# Exceptional points of any order in a single, lossy, waveguide beamsplitter by photon-number-resolved detection

## TL;DR

This paper demonstrates that a single lossy waveguide beamsplitter, when excited by N indistinguishable photons and post-selected, can exhibit an exceptional point of order N+1, enabling the realization of high-order EPs in a simple device.

## Contribution

It analytically shows how to generate arbitrary-order exceptional points in a single, lossy waveguide beamsplitter using photon-number-resolved detection, overcoming previous symmetry constraints.

## Key findings

- EP of order N+1 observed in N-photon dynamics
- Analytical model mapping beamsplitter to perfect state transfer
- Numerical simulations confirm high-order EPs in post-selected states

## Abstract

Exceptional points (EPs) are degeneracies of non-Hermitian operators where, in addition to the eigenvalues, corresponding eigenmodes become degenerate. Classical and quantum photonic systems with EPs have attracted tremendous attention due to their unusual properties, topological features, and an enhanced sensitivity that depends on the order of the EP, i.e. the number of degenerate eigenmodes. Yet, experimentally engineering higher-order EPs in classical or quantum domains remains an open challenge due to the stringent symmetry constraints that are required for the coalescence of multiple eigenmodes. Here we analytically show that the number-resolved dynamics of a single, lossy, waveguide beamsplitter, excited by $N$ indistinguishable photons and post-selected to the $N$-photon subspace, will exhibit an EP of order $N+1$. By using the well-established mapping between a beamsplitter Hamiltonian and the perfect state transfer model in the photon-number space, we analytically obtain the time evolution of a general $N$-photon state, and numerically simulate the system's evolution in the post-selected manifold. Our results pave the way towards realizing robust, arbitrary-order EPs on demand in a single device.