Statistical mechanics is founded on the assumption that a system can reachthermal equilibrium, regardless of the starting state. Interactions betweenparticles facilitate thermalization, but, can interacting systems alwaysequilibrate regardless of parameter values\,? The energy spectrum of a systemcan answer this question and reveal the nature of the underlying phases.However, most experimental techniques only indirectly probe the many-bodyenergy spectrum. Using a chain of nine superconducting qubits, we implement anovel technique for directly resolving the energy levels of interactingphotons. We benchmark this method by capturing the intricate energy spectrumpredicted for 2D electrons in a magnetic field, the Hofstadter butterfly. Byincreasing disorder, the spatial extent of energy eigenstates at the edge ofthe energy band shrink, suggesting the formation of a mobility edge. At strongdisorder, the energy levels cease to repel one another and their statisticsapproaches a Poisson distribution - the hallmark of transition from thethermalized to the many-body localized phase. Our work introduces a newmany-body spectroscopy technique to study quantum phases of matter.
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