Simultaneous continuous measurement of photon-counting and homodyne detection on a free photon field: dynamics of state reduction and mutual influence of measurement backaction

TL;DR

This paper develops a theoretical framework for the simultaneous measurement of photon-counting and homodyne detection, analyzing their combined effects on quantum state dynamics and measurement backaction.

Contribution

It provides analytic expressions for conditioned wave functions and measurement records, including the backaction effects, and applies these to various initial quantum states.

Findings

Derived explicit formulas for conditioned wave functions and measurement distributions.

Quantified measurement backaction effects on homodyne detection.

Simulated measurement dynamics for different quantum states.

Abstract

We analyze a simultaneous continuous measurement of photon-counting and homodyne detection. The stochastic master equation or stochastic Schr\"odinger equation describing the measurement process includes both jump-type and diffusive-type stochastic increments. Analytic expressions of the wave function conditioned on homodyne and photon-counting records are obtained, yielding the probability density distributions and generating functions of the measurement records. Formula for the expectation values of the homodyne records conditioned on a photon-counting event is also derived which quantitatively describes the measurement backaction of photon-counting on the homodyne output. The obtained results are applied to typical initial states --- coherent, number, thermal, and squeezed states. Monte Carlo simulations of the measurement processes are also presented to demonstrate the dynamics of the combined measurement process.