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Mesoscopic Fluctuations of the Loschmidt Echo

Cyril Petitjean·Philippe Jacquod
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Abstract

We investigate the time-dependent variance of the fidelity with which an initial narrow wavepacket is reconstructed after its dynamics is time-reversed with a perturbed Hamiltonian. In the semiclassical regime of perturbation, we show that the variance first rises algebraically up to a critical time $t_c$, after which it decays. To leading order in the effective Planck's constant $\hbar_{\rm eff}$, this decay is given by the sum of a classical term $\simeq \exp[-2 λt]$, a quantum term $\simeq 2 \hbar_{\rm eff} \exp[-Γt]$ and a mixed term $\simeq 2 \exp[-(Γ+λ)t]$. Compared to the behavior of the average fidelity, this allows for the extraction of the classical Lyapunov exponent $λ$ in a larger parameter range. Our results are confirmed by numerical simulations.

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Cyril Petitjean, Philippe Jacquod. 2005-03-17. Mesoscopic Fluctuations of the Loschmidt Echo. https://doi.org/10.1103/physreve.71.036223

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