  Spontaneous breaking of continuous time translation symmetry into a discrete
one is related to time crystal formation. While the phenomenon is not possible
in the ground state of a time-independent many-body system, it can occur in an
excited eigenstate. Here, we concentrate on bosons on a ring with attractive
contact interactions and analyze a quantum quench from the time crystal regime
to the non-interacting regime. We show that dynamical quantum phase transitions
can be observed where the return probability of the system to the initial state
before the quench reveals a non-analytical behavior in time. The problem we
consider constitutes an example of the dynamical quantum phase transitions in a
system where both time and space continuous translation symmetries are broken.
