Semiclassical asymptotics of multiphotonic scattering probabilities with partial indistinguishability

Abstract

We propose a framework for computing multiphotonic scattering probabilities in a lossless multiport interferometer for arbitrary photon numbers and degrees of indistinguishability. By exploiting a toroidal expansion of multiphotonic states in tensor powers of single-particle states, the framework defines a map from a torus of relative phases to the probability simplex that governs the asymptotic behavior of scattering probabilities in the large-photon limit. Specifically, the probabilities concentrate on the "classically allowed region" defined by the map, and the slowly-varying part of the multiphotonic distribution reproduces a classical measure induced by the map. As a result, we are able to establish a new asymptotic formula for the multiphotonic probabilities in a general scenario of partially indistinguishable photons, while also providing a single-particle picture to explain the asymptotics of known multiphotonic transition amplitudes in the fully indistinguishable case. More broadly, our framework yields new, directly testable consequences in relation to asymptotic photon bunching patterns: it translates features of the classical map -- such as caustics and voids -- into direct predictions about regions of large or exponentially suppressed photon-distribution probability.

The paper studies multiphoton scattering in a lossless $M$-port interferometer as $N\to\infty$ with $M$ and normalized input/output occupations fixed. It treats “correlated mode states”: photons entering each port share a pure internal state, while states associated with different ports have overlaps encoded by a Gram matrix $\Gamma$. A Fourier projection expands the input state into tensor powers of single-particle states parametrized by relative phases. This yields an exact double-torus integral for scattering probabilities and a classical map $\phi\mapsto\operatorname{Diag}(Ut_\phi\rho t_\phi^\dagger U^\dagger)$, where $\rho=[\bar n]^{1/2}\Gamma[\bar n]^{1/2}$. Its image $\mathcal R$ controls the large-$N$ distribution. In particular, $P(n'|U,n,\Gamma)\leq(N+1)^M e^{-N\min_{\nu\in\mathcal R}D(\bar n'\|\nu)}$, establishing exponential suppression outside the classically allowed region. At regular points, complex Laplace analysis shows that $N^{M-1}P_{\mathrm{slow}}$ converges to the pushforward of uniform phase measure, with density given by a sum of inverse absolute Jacobian determinants. For generic interferometers and full-rank $\rho$, off-diagonal interference contributions vanish asymptotically, so this density describes the full probability. Perfect indistinguishability instead recovers known semiclassical amplitude formulas with interference phases. Three-port examples at $N=24$ and $48$ illustrate suppressed-probability voids, enhanced-probability caustics, and interference fringes. The analysis excludes caustics from its pointwise asymptotic formulas. Stated open extensions include sharp forbidden-region exponential rates, uniform treatment of caustics and allowed–forbidden transitions, intermediate-rank Gram matrices, and higher-port geometry. The fixed-$M$ regime and restricted input family do not cover general partially distinguishable boson-sampling inputs.