Exact fidelity and full fidelity statistics in regular and chaotic surroundings

For a prepared state exact expressions for the time-dependent mean fidelity as well as for the mean inverse participation ratio are obtained analytically. The prepared state is taken as an eigenstate of the unperturbed system, and the studied fidelity is identical to the survival probability. The full distribution functions of fidelity in the long-time limit and of inverse participation ratio are studied numerically and analytically. Surprising features such as fidelity revival and nonergodicity are observed. The roles of the coupling coefficients and of complexity of background are studied as well.

Figure 1

(Color online) Evolution in time of fidelity for coupling strength $\lambda =0.1$. The full lines describe from top to bottom the fidelity decay for real coupling to a Poissonian (green), to a GOE (red), and to a GUE (black) background. The dashed lines describe from top to bottom the corresponding fidelity decay for complex coupling. The inset shows the average fidelity versus time (from top to bottom) for $\lambda =0.1$ (green), 0.2 (black), 0.3 (blue), and 0.4 (red), for complex coupling to Poisson background. For comparison the FGR result is shown by thinner dotted lines.

Figure 2

(Color online) Fidelity distributions at times $t=1$ (pink), 2.5 (blue), 5 (red), 10 (green), and 20 (black) for coupling strength $\lambda =0.1$ of real matrix elements to a GOE background. For times $t>20$ the distribution is stable.

Figure 3

(Color online) Comparison of the distributions $P_0$ obtained from Eq. (15) [red (dashed) line] with $P_{\text{IPR}}$ [full green (light gray) line] and $P_{\text{F}}\left(c,\infty \right)$ [full black line] obtained from Monte Carlo simulations for a coupling constant $\lambda =0.05$.