Computer simulations of nonlinear quantum-dynamical systems have presented several unexpected difficulties, such as divergences. Using s-order positive-quasiprobability representations for the problem of a nonlinearly damped interferometer, it is found that the divergences disappear for s=2/3.
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.
Marginal properties of arbitrary s-ordered quasiprobability distribution fucntions are investigated. The positivity of marginal distributions for all quasiprobabilities, interpolating between the Wigner function and the Q representation of the density operator (s\ensuremath{\le}0), is proven. We also show that, on the contrary, marginal distributions of the remaining quasiprobabilities, interpolating between the Wigner function and the P representation (s>0), can take on negative values. General formulas for the marginal distributions are given, and their relations to the actual quantum-mechanical probability distributions for position and momentum are established. Our results provide an interesting generalization of some recent results concerning marginal properties of the P representation [G. S. Agarwal, Opt. Commun. 95, 109 (1993)].
The Lindblad master equation for the damped quantum harmonic oscillator is transformed into Fokker–Planck equations for quasiprobability distributions. A comparative study is made for the Glauber P representation, the antinormal-ordering Q representation and the Wigner W representation. It will be proven that the variances for the damped harmonic oscillator found with these representations are the same. By solving the Fokker–Planck equations in the steady state, it will be shown that the quasiprobability distributions are two-dimensional Gaussians with widths determined by the diffusion coefficients.
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.
A review of quasiprobability methods for transforming chemical and quantum-optical master equations into Fokker-Planck equations is presented. For cases where conventional representations lead to Fokker-Planck equations with non-positive-definite diffusion coefficients; e.g., sub-Poissonian statistics, a generalization of the representation involving an extension to the complex plane enables analytic results to be obtained for certain nonlinear chemical and optical processes. Alternatively, a different integration measure may be chosen which ensures a positive distribution and Fokker-Planck equation with positive-semidefinite diffusion coefficients. This enables stochastic differential equations to be defined. These methods are applied to two-photon absorption and dispersive bistability in quantum optics where nonclassical photon statistics arise and to two models of nonlinear chemical reactions where sub-Poissonian statistics occur.
Stochastic nonlinear dynamics underlie many models in engineering and computational physics, yet accurate high-dimensional simulation remains challenging. We present a quantum algorithm for a broad class of N-dimensional stochastic differential equations with dissipation and quadratic drift. The algorithm applies to strongly nonlinear systems with all-to-all interactions, thereby extending the scope of previously known quantum algorithms that were limited to weak nonlinearity and sparse systems. For norm-preserving drifts, a condition satisfied by key fluid dynamics discretizations, our method approximates expectation values of low-order correlation functions with rigorous error bounds at a cost polynomial in log(N) and linear in the evolution time. Our main technical advance is a subroutine for simulating an auxiliary system of N interacting quantum harmonic oscillators with cost polylogarithmic in N. Finally, we formulate turbulence models, including Navier-Stokes and damped Euler equations, within this framework, opening a route to quantum simulation of strongly nonlinear SDEs governing turbulence and nonlinear wave dynamics.