Emergence of topological and strongly correlated ground states in trapped Rashba spin-orbit-coupled Bose gases

We theoretically study an interacting few-body system of Rashba spin-orbit-coupled two-component Bose gases confined in a harmonic trapping potential. We solve the interacting Hamiltonian at large Rashba coupling strengths using an exact-diagonalization scheme, and obtain the ground-state phase diagram for a range of interatomic interactions and particle numbers. At small particle numbers, we observe that the bosons condense to an array of topological states with $n+1/2$ quantum angular momentum vortex configurations, where $n=0,1,2,3,...$. At large particle numbers, we observe two distinct regimes: at weaker-interaction strengths, we obtain ground states with topological and symmetry properties that are consistent with mean-field theory computations; at stronger-interaction strengths, we report the emergence of strongly correlated ground states.

Figure 1

Plots (a) and (b) show wave functions ${\varphi }_{\uparrow }\left(\rho \right)$ of single-particle states in the $n=0$ manifold at small and large SO-coupling strengths, respectively. $m=0$ (solid black line), $m=1$ (dotted red line), $m=2$ (dash-dotted black line), and $m=3$ (dashed red line).

Figure 2

Plots (a) and (b) show the energy spectrum of single-particle states at small and large SO-coupling strengths, respectively: $n=0\to 7$ (bottom$\to$top) and $m=-16\to +15$. While energies of states within each $n$ are represented by a specific symbol, it is evident that states with higher $n$ have progressively higher energies.

Figure 3

Energy spectrum for extremely weak-interaction strengths with ${\lambda }_{\mathrm{SO}}=20$ and $N=2$. Here, each marker (red) represents the lowest-energy eigenvalue of a specific block diagonal with a fixed value of $J_z$. Since energy eigenvalues are very close, we identify the ground-state energies by circled (black) markers and, further, show the zoomed-in plots in the inset.

Figure 4

Plots of (a) ground-state $J_z/N$ manifolds and (b) entanglement entropy, as a function of interaction strength $g$ with ${\lambda }_{\mathrm{SO}}=20,N=2,g_{\uparrow \downarrow }/g=0.5$. For representative interaction strengths, denoted by circled (black) markers, we illustrate the ground-state properties in Fig. 6.

Figure 5

Plots of (a) ground-state $J_z/N$ manifolds and (b) entanglement entropy, as a function of interaction strength $g$ with ${\lambda }_{\mathrm{SO}}=20,N=2,g_{\uparrow \downarrow }/g=1.5$.

Figure 6

Plots in each row illustrate the ground-state properties at a representative interaction strength of Fig. 4a. In the first column (from left), we show density distributions of spin-up component $n_{\uparrow }\left(\rho \right)$ (solid green line) and of spin-down component $n_{\downarrow }\left(\rho \right)$ (dashed red line). In the second column, we show eigenvalues $O_i$ of the single-particle density matrix as a function of angular momentum $j_z$ of the single-particle states $|{\varphi }_i\left(\mathbf{r}\right)\rangle$. In the third column, we show corresponding OES plots of entanglement pseudoenergies ${\xi }_i$ as a function of $J_z^A/N$, the average angular momentum of subsystem $A$. In the last column, we show contour plots (a4) and (b4) that are normalized pair-correlation functions $\langle n_{\uparrow }\left({\mathbf{r}}_0\right)n_{\downarrow }\left(\mathbf{r}\right)\rangle$, with ${\mathbf{r}}_0$ denoted by a (yellow) marker. Phase plots (c4) and (d4) are derived from reduced wave function ${\psi }_{c,\downarrow }\left(\mathbf{r}\right)$, which is computed by fixing one of the two particles at their most probable locations, and their corresponding radii are indicated by (yellow) markers. The closed dashed (blue) contour is a guide to the eye, which allows us to count the number of phase slips.

Figure 7

Plots of (a) ground state $J_z/N$ manifolds and (b) entanglement entropy, as a function of interaction strength $g$ with ${\lambda }_{\mathrm{SO}}=20,N=8,g_{\uparrow \downarrow }/g=0.5$. For representative interaction strengths denoted by circled (black) markers, we illustrate the ground-state properties in Fig. 9.

Figure 8

Plots of (a) ground state $J_z/N$ manifolds and (b) entanglement entropy, as a function of interaction strength $g$ with ${\lambda }_{\mathrm{SO}}=20,N=8,g_{\uparrow \downarrow }/g=1.5$. For the representative interaction strength, denoted by a circled (black) marker, we illustrate the ground-state properties in Fig. 9.

Figure 9

Plots in each row illustrate the ground-state properties at a representative interaction strength in Fig. 7a or 8a. In the first column (from left), we show density distributions of the spin-up component $n_{\uparrow }\left(\rho \right)$ (solid green line) and of the spin-down component $n_{\downarrow }\left(\rho \right)$ (dashed red line). In the second column, we show eigenvalues $O_i$ of the single-particle density matrix as a function of angular momentum $j_z$ of the single-particle states $|{\varphi }_i\left(\mathbf{r}\right)\rangle$. In the third column, we show corresponding OES plots of entanglement pseudoenergies ${\xi }_i$ as a function of $J_z^A/N$, the average angular momentum of subsystem $A$. In the last column, we show contour plots (a4), (c4), and (d4) that are normalized pair-correlation functions $\langle n_{\uparrow }\left({\mathbf{r}}_0\right)n_{\downarrow }\left(\mathbf{r}\right)\rangle$, with ${\mathbf{r}}_0$ denoted by a (yellow) marker. Phase plot (b4) is derived from reduced wave function ${\psi }_{c,\downarrow }\left(\mathbf{r}\right)$, which is computed by fixing seven of the eight particles at their most probable locations, and their corresponding radii are indicated by (yellow) markers. The closed dashed (blue) contour is a guide to the eye, which allows us to count the number of phase slips.