  We show that an ultra-cold atomic cloud bouncing on an oscillating mirror can
reveal spontaneous breaking of a discrete time translation symmetry. In
many-body simulations we illustrate the process of the symmetry breaking that
can be induced by atomic losses or by a measurement of particle positions. The
results pave the way for understanding and realization of the time crystal idea
where crystalline structures form in the time domain due to spontaneous
breaking of continuous time translation symmetry.
