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We study the charge transfer dynamics following the formation of a phase orvoltage biased su- perconducting nano-junction using a full counting statisticsanalysis. We demonstrate that the evolution of the zeros of the generatingfunction allows one to identify the population of different many body statesmuch in the same way as the accumulation of Yang-Lee zeros of the partitionfunction in equilibrium statistical mechanics is connected to phasetransitions. We give an exact expression connecting the dynamical zeros to thecharge transfer cumulants and discuss when an approximation based on "dominant"zeros is valid. We show that, for generic values of the parameters, the systemgets trapped into a metastable state characterized by a non-equilibriumpopulation of the many body states which is dependent on the initialconditions. We study in particular the effect of the switching rates in thedynamics showing that, in contrast to intuition, the deviation from thermalequilibrium increases for the slower rates. In the voltage biased case thesteady state is reached independently of the initial conditions. Our methodallows us to obtain accurate results for the steady state current and noise inquantitative agreement with steady state methods developed to describe themultiple Andreev reflections regime. Finally, we discuss the system dynamicsafter a sudden voltage drop showing the possibility of tuning the many bodystates population by an appropriate choice of the initial voltage, providing afeasible experimental way to access the quench dynamics and control the stateof the system.

Quench dynamics in superconducting nanojunctions: metastability and dynamical Yang-Lee zeros