Analysis & Computation for Ground States of Degenerate Quantum Gas

Weizhu Bao

Department of Mathematics National University of Singapore Email: matbaowz@nus.edu.sg

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

Outline

Motivation and models –GPE/NLSE Ground states -- Mathematical theory – Existence, uniqueness & non-existence

Numerical methods and results – Normalized gradient flow method – Regularized Newton method

Extension to – Rotation frame, nonlocal interaction & system

Conclusions

BEC@JILA, 95’

Vortex @ENS

Degenerate Quantum Gas

Typical degenerate quantum gas – Liquid Helium 3 & 4 – Bose-Einstein condensation (BEC)

• Boson vs Fermion condensation • One component, two-component & spin-1 • Dipolar, spin-orbit, ….

Typical properties – Low (mK) or ultracold (nK) temperature – Quantum phase transition & closely related to nonlinear wave – Superfluids – flow without friction & quantized vortices

Model for Degenerate Quantum Gas (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 – A. Einstein, 1924’ (using S. Bose statistics)

Experimental realization – JILA 1995’

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

GPE for a BEC – with N identical bosons

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

Hartree ansatz Fermi interaction Dilute 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 ≈∫

GPE 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

3

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

2 2NgE V x dx x t

mψ ψ ψ ψ ψ ψ

= ∇ + + =

∫

222 3( )( , ) ( ) ,

2tEi x t V x Ng x

mδ ψψ ψ ψδψ

∂ = = − ∇ + + ∈

2 21 4( , ) ( ) | | , with 2

d st

s

Nai x t V x xxπψ ψ ψ β ψ ψ β∂ = − ∇ + + ∈ =

NLSE / GPE

The nonlinear Schrodinger equation (NLSE) ---1925

– t : time & : spatial coordinate (d=3,2,1) – : 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

Other applications

In nonlinear (quantum) optics 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 – electron structure computation

– First principle computation – Semiconductor industry

In biology – protein folding In superfluids – flow without friction

Conservation laws

Dispersive Mass conservation

Energy conservation

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

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

d d

N t N t x t d x x d xψ ψ ψ= = ≡ =∫ ∫

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

E t E t V x d x Eβψ 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 -- constrained minimization problem -- nonconvex

– Euler-Lagrange equation (or first-order optimality condition) nonlinear eigenvalue problem with a constraint

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= ∇ + +∫

arg min ( )gS

Eφ

φ φ∈

= 2 2| ( ) 1, ( )d

S x dx Eφ 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≠ ∈

∇=

Different numerical methods

Methods via nonlinear eigenvalue problems – Runge-Kutta method: M. Edwards and K. Burnett, Phys. Rev. A, 95’, …

– Analytical expansion: R. Dodd, J. Res. Natl. Inst. Stan., 96’,…

– Gauss-Seidel iteration method: W.W. Lin et al., JCP, 05’, …

– Continuation method: Chang, Chien & Jeng, JCP, 07’; ……

Methods via constrained minimization problem – Explicit imaginary time method: S. Succi, M.P. Tosi et. al., PRE, 00’; Aftalion & Du, PRA, 01’,

– FEM discretization+ nonlinear solver: Bao & Tang, JCP, 03’, …

– Normalized gradient flow via BEFD Bao&Du, SIAM Sci. Comput., 03’; Bao, Chern& Lim, JCP, 06,...

– Sobolev gradient method : Garcia-Ripoll,& Perez-Garcia, SISC, 01’; Danaila & Kazemi, SISC, 10’

– Regularized Newton method via trust-region strategy: Xu, Wen & Bao, 15’, …

Computing ground states

Idea: Steepest decent method + Projection – normalized gradient flow

– 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 flow

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’)

• GPRLab – Antoine & Duboscq, CPC, 14’

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= = ≤ ≥

Regularized Newton Method

Discretization--- Xu, Wen & Bao, 15’

– Construct initial solutions via feasible gradient type method – A regularized Newton method via trust-region strategy --- MJD

Powell, Y.X. Yuan, RH Byrd, RB Schnabel, GA, Schultz, W. Sun, Z. Wen, ……..

– Use multigrid method to accelerate convergence – Similar method has been developed and used for electronic structure computation –

Z. Wen & W. Yin, 13’; Z. Wen, A. Milzarek, Ulbrich & Zhang, SISC, 13’, Z Wen & A. Zhou, SISC 14’, …..

– Converge very fast in strong interaction regime & fast rotation

4

& 1 1

1arg min ( ) min ( ) :2N

NT

g jX XS jE F X X AX X

φφ φ α

∈ =∈ =

= ⇒ = + ∑

Ground states in 1D & 3D

Extension to GPE with rotation

GPE / NLSE with an angular momentum rotation

Ground states

Energy functional

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 xβφ φ φ φ φ φΩ = ∇ + −Ω + ∫

Vortex @MIT

arg min ( )gS

Eφ

φ φΩ∈

=

Ground states

Constrained Minimization problem – Energy functional is non-convex – Constraint is non-conex – Minimizer is complex instead of real

Existence & uniqueness – Seiringer, CMP, 02’; Bao,Wang & Markowich, CMS, 05’;

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

Numerical methods --- NGF via BEFD or BEFP or Newton method

0 & min ,x yβ γ γ≥ Ω ≤

( )c βΩ < Ω

( )c βΩ ≥ Ω

Ground states with different Ω

Ground states of rapid rotation

BEC@MIT

Dipolar quantum gas (BEC)

GPE with long-range anisotropic DDI

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

– Energy

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− ⋅ −

= =

( )3

2 2 4 2 2dip

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

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

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’)

Ground state Model in 2D

2 2

2 2 3

| |

1( , ) ( ) ( ) | | 32

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

t n n

x

i x t V x

x t x t x x t

ψ β λ ψ λ ϕ ψ

ϕ ψ ϕ→∞

∂ = − ∇ + + − − ∂ −∇ = ∈ =

3

2 2 4 2 2 21 3( ) : | | ( ) | | | | | | with | |2 2 2 nE V x d xβ λ λφ φ φ φ ϕ ϕ φ− = ∇ + + + ∂ ∇ −∇ = ∫

21/2 2 2

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

D

xx t x t x x tϕ ψ ϕ⊥ →∞

→ −∆ = ∈ =

arg min ( )gS

Eφ

φ φ∈

=

Ground State Results

Theorem (Existence, uniqueness & nonexistence) (Carles, Markowich& Sparber, 08’; 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:

Numerical method for DDI via NUFFT (Jiang, Greengard, Bao, SISC,14’; Bao, Tang& Zhang, 15’)

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≥ > < −

3

2dip dip dip( ) * ( ) ( ) ( ) [| | ( )]... | |i x i xP x U e U d e U dξ ξρ ρ ξ ξ ξ ρ ξ ξ ξ ξ= = =∫ ∫

2

dip 23( )( ) 1

| |nU ξξξ⋅

= − +

Degenerate quantum gas (BEC) with spin-orbit-coupling

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

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

Applications ----Topological insulator

Ground states (Bao & Cai, SIAP,15’) --- existence, asymptotic behavior, numerical methods

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∂

arg min ( )gS

EΦ∈

Φ = Φ

( ) ( ) ( ) ( )

2 2 21 2 1 2

2 2 2 2 2 4 2 2 41 2 0 1 1 2 2 1 2 11 1 12 1 2 22 2

1

( , ) | 1, ( )

1 1( ) [ ( ) + Re 2 ]2 2 2d

j j j x xj

S E

E V x ik dx

φ φ φ φ

δφ φ φ φ φ φ φ φ φφ β φ β φ φ β φ=

= Φ = Φ = + = Φ < ∞

Φ = ∇ + + − ∂ − ∂ +Ω + + +

∑∫

Spinor (F=1) degenerate quantum gas

Coupled GPEs

Ground states – with

– Theory – Cao & Wei, 08’; Chern & Lin, 13’

– Numerical methods and results –Zhang, Yi & You, PRA, 05’; Bao & Wang, SIAM J. Numer. Anal., 07’; Bao & Lim, SISC, 08’; PRE, 08’, Ueda, 10’

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

− −

− −

− − −

∂= − ∇ + + + + − +

∂∂

= − ∇ + + + + +∂∂

= − ∇ + + + + − +∂

arg min ( )gS

EΦ∈

Φ = Φ

( ) ( )( ) ( )

22 2 21 0 1 1 1 1 0 1

1 2 2 22 2 21 1 0 1 1 s 1 0 1 1 0 1

1

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

1( ) [ ( ) + 2 ]2 2 2d

j j

n sj j

j

S M E

E V x dx

φ φ φ φ φ ρ φ ρ ρ ρ ρ

β βφ φ ρ ρ ρ ρ ρ ρ β φ φ φ φ φ φ

− − −

− − − −=−

= Φ = Φ = − = Φ < ∞ = = + +

Φ == ∇ + + − + + + +

∑∫

Conclusions & Future Challenges

Conclusions: – NLSE / GPE – brief derivation

– Ground states • Existence, uniqueness, non-existence • Numerical methods: Gradiend flow method via BEFD or Regularized Newton

method via trust-region strategy

Future Challenges – Nonlocal high-order interaction & system, e.g. spin-2 BEC

– Bogoliubov excitation --- linear response; – Excited states – constrained min-max problems

Modeling + Numerical PDE + Optimization materials science & quantum physics, chemistry

Acknowledgement

Collaborators – Q. Du (Columbia); P. A. Markowich (KAUST, Vienna,

Cambridge); I.-L. Chern (NTU); Z. Wen (Peking); X. Wu (Fudan),

– Y. Cai (Purdue); F. Y. Lim (IHPC); H. Wang (Yunan); Q. Tang (France); Y Zhang (Vienna)

Funding support – Singapore MOE Tier 2 – Singapore MOE Tie 1 – Singapore A*STAR PSF