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arXiv:1605.02136 · Physical Review A

Superfluid density of a spin-orbit-coupled Bose gas

Yi‐Cai Zhang, Zeng-Qiang Yu, Tai Kai Ng, Shizhong Zhang, Лев П. Питаевский, S. Stringari

Published Sep 30, 2016• 53 citations
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Abstract

We discuss the superfluid properties of a uniform, weakly interacting Bose-Einstein condensed gas with spin-orbit coupling, realized recently in experiments. We find a finite normal fluid density ${\ensuremath{\rho}}_{n}$ at zero temperature which turns out to be a function of the Raman coupling. In particular, the entire fluid becomes normal at the transition point from the zero momentum to the plane wave phase, even though the condensate fraction remains finite. We emphasize the crucial role played by the breaking of Galilean invariance and by the gapped branch of the elementary excitations whose contribution to various sum rules is discussed explicitly. Our predictions for the superfluid density are successfully compared with the available experimental results based on the measurement of the sound velocities.

DOI
10.1103/physreva.94.033635

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We discuss the superfluid properties of a uniform, weakly interacting Bose-Einstein condensed gas with spin-orbit coupling, realized recently in experiments. We find a finite normal fluid density \ensuremathρn{\ensuremath{\rho}}_{n}\ensuremathρn​ at zero temperature which turns out to be a function of the Raman coupling. In particular, the entire fluid becomes normal at the transition point from the zero momentum to the plane wave phase, even though the condensate fraction remains finite. We emphasize the crucial role played by the breaking of Galilean invariance and by the gapped branch of the elementary excitations whose contribution to various sum rules is discussed explicitly. Our predictions for the superfluid density are successfully compared with the available experimental results based on the measurement of the sound velocities.
We investigate the superfluidity of a three-dimensional weakly interacting Bose gas with a one-dimensional Raman-type spin-orbit coupling at both zero and finite temperatures. Using the imaginary-time Green's function within the Bogoliubov approximation, we explicitly derive analytic expressions of the current-current response functions in the plane-wave and zero-momentum phases, from which we extract the superfluid density in the limits of long wavelength and zero frequency. At zero temperature, we check that the resultant superfluid density agrees exactly with our previous analytic prediction obtained from a phase-twist approach. Both results also satisfy a generalized Josephson relation in the presence of spin-orbit coupling. At finite temperature, we find a significant nonmonotonic temperature dependence of superfluid density near the transition from the plane-wave phase to the zero-momentum phase. We show that this nontrivial behavior might be understood from the sound velocity, which has a similar temperature dependence. The nonmonotonic temperature dependence is also shared by Landau critical velocity, above which the spin-orbit-coupled Bose gas loses its superfluidity. Our results would be useful for further theoretical and experimental studies of superfluidity in exotic spin-orbit-coupled quantum gases.
We theoretically study a three-dimensional weakly interacting Bose gas with Raman-induced spin-orbit coupling at finite temperature. By employing a generalized Hartree-Fock-Bogoliubov theory with Popov approximation, we determine a complete finite-temperature phase diagram of three exotic condensation phases (i.e., the stripe, plane-wave, and zero-momentum phases), against both quantum and thermal fluctuations. We find that the plane-wave phase is significantly broadened by thermal fluctuations. The phonon mode and sound velocity at the transition from the plane-wave phase to the zero-momentum phase are thoughtfully analyzed. At zero temperature, we find that quantum fluctuations open an unexpected gap in sound velocity at the phase transition, in stark contrast to the previous theoretical prediction of a vanishing sound velocity. At finite temperature, thermal fluctuations continue to significantly enlarge the gap, and simultaneously shift the critical minimum. For a Bose gas of 87Rb^{87}\mathrm{Rb}87Rb atoms at the typical experimental temperature, T=0.3T0T=0.3{T}_{0}T=0.3T0​, where T0{T}_{0}T0​ is the critical temperature of an ideal Bose gas without spin-orbit coupling, our results of gap opening and critical minimum shifting in the sound velocity are qualitatively consistent with the recent experimental observation [Ji et al., Phys. Rev. Lett. 114, 105301 (2015)].
We study systematically the quantum corrections to a weakly interacting Bose-Einstein condensate with spin-orbit coupling. We show that quantum fluctuations, enhanced by the spin-orbit coupling, modify quantitatively the mean-field properties such as the superfluid density, spin polarizability, and sound velocity. We find that the phase boundary between the plane wave and zero momentum phases is shifted to a smaller transverse field. We also calculate the Beliaev and Landau damping rates and find that the Landau process dominates the quasiparticle decay even at low temperature.
By calculating the density response function we identify the excitation spectrum of a Bose-Einstein condensate with equal Rashba and Dresselhaus spin-orbit coupling. We find that the velocity of sound along the direction of spin-orbit coupling is deeply quenched and vanishes when one approaches the second-order phase transition between the plane-wave and the zero momentum quantum phases. We also point out the emergence of a roton minimum in the excitation spectrum for small values of the Raman coupling, providing the onset of the transition to the stripe phase. Our findings point out the occurrence of a strong anisotropy in the dynamic behavior of the gas. A hydrodynamic description accounting for the collective oscillations in both uniform and harmonically trapped gases is also derived.
In this paper we investigate the properties of Bose gases with Raman-induced spin-orbit(SO) coupling. It is found that the SO coupling can greatly modify the single particle density-of-state, and thus lead to non-monotonic behavior of the condensate depletion, the Lee-Huang-Yang correction of ground-state energy and the transition temperature of a non-interacting Bose-Einstein condensate. The presence of the SO coupling also breaks the Galilean invariance, and this gives two different critical velocities, corresponding to the movement of the condensate and the impurity respectively. Finally, we show that with SO coupling, the interactions modify the BEC transition temperature even at Hartree-Fock level, in contrast to the ordinary Bose gas without SO coupling. All results presented here can be directly verified in the current cold atom experiments using Raman laser-induced gauge field.
We theoretically investigate a three-dimensional weakly interacting Bose gas with one-dimensional Raman-type spin-orbit coupling at zero temperature. By employing an improved ansatz, including high-order harmonics in the stripe phase, we show that the critical transition from the stripe to the plane-wave phases is shifted to a relatively larger Rabi frequency compared to the prediction by previous work [Li et al., Phys. Rev. Lett. 108, 225301 (2012)] using a first-order stripe ansatz. We also determine the quantum depletion and superfluid density over a large range of Rabi frequency in different phases. The depletion exhibits an intriguing behavior with a discontinuous jump at the transition between the stripe and plane-wave phases, and a maximum at the transition between the plane-wave and zero-momentum phases. The superfluid density is derived through a phase-twist method. In the plane-wave and zero-momentum phases, it is significantly suppressed along the spin-orbit-coupling direction and vanishes at the transition, consistent with a recent work [Zhang et al., Phys. Rev. A 94, 033635 (2016)], while in the stripe phase, it smoothly decreases with increasing Rabi frequency. Our predictions would be useful for further theoretical and experimental studies of the exotic supersolid stripe phase.