Multiphoton Interference with a symmetric SU(N) beam splitter and the generalization of the extended Hong-Ou-Mandel effect

TL;DR

This paper generalizes the Hong-Ou-Mandel effect to symmetric SU(N) beam splitters, analyzing multiphoton interference and the conditions for zero output probability, revealing symmetry properties and interference mechanisms.

Contribution

It introduces a comprehensive framework for understanding multiphoton interference in SU(N) beam splitters, extending the eHOM effect and deriving analytical conditions for zero amplitudes.

Findings

Derived symmetry properties for generalized eHOM effect.

Established analytical constraints for zero output amplitudes.

Extended the concept of central nodal line to higher N with odd N'.

Abstract

We examine multiphoton interference with a symmetric $SU(N)$ beam splitter $S_N$, an extension of features of the $SU(2)$ 50/50 beam splitter extended Hong-Ou-Mandel (eHOM) effect, whereby one obtains a zero amplitude (probability) for the output coincidence state (defined by equal number of photons $n/N$ in each output port), when a total number $n$ of photons impinges on the $N$-port device. These are transitions of the form $|n_1,n_2,\ldots,n_N\rangle\overset{S_N}{\to}|n/N\rangle^{\otimes N}$, where $n=\sum_{i=1}^N n_i$, which generalize the Hong-Ou-Mandel (HOM) effect $|1,1\rangle \overset{S_2}{\to}|1,1\rangle $, the eHOM effect $|n_1,n_2\rangle \overset{S_2}{\to}|\tfrac{n_1+n_2}{2},\tfrac{n_1+n_2}{2}\rangle $, and the generalized HOM effect (gHOM) $|1\rangle^{\otimes N}\overset{S_N}{\to}|1\rangle^{\otimes N}$, which have previously been studied in the literature. The emphasis of this work is on illuminating how the overall destructive interference occurs in separate groups of destructive interferences of sub-amplitudes of the total zero amplitude. We develop symmetry properties for the generalized eHOM effect (geHOM) $|n_1,n_2,\ldots,n_N\rangle\overset{S_N}{\to}|n/N\rangle^{\otimes N}$ involving a zero amplitude governed by Perm($\Lambda$)=0, for an appropriately constructed matrix $\Lambda(S_N)$ built from the matrix elements of $S_N$. We develop an analytical constraint equation for Perm$(\Lambda)$ for arbitrary $N$ that allows us to determine when it is zero. We generalize the SU(2) beam splitter feature of central nodal line (CNL), which has a zero diagonal along the output probability distribution when one of the input states is of odd parity (containing only odd number of photons), to the general case of $N = 2 * N'$ where $N'\in odd$.