  The Kibble-Zurek (KZ) hypothesis identifies the relevant time scales in
out-of-equilibrium dynamics of critical systems employing concepts valid at
equilibrium: It predicts the scaling of the defect formation immediately after
quenches across classical and quantum phase transitions as a function of the
quench speed. Here we study the crossover between the scaling dictated by a
slow quench, which is ruled by the critical properties of the quantum phase
transition, and the excitations due to a faster quench, where the dynamics is
often well described by the classical model. We estimate the value of the
quench rate that separates the two regimes and support our argument using
numerical simulations of the out-of-equilibrium many-body dynamics. For the
specific case of a $\phi^4$ model we demonstrate that the two regimes exhibit
two different power-law scalings, which are in agreement with the KZ theory
when applied to the quantum and to the classical case. This result contributes
to extending the prediction power of the Kibble-Zurek mechanism and to provide
insight into recent experimental observations in systems of cold atoms and
ions.
