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

Optical down-conversion 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 nondepleted and nonentangled pump. By finding an approximate solution to the Schrödinger equation of the down-conversion process, we obtain an improved analytical model beyond the parametric one, which accounts for pump depletion and pump-signal entanglement. Our 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>r</a:mi><a:mo>≈</a:mo><a:mn>1.8</a:mn></a:mrow></a:math> and a coherent pump amplitude of <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>α</b:mi><b:mo>≈</b:mo><b:mn>2</b:mn><b:mo>×</b:mo><b:msup><b:mn>10</b:mn><b:mn>6</b:mn></b:msup></b:mrow></b:math>, 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>α</c:mi><c:mo>≈</c:mo><c:mn>2</c:mn><c:mo>×</c:mo><c:msup><c:mn>10</c:mn><c:mn>6</c:mn></c:msup></c:mrow></c:math>, the parametric approximation breaks down when the squeezing parameter exceeds <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>r</d:mi><d:mo>≈</d:mo><d:mn>4.5</d:mn></d:mrow></d:math>, whereas the new approximation remains valid up to <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi>r</e:mi><e:mo>≈</e:mo><e:mn>6</e:mn></e:mrow></e:math> with an acceptable error of 1%.