The quantum-classical crossover in the adiabatic response of chaotic systems

The autocorrelation function of the force acting on a slow classical system, resulting from interaction with a fast quantum system is calculated following Berry-Robbins, Wilkinson and Jarzynski within the leading order correction to the adiabatic approximation. The time integral of the autocorrelation function is often proportional to the rate of energy transfer between the systems. The fast quantum system is assumed to be chaotic in the classical limit for each configuration of the slow system. An analytic formula is obtained for the finite-time integral of the correlation function, in the framework of random matrix theory (RMT), for a specific dependence on the adiabatically varying parameter. Extension to a wider class of RMT models is discussed. For the Gaussian unitary and symplectic ensembles for long times the time integral of the correlation function vanishes or falls off as a Gaussian with a characteristic time that is proportional to the Heisenberg time, depending on the details of the model. The fall-off is inversely proportional to time for the Gaussian orthogonal ensemble. The correlation function is found to be dominated by the nearest-neighbour level spacings. It was calculated for a variety of nearest-neighbour level spacing distributions, including ones that do not originate From RMT ensembles. The various approximate formulae obtained are tested numerically in RMT. The results shed light on the quantum to classical crossover for chaotic systems. The implications on the possibility to experimentally observe deterministic friction are discussed.