Hydrodynamics of two-component Bose mixtures Jogundas Armaitis - PowerPoint PPT Presentation

Hydrodynamics of two-component Bose mixtures Jogundas Armaitis Rembert Duine and Henk Stoof Utrecht University, The Netherlands PRL 110, 260404 (2013) PRA 91, 043641 (2015) Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 1 / 18

Main results in pictures 14 12 c 2 �� g n 0 � 3 m � 10 8 6 4 2 0 0.2 0.4 0.6 0.8 1.0 T � T c Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 2 / 18

Hydrodynamic two-component Bose mixtures They exist! Image courtesy Peter van der Straten group (UU) The Lagrangian describing this system is � � � ∂ τ − � 2 ∇ 2 � � φ σ + 1 φ ∗ 2 g φ ∗ σ φ ∗ + g ↑↓ φ ∗ ↑ φ ∗ L = − µ σ φ σ φ σ ↓ φ ↓ φ ↑ , σ 2 m σ where g > 0 and g ↑↓ > 0 are repulsive interactions. Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 3 / 18

Two main regimes Immiscible (ferromagnetic) Miscible (non-ferromagnetic) g < g ↑↓ g > g ↑↓ Only one component may condense Both components may condense PM, FM, BEC+FM PM, BEC Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 4 / 18

Ferromagnetic (immiscible) case based on J. Armaitis, H. T. C. Stoof, and R. A. Duine Magnetization Relaxation and Geometric Forces in a Bose Ferromagnet Phys. Rev. Lett. 110, 260404 (2013)

Mean-field phase diagram (immiscible) 1.5 1.4 PM FM 1.3 T � T c 1.2 1.1 FM � BEC 1.0 0.00 0.05 0.10 0.15 0.20 n 1 � 3 a �� C.f. Q. Gu and R. A. Klemm, PRA 68, 031604R (2003) Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 6 / 18

Hydrodynamic equations Conservation of order parameters ◮ superfluid density ◮ magnetization Other conserved quantities ◮ density ◮ momentum ◮ energy Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 7 / 18

Example: momentum conservation m ∂ t j + ∇ p + n ∇ V − Pn E − P j × B = 0 Effective electric field E = � ε αβγ Ω α ( ∂ t Ω β )( ∇ Ω γ ) / 2 Effective magnetic field B = − � ε αβγ Ω α ( ∇ Ω β ) × ( ∇ Ω γ ) / 4 Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 8 / 18

Application: (rigid) skyrmion Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 9 / 18

Skyrmion: experiment Source: Seoul group, PRL 111, 245301 (2013) Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 10 / 18

Non-ferromagnetic (miscible) case based on J. Armaitis, H. T. C. Stoof, and R. A. Duine Hydrodynamic Modes of Partially-Condensed Bose Mixtures Phys. Rev. A 91, 043641 (2015) Hydrodynamic equations Microscopic theory Collective modes

Microscopic theory: Popov Fluctuation expansion ( mean-field #1 ) φ σ ( x , τ ) = φ 0 σ ( x ) + φ ′ σ ( x , τ ) i.e. � φ σ � = φ 0 σ Finite thermal particle density ( mean-field #2 ) ∗ φ ′ � φ ′ σ � = n ′ σ σ Partition function n 2 0 ↓ + n 2 � 0 ↑ g ↑↓ n ↑ n ↓ + g ( n 2 ↓ + n 2 � � Z = exp β ↑ ) − g 2 − 1 � � ln(1 − e − β E s , k ) V s = ± , k � =0 Quasiparticle dispersion E 2 ± , k = ε k ( ε k + 2 n 0 [ g ± g ↑↓ ]) Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 12 / 18

Hydrodynamic equations (miscible case) Nomenclature nf normal fluid sf superfluid x tot x ↑ + x ↓ x diff x ↑ − x ↓ “Spin” Density ∂ t n nf diff + ∇ · j nf diff = 0 ∂ t n tot + ∇ · j tot = 0 ∂ t n sf diff + ∇ · j sf ∂ t ( n tot s ) + n tot s ∇ · v nf diff = 0 tot = 0 ∂ t j nf diff + n nf m ∂ t j tot + ∇ p = 0 tot ∇ µ diff / 2 m = 0 m ∂ t v sf ∂ t j sf diff + n sf tot + ∇ µ tot = 0 tot ∇ µ diff / 2 m = 0 Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 13 / 18

Hydrodynamic equations (miscible case) Density equations m ∂ 2 t n tot = ∇ 2 p t s = n sf m ∂ 2 tot s 2 ∇ 2 T n nf tot “Spin” equations diff = n nf ∂µ diff ∂ 2 t n nf tot ∇ 2 ( n nf diff + n sf diff ) 2 m ∂ n diff diff = n sf ∂µ diff ∂ 2 t n sf tot ∇ 2 ( n nf diff + n sf diff ) 2 m ∂ n diff Travelling wave ansatz for each quantity, e.g., n tot = n tot , eq + δ n tot exp( − i [ ω t − k · x ]) Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 14 / 18

Collective mode spectrum 14 first sound 12 c 2 �� g n 0 � 3 m � 10 8 6 4 second sound 2 spin sounds 0 0.2 0.4 0.6 0.8 1.0 T � T c Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 15 / 18

Collective mode character 1 second sound 0 ∆ T � T ∆ n � n � 1 first sound � 2 � 3 0.0 0.2 0.4 0.6 0.8 1.0 T � T c Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 16 / 18

Summary Hydrodynamic theories for binary Bose mixtures Microscopic details: thermodynamic functions and kinetic timescales Input for experiments and some verification already Outlook: spin-orbit coupling, strong interactions, higher spin. . . Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 17 / 18

Main results in pictures 14 12 c 2 �� g n 0 � 3 m � 10 8 6 4 2 0 0.2 0.4 0.6 0.8 1.0 T � T c Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 18 / 18