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 ρ̂ or a phase-space distribution which is, in general, a quasi-probability distribution. Expansions of the density matrix ρ̂ in terms of a given complete set of orthogonal operators establishes a one-to-one correspondence between the expanded operator ρ̂ 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 ρ̂ 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 ρ̂ operator. With the statistical information vested in the phase-space distribution, in lieu of the density matrix ρ̂ the expected value of an observable F̂ is given by an integral of the phase-space distribution multiplied by a weighting function which is representative of the observable F̂. This is essentially the method first introduced by Wigner[l] and Moyal[2].

Work supported by the Office of Naval Research