Non-Abelian Vortices in Non-Abelian Vortices in Spinor Spinor - - PowerPoint PPT Presentation

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

Michikazu Kobayashi Michikazu Kobayashia

a

University of Tokyo University of Tokyoa

a and Keio University

and Keio Universityb

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 Collaborators: Yuki Kawaguchi Collaborators: Yuki Kawaguchia

a, Muneto Nitta

, Muneto Nittab

b, and Masahito Ueda

, and Masahito Uedaa

a

Quantized Vortex and Topological Charge Quantized Vortex and Topological Charge

Single component BEC :

Topological charge of a vortex can be considered how Topological charge of a vortex can be considered how

• rder parameter changes around the vortex core
• rder parameter changes around the vortex core

Topological charge can be expressed by integer Topological charge can be expressed by integer n n

• K. W. Madison et al.

PRL 86, 4443 (2001)

vortex in 87Rb BEC vortex in 4He

• G. P. Bewley et al.

Nature 441, 588 (2006)

Spin-2 BEC Spin-2 BEC

Bose-Einstein condensate in optical trap (spin degrees of freedom is alive)

Hyperfine coupling （F = I + S）

87Rb（I = 3/2）

BEC characterized by mF

Mean Field Approximation for BEC at Mean Field Approximation for BEC at T T = 0 = 0

ntot : total density F : magnetization A00 : singlet pair amplitude

Case of Spin-2

Spin-2 BEC Spin-2 BEC

1. c1 < 0 , c2 > 20 c1 → ferromagnetic phase : F ≠ 0

• 2. c1 > c2/20, c2 < 0 → nematic phase : F = 0, A00 ≠ 0
• 3. c1 > 0, c2 > 0 → cyclic phase : F = A00 = 0

ferromagnetic nematic cyclic

Spin-2 BEC Spin-2 BEC

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

Cyclic phase

headless triad

4¼/3 2¼/3

Vortices in Spinor BEC Vortices in Spinor BEC



S = 1 Polar phase

¼ gauge transformation

Half quantized vortex : spin & gauge rotate Half quantized vortex : spin & gauge rotate by by ¼ ¼ around vortex core around vortex core Topological charge can be expressed by Topological charge can be expressed by integer and half integer (Abelian vortex) integer and half integer (Abelian vortex) headless vector

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 ex, ey, ez

mass circulation : 0 spin circulation : 1/2

Vortices in Spin-2 BEC Vortices in Spin-2 BEC

1/3 vortex : triad rotate by 2¼/3 around four axis e1, e2, e3 , e4 and 2¼/3 gauge transformation

2¼/3 gauge transformation

2/3 4/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 4¼/3 around four axis e1, e2, e3 , e4 and 4¼/3 gauge transformation

4¼/3 gauge transformation

2/3 4/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

→Numerical simulation of the Gross-Pitaevskii equation

Initial state：two straight vortices in oblique angle, linked vortices

“Non-Abelian Non-Abelian” character becomes remarkable when two vortices collide with each other

Collision Dynamics of Vortices Collision Dynamics of Vortices

Commutative (e1 & e1) reconnection reconnection Non-commutative (e1 & e2) rung formation rung formation

Collision Dynamics of Linked Vortices Collision Dynamics of Linked Vortices

Non-commutative (e1 & e2) not untangle not untangle Commutative (e1 & e1) untangle untangle

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

○

× ×

reconnection

Energetically unfavorable

Collision of Different Non-commutative Vortices Collision of Different Non-commutative Vortices

Topologically Topologically forbidden 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

Linked Vortices Linked Vortices

B A A ABA-1 ABA-1 AB-1ABA-1 B ABA-1 A AB-1ABA-1 AB-1A-1B non-commutative B B A A commutative

Linked vortices Linked vortices cannot untangle cannot untangle

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!