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)
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Modeling, Analysis and Simulation for Degenerate Dipolar Quantum Gas
Weizhu Bao Department of Mathematics
& Center for Computational Science and Engineering National University of Singapore
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)
Experimental setup – Molecules meet to form dipoles – Cool down dipoles to ultracold – Hold in a magnetic trap – Dipolar condensation – Degenerate dipolar quantum gas
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− ⋅ = ∇ + + + ∂ ∇ ∫
• 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???
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
−− ≤ ≤ ≤
2
1: x
z
γγγ 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!!