S-order nondiagonal quasiprobabilities.

Abstract

This paper presents a correspondence between the annihilation a^ and creation a${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ operators and two independent complex variables \ensuremath{\alpha} and \ensuremath{\beta}, which makes possible the definition of positive nondiagonal quasiprobabilities for any Cahill-Glauber s order. A generalized version of the Cahill and Glauber s-ordered displacement operator displacing the annihilation and creation operators by \ensuremath{\alpha} and \ensuremath{\beta} is defined. Corresponding to this n/Inondiagonal displacement, a nondiagonal ordering operator is introduced so that a map of s-ordered operators into c numbers a^\ensuremath{\rightarrow}\ensuremath{\alpha}, a${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$\ensuremath{\rightarrow}\ensuremath{\beta} can be defined. The Drummond-Gardiner projector and the Cahill-Glauber diagonal ordering operator are obtained as particular cases. In order to use the nondiagonal correspondence in the determination of quantum expectation values of observables, several families of quasiprobabilities are defined, generalizing Drummond-Gardiner (normal order and nondiagonal) positive-P-function and Cahill-Glauber (s-order and diagonal) quasiprobabilities. It is demonstrated that these nondiagonal quasiprobabilities exist as well-behaved functions where at least one is non-negative, for any state and any order. The time evolution of these quasiprobabilities is discussed both in the Schr\odinger and in the Heisenberg pictures. In the Heisenberg picture, a method for obtaining c-number stochastic differential equations (SDE's) directly from operator equations using the s-order nondiagonal correspondence is described. The main difference between this method and the Langevin approach is that in the latter a diagonal correspondence is used, leading eventually to wrong results. The use of these SDE's to solve quantum optical problems is discussed, and an application to the nonlinearly damped degenerate parametric oscillator is made. In order to obtain the SDE, various (inequivalent) truncation schemes are necessary.