Robust Algorithms for Solving Stochastic Partial Differential Equations

A robust semi-implicit central partial difference algorithm for the numerical solution of coupled stochastic parabolic partial differential equations (PDEs) is described. This can be used for calculating correlation functions of systems of interacting stochastic fields. Such field equations can arise in the description of Hamiltonian and open systems in the physics of nonlinear processes, and may include multiplicative noise sources. The algorithm can be used for studying the properties of nonlinear quantum or classical field theories. The general approach is outlined and applied to a specific example, namely the quantum statistical fluctuations of ultra-short optical pulses in ?(2)parametric waveguides. This example uses a non-diagonal coherent state representation, and correctly predicts the sub-shot noise level spectral fluctuations observed in homodyne detection measurements. It is expected that the methods used will be applicable for higher-order correlation functions and other physical problems as well. A stochastic differencing technique for reducing sampling errors is also introduced. This involves solving nonlinear stochastic parabolic PDEs in combination with a reference process, which uses the Wigner representation in the example presented here. A computer implementation on MIMD parallel architectures is discussed.

[1]  William H. Press,et al.  Book-Review - Numerical Recipes in Pascal - the Art of Scientific Computing , 1989 .

[2]  Lugiato,et al.  Quantum spatial correlations in the optical parametric oscillator with spherical mirrors. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[3]  Francesco Petruccione,et al.  Numerical integration of stochastic partial differential equations , 1993 .

[4]  H. Risken.  The Fokker-Planck equation : methods of solution and applications , 1985 .

[5]  Werner.  Quantum statistics of fundamental and higher-order coherent quantum solitons in Raman-active waveguides. , 1996, Physical Review A. Atomic, Molecular, and Optical Physics.

[6]  K. B. Davis,et al.  An analytical model for evaporative cooling of atoms , 1995 .

[7]  Werner,et al.  Quasiprobability distributions for the cavity-damped Jaynes-Cummings model with an additional Kerr medium. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[8]  George Marsaglia,et al.  Toward a universal random number generator , 1987 .

[9]  K. B. Davis,et al.  Bose-Einstein Condensation in a Gas of Sodium Atoms , 1995, EQEC'96. 1996 European Quantum Electronic Conference.

[10]  Reid,et al.  Squeezing of quantum solitons. , 1987, Physical review letters.

[11]  P. D. Drummond,et al.  Central partial difference propagation algorithms , 1983 .

[12]  R. Shelby,et al.  Time dependence of quantum fluctuations in solitons. , 1989, Optics letters.

[13]  Peter D. Drummond,et al.  Simulation of Quantum Effects in Raman-Active Waveguides , 1993 .

[14]  Peter D. Drummond,et al.  Computer simulations of multiplicative stochastic differential equations , 1991 .

[15]  R. Shelby,et al.  Quantum solitons in optical fibres , 1993, Nature.

[16]  Jack Dongarra,et al.  PVM: Parallel virtual machine: a users' guide and tutorial for networked parallel computing , 1995 .

[17]  Drummond,et al.  Squeezed quantum solitons and Raman noise. , 1991, Physical review letters.

[18]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[19]  Heleno Bolfarine,et al.  Population variance prediction under normal dynamic superpopulation models , 1989 .

[20]  C. Wieman,et al.  Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor , 1995, Science.

[21]  K. Burnett.  Bose-Einstein condensation with evaporatively cooled atoms , 1996 .

[22]  Shang‐keng Ma.  Modern Theory of Critical Phenomena , 1976 .

[23]  P. Drummond,et al.  Limits to wideband pulsed squeezing in a traveling-wave parametric amplifier with group-velocity dispersion. , 1991, Optics letters.

[24]  P. Drummond,et al.  Quantum-field theory of squeezing in solitons , 1987 .

[25]  Bradley,et al.  Evidence of Bose-Einstein Condensation in an Atomic Gas with Attractive Interactions. , 1995, Physical review letters.

[26]  C. Gardiner,et al.  Generalised P-representations in quantum optics , 1980 .

[27]  G. N. Mil’shtejn.  Approximate Integration of Stochastic Differential Equations , 1975 .

[28]  C. W. Gardiner,et al.  Handbook of stochastic methods - for physics, chemistry and the natural sciences, Second Edition , 1986, Springer series in synergetics.

[29]  Noriaki Nagase.  Remarks on Nonlinear Stochastic Partial Differential Equations: An Application of the Splitting-Up Method , 1995 .

[30]  Machida,et al.  Observation of Optical Soliton Photon-Number Squeezing. , 1996, Physical review letters.