On the correspondence between classical and quantum measurements on a bosonic field

Abstract

We study the correspondence between classical and quantum measurements on a harmonic oscillator that describes a one-mode bosonic field. We connect the quantum measurement of an observable of the field with the possibility of amplifying the observable ideally through a quantum amplifier. The ``classical'' measurement corresponds to the joint measurement of the position $q$ and momentum $p$ of the harmonic oscillator, with following evaluation of a function $f$ of the outcome $α=q+ip$. For the electromagnetic field the joint measurement is achieved by a heterodyne detector. The quantum measurement of an observable $\hat O$ is obtained by preamplifying the heterodyne detector through an ideal amplifier of $\hat O$, and rescaling the outcome by the gain $g$. We give a general criterion which states when this preamplified heterodyne detection scheme approaches the ideal quantum measurement of $\hat O$ in the limit of infinite gain. We show that this criterion is satisfied and the ideal measurement is achieved for the case of the photon number operator and for the quadrature. For both operators the method is robust to nonunit quantum efficiency of the heterodyne detector. On the other hand, we show that the preamplified heterodyne detection scheme does not work for arbitrary observable of the field. As a counterexample, we prove that the simple quadratic function of the field $\hat K=i(a^{†2}-a^2)/2$ has no corresponding polynomial function $f(α,\bar α)$---including the obvious choice $f=\hbox{Im}(α^2)$---that allows the measurement of $\hat K$ through the preamplified heterodyne measurement scheme.