  Time crystals are quantum many-body systems which are able to self-organize
their motion in a periodic way in time. Discrete time crystals have been
experimentally demonstrated in spin systems. However, the first idea of
spontaneous breaking of discrete time translation symmetry, in ultra-cold atoms
bouncing on an oscillating mirror, still awaits experimental demonstration.
Here, we perform a detailed analysis of the experimental conditions needed for
the realization of such a discrete time crystal. Importantly, the considered
system allows for the realization of dramatic breaking of discrete time
translation symmetry where a symmetry broken state evolves with a period tens
of times longer than the driving period. Moreover, atoms bouncing on an
oscillating mirror constitute a suitable system for the realization of
dynamical quantum phase transitions in discrete time crystals and for the
demonstration of various non-trivial condensed matter phenomena in the time
domain. We show that Anderson localization effects, which are typically
associated with spatial disorder and exponential localization of eigenstates of
a particle in configuration space, can be observed in the time domain when
ultra-cold atoms are bouncing on a randomly moving mirror.
