Multiphoton approximation error of parametric down conversion source detection

Biphotons are fundamental to applications in the fields of quantum information and optics. For this reason, biphoton sources are well studied and widely implemented in a variety of experimental settings. Parametric down converters (PDC) are an example of a source for these photons. The accepted method of deriving the PDC state is to create a squeezed state from the vacuum with creation operators formed from an assumed interaction Hamiltonian, where the output state is an infinite sum over biphoton states. Because of the trivially small probability of nonlinear interactions within the PDC crystal, the occurrence probability of photon pairs is likewise trivial, and the series is typically truncated. Naturally, this procedure introduces error into subsequent calculations. In this paper, we examine the error introduced into the probability of detection when neglecting these higher order photon contributions. We justify the relaxation of the common assumption that the squeezed state parameter must be small, and compare our calculations to the usual treatment of the system, finding a nontrivial discrepancy for reasonable experimental parameters. This provides insight that may be useful to mitigate error in future experiments and quantum optical setups.