Spin-orbit-coupled Bose gases with nonlocal Rydberg interactions held under a toroidal trap

We consider a spin-orbit-coupled Bose-Einstein condensate with nonlocal Rydberg interactions confined in a toroidal trap. Our results show that the interplay among nonlocal Rydberg interactions, contact interactions, spin-orbit coupling, and the external toroidal trap can give rise to a rich variety of ground-state phases, such as lump necklace, chiral supersolid, standing- and plane-wave supersolid, and so on. Furthermore, we clarify the different breaking of symmetry induced by system parameters. Finally, the limiting case of a quasi-one-dimensional ring has also been investigated, and our analytical results agree well with the Gross-Pitaevskii numerical simulations. We also give an experimental protocol to observe these states in future experiments.

Figure 1

Typical ground-state density and phase distributions of the system in the absence of spin-orbit coupling and contact interactions. The nonlocal Rydberg interactions are $C_6^{(\uparrow \uparrow )}=C_6^{(\downarrow \downarrow )}=C_6^{(\uparrow \downarrow )}\equiv C_6=200,800,1600$, in (a)–(c) and (d)–(f), respectively. The Rydberg blockade radius is chosen as $R_c=2$ for (a)–(c) and $R_c=5$ for (d)–(f). (g–i) Corresponding phase distributions of component ${\psi }_{\uparrow }$ in (a)–(c), respectively. Other parameters are $\kappa =0, g_{\uparrow \uparrow }=g_{\downarrow \downarrow }=g_{\uparrow \downarrow }=0, I_0=10$, and $R=5$. The field of view is $20\times 20$ in units of $\sqrt{\hslash /M{\omega }_{\perp }}$.

Figure 2

Typical ground-state density distributions of the system in the presence of spin-orbit coupling and contact interactions. The nonlocal Rydberg interactions are $C_6^{(\uparrow \uparrow )}=C_6^{(\downarrow \downarrow )}=C_6^{(\uparrow \downarrow )}\equiv C_6=0,500,1200,1500$, and 1800, in (a), (b), (c), (d), and (e), respectively. Other parameters are $\kappa =5, g_{\uparrow \uparrow }=g_{\downarrow \downarrow }=g_{\uparrow \downarrow }=100, I_0=10, R=5$, and $R_c=2$. The field of view is $20\times 20$ in units of $\sqrt{\hslash /M{\omega }_{\perp }}$.

Figure 3

Typical ground-state density and phase distributions of the system for equal nonlocal Rydberg interactions $C_6^{(\uparrow \uparrow )}=C_6^{(\downarrow \downarrow )}=C_6^{(\uparrow \downarrow )}=1000$. The intracomponent contact interactions are $g_{\uparrow \uparrow }=g_{\downarrow \downarrow }=50$, while intercomponent interaction is $g_{\uparrow \downarrow }=g_{\downarrow \uparrow }=35$ for (a) and $g_{\uparrow \downarrow }=g_{\downarrow \uparrow }=55$ for (b) and (c). The spin-orbit coupling is chosen as (a) $\kappa =5$, (b) $\kappa =3$, and (c) $\kappa =5$. Other parameters are $I_0=10, R=5$, and $R_c=2$. The field of view is $20\times 20$ in units of $\sqrt{\hslash /M{\omega }_{\perp }}$.

Figure 4

Typical ground-state density and phase distributions of the system for unequal nonlocal Rydberg interactions $C_6^{(\uparrow \uparrow )}=2C_6^{(\downarrow \downarrow )}=2C_6^{(\uparrow \downarrow )}=2000$. The intracomponent contact interactions are $g_{\uparrow \uparrow }=g_{\downarrow \downarrow }=50$, while intercomponent interaction is $g_{\uparrow \downarrow }=g_{\downarrow \uparrow }=35$ for (a) and $g_{\uparrow \downarrow }=g_{\downarrow \uparrow }=60$ for (b). The panels in the second row are the magnified cells of the density and phase distributions of ${\psi }_{\uparrow }$ and the phase of ${\psi }_{\downarrow }$, respectively, where the black arrows indicate the particle current and the green circles indicate the vortex cores within the cell. Other parameters are $\kappa =1, I_0=10, R=5$, and $R_c=2$. The field of view is $20\times 20$ in units of $\sqrt{\hslash /M{\omega }_{\perp }}$.

Figure 5

Total energy $E_{\mathrm{kin}}+E_{\mathrm{ryd}}$ in Eqs. (12) and (13) as a function of $n$ for $R=5$. (a) $R_c=2$ and (b) $R_c=5$ for $C_6=200$ and 800. The horizontal lines are the guides to the eyes for comparison of the energies for $n=0$ and other $n$. The energy is measured in units of $\hslash {\omega }_{\perp }$.

Figure 6

(a) Comparison between the variational wave function $|{\varphi }_{\uparrow }{|}^2={\left|{\varphi }_{\downarrow }\right|}^2$ in Eq. (14) and $|{\varphi }_{\uparrow }{|}^2$ and $|{\varphi }_{\downarrow }{|}^2$ in Eq. (19) obtained by the GP equation for $g_{\uparrow \uparrow }=g_{\downarrow \downarrow }=50$ and $g_{\uparrow \downarrow }=35$. The lower panels are the density distribution and the phase distribution near $r=R$ obtained by the GP equation. The phases $\arg \left({\psi }_{\uparrow }\right)$ and $\arg \left({\psi }_{\downarrow }\right)$ change by $24\times 2\pi$ and $25\times 2\pi$ around the circle, respectively. (b) Comparison between $|{\varphi }_{\uparrow }{|}^2$ in Eq. (15) and $|{\varphi }_{\uparrow }{|}^2$ in Eq. (19) obtained by the GP equation for $g_{\uparrow \uparrow }=g_{\downarrow \downarrow }=35$ and $g_{\uparrow \downarrow }=50$. The lower panels show the density distributions obtained by the GP equation. Other parameters are $R=5, I_0=10, \kappa =5, R_c=5$, and $C_6=800$. The integers in Eqs. (14) and (15) are taken to be $\ell =24$ and $n=4$, which minimize the variational energy. In (a) and (b), the field of view of the density and phase distributions are $20\times 20$ in units of $\sqrt{\hslash /M{\omega }_{\perp }}$.