Superfluidity of a Raman spin-orbit-coupled Bose gas at finite temperature

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.

Figure 1

The longitudinal and transverse (i.e., dotted red and dashed blue lines, respectively) static current-current response functions $\frac{m}{N}{\chi }_{JJ}^{L,T}(\mathbf{q};0)$ in the $\mathbf{q}\to 0$ limit, as a function of Rabi frequency $\mathrm{\Omega }$. Their difference (i.e., solid black line) reveals the superfluid density fraction. Here we take the parameters as in the experiment [37], with a total density $n=0.46k_{\mathrm{r}}^3$, $gn=0.38E_{\mathrm{r}}$ and $g_{{}_{\uparrow \downarrow }}/g=100.99/101.20$, i.e., $G_1=0.19E_{\mathrm{r}}$ and $G_2\approx 0$, giving rise to ${\mathrm{\Omega }}_{c1}=0$ and ${\mathrm{\Omega }}_{c2}=4E_{\mathrm{r}}$ (i.e., the vertical dotted line).

Figure 2

Superfluid density fraction ${\rho }_{\mathrm{s}}^{\left(\mathrm{SOC}\right)}/\rho$ in the SOC direction as a function of Rabi frequency at (a) equal and (b) unequal intra- and interspin interactions. Here the predictions given by the current-current response function and by the generalized Josephson relation are denoted by red circles and blue crosses, respectively. The black lines indicate the analytic prediction using a phase-twist approach in Ref. [34]. The vertical dashed and dotted curves indicate the critical ${\mathrm{\Omega }}_{c1}$ and ${\mathrm{\Omega }}_{c2}$, respectively, i.e., Eqs. (11a) and (11b). Here, we take the interaction energies (a) [$G_1$, $G_2]/E_{\mathrm{r}}=[0.8,0]$ and (b) [$G_1$, $G_2]/E_{\mathrm{r}}=[0.7,0.1]$.

Figure 3

Superfluid density fraction ${\rho }_{\mathrm{s}}^{\left(\mathrm{SOC}\right)}/\rho$ in the SOC direction as a function of Rabi frequency at various values of temperature $T/T_0={10}^{-4}$, 0.1, 0.3, and 0.5 (i.e., lines) for a SOC ${}^{87}\mathrm{Rb}$ gas. Here $T_0$ is the critical BEC temperature of an ideal spinless Bose gas with density $n$, i.e., $T_0=2\pi {\hslash }^2{[n/\zeta (3/2)]}^{2/3}/\left(m{k}_B\right)$, and the circles denote the zero-temperature analytic result from Ref. [34]. The zoomed-in inset focuses on the regime near the critical Rabi frequency. Other parameters are the same as in Fig. 1.

Figure 4

(a) Superfluid density fraction ${\rho }_{\mathrm{s}}^{\left(\mathrm{SOC}\right)}/\rho$ in the SOC direction, (b) sound velocities $c_s^{\pm }$, and (c) critical velocities $v_c^{\pm }$ in the $\pm x$ directions, as a function of temperature at three sets of Rabi frequency $\mathrm{\Omega }/E_{\mathrm{r}}=3$, 4, and 5 for a SOC ${}^{87}\mathrm{Rb}$ gas, locating in the PW phase, at the PW-ZM transition point and in the ZM phase, respectively. Other parameters are the same as in Fig. 1.

Figure 5

Superfluid density fraction ${\rho }_{\mathrm{s}}^{\left(\mathrm{SOC}\right)}/\rho$ in the SOC direction as a function of temperature at various values of Rabi frequency $\mathrm{\Omega }/E_{\mathrm{r}}=3.4$, 3.6, 3.8, 4, 4.2, 4.4, and 4.6 (from top to bottom) for a SOC ${}^{87}\mathrm{Rb}$ gas. Two vertical dotted lines denote the ZM-PW transition points at $T^{\text{(ZM-PW)}}/T_0\simeq 0.225$ and 0.55 for $\mathrm{\Omega }/E_{\mathrm{r}}=4.2$ and 4.4, respectively. Other parameters are the same as in Fig. 1.