Ring model for trapped condensates with synthetic spin-orbit coupling

We derive an effective ring model in momentum space for trapped bosons with synthetic spin-orbit coupling. This effective model is characterized by a peculiar form of the interparticle interactions, which is crucially modified by the external confinement. The ring model allows for an intuitive understanding of the phase diagram of trapped condensates with isotropic spin-orbit coupling and, in particular, for the existence of skyrmion lattice phases. The model, which may be generally applied for spinor condensates of arbitrary spin and spin-dependent interactions, is illustrated for the particular cases of spin-$\frac{1}{2}$ and spin-1 condensates.

Figure 1

Functions $V\left(\Delta \varphi \right)$ (dashed curve) and $U_R\left(\Delta \varphi \right)\equiv \text{Re}\left(U\left(\Delta \varphi \right)\right)$ (bold dashed curve) for a spin-$\frac{1}{2}$ BEC with $\tilde{\kappa }=20$. The final expressions are of the form $V\left(\Delta \varphi \right)=V_0\left(\Delta \varphi \right)f_0\left(\Delta \varphi \right)+0.59V_0(\pi -\Delta \varphi )f_{\pi }\left(\Delta \varphi \right)+V_a\left(\Delta \varphi \right)[1-f_0\left(\Delta \varphi \right)-f_{\pi }\left(\Delta \varphi \right)]$, where $f_0\left(\varphi \right)=e^{-{(\varphi /0.5\pi )}^4}$ and $f_{\pi }\left(\varphi \right)=e^{-{[(\pi -\varphi )/0.5\pi ]}^4}$ are interpolating functions. Similarly for $U,\text{Re}\left[U\left(\Delta \varphi \right)\right]=0.82V_0\left(\Delta \varphi \right)f_0\left(\Delta \varphi \right)+0.82V_0(\pi -\Delta \varphi )f_{\pi }\left(\Delta \varphi \right)+\text{Re}\left[U_a\left(\Delta \varphi \right)\right][1-f_0\left(\Delta \varphi \right)-f_{\pi }\left(\Delta \varphi \right)]$. In the figure we depict as well the analytic expressions $V_a\left(\Delta \varphi \right)$ (solid curve) and $\text{Re}\left[U_a\left(\Delta \varphi \right)\right]$ (bold solid curve).

Figure 2

Comparison of the results for the average angular momentum $|\langle {\widehat{l}}_z-\frac{1}{2}\rangle |$ versus $\tilde{g}$ for $\tilde{\kappa }=20$ between the 2D model (solid) and the effective 1D ring model (dashed). The transitions from $\text{HV}\left(\frac{1}{2}\right)$ to $\text{HV}\left(\frac{3}{2}\right)$ and $\text{HV}\left(\frac{3}{2}\right)$ to triangular-lattice phases are represented by black solid vertical lines for the 2D model and pink dashed lines for the 1D model.

Figure 3

(top) Momentum distribution (as a function ${\tilde{k}}_x\equiv k_x{l}_0$ and ${\tilde{k}}_y\equiv k_y{l}_0)$ of a spin-$\frac{1}{2}$ BEC with isotropic SOC for $\tilde{\kappa }=20$ and $\tilde{g}=0.19$ obtained from a direct numerical simulation of Eq. (4) and (bottom) of the effective ring model. A very similar triangular momentum distribution is observed with both models.

Figure 4

Same as Fig. 3 but under different initial conditions for the imaginary time evolution. A very similar hexagonal pattern appears. The energy of the triangular pattern of Fig. 3 and of the hexagonal pattern of this figure is, within our numerical accuracy, basically identical (see discussion in main text).

Figure 5

Functions $V\left(\Delta \varphi \right)$ (dashed curve) and $U\left(\Delta \varphi \right)$ (bold dashed curve) for a spin-1 BEC with $\tilde{\kappa }=20$. In the figure we depict as well the analytical expressions $V_a\left(\Delta \varphi \right)$ (solid curve) and $U_a\left(\Delta \varphi \right)$ (bold solid curve).

Figure 6

Mean value of the angular momentum $|\langle {\widehat{l}}_z\rangle |$ as a function of $\tilde{g}$ for a spin-1 BEC with $\tilde{\kappa }=20$. We compare the results obtained from the 2D GPE (4) (solid) and from the ring model (dashed). The transitions from HV(0) to HV(1) and from HV(1) to a triangular-lattice phase are represented by black solid vertical lines for the 2D model and pink dashed lines for the 1D model.

Figure 7

(top) Momentum distribution (as a function of ${\tilde{k}}_x\equiv k_x{l}_0$ and ${\tilde{k}}_y\equiv k_y{l}_0)$ of a spin-1 BEC with isotropic SOC for $\tilde{\kappa }=20$ and $\tilde{g}=0.2$ obtained from a direct simulation of Eq. (4) and (bottom) of the effective ring model. A very similar triangular-lattice phase is observed in both cases.

Figure 8

(top) Momentum distribution (as a function of ${\tilde{k}}_x\equiv k_x{l}_0$ and ${\tilde{k}}_y\equiv k_y{l}_0)$ of a spin-1 BEC with isotropic SOC for $\tilde{\kappa }=20$ and $\tilde{g}=0.8$ obtained from a direct simulation of Eq. (4) and (bottom) of the effective ring model. A very similar square-lattice phase is observed in both cases.