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Physical Review A

Positive P representation: Application and validity

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

Published Apr 1, 1997• 149 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.55.3014

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Physical Review A•Apr 1, 1990•17 citations
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.
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.
This paper considers the ``positive-P'' description of nonlinear optical damping where one-photon and two-photon loss mechanisms are allowed. An exact solution of the corresponding stochastic differential equations is presented that graphically shows the breakdown of the positive-P representation found in earlier numerical-simulation work. The problem of ``spiking'' in simulations of nonlinear optical processes is then addressed, using the nonlinear damping process as an example. It is pointed out that although ``spiking'' can be exascerbated at high levels of quantum noise by numerical inaccuracy, the tendency to spike is in fact an analytic property of the stochastic variables.
PPP-representation techniques, which have been very successful in quantum optics and in other fields, are also useful for general bosonic quantum-dynamical many-body calculations such as Bose-Einstein condensation. We introduce a representation called the gauge P representation, which greatly widens the range of tractable problems. Our treatment results in an infinite set of possible time evolution equations, depending on arbitrary gauge functions that can be optimized for a given quantum system. In some cases, previous methods can give erroneous results, due to the usual assumption of vanishing boundary conditions being invalid for those particular systems. Solutions are given to this boundary-term problem for all the cases where it is known to occur: two-photon absorption and the single-mode laser. We also provide some brief guidelines on how to apply the stochastic gauge method to other systems in general, quantify the freedom of choice in the resulting equations, and make a comparison to related recent developments.
We present here an extensive study of the technique of numerical simulations, applied to the solution of a system of equations obtained by the positive P representation. The model we introduce to aid in these investigations has the distinct advantage of an immediate physical interpretation. In these simulations we experience the divergent trajectory or ``spike,'' noted by other authors; however, due to the simplicity of our equations, we are able to categorize and explain these spikes. Finally, we report a disturbing anomaly in some of the simulation results which indicates a possible problem with the use of the positive P representation.
Deriving the time evolution of a distribution of probability (or a probability density matrix) is a problem encountered frequently in a variety of situations: for physical time, it could be a kinetic reaction study, while identifying time with the number of computational steps gives a typical picture of algorithms routinely used in quantum impurity solvers, density functional theory, etc. Using a truncation scheme for the expansion of the exact quantity is necessary due to constraints of the numerical implementation. However, this leads in turn to serious complications such as the Fermion Sign Problem (essentially, density or weights will become negative). By integrating angular degrees of freedom and reducing the dynamics to the radial component, the time evolution is reformulated as a nonlinear integral transform of the distribution function. A canonical decomposition into orthogonal polynomials leads back to the original sign problem, but using a characteristic-function representation allows to extract the asymptotic behavior, and gives an exact large-time limit, for many initial conditions, with guaranteed positivity.
The problem of the formulation and simulation of quantized optical systems using the positive-P representation is considered. We show that feasible computational strategies are determined mainly by the eigenstructure of the system of stochastic differential equations that correspond to the positive-P Fokker-Planck equation. We demonstrate analytically that the stability and accuracy of numerical solutions are generally much improved by adopting time-centered implicit schemes. These findings are then related to the specific problem of second-harmonic generation. It is shown that the eigenstructure of the stochastic system possesses interrelationships that can lead to severe difficulties in numerical simulations. In particular we show how the problem of ``spiking'' encountered at high levels of quantum noise can be related to the numerical ``stiffness'' of the problem. We also demonstrate the advantages of time-centered implicit methods by highlighting the distortions that can occur in the Euler-type simulation of limit cycle solutions. Of physical interest is the fact that the phase-space morphology of the quantized second-harmonic generator admits families of limit cycles. This is in marked contrast to the purely classical picture in which, for a given parameter set, there exists only a single limit cycle. Finally we demonstrate by direct analytic solution that, under the conditions of adiabatic elimination, the positive-P representation for the second-harmonic generator with zero driving field fails, in the sense that it produces nonphysical results. This failure is in all respects analogous to the breakdown encountered in the problem of nonlinear optical damping.