Geometric Phase in Open Systems

We calculate the geometric phase associated with the evolution of a system subjected to decoherence through a quantum-jump approach. The method is general and can be applied to many different physical systems. As examples, two main sources of decoherence are considered: dephasing and spontaneous decay. We show that the geometric phase is completely insensitive to the former, i.e., it is independent of the number of jumps determined by the dephasing operator.

Figure 1

Evolution of the state along a “one-jump” trajectory on the Bloch sphere, in the case of phase diffusion decoherence ($\Gamma \propto {\sigma }_z$). From time $t_0$ to time $t_1$ the state evolves under the no-jump Hamiltonian $\tilde{H}$ alongside the parallel of the sphere. At time $t_1$ a jump occurs, flipping (instantaneously) the Bloch vector about the $z$ to the point $t_1^{\prime }$, and the no-jump evolution starts again. At time $t_2=t_0+2\pi /\omega$ the geometric phase $\gamma =\pi (1-\cos \theta )$ is recovered. The geometric phase is half the area enclosed in the path spanned by the Bloch vector. This is given by two contribution, the surface $t_1-t_0-t_1^{\prime }$ and the surface $t_0-t_1-t_2$. This picture also represents the evolution of a spin-$J$ particle subjected to dephasing, in a generalized Bloch sphere.

Figure 2

Evolution of the state along the no-jump trajectory on the Bloch sphere, for the spontaneous decay case (${\Gamma }_1=\lambda {\sigma }_z$, ${\Gamma }_2=\alpha {\sigma }_{-}$). The evolution is a smooth spiral converging to the lower state: while the state rotates about the $z$ axis, it is smoothly brought towards the lower state with a velocity given by the spontaneous decay rate $\alpha$. The geometric phase is given by the area enclosed in the path shown, where the last segment is the geodesic connecting initial and final point of the evolution.