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Mathematical Analysis and Numerical Simulation for Bose-Einstein Condensation
Weizhu Bao
Department of Mathematics& Center of Computational Science and Engineering
National University of SingaporeEmail: bao@math.nus.edu.sg
URL: http://www.math.nus.edu.sg/~bao
Collaborators
External – P.A. Markowich, Institute of Mathematics, University of Vienna, Austria – D. Jaksch, Department of Physics, Oxford University, UK– Q. Du, Department of Mathematics, Penn State University, USA– J. Shen, Department of Mathematics, Purdue University, USA– L. Pareschi, Department of Mathematics, University of Ferarra, Italy– I-Liang Chern, Department of Mathematics, National Taiwan University, Taiwan– C. Schmeiser & R.M. Weishaeupl, University of Vienna, Austria – W. Tang & L. Fu, Beijing Institute of Appl. Phys. & Comput. Math., China
Internal– Yanzhi Zhang, Hanquan Wang, Fong Ying Lim, Ming Huang Chai– Yunyi Ge, Fangfang Sun, etc.
Outline
Part I: Predication & Mathematical modeling– Theoretical predication– Physical experiments and results– Applications– Gross-Pitaevskii equation
Part II: Analysis & Computation for Ground states – Existence & uniqueness – Energy asymptotics & asymptotic approximation– Numerical methods– Numerical results
Outline
Part III: Analysis & Computation for Dynamics in BEC– Dynamical laws– Numerical methods– Vortex stability & interaction
Part IV: Rotating BEC & multi-component BEC– BEC in a rotational frame– Two-component BEC– Spinor BEC– BEC at finite temperature– Conclusions & Future challenges
Part I
Predication &
Mathematical modeling
Theoretical predication
Particles be divided into two big classes– Bosons: photons, phonons, etc
• Integer spin• Like in same state & many can occupy one obit• Sociable & gregarious
– Fermions: electrons, neutrons, protons etc• Half-integer spin & each occupies a single obit• Loners due to Pauli exclusion principle
Theoretical predication
For atoms, e.g. bosons– Get colder:
• Behave more like waves & less like particles– Very cold:
• Overlap with their neighbors
– Extremely cold: • Most atoms behavior in the same way, i.e gregarious • quantum mechanical ground state, • `super-atom’ & new matter of wave & fifth state
Theoretical predication
S.N. Bose: Z. Phys. 26 (1924) – Study black body radiation: object very hot– Two photons be counted up as either identical or different – Bose statistics or Bose-Einstein statistics
A. Einstein: Sitz. Ber. Kgl. Preuss. Adad. Wiss. 22 (1924) – Apply the rules to atoms in cold temperatures– Obtain Bose-Einstein distribution in a gas
1
1)(
/ Tki Bie
f
Experimental results
JILA (95’, Rb, 5,000): Science 269 (1995)
–Anderson et al., Science, 269 (1995),
198: JILA Group; Rb
–Davis et al., Phys. Rev. Lett., 75 (1995),
3969: MIT Group; Rb
–Bradly et al., Phys. Rev. Lett., 75 (1995),
1687, Rice Group; Li
Experimental results
Experimental implementation– JILA (95’): First experimental realization of BEC in a gas – NIST (98’): Improved experiments– MIT, ENS, Rice, – ETH, Oxford, – Peking U., …
2001 Nobel prize in physics:– C. Wiemann: U. Colorado– E. Cornell: NIST– W. Ketterle: MIT ETH (02’, Rb, 300,000)
Experimental difficulties
Low temperatures absolutely zero (nK)Low density in a gas
Experimental techniques
Laser coolingMagnetic trappingEvaporative Cooling
($100k—300k)
Possible applications
Quantized vortex for studying superfluidity
Test quantum mechanics theoryBright atom laser: multi-componentQuantum computingAtom tunneling in optical lattice trapping, …..
Square Vortex lattices in spinor BECs
Giant vortices
Vortex latticedynamics
Mathematical modeling
N-body problem– (3N+1)-dim linear Schroedinger equation
Mean field theory: – Gross-Pitaevskii equation (GPE): – (3+1)-dim nonlinear Schroedinger equation (NLSE)
Quantum kinetic theory – High temperature: QBME (3+3+1)-dim – Around critical temperature: QBME+GPE– Below critical temperature: GPE
(nK)cT T O
Gross-Pitaevskii equation (GPE)
Physical assumptions– At zero temperature– N atoms at the same hyperfine species (Hartree ansatz)
– The density of the trapped gas is small
– Interatomic interaction is two-body elastic and in Fermi form
t),ψ(t),,,,(N
iiNN xxxxΨ
1
21
.1,1|| ||
3 aa
sas
Second Quantization model
The second quantized Hamiltonian:– A gas of bosons are condensed into the same single-particle state – Interacting by binary collisions – Contained by an external trapping potential
† † †0 int
1( ) ( ') ( ') ' ( ') ( ) ( ', ) ( ') ( ) '
2H x H x dx x x V x x x x dx dx
2
0 ext
†
( ) : ( , ) : Bose field operator
( ') : single particle Hamiltonian2
( , , ) : moentum operator
( ) & ( ) : creation & annihilation of a particle at position
Tx y z
x x t
PH V x
m
P i p p p
x x x
Second quantization model
– Crucial Bose commutation rules:
– Atomic interactions are low-energy two-body s-wave collisions, i.e. essentially elastic & hard-sphere collisions
– The second quantized Hamiltonian
† † †( '), ( ) ( ' ), ( '), ( ) ( '), ( ) 0x x x x x x x x
2int 0 0( ', ) ( ') with 4 /sV x x U x x U a m
† † †00( ) ( ') ( ') ' ( ') ( ') ( ') ( ') '
2
UH x H x dx x x x x dx
Second quantization model
The Heisenberg equation for motion:
For a single-particle state with macroscopic occupation
– Plugging, taking only the leading order term – neglecting the fluctuation terms (i.e., thermal and quantum depletion of the
condensate) – Valid only when the condensate is weakly-interacting & low tempertures
( ) ( , ) ( , )
( , ) : marcoscopic wave function (= ( ) / : expectation value of ( ))
( , ) : fluctation operator satisfies ( , ) 0
x N x t x t
x t x N x
x t x t
†0 0( ) ( ), ( ) : ( ) ( ) ( )i x x H H U x x x
t
Gross-Pitaevskii equation
The Schrodinger equation (Gross, Nuovo. Cimento., 61; Pitaevskii, JETP,61 )
– The Hamiltonian:
– The interaction potential is taken as in Fermi form
22
( ) *( , ) [ ( , )] ( , )2
1*( , ) *( , ) ( ) ( , ) ( , )
2
H x t V x t x t dxm
x t x t x x x t x t dx dx
( ), ( , , )
*( , )
ψ(x, t) Hi x x y z
t x t
.4),()1()(2
00 mxNx aUU s
Gross-Pitaevskii equation
The 3d Gross-Pitaevskii equation ( )
– V is a harmonic trap potential
– Normalization condition
),(|),(|),()(),(2
),( 20
2
2
txtxUNtxxVtxm
txt
i
)(2
)( 222222zyx
mxV
zyx
.1|),(| 2
R3 xdtx
),,( zyxx
Gross-Pitaevskii equation
Scaling (w.l.o.g. )– Dimensionless variables
– Dimensionless Gross-Pitaevskii equation
– With
3/ 2
0 00
, , ( , ) ( , ),x
x
xt t x x t x t
a ma a
),(|),(|),()(),(2
1),( 22 txtxtxxVtxtx
ti
0
22222 4,,),(
2
1)(
a
aNzyxxV s
x
z
zx
y
yzy
x y z
Gross-Pitaevskii equation
Typical parameters ( )– Used in JILA
– Used in MIT
]Js[1005.1 34
Rb87
NN
m
m
aaaa s
xs
xzyx
01881.04
],m[103407.0],nm[1.5
8],s/1[210],kg[1044.1
0
5
0
25
Na23
NN
m
m
aaaa s
zs
zyx
003083.04
],m[101209.1],nm[75.2
s]/1[25.3],s/1[2360],kg[108.3
0
5
0
26
Gross-Pitaevskii equation
Reduction to 2d (disk-shaped condensation)– Experimental setup– Assumption: No excitations along z-axis due to large energy
2d Gross-Pitaevskii equation ( )
1,1, zyxzyx
2
1
12 3
1/ 4/ 22 1/ 2
3 hoR
( , , , ) ( , , ) ( ) with
( ) ( | ( , , ) | ) ( ) z zzg
x y z t x y t z
z x y z dxdy z e
2 2 22
2
4 42 3 ho 2
1( , ) | | ,
2 2
( ) ( ) :2
y
az
x yi x t
t
z dz z dz
12),,( yxx
Numerical Verification
Numerical Results
Bao, Y. Ge, P. Markowich & R. Weishaupl, 06’
Gross-Pitaevskii equation
General form of GPE ( )
with
Normalization condition
),(|),(|),()(),(2
1),( 22 txtxtxxVtxtx
ti dd
dx R
3
2
1
),(2
1
),(2
1
,2
1
)(
,
,2
)(
,2
),(
22222
222
2
4
3
R
4
232
d
d
d
zyx
yx
x
xVdzz
dydzzy
zy
ydz
zy
d
.1|),(| 2
R xdtx
d
Gross-Pitaevskii equation
Two kinds of interaction– Repulsive (defocusing) interaction
– Attractive (focusing) interaction
Four typical interaction regimes:– Linear regime: one atom in the condensation
– Weakly interacting condensation1|| d
00 dsa
00 dsa
0d
Gross-Pitaevskii equation
– Strongly repulsive interacting condensation
– Strongly attractive interaction in 1D
Other potentials– Box potential– Double-well potential– Optical lattice potential– On a ring or torus
1d
1 10 & | | 1
Gross-Pitaevskii equation
Conserved quantities– Normalization of the wave function
– Energy
Chemical potential
d
NxdtxtNR
2 1))0((|),(|))((
2 2 4
R
1( ( )) [ | ( , ) | ( )| ( , ) | | ( , ) | ]
2 2( (0))
d
ddt x t V x x t x t dxE
E
xdtxtxxVtxt ddd
]|),(||),(|)(|),(|
2
1[))(( 422
R
Semiclassical scaling
When , re-scaling
With
Leading asymptotics (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci., 05’)
1d 1/ 2 / 4 2 /( 2)1/d ddx x
),(|),(|),()(),(2
),( 222
txtxtxxVtxtxt
i d
)1(]||2
1||)(||
2[)( 422
R
2
OxdxVE dd
1 1 2/( 2)
1 1 2/( 2)
( ) ( )
( ) ( )
dd
dd
E E O O
O O
Comparison of two scaling
1/ 2 1/( 2) 1/( 2)0 0 0 0
0
3/ 2 3/ 2 / 40 0
21
2
2/( 2)
2 /( 2)
Quanties Thomas-Fermi scaling Semiclassical scaling
1/ 1/
4/ ( )
:
Energy (1)
Chemical potential (1
s x x
d dss x d
ds
ss x x
s
dd
dd
t
Nax a m a a a
a
a a
mxE
t
E O O
O O
1/( 2)
/ 2( 2)
)
length of wave function ( 2 ) (1)
height of wave function ( / ) (1)
dd
d dd d
O O O
O O O
Quantum Hydrodynamics
Set
Geometrical Optics: (Transport + Hamilton-Jacobi)
Quantum Hydrodynamics (QHD): (Euler +3rd dispersion)
)2/(2/ /1,,, dd
iS vJSve
22
t
( ) 0,
1 1( )
2 2
t
d
S
S S V x
2
2
( ) 0
( ) ( ) ( ) ( ln )4
( ) / 2
t
t d
v
J JJ P V
P
Part II
Analysis & Computation
for
Ground states
Stationary states
Stationary solutions of GPE
Nonlinear eigenvalue problem with a constraint
Relation between eigenvalue and eigenfunction
)(),( xtx eti
1|)x(|
R),(|)(|)()()(2
1)(
2
R
22
xd
xxxxxVxx
d
ddd
xdEd
d 4
R|)x(|
2)()(
Stationary states
Equivalent statements:– Critical points of over the unit sphere – Eigenfunctions of the nonlinear eigenvalue problem– Steady states of the normalized gradient flow:(Bao & Q. Du, SIAM J. Sci. Compu., 03’)
Minimizer/saddle points over the unit sphere : – For linear case (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci., 05’)
• Global minimizer vs saddle points– For nonlinear case
• Global minimizer, local minimizer (?) vs saddle points
( )E
2 22
0 0
( )1( , ) [ ( ) | | ] ,
2 || ||
( ,0) ( ) with || ( ) || 1.
t x t V x
x x x
0d
0d
2| 1, ( )L
S E
Ground state
Ground state:
Existence and uniqueness of positive solution :– Lieb et. al., Phys. Rev. A, 00’
Uniqueness up to a unit factor
Boundary layer width & matched asymptotic expansion– Bao, F. Lim & Y. Zhang, Bull. Institute of Math., Acad. Scinica , 07’
4gR|| || 1
( ) min ( ), ( ) ( ) | ( ) |2 d
dg g g gE E E x dx
0d
00with any constant i
g g e
Excited & central vortex states
Excited states:Central vortex states:
Central vortex line states in 3D:Open question: (Bao & W. Tang, JCP, 03’; Bao, F. Lim & Y. Zhang, TTSP, 06’)
,,, 321
???????)()()(
)()()(
,,,
21
21
21
g
g
g
EEE
immm eryxtyx m
ti
m
ti
ee )(),(),,(
2 22
22
2
0
( )1( ) ( ) | | , 0 r
2 2 2
2 ( ) 1, (0) 0
mm m m m m
m m
d rd m rr r r
r dr dr r
r r dr
Approximate ground states
Three interacting regimes– No interaction, i.e. linear case– Weakly interacting regime– Strongly repulsive interacting regime
Three different potential– Box potential– Harmonic oscillator potential– BEC on a ring or torus
Energies revisited
Total energy:
– Kinetic energy:– Potential energy:– Interaction energy:
Chemical potential
2 2 4
kin pot intR
1( ) [ | ( ) | ( )| ( ) | | ( ) | ] : ( ) ( ) ( )
2 2d
ddx V x x x dxE E E E
2
kin R
1( ) | ( ) |
2 dx dxE
2
pot R( ) ( )| ( ) |
d dV x x dxE
4
int R( ) | ( ) |
2 d
d x dxE
2 2 4
R
int kin pot int
1( ) [ | ( ) | ( )| ( ) | | ( ) | ]
2 = ( ) ( ) ( ) ( ) 2 ( )
d d dx V x x x dx
E E E E E
Box Potential in 1D
The potential:The nonlinear eigenvalue problem
Case I: no interaction, i.e. – A complete set of orthonormal eigenfunctions
0, 0 1,( )
, otherwise.
xV x
2
12
0
1( ) ( ) | ( ) | ( ), 0 1,
2
(0) (1) 0 with | ( ) | 1
x x x x x
x dx
0
2 21( ) 2 sin( ), , 1, 2,3,
2l lx l x l l
Box Potential in 1D
– Ground state & its energy:
– j-th-excited state & its energy
Case II: weakly interacting regime, i.e.– Ground state & its energy:
– j-th-excited state & its energy
20 0 0
0 0( ) ( ) 2 sin( ), : ( ) : ( )2g g g g g gx x x E E
2 20 0 0
0 0
( 1)( ) ( ) 2 sin(( 1) ), : ( ) : ( )
2j j j j j j
jx x j x E E
| | (1)o
2 20 0 03
( ) ( ) 2 sin( ), : ( ) ( ) , : ( ) ( ) 32 2 2g g g g g g g gx x x E E E
2 20 0
2 20
( 1) 3( ) ( ) 2 sin(( 1) ), : ( ) ( ) ,
2 2
( 1): ( ) ( ) 3
2
j j j j j
j j j
jx x j x E E E
j
Box Potential in 1D
Case III: Strongly interacting regime, i.e.– Thomas-Fermi approximation, i.e. drop the diffusion term
• Boundary condition is NOT satisfied, i.e. • Boundary layer near the boundary
1
TFTF TF TF 2 TF TF
1TF 2
0
TF TF TFg g g
( ) | ( ) | ( ), 0 1, ( )
| ( ) | 1
(x) ( ) 1, E E , , 2
gg g g g g
g
g g g
x x x x x
x dx
x
TF TF(0) (1) 1 0g g
Box Potential in 1D
– Matched asymptotic approximation• Consider near x=0, rescale• We get
• The inner solution
• Matched asymptotic approximation for ground state
1, ( ) ( )g
g
x X x x
31( ) ( ) ( ), 0 ; (0) 0, lim ( ) 1
2 XX X X X X
( ) tanh( ), 0 ( ) tanh( ), 0 (1)gg gX X X x x x o
MAMA MA MA MA
1MA 2 MA TF
0
( ) ( ) tanh( ) tanh( (1 )) tanh( ) , 0 1
1 | ( ) | 2 1 2 2 1 2, 1.
gg g g g g
g g g g
x x x x x
x dx
Box Potential in 1D
• Approximate energy
• Asymptotic ratios:
• Width of the boundary layer:
MA MAint, int,
MAkin, kin,
4 21 2, 1,
2 3 2 32
1 23
g g g g
g g
E E E E
E E
(1/ )O
int, kin,1lim , lim 1, lim 0,
2g g g
g g g
E E E
E E
Numerical observations:
2
3 / 2 3 / 2 3 / 2MA MA MA
MA MA MAkin, kin, int, int,
( ), ( ), ( )
(1/ ), (1/ ), (1/ )
g g g g g gL L
g g g g g g
O e O e O e
E E O E E O E E O
Box Potential in 1D
• Matched asymptotic approximation for excited states
• Approximate chemical potential & energy
MA [( 1) / 2]MA MA
0
[ / 2]MA MA
0
2( ) ( ) [ tanh( ( ))
1
2 1tanh( ( )) tanh( )]
1
jj
j j gl
j
g j gl
lx x x
j
lx C
j
MA 2 2
MA 2
MA 2int, int,
MA 2 2kin, kin,
2( 1) ( 1) 2( 1) ,
4( 1) ( 1) ,
2 32
( 1) ( 1) ,2 32
( 1) ( 1) 2( 1)3
j j
j j
j j
j j
j j j
E E j j
E E j j
E E j j j
Fifth excited states
Energy & Chemical potential
Box potential in 1D
• Boundary layers & interior layers with width
• Observations: energy & chemical potential are in the same order
• Asymptotic ratios:
• Extension to high dimensions
1 2 1 2( ) ( ) ( ) ( ) ( ) ( )g gE E E
int, kin,1lim , lim 1, lim 0,
2
lim 1, lim 1, lim 0,
j j j
j j j
j j j
g g g
E E E
E E
E E
E
(1/ )O
Harmonic Oscillator Potential in 1D
The potential:The nonlinear eigenvalue problem
Case I: no interaction, i.e. – A complete set of orthonormal eigenfunctions
2
( )2
xV x
2 21( ) ( ) ( ) ( ) | ( ) | ( ), with | ( ) | 1
2x x V x x x x x dx
0
2
2
2
1/ 2 / 21/ 4
20 1 2
1 1( ) (2 !) ( ), , 0,1,2,3,
2
( ) ( 1) : Hermite polynomials with
( ) 1, ( ) 2 , ( ) 4 2,
l xl l l
l xl x
l l
lx l e H x l
d eH x e
dx
H x H x x H x x
Harmonic Oscillator Potential in 1D
– Ground state & its energy:
– j-th-excited state & its energy
Case II: weakly interacting regime, i.e.– Ground state & its energy:
– j-th-excited state & its energy
20 / 2 0 00 01/ 4
1 1( ) ( ) , : ( ) : ( )
2x
g g g g g gx x e E E
20 1/ 2 / 2 0 00 01/ 4
1 ( 1)( ) ( ) (2 !) ( ), : ( ) : ( )
2j x
j j j j j j j
jx x j e H x E E
| | (1)o
20 / 2 0 00 01/ 4
1 1 1( ) ( ) , : ( ) ( ) , : ( ) ( )
2 2 2x
g g g g g g g gx x e E E E C C
0 0
0 0 4j j
-
( 1)( ) ( ), : ( ) ( ) ,
2 2
( 1): ( ) ( ) with C = | ( ) |
2
j j j j j j
j j j j
jx x E E E C
jC x dx
Harmonic Oscillator Potential in 1D
Case III: Strongly interacting regime, i.e.– Thomas-Fermi approximation, i.e. drop the diffusion term
– Characteristic length:– It is NOT differentiable at– The energy is infinite by direct definition:
1
TF 2 TFTF TF TF TF 2 TF TF
TF 3/ 2TF 2 TF 2/3
-
( / 2) / , | | 2( ) ( ) ( ) | ( ) | ( ) ( )
0, otherwise
2(2 ) 1 3 1 | ( ) | ( )
3 2 2
g gg g g g g g
gg g g
x xx V x x x x x
x dx
1/3( )O TF2 gx
TF TFkin( ) , ( )g gE E
Harmonic Oscillator Potential in 1D
– A new way to define the energy
– Asymptotic ratios
TF TF 2/3int, int, int
TF TF 2/3pot, pot, pot
TF TF 2/3 TFint, int,
TF TF TFkin, int, pot,
1 3( ) ( ) ,
5 21 3
( ) ( )10 2
3 3( ) : ,
10 2
0
g g g
g g g
g g g g g g
g g g g
E E E
E E E
E E E E
E E E E
int, pot , kin,3 2 1lim , lim , lim , lim 0,
5 3 3g g g g
g g g g
E E E E
E E E
Numerical observations:
2
TF MA MA2/5 2/5 2 /3
MA MA MAkin, kin, int, int,2 /3 2 /3 2 /3
ln ln ln( ), ( ), ( )
ln ln ln( ), ( ), ( )
g g g g g gL L
g g g g g g
O O O
E E O E E O E E O
Harmonic Oscillator Potential in 1D
– Thomas-Fermi approximation for first excited state
• Jump at x=0!• Interior layer at x=0
TF TF TF TF 2 TF1 1 1 1 1
TF 2 TFTF 1 1
1
TF 3/ 2TF 2 TF 2/31
1 1 1
-
( ) ( ) ( ) | ( ) | ( )
sign( ) ( / 2) / , 0 | | 2( )
0, otherwise
2(2 ) 1 3 1 | ( ) | ( )
3 2 2
x V x x x x
x x xx
x dx
Harmonic Oscillator Potential in 1D
– Matched asymptotic approximation
– Width of interior layer:
– Ordering:
MA MA 2 MAMA MA1 1 1
1 1
sign( ) ( / 2) / , 0 | | 2 ( ) [tanh( ) sign( )]
0, otherwise
x x xx x x
MA 1/3 MA 2/31 1(1/ ) (1/ ) ( )O O O
1 1( ) ( ) ( ) ( )g gE E
Harmonic Oscillator Potential
Extension to high dimensionsIdentity of energies for stationary states in d-dim.
– Scaling transformation
– Energy variation vanishes at first order in
kin pot int2 2 0E E d E
/ 20 0( ) (1 ) ((1 ) ) with ( ): a stationary statedx x x
2 2kin 0 pot 0 int 0
0
( ( )) (1 ) ( ) (1 ) ( ) (1 ) ( )
( ( )) | 0
dE x E E E
dE x
d
BEC on a ring
The potential:The nonlinear eigenvalue problem
For linear case, i.e.– A complete set of orthonormal eigenfunctions
( ) 0 on an interalV x
2
22
0
1( ) ( ) | ( ) | ( ), 0 2 ,
2
( 2 ) ( ), with | ( ) | 1d
0
0 2 2 1
2
0 2 2 1
1 1 1( ) , ( ) cos( ), ( ) sin( );
2
0, , 1, 2,3,2
l l
l l
l l
ll
BEC on a ring
– Ground state & its energy:
– j-th-excited state & its energy
Some properties– Ground state & its energy – With a shift:
– Interior layer can be happened at any point in excited states
0 0 00 0
1( ) ( ) , : ( ) 0 : ( )
2g g g g g gx x E E
20 0 0
0 0
1( ) ( ) cos( ), : ( ) : ( )
2j j j j j j
jx x l E E
| | (1)o
0 0 01( ) ( ) , : ( ) , : ( )
8 42g g g g g gx x E E
0( ) is a solution ( ) is also a solution
Numerical methods for ground states
Runge-Kutta method: (M. Edwards and K. Burnett, Phys. Rev. A, 95’)
Analytical expansion: (R. Dodd, J. Res. Natl. Inst. Stan., 96’)
Explicit imaginary time method: (S. Succi, M.P. Tosi et. al., PRE, 00’)
Minimizing by FEM: (Bao & W. Tang, JCP, 02’)
Normalized gradient flow: (Bao & Q. Du, SIAM Sci. Comput., 03’)
– Backward-Euler + finite difference (BEFD)– Time-splitting spectral method (TSSP)
Gauss-Seidel iteration method: (W.W. Lin et al., JCP, 05’) Spectral method + stabilization: (Bao, I. Chern & F. Lim, JCP, 06’)
( )E
Imaginary time method
Idea: Steepest decent method + Projection
Physical institutive in linear case– Solution of GPE: – Imaginary time dynamics:
.1||)(|| with )x()0,(
,2,1,0,||),(||
),(),(
,||)(2
1)(
2
1),(
00
1
11
122
xx
nx
xtx
tttxVE
tx
ttn
nn
nnt
0
1
2 1̂
??)()(
)()ˆ(
)()ˆ(
01
11
01
EE
EE
EE
g
00 0
( , ) ( ) with ( ,0) ( ) ( )ji t
j j j jj j
x t a e x x x a x
i t 0 1
0 00 0
( , )( , ) ( , ) ( ) ( ) : grond state j
j jj
xx x t a e x x
a e
Mathematical justification
For gradient flow (Bao & Q. Du, SIAM Sci. Comput., 03’)
For linear case: (Bao & Q. Du, SIAM Sci. Comput., 03’)
For nonlinear case: ???))0(.,())(.,())(.,( 0010 EtEtE nn
Mathematical justification
Normalized gradient glow
Idea: (Bao & Q. Du, SIAM Sci. Comput., 03’)
– The projection step is equivalent to solve an ODE
– Gradient flow with discontinuous coefficients:
– Letting time step go to 0
– Mass conservation & Energy diminishing
2 22
0 0
( (., ))1( , ) ( ) | | , 0,
2 || (., ) ||
( ,0) ( ) with || ( ) || 1.
t
tx t V x t
t
x x x
0,0))(.,(,1||||||)(.,|| 0 ttEtd
dt
2
1 1 1
1( , ) ( , ) ( , ), with ( , ) ln ( , ) & ( , ) ( , )
2t n n n n n nn
x t t t x t t t t t k x t x t x tt
2 21( , ) ( ) | | ( , ) , 0,
2t nx t V x t t t
Fully discretization
Consider in 1D:
Different Numerical Discretizations– Physics literatures: Crank-Nicolson FD or Forward Euler FD – 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’)
– Crank-Ncolson FD for normalized gradient flow
21
11 0 0
1
1( , ) ( ) | | , ( , ), , ( , ) ( , ) 0
2
( , )( , ) , ( ,0) (x) with || ( ) || 1.
|| ( , ) ||
t xx n n
nn
n
x t V x x a b t t t a t b t
xx t x x
xtt
Backward Euler Finite Difference
Mesh and time steps:
BEFD discretization
2nd order in space; unconditional stable; at each step, only a linear system with sparse matrix to be solved!
; 0;b a
h x k tM
j n , 0,1, , ; , k=0,1,2, ; (x ,t ) nj n jx a j h j M t n k
Backward Euler Spectral method
Discretization
Spectral order in space; efficient & accurate
Ground states
Numerical results (Bao&W. Tang, JCP, 03’; Bao, F. Lim & Y. Zhang, TTSP, 06’)
– In 1d• Box potential:
– Ground state; excited states: first fifth
• Harmonic oscillator potential: – ground & first excited & Energy and chemical potential
• Double well potential :– Ground & first excited state
• Optical lattice potential:– Ground & first excited state with potential next
otherwise;100)( xxV
2/xV(x) 2
2/)4()( 22xxV
)4(sin122/)( 22 xxxV
back
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1)(
)(lim1
)(
)(lim
fixedany for ,)()()(,)()()(
nsObservatio
419.32691.31971.30259.30383.20629.19891.18171.1871.313
252.21512.20784.19070.19719.13944.12191.12464.11855.156
8349.80432.82802.75527.64043.65573.57438.49810.3371.31
1919.42590.33578.25266.18505.38865.29414.10441.11371.3
500.35000.25000.15000.0500.35000.25000.15000.00
)()()()()()()()(
11
12121
3213211
g
j
g
j
gg
gg
E
E
EEE
EEEE
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Ground states
Numerical results (Bao&W. Tang, JCP, 03’; Bao, F. Lim & Y. Zhang, BIM, 07’)
– In 2d• Harmonic oscillator potentials:
– ground
• Optical lattice potential:– Ground & excited states
– In 3D • Optical lattice potential: ground excited states next
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Part III
Analysis & Computation
for
Dynamics in BEC
Dynamics of BEC
Time-dependent Gross-Pitaevskii equation
Dynamical laws – Time reversible & time transverse invariant– Mass & energy conservation– Angular momentum expectation– Condensate width– Dynamics of a stationary state with its center shifted
)()0,(
),(|),(|),()(),(2
1),(
0
22
xx
txtxtxxVtxtxt
i dd
Angular momentum expectation
Definition:
Lemma Dynamical laws (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci, 05’)
For any initial data, with symmetric trap, i.e. , we have
Numerical test next
0,)(**:)( txdxyixdLtLdd R
yx
R
zz
0,|),(|)()( 222 txdtxxy
dt
tLd
dR
yxz
yx
,0 ,0 0( ) (0), ( ) ( ), 0.z zL t L E E t
back
Angular momentum expectation
Energy
Dynamics of condensate width
Definition:
Dynamic laws (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci, 05’)
– When for any initial data: – When with initial data Numerical Test– For any other cases:
xdtxtxdtxyxtdd RR
r
22222 |,(|)(,|,(|)()(
22
02
( )4 ( ) 4 ( ), 0r
x r
d tE t t
dt
yxd &2
yxd &2 imerfyx )(),(0
0),(2
1)()( tttt ryx
22
02
( )4 ( ) 4 ( ) ( ), 0
d tE t f t t
dt
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Symmetric trap Anisotropic trap
Dynamics of Stationary state with a shift
Choose initial data as: The analytical solutions is: (Garcia-Ripoll el al., Phys. Rev. E, 01’)
– In 2D:
– In 3D, another ODE is added
)()( 00 xxx s
( , )0( , ) ( ( )) , ( , ) 0, (0)si t iw x t
sx t e x x t e w x t x x
2
2
0 0
( ) ( ) 0,
( ) ( ) 0,
(0) , (0) , (0) 0, (0) 0
x
y
x t x t
y t y t
x x y y x y
20( ) ( ) 0, (0) , (0) 0zz t z t z z z
Solution of the center of mass
Center of mass: Bao & Y. Zhang, Appl. Numer. Math., 2006
In a non-rotating BEC:
– Trajectory of the center Motion of the solution– Pattern Classification:
• Each component of the center is a periodic function • In a symmetric trap, the trajectory is a straight segment• If is a rational #, the center moves periodically with period • If is an irrational #, the center moves chaotically, envelope is a rectangle
2 2( ) : | ( , ) | | ( ( )) | ( )d d
sx t x x t dx x x x t dx x t
0 0( ) cos( ), ( ) cos( ), 0x yx t x t y t y t t
/y x /y x
2p
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Numerical methods for dynamics
Lattice Boltzmann Method (Succi, Phys. Rev. E, 96’; Int. J. Mod. Phys., 98’)
Explicit FDM (Edwards & Burnett et al., Phys. Rev. Lett., 96’)
Particle-inspired scheme (Succi et al., Comput. Phys. Comm., 00’) Leap-frog FDM (Succi & Tosi et al., Phys. Rev. E, 00’)
Crank-Nicolson FDM (Adhikari, Phys. Rev. E 00’) Time-splitting spectral method (Bao, Jaksch&Markowich, JCP, 03’)
Runge-Kutta spectral method (Adhikari et al., J. Phys. B, 03’)
Symplectic FDM (M. Qin et al., Comput. Phys. Comm., 04’)
Time-splitting spectral method (TSSP)
Time-splitting:
For non-rotating BEC – Trigonometric functions (Bao, D. Jaksck & P. Markowich, J. Comput. Phys., 03’)
– Laguerre-Hermite functions (Bao & J. Shen, SIAM Sci. Comp., 05’)
2
2
2
( ( ) | ( , )| )1
1 Step 1: ( , ) ,
2
Step 2: ( , ) ( ) ( , ) | ( , ) | ( , )
| ( , ) | | ( , ) |
( , ) ( , )d d n
t
t d d
n
i V x x t tn n
i x t
i x t V x x t x t x t
x t x t
x t e x t
Time-splitting spectral method
Properties of TSSP
– Explicit, time reversible & unconditionally stable– Easy to extend to 2d & 3d from 1d; efficient due to FFT– Conserves the normalization– Spectral order of accuracy in space – 2nd, 4th or higher order accuracy in time– Time transverse invariant
– ‘Optimal’ resolution in semicalssical regime
unchanged|),(|)()( 2txxVxV dd
)2/(2/1,, ddOkOh
Dynamics of Ground states
1d dynamics: 2d dynamics of BEC (Bao, D. Jaksch & P. Markowich, J. Comput. Phys., 03’)
– Defocusing: – Focusing (blowup):
3d collapse and explosion of BEC (Bao, Jaksch & Markowich,J. Phys B, 04’)
– Experiment setup leads to three body recombination loss
– Numerical results: • Number of atoms , central density & Movie
xx1 40,at t 100
yy 2,2 0at t,20 xx2
5040 0At t 2
next
420
22 ||||)(2
1),( ixVtx
ti
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Collapse and Explosion of BEC
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Number of atoms in condensate
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Central density
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Central quantized vortices
Central vortex states in 2D:
with
Vortex Dynamics – Dynamical stability – Interaction
• Pattern I• Pattern II
immm eryxtyx m
ti
m
ti
ee )(),(),,(
1)(2
r0,||)(22
)(
2
1)(
0
2
22
2
2
2
drrr
rr
r
m
dr
rdr
dr
d
rr
m
mmmm
mm
01
( , ) ( , ) / || ||j
N
n j jj
x y x x y y
),()()( txWxVxV
01
( , ) ( , ) / || ||j
N
n j jj
x y x x y y
Central Vortex states
Central Vortex states
Vortex stability & interaction
Dynamical stability (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci., 05’)
– m=1: stable velocity– m=2: unstable velocity
Interaction (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci., 05’)
– N=2: Pair: velocity trajectory phase phase2 Anti-pair: phase trajectory angular trajectory2 – N=3: velocity trajectory – Pattern II: Linear nonlinear
Interaction laws: – On-going with Prof. L. Fu & Miss Y. Zhang next
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Linear case
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Noninear case: BEC
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Linear case
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Linear case
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Linear case
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Some Open Questions
Dynamical laws for vortex interaction
With a quintic damping, mass goes to constant
Convergence & error estimate of the TSSP?Energy diminishing of the gradient flow in nonlinear case & error estimate ?
( )?????jdx t
dt
20( ) | ( , ) | 0 ????
d
t
N t x t d x C
Part IV
Rotating BEC
&
multi-component BEC
Rotating BEC
The Schrodinger equation ( )
– The Hamiltonian:
– The interaction potential is taken as in Fermi form
xdxdtxtxxxtxtx
xdtxLxVm
txH z
),(),()(),(*),(*2
1
),(])(2
[),(*)(22
*
)(
H
t
t),xψ(i
.4),()1()(2
00 mxNx aUU s
),,( zyxx
Rotating BEC
The 3D Gross-Pitaevskii equation ( )
– Angular momentum rotation
– V is a harmonic trap potential
– Normalization condition
]||)(2
[),( 20
2
2
UNLxVm
txt
i z
)(2
)( 222222zyx
mxV
zyx
1|),(| 2
R3 xdtx
),,( zyxx
iPPxLiyxiypxpL xyxyz ,,)(:
Rotating BEC
General form of GPE ( )
with
Normalization condition
),(]||)(2
1[),( 22 txLxVtx
ti dzd
dx R
3),(
2
1
2),(2
1
)(
,
,2
)(22222
2224
3
dzyx
dyxxV
dzz
zy
y
d
z
d
.1|),(| 2
R xdtx
d
iyxiL xyz )(:
Rotating BEC
Conserved quantities– Normalization of the wave function
– Energy
Chemical potential
d
NxdtxtNR
2 1))0((|),(|))((
))0((
]||2
*||)(||2
1[))((
,
422
R,
E
E xdLxVt dzdd
xdLxVt dzdd
]||*||)(||
2
1[))(( 422
R,
Semiclassical scaling
When , re-scaling
With
Leading asymptotics
1d)2/(24/2/1 /1 d
ddxx
]||)(2
[),( 222
zd LxVtxt
i
)1(
]||2
1*)(||)(||
2[)( 422
R
2
,
O
xdLxVE zdd
)2/(2,
)2/(21,
1, )(,)()(
dd
dd OOOEE
Quantum Hydrodynamics
Set
Geometrical Optics: (Transport + Hamilton-Jacobi)
Quantum Hydrodynamics (QHD): (Euler +3rd dispersion)
)2/(2/ /1,,, dd
iS vJSve
1
2ˆ)(
2
1
)(:ˆ,0ˆ)(2
2
t SLxVSS
yxLLS
zd
xyzzt
01
10A ,J , 2/)(
)ln(4
ˆ)()()(
0ˆ)(
2
2
vP
JAJLVPJJ
J
Lv
zdt
zt
Stationary states
Stationary solutions of GPE
Nonlinear eigenvalue problem with a constraint
Relation between eigenvalue and eigenfunction
)(),( xtx eti
1|)x(|
),(]|)(|)(2
1[)(
2
R
22
xd
xxLxVx
d
dzd
xdEd
d 4
R,, |)x(|2
)()(
Stationary states
Equivalent statements:– Critical points of over the unit sphere– Eigenfunctions of the nonlinear eigenvalue problem– Steady states of the normalized gradient flow:
Minimizer/saddle points over the unit sphere : – For linear case
• Global minimizer vs saddle points– For nonlinear case
• Global minimizer, local minimizer (?) vs saddle points
)(, E
.1||)(||with)()0,(
,]||||
)(||)(
2
1[),(
00
2
,22
xxx
LxVtx zt
1&0 d
1&0 d
Ground state
Ground state:
Existence:– Seiringer (CMP, 02’)
Uniqueness of positive solution:– Lieb et al. (PRA, 00’)
Energy bifurcation:– Aftalion & Du (PRA, 01’); B., Markowich & Wang 04’
xdxEEEd
dgggg
4
R g,,,1||||
, |)(|2
)()(),(min)(
0&1|| d
0&0 d
10 c
Numerical results
– Ground states: in 2D in 3D isosurface – Quantized vortex generation in 2D
• surface contour
– Vortex lattice• Symmetric trapping anisotropic trapping
– Giant vortex generation in 2D• surface contour
– Giant vortex• In 2D In 3D
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Numerical & Asymptotical results
Critical angular frequency: symmetric state vs quantized vortex state
Asymptotics of the energy:
Ratios between energies of different states
Rank according to energy and chemical potential– Stationary states are ranked according to their energy, then their chemical
potential are in the same order. Next
)()(,10 00,10, EEcc
),()(;1),1( )2/(2
,1,,d
sgg OEOEE
1||0,constlim.,constlim;0,1lim,1lim,
,
,
,
,
,
,
,
g
s
g
s
g
s
g
s
dddd E
E
E
E
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Dynamical laws of rotating BEC
Time-dependent Gross-Pitaevskii equation
Dynamical laws – Time reversible & time transverse invariant– Conservation laws– Angular momentum expectation– Condensate width– Dynamics of a stationary state with its center shifted
iyxiLxx
LxVtxt
i
xyz
dzd
)(:),()0,(
]||)(2
1[),(
0
22
Conservation laws
Conserved quantities– Normalization of the wave function
– Energy
Chemical potential
d
NxdtxtNR
2 1))0((|),(|))((
))0((
]||2
*||)(||2
1[))((
,
422
R,
E
E xdLxVt dzdd
xdLxVt dzdd
]||*||)(||
2
1[))(( 422
R,
Angular momentum expectation
Definition:Lemma The dynamics of satisfies
For any initial data, with symmetric trap, i.e. , we have
Numerical test next
Bao, Du & Zhang, SIAM J. Appl. Math., 66 (2006), 758
0,)(**:)( txdxyixdLtLdd R
yx
R
zz
0,|),(|)()( 222 txdtxxy
dt
tLd
dR
yxz
yx
( )zL t
,0 ,0 0( ) (0), ( ) ( ), 0.z zL t L E E t
back
Angular momentum expectation
Energy
Dynamics of condensate width
Definition: Bao, Du & Zhang, SIAM J. Appl. Math., 66 (2006), 758
Dynamic laws– When for any initial data: – When with initial data Numerical Test– For any other cases:
xdtxtxdtxyxtdd RR
r
22222 |,(|)(,|,(|)()(
0),(4)(4)( 2
0,2
2
ttEdt
tdrx
r
yxd &2
yxd &2 imerfyx )(),(0
0),(2
1)()( tttt ryx
0),()(4)(4)( 2
0,2
2
ttftEdt
td
next
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Symmetric trap Anisotropic trap
Dynamics of Stationary state with a shift
Choose initial data as: The analytical solutions is: Bao, Du & Zhang, SIAM J. Appl.Math., 2006
– In 2D:
– In 3D, another ODE is added
)()( 00 xxx s
( , )0( , ) ( ( )) , ( , ) 0, (0)si t iw x t
sx t e x x t e w x t x x
2 2
2 2
0 0 0 0
( ) 2 ( ) ( ) ( ) 0,
( ) 2 ( ) ( ) ( ) 0,
(0) , (0) , (0) , (0)
x
y
x t y t x t
y t x t y t
x x y y x y y x
20( ) ( ) 0, (0) , (0) 0zz t z t z z z
Solution of the center of mass
Center of mass: Bao & Zhang, Appl. Numer. Math., 2006
In a non-rotating BEC:
– Pattern Classification: • Each component of the center is a periodic function • In a symmetric trap, the trajectory is a straight segment• If is a rational #, the center moves periodically with period • If is an irrational #, the center moves chaotically, envelope is a rectangle
2 2( ) : | ( , ) | | ( ( )) | ( )d d
sx t x x t dx x x x t dx x t
00 0( ) cos( ), ( ) cos( ), 0x yx t x t y t y t t
/y x /y x
2p
Solution of the center of mass
In a rotating BEC with a symmetric trap:
– Trajectory of the center – Distance between the center and trapping center – Motion of the solution: 0.5 1 2 4– Pattern Classification:
0 0
0 0
2 2 2 20 0
| |( ) cos( ) cos( ) sin( ) sin( ) , | |
2 2| |
( ) cos( ) cos( ) sin( ) sin( ) , | |2 2
| ( ) |: ( ) ( ) | cos( ) |
x
x
x
x yx t a t b t a t b t a
y xy t a t b t a t b t b
x t x t y t x y t
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1/5, 4/5, 1
3/2, 6, Pi
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Pattern Classification
Pattern Classification: Bao & Zhang, Appl. Numer. Math., 2006
– The distance between the center and trap center is periodic function– When is a rational #
• The center moves periodically• The graph of the trajectory is unchanged under a rotation
– When is an irrational #, • The center moves chaotically• The envelope of the trajectory is a circle
– The solution of GPE agrees very well with those from the ODE system
/ qp
/ qp
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Solution of the center of mass
In a rotating BEC with an anisotropic trap– When results
• The trajectory is a spiral coil to infinity • The trajectory is an ellipse
– Otherwise result1 result2• The center moves chaotically & graph is a bounded set
• The center moves along a straight line to infinity
||or xy
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Total density with dissipation
Time-dependent Gross-Pitaevskii equation
Lemma The dynamics of total density satisfies
– The total density decreases when density function energy next
2 2
0
1( , ) [ ( ) ( , ) | | ]
2
( ,0) ( ), : ( )
d z d
z y x
i x t V x W x t Lt
x x L i x y i
2, 2
2()()|(,)|()0,0
1 d
dNtxtdxt
dt
0& | | min{ , }x y
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Numerical Methods
Time-splitting pseudo-spectral method (TSSP)
– Use polar coordinates (B., Q. Du & Y. Zhang, SIAP 06’)– Time-splitting + ADI technique (B. & H. Wang, JCP, 06’)– Generalized Laguerre-Hermite functions (B., J. Shen & H. Wang, 06’)
|),(||),(|),(|),(|),()(),( :2 Step
)(:,2
1),( :1 Step
2
2
nddt
xyzzt
txtxtxtxtxxVtxi
iyxiLLtxi
Numerical methods for rotating BEC
Numerical Method one: (Bao, Q. Du & Y. Zhang, SIAM, Appl. Math. 06’)
– Ideas• Time-splitting • Use polar coordinates: angular momentum becomes constant coefficient • Fourier spectral method in transverse direction + FD or FE in radial direction• Crank-Nicolson in time
– Features• Time reversible• Time transverse invariant• Mass Conservation in discretized level• Implicit in 1D & efficient to solve• Accurate & unconditionally stable
Numerical methods for rotating BEC
Numerical Method two: (Bao & H. Wang, J. Comput. Phys. 06’)
– Ideas• Time-splitting • ADI technique: Equation in each direction become constant coefficient • Fourier spectral method
– Features• Time reversible• Time transverse invariant• Mass Conservation in discretized level• Explicit & unconditionally stable• Spectrally accurate in space
Dynamics of ground state
Choose initial data as: : ground state
Change the frequency in the external potential:– Case 1: symmetric: surface contour– Case 2: non-symmetric: surface contour– Case 3: dynamics of a vortex lattice with 45 vortices: image contour next
)()(0 xx g
21:&21: yx
2.21:&8.11: yx
1,8.0,100 zy
canisotropi :),(,9.0,1000 txV
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Interaction of two vortices in linear
0
1/ 3
Interaction of two vortices in linear
1/ 2
1
Interaction of two vortices in linear
1/
Interaction of vortices in nonlinear
1/
Interaction of vortices in nonlinear
0
1/ 2
Interaction of vortices in nonlinear
1
4
Interaction of vortices in nonlinear
1/
Some Open Questions
Dynamical laws for vortex interaction
With a quintic damping, mass goes to constant
Semiclassical limit when initial data has vortices??? Vortex line interaction laws, topological change?What is a giant vortex?
( )?????jdx t
dt
20( ) | ( , ) | 0 ????
d
t
N t x t d x C
Two-component BEC
The 3D coupled Gross-Pitaevskii equations
Normalization conditions
Intro- & inter-atom Interactions
2
2 2 21 11 1 12 2 1 2
2
2 2 22 21 1 22 2 2 1
( , ) [ ( ) | | | | ]2
( , ) [ ( ) | | | | ]2
z
z
i x t V x L U Ut m
i x t V x L U Ut m
3 3
22 0 0 0 2
1 21
( ) | ( , ) | : with | ( ,0) | ,j j jj
N t x t dx N N N N x dx
2
12 21
4 with jl
jl
aU a a
m
Two-component BEC
Nondimensionalization
Normalization conditions– There is external driven field
– No external driven field
2 2 21 11 1 12 2 1 2
2 2 22 21 1 22 2 2 1
1( , ) [ ( ) | | | | ]
2
1( , ) [ ( ) | | | | ]
2
z
z
i x t V x Lt
i x t V x Lt
3 3
2 21 2( ) | ( , ) | | ( , ) | 1N t x t dx x t dx
3 3
0 02 21 2
1 2| ( , ) | , | ( , ) |N N
x t dx x t dxN N
0
0
Two-component BEC
Energy
Reduction to one-component:
2 22 2 2 2 *
1 2Rj=1 1
1( ) [ ( | | ( )| | * | | | | ) 2 Re( )]
2 2d
jlj j j z j j l
l
E V x L dx
0 0 01 2 10, , ( )N N N O N
3 3
0 02 22 1
2 2 1 1( ) | ( , ) | : 1, ( ) | ( , ) | : 1 1N N
N t x t dx N t x t dxN N
2 2
0 01 1 1 11
1( , ) [ ( ) | | ] ( , ),
2
| ( ) ( ) |( , ) / ( , ) & = / ( )
( )
z
s
s
i x t V x L x tt
E Ex t N N x t N N O
E
Two-Component BEC
Semiclassical scaling
Semiclassical limit– No external field:
• WKB expansion, two-fluid model– With external field:
• WKB expansion doesn’t work, Winger transform
22 2 2
1 11 1 12 2 1 2
22 2 2
2 21 1 22 2 2 1
( , ) [ ( ) | | | | ]2
( , ) [ ( ) | | | | ]2
z
z
i x t V x Lt
i x t V x Lt
0
0
Ground state
No external field:
Nonlinear eigenvalue problem
Existence & uniqueness of positive solutionNumerical methods can be extended
0
1 21 2|| || ,|| ||
min ( , ) with 1E
2 2 21 1 11 1 12 2 1
2 2 22 2 21 1 22 2 2
1( ) [ ( ) | | | | ]
21
( ) [ ( ) | | | | ]2
z
z
x V x L
x V x L
Ground states
crater
Ground state
With external field:
Nonlinear eigenvalue problem
Existence & uniqueness of positive solution ???Numerical methods can be extended????
0
2 21 2
1 2|| || || || 1
min ( , )E
2 2 21 11 1 12 2 1 2
2 2 22 21 1 22 2 2 1
1( ) [ ( ) | | | | ]
21
( ) [ ( ) | | | | ]2
z
z
x V x L
x V x L
Dynamics
Dynamical laws:– Conservation of Angular momentum expectation– Dynamics of condensate width– Dynamics of a stationary state with a shift– Dynamics of mass of each component, they are periodic
function when – Vortex can be interchanged!
Numerical methods– Time-splitting spectral method
11 12 22
Dynamics
Dynamics
Spinor BEC
Spinor F=1 BEC
With
2
2 * 21 1 1 0 1 1 1 0
2
2 *0 0 1 1 1 1 1 0
2
2 * 21 1 1 0 1 1 1 0
[ ( ) ] ( )2
[ ( ) ] ( ) 22
[ ( ) ] ( )2
z n s s
z n s s
z n s s
i V x L g g gt m
i V x L g g gt m
i V x L g g gt m
2 22 0 2 2 0
1 0 1
0 2
24 4, | | , ,
3 3, : s-wave scattering length with the total spin 0 and 2 channels
j j n s
a a a ag g
m ma a
Spinor BEC
Total mass conservation
Total magnetization conservation
Energy conservation
3 3
12 0 0 0 0 2
1 0 11
( ) | ( , ) | : with | ( ,0) | ,j j jj
N t x t dx N N N N N x dx
3 3
2 2 0 01 1 1 1( ) | ( , ) | | ( , ) | :M t x t dx x t dx N N M
212 2 2
Rj=-1
2 2 * 2 * * 21 1 1 0 1 0 1 1 1 0 1 1 0 1
( ) [ ( | | ( )| | * )2m 2
( 2 2 2 ) ( ( ) )]2
d
nj j j z j
ss
gE V x L
gg dx
Spinor BEC
Dimension reductionGround state– Existence & uniqueness of positive solution??– Numerical methods ???
Dynamics – Dynamical laws – Numerical methods: TSSP
Semiclassical limit & hydrodynamics equation??
BEC at Finite Temperature
Condensate coexists with non-condensed thermal cloud Coupled equations of motion for condensate and thermal cloudMean-field theory in collisionless regimeZGN theory in collision dominated regime
Mean-field Theory
Evolution of quantum field operator
where is the annihilation field operatorand is the creation field operatorMean-field description
Condensate wavefunction
ˆˆˆˆ
2
ˆ†2
2
gVmt
i ext
0),(~
),(),(
),(~
ˆ),(),( 0
tx
txtx
txatxtx
mag s /4 2
~~~~2
2†*2
2
gmggngnVmt
i Tcext
),(ˆ tx
),(ˆ † tx
Mean-field Theory
Generalized GPE for condensate wavefunction
Temperature-dependent fluctuation field for non-condensate
functionn correlatio field-hree t ~~~
density condensate-non diagonal-off ~~
),(~
density condensate-non ~~
),(
density condensate ),(
~~~~22
†
†
2
†*22
txm
txn
txn
gmggngnVmt
i
T
c
Tcext
~~~~~
22
~††2
2
ggmgnVmt
i ext
†††
††
~~~~~~2
~~~
~~~~
~~~~
),(~),(),(
),(),(),(2 txmtxtxm
txntxntxn Tc
Hartree-Fock Bogoliubov Theory
Ignore the three-field correlation function
Bogoliubov transformation
where creates (annihilates) a Bogoliubov quasiparticle of energy εj
The quasiparticles are non-interacting
*22
~22
mgnngVmt
i Tcext
~~~ †
jjjj
jjjj
txvtxutx
txvtxutx
ˆ),(ˆ),(),(~
ˆ),(ˆ),(),(~
†j
*†
†j
*
†22 ~~
22
~
gmgnVmt
i ext
)ˆ(ˆ †j j
Hartree-Fock Bogoliubov Theory
Bogoliubov equations for non-condensate
where
jjextj
jjextj
ugmvgnVmt
vi
gmvugnVmt
ui
*22
22
22
22
1)/exp(
1ˆˆ
)21(),(
)1(),(
),(
†
*2
22
2
kTN
Nvutxm
NvNutxn
txn
jjjj
jjjj
jjjjjT
c
Time-independent Hartree-Fock Bogoliubov Theory
Stationary states
Time-independent generalized GPE and Bogoliubov equations
//
//
/
)(),(
)(),(
)(),(
titijj
titijj
ti
eexvtxv
eexutxu
extx
j
j
*2
2~2
2mgnngV
m Tcext
jjjjext
jjjjext
vugmvgnVm
ugmvugnVm
*22
22
22
22
HFB-Popov Approximation
HFB produces an energy gap in the excitation spectrumSolution: leave outGeneralized GPE and Bogoliubov equations within Popov approximation (gapless spectrum)
*22
22
Tcext nngVmt
i
jjextj
jjextj
ugvgnVmt
vi
vgugnVmt
ui
2*22
222
22
22
m~
Hartree-Fock Approximation
Approximate Bogoliubov excitations with single-particle excitations, i.e. let
Restricted to finite temperature close to Tc, where the non-condensed particles have higher energies
0jv
*22
22
Tcext nngVmt
i
jextj gnV
mti
22
22
jjjT
c
Ntxn
txn2
2
),(
),(
ZGN Theory
Mean-field theory deals with BEC in collisionless region (low density thermal cloud):
l >> l is the collisonal mean-free-path of excited particlesis the wavelength of excitations
In collision-dominated region l << (higher density thermal cloud)the problem becomes hydrodynamic in nature
ZGN theory (E. Zaremba, A. Griffin, T. Nikuni, 1999) describes finite-T BEC with interparticle collisions in the semi-classical limit
kBT >> ħ: trap frequency) kBT >> gn
ZGN Theory
Apply Popov approximation (ignore ) but include the three-field correlation functionGPE for condensate wavefunction
Quantum Boltzmann equation for phase-space distribution function of non-condensate
~~~
22
†22
gnngVmt
i Tcext
][2
22 12
22
fn
inngV
mti
cTcext
][][),,(
),,(2 2212 fCfCt
tpxftpxfgnV
m
p
t collisionpextx
][)2(
~~~Im
2][ termsource 123
†*12 fC
pdgf
~~~ †
m~
ZGN Theory
Thermal cloud density
Collision between condensate and non-condensate-- transfer atoms from/to the condensate
Collision between non-condensate particles
),,(2
),( 3 tpxfpd
txnT
)1)(1()1(
)()()()(
)()2(
2][
321321
321321
32132142
2
12
ffffff
pppppp
pppvmpdpdpdng
fC
pppc
sc
)1)(1()1)(1()(
)()2(
2][
321321321
32132175
2
22
ffffffff
pppppdpdpdg
fC
pppp
ZGN Theory
Energy of condensate atoms
Local chemical potential
Superfluid velocity
Energy of non-condensate atoms – Hartree-Fock energy
Limited to high temperature (close to Tc)For lower temperature, the spectrum of excited atoms should be described by Bogoliubov approximation
gnVm
ptx extp 2
2),(
2
),(2
1),(),( 2 txmvtxtx scc
Tcext
c
cc gngnV
n
n
mtx 2
2),(
22
),(),( txm
txvs
),(exp),(),( txitxntx c
Open questions
Mathematical theory– Quantum Boltzmann Master equation (QBE)– GPE with damping term– Coupling QBE +GPE
Numerical methods– For QBE: P. Markowich & L. Pareschi (Numer. Math., 05’)
– For QBE+GPE – Comparison with experiments– Rotational frame
Conclusions
– Review of BEC– Experiment progress– Mathematical modeling– Efficient methods for computing ground & excited states– Efficient methods for dynamics of GPE– Comparison with experimental results– Vortex dynamics– Quantized vortex stability & interaction
Future Challenges
– Multi-component BEC for bright laser– Applications of BEC in science and engineering– Precise measurement– Fermions condensation, BEC in solids & waveguide– Dynamics in optical lattice, atom tunneling– Superfluidity & dissipation, quantized vortex lattice – Coupling GPE & QBE for BEC at finite temperature– Mathematical theory for BEC– Interdisciplinary research: experiment,physics, mathematics, computation, ….
References
[1] M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman and E.A. Cornell, Science 269 (1995) 198-201.
[2] W. Bao, J. Shi and P.A. Markowich, J. Comput. Phys. , Vol. 175, pp. 487-524, 2002. [3] W. Bao and W.J. Tang, J. Comput. Phys., Vol. 187, No. 1, pp. 230 - 254, 2003. [4] W. Bao, D. Jaksch and P.A. Markowich, J. Comput. Phys., Vol. 187, No. 1, pp. 318 - 342,
2003.[5] W. Bao, S. Jin and P.A. Markowich, SIAM J. Sci. Comput., Vol. 25, No. 1. pp. 27-64, 2003. [6] W. Bao and D. Jaksch, SIAM J. Numer. Anal., Vol. 41, No. 4. pp. 1406-1426, 2003. [7] W. Bao, D. Jaksch and P.A. Markowich, J. Phys. B: At. Mol. Opt. Phys., Vol. 37, No. 2, pp.
329-343, 2004. [8] W. Bao, Multiscale Modeling and Simulation: a SIAM Interdisciplinary Journal, Vol. 2, No. 2.
pp. 210-236, 2004. [9] W. Bao and Q. Du, SIAM J. Sci. Comput. , Vol. 25, No. 5. pp. 1674-1697, 2004.
References
[9] W. Bao and Q. Du, SIAM J. Sci. Comput. , Vol. 25, No. 5. pp. 1674-1697, 2004.[10] W. Bao, H.Q. Wang and P.A. Markowich, Comm. Math. Sci. , Vol. 3, No. 1, pp. 57-88,
2005. [11] W. Bao, P.A. Markowich, C. Schmeiser and R. M. Weishaupl, Math. Mod. Meth. Appl.
Sci. , Vol. 15, No. 5, pp. 767-782, 2005. [12] W. Bao and J. Shen, SIAM J. Sci. Comput. , Vol. 26 , No. 6, pp. 2010-2028, 2005. [13] W. Bao and Y.Z. Zhang, Math. Mod. Meth. Appl. Sci. , Vol. 15 , No. 12, pp. 1863-1896,
2005. [14] W. Bao, Qiang Du and Yanzhi Zhang, SIAM J. Appl. Math., Vol. 66 , No. 3, pp. 758-
786, 2006. [15] W. Bao and H. Wang, J. Comput. Phys., Vol. 217, No. 2, pp. 612-626, 2006.[16] W. Bao, I-L. Chern and F. Y. Lim, J. Comput. Phys., Vol. 219, No. 2, pp. 836-854, 2006[17] W. Bao and Y. Zhang, Appl. Numer. Math., Vol. 57, No. 5-7, pp. 697-709, 2007.
References
[18] W. Bao, F. Y. Lim and Y. Zhang, Bulletin of the Institute of Mathematics, Academia Sinica, Vol. 2, No. 2, pp. 495-532, 2007.
[19] W. Bao, H.L. Li and Y. Zhang, Physica D: Nonlinear Phenomena, Vol. 234, pp. 49-69, 2007.[20] W. Bao, Y. Ge, D. Jaksch, P. A. Markowich and R. M. Weishaeupl, Comput. Phys. Comm.,
Vol. 177, No. 11, pp. 832-850, 2007. [21] W. Bao and H. Wang, A mass and magnetication conservative and energy diminishing
numerical method for computing ground state of spin-1 Bose-Einstein condensates, SIAM J. Numer. Anal., Vol. 45, No. 5, pp. 2177-2200, 2007.
[22]. A. Klein, D. Jaksch, Y. Zhang and W. Bao, Dynamics of vortices in weakly interacting Bose-Einstein condensates, Phys. Rev. A, Vol. 76, article 043602, 2007.
[23]. W. Bao and M.-H. Chai, A uniformly convergent numerical method for singularly perturbed nonlinear eigenvalue problems, Commun. Comput. Phys., to appear.
[24]. W. Bao and F. Y. Lim, Computing Ground States of Spin-1 Bose-Einstein Condensates by the Normalized Gradient Flow, arXiv: 0711.0568.
References
[25] Bradly et al., Phys. Rev. Lett., 75 (1995), 1687.[26] Davis et al., Phys. Rev. Lett., 75 (1995), 3969.[27] A.L. Fetter and A. A. Svidzinsky, Vortices in a trapped dilute Bose-Einstein condensate
(topical review), J. Phys.: Condens. Matter 13 (2001), 135-194. [28] A.J. Leggett, Bose-Einstein condensation in the alkali gases: some fundamental concepts,
Rev. Modern Phys., 73 (2001), 307-356.[29] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford University Press, 2003.[30]E.H. Lieb, R. Seiringer, J.P. Solovej and J. Yngvason, The Mathematics of the Bose Gas and
its Condensation, Birkhauser, 2000.[31] A. Aftalion, Vortices in Bose-Einstein Condensates, Birkhauser, 2006.[32] F. Dalfovo, S. Giorgini, L.P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation
in trapped gases, Rev. Modern. Phys., 71 (1999), 463-512.[33] C.J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge
University Press, 2002.
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