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

TL;DR

This paper investigates the spectrum and eigenstates of field quadrature operators limited to a finite photon number, establishing a connection to the infinite-dimensional case and analyzing the spectral limits as photon number increases.

Contribution

It introduces a framework for understanding the spectrum of finite-photon quadratures and defines approximate eigenstates that bridge finite and infinite-dimensional spectra.

Findings

Finite-photon quadrature spectra converge to the infinite-dimensional spectrum.

Approximate eigenstates are highly localized wavefunctions with up to N photons.

Zeros of the Christoffel-Darboux kernel exhibit a regular structure.

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.