  While equilibrium phase transitions are well described by a free-energy
landscape, there are few tools to describe general features of their
non-equilibrium counterparts. On the other hand, near-equilibrium free-energies
are easily accessible but their full geometry is only explored in
non-equilibrium, e.g. after a quench. In the particular case of a
non-stationary system, however, the concepts of an order parameter and free
energy become ill-defined, and a comprehensive understanding of non-stationary
(transient) phase transitions is still lacking. Here, we probe transient
non-equilibrium dynamics of an optically pumped, dye-filled microcavity which
exhibits near-equilibrium Bose-Einstein condensation under steady-state
conditions. By rapidly exciting a large number of dye molecules, we quench the
system to a far-from-equilibrium state and, close to a critical excitation
energy, find delayed condensation, interpreted as a transient equivalent of
critical slowing down. We introduce the two-time, non-stationary, second-order
correlation function as a powerful experimental tool for probing the
statistical properties of the transient relaxation dynamics. In addition to
number fluctuations near the critical excitation energy, we show that transient
phase transitions exhibit a different form of diverging fluctuations, namely
timing jitter in the growth of the order parameter. This jitter is seeded by
the randomness associated with spontaneous emission, with its effect being
amplified near the critical point. The general character of our results are
then discussed based on the geometry of effective free-energy landscapes. We
thus identify universal features, such as the formation timing jitter, for a
larger set of systems undergoing transient phase transitions. Our results carry
immediate implications to diverse systems, including micro- and nano-lasers and
growth of colloidal nanoparticles.
