Numerical Methods for the Dynamics of the Nonlinear Schrodinger / Gross-Pitaevskii Equations. Weizhu Bao.

Transcription

1 Numerical Methods for the Dynamics of the Nonlinear Schrodinger / Gross-Pitaevskii Equations Weizhu Bao Department of Mathematics National University of Singapore matbaowz@nus.edu.sg URL: Vortex@ENS

2 Outline Nonlinear Schroedinger / 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

3 NLSE / GPE The nonlinear Schroedinger equation (NLSE) ε iε tψ(,) xt = ψ + V( x) ψ + β ψ ψ t : time & d x ( ) : spatial coordinate ψ (,) xt V( x ) : complex-valued wave function : real-valued external potential β : given interaction constant (=0: linear; >0: repulsive & <0: attractive) 0< ε 1 : scaled Planck constant ε =1 0 < ε 1& β =± 1 ( : standard; : semiclassical)

4 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 194 Experimental realization JILA Nobel 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

5 Laser beam propagation Nonlinear wave (or Maxwell) equations Helmholtz equation time harmonic In a Kerr medium Paraxial (or parabolic) approximation -- NLSE

6 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

7 Dispersive Time symmetric: Conservation laws ε i xt V x ε tψ(,) = ψ + ( ) ψ + β ψ ψ & take conjugate Time transverse (gauge) invariant t t V x V x e α ε Mass (or wave energy) conservation ψ ψ ψ d d Energy (or Hamiltonian) conservation unchanged!! / ( ) ( ) + α ψ ψ i t ρ = ψ --unchanged!! N(): t = N( (,)) t = ( x,) t dx ( x,0) dx= 1, t 0 4 E( t) : E( (, t)) ε β = ψ = ψ + V( x) ψ + ψ dx E(0), t 0 d

8 ψ Dynamics with no potential V( x) 0, x Momentum conservation Dispersion relation ψ(,) Soliton solutions in 1D: ε = 1 Bright soliton when β < decaying to zero at far-field Dark (or gray) soliton β > nonzero &oscillatory at far-field d J( t) : = Im ψ ψ dx J(0) t 0 d ε β i( k x ω t) x t = Ae ω = k + A ε 1 a ψ x t = a x vt x0 e x t β 1 ( ( ) 0 ) (, ) sech( ( )) i vx v a t + θ,, 0 [ ] i( kx ( k + a + ( v k ) ) t+ θ0 ) ( x, t) = a tanh( a( x vt x )) + i( v k) e, x, t 0 β 0 1

9 Interaction of two bright solitons

10 Interaction of two dark solitons

11 ψ Dynamics with harmonic potential ε = 1 Harmonic potential Center-of-mass: = d x t x t t An analytical solution if γ x x d = 1 1 V( x) = γ x x + γ y y d = γ x x + γ y y + γ z z d = 3 xc() t xψ ( x,) t dx c( ) + diag( γ x, γ y, γ z) c( ) = 0, > 0 each component is periodic!! ψ ( x ) = φ ( x x ) 0 s 0 (,) (,) i s t ( ()) iw x µ xt = e φ t s x xc t e, xc(0) = x0 & wxt (,) = 0 ρ = ψ = φ ( xt, ) : ( xt, ) s( x xc( t)) -- moves like a particle!! ε µφ s s( x) = φs + V( x) φs + β φs φs

12 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 ε d iε t ψ( xt,) = ψ + V( x) ψ + β ψ ψ, x, t> 0 with ψ( x,0) = ψ ( x) Design efficient & accurate numerical algorithms Explicit vs implicit (or computation cost) Spatial/temporal accuracy, Stability Resolution in strong interaction regime: β 1& ε = 1 or 0 < ε 1& β = ± 1 0

13 Crank-Nicolson finite difference (CNFD) Crank-Nicolson finite difference (CNFD) method (Chan&Shen, SINUM, 86 &87 ; Guo, JCM, 86 ; Glassey, JCP, 9 ; Chan, Guo & Jiang, Math. Comp., 94 ; Chang & Sun, JCP 03 ; Bao&Cai, Math. Comp., 13 ;...) ψ ψ ψ ψ + ψ ψ ψ + ψ iε τ 4 h h V x n+ 1 n n+ 1 n+ 1 n+ 1 n n n j j ε j+ 1 j j 1 j+ 1 j j 1 = + Implicit: need solve a fully nonlinear system per time step Time reversible: Yes ε iε t ψ xt = ψ + V xψ + β ψ ψ a< x< b t> ψ( at, ) = ψ( bt, ) = 0, with ψ( x,0) = ψ ( x), a x b Time transverse invariant: No Mass conservation: Yes (,) xx ( ),, 0 ( j ) ( n+ 1 n β ) ( n+ 1 n )( n+ 1 n ψ j ψ j ψ j ψ j ψ j ψ j ) 0

14 CNFD for NLSE Stability: Yes Energy conservation: Yes nonlinear system must be solved very accurately Dispersion relation without potential: No Accuracy Spatial: nd order Temporal: nd order Resolution in semiclassical regime (Markowich, Poala & Mauser, SINUM, 0 ) h = o = o h = O = O ( ε) & τ ( ε) ( ε ) & τ ( ε )

15 CNFD for NLSE Error estimate in H^1-norm when : In 1D for CNFD --- Glassey, Math. Comp. 9 ; Chang,Guo&Jiang, Math. Comp., 95 : Energy method Key inequality In D&3D & for non-energy conservative: -- Bao&Cai, Math. Comp. 13 Energy method Inverse inequality For CNFD ---- cut-off function techniques + + e n e n Ch ( τ ) h l l ε = 1 1 L L L 0 f f f, f H ( ) Ω n ψ ψ l L + For SIFD --- the method of mathematical induction e n l

16 Time-splitting spectral method (TSSP) For [ tn, t n + 1], apply time-splitting technique Step 1: Discretize by spectral method & integrate in phase space exactly i ε tψ(,) xt = ψ Step : solve the nonlinear ODE analytically iε tψ(,) xt = V( x) ψ(,) xt + β ψ(,) xt ψ(,) xt t( ψ( xt, ) ) = 0 ψ( xt, ) = ψ( xt, n) iε tψ( xt, ) = V( x) ψ( xt, ) + β ψ( xt, n) ψ( xt, ) i( t tn)[ V( x) + βψ ( xt, n) ]/ ε ψ( xt, ) = e ψ( xt, ) Use nd or 4 th order splitting (Bao,Jin & Markowich, JCP, 0 ; citations >300!!) ε = 1: ε (Hardin & Tarppent, SIAM 73 ; Taha&Ablowitz, JCP, 84 ; Weideman&Herbst, SINUM, 86 ; Pathria&Moris, JCP, 90 ; Bao,Jin&Markowich, JCP, 0, SISC, 03,..) n

17 Properties of the method Explicit & computational cost per time step: Time reversible: yes Time transverse invariant: yes Mass conservation: yes M 1 M 1 n n 0 = j = 0 = l 0 j = Stability: yes OM ( ln M) n+ 1 n & τ τ scheme unchanged!! V x V x j M e n n inτα/ ε n ( j) ( j) + α (0 ) ψ j ψ j ψ j unchanged!!! ψ : h ψ ψ ψ : h ψ ( x ), n 0,1, for any h& k l l j= 0 j= 0

18 Properties of the method Dispersion relation without potential: yes ψ 0 j ikxj n i( kxj ωtn/ ε) ψ j = ae (0 j M) = ae (0 j M& n 0) ε = k + a M > k with ω β if Exact for plane wave solution Energy conservation (Bao, Jin & Markowich, JCP, 0 ): cannot prove analytically Conserved very well in computation

19 Properties of the method Accuracy----- Spatial: spectral order & Temporal: nd order Resolution in semiclassical regime (Bao, Jin & Markowich, JCP, 0 ) Linear case: β = 0 h = O( ε) & τ independent of ε Weakly nonlinear case: Strongly repulsive case: β = O( ε) h = O( ε) & τ independent of ε n m 0 < β = O(1) e Ch ( + τ ) h = O( ε) & τ = O( ε) ε = 1 Error estimate in L^-norm and/or H^1 when : Besse,Bidegary&Descombes, 0 ; Lubich, 08 ; Koch&Thalhammer, M. Caliari, C. Neuhauser, Faou, 1 ; A. Debussche, L. Gauckler, E. Hairer, Shen&Wang, 13 ; Bao&Cai, 13 ; etc.

20 Other approaches Time discretization Leap-frog or explicit scheme --- severe stability constraint!! Multi-sympletic scheme --- Hong et al., 06 10,. 4 th -order Runge-Kutta (RK4) --- not time symmetric,. Spatial discretization Finite element method Akrivis, 91 ; Akrivis, Dugalis & Karakashan, 91 Finite volume method Compact finite difference method,. Time + spatial discretization different methods for NLSE Bao & Cai, Kinet. Relat. Mod, 6 (013), pp Review paper. Antonie, Bao & Besse, Comput. Phys. Commun., 013(arXiv: math.na ) Feature article.

21 Comparison

22 Interaction of 3 like vortices

23 Interaction of a lattice

24 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 i Mathematical model (Duine & Stoof, PRL, 01 ) 1 ψ ( x, t) = ψ + V ( x) ψ + β ψ ψ iδ 0β t ψ 4 ψ β = 4π Nas x s

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

26 3D Collapse and explosion in BEC β = 4π Nas x s

27 3D Collapse and explosion in BEC

28 GPE with angular rotation GPE / NLSE with an angular momentum rotation 1 d i tψ( xt,) = [ + V( x) Ω Lz + β ψ ] ψ, x, t> 0 L : = xp yp = i( x y ) i, L = x P, P = i z y x y x θ

29 Numerical methods Time-splitting + polar (cylindrical) coordinates Bao, Du & Zhang, SIAP, 05 1 ψ i ψ xt = V x + β ψ ψ Step 1: i tψ( xt, ) = [ ΩLz] Step : t (, ) [ ( ) ] Time-splitting + ADI -- Bao & Wang, JCP, 06 1 Step 1: i t ψ( xt, ) = [ xx Ω i y x ] ψ 1 Step : i t ψ( xt, ) = [ yy +Ω i x y ] ψ ψ β ψ ψ Step 3: i t ( xt, ) = [ V( x) + ] Time-splitting + Laguerre-Hermite functions Bao, Li &Shen, SISC, 09 1 Step 1: i tψ( xt, ) = [ Ω Lz + x / ] ψ : = Lψ Step : i ψ( xt, ) = [ W( x) + β ψ ] ψ t

30 A simple & efficient method Ideas Bao, Marahrens,Tang & Zhang, SISC, 13 ; Bao & Cai, KRM, 13 ; A rotating Lagrange coordinate: x = At x φ xt = ψ xt = ψ Atxt 1 () & (,): (,) ( (),) cos( Ωt) sin( Ωt) 0 cos( Ωt) sin( Ωt) At ( ) = for d ; At ( ) sin( t) cos( t) 0 for d 3 sin( t) cos( t) = = Ω Ω = Ω Ω GPE in rotating Lagrange coordinates 1 d i t φ( xt,) = [ + V( At () x ) + β φ ] φ, x, t> 0 TSSP method 1 φ φ β φ φ Step 1: i t φ( xt, ) =, Step : i t ( xt, ) = [ V( At ( ) x) + ],

31 Dynamics of a vortex lattice

32 Extension to dipolar quantum gas Gross-Pitaevskii equation (re-scaled) 1 ( (, ) ( ) ) i tψ xt = + V x Ω Lz + β ψ + λ Udip ψ ψ U Trap potential Interaction constants 1 V( x) = x + y + z s Long-range dipole-dipole interaction kernel 3 1 3( n x) / x 3 1 3cos ( θ ) ( x) = = 4 π x 4 π x References: dip 3 3 L. Santos, et al. PRL 85 (000), S. Yi & L. You, PRA 61 (001), (R); D. H. J. O Dell, PRL 9 (004), ( γ x γ y γ z ) ψ = ψ( xt,) x 4π Na mnµµ s β = (short-range), λ = x 3 0 dip xs (long-range) 3

33 A New Formulation Using the identity (O Dell et al., PRL 9 (004), 50401, Parker et al., PRA 79 (009), ) 3 3( n x) 1 Udip( x) = 1 = δ ( x) 3 3 nn 4πr r 4πr 3( n ξ ) Udip( ξ ) = 1+ ξ Dipole-dipole interaction becomes U r = x & = n & = ( ) dip = 3 nn ψ ψ ϕ 1 ϕ = ψ ϕ = ψ 4π r n nn n n BEC@Stanford

34 A New Formulation Gross-Pitaevskii-Poisson type equations (Bao,Cai & Wang, JCP, 10 ) 1 i tψ( xt, ) = + V( x) Ω Lz + ( β λ) ψ 3λ nnϕ ψ 3 ϕ( xt,) = ψ( xt,), x, lim ϕ( xt,) = 0 Energy 1 β λ 3λ ( ψ(, )): ψ ( ) ψ ψ ψ ψ ϕ Model in D x 4 E t = + V x Ω Lz + + n dx 3 D = = 1/ ( ) ϕ( xt,) ψ( xt,), x, lim ϕ( xt,) 0 x Numerical methods --- TSSP with sine basis instead of Fourier basis Bao, Cai & Wang, JCP, 10 ; Bao & Cai, KRM, 13 Bao, Marahrens,Tang & Zhang, SISC, 13 ; Bao & Cai, KRM, 13 ; Bao, Greengard &Jiang, nonuniform FFT

35 New numerical methods for DDI How to compute nonlocal DDI FFT (fast Fourier transform) DST (discrete sine transform) φ: = Udip ψ 3( n ξ ) ( ξ ) = 1+ ξ dip φ = ψ 3 nnϕ & ϕ( xt,) = ψ( xt,) Nonuniform FFT (Bao, Jiang, Greengard, SISC,14 ; Bao, Tang & Zhang, CiCP, 16 ) ix ξ φ(,) xt = U ( ξ) ρζ (,) te dξ ρ= ψ sphere coordinate = 3 S U dip + U dip( ξ) ξ ρζ (, ) te ix ξ

36 Dynamics of a vortex lattice

37 Coupled GPEs Spinor F=1 BEC With 1 * i ψ1 = [ + V( x) + βnρ] ψ1+ βs( ρ1+ ρ0 ρ 1) ψ1+ βψ s 1ψ0 t 1 * i ψ0 = [ + V( x) + βnρ] ψ0 + βs( ρ1+ ρ 1) ψ0 + βψψ s 1 1ψ 0 t 1 * i ψ 1 = [ + V( x) + βnρ] ψ 1+ βs( ρ 1+ ρ0 ρ1) ψ 1+ βψ s 1ψ0 t 4 πna ( + a) 4 πna ( a) ρ = ρ + ρ + ρ, ρ = ψ, β =, g = Numerical methods ---- TSSP with 3 steps Bao, Markowich, Schmeiser & Weishaupl, M3AS, 05 Bao & Cai, KRM, 13, 0 0 j j n s 3xs 3xs a, a : s-wave scattering length with the total spin 0 and channels

38 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

39 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 (Purdue), 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