Nonmonotonic short-time decay of the Loschmidt echo in quasi-one-dimensional systems

Origin of the exponential decay of the Loschmidt echo in integrable systems

We address the time decay of the Loschmidt echo, measuring the sensitivity of quantum dynamics to small Hamiltonian perturbations, in one-dimensional integrable systems. Using a semiclassical analysis, we show that the Loschmidt echo may exhibit a well-pronounced regime of exponential decay, similar to the one typically observed in quantum systems whose dynamics is chaotic in the classical limit. We derive an explicit formula for the exponential decay rate in terms of the spectral properties of the unperturbed and perturbed Hamilton operators and the initial state. In particular, we show that the decay rate, unlike in the case of the chaotic dynamics, is directly proportional to the strength of the Hamiltonian perturbation. Finally, we compare our analytical predictions against the results of a numerical computation of the Loschmidt echo for a quantum particle moving inside a one-dimensional box with Dirichlet-Robin boundary conditions, and find the two in good agreement.

Initial-value representation for the Loschmidt echo

We obtain an initial-value representation for the quantum Loschmidt echo from the semiclassical theory of Wigner function evolution, together with the classical first-order perturbation theory. In the limit of small actions, the amplitude of each trajectory reduces to unity, just as in the dephasing representation introduced by Vaníček [Phys. Rev. E 70, 055201(R) (2004)], but these trajectories are generated here by the mean Hamiltonian for both the forward and the backward motion. This slight change of action may substantially alter the phase. The amplitude correction depends on the second derivative of the action. This improved dephasing approximation is verified to work even for quadratic Hamiltonians, for which the semiclassical evolution is exact, thus extending the range of application beyond its original scope in quantum chaos.