  Discrete time crystals (DTCs) refer to a novel many-body steady state that
spontaneously breaks the discrete time-translational symmetry in a
periodically-driven quantum system. Here, we study DTCs in a Bose-Einstein
condensate (BEC) bouncing resonantly on an oscillating mirror, using a two-mode
model derived from a standard quantum field theory. We investigate the validity
of this model and apply it to study the long-time behavior of our system. A
wide variety of initial states based on two Wannier modes are considered. We
find that in previous studies the investigated phenomena in the evolution
time-window ($\lessapprox$2000 driving periods) are actually "short-time"
transient behavior though DTC formation signaled by the sub-harmonic responses
is still shown if the inter-boson interaction is strong enough. After a much
longer (about 20 times) evolution time, initial states with no "long-range"
correlations relax to a steady state, where time-symmetry breaking can be
unambiguously defined. Quantum revivals also eventually occur. This long-time
behavior can be understood via the many-body Floquet quasi-eigenenergy spectrum
of the two-mode model. A symmetry-breaking edge for DTC formation appears in
the spectrum for strong enough interaction, where all quasi-eigenstates below
the edge are symmetry-breaking while those above the edge are symmetric. The
late-time steady state's time-translational symmetry depends solely on whether
the initial energy is above or below the symmetry-breaking edge. A phase
diagram showing regions of symmetry-broken and symmetric phases for differing
initial energies and interaction strengths is presented. We find that according
to this two-mode model, the discrete time crystal survives for times out to at
least 250,000 driving periods.
