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Dynamical quantum phase transitions can occur following quenches in quantumsystems when the rate function, a dynamical analogue of the free energy,becomes non-analytic at critical times. Here we exhaustively investigate in anexemplary model how the dynamically evolving state responds to a second quench.We demonstrate that for quenches where the initial and final Hamiltonian belongto different phases always result in dynamical quantum phase transitions,irrespective of the intermediate quench and dynamics or the time of the secondquench. However, if the initial and final Hamiltonian belong to the sameequilibrium phase then the intermediate Hamiltonian must belong to a differentphase. In this case, the second quench time in relation to the critical timesof the first quench becomes crucial to the existence of dynamical quantum phasetransitions.

Dynamical Quantum Phase Transitions Following Double Quenches: Persistence of the Initial State vs Dynamical Phases