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arXiv:quant-ph/9702007 · Reviews of Modern Physics

The quantum-jump approach to dissipative dynamics in quantum optics

Martin Bodo Plenio, P. L. Knight

Published Jan 1, 1998• 1,514 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/revmodphys.70.101

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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 present a wave-function approach to the study of the evolution of a small system when it is coupled to a large reservoir. Fluctuations and dissipation originate in this approach from quantum jumps that occur randomly during the time evolution of the system. This approach can be applied to a wide class of relaxation operators in the Markovian regime, and it is equivalent to the standard master-equation approach. For systems with a number of states N much larger than unity this Monte Carlo wave-function approach can be less expensive in terms of calculation time than the master-equation treatment. Indeed, a wave function involves only N components, whereas a density matrix is described by N2 terms. We evaluate the gain in computing time that may be expected from such a formalism, and we discuss its applicability to several examples, with particular emphasis on a quantum description of laser cooling.
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.
In many problems in quantum optics, damping plays an important role. These include, for example, the decay of an atom in an excited state to a lower state and the decay of the radiation field inside a cavity with partially transparent mirrors. In general, damping of a system is described by its interaction with a reservoir with a large number of degrees of freedom. We are interested, however, in the evolution of the variables associated with the system only. This requires us to obtain the equations of motion for the system of interest only after tracing over the reservoir variables. There are several different approaches to deal with this problem. In this chapter, we present a theory of damping based on the density operator in which the reservoir variables are eliminated by using the reduced density operator for the system in the Schrödinger (or interaction) picture. We also present a ‘quantum jump’ approach to damping. In the next chapter, the damping of the system will be considered using the noise operator method in the Heisenberg picture. An insight into the damping mechanism is obtained by considering the decay of an atom in an excited state inside a cavity. The atom may be considered as a single system coupled to the radiation field inside the cavity. Even in the absence of photons in the cavity, there are quantum fluctuations associated with the vacuum state. As discussed in Chapter 1, the field may be visualized as a large number of harmonic oscillators, one for each mode of the cavity.
We propose physical interpretations, also valid for temperatures different from zero, for stochastic methods which have been developed recently to describe the evolution of a quantum system interacting with a reservoir. As opposed to the usual reduced density operator approach, which refers to ensemble averages, these methods deal with the dynamics of single realizations, and involve the solution of stochastic Schrödinger equations. These procedures have been shown to be completely equivalent to the master equation approach when ensemble averages are taken over many realizations. We show that these techniques are not only convenient mathematical tools for dissipative systems, but may actually correspond to concrete physical processes, for any temperature of the reservoir. We consider a mode of the electromagnetic field in a cavity interacting with a beam of two- or three-level atoms, the field mode playing the role of a small system and the atomic beam standing for a reservoir at finite temperature, the interaction between them being given by the Jaynes-Cummings model. We show that the evolution of the field states, under continuous monitoring of the state of the atoms which leave the cavity, can be described in terms of either the Monte Carlo wavefunction (quantum jump) method or a stochastic Schrödinger equation, depending on the system configuration. We also show that the Monte Carlo wavefunction approach leads, for finite temperatures, to localization into jumping Fock states, while the diffusion equation method leads to localization into states with a diffusing average photon number, which for sufficiently small temperatures are close approximations to mildly squeezed states. We prove analytically that, in the quantum jump situation, the system evolves in the mean towards a Fock state, even if an infinite number of photon-number amplitudes is present in the initial state.
In the quantum trajectory approach an open quantum system is represented by a stochastic pure-state wave function. The mixed-state density operator that satisfies the usual master equation is recovered from the quantum trajectories by performing either an ensemble average or, for stationary systems, an average over time. This unraveling of the master equation dynamics into pure-state trajectories provides new insight into the decay of quantum coherence in systems that are open to the environment. Under some conditions macroscopic superposition states are preserved as macroscopic superposition states along individual trajectories, but are reduced to mixtures by the trajectory average. Under other conditions one of the states in a macroscopic superposition becomes dominant (in amplitude) over the other along each quantum trajectory. We discuss the mechanisms that produce these dynamics and assess the implications for the observation of Schrodinger cat states in optics experiments. We present results for some specific examples of macroscopic superposition states generated by a cavity QED system. The system involves a small collection of atoms in an optical cavity driven by a coherent laser field. Under strong-coupling conditions this system produces a variety of Schrodinger cat states whose precise form depends on the number of atoms, the method of excitation (through a cavity mirror or from the side), the initial state of the atoms, and the position of the atoms in the cavity.