Drag force and superfluidity in the supersolid striped phase of a spin-orbit-coupled Bose gas

/ 6 Drag force and superfluidity in the supersolid striped phase of a spin-orbit-coupled Bose gas Giovanni Italo Martone with G. V. Shlyapnikov Worhshop on Exploring Nuclear Physics with Ultracold Atoms ECT*, Trento, June 9th, 8

/ 6 Introduction Ultracold atomic gases: ideal tool to explore new phenomena

/ 6 Introduction Ultracold atomic gases: ideal tool to explore new phenomena Main advantages: can be formed by bosons, fermions, or mixtures of both;

/ 6 Introduction Ultracold atomic gases: ideal tool to explore new phenomena Main advantages: can be formed by bosons, fermions, or mixtures of both; tunability of interparticle interactions;

/ 6 Introduction Ultracold atomic gases: ideal tool to explore new phenomena Main advantages: can be formed by bosons, fermions, or mixtures of both; tunability of interparticle interactions; large variety of possible energy landscapes: harmonic, periodic, quasiperiodic, disordered,...

/ 6 Introduction Ultracold atomic gases: ideal tool to explore new phenomena Main advantages: can be formed by bosons, fermions, or mixtures of both; tunability of interparticle interactions; large variety of possible energy landscapes: harmonic, periodic, quasiperiodic, disordered,... Interesting question: can one use quantum gases to simulate phenomena related to orbital magnetism or spin-orbit coupling?

/ 6 Introduction Ultracold atomic gases: ideal tool to explore new phenomena Main advantages: can be formed by bosons, fermions, or mixtures of both; tunability of interparticle interactions; large variety of possible energy landscapes: harmonic, periodic, quasiperiodic, disordered,... Interesting question: can one use quantum gases to simulate phenomena related to orbital magnetism or spin-orbit coupling? YES!

Y.-J. Lin et al., Nature 47, 83 () 3 / 6 Light-induced spin-orbit coupling on BECs - I NIST experimental setup: 87 Rb BEC in F = electronic manifold

Y.-J. Lin et al., Nature 47, 83 () 3 / 6 Light-induced spin-orbit coupling on BECs - I NIST experimental setup: 87 Rb BEC in F = electronic manifold Bias magnetic field: Zeeman shift

Y.-J. Lin et al., Nature 47, 83 () 3 / 6 Light-induced spin-orbit coupling on BECs - I NIST experimental setup: 87 Rb BEC in F = electronic manifold Bias magnetic field: Zeeman shift Two counterpropagating Raman lasers Frequencies ω L and ω L + ω L Wavevector difference k r = k r ê x

Y.-J. Lin et al., Nature 47, 83 () 3 / 6 Light-induced spin-orbit coupling on BECs - I NIST experimental setup: 87 Rb BEC in F = electronic manifold Bias magnetic field: Zeeman shift Two counterpropagating Raman lasers Frequencies ω L and ω L + ω L Wavevector difference k r = k r ê x Lasers induce transitions between internal states with strength Ω

Y.-J. Lin et al., Nature 47, 83 () 3 / 6 Light-induced spin-orbit coupling on BECs - I NIST experimental setup: 87 Rb BEC in F = electronic manifold Bias magnetic field: Zeeman shift Two counterpropagating Raman lasers Frequencies ω L and ω L + ω L Wavevector difference k r = k r ê x Lasers induce transitions between internal states with strength Ω close to Raman resonance, δ = ω L ω Z

Y.-J. Lin et al., Nature 47, 83 () 3 / 6 Light-induced spin-orbit coupling on BECs - I NIST experimental setup: 87 Rb BEC in F = electronic manifold Bias magnetic field: Zeeman shift Two counterpropagating Raman lasers Frequencies ω L and ω L + ω L Wavevector difference k r = k r ê x Lasers induce transitions between internal states with strength Ω close to Raman resonance, δ = ω L ω Z + detuned from Raman resonance due to quadratic shift ω q

Y.-J. Lin et al., Nature 47, 83 () 3 / 6 Light-induced spin-orbit coupling on BECs - I NIST experimental setup: 87 Rb BEC in F = electronic manifold Bias magnetic field: Zeeman shift Two counterpropagating Raman lasers Frequencies ω L and ω L + ω L Wavevector difference k r = k r ê x Lasers induce transitions between internal states with strength Ω close to Raman resonance, δ = ω L ω Z + detuned from Raman resonance due to quadratic shift ω q Effective two-level system

4 / 6 Light-induced spin-orbit coupling on BECs - II Single-particle Hamiltonian in rotating-wave approximation h = p m + Ω [ ] σ + e i(krx ωlt) + H.c. ω Z σ z Invariant under helicoidal translations T h (d) = exp [id (p x + k r σ z )]

4 / 6 Light-induced spin-orbit coupling on BECs - II Single-particle Hamiltonian in rotating-wave approximation h = p m + Ω [ ] σ + e i(krx ωlt) + H.c. ω Z σ z Invariant under helicoidal translations T h (d) = exp [id (p x + k r σ z )] Remove space and time dependence through U = e i(krx ω Lt)σ z/ h SO = [ ] (p x k r σ z ) + p + Ω m σ x + δ σ z

4 / 6 Light-induced spin-orbit coupling on BECs - II Single-particle Hamiltonian in rotating-wave approximation h = p m + Ω [ ] σ + e i(krx ωlt) + H.c. ω Z σ z Invariant under helicoidal translations T h (d) = exp [id (p x + k r σ z )] Remove space and time dependence through U = e i(krx ω Lt)σ z/ h SO = [ ] (p x k r σ z ) + p + Ω m σ x + δ σ z Equal-weighted Rashba and Dresselhaus SOCs + Rabi coupling or External spin-dependent gauge potential

5 / 6 Spin-orbit-coupled BEC (single-particle picture) Single-particle Hamiltonian h SO = [ ] (p x k r σ z ) + p + Ω m σ x + δ σ z

5 / 6 Spin-orbit-coupled BEC (single-particle picture) Single-particle Hamiltonian h SO = [ ] (p x k r σ z ) + p + Ω m σ x + δ σ z

5 / 6 Spin-orbit-coupled BEC (single-particle picture) Single-particle Hamiltonian h SO = [ ] (p x k r σ z ) + p + Ω m σ x + δ σ z Energy dispersion Ω = ε± (px) /Er p x /k r

5 / 6 Spin-orbit-coupled BEC (single-particle picture) Single-particle Hamiltonian h SO = [ ] (p x k r σ z ) + p + Ω m σ x + δ σ z Energy dispersion two regimes Ω Ω < 4E r : two degenerate minima ε± (px) /Er ( ) Ω ±k = ±k r 4E r p x /k r

5 / 6 Spin-orbit-coupled BEC (single-particle picture) Single-particle Hamiltonian h SO = [ ] (p x k r σ z ) + p + Ω m σ x + δ σ z Energy dispersion two regimes ε± (px) /Er Ω p x /k r Ω < 4E r : two degenerate minima ( ) Ω ±k = ±k r 4E r Ω 4E r : single minimum k =

6 / 6 Many-body ground state A unique many-body ground state is selected by interactions T.-L. Ho and S. Zhang, PRL 7, 543 () Y. Li, L. P. Pitaevskii, S. Stringari, PRL 8, 53 ()

6 / 6 Many-body ground state A unique many-body ground state is selected by interactions T.-L. Ho and S. Zhang, PRL 7, 543 () Y. Li, L. P. Pitaevskii, S. Stringari, PRL 8, 53 () Look for mean-field ground state: variational ansatz Ψ = ( ) ψ ψ = [ ( ) ( ) ] N cos θ C + e ikx sin θ + C V sin θ e ik x cos θ

6 / 6 Many-body ground state A unique many-body ground state is selected by interactions T.-L. Ho and S. Zhang, PRL 7, 543 () Y. Li, L. P. Pitaevskii, S. Stringari, PRL 8, 53 () Look for mean-field ground state: variational ansatz Ψ = ( ) ψ ψ = [ ( ) ( ) ] N cos θ C + e ikx sin θ + C V sin θ e ik x cos θ Minimization of energy E = { dr Ψ h SO Ψ + g + (Ψ Ψ) + g } (Ψ σ z Ψ), g ± = g ± g

Y. Li, L. P. Pitaevskii, S. Stringari, PRL 8, 53 () 7 / 6 Zero-temperature phase diagram 4 3.5.5.8.6.4. Color: k /k r

Y. Li, L. P. Pitaevskii, S. Stringari, PRL 8, 53 () 7 / 6 Zero-temperature phase diagram 4 3.5.5.8.6.4. Striped phase I Two counter-propagating waves interfere, making a standing wave Color: k /k r Ψ ST = [( ) ( ) ] n cos θ e ikx + e iϕ sin θ e ik x sin θ cos θ

Y. Li, L. P. Pitaevskii, S. Stringari, PRL 8, 53 () 7 / 6 Zero-temperature phase diagram 4 3.5.5.8.6.4. Striped phase I Two counter-propagating waves interfere, making a standing wave Plane-wave phase II Single propagating wave Spin polarization σ z = k /k r Color: k /k r Ψ + PW = n ( ) cos θ e ik x sin θ or Ψ PW = n ( ) sin θ e ik x cos θ

Y. Li, L. P. Pitaevskii, S. Stringari, PRL 8, 53 () 7 / 6 Zero-temperature phase diagram 4 3.5.5 Color: k /k r.8.6.4. Striped phase I Two counter-propagating waves interfere, making a standing wave Plane-wave phase II Single propagating wave Spin polarization σ z = k /k r Zero-momentum phase III Uniform wavefunction Ψ SM = ( ) n

Y. Li, G. I. Martone, L. P. Pitaevskii, S. Stringari, PRL, 353 (3) 8 / 6 Striped phase: ground state properties Periodic structure arising from SO coupl. + nonlinearity of GP theory

Y. Li, G. I. Martone, L. P. Pitaevskii, S. Stringari, PRL, 353 (3) 8 / 6 Striped phase: ground state properties Periodic structure arising from SO coupl. + nonlinearity of GP theory Expand stationary wavefunction into Bloch waves: Ψ(r) = ( ak n + K K b k + K ) e i(k + K)x K reciprocal lattice vectors.7 n, / n.5.3 k x/π

Y. Li, G. I. Martone, L. P. Pitaevskii, S. Stringari, PRL, 353 (3) 8 / 6 Striped phase: ground state properties Periodic structure arising from SO coupl. + nonlinearity of GP theory Expand stationary wavefunction into Bloch waves: Ψ(r) = ( ak n + K K b k + K ) e i(k + K)x K reciprocal lattice vectors.7 λ n, / n.5 C Period of stripes: λ = π/k Contrast C Ω/E r.3 k x/π

Y. Li, G. I. Martone, L. P. Pitaevskii, S. Stringari, PRL, 353 (3) 8 / 6 Striped phase: ground state properties Periodic structure arising from SO coupl. + nonlinearity of GP theory Expand stationary wavefunction into Bloch waves: Ψ(r) = ( ak n + K K b k + K ) e i(k + K)x K reciprocal lattice vectors.7 Spontaneously broken symmetries: n, / n.5.3 k x/π U() superfluidity translational crystalline order

9 / 6 Excitation spectrum in Striped phase 3.5.5

9 / 6 Excitation spectrum in Striped phase 3 Double gapless band structure.5.5

9 / 6 Excitation spectrum in Striped phase 3 Double gapless band structure Two sound modes at q.5.5

9 / 6 Excitation spectrum in Striped phase 3 q B = k.5.5 Double gapless band structure Two sound modes at q Frequency of lowest modes vanishes at Brillouin point q B

/ 6 Static structure factors in Striped phase.8.6.4..5.5.8.6.4..5.5

/ 6 Static structure factors in Striped phase.8.6.4..5.5.8.6.4..5.5 Lower branch: spin mode at low q x, strong density character at q x q B

/ 6 Static structure factors in Striped phase.8.6.4..5.5.8.6.4..5.5 Lower branch: spin mode at low q x, strong density character at q x q B Upper branch: density mode at q x

/ 6 Static structure factors in Striped phase.8.6.4..5.5.8.6.4..5.5 Lower branch: spin mode at low q x, strong density character at q x q B Upper branch: density mode at q x Density structure factor diverges at Brillouin point

G. E. Astrakharchik and L. P. Pitaevskii, PRA 7, 368 (4) / 6 Moving impurity in a BEC. Drag force Consider heavy pointlike impurity moving with velocity v U imp (r, t) = g imp δ(r vt)

G. E. Astrakharchik and L. P. Pitaevskii, PRA 7, 368 (4) / 6 Moving impurity in a BEC. Drag force Consider heavy pointlike impurity moving with velocity v U imp (r, t) = g imp δ(r vt) Assume small g imp, expand condensate order parameter Ψ(r, t) = Ψ (r) + δψ(r, t) Calculate δψ(r, t) by solving linearized Gross-Pitaevskii equation

G. E. Astrakharchik and L. P. Pitaevskii, PRA 7, 368 (4) / 6 Moving impurity in a BEC. Drag force Consider heavy pointlike impurity moving with velocity v U imp (r, t) = g imp δ(r vt) Assume small g imp, expand condensate order parameter Ψ(r, t) = Ψ (r) + δψ(r, t) Calculate δψ(r, t) by solving linearized Gross-Pitaevskii equation Evaluate drag force from standard definition F = dr Ψ (r, t) [ U imp (r, t)] Ψ(r, t) = [Ψ (r, t)ψ(r, t)] r=vt

G. E. Astrakharchik and L. P. Pitaevskii, PRA 7, 368 (4) / 6 Drag force in a standard BEC Drag force in standard BEC (Astrakharchik Pitaevskii s formula) ( ) F C = 4π nb mv c v Θ(v c)ˆv

G. E. Astrakharchik and L. P. Pitaevskii, PRA 7, 368 (4) / 6 Drag force in a standard BEC Drag force in standard BEC (Astrakharchik Pitaevskii s formula) ( ) F C = 4π nb mv c v Θ(v c)ˆv Vanishes at v < v c = c: consistency with Landau criterion Antiparallel to ˆv, proportional to v for large v

G. E. Astrakharchik and L. P. Pitaevskii, PRA 7, 368 (4) / 6 Drag force in a standard BEC Drag force in standard BEC (Astrakharchik Pitaevskii s formula) ( ) F C = 4π nb mv c v Θ(v c)ˆv Vanishes at v < v c = c: consistency with Landau criterion Antiparallel to ˆv, proportional to v for large v Two-component BEC: pure spin and pure density Bogoliubov modes ω d,s (q) = [ q m ( q m + mc d,s )] / Spin mode not excited by moving impurity: zero contribution to F C F C identical to F C with c c d

3 / 6 Drag force in SO-coupled BECs General formula (after time averaging in stripe phase) F = πg imp V q l, q δρ q δ (ω l,q q v) l,q Same structure as in other systems with multiple bands in spectrum

3 / 6 Drag force in SO-coupled BECs General formula (after time averaging in stripe phase) F = πg imp V q l, q δρ q δ (ω l,q q v) l,q Same structure as in other systems with multiple bands in spectrum Critical velocity for exciting l-th branch ω l,q v c,l (ˆv) = min q ˆv> q ˆv Reproduces Landau criterion for anisotropic systems

Drag force in SO-coupled BECs General formula (after time averaging in stripe phase) F = πg imp V q l, q δρ q δ (ω l,q q v) l,q Same structure as in other systems with multiple bands in spectrum Critical velocity for exciting l-th branch ω l,q v c,l (ˆv) = min q ˆv> q ˆv Reproduces Landau criterion for anisotropic systems Noncollinearity of drag force with velocity v = v(cos θ v ˆx + sin θ v ŷ) F = F (cos θ F ˆx + sin θ F ŷ), π/ θ F θ v π/ Collinearity restored (θ F = θ v ) only if θ v =, π/ 3 / 6

G. I. Martone and G. V. Shlyapnikov, arxiv:85.55 4 / 6 Drag force in striped phase v c,l (ˆv) vanishes if ˆv does not lie in yz-plane θf /π 8 8 6 θ v = θ v = π/6 6 8 4 θ v = π/4 6 θ v = π/3 4 θ v = π/ 4 Ω/E r =. Ω/E r =. Ω/E r =...4.6.8..4.6.8.4.4..4.6.8.3.3.3 F........4.6.8 v..4.6.8 v..4.6.8 v

Drag force in striped phase v c,l (ˆv) vanishes if ˆv does not lie in yz-plane θf /π 8 8 6 θ v = θ v = π/6 6 8 4 θ v = π/4 6 θ v = π/3 4 θ v = π/ 4 Ω/E r =. Ω/E r =. Ω/E r =...4.6.8..4.6.8.4.4..4.6.8.3.3.3 F....4.6.8 v....4.6.8 v....4.6.8 v At low v main contribution from l =, modes with q q B F x 6π k b n f ( ) m c v x + v y c, F y 6π k b n f m c 4 v xv y G. I. Martone and G. V. Shlyapnikov, arxiv:85.55 4 / 6

Time scale for energy dissipation τ τ 8 6 4 θ v = θ v = π/6 θ v = π/4 θ v = π/3 θ v = π/ Ω/E r =...4.6.8.5 Define characteristic time for energy dissipation of impurity τ = Ė E = ε χ F v, χ = N imp N.5 Ω/E r =...4.6.8.5 τ.5 Ω/E r =...4.6.8 v G. I. Martone and G. V. Shlyapnikov, arxiv:85.55 5 / 6

Time scale for energy dissipation τ τ 8 6 4 θ v = θ v = π/6 θ v = π/4 θ v = π/3 θ v = π/ Ω/E r =...4.6.8.5.5 Ω/E r =...4.6.8.5 Define characteristic time for energy dissipation of impurity τ = Ė E = ε χ F v, χ = N imp N τ comparable or larger than typical duration of an experiment for a reasonable range of velocities τ.5 Ω/E r =...4.6.8 v G. I. Martone and G. V. Shlyapnikov, arxiv:85.55 5 / 6

6 / 6 Conclusions Main results on spin-orbit-coupled Bose gases: appearance of new quantum phases (striped, spin-polarized) and new phase transitions; striped phase exhibits typical features of supersolidity, including double gapless band structure of excitation spectrum; emergence of a finite drag force for an arbitrarily small velocity of impurity in striped phase; drag force not collinear with direction of the motion; very large characteristic time for energy dissipation.