numerical method for computing ground states of spin-1 bose-einstein condensates
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Numerical Method for Computing Ground States of Spin-1 Bose-Einstein Condensates. Fong Yin Lim Department of Mathematics and Center for Computational Science & Engineering National University of Singapore Email: [email protected] - PowerPoint PPT PresentationTRANSCRIPT
Numerical Method for Computing Numerical Method for Computing Ground States of Spin-1 Ground States of Spin-1
Bose-Einstein CondensatesBose-Einstein Condensates
Fong Yin LimFong Yin Lim
Department of MathematicsDepartment of Mathematicsandand
Center for Computational Science & EngineeringCenter for Computational Science & EngineeringNational University of SingaporeNational University of SingaporeEmail: Email: [email protected]@nus.edu.sg
Collaborators: Weizhu Bao (National University of Singapore)Collaborators: Weizhu Bao (National University of Singapore) I-Liang Chern (National Taiwan University)I-Liang Chern (National Taiwan University)
OutlineOutline
Introduction Introduction
The Gross-Pitaevskii equation The Gross-Pitaevskii equation
Numerical method for single component BEC Numerical method for single component BEC ground stateground state
Gradient flow with discrete normalizationGradient flow with discrete normalizationBackward Euler sine-pseudospectral methodBackward Euler sine-pseudospectral method
Spin-1 BEC and the coupled Gross-Pitaevskii Spin-1 BEC and the coupled Gross-Pitaevskii equationsequations
Numerical method for spin-1 condensate ground Numerical method for spin-1 condensate ground statestate
ConclusionsConclusions
Single Component BECSingle Component BEC
Hyperfine spin Hyperfine spin FF = = I I + + SS
2F+12F+1 hyperfine components: hyperfine components: mmFF = -F, -F+1, …., F-1, F = -F, -F+1, …., F-1, F
Experience different potentials under external magnetic Experience different potentials under external magnetic fieldfieldEarlier BEC experiments: Cooling of magnetically trapped Earlier BEC experiments: Cooling of magnetically trapped atomic vapor to nanokelvins temperature atomic vapor to nanokelvins temperature single single component BECcomponent BEC
ETH (02’,87Rb)
Spinor BECSpinor BEC
Optical trap provides equal confinement for all Optical trap provides equal confinement for all hyperfine componentshyperfine components
mmFF-independent multicomponent BEC-independent multicomponent BEC
Hamburg (03’,87Rb, F=1) Hamburg (03’,87Rb, F=2)
Gross-Pitaevskii EquationGross-Pitaevskii Equation
GPE describes BEC at GPE describes BEC at T <<TT <<Tcc
Non-condensate fraction can be neglectedNon-condensate fraction can be neglectedInteraction between particles is treated by mean field Interaction between particles is treated by mean field approximationapproximation
where interparticle mean-field interactionwhere interparticle mean-field interaction
Number densityNumber density
txtxnctxxVtxm
txt
i ,,,,2
, 02
2
Nxdtx 2
,
2,, txtxn
mac s /4 20
Dimensionless GPEDimensionless GPE
Non-dimensionalization of GPENon-dimensionalization of GPE
Conservation of total number of particlesConservation of total number of particles
Conservation of energyConservation of energy
Time-independent GPETime-independent GPE
12 N
22 )(2
1
xVt
i
xdtxtxxVtxtE 422
,2
,,2
1
)()()()()(2
1)(
22 xxxxVxx
tiextx )(),(
BEC Ground StateBEC Ground State
Boundary eigenvalue methodBoundary eigenvalue method Runge-Kutta space-marchingRunge-Kutta space-marching (Edward & Burnett, (Edward & Burnett, PRAPRA, 95’), 95’)
(Adhikari, (Adhikari, Phys. Lett. APhys. Lett. A, 00’), 00’) Variational methodVariational method
Direct minimization of energy functional with FEM approachDirect minimization of energy functional with FEM approach (Bao & Tang, (Bao & Tang, JCPJCP, 02’), 02’)
Nonlinear algebraic eigenvalue problem approachNonlinear algebraic eigenvalue problem approachGauss-Seidel type iterationGauss-Seidel type iteration (Chang et. al., (Chang et. al., JCPJCP, 05’), 05’)
Continuation methodContinuation method (Chang et. al., (Chang et. al., JCPJCP, 05’), 05’) (Chien et. al., (Chien et. al., SIAM J. SIAM J.
Sci. ComputSci. Comput., 07’)., 07’)
Imaginary time methodImaginary time methodExplicit imaginary time algorithm via Visscher schemeExplicit imaginary time algorithm via Visscher scheme (Chiofalo et. al., (Chiofalo et. al., PREPRE, 00’), 00’)
Backward Euler finite difference (BEFD) and time-splitting Backward Euler finite difference (BEFD) and time-splitting sine-pseudospectral methodsine-pseudospectral method (TSSP) (TSSP) (Bao & Du, (Bao & Du, SIAM J. Sci. ComputSIAM J. Sci. Comput., 03’)., 03’)
--
Imaginary Time MethodImaginary Time Method
Replace Replace in the time-dependent GPE in the time-dependent GPE (imaginary time method) and form gradient flow with (imaginary time method) and form gradient flow with discrete normalization (GFDN) in each time intervaldiscrete normalization (GFDN) in each time interval
BEFD – BEFD – implicit, unconditionally stable, energy diminishing, second order accuracy in space second order accuracy in space
TSSP – explicit, conditionally stable, TSSP – explicit, conditionally stable, spectral accuracy in space
1
22 ,)(2
1
nndd tttxVt
itt
),(
),(),(
1
11
n
nn
tx
txtx
--
Discretization SchemeDiscretization Scheme
The problem is truncated into bounded domain with The problem is truncated into bounded domain with zero boundary conditionszero boundary conditions
Backward Euler sine-pseudospectral method (BESP)Backward Euler sine-pseudospectral method (BESP)( Bao, Chern ( Bao, Chern
& Lim, & Lim, JCPJCP, 06’), 06’)
Backward Euler scheme is applied, except the non-Backward Euler scheme is applied, except the non-linear interaction termlinear interaction term
Sine-pseudospectral method for discretization in spaceSine-pseudospectral method for discretization in space
Consider 1D gradient flow in Consider 1D gradient flow in 1 nn ttt
1,,2,1,)(2
1 *2***
MjxVD
t jnjjjxx
sxx
njj
j
0**0 M
Mjjnj ,,1,0,
*
*1
Backward Euler Sine-pseudospectral Backward Euler Sine-pseudospectral Method (BESP)Method (BESP)
At every time step, a linear system is solved iterativelyAt every time step, a linear system is solved iteratively
Differential operator of the second order spatial derivative Differential operator of the second order spatial derivative of vector of vector U=(UU=(U00, U, U11, …, U, …, UMM))TT
; ; UU satisfying satisfying UU00 = U = UMM = 0 = 0
The sine transform coefficientsThe sine transform coefficients
1
1
2 1,,2,1,)(sin)ˆ(2 M
ljlllxx
sxx MjaxU
MUD
j
1
1
1,,2,1,,)(sin)ˆ(M
jljljl Ml
ab
laxUU
mj
njjxx
msxx
nj
m
xVDt j
*,21*,1*,
)(2
1
Backward Euler Sine-pseudospectral Backward Euler Sine-pseudospectral Method (BESP)Method (BESP)
A stabilization parameter A stabilization parameter is introduced to ensure the is introduced to ensure the convergence of the numerical schemeconvergence of the numerical scheme
Discretized gradient flow in position spaceDiscretized gradient flow in position space
Discretized gradient flow in phase spaceDiscretized gradient flow in phase space
mj
njj
mjxx
msxx
nj
m
xVDt j
*,21*,1*,1*,
)(2
1
Mjnjj ,,1,0,0*,
MjxVG mj
njj
mj ,,1,0,)( *,2
1,,2,1,)ˆ()ˆ(22
2)ˆ(
21*,
MlGt
t lm
ln
ll
m
Backward Euler Sine-pseudospectral Backward Euler Sine-pseudospectral Method (BESP)Method (BESP)
Stabilization parameterStabilization parameter
guarantees the convergence of the iterative method guarantees the convergence of the iterative method and gives the optimal convergence rateand gives the optimal convergence rateBESP is spectrally accurate in space and is BESP is spectrally accurate in space and is unconditionally stable, thereby allows larger mesh size unconditionally stable, thereby allows larger mesh size and larger time-step to be usedand larger time-step to be used
minmax2
1bb
2
11min
2
11max )(min,)(max n
jjMj
njj
MjxVbxVb
BEC in 1D PotentialsBEC in 1D Potentials
-16 -8 0 8 160
0.1
0.2
0.3
0.4
g(x)
-16 -8 0 8 160
35
70
105
140
V(x
)
x
400,2
)(2
xxV
-16 -8 0 8 160
0.1
0.2
0.3
0.4
g(x)
-16 -8 0 8 160
35
70
105
140
V(x
)
x
250,4
sin252
)( 22
xx
xV
BEC in 1D Optical LatticeBEC in 1D Optical Lattice
Comparison of spatial accuracy of BESP and backward-Euler Comparison of spatial accuracy of BESP and backward-Euler finite difference (BEFD)finite difference (BEFD)
BEC in 3D Optical LatticeBEC in 3D Optical Lattice
Multiscale structures due to the oscillatory nature of trapping Multiscale structures due to the oscillatory nature of trapping potentialpotential
High spatial accuracy is required, especially for 3D problemsHigh spatial accuracy is required, especially for 3D problems
6400,800,100
4sin
4sin
4sin50
2
1)( 222222
zyxzyxxV
Spin-1 BECSpin-1 BEC
3 hyperfine components: 3 hyperfine components: mmFF = -1, 0 ,1 = -1, 0 ,1
Coupled Gross-Pitaevskii equations (CGPE)Coupled Gross-Pitaevskii equations (CGPE)
ββnn -- -- spin-independent mean-field interaction spin-independent mean-field interaction
ββss -- s -- spin-exchange interactionpin-exchange interaction
Number densityNumber density nnnnn ii 0
2,
200
2
00020
200
2
)()(2
1
2)()(2
1
)()(2
1
ssn
ssn
ssn
nnnnxVt
i
nnnxVt
i
nnnnxVt
i
Spin-1 BECSpin-1 BEC
Conservation of total number of particlesConservation of total number of particles
Conservation of energyConservation of energy
Conservation of total magnetizationConservation of total magnetization
122
0
2 N
22
M
xd
nnnnnnn
nnxVE
s
snsn
n
20
20
022
222
0
2
)(22
2)(
2
10
Spin-1 BECSpin-1 BEC
Time-independent CGPETime-independent CGPE
Chemical potentialsChemical potentials
Lagrange multipliers, Lagrange multipliers, µ µ and and λλ,, are introduced to the are introduced to the free energy to satisfy the constraints N and Mfree energy to satisfy the constraints N and M
200
2
0002
00
200
2
)()(2
1
2)()(2
1
)()(2
1
ssn
ssn
ssn
nnnnxV
nnnxV
nnnnxV
,, 0
tiii
iextx )(),(
Spin-1 BEC Ground StateSpin-1 BEC Ground State
Imaginary time propagation of CGPE with initial Imaginary time propagation of CGPE with initial complex Gaussian profiles with constant speedcomplex Gaussian profiles with constant speed
Continuous normalized gradient flow (CNGF)Continuous normalized gradient flow (CNGF)
-- -- -- N-- N and and MM conserved, energy diminishing conserved, energy diminishing-- Involve -- Involve and implicitlyand implicitly
200
2
00020
200
2
)()()(2
1
2)()(2
1
)()()(2
1
ssn
ssn
ssn
nnnnxVt
nnnxVt
nnnnxVt
(Zhang, Yi & You, (Zhang, Yi & You, PRA,PRA, 02’) 02’)
(Bao & Wang, (Bao & Wang, SIAM J. Numer. AnalSIAM J. Numer. Anal., 07’)., 07’))(),( tt
Normalization ConditionsNormalization Conditions
Numerical approach with GFDN by introducing third Numerical approach with GFDN by introducing third normalization conditionnormalization condition
Time-splitting scheme to CNGF in Time-splitting scheme to CNGF in 1.1. Gradient flowGradient flow
2.2. Normalization/ ProjectionNormalization/ Projection
200
2
00020
200
2
)()(2
1
2)()(2
1
)()(2
1
ssn
ssn
ssn
nnnnxVt
nnnxVt
nnnnxVt
)(,,)( 00
ttt
1 nn ttt
Normalization ConditionsNormalization Conditions
Normalization stepNormalization step
Third normalization conditionThird normalization condition
Normalization constantsNormalization constants
*)(
*0
*)(
0
t
t
t
e
e
e
20
2
1
2*
2*0
22
1
2*
2*0
2
2
1
4*0
22*2*22*0
2
0
2
1,
2
1
)1(4
1
00
MM
MM
M
*
*00
*
0
Discretization SchemeDiscretization Scheme
Backward-forward Euler sine-pseudospectral method Backward-forward Euler sine-pseudospectral method (BFSP) (BFSP)
Backward Euler scheme for the Laplacian; forward Euler Backward Euler scheme for the Laplacian; forward Euler scheme for other termsscheme for other terms
Sine-pseudospectral method for discretization in spaceSine-pseudospectral method for discretization in space
1D gradient flow for 1D gradient flow for mmFF = +1 = +1
ExplicitExplicit
Computationally efficientComputationally efficient
nj
njs
nj
nj
nj
njs
njnj
jxx
sxx
njj
nnnnxV
Dt j
,
2
,0,,,0,
*,
*,*
,
)(
2
1
8787Rb in 1D harmonic potentialRb in 1D harmonic potential
Repulsive and ferromagnetic interaction (Repulsive and ferromagnetic interaction (ββnn >0 >0 , , ββss < 0) < 0)
Initial conditionInitial condition
2/4/1
02/4/1
00
2/4/1
0 222 )1(5.0,,
)1(5.0 xxx eM
eeM
8787Rb in 1D harmonic potentialRb in 1D harmonic potential
-16 -8 0 8 160
0.05
0.1
0.15
0.2
0.25
-16 -8 0 8 160
30
60
90
120
150
x
M=0.2
42
10,2
)( Nx
xV
-16 -8 0 8 160
0.05
0.1
0.15
0.2
0.25M=0.7
-16 -8 0 8 160
30
60
90
120
150
x
V(x
)
Repulsive and ferromagnetic interaction (Repulsive and ferromagnetic interaction (ββnn >0 >0 , , ββss < 0) < 0)
-16 -8 0 8 160
0.05
0.1
0.15
0.2
0.25
x
(x)
-16 -8 0 8 160
30
60
90
120
150M=0
|+|
|0|
|-|
2323Na in 1D harmonic potentialNa in 1D harmonic potential
-12 -6 0 6 120
0.1
0.2
0.3
-12 -6 0 6 120
30
60
90
x
M=0.2
42
10,2
)( Nx
xV
-12 -6 0 6 120
0.1
0.2
0.3
x-12 -6 0 6 12
0
30
60
90
V(x
)
M=0.7
Repulsive and antiferromagnetic interaction (Repulsive and antiferromagnetic interaction (ββnn >0 >0 , , ββss > 0) > 0)
Initial conditionInitial condition
-12 -6 0 6 120
0.1
0.2
0.3
x
(x)
-12 -6 0 6 120
30
60
90M=0
|+|
|0|
|-|
2/4/1
02/4/1
00
2/4/1
0 222 )1(5.0,,
)1(5.0 xxx eM
eeM
8787Rb in 3D optical latticeRb in 3D optical lattice
)
2(sin)
2(sin)
2(sin100
2
1)( 222222 zyx
zyxxV
5.0,104 MN
2323Na in 3D optical latticeNa in 3D optical lattice
)
2(sin)
2(sin)
2(sin100
2
1)( 222222 zyx
zyxxV
5.0,104 MN
Relative PopulationsRelative Populations
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
M
||+||2
||0||2
||-||2
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
M
Relative populations of each componentRelative populations of each component
Same diagrams are obtained for all kind of trapping potential Same diagrams are obtained for all kind of trapping potential in the absence of magnetic fieldin the absence of magnetic field
ββss < 0 ( < 0 (8787Rb)Rb) ββss > 0 ( > 0 (2323Na)Na)
Chemical PotentialsChemical Potentials
Weighted errorWeighted error
Minimize Minimize ee with respect to with respect to µµ and and λλ
2220
2
022
e
dxnnnnnVx
dxnnnnVx
dxnnnnnVx
ssn
ssn
ssn
200
2
2
200
2
02
0
0
200
2
2
)(2
11
2)(2
11
)(2
11
222222
22
,M
AMB
M
BMA
222
0
2
0
2, BA
Spin-1 BEC in 1D Harmonic PotentialSpin-1 BEC in 1D Harmonic Potential
8787Rb (N=10Rb (N=1044))• E, µ are E, µ are
independent of Mindependent of M• λλ = 0 for all M = 0 for all M
(You et. al., (You et. al., PRAPRA, 02’), 02’)
2323Na (N=10Na (N=1044))
0 0.2 0.4 0.6 0.8 115.2
15.4
15.6
E
0 0.2 0.4 0.6 0.8 1
25.2
25.3
25.4
0 0.2 0.4 0.6 0.8 10
0.4
0.8
M
Spin-1 BEC in Magnetic FieldSpin-1 BEC in Magnetic Field
Spin-1 BEC subject to external magnetic field Spin-1 BEC subject to external magnetic field
Straightforward to include magnetic field in the Straightforward to include magnetic field in the numerical schemenumerical scheme
Stability??Stability??
200
2
000020
200
2
)()()(2
1
2)()()(2
1
)()()(2
1
ssn
ssn
ssn
nnnnxExVt
i
nnnxExVt
i
nnnnxExVt
i
)(xB
ConclusionsConclusions
Spectrally accurate and unconditionally stable method Spectrally accurate and unconditionally stable method for single component BEC ground state computationfor single component BEC ground state computation
Extension of normalized gradient flow and sine-Extension of normalized gradient flow and sine-pseudospectal method to spin-1 condensatepseudospectal method to spin-1 condensate
Introduction of the third normalization condition in Introduction of the third normalization condition in addition to the existing conservation of N and Maddition to the existing conservation of N and M
Future works:Future works:-- -- Extension of the method to spinor condensates with Extension of the method to spinor condensates with higher spin degrees of freedomhigher spin degrees of freedom---- Finite temperature effectFinite temperature effect