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: International Symposium on Quantum Fluids and Solids, August 6, 2010
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. Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Vortices in Bose-Einstein Condensates Integer vortex (Scalar BEC or 4 He) v = ( n / m ) q : superfluid Y e in q velocity v dl = nh / m : circulation Around the vortex core 1. Phase winds by integer multiple of 2 p . 2. Circulation takes integer multiple of h / m Topological charge of vortex is characterized by additive group of integers Abelian vortex. Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Vortices in Bose-Einstein Condensates Half-quantized vortex Polar phase in spin-1 3 He-A 3 He-B in low T & P spinor BEC Non-Abelian vortices are realized in the cyclic phase of spin-2 BEC Double core of half- quantized vortices reverse d -vector Topological charge is characterized by 2nd cyclic group Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Spin-2 Spinor BEC 5 - component BEC : Y = (Y 2 , Y 1 , Y 0 , Y -1 , Y -2 ) T F = 2 87 Rb BEC and its spin dynamics is observed H. Schmaljohann et al. PRL 92 , 040402 (2004) Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Ground State of Spin-2 Spinor BEC c 1 Nematic Cyclic 87 Rb A. Widera et al. NJP 8 , 152 (2006) c 2 Ferromagnetic C. V. Cionabu et al. PRA 61 , 033607 (2000) Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Spherical Harmonics Expression of Cy clic Phase cyclic phase Y 2,2 Y 2,1 Y 2,0 Y 2,-1 Y 2,-2 + + + + 0 1. Cyclic state can be expressed - 2 p /3 as a headless triad 2. Phase difference between each lobe is 2 p /3 2 p /3 - ¼ ¼ Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Invariant Spin or Spin – Gauge Transfor mation p – spin rotation 0 Including identity, there are 12 transformations - 2 p /3 keeping headless triad invariant. 2 p /3 – spin & gauge transformation 2 p /3 - ¼ ¼ H. Mäkelä et al. J. Phys. A 36 , 8555 (2003), G. W. Semenoff et al. PRL 98 , 100401 (2007) Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Invariant Spin or Spin – Gauge Transfor mation 12 transformations form non- Abelian tetrahedral group Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Vortices Invariant transformations define vortices 1/2 spin vortex 1/3 vortex 2 p /3 ¼ Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Topological Charge of Vortices Cyclic phase in spin-2 spinor BEC Scalar BEC Y e in q Topological charge : Additive group of integer n Abelian vortices Topological charge : Tetrahedral group Non-Abelian vortices Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Collision Dynamics of Non-Abelian Vorti ces Non-Abelian property of vortices becomes remarkable in their collision dynamics → Numerical simulation of Gross-Pitaevskii equation Initial state : two straight vortices, linked vortex rings Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Collision Dynamics of Non-Abelian Vorti Collision of vortices with non-commutative charge forms a new “ rung ” vortex connecting two vortices ces Same charge Reconnection Commutative Non-commutative charges charges Rung vortex Passing 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. Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Algebra B A A ABA -1 Topological charge of vortex can be fixed by a closed path encircling the vortex Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Collision of Vortex B A B A BA -1 A ABA -1 A ABA -1 Rung BA -1 is formed through the collision. Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Collision of Vortex A A B A A = B AA -1 = 1 A ABA -1 A A Rung disappears for the same charge resulting reconnection. Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Collision of Vortex B B A A AB = BA A A B ABA -1 Passing dynamics is also possible for commutative case Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Linked Vortex Rings B A B A AB -1 A -1 B B A AB = BA Linked vortex rings with non-commutative charges never unravel. Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Application of Non-Abelian Vortices : N on-Abelian Turbulence Preliminary simulation Abelian turbulence ↓ Cascade of vortices through reconnections Non-Abelian turbulence ↓ Large-scale networking structure of vortices through formation of rungs New type of turbulence 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) Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Cyclic State vs. Singlet-trio Condensed State For c 1 >0, c 2 >0 M. Koashi, and M. Ueda. PRL 84 , 1066 (2000) Singlet-trio condensed state (only U(1) is broken) Transition occurs under ~1 µ G Cyclic state ( U(1) SO(3) is broken) Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Nematic State vs. Singlet-pair Condensed State For c 1 >0, c 2 <0 M. Koashi, and M. Ueda. PRL 84 , 1066 (2000) Singlet-pair condensed state (only U(1) is broken) Transition occurs under ~1 µ G Nematic state ( U(1) SO(3) is broken) 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) Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Hamiltonian of Spin-2 Spinor BEC Mean-field approximation density spin density singlet – pair amplitude Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Breaking of U (1) G SO (3) S Fixed from Hamiltonian Spin rotation ： SO (3) S Gauge transformation ： U (1) G Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Gross-Pitaevskii Equation Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Ground State Phase Diagram S. Uchino, M. Kobayashi, and M. Ueda. PRA 81 , 063632 (2010) Cyclic : T Uniaxial Nematic : D Ferromagnetic : SO (2) / 2 Biaxial Nematic : D 4 - ¼ ¼ Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Algebra Path d defines vortex B as ABA -1 (same conjugacy class) Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Y – Shaped Structure B AB A Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Collision of Vortex B A B A AB A ABA -1 A ABA -1 B A Only Abelian B B -1 AB BA -1 A B A ABA -1 Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Same Charge A A A A × A 2 Energetically unfavorable A A A A A A ○ △ A A 1 reconnection A A A A Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Commutative Charge B A B A × AB Energetically unfavorable A ABA -1 A ABA -1 × B A ○ B B -1 AB BA -1 Passing A B A ABA -1 Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Non-Commutative Charge B A B A AB ○ A ABA -1 A ABA -1 B A ○ × Topologically forbidden B B -1 AB BA -1 rung A B A ABA -1 Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Linked Rings Non-Commutative B A AB -1 A -1 B B A AB -1 ABA -1 ABA -1 A ABA -1 B A Commutative ABA -1 AB -1 ABA -1 A B Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Characteristic Form of Wave Function 1/2 – spin vortex 1/3 vortex Mass circulation Spin circulation Vortex Core ( h / m ) ( h / m ) 1/2 – spin 0 1/2 Nematic 1/3 1/3 1/3 Ferromagnetic Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Homotopy Group of Cyclic State Order-parameter manifold First homotopy group (vortex) Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Non-Abelian Quantum Turbulence Decaying turbulence 100 t -1/3 Total vortex line length 10 Abelian Non-Abelian 1 Turbulence with non-Abelian t -3/2 vortices does not seem to 0.1 have classical analog. 0.1 1 10 100 1000 Time Non-Abelian Vortices in Spinor Bose-Einstein Condensates
Non-Abelian Quantum Turbulence Decaying turbulence 100 Total vortex line length 10 Abelian Non-Abelian 1 Turbulence with non-Abelian vortices does not seem to 0.1 have classical analog. 0.1 1 10 100 1000 Time Non-Abelian Vortices in Spinor Bose-Einstein Condensates
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