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arXiv:nlin/0203052 · Physical Review Letters

Decay of the Loschmidt Echo for Quantum States with Sub-Planck-Scale Structures

Philippe Jacquod, İnanç Adagideli, C. W. J. Beenakker

Published Sep 20, 2002• 82 citations
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Abstract

Quantum states extended over a large volume in phase space have oscillations from quantum interferences in their Wigner distribution on scales smaller than variant Planck's over 2pi [W. H. Zurek, Nature (London) 412, 712 (2001)]]. We investigate the influence of those sub-Planck-scale structures on the sensitivity to an external perturbation of the state's time evolution. While we do find an accelerated decay of the Loschmidt Echo for an extended state in comparison to a localized wave packet, the acceleration is described entirely by the classical Lyapunov exponent and hence cannot originate from quantum interference.

DOI
10.1103/physrevlett.89.154103

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Quantum states extended over a large volume in phase space have oscillations from quantum interferences in their Wigner distribution on scales smaller than variant Planck's over 2pi [W. H. Zurek, Nature (London) 412, 712 (2001)]]. We investigate the influence of those sub-Planck-scale structures on the sensitivity to an external perturbation of the state's time evolution. While we do find an accelerated decay of the Loschmidt Echo for an extended state in comparison to a localized wave packet, the acceleration is described entirely by the classical Lyapunov exponent and hence cannot originate from quantum interference.
The Loschmidt echo measures the sensitivity to perturbations of quantum evolutions. We study its short-time decay in classically chaotic systems. Using perturbation theory and throwing out all correlation imposed by the initial state and the perturbation, we show that the characteristic time of this regime is well described by the inverse of the width of the local density of states. This result is illustrated and discussed in a numerical study in a two-dimensional chaotic billiard system perturbed by various contour deformations and using different types of initial conditions. Moreover, the influence to the short-time decay of sub-Planck structures developed by time evolution is also investigated.
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.
Loschmidt echo (LE) is a measure of reversibility and sensitivity to perturbations of quantum evolutions. For weak perturbations its decay rate is given by the width of the local density of states (LDOS). When the perturbation is strong enough, it has been shown in chaotic systems that its decay is dictated by the classical Lyapunov exponent. However, several recent studies have shown an unexpected nonuniform decay rate as a function of the perturbation strength instead of that Lyapunov decay. Here we study the systematic behavior of this regime in perturbed cat maps. We show that some perturbations produce coherent oscillations in the width of LDOS that imprint clear signals of the perturbation in LE decay. We also show that if the perturbation acts in a small region of phase space (local perturbation) the effect is magnified and the decay is given by the width of the LDOS.
The Loschmidt echo (LE) is a magnitude that measures the sensitivity of quantum dynamics to perturbations in the Hamiltonian. For a certain regime of the parameters, the LE decays exponentially with a rate given by the Lyapunov exponent of the underlying classically chaotic system. We develop a semiclassical theory, supported by numerical results in a Lorentz gas model, which allows us to establish and characterize the universality of this Lyapunov regime. In particular, the universality is evidenced by the semiclassical limit of the de Broglie wavelength going to zero, the behavior for times longer than Ehrenfest time, the insensitivity with respect to the form of the perturbation, and the behavior of individual (nonaveraged) initial conditions. Finally, by elaborating a semiclassical approximation to the Wigner function, we are able to distinguish between classical and quantum origin for the different terms of the LE. This approach renders an understanding for the persistence of the Lyapunov regime after the Ehrenfest time, as well as a reinterpretation of our results in terms of the quantum-classical transition.
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.