Orthogonal Operators and Phase Space Distributions in Quantum Optics

Abstract

Electromagnetic fields at optical frequencies are excited by indeterministic sources. Their statistical description is usually given by means of the density operator ρ or a phase-space distribution which is, in general, a quasi-probability distribution. Expansions of the density matrix ρ in terms of a given complete set of orthogonal operators establishes a one-to-one correspondence between the expanded operator ρ and the quasi-probability phase-space distribution that appears in said expansion. The time evolution of the field’s statistics are obtained then as a solution of the equation of motion of the density operator ρ or the differential equation of the corresponding phase-space distribution. The latter method has the advantage of being unburdened by problems of commutativity associated with the ρ operator. With the statistical information vested in the phase-space distribution, in lieu of the density matrix ρ the expected value of an observable F is given by an integral of the phase-space distribution multiplied by a weighting function which is representative of the observable F. This is essentially the method first introduced by Wigner[l] and Moyal[2].