( , ) [ ( ) | | ]

2 zi x t V x Lt

ε ε εεε ψ ε ψ ψ∂= − ∇ + − Ω +

∂

22 2 4

, R

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

(1)

d zE V x L dx

O

ε ε ε ε ε εε

εψ ψ ψ ε ψ ψ ψΩ = ∇ + − Ω +

=

∫

( ) ( ) ( )1 1 2/( 2) 2/( 2),( ) ( ) , ( )d dE E O O Oε

εψ ε ψ ε β µ ψ β− − + +Ω Ω Ω= = = =

Quantum Hydrodynamics

SetGeometrical Optics: (Transport + Hamilton-Jacobi)

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

/ 2/( 2), , , 1 /iS de v S J vεε ε ε ε ε ε ε εψ ρ ρ ε β += = ∇ = =

22

t

ˆ ˆ( ) 0, : ( )

1 1ˆ( )2 2

t z z y x

z

S L L x y

S S V x L S

ε ε ε εθ

ε ε ε ε ε

ε

ρ ρ ρ

ερ ρρ

∂ +∇• ∇ +Ω = = ∂ − ∂ ≡ ∂

∂ + ∇ + + +Ω = ∆

2

2

ˆ( ) 0

ˆ( ) ( ) ( ) ( ln )4

0 1( ) / 2 , J , A

1 0

t z

t z

v L

J JJ P V L J A J

P v

ε ε ε ε

ε εε ε ε ε ε ε ε

ε

ε ε ε

ρ ρ ρ

ερ ρ ρ ρρ

ρ ρ ρ

∂ +∇• +Ω =

⊗∂ +∇• +∇ + ∇ +Ω +Ω = ∇ ∆

= = = −

Dipolar Quantum Gas

Experimental setup– Molecules meet to form dipoles – Cool down dipoles to ultracold – Hold in a magnetic trap – Dipolar condensation – Degenerate dipolar quantum gas

Experimental realization– Chroimum (Cr52)– 2005@Univ. Stuttgart, Germany– PRL, 94 (2005) 160401

Big-wave in theoretical studyA. Griesmaier,et al., PRL, 94 (2005)160401

BEC with strong DDI

Lu, Burdick, Youn & Lev, PRL 107 (2011), 190401.

164 Dy

Mathematical Model

Gross-Pitaevskii equation (re-scaled)

– Trap potential– Interaction constants– Long-range dipole-dipole interaction kernel

References:– L. Santos, et al. PRL 85 (2000), 1791-1797– S. Yi & L. You, PRA 61 (2001), 041604(R); D. H. J. O’Dell, PRL 92 (2004), 250401

( )2 2ext dip

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

i x t V x U x ttψ β ψ λ ψ ψ∂ = − ∆ + + + ∗ ∂

3( , )x t xψ ψ= ∈

( )2 2 2 2 2 2ext

1( )2 x y zV z x y zγ γ γ= + +

20 dip

20 0

4 (short-range), (long-range)3

s mNN aa a

µ µπβ λ= =

2 2 23

dip 3 33 1 3( ) / | | 3 1 3cos ( )( ) , fixed & satisfies | | 1

4 | | 4 | |n x xU x n n

x xθ

π π− ⋅ −

= = ∈ =

Mathematical Model

Mass conservation (Normalization condition)

Energy conservation

Long-range interaction kernel:– It is highly singular near the origin !! At singularity near the origin !! – Its Fourier transform reads

• No limit near origin in phase space !! • Bounded & no limit at far field too !!• Physicists simply drop the second singular term in phase space near origin!!• Locking phenomena in computation !!

3 3

2 22( ) : ( , ) ( , ) ( ,0) 1N t t x t d x x d xψ ψ ψ= ⋅ = ≡ =∫ ∫

3

2 2 4 2 2ext dip 0

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

E t V x U d x Eβ λψ ψ ψ ψ ψ ψ ψ ⋅ = ∇ + + + ∗ ≡ ∫

23

dip 23( )( ) 1

| |nU ξξ ξξ⋅

= − + ∈

31

| |O

x

A New Formulation

Using the identity (O’Dell et al., PRL 92 (2004), 250401, Parker et al., PRA 79 (2009), 013617)

Dipole-dipole interaction becomes

2

dip 3 2

2

dip 2

3 3( ) 1( ) 1 ( ) 3 4 4

3( ) ( ) 1| |

n nn xU x x

r r r

nU

δπ π

ξξξ

⋅ = − = − − ∂

⋅⇒ = − +

2 2dip

2 2

| | | | 3

1 | | | |4

n nU

r

ψ ψ ϕ

ϕ ψ ϕ ψπ

∗ = − − ∂

= ∗ ⇔ −∆ =

| | & & ( )n n n n nr x n= ∂ = ⋅∇ ∂ = ∂ ∂

A New Formulation

Gross-Pitaevskii-Poisson type equations (Bao,Cai & Wang, JCP, 10’)

Energy

2ext

2

| |

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

( , ) | ( , ) | , lim ( , ) 0

n n

x

i x t V x x tt

x t x t x t

ψ β λ ψ λ ϕ ψ

ϕ ψ ϕ→∞

∂ = − ∆ + + − − ∂ ∂ − ∆ = =

3

2 2 4 2ext

1 3( ( , )) : | | ( ) | | | | | |2 2 2 nE t V x d xβ λ λψ ψ ψ ψ ϕ− ⋅ = ∇ + + + ∂ ∇ ∫

Ground State

Non-convex minimization problem

Nonlinear Eigenvalue problem (Euler-Language eq.)

Chemical potential

2ext

2

| |

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

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

n n

x

x V x x

x x x

µ φ β λ φ λ ϕ φ

ϕ φ ϕ φ→∞

= − ∆ + + − − ∂ − ∆ = = =

( ) : min ( ) with | 1& ( )g SE E S E

φφ φ φ φ φ

∈= = = < ∞

3

3

2 2 4 2ext

4 2 2

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

3( ) | | | | , & | |2 2

n

n

V x d x

E d x

µ φ φ β λ φ λ ϕ

β λ λφ φ ϕ ϕ φ

= ∇ + + − + ∂ ∇

− = + + ∂ ∇ − ∆ =

∫

∫

Ground State Results

Theorem (Existence, uniqueness & nonexistence) (Bao, Cai & Wang, JCP, 10’)

– Assumptions

– Results• There exists a ground state if

• Positive ground state is unique

• Nonexistence of ground state, i.e.– Case I: – Case II:

3ext ext| |

( ) 0, & lim ( ) (confinement potential)x

V x x V x→∞

≥ ∀ ∈ = +∞

g Sφ ∈ 0 &2ββ λ β≥ − ≤ ≤

00| | with i

g ge θφ φ θ= ∈

lim ( )S

Eφ

φ∈

= −∞0β <

0 & or 2ββ λ β λ≥ > < −

Key Techniques in Proof

Estimate on the Poisson equation

Positivity & semi-lower continuous

The energy is strictly convex in if

Confinement potentialNon-existence result

2( ) (| |) ( ), with | |E E E Sφ φ ρ φ ρ φ≥ = ∀ ∈ =

224| |

| | : & lim ( ) 0 ( )nxxϕ φ ρ ϕ ϕ ϕ ϕ ρ φ

→∞−∆ = = = ⇒ ∂ ∇ ≤ ∇ ∇ = ∆ = =

( )E ρ ρ0 &

2ββ λ β≥ − ≤ ≤

1 2

2 2 23

, 1/2 1/41 2 1 2

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

x y zx xε εφπε πε ε ε

+= − − ∈

Numerical Method for Ground State

Gradient flow with discrete normalization

Full discretization– Backward Euler sine pseudospectal (BESP) method – Avoid to use zero-mode in phase space via DST !!

2ext

21| |

11 1

1

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

( , ) | ( , ) | , lim ( , ) 0, & ,

( , )( , ) : ( , ) , & 0,( , )

( , ) | ( , ) | 0, 0

n n

n nx

nn n

n

x x

x t V x x tt

x t x t x t x t t t

x tx t x t x nx t

x t x t t

φ β λ φ λ ϕ φ

ϕ φ ϕ

φφ φφ

φ ϕ

+→∞

−+ +

+ + −+

∈∂Ω ∈∂Ω

∂ = ∆ − − − + ∂ ∂ − ∆ = = ∈Ω ≤ <

= = ∈Ω ≥

= = ≥

0 0; ( ,0) ( ) 0, , with 1.x x xφ φ φ= ≥ ∈Ω =

Dynamics and its Computation

The Problem

Mathematical questions– Existence & uniqueness & finite time blow-up???

Existing results– Carles, Markowich & Sparber, Nonlinearity, 21 (2008), 2569-2590– Antonelli & Sparber, 09, preprint --- existence of solitary waves.

2ext

2 3

| |

30

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

( , ) | ( , ) | , lim ( , ) 0, , 0

( ,0) ( ), ,

n n

x

i x t V x x tt

x t x t x t x t

x x x


ϕ ψ ϕ

ψ ψ→∞

∂ = − ∆ + + − − ∂ ∂ − ∆ = = ∈ >

= ∈

Well-posedenss Results

Theorem (well-posedness) (Bao, Cai & Wang, JCP, 10’)

– Assumptions

– Results• Local existence, i.e.

• If global existence, i.e.

2 2

3 3 3ext ext ext

2 2 21 30 ext

3

(i) ( ) ( ), ( ) 0, & ( ) ( ) | | 2

(ii) ( ) | ( ) ( )X L L

V x C V x x D V x L

X u H u u u V x u x d x

α α

ψ

∞ ∞∈ ≥ ∀ ∈ ∈ ≥

∈ = ∈ = + ∇ + < ∞

∫

0 &2ββ λ β≥ − ≤ ≤

max max(0, ], s. t. the problem has a unique solution ([0, ), )T C T Xψ∃ ∈ ∞ ∈

maxT = +∞

Finite Time Blowup Results

Theorem (finite time blowup) (Bao, Cai & Wang, JCP, 10’)

– Assumptions

– Results: • For any , there exists finite time blowup, i.e. • If one of the following conditions holds

3ext ext

(i) <0 or 0 & or >2

(ii) 3 ( ) ( ) 0,V x x V x x

ββ β λ λ β≥ < −

+ ⋅∇ ≥ ∀ ∈

maxT < +∞0( )x Xψ ∈

3

2

3

0

0 0 0

0 0 0 0 0

(i) ( ) 0

(ii) ( ) 0 & Im ( ) ( ( )) 0

(iii) ( ) 0 & Im ( ) ( ( )) 3 ( )L

E

E x x x d x

E x x x d x E x

ψ

ψ ψ ψ

ψ ψ ψ ψ ψ

<

= ⋅∇ <

> ⋅∇ < −

∫

∫

Numerical Method for dynamics

Time-splitting sine pseudospectral (TSSP) method, – Step 1: Discretize by spectral method & integrate in phase space exactly

– Step 2: solve the nonlinear ODE analytically

1[ , ]n nt t +

21( , )2ti x tψ ψ∂ = − ∇

2ext

2

2

2ext

( , ) ( ) ( ) | ( , ) | 3 ( , ) ( , )

( , ) | ( , ) | , (| ( , ) | ) 0 | ( , ) | | ( , ) | & ( , ) ( , )

( , ) ( ) ( ) | ( , ) | 3 ( , ) (

t n n

t n n

t n n n n

i x t V x x t x t x t

x t x tx t x t x t x t x t

i x t V x x t x t


ϕ ψ

ψ ψ ψ ϕ ϕ


∂ = + − − ∂ − ∆ =

⇓ ∂ = ⇒ = =

∂ = + − − ∂

2ext

2

( )[ ( ) ( )| ( , )| 3 ( , )]

, )

( , ) | ( , ) | ,

( , ) ( , )n n n n n

n n

i t t V x x t x tn

x t

x t x t

x t e x tβ λ ψ λ ϕ

ϕ ψ

ψ ψ− − + − − ∂

− ∆ =

⇒ =

New numerical methods for DDI

How to compute nonlocal DDI– FFT (fast Fourier transform)

– DST (discrete sine transform)

– Nonuniform FFT

2dip

2

dip 2

: | |

3( )( ) 1| |

U

nU

φ ψ

ξξξ

= ∗

⋅= − +

2 2| | 3 & ( , ) | ( , ) |nn x t x tφ ψ ϕ ϕ ψ= − − ∂ − ∆ =

3

2

2dip

sphere coordinate2

dip

( ) ( , ) =| |

( ) | | ( , )S

U t d

U t

φ ξ ρ ζ ξ ρ ψ

ξ ξ ρ ζ+×

=

=

∫

∫

Dynamics of a vortex lattice

Dimension Reduction (3D 2D)

Assumptions

Decomposition of the linear operator

Ansatz

2

ext 2 41& (1) & ( ) ( , ) , :

2z x y Dz

zO V x V x yγ γ γ εε γ

= = + =

22

22 1/4 21( , , , ) ( , , ) ( ) & ( ) exp

( ) 2

i t zx y z t e x y t z zεε εψ ψ ω ω

ε π ε−

≈ = −

2 2

4 21 1 12 2 2 2z zz zz

z zLε ε

= − ∂ + = − ∂ +

ext 21 1: ( ) ( , )2 2 D zL V x V x y L⊥= − ∆ + = − ∆ + +

Dimension Reduction (3D 2D)

2D equations when (Bao, Cai, Lei, Rosenkranz, PRA, 10’)

Energy

223

2

23

1/2 2

|( , )|

1 3( , , ) [ ( , ) | |2 2

3 ( ) ] ( , , )2

( ) ( , , ) | ( , , ) | , lim ( , , ) 0

D

n n

x y

ni x y t V x yt

n x y t

x y t x y t x y t

β λ λψ ψε π

λ ϕ ψ

ϕ ψ ϕ

⊥ ⊥

⊥

⊥

⊥ →∞

∂ − += − ∆ + +

∂

− ∂ − ∆

−∆ = =

0ε →

3

2 2 2 42 3

2 21/4 2 1/43

1 1( ( , )) : | | ( ) | | ( 3 ) | |2 2 2

3 + [ ( ) ( ) ]4

D

n

E t V x n

n dx

ψ ψ ψ β λ λ ψε π

λ ϕ ϕ⊥

⊥

⊥ ⊥ ⊥

⋅ = ∇ + + − +

∂ −∆ − ∇ −∆

∫

Two-component BEC

The 3D coupled Gross-Pitaevskii equations

Normalization conditions

Intro- & inter-atom Interactions

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 L U Ut m

i x t V x L U Ut m

ψ ψ ψ ψ λ ψ

ψ ψ ψ ψ λ ψ

∂= − ∇ + −Ω + + −

∂

∂= − ∇ + −Ω + + −

∂

3 3

22 0 0 0 2

1 21

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

N t x t dx N N N N x dxψ ψ=

= = + = =∑ ∫ ∫

2

12 21

4 with jl

jl

aU a a

mπ

= =

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( , ) [ ( ) | | | | ]21( , ) [ ( ) | | | | ]2

z

z

i x t V x Lt

i x t V x Lt

ψ β ψ β ψ ψ λψ

ψ β ψ β ψ ψ λψ

∂= − ∇ + −Ω + + −

∂∂

= − ∇ + −Ω + + −∂

3 3

2 21 2( ) | ( , ) | | ( , ) | 1N t x t dx x t dxψ ψ= + =∫ ∫

3 3

0 02 21 2

1 2| ( , ) | , | ( , ) |N Nx t dx x t dxN N

ψ ψ= =∫ ∫

0λ ≠

0λ =

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

lE V x L dx

βψ ψ ψ ψ ψ ψ λ ψ ψ

=

Ψ = ∇ + −Ω + −∑ ∑∫

0 0 01 2 10, , ( )N N N O Nλ = =

3 3

0 02 22 1

2 2 1 1( ) | ( , ) | : 1, ( ) | ( , ) | : 1 1N NN t x t dx N t x t dxN N

ψ ε ψ ε= = = = = = − ≈∫ ∫

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 OE

ψ β ψ ψ

ψψ ψ β β εψ

∂= − ∇ + −Ω +

∂Ψ −

= =

Ground state

No external field:

Nonlinear eigenvalue problem

Existence & uniqueness of positive solutionNumerical methods can be extended

0λ =

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

min ( , ) with 1Eφ α φ β

φ φ α β= =

+ =

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

µ φ β φ β φ φ

µ φ β φ β φ φ

= − ∇ + −Ω + +

= − ∇ + −Ω + +

Ground states

crater

Ground state

With external field:

Nonlinear eigenvalue problem

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

0λ ≠

2 21 2

1 2|| || || || 1

min ( , )Eφ φ

φ φ+ =

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

µ φ β φ β φ φ λφ

µ φ β φ β φ φ λφ

= − ∇ + −Ω + + −

= − ∇ + −Ω + + −

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 22β β β= =

Dynamics

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

εε ψ ε α ψ α ψ ψ ελψ

εε ψ ε α ψ α ψ ψ ελψ

∂= − ∇ + − Ω + + −

∂

∂= − ∇ + − Ω + + −

∂

0λ =

0λ ≠

Spin-orbit coupled BEC

Coupled GPE with a spin-orbit coupling & internal Josephson junction

Experiments: Lin, et al, Nature, 471( 2011), 83.

Applications ----Topological insulator

Analysis & numerical methods:– For ground state & dynamics (Bao & Cai, 14’)

2 2 21 0 11 1 12 2 1 1

2 2 21 0 21 1 22 2 1 1

1[ ( ) ( | | | | )]21[ ( ) ( | | | | )]2

x

x

i V x ikt

i V x ikt

ψ δ β ψ β ψ ψ ψ

ψ δ β ψ β ψ ψ ψ

−

− −

∂= − ∇ + + ∂ + + + +Ω

∂∂

= − ∇ + − ∂ + + + +Ω∂

Spinor BEC

Spinor F=1 BEC

With

22 * 2

1 1 1 0 1 1 1 0

22 *

0 0 1 1 1 1 1 0

22 * 2

1 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

ψ ρ ψ ρ ρ ρ ψ ψ ψ

ψ ρ ψ ρ ρ ψ ψ ψ ψ

ψ ρ ψ ρ ρ ρ ψ ψ ψ

− −

− −

− − −

∂= − ∇ + −Ω + + + − +

∂

∂= − ∇ + −Ω + + + +

∂

∂= − ∇ + −Ω + + + − +

∂

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 sa a a ag g

m ma a

π πρ ρ ρ ρ ρ ψ−

+ −= + + = = =

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 dxψ ψ−=−

= ≡ + + = =∑ ∫ ∫

3 3

2 2 0 01 1 1 1( ) | ( , ) | | ( , ) | :M t x t dx x t dx N N Mψ ψ−= − ≡ − =∫ ∫

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

dn

j j j z j

ss

gE V x L

g g dx

ψ ψ ψ ψ ρ

ρ ρ ρ ρ ρ ρ ρ ρ ψ ψ ψ ψ ψ ψ− − − − −

Ψ = ∇ + −Ω + +

+ + + − + +

∑∫

Spinor BEC

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

Dynamics – Dynamical laws – Numerical methods: TSSP

Semiclassical limit & hydrodynamics equation??

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– Rotating BEC & dipolar BEC– Multi-component BEC – two-component, spin-1 BEC, spin-orbit-coupled BEC

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, random potential– Superfluidity & dissipation, quantized vortex lattice – Coupling GPE & QBE for BEC at finite temperature– Mathematical theory for BEC– Interdisciplinary research: experiment,physics, mathematics, computation, ….

References

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References

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References

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[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.

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[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., Vol 4, pp.135-160, 2007. .

[24]. W. Bao and F. Y. Lim, Computing Ground States of Spin-1 Bose-Einstein Condensates by the Normalized Gradient Flow, SIAM J. Sci. Comput., Vol 30, 2009, pp1925--1948.

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