Coupling Classical and Quantum Variables using Continuous Quantum Measurement Theory

We propose a system of equations to describe the interaction of a quasiclassical variable X with a set of quantum variables x that goes beyond the usual mean field approximation. The idea is to regard the quantum system as continuously and imprecisely measured by the classical system. The effective equations of motion for the classical system therefore consist of treating the quantum variable x as a stochastic c-number x̄(t) the probability distibution for which is given by the theory of continuous quantum measurements. The resulting theory is similar to the usual mean field equations (in which x is replaced by its quantum expectation value) but with two differences: a noise term, and more importantly, the state of the quantum subsystem evolves according to the stochastic non-linear Schrödinger equation of a continuously measured system. In the case in which the quantum system starts out in a superposition of well-separated localized states, the classical system goes into a statistical mixture of trajectories, one trajectory for each individual localized state. † Visiting Research Fellow at: Theory Group, Blackett Laboratory, Imperial College, London, SW7 2BZ, UK. Email address: [email protected] ∗ Email address: [email protected] A variety of problems in a number of different fields involve coupling quantum variables to variables that are effectively classical. A case of particular interest is quantum field theory in curved space time, where one would often like to understand how a quantized matter field affects a classical gravitational field. The most commonly postulated way of modeling this situation is the semiclassical Einstein equations [1,2]: