Quantized Vortex Stability and Dynamics in Superfluidity and Superconductivity

Weizhu Bao

Department of Mathematics National University of SingaporeEmail: matbaowz@nus.edu.sg

URL: http://blog.nus.edu.sg/matbwz/Collaborators:

– Qiang Du (Columbia); Yanzhi Zhang (MUST); Alexander Klein, Dieter Jaksch (Oxford)

– Qinglin Tang (Sichuan Univ); Zhihuo Xu & Shaoyun Shi (Jida); Teng Zhang (NUS)

Cloud vortex – Saturn hexagon at the north pole of the planet Saturnhttp://en.wikipedia.org/wiki/Saturn%27s_hexagon

Vortex: From macroscale to microscale

Vortex galaxy – vortex in cosmos

Vortex: From macroscale to microscale

Tornado –Vortex in air

Vortex: From macroscale to microscale

Vortex in water – generated by a boat

Vortex: From macroscale to microscale

Vortex in water – generated by an airplane

Vortex: From macroscale to microscale

Magnetic vortex –vortex in plasma

Vortex: From macroscale to microscale

Vortex in plant

Vortex: From macroscale to microscale

Quantized Vortex in liquid Helium 3

Vortex: From macroscale to microscale

Quantized Vortex in a type-II superconductor

Vortex: From macroscale to microscale

Quantized Vortex in Bose-Einstein condensation (BEC)

Vortex: From macroscale to microscale

Outline

MotivationMathematical models Central vortex states and stabilityVortex interaction and reduced dynamic laws Numerical methods Numerical results – whole space, bounded domain and in BEC

Conclusions

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

nddexx i πφψρψψ φ

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)

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; cosmology

– Here

( )2 22

1 ( ) , , 0,t V x x tψ ψ ψ ψε

= ∆ + − ∈ >

: complex-valued wave function or order parameter,0: dimensionless constant,

( ) : real-valued external potentialV x

ψε >

( )2 22

1 ( ) , , 0,ti V x x tψ ψ ψ ψε

− = ∆ + − ∈ >

( )2 22

1 ( ) , , 0,tt V x x tψ ψ ψ ψε

= ∆ + − ∈ >

Mathematical models

• `Free Energy’ 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 22

1( ) ( ) ,2

E V x dxψ ψ ψε

= ∇ + − ∫

0),()( 0 ≥≡ tEE ψψ

( )2

2 20| ( , ) | | ( ) | 0, 0x t x dx tψ ψ− ≡ ≥∫

( )*

Eitψ δ ψ

δψ∂

− = −∂

( )*

Etψ δ ψ

δψ∂

= −∂

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 e θψ ψ→ →∞ ⇒ → →∞ →

1)(,1 ≡= xV εθφψ in

nn erfxx )()()( ==

.1)(,0)0(

,0,0)())(1()()(1)( 22

2

=∞=

∞<<=−+−′+′′

nn

nnnnn

ff

rrfrfrfrnrf

rrf

∞→+−→+

≈+

.),/1(2/1,0),(

)( 422

2||||

rrOrnrrOar

rfnn

n

`Bright-tail’ vortex

Two kinds of vortices

`Dark-tail’ vortex: BEC (wave function)

– Center vortex states:

– Asymptotic results

( , ) ( ) ( )n ni t i t inn nx t e x e f r eµ µ θψ φ− −= =

2 23

2

2

0

( )( ) ( ) ( ), 0 ,2

(0) 0, ( ) 0, 2 ( ) 1

nn n n n

n n n

d df r n rf r r f r f r rdr dr r

f f f r r dr

µ β

π∞

= − + + + < < ∞

= ∞ = =∫

2

| | | | 2

| |

( ), 0,( )

, .

n n

n n r

a r O r rf r

b r e r

+

−

+ →≈ →∞

222 2 2| | | | , , 0, | | 1

2txi x R t dxψ ψ ψ β ψ ψ ψ ψ= −∆ + + ∈ > = =∫

`Dark-tail’ vortex

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

• Results

)noise ()()()),,((1),(,1 0 +=+≡= xxtxWtxV hn φψε

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Summary of Stability

• For GLE or 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 NLSE or GPE:– under perturbation in initial data

• dynamically stable: Angular momentum expectation is conserved!!

– under perturbation in external potential• |n|=1: dynamically stable, • |n|>1: dynamically unstable. Vortex centers can NOT move out of core size.

Vortex interaction

• For `bright-tail’ vortex:

• For `dark-tail’ vortex– Well-separate

– Overlapped

0 0 00

1 1

( ) ( ) ( , )j j

N N

m j m j jj j

x x x x x y yψ φ φ= =

= − = − −∏ ∏

01

( , ) ( , ) / || ||j

N

n j jj

x y x x y yψ φ=

= − − •∑

Reduced dynamic laws

• For `bright-tail’ vortex– Take ansatz

– Plug into GLE or GPE or NLWE– Using matched asymptotic techniques– Prove rigorously

1

1

( , ) ( ( )) high order terms

( ( ), ( ))+high order terms

j

j

N

m jj

N

m j jj

x t x x t

x x t y y t

ψ φ

φ

=

=

= − +

= − −

∏

∏

Reduced dynamic laws

• For `bright-tail’ vortex– For GLE

– For GPE

– For NLWE

21,

0

( ) ( ) ( )( ) : 2 , 0

( ) ( )

(0) , 1 .

Nj j l

j j ll l j j l

j j

d x t x t x tv 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

Nj j l

j ll l j j l

j j

d x t J x t x tv t m t

dt x t x t

x x j N J

= ≠

−= = ≥

−

= ≤ ≤ = −

∑

2

221,

0 ' 0

( ) ( ) ( )2 , 0

( ) ( )

(0) , (0) , 1 .

Nj j l

j ll l j j l

j j j j

d x t x t x tm m t

dt x t x t

x x x v j N

κ= ≠

−= ≥

−

= = ≤ ≤

∑

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

– Dynamical properties: Xu, Bao & Shi – DCDS-B,18’

• 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.

– Global existence???

Conservation laws

• Mass centerLemma 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 is nonzero

1

1( ) : ( )N

jj

x t x tN =

= ∑

0

1 1 1

1 1 1( ) : ( ) (0) : (0)N N N

j j jj j j

x t x t x x xN N N= = =

= ≡ = =∑ ∑ ∑

Conservation laws

• Signed mass centerLemma The signed mass center of the N vortices for NLSE is conserved

1

1( ) : ( )N

j jj

x t m x tN =

= ∑

0

1 1 1

1 1 1( ) : ( ) (0) : (0)N N N

j j j j j jj j j

x t m x t x m x m xN N N= = =

= ≡ = =∑ ∑ ∑

Analytical Solutions

• like vortices on a circle (Bao, Du & Zhang, SIAP, 07’)

• Analytical solutions for GLE figure

• Analytical solutions for NLSE figure next

( 2)N ≥0

02 2cos , sin , 1, 1j j

j jx a m m j NN Nπ π = = = ± ≤ ≤

2 2( 1) 2 2( ) cos , sinjN j jx t a t

N Nπ π

κ− = +

2 2

2 1 2 1( ) cos , sinjj N j Nx t a t tN a N aπ π − − = + +

N=2 N=3

backN=4

N=2 N=3back

Analytical Solutions

• like vortices on a circle 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 11 1N j

j jx x a j NN Nπ π = = ≤ ≤ − − −

2 2

2 2 2 2( ) 0; ( ) cos , sin1 1N j

j N j Nx t x t a t tN a N aπ π − − = = + + − −

2 2 2 2( ) 0; ( ) cos , sin , 1 11 1N j

N j jx t x t a t j NN Nπ π

κ = = + ≤ ≤ − − −

N=3 N=4back

N=3 N=4back

Analytical Solutions

• Two opposite vortices (Bao, Du & Zhang, SIAP, 07’)

• Analytical solutions for GLE figure

• Analytical solutions for NLSE figure next

( ) ( )( )0 01 2 0 0 1 2cos , sin , 1x x a m mθ θ= − = = − =

( ) ( )( )21 2 0 0

2( ) ( ) cos , sinx t x t a t θ θκ

= − = −

( ) ( )( )00 0( ) sin , cos , 1, 2j j

tx t x ja

θ θ= + − =

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Analytical Solutions

• opposite vortices on a circle and its center

• Analytical solutions for GLE figure

• Analytical solutions for NLSE figure next

( 3)N ≥0 0 2 20 ( ); cos , sin ( ), 1 1

1 1N jj jx x a j N

N Nπ π = − = + ≤ ≤ − − −

2 2

2 4 2 4( ) 0; ( ) cos , sin1 1N j

j N j Nx t x t a t tN a N aπ π − − = = + + − −

2 2( 4) 2 2( ) 0; ( ) cos , sin , 1 11 1N j

N j jx t x t a t j NN Nπ π

κ− = = + ≤ ≤ − − −

N=3 N=4

backN=5

N=3 N=4

backN=5

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

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

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

• Summary

∑∏==

=−=≡=N

jj

N

jjn nmxxxtxV

j11

0 ,)()(,1),(,1 φψε

)0,(),0,(,1,1,2 2121 axaxnnN −=====

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backGLE

GPE NLWE

Summary 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

For NLWE (Bao & Zhang, Physica D 07’)– Similar as GLE but with different speed

∞→→ txEtxE )),((2)),(( 1 ψψ

Vortex dynamics & interaction

• Two vortices with opposite winding numbers

– For GLE: velocity density– For NLSE:

• Case 1: velocity density• Case 2: velocity density

– Trajectory• Summary

)0,(),0,(,1,1,2 2121 axaxnnN −==−===

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NLSEGLE NLSE NLWE

Summary 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

For NLWE (Bao & Zhang, Physica D 07’)– Undergo an attractive interaction & merge at finite time

∞→→ ttxE ,0)),(( ψ

Vortex-pair on bounded domain – Bao & Tang, SIAM MMS, 2014

Vortex-dipole on bounded domain– Bao & Tang, SIAM MMS, 2014

`dark-tail’ vortex-pair in BEC – well-separate

`dark-tail’ vortex-dipole in BEC

Ground states of rapid rotation

BEC@MIT

`dark-tail’ vortex lattice in BEC

Impurities Bound by Vortex lattice

Conclusions • Analytical results for reduced dynamical laws

– Mass/signed mass center is conserved in GLE/NLSE– 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, ….• Vortex interaction on bounded domain with different BCs• On bounded domain and applications in BEC

Future works: – Other models, e.g. Klein-Gordon equation, GPE with nonlocal interaction– In 3D and Quantum turbulence– Compare with experimental results