Geometric Nature of the Environment-Induced Berry Phase and Geometric Dephasing

We investigate the geometric phase or Berry phase acquired by a spin half which is both subject to a slowly varying magnetic field and weakly coupled to a dissipative environment (either quantum or classical). We study how this phase is modified by the environment and find that the modification is of a geometric nature. While the original Berry phase (for an isolated system) is the flux of a monopole field through the loop traversed by the magnetic field, the environment-induced modification of the phase is the flux of a quadrupolelike field. We find that the environment-induced phase is complex, and its imaginary part is a geometric contribution to dephasing. Its sign depends on the direction of the loop. Unlike the Berry phase, this geometric dephasing is gauge invariant for open paths of the magnetic field.

Figure 1

(a) The field $\delta \mathbf{a}(\mathbf{B})$, see Eq. (12), in the vicinity of an open path of $\mathbf{B}(t)$ (dashed line). The environment-induced modification of the Berry phase, $\delta {\Phi }_{\mathrm{B}\mathrm{P}}$, is the integral of $\delta \mathbf{a}(\mathbf{B})$ along this path. For such open paths, the geometric dephasing is gauge invariant (unlike the phase). (b) The quadrupolelike field $\delta \mathbf{b}(\mathbf{B})$, see Eqs. (13) [$\delta \mathbf{b}(\mathbf{B})$ is cylindrically symmetric about the $B_z$ axis]. For a closed path of $\mathbf{B}(t)$, $\delta {\Phi }_{\mathrm{B}\mathrm{P}}$ is the flux of $\delta \mathbf{b}(\mathbf{B})$ through the path.

Figure 2

The laboratory $(x,y,z)$ and rotating $(\tilde{x},\tilde{y},\tilde{z})$ frames.