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arXiv:1706.01170 · Physical Review A

Quantum and thermal fluctuations in a Raman spin-orbit-coupled Bose gas

Xiao-Long Chen, Xia-Ji Liu, Hui Hu

Published Jul 25, 2017• 17 citations
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Abstract

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 $^{87}\mathrm{Rb}$ atoms at the typical experimental temperature, $T=0.3{T}_{0}$, where ${T}_{0}$ 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)].

DOI
10.1103/physreva.96.013625

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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)].
Abstract We implement the Hartree–Fock–Bogoliubov theory with the Popov approximation for a homogeneous Raman-induced spin–orbit-coupled spin-1 Bose–Einstein condensate and investigate the effects of finite temperature (T) on the ground-state phase diagram. We calculate the roton gap as a function of Raman coupling (Ω) or quadratic Zeeman field strength (ε) to extract the critical points separating the supersolid stripe phase from the plane wave or zero-momentum phase at finite temperatures. We present a few representative finite-temperature phase diagrams for the system in the T − Ω and T − ϵ planes. Our observations indicate that the supersolid stripe phase melts at finite temperatures. We also discuss the contrasting roles of quantum and thermal fluctuations in shifting the phase boundary separating the supersolid stripe from the plane-wave phase.
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 study two-component Bose gases with Raman induced spin-orbit coupling via a perturbation approach at finite temperature. For weak coupling, free energy is expanded in terms of Raman coupling strength up to the second order, where the coefficient (referred to as Raman susceptibility) is determined according to linear response theory. The equation of state for the stripe phase and the plane-wave phase are obtained in Popov approximation, and the first order transition between these two phases is investigated. As temperature increases, we find the phase boundary bends toward the stripe phase side in most temperature regions, which implies the ferromagnetic order is more robust than the crystalline order in presence of thermal fluctuations. Our results qualitatively agree with the recent experimental observation in rubidium atomic gases. A method to measure Raman susceptibility through the two-photon Bragg scattering experiment is also discussed.
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.
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.
Close to the superfluid plane-wave (PW)--supersolid stripe (ST) phase transition point of a zero-temperature quasi-one-dimensional spin-orbit-coupled Bose gas, we find that an increase in temperature induces a phase transition to the supersolid phase with a broken translational symmetry from the superfluid plane-wave phase. We use the Hartree-Fock-Bogoliubov theory with the Popov approximation to investigate the effect of thermal fluctuations on the collective excitation spectrum and investigate the softening of the spin-dipole mode corresponding to the shift in the quantum critical point. This is in stark contrast to the PW-ST phase transition in a homogeneous system where nonzero temperatures facilitate the melting of the stripe phase.