Broadband complex two-mode quadratures for quantum optics

TL;DR

This paper introduces a frequency-agnostic formalism for broadband two-mode quadratures in quantum optics, linking them to EPR variables and measurable physical observables, enhancing understanding of broadband squeezing.

Contribution

It proposes a modified, frequency-agnostic two-mode quadrature formalism that simplifies transformations and directly relates to measurable broadband squeezing properties.

Findings

Quadrature power generates the $SU(1,1)$ algebra of squeezing Hamiltonians.

Real and imaginary parts of quadratures coincide with EPR variables.

Formalism is broadband and agnostic to frequency, enabling new measurement approaches.

Abstract

In their seminal paper, Caves and Schumaker presented a new formalism for quantum optics, intended to serve as a building block for describing two-photon processes, in terms of new, generalized qudratures. The important, revolutionary concept in their formalism was that it was fundamentally two-mode, i.e. the related observables could not be attributed to any single one of the comprising modes, but rather to a generalized complex quadrature that could only be attributed to both of them. Here, we propose a subtle, but fundamentally meaningful modification to their important work. Unlike the above proposal, we deliberately choose a frequency-agnostic definition of the two-mode quadrature, that we motivate on physical grounds. This simple modification has far-reaching implications to the formalism -- the real and imaginary parts of the quadratures now coincide with the famous EPR variables, and our two-mode operators transform trivially under two-mode and single-mode squeezing operations. Their quadratic forms, which we call the "quadrature power" are shown to succinctly generate the $SU(1,1)$ algebra of squeezing Hamiltonians, and correspond directly to important, broadband physical observables, that have been directly measured in experiment and are explicitly related to properties like squeezing and entanglement. This new point of view gives a fresh perspective on two-mode processes that is completely agnostic to the bandwidth, and reveals intriguing new ways for understanding and measuring broadband two-mode squeezing.