Observing the emergence of a velocity hierarchy in matter waves
Classical waves in dispersive media naturally exhibit distinct phase and group velocities. Whether an analogous separation of velocities can emerge in matter waves under strong many-body interactions has remained experimentally unexplored. Here, we demonstrate the emergence of a velocity hierarchy in a strongly interacting lattice gas. Using quench spectroscopy together with time-resolved correlation measurements, we independently determine the sound, group, and phase velocities across the superfluid-to-Mott-insulator transition. These velocities are nearly degenerate close to the transition, but progressively separate as the Mott gap opens and the quasiparticle dispersion acquires a massive relativistic-like form. Strikingly, phase-coherence fronts propagate faster than the Lieb-Robinson velocity scale while remaining fully consistent with locality. The measured velocities satisfy a relativistic-like invariance relation in the insulating regime. Our results establish propagation-velocity hierarchies as emergent signatures of strongly correlated quantum dynamics.