Isoenergetic model for optical downconversion and error-specific limits of the parametric approximation

TL;DR

The paper addresses the breakdown of the standard parametric approximation in optical downconversion at high squeezing due to pump depletion and pump–signal entanglement. It introduces an isoenergetic, energy-conserving solution that operates within invariant subspaces and provides an improved description beyond the fixed-pump model. The main contributions are (i) a perturbative analysis linking to the Hillery–Zubairy corrections, (ii) a nonperturbative isoenergetic formulation that preserves norm within each energy subspace, and (iii) quantified, error-specific validity domains showing broader applicability than the traditional approach, with practical implications for Gaussian boson sampling and multi-squeezer interferometers. The findings offer a principled way to assess when the Gaussian (parametric) description remains accurate and when a more complete, energy-conserving model is required, thereby affecting the interpretation of high-squeezing experiments and the design of quantum-optical protocols.

Abstract

Optical downconversion is widely used for generating photon pairs, squeezed and entangled states of light, making it an indispensable tool in quantum optics and quantum information. In the regime where the pump is much stronger than the generated field, the standard parametric approximation treats the pump amplitude as a fixed parameter of the model. This approximation has a limited domain of validity since it assumes a non-depleted and non-entangled pump. By finding an approximate solution to the Schrödinger equation of the downconversion process, we obtain an improved analytical model beyond the parametric one, which accounts for pump depletion and pump-signal entanglement. The new model is advantageous, first, because it allows one to compute averages of field operators far beyond the domain of validity of the parametric approximation, and second, because it allows one to establish error-specific limits of the latter domain. For a given pump amplitude, we find a maximum squeezing parameter, up to which the approximation remains valid within a specified acceptable error. Our results confirm that recent experiments on Gaussian boson sampling, with a squeezing parameter of $r\approx 1.8$ and a coherent pump amplitude of $α\approx 2\cdot10^6$, can still be accurately described by the standard parametric approximation. However, we observe a sharp decline in validity as the squeezing parameter increases. For pump amplitudes of $α\approx 2\cdot10^6$, the parametric approximation breaks down when the squeezing parameter exceeds $r\approx 4.5$, whereas the new approximation remains valid up to $r\approx 6$ with an acceptable error of 1%.

Figures (3)

• Figure 1: Domains of validity of the parametric approximation. The area above the red dashed line corresponds to $V_\text{mean}(r,\alpha,\epsilon)<1$; in this area, the mean number of signal photons given by the parametric approximation has an error below $\epsilon$. The area above the orange solid line corresponds to $V_\text{sq}(r,\alpha,\epsilon)<1$; in this area, the squeezed quadrature variance given by the parametric approximation has an error below $\epsilon$. The area above the blue dash-dotted line corresponds to $V_{HZ}(r,\alpha,\epsilon)<1$; in this area, any average of the field operators given by the parametric approximation has an error below $\epsilon$. The two points correspond to the experimental conditions of the Jiuzhang boson sampler Zhong20 and to the FLC experiment Florez20.
• Figure 2: Domains of validity of the parametric and isoenergetic approximations. The area above the green solid line corresponds to $V_{IE}(r,\alpha,\epsilon)<1$; in this area, the isoenergetic approximation is valid up to error $\epsilon$. The area above the magenta dashed line corresponds to $V_{P}(r,\alpha,\epsilon)<1$; in this area, the parametric approximation is valid up to error $\epsilon$ according to the isoenergetic approach. The area above the blue dot-dashed line corresponds to $V_{HZ}(r,\alpha,\epsilon)<1$; in this area, the parametric approximation is valid up to error $\epsilon$ according to the perturbative path integration of Hillery and Zubairy Hillery84. The two points correspond to the experimental conditions of the Jiuzhang boson sampler Zhong20 and to the FLC experiment Florez20, similar to Fig. \ref{['fig:ValidityPert']}.
• Figure 3: Dependence of the indicator function on the squeezing parameter at a fixed pump amplitude. The indicator function for the isoenergetic approximation, $V_{IE}(r,\alpha,\epsilon)$, crosses the value 1 at a higher $r$ than both indicator functions for the parametric approximation.