Numerical integration methods for stochastic wave function equations

Different methods for the numerical solution of stochastic differential equations arising in the quantum mechanics of open systems are discussed. A comparison of the stochastic Euler and Heun schemes, a stochastic variant of the fourth order Runge-Kutta scheme, and a second order scheme proposed by Platen is performed. By employing a natural error measure the convergence behaviour of these schemes for stochastic differential equations of the continuous spontaneous localization type is investigated. The general theory is tested by two examples from quantum optics. The numerical tests confirm the expected convergence behaviour in the case of the Euler, the Heun and the second order scheme. On the contrary, the heuristic Runge-Kutta scheme turns out to be a first order scheme such that no advantage over the simple Euler scheme is obtained. The results also clearly reveal that the second order scheme is superior to the other methods with regard to convergence behaviour and numerical performance. (C) 2000 Elsevier Science B.V. All rights reserved.