  The dynamics of quantum phase transitions poses one of the most challenging
problems in modern many-body physics. Here, we study a prototypical example in
a clean and well-controlled ultracold atom setup by observing the emergence of
coherence when crossing the Mott insulator to superfluid quantum phase
transition. In the one-dimensional Bose-Hubbard model, we find perfect
agreement between experimental observations and numerical simulations for the
resulting coherence length. We thereby perform a largely certified analogue
quantum simulation of this strongly correlated system reaching beyond the
regime of free quasiparticles. Experimentally, we additionally explore the
emergence of coherence in higher dimensions where no classical simulations are
available, as well as for negative temperatures. For intermediate quench
velocities, we observe a power-law behaviour of the coherence length,
reminiscent of the Kibble-Zurek mechanism. However, we find exponents that
strongly depend on the final interaction strength and thus lie outside the
scope of this mechanism.
