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

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, $T=0.3T_0$, 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 [Ji et al., Phys. Rev. Lett. 114, 105301 (2015)].

Figure 1

Phase diagram of a two-component Bose gas with Raman-induced spin-orbit coupling in the plane of temperature $T$ and Raman Rabi frequency $\mathrm{\Omega }$. The empty circles show a low bound for the first-order ST-PW transition, determined from the vanishing roton gap, while the empty diamonds give the second-order PW-ZW transition. The temperature is measured in units of the critical Bose-Einstein condensate temperature of an ideal spinless Bose gas with density $n/2$, i.e., $T_0=2\pi {\hslash }^2{[(n/2)/\zeta (3/2)]}^{2/3}/\left(m{k}_B\right)$. We take the total density of our SOC Bose gas, $n=1.0k_{\mathrm{r}}^3$, the intraspecies interaction energy, $gn=0.8E_{\mathrm{r}}$, and the interspecies interaction energy, $g_{{}_{\uparrow \downarrow }}n=0.5E_{\mathrm{r}}$. Here, $k_{\mathrm{r}}$ and $E_{\mathrm{r}}={\hslash }^2{k}_{\mathrm{r}}^2/\left(2m\right)$ are Raman wave vector and the recoil energy, respectively.

Figure 2

(a) The Bogoliubov excitation spectra ${\varepsilon }_{\mathbf{q},\tau =-}$ at temperature $T=0.4T_0$ and at various Rabi frequencies. Here, we set $q_{{}_y}=q_z=0$. The roton gap at $\mathrm{\Omega }=2.5E_{\mathrm{r}}$ is explicitly indicated. (b) The roton gap $\mathrm{\Delta }$ as a function of Rabi frequency $\mathrm{\Omega }$ at different temperatures, fitted with second-order polynomials (solid lines).

Figure 3

(a) The contour plot of the condensate momentum $P_x$, in units of $k_{\mathrm{r}}$, as functions of $T$ and $\mathrm{\Omega }$. The boundary $P_x=0$ between PW and ZM phases is highlighted by the white dashed line. (b) The Bogoliubov excitation spectra ${\varepsilon }_{\mathbf{q},\tau =-}$ at various temperatures, with $q_{{}_x}=q_y=0$. The inset shows the temperature dependence of the effective Rabi frequency $\delta {\mathrm{\Omega }}_{\mathrm{eff}}=2g_{\uparrow \downarrow }{\varphi }_{\uparrow }{\varphi }_{\downarrow }$. Here, we take $\mathrm{\Omega }=4.0E_{\mathrm{r}}$.

Figure 4

Sound velocities $c_{+}$ and $c_{-}$ as a function of Rabi frequency $\mathrm{\Omega }$ at $T=0$ (black lines: without quantum fluctuations; blue lines with crosses: with quantum fluctuations), $T=0.2T_0$ (yellow lines with squares), and $T=0.4T_0$ (purple lines with triangles). The sound velocities at $T=0$ near the transition is highlighted in the inset (b). The inset (a) shows the sound velocities in a ${}^{87}\mathrm{Rb}$ SOC gas at $T=0$ (black lines) and $T=0.3T_0$ (purple dotted lines). The symbols are the experimental data [26]. To simulate the experiment, we take a total density $n=0.46k_{\mathrm{r}}^3,gn=0.38E_{\mathrm{r}}$, and $g_{{}_{\uparrow \downarrow }}/g=100.99/101.20$.