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Annals of the Israel Physical Society

Classical phase-space distribution for approximating quantum states of the harmonic oscillator

David Agmon, I. R. Senitzky

Published Jan 1, 1978• 0 citations
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We investigate systems of few harmonic oscillators with mutual nonlinear coupling. Using classical trajectories---the solutions of Hamiltonian equations of motion for a given nonlinear system---we construct the approximate quasiprobability distribution function in phase space that enables a quantum description. The nonclassical effects (quantum noise reduction) and their scaling laws can be so studied for high excitation numbers. In particular, the harmonic oscillators represent modes of the electromagnetic field and the Hamiltonians under consideration describe representative nonlinear optical processes (multiwave mixings). The range of the validity of the approximation for Wigner and Husimi functions evolved within the classical Liouville equation is discussed for a diverse class of initial conditions, including those without classical counterparts, e.g., Fock states.
An observable criterion is derived that allows one to distinguish nonclassical states of the harmonic oscillator from those having a classical counterpart. A quantum state is shown to have no classical counterpart if and only if the characteristic functions of the quadrature distributions or the s-parametrized phase-space distributions exhibit a slower decay than for the ground state of the oscillator. This renders it possible to experimentally check the failure of the P function to be a probability measure.