Modeling, Analysis and Simulation for Degenerate Dipolar Quantum Gas

Transcription

1 Modeling, Analysis and Simulation for Degenerate Dipolar Quantum Gas Weizhu Bao Department of Mathematics National University of Singapore URL: Collaborators: Y. Cai (Postdoc, Purdue), M. Rosenkranz (Postdoc), N. Ben Abdallah (UPS, France), Z. Lei (Fudan University, China), H. Wang (Yunan Univ. Economics and Finance,)

2 Outline Motivation---dipolar BEC Mathematical models Ground state and its theory Dynamics and its efficient computation Dimension reduction Conclusion & future challenges

3 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 Boson-fermion mixture Typical properties Low (mk) or ultracold (nk) temperature Quantum phase transition & closely related to nonlinear wave Superfluids flow without friction & quantized vortices

4 Recent Developments Quantum transport Move a BEC in an optical lattice Atomic circuit, Quantum computing Interaction of BEC and particles Quantized vortices for superfluidity Vortex states Vortex lattice patterns Interaction between vortices Fermion condensate, Boson-fermion, atom-molecule,

5 Dipolar Quantum Gas Experimental setup Molecules meet to form dipoles Cool down dipoles to ultracold Hold in a magnetic trap Dipolar condensation Degenerate dipolar quantum gas Experimental realization Chroimum (Cr5) 005@Univ. Stuttgart, Germany PRL, 94 (005) Big-wave in theoretical study A. Griesmaier,et al., PRL, 94 (005)160401

6 BEC with strong DDI 164 Dy Lu, Burdick, Youn & Lev, PRL 107 (011),

7 Mathematical Model Gross-Pitaevskii equation (re-scaled) 1 i xt V x ( U ) dip xt t ψ β ψ λ ψ ψ (,) = + ext( ) + + (,) Trap potential Interaction constants 1 Vext( z) = x + y + z ( γ x γ y γ z ) Long-range dipole-dipole interaction kernel ψ = ψ( xt,) x 4π Na s mnµµ β = = a 0 dip (short-range), λ 0 3 a0 (long-range) 3 1 3( n x ) / x 3 1 3cos ( θ ) Udip( x) = =, π x 4 π x 3 n fixed & satisfies n = 1 References: L. Santos, et al. PRL 85 (000), S. Yi & L. You, PRA 61 (001), (R); D. H. J. O Dell, PRL 9 (004),

8 Mathematical Model Mass conservation (Normalization condition) N(): t = ψ(,) t = ψ(,) xt dx ψ(,0) x dx= 1 Energy conservation Long-range interaction kernel: It is highly singular near the origin!! At O 3 singularity near the origin!! x Its Fourier transform reads 3( n ξ ) No limit near origin in phase space!! Udip( ξ) = 1 + ξ ξ Bounded & no limit at far field too!! β 4 λ E( ψ(, t)): = ψ + Vext( x) ψ + ψ + ( Udip ψ ) ψ dx E( ψ0) 3 Physicists simply drop the second singular term in phase space near origin!! Locking phenomena in computation!! 1 3

9 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 n nn n n ψ ψ ϕ 1 ϕ = ψ ϕ = ψ 4π r

10 A New Formulation Gross-Pitaevskii-Poisson type equations (Bao,Cai & Wang, JCP, 10 ) 1 i ψ xt V x β λ ψ λ ϕ ψ xt t ϕ = ψ ϕ = (,) = + ext( ) + ( ) 3 nn (,) ( xt,) ( xt,), lim ( xt,) 0 x Energy 1 β λ 4 3λ E( ψ(, t)): ψ Vext( x) ψ = + + ψ + n ϕ dx 3

11 Ground State Non-convex minimization problem Nonlinear Eigenvalue problem (Euler-Language eq.) 1 µφ β λ φ λ ϕ φ ϕ = φ ϕ = φ = ( x) = + Vext( x) + ( ) 3 nn ( x) Chemical potential { } E( φ ) : = min E( φ) with S = φ φ = 1& E( φ) < g φ S ( x) ( x), lim ( x) 0, 1 x 1 4 µ : = φ + Vext( x) φ + ( β λ) φ + 3 λ n ϕ dx 3 β λ 3λ = E( φ) + φ + ϕ dx, & ϕ = φ 3 4 n

12 Ground State Results Theorem (Existence, uniqueness & nonexistence) (Bao, Cai & Wang, JCP, 10 ) Assumptions V x x V x = + 3 ext( ) 0, & lim ext( ) (confinement potential) x Results There exists a ground state Positive ground state is unique Nonexistence of ground state, i.e. Case I: Case II: β < 0 β φg S if β 0 & λ β φ S i 0 φ e θ φ with θ g = lim E ( φ) = β β 0 & λ > β or λ < g 0

13 Key Techniques in Proof Estimate on the Poisson equation ϕ = φ = ρ ϕ = ϕ ϕ = ϕ = ρ = φ : & lim ( x) 0 n ( ) x Positivity & semi-lower continuous E( φ) E( φ ) = E( ρ), φ S with ρ = φ E( ρ ) ρ The energy is strictly convex in if Confinement potential Non-existence result β β 0 & λ β 1 1 x + y z φ ε 1, ε x 1/ 1/4 ( πε1) ( πε ) ε1 ε ( ) = exp exp, x 3 4

14 Numerical Method for Ground State Gradient flow with discrete normalization 1 φ( xt,) = Vext( x) ( β λ) φ + 3 λ nnϕ φ( xt,), t ϕ xt = φ xt ϕ xt = x Ω t t< t φ( xt, ) φ (,) (,), lim (,) 0, & n n+ 1, x + n+ 1 ( xt, n+ 1): = φ( xt, n+ 1) =, x Ω& n 0, φ( xt, n+ 1) φ( xt, ) = ϕ( xt, ) = 0, t 0; φ( x,0) = φ ( x) 0, x Ω, with φ = 1. x Ω x Ω 0 0 Full discretization Backward Euler sine pseudospectal (BESP) method Avoid to use zero-mode in phase space via DST!!

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17 Dynamics and its Computation The Problem 1 i ψ( xt,) = + Vext( x) + ( β λ) ψ 3 λ nnϕ ψ( xt,) t ϕ = ψ ϕ = > ψ Mathematical questions Existence & uniqueness & finite time blow-up??? Existing results 3 ( xt,) ( xt,), lim ( xt,) 0, x, t 0 x ψ 3 ( x,0) = 0( x), x, Carles, Markowich & Sparber, Nonlinearity, 1 (008), Antonelli & Sparber, Physica D (009) --- existence of solitary waves.

18 Well-posedenss Results Theorem (well-posedness) (Bao, Cai & Wang, JCP, 10 ) Assumptions 3 3 α 3 (i) Vext( x) C ( ), Vext( x) 0, x & D Vext( x) L ( ) α 1 3 (ii) ψ 0 X= u H ( ) u = u + u + Vext( x) u( x) dx< X L L 3 Results Local existence, i.e. T (0, ], s. t. the problem has a unique solution ψ C([0, T ), X) max If β β 0 & λ β global existence, i.e. max T max = +

19 Finite Time Blowup Results Theorem (finite time blowup) (Bao, Cai & Wang, JCP, 10 ) Assumptions Results: β (i) β<0 or β 0 & λ < or λ> β 3 (ii) 3 V ( x) + x V ( x) 0, x For any ψ ( x ) X, there exists finite time blowup, i.e. 0 If one of the following conditions holds (i) E( ψ 0) < 0 (ii) E( ψ ) = 0 & Im ψ ( x) ( x ψ ( x)) dx< 0 ext ext T max < + (iii) E( ψ0) > 0 & Im ψ0( x) ( x ψ0( x)) dx< 3 E( ψ0) xψ0 L 3

20 Numerical Method for dynamics Time-splitting sine pseudospectral (TSSP) method, [ tn, t n + 1] Step 1: Discretize by spectral method & integrate in phase space exactly 1 i tψ(,) xt = ψ Step : solve the nonlinear ODE analytically i tψ( xt,) = Vext( x) + ( β λ) ψ( xt,) 3 λ nnϕ( xt,) ψ( xt,) ϕ( xt,) = ψ( xt,), ψ xt = ψ xt = ψ xt ϕ xt = ϕ xt i tψ( xt, ) = Vext( x) + ( β λ) ψ( xt, n) 3 λ nnϕ( xt, n) ψ( xt,) ϕ( xt, n) = ψ( xt, n), i( t tn)[ Vext ( x) + ( β λψ ) ( xt, n) 3 λ nnϕ( xt, n)] ψ( xt, ) = e ψ( xt, ) t( (, ) ) 0 (, ) (, n) & (, ) (, n) n

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25 New numerical methods for DDI How to compute nonlocal DDI FFT (fast Fourier transform) DST (discrete sine transform) dip Nonuniform FFT (Bao, Jiang, Greengard, SISC,14 ) φ= U ( ξρζ ) (, td ) ξ ρ= ψ U φ: = Udip ψ 3( n ξ ) ( ξ ) = 1+ ξ φ = ψ 3 nnϕ & ϕ( xt,) = ψ( xt,) 3 dip sphere coordinate = S + U ξ ξ ρζ dip( ) (, t)

26 Dimension Reduction Gross-Pitaevskii-Poisson equations 1 i ψ( xt,) = + Vext( x) + ( β λ) ψ 3 λ nnϕ ψ( xt,) t ϕ xt = ψ xt ϕ xt = x t> Strongly anisotropic potential 1 Vext( x) = γ x x + γ y y + γ z z ( ) Case I: 3D D T z x y & n = ( n1, n, n3), n = n1 + n + n3 = 1 Case II: 3D 1D γ γ γ 3 (,) (,), lim (,) 0,, 0 x γ γ & γ γ z x y x

27 Dimension Reduction Existing results BEC without dipole-dipole interaction: λ = 0 Formal asymptotic (Bao, Markowich, Schmeiser & Weishaupl, M3AS, 05 ) Numerical results (Bao, Ge, Jaksch, Markowich & Weishaeupl, CPC, 07 ) Rigorous proof (Ben Abdallah, Mehats et al., SIMA, 05; JDE 08 ) From N-body to mean field theory (Lieb, Seiringer & Yngvason, CMP, 04 ; Erdos, Schlein & Yau, Ann. Math., 10 ) Dipolar BEC (Carles, Markowich & Sparber, Nonlinearity, 08 ) formal result

28 Dimension Reduction (3D D) L Assumptions z γ z γ x & γ y = O(1) & Vext( x) = V D( xy, ) +, ε : = 4 ε Decomposition of the linear operator 1 1 L: = + Vext ( x) = + V D( xy, ) + L 1 z 1 1 z = + = + ε ε Ansatz z zz 4 zz it 1 z ψ( xyzt,,,) e ε ψ(, xyt,) ωε() z & ωε() z = exp 1/4 ( επ) ε z 1 γ z

29 Dimension Reduction (3D D) D equations (Bao, Cai, Lei, Rosenkranz, PRA, 10 ) 1 β λ(1 3 n3 ) i ψ( xyt,, ) = [ + V D( xy, ) + ψ t ε π ϕ D ( xyt,,) = Uε ψ, 3 λ ( n n n3 ) ϕ ] ψ ( xyt,,) ( s ) 1 exp / U (, x y) = U () r = ds, r = x + y D D ε ε 3/ π r + ε s

30 Asymptotic of D Kernel For fixed U D ε () r When π ε > 0 3/ ε 0 1 ε D 1 ε ( ε ) ln r+ ln + C, r 0 1, π r U () r, r 0 π r > r

31 Fourier Transform of D Kernel Fourier transform ( ) 1 exp ε s / D D Uε ( ξ1, ξ) = Uε ( ξ ) = ds π ξ + s Asymptotic For fixed When D U ε ε > 0 1, ξ 0 ξ ( ξ ) 1/ 1, ξ π ε ξ ε 0 1 U ε ( ξ ), ξ ξ D

32 Ground State Results for quais-d f f L ( ) L ( ) Cb : = inf ---- Gagliardo-Nirenberg inequality f H ( ) f 4 L ( ) Theorem (Existence & uniqueness) (Bao, Ben Abdallah, Cai, SIMA, 1 ) V ( x) 0, x & lim V ( x) = + (confinement potential) Results D There exists a ground state Case I: Or case II x i 0 Positive ground state is unique g e θ g with Case I: λ 0 & β λ 0 Or case II λ ( λ < 0 & β n ) No ground state if λ ( β + 1 3n ) 3 < ε πc b D φ S λ 0& β λ > ε πcb g λ λ < 0 & β + ( 1+ 3 n3 1 ) > ε πc b if φ = φ θ 0

33 γ γ = = γ x : ε 0 z

34 Dimension Reduction (3D D) D equations when ε 0 (Bao, Cai, Lei, Rosenkranz, PRA, 10 ) 1 β λ + 3λn3 i ψ( xyt,, ) = [ + V D( xy, ) + ψ t ε π 3 λ ( n n n3 ) ϕ ] ψ ( xyt,,) ϕ xyt = ψ xyt ϕ xyt = 1/ ( ) (,,) (,,), lim (,,) 0 ( xy, ) Energy 1 1 E( ψ(, t)): = { ψ + V ( x) ψ + ( β λ + 3 λn ) ψ 3 ε π 4 D 3 3λ 1/4 1/4 + [ n ( ) ϕ n3 ( ) ϕ ]} dx 4

35 Ground State Results for quais-d V x x V x D( ) 0, & lim D( ) = + (confinement potential) x Theorem (Existence & uniqueness) (Bao, Ben Abdallah, Cai, SIMA, 1 ) There exists a ground state Case I: Or case II Or case III Positive ground state is unique Case I: λ = 0 & β 0 Or case II Or case III φ S λ < 0, n3 1 / & β λ 1 3n3 0 No ground state > 0& n3 0 or < 0&n3 < 1 or = 0& < Cb g λ = 0& β > ε πcb λ > = β λ > ε π 0, n3 0 & Cb λ > 0, n = 0 & β λ 3 if ( ) λ < n β λ n > ε πc 0, 3 1 / & b i 0 φ e θ φ with θ g = g ( ) λ λ λ β ε π 0

36 Well-posedness & convergence rate Well-posedness of the Cauchy problem related to the D equations Finite time blow-up may happen!! Theorem (convergence rate) (Bao, Ben Abdallah, Cai, SIMA, 1 ) β Assume β 0, λ β, β = O( ε), λ = O( ε) Then we have it ε ψ(, xyzt,,) e ψ( xyt,,) ω () z Cε, 0 t T ε L T

37 Dimension Reduction (3D 1D) Assumptions x + y γ x = γ y γ z = O(1) & Vext( x) = V1 D( z) +, ε : = 4 ε Decomposition of the linear operator 1 1 L: = + V ( x) = + V ( z) + L ext zz 1D xy 1 γ x L Ansatz 1 x + y 1 1 x + y = + = + ε ε xy xy 4 xy it 1 x ε ψ( xyzt,,, ) e ψ( zt, ) ωε( xy, ) & ωε( xy, ) = exp πε + y ε

38 Dimension Reduction (3D 1D) 1D equations (Bao, Cai, Lei, Rosenkranz, PRA, 10 ) 1 β + λ(1 3 n3 ) i ψ(,) zt = [ zz + V1 D() z + ψ t 4πε Linear case if 3 λ(1 3 n3 ) + 8ε π ϕ ] ψ(,) zt z /ε 1D 1 D e s /ε ϕ( z,) t = Uε ψ, Uε ( z) = e ds, π z n β λ = 1/3& = 0& 0 3 zz

39 Asymptotic of 1D Kernel For fixed U 1D ε When ε > 0 1 z + Oz ( ), z 0 πε ( z) ε, z π z ε 0 1D 1, z = 0 Uε ( z) 0, z 0

40 Fourier Transform of 1D Kernel Fourier transform Asymptotic For fixed When 1D U ε ε > 0 U 1D ε ε ε γe ln ξ ln, ξ 0 π ( ξ ), ξ ε π ξ ε 0 ( ε s ) ε exp / ( ξ ) = ds, 0 π ξ + s 1 D U ε ( ξ )??, ξ

41 Ground State Results for quais-1d V ( z) 0, z & lim V ( z) = + (confinement potential) 1D Theorem (Existence & uniqueness) (Bao, Ben Abdallah, Cai, SIMA, 1 ) There exists a ground state Positive ground state is unique Case I: ( n ) Or case II for any Dynamics results global well-posedness of the Cauchy problem Convergence rate if it z 1D φ S λ & β λ(1 3 n ) g i 0 φ φ θ ( n3) ( n3) g βλε,,,n 1 = e θ with λ 1 3 < 0 & β + λ 1 3 / 0 β = O ε λ = ε ( )& O( ) ε ψ( xyzt,,,) e ψ(,) ztω (, xy) Cε, 0 t T ε L T g 0

42 γ x 1 γ : = γ = ε z

43 Reduction in Multilayered Potential With multilayered potential in z-direction V ( x) = V ( xy, ) + Vsin ( πz) Vω ext D 0 0

44 Reduction in Multilayered Potential GPEs with infinite many equations; Effective single-mode approximation; Bogoliubov enegies (Rosenkranz, Cai &Bao, PRA, 88 (013) )

45 Conclusion & future challenges Conclusion Ground state in 3D existence, uniqueness & nonexistence Dynamics in 3D well-posedness & finite time blowup Efficient numerical methods via DST Dimension Reduction --- 3D D & 3D1D Ground states and dynamics in quasi-d & quasi-1d Future challenges Convergence rate for reduction in O(1) regime In rotating frame & multi-component & spin-1 Dipolar BEC with random potential disorder!!