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arXiv:0906.4723 · Physical Review A

Wave-function Monte Carlo method for simulating conditional master equations

Kurt A. Jacobs

Published Apr 12, 2010• 9 citations
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Abstract

Wave-function Monte Carlo methods are an important tool for simulating quantum systems, but the standard method cannot be used to simulate decoherence in continuously measured systems. Here I present a Monte Carlo method for such systems. This was used to perform the simulations of a continuously measured nanoresonator in [Phys. Rev. Lett. 102, 057208 (2009)].

DOI
10.1103/physreva.81.042106

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We discuss Monte Carlo simulation of the quantum master equation in terms of system wave functions undergoing a sequence of quantum jumps based on the delay function for the occurrence of the “next jump”.1-3 The formalism can be applied to calculate the density operator and correlation functions (spectra) in quantum systems with a large number of degrees of freedom. We present the formalism and applications to (i) a multilevel atom undergoing spontaneous emission including mechanical light effects (in particular, laser cooling in the quantum limit), (ii) coupling of an atomic system to a squeezed vacuum, and (iii) cavity QED problems with coupling to thermal and squeezed baths.
A Monte Carlo simulation of the atomic master equation for spontaneous emission in terms of atomic wave functions is developed. Realizations of the time evolution of atomic wave functions are constructed that correspond to an ensemble of atoms driven by laser light undergoing a sequence of spontaneous emissions. The atomic decay times are drawn according to the photon count distribution of the driven atom. Each quantum jump of the atomic electron projects the atomic wave function to the ground state of the atom. Our theory is based on a stochastic interpretation and generalization of Mollow's pure-state analysis of resonant light scattering, and the Srinivas-Davies theory of continuous measurements in photodetection. An extension of the theory to include mechanical light effects and a generalization to atomic systems with Zeeman substructure are given. We illustrate the method by simulating the solutions of the optical Bloch equations for two-level systems, and laser cooling of a two-level atom in an ion trap where the center-of-mass motion of the atom is described quantum mechanically.
We demonstrate a scaling method for Monte Carlo wave-function simulations used to study the dynamics of non-Markovian open quantum systems weakly coupled to their environments. We derive a scaling equation, from which the result for the expectation values of arbitrary operators of interest can be calculated, all the quantities in the equation being easily obtainable from the scaled Monte Carlo simulations. In the optimal case, the scaling method can be used, within the weak coupling approximation, to reduce the size of the generated Monte Carlo ensemble by several orders of magnitude. Thus, the developed method allows faster simulations and makes it possible to solve the dynamics of the certain class of non-Markovian systems whose simulation would be otherwise too tedious because of the requirement for large computational resources.
The Monte Carlo wave function method or the quantum-trajectory-jump approach is a powerful tool to study dissipative dynamics governed by the Markovian master equation, in particular for high-dimensional systems and when it is difficult to simulate directly. We extend this method to the non-Markovian case described by the generalized Lindblad master equation. Two examples to illustrate the method are presented and discussed. The results show that the method can correctly reproduce the dissipative dynamics for the system. The difference between this method and the traditional Markovian jump approach and the computational efficiency of this method is also discussed.