  We study the charge transfer dynamics following the formation of a phase or
voltage biased su- perconducting nano-junction using a full counting statistics
analysis. We demonstrate that the evolution of the zeros of the generating
function allows one to identify the population of different many body states
much in the same way as the accumulation of Yang-Lee zeros of the partition
function in equilibrium statistical mechanics is connected to phase
transitions. We give an exact expression connecting the dynamical zeros to the
charge transfer cumulants and discuss when an approximation based on "dominant"
zeros is valid. We show that, for generic values of the parameters, the system
gets trapped into a metastable state characterized by a non-equilibrium
population of the many body states which is dependent on the initial
conditions. We study in particular the effect of the switching rates in the
dynamics showing that, in contrast to intuition, the deviation from thermal
equilibrium increases for the slower rates. In the voltage biased case the
steady state is reached independently of the initial conditions. Our method
allows us to obtain accurate results for the steady state current and noise in
quantitative agreement with steady state methods developed to describe the
multiple Andreev reflections regime. Finally, we discuss the system dynamics
after a sudden voltage drop showing the possibility of tuning the many body
states population by an appropriate choice of the initial voltage, providing a
feasible experimental way to access the quench dynamics and control the state
of the system.
