The quantum beam splitter with many partially indistinguishable photons: multiphotonic interference and asymptotic classical correspondence

We present the asymptotic analysis of the quantum two-port interferometer in the $n\to \mathrm{\infty }$ limit of $n$ partially indistinguishable photons. Using the unitary-unitary duality between port and inner-mode degrees of freedom, the probability distribution of output port counts can be decomposed as a sum of contributions from independent channels, each associated to a spin-$j$ representation of $SU(2)$ and, in this context, to $2j$ effectively indistinguishable photons in the channel. Our main result is that the asymptotic output distribution is dominated by the $O(\sqrt{n})$ channels around a certain $j^{\ast }$ that depends on the degree of indistinguishability. The asymptotic form is essentially the doubly-humped semi-classical envelope of the distribution that would arise from $2j^{\ast }$ indistinguishable photons, and which reproduces the corresponding classical intensity distribution.