Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Ein - - PowerPoint PPT Presentation

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Ein - - PowerPoint PPT Presentation

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Ein stein Condensates Department of Basic Science, University of Tokyo Michikazu Kobayashi Collaborator : Yuki Kawaguchi (Univ. of Tokyo), Muneto Nitta (Keio Univ.), Masahito

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Ein stein Condensates

Department of Basic Science, University of Tokyo Michikazu Kobayashi

Yuki Kawaguchi (Univ. of Tokyo), Muneto Nitta (Keio Univ.), Masahito Ueda(Univ. of Tokyo) 19th International Laser Physics Workshop, July 8, 2010 Collaborator :

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Conclusion

  • 1. Non-Abelian vortices are realized in the cyclic

phase of spin-2 spinor Bose-Einstein condensates.

  • 2. Non-Abelian character becomes remarkable in

collision dynamics of two vortices. I. We numerically show.

  • II. We algebraically confirm.
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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Vortices in Bose-Einstein Condensates

v = (n/m)q : superfluid velocity  v  dl = nh/m : circulation

Y  einq

Around the vortex core

  • 1. Phase changes by integer multiple of 2p.
  • 2. Circulation takes integer multiple of h/m

Topological charge of vortex is characterized by additive group of integers  Abelian vortex.

Integer vortex (Scalar BEC or 4He)

Non-Abelian vortices are realized in the cyclic phase of spin-2 BEC

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Spin-2 Spinor BEC

5 - component BEC : Y = (Y2, Y1, Y0, Y-1, Y-2)T

F = 2 87Rb BEC and its spin dynamics is observed

  • H. Schmaljohann et al. PRL 92, 040402 (2004)
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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Ground State of Spin-2 Spinor BEC

Cyclic Ferromagnetic Nematic

c1 c2

87Rb

  • C. V. Cionabu et al. PRA 61, 033607 (2000)
  • A. Widera et al. NJP 8, 152 (2006)
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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Spherical Harmonics Expression of Cy clic Phase

cyclic phase

Y2,2 Y2,1 Y2,0 Y2,-1 Y2,-2 + + + +

  • 2p/3

2p/3

  • 1. Cyclic state can be expressed

as a headless triad

  • 2. Phase difference between

each lobe is 2p/3

  • ¼

¼

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Invariant Spin or Spin – Gauge Transfor mation

  • 2p/3

2p/3

  • ¼

¼

p – spin rotation 2p/3 – spin & gauge transformation

Including identity, there are 12 transformations keeping headless triad invariant.

  • H. Mäkelä et al. J. Phys. A 36, 8555 (2003), G. W. Semenoff et al. PRL 98, 100401 (2007)
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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Invariant Spin or Spin – Gauge Transfor mation

12 transformations form non- Abelian tetrahedral group

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Vortices

Invariant transformations define vortices

¼ 2p/3

1/2 spin vortex 1/3 vortex

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Topological Charge of Vortices

Scalar BEC Topological charge : Additive group of integer n  Abelian vortices Cyclic phase in spin-2 spinor BEC Topological charge : Tetrahedral group  Non-Abelian vortices

Y  einq

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Initial state : two straight vortices, linked vortex rings

Collision Dynamics of Non-Abelian Vorti ces

Non-Abelian property of vortices becomes remarkable in their collision dynamics

→ Numerical simulation of Gross-Pitaevskii equation

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Hamiltonian of Spin-2 Spinor BEC

density spin density singlet – pair amplitude

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Gross-Pitaevskii Equation

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Collision Dynamics of Non-Abelian Vorti ces

Same charge Commutative charges Non-commutative charges Reconnection Passing Rung vortex

Collision of vortices with non-commutative charge forms a new “rung” vortex connecting two vortices

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Collision Dynamics of Non-Abelian Vorti ces

Same Commutative Non-commutative Large ring Unraveling of link Rung vortex

Linked vortices with non-commutative charges cannot unravel because of the formation of the rung vortex.

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Algebra

Topological charge of vortex can be fixed by a closed path encircling the vortex B A A ABA-1

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Collision of Vortex

B A A ABA-1 B A ABA-1 BA-1 A

Rung BA-1 is formed through the collision.

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Collision of Vortex

B A A ABA-1 A A A AA-1 = 1 A A = B

Rung disappears for the same charge resulting reconnection.

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Collision of Vortex

B A A ABA-1

Passing dynamics is also possible for commutative case

AB = BA B A B A

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Linked Vortex Rings

B A B A AB -1A-1B B A AB = BA

Linked vortex rings with non-commutative charges never unravel.

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Application of Non-Abelian Vortices : N

  • n-Abelian Turbulence

Abelian turbulence ↓ Cascade of vortices through reconnections Non-Abelian turbulence ↓ Large-scale networking structure of vortices through formation of rungs

New type of turbulence

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Conclusion

  • 1. Non-Abelian vortices are realized in the cyclic

phase of spin-2 spinor Bose-Einstein condensates.

  • 2. Non-Abelian character becomes remarkable in

collision dynamics of two vortices. I. Rung vortex is formed after the collision.

  • II. Linked vortex rings never unravel
  • M. Kobayashi, Y. Kawaguchi, M. Nitta, and M. Ueda. PRL 103, 115301(2009)
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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Cyclic State vs. Singlet-trio Condensed State

Cyclic state (U(1)  SO(3) is broken)

For c1>0, c2>0

Singlet-trio condensed state (only U(1) is broken)

Transition occurs under ~1µG

  • M. Koashi, and M. Ueda. PRL 84, 1066 (2000)
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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Nematic State vs. Singlet-pair Condensed State

For c1>0, c2<0

Singlet-pair condensed state (only U(1) is broken)

Transition occurs under ~1µG

  • M. Koashi, and M. Ueda. PRL 84, 1066 (2000)

Nematic state (U(1)  SO(3) is broken)

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Hamiltonian of Spin-2 Spinor BEC

Bose system with spin degrees of freedom Low energy contact interaction (l = 0)

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Hamiltonian of Spin-2 Spinor BEC

Mean-field approximation

density spin density singlet – pair amplitude

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Breaking of U(1)GSO(3)S

Gauge transformation： U(1)G Spin rotation：SO(3)S Fixed from Hamiltonian

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Ground State Phase Diagram

Biaxial Nematic : D4 Cyclic : T Ferromagnetic : SO(2) / 2 Uniaxial Nematic : D

  • ¼

¼

  • S. Uchino, M. Kobayashi, and M. Ueda. PRA 81, 063632 (2010)
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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Algebra

Path d defines vortex B as ABA-1 (same conjugacy class)

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Y – Shaped Structure

B A AB

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Collision of Vortex

Only Abelian B A A ABA-1

B

A B B-1AB B A AB A ABA-1 B A A ABA-1 BA-1

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Same Charge

A A A A A A A A A A A2 A A A A A A 1

○

×

△

reconnection Energetically unfavorable

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Commutative Charge

B A A ABA-1 B A B B-1AB B A AB A ABA-1 B A A ABA-1 BA-1

× ×

○

Passing Energetically unfavorable

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Non-Commutative Charge

Topologically forbidden B A A ABA-1 B A B B-1AB B A AB A ABA-1 B A A ABA-1 BA-1

×

○ ○

rung

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Linked Rings

B A A ABA-1 ABA-1 B ABA-1 A AB -1ABA-1 AB -1A-1B Non-Commutative B B A A Commutative AB -1ABA-1

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Characteristic Form of Wave Function

1/2 – spin vortex 1/3 vortex Vortex Mass circulation (h/m) Spin circulation (h/m) Core 1/2 – spin 1/2 Nematic 1/3 1/3 1/3 Ferromagnetic

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Collision Dynamics of Non-Abelian Vortices in Spin-2 Spinor Bose-Einstein Condensates

Homotopy Group of Cyclic State

Order-parameter manifold First homotopy group (vortex)