  In this paper we address the question how the Kibble-Zurek mechanism, which
describes the formation of topological defects in quantum systems subjected to
a quench across a critical point, is generalized to the same scenario but for
driven-dissipative quantum critical systems. In these out of equilibrium
systems, the critical behavior is manifested in the steady states rather than
in the ground states as in quantum critical models of closed systems. To give
the generalization we need to establish what is meant by adiabaticity in open
quantum systems. Another most crucial concept that we clarify is the
characterization of non-adiabatic excitations. It is clear what these are for a
Hamiltonian systems, but question is more subtle for driven-dissipative
systems. In particular, the important observation is that the instantaneous
steady states serve as reference states. From these the excitations can then be
extracted by comparing the distance from the evolved state to the instantaneous
steady state. With these issues resolved we demonstrate the applicability of
the generalized Kibble-Zurek mechanism to an open Landau-Zener problem and show
universal Kibble-Zurek scaling for an open transverse Ising model. Thus, our
results support any assumption that non-equilibrium quantum critical behavior
can be understood from universal features.
