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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Numerical solution methods for quantum stochastic processes

Eli Pollak

Published May 8, 2003• 1 citation
Publisher

Abstract

The study of quantum stochastic processes presents severe difficulties, both on the theory level as well as on technical grounds. The numerically exact solution remains prohibitive even today. In this paper we review and present new results for three different methods used for the modelling of quantum stochastic processes. These include a mixed quantum classical approach, semiclassical initial value representations of the quantum propagator and the reduced density matrix approach as typified by the quantum Wigner-Fokker-Planck equation. A new semiclassical initial value representation that does away with cumbersome prefactors which depend on the monodromy matrix elements but is exact for a harmonic oscillator is presented and its properties analysed. A recently proposed systematic method for improving semiclassical initial value representations is reviewed. The generalization of the Wigner-Fokker-Planck equation to stochastic processes with memory is obtained by using a novel integral equation representation.

DOI
10.1117/12.488566

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The study of quantum stochastic processes presents severe difficulties, both on the theory level as well as on technical grounds. The numerically exact solution remains prohibitive even today. In this paper we review and present new results for three different methods used for the modelling of quantum stochastic processes. These include a mixed quantum classical approach, semiclassical initial value representations of the quantum propagator and the reduced density matrix approach as typified by the quantum Wigner-Fokker-Planck equation. A new semiclassical initial value representation that does away with cumbersome prefactors which depend on the monodromy matrix elements but is exact for a harmonic oscillator is presented and its properties analysed. A recently proposed systematic method for improving semiclassical initial value representations is reviewed. The generalization of the Wigner-Fokker-Planck equation to stochastic processes with memory is obtained by using a novel integral equation representation.
We demonstrate a method which allows the stochastic modelling of quantum systems for which the generalised Fokker-Planck equation in the phase space contains derivatives of higher than second order. This generalises quantum stochastics far beyond the quantum-optical paradigm of three and four-wave mixing problems to which these techniques have so far only been applicable. To verify our method, we model a full Wigner representation for the optical parametric oscillator, a system where the correct results are well known and can be obtained by other methods. 1 I.
We demonstrate a method which allows the stochastic modelling of quantum systems for which the generalised Fokker-Planck equation in the phase space contains derivatives of higher than second order. This generalises quantum stochastics far beyond the quantum-optical paradigm of three- and four-wave mixing problems to which these techniques have so far only been applicable. To verify our method, we model a full Wigner representation for the optical parametric oscillator, a system where the correct results are well known and can be obtained by other methods.
The quantum Liouville equation in the Wigner representation is solved numerically by using Monte Carlo methods. For incremental time steps, the propagation is implemented as a classical evolution in phase space modified by a quantum correction. The correction, which is a momentum jump function, is simulated in the quasi-classical approximation via a stochastic process. In this paper the technique, which is developed and validated in two- and three-dimensional momentum space, extends an earlier one-dimensional work. Also, by developing a new algorithm, the application to bound state motion in an anharmonic quartic potential shows better agreement with exact solutions in two-dimensional phase space. I. Introduction Nuclear interaction theory is formulated in the language of quantum mechanics, and hence the development of general methods of solutions to quantum dynamics will provide useful tools for application to a large class of problems in nuclear many-body theory. Different approaches...
In this paper, we propose efficient quantum algorithms for solving nonlinear stochastic differential equations (SDE) via the associated Fokker-Planck equation (FPE). We discretize the FPE in space and time using two well-known numerical schemes, namely Chang-Cooper and implicit finite difference. We then compute the solution of the resulting system of linear equations using the quantum linear systems algorithm. We present detailed error and complexity analyses for both these schemes and demonstrate that our proposed algorithms, under certain conditions, provably compute the solution to the FPE within prescribedϵ\epsilonϵerror bounds with polynomial dependence on state dimensionddd. Classical numerical methods scale exponentially with dimension, thus, our approach, under the aforementioned conditions, provides an exponential speed-up over traditional approaches.
The quantum molecular trajectory is the deterministic trajectory, arising from the Bohm theory, that describes the instantaneous positions of the nuclei of molecules by assuring the agreement with the predictions of quantum mechanics. Therefore, it provides the suitable framework for representing the geometry and the motions of molecules without neglecting their quantum nature. However, the quantum molecular trajectory is extremely demanding from the computational point of view, and this strongly limits its applications. To overcome such a drawback, we derive a stochastic representation of the quantum molecular trajectory, through projection operator techniques, for the degrees of freedom of an open quantum system. The resulting Fokker-Planck operator is parametrically dependent upon the reduced density matrix of the open system. Because of the pilot role played by the reduced density matrix, this stochastic approach is able to represent accurately the main features of the open system motions both at equilibrium and out of equilibrium with the environment. To verify this procedure, the predictions of the stochastic and deterministic representation are compared for a model system of six interacting harmonic oscillators, where one oscillator is taken as the open quantum system of interest. The undeniable advantage of the stochastic approach is that of providing a simplified and self-contained representation of the dynamics of the open system coordinates. Furthermore, it can be employed to study the out of equilibrium dynamics and the relaxation of quantum molecular motions during photoinduced processes, like photoinduced conformational changes and proton transfers.
A numerical technique for the solution of the Quantum Liouville equation is discussed which is expected to be useful for mixed states describing highly excited many body systems.;The equation is represented in phase-space, where the density operator becomes the Wigner function. The Liouville operator which determines its time evolution, separates into a classical and quantum term. When the Wigner function varies slowly in momentum, the quasi-classical approximation is valid in which one keeps only the first order quantum term in the Liouville equation.;The Wigner function is represented by a sample set of gaussian smeared points. In an iterative time development scheme, the points are propagated first along a classical phase-space trajectory, which is then followed by a quantum development. The latter is represented by a stochastic process incorporated into the numerical procedure as a generalized Markovian process.;This method is numerically tested for an anharmonic quartic potential. The contours of the Wigner function at different time intervals are compared with the purely classical and exact results. It is seen that it provides a definite improvement over the purely classical approximation. Various averages are also compared and sources of error discussed.;In addition it was discovered that intermediate mixed states show a 'quantum focusing' effect, wherein the function peaks to values greater than the original maximum. This quantum-mechanical effect did not appear in the pure case, and disappears because of Liouville's theorem for highly mixed states which are essentially classical. There are other numerical schemes, which treat the points as moving in an effective potential, wherein the points retain their identity. These again by Liouville's theorem cannot show this focusing effect.