Quantization of H�non's map with dissipation

Abstract

Henon's map with dissipation is suspended to the nonlinearly kicked damped harmonic oscillator and then quantized. The ensuing master equation between two subsequent kicks is solved exactly in the representation by the Wigner distribution, resulting in a quantized version of Henon's dissipative map. The semi-classical limit of the map is studied. The leading quantum corrections are shown to be associated with dissipation and can be formulated as a classical map with classical stochastic perturbations. The next-to-leading quantum corrections, arising from the nonlinearity of the kicks, are similar as in the area conserving map and cannot be described within the framwork of classical statistics. The Wigner distribution in the steady state is investigated in the limit of strong dissipation, where Henon's map is reduced to the logistic map. The insensitivity of the main results against details of the quantization procedure is demonstrated by comparing with the results of a different phenomenological quantization procedure.