Detecting non-Abelian geometric phases with three-level $\Lambda$ systems

We show that a non-Abelian gauge potential in two nearly degenerated dressed states may be induced by two laser beams interacting with a three-level $\Lambda$ atomic system. We demonstrate that the populations of the atomic states at the end of a composed path formed by two closed loops are dependent on the order of those two loops, showing an unambiguous signature of the non-Abelian geometric phase. Through numerical calculations, we show that the non-Abelian feature of the geometric phases can be tested under realistic conditions.

Figure 1

(a) Three-level $\Lambda$ atomic system coupled with two laser beams characterized by the Rabi frequencies ${\Omega }_1$ and ${\Omega }_2$ with a large single-photon detuning $\Delta$. (b) The dressed states and the corresponding energy levels.

Figure 2

The population difference $P_d$ versus different parameters of $\alpha$ and $\beta$. The population difference is induced by the entire phase which contains geometric phase as well as dynamic phase. (b) The population difference $P_d$ versus parameter $\alpha$ when $\beta =0.8$.

Figure 3

Schematic representation of the multiloop method to cancel the effects of dynamic phases. The cold atoms are interacting with lasers; its Hamiltonian evolves in the parameter space as a spin particle subject to an effective magnetic field (denoted as $\mathbf{B}$).

Figure 4

The population difference $P_d$ versus the parameters of $\alpha$ and $\beta$. The population difference is induced by the pure geometric phase after canceling the dynamic phase. (b) The popul-ation difference $P_d$ versus parameter $\beta$ when $\alpha =0.8$.