About Quantum Revivals, Quantum Fidelity, A semiclassical Approach

In this paper we develop the topics of Quantum Recurrences and of Quantum Fidelity which have attracted great interest in recent years. The return probability is given by the square 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 initial wavepacket a coherent state located on a closed orbit of the corresponding classical motion. The "quantum fidelity" (or Loschmidt Echo) is the square modulus of the overlap of an evoloved quantum state with the same evoloved by a slightly perturbed Hamiltonian. Its decrease in time measures the sensitivity of Quantum Evolution with respect to small perturbations. It is believed to have significantly different behavior in time when the underlying classical motion is chaotic or regular. 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

The quantum fidelity for the time-dependent singular quantum oscillator

In this paper we perform an exact study of ``Quantum Fidelity'' (also called Loschmidt Echo) for the time-periodic quantum Harmonic Oscillator of Hamiltonian : $$\hat H\_{g}(t):=\frac{P^2}{2}+ f(t)\frac{Q^2}{2}+\frac{g^2}{Q^2}$$ when compared with the quantum evolution induced by $\hat H\_{0}(t)$ ($g=0$), in the case where $f$ is a $T$-periodic function and $g$ a real constant. The reference (initial) state is taken to be an arbitrary ``generalized coherent state'' in the sense of Perelomov. We show that, starting with a quadratic decrease in time in the neighborhood of $t=0$, this quantum fidelity may recur to its initial value 1 at an infinite sequence of times {$t\_{k}$}. We discuss the result when the classical motion induced by Hamiltonian $\hat H\_{0}(t)$ is assumed to be stable versus unstable. A beautiful relationship between the quantum and the classical fidelity is also demonstrated