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.
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