Upload
phamdiep
View
224
Download
0
Embed Size (px)
Citation preview
Modeling, Analysis and Simulation for Degenerate Dipolar Quantum Gas
Weizhu Bao Department of Mathematics
& Center for Computational Science and Engineering National University of Singapore
Email: [email protected] URL: http://www.math.nus.edu.sg/~bao
Collaborators: Y. Cai (Postdoc, NUS), M. Rosenkranz (Postdoc, NUS), N. Ben Abdallah (UPS, France), Z. Lei (Fudan University, China), H. Wang (Yunan Univ. Economics and Finance, China & NUS)
Outline
Motivation---dipolar BEC
Mathematical models Ground state and its theory Dynamics and its efficient computation Dimension reduction Conclusion & future challenges
Degenerate Quantum Gas
Typical degenerate quantum gas – Liquid Helium 3 & 4 – Bose-Einstein condensation (BEC)
• Boson vs Fermion condensation • One component, two-component & spin-1 • Boson-fermion mixture
Typical properties – Low (mK) or ultracold (nK) temperature – Quantum phase transition & closely related to nonlinear wave – Superfluids – flow without friction & quantized vortices
Recent Developments
Quantum transport – Move a BEC in an optical lattice – Atomic circuit, Quantum computing
Interaction of BEC and particles Quantized vortices for superfluidity – Vortex states – Vortex lattice patterns – Interaction between vortices
Fermion condensate, Boson-fermion, atom-molecule, …
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 study A. Griesmaier,et al., PRL, 94 (2005)160401
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 potential Non-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− − + − − ∂
− ∆ =
⇒ =
Dimension Reduction
Gross-Pitaevskii-Poisson equations Strongly anisotropic potential
– Case I: 3D 2D – Case II: 3D 1D
( )2 2 2 2 2 2ext
1( )2 x y zV x x y zγ γ γ= + +
2 2 2 21 2 3 1 2 3& ( , , ) , | | 1T
z x y n n n n n n n nγ γ γ≈ = = + + =
&z x y xγ γ γ γ
2ext
2 3
| |
1( , ) ( ) ( ) | | 3 ( , )2
( , ) | ( , ) | , lim ( , ) 0, , 0
n n
x
i x t V x x tt
x t x t x t x t
ψ β λ ψ λ ϕ ψ
ϕ ψ ϕ→∞
∂ = − ∆ + + − − ∂ ∂ − ∆ = = ∈ >
Dimension Reduction
Existing results – BEC without dipole-dipole interaction:
• Formal asymptotic (Bao, Markowich, Schmeiser & Weishaupl, M3AS, 05’)
• Numerical results (Bao, Ge, Jaksch, Markowich & Weishaeupl, CPC, 07’)
• Rigorous proof (Ben Abdallah, Mehats et al., SIMA, 05; JDE 08’)
• From N-body to mean field theory (Lieb, Seiringer & Yngvason, CMP, 04’; Erdos, Schlein & Yau, Ann. Math., 10’)
– Dipolar BEC (Carles, Markowich & Sparber, Nonlinearity, 08’) – formal result
0λ =
Dimension Reduction (3D 2D)
Assumptions
Decomposition of the linear operator Ansatz
2
ext 2 4
1& (1) & ( ) ( , ) , :2z x y D
z
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 2
1 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 (Bao, Cai, Lei, Rosenkranz, PRA, 10’)
( )
223
2
23
2 2
22 2 2 2
3/2 2 2 2
1 (1 3 )( , , ) [ ( , ) | |2 2
3 ( ) ] ( , , )2
( , , ) | | ,
exp / 21 ( , ) ( ) , 2 2
D
n n
D
D D
ni x y t V x yt
n x y t
x y t U
sU x y U r ds r x y
r s
ε
ε ε
β λψ ψε π
λ ϕ ψ
ϕ ψ
π ε
⊥ ⊥
⊥
⊥
∂ − −= − ∆ + +
∂
− ∂ − ∆
= ∗
−= = = +
+∫
Asymptotic of 2D Kernel
For fixed When
0ε >( )3/2
2
1 ln ln 2 , 02( )
1 ,2
Dr C r
U rr
r
ε
επ ε
π
− + + →≈ → ∞
0ε →
2 1( ) , 02
DU r rrε π
≈ >
Fourier Transform of 2D Kernel
Fourier transform Asymptotic – For fixed
– When
21/2
2
1 , | | 0| |
(| |)2 1 , | |
| |
DUε
ξξ
ξξ
π ε ξ
→≈ → ∞
( )2 22 2
1 2 2 2
exp / 21( , ) (| |)| |
D D sU U ds
sε ε
εξ ξ ξ
π ξ−
= =+∫
0ε >
0ε →2 21(| |) ,
| |DUε ξ ξ
ξ≈ ∈
Ground State Results for quais-2D
Theorem (Existence & uniqueness) (Bao, Ben Abdallah, Cai, SIMA, 12’) – Results
• There exists a ground state if – Case I: – Or case II
• Positive ground state is unique – Case I: – Or case II
• No ground state if
0 & 0λ β λ≥ − ≥
22D 2D| |
( ) 0, & lim ( ) (confinement potential)x
V x x V x→∞
≥ ∀ ∈ = +∞
g Sφ ∈
( )230 & 1 3 | 2 1 | 2
2 bn Cλλ β ε π< + + − > −
00| | with i
g ge θφ φ θ= ∈
0 & 2 bCλ β λ ε π≥ − > −
( )230 & 1 3 | 2 1 | 0
2nλλ β< + + − ≥
2 2 2 2
1 24 2
2 2
( ) ( )40 ( )
( )
: inf ---- Gagliardo-Nirenberg inequalityL Lb f H
L
f fC
f≠ ∈
∇ ⋅=
( )231 3 2
2 bn Cλβ ε π+ − < −
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⊥
⊥
⊥ ⊥ ⊥
⋅ = ∇ + + − +
∂ −∆ − ∇ −∆
∫
Ground State Results for quais-2D
Theorem (Existence & uniqueness) (Bao, Ben Abdallah, Cai, SIMA, 12’) • There exists a ground state if
– Case I: – Or case II – Or case III
• Positive ground state is unique – Case I: – Or case II – Or case III
• No ground state
0 & 0λ β= ≥
22D 2D| |
( ) 0, & lim ( ) (confinement potential)x
V x x V x→∞
≥ ∀ ∈ = +∞
g Sφ ∈
30, 0 & 2 bn Cλ β λ ε π> = − > −
00| | with i
g ge θφ φ θ= ∈
0 & 2 bCλ β ε π= > −
30, 0 &nλ β λ> = ≥
( )2 23 30, 1 / 2 & 1 3 2 bn n Cλ β λ ε π< ≥ − − > −
( )2 23 30, 1 / 2 & 1 3 0n nλ β λ< ≥ − − ≥
23 30 & 0 or 0 & 2 1 or 0 & 2 bn n Cλ λ λ β ε π> ≠ < < = < −
Well-posedness & convergence rate
Well-posedness of the Cauchy problem related to the 2D equations Finite time blow-up may happen!!
Theorem (convergence rate) (Bao, Ben Abdallah, Cai, SIMA, 12’) Assume Then we have
0, , ( ), ( )2
O Oββ λ β β ε λ ε≥ − ≤ ≤ = =
2
2
2( , , , ) ( , , ) ( ) , 0i t
T
L
x y z t e x y t z C t Tεεψ ψ ω ε
−− ≤ ≤ ≤
Dimension Reduction (3D 1D)
Assumptions Decomposition of the linear operator Ansatz
2 2
ext 1 4
1(1) & ( ) ( ) , :2x y z D
x
x yO V x V zγ γ γ εε γ+
= = = + =
22 2
21( , , , ) ( , ) ( , ) & ( , ) exp
2
i t x yx y z t e z t x y x yεε εψ ψ ω ω
επε
− +≈ = −
2 2 2 2
4 2
1 1 12 2 2 2xy xy xy
x y x yLε ε
+ += − ∆ + = − ∆ +
ext 11 1: ( ) ( )2 2 zz D xyL V x V z L= − ∆ + = − ∂ + +
Dimension Reduction (3D 1D)
1D equations (Bao, Cai, Lei, Rosenkranz, PRA, 10’)
– Linear case if
2 22 2
223
1 2
23
/21 2 1 /2
| |
1 2 (1 3 )( , ) [ ( ) | |2 4
3 (1 3 ) ] ( , )8 2
2 ( , ) | | , ( ) ,
zz D
zz
zD D s
z
ni z t V zt
n z t
ez t U U z e dsε
εε ε
β λψ ψπε
λ ϕ ψε π
ϕ ψπ
∞ −
∂ + −= − ∂ + +
∂
−+ ∂
= ∗ = ∫23 1 / 3& 0 & 0n β λ= = ≠
Asymptotic of 1D Kernel
For fixed When
0ε >2
1
21 | | ( ), 0( )
2 ,
D
z O z zU z
zz
επ ε
επ
− + →
≈ → ∞
0ε →
1 1, 0( )
0, 0D z
U zzε
=≈ ≠
Fourier Transform of 1D Kernel
Fourier transform Asymptotic – For fixed
– When
1
2
2 2 ln | | ln , | | 02
( )2 2 , | |
| |
eDUε
ε εγ ξ ξπ
ξξ
ε π ξ
− − − → ≈
→ ∞
( )21
20
exp / 22( ) ,D sU ds
sε
εεξξπ
∞ −=
+∫
0ε >
0ε →1 (| |) ??,DUε ξ ξ≈ ∈
Ground State Results for quais-1D
Theorem (Existence & uniqueness) (Bao, Ben Abdallah, Cai, SIMA, 12’) • There exists a ground state for any • Positive ground state is unique
– Case I: – Or case II
Dynamics results – global well-posedness of the Cauchy problem Convergence rate if
( )2 23 31 3 0 & (1 3 ) 0n nλ β λ− ≥ − − ≥
1D 1D| |( ) 0, & lim ( ) (confinement potential)
zV z z V z
→∞≥ ∀ ∈ = +∞
g Sφ ∈0
0| | with ig ge θφ φ θ= ∈
( ) ( )2 23 31 3 0 & 1 3 / 2 0n nλ β λ− < + − ≥
1, , ,nβ λ ε
2 2( ) & ( )O Oβ ε λ ε= =
2
2
( , , , ) ( , ) ( , ) , 0i t
T
L
x y z t e z t x y C t Tεεψ ψ ω ε
−− ≤ ≤ ≤
Reduction in Multilayered Potential
With multilayered potential in z-direction 2
ext 2 0 0( ) ( , ) sin ( ) DV x V x y V z Vπ ω= +
Reduction in Multilayered Potential
GPEs with infinite many equations; Effective single-mode approximation; Bogoliubov enegies (Rosenkranz, Cai &Bao, PRA, 88 (2013) 013616)
Conclusion & future challenges
Conclusion – Ground state in 3D – existence, uniqueness & nonexistence
– Dynamics in 3D – well-posedness & finite time blowup – Efficient numerical methods via DST – Dimension Reduction --- 3D 2D & 3D1D – Ground states and dynamics in quasi-2D & quasi-1D
Future challenges – Convergence rate for reduction in O(1) regime – In rotating frame & multi-component & spin-1 – Dipolar BEC with random potential – disorder!!