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Hybrid quantum-classical master equations

Lajos Diósi

Published Jan 1, 2014• 30 citations
OpenAlex

Abstract

We discuss hybrid master equations of composite systems which are hybrids of classical and quantum subsystems. A fairly general form of hybrid master equations is suggested, its consistency is derived from the consistency of Lindblad quantum master equations. We emphasize that quantum measurement is a natural example of exact hybrid systems. We derive a heuristic hybrid master equation of time-continuous position measurement (monitoring).

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•Nov 6, 2006•262 citations
We discuss hybrid master equations of composite systems which are hybrids of classical and quantum subsystems. A fairly general form of hybrid master equations is suggested, its consistency is derived from the consistency of Lindblad quantum master equations. We emphasize that quantum measurement is a natural example of exact hybrid systems. We derive a heuristic hybrid master equation of time-continuous position measurement (monitoring).
A concise and self-contained derivation of hybrid quantum-classical dynamics is given in terms of Markovian master equations. Many previously known results are rederived and revised and some of them are completed or corrected. Using a method as simple as possible, our goal is a brief introduction to the state of the art of hybrid dynamics, with a limited discussion of the implications for foundations and without discussion of further relevance in the measurement problem, quantum gravity, chemistry, numeric methods, etc. Hybrid dynamics is defined as a special case of composite quantum dynamics where the observables of one of the two subsystems are restricted to the commuting set of diagonal operators in a fixed basis. With this restriction, the derivation of hybrid dynamical equations is clear conceptually and simple technically. Jump and diffusive dynamics follow in the form of hybrid master equations. Their stochastic interpretation (called unravelings) is derived. We discuss gauge-type ambiguities, problems of uniqueness, and covariance of the diffusive master equation. Also conditions of minimum noise and of monitoring the quantum trajectory are derived. We conclude that the hybrid formalism is equivalent to the standard Markovian theory of time-continuous quantum measurement (monitoring) on the one hand and is a motivating alternative formalism on the other.
Markovian quantum master equations are generally thought to satisfy the following three conditions: (i) they describe the dynamics of time scales that are larger than the reservoir correlation time scale, (ii) these stationary solutions give the thermal equilibrium states with a reservoir, and (iii) they can be written in the Lindblad form. In fact, for single systems such as a single harmonic oscillator and a single two-level system, Markovian quantum master equations satisfying all these conditions can be obtained. However, for composite systems, which consist of more than one subsystem coupled with each other, such equations have not been obtained thus far. In this study, we found that we cannot use the Born–Markov approximation for obtaining quantum master equations for composite systems in the strong-coupling regime, no matter how short the reservoir correlation time is.
Quantum stochastic master equations of jump type are formulated in a general way and connections with quantum/classical hybrid systems and quantum filtering theory are discussed. By introducing the notion of ``typical trajectory", we show how to recursively construct the solution of the non-linear stochastic master equation (the conditional state). Moreover, by the notion of ``exclusive probability densities" we can describe all the probabilities related to the jumps, in particular, the waiting times of the jumps and their probability distributions. This general formulation and the idea of hybrid system allow to unify and generalize different fields: evolutions under non-Hermitian Hamiltonians, unitary dynamics interspersed by quantum channels at random times, quantum renewal processes, continuous time open quantum walks, Lindblad rate equation, ...
We present a pedagogical treatment of the formalism of continuous quantum measurement. Our aim is to show the reader how the equations describing such measurements are derived and manipulated in a direct manner. We also give elementary background material for those new to measurement theory, and describe further various aspects of continuous measurements that should be helpful to those wanting to use such measurements in applications. Specifically, we use the simple and direct approach of generalized measurements to derive the stochastic master equation describing the continuous measurements of observables, give a tutorial on stochastic calculus, treat multiple observers and inefficient detection, examine a general form of the measurement master equation, and show how the master equation leads to information gain and disturbance. To conclude, we give a detailed treatment of imaging the resonance fluorescence from a single atom as a concrete example of how a continuous position measurement arises in a physical system.