  In conventional Bardeen-Cooper-Schrieffer (BCS) superconductors, electrons
with opposite momenta bind into Cooper pairs due to an attractive interaction
mediated by phonons in the material. While superconductivity naturally emerges
at thermal equilibrium, it can also emerge out of equilibrium when the system's
parameters are abruptly changed. The resulting out-of-equilibrium phases are
predicted to occur in real materials and ultracold fermionic atoms but have not
yet all been directly observed. Here we realize an alternate way to generate
the proposed dynamical phases using cavity quantum electrodynamics (cavity
QED). Our system encodes the presence or absence of a Cooper pair in a
long-lived electronic transition in $^{88}$Sr atoms coupled to an optical
cavity and represents interactions between electrons as photon-mediated
interactions through the cavity. To fully explore the phase diagram, we
manipulate the ratio between the single-particle dispersion and the
interactions after a quench and perform real-time tracking of subsequent
dynamics of the superconducting order parameter using non-destructive
measurements. We observe regimes where the order parameter decays to zero
(phase I), assumes a non-equilibrium steady-state value (phase II), or exhibits
persistent oscillations (phase III). This opens up exciting prospects for
quantum simulation, including the potential to engineer unconventional
superconductors and to probe beyond mean-field effects like the spectral form
factor, and for increasing coherence time for quantum sensing.
