Quantum depletion and superfluid density of a supersolid in Raman spin-orbit-coupled Bose gases

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.

Figure 1

(Upper panel) The high-order stripe density profile $n$ for spin-up atoms, spin-down atoms, and total atoms along the SOC direction at two Rabi frequencies $\mathrm{\Omega }=0.1E_{\mathrm{r}}$ (a) and $\mathrm{\Omega }=1.0E_{\mathrm{r}}$ (b). (Lower panel) The corresponding excitation spectrum ${\varepsilon }_j$ for the lowest five branches. Here, we take $G_1=0.5E_{\mathrm{r}}$ and $G_2=0.1E_{\mathrm{r}}$. $d=\pi /P_x$ is the spatial periodicity of stripes. The two dashed lines in (c) show the phonon dispersion of a conventional two-component Bose gas in the limit of $\mathrm{\Omega }=0$.

Figure 2

The ground-state energy as a function of the Rabi frequency in the first-order (dashed-blue), high-order (solid-black) stripe ansatz and the plane-wave ansatz (dotted-red). The interaction parameters are the same as in Fig. 1.

Figure 3

(a) Contour plot of the shift $\delta {\mathrm{\Omega }}_1={\mathrm{\Omega }}_{c1}^{\left(\mathrm{new}\right)}-{\mathrm{\Omega }}_{c1}$ for the ST-PW transition as functions of the interaction energy strengths $G_1$ and $G_2$. (b) The dependence of $\delta {\mathrm{\Omega }}_1$ on $G_2$ at $G_1=0.5E_{\mathrm{r}}$.

Figure 4

Quantum depletion $n_{\mathrm{qd}}/\overline{n}$ as a function of the Rabi frequency $\mathrm{\Omega }$ in the ST phase (red dotted line) and in the PW and ZM phases (blue dashed line). The blue diamonds show the depletion of a uniform single-component weakly interacting Bose gas with the same interaction parameters, while the red circles give the depletion of a two-component Bose gas. The vertical dashed and dotted curves indicate the critical ${\mathrm{\Omega }}_{c1}^{\left(\mathrm{new}\right)}$ and ${\mathrm{\Omega }}_{c2}$, respectively. Here, we take the interaction energies $G_1=0.5E_{\mathrm{r}}$ and $G_2=0.01E_{\mathrm{r}}$ (a), $0.04E_{\mathrm{r}}$ (b), and $0.07E_{\mathrm{r}}$ (c).

Figure 5

Superfluid fraction $n_{\mathrm{s}}^{\left(x\right)}/\overline{n}$ as a function of the Rabi frequency $\mathrm{\Omega }$ in the ST phase (red dashed line – first-order ansatz; green dotted line – high-order ansatz), the PW and ZM phases (blue dashed line). The solid-black lines are the component $n_{\mathrm{s}}^{(\perp )}/\overline{n}$ in the perpendicular plane. The two vertical lines indicate the critical Rabi frequency of the phase transition. The interaction parameters are the same as in Fig. 4.

Figure 6

Quantum depletion in the limit of $\mathrm{\Omega }\to 0$ as a function of $N_{\mathrm{L}}=N_{\mathrm{M}}$. The anticipated result of a two-component Bose gas is shown by a dashed line. The interaction parameters are the same as in Fig. 1.