 05326 Monique Combescure; Didier Robert
 A phasespace study of the quantum Loschmidt Echo in the semiclassical limit
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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 socalled ``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
phasespace, 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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