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arXiv:nlin/0409007

Fidelity Decay for Phase Space Displacements

Diego Vaz Bevilaqua, Eric J. Heller

Published Sep 3, 2004• 2 citations
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Abstract

In this letter we analyse the behavior of fidelity decay under a very specific kind of perturbation: phase space displacements. Under these perturbations, systems will decay following the Lyapunov regime only. Others universal regimes discussed in the literature are not presented in this case; instead, for small values of the perturbation we observe quantum freeze of the fidelity. We also show that it is possible to connect this result with the incoherent neutron scattering problem

DOI
10.48550/arxiv.nlin/0409007

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In this letter we analyse the behavior of fidelity decay under a very specific kind of perturbation: phase space displacements. Under these perturbations, systems will decay following the Lyapunov regime only. Others universal regimes discussed in the literature are not presented in this case; instead, for small values of the perturbation we observe quantum freeze of the fidelity. We also show that it is possible to connect this result with the incoherent neutron scattering problem
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.
We discuss quantum fidelity decay of classically regular dynamics, in particular for an important special case of a vanishing time-averaged perturbation operator, i.e. vanishing expectation values of the perturbation in the eigenbasis of unperturbed dynamics. A complete semiclassical picture of this situation is derived in which we show that the quantum fidelity of individual coherent initial states exhibits three different regimes in time: (i) first it follows the corresponding classical fidelity up to time , (ii) then it freezes on a plateau of constant value, (iii) and after a timescale it exhibits fast ballistic decay as where is a strength of perturbation. All the constants are computed in terms of classical dynamics for sufficiently small effective value of the Planck constant. A similar picture is worked out also for general initial states, and specifically for random initial states, where , and . This prolonged stability of quantum dynamics in the case of a vanishing time-averaged perturbation could prove to be useful in designing quantum devices. Theoretical results are verified by numerical experiments on the quantized integrable kicked top.
We discuss quantum fidelity decay of classically regular dynamics, in particular for an important special case of vanishing time averaged perturbation operator. We show that quantum fidelity of individual initial states exhibits three different regimes in time: (i) for short times t < t_1 it follows the corresponding classical fidelity, (ii) then it freezes at a constant value - the plateau, (iii) only after much longer time t_2 it again starts to decay. This freezing of fidelity is purely quantum phenomenon and could find applications in improving stability of quantum devices.
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.
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 show, via numerical simulations, that the fidelity decay behavior of quasi-integrable systems is strongly dependent on the location of the initial coherent state with respect to the underlying classical phase space. In parallel to classical fidelity, the quantum fidelity generally exhibits Gaussian decay when the perturbation affects the frequency of periodic phase space orbits and power-law decay when the perturbation changes the shape of the orbits. For both behaviors the decay rate also depends on initial state location. The spectrum of the initial states in the eigenbasis of the system reflects the different fidelity decay behaviors. In addition, states with initial Gaussian decay exhibit a stage of exponential decay for strong perturbations. This elicits a surprising phenomenon: a strong perturbation can induce a higher fidelity than a weak perturbation of the same type.