ON P-REPRESENTATION FOR PARAMETRIC AMPLIFIER AND PARAMETRIC FREQUENCY CONVERTER

The statistical properties of a parametric amplifier and a frequency converter are studied by means of quantum mechanical methods. The Schrödinger picture and the P-representation of the density matrix are used. Carrying out the Fourier transformation of the Liouville equation a partial differential equation for a generating function is obtained. The inverse Fourier transform of a solution of this equation is a time-dependent P-representation PN(α1, α2, t). For the parametric amplifier a relation is derived which enables us to compute the function PA(α1, α2, t) = =1 π<α1, α2ρ{variant}/α1> is shown that PA is classical distribution if PN(α1, α2,0) is a positive distribution, while the P-representation PN(α1, α2, t) need not exist as a distribution and the P-representation PN(α1, α2, t) for the parametric frequency converter is constant along classical trajectories. © 1969 Nakladatelstvi Československé akademie véd.