Infinite lattices of vortex molecules in Rabi-coupled condensates

Vortex molecules can form in a two-component superfluid when a Rabi field drives transitions between the two components. We study the ground state of an infinite system of vortex molecules in two dimensions, using a numerical scheme which makes no use of the lowest Landau level approximation. We find the ground state lattice geometry for different values of intercomponent interactions and strength of the Rabi field. In the limit of large field when molecules are tightly bound, we develop a complementary analytical description. The energy governing the alignment of molecules on a triangular lattice is found to correspond to that of an infinite system of two-dimensional quadrupoles, which may be written in terms of an elliptic function $\mathcal{Q}(z_{ij};{\omega }_1,{\omega }_2)$. This allows for a numerical evaluation of the energy which enables us to find the ground state configuration of the molecules.

Figure 1

Schematic view of a pair of free vortices, one in each component, in the absence of an external field. There is no energy cost for having phases misaligned, and the winding is smooth to minimize the kinetic energy.

Figure 2

Vortex pair in the presence of the external field. Because of the energy cost for misaligning phases Eq. (3), the misalignment region (in orange) is confined in a region. Out of this region the phases are aligned.

Figure 3

Schematic view of the computational unit cell. From left to right, we show the computational unit cell for ${\mathrm{\Omega }}_{\text{R}}/n_v=0$ and $\alpha =1,0.5,0.2$, respectively. The aspect ratios are $\mathcal{R}=1/\sqrt{3},0.5,1/\sqrt{3}$. The ground state aspect ratios are the given ones, only within integration error, i.e., there is a set of $\mathcal{R}$ within integration error that fulfill the doubling-halving criterion. Our finding agrees qualitatively with what was found in Ref. [29].

Figure 4

The color code shows the relative phase $\phi ={\theta }_b-{\theta }_a$. From left to right, we show the computational unit cell for ${\mathrm{\Omega }}_{\text{R}}/n_v=0.5,0.25,0.1$ and $\alpha =1,0.5,0.2$, respectively. The aspect ratios are $\mathcal{R}=0.5,0.64,0.79$. The ground state aspect ratios are the given ones, only within integration error, i.e., there is a set of $\mathcal{R}$ within integration error that fulfill the doubling-halving criterion. Our finding agrees qualitatively with what was found in Ref. [20] (except $\alpha =0.2$), in the center of the trap. Note that for $\alpha =1$ repeated computations yield different axes along which molecules are antialigned.

Figure 5

From left to right, we show the total density $\rho$ and $S_z$, for ${\mathrm{\Omega }}_{\text{R}}/n_v=0.5,0.25,0.1$ and $\alpha =1,0.5,0.2$, respectively. The aspect ratios are $\mathcal{R}=0.5,0.64,0.79$. For $\alpha =1$, $S_z$ winds at constant rate around the center of each molecule, whereas for $\alpha <1$ the winding is concentrated around the vortex cores. Consequently, for $\alpha =1$, the density profile around the molecules is circular (at distances close enough from the center). On the other hand, for $\alpha <1$ it is elongated.

Figure 6

We show the two inequivalent configurations with the ground state unit cell we find.

Figure 7

A schematic square lattice of molecules with $N_{\text{mol}}=4$. Even though our numerical calculations are done for the triangular lattice, we show a square lattice to emphasize that our result of the general expression of $V_{\text{inf}}/N_{\text{uc}}$ is valid for any lattice geometry.