Non-Abelian Vortices in Non-Abelian Vortices in Spinor Spinor Bose-Einstein Condensates Bose-Einstein Condensates Michikazu Kobayashi a Michikazu Kobayashi a Collaborators: Yuki Kawaguchi a , Muneto Nitta b , and Masahito Ueda a Collaborators: Yuki Kawaguchi a , Muneto Nitta b , and Masahito Ueda a University of Tokyo a and Keio University b University of Tokyo a and Keio University b July. 27 – Aug. 7, 2009, Eleventh J. J. Gianbiagi Winter School: The Quantum July. 27 – Aug. 7, 2009, Eleventh J. J. Gianbiagi Winter School: The Quantum Mechanics of the XXI Century: Manipulation of Coherent Atomic Matter Mechanics of the XXI Century: Manipulation of Coherent Atomic Matter
Quantized Vortex and Topological Charge Quantized Vortex and Topological Charge Topological charge of a vortex can be considered how Topological charge of a vortex can be considered how order parameter changes around the vortex core order parameter changes around the vortex core Single component BEC : Topological charge can be expressed by integer n Topological charge can be expressed by integer n vortex in 4 He vortex in 87 Rb BEC K. W. Madison et al. G. P. Bewley et al. PRL 86 , 4443 (2001) Nature 441 , 588 (2006)
Spin-2 BEC Spin-2 BEC Bose-Einstein condensate in optical trap (spin degrees of freedom is alive) 87 Rb （ I = 3/2 ） Hyperfine coupling （ F = I + S ） BEC characterized by m F
Mean Field Approximation for BEC at T = 0 = 0 Mean Field Approximation for BEC at T Case of Spin-2 n tot : total density F : magnetization A 00 : singlet pair amplitude
Spin-2 BEC Spin-2 BEC 1. c 1 < 0 , c 2 > 20 c 1 → ferromagnetic phase : F ≠ 0 2. c 1 > c 2 /20, c 2 < 0 → nematic phase : F = 0, A 00 ≠ 0 3. c 1 > 0, c 2 > 0 → cyclic phase : F = A 00 = 0 ferromagnetic nematic cyclic
Spin-2 BEC Spin-2 BEC Cyclic phase Y 2,2 Y 2,1 Y 2,0 0 4 ¼ /3 + + Y 2,-1 Y 2,-2 2 ¼ /3 + + headless triad
Vortices in Spinor BEC Vortices in Spinor BEC S = 1 Polar phase ¼ gauge transformation headless vector Half quantized vortex : spin & gauge rotate Half quantized vortex : spin & gauge rotate by ¼ around vortex core by ¼ around vortex core 0 Topological charge can be expressed by Topological charge can be expressed by integer and half integer (Abelian vortex) integer and half integer (Abelian vortex)
Vortices in Spin-2 BEC Vortices in Spin-2 BEC There are 5 types of vortices in the cyclic phase gauge vortex integer spin vortex mass circulation : 1 spin circulation : 0 mass circulation : 0 spin circulation : 1
Vortices in Spin-2 BEC Vortices in Spin-2 BEC 1/2-spin vortex : triad rotate by ¼ around three axis e x , e y , e z mass circulation : 0 spin circulation : 1/2
Vortices in Spin-2 BEC Vortices in Spin-2 BEC 1/3 vortex : triad rotate by 2 ¼ /3 around four axis e 1 , e 2 , e 3 , e 4 and 2 ¼ /3 gauge transformation 0 4 /3 2 ¼ /3 gauge transformation 2 /3 mass circulation : 1/3 spin circulation : 1/3
Vortices in Spin-2 BEC Vortices in Spin-2 BEC 4, 2/3 vortex : triad rotate by 4 ¼ /3 around four axis e 1 , e 2 , e 3 , e 4 and 4 ¼ /3 gauge transformation 0 4 /3 4 ¼ /3 gauge transformation 2 /3 mass circulation : 2/3 spin circulation : 2/3
Topological Charge of Vortices is Non-Abelian Topological Charge of Vortices is Non-Abelian There are 12 There are 12 rotations for rotations for vortices vortices
Collision Dynamics of Vortices Collision Dynamics of Vortices “ Non-Abelian Non-Abelian ” character becomes remarkable when two vortices collide with each other →Numerical simulation of the Gross-Pitaevskii equation Initial state ： two straight vortices in oblique angle, linked vortices
Collision Dynamics of Vortices Collision Dynamics of Vortices Commutative ( e 1 & e 1 ) reconnection reconnection Non-commutative ( e 1 & e 2 ) rung formation rung formation
Collision Dynamics of Linked Vortices Collision Dynamics of Linked Vortices untangle Commutative ( e 1 & e 1 ) untangle not untangle not untangle Non-commutative ( e 1 & e 2 )
Algebraic Approach Algebraic Approach Path d defines vortex B as ABA -1 (same conjugacy class)
Collision of Same Vortices Collision of Same Vortices A A A A × A 2 A A A A A A ○ × Energetically unfavorable A A 1 reconnection A A A A
Collision of Different Non-commutative Vortices Collision of Different Non-commutative Vortices B A B A AB ○ ABA -1 A A ABA -1 B A ○ × Topologically Topologically forbidden forbidden B B -1 AB BA -1 rung A A B ABA -1
Linked Vortices Linked Vortices non-commutative B A B A AB -1 A -1 B ABA -1 A AB -1 ABA -1 ABA -1 ABA -1 AB -1 ABA -1 B A commutative Linked vortices Linked vortices cannot untangle cannot untangle A B
Summary 1. Vortices with non-commutative circulations are defined as non-Abelian vortices. 2. Non-Abelian vortices can be realized in the cyclic phase of spin-2 BEC 3. Collision of two non-Abelian vortices create a new vortex between them as a rung (networking structure).
Future: Network Structure in Quantum Future: Network Structure in Quantum Turbulence Turbulence Turbulence with Abelian vortices ↓ • Cascade of vortices Turbulence with non-Abelian vortices ↓ • Large-scale networking structures among vortices with rungs • Non-cascading turbulence New turbulence! New turbulence!
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