Anomalous Slow Fidelity Decay for Symmetry-Breaking Perturbations

Symmetries as well as other special conditions can cause anomalous slowing down of fidelity decay. These situations will be characterized, and a family of random matrix models to emulate them generically presented. An analytic solution based on exponentiated linear response will be given. For one representative case the exact solution is obtained from a supersymmetric calculation. The results agree well with dynamical calculations for a kicked top.

Figure 1

The complement of the average fidelity amplitude, for a weak perturbation, $\epsilon =5\times {10}^{-5}$. Part (a) shows the GOE case with a Gaussian random perturbation of type (i). Part (b) shows the GUE case, with an independent random perturbation of type (iii). The exponentiated linear response approximations are plotted with short dashed lines (for comparison, the exact results for full GOE and GUE perturbations are shown with chain curves). The results from time-independent perturbation theory are plotted with long dashed lines. In the GOE case, the exact analytical result, Eq. (12) is plotted with a solid line. The random matrix simulations are plotted with points.

Figure 2

Fidelity freeze for quantized kicked tops. (a) The kicked top of Eq. (13) (possessing an antiunitary symmetry) with a perturbation of type (i). (b) The kicked top of Eq. (14) (without any symmetry) with a perturbation of type (iii). The dashed lines give the numerical simulations, while the solid lines give the square of the theoretical prediction (12) in (a) and RMT simulations in (b).