On the Spectrum of Field Quadratures for a Finite Number of Photons

Abstract

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 \textit{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.

16 pages, 11 figures

References in corpus (1)

• Quantum mechanics in finite dimensional Hilbert space

Cited by in corpus (1)

• Antisymmetric Wilson loops in N = 4 SYM: from exact results to non-planar corrections