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 held with annihilation and creation operators a and a dagger with commutation [a, a dagger] = 1. We connect the quantum measurement of an observable (O) over cap = (O) over cap(a, a dagger) of the field with the possibility of amplifying the observable (O) over cap ideally through a quantum amplifier which achieves the Heisenberg-picture evolution (O) over cap --> g (O) over cap, where g is the gain of the amplifier. The "classical" measurement of (O) over cap corresponds to the joint measurement of the position (q) over cap = 1/2(a dagger + a) and momentum (p) over cap = i/2(a dagger - a) of the harmonic oscillator, with following evaluation of a function f(alpha, <(alpha)over bar>) of the outcome alpha = a + ip. For the electromagnetic field the joint measurement is achieved by a heterodyne detector. The quantum measurement of (O) over cap is obtained by preamplifying the heterodyne detector through an ideal amplifier of (O) over cap 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 (O) over cap 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 dagger a and for the quadrature (X) over cap phi = (a dagger e(i phi) + ae(-i phi))/2, where one measures the functions f(alpha, <(alpha)over bar>) = \alpha\(2) and f(alpha, <(alpha)over bar>) = Re(alpha e(-i phi)) of the field, respectively. For the photon number operator a dagger a the amplification scheme also achieves the transition from the continuous spectrum \alpha\(2) is an element of R to the discrete one n is an element of N of the operator a dagger a. Moreover, for both operators ata and (X) over cap(phi) 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 (K) over cap = i(a dagger(2) - alpha(2))/2 has no corresponding polynomial function f(alpha, <(alpha)over bar>) - including the obvious choice f = Im(alpha 2) - that allows the measurement of (K) over cap through the preamplified heterodyne measurement scheme.