Fig. 3: Band structure of multiphoton bound states.

a, Schematic of circuit used for many-body spectroscopy. n_ph adjacent qubits are prepared in the \(| +\rangle \) state, before evolving the state with a variable number of fSim gates. The phase of the bound state is probed by measuring the correlator \(\langle {\sigma }_{i}^{+}..{\sigma }_{i+{n}_{{\rm{ph}}}-1}^{+}\rangle \) for all sets of n_ph adjacent qubits. b, Real (top, Re) and imaginary (bottom, Im) parts of the n_ph = 2 correlator. c, Band structure for n_ph = 1–5 (top to bottom), obtained via a 2D Fourier transform in space and time of the n_ph correlators. Colour scale: absolute square of the Fourier transform, ∣A_k,ω∣². Dashed curves: theoretical prediction in equation (3). d, Band structure for n_ph = 2 in the weakly interacting (ϕ < 2θ) regime, displaying the emergence of a bound state only at momenta near k = ±π. Dashed black lines: theoretically predicted momentum threshold for the existence of the bound state (see Supplementary Information). e, Flux dependence of the n_ph = 2 band structure, displaying a gradual momentum shift as the flux increases (Φ₀ = 2πN_Q). Orange circles and dashed line indicate the peak position of the band. f, Extracted momentum shifts as a function of flux for n_ph = 1–5 (top to bottom), indicating that the rate of shifting scales linearly with the photon number of the bound states, that is, the pseudo-charge q of each bound state is proportional to its number of photons. Coloured lines: theoretical prediction.