Supersolid with nontrivial topological spin textures in spin-orbit-coupled Bose gases

Supersolid is a long-sought exotic phase of matter, which is characterized by the coexistence of a diagonal long-range order of solid and an off-diagonal long-range order of superfluid. Possible candidates to realize such a phase have been previously considered, including hard-core bosons with long-range interaction and soft-core bosons. Here we demonstrate that an ultracold atomic condensate of hard-core bosons with contact interaction can establish a supersolid phase when simultaneously subjected to spin-orbit coupling and a spin-dependent periodic potential. This supersolid phase is accompanied by topologically nontrivial spin textures, and is signaled by the separation of momentum distribution peaks, which can be detected via time-of-flight measurements. We also discuss possibilities to produce and observe the supersolid phase for realistic experimental situations.

Figure 1

Spin-orbit-coupling induced transition from superfluid to supersolid. Various ground-state properties of Rashba SO-coupled Bose gases loaded in a 1D spin-dependent periodic potential are shown, with the SO-coupling strengths $\kappa =3\pi \hslash /ma$ (top), $4\pi \hslash /ma$ (middle), and $5\pi \hslash /ma$ (bottom). With increasing SO-coupling strength, the system undergoes a phase transition from the superfluid phase (top) to a supersolid phase (middle and bottom) characterized by density modulation along the $y$ direction (first and second columns). The supersolid phase features a triangular lattice (middle) and a rectangular lattice (bottom) at intermediate and high SO-coupling strengths, respectively. The supersolid is accompanied by a spontaneous generation of vortex and antivortex chains in the spin-up and spin-down domains, respectively, as can be seen from the condensate phase modulation (third column). The occurrence of the supersolid phase is signaled by the separation of the momentum distribution peaks (fourth column), and can be readily observed via time-of-flight measurements. Other parameters used in these plots are $V_0=20{\pi }^2{\hslash }^2/m{a}^2$ and $\tilde{g}=6000$.

Figure 2

Line density of vortices as a function of periodic potential depth. With increasing the depth $V_0$ of the periodic potential, the line density $n_v$ of the vortices decreases gradually, and drops to zero suddenly at the transition point $V_0=35{\pi }^2{\hslash }^2/m{a}^2$ between the supersolid and superfluid phases. The insets depict the change of vortex density and atomic momentum distributions. The line density $n_v$ of the vortices is proportional to the separation $\delta$ of the momentum distribution peaks, and can be expressed as $n_v=\delta /\pi$. The Rashba SO-coupling strength is fixed at $\kappa =4\pi \hslash /ma$, and the dimensionless interaction parameter is taken as $\tilde{g}=6000$.

Figure 3

The ground-state phase diagram spanned by the Rashba SO-coupling strength $\kappa$ and the periodic potential depth $V_0$. Three phases can be identified on this phase diagram, including the superfluid (SF) phase, the triangular supersolid (TSS) phase, and the rectangular supersolid (RSS) phase. The dimensionless interaction parameter is taken as $\tilde{g}=6000$.

Figure 4

Topological spin textures. (a), (b) Spin configurations of the triangular and rectangular supersolid phases showing the spontaneous emergence of meron pairs and antimeron pairs. In these plots, the arrows represent the directions of spin vector $\left(S_z,S_y\right)$ and the colors ranging from blue to red describe the values of $S_x$ from $-1$ to 1. (c), (d) Density distributions $q\left(\mathbf{r}\right)$ of topological charge for the spin textures shown in (a) and (b). The pink and green bubbles indicate that the meron pairs and antimeron pairs carry positive and negative topological charges, respectively. (e), (f) Schematic spin configurations of a meron pair (e) and an antimeron pair (f). Both the meron pairs and antimeron pairs have the same “circular-hyperbolic” structure, but with opposite spin orientations. Parameters used in these plots are identical to those used in Figs. 1–1 and 1–1.

Figure 5

Domain wall chirality. (a) Section view of the relative phase ${\theta }_{\uparrow }\text{−}{\theta }_{\downarrow }$ along the $x$ axis. The presence of Rashba SO coupling locks the relative phase at $\pm \pi /2$, and the periodic change of the densities along the $x$ direction induces a periodic modulation of the relative phase between $\pi /2$ and $-\pi /2$. By tuning the sign of the Rashba SO coupling from positive $\left(\kappa >0\right)$ to negative $\left(\kappa <0\right)$, the relative phase will be changed from $\pm \pi /2$ (red thick line) to $\mp \pi /2$ (blue thick dashed line). (b) An illustration of right-handed chiral Bloch walls stabilized by positive Rashba SO coupling with $\kappa >0$. (c) An illustration of left-handed chiral Bloch walls stabilized by negative Rashba SO coupling with $\kappa <0$. By crossing a right-handed (left-handed) chiral Bloch wall, the spin vector flips like a right-rotating (left-rotating) spiral.

Figure 6

Effects of asymmetric interaction and anisotropic spin-orbit coupling. (a)–(c) Density distributions of Bose condensates with Rashba SO coupling and asymmetric interaction. While the intracomponent interaction is fixed with ${\tilde{g}}_{\uparrow \uparrow }={\tilde{g}}_{\downarrow \downarrow }=6000$, the intercomponent interactions are varied to be (a) ${\tilde{g}}_{\uparrow \downarrow }=3000$, (b) ${\tilde{g}}_{\uparrow \downarrow }=7000$, and (c) ${\tilde{g}}_{\uparrow \downarrow }=18000$. The Rashba SO-coupling strength is taken as $\kappa =4\pi \hslash /ma$ in these plots. Notice that a sufficiently strong intercomponent interaction can drive the system from the supersolid phase to the superfluid phase. (e), (f) Density distributions of Bose condensates in the presence of anisotropic SO interaction with ${\kappa }_x\ne {\kappa }_y$. Parameters used in these two plots are (e) ${\kappa }_x=3\pi \hslash /ma,{\kappa }_y=4\pi \hslash /ma$ and (f) ${\kappa }_x=2\pi \hslash /ma,{\kappa }_y=4\pi \hslash /ma$. Here, the interatomic interactions are considered to be $\text{SU}\left(2\right)$ symmetric with $\tilde{g}=6000$. By increasing the SO-coupling anisotropy beyond a certain value, the system will undergo a phase transition and become a superfluid which can be regarded as a plane-wave state characterized by phase modulation shown in (g). We emphasize that the superfluid phases driven by asymmetric interaction (c) and by anisotropic SO coupling (f) acquire different properties, as can be easily distinguished from the momentum distribution depicted in (d) and (h), respectively. In this figure, the periodic potential depth is taken as $V_0=20{\pi }^2{\hslash }^2/m{a}^2$.

Figure 7

Experimental setup for creating spin-orbit-coupled Bose gases in a spin-dependent periodic potential. (a) The cloud of atoms is situated tens of micrometers above the surface of an atom chip. Two pairs of parallel microwires with amplitude modulated rf current are embedded in the chip, and produce periodic pulsed magnetic field gradients along perpendicular directions. Another pulsed uniform magnetic field oriented along the $x$ axis is added, with the sum frequency of the two magnetic fields equal to the magnetic splitting induced by a strong bias field $B_0{\mathbf{e}}_z$ between the sublevels $m_F=\pm 1$. This provides a two-photon coupling between the hyperfine states, and induces an effective 2D SO coupling in the first-order approximation to the pulse duration $\tau$. Two counterpropagating linearly polarized laser beams with the same frequency but perpendicular polarization vectors create a spin-dependent periodic potential. (b) The pulse sequence used to implement SO coupling and spin-dependent periodic potential. The parameters ${\gamma }_x$ and ${\gamma }_y$ characterize the strength of the magnetic field gradient, and ${\beta }_1\left(t\right)$ and ${\beta }_2\left(t\right)$ define the temporal shape of the magnetic fields. (c) Two polarized standing wave laser fields ${\sigma }^{+}$ (purple) and ${\sigma }^{-}$ (cyan) are produced by the counterpropagating linearly polarized lasers in (a). Due to the polarization-dependent ac Stark shift, the internal state $|1,1\rangle$ is affected by the standing wave laser field ${\sigma }^{+}$, while the internal state $|1,-1\rangle$ experiences the standing wave laser field ${\sigma }^{-}$.

Figure 8

Quantum depletion as a function of the spin-orbit-coupling strength (a) and the periodic potential depth (b). The periodic potential depth is fixed at $V_0=20{\pi }^2{\hslash }^2/m{a}^2$ in (a), and the spin-orbit-coupling strength is fixed at $\kappa =4\pi \hslash /ma$ in (b). The dimensionless interaction parameter is taken as $\tilde{g}=6000$. The quantum depletion is always less than $0.1\%$, thereby confirming the validity of the mean-field approach.