Non-Abelian Vortices in Spinor Bos e-Einstein Condensates - - PowerPoint PPT Presentation

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Non-Abelian Vortices in Spinor Bos e-Einstein Condensates - - PowerPoint PPT Presentation

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Non-Abelian Vortices in Spinor Bos e-Einstein Condensates Department of Basic Science, University of Tokyo Michikazu Kobayashi Collaborator : Yuki Kawaguchi (Univ. of Tokyo), Muneto Nitta (Keio Univ.), Masahito Ueda(Univ. of Tokyo) QFS2010:

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Non-Abelian Vortices in Spinor Bos e-Einstein Condensates

Department of Basic Science, University of Tokyo Michikazu Kobayashi

Yuki Kawaguchi (Univ. of Tokyo), Muneto Nitta (Keio Univ.), Masahito Ueda(Univ. of Tokyo) QFS2010: International Symposium on Quantum Fluids and Solids, August 6, 2010 Collaborator :

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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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 winds 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)

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Non-Abelian Vortices in Spinor Bose-Einstein Condensates

Vortices in Bose-Einstein Condensates

Half-quantized vortex

Polar phase in spin-1 spinor BEC

reverse d -vector

3He-A 3He-B in low T & P

Double core of half- quantized vortices

Topological charge is characterized by 2nd cyclic group  Non-Abelian vortices are realized in the cyclic phase of spin-2 BEC

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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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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Non-Abelian Vortices in Spinor Bose-Einstein Condensates

Invariant Spin or Spin – Gauge Transfor mation

12 transformations form non- Abelian tetrahedral group

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Non-Abelian Vortices in Spinor Bose-Einstein Condensates

Vortices

Invariant transformations define vortices

¼ 2p/3

1/2 spin vortex 1/3 vortex

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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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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SLIDE 17

Non-Abelian Vortices in 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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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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

Preliminary simulation

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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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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Non-Abelian Vortices in Spinor Bose-Einstein Condensates

Hamiltonian of Spin-2 Spinor BEC

Mean-field approximation

density spin density singlet – pair amplitude

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Non-Abelian Vortices in 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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Non-Abelian Vortices in Spinor Bose-Einstein Condensates

Gross-Pitaevskii Equation

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Non-Abelian Vortices in 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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Non-Abelian Vortices in Spinor Bose-Einstein Condensates

Algebra

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

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Non-Abelian Vortices in Spinor Bose-Einstein Condensates

Y – Shaped Structure

B A AB

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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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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Non-Abelian Vortices in 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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Non-Abelian Vortices in Spinor Bose-Einstein Condensates

Homotopy Group of Cyclic State

Order-parameter manifold First homotopy group (vortex)

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Non-Abelian Vortices in Spinor Bose-Einstein Condensates

Non-Abelian Quantum Turbulence

0.1 1 10 100 0.1 1 10 100 1000 Decaying turbulence Abelian Non-Abelian Total vortex line length Time t -3/2 t -1/3

Turbulence with non-Abelian vortices does not seem to have classical analog.

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Non-Abelian Vortices in Spinor Bose-Einstein Condensates

Non-Abelian Quantum Turbulence

Turbulence with non-Abelian vortices does not seem to have classical analog.

0.1 1 10 100 0.1 1 10 100 1000 Decaying turbulence Abelian Non-Abelian Total vortex line length Time