Stability of quantum motion and correlation decay

Abstract

We derive a simple and general relation between the fidelity of quantum motion, characterizing the stability of quantum dynamics with respect to arbitrary static perturbation of the unitary evolution propagator, and the integrated time auto-correlation function of the generator of perturbation. Surprisingly, this relation predicts the slower decay of fidelity the faster decay of correlations is. In particular, for non-ergodic and non-mixing dynamics, where asymptotic decay of correlations is absent, a qualitatively different and faster decay of fidelity is predicted on a time scale ∝ 1/δ as opposed to mixing dynamics where the fidelity is found to decay exponentially on a time-scale ∝ 1/δ 2 , where δ is a strength of perturbation. A detailed discussion of a semi-classical regime of small effective values of Planck constanth is given where classical correlation functions can be used to predict quantum fidelity decay. Note that the correct and intuitively expected classical stability behavior is recovered in the classical limith → 0, as the two limits δ → 0 andh → 0 do not commute. In addition we also discuss non-trivial dependence on the number of degrees of freedom. All the theoretical results are clearly demonstrated numerically on a celebrated example of a quantized kicked top.

Quantum chaos, dynamical stability and decoherence

“…Due to the existence of such time scales, what may be different, and indeed it is, is the degree of stability of dynamical motion. Indeed, as clearly illustrated in the analysis of the Loschmidt echoes with respect to variation of the wavefunction [4] or variation of the Hamiltonian [7], quantum motion turns out to be more stable than the classical motion.…”

“…d xρ 2 ( x, 0) = 1) as determined by the t-th iteration of the Perron-Frobenius operators U 0 and U ε , corresponding to the Hamiltonians H 0 and H ε , respectively. The above definition can be shown to correspond to the classical limit of quantum fidelity [7,15]. For some other applications in the context of classical mechanics see Ref.…”

“…The exponent Γ which can be understood also as the width of the Local density of states [12], can be computed [7] in terms of a 2-point time-correlation function of the perturbation…”

“…Comparing the inverse decay rate 1/Γ of the formula (9) with the Heisenberg time, we obtain the perturbative border [7,12,13] …”

“…Confining ourselves to classically chaotic systems, the emerging picture which results from analytical and numerical investigations [7,[10][11][12][13][14][15][16][17][18] is that both exponential and Gaussian decays are present in the time behavior of fidelity. The strength of the perturbation determines which of the two regimes prevails.…”