Ground States and Dynamics of the 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 Ground states – Existence, uniqueness & non-existence – Numerical methods & results

Dynamics – Well-posedness & dynamical laws – Numerical methods & results

Applications --- collapse & explosion of a BEC

Extension to rotation, nonlocal interaction & system Conclusions

NLSE / GPE

The nonlinear Schrodinger equation (NLSE) ---1925

– 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 – Gross-Pitaevskii equation (GPE) :

• E. Schrodinger 1925’; • 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

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 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 in Using 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 ≥

Phase transition –symmetry breaking

Attractive interaction with double-well potential in 1D 2 2

2 2

1( ) ''( ) ( ) ( ) | ( ) | ( ), with | ( ) | 12

( ) ( 1) & : positive 0 negative

x x V x x x x x dx

V x x

µ φ φ φ β φ φ φ

β

∞

−∞

= − + + =

= − → →

∫

Excited states

Excited states: Open question: (Bao & W. Tang, JCP, 03’; Bao, F. Lim & Y. Zhang, Bull Int. Math, 06’)

Gaps between ground and first excited states – Linear case – fundamental gap conjecture (B. Andrews & J. Clutterbuck, JAMS 11’)

– Nonlinear case ?????

,,, 321 φφφ

1 2

1 2

1 2

, , ,

( ) ( ) ( )

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

g

g

g

E E Eϕ ϕ ϕ

ϕ ϕ ϕ

µ ϕ µ ϕ µ ϕ

< ≤ ≤

< ≤ ≤

1 1( ) : ( ) ( ) 0, ( ) : ( ) ( ) 0g E gE Eβ β β βµδ β µ φ µ φ δ β φ φ= − > = − >

23: (0) (0) on bounded domain | |

dE D

Dµπδ δ δ= = ≥ ⊂

1 2( ) 0, ( ) 0, 0????EC Cµδ β δ β β≥ > ≥ > ≥

Computing ground states

Idea: Steepest decent method + Projection

– The first equation can be viewed as choosing in NLS – For linear case: (Bao & Q. Du, SIAM Sci. Comput., 03’)

– For nonlinear case with small time step, CNGF

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

+

−

++ −

+

∂ = − = ∇ − − ≤ <

= =

= =

))0(.,())(.,())(.,( 0010 φφφ EtEtE nn ≤≤≤+

τit =

0φ

1φ

2φ 1φ

??)()()()ˆ(

)()ˆ(

01

11

01

φφφφ

φφ

EEEE

EE

<<

<

gφ

Normalized gradient glow

Idea: letting time step go to 0 (Bao & Q. Du, SIAM Sci. Comput., 03’)

– Mass conservation & energy diminishing

– Numerical discretizations • 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’)

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= = ≤ ≥

Ground states in 1D & 3D

Dynamics

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 conservation Dispersion relation Soliton solutions in 1D: – Bright soliton ----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( ) 21( , )2

i k x tx t Ae k Aωψ ω β−= ⇒ = +

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− + + − +

= − − + − ∈ ≥

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= − ∇ + +

Numerical difficulties

Dispersive & nonlinear Solution and/or potential are smooth but may oscillate wildly Keep 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

Different methods – Crank-Nicolson finite difference method (CNFD) – Time-splitting spectral method (TSSP) – Leap-frog (or RK4) + FD (or spectral) methods – ……..

Time-splitting spectral method (TSSP) 2 2

2

( ) / 2 / 2 31

5

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

( , ) (( ) )( , ) ( , ) ( , ) (( ) )

(( ) )

t

i A t i B tn

i A B t i A t i B t i A tn n n

i x t A B A B V x

e e x t O tx t e x t e e e x t O t

O t

ψ ψ β ψ

ψψ ψ ψ

− ∆ − ∆

− + ∆ − ∆ − ∆ − ∆+

∂ = + = − ∇ = +

+ ∆

= ≈ + ∆ + ∆

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 order Strang splitting (or 4th order time-splitting)

1[ , ]n nt t +

21( , )2ti x tψ 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

ψ ψ β ψ ψ

ψ ψ ψ

ψ ψ β ψ ψ

ψ ψ− − +

∂ = +

⇓ ∂ = ⇒ =

∂ = +

⇒ =

Properties of TSSP

Explicit & computational cost per time step: Time symmetric: yes Time transverse invariant: yes Mass conservation: yes Stability: yes Dispersion relation without potential: yes Accuracy – Spatial: spectral order; Temporal: 2nd or 4th order

Best resolution in strong interaction regime:

( ln )O M M

1β

Interaction of 3 like vortices

Interaction of 3 opposite vortices

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

Extension to GPE with rotation

GPE / NLSE with an angular momentum rotation

Mass conservation

Energy conservation

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 + ∈ >

= − = − ∂ − ∂ ≡ − ∂ = × = − ∇

2 2 42

R

1( ) : ( )2d

zE V x L d xβψ ψ ψ ψ ψ ψΩ = ∇ + −Ω + ∫

Vortex @MIT

2( ) ( , )

d

N t x t d xψ= ∫

Ground states

Ground states – Seiringer, CMP, 02’; Bao,Wang & Markowich, CMS, 05’; …..

Existence & uniqueness – Exists a ground state when – Uniqueness when – Quantized vortices appear when – Phase transition & bifurcation in energy diagram

Numerical methods --- GFDN & BEFD or BEFP

0 & min ,x yβ γ γ≥ Ω ≤

( )c βΩ < Ω

( )c βΩ ≥ Ω

min ( )S

Eφ

φΩ∈

Ground states with different Ω

Ground states of rapid rotation

BEC@MIT

Dynamics – Bao, Du & Zhang, SIAP, 05’; Bao & Cai, KRM, 13’; ……

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 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= ∂ = ⋅∇ ∂ = ∂ ∂

A New Formulation

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

– Energy

– Model in 2D

Ground state – Bao, Cai & Wang, JCP, 10’; Bao, Ben Abdallah & Cai, SIMA, 12’

Dynamics – Bao, Cai & Wang, JCP, 10’; Bao & Cai, KRM, 13’

Dimension reduction – Cai, Rosenkranz, Lei & Bao, PRA, 10’; 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 + + ∂ ∇ ∫

0 &2ββ λ β≥ − ≤ ≤

21/2 2 2

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

D

xx t x t x x tϕ ψ ϕ⊥ →∞

→ −∆ = ∈ =

Dynamics of a vortex lattice

Coupled GPEs

Spinor F=1 BEC With Analysis & numerical methods: – For ground state (Bao & Wang, SIAM J. Numer. Anal., 07’; Bao & Lim, SISC, 08’; PRE, 08’)

– For Dynamics (Bao, Markowich, Schmeiser & Weishaupl, M3AS, 05’)

2 * 21 1 1 0 1 1 1 0

2 *0 0 1 1 1 1 1 0

2 * 21 1 1 0 1 1 1 0

1[ ( ) ] ( )21[ ( ) ] ( ) 221[ ( ) ] ( )2

n s s

n s s

n s s

i V xt

i V xt

i V xt

ψ β ρ ψ β ρ ρ ρ ψ βψ ψ

ψ β ρ ψ β ρ ρ ψ βψψ ψ

ψ β ρ ψ β ρ ρ ρ ψ βψ ψ

− −

− −

− − −

∂= − ∇ + + + + − +

∂∂

= − ∇ + + + + +∂∂

= − ∇ + + + + − +∂

2 0 2 2 01 0 1

0 2

4 ( 2 ) 4 ( ), | | , ,3 3

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

j j n ss s

N a a N a agx x

a a

π πρ ρ ρ ρ ρ ψ β−+ −

= + + = = =

Conclusions & Future Challenges

Conclusions: – NLSE / GPE – brief derivation

– Ground states • Existence, uniqueness, non-existence • Numerical methods -- BEFD

– Dynamics • Well-posedness & dynamical laws • Numerical methods -- TSSP

Future Challenges – System of NLSE/GPE; with random potential; high dimensions – Coupling GPE & Quantum Boltzmann equation (QBE)

Collaborators

In Mathematics – External: P. Markowich (KAUST, Vienna, Cambridge); Q. Du (PSU); J. Shen

(Purdue); S. Jin (UW-Madison); L. Pareshi (Italy); P. Degond (France); N. Ben Abdallah (Toulouse), W. Tang (Beijing), I.-L. Chern (Taiwan), Y. Zhang (MUST), H. Wang (China), Y. Cai (UW/UM), H.L. Li (Beijing), T.J. Li (Peking), …….

– Local: X. Dong, Q. Tang, X. Zhao, …….

In Physics – External: D. Jaksch (Oxford); A. Klein (Oxfrod); M. Rosenkranz (Oxford);

H. Pu (Rice), Donghui Zhang (Dalian), W. M. Liu (IOP, Beijing), X. J. Zhou (Peking U), …..

– Local: B. Li, J. Gong, B. Xiong, W. Ji, F. Y. Lim (IHPC), M.H. Chai (NUSHS), ………

Fund support: ARF Tier 1 & Tier 2