Chaos and energy spreading for time-Dependent Hamiltonians, and the various Regimes in the theory of Quantum Dissipation

TL;DR

This paper develops a general theory for energy spreading and quantum dissipation in time-dependent Hamiltonians, connecting classical chaos, quantum regimes, and fluctuation-dissipation relations, with implications for nuclear and mesoscopic physics.

Contribution

It introduces a unified framework for understanding energy spreading and dissipation across classical and quantum regimes, including the crossover from perturbative to semiclassical behavior.

Findings

Classical chaos leads to a crossover from ballistic to diffusive energy spreading.

Perturbation theory applies in certain regimes, while semiclassical considerations are valid in others.

A universal fluctuation-dissipation relation is established during the crossover.

Abstract

We make the first steps towards a generic theory for energy spreading and quantum dissipation. The Wall formula for the calculation of friction in nuclear physics and the Drude formula for the calculation of conductivity in mesoscopic physics can be regarded as two special results of the general formulation. We assume a time-dependent Hamiltonian $H(Q,P;x(t))$ with $x(t)=Vt$, where $V$ is slow in a classical sense. The rate-of-change $V$ is not necessarily slow in the quantum-mechanical sense. Dissipation means an irreversible systematic growth of the (average) energy. It is associated with the stochastic spreading of energy across levels. The latter can be characterized by a transition probability kernel $P_t(n|m)$ where $n$ and $m$ are level indices. This kernel is the main object of the present study. In the classical limit, due to the (assumed) chaotic nature of the dynamics, the second moment of $P_t(n|m)$ exhibits a crossover from ballistic to diffusive behavior. We define the $V$ regimes where either perturbation theory or semiclassical considerations are applicable in order to establish this crossover in the quantal case. In the limit $\hbar\to 0$ perturbation theory does not apply but semiclassical considerations can be used in order to argue that there is detailed correspondence, during the crossover time. In the perturbative regime there is a lack of such correspondence. Namely, $P_t(n|m)$ is characterized by a perturbative core-tail structure that persists during the crossover time. In spite of this lack of (detailed) correspondence there may be still a restricted correspondence as far as the second-moment is concerned. Such restricted correspondence is essential in order to establish the universal fluctuation-dissipation relation.