Uniform semiclassical approach to fidelity decay: From weak to strong perturbation

Abstract

We study fidelity decay by a uniform semiclassical approach, in the three perturbation regimes, namely, the perturbative regime, the Fermi-golden-rule (FGR) regime, and the Lyapunov regime. A semiclassical expression is derived for fidelity of initial Gaussian wave packets with width of the order √h (h being the effective Planck constant). Short time decay of fidelity of initial Gaussian wave packets is also studied, with respect to two time scales introduced in the semiclassical approach. In the perturbative regime, it is confirmed numerically that fidelity has the FGR decay before the Gaussian decay sets in. An explanation is suggested to a non-FGR decay in the FGR regime, which has been observed in a system with weak chaos in the classical limit, by using the Levy distribution as an approximation for the distribution of action difference. In the Lyapunov regime, it is shown that the average of the logarithm of fidelity may have roughly the Lyapunov decay within some time interval, in systems possessing large fluctuation of the finite-time Lyapunov exponent in the classical limit.

Dynamical Stability in a Non-Hermitian Kicked Rotor Model

“…Quantum mapping models provide ideal platforms for investigating quantum chaoticity from different prospectives, such as the eigenenergy level spacing and the wavepacket dynamics. A paradigm model of quantum mapping systems is the quantum kicked rotor (QKR), which has been widely employed in the study of the fundamental problems, for instance quantum-classical transition [2], quantum irreversibility [3], ergodicity [4], and prethermalization [5]. The landmark study by Peres shown that for classically chaotic systems, the small perturbation on the Hamiltonian leads to the exponential divergence of the fidelity, i.e., Loschmidt echo, between two nearby quantum states [6], which is a solid evidence of exponential instability of quantum chaos.…”

Sensitivity of energy eigenstates to perturbation in quantum integrable and chaotic systems

“…Loosely speaking, the LE decays exponentially in quantum chaotic systems under perturbations neither very weak nor very strong (see, e.g. [9,10]); here, under relatively strong perturbations, the decay rate is given by a Lyapunov-exponenttype quantity of the underlying classical dynamics [9]. In contrast, in integrable systems, the LE shows a Gaussian decay within some initial times [11], followed by a power-law decay at long times [12].…”

Path integral approach to the quantum fidelity amplitude

“…Vaníček & Heller [20] avoided the search for stationary-phase points and obtained a uniform expression for fidelity by combining Miller's initial value representation [21,22] with the semiclassical perturbation approximation [23]. This surprisingly simple and accurate expression, although limited to wave packets localized in position, was successfully applied as a starting point to derive the decay of fidelity in the deep Lyapunov regime [24] and the plateau of fidelity in neutron scattering [25]. By linearizing the semiclassical initial value representation of the fidelity amplitude, Vaníček later obtained [26,27] a more general and accurate approximation, the so-called dephasing representation,…”