The Loschmidt echo in classically chaotic systems: Quantum chaos, irreversibility and decoherence

摘要

The Loschmidt echo (LE) is a measure of the sensitivity of quantum mechanicsto perturbations in the evolution operator. It is defined as the overlap of twowave functions evolved from the same initial state but with slightly differentHamiltonians. Thus, it also serves as a quantification of irreversibility inquantum mechanics. In this thesis the LE is studied in systems that have a classical counterpartwith dynamical instability, that is, classically chaotic. An analyticaltreatment that makes use of the semiclassical approximation is presented. It isshown that, under certain regime of the parameters, the LE decaysexponentially. Furthermore, for strong enough perturbations, the decay rate isgiven by the Lyapunov exponent of the classical system. Some particularlyinteresting examples are given. The analytical results are supported by thorough numerical studies. Inaddition, some regimes not accessible to the theory are explored, showing thatthe LE and its Lyapunov regime present the same form of universality ascribedto classical chaos. In a sense, this is evidence that the LE is a robusttemporal signature of chaos in the quantum realm. Finally, the relation between the LE and the quantum to classical transitionis explored, in particular with the theory of decoherence. Using two differentapproaches, a semiclassical approximation to Wigner functions and a masterequation for the LE, it is shown that the decoherence rate and the decay rateof the LE are equal. The relationship between these quantities results mutuallybeneficial, in terms of the broader resources of decoherence theory and of thepossible experimental realization of the LE.