  In analogy to spontaneous breaking of continuous space translation symmetry
in the process of space crystal formation, it was proposed that spontaneous
breaking of continuous time translation symmetry could lead to time crystal
formation. In other words, a time-independent system prepared in the energy
ground state is expected to reveal periodic motion under infinitely weak
perturbation. In the case of the system proposed originally by Frank Wilczek,
spontaneous breaking of time translation symmetry can not be observed if one
starts with the ground state. We point out that the symmetry breaking can take
place if the system is prepared in an excited eigenstate. The latter can be
realized experimentally in ultra-cold atomic gases. We simulate the process of
the spontaneous symmetry breaking due to measurements of particle positions and
analyze the lifetime of the resulting symmetry broken state.
