Quantum freeze of fidelity decay for a class of integrable dynamics

Abstract

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 t1 ∼ ℏ-1/2, (ii) then it freezes on a plateau of constant value, (iii) and after a timescale t2 ∼ min{ℏ 1/2δ-2, ℏ-1/2δ-1} it exhibits fast ballistic decay as exp(-constant x δ4t 2/ℏ) where δ is a strength of perturbation. All the constants are computed in terms of classical dynamics for sufficiently small effective value ℏ of the Planck constant. A similar picture is worked out also for general initial states, and specifically for random initial states, where t1 ∼ 1, and t2 ∼ δ-1. 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.