numerical simulations for bose-einstein condensation (bec) · 2019. 5. 5. · existence &...
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Mathematical Models & Numerical Simulation for Bose-Einstein Condensation
Weizhu Bao
Department of Mathematics National University of SingaporeEmail: [email protected]
URL: http://www.math.nus.edu.sg/~bao
BEC@JILA, 95’
Vortex@ENS
Outline
Part I: Predication, Experiment & Mathematical Models– Theoretical predication– Physical experiments and results– Applications– Gross-Pitaevskii equation (GPE)
Part II: Analysis & Computation for Ground States – Existence & uniqueness – Energy asymptotics & asymptotic approximation– Numerical methods– Numerical results
BEC@ETH, 98’
Outline
Part III: Analysis & Computation for Dynamics of BEC– Dynamical laws– Numerical methods– Numerical results
Part IV: Rotating BEC & Multi-component BEC– BEC in a rotational frame– BEC with nonlocal dipole-dipole interaction– Two-component BEC– Spinor BEC– Conclusions & Future challenges BEC@MIT, 00’
Part I
Predication, Experiment &
Mathematical models
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 of BEC
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 cold gas
( ) 0 1/
1 : ( ) , 0,1,2, ,1i Bi ik Tn f j
e ε µ ε µ ε ε−= = = < < <−
Einstein’s prediction on BEC: At zero or `low’ temperatures, most particles behavior in the same way (at quantum mechanical ground state)! –gregarious behavior----- quantum phase transition happens at extremely low temperatures --- degenerate quantum gas -- `super atom’ –--fifth matter of state.
Experimental difficulties
Low temperatures, almost absolutely zero (nK)Low density in a gas
Experimental techniques
Laser coolingMagnetic trappingEvaporative Cooling
($100k—300k)
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., NUS, …
2001 Nobel prize in physics:– C. Wiemann: U. Colorado– E. Cornell: NIST– W. Ketterle: MIT
ETH (02’, Rb, 300,000)
Properties of BEC in experiments
87 7 23 52 4
11 15 3
Kinds of atom Rb, Li, Na, Cr, He, Temperatures 50nK 2 K
Density 10 10 cm µ
−
− − −− − −
7 6 23 9 4# of atoms 100( Li) 10 ( Na) 10 ( He) Sphere at diameter 10--15 m
Spatial sizecigar-shaped at length 300 m & diameter 15μm
Life span A few seconds to several minutes
µµ
− − − − −
Brief BEC research history
Milestones– 1924 ----- Prediction of BEC by A. Einstein
– 1938-----London & Tisza linked BEC & superfluidity in liquid helium 4
– 1995 --- Experiments of BEC in ultracold gas of alkali atoms
– Since then--- BEC begins a new era in atomic, molecular & optical (AMO) physics & quantum optics; attracts interests of computational, applied & pure mathematicians.
Some famous physicists in BEC research – Nobel Laureates: A. Einstein (1921), L. D. Landau (1961), L. Onsager (1968), R. Feynmann
(1965),T.-D. Lee&C.N. Yang (1957), A. Leggett (2003), E. Cornell, C. Weimann&W. Ketterle (2001),
– Others: S. Bose, F. London, L. Tisza, N. Bogoliubov, R. Penrose, K. Huang, D. Beliaev, L. Pitaevskii, E. Gorss, A. Fetter, P. Zoller, D. Jaksch, I. Bloch, R. Hulet, J. Yngvason, S. Stringari, E.H.. Lieb, …..
Mathematical models
N-body problem– (3N+1)-dim linear Schroedinger equation
Mean field theory--zeor or `extremely’ low temperatures
– Gross-Pitaevskii equation (GPE): – (3+1)-dim nonlinear Schroedinger equation (NLSE)
Quantum kinetic theory-- high temperatures
– High temperature: QBME (3+3+1)-dim – Around critical temperature: QBME+GPE– Below critical temperature: GPE
(nK)cT T O< =
Model for a BEC at zero temperature– with N identical bosons
N-body problem –3N+1 dim. (linear) Schrodinger equation
Hartree ansatzFermi interactionDilute quantum gas -- two-body elastic interaction
1 2 1 2
22
int1 1
( , , , , ) ( , , , , ), with
( ) ( )2
t N N N N N
N
N j j j kj j k N
i x x x t H x x x t
H V x V x xm= ≤ < ≤
∂ Ψ = Ψ
= − ∇ + + −
∑ ∑
31 2
1( , , , , ) ( , ),
N
N N j jj
x x x t x t xψ=
Ψ = ∈∏
2
int4( ) ( ) with s
j k j kaV x x g x x g
mπδ− = − =
31( ) : ( )--energy per particle
NN N N N N NE H d x d x N E ψΨ = Ψ Ψ ≈∫
Model for a BEC – with N identical bosons
Energy per particle – mean field approximation (Lieb et al, 00’)
Dynamics (Gross, Pitaevskii 1961’; Erdos, Schlein & Yau, Ann. Math. 2010’)
Proper non-dimensionalization & dimension reduction– GPE/NLSE
3
22 2 4( 1)( ) ( ) with : ( , )
2 2N gE V x dx x t
mψ ψ ψ ψ ψ ψ
−= ∇ + + =
∫
222 3( )( , ) ( ) ( 1) ,
2tEi x t V x N g x
mδ ψψ ψ ψδψ
∂ = = − ∇ + + − ∈
2 21( , ) ( ) | | , 2
dti x t V x xψ ψ ψ β ψ ψ∂ = − ∇ + + ∈
4 ( 1) 4 s s
s s
N a Nax x
π πβ −= ≈
Gross-Pitaevskii equation (GPE)
The 3D Gross-Pitaevskii equation ( )
– V is a harmonic trap potential
– Normalization condition
22 2( , ) ( , ) ( ) ( , ) ( 1) | ( , ) | ( , )
2i x t x t V x x t N g x t x t
t mψ ψ ψ ψ ψ∂
= − ∇ + + −∂
)(2
)( 222222zyxmxV zyx ωωω ++=
3
2| ( , ) | 1.x t dxψ =∫
),,( zyxx =
Gross-Pitaevskii equation
Scaling (w.l.o.g. )– Dimensionless variables
– Dimensionless Gross-Pitaevskii equation
– With
3/2, , ( , ) ( , ),x s ss x
xt t x x t x tx mx xψ ψω ω
= = = =
),(|),(|),()(),(21),( 22 txtxtxxVtxtx
ti ψψβψψψ ++∇−=∂∂
2 22 2 21 4 ( 1) 4( ) ( ), , ,2
y s szy z y z
s sx x
N a N aV x x y zx x
π πβω ωγ γ γ γω ω−
= + + = = = ≈
x y zω ω ω≤ ≤
Gross-Pitaevskii equation
Typical parameters ( )– Used in JILA
– Used in MIT
]Js[1005.1 34−×=
Rb87
25
5
1.44 10 [kg], 10 2 [1 / s], 8
45.1[nm], 0.3407 10 [m], 0.01881
x y z x
ss s
x s
m
NN
maa x x
π
πβ
ω ω ω ω
ω
−
−
= × = = × =
= = = × = =
Na23
26
5
3.8 10 [kg], 360 2 [1 / s], 3.5 2 [1 / s]
42.75 [nm], 1.1209 10 [m], 0.003083
x y z
ss s
z s
m
NN
maa x x
π π
πβ
ω ω ω
ω
−
−
= × = = × = ×
= = = × = =
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 γγωωωω
2
2
12 31/4
/22 1/23 ho
( , , , ) ( , , ) ( ) with
( ) ( | ( , , ) | ) ( ) z zzg
x y z t x y t z
z x y z dxdy z e γ
ψ ψ φ
γφ φ φπ
−
=
= ≈ = ∫
2 2 22
2
4 42 3 ho 2
1( , ) | | ,2 2
( ) ( ) :2
y
az
x yi x t
t
z dz z dz
γψ ψ ψ β ψ ψ
γβ φ φ β βπ
∞ ∞
−∞ −∞
+∂= − ∆ + +
∂
= ≈ = =∫ ∫
12),,( ψψ == yxx
Mathematical model for BEC—mean field theory
The Gross-Pitaevskii equation (GPE/NLSE)
– t : time & : spatial coordinate (d=1,2,3)– : complex-valued wave function– : real-valued external potential– : dimensionless interaction constant
• =0: linear; >0(<0): repulsive (attractive) interaction– :Schrodinger equation (E. Schrodinger 1925’)
– :GPE (E.P. Gross 1961’; L.P. Pitaevskii 1961’)
2 21( , ) ( ) | |2ti x t V xψ ψ ψ β ψ ψ∂ = − ∇ + +
( )dx ∈
( , )x tψ
( )V x
β
0β =0β ≠
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 condensation| | 1β <<
0 0sa β> ⇒ ≥
0 0sa β< ⇒ ≤
0β =
Gross-Pitaevskii equation
– Strongly repulsive interacting condensation
– Strongly attractive interaction
Other potentials– Box potential– Double-well potential– Optical lattice potential– On a ring or torus
1β >>
0 & | | 1β β< >>
Other applications of GPE/NLSE
For laser beam propagation In plasma physics: wave interaction between electrons and ions
– Zakharov system---NLSE +wave equation, …..
In quantum chemistry: chemical interaction based on the first principle– Schrodinger-Poisson system
In materials science: – First principle computation, DFT, …– Semiconductor industry
In nonlinear (quantum) opticsIn biology – protein foldingIn superfluids – flow without friction, liquid
2 2 22 2H O H O+ →
4 He
Part II
Analysis & Computation
for
Ground states
Conservation laws
DispersiveTime symmetric: Time transverse (gauge) invariant
Mass conservation
Energy conservation
2 21( , ) ( ) | |2ti x t V xψ ψ ψ β ψ ψ∂ = − ∇ + +
& take conjugate unchanged!!t t→ − ⇒
2( ) ( ) --unchanged!!i tV x V x e αα ψ ψ ρ ψ−→ + ⇒ → ⇒ =
2 2( ) : ( ( , )) ( , ) ( ,0) 1, 0
d d
N t N t x t d x x d x tψ ψ ψ= = ≡ = ≥∫ ∫
2 2 41( ) : ( ( , )) ( ) (0), 02 2d
E t E t V x d x E tβψ ψ ψ ψ = = ∇ + + ≡ ≥ ∫
Stationary states
Stationary states (ground & excited states)
Nonlinear eigenvalue problems: Find
Time-independent NLSE or GPE:Eigenfunctions are– Orthogonal in linear case & Superposition is valid for dynamics!!– Not orthogonal in nonlinear case !!!! No superposition for dynamics!!!
2 2
2 2
1( ) ( ) ( ) ( ) | ( ) | ( ),2
with : | (x) | 1d
dx x V x x x x x
d x
µ φ φ φ β φ φ
φ φ
= − ∇ + + ∈
= =∫
( , ) s.t.µ φ( , ) ( ) i tx t x e µψ φ −=
2 21( , ) ( ) | |2ti x t V xψ ψ ψ β ψ ψ∂ = − ∇ + +
Ground states
The eigenvalue is also called as chemical potential
– With energy
Ground states -- nonconvex minimization problem
– Euler-Lagrange equation nonlinear eigenvalue problem
4: ( ) ( ) | (x) |2 d
E d xβµ µ φ φ φ= = + ∫
2 2 41( ) [ | ( ) | ( ) | ( ) | | ( ) | ]2 2d
E x V x x x d xβφ φ φ φ= ∇ + +∫
( ) min ( ) | 1, ( )g SE E S E
φφ φ φ φ φ
∈= = = < ∞
Existence & uniqueness
Theorem (Lieb, etc, PRA, 02’; Bao & Cai, KRM, 13’) If potential is confining
– There exists a ground state if one of the following holds
– The ground state can be chosen as nonnegative – Nonnegative ground state is unique if– The nonnegative ground state is strictly positive if– There is no ground stats if one of the following holds
| |( ) 0 for & lim ( )d
xV x x V x
→∞≥ ∈ = ∞
( ) 3& 0; ( ) 2 & ; ( ) 1&bi d ii d C iii dβ β β= ≥ = > − = ∈0, . . i
g g gi e e θφ φ φ=
0β ≥2loc( )V x L∈
( ) ' 3& 0; ( ) ' 2 & bi d ii d Cβ β= < = ≤ −
2 2 2 2
1 24 2
2 2
( ) ( )40 ( )
( )
inf L Lb f H
L
f fC
f≠ ∈
∇=
Key Techniques in Proof
Positivity & semi-lower continuous
The energy is bounded below if conditons (i) or (ii) or (iii) and strictly convex if Confinement potential implies decay at far field The set is convex inUsing convex minimization theorem Non-existence result
2( ) (| |) ( ), with | |E E E Sφ φ ρ φ ρ φ≥ = ∀ ∈ =
( ) : ( )E Eρ ρ=
2
/41 | |( ) exp , with 0
(2 ) 2d
dxx xεφ ε
πε ε
= − ∈ →
| ( ) =1 & ( )d
S x d x Eρ ρ ρ = < ∞
∫
ρ
0β ≥
0( )E
ε
εφ→
→−∞
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’)
Fundamental gaps
,,, 321 φφφ
1 2
1 2 1 2
, , ,( ) ( ) ( ) ( ) ( ) ( ) ???????
g
g gE E Eφ φ φ
φ φ φ µ φ µ φ µ φ< < < ⇒ < < <
θφφψ µµ immm eryxtyx mti
mti ee )(),(),,( −− ==
2 22
22
2
0
( )1( ) ( ) | | , 0 r2 2 2
2 ( ) 1, (0) 0
mm m m m m
m m
d rd m rr r rr dr dr r
r r dr
φµ φ φ β φ φ
π φ φ∞
= − + + + < < ∞
= =∫
1 1( ) : ( ) ( ) 0, ( ) : ( ) ( ) 0, 0E g gE E C C C Cβ µ βδ β φ φ δ β µ φ µ φ β= − > ≥ > = − > ≥ > ≥
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 4kin pot int
1( ) [ | ( ) | ( ) | ( ) | | ( ) | ] : ( ) ( ) ( )2 2d
E x V x x x dx E E Eβφ φ φ φ φ φ φ= ∇ + + = + +∫
2kin
1( ) | ( ) |2 d
x dxE φ φ= ∇∫
2pot( ) ( ) | ( ) |
dV x x dxE φ φ= ∫
4int( ) | ( ) |
2 dx dxE βφ φ= ∫
2 2 4
int kin pot int
1( ) [ | ( ) | ( ) | ( ) | | ( ) | ]2
= ( ) ( ) ( ) ( ) 2 ( )
dx V x x x dx
E E E E E
µ φ φ φ β φ
φ φ φ φ φ
= ∇ + +
+ = + +
∫
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.x
V x≤ ≤
= ∞
2
12
0
1( ) ( ) | ( ) | ( ), 0 1,2
(0) (1) 0 with | ( ) | 1
x x x x x
x dx
µ φ φ β φ φ
φ φ φ
′′= − + < <
= = =∫0β =
2 21( ) 2 sin( ), , 1, 2,3,2l lx l x l lφ π µ π= = =
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( ) ( ) 2 sin( ), : ( ) : ( )
2g g g g g gx x x E E πφ φ π φ µ µ φ= = = = = =
2 20 0 0( 1)( ) ( ) 2 sin(( 1) ), : ( ) : ( )
2j j j j j jjx x j x E E πφ φ π φ µ µ φ+
= = + = = = =
| | (1)oβ =
2 20 0 03( ) ( ) 2 sin( ), : ( ) ( ) , : ( ) ( ) 3
2 2 2g g g g g g g gx x x E E E π β πφ φ π φ φ µ µ φ µ φ β≈ = = ≈ = + = ≈ = +
2 20 0
2 20
( 1) 3( ) ( ) 2 sin(( 1) ), : ( ) ( ) ,2 2
( 1): ( ) ( ) 32
j j j j j
j j j
jx x j x E E E
j
π βφ φ π φ φ
πµ µ φ µ φ β
+≈ = + = ≈ = +
+= ≈ = +
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
1β
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
µµ φ β φ φ φ
β
φ
βφ φ µ µ β
= < < ⇒ =
⇓ =
≈ = ≈ = ≈ =
∫
TF TF(0) (1) 1 0g gφ φ= = ≠
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 xµ
φβµ
= = Φ
31( ) ( ) ( ), 0 ; (0) 0, lim ( ) 12 X
X X X X X→∞
′′Φ = − Φ +Φ ≤ < ∞ Φ = Φ =
( ) tanh( ), 0 ( ) tanh( ), 0 (1)gg gX X X x x x o
µφ µ
βΦ = ≤ < ∞ ⇒ ≈ ≤ =
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
µφ φ µ µ µ
β
φ µ µ β β µ β β
≈ = + − − ≤ ≤
= ⇒ ≈ = + + + = + + +∫
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 3
2 1 23
g g g g
g g
E E E E
E E
β ββ β
β
≈ = + + + ≈ = + +
≈ = + +
(1/ )O β
int, kin,1lim , lim 1, lim 0,2
g g g
g g g
E E EE Eβ β βµ→∞ →∞ →∞
= = =
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
β β βφ φ φ φ µ µ
β β β
∞
− − −− = − = = +
= + = + = +
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 xj
l x Cj
µφ φ µ
β
µ µ
+
=
=
≈ = − ++
+− −
+
∑
∑
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
µ µ β β
β β
β β
β
≈ = + + + + + +
≈ = + + + +
≈ = + + + +
≈ = + + + + +
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 Eφ φ φ µ φ µ φ µ φ< < < ⇒ < < <
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 EE E
E EE
β β β
β β β
µ
µµ µ
→∞ →∞ →∞
→∞ →∞ →∞
= = =
= = =
(1/ )O β
Harmonic Oscillator Potential in 1D
The potential:The nonlinear eigenvalue problem
Case I: no interaction, i.e. – A complete set of orthonormal eigenfunctions
2
( )2xV x =
2 21( ) ( ) ( ) ( ) | ( ) | ( ), with | ( ) | 12
x x V x x x x x dxµ φ φ φ β φ φ φ∞
−∞
′′= − + + =∫0β =
2
22
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
x l e H x l l
d eH x edx
H x H x x H x x
φ µπ
− −
−
= = + =
= −
= = = −
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 01/41 1( ) ( ) , : ( ) : ( )
2x
g g g g g gx x e E Eφ φ φ µ µ φπ
−= = = = = =
20 1/2 /2 0 01/41 ( 1)( ) ( ) (2 !) ( ), : ( ) : ( )
2j x
j j j j j j jjx x j e H x E Eφ φ φ µ µ φ
π− − +
= = = = = =
| | (1)oβ =
20 /2 0 00 01/4
1 1 1( ) ( ) , : ( ) ( ) , : ( ) ( )2 2 2
xg g g g g g g gx x e E E E C Cβφ φ φ φ µ µ φ µ φ β
π−≈ = = ≈ = + = ≈ = +
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
j C x dx
βφ φ φ φ
µ µ φ µ φ β φ∞
∞
+≈ = ≈ = +
+= ≈ = + ∫
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:
1β
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
µ β µµ φ φ β φ φ φ
µ βφ µ µβ
∞
∞
− ≤= + ⇒ =
= = ⇒ ≈ =∫
1/3( )O βTF2 gx µ= ±
TF TFkin( ) , ( )g gE Eφ φ= ∞ = ∞
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 2
1 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
βφ
βφ
βµ µ
≈ = =
≈ = =
= − ≈ − = =
≈ − − =
int, pot, kin,3 2 1lim , lim , lim , lim 0,5 3 3
g g g g
g g g g
E E E EE E Eβ β β βµ→∞ →∞ →∞ →∞
= = = =
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
β β βφ φ φ φ µ µβ β ββ β β
β β β
∞− = − = = +
= + = + = +
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
µ φ φ β φ φ
µ β µφ
µ βφ µ µβ
∞
∞
= +
− < ≤=
= = ⇒ ≈ =∫
Harmonic Oscillator Potential in 1D
– Matched asymptotic approximation
– Width of interior layer:
– Ordering:
MA MA 2 MAMA MA1 1 1
1 1sign( ) ( / 2) / , 0 | | 2 ( ) [tanh( ) sign( )]
0, otherwisex x xx x xµ µ β µφ µ
β − < ≤= − +
MA 1/3 MA 2/31 1(1/ ) (1/ ) ( )O O Oµ β µ β= ⇐ =
1 1( ) ( ) ( ) ( )g gE Eφ φ µ φ µ φ< ⇒ <
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 xψ ν ψ ν ψ= + +
2 2kin 0 pot 0 int 0
0
( ( )) (1 ) ( ) (1 ) ( ) (1 ) ( )
( ( )) | 0
dE x E E Ed E x
d ν
ψ ν ψ ν ψ ν ψ
ψν
−
=
= + + + + +
=
ν
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 | ( ) | 1dπ
µ φ θ φ θ β φ θ φ θ θ π
φ θ π φ θ φ θ θ
′′= − + ≤ ≤
+ = =∫0β =
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
l l
φ θ φ θ θ φ θ θπ π π
µ µ µ
+
+
= = =
= = = =
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 01( ) ( ) , : ( ) 0 : ( )2g g g g g gx x E Eφ φ φ µ µ φπ
= = = = = =
20 0 01( ) ( ) cos( ), : ( ) : ( )
2j j j j j jjx x l E Eφ φ θ φ µ µ φ
π= = = = = =
| | (1)oβ =
0 0 01( ) ( ) , : ( ) , : ( )8 42g g g g g gx x E E β βφ φ φ µ µ φπ ππ
= = = = = =
0( ) is a solution ( ) is also a solutionφ θ φ θ θ⇒ +
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)– Backward Euler + spectral method (BESP): (Bao, Chern & Lim, JCP, 06’)
Gauss-Seidel iteration method: (W.W. Lin et al., JCP, 05’)
Spectral method + stabilization: (Bao, I. Chern & F. Lim, JCP, 06’)
Reguarized Newton method: (Bao, Wen & Xu, 15’)
( )E φ
Imaginary time method
Idea: Steepest decent method + Projection
Physical institutive in linear case– Solution of GPE: – Imaginary time dynamics:
2 21
11
1
0 0
1 ( ) 1( , ) ( ) | | ,2 2
( , )( , ) , 0,1,2,
|| ( , ) ||( ,0) (x) with || ( ) || 1.
t n n
nn
n
Ex t V x t t t
xx t n
xx x
tt
δ φφ φ φ β φ φδ φ
φφ
φ
φ φ φ
+
−
++ −
+
= − = ∇ − − ≤ <
= =
= =
0φ
1φ
2φ 1φ
??)()()()ˆ(
)()ˆ(
01
11
01
φφφφ
φφ
EEEE
EE
<<
<
gφ
00 0
( , ) ( ) with ( ,0) ( ) ( )ji tj j j j
j jx t a e x x x a xµψ φ ψ ψ φ
∞ ∞−
= =
= = =∑ ∑
i tτ =0 1
0 00 0
( , )( , ) ( , ) ( ) ( ) : grond state jj j
j
xx x t a e x xa e
µ µ τµ τµ τ
φ τφ τ ψ φ φ∞ < < →∞
−−
=
= = ⇒ →∑
Normalized gradient flow
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.
ttx t V x t
tx x x
µ φφ φ φ β φ φ φφ
φ φ φ
= ∇ − − + ≥
= =
0|| (., ) || || || 1, ( (., )) 0, 0dt E t td t
φ φ φ= = ≤ ≥
2
1 1 11( , ) ( , ) ( , ), 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 ttφ φφ µ φ µ φ φ φ− + −
+ + += ∆ < < = − =∆
2 21( , ) ( ) | | ( , ) , 0,2t nx t V x t t tφφ φ φ β φ φ µ φ= ∇ − − + ∆ ≥
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’)
– BESP: Spectral accuracy in space & stable (Bao, I. Chern & F. Lim, JCP, 06’)
– Crank-Ncolson FD for continuous normalized gradient flow
21
11 0 0
1
1( , ) ( ) | | , ( , ), , ( , ) ( , ) 02
( , )( , ) , ( ,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
φ φ φ β φ φ φ φ
φφ φ φ φ
φ
+
−
++ −
+
= − − ∈Ω = ≤ < = =
= = =
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 ah x k tM−
= ∆ = = ∆ >
j n , 0,1, , ; , k=0,1,2, ; (x ,t ) nj n jx a j h j M t n k φ φ= + = = ≈
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 & asymptotic excited state• Optical lattice potential:
– Ground & asymptotic excited state with potential next
otherwise;100)( ∞≤≤= xxV
2/xV(x) 2=
2/)4()( 22xxV −=
)4(sin122/)( 22 xxxV +=
back
back
back
back
back
1 2 3 1 2 3( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 0.5000 1.5000 2.5000 3.500 0.5000 1.5000 2.5000 3.500
3.1371 1.0441 1.9414 2.8865 3.8505 1.5266 2.3578 3.2590 4.191931.371 3.9810 4.7438 5.5573 6.4043 6.5527 7.2802 8.0432 8.8349
156.
g gE E E Eβ φ φ φ φ µ φ µ φ µ φ µ φ
1 2 1 2
855 11.464 12.191 12.944 13.719 19.070 19.784 20.512 21.252313.71 18.171 18.891 19.629 20.383 30.259 30.971 31.691 32.419Observations
( ) ( ) ( ) ( ) ( ) ( ) , for any fixed ( )
lim 1 lim( )
g g
j
g
E E EEEβ β
φ φ φ µ φ µ φ µ φ β
φ µφ→+∞ →+∞
< < < ⇒ < < <
=
( )1
( )j
g
φµ φ
=
back
back
back
back
back
back
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 & asymptotic excited states
– In 3D • Optical lattice potential:
ground asymptotic excited states next
back
back
back
back
Part III
Analysis & Computation
for
Dynamics in BEC
Dynamics of a BEC
Time-dependent NLSE / GPE
Well-posedness & dynamical laws – Well-posedness & finite time blow-up– Dynamical laws
• Soliton solutions• Center-of-mass • An exact solution under special initial data
– Numerical methods and applications
2 2
0
1( , ) ( ) | | , , 02
( ,0) ( )
dti x t V x x t
x x
ψ ψ ψ β ψ ψ
ψ ψ
∂ = − ∇ + + ∈ >
=
Dynamics with no potential
Momentum conservationDispersion relationBright soliton in 1D:
( ) 0, dV x x≡ ∈
( ) : Im (0) 0d
J t d x J tψ ψ= ∇ ≡ ≥∫
2( ) 21( , )2
i k x tx t Ae k Aωψ ω β−= ⇒ = +
2 20
1( ( ) )2
0( , ) sech( ( ))i vx v a tax t a x vt x e
θψ
β− − +
= − −−
Dynamics with harmonic potential
Harmonic potential
Center-of-mass:
An analytical solution if
2 2
2 2 2 2
2 2 2 2 2 2
11( ) 22
3
x
x y
x y z
x dV x x y d
x y z d
γγ γ
γ γ γ
== + = + + =
2( ) ( , )
dcx t x x t d xψ= ∫
2 2 2( ) diag( , , ) ( ) 0, 0 each component is periodic!!c x y z cx t x t tγ γ γ+ = > ⇒
0 0( ) ( )sx x xψ φ= −
( , )0
2 2
( , ) ( ( )) , (0) & ( , ) 0( , ) : | ( , ) | | ( ( )) | -- moves like a particle!!
si t iw x ts c c
s c
x t e x x t e x x w x tx t x t x x t
µψ φ
ρ ψ φ
−= − = ∆ =
⇒ = = −
2 21( ) ( ) | |2s s s s s sx V xµ φ φ φ β φ φ= − ∇ + +
Well-posedenss
Theorem (T. Cazenave, 03’; C. Sulem & P.L. Sulem, 99’’) Assumptions
– Local existence, i.e.
– Global existence, i.e. if
2 2
2 2 210
(i) ( ) ( ), ( ) 0, & ( ) ( ) | | 2
(ii) ( ) | ( ) ( )d
d d d
dX L L
V x C V x x D V x L
X u H u u u V x u x d x
α α
ψ
∞ ∞∈ ≥ ∀ ∈ ∈ ≥
∈ = ∈ = + ∇ + < ∞
∫
2
20 ( )
1 or 2 with / or 3& 0db Ld d C dβ ψ β= = ≥ − = ≥
max max(0, ], s. t. the problem has a unique solution ([0, ), )T C T Xψ∃ ∈ ∞ ∈
maxT = +∞
Finite time blowup
Theorem (T. Cazenave, 03’; C. Sulem & P.L. Sulem, 99’) Assumptions
– There exists finite time blowup, i.e. if one of the following holds
– Proof:
2 20 V 0
<0 & ( ) ( ) 0, with 2,3
with finite variance (0):= | ( ) |d
dV x d x V x x d
X x x d x
β
ψ δ ψ
+ ⋅∇ ≥ ∀ ∈ =
∈ < ∞∫
maxT < +∞
2
0
0 0 0
0 0 0 0 0
(i) ( ) 0
(ii) ( ) 0 & Im ( ) ( ( )) 0
(iii) ( ) 0 & Im ( ) ( ( )) ( )
d
dL
E
E x x x d x
E x x x d x E d x
ψ
ψ ψ ψ
ψ ψ ψ ψ ψ
<
= ⋅∇ <
> ⋅∇ < −
∫
∫
2 20
2 '0
( ) : | ( , )| ( ) 2 ( ), 0, 2,3
( ) ( ) (0) (0) 0 * & ( *) 0!!d
V V
V V V V
t x x t d x t d E t d
t d E t t t t
δ ψ δ ψ
δ ψ δ δ δ
= ⇒ ≤ ≥ =
⇒ ≤ + + ⇒ ∃ < < ∞ =
∫
Numerical difficulties for dynamics
Dispersive & nonlinearSolution and/or potential are smooth but may oscillate wildlyKeep the properties of NLS on the discretized level– Time reversible & time transverse invariant – Mass & energy conservation– Dispersion relation
In high dimensions: many-body problems
Design efficient & accurate numerical algorithms– Explicit vs implicit (or computation cost)– Spatial/temporal accuracy, Stability– Resolution in strong interaction regime: 1β
2 2
0
1( , ) ( ) | | , , 02
with ( ,0) ( )
dti x t V x x t
x x
ψ ψ ψ β ψ ψ
ψ ψ
∂ = − ∇ + + ∈ >
=
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: ( , ) ( ) ( , ) | ( , ) | ( , )
| ( , ) | | ( , ) |
( , ) ( , )n
t
t
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 tβψ
ψ ψ
ψ ψ β ψ ψ
ψ ψ
ψ ψ− + ∆+
= − ∇
= +
⇓ =
=
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
2( ) ( ) | ( , ) | unchangedV x V x x tα ψ→ + ⇒
( ) ( ) 2/(2 ), , 1 / dh O k Oε ε ε β += = =
Crank-Nicolson finite difference (CNFD) method
Crank-Nicolson finite difference (CNFD)method
– Implicit: need solve a fully nonlinear system per time step via iterations!!!– Time reversible: Yes– Time transverse invariant: No– Mass conservation: Yes
CNFD
– Stability: Yes– Energy conservation: Yes– Dispersion relation without potential: No– Accuracy
• Spatial: 2nd order• Temporal: 2nd order
– Resolution in semiclassical regime (Markowich, Poala & Mauser, SINUM, 02’)
– Error estimate: Yes
2 2( ) & ( ) ( ) & ( )h o o h O Oε τ ε ε τ ε= = ⇐ = =
Error estimates for CNFD
Assume
Theorem: Assume , there exist sufficiently small, when , we have the following error estimate
In addition, if either , we have
3 1, 2 3, 0 5, 1 10([0, ]; ) ([0, ]; ) ([0, ]; ) &C T W C T W C T W H V Cψ ∞ ∞ ∞∈ ∩ ∩ ∩ ∈
( , )n nj j n je x tψ ψ= −
0C hτ ≤ 0 00 & 0h τ> >
0 00 & 0h h τ τ< ≤ < <
2 2 3/2 3/2[ ] & [ ], 0 /n ne C h e C h n Tτ δ τ τ+≤ + ≤ + ≤ ≤
2 2[ ], 0 /n ne e C h n Tδ τ τ++ ≤ + ≤ ≤
0 20( ) | 0 or ([0, ]; )nV x C T Hψ∂Ω∂ = ∈
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 Jets formation & Movie
x x100 at t 0, 4β ω ω= = →
x x20, at t 0 2 , 2y yβ ω ω ω ω= = → →
At t 0 40 50β= = → −
next
ψψβδψψβψψψ 420
22 ||||)(21),( ixVtx
ti −++∇−=∂∂
back
back
back
Collapse and Explosion of BEC
back
Number of atoms in condensate
back
Central density
back
back
Numerical results (Bao et., J Phys. B, 04)
Jet formation back
Semiclassical scaling
When , re-scaling
With
Leading asymptotics (Bao & Y. Zhang, Math. Mod. Meth. Appl. Sci., 05’)
1β >> 1/2 /4 2/( 2)1 /d dx x εε ψ ψ ε ε β− +→ = =
22 2( , ) ( , ) ( ) ( , ) | ( , ) | ( , )
2i x t x t V x x t x t x t
tε ε ε ε εεε ψ ψ ψ ψ ψ∂
= − ∇ + +∂
22 2 41( ) [ | | ( ) | | | | ] (1)
2 2dE V x dx Oε ε ε ε εεψ ψ ψ ψ= ∇ + + =∫
( ) ( )( ) ( )
1 1 2/( 2)
1 1 2/( 2)
( ) ( )
( ) ( )
d
d
E E O O
O O
ε ε
ε ε
ψ ε ψ ε β
µ ψ ε µ ψ ε β
− − +
− − +
= = =
= = =
Comparison of two scaling
( )( )
1/2 1/( 2) 1/( 2)
3/2 3/2 /4
21
2
2/( 2)
2/( 2)
Quanties Thomas-Fermi scaling Semiclassical scaling1 / 1 /
4/ ( )
:
Energy (1)
Chemical potential (1)
le
s x x
d dss s x s s s
sd
s s s
ss x x
s
d
d
tNax x m x x xx
x xmxEt
E O O
O O
ω ωπω ε β
ψ ε
ω ω ε
β
µ β
− + +
− −
−
+
+
= = =
=
( )( )
1/( 2)
/2( 2)
ngth of wave function ( 2 ) (1)
height of wave function ( / ) (1)
d
d d
O O O
O O O
β µ
β µ β
+
− +
=
=
Semiclassical limits
WKB analysis -- Gregor Wentzel, Hans Kramers & Leon Brillouin, 1926
– Formally assume
– Geometrical Optics: Transport + Hamilton-Jacobi
0
22 2
( ) /0 0
( , ) ( ) | |2
( ,0) : ( ) ( )
t
i S x
i x t V x
x x x eε
ε ε ε ε ε
εε ε ε
εε ψ ψ ψ β ψ ψ
ψ ψ ρ
∂ = − ∇ + +
= =
/ , ,iSe v S J vεε ε ε ε ε ε ε εψ ρ ρ= = ∇ =
22
t
( ) 0,1 1( )2 2
t
d
S
S S V x
ε ε ε
ε ε ε ε
ε
ρ ρ
ερ ρρ
∂ +∇• ∇ =
∂ + ∇ + + = ∆
0 1: ε
ε
ψ ψ
<
=
From QM to fluid dynamics
– Quantum Hydrodynamics (QHD): Euler +3rd dispersion
– Formal Limits --- Euler equations for fluids
Mathematical justification: G. B. Whitman, E. Madelung, E. Wigner, P.L. Lious, P. A. Markowich, F.-H. Lin, P. Degond, C. D. Levermore, D. W. McLaughlin, E. Grenier, F. Poupaud, C. Ringhofer, N. J. Mauser, P. Gerand, R. Carles, P. Zhang, P. Marcati, J. Jungel, C. Gardner, S. Kerranni, H.L. Li, C.-K. Lin, C. Sparber, …..
– Linear case – NLSE before caustics
2
2
( ) 0 ( ) / 2
( ) ( ) ( ) ( ln ) 4
t
t
v P
J JJ P V
ε ε ε
ε εε ε ε ε ε
ε
ρ ρ ρ β ρ
ερ ρ ρ ρρ
∂ +∇• = =
⊗∂ +∇• +∇ + ∇ = ∇ ∆
0 0 0 2
0 00 0 0
0
( ) 0 ( ) / 2
( ) ( ) ( ) 0
t
t
v P
J JJ P V
ρ ρ ρ β ρ
ρ ρρ
∂ +∇• = =
⊗∂ +∇• +∇ + ∇ =
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
′′′−Φ′+
Ω−+−=
∫
∇∫
),(),()(),(*),(*21
),(])(2
[),(*)(22
ψψψψ
ψψψ
*)(
ψδψδ H
tt),xψ(i =
∂∂
.4),()1()( 200 mxNx aUU s
πδ =−=Φ
),,( zyxx =
Rotating BEC
The 3D Gross-Pitaevskii equation ( )
– Angular momentum rotation
– V is a harmonic trap potential
– Normalization condition
ψψψ ]||)(2
[),( 20
22
UNLxVm
txt
i z +Ω−+∇−=∂∂
)(2
)( 222222zyxmxV zyx ωωω ++=
3
2| ( , ) | 1x t dxψ =∫
),,( zyxx =
∇−=×=∂−≡∂−∂−=−=
iPPxLiyxiypxpL xyxyz ,,)(: θ
Rotating BEC
General form of GPE ( )
with
Normalization condition
2 21( , ) [ ( ) | | ] ( , )2 zi x t V x L x t
tψ β ψ ψ∂
= − ∇ + −Ω +∂
dx R∈
22 24
32 22 2 2
1 ( ), 2( ) , 2( )21 ( ), 3, 2
z y
y z
x y dz dzV x
x y z d
γβ ββ π
β
γϕγ γ
∞
−∞
+ =≈ = = + + =
∫
2| ( , ) | 1.d
x t dxψ =∫
θ∂−≡∂−∂−= iyxiL xyz )(:
Rotating BEC
Conserved quantities– Normalization of the wave function
– Energy
Chemical potential
∫ ===d
NxdtxtNR
2 1))0((|),(|))(( ψψψ
2 2 41( ( )) [ | | ( ) | | * | | ]2 2
( (0))
d zt V x L dxEE
βψ ψ ψ ψ ψ ψ
ψ
Ω
Ω
= ∇ + −Ω +
=
∫
2 2 41( ( )) [ | | ( ) | | * | | ]2d zt V x L dxψ ψ ψ ψ ψ β ψµ
Ω= ∇ + −Ω +∫
Stationary states
Stationary solutions of GPE
Nonlinear eigenvalue problem with a constraint
Relation between eigenvalue and eigenfunction
)(),( xtx e ti φψ µ−=
2 2
2
1( ) [ ( ) | ( ) | ] ( ),2
| (x) | 1d
zx V x L x x
dx
µ φ β φ φ
φ
= − ∇ + −Ω +
=∫
4( ) ( ) | (x) |2 d
E dxβµ µ φ φ φΩ Ω= = + ∫
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’
4g|| || 1
( ) min ( ), ( ) ( ) | ( ) |2 dg g g gE E E x dx
φ
βφ φ µ µ φ φ φΩ Ω Ω Ω== = = + ∫
0&1|| ≥<Ω dβ
0&0 ≥=Ω dβ
10 ≤Ω< cβ
Numerical results
– Ground states: in 2D in 3D isosurface
– Vortex lattice• Symmetric trapping anisotropic trapping
– Giant vortex• In 2D In 3D
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Ground states of rapid rotation
BEC@MIT
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Dynamical laws of rotating BEC
Time-dependent Gross-Pitaevskii equation
Dynamical laws – Time reversible & time transverse invariant– Conservation laws– Well-posedness & finite-time – Dynamics of a stationary state with its center shifted
Numerical methods & results
2 2
0
1( , ) [ ( ) | | ]2
( ,0) ( ), : ( )
z
z y x
i x t V x Ltx x L i x y i θ
ψ β ψ ψ
ψ ψ
∂= − ∇ + −Ω +
∂= = − ∂ − ∂ = − ∂
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., H. Li &J . Shen, SISC 09’)
A method via rotating Lagrange coordinate (B., et al. SISC, 13’)
|),(||),(|),(|),(|),()(),( :2 Step
)(:,21),( :1 Step
2
2
nddt
xyzzt
txtxtxtxtxxVtxi
iyxiLLtxi
ψψψψβψψ
ψψψ θ
=⇒+=
∂−=∂−∂−=Ω−∇−=
TSFP 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
TSFP 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
TSLHP for rotating BEC
Numerical Method three: (Bao, LI & Shen, SISC 09’)
– Ideas• Time-splitting • Polar/cylindrical coordinates in 2D/3D, respectively• Lagurre + Fourier basis in 2D• Lagurre + Fourier +Hermite in 3D
– Features• Time reversible• Time transverse invariant• Mass Conservation in discretized level• Explicit & unconditionally stable• Spectrally accurate in space
Numerical methods A new formulation – A rotating Lagrange coordinate:
– GPE in rotating Lagrange coordinates
– Analysis & numerical methods -- Bao & Wang, JCP6’; Bao, Li & Shen, SISC, 09’; Bao, Marahrens, Tang & Zhang, 13’, …….
cos( ) sin( ) 0cos( ) sin( )
( ) for 2; ( ) sin( ) cos( ) 0 for 3sin( ) cos( )
0 0 1
t tt t
A t d A t t t dt t
Ω Ω Ω Ω = = = − Ω Ω = − Ω Ω
1( ) &
( , ) : ( , ) ( ( ) , )
x A t x
x t x t A t x tφ ψ ψ
−=
= =
2 21( , ) [ ( ( ) ) | | ] , , 02
dti x t V A t x x tφ β φ φ∂ = − ∇ + + ∈ >
A Method via Rotating Lagrange Coordinate
Numerical methods A new formulation – A rotating Lagrange coordinate:
– GPE in rotating Lagrange coordinates
– Analysis & numerical methods -- Bao & Wang, JCP6’; Bao, Li & Shen, SISC, 09’; Bao, Marahrens, Tang & Zhang, 13’, …….
cos( ) sin( ) 0cos( ) sin( )
( ) for 2; ( ) sin( ) cos( ) 0 for 3sin( ) cos( )
0 0 1
t tt t
A t d A t t t dt t
Ω Ω Ω Ω = = = − Ω Ω = − Ω Ω
1( ) &
( , ) : ( , ) ( ( ) , )
x A t x
x t x t A t x tφ ψ ψ
−=
= =
2 21( , ) [ ( ( ) ) | | ] , , 02
dti x t V A t x x tφ β φ φ∂ = − ∇ + + ∈ >
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 φψ =
21:&21: →→ yx γγ
2.21:&8.11: →→ yx γγ
1,8.0,100 ===Ω= zy γγβ
canisotropi :),(,9.0,1000 txV
=Ω=β
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Semiclassical scaling
When , re-scaling
With
Leading asymptotics
1β >> 1/2 /4 2/( 2)1 /d dx x εε ψ ψ ε ε β− +→ = =
22 2( , ) [ ( ) | | ]
2 zi x t V x Lt
ε ε εεε ψ ε ψ ψ∂= − ∇ + − Ω +
∂
22 2 4
, R
1( ) [ | | ( ) | | ( ) * | | ]2 2
(1)
d zE V x L dx
O
ε ε ε ε ε εε
εψ ψ ψ ε ψ ψ ψΩ = ∇ + − Ω +
=
∫
( ) ( ) ( )1 1 2/( 2) 2/( 2),( ) ( ) , ( )d dE E O O Oε
εψ ε ψ ε β µ ψ β− − + +Ω Ω Ω= = = =
Quantum Hydrodynamics
SetGeometrical Optics: (Transport + Hamilton-Jacobi)
Quantum Hydrodynamics (QHD): (Euler +3rd dispersion)
/ 2/( 2), , , 1 /iS de v S J vεε ε ε ε ε ε ε εψ ρ ρ ε β += = ∇ = =
22
t
ˆ ˆ( ) 0, : ( )
1 1ˆ( )2 2
t z z y x
z
S L L x y
S S V x L S
ε ε ε εθ
ε ε ε ε ε
ε
ρ ρ ρ
ερ ρρ
∂ +∇• ∇ +Ω = = ∂ − ∂ ≡ ∂
∂ + ∇ + + +Ω = ∆
2
2
ˆ( ) 0
ˆ( ) ( ) ( ) ( ln )4
0 1( ) / 2 , J , A
1 0
t z
t z
v L
J JJ P V L J A J
P v
ε ε ε ε
ε εε ε ε ε ε ε ε
ε
ε ε ε
ρ ρ ρ
ερ ρ ρ ρρ
ρ ρ ρ
∂ +∇• +Ω =
⊗∂ +∇• +∇ + ∇ +Ω +Ω = ∇ ∆
= = = −
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 (Cr52)– 2005@Univ. Stuttgart, Germany– PRL, 94 (2005) 160401
Big-wave in theoretical studyA. Griesmaier,et al., PRL, 94 (2005)160401
BEC with strong DDI
Lu, Burdick, Youn & Lev, PRL 107 (2011), 190401.
164 Dy
Mathematical Model
Gross-Pitaevskii equation (re-scaled)
– Trap potential– Interaction constants– Long-range dipole-dipole interaction kernel
References:– L. Santos, et al. PRL 85 (2000), 1791-1797– S. Yi & L. You, PRA 61 (2001), 041604(R); D. H. J. O’Dell, PRL 92 (2004), 250401
( )2 2ext dip
1( , ) ( ) | | | | ( , )2
i x t V x U x ttψ β ψ λ ψ ψ∂ = − ∆ + + + ∗ ∂
3( , )x t xψ ψ= ∈
( )2 2 2 2 2 2ext
1( )2 x y zV z x y zγ γ γ= + +
20 dip
20 0
4 (short-range), (long-range)3
s mNN aa a
µ µπβ λ= =
2 2 23
dip 3 33 1 3( ) / | | 3 1 3cos ( )( ) , fixed & satisfies | | 1
4 | | 4 | |n x xU x n n
x xθ
π π− ⋅ −
= = ∈ =
Mathematical Model
Mass conservation (Normalization condition)
Energy conservation
Long-range interaction kernel:– It is highly singular near the origin !! At singularity near the origin !! – Its Fourier transform reads
• No limit near origin in phase space !! • Bounded & no limit at far field too !!• Physicists simply drop the second singular term in phase space near origin!!• Locking phenomena in computation !!
3 3
2 22( ) : ( , ) ( , ) ( ,0) 1N t t x t d x x d xψ ψ ψ= ⋅ = ≡ =∫ ∫
3
2 2 4 2 2ext dip 0
1( ( , )) : | | ( ) | | | | ( | | ) | | ( )2 2 2
E t V x U d x Eβ λψ ψ ψ ψ ψ ψ ψ ⋅ = ∇ + + + ∗ ≡ ∫
23
dip 23( )( ) 1
| |nU ξξ ξξ⋅
= − + ∈
31
| |O
x
A New Formulation
Using the identity (O’Dell et al., PRL 92 (2004), 250401, Parker et al., PRA 79 (2009), 013617)
Dipole-dipole interaction becomes
2
dip 3 2
2
dip 2
3 3( ) 1( ) 1 ( ) 3 4 4
3( ) ( ) 1| |
n nn xU x x
r r r
nU
δπ π
ξξξ
⋅ = − = − − ∂
⋅⇒ = − +
2 2dip
2 2
| | | | 3
1 | | | |4
n nU
r
ψ ψ ϕ
ϕ ψ ϕ ψπ
∗ = − − ∂
= ∗ ⇔ −∆ =
| | & & ( )n n n n nr x n= ∂ = ⋅∇ ∂ = ∂ ∂
A New Formulation
Gross-Pitaevskii-Poisson type equations (Bao,Cai & Wang, JCP, 10’)
Energy
2ext
2
| |
1( , ) ( ) ( ) | | 3 ( , )2
( , ) | ( , ) | , lim ( , ) 0
n n
x
i x t V x x tt
x t x t x t
ψ β λ ψ λ ϕ ψ
ϕ ψ ϕ→∞
∂ = − ∆ + + − − ∂ ∂ − ∆ = =
3
2 2 4 2ext
1 3( ( , )) : | | ( ) | | | | | |2 2 2 nE t V x d xβ λ λψ ψ ψ ψ ϕ− ⋅ = ∇ + + + ∂ ∇ ∫
Ground State
Non-convex minimization problem
Nonlinear Eigenvalue problem (Euler-Language eq.)
Chemical potential
2ext
2
| |
1( ) ( ) ( ) | | 3 ( )2
( ) | ( ) | , lim ( ) 0, 1
n n
x
x V x x
x x x
µ φ β λ φ λ ϕ φ
ϕ φ ϕ φ→∞
= − ∆ + + − − ∂ − ∆ = = =
( ) : min ( ) with | 1& ( )g SE E S E
φφ φ φ φ φ
∈= = = < ∞
3
3
2 2 4 2ext
4 2 2
1: | | ( ) | | ( ) | | 3 | |2
3( ) | | | | , & | |2 2
n
n
V x d x
E d x
µ φ φ β λ φ λ ϕ
β λ λφ φ ϕ ϕ φ
= ∇ + + − + ∂ ∇
− = + + ∂ ∇ − ∆ =
∫
∫
Ground State Results
Theorem (Existence, uniqueness & nonexistence) (Bao, Cai & Wang, JCP, 10’)
– Assumptions
– Results• There exists a ground state if
• Positive ground state is unique
• Nonexistence of ground state, i.e.– Case I: – Case II:
3ext ext| |
( ) 0, & lim ( ) (confinement potential)x
V x x V x→∞
≥ ∀ ∈ = +∞
g Sφ ∈ 0 &2ββ λ β≥ − ≤ ≤
00| | with i
g ge θφ φ θ= ∈
lim ( )S
Eφ
φ∈
= −∞0β <
0 & or 2ββ λ β λ≥ > < −
Key Techniques in Proof
Estimate on the Poisson equation
Positivity & semi-lower continuous
The energy is strictly convex in if
Confinement potentialNon-existence result
2( ) (| |) ( ), with | |E E E Sφ φ ρ φ ρ φ≥ = ∀ ∈ =
224| |
| | : & lim ( ) 0 ( )nxxϕ φ ρ ϕ ϕ ϕ ϕ ρ φ
→∞−∆ = = = ⇒ ∂ ∇ ≤ ∇ ∇ = ∆ = =
( )E ρ ρ0 &
2ββ λ β≥ − ≤ ≤
1 2
2 2 23
, 1/2 1/41 2 1 2
1 1( ) exp exp ,(2 ) (2 ) 2 2
x y zx xε εφπε πε ε ε
+= − − ∈
Numerical Method for Ground State
Gradient flow with discrete normalization
Full discretization– Backward Euler sine pseudospectal (BESP) method – Avoid to use zero-mode in phase space via DST !!
2ext
21| |
11 1
1
1( , ) ( ) ( ) | | 3 ( , ),2
( , ) | ( , ) | , lim ( , ) 0, & ,
( , )( , ) : ( , ) , & 0,( , )
( , ) | ( , ) | 0, 0
n n
n nx
nn n
n
x x
x t V x x tt
x t x t x t x t t t
x tx t x t x nx t
x t x t t
φ β λ φ λ ϕ φ
ϕ φ ϕ
φφ φφ
φ ϕ
+→∞
−+ +
+ + −+
∈∂Ω ∈∂Ω
∂ = ∆ − − − + ∂ ∂ − ∆ = = ∈Ω ≤ <
= = ∈Ω ≥
= = ≥
0 0; ( ,0) ( ) 0, , with 1.x x xφ φ φ= ≥ ∈Ω =
Dynamics and its Computation
The Problem
Mathematical questions– Existence & uniqueness & finite time blow-up???
Existing results– Carles, Markowich & Sparber, Nonlinearity, 21 (2008), 2569-2590– Antonelli & Sparber, 09, preprint --- existence of solitary waves.
2ext
2 3
| |
30
1( , ) ( ) ( ) | | 3 ( , )2
( , ) | ( , ) | , lim ( , ) 0, , 0
( ,0) ( ), ,
n n
x
i x t V x x tt
x t x t x t x t
x x x
ψ β λ ψ λ ϕ ψ
ϕ ψ ϕ
ψ ψ→∞
∂ = − ∆ + + − − ∂ ∂ − ∆ = = ∈ >
= ∈
Well-posedenss Results
Theorem (well-posedness) (Bao, Cai & Wang, JCP, 10’)
– Assumptions
– Results• Local existence, i.e.
• If global existence, i.e.
2 2
3 3 3ext ext ext
2 2 21 30 ext
3
(i) ( ) ( ), ( ) 0, & ( ) ( ) | | 2
(ii) ( ) | ( ) ( )X L L
V x C V x x D V x L
X u H u u u V x u x d x
α α
ψ
∞ ∞∈ ≥ ∀ ∈ ∈ ≥
∈ = ∈ = + ∇ + < ∞
∫
0 &2ββ λ β≥ − ≤ ≤
max max(0, ], s. t. the problem has a unique solution ([0, ), )T C T Xψ∃ ∈ ∞ ∈
maxT = +∞
Finite Time Blowup Results
Theorem (finite time blowup) (Bao, Cai & Wang, JCP, 10’)
– Assumptions
– Results: • For any , there exists finite time blowup, i.e. • If one of the following conditions holds
3ext ext
(i) <0 or 0 & or >2
(ii) 3 ( ) ( ) 0,V x x V x x
ββ β λ λ β≥ < −
+ ⋅∇ ≥ ∀ ∈
maxT < +∞0( )x Xψ ∈
3
2
3
0
0 0 0
0 0 0 0 0
(i) ( ) 0
(ii) ( ) 0 & Im ( ) ( ( )) 0
(iii) ( ) 0 & Im ( ) ( ( )) 3 ( )L
E
E x x x d x
E x x x d x E x
ψ
ψ ψ ψ
ψ ψ ψ ψ ψ
<
= ⋅∇ <
> ⋅∇ < −
∫
∫
Numerical Method for dynamics
Time-splitting sine pseudospectral (TSSP) method, – Step 1: Discretize by spectral method & integrate in phase space exactly
– Step 2: solve the nonlinear ODE analytically
1[ , ]n nt t +
21( , )2ti x tψ ψ∂ = − ∇
2ext
2
2
2ext
( , ) ( ) ( ) | ( , ) | 3 ( , ) ( , )
( , ) | ( , ) | , (| ( , ) | ) 0 | ( , ) | | ( , ) | & ( , ) ( , )
( , ) ( ) ( ) | ( , ) | 3 ( , ) (
t n n
t n n
t n n n n
i x t V x x t x t x t
x t x tx t x t x t x t x t
i x t V x x t x t
ψ β λ ψ λ ϕ ψ
ϕ ψ
ψ ψ ψ ϕ ϕ
ψ β λ ψ λ ϕ ψ
∂ = + − − ∂ − ∆ =
⇓ ∂ = ⇒ = =
∂ = + − − ∂
2ext
2
( )[ ( ) ( )| ( , )| 3 ( , )]
, )
( , ) | ( , ) | ,
( , ) ( , )n n n n n
n n
i t t V x x t x tn
x t
x t x t
x t e x tβ λ ψ λ ϕ
ϕ ψ
ψ ψ− − + − − ∂
− ∆ =
⇒ =
New numerical methods for DDI
How to compute nonlocal DDI– FFT (fast Fourier transform)
– DST (discrete sine transform)
– Nonuniform FFT
2dip
2
dip 2
: | |
3( )( ) 1| |
U
nU
φ ψ
ξξξ
= ∗
⋅= − +
2 2| | 3 & ( , ) | ( , ) |nn x t x tφ ψ ϕ ϕ ψ= − − ∂ − ∆ =
3
2
2dip
sphere coordinate2
dip
( ) ( , ) =| |
( ) | | ( , )S
U t d
U t
φ ξ ρ ζ ξ ρ ψ
ξ ξ ρ ζ+×
=
=
∫
∫
Dynamics of a vortex lattice
Dimension Reduction (3D 2D)
Assumptions
Decomposition of the linear operator
Ansatz
2
ext 2 41& (1) & ( ) ( , ) , :
2z x y Dz
zO V x V x yγ γ γ εε γ
= = + =
22
22 1/4 21( , , , ) ( , , ) ( ) & ( ) exp
( ) 2
i t zx y z t e x y t z zεε εψ ψ ω ω
ε π ε−
≈ = −
2 2
4 21 1 12 2 2 2z zz zz
z zLε ε
= − ∂ + = − ∂ +
ext 21 1: ( ) ( , )2 2 D zL V x V x y L⊥= − ∆ + = − ∆ + +
Dimension Reduction (3D 2D)
2D equations when (Bao, Cai, Lei, Rosenkranz, PRA, 10’)
Energy
223
2
23
1/2 2
|( , )|
1 3( , , ) [ ( , ) | |2 2
3 ( ) ] ( , , )2
( ) ( , , ) | ( , , ) | , lim ( , , ) 0
D
n n
x y
ni x y t V x yt
n x y t
x y t x y t x y t
β λ λψ ψε π
λ ϕ ψ
ϕ ψ ϕ
⊥ ⊥
⊥
⊥
⊥ →∞
∂ − += − ∆ + +
∂
− ∂ − ∆
−∆ = =
0ε →
3
2 2 2 42 3
2 21/4 2 1/43
1 1( ( , )) : | | ( ) | | ( 3 ) | |2 2 2
3 + [ ( ) ( ) ]4
D
n
E t V x n
n dx
ψ ψ ψ β λ λ ψε π
λ ϕ ϕ⊥
⊥
⊥ ⊥ ⊥
⋅ = ∇ + + − +
∂ −∆ − ∇ −∆
∫
Two-component BEC
The 3D coupled Gross-Pitaevskii equations
Normalization conditions
Intro- & inter-atom Interactions
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 L U Ut m
i x t V x L U Ut m
ψ ψ ψ ψ λ ψ
ψ ψ ψ ψ λ ψ
∂= − ∇ + −Ω + + −
∂
∂= − ∇ + −Ω + + −
∂
3 3
22 0 0 0 2
1 21
( ) | ( , ) | : with | ( ,0) | ,j j jj
N t x t dx N N N N x dxψ ψ=
= = + = =∑ ∫ ∫
2
12 21
4 with jl
jl
aU a a
mπ
= =
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( , ) [ ( ) | | | | ]21( , ) [ ( ) | | | | ]2
z
z
i x t V x Lt
i x t V x Lt
ψ β ψ β ψ ψ λψ
ψ β ψ β ψ ψ λψ
∂= − ∇ + −Ω + + −
∂∂
= − ∇ + −Ω + + −∂
3 3
2 21 2( ) | ( , ) | | ( , ) | 1N t x t dx x t dxψ ψ= + =∫ ∫
3 3
0 02 21 2
1 2| ( , ) | , | ( , ) |N Nx t dx x t dxN N
ψ ψ= =∫ ∫
0λ ≠
0λ =
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
lE V x L dx
βψ ψ ψ ψ ψ ψ λ ψ ψ
=
Ψ = ∇ + −Ω + −∑ ∑∫
0 0 01 2 10, , ( )N N N O Nλ = =
3 3
0 02 22 1
2 2 1 1( ) | ( , ) | : 1, ( ) | ( , ) | : 1 1N NN t x t dx N t x t dxN N
ψ ε ψ ε= = = = = = − ≈∫ ∫
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 OE
ψ β ψ ψ
ψψ ψ β β εψ
∂= − ∇ + −Ω +
∂Ψ −
= =
Ground state
No external field:
Nonlinear eigenvalue problem
Existence & uniqueness of positive solutionNumerical methods can be extended
0λ =
1 21 2|| || ,|| ||
min ( , ) with 1Eφ α φ β
φ φ α β= =
+ =
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
µ φ β φ β φ φ
µ φ β φ β φ φ
= − ∇ + −Ω + +
= − ∇ + −Ω + +
Ground states
crater
Ground state
With external field:
Nonlinear eigenvalue problem
Existence & uniqueness of positive solution ???Numerical methods can be extended????
0λ ≠
2 21 2
1 2|| || || || 1
min ( , )Eφ φ
φ φ+ =
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
µ φ β φ β φ φ λφ
µ φ β φ β φ φ λφ
= − ∇ + −Ω + + −
= − ∇ + −Ω + + −
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 22β β β= =
Dynamics
Dynamics
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
εε ψ ε α ψ α ψ ψ ελψ
εε ψ ε α ψ α ψ ψ ελψ
∂= − ∇ + − Ω + + −
∂
∂= − ∇ + − Ω + + −
∂
0λ =
0λ ≠
Spin-orbit coupled BEC
Coupled GPE with a spin-orbit coupling & internal Josephson junction
Experiments: Lin, et al, Nature, 471( 2011), 83.
Applications ----Topological insulator
Analysis & numerical methods:– For ground state & dynamics (Bao & Cai, 14’)
2 2 21 0 11 1 12 2 1 1
2 2 21 0 21 1 22 2 1 1
1[ ( ) ( | | | | )]21[ ( ) ( | | | | )]2
x
x
i V x ikt
i V x ikt
ψ δ β ψ β ψ ψ ψ
ψ δ β ψ β ψ ψ ψ
−
− −
∂= − ∇ + + ∂ + + + +Ω
∂∂
= − ∇ + − ∂ + + + +Ω∂
Spinor BEC
Spinor F=1 BEC
With
22 * 2
1 1 1 0 1 1 1 0
22 *
0 0 1 1 1 1 1 0
22 * 2
1 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
ψ ρ ψ ρ ρ ρ ψ ψ ψ
ψ ρ ψ ρ ρ ψ ψ ψ ψ
ψ ρ ψ ρ ρ ρ ψ ψ ψ
− −
− −
− − −
∂= − ∇ + −Ω + + + − +
∂
∂= − ∇ + −Ω + + + +
∂
∂= − ∇ + −Ω + + + − +
∂
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 sa a a ag g
m ma a
π πρ ρ ρ ρ ρ ψ−
+ −= + + = = =
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 dxψ ψ−=−
= ≡ + + = =∑ ∫ ∫
3 3
2 2 0 01 1 1 1( ) | ( , ) | | ( , ) | :M t x t dx x t dx N N Mψ ψ−= − ≡ − =∫ ∫
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
dn
j j j z j
ss
gE V x L
g g dx
ψ ψ ψ ψ ρ
ρ ρ ρ ρ ρ ρ ρ ρ ψ ψ ψ ψ ψ ψ− − − − −
Ψ = ∇ + −Ω + +
+ + + − + +
∑∫
Spinor BEC
Dimension reductionGround state– Existence & uniqueness of positive solution??– Numerical methods ???
Dynamics – Dynamical laws – Numerical methods: TSSP
Semiclassical limit & hydrodynamics equation??
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– Rotating BEC & dipolar BEC– Multi-component BEC – two-component, spin-1 BEC, spin-orbit-coupled BEC
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, random potential– Superfluidity & dissipation, quantized vortex lattice – Coupling GPE & QBE for BEC at finite temperature– Mathematical theory for BEC– Interdisciplinary research: experiment,physics, mathematics, computation, ….
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[23]. W. Bao and M.-H. Chai, A uniformly convergent numerical method for singularly perturbed nonlinear eigenvalue problems, Commun. Comput. Phys., Vol 4, pp.135-160, 2007. .
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