Coupling Classical and Quantum Variables using Continuous Quantum Measurement Theory

Continuous quantum measurement theory is used to construct a phenomenological description of the interaction of a quasiclassical variable $X$ with a quantum variable $x$, where the quasiclassical nature of $X$ is assumed to have come about as a result of decoherence. The state of the quantum subsystem evolves according to the stochastic nonlinear Schrödinger equation of a continuously measured system, and the classical system couples to a stochastic $c$ number $\overline{x}\left(t\right)$ representing the imprecisely measured value of $x$. The theory gives intuitively sensible results even when the quantum system starts out in a superposition of well-separated localized states.