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Annual Meeting Optical Society of America

Numerical difficulties with the positive P representation: an Illustrative example

Murray Wolinsky, H. J. Carmichael

Published Jan 1, 1988• 0 citations
Publisher

Abstract

Nonclassical states of the electromagnetic field are states for which the Glauber-Sudarshan P function does not exist as a positive-definite nonsingular probability density. A classical stochastic description for systems that produce nonclassical states can be recovered, however, using the positive P representation. But difficulties can arise in numerical simulations of nonlinear stochastic models derived using this representation.1 This paper addresses the unresolved question of whether the difficulties are purely numerical or fundamental to the representation. Recently we used the positive P representation to provide an appealing picture of the nonclassical states produced by a degenerate parametric oscillator.2 The resolution of difficulties in that problem by a reflecting boundary condition seemed rather model dependent. We analyze the quantized anharmonic oscillator in the positive P representation. Correct operator averages are derived from the stochastic model analytically. However, an explicit construction of the positive P function shows that there are inherent difficulties for numerical simulations. Unconstrained diffusion in phase space occurs, and for long times the important trajectories for recovering correct moments are those that diffuse far from the origin. Even if numerical instability can be avoided, an enormous number of trajectories would be required to cover phase space sufficiently to obtain accurate averages.

DOI
10.1364/oam.1988.tua2

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Nonclassical states of the electromagnetic field are states for which the Glauber-Sudarshan P function does not exist as a positive-definite nonsingular probability density. A classical stochastic description for systems that produce nonclassical states can be recovered, however, using the positive P representation. But difficulties can arise in numerical simulations of nonlinear stochastic models derived using this representation.1 This paper addresses the unresolved question of whether the difficulties are purely numerical or fundamental to the representation. Recently we used the positive P representation to provide an appealing picture of the nonclassical states produced by a degenerate parametric oscillator.2 The resolution of difficulties in that problem by a reflecting boundary condition seemed rather model dependent. We analyze the quantized anharmonic oscillator in the positive P representation. Correct operator averages are derived from the stochastic model analytically. However, an explicit construction of the positive P function shows that there are inherent difficulties for numerical simulations. Unconstrained diffusion in phase space occurs, and for long times the important trajectories for recovering correct moments are those that diffuse far from the origin. Even if numerical instability can be avoided, an enormous number of trajectories would be required to cover phase space sufficiently to obtain accurate averages.
The positive Prepresentation has proved a useful tool for analyzing optical systems which exhibit squeezing. The Glauber-Sudarshan phase-space representation gives a Fokker-Planck equation without positive semidefinite diffusion for systems which generate nonclassical fields. The positive Prepresentation provides a method for guaranteeing positive semidefinite diffusion and enables a quantum- dynamical model to be formulated in terms of classical stochastic differential equations. In previous work we had some success in applying this approach to describe the decay of the unstable state above threshold in a degenerate parametric oscillator.1 Below threshold a linearized analysis of fluctuations recovers the predictions of squeezing obtained by alternative methods.
Non-Gaussian correlations in a pure state are inextricably linked with certain nonclassical features, such as a non-positive-definite Wigner function. In a commonly used simulation technique in ultracold atoms and quantum optics, known as the truncated Wigner method, the quantum dynamics is mapped to stochastic trajectories in phase space, governed by a positive approximation to the true Wigner distribution. The question thus arises: How accurate is this approach in predicting truly nonclassical behavior? In this article, we benchmark the ability of the truncated Wigner phase-space method to reproduce the non-Gaussian statistics of the single-mode anharmonic oscillator. We find that the this method can reliably predict departures from Gaussian statistics over a wide range of particle numbers, whereas the positive-PPP representation, which involves no approximations, is limited by rapidly growing statistical uncertainty. The truncated Wigner function, furthermore, is able to reproduce the non-Gaussian correlations while satisfying the condition for purity.
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.
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.
We present dynamical calculations for the quantum parametric oscillator using both number-state and coherent-state bases. The coherent-state methods use the positive-P representation, which has a nonclassical phase space---an essential requirement in obtaining an exact stochastic representation of this nonlinear problem. This also provides a way to directly simulate quantum tunneling between the two above-threshold stable states of the oscillator. The coherent-state methods provide both analytic results at large photon numbers, and numerical results for any photon number, while our number-state calculations are restricted to numerical results in the low-photon-number regime. The number-state and coherent-state methods give precise agreement within the accuracy of the numerical calculations. We also compare our results with methods based on a truncated Wigner representation equivalent to stochastic electrodynamics, and find that these are unable to correctly predict the tunneling rate given by the other methods. An interesting feature of the results is the much faster tunneling predicted by the exact quantum-theory methods compared with earlier semiclassical calculations using an approximate potential barrier. This is similar to the faster tunneling found when comparing quantum penetration of a barrier to classical thermal activation. The quantum parametric oscillator, which has an exact steady-state solution, therefore provides a useful and accessible system in which nonlinear quantum effects can be studied far from thermal equilibrium.
Phase-space representations based on coherent states (PPP,QQQ, Wigner) have been successful in the creation of stochastic differential equations (SDEs) for the efficient stochastic simulation of high-dimensional quantum systems. However, many problems using these techniques remain intractable over long integration times. We present a number-phase Wigner representation that can be unraveled into SDEs. We demonstrate convergence to the correct solution for an anharmonic oscillator with small dampening for significantly longer than other phase-space representations. This process requires an effective sampling of a nonclassical probability distribution. We describe and demonstrate a method of achieving this sampling using stochastic weights.