Fig. 1. Two-photon one-dimensional quantum problem and its topolectrical circuit simulator.

a Artistic view of two-photon excitations in the array of microresonators with tunneling couplings. The depicted state is ${â}_1^{†}{â}_3^{†}∣0⟩$. b Extended version of Bose–Hubbard model considered in the present article. Single-photon tunnelings J are shown by blue solid lines, direct two-photon tunnelings P are indicated by purple wavy lines. c Top view of the equivalent two-dimensional topolectrical circuit with a voltage at the site (m, n) corresponding to probability amplitude β_mn for one photon to be located at the mth resonator of the array with another one located at the nth resonator [cf. Eq. (2)]. Colored regions show characteristic voltage patterns for two-photon scattering states (green), doublons (red), and doublon edge state (blue). External voltage source applied for the system excitation and voltmeter are shown to the right. Side view of the diagonal (lower inset) and off-diagonal (upper inset) sites of the topolectrical circuit, where grounding elements are shown. d The photograph of experimental setup having the size of 15 × 15 nodes. Inset shows the enlarged fragment of the circuit which includes two unit cells.