Analysis & Computation for Ground States of Degenerate Quantum Gas
Weizhu Bao
Department of Mathematics National University of Singapore Email: [email protected]
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