Wigner Quasiprobability Distribution Function Research Articles

A semiclassical theory of quantum noise in open chaotic systems

We consider the quantum evolution of classically chaotic systems in contact with the surroundings. Based on ℏ-scaling of an equation for time evolution of the Wigner's quasi-probability distribution function in presence of dissipation and thermal diffusion we derive a semiclassical equation for quantum fluctuations. This identifies an early regime of evolution dominated by fluctuations in the curvature of the potential due to classical chaos and dissipation. A stochastic treatment of this classical fluctuations leads us to a Fokker-Planck equation which is reminiscent of Kramers' equation for thermally activated processes. This reveals an interplay of three aspects of evolution of quantum noise in weakly dissipative open systems; the reversible Liouville flow, the irreversible chaotic diffusin which is characteristic of the system itself, and irreversible dissipation induced by the external reservoir. It has been demonstrated that in the dissipation-free case a competition between Liouville flow in the contracting direction of phase space and chaotic diffusion sets a critical width in the Wigner function for quantum fluctuations. We also show how the initial quantum noise gets amplified by classical chaos and ultimately equilibrated under the influence of dissipation. We establish that there exists a critical limit to the expansion of phase space. The limit is determined by chaotic diffusion and dissipation. Making use of appropriate quantum-classical correspondence we verify the semiclassical analysis by the fully quantum simulation in a chaotic quartic oscillator.

Quantum distribution functions for radial observables

For quantum systems with two-dimensional configuration space we construct a physical radial momentum observable. Rescaling the radius we find that the dilation degrees of freedom form a Weyl algebra. With this we construct a radial Wigner quasiprobability distribution function.

Wigner function for a generalized model of a parametric oscillator: phase-space tristability, competition and nonclassical effects

We present an exact quantum treatment of the generalized model of a degenerate parametric oscillator, in which we allow for self-phase modulation of the signal mode. Using the steady-state solution of the Fokker-Planck equation in the complex P-representation we obtain an exact analytical result for the Wigner quasiprobability distribution function. The obtained Wigner function allows us to give an explicit phase-space description of the nonlinear system under consideration, including the critical transition behavior in the monostable and bistable (with respect to the signal mode intensity versus the pump field intensity) operation regimes and in the threshold region. The competitive effects influenced by the self-phase modulation and the phase-space tristability are analyzed. Nonclassical effects of quadrature squeezing and quantum superposition are discussed as well.