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

0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 14 T Tc c2g n03m

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 L =

• φ∗

σ

• ∂τ − 2∇2

2mσ − µ

• φσ + 1

2gφ∗

σφ∗ σφσφσ

• + g↑↓φ∗

↑φ∗ ↓φ↓φ↑,

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)

0.00 0.05 0.10 0.15 0.20 1.0 1.1 1.2 1.3 1.4 1.5 n13a T Tc PM FM FMBEC

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∂tj + ∇p + n∇V − PnE − Pj × B = 0 Effective electric field E = εαβγΩα(∂tΩβ)(∇Ωγ)/2 Effective magnetic field B = −εαβγΩα(∇Ωβ) × (∇Ωγ)/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)

Collective modes Microscopic theory Hydrodynamic equations

Microscopic theory: Popov

Fluctuation expansion ( mean-field #1 ) φσ(x, τ) = φ0σ(x) + φ′

σ(x, τ)

i.e. φσ = φ0σ Finite thermal particle density ( mean-field #2 ) φ′

σ ∗φ′ σ = n′ σ

Partition function Z = exp

• β
• g↑↓n↑n↓ + g(n2

↓ + n2 ↑) − g

n2

0↓ + n2 0↑

2

• − 1

V

• s=±,k=0

ln(1 − e−βEs,k)

• Quasiparticle dispersion

E 2

±,k = εk(εk + 2n0[g ± g↑↓])

Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 12 / 18

Hydrodynamic equations (miscible case)

Nomenclature nf normal fluid sf superfluid xtot x↑ + x↓ xdiff x↑ − x↓ Density ∂tntot + ∇ · jtot = 0 ∂t(ntots) + ntots∇ · vnf

tot = 0

m∂tjtot + ∇p = 0 m∂tvsf

tot + ∇µtot = 0

“Spin” ∂tnnf

diff + ∇ · jnf diff = 0

∂tnsf

diff + ∇ · jsf diff = 0

∂tjnf

diff + nnf tot∇µdiff/2m = 0

∂tjsf

diff + nsf tot∇µdiff/2m = 0

Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 13 / 18

Hydrodynamic equations (miscible case)

Density equations m∂2

t ntot = ∇2p

m∂2

t s = nsf tot

nnf

tot

s2∇2T “Spin” equations ∂2

t nnf diff = nnf tot

2m ∂µdiff ∂ndiff ∇2(nnf

diff + nsf diff)

∂2

t nsf diff = nsf tot

2m ∂µdiff ∂ndiff ∇2(nnf

diff + nsf diff)

Travelling wave ansatz for each quantity, e.g., ntot = ntot,eq + δntot exp(−i[ωt − k · x])

Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 14 / 18

Collective mode spectrum

0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 14 T Tc c2g n03m

first sound second sound spin sounds

Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 15 / 18

Collective mode character

0.0 0.2 0.4 0.6 0.8 1.0 3 2 1 1 T Tc ∆ n n ∆ T T

first sound second sound

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

0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 14 T Tc c2g n03m

Jogundas Armaitis (Utrecht) Bose mixture hydrodynamics 2015 06 25 18 / 18