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”

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 Scalar BEC without internal degrees of freedom y ( x )=| y ( x )| exp[ i  ( x )] : broken U (1) symmetry of global phase shift 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. 87 Rb and 41 K BECs or different hyperfine level) 2. Spinor BEC (ex. 87 Rb → spin -1 and spin-2 BECs) 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 F =1, 2 87 Rb, 23 Na, 7 Li, 41 K I : nuclear spin F =2, 3 85 Rb S : electron spin F =3, 4 133 Cs F =3 52 Cr Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Symmetry and Topological Excitation in BEC Order parameter manifold Topological excitation ( 𝐻/𝐼 ) Scalar BEC 𝑉(1)/1 ≃ 𝑉(1) vortex 2-component BEC (miscible) 𝑉 1 × 𝑉 1 /ℤ 2 vortex 𝑉 1 × 𝑉 1 /(𝑉(1)/ℤ 2 ) 2-component BEC (inmiscible) vortex & domain wall ≃ 𝑃(2) ≃ 𝑉(1) ⋊ ℤ 2 Spin-1 BEC (ferro) (𝑉(1) × 𝑇𝑃(3))/𝑉(1) ≃ 𝑇𝑃(3) vortex (𝑉(1) × 𝑇𝑃(3))/(𝑉(1) ⋊ ℤ 2 ) Spin-1 BEC (polar) vortex & monopole ≃ (𝑉(1) × 𝑇 2 )/ℤ 2 (𝑉(1) × 𝑇𝑃(3))/(𝑉(1) × ℤ 2 ) Spin-2 BEC (ferro) vortex ≃ 𝑇𝑃(3)/ℤ 2 𝑉(1) × 𝑇𝑃(3)/(𝑉(1) ⋊ ℤ 2 ) Spin-2 BEC (uniaxial nematic) vortex & monopole ≃ 𝑉(1) × ℝℙ 2 Spin-2 BEC (biaxial nematic) (𝑉 1 × 𝑇𝑃 3 )/𝐸 4 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 Topological excitation ( 𝐻/𝐼 ) Scalar BEC 𝑉(1)/1 ≃ 𝑉(1) vortex 2-component BEC (miscible) 𝑉 1 × 𝑉 1 /ℤ 2 vortex 𝑉 1 × 𝑉 1 /(𝑉(1)/ℤ 2 ) 2-component BEC (inmiscible) vortex & domain wall ≃ 𝑃(2) Spin-1 BEC (ferro) (𝑉(1) × 𝑇𝑃(3))/𝑉(1) ≃ 𝑇𝑃(3) vortex (𝑉(1) × 𝑇𝑃(3))/(𝑉(1) ⋊ ℤ 2 ) Spin-1 BEC (polar) vortex & monopole ≃ (𝑉(1) × 𝑇 2 )/ℤ 2 (𝑉(1) × 𝑇𝑃(3))/(𝑉(1) × ℤ 2 ) Spin-2 BEC (ferro) vortex ≃ 𝑇𝑃(3)/ℤ 2 𝑉(1) × 𝑇𝑃(3)/(𝑉(1) ⋊ ℤ 2 ) Spin-2 BEC (uniaxial nematic) vortex & monopole ≃ 𝑉(1) × ℝℙ 2 Spin-2 BEC (biaxial nematic) (𝑉 1 × 𝑇𝑃 3 )/𝐸 4 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 ℏ 2 † (𝒚 1 )𝛼Ψ 𝑛 (𝑦 1 ) 𝐼 = 𝑒𝒚 1 2𝑁 𝛼Ψ 𝑛 + 1 † † 2 𝑒𝒚 2 Ψ 𝑛 1 𝒚 1 Ψ 𝑛 2 𝒚 2 𝑊 𝑛 1 𝑛 2 𝑛 1′ 𝑛 2′ (𝒚 1 − 𝒚 2 )Ψ 𝑛 2′ (𝒚 2 )Ψ 𝑛 1′ (𝒚 1 ) Low energy contact interaction ( 𝑚 = 0 ) ∗ 𝐺𝑁 𝐺𝑁 𝑊 𝑛 1 𝑛 1 𝑛 1′ 𝑛 2′ 𝒚 1 − 𝒚 2 = 𝜀 𝒚 1 − 𝒚 2 𝑕 𝐺 𝑃 𝑛 1 𝑛 2 𝑃 𝑛 1 ′ 𝑛 2 ′ 𝐺=even 𝑛 1 𝑛 2 𝑛 1′ 𝑛 2′ 𝑁 Coupling constant depends on total spin of two colliding particles Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Theory of Spinor BEC For spin-2 case ℏ 2 † 𝛼Ψ 𝑛 + 𝑑 0 2 : 𝑜 2 : + 𝑑 1 2 : 𝑮 2 : + 𝑑 2 2 † 𝐵 20 2 𝐼 = 𝑒𝒚 2𝑁 𝛼Ψ 𝑛 2 𝐵 20 𝑑 0 = 4𝑕 2 + 3𝑕 4 𝑑 0 = 𝑕 4 − 𝑕 2 𝑑 0 = 7𝑕 0 − 10𝑕 2 + 3𝑕 4 , , 7 7 35 † Ψ 𝑛 : number density operator 𝑜 = Ψ 𝑛 † 𝑮 𝑛𝑜 Ψ 𝑜 : spin density operator 𝑮 = Ψ 𝑛 𝐵 20 = (−1) 𝑛 Ψ 𝑛 Ψ −𝑛 : time reversal operator (singlet-pair amplitude) Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Theory of Spinor BEC 𝑂 𝜔 = 𝜔 𝑛 𝑏 𝑛,𝒍=0 0 Mean-field theory at 𝑈 = 0 : : all particles condense into a single-particle ground state ℏ 2 † 𝛼𝜔 𝑛 + 𝑑 0 2 𝑜 2 + 𝑑 1 2 𝑮 2 + 𝑑 2 2 † 𝐵 20 2 𝐼 = 𝑒𝒚 2𝑁 𝛼𝜔 𝑛 2 𝐵 20 † 𝜔 𝑛 : number density 𝑜 = 𝜔 𝑛 † 𝑮 𝑛𝑜 𝜔 : spin density 𝑮 = 𝜔 𝑛 𝐵 20 = (−1) 𝑛 𝜔 𝑛 𝜔 −𝑛 : singlet-pair amplitude Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Phase Diagram for Ground State ℏ 2 † 𝛼𝜔 𝑛 + 𝑑 0 2 𝑜 2 + 𝑑 1 2 𝑮 2 + 𝑑 2 2 † 𝐵 20 2 𝐼 = 𝑒𝒚 2𝑁 𝛼𝜔 𝑛 2 𝐵 20 𝑑 1 Uniaxial Nematic: Cyclic: C = 1 U = 0 0 𝑈 𝜔 𝑛 𝑗 𝑈 0 1 0 𝜔 𝑛 2 𝑗 0 2 0 87 Rb degenerate 𝑑 2 Biaxial Nematic: Ferromagnetic: B = 1 1 𝑈 𝜔 𝑛 2 1 0 0 0 F = 1 0 𝑈 𝜔 𝑛 0 0 0 𝑑 2 = 4𝑑 1 A. Widera et al. NJP 8 , 152 (2006) Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Spherical Harmonic Representation 2 𝜔 𝜄, 𝜚 = 𝜔 𝑛 𝑍 2,𝑛 (𝜄, 𝜚) 𝑛=−2 Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Phase Diagram for Ground State ℏ 2 † 𝛼𝜔 𝑛 + 𝑑 0 2 𝑜 2 + 𝑑 1 2 𝑮 2 + 𝑑 2 2 † 𝐵 20 2 𝐼 = 𝑒𝒚 2𝑁 𝛼𝜔 𝑛 2 𝐵 20 𝑑 1 Uniaxial Nematic: Cyclic: 𝑈 : tetrahedral symmetry U = 0 0 𝑈 𝜔 𝑛 0 1 0 C = 1 𝑗 𝑈 𝜔 𝑛 2 𝑗 0 2 0 𝐸 ∞ : cylindrical symmetry 87 Rb 𝑑 2 Biaxial Nematic: Ferromagnetic: B = 1 1 𝑈 F = 1 𝜔 𝑛 2 1 0 0 0 0 𝑈 𝜔 𝑛 0 0 0 𝐸 4 : square symmetry 𝑉 1 × ℤ 2 : oriented toroidal symmetry 𝑑 2 = 4𝑑 1 Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Symmetry of cyclic state Spin rotates by p 0 - 2 p /3 Phase shift by  2 p /3 and 2 p /3 spin rotates by  2 p /3 - p p 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 ℏ 2 † 𝛼𝜔 𝑛 + 𝑑 0 2 𝑜 2 + 𝑑 1 2 𝑮 2 + 𝑑 2 2 † 𝐵 20 2 𝐼 = 𝑒𝒚 2𝑁 𝛼𝜔 𝑛 2 𝐵 20 𝑑 1 Uniaxial Nematic: Cyclic: 𝑈 : tetrahedral symmetry U = 0 0 𝑈 𝜔 𝑛 0 1 0 C = 1 𝑗 𝑈 𝜔 𝑛 2 𝑗 0 2 0 𝐸 ∞ : cylindrical symmetry 𝑑 2 Biaxial Nematic: Ferromagnetic: B = 1 1 𝑈 F = 1 𝜔 𝑛 2 1 0 0 0 0 𝑈 𝜔 𝑛 0 0 0 𝐸 4 : square symmetry 𝑉 1 × ℤ 2 : oriented toroidal symmetry 𝑑 2 = 4𝑑 1 Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Phase Diagram for Ground State ℏ 2 † 𝛼𝜔 𝑛 + 𝑑 0 2 𝑜 2 + 𝑑 1 2 𝑮 2 + 𝑑 2 2 † 𝐵 20 2 𝐼 = 𝑒𝒚 2𝑁 𝛼𝜔 𝑛 2 𝐵 20 𝑑 1 Uniaxial Nematic: Cyclic: 𝑈 : tetrahedral symmetry Non-Abelian vortices appear due C = 1 to non-Abelian discrete symmetry 𝑗 𝑈 𝜔 𝑛 2 𝑗 0 2 0 𝑑 2 Biaxial Nematic: Ferromagnetic: B = 1 1 𝑈 𝜔 𝑛 2 1 0 0 0 𝐸 4 : square symmetry 𝑑 2 = 4𝑑 1 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 Quantized vortex for m = +1 | y | For scalar BEC : y =| y | e im  Topological charge can be characterized by widing number m (additive group of integers) - p p Arg( y ) Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems

Non-Abelian Vortex Topological charge of vortices Scalar BEC Cyclic phase in spin-2 spinor BEC Integer (winding of phase by 2 p multiple) Component of tetrahedral group Topological excitations and dynamical behavior in Bose-Einstein condensates and other systems