Fig. 1. 1D array of M identical nearest-neighbour evanescently coupled waveguides with coupling coefficients κm,m+1.

Fig. 2. Probability distribution |⟨m,N−m|U^(z)|ψ(0)⟩|2 for the initial state |ψ(0)⟩=|5,5⟩ propagating through a waveguide beam splitter with (a) β1=β2=1 (discrete “diffraction” in state space) and (b) β1=0 and β2=4 (“Bloch oscillations” in state space).

Fig. 3. Probability distribution |⟨n1,n2,n3|U^(z)|ψ(0)⟩|2 for the initial state |ψ(0)⟩=|1,0,1⟩ propagating through a balanced three-waveguide beam splitter (κ1=κ2=1) with (a) β1=β2=β3=0 and (b) β1=β3=0 and β2=2. At the dotted horizontal line, the state has evolved almost exactly into a two-photon NOON state in state space.

Fig. 4. Pseudo-energy term diagrams for (a) N=1 photon in M=3 coupled waveguides, (b) N=2 photons in M=2 coupled waveguides, and (c) N=2 photons in M=3 waveguides. Horizontal lines symbolize the different Fock states; vertical arrows indicate allowed transitions along with the corresponding pseudo-exchange energy.

Fig. 5. Matrix components of the effective Hamiltonian Hμν for N=2 photons propagating in M=3 identical, nearest-neighbor-coupled waveguides (β1=β2 and κ1=κ2=1).

Fig. 6. (a) 2D Fock graph for M=3 waveguides excited by N=2 indistinguishable photons. The corresponding adjacency matrix is induced by the effective Hamiltonian in Fig. 5 according to Eq. (31). (b) Sample trial implementation of the (M=3,N=2) Fock graph for a single photon and six waveguides arranged in 2D. Dotted lines indicate additional crosstalk between the waveguides, which is topologically unavoidable in this and any other real-space configuration that we have considered. Therefore, to the best of our knowledge, the synthetic coupled structure in (a) cannot be implemented in the single-photon regime.

Fig. 7. (a) Overview of several 2D and 3D embeddings of Fock graphs Aμ,ν(N,M) for M=2,…,6 waveguides excited by N=1,…,5 indistinguishable photons. Different node colors indicate layer-like structures that emerge for N≥3,M≥4 (all nodes in the same layer feature the same color). For readability, we have omitted the node labels as well as the graphs for M≥5,N≥4. (b) Smallest example of an isomorphic pair of planar Fock graphs with N=2,M=4 and N=3,M=3, respectively.

Fig. 8. Evolution of the probabilities |⟨Kν|U^(z)|ψ⟩|2 of the state |ψ⟩ as defined in Eq. (37).

Table 1. Possible Lattice Configurations for States Arising in a Waveguide Trimer Excited by Two Photons