  Non-equilibrium aspects of the BCS model have fascinated physicists for
decades, from the seminal works of Eliashberg to modern realizations in cold
atom experiments. The latter scenarios have lead to a great deal of interest in
the quench dynamics of fermions with pairing interactions. The recently
introduced notion of a dynamical quantum phase transition is an attempt to
classify the myriad of possible phenomena which can result in such far from
equilibrium systems. These are defined as non-analytic points of the logarithm
of the Loschmidt echo and are linked to oscillations in the dynamics a systems
order parameter. In this work we analytically investigate the relation between
DQPTs and oscillation of the superconducting order parameter in quenches of the
BCS model. We find that each oscillation of the order parameter is accompanied
by a DQPT which is first order in nature. We show this for a variety of initial
states and furthermore find that when the order parameter attains a constant
steady state then no DQPTS occur.
