Bloch-Messiah decomposition and Magnus expansion for parametric down-conversion with monochromatic pump

TL;DR

This paper analyzes the broadband squeezed light generated by type-I parametric down-conversion with a monochromatic pump, using Bloch-Messiah decomposition and Magnus expansion to evaluate squeezing properties and approximation accuracy.

Contribution

It introduces an exact solution for the process, evaluates the effectiveness of Magnus expansion orders, and proposes criteria for ultra-high-gain regimes in parametric down-conversion.

Findings

First-order Magnus approximation suffices below 12.5 dB squeezing.

Third-order Magnus series converges rapidly for higher squeezing.

Higher-order Magnus terms are necessary in ultra-high-gain regimes.

Abstract

We discuss the Bloch-Messiah decomposition for the broadband squeezed light generated by type-I parametric down-conversion with monochromatic pump. Using an exact solution for this process, we evaluate the squeezing parameters and the corresponding squeezing eigenmodes. Next, we consider the Magnus expansion of the quantum-mechanical evolution operator for this process and obtain its first three approximation orders. Using these approximated solutions, we evaluate the corresponding approximations for the Bloch-Messiah decomposition. Our results allow us to conclude that the first-order approximation of the Magnus expansion is sufficient for description of the broadband squeezed light for squeezing values below 12.5 dB. For higher degrees of squeezing we show fast convergence of the Magnus series providing a good approximation for the exact solution already in the third order. We propose a quantitative criterion for this ultra-high-gain regime of parametric down-conversion when the higher-orders terms of the Magnus expansion, known in the literature as the operator-ordering effects, become necessary.