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

We study the correspondence between classical and quantum measurements on a harmonic oscillator that describes a one-mode bosonic field with annihilation and creation operators a and a† with commutation [a, a†] = 1. We connect the quantum measurement of an observable Ô = Ô(a, a†) of the field with the possibility of amplifying the observable Ô ideally through a quantum amplifier which achieves the Heisenberg-picture evolution Ô → gÔ, where g is the gain of the amplifier. The “classical” measurement of Ô corresponds to the joint measurement of the position q̂ = 12(a †+a) and momentum p̂ = i 2 (a † − a) of the harmonic oscillator, with following evaluation of a function f(α, ᾱ) of the outcome α = q + ip. For the electromagnetic field the joint measurement is achieved by a heterodyne detector. The quantum measurement of Ô is obtained by preamplifying the heterodyne detector through an ideal amplifier of Ô, 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 Ô 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 a†a and for the quadrature X̂φ = (a † eiφ+a e−iφ)/2, where one measures the functions f(α, ᾱ) = |α|2 and f(α, ᾱ) = Re(αe−iφ) of the field, respectively. For the photon number operator a†a the amplification scheme also achieves the transition from the continuous spectrum |α|2 ∈ R to the discrete one n ∈ N of the operator a†a. Moreover, for both operators a†a and X̂φ 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 1 quadratic function of the field K̂ = i(a†2−a2)/2 has no corresponding polynomial function f(α, ᾱ)—including the obvious choice f = Im(α2)—that allows the measurement of K̂ through the preamplified heterodyne measurement scheme. 1999 PACS number(s): 03.65.-w, 03.65.Bz, 42.50.Dv, 42.50-p