Linear stochastic wave-equations for continuously measured quantum systems

TL;DR

This paper derives and analyzes a linear stochastic wave-equation for quantum systems under continuous measurement, highlighting how superposition and linearity are preserved despite measurement backaction.

Contribution

It provides a first-principles derivation of the linear stochastic wave-equation for systems coupled to Markovian reservoirs, applicable to various measurement schemes.

Findings

Exact analytical solutions are possible in simple cases.

The superposition principle holds for the stochastic wave-function.

The derivation clarifies the physical content of the wave-equation.

Abstract

While the linearity of the Schr\"odinger equation and the superposition principle are fundamental to quantum mechanics, so are the backaction of measurements and the resulting nonlinearity. It is remarkable, therefore, that the wave-equation of systems in continuous interaction with some reservoir, which may be a measuring device, can be cast into a linear form, even after the degrees of freedom of the reservoir have been eliminated. The superposition principle still holds for the stochastic wave-function of the observed system, and exact analytical solutions are possible in sufficiently simple cases. We discuss here the coupling to Markovian reservoirs appropriate for homodyne, heterodyne, and photon counting measurements. For these we present a derivation of the linear stochastic wave-equation from first principles and analyze its physical content.