  We study a Bose-Einstein condensate at the low energy limit and show that
their collective dynamics exhibit interesting quantum dynamical behavior. The
system undergoes a dynamical quantum phase transition after a sudden quench
into a properly distributed static force, provided its dispersion relation is
linear. We corroborate the occurrence of the dynamical phase transition by
calculating Fisher zeros of the Loschmidt amplitude and showing that they cross
the real time axis in thermodynamic limit. A connection is established between
the order of nonanalycity in the return rate function and the spectral density
of the force. Furthermore, it is shown that a logarithmic or power law scaling
holds at the critical times depending on the displacement spectrum. The scaling
behaviors are studied for three different cases. Eventually, a scheme for the
quantum simulation of such dynamical phase transition and its verification is
proposed. We show that the behavior remains observable even by taking into
account the environmental effects.
