Harmonic Analysis of the Semi-Classical Limit

Sándor Varró

Abstract We discuss the general matrix elements of the squeezing operator between number eigenstates of a harmonic oscillator (which may also represent a quantized mode of the electromagnetic radiation). These matrix elements have first been used by Popov and Perelomov (1969) long ago, in their thorough analysis of the parametric excitation of harmonic oscillators. They expressed the matrix elements in terms of transcendental functions, the associated Legendre functions. In the present paper we will show that these matrix elements can also be derived in an different form, expressed by the classical Gegenbauer polynomials. This new expression makes it possible to determine coherent and incoherent superpositions of these matrix elements in closed analytic forms. As an application, we describe multiphoton transitions in the system “charged particle + electromagnetic radiation”, induced by a (strong) coherent field or by a black-body radiation component (with a Planck-Bose photon number distribution). The exact results are compared with the semi-classical ones. We will show that in case of interaction with a thermal field, the semi-classical result (with a Gaussian stochastic field amplitude) yields an acceptable approximation only in the Rayleigh-Jeans limit, however, in the Wien limit it completely fails.

UMBERTO LOMBARDO, G. Giuliani, Yifei Niu

Jan Mostowski, Joanna Pietraszewicz

The Wigner function is a quantum analogue of the classical joined distribution of position and momentum. As such is should be a good tool to study quantum-classical correspondence. In this paper, the classical limit of the Wigner function is shown using the quantum harmonic oscillator as an example. The Wigner function is found exactly for all states. The semi-classical wavefunctions for highly excited states are used as the approach to the classical limit. Therefore, one can found the classical limit of the Wigner function for highly excited states and shown that it gives the classical microcanonical ensemble.

Michael Dörfle, R. Graham

Second harmonic generation and subharmonic generation in a cavity are described by the usual master equation (1) for the statistical operator of the system of two modes (fundamental and second harmonic). The master equation is equivalent to a partial differential equation for the Wigner function which is a generalized Fokker-Planck equation involving partial derivatives of the first, second and third order (2). In the semi-classical limit the third order derivatives are negligible and the Wigner distribution satisfies the Fokker-Planck equation equivalent to the Langevin equation with the formally classical Gaussian white noise ξ1, ξ2 with the only non-vanishing correlation coefficients β1, β2 are the mode amplitudes (normalized to photon numbers), g is the coupling constant, Δ1, 2 the frequency mismatch, x is the damping rate, Fp the amplitude of the pump field.