On the spectrum of field quadratures for a finite number of photons

The spectrum and eigenstates of any field quadrature operator restricted to a finite number N of photons are studied, in terms of the Hermite polynomials. By (naturally) defining approximate eigenstates, which represent highly localized wavefunctions with up to N photons, one can arrive at an appropriate notion of limit for the spectrum of the quadrature as N goes to infinity, in the sense that the limit coincides with the spectrum of the infinite-dimensional quadrature operator. In particular, this notion allows the spectra of truncated phase operators to tend to the complete unit circle, as one would expect. A regular structure for the zeros of the Christoffel-Darboux kernel is also shown.