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Quantum Monte Carlo calculations

Eric Dale Lieser

Published Jan 1, 2003• 2 citations
OpenAlex

Abstract

A review of the density operator and master equation leads to a derivation of the method of quantum jumps, or quantum trajectories. This method of quantum trajectories is used to describe the evolution of a two level atom driven by a laser field and damped by a quantized radiation field. The quantum trajectory method is then compared to the master equation via numerical simulations of this two level atom system.

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A review of the density operator and master equation leads to a derivation of the method of quantum jumps, or quantum trajectories. This method of quantum trajectories is used to describe the evolution of a two level atom driven by a laser field and damped by a quantized radiation field. The quantum trajectory method is then compared to the master equation via numerical simulations of this two level atom system.
Very recently there have been a number of papers discussing the applicability of stochastic methods as an alternative to the usual treatment of master equations in quantum mechanics.1-5 In this paper we present a wave function approach including a stochastic element to study the evolution of a small system when it is coupled to a reservoir. We show how fluctuations and dissipation originate from quantum jumps occuring randomly during the time evolution of the system, and we present a physical interpretation of the procedure. We demonstrate how this treatment may replace a wide class of master equations, and its use in problems with a large number of states N is discussed. Since the density matrix has N ×N elements, where a wave function is determined by only N variables, there is a computational advantage in using the wave function procedure. This is illustrated by calculations on cooling atoms in a laser field.
Summary form only given. Quantum trajectories of state vectors can be used to describe open systems that otherwise can only be described using a density matrix. Besides the numerical advantage, quantum trajectories provide insight in the actual evolution of a system under continuous observation. Two distinct methods are usually considered, the quantum-jump method, associated with direct photodetection, and the quantum state diffusion (QSD) method, associated with homodyne or heterodyne mixing. We derive a general stochastic equation for quantum trajectories.
Dissipation, the irreversible loss of energy and coherence, from a microsystem is the result of coupling to a much larger macrosystem (or reservoir) that is so large that one has no chance of keeping track of all of its degrees of freedom. The microsystem evolution is then described by tracing over the reservoir states, which results in an irreversible decay as excitation leaks out of the initially excited microsystems into the outer reservoir environment. Earlier treatments of this dissipation used density matrices to describe an ensemble of microsystems, either in the Schr\"odinger picture with master equations, or in the Heisenberg picture with Langevin equations. The development of experimental techniques to study single quantum systems (for example, single trapped ions, or cavity-radiation-field modes) has stimulated the construction of theoretical methods to describe individual realizations conditioned on a particular observation record of the decay channel. These methods, variously described as quantum-jump, Monte Carlo wave function, and quantum-trajectory methods, are the subject of this review article. We discuss their derivation, apply them to a number of current problems in quantum optics, and relate them to ensemble descriptions.
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.