Title: Quantum and thermal fluctuations in a Raman spin-orbit coupled Bose gas,
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 $^{87}$Rb atoms at the typical experimental temperature, $T=0.3T_{0}$, 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
{[}S.-C. Ji \textit{et al.}, Phys. Rev. Lett. \textbf{114}, 105301 (2015){]}.