General relation between quantum ergodicity and fidelity of quantum dynamics

A general relation is derived, which expresses the fidelity of quantum dynamics, measuring the stability of time evolution to small static variation in the Hamiltonian, in terms of ergodicity of an observable generating the perturbation as defined by its time correlation function. Fidelity for ergodic dynamics is predicted to decay exponentially on time scale $\propto {\delta }^{-2},$ $\delta \sim$ strength of perturbation, whereas faster, typically Gaussian decay on shorter time scale $\propto {\delta }^{-1}$ is predicted for integrable, or generally nonergodic dynamics. This result needs the perturbation $\delta$ to be sufficiently small such that the fidelity decay time scale is larger than any (quantum) relaxation time, e.g., mixing time for mixing dynamics, or averaging time for nonergodic dynamics (or Ehrenfest time for wave packets in systems with chaotic classical limit). Our surprising predictions are demonstrated in a quantum Ising spin-(1/2) chain periodically kicked with a tilted magnetic field where we find finite parameter-space regions of nonergodic and nonintegrable motion in the thermodynamic limit.