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 $N'\in odd$.
Categories: quant-ph