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.

Figure 1

Particle in a one-dimensional box with Dirichlet-Robin boundary conditions.

Figure 2

Time decay of the LE for a particle moving inside a one-dimensional box with Dirichlet-Robin boundary conditions. The system, perturbation, and the initial state are characterized by the following set of parameters: $\lambda =1$, $\epsilon =0.01$, $x_0=0.5$, $\sigma =0.01$, and $p_0=300$ (red), 400 (green), 500 (blue). Solid curves represent the numerically exact LE. Straight dashed lines of the corresponding color show the trend of the exponential decay $e^{-\gamma t}$, with $\gamma$ computed in accordance with Eqs. (37). See text for details.