Fig. 1. (a) Schematic of a single, lossy waveguide beam splitter excited with N indistinguishable photons prepared in the state |m)≔|N−m,m⟩=|N−m⟩a|m⟩b, where a represents the neutral (gray) waveguide and b is the lossy (red) waveguide. (b) Mapping onto the N-photon subspace spanned by (N+1) multiphoton states |m), represented as a tight-binding lattice model. The coupling between adjacent “modes” is given by matrix elements of J^x; the linearly increasing loss is also shown. (c) Flow of eigenvalues of H^N for N=4. R(λr) shows level attraction with an EP of order five at Γ=2κ; I(λr) shows the emergence of slow modes past the transition. (d) Intensity I(z) shows the fraction of trials where the system remains in the N-photon subspace, i.e., the post-selection probability. It reflects the order of the exceptional point. The beam splitter parameters are ω0=κ=1  cm−1, and the initial state is |ψ(0)⟩=|0).

Fig. 2. Evolution of spin-projections Jr for the N+1 eigenmodes in the post-selected manifold with N=4 (left column) and N=5 (right column) photons, considering different values of the dissipation coefficient: (a), (b) Γ=Γc/2, (c), (d) Γ=0.99Γc, and (e), (f) Γ=1.5Γc. The vector coordinates in the (Jx,Jy,Jz) space are defined by the expectation values of the J^α operators in each eigenstate. When Γ<Γc, the spin projections are in the x−y plane; at the EP, they coalesce along the positive y axis; and when Γ>Γc, they are in the x−z plane.

Fig. 3. Mode occupation dynamics in the post-selected manifold with NOON state input. (a) For N=5 and small loss, the dynamics show asymmetric oscillations. (b) At the EP, P(|m),z) reaches a steady state with most of the weight localized in the low-loss region. (c) After the transition, the steady-state is reached more slowly. (d)–(f) show qualitatively similar results for an N=8 NOON state input. The waveguide beam splitter parameters are set to ω0=κ=1  cm−1.