05-326 Monique Combescure; Didier Robert

A phase-space study of the quantum Loschmidt Echo in the semiclassical limit (246K, pdf) Sep 19, 05

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Abstract. The notion of Loschmidt echo (also called ``quantum fidelity'') has been introduced in order to study the (in)-stability of the quantum dynamics under perturbations of the Hamiltonian. It has been extensively studied in the past few years in the physics literature, in connection with the problems of ``quantum chaos'', quantum computation and decoherence.\\ In this paper, we study this quantity semiclassically (as $\hbar \to 0$), taking as reference quantum states the usual coherent states. The latter are known to be well adapted to a semiclassical analysis, in particular with respect to semiclassical estimates of their time evolution. For times not larger than the so-called ``Ehrenfest time'' $C \vert \log \hbar \vert$, we are able to estimate semiclassically the Loschmidt Echo as a function of $t$ (time), $\hbar$ (Planck constant), and $\delta$ (the size of the perturbation). The way two classical trajectories merging from the same point in classical phase-space, fly apart or come close together along the evolutions governed by the perturbed and unperturbed Hamiltonians play a major role in this estimate.\\ We also give estimates of the ``return probability'' (again on reference states being the coherent states) by the same method, as a function of $t$ and $\hbar$.

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