- 05-326 Monique Combescure; Didier Robert
- A phase-space study of the quantum Loschmidt Echo in the semiclassical limit
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 $\hbar$.