Statistical approach to quantum chaotic ratchets: First results and open problems

This paper is a brief review of a new approach to the quantum-chaotic ratchet effect, introduced recently to address for the first time the sensitivity of the effect of the initial state in a global fashion. This is done by studying statistical properties of the ratchet current over well-defined sets of initial states. First results concern the semiclassical full-chaos regime, where the current is strongly sensitive to the initial state. Natural initial states in this regime are those that are phase-space uniform with the maximal possible resolution of the one Planck cell. General arguments, for a class of paradigmatic model systems and for special quantum-resonance values of a scaled Planck constant , predict that the distribution of the momentum current over all such states is a zero-mean Gaussian with variance D ²2π², where D is the chaotic-diffusion coefficient. This prediction is well supported by extensive numerical evidence. The average strength of the effect, measured by the variance above, is significantly larger than that for the usual momentum states and other states. Open problems, concerning extensions of these first results in different directions, are discussed.