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arXiv:quant-ph/0612003 · Physical Review Letters

Displacement Echoes: Classical Decay and Quantum Freeze

Cyril Petitjean, Diego Vaz Bevilaqua, Eric J. Heller, Philippe Jacquod

Published Apr 18, 2007• 23 citations
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Abstract

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.

DOI
10.1103/physrevlett.98.164101

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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 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 ${\ensuremath{\hbar}}_{\mathrm{eff}}$, 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.
The existence of a quantum butterfly effect in the form of exponential sensitivity to small perturbations has been under debate for a long time. Lately, this question has gained increased interest due to the proposal to probe chaotic dynamics and scrambling using out-of-time-order correlators. In this work we study echo dynamics in the Sachdev-Ye-Kitaev model under effective time reversal in a semiclassical approach using the truncated Wigner approximation, which accounts for nonvanishing quantum fluctuations that are essential for the dynamics. We demonstrate that small imperfections introduced in the time-reversal procedure result in an exponential divergence from the perfect echo, which allows us to identify a Lyapunov exponent ${\ensuremath{\lambda}}_{L}$. In particular, we find that ${\ensuremath{\lambda}}_{L}$ is twice the Lyapunov exponent of the semiclassical equations of motion. This behavior is attributed to the growth of an out-of-time-order double commutator that resembles an out-of-time-order correlator.
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.
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.
Motivated by atomic optics experiments, we investigate a class of fidelity functions describing the reconstruction of quantum states by time-reversal operations as ${M}_{\mathrm{Da}}(t)=|\ensuremath{\langle}\ensuremath{\psi}|{e}^{i{H}_{2}t/2}{e}^{i{H}_{1}t/2}{e}^{\ensuremath{-}i{H}_{2}t/2}{e}^{\ensuremath{-}i{H}_{1}t/2}|\ensuremath{\psi}\ensuremath{\rangle}|{}^{2}$. We show that the decay of MDa{M}_{\mathrm{Da}}MDa​ is quartic in time at short times and that it freezes well above the ergodic value at long times, when ${H}_{2}\ensuremath{-}{H}_{1}$ is not too large. The long-time saturation value of MDa{M}_{\mathrm{Da}}MDa​ contains easily extractable information on the strength of decoherence in these systems.