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

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 $|n_1,n_2,\dots ,n_N⟩\stackrel{S_N}{\to }|n/N{⟩}^{\otimes N}$, where $n={\sum }_{i=1}^N{n}_i$, which generalize the Hong-Ou-Mandel (HOM) effect $|1,1⟩\stackrel{S_2}{\to }|1,1⟩$, the eHOM effect $|n_1,n_2⟩\stackrel{S_2}{\to }|\frac{n_1+n_2}{2},\frac{n_1+n_2}{2}⟩$, and the generalized HOM effect (gHOM) $|1{⟩}^{\otimes N}\stackrel{S_N}{\to }|1{⟩}^{\otimes N}$, 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) $|n_1,n_2,\dots ,n_N⟩\stackrel{S_N}{\to }|n/N{⟩}^{\otimes N}$ involving a zero amplitude governed by Perm($\hat{I}›$)=0, for an appropriately constructed matrix $\hat{I}›(S_N)$ built from the matrix elements of $S_N$. We develop an analytical constraint equation for Perm$(\hat{I}›)$ 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 $N=2\ast N^{\prime }$ where $N^{\prime }\in odd$.

31 pages including 2 source code, and10 figures