P-representations in quantum optics

Yingying Tuo

The phase characteristic and phase variance of the non-classical optical field with negative P representation according to P-B quantum phase theory was studied in this paper.It is shown that the non-classical optical field with negative P representation is a totally stochastically phase state.The fluctuation of the phase operators is a constant(π2/3).

Z. Kis, József Janszky, Петер Адам

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.

H. J. Carmichael, Roy J. Glauber, Marlan O. Scully

H. J. Carmichael, Roy J. Glauber, Marlan O. Scully

Alexei Gilchrist, Crispin W. Gardiner, P. D. Drummond

The positive P representation is a very successful tool in quantum optics. However, the usual assumption of negligible boundary terms in the time-evolution equations is not always valid. We explore the range of validity of the time-evolution equations both analytically and by numerical investigation of a number of specific examples. We present practical ways of verifying the validity of the use of the positive P representation and find that the standard time-evolution equation can become invalid when nonlinear terms (at unit photon number) are large relative to the damping rate. This is very much larger than is normally the case in nonlinear optics, except possibly near resonances. We are able to show that when the positive P representation is invalid, the boundary terms, normally neglected in an integration by parts, become non-negligible. When numerical simulations are carried out using the positive P representation, specific checks given in this paper should be carried out to verify the compactness of the distribution. In conclusion, we find that (apart from special cases) this technique of quantum time evolution is typically asymptotically valid in the limit of small nonlinearity, rather than exact.