Numerical simulation of the dynamics of the rotating dipolar Bose-Einstein condensates

Transcription

1 Numerical simulation of the dynamics of the rotating dipolar Bose-Einstein condensates Qinglin TANG Inria, University of Lorraine Joint work with: Prof. Weizhu BAO, Daniel MARAHRENS, Yanzhi ZHANG and Yong ZHANG

2 Outline 1 Brief introduction 2 Numerical method 3 Numerical examples 4 Summary

3 Brief introduction Outline 1 Brief introduction 2 Numerical method 3 Numerical examples 4 Summary Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 3 / 28

4 Brief introduction BEC Bose-Einstein condensate (BEC): many bosons occupy same quantum state if T < T c. Prediction: Einstein 1924 Experiments: JILA, BEC (1995), quantized vortex (1999) Interaction between particles: short-range s-wave contact interaction 1 Griesmaier, Werner, Hensler, Stuhler & Pfau, PRL 94 (2005) 2 Shuman, Barry & Demile, Nature 467 (2010) Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 4 / 28

5 Brief introduction BEC Bose-Einstein condensate (BEC): many bosons occupy same quantum state if T < T c. Prediction: Einstein 1924 Experiments: JILA, BEC (1995), quantized vortex (1999) Interaction between particles: short-range s-wave contact interaction Realization of BECs of 52 Cr (Chromium-52) 1 and creation of ultracold molecules 2 : long-range dipole-dipole interaction (DDI) between particle besides the contact interaction leading to fascinating and sometimes completely unexpected effects. 1 Griesmaier, Werner, Hensler, Stuhler & Pfau, PRL 94 (2005) 2 Shuman, Barry & Demile, Nature 467 (2010) Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 4 / 28

6 Brief introduction Collapse even if the contact interaction are repulsive, dipole orientation (0,0,1): Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 5 / 28

7 Brief introduction Tunability of the dipole: dipole orientation (cos(0.2t),0,sin(0.2t)) Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 6 / 28

8 Brief introduction Gross-Pitaevskii equation (GPE): S. Yi and H. Pu, PRA 73 (2006); O Dell, et al., PRA 80 (2009) i tψ(x,t) = [ 12 ] 2 +V(x)+β ψ 2 +λφ(x,t) ΩL z ψ(x,t), x R d. (1) d = 2,3; V(x): trapping potential; L z = i(x y y x) = i θ. 3 O Dell et al., PRL 92 (2004); Bao et al., PRA 82 (2010) Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 7 / 28

9 Brief introduction Gross-Pitaevskii equation (GPE): S. Yi and H. Pu, PRA 73 (2006); O Dell, et al., PRA 80 (2009) i tψ(x,t) = [ 12 ] 2 +V(x)+β ψ 2 +λφ(x,t) ΩL z ψ(x,t), x R d. (1) d = 2,3; V(x): trapping potential; L z = i(x y y x) = i θ. Φ(x,t): dipole-dipole interaction (DDI) potential, reads as 3 Φ(x,t) = U d dip ψ 2, (2) with Udip d (x,t) = 3 2 ( ) δ(x) 3 1 nn, d = 3, 4π x ( n n n ) ( 1 2π x ), d = 2, (3) where n = (n 1 (t),n 2 (t),n 3 (t)), is the time (in)-dependent dipole axis, n = (n 1 (t),n 2 (t)). 3 O Dell et al., PRL 92 (2004); Bao et al., PRA 82 (2010) Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 7 / 28

10 Brief introduction Numerics: Ω = λ = 0, time-splitting spectral method(tssp) 4 Step1 : i tψ(x,t) = ψ(x,t), Step2 : i tψ(x,t) = ( V(x)+β ψ 2) ψ(x,t). (4) Step 1: discretised by spectral method and integrated in space exactly. Step 2: nonlinear ODE solved analytically Spectral in space, easy to implement. 4 Bao, Jaksch & Markowich, JCP 03; Bao, Jin, Markowich, SIAM J. Sci. Comput., 03; etc Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 8 / 28

11 Brief introduction Numerics: Ω = λ = 0, time-splitting spectral method(tssp) 4 Step1 : i tψ(x,t) = ψ(x,t), Step2 : i tψ(x,t) = ( V(x)+β ψ 2) ψ(x,t). (4) Step 1: discretised by spectral method and integrated in space exactly. Step 2: nonlinear ODE solved analytically Spectral in space, easy to implement. Cannot be simply extended to rotating system: Ω 0. Numerical difficulties: Ω 0, λ 0 (1). rotating term: L zψ. (2). nonlocal DDI term: Φ(x,t) =: U d dip ψ 2. 4 Bao, Jaksch & Markowich, JCP 03; Bao, Jin, Markowich, SIAM J. Sci. Comput., 03; etc Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 8 / 28

12 Numerical method Outline 1 Brief introduction 2 Numerical method 3 Numerical examples 4 Summary Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 9 / 28

13 Consider only the rotational term first: i.e., λ = 0, Ω 0 L z ψ(x,t) = i(y x x y )ψ(x,t), x R d,t 0

14 Numerical method Existing numerical methods: Ω 0, λ = 0, L z = i(y x x y ) = i θ Time splitting + ADI: Bao and Wang, JCP, Time splitting + polar/cylindrical coordinates: Bao, Du and Zhang, SIAM J. Appl. Math., Time splitting + Laguerre-Fourier-Hermite: Bao, Li and Shen, SIAM J. Sci. Comupt., 2009.: Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 11 / 28

15 Numerical method Existing numerical methods: Ω 0, λ = 0, L z = i(y x x y ) = i θ Time splitting + ADI: Bao and Wang, JCP, extra error for the splitting, not trivial to extend to higher order (in time) scheme. Step1 : i tψ(x,t) = [ 12 xx 14 ] zz iω y x ψ(x,t). (5) Step2 : i tψ(x,t) = [ 12 yy 14 ] zz +iω x y ψ(x,t). (6) Step3 : i tψ(x,t) = [ V(x)+β ψ 2] ψ(x,t). (7) Time splitting + polar/cylindrical coordinates: Bao, Du and Zhang, SIAM J. Appl. Math., Time splitting + Laguerre-Fourier-Hermite: Bao, Li and Shen, SIAM J. Sci. Comupt., 2009.: Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 11 / 28

16 Numerical method Existing numerical methods: Ω 0, λ = 0, L z = i(y x x y ) = i θ Time splitting + ADI: Bao and Wang, JCP, Time splitting + polar/cylindrical coordinates: Bao, Du and Zhang, SIAM J. Appl. Math., only 2nd or 4th order accuracy in radial direction. Step1 : i tψ(x,t) = [ 12 ] 2 ΩL z ψ(x,t). (8) Step2 : i tψ(x,t) = [ V(x)+β ψ 2] ψ(x,t). (9) Time splitting + Laguerre-Fourier-Hermite: Bao, Li and Shen, SIAM J. Sci. Comupt., 2009.: Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 11 / 28

17 Numerical method Existing numerical methods: Ω 0, λ = 0, L z = i(y x x y ) = i θ Time splitting + ADI: Bao and Wang, JCP, Time splitting + polar/cylindrical coordinates: Bao, Du and Zhang, SIAM J. Appl. Math., Time splitting + Laguerre-Fourier-Hermite: Bao, Li and Shen, SIAM J. Sci. Comupt., 2009.: implementation of the code is quite involved. Step1 : i tψ(x,t) = [ 12 ] 2 ΩL z + x 2 ψ(x,t). (10) 2 Step2 : i tψ(x,t) = [ U(x)+β ψ 2] ψ(x,t). (11) Q. Tang (Inria, UdL) 2 Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 11 / 28

18 Numerical method A rotating Lagrangian coordinate transformation Our method 5 : rotating Lagrangian coordinates transformation = relax ΩL z = TSSP. 5 Bao, Marahrens, Tang & Zhang, SIAM J. Sci. Comput., 13; Ming, Tang, & Zhang, J. Comput. Phys., 13 Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 12 / 28

19 Numerical method A rotating Lagrangian coordinate transformation Our method 5 : rotating Lagrangian coordinates transformation = relax ΩL z = TSSP. Rotating Lagrangian coordinates x: x = A 1 (t)x = A T (t)x x = A(t) x, x R d. (12) where A(t): orthogonal rotational matrix defined in 3-d,respectively, 2-d cos(ωt) sin(ωt) 0 ( A(t) = cos(ωt) sin(ωt) sin(ωt) cos(ωt) 0, A(t) = sin(ωt) cos(ωt) Geometrical relation: Cartesian vs rotating Lagrangian coordinates. ). (13) 5 Bao, Marahrens, Tang & Zhang, SIAM J. Sci. Comput., 13; Ming, Tang, & Zhang, J. Comput. Phys., 13 Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 12 / 28

20 Numerical method New formation of GPE, simple and efficient numerical method (Recall) GPE in Cartesian coordinate: i tψ(x,t) = [ 12 ] 2 +V(x)+β ψ 2 ψ(x,t) ΩL zψ(x,t). (14) Take rotating Lagrangian transformation, set φ( x,t) := ψ(x,t) = ψ(a(t) x,t), GPE with rotational term in rotating Lagrangian coordinate i tφ( x,t) = [ 12 ] 2 +V(A(t) x)+β φ 2 φ( x,t). (15) TSSP methods: Step1 : i tφ( x,t) = φ( x,t). (16) Step2 : i tφ( x,t) = ( V(A(t) x)+β φ 2) φ( x,t). (17) Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 13 / 28

21 Numerical method Taking back the nonlocal dipolar term Reformulate GPE and truncated into bounded domain problem, D R d bounded: i tφ( x,t) = [ 12 ] 2 + V(A(t) x) + β φ 2 + λũddip φ( x,t) 2 φ( x, t), x D (18) with Ũd dip ( x,t) = 3 2 ( ) δ( x) 3 1 mm 4π x, m(t) = A(t)n(t), d = 3, ( m m m 2 ) ( π x ), m (t) = A(t)n (t), d = 2, (19) Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 14 / 28

22 Numerical method Taking back the nonlocal dipolar term Reformulate GPE and truncated into bounded domain problem, D R d bounded: i tφ( x,t) = [ 12 ] 2 + V(A(t) x) + β φ 2 + λũddip φ( x,t) 2 φ( x, t), x D (18) with Ũd dip ( x,t) = 3 2 ( ) δ( x) 3 1 mm 4π x, m(t) = A(t)n(t), d = 3, ( m m m 2 ) ( π x ), m (t) = A(t)n (t), d = 2, (19) TSSP methods: for t n < t < t n+1 Step1 : i tφ( x,t) = φ( x,t) (20) [ Step2 : i tφ( x,t) = V(A(t) x) + β φ( x,t) 2 + λũd dip φ( x,t) 2] φ( x,t) (21) Step 2: density ρ(x,t) =: φ( x,t) 2 = φ( x,t n) 2 =: ρ n ( x) integrate analytically { [ t ( ) ]} φ( x,t) = exp i βρ n ( x)(t t n) + V (A(τ) x) + Ũd dip ( x,τ) ρn ( x) dτ tn (22) Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 14 / 28

23 Numerical method Taking back the nonlocal dipolar term Reformulate GPE and truncated into bounded domain problem, D R d bounded: i tφ( x,t) = [ 12 ] 2 + V(A(t) x) + β φ 2 + λũddip φ( x,t) 2 φ( x, t), x D (18) with Ũd dip ( x,t) = 3 2 ( ) δ( x) 3 1 mm 4π x, m(t) = A(t)n(t), d = 3, ( m m m 2 ) ( π x ), m (t) = A(t)n (t), d = 2, (19) TSSP methods: for t n < t < t n+1 Step1 : i tφ( x,t) = φ( x,t) (20) [ Step2 : i tφ( x,t) = V(A(t) x) + β φ( x,t) 2 + λũd dip φ( x,t) 2] φ( x,t) (21) Step 2: density ρ(x,t) =: φ( x,t) 2 = φ( x,t n) 2 =: ρ n ( x) integrate analytically { [ t ( ) ]} φ( x,t) = exp i βρ n ( x)(t t n) + V (A(τ) x) + Ũd dip ( x,τ) ρn ( x) dτ tn Question: How to evaluate the nonlocal DDI term: Φ = Ũd dip ρn ( x)? (22) Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 14 / 28

24 Evaluation of nonlocal potential: Φ = Ũd dip ρn ( x)

25 Numerical method Numerical methods to evaluate the nonlocal term Nonlocal DDI potential Φ = Ũd dip ρn ( x) (23) with ( ) Ũ d dip = δ( x) 3 1 mm 4π x, 3 ( 2 m m n ) ( 1 2π x ), Ũ d 1 + 3(m k)2 dip = k 2, d = 3, 3 [(m k) 2 m 2 3 k 2], d = 2, 2 k where Ũ d } dip {Ũd (k,t) =: F dip ( x,t) denotes the Fourier transform of Ũd dip (24) Naturally, convolution theorem + standard fast Fourier transform (FFT) 6 { Φ( x,t) = F 1 ρ n Ũ d } dip (25) standard FFT: uniform grid points in Cartesian coordinates, require value of 0-mode in Fourier space singularity of Ũ d dip (k,t) at k = 0 = locking phenomena 6 Lahaye, Metz, Fröhlich, Koch, Meister, Griesmaier, Pfau, Saito, Kawaguchi & Ueda, Phys. Rev. Lett., 08 Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 16 / 28

26 Numerical method Numerical methods to evaluate the nonlocal term: Φ = Ũ d dip ρn ( x) Avoid the use of 0-mode: reformulate problem to equivalent poisson/fractional-poisson 7 : Ũd dip = with ( ) 1 δ( x) 3 mm, 4π x 3 ( 2 m m m d = 3 : d = 2 : ) ( 1 2π x ), { ρ n Φ( x,t) ( x) + 3 mmϕ, = ( 3 2 m m m 2 (26) 3 ) 2 ϕ, }ϕ( x,t) = ρ n ( x) with lim ϕ( x,t) = 0. (27) x homogeneous Dirichlet BC + discrete sine spectral (DST) Polynomial decay of ϕ( x, t) large computational domain to reduce the boundary truncation error: 7 Bao, Cai, & Wang, J. Comput. Phys., 10; Bao, Marahrens, Tang & Zhang, SIAM J. Sci. Comput., 13; Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 17 / 28

27 Numerical method Numerical methods to evaluate the nonlocal term: Φ = Ũ d dip ρn ( x) For example, we take d = 3, and ρ n ( x) = e x 2 /c 2 = Φ = ρ n ( x) 3 m T D m (28) here, the Hessian matrix D is given as follows: ( c 2 x 2 D ij = δ ij 2 x 2e c 2 c3 ( ) π x ) 4 x Erf 3 c ( +x ix j 3 c2 x 2 2 x 4e c 2 1 x 2 x 2e c c3 π ( x )) 4 x Erf. 5 c where δ ij is the dirac function and Erf(r) is the error function defined as Erf(r) = 2 π r 0 e t2 dt. Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 18 / 28

28 Numerical method Numerical methods to evaluate the nonlocal term: Φ = Ũ d dip ρn ( x) For example, we take d = 3, and ρ n ( x) = e x 2 /c 2 = Φ = ρ n ( x) 3 m T D m (28) here, the Hessian matrix D is given as follows: ( c 2 x 2 D ij = δ ij 2 x 2e c 2 c3 ( ) π x ) 4 x Erf 3 c ( +x ix j 3 c2 x 2 2 x 4e c 2 1 x 2 x 2e c c3 π ( x )) 4 x Erf. 5 c where δ ij is the dirac function and Erf(r) is the error function defined as Erf(r) = 2 π r 0 e t2 dt. Let m = (0,0,1) T, c = 1.4, solve Φ on computational domain D = [ L,L] 3 with h the mesh size via DST and compare with the exact solution (28). Table: l 2 -errors of Φ by DST on D[ L,L] 2. h = 1 h = 1/2 h = 1/4 h = 1/8 L = E E E E-02 L = E E E E-02 L = E E E E-02 Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 18 / 28

29 Numerical method Numerical methods to evaluate the nonlocal term: Φ = Ũ d dip ρn ( x) Convolution theorem +coordinate transform +non-uniform fast Fourier transform (NUFFT) 8 { Φ( x,t) = F 1 ρ n Ũ d } dip, with Ûdip d = 3[(m k) 2 m 2 3 k 2 ] 2 k, d = (m k)2 k 2, d = 3 (29) Singularity at k = 0 removable: adopt spherical/polar coordinate transform in 3-d/2-d Φ( x, t) = = eik x Ûd dip (k,t) ρn (k) R d (2π) d 1 4π 2 R 0 1 8π 3 R 0 e ik xûd dip dk (k,t) ρ n (k) k <R (2π) d dk 2π 0 e ik x k Û d dip ρ n drdφ, d = 2 π 0 2π 0 e ik x k 2Ûd dip ρ n sinθdrdθdφ, d = 3 (30) Further discretized by high-order Gauss quadrature: Azimuthal φ-direction: trapezoidal rule radial r-direction (and inclination θ-direction in 3d): (shifted and scaled) Gauss-Legendre quadrature 8 Bao, Greengard & Jiang, SIAM J. Sci. Comput., 2014, Bao, Tang & Zhang, preprint, 2014 Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 19 / 28

30 Numerical method NUFFT to evaluate the nonlocal term: Φ = Ũ d dip ρn ( x) Same example as the DST one, now we solve Φ by NUFFT. Table: l 2 -errors of Φ by NUFFT on D[ L,L] 2. h = 1 h = 1/2 h = 1/4 h = 1/8 L = E E E-14 <1E-14 L = E E E-15 <1E-15 Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 20 / 28

31 Extension to rotating two-component dipolar BEC

32 Numerical method Extension to rotating two-component dipolar BEC 9 Rotating two-component GPE: j = 1, 2 [ ] i tψ j (x,t) = V j (x) ΩL z + (β jl ψ l 2 +λ jl Φ l ) ψ j (x,t) κψ 3 j,(31) l=1 Φ l (x,t) = Udip d ψ l(x,t) 2, l = 1, 2, x R d, t 0 (32) κ: Rabi frequency represents the internal atomic Josephson Junction effect (JJE). 9 Saito, Kawaguchi and Ueda, PRL 102 (2009), Ming, Tang & Zhang, JCP 258 (2013), Bao, Mauser, Tang & Yong, preprint, 2014 Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 22 / 28

33 Numerical method Extension to rotating two-component dipolar BEC 9 Rotating two-component GPE: j = 1, 2 [ ] i tψ j (x,t) = V j (x) ΩL z + (β jl ψ l 2 +λ jl Φ l ) ψ j (x,t) κψ 3 j,(31) l=1 Φ l (x,t) = Udip d ψ l(x,t) 2, l = 1, 2, x R d, t 0 (32) κ: Rabi frequency represents the internal atomic Josephson Junction effect (JJE). Take rotating Lagrangian transformation: φ j ( x,t) := ψ(a(t)x) i φ j( x,t) t = [ V j (A(t) x,t)+ 2 ( βjl φ l 2 ) ] +λ jl Φ j φ j κφ 3 j ( x,t),(33) l=1 Φ l (x,t) = Ũd dip φ l( x,t) 2, l = 1, 2, x R d, t 0 (34) 9 Saito, Kawaguchi and Ueda, PRL 102 (2009), Ming, Tang & Zhang, JCP 258 (2013), Bao, Mauser, Tang & Yong, preprint, 2014 Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 22 / 28

34 Numerical method Extension to rotating two-component BEC TSSP methods: ( ) 2 Step1 : i tφ j ( x,t) = V j (A(t) x,t)+ (β jl φ l 2 +λ jl Φj ) φ j (35) l=1 Step2 : i tφ j ( x,t) = φ j ( x,t) κψ 3 j (36) Step 2: set ϕ 1 ( x,t) = φ 1 +φ 2 and ϕ 2 ( x,t) = φ 1 φ 2 φ 1 = ϕ 1+ϕ 2 and φ 2 2 = ϕ 1 ϕ 2 2 i tϕ 1 ( x,t) = ϕ 1 ( x,t) κϕ 1, (37) i tϕ 2 ( x,t) = ϕ 2 ( x,t)+κϕ 2 (38) Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 23 / 28

35 Numerical examples Outline 1 Brief introduction 2 Numerical method 3 Numerical examples 4 Summary Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 24 / 28

36 Numerical examples Test of accuracy: one-component BEC Let d = 3, dipole axis n = m = (0,0,1) T, computational domain D = [ 8,8] 3, let initial data be ψ 0 (x) = 1 π 3/4e (x2 +y 2 +z 2 )/2, (39) and take γ x = γ y = γ z = 1, Ω = 0. We compute the exact solution at time t = 0.28 with very small mesh size h = 1 16 and τ = , and λ = β 2. Table: l 2 -error at t = 0.28 in spatial direction (upper parts) and temporal direction (lower parts h = 1/2 1/4 1/8 1/16 β = E E E E-12 β = E E E E-12 β = E E E E-11 τ = τ/2 τ/4 τ/8 β = E E E E-8 β = E E E E-7 β = E E E E-6 Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 25 / 28

37 Numerical examples Application to two-component dipolar BEC Initial data: ground state under parameters n = (0,0,1),γ x = γ y = γ z = 1,Ω = 0,κ = 0, (40) β 12 = β 21 = λ 12 = λ 21 = 0,β 11 = β 22 = ,λ 11 = λ 22 = (41) Dynamics: change parameters: β 12 = β 21 = 100 and λ 22 = 0. Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 26 / 28

38 Summary Outline 1 Brief introduction 2 Numerical method 3 Numerical examples 4 Summary Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 27 / 28

39 Summary Numerical methods on dynamic simulation of rotating dipolar BEC was presented: Rotating Lagrangian Coordinate transformation to eliminate the constrain of rotation term NUFFT to evaluate non-local DDI potential Easy to apply to other system: rotating multi-component BECs, rotating spin-orbit coupling BEC, etc... Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 28 / 28

40 Summary Numerical methods on dynamic simulation of rotating dipolar BEC was presented: Rotating Lagrangian Coordinate transformation to eliminate the constrain of rotation term NUFFT to evaluate non-local DDI potential Easy to apply to other system: rotating multi-component BECs, rotating spin-orbit coupling BEC, etc... Thank You For Attention! Q. Tang (Inria, UdL) Dynamics simulation of rotating dipolar BEC 10/12/2014, Paris 28 / 28