Quantum Chaos: Degree of Reversibility of Quantum Dynamics of Classically Chaotic Systems

We present a quantitative analysis of the reversibility properties of classically chaotic quantum motion by relating the degree of reversibility to the rate at which a quantum state acquires a more and more complicated structure during its time evolution. This complexity can be characterized by the number M(t) of harmonics of the (initially isotropic, i.e. M(0) = 0) Wigner function, which are generated by the time t. We show that, in contrast to the classical exponential increase, this number can grow after the Ehrenfest time tE not faster than linearly. It follows that if the motion is reversed at some arbitrary moment T immediately after applying an instant perturbation with intensity described by the parameter ξ, then there exists a critical perturbation strength, ξc ≈ √ 2/M(T ), below which the initial state is well recovered, whereas reversibility disappears when ξ & ξc(T ). In the classical limit the number of harmonics proliferates exponentially with time and the motion becomes practically irreversible. The above results are illustrated in the example of the kicked quartic oscillator model.