Bose-Einstein Condensates and Other Systems Michikazu Kobayashi - - PowerPoint PPT Presentation

Topological Excitations and Dynamical Behavior in Bose-Einstein Condensates and Other Systems Michikazu Kobayashi Kyoto University

• Oct. 24th, 2013 in Okinawa

“International Workshop for Young Researchers on Topological Quantum Phenomena in Condensed Matter with Broken Symmetries 2013”

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Contents

• 1. Bose-Einstein condensates with internal degrees of

freedom

• 2. Spin-2 spinor BEC
• 3. Vortices in spinor BEC
• 4. Dynamics of vortices in spinor BEC
• 5. Summary

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Contents

• 1. Bose-Einstein condensates with internal degrees of

freedom

• 2. Spin-2 spinor BEC
• 3. Vortices in spinor BEC
• 4. Dynamics of vortices in spinor BEC
• 5. Summary

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Bose-Einstein Condensate with Internal Degrees of Freedom

y(x)=|y(x)| exp[i(x)] : broken U(1) symmetry

• f global phase shift

Scalar BEC without internal degrees of freedom

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Bose-Einstein Condensate with Internal Degrees of Freedom

BEC with internal degrees of freedom

1. Multi-component BEC (ex. 87Rb and 41K BECs or different hyperfine level) 2. Spinor BEC (ex. 87Rb → spin-1 and spin-2 BECs) I : nuclear spin S : electron spin magnetic trap : spin degrees of freedom is frozen  scalar BEC laser trap : spin degrees of freedom is alive  spinor BEC Hyperfine spin : F = I + S

87Rb, 23Na, 7Li, 41K

F=1, 2

85Rb

F=2, 3

133Cs

F=3, 4

52Cr

F=3

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Symmetry and Topological Excitation in BEC

Order parameter manifold (𝐻/𝐼) Topological excitation Scalar BEC 𝑉(1)/1 ≃ 𝑉(1) vortex 2-component BEC (miscible) 𝑉 1 × 𝑉 1 /ℤ2 vortex 2-component BEC (inmiscible) 𝑉 1 × 𝑉 1 /(𝑉(1)/ℤ2) ≃ 𝑃(2) ≃ 𝑉(1) ⋊ ℤ2 vortex & domain wall Spin-1 BEC (ferro) (𝑉(1) × 𝑇𝑃(3))/𝑉(1) ≃ 𝑇𝑃(3) vortex Spin-1 BEC (polar) (𝑉(1) × 𝑇𝑃(3))/(𝑉(1) ⋊ ℤ2) ≃ (𝑉(1) × 𝑇2)/ℤ2 vortex & monopole Spin-2 BEC (ferro) (𝑉(1) × 𝑇𝑃(3))/(𝑉(1) × ℤ2) ≃ 𝑇𝑃(3)/ℤ2 vortex Spin-2 BEC (uniaxial nematic) 𝑉(1) × 𝑇𝑃(3)/(𝑉(1) ⋊ ℤ2) ≃ 𝑉(1) × ℝℙ2 vortex & monopole Spin-2 BEC (biaxial nematic) (𝑉 1 × 𝑇𝑃 3 )/𝐸4 vortex (non-Abelian) Spin-2 BEC (cyclic) (𝑉 1 × 𝑇𝑃 3 )/𝑈 vortex (non-Abelian)

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Symmetry and Topological Excitation in BEC

Order parameter manifold (𝐻/𝐼) Topological excitation Scalar BEC 𝑉(1)/1 ≃ 𝑉(1) vortex 2-component BEC (miscible) 𝑉 1 × 𝑉 1 /ℤ2 vortex 2-component BEC (inmiscible) 𝑉 1 × 𝑉 1 /(𝑉(1)/ℤ2) ≃ 𝑃(2) vortex & domain wall Spin-1 BEC (ferro) (𝑉(1) × 𝑇𝑃(3))/𝑉(1) ≃ 𝑇𝑃(3) vortex Spin-1 BEC (polar) (𝑉(1) × 𝑇𝑃(3))/(𝑉(1) ⋊ ℤ2) ≃ (𝑉(1) × 𝑇2)/ℤ2 vortex & monopole Spin-2 BEC (ferro) (𝑉(1) × 𝑇𝑃(3))/(𝑉(1) × ℤ2) ≃ 𝑇𝑃(3)/ℤ2 vortex Spin-2 BEC (uniaxial nematic) 𝑉(1) × 𝑇𝑃(3)/(𝑉(1) ⋊ ℤ2) ≃ 𝑉(1) × ℝℙ2 vortex & monopole Spin-2 BEC (biaxial nematic) (𝑉 1 × 𝑇𝑃 3 )/𝐸4 vortex (non-Abelian) Spin-2 BEC (cyclic) (𝑉 1 × 𝑇𝑃 3 )/𝑈 vortex (non-Abelian)

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Contents

• 1. Bose-Einstein condensates with internal degrees of

freedom

• 2. Spin-2 spinor BEC
• 3. Vortices in spinor BEC
• 4. Dynamics of vortices in spinor BEC
• 5. Summary

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Theory of Spinor BEC

Hamiltonian of Bose system with spin Low energy contact interaction (𝑚 = 0)

Coupling constant depends on total spin of two colliding particles

𝐼 = 𝑒𝒚1 ℏ2 2𝑁 𝛼Ψ𝑛

† (𝒚1)𝛼Ψ𝑛(𝑦1)

+ 1 2 𝑒𝒚2Ψ𝑛1

†

𝒚1 Ψ𝑛2

†

𝒚2 𝑊

𝑛1𝑛2𝑛1′𝑛2′(𝒚1 − 𝒚2)Ψ𝑛2′(𝒚2)Ψ𝑛1′(𝒚1)

𝑊

𝑛1𝑛1𝑛1′𝑛2′ 𝒚1 − 𝒚2 = 𝜀 𝒚1 − 𝒚2 𝐺=even

𝑕𝐺

𝑛1𝑛2𝑛1′𝑛2′𝑁

𝑃𝑛1𝑛2

𝐺𝑁

𝑃𝑛1

′ 𝑛2 ′

𝐺𝑁 ∗

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Theory of Spinor BEC

For spin-2 case

𝐼 = 𝑒𝒚 ℏ2 2𝑁 𝛼Ψ𝑛

† 𝛼Ψ𝑛 + 𝑑0

2 : 𝑜2: + 𝑑1 2 : 𝑮2: + 𝑑2 2 𝐵20

2 †𝐵20 2

𝑑0 = 4𝑕2 + 3𝑕4 7 , 𝑑0 = 𝑕4 − 𝑕2 7 , 𝑑0 = 7𝑕0 − 10𝑕2 + 3𝑕4 35

𝑜 = Ψ𝑛

† Ψ𝑛 : number density operator

𝑮 = Ψ𝑛

† 𝑮𝑛𝑜Ψ𝑜 : spin density operator

𝐵20 = (−1)𝑛Ψ𝑛Ψ−𝑛 : time reversal operator (singlet-pair amplitude)

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Theory of Spinor BEC

Mean-field theory at 𝑈 = 0 : 𝜔 = 𝜔𝑛𝑏𝑛,𝒍=0

𝑂

: all particles condense into a single-particle ground state

𝐼 = 𝑒𝒚 ℏ2 2𝑁 𝛼𝜔𝑛

† 𝛼𝜔𝑛 + 𝑑0

2 𝑜2 + 𝑑1 2 𝑮2 + 𝑑2 2 𝐵20

2 †𝐵20 2

𝑜 = 𝜔𝑛

† 𝜔𝑛 : number density

𝑮 = 𝜔𝑛

† 𝑮𝑛𝑜𝜔: spin density

𝐵20 = (−1)𝑛𝜔𝑛𝜔−𝑛 : singlet-pair amplitude

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Phase Diagram for Ground State

Biaxial Nematic: Cyclic: Ferromagnetic: Uniaxial Nematic:

87Rb

• A. Widera et al. NJP 8, 152 (2006)

𝐼 = 𝑒𝒚 ℏ2 2𝑁 𝛼𝜔𝑛

† 𝛼𝜔𝑛 + 𝑑0

2 𝑜2 + 𝑑1 2 𝑮2 + 𝑑2 2 𝐵20

2 †𝐵20 2

𝑑1 𝑑2 𝑑2 = 4𝑑1 𝜔𝑛

U = 0

1 0 𝑈 𝜔𝑛

B = 1

2 1 1 𝑈 𝜔𝑛

C = 1

2 𝑗 2 𝑗 𝑈 𝜔𝑛

F = 1

0 𝑈 degenerate

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Spherical Harmonic Representation

𝜔 𝜄, 𝜚 =

𝑛=−2 2

𝜔𝑛𝑍

2,𝑛(𝜄, 𝜚)

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Phase Diagram for Ground State

Biaxial Nematic: Cyclic: Ferromagnetic: Uniaxial Nematic:

87Rb

𝐼 = 𝑒𝒚 ℏ2 2𝑁 𝛼𝜔𝑛

† 𝛼𝜔𝑛 + 𝑑0

2 𝑜2 + 𝑑1 2 𝑮2 + 𝑑2 2 𝐵20

2 †𝐵20 2

𝑑1 𝑑2 𝑑2 = 4𝑑1 𝜔𝑛

U = 0

1 0 𝑈 𝜔𝑛

B = 1

2 1 1 𝑈 𝜔𝑛

C = 1

2 𝑗 2 𝑗 𝑈 𝜔𝑛

F = 1

0 𝑈 𝐸∞ : cylindrical symmetry 𝐸4 : square symmetry 𝑈 : tetrahedral symmetry 𝑉 1 × ℤ2 : oriented toroidal symmetry

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Symmetry of cyclic state

• 2p/3

2p/3

• p p

Spin rotates by p Phase shift by 2p/3 and spin rotates by  2p/3

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Symmetry of cyclic state

Spin rotations keeping cyclic state invariant form a non Abelian tetrahedral symmetry

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Phase Diagram for Ground State

Biaxial Nematic: Cyclic: Ferromagnetic: Uniaxial Nematic: 𝐼 = 𝑒𝒚 ℏ2 2𝑁 𝛼𝜔𝑛

† 𝛼𝜔𝑛 + 𝑑0

2 𝑜2 + 𝑑1 2 𝑮2 + 𝑑2 2 𝐵20

2 †𝐵20 2

𝑑1 𝑑2 𝑑2 = 4𝑑1 𝜔𝑛

U = 0

1 0 𝑈 𝜔𝑛

B = 1

2 1 1 𝑈 𝜔𝑛

C = 1

2 𝑗 2 𝑗 𝑈 𝜔𝑛

F = 1

0 𝑈 𝐸∞ : cylindrical symmetry 𝐸4 : square symmetry 𝑈 : tetrahedral symmetry 𝑉 1 × ℤ2 : oriented toroidal symmetry

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Phase Diagram for Ground State

Biaxial Nematic: Cyclic: Ferromagnetic: Uniaxial Nematic: 𝐼 = 𝑒𝒚 ℏ2 2𝑁 𝛼𝜔𝑛

† 𝛼𝜔𝑛 + 𝑑0

2 𝑜2 + 𝑑1 2 𝑮2 + 𝑑2 2 𝐵20

2 †𝐵20 2

𝑑1 𝑑2 𝑑2 = 4𝑑1 𝜔𝑛

B = 1

2 1 1 𝑈 𝜔𝑛

C = 1

2 𝑗 2 𝑗 𝑈 𝐸4 : square symmetry 𝑈 : tetrahedral symmetry

Non-Abelian vortices appear due to non-Abelian discrete symmetry

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Contents

• 1. Bose-Einstein condensates with internal degrees of

freedom

• 2. Spin-2 spinor BEC
• 3. Vortices in spinor BEC
• 4. Dynamics of vortices in spinor BEC
• 5. Summary

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Quantized Vortices in BEC

|y|

• p

p Arg(y)

Quantized vortex for m = +1 Topological charge can be characterized by widing number m (additive group of integers)

For scalar BEC : y=|y|eim

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Non-Abelian Vortex

Topological charge of vortices

Scalar BEC Integer (winding of phase by 2p multiple) Cyclic phase in spin-2 spinor BEC Component of tetrahedral group

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Vortices in cyclic state

p 2p/3

1/2 spin vortex 1/3 vortex

𝜔 = 1 2 𝑗𝑓𝑗𝜒 2 𝑗𝑓−𝑗𝜒 𝑈 𝜔 = 1 3 𝑓𝑗𝜒 2 0 𝑈

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Vortices in biaxial nematic state

1/2 spin vortex

1 2 𝑓𝑗𝜒 𝑓−𝑗𝜒 𝑈 1 2 𝑓𝑗𝜒 1 𝑈

1/4 vortex 1/2 vortex

1 2 0 𝑓𝑗𝜒 1 0 𝑈

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Contents

• 1. Bose-Einstein condensates with internal degrees of

freedom

• 2. Spin-2 spinor BEC
• 3. Vortices in spinor BEC
• 4. Dynamics of vortices in spinor BEC
• 5. Summary

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Gross-Pitaevskii Equation

Coherent dynamics of mean-field : Gross-Pitaevskii equation

𝑗ℏ 𝜖𝜔±2 𝜖𝑢 = − ℏ2 2𝑁 𝛼2𝜔±2 + 𝑑0𝑜𝜔±2 + 𝑑1 𝐺∓𝜔1 ± 2𝐺

𝑨𝜔±2 + 𝑑2

5 𝐵00𝜔∓2

∗

𝑗ℏ 𝜖𝜔±1 𝜖𝑢 = − ℏ2 2𝑁 𝛼2𝜔±1 + 𝑑0𝑜𝜔±1 + 𝑑1 6 2 𝐺

∓0𝜔0 + 𝐺±𝜔±2 ± 𝐺 𝑨𝜔±1

− 𝑑2 5 𝐵00𝜔∓1

∗

𝑗ℏ 𝜖𝜔0 𝜖𝑢 = − ℏ2 2𝑁 𝛼2𝜔0 + 𝑑0𝑜𝜔0 + 6 2 𝑑1 𝐺

−𝜔−1 + 𝐺+ + 𝑑2

5 𝐵00𝜔0

∗

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Collision Dynamics

For Abelian vortex

Reconnect : All Abelian vortices such as those in scalar BEC Pass through : Rarely seen for quantized vortices, and sometimes seen for disclination in liquid crystals or cosmic strings

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Collision Dynamics for Cyclic Phase

There are 12 kinds of vortices in cyclic phase For same kinds of vortices : +2p/3 & +2p/3 →reconnection For different and commutative vortices : +2p/3 & -2p/3 →pass through What happens for non-commutative vortices for different spin rotations?

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Collision Dynamics for Cyclic Phase

New “rung” vortex appears bridging two colliding vortices

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Collision of Vortex

AB=BA B A A ABA-1 B A B B-1AB B A A ABA-1 A=B A A ABBA BA-1

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Monopole Confined in Vortex Junction

Monopole (div𝑮 ≠ 0) appears at the Y-shape junction point Magnetization and its divergence is confined to only vortex lines → confined monopole (charge is classified by the tetrahedral symmetry)

Vortex core has usually internal structure different from cyclic state depending on the charge of vortex (ex. ferromagnetic state with 𝑮 ≠ 0)

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Monopole Confined in Vortex Junction

• Each arrow shows the direction of

magnetization

Confined monopoles appear at the junctions points of vortices as a form of monopole-antimonopole pair

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Collision Dynamics in Biaxial Nematic Phase

Rung vortex burst and disappears →Non-Abelian property cannot be seen

1/4 vortex 1/2 vortex

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Degeneration between Uniaxial and Biaxial Nematic Phases

Biaxial Nematic: Uniaxial Nematic: 𝐼 = 𝑒𝒚 ℏ2 2𝑁 𝛼𝜔𝑛

† 𝛼𝜔𝑛 + 𝑑0

2 𝑜2 + 𝑑1 2 𝑮2 + 𝑑2 2 𝐵20

2 †𝐵20 2

𝑑1 𝑑2 𝑑2 = 4𝑑1 𝜔𝑛

U = 0

1 0 𝑈 𝜔𝑛

B = 1

2 1 1 𝑈 𝐸∞ : cylindrical symmetry 𝐸4 : square symmetry degeneration with another continuous degree of freedom 1 2 cos𝜃 2sin𝜃 cos𝜃

𝑈

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Degeneration between Uniaxial and Biaxial Nematic Phases

Biaxial Nematic: Uniaxial Nematic: 1 2 cos𝜃 2sin𝜃 cos𝜃

𝑈

Very large order-parameter manifold : (𝑉 1 × 𝑇4)/ℤ2 →Several vortices in both phases are topologically unstable due to 𝜃

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Quasi-Nambu-Goldstone Mode

1 2 cos𝜃 2sin𝜃 cos𝜃

𝑈

𝜃 is not the symmetry of the Hamiltonian (accidental symmetry) →Gapless excitation mode due to 𝜃 is not the true Nambu-Goldstone mode (Quasi-Nambu-Goldstone mode)

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Thermal Phase Diagram

Quasi-Nambu-Goldstone mode easily becomes gapful through (quantum or) thermal fluctuation Biaxial Nematic: Cyclic: Ferromagnetic: Uniaxial Nematic:

87Rb

𝑑1 𝑑2 𝑑2 = 4𝑑1 𝜔𝑛

U = 0

1 0 𝑈 𝜔𝑛

B = 1

2 1 1 𝑈 𝜔𝑛

C = 1

2 𝑗 2 𝑗 𝑈 𝜔𝑛

F = 1

0 𝑈 phase boundary at finite 𝑈

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

BEC at Finite Temperature

𝑗ℏ 𝜖𝜔𝑛 𝜖𝑢 = 1 − 𝛿 𝜀𝐼 𝜀𝜔𝑛 + 𝜊 𝜊 𝒚, 𝑢 𝜊(𝒚′, 𝑢′) = 𝑙B𝑈𝜀 𝑦 − 𝑦′ 𝜀(𝑢 − 𝑢′)

Stochastic Gross-Pitaevskii equation (complex Langevin equation)

Condensate fraction and its fluctuation

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Collision Dynamics at Finite Temperature

𝑈 = 0.1𝑈

c

𝑈 = 0.4𝑈

c

Non-Abelian property is restored by massive quasi- Nambu-Goldstone mode due to thermal fluctuation

Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Summary

• 1. Spin-2 spinor BEC can have exotic non-Abelian vortex due

to non-Abelian discrete symmetry.

• 2. Collision of non-Abelian vortex in the cyclic phase show the

formation of rung vortex bridging colliding vortices.

• 3. After the collision, there appears a monopole confined in the

Y-shaped junction.

• 4. Non-Abelian property disappears in the biaxial nematic

phase due to the quasi-Nambu-Goldstone mode, and is restored by thermal fluctuations at finite temperatures.