Computational Methods for the Dynamics of Nonlinear Schrodinger / Gross-Pitaevskii Equations

Weizhu Bao

Department of Mathematics

& Center for Computational Science and Engineering National University of Singapore

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

& Beijing Computational Science Research Center (CSRC)

URL: http://www.csrc.ac.cn

Outline

Nonlinear Schrodinger / Gross-Pitaevskii equations Dynamical properties – Conserved quantities – Center-of-mass & an analytical solution – Specific solutions – soliton in 1D

Numerical methods – Finite difference time domain (FDTD) methods – Time-splitting spectral (TSSP) method – Applications – collapse & explosion of a BEC, vortex lattice dynamics

Extension to -- rotation, nonlocal interaction

Conclusions

NLSE / GPE

The nonlinear Schrodinger equation (NLSE) --1925

– t : time & : spatial coordinate – : complex-valued wave function – : real-valued external potential – : given interaction constant

• (=0: linear; >0: repulsive & <0: attractive)

– : scaled Planck constant • ( : standard; : semiclassical)

22 2( , ) ( ) | |

2ti x t V xεε ψ ψ ψ β ψ ψ∂ = − ∇ + +

( )dx ∈

( , )x tψ

( )V x

β

0 1ε< ≤0 1& 1ε β< = ±1ε =

Model for BEC

Bose-Einstein condensation (BEC): – Bosons at nano-Kevin temperature – Many atoms occupy in one obit -- at quantum mechanical ground state – Form like a `super-atom’, New matter of wave --- fifth state

Theoretical prediction – S. Bose & E. Einstein 1924’

Experimental realization – JILA 1995’

2001 Noble prize in physics – E. A. Cornell, W. Ketterle, C. E. Wieman

Mean-field approximation – Gross-Pitaevskii equation (GPE) :

• E.P. Gross 1961’; L.P. Pitaevskii 1961’

BEC@ JILA

Laser beam propagation

Nonlinear wave (or Maxwell) equations Helmholtz equation – time harmonic

In a Kerr medium Paraxial (or parabolic) approximation -- NLSE

Other applications

In plasma physics: wave interaction between electrons and ions – Zakharov system, …..

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

In materials science: – First principle computation – Semiconductor industry

In nonlinear (quantum) optics In biology – protein folding In superfluids – flow without friction

Conservation laws

Dispersive Time symmetric: Time transverse (gauge) invariant Mass (or wave energy) conservation

Energy (or Hamiltonian) conservation

22 2( , ) ( ) | |

2ti x t V xεε ψ ψ ψ β ψ ψ∂ = − ∇ + +

& take conjugate unchanged!!t t→ − ⇒

2/( ) ( ) --unchanged!!i tV x V x e α εα 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= = ≡ = ≥∫ ∫

22 2 4( ) : ( ( , )) ( ) (0), 0

2 2d

E t E t V x d x E tε βψ ψ ψ ψ

= = ∇ + + ≡ ≥ ∫

Dynamics with no potential

Momentum conservation Dispersion relation Soliton solutions in 1D: – Bright soliton when ---- decaying to zero at far-field

– Dark (or gray) soliton ---- nonzero &oscillatory at far-field

( ) 0, dV x x≡ ∈

( ) : Im (0) 0d

J t d x J tψ ψ= ∇ ≡ ≥∫

2( ) 2( , )2

i k x tx t Ae k Aω ε βψ ωε

−= ⇒ = +

0β <

0β >

2 20

1( ( ) )2

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

θψ

β− − +

= − − ∈ ≥−

[ ]2 2 2

01( ( 2 2( ) ) )2

01( , ) tanh( ( )) ( ) , , 0

i kx k a v k tx t a a x vt x i v k e x t

θψ

β− + + − +

= − − + − ∈ ≥

1ε =

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

−= − = ∆ =

⇒ = = −

1ε =

22 2( ) ( ) | |

2s s s s s sx V xεµ φ φ φ β φ φ= − ∇ + +

Numerical methods for dynamics

Dispersive & nonlinear Solution and/or potential are smooth but may oscillate wildly Keep the properties of NLSE 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: 1& 1 or 0 1& 1β ε ε β= < = ±

22 2

0

( , ) ( ) | | , , 02

with ( ,0) ( )

dti x t V x x t

x x

εε ψ ψ ψ β ψ ψ

ψ ψ

∂ = − ∇ + + ∈ >

=

Numerical difficulties

Explicit vs implicit (or computation cost)

Spatial/temporal accuracy Stability Keep the properties of NLSE in the discretized level – Time reversible & time transverse invariant – Mass & energy conservation – Dispersion conservation

Resolution in the semiclassical regime:

0 1ε<

/ (solution has wavelength of ( ))i Se Oεψ ρ ε=

Crank-Nicolson finite difference (CNFD)

Crank-Nicolson finite difference (CNFD) method (Glassey, JCP, 92’; Chan, Guo & Jiang, Math. Comp., 94’; Chan& Shen, SINUM, 88’; Chang & Sun, JCP 03’; ....)

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

1 1 1 121 1 1 1

2 2

1 1 2 2 1

2 24

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

2 4

n n n n n n n nj j j j j j j j

j n n n n n nj j j j j j

ih h

V x

ψ ψ ψ ψ ψ ψ ψ ψεετ

βψ ψ ψ ψ ψ ψ

+ + + ++ − + −

+ + +

− − + − += − +

+ + + + +

22

0

( , ) ( ) | | , , 02

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

t xxi x t V x a x b t

a t b t x x a x b

εε ψ ψ ψ β ψ ψ

ψ ψ ψ ψ

∂ = − ∂ + + < < >

= = = ≤ ≤

CNFD for NLSE

– Stability: Yes – Energy conservation: Yes – nonlinear system must be solved very accurately

– Dispersion relation without potential: No – Accuracy

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

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

– Error estimate in H^1-norm: • In 1D: R.T. Glassey, Q. Chang, B. Guo, H. Jiang, etc. • In 2D&3D: W. Bao and Y. Cai , Math. Comp. 13’

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

2 2( )ne C h τ≤ +

Time-splitting spectral method (TSSP)

For , apply time-splitting technique – Step 1: Discretize by spectral method & integrate in phase space exactly

– Step 2: solve the nonlinear ODE analytically

Use 2nd or 4th order splitting (Bao,Jin & Markowich, JCP, 02’; citations >200 !!) – (Tarppent, SIAM 78’; Moris etc., JCP, 88’; Ablowitz etc. JCP, 84’,…..)

1[ , ]n nt t +

22( , )

2ti x t εε ψ ψ∂ = − ∇

2

2

2

2

( )[ ( ) | ( , )| ]/

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

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

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

( , ) ( , )n n

t

t n

t n

i t t V x x tn

i x t V x x t x t x t

x t x t x ti x t V x x t x t x t

x t e x tβ ψ ε

ε ψ ψ β ψ ψ

ψ ψ ψ

ε ψ ψ β ψ ψ

ψ ψ− − +

∂ = +

⇓ ∂ = ⇒ =

∂ = +

⇒ =

1:ε =

Properties of the method

Explicit & computational cost per time step: Time reversible: yes Time transverse invariant: yes Mass conservation: yes Stability: yes

( ln )O M M

1 & scheme unchanged!!n n τ τ+ ↔ ↔ − ⇒

/( ) ( ) (0 ) | | unchanged!!!n n i n nj j j j jV x V x j M e τ α εα ψ ψ ψ−→ + ≤ ≤ ⇒ → ⇒

22 2

1 12 0 2

0 00 0

: | | : | ( ) | , 0,1, for any &M M

n nj jll l

j jh h x n h kψ ψ ψ ψ ψ

− −

= =

= ≡ = = =∑ ∑

Properties of the method

Dispersion relation without potential: yes

– Exact for plane wave solution

Energy conservation (Bao, Jin & Markowich, JCP, 02’): – cannot prove analytically – Conserved very well in computation

( / )0

22 2

(0 ) (0 & 0)

with | | if2

j j ni k x i k x tnj ja e j M a e j M n

k a M k

ω εψ ψ

εω β

−= ≤ ≤ ⇒ = ≤ ≤ ≥

= + >

Properties of the method

Accuracy----- Spatial: spectral order & Temporal: 2nd order

Resolution in semiclassical regime (Bao, Jin & Markowich, JCP, 02’)

– Linear case: – Weakly nonlinear case: – Strongly repulsive case:

Error estimate in L^2-norm and/or H^1:

– C. Besse, C. Lubich, O. Koch, M. Thalhammer, M. Caliari, C. Neuhauser, E. Faou, A. Debussche, L. Gauckler, E. Hairer, J. Shen&Z.Q. Wang, W. Bao&Y. Cai, etc.

0β =

( ) & independent of h O ε τ ε= −( )Oβ ε=

( ) & independent of h O ε τ ε= −0 (1)Oβ< =

( ) & ( )h O Oε τ ε= = 2( )n me C h τ≤ +

Comparison

Interaction of a lattice

3D collapse & explosion of BEC

Experiment (Donley et., Nature, 01’) – Start with a stable condensate (as>0) – At t=0, change as from (+) to (-) – Three body recombination loss

Mathematical model (Duine & Stoof, PRL, 01’)

ψψβδψψβψψψ 420

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

ti −++∇−=∂∂

4 s

s

N ax

πβ =

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

Jet formation

3D Collapse and explosion in BEC

4 s

s

N ax

πβ =

3D Collapse and explosion in BEC

GPE with angular rotation

GPE / NLSE with an angular momentum rotation

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

: ( ) , ,

dt z

z y x y x

i x t V x L x t

L xp yp i x y i L x P P iθ

ψ β ψ ψ∂ = − ∇ + −Ω + ∈ >

= − = − ∂ − ∂ ≡ − ∂ = × = − ∇

Vortex @MIT

Numerical methods

Time-splitting + polar (cylindrical) coordinates –Bao, Du & Zhang, SIAP, 05’

Time-splitting + ADI -- Bao & Wang, JCP, 06’

Time-splitting + Laguerre-Hermite functions –Bao, Li &Shen, SISC, 09’

2

2

1Step 1: ( , ) [ ]2

Step 2: ( , ) [ ( ) | | ]

t z

t

i x t L

i x t V x

ψ ψ

ψ β ψ ψ

∂ = − ∇ −Ω

∂ = +

2

1Step 1: ( , ) [ ]21Step 2: ( , ) [ ]2

Step 3: ( , ) [ ( ) | | ]

t xx x

t yy y

t

i x t i y

i x t i x

i x t V x

ψ ψ

ψ ψ

ψ β ψ ψ

∂ = − ∂ − Ω ∂

∂ = − ∂ + Ω ∂

∂ = +

22

2

1Step 1: ( , ) [ / 2] :2

Step 2: ( , ) [ ( ) | | ]

t z

t

i x t L x L

i x t W x

ψ ψ ψ

ψ β ψ ψ

∂ = − ∇ −Ω + =

∂ = +

A simple & efficient method

Ideas – Bao, Marahrens,Tang & Zhang, 13’; Bao & Cai, KRM, 13’; ……

– A rotating Lagrange coordinate:

– GPE in rotating Lagrange coordinates – TSSP method

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∂ = − ∇ + + ∈ >

2

2

1Step 1: ( , ) ,2

Step 2: ( , ) [ ( ( ) ) | | ] ,

t

t

i x t

i x t V A t x

φ φ

φ β φ φ

∂ = − ∇

∂ = +

Dynamics of a vortex lattice

Extension to dipolar quantum gas

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 2 2dip

1( , ) ( ) | | | |2t zi x t V x L Uψ β ψ λ ψ ψ ∂ = − ∇ + −Ω + + ∗

3( , )x t xψ ψ= ∈

( )2 2 2 2 2 21( )2 x y zV x x y zγ γ γ= + +

20 dip

2

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

s

s s

mNN ax x

µ µπβ λ= =

2 2 2

dip 3 33 1 3( ) / | | 3 1 3cos ( )( )

4 | | 4 | |n x xU x

x xθ

π π− ⋅ −

= =

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 2

| | | | 3

1 | | | |4

n nU

r

ψ ψ ϕ

ϕ ψ ϕ ψπ

∗ = − − ∂

= ∗ ⇔ −∇ =

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

BEC@Stanford

A New Formulation

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

– Energy

– Model in 2D

Numerical methods --- TSSP with sine basis instead of Fourier basis – Bao, Cai & Wang, JCP, 10’; Bao & Cai, KRM, 13’ – Bao, Marahrens,Tang & Zhang, 13’’; Bao & Cai, KRM, 13’; ……

2 2

2 2 3

| |

1( , ) ( ) ( ) | | 32

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

t z n n

x

i x t V x L

x t x t x x t

ψ β λ ψ λ ϕ ψ

ϕ ψ ϕ→∞

∂ = − ∇ + −Ω + − − ∂ −∇ = ∈ =

3

2 2 4 21 3( ( , )) : | | ( ) | | | | | |2 2 2z nE t V x L d xβ λ λψ ψ ψ ψ ψ ψ ϕ− ⋅ = ∇ + −Ω + + ∂ ∇ ∫

21/2 2 2

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

D

xx t x t x x tϕ ψ ϕ⊥ →∞

→ −∆ = ∈ =

Dynamics of a vortex lattice

Conclusions & Future Challenges

Conclusions: – NLSE / GPE – brief motivation

– Dynamical properties – Numerical methods

• Time-splitting spectral (TSSP) method & Applications – Extensions – rotations, nonlocal, system

Future Challenges – With random potential or high dimensions – Coupling GPE & Quantum Boltzmann equation (QBE) – NLSE / GPE coupled with other equations – BEC at finite temperature & quantum turbulence