Wigner function and Schroedinger equation in phase space representation

We discuss a family of quasi-distributions (s-ordered Wigner functions of Agarwal and Wolf) and its connection to the so called phase space representation of the Schroedinger equation. It turns out that although Wigner functions satisfy the Schroedinger equation in phase space they have completely different interpretation.Comment: 6 page

Quantum versus classical phase-locking transition in a driven-chirped oscillator

Classical and quantum-mechanical phase locking transition in a nonlinear oscillator driven by a chirped frequency perturbation is discussed. Different limits are analyzed in terms of the dimensionless parameters $/\sqrt{2m\hbar \omega_{0}\alpha}$ and $P_{2}=(3\hbar \beta)/(4m\sqrt{\alpha})$ ($\epsilon,$ $\alpha,$ $\beta$ and $\omega_{0}$ being the driving amplitude, the frequency chirp rate, the nonlinearity parameter and the linear frequency of the oscillator). It is shown that for $P_{2}\ll P_{1}+1$, the passage through the linear resonance for $P_{1}$ above a threshold yields classical autoresonance (AR) in the system, even when starting in a quantum ground state. In contrast, for $$, the transition involves quantum-mechanical energy ladder climbing (LC). The threshold for the phase-locking transition and its width in $P_{1}$ in both AR and LC limits are calculated. The theoretical results are tested by solving the Schrodinger equation in the energy basis and illustrated via the Wigner function in phase space

Optical second harmonic generation near a black hole horizon as possible source of experimental information on quantum gravitational effects

Optical second harmonic generation near a black hole horizon is suggested as a source of experimental information on quantum gravitational effects. While absent in the framework of general relativity, second harmonic generation appears in the toy models of sonic and electromagnetic black holes, where spatial dispersion at high frequencies for waves boosted towards the horizon is introduced. Localization effects in the light scattering from random fluctuations of matter fields and space-time metric near the black hole horizon produce a pronounced peak in the angular distribution of second harmonics of light in the direction normal to the horizon. Such second harmonic light has the best chances to escape the vicinity of the black hole. This phenomenon is similar to the well-known strong enhancement of diffuse second harmonic emission from a randomly rough metal surface in the direction normal to the surface.Comment: 4 pages, 1 figur