weizhu bao- quantized vortex stability and dynamics in superfluidity and superconductivity
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8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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Quantized Vortex Stability and Dynamics
in Superfluidity and Superconductivity
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:
– Qiang Du: Penn State U.; Yanzhi Zhang: FSU
– Alexander Klein, Dieter Jaksch: Oxford
8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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Outline
MotivationMathematical models
Central vortex states andstabilityVortex interactionand reduced dynamic laws
Numerical methodsNumerical results for vortex interaction
Conclusions
8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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Motivation
• Quantized Vortex: Particle-like (topological) defect – Zero of the complex scalar field,
– localized phase singularities with integer topological charge:
– Key of superfluidity: ability to support dissipationless flow
02)(arg,)(,0)( 0 ≠==== ∫ ∫ Γ Γ
nd d e x x i π φ ψ ρ ψ ψ φ
8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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Motivation
• Existing in – Superconductors:
• Ginzburg-Landau equations (GLE)
– Liquid helium:• Two-fluid model
• Gross-Pitaevskii equation (GPE)
– Bose-Einstein condensation (BEC):• Nonlinear Schroedinger equation (NLSE) or GPE
– Nonlinear optics & propagation of laser beams
• Nonlinear Schroedinger equation (NLSE)• Nonlinear wave equation (NLWE)
8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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Mathematical models
• Ginzburg-Landau equation (GLE):Superconductivity, nonlinear heat flow, etc.
• Gross-Pitaevskii equation (GPE): nonlinear optics, BEC, superfluidity
• Nonlinear wave equation (NLWE): wave motion
– Here
( )2 2
2
1( ) , , 0,t V x x R t ψ ψ ψ ψ
ε = Δ + − ∈ >
: complex-valued wave function,
0: constant,( ) : real-valued external potentialV x
ψ
ε >
( )2 2
21 ( ) , , 0,t i V x x R t ψ ψ ψ ψ ε − = Δ + − ∈ >
( )2 2
2
1( ) , , 0,
tt V x x R t ψ ψ ψ ψ
ε = Δ + − ∈ >
8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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Mathematical models
• `FreeEnergy’ or Lyapunov functional:
• Dispersive system(GPE): – Energy conservation:
– Density conservation:
– Admits particle like solutions: solitons, kinks & vortices
• Dissipative system(GLE):
– Energy diminishing, etc.
( )2
22 2
2
1( ) ( ) ,
2 E V x dxψ ψ ψ
ε ℜ
⎡ ⎤= ∇ + −⎢ ⎥⎣ ⎦
∫
0),()( 0 ≥≡ t E E ψ ψ
( ) 0,0|)(||),(|2
2
0
2 ≥≡−∫ ℜ
t xd xt x
ψ ψ
( )
*
E
i t
ψ δ ψ
δψ
∂− = −
∂
( )
*
E
t
ψ δ ψ
δψ
∂= −
∂
8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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Two kinds of vortices
`Bright-tail’ vortex: GLE, GPE & NLWE (̀order parameter’)
– Time independent case (Neu, 90’):
– Vortex solutions:
– Asymptotic results
( ) 1, when , ( , ) 1, , ( . . ),imV x x x t x e g eθ ψ ψ → → ∞ ⇒ → → ∞ →
1)(,1 ≡= xV
ε
θ φ ψ in
nn er f x x )()()( ==
.1)(,0)0(
,0,0)())(1()()(1
)( 2
2
2
=∞=
∞<<=−+−′+′′
nn
nnnnn
f f
r r f r f r f
r
nr f
r
r f
⎩⎨
⎧
∞→+−
→+
≈
+
.), / 1(2 / 1
,0),(
)( 422
2||||
r r Or n
r r Oar
r f
nn
n
8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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`Bright-tail’ vortex
8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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Two kinds of vortices
`Dark-tail’ vortex: BEC (wave function)
– Center vortex states:
– Asymptotic results
( , ) ( ) ( )n ni t i t in
n n x t e x e f r e
μ μ θ ψ φ − −= =
2 23
2
2
0
( )( ) ( ) ( ), 0 ,
2
(0) 0, ( ) 0, 2 ( ) 1
nn n n n
n n n
d df r n r f r r f r f r r
dr dr r
f f f r r dr
μ β
π
∞
⎛ ⎞⎛ ⎞= − + + + < < ∞⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
= ∞ = =∫
2
| | | | 2
| |
( ), 0,( )
, .
n n
n n r
a r O r r f r
b r e r
+
−
⎧ + →⎪≈
⎨ → ∞⎪⎩
222 2 2| |
| | , , 0, | | 12
t
xi x R t dxψ ψ ψ β ψ ψ ψ ψ = −Δ + + ∈ > = =∫
8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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`Dark-tail’ vortex
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`Dark-tail’ vortex line in 3D
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Stability of vortices
• Data chosen
• Results (Bao,Du & Zhang, Eur. J. Appl. Math., 07’)
– For GLE or NLWE with initial data perturbed
• n=1: velocity density
• N=3: velocity density
– For NLSE with external potential perturbed• n=1: velocity density
• N=3: velocity density
• Conclusion next
)noise()()()),,((1),(,1 0 +=+≡= x xt xW t xV h
n
φ ψ ε
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Conclusion of Stability
• For GLEor NLWE: – under perturbation either in initial data or in external potential
• |n|=1: dynamically stable,
• |n|>1: dynamically unstable. Splitting pattern depends on perturbation,when t >>1, it becomes n well-separated vortices with index 1 or –1.
• For NLSEor GPE:
– under perturbation in initial data• dynamically stable: Angular momentum expectation is conserved!!
– under perturbation in external potential
• |n|=1: dynamically stable, back• |n|>1: dynamically unstable. Vortex centers can NOT move out of core size.
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Vortex interaction
• For ̀ bright-tail’ vortex:
• For ̀ dark-tail’ vortex
– Well-separate
– Overlapped
0 0 0
0
1 1
( ) ( ) ( , ) j j
N N
m j m j j
j j
x x x x x y yψ φ φ = =
= − = − −∏ ∏
0
1
( , ) ( , ) / || || j
N
n j j
j
x y x x y yψ φ =
= − − •∑
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Reduced dynamic laws
• For `bright-tail’ vortex – For GLE
– For GPE
– For NLWE
21,
0
( ) ( ) ( )( ) : 2 , 0
( ) ( )
(0) , 1 .
N j j l
j j l
l l j j l
j j
d x t x t x t v t m m t
dt x t x t
x x j N
κ κ = ≠
−= = ≥
−
= ≤ ≤
∑
21,
0
( ) ( ( ) ( ))( ) : 2 0,
( ) ( )
0 1(0) 1 ;
1 0
N j j l
j l
l l j j l
j j
d x t J x t x t v t m t
dt x t x t
x x j N J
= ≠
−= = ≥
−
⎛ ⎞= ≤ ≤ = ⎜ ⎟
−⎝ ⎠
∑
2
221,
0 ' 0
( ) ( ) ( )2 , 0
( ) ( )
(0) , (0) , 1 .
N j j l
j l
l l j j l
j j j j
d x t x t x t m m t
dt x t x t
x x x v j N
κ = ≠
−= ≥
−
= = ≤ ≤
∑
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Existing mathematical results
• For GLE (or NLHE): – Pairs of vortices with like (opposite) index undergo a replusive (attractive)
interaction: Neu 90, Pismen & Rubinstein 91, W. E, 94, Bethual, etc.
– Energy concentrated at vortices in 2D & filaments in 3D: Lin 95--
– Vortices are attracted by impurities: Chapman & Richardson 97, Jian 01
• For NLSE: – Vortices behave like point vortices in ideal fluid: Neu 90
– Obeys classical Kirchhoff law for fluid point vortices: Lin & Xin 98
– Equations for vortex dynamics & radiation: Ovchinnikov& Sigal 98--; Jerrard,Bethual, etc.
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Conservation laws
• Mass center Lemma The mass center of the N vortices for GLE is conserved
Lemma The mass center of the N vortices for NLWE is conserved
if initial velocity is zero and it moves linearly if initial velocity isnonzero
1
1
( ) : ( )
N
j j x t x t N =
=
∑
0
1 1 1
1 1 1( ) : ( ) (0) : (0)
N N N
j j j j j j
x t x t x x x N N N = = =
= ≡ = =
∑ ∑ ∑
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Conservation laws
• Signed mass center
Lemma The signed mass center of the N vortices for NLSE is conserved
1
1( ) : ( )
N
j j
j
x t m x t N =
= ∑
0
1 1 1
1 1 1( ) : ( ) (0) : (0)
N N N
j j j j j j
j j j
x t m x t x m x m x N N N = = =
= ≡ = =∑ ∑ ∑
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Analytical Solutions
• like vortices on acircle (Bao, Du & Zhang, SIAP, 07’)
• Analytical solutions for GLE figure
• Analytical solutions for NLSEfigure next
( 2) N ≥0
0
2 2cos , sin , 1, 1 j j
j j x a m m j N
N N
π π ⎛ ⎞⎛ ⎞ ⎛ ⎞= = = ± ≤ ≤⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠⎝ ⎠
2 2( 1) 2 2( ) cos , sin j
N j j x t a t
N N
π π
κ
− ⎛ ⎞⎛ ⎞ ⎛ ⎞= + ⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠⎝ ⎠
2 2
2 1 2 1( ) cos , sin j
j N j N x t a t t
N a N a
π π ⎛ − − ⎞⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠⎝ ⎠
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N=2 N=3
back
N=4
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N=2 N=3back
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Analytical Solutions
• like vortices on acircle and its center (Bao,Du&Zhang, SIAP, 07’)
• Analytical solutions for GLE figure
• Analytical solutions for NLSE figure next
( 3) N ≥
0 0 2 20; cos , sin , 1 1
1 1 N j
j j x x a j N
N N
π π ⎛ ⎞⎛ ⎞ ⎛ ⎞= = ≤ ≤ −⎜ ⎟ ⎜ ⎟⎜ ⎟− −⎝ ⎠ ⎝ ⎠⎝ ⎠
2 2
2 2 2 2( ) 0; ( ) cos , sin
1 1 N j
j N j N x t x t a t t
N a N a
π π ⎛ − − ⎞⎛ ⎞ ⎛ ⎞= = + +⎜ ⎟ ⎜ ⎟⎜ ⎟− −⎝ ⎠ ⎝ ⎠
⎝ ⎠
2 2 2 2( ) 0; ( ) cos , sin , 1 1
1 1 N j
N j j x t x t a t j N
N N
π π
κ
⎛ ⎞⎛ ⎞ ⎛ ⎞= = + ≤ ≤ −⎜ ⎟ ⎜ ⎟⎜ ⎟− −⎝ ⎠ ⎝ ⎠⎝ ⎠
8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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N=3 N=4back
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N=3 N=4back
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Analytical Solutions
• Twooppositevortices (Bao, Du & Zhang, SIAP, 07’)
• Analytical solutions for GLEfigure
• Analytical solutions for NLSE figure next
( ) ( )( )0 0
1 2 0 0 1 2cos , sin , 1 x x a m mθ θ = − = = − =
( ) ( )( )2
1 2 0 0
2( ) ( ) cos , sin x t x t a t θ θ
κ = − = −
( ) ( )( )0
0 0( ) sin , cos , 1, 2 j j
t x t x j
aθ θ = + − =
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Analytical Solutions
• opposite vortices on acircle and its center
• Analytical solutions for GLEfigure
• Analytical solutions for NLSE figure next
( 3) N ≥0 0 2 20 ( ); cos , sin ( ), 1 1
1 1 N j
j j x x a j N
N N
π π ⎛ ⎞⎛ ⎞ ⎛ ⎞= − = + ≤ ≤ −⎜ ⎟ ⎜ ⎟⎜ ⎟− −⎝ ⎠ ⎝ ⎠⎝ ⎠
2 2
2 4 2 4( ) 0; ( ) cos , sin
1 1 N j
j N j N x t x t a t t
N a N a
π π ⎛ − − ⎞⎛ ⎞ ⎛ ⎞= = + +⎜ ⎟ ⎜ ⎟⎜ ⎟− −⎝ ⎠ ⎝ ⎠⎝ ⎠
2 2( 4) 2 2( ) 0; ( ) cos , sin , 1 1
1 1 N j
N j j x t x t a t j N
N N
π π
κ
− ⎛ ⎞⎛ ⎞ ⎛ ⎞= = + ≤ ≤ −⎜ ⎟ ⎜ ⎟⎜ ⎟− −⎝ ⎠ ⎝ ⎠⎝ ⎠
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N=3 N=4
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N=5
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N=3 N=4
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N=5
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Numerical difficulties
• Numerical difficulties: – Highly oscillatory nature in the transverse direction: resolution
– Quadratic decay rate in radial direction: large domain
– For NLSE, more difficulties:
• Time reversible
• Time transverse invariant
• Dispersive equation
• Keep conservation laws in discretized level
• Radiation
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Numerical methods
• For ̀ bright-tail’ vortex(Bao,Du & Zhang, Eur. J. Appl. Math., 07’)
– Apply a time-splitting technique: decouple nonlinearity
– Adapt polar coordinate: resolve solution in transverse better
– Use Fourier pesudo-spectral discretization in transverse – Use 2nd or 4th order FEM or FDM in radial direction
For ̀ dark-tail’ vortex (Bao & Zhang, M3AS, 05’)
– Apply a time-splitting technique: decouple nonlinearity
– Use Fourier pseudo-spectral discretization
– Use generalized Laguerre-Hermite pseudo-spectral method
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Vortex dynamics & interaction
• Data chosen (Bao, Du & Zhang, SIAM, J. Appl. Math., 07’)
• Two vortices with like winding numbers
– For GLE: velocity density
– For GPE: velocity density
– Trajectory
• Conclusion next
∑∏==
=−=≡= N
j
j
N
j
jn nm x x xt xV j
11
0 ,)()(,1),(,1
φ ψ ε
)0,(),0,(,1,1,2 2121 a xa xnn N −=====
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GLE GPE NLWE
8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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Conclusion of interaction
• For GLE(Bao, Du & Zhang, SIAP, 07’)
– Undergo a repulsive interaction
– Core size of the two vortices will well separated
– Energy decreasing:
• For NLSE (Bao, Du & Zhang, SIAP, 07’)
– Behave like point vortices in ideal fluid
– The two vortex centers move almost along a circle
– Energy conservation & radiation back
• For NLWE (Bao & Zhang, 07’)
– Similar as GLE but with different speed
∞→→ t x E t x E )),((2)),(( 1
ψ ψ
8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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Vortex dynamics & interaction
• Two vortices with opposite winding numbers
– For GLE: velocity density – For NLSE:
• Case 1: velocity density
• Case 2: velocity density
– Trajectory
• Conclusion next
)0,(),0,(,1,1,2 2121 a xa xnn N −==−===
8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
http://slidepdf.com/reader/full/weizhu-bao-quantized-vortex-stability-and-dynamics-in-superfluidity-and-superconductivity 52/59
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8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
http://slidepdf.com/reader/full/weizhu-bao-quantized-vortex-stability-and-dynamics-in-superfluidity-and-superconductivity 53/59
back
8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
http://slidepdf.com/reader/full/weizhu-bao-quantized-vortex-stability-and-dynamics-in-superfluidity-and-superconductivity 54/59
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8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
http://slidepdf.com/reader/full/weizhu-bao-quantized-vortex-stability-and-dynamics-in-superfluidity-and-superconductivity 55/59
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NLSEGLENLSE NLWE
8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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Conclusion of interaction
• For GLE(Bao, Du & Zhang, SIAP, 07’)
– Undergo an attractive interaction & merge at finite time
– Energy decreasing:
• For GPE (Bao, Du & Zhang, SIAP, 07’)
– depends on initial distance of the two centers
• Small: merge at finite time & generate shock wave
• Large: move almost paralleling & don’t merge, solitary wave – Energy conservation & radiation
– Sound wave generation back
For NLWE (Bao & Zhang, 07’)
– Undergo an attractive interaction & merge at finite time
∞→→ t t x E ,0)),((
ψ
8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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Reduced dynamic laws
For ̀ dark-tail’ vortex in BEC(Klein, Jaksch, Zhang & Bao, PRA, 07’)
– For linear with harmonic potential
• N=1:
• Vortex pair:• Vortex dipole:
8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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Reduced dynamic laws
– For weak nonlinear case
• Vortex pair
• Vortex dipole
• Vortex tripole
– For strong interaction: ???
8/3/2019 Weizhu Bao- Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity
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Conclusions
• Analytical results for reduced dynamical laws – Mass center is conserved in GLE
– Signed mass center is conserved in GPE
– Solve analytically for a few types initial data
• Numerical results – Study numerically vortex dynamics in GLE, GPE, NLWE & NLSE
• Stability of a vortex with different winding number
• Interaction of vortex pair, vortex dipole, vortex tripole, ….