Superstripes and the excitation spectrum of a spin-orbit-coupled BEC

Transcription

1 INO ISTITUTO NAZIONALE DI OTTICA UNIVERSITA DEGLI STUDI DI TRENTO Superstripes and the excitation spectrum of a spin-orbit-coupled BEC Yun Li, Giovanni I. Martone, Lev P. Pitaevskii, and Sandro Stringari Trieste, 3 May

2 Breaking of two symmetries Bose condensation of defectons A. F. Andreev and I. M. Lifshitz JETP 9, 7 (969)

3 Breaking of two symmetries Soft-core interactions (a).5 (b) y Bose condensation of defectons y (c).5 (d) A. F. Andreev and I. M. Lifshitz JETP 9, 7 (969).5 x.5 x N. Henkel, et al., PRL 4, 953 () F. Cinti, et al., PRL 5, 353 ()

4 Spin-orbit coupled BEC (single particle picture) bulk, no interactions, Ω = BEC E(p) / k p / k Single-particle Hamiltonian h = ˆ(px k σ z) + p + Ω σx + δ σz = U h lab U U = e iθ(x)σz/ Equal Rashba and Dresselhaus couplings

5 Spin-orbit coupled BEC (single particle picture) bulk, no interactions, Ω BEC E(p) / k p / k Single-particle Hamiltonian h = ˆ(px k σ z) + p + Ω σx + δ σz = U h lab U U = e iθ(x)σz/ ±k = ±k s Ω 4k 4 Equal Rashba and Dresselhaus couplings

6 Spin-orbit coupled BEC (single particle picture) bulk, no interactions, Ω BEC E(p) / k p / k Single-particle Hamiltonian h = ˆ(px k σ z) + p + Ω σx + δ σz = U h lab U U = e iθ(x)σz/ ±k = ±k s Ω 4k 4 Equal Rashba and Dresselhaus couplings

7 Spin-orbit coupled BEC (single particle picture) bulk, no interactions, Ω BEC E(p) / k p / k Single-particle Hamiltonian h = ˆ(px k σ z) + p + Ω σx + δ σz = U h lab U U = e iθ(x)σz/ ±k = ±k s Ω 4k 4 Equal Rashba and Dresselhaus couplings

8 Spin-orbit coupled BEC (single particle picture) bulk, no interactions, Ω BEC E(p) / k p / k Single-particle Hamiltonian h = ˆ(px k σ z) + p + Ω σx + δ σz = U h lab U U = e iθ(x)σz/ ±k = ±k s Ω 4k 4 Equal Rashba and Dresselhaus couplings

9 Spin-orbit coupled BEC (single particle picture) bulk, no interactions, Ω BEC E(p) / k p / k Single-particle Hamiltonian h = ˆ(px k σ z) + p + Ω σx + δ σz = U h lab U U = e iθ(x)σz/ ±k = ±k s Ω 4k 4 Equal Rashba and Dresselhaus couplings

10 Spin-orbit coupled BEC (single particle picture) bulk, no interactions, Ω BEC E(p) / k p / k Single-particle Hamiltonian h = ˆ(px k σ z) + p + Ω σx + δ σz Equal Rashba and Dresselhaus couplings R aman Coupling /E L Minima location in units of k L ±k = ±k s Ω 4k 4 Lin et al, Nature 83, 53 ()

11 Many-body ground state Three quantum phases γ = (g g ) /(g + g ) >, n (c) = k / (γg) (I). k, C + = C, σ z = (II). k, C = or C + =, σ z (III). k =, σ z = ψ = ψ ψ «cos θ = k k, = n» ««cos θ sin θ C + e ikx + C e ik x sin θ cos θ σ z = k k ` C+ C LY, Pitaevskii, Stringari, PRL 8, 53 ()

12 Many-body ground state Three quantum phases γ = (g g ) /(g + g ) >, n (c) = k / (γg) (I). k, C + = C, σ z = (II). k, C = or C + =, σ z (III). k =, σ z = Ω (I-II) cr = k p γ/( + γ) small for 87 Rb Ω (II-III) cr = k P hase s separation,..8 Dynamics at Ω =.6 E L. τ =.4(3) s Hold time t h (s) ΩC =.() E L Raman Coupling Ω /EL

13 Many-body ground state Three quantum phases γ = (g g ) /(g + g ) >, n (c) = k / (γg).4 x 3 ψ, (x) [cm 3 ]..8.6 k x / π Ω (I-II) cr = k p γ/( + γ) small for 87 Rb Ω (II-III) cr = k To increase the effect of the contrast, choose larger values of γ

14 Excitation spectrum in phase II ω + / k 3 ω / k q / k G /k =, G /k =. Ω/k =.,.33,.46 Despite spinor nature, occurrence of Raman coupling gives rise to a single gapless branch Emergence of a roton minimum at finite q: a tendency of the system towards crystallization Martone, LY, Pitaevskii and Stringari, PRA 86, 636()

15 Excitation spectrum in phase II.4 E(p). ω / k q / k G /k =, G /k =. Ω/k =.,.33,.46 Despite spinor nature, occurrence of Raman coupling gives rise to a single gapless branch Emergence of a roton minimum at finite q: a tendency of the system towards crystallization Martone, LY, Pitaevskii and Stringari, PRA 86, 636()

16 Excitation spectrum in phase II.4 E(p). ω / k q / k G /k =, G /k =. Ω/k =.,.33,.46 Despite spinor nature, occurrence of Raman coupling gives rise to a single gapless branch Emergence of a roton minimum at finite q: a tendency of the system towards crystallization Martone, LY, Pitaevskii and Stringari, PRA 86, 636()

17 Excitation spectrum in phase I Equilibrium state.4 x 3 ψ, (x) [cm 3 ]..8.6 k x / π ψ G /k =.3, G /k =.8, Ω/k = ψ Translational invariance symmetry breaking U() symmetry breaking «= P «a k + K K e i( K k )x, b k + K K is the reciprocal lattice vector

18 Excitation spectrum in phase I Bogoliubov + Bloch theory ψ = e iµt ψ ψ ψ «+ «u (r) e iωt + u (r) «v (r) v (r) e iωt u q, (r) = e ik x X K iq r+i Kx U q, K e v q, (r) = e ik x X K iq r i Kx V q, K e ω / k / k Emergence of two gapless bands Vanishing of the frequency at the edge of the Brillouin zone Divergent behavior of static structure factor in density channel

19 Excitation spectrum in phase I Superfluid Supersolid 5 a) b) 5 Soft-core interactions ω ω ka/π..4 ka/π.6.8 S. Saccani et al., PRL 8, 753 ().6 Emergence of two gapless bands Vanishing of the frequency at the edge of the Brillouin zone ω / k Divergent behavior of static structure factor in density channel qx / k

20 Density structure factor.6. ω / k.8 S( ) / k / k S(ω, ) / k ω / k..6

21 Density structure factor.6. ω / k.8 S( ) / k / k S(ω, ) Upper branch is a density wave at small / k ω / k..6

22 Density structure factor.6. ω / k.8 S( ) / k / k S(ω, ) Upper branch is a density wave at small / k ω / k..6 S() for the lower branch diverges

23 Spin structure factor.6. ω / k.8.4 S σ ( ).5.5 / k / k S σ (ω, ) / k ω / k..6

24 Spin structure factor.6. ω / k.8.4 S σ ( ).5.5 / k / k S σ (ω, ) Lower branch is a spin wave at small / k ω / k..6

25 Sum rule approach Define -component density operator F and ( q B)-component momentum density operator G F = X e iqxx j, [F, G] = N e iqbx j G = X i hp xj e i(qx q B)x j + e i(qx q B)x j p xj / j p th moments: m p(o) = P l ( O l + O l )ω p l q B F G m qx ( q B) m S() q B m χ() Bogoliubov inequality: m (F)m (G) [F, G] χ() /( q B) Uncertainty inequality: m (F)m (G) [F, G] S() / q B

26 Sound velocity c / k Upper branch Lower branch I II phase transition Ω / k (d, s) c > c c (d) (d, s) x, inertia of flow caused by stripes (g + g ) n/, well reproduced by usual Bogoliubov sound velocity

27 Conclusions Excitation spectrum in the stripe phase: double gapless band structure At small wave vector the lower and upper branches have, respectively, a spin and density nature Close to the first Brillouin zone the lower branch acquires an important density character, S() diverges LY, Martone, Pitaevskii, Stringari arxiv: ω / k ω / k.5.5 / k.5.5 / k

28 Thank you!