Stochastic wave-function approach to the calculation of multitime correlation functions of open quantum systems

Abstract

Within the framework of probability distributions on projective Hilbert space, a scheme for the calculation of multitime correlation functions is developed. The starting point is the Markovian stochastic wave-function description of an open quantum system coupled to an environment consisting of an ensemble of harmonic oscillators in arbitrary pure or mixed states. It is shown that matrix elements of reduced Heisenberg picture operators and general time-ordered correlation functions can be expressed by time-symmetric expectation values of extended operators in a doubled Hilbert space. This representation allows the construction of a stochastic process in the doubled Hilbert space which enables a determination of arbitrary matrix elements and correlation functions. The numerical efficiency of the resulting stochastic simulation algorithm is investigated and compared with an alternative Monte Carlo wave function method proposed first by Dalibard, Castin, and Mo "lmer ͓Phys. Rev. Lett. 68, 580 ͑1992͔͒. By means of a standard example the suggested algorithm is shown to be more efficient numerically and to converge faster. Finally, some specific examples from quantum optics are presented in order to illustrate the proposed method, such as the coupling of a system to a vacuum, a squeezed vacuum within a finite solid angle, and a thermal mixture of coherent states.

Stochastic wave-function simulation of two-time correlation functions

“…The scheme proposed by Castin et al [7], for example, is threatened by numerical instability as it relies on the subtraction of possibly large numbers. The particular scheme proposed by Gardiner and Zoller [3], and Gisin [8], on the other hand, is exponentially inefficient, that is the number of trajectories which are needed for a reliable estimate of the desired correlation function is bound from below by an increasing exponential function of the correlator's time t. The somewhat more intricate algorithm which may be extracted from the work of Breuer and collaborators [9] does not suffer from this particular kind of ineffeciency, but it is not yet optimized. To date no systematic investigation has addressed the issue of how to tailor an algorithm which is both effective and efficient.…”

“…In the past, several stochastic wave function algorithms have been proposed which support the numerical simulation of a two-time correlation function [3,[7][8][9]. Since all these algorithms are build to yield the correct result in the limit of infinitely many runs, they are effective, but they are usually not very efficient.…”

“…As a non-trivial example, we apply our algorithm to the problem of tunneling in the degenerate optical parametric oscillator. The methods of Castin and collaborators [7], and of Breuer and collaborators [9] are revied in Appendices A and B.…”

“…III, we extend our analysis for the stochastic representation of the dynamics of skew-symmetric state operators, and we indicate an optimal algorithm. Using the simple example of spontaneous emission of a two-state system we analyse the efficiency of alternative algorithm, including the Gardiner-Zoller [3] and Breuer-Kappler-Petruccione [9] method. As a non-trivial example, we apply our algorithm to the problem of tunneling in the degenerate optical parametric oscillator.…”

Stochastic unraveling of time-local quantum master equations beyond the Lindblad class

“…[15] extended a scheme which they had used to calculate multi-time correlation functions [19] to the unraveling of QMEs. Their technique is based on doubling the Hilbert space.…”