The Positive P Representation

CW Gardiner, Alexei Gilchrist, P. D. Drummond

Crispin W. Gardiner, P. D. Drummond

Hong-Yi Fan, Min Xiao

Sarben Sarkar, J.S. Satchell, H. J. Carmichael

The drift in the Fokker-Planck equation in the positive P representation describing a gas laser is shown to have a continuum of stable steady states. Only a small subset is physically acceptable. It is demonstrated that the diffusion does manage to average over the unphysical steady states to give correct correlation properties in the lasing region. However, in the chaotic region, the positive P representation equations have additional instabilities. As a consequence the positive P representation in the chaotic region leads to behaviour either different from known solutions or breaks down.

P. D. Drummond, Crispin W. Gardiner

A class of normal ordering representations of quantum operators is introduced, that generalises the Glauber-Sudarshan P-representation by using nondiagonal coherent state projection operators. These are shown to have practical application to the solution of quantum mechanical master equations. Different representations have different domains of integration, on a complex extension of the usual canonical phase-space. The 'complex P-representation' is the case in which analytic P-functions are defined and normalised on contours in the complex plane. In this case, exact steady-state solutions can often be obtained, even when this is not possible using the Glauber-Sudarshan P-representation. The 'positive P-representation' is the case in which the domain is the whole complex phase-space. In this case the P-function may always be chosen positive, and any Fokker-Planck equation arising can be chosen to have a positive-semidefinite diffusion array. Thus the 'positive P-representation' is a genuine probability distribution. The new representations are especially useful in cases of nonclassical statistics.

Hong-Yi Fan, Yue Fan, Junhua Chen

Based on the newly constructed overcompleteness relation of the nonlinear coherent states (NLCSs) and corresponding 𝒫-representation we derive: (1) a new antinormally ordered expansion of density matrices; and (2) the corresponding positive 𝒫-representation. Our derivation employs the technique of integration within an ordered product of nonlinear Bose operator functions. A relationship between the positive 𝒫-representation and the generalized characteristic function in the context of NLCSs is also obtained.