Vortex lattices for ultracold bosonic atoms in a non-Abelian gauge potential

The use of coherent optical dressing of atomic levels allows the coupling of ultracold atoms to effective nondynamical gauge fields. These can be used to generate effective magnetic fields and have the potential to generate non-Abelian gauge fields. We consider a model of a gas of bosonic atoms coupled to a gauge field with $\text{U}\left(2\right)$ symmetry and with constant effective magnetic field. We include the effects of weak contact interactions by applying Gross-Pitaevskii mean-field theory. We study the effects of a $\text{U}\left(2\right)$ non-Abelian gauge field on the vortex lattice phase induced by a uniform effective magnetic field, generated by an Abelian gauge field or, equivalently, by rotation of the gas. We show that, with increasing non-Abelian gauge field, the nature of the ground state changes dramatically, with structural changes of the vortex lattice. We show that the effect of the non-Abelian gauge field is equivalent to the introduction of effective interactions with nonzero range. We also comment on the consequences of the non-Abelian gauge field for strongly correlated fractional quantum Hall states.

Figure 1

The energy levels (6) for $n=0,1,2,3$ as functions of $\beta$. We find that $E_1$ is the minimum energy for $0<\beta <\sqrt{3}$, while $E_2$ is the minimum for $\sqrt{3}<\beta <\sqrt{5}$.

Figure 2

Contour plots for the particle density of a triangular vortex lattice for $\beta =1.0$ (aspect ratio $a_r=\sqrt{3}$). We present one cell which contains $N_{\mathrm{V}}=2$ vortices. (a) The total particle density (in units of the mean particle density). The particle density is not zero anywhere since the single-particle wave functions involve contributions from more than one Landau level, as shown in Eqs. (7) and (12). (b) The density for particles in the lowest Landau level (in units of the mean particle density in the lowest Landau level). (c) The density for particles in the first Landau level (in units of the mean particle density in the first Landau level).

Figure 3

Contour plots for the particle density of a square vortex lattice for $\beta =1.5$ (aspect ratio $a_r=1$). We present one cell which contains $N_{\mathrm{V}}=1$ vortex. (a) The total particle density. The particle density is not zero anywhere since the single-particle wave functions involve contributions from more than one Landau level. (b) The density for particles in the lowest Landau level. (c) The density for particles in the first Landau level.

Figure 4

Contour plots for the particle density of a $q=4$ bubble crystal lattice for $\beta =2.0$ (aspect ratio $a_r=\sqrt{3}$). We present a cell which contains $N_{\mathrm{V}}=8$ vortices. (a) The total particle density. The particle density is zero on the sites of a triangular lattice. (b) The density for particles in the first Landau level. (c) The density for particles in the second Landau level.