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arXiv:1106.4206 · New Journal of Physics

Semiclassical approach to fidelity amplitude

Ignacio García-Mata, Raúl O Vallejos, Diego A. Wisniacki

Published Oct 31, 2011• 8 citations
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Abstract

The fidelity amplitude (FA) is a quantity of paramount importance in echo-type experiments.We use semiclassical theory to study the average FA for quantum chaotic systems under external perturbation.We explain analytically two extreme cases: the random dynamics limit-attained approximately by strongly chaotic systems-and the random perturbation limit, which shows a Lyapunov decay.Numerical simulations help us to bridge the gap between both the extreme cases.

DOI
10.1088/1367-2630/13/10/103040

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The fidelity amplitude (FA) is a quantity of paramount importance in echo-type experiments.We use semiclassical theory to study the average FA for quantum chaotic systems under external perturbation.We explain analytically two extreme cases: the random dynamics limit-attained approximately by strongly chaotic systems-and the random perturbation limit, which shows a Lyapunov decay.Numerical simulations help us to bridge the gap between both the extreme cases.
The fidelity amplitude (FA) is a quantity of paramount importance in echo-type experiments. We use semiclassical theory to study the average FA for quantum chaotic systems under external perturbation. We explain analytically two extreme cases: the random dynamics limit-attained approximately by strongly chaotic systems-and the random perturbation limit, which shows a Lyapunov decay. Numerical simulations help us to bridge the gap between both the extreme cases. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.
We present a numerically feasible semiclassical (SC) method to evaluate fidelity decay (Loschmidt echo, FD) in a classically chaotic system. It was thought that such evaluation would be intractable, but instead we show that a uniform SC expression not only is tractable but it gives remarkably accurate numerical results for the standard map in both the Fermi-golden-rule and Lyapunov regimes. Remarkably, it also explicitly contains the ``building blocks'' of analytical theories of recent literature, and thus permits a direct test of the approximations made by other authors in these regimes, rather than a post facto comparison with numerical results. We explain in more detail the extended validity of the classical perturbation approximation (CPA) and show that within this approximation,``diagonal approximation'' of Jalabert and Pastawski [1] is automatic and does not require ensemble averaging.
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.
The Loschmidt echo--also known as fidelity--is a very useful tool to study irreversibility in quantum mechanics due to perturbations or imperfections. Many different regimes, as a function of time and strength of the perturbation, have been identified. For chaotic systems, there is a range of perturbation strengths where the decay of the Loschmidt echo is perturbation independent, and given by the classical Lyapunov exponent. But observation of the Lyapunov decay depends strongly on the type of initial state upon which an average is carried out. This dependence can be removed by averaging the fidelity over the Haar measure, and the Lyapunov regime is recovered, as has been shown for quantum maps. In this work, we introduce an analogous quantity for systems with infinite dimensional Hilbert space, in particular the quantum stadium billiard, and we show clearly the universality of the Lyapunov regime.
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.
We show that the mechanism of quantum freeze of fidelity decay for perturbations with a zero time average, recently discovered for a specific case of integrable dynamics [New J. Phys. 5, 109 (2003)], can be generalized to arbitrary quantum dynamics. We work out explicitly the case of a chaotic classical counterpart, for which we find semiclassical expressions for the value and the range of the plateau of fidelity. After the plateau ends, we find explicit expressions for the asymptotic decay, which can be exponential or Gaussian depending on the ratio of the Heisenberg time to the decay time. Arbitrary initial states can be considered; e.g., we discuss coherent states and random states.