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arXiv:quant-ph/0410099 · Physical Review E

Mesoscopic fluctuations of the Loschmidt echo

Cyril Petitjean, Philippe Jacquod

Published Mar 29, 2005• 14 citations
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Abstract

We investigate the time-dependent variance of the fidelity with which an initial narrow wave packet 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 ${\ensuremath{\hbar}}_{\mathrm{eff}}$, this decay is given by the sum of a classical term $\ensuremath{\simeq}\mathrm{exp}[\ensuremath{-}2\ensuremath{\lambda}t]$, a quantum term $\ensuremath{\simeq}2{\ensuremath{\hbar}}_{\mathrm{eff}}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[\ensuremath{-}\ensuremath{\Gamma}t]$, and a mixed term $\ensuremath{\simeq}2\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[\ensuremath{-}(\ensuremath{\Gamma}+\ensuremath{\lambda})t]$. Compared to the behavior of the average fidelity, this allows for the extraction of the classical Lyapunov exponent $\ensuremath{\lambda}$ in a larger parameter range. Our results are confirmed by numerical simulations.

DOI
10.1103/physreve.71.036223

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•Apr 3, 2008•0 citations
We investigate the time-dependent variance of the fidelity with which an initial narrow wave packet 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 tc{t}_{c}tc​, after which it decays. To leading order in the effective Planck's constant \ensuremathℏeff{\ensuremath{\hbar}}_{\mathrm{eff}}\ensuremathℏeff​, this decay is given by the sum of a classical term \ensuremath≃exp[\ensuremath−2\ensuremathλt]\ensuremath{\simeq}\mathrm{exp}[\ensuremath{-}2\ensuremath{\lambda}t]\ensuremath≃exp[\ensuremath−2\ensuremathλt], a quantum term \ensuremath≃2\ensuremathℏeffexp[\ensuremath−\ensuremathΓt]\ensuremath{\simeq}2{\ensuremath{\hbar}}_{\mathrm{eff}}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[\ensuremath{-}\ensuremath{\Gamma}t]\ensuremath≃2\ensuremathℏeff​exp[\ensuremath−\ensuremathΓt], and a mixed term \ensuremath≃2exp[\ensuremath−(\ensuremathΓ+\ensuremathλ)t]\ensuremath{\simeq}2\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[\ensuremath{-}(\ensuremath{\Gamma}+\ensuremath{\lambda})t]\ensuremath≃2exp[\ensuremath−(\ensuremathΓ+\ensuremathλ)t]. Compared to the behavior of the average fidelity, this allows for the extraction of the classical Lyapunov exponent \ensuremathλ\ensuremath{\lambda}\ensuremathλ in a larger parameter range. Our results are confirmed by numerical simulations.
Motivated by neutron scattering experiments, we investigate the decay of the fidelity with which a wave packet is reconstructed by a perfect time-reversal operation performed after a phase-space displacement. In the semiclassical limit, we show that the decay rate is generically given by the Lyapunov exponent of the classical dynamics. For small displacements, we additionally show that, following a short-time Lyapunov decay, the decay freezes well above the ergodic value because of quantum effects. Our analytical results are corroborated by numerical simulations.
The Loschmidt echo (LE) is a measure of the stability and reversibility of quantum evolution under perturbations of the Hamiltonian. One of the expected and most relevant characteristics of this quantity for chaotic systems is an exponential decay with a perturbation-independent decay rate given by the classical Lyapunov exponent. However, a non-uniform decay—instead of the Lyapunov regime—has been reported in several systems. In this work, we find an analytical semiclassical expression for the averaged fidelity amplitude that can be directly related to the anomalous—unexpected—behaviour of the LE.
We study the decay rate of the Loschmidt echo or fidelity in a chaotic system under a time-dependent perturbation V(q,t)V(\mathbf{q},t)V(q,t) with typical strength $\ensuremath{\hbar}∕{\ensuremath{\tau}}_{V}$. The perturbation represents the action of an uncontrolled environment interacting with the system, and is characterized by a correlation length ${\ensuremath{\xi}}_{0}$ and a correlation time ${\ensuremath{\tau}}_{0}$. For small perturbation strengths or rapid fluctuating perturbations, the Loschmidt echo decays exponentially with a rate predicted by the Fermi ``golden rule,'' $1∕\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}={\ensuremath{\tau}}_{c}∕{\ensuremath{\tau}}_{V}^{2}$, where ${\ensuremath{\tau}}_{c}\ensuremath{\sim}\mathrm{min}[{\ensuremath{\tau}}_{0},{\ensuremath{\xi}}_{0}∕v]$ and vvv is the typical particle velocity. Whenever the rate $1∕\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}$ is larger than the Lyapunov exponent of the system, a perturbation independent Lyapunov decay regime arises. We also find that by speeding up the fluctuations (while keeping the perturbation strength fixed) the fidelity decay becomes slower, and hence one can protect the system against decoherence.
Classical chaotic dynamics is characterized by exponential sensitivity to initial conditions. Quantum mechanics, however, does not show this feature. We consider instead the sensitivity of quantum evolution to perturbations in the Hamiltonian. This is observed as an attenuation of the Loschmidt echo M(t), i.e., the amount of the original state (wave packet of width sigma) which is recovered after a time reversed evolution, in the presence of a classically weak perturbation. By considering a Lorentz gas of size L, which for large L is a model for an unbounded classically chaotic system, we find numerical evidence that, if the perturbation is within a certain range, M(t) decays exponentially with a rate 1/tau(phi) determined by the Lyapunov exponent lambda of the corresponding classical dynamics. This exponential decay extends much beyond the Eherenfest time t(E) and saturates at a time t(s) approximately equal to lambda(-1)ln[N], where N approximately (L/sigma)(2) is the effective dimensionality of the Hilbert space. Since tau(phi) quantifies the increasing uncontrollability of the quantum phase (decoherence) its characterization and control has fundamental interest.
We reexamine the problem of the ``Loschmidt echo,'' that measures the sensitivity to perturbation of quantum-chaotic dynamics. The overlap squared M(t)M(t)M(t) of two wave packets evolving under slightly different Hamiltonian is shown to have the double-exponential initial decay $\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}\mathrm{constant}\ifmmode\times\else\texttimes\fi{}{e}^{2{\ensuremath{\lambda}}_{0}t})$ in the main part of the phase space. The coefficient ${\ensuremath{\lambda}}_{0}$ is the self-averaging Lyapunov exponent. The average decay $\overline{M}\ensuremath{\propto}{e}^{\ensuremath{-}{\ensuremath{\lambda}}_{1}t}$ is single exponential with a different coefficient ${\ensuremath{\lambda}}_{1}.$ The volume of phase space that contributes to M‾\overline{M}M vanishes in the classical limit $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Elzxh}}0$ for times less than the Ehrenfest time ${\ensuremath{\tau}}_{E}=\frac{1}{2}{\ensuremath{\lambda}}_{0}^{\ensuremath{-}1}|\mathrm{ln}\ensuremath{\Elzxh}|.$ It is only after the Ehrenfest time that the average decay is representative for a typical initial condition.
We address the sensitivity of quantum mechanical time evolution by considering the time decay of the Loschmidt echo (LE) (or fidelity) for local perturbations of the Hamiltonian. Within a semiclassical approach we derive analytical expressions for the LE decay for chaotic systems for the whole range from weak to strong local perturbations and identify different decay regimes which complement those known for the case of global perturbations. For weak perturbations a Fermi-golden-rule (FGR) type behavior is recovered. For strong perturbations the escape-rate regime is reached, where the LE decays exponentially with a rate independent of the perturbation strength. The transition between the FGR regime and the escape-rate regime is non-monotonic, i.e. the rate of the exponential time-decay of the LE oscillates as a function of the perturbation strength. We further perform extensive quantum mechanical calculations of the LE based on numerical wave packet evolution which strongly support our semiclassical theory. Finally, we discuss in some detail possible experimental realizations for observing the predicted behavior of the LE.