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

TL;DR

This work generalizes the extended Hong-Ou-Mandel effect to symmetric SU(N) beam splitters, analyzing transitions of the form |n1,n2,...,nN⟩ → |n/N⟩^{⊗N} and identifying when the output amplitude vanishes via a constructed Λ matrix. By combining a K-matrix combinatorial framework with Scheel’s permanent approach, the authors reveal that zero amplitudes arise from grouping sub-amplitudes into sectors whose phases align with the root-of-unity structure; they show how destructive interference can be organized into groups and extend the central nodal line phenomenon to even N with odd input parity. A key analytic result is a symmetry constraint on Perm(Λ) that explains the observed patterns for odd and even N, and they demonstrate CNLs for a broad class of geHOM transitions. The paper provides extensive symbolic and numerical verification up to N = 16, yielding practical insights for higher-dimensional quantum interference and potential applications in boson sampling and photonic quantum information processing.

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($Λ$)=0, for an appropriately constructed matrix $Λ(S_N)$ built from the matrix elements of $S_N$. We develop an analytical constraint equation for Perm$(Λ)$ 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 $N'\in odd$.

Figures (10)

• Figure 1: (left) The zero amplitude $A=0$ two-photon HOM transition $|1,1\rangle\overset{S_2}{\to}|1,1\rangle$. (right) The zero amplitude $A=0$ four-photon eHOM transition $|1,3\rangle\overset{S_2}{\to}|2,2\rangle$. In both cases the total amplitude is given by $A = A_{k=0}+A_{k=1}$ where $k$ indicates the number $n_1$ of input photons in port-$1$ that are transmitted to output port-$2$. Both sub-amplitudes $A_{k=0}$ and $A_{k=1}$ have equal amplitudes, but opposite signs, and thus the pair cancels. to produce a total amplitude of $A=0$. Here, $t=r=1/\sqrt{2}$ in the general $SU(2)$ beam splitter $S_2 = \scriptsize{(trr-t)}$.
• Figure 2: The zero amplitude $A=0$, 8-photon eHOM transition $|3,5\rangle\overset{S_2}{\to}|4,4\rangle$ illustrating the two pairs of scattering amplitudes, $(A_{k=0}, A_{k=3})$, and $(A_{k=1}, A_{k=2})$, each with with equal $k$-dependent magnitudes and opposite signs, that cancel in pairs, and contribute to the complete destructive interference on the eHOM coincident output state $|4,4\rangle$ via $A =C_0\,(A_{k=0} +A_{k=3}) + C_1\,(A_{k=1}+A_{k=2})=0+0=0$. The coefficients $C_k$ are combinatorial factors with $C_0=C_3$ and $C_1=C_2$.
• Figure 3: Joint output probability $P(m_1, m_2)$ to measure $m_1$ photons in mode-1 and $m_2$ photons in mode-2 from a 50:50 BS for input Fock number states (FS) $|n\rangle_1$ in mode-1, for $n_1=\{0,1,2,3\}$ (top row, left to right), and an input coherent state (CS) $|\beta\rangle_2$ in mode-2, with mean number of photons with $|\beta|^2=9$. An output central nodal line (CNL) of zeros for the input states $|n,\beta\rangle$ is observed for odd $n=\{1,3\}$ indicating destructive interference of coincidence detection on all output FS/FS $|m,m\rangle$. No CNL is observed for the input states with even $n=\{0,2\}$, indicating non-zero coincidence detection. (bottom row) Same as top row, but now with the CS mode-2 input state replaced by a mixed thermal state $\rho^{\textrm{thermal}}_2$ of average photon number $\bar{n} =9$.
• Figure 4: $N=4$ Scattering diagrams for the transition $|1111\rangle\overset{S_4}{\to}|1111\rangle$ for the $K$ matrices in (top row) Eq.(\ref{['N4:om0:1111:1111:om:0']}) associated with factor $\omega^0$, and (bottom row) Eq.(\ref{['N4:om0:1111:1111:om:2']}) associated with factor $\omega^2$. Any pair of diagrams, one from each row, contributes a pair of sub-amplitude (with equal coefficients) which sums to $1+\omega^2 = 0$, since $\omega^2 = (e^{i 2 \pi/4})^2 = -1$ for $N=4$.. The two groups (top and bottom row) can be said to cancel as a 4-bipartite group.
• Figure 5: The three scattering diagrams for the $N=3$ transition $|012\rangle\overset{S_3}{\to}|111\rangle$ for the $K$ matrices in Eq.(\ref{['N3:012:111']}) associated with factors $(\omega^0, \omega^1, \omega^2)$. This group can be said to cancel as a 3-element group.