Analysis and Computation for Analysis and Computation for Nonlinear - - PowerPoint PPT Presentation

Analysis and Computation for Analysis and Computation for Nonlinear Nonlinear Eigenvalue Eigenvalue Problems Problems Under Constraints Under Constraints

Fong Yin Lim Fong Yin Lim

Department of Mathematics Department of Mathematics and and Center for Computational Science & Engineering Center for Computational Science & Engineering National University of Singapore National University of Singapore Email: Email: fongyin.lim@nus.edu.sg fongyin.lim@nus.edu.sg

Joint work: Weizhu Bao (National University of Singapore) Joint work: Weizhu Bao (National University of Singapore) I I-

• Liang

Liang Chern Chern (National Taiwan University) (National Taiwan University) Ming Huang Ming Huang Chai Chai, , Hanquan Hanquan Wang, Wang, Yanzhi Yanzhi Zhang Zhang

Outline Outline

Introduction Introduction

Bose Bose-

• Einstein condensation (BEC)

Einstein condensation (BEC) Gross Gross-

• Pitaevskii

Pitaevskii equation equation

Singularly perturbed nonlinear Singularly perturbed nonlinear eigenvalue eigenvalue problems problems Asymptotic approximations Asymptotic approximations Numerical methods and numerical results Numerical methods and numerical results Extension to other BEC systems Extension to other BEC systems Conclusions Conclusions

Introduction Introduction

Bose Bose-

• Einstein condensation

Einstein condensation

• - Bose

Bose-

• Einstein statistical mechanics

Einstein statistical mechanics for bosons (spin 0, for bosons (spin 0, 1, 2, 1, 2, … …) )

• - below transition temperature

below transition temperature T Tc

c, a significant fraction of

, a significant fraction of bosonic bosonic particles will be in their single particles will be in their single-

• particle lowest energy

particle lowest energy state state

JILA (95’,87Rb)

( )

1 1 ) (

/

− =

− T k i

B i

e f

µ ε

ε

Experiments: Cooling of magnetically trapped dilute Experiments: Cooling of magnetically trapped dilute atomic vapor to atomic vapor to nanokelvins nanokelvins temperature temperature

Gross Gross-

• Pitaevskii

Pitaevskii equation equation

Gross Gross-

• Pitaevskii

Pitaevskii equation (GPE) equation (GPE)

V(x V(x) ) – – trapping potential trapping potential β β --

• - mean

mean-

• field interaction

field interaction

Normalization conservation Normalization conservation Energy conservation Energy conservation Stationary state Stationary state

µ µ --

• - chemical potential

chemical potential

Probability density of finding a particle Probability density of finding a particle

t i

e x t x

µ

φ ψ

−

= ) ( ) , ( r r

d

x x x x x V x t x i ℜ ∈ + + ∇ − = ∂ ∂ r r r r r r r ), ( ) ( ) ( ) ( ) ( 2 1 ) (

2 2

ψ ψ β ψ ψ ψ

2 2

| ) ( | | ) , ( | x t x r r φ ψ =

1 ) , (

2 =

= t x N r ψ

( ) [ ] ( ) ( ) ( ) ( )

x d t x t x x V t x t E r r r r r

4 2 2

, 2 , , 2 1 ψ β ψ ψ ψ + + ∇ = ∫

Stationary States Stationary States

Nonlinear Nonlinear eigenvalue eigenvalue problem (time problem (time-

• independent GPE)

independent GPE) Eigenvalue Eigenvalue (chemical potential) (chemical potential) Energy Energy Other applications Other applications

Nonlinear optics Nonlinear optics Complex quantum system Complex quantum system

x d x E

d

r r

4

) ( 2 ) ( ) ( φ β φ φ µ µ

β β

∫

ℜ

+ = =

∫

ℜ

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + ∇ =

d

x d x x x V x E r r r r r

4 2 2

) ( 2 ) ( ) ( ) ( 2 1 ) ( φ β φ φ φ

β d

x x x x x V x x ℜ ∈ + + ∇ − = r r r r r r r ), ( ) ( ) ( ) ( ) ( 2 1 ) (

2 2

φ φ β φ φ φ µ 1 ) (

2 2

= = ∫

ℜ

x d x

d

r r φ φ

Stationary States Stationary States

Non Non-

• convex minimization problem: ground state

convex minimization problem: ground state φ φg

g

Excited states Excited states Open question Open question

{ }

( ) min ( ) | 1, | 0, ( )

g x S

E E S E

β β φ

φ φ φ φ φ φ

∈Γ ∈

= = = = < ∞

r

L , , ,

3 2 1

φ φ φ

??????? ) ( ) ( ) ( ) ( ) ( ) ( , , ,

2 1 2 1 2 1

L L L < < < < < < φ µ φ µ φ µ φ φ φ φ φ φ

β β β β β g g g

E E E

(Bao & W. Tang, JCP, 03’; Bao, F. Lim & Y. Zhang, Bull Int. Math, 06’)

Singularly Perturbed NEP Singularly Perturbed NEP

Three typical parameter regimes: Three typical parameter regimes:

Linear: Linear: Weak interaction: Weak interaction: Strong repulsive interaction: Strong repulsive interaction:

, reformulate GPE into singularly perturbed NEP , reformulate GPE into singularly perturbed NEP

= β ) 1 (

• =

β 1 >> β

1 >> β

∫

Ω

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + ∇ = x d x x x V x E r r r r r

4 2 2 2

) ( 2 1 ) ( ) ( ) ( 2 ) (

ε ε ε ε ε

φ φ φ ε φ x d x E r r

4

) ( 2 1 ) ( ) (

ε ε ε ε ε ε

φ φ φ µ µ

∫

Ω

+ = = 1 ) ( : ; , ) ( ), ( ) ( ) ( ) ( ) ( 2 ) (

2 2 2

= = Ω ∂ ∈ = ℜ ⊂ Ω ∈ + + ∇ − =

∫

Ω

x d x x x x x x x x V x x

d

r r r r r r r r r r

ε ε ε ε ε ε ε ε ε

φ φ φ φ φ φ φ ε φ µ

Singularly Perturbed NEP Singularly Perturbed NEP

Bounded Bounded Ω Ω with box potential with box potential Rescaled energy and Rescaled energy and eigenvalue eigenvalue Leading asymptotic approximation Leading asymptotic approximation

⎩ ⎨ ⎧ ∞ Ω ∈ =

• therwise

, , ) ( x x V r r

) ( ) ( , , 1

2

x x r r φ φ µ ε µ β ε

ε ε

= = =

1 ), 1 ( ) ( 2 1

4

<< < = + =

∫

Ω

ε φ µ

ε ε ε

O x d x E r r ) 1 ( ) ( 2 1 ) ( 2

4 2 2

O x d x x E = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ∇ = ∫

Ω

r r r

ε ε ε

φ φ ε 1 ), ( ), ( >> = = = = β β βµ µ β β

ε ε

O O E E

Singularly Perturbed NEP Singularly Perturbed NEP

Whole space with harmonic potential Whole space with harmonic potential Semiclassical Semiclassical scaling scaling Rescaled energy and Rescaled energy and eigenvalue eigenvalue Leading asymptotic approximation Leading asymptotic approximation

( )

) ( ) ( lim ), ( 2 1 ) (

2 2 2 1 2 1

= + + + =

∞ →

x V x W x W x x x V

x d d

r r r L r

r

γ γ

) ( ) ( , , ,

1 2 / 1 ) 2 /( 2

x x x x

d

r r r r φ φ µ ε µ ε β ε

ε ε

= ′ = = ′ =

− − + −

( )

) 1 ( ) ( 2 1 ) ( ) ( ) ( ) ( 2

4 2 2 2

O x d x x x W x V x E = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + + ∇ = ∫

Ω

r r r r r r

ε ε ε ε

φ φ φ ε 1 ), 1 ( ) ( 2 1

4

<< < = + =

∫

Ω

ε φ µ

ε ε ε

O x d x E r r 1 ), ( ), (

) 2 /( 2 ) 2 /( 2

>> = = = =

+ +

β β βµ µ β β

ε ε d d

O O E E

Asymptotic Approximation Asymptotic Approximation

Thomas Thomas-

• Fermi regime

Fermi regime --

• - drop the diffusion term in the

drop the diffusion term in the GPE for GPE for β β >>1 ( >>1 ( ε ε << 1 ) << 1 ) Ground state in box potential Ground state in box potential Boundary conditions not satisfied Boundary conditions not satisfied Existence of boundary layers Existence of boundary layers

Ω ∈ + = x x x x x V x r r r r r r ), ( ) ( ) ( ) ( ) (

2φ

φ φ φ µ

1 ) ( , 1 1 ) (

TF TF 2 TF

= = ⇒ =

∫Ω

x x d x

g g g

r r r φ µ φ

TF TF

) (

g g

x µ φ = r

0.2 0.4 0.6 0.8 1 0.5 1 1.5 x φg(x)

⎩ ⎨ ⎧ ∞ = < < =

• therwise

, to 1 , 1 , ) ( d i x x V

i

r

Matched Asymptotic Method Matched Asymptotic Method

For boundary layer at For boundary layer at x x = 0, rescale = 0, rescale Solution Solution Match the solutions at intermediate region Match the solutions at intermediate region

) ( ) ( , X x X x

g g

Φ = = µ φ µ ε

∞ < ≤ Φ + Φ − = Φ X X X X

XX

), ( ) ( 2 1 ) (

3

1 ) ( lim , ) ( = Φ = Φ

∞ →

X

X

2 / 1 ), tanh( ) ( < ≤ ≈ x x x

g g g

ε µ µ φ

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ε µ ε µ ε µ µ φ

MA MA MA MA MA

tanh ) 1 ( tanh tanh ) (

g g g g g

x x x

⇒ =

∫

1 ) (

1 2 MA

dx x

g

φ

2 2 MA

2 1 2 1 ε ε ε µ + + + =

g

2 2 MA

2 1 3 4 2 1 ε ε ε + + + =

g

E

Matched Asymptotic Method Matched Asymptotic Method

k kth th excited state excited state --

• - k

k interior layers at interior layers at x = j x = j /( /(k k+1) ( +1) (j j = 1, = 1, … …, , k k) ) Matched asymptotic approximation of the Matched asymptotic approximation of the k kth th excited state excited state

⎣ ⎦ ⎣ ⎦

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − =

∑ ∑

= + = 2 / MA MA 2 / ) 1 ( MA MA MA

tanh 1 1 2 tanh 1 2 tanh ) (

k j k k k k j k k k

C x k j k j x x ε µ ε µ ε µ µ φ ⎩ ⎨ ⎧ = k k Ck

• dd

0, even , 1

2 2 2 2 MA

) 1 ( 2 ) 1 ( 1 ) 1 ( 2 1 ε ε ε µ + + + + + + = k k k

k 2 2 2 2 MA

) 1 ( 2 ) 1 ( 1 ) 1 ( 3 4 2 1 ε ε ε + + + + + + = k k k Ek

0.2 0.4 0.6 0.8 1 0.5 1 1.5 x φg(x) 0.2 0.4 0.6 0.8 1

• 1.5
• 1
• 0.5

0.5 1 1.5 x φ1(x)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

• 1.5
• 1
• 0.5

0.5 1 1.5 x φ5(x)

BEC in 1D Box Potential BEC in 1D Box Potential

Energies and chemical potentials of the stationary Energies and chemical potentials of the stationary states are ranked in the same order states are ranked in the same order Width of boundary layers Width of boundary layers Width of interior layers Width of interior layers Comparisons between asymptotic approximation and Comparisons between asymptotic approximation and numerical results numerical results

L L < < < < < < ) ( ) ( ) ( , ) ( ) ( ) (

2 1 2 1

φ µ φ µ φ µ φ φ φ

g g

E E E ε 2 ~

ε 4 ~

( ))

(

2 / 3 MA

2

ε

φ φ

−

= − e O

L k k

) (

MA

ε O E E

k k

+ =

( ))

(

2 / 3 MA ε

µ µ

−

+ = e O

k k

BEC in 1D Harmonic Potential BEC in 1D Harmonic Potential

Harmonic Harmonic Thomas Thomas-

• Fermi approximation

Fermi approximation

⎪ ⎩ ⎪ ⎨ ⎧ ≤ − =

• therwise

, ) ( , ) ( ) (

TF TF TF g g g

x V x V x µ µ φ

3 / 2 TF 2 TF

2 3 2 1 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⇒ =

∫

∞ ∞ − g g

dx µ φ

3 / 2 TF

2 3 10 3 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =

g

E

Ω ∈ + = x x x x x V x r r r r r r ), ( ) ( ) ( ) ( ) (

2φ

φ φ φ µ

2

2 1 ) ( x x V =

BEC in 1D Harmonic Potential BEC in 1D Harmonic Potential

Matched asymptotic approximation for the first excited Matched asymptotic approximation for the first excited state state Width of interior layer: Width of interior layer:

MA 1 MA MA 1 1 MA MA 2 MA 1 1 1

| | tanh( ) 0 | | 2 ( ) 2 / 2

• therwise

x x x x x x µ µ µ ε φ µ µ ⎧ ⎪ − < ≤ ⎪ ⎡ ⎤ = − + ⎨ ⎣ ⎦ ⎪ ⎪ ⎩

( ) O ε

• 16
• 12
• 8
• 4

4 8 12 16 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x φg(x)

• 16
• 12
• 8
• 4

4 8 12 16

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 x φ1(x)

Numerical Methods for Ground States Numerical Methods for Ground States

Boundary Boundary eigenvalue eigenvalue method method

Runge Runge-

• Kutta

Kutta space space-

• marching

marching

(Edward & Burnett, (Edward & Burnett, PRA PRA, 95 , 95’ ’) ) ( (Adhikari Adhikari, , Phys.

• Phys. Lett
• Lett. A

. A, 00 , 00’ ’) )

Variational method Variational method

Direct minimization of energy functional with FEM approach Direct minimization of energy functional with FEM approach

(Bao & Tang, (Bao & Tang, JCP JCP, 02 , 02’ ’) )

Nonlinear algebraic Nonlinear algebraic eigenvalue eigenvalue problem approach problem approach

Gauss Gauss-

• Seidel type iteration

Seidel type iteration (Chang et. al.,

(Chang et. al., JCP JCP, 05 , 05’ ’) )

Continuation method Continuation method

(Chang et. al., (Chang et. al., JCP JCP, 05 , 05’ ’) ) ( (Chien Chien et. al.,

• et. al., SIAM J.

SIAM J. Sci Sci. . Comput Comput., 07 ., 07’ ’) )

Imaginary time method Imaginary time method

Explicit imaginary time algorithm via Explicit imaginary time algorithm via Visscher Visscher scheme scheme

( (Chiofalo Chiofalo et. al.,

• et. al., PRE

PRE, 00 , 00’ ’) )

Backward Euler finite difference (BEFD) and time-splitting sine-pseudospectral method (TSSP)

(Bao & (Bao & Du Du, , SIAM J. SIAM J. Sci Sci. . Comput Comput., 04 ., 04’ ’) )

Backward Euler sine-pseudospectral method (BESP)

(Bao et. al., (Bao et. al., JCP JCP, 06 , 06’ ’) )

Imaginary Time Method Imaginary Time Method

Replace Replace in the time in the time-

• dependent GPE

dependent GPE (imaginary time method) and form gradient flow with (imaginary time method) and form gradient flow with discrete normalization (GFDN) in each time interval discrete normalization (GFDN) in each time interval Linear case Linear case – – energy diminishing energy diminishing

it t − →

• 1

2 2

, ), , ( ) , ( ) ( 2 1 ) , (

+

< ≤ ℜ ∈ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − ∇ = ∂ ∂

n n d

t t t x t x t x x V t t x r r r r φ φ β φ

d L n n n n

x t x t x t x t x ℜ ∈ = =

− + − + + + +

, ) , ( ) , ( ) , ( ) , (

2

1 1 1 1

r r r r φ φ φ φ 1 with ), ( ) , ( = ℜ ∈ = φ φ φ

d

x x x r r r

( ) ( ) ( ) ( ) ( ) ( )

, , ,

1

⋅ ≤ ≤ ⋅ ≤ ⋅

+

φ φ φ E t E t E

n n

L

d

x x V ℜ ∈ ∀ ≥ = r r ) ( and β

Imaginary Time Method Imaginary Time Method

Nonlinear case Nonlinear case – – consider GFDN as 1 consider GFDN as 1st

st order time splitting

• rder time splitting

to the continuous normalized gradient flow (CNGF) to the continuous normalized gradient flow (CNGF) Normalization/ projection Normalization/ projection For For , the CNGF is normalization , the CNGF is normalization conserved and energy diminishing conserved and energy diminishing

, ), , ( ) ( ) , ( ) ( 2 1 ) , (

2 2

≥ ℜ ∈ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − − ∇ = ∂ ∂ t x t x t t x x V t t x

d

r r r r r φ µ φ β φ

φ

1 with ), ( ) , ( = ℜ ∈ = φ φ φ

d

x x x r r r

1

, ), , ( ) , (

+

≤ ≤ ℜ ∈ = ∂ ∂

n n d

t t t x t x t t x r r r φ µ φ

φ

d

x x V ℜ ∈ ∀ ≥ ≥ r r ) ( and β

Discretization Discretization Scheme (I) Scheme (I)

The problem is truncated into bounded domain with The problem is truncated into bounded domain with zero boundary conditions zero boundary conditions Backward Euler finite difference (BEFD) + 2 Backward Euler finite difference (BEFD) + 2nd

nd-

• order
• rder

centered difference for spatial derivative centered difference for spatial derivative Consider 1D gradient flow in Consider 1D gradient flow in Implicit, unconditionally stable, energy diminishing, Implicit, unconditionally stable, energy diminishing, second order accuracy in space

1 +

< ≤

n n

t t t

1 , , 2 , 1 , ) ( 2 1

* 2 * * *

− = − − = ∆ −

=

M j x V D t

j n j j j x x s xx n j j

j

L ϕ ϕ β ϕ ϕ ϕ ϕ

* *

= =

M

ϕ ϕ

M j

j n j

, , 1 , ,

* * 1

L = =

+

ϕ ϕ ϕ

second order accuracy in space

Discretization Discretization Scheme (II) Scheme (II)

Time Time-

• splitting sine

splitting sine-

• pseudospectral

pseudospectral method (TSSP) method (TSSP)

• - discretize

discretize with sine with sine-

• pseudospectral

pseudospectral method and method and integrated in time exactly integrated in time exactly

• - solve analytically

solve analytically

2 2nd

nd-

• order
• rder Strang

Strang splitting splitting Explicit, spectral accuracy in space, time Explicit, spectral accuracy in space, time-

• splitting error

splitting error

1

, ), , ( ) , (

+

< ≤ Ω ∈ = ∂ ∂

n n xx

t t t x t x t t x φ φ

( )

1 2

, ), , ( ) , ( ) ( ) , (

+

< ≤ Ω ∈ + − = ∂ ∂

n n

t t t x t x t x x V t t x φ φ β φ

Discretization Discretization Scheme (III) Scheme (III)

Backward Euler sine Backward Euler sine-

• pseudospectral

pseudospectral method (BESP) method (BESP) Backward Euler scheme for Backward Euler scheme for discretization discretization in time in time Sine Sine-

• pseudospectral

pseudospectral method for method for discretization discretization in space in space At every time step, a linear system is solved iteratively At every time step, a linear system is solved iteratively

1 , , 2 , 1 , ) ( 2 1

* 2 * * *

− = − − = ∆ −

=

M j x V D t

j n j j j x x s xx n j j

j

L ϕ ϕ β ϕ ϕ ϕ ϕ

* *

= =

M

ϕ ϕ

M j

j n j

, , 1 , ,

* * 1

L = =

+

ϕ ϕ ϕ

m j n j j x x m s xx n j m

x V D t

j

*, 2 1 *, 1 *,

) ( 2 1 ϕ ϕ β ϕ ϕ ϕ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − = ∆ −

= + +

(Bao, (Bao, Chern Chern & Lim, & Lim, JCP JCP, 06 , 06’ ’) )

BESP BESP

A stabilization parameter A stabilization parameter α α is introduced to ensure the is introduced to ensure the convergence of the numerical scheme convergence of the numerical scheme α α guarantees the convergence of the iterative method guarantees the convergence of the iterative method and gives the optimal convergence rate and gives the optimal convergence rate Unconditionally stable, energy diminishing, spectral Unconditionally stable, energy diminishing, spectral accuracy in space accuracy in space Larger mesh size and larger time Larger mesh size and larger time-

• step can be used

step can be used

m j n j j m j x x m s xx n j m

x V D t

j

*, 2 1 *, 1 *, 1 *,

) ( 2 1 ϕ ϕ β α αϕ ϕ ϕ ϕ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − + − = ∆ −

+ = + +

( )

min max

2 1 b b + = α ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + =

− ≤ ≤ − ≤ ≤ 2 1 1 min 2 1 1 max

) ( min , ) ( max

n j j M j n j j M j

x V b x V b ϕ β ϕ β

Numerical Results in 1D Numerical Results in 1D

• 16
• 8

8 16 0.1 0.2 0.3 0.4 φg(x)

• 16
• 8

8 16 35 70 105 140 V(x) x

• 16
• 8

8 16

• 0.4
• 0.2

0.2 0.4 φ1(x) x

• 16
• 8

8 16 35 70 105 140 V(x)

400 2 ) (

2

= = β x x V

• 16
• 8

8 16 0.1 0.2 0.3 0.4 φg(x)

• 16
• 8

8 16 35 70 105 140 V(x) x

• 16
• 8

8 16

• 0.4
• 0.2

0.2 0.4 φ1(x) x

• 16
• 8

8 16 35 70 105 140 V(x)

250 4 sin 25 2 ) (

2 2

= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = β πx x x V

2D Box Potential 2D Box Potential

Uniformly convergent method Uniformly convergent method – – GFDN + piecewise uniform mesh GFDN + piecewise uniform mesh

(Bao & (Bao & Chai Chai, , Comm.

• Comm. Comput
• Comput. Phys

. Phys, 07 , 07’ ’) )

3D Optical Lattice 3D Optical Lattice

( )

100 , 4 sin 4 sin 4 sin 50 2 1 ) (

2 2 2 2 2 2

= ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + + = β π π π z y x z y x x V

3D Optical Lattice 3D Optical Lattice

( )

6400 , 800 , 100 4 sin 4 sin 4 sin 50 2 1 ) (

2 2 2 2 2 2

= ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + + = β π π π z y x z y x x V

Multiscale Multiscale structures due to the oscillatory nature of trapping structures due to the oscillatory nature of trapping potential potential High spatial accuracy is required, especially for 3D problems High spatial accuracy is required, especially for 3D problems

Extension to Rotating BEC Extension to Rotating BEC

GPE in rotational frame, with rotating speed GPE in rotational frame, with rotating speed Ω Ω z z-

• component of angular momentum

component of angular momentum Backward Euler finite difference Backward Euler finite difference

(Bao, Wang & (Bao, Wang & Markowich Markowich, , Comm. Math.

• Comm. Math. Sci

Sci., 05 ., 05’ ’) )

Backward Backward-

• forward Euler Fourier

forward Euler Fourier-

• pseudospectral

pseudospectral method method

(Bao & Zhang, 06 (Bao & Zhang, 06’ ’)

d z z

x x L x x V x ℜ ∈ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ Ω − + + ∇ − = r r r r r ), ( ) ( ) ( 2 1 ) (

2 2

φ φ β φ µ

1 ) (

2 2

= = ∫

ℜ

x d x

d

r r φ φ ) ( :

x y x y z

y x i yp xp L ∂ − ∂ − = − =

)

Central Vortex State Central Vortex State

Surface plot Phase plot

2 ) (

2

x x V =

θ

φ φ

im m

e r y x ) ( ) , ( =

Fast Rotating BEC Fast Rotating BEC

2 2 4 2

, 4 ) 1 ( ) ( y x r r k r r V + = + − = α

3D Fast Rotating BEC 3D Fast Rotating BEC

2 2 2 2 4 2

, 4 ) 1 ( ) ( y x r z r k r r V

z

+ = + + − = γ α

Extension to Multi Extension to Multi-

• component BEC

component BEC

Coupled GPE for two Coupled GPE for two-

• component condensate

component condensate Normalizations are individually conserved Normalizations are individually conserved Direct extension of GFDN with BEFD

2 2 2 22 2 1 21 2 2 2 1 2 2 12 2 1 11 2 1 1

) ( 2 1 ) ( ) ( 2 1 ) ( φ φ β φ β φ µ φ φ β φ β φ µ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + + ∇ − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + + ∇ − = x V x x V x r r r r

1 ) ( , 1 ) (

2 2 2 2 2 1 2 1

= = = =

∫ ∫

ℜ ℜ

x d x x d x

d d

r r r r φ φ φ φ

Direct extension of GFDN with BEFD

Two Two-

• component BEC

component BEC

N N1

1 + N

+ N2

2 = N, for different

= N, for different N N1

1/N

/N2

2

3D trap with cylindrical 3D trap with cylindrical symmetry

21 12 1 21 21 2 12 12 2 22 22 1 11 11

, , a a N a N a N a N a = = = = = β β β β

2 12 22 11

< − a a a

symmetry

Extension to Extension to Spinor Spinor BEC BEC

Hamburg (03’,87Rb, F=1) Hamburg (03’,87Rb, F=2)

Hyperfine spin Hyperfine spin F F = = I I + + S S 2F+1 2F+1 hyperfine components: hyperfine components: m mF

F =

= -

• F,

F, -

• F+1,

F+1, … …., F ., F-

• 1, F

1, F Experience different potentials under external magnetic Experience different potentials under external magnetic field field Magnetic trap freezes the spin degree of freedom Magnetic trap freezes the spin degree of freedom single single component BEC component BEC Optical trap provides equal confinement for all components Optical trap provides equal confinement for all components m mF

F-

• independent

independent multicomponent multicomponent BEC BEC

Spin Spin-

• 1 BEC

1 BEC

3 hyperfine components: 3 hyperfine components: m mF

F =

= -

• 1, 0 ,1

1, 0 ,1 Coupled Gross Coupled Gross-

• Pitaevskii

Pitaevskii equations (CGPE) equations (CGPE)

β βn

n --

• - spin

spin-

• independent mean

independent mean-

• field interaction

field interaction β βs

s --

• - s

spin pin-

• exchange interaction

exchange interaction

Number density

[ ] [ ] [ ]

+ − + − − − − + − + − + − + + +

+ − + + + + ∇ − = ∂ ∂ + + + + + ∇ − = ∂ ∂ + − + + + + ∇ − = ∂ ∂ ψ ψ β ψ β β ψ ψ ψ ψ ψ β ψ β β ψ ψ ψ ψ β ψ β β ψ ψ

2 2 2 2 2

) ( ) ( 2 1 2 ) ( ) ( 2 1 ) ( ) ( 2 1

s s n s s n s s n

n n n n x V t i n n n x V t i n n n n x V t i r r r

Number density

− +

+ + = = n n n n n

i i 2,

ψ

Spin Spin-

• 1 BEC

1 BEC

Conservation of total mass Conservation of total mass Conservation of energy Conservation of energy Conservation of total magnetization/ total spin Conservation of total magnetization/ total spin

1

2 2 2

= + + =

− +

ψ ψ ψ N

2 2 − +

− = ψ ψ M

[ ]

[ ] (

) ( )

[ ]}

x d n n n n n n n n n x V E

s s n s n n

r r

2 2 2 2 2 2 2 2

) ( 2 2 2 ) ( 2 1 ψ ψ ψ ψ ψ ψ β β β β β β ψ ψ ψ

− + − + − + − + − + − +

+ + − + + + + + + ⎩ ⎨ ⎧ + + ∇ + ∇ + ∇ = ∫

Spin Spin-

• 1 BEC

1 BEC

Time Time-

• independent CGPE

independent CGPE Chemical potentials Chemical potentials Lagrange multipliers, Lagrange multipliers, µ µ and and λ λ, , are introduced to the are introduced to the free energy to satisfy the constraints N and M

t i i i

i

e x t x

µ

ϕ ψ

−

= ) ( ) , ( r r

[ ] [ ] [ ]

+ − + − − − − − + − + − + − + + + +

+ − + + + + ∇ − = + + + + + ∇ − = + − + + + + ∇ − = ϕ ϕ β ϕ β β ϕ ϕ µ ϕ ϕ ϕ β ϕ β β ϕ ϕ µ ϕ ϕ β ϕ β β ϕ ϕ µ

2 2 2 2 2

) ( ) ( 2 1 2 ) ( ) ( 2 1 ) ( ) ( 2 1

s s n s s n s s n

n n n n x V n n n x V n n n n x V r r r

λ µ µ µ µ λ µ µ − = = + =

− +

, ,

free energy to satisfy the constraints N and M

Numerical Method Numerical Method

Continuous normalized gradient flow (CNGF) Continuous normalized gradient flow (CNGF) N and M conserved; energy diminishing N and M conserved; energy diminishing

[ ] [ ] [ ]

+ − Φ Φ + − − − − + Φ − + − + Φ Φ − + + +

− − − − + + + − ∇ = ∂ ∂ − − + + + − ∇ = ∂ ∂ − + − − + + + − ∇ = ∂ ∂ ϕ ϕ β ϕ λ µ β β ϕ ϕ ϕ ϕ ϕ β ϕ µ β β ϕ ϕ ϕ ϕ β ϕ λ µ β β ϕ ϕ

2 2 2 2 2

) ( ) ( ) ( 2 1 2 ) ( ) ( 2 1 ) ( ) ( ) ( 2 1

s s n s s n s s n

n n n n x V t n n n x V t n n n n x V t r r r

(Bao & Wang, (Bao & Wang, SIAM J. SIAM J. Numer

• Numer. Anal

. Anal., 07 ., 07’ ’) )

) ( ), ( t t

Φ Φ Φ Φ

= = λ λ µ µ

CNGF for Spin CNGF for Spin-

• 1 BEC

1 BEC

( )

2 2 2 2 2 2 2 2

, M AM B M BM A − + − = − + − + =

− + − + − +

ϕ ϕ λ ϕ ϕ ϕ ϕ µ

− − + + − − + +

− = + + = µ ϕ µ ϕ µ ϕ µ ϕ µ ϕ

2 2 2 2 2

, B A

( ) ( ) ( ) ( )

dx n n n n n V x dx n n n n V x dx n n n n n V x

s s n s s n s s n − + + + − − − − − + − + − + + − + + + +

+ − + + + + ∂ ∂ = + + + + + ∂ ∂ = + − + + + + ∂ ∂ =

∫ ∫ ∫

ϕ ϕ ϕ β β β ϕ ϕ µ ϕ ϕ ϕ β β β ϕ ϕ µ ϕ ϕ ϕ β β β ϕ ϕ µ

2 2 2 2 2 2 2 2 2

) ( 2 1 1 2 ) ( 2 1 1 ) ( 2 1 1

Crank Crank-

• Nicolson

Nicolson finite difference finite difference Simpler numerical approach with GFDN by introducing Simpler numerical approach with GFDN by introducing third normalization condition third normalization condition

Numerical Method Numerical Method

Time Time-

• splitting scheme to CNGF in

splitting scheme to CNGF in

1. 1.

Gradient flow Gradient flow

2. 2.

Normalization/ Projection Normalization/ Projection

[ ] [ ] [ ]

+ − + − − − − + − + − + − + + +

− − + + + − ∇ = ∂ ∂ − + + + − ∇ = ∂ ∂ − − + + + − ∇ = ∂ ∂ ϕ ϕ β ϕ β β φ ϕ ϕ ϕ ϕ β ϕ β β φ ϕ ϕ ϕ β ϕ β β ϕ ϕ

2 2 2 2 2

) ( ) ( 2 1 2 ) ( ) ( 2 1 ) ( ) ( 2 1

s s n s s n s s n

n n n n x V t n n n x V t n n n n x V t r r r

− Φ Φ − Φ + Φ Φ +

− = ∂ ∂ = ∂ ∂ + = ∂ ∂ ϕ λ µ ϕ ϕ µ ϕ ϕ λ µ ϕ ) ( , , ) ( t t t

1 +

< ≤

n n

t t t

Numerical Method Numerical Method

Normalization step Normalization step Third normalization condition Third normalization condition Normalization constants Normalization constants Backward Backward-

• forward Euler sine

forward Euler sine-

• pseudospectral

pseudospectral method method

* ) ( * * ) (

− Φ Φ Φ + Φ Φ

∆ − − ∆ ∆ + +

= = = ϕ ϕ ϕ ϕ ϕ ϕ

λ µ µ λ µ t t t

e e e

2

σ σ σ =

− + * * *

− +

− − + +

= = = ϕ σ ϕ ϕ σ ϕ ϕ σ ϕ

2 1 2 * 2 * 2 2 1 2 * 2 * 2 2 1 4 * 2 2 * 2 * 2 2 * 2

2 1 , 2 1 ) 1 ( 4 1 ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − − = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − + = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + − + − =

− − + + − +

ϕ ϕ σ σ ϕ ϕ σ σ ϕ ϕ ϕ ϕ σ M M M M M

87 87Rb in 1D Harmonic Potential

Rb in 1D Harmonic Potential

Repulsive and ferromagnetic interaction ( Repulsive and ferromagnetic interaction (β βn

n >0

>0 , , β βs

s < 0)

< 0) Initial condition Initial condition

2 / 4 / 1 2 / 4 / 1 2 / 4 / 1

2 2 2

) 1 ( 5 . , , ) 1 ( 5 .

x x x

e M e e M

− − − − +

− − = = − + = π κ ϕ π κ ϕ π κ ϕ

• 16
• 8

8 16 0.05 0.1 0.15 0.2 0.25 M=0.7

• 16
• 8

8 16 30 60 90 120 150 x V(x)

• 16
• 8

8 16 0.05 0.1 0.15 0.2 0.25

• 16
• 8

8 16 30 60 90 120 150 x M=0.2

• 16
• 8

8 16 0.05 0.1 0.15 0.2 0.25 x

φ(x)

• 16
• 8

8 16 30 60 90 120 150 M=0 |φ+| |φ0| |φ-|

4 2

10 , 2 ) ( = = N x x V

23 23Na in 1D Harmonic Potential

Na in 1D Harmonic Potential

Repulsive and Repulsive and antiferromagnetic antiferromagnetic interaction ( interaction (β βn

n >0

>0 , , β βs

s > 0)

> 0) Initial condition Initial condition

2 / 4 / 1 2 / 4 / 1 2 / 4 / 1

2 2 2

) 1 ( 5 . , , ) 1 ( 5 .

x x x

e M e e M

− − − − +

− − − = = − + = π κ ϕ π κ ϕ π κ ϕ

• 12
• 6

6 12 0.1 0.2 0.3 x

φ(x)

• 12
• 6

6 12 30 60 90 M=0 |φ+| |φ0| |φ-|

• 12
• 6

6 12 0.1 0.2 0.3 x

• 12
• 6

6 12 30 60 90 V(x) M=0.7

• 12
• 6

6 12 0.1 0.2 0.3

• 12
• 6

6 12 30 60 90 x M=0.2

4 2

10 , 2 ) ( = = N x x V

87 87Rb in 3D Optical Lattice

Rb in 3D Optical Lattice

( (β βn

n > 0 ,

> 0 , β βs

s < 0)

< 0)

( )

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + + + + = ) 2 ( sin ) 2 ( sin ) 2 ( sin 100 2 1 ) (

2 2 2 2 2 2

z y x z y x x V π π π

5 . , 104 = = M N

23 23Na in 3D Optical Lattice

Na in 3D Optical Lattice

( (β βn

n > 0 ,

> 0 , β βs

s > 0)

> 0)

( )

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + + + + = ) 2 ( sin ) 2 ( sin ) 2 ( sin 100 2 1 ) (

2 2 2 2 2 2

z y x z y x x V π π π

5 . , 104 = = M N

Relative Populations Relative Populations

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 M 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 M ||φ+||2 ||φ0||2 ||φ-||2

β βs

s < 0 (

< 0 (87

87Rb)

Rb) β βs

s > 0 (

> 0 (23

23Na)

Na)

Relative populations of each component Relative populations of each component Same diagrams are obtained for all kind of trapping Same diagrams are obtained for all kind of trapping potential in the absence of magnetic field potential in the absence of magnetic field

Conclusions Conclusions

Analytical study Analytical study

Leading Leading asymptotics asymptotics of energy and chemical potential

• f energy and chemical potential

for strong repulsive interacting regime for strong repulsive interacting regime Thomas Thomas-

• Fermi approximation and matched

Fermi approximation and matched asymptotic approximation asymptotic approximation

Numerical study Numerical study

Imaginary time method and normalized gradient flow Imaginary time method and normalized gradient flow Different Different discretization discretization schemes schemes Extension to rotating BEC, multi Extension to rotating BEC, multi-

• component BEC, and

component BEC, and spin spin-

• 1 BEC

1 BEC

Future works: Future works:

Finite temperature effect Finite temperature effect