Quasiprobability distributions in stochastic wave-function methods

Abstract

Quasiprobability distributions emerging in the stochastic wave-function method of Carusotto et al. [Phys. Rev. A 63, 023606 (2001)] are investigated. We show that there are actually two types of quasiprobabilities. The first one, the “diagonal Hartree-Fock state projection” representation, is useful in representing the initial conditions for stochastic simulation in the most compact form. It defines antinormally ordered expansion of the density operator and normally ordered mapping of the observables to be averaged. We completely characterize the equivalence classes of this phase-space representation. The second quasiprobability distribution, the “nondiagonal Hartree-Fock state projection” representation, extends the first one in order to achieve stochastic representation of the quantum dynamics. We demonstrate how the differential identities of the stochastic ansatz generate the automorphisms of this phase-space representation. These automorphisms turn the stochastic representation into a gauge theory. The gauge transformations of the quasiprobability master equation are described. In particular, it is the analyticity of the stochastic ansatz that allows one to transform the master equation into the genuine Fokker-Planck equation. We demonstrate how the different variants of the stochastic wave-function method can be constructed, first by choosing a certain optimality criteria or constraints, and then by satisfying these criteria with a suitable choice of gauge. The problem of boundary terms is considered. It is demonstrated that the simple scheme with Fock states of Carusotto et al. is not subjected to this problem.