Effects of an optically induced non-Abelian gauge field in cold atoms

We show that $N-1$ degenerate dark states can be generated by coupling $N$-fold degenerate ground states and a common excited state with $N$ laser fields. Interferences between light waves with different frequencies can produce laser fields with time-dependent amplitudes, which can induce not only $\mathrm{U}\left(N\right)$ non-Abelian vector fields but also the scalar ones for the adiabatic motion of atoms in such laser fields. As an example, a time-periodic gauge potential is produced by applying specific laser fields to a tripod system. Some features of the Landau levels and the ground-state phase diagram of a rotating Bose-Einstein condensate for a concrete gauge field are also discussed.

Figure 1

(Color online) Ground-state phase diagrams with different choices of the parameters for a rotating BEC in the presence of gauge potentials in the $\xi -\Omega ∕{\omega }_{\perp }$ plane. The boundaries separating different phases are plotted. ${\sigma }_i$, ${\sigma }_{12}$, $\Omega ∕{\omega }_{\perp }$, and $\xi$ are all dimensionless quantities.

Figure 2

(Color online) The curve surfaces are the derivatives (in units of $\hslash {\omega }_{\perp }$) of the ground-state energy along $(1,-1)$ direction versus $(\xi ,\Omega ∕{\omega }_{\perp })$ for parameters ${\sigma }_1=0.2$, ${\sigma }_2=0.4$, ${\sigma }_{12}=0.3$ (a) and ${\sigma }_1={\sigma }_2={\sigma }_{12}=0.4$ (b), respectively. The discontinuous steps occur precisely at the boundaries separating different phases illustrated in Fig. 1.