On quantum revivals and quantum fidelity. A semiclassical approach

The aim of this paper is three-fold: first, to establish in a clear and rigorous way a formula proposed heuristically by Mehlig and Wilkinson for the metaplectic operators corresponding to a given symplectic transformation in classical phase space. Second, this formula is applied to the study of quantum recurrences, which has attracted a great deal of interest in recent years (see [35] for a complete account of the recent approaches). The return probability is given by the squared modulus of the overlap between a given initial wavepacket and the corresponding evolved one; quantum recurrences in time can be observed if this overlap is unity. We provide some conditions under which this is semiclassically achieved taking as the initial wavepacket a coherent state localized on a closed orbit of the corresponding classical motion. Third, we start a rigorous approach of the 'quantum fidelity' (or the Loschmidt echo): it is the squared modulus of the overlap of an evolved quantum state with the same state evolved by a slightly perturbed Hamiltonian. It has attracted a great deal of interest in the last decade, for the purpose of 'quantum chaos' problems, and in quantum computation analysis. However, the results are most of the time not entirely conclusive, and sometimes even contradictory. Thus it is useful to start a rigorous approach to this problem. The decrease in time of the quantum fidelity measures the sensitivity of quantum evolution with respect to small perturbations. Starting with suitable initial quantum states, we develop a semiclassical estimate of this quantum fidelity in the linear response framework (appropriate for the small perturbation regime), assuming some ergodicity conditions on the corresponding classical motion.