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We study the crossover between the sudden quench limit and the adiabaticdynamics of superconducting states in the attractive Hubbard model. We focus onthe dynamics induced by the change of the attractive interaction during afinite ramp time which is varied in order to track the evolution of thedynamical phase diagram from the sudden quench to the equilibrium limit. Twodifferent dynamical regimes are realized for quenches towards weak and strongcoupling interactions. At weak coupling the dynamics depends only on the energyinjected into the system, whereas a dynamics retaining memory of the initialstate takes place at strong coupling. We show that this is related to a sharptransition between a weak and a strong coupling quench dynamical regime, whichdefines the boundaries beyond which a dynamics independent from the initialstate is recovered. Comparing the dynamics in the superconducting andnon-superconducting phases we argue that this is due to the lack of anadiabatic connection to the equilibrium ground state for non-equilibriumsuperconducting states in the strong coupling quench regime.

From sudden quench to adiabatic dynamics in the attractive Hubbard model