  We study the quantum dynamics of many-body systems, in the presence of
dissipation due to the interaction with the environment, under Kibble-Zurek
(KZ) protocols in which one Hamiltonian parameter is slowly, and linearly in
time, driven across the critical value of a zero-temperature quantum
transition. In particular we address whether, and under which conditions, open
quantum systems can develop a universal dynamic scaling regime similar to that
emerging in closed systems. We focus on a class of dissipative mechanisms whose
dynamics can be reliably described through a Lindblad master equation governing
the time evolution of the system's density matrix. We argue that a dynamic
scaling limit exists even in the presence of dissipation, whose main features
are controlled by the universality class of the quantum transition. This
requires a particular tuning of the dissipative interactions, whose decay rate
$u$ should scale as $u\sim t_s^{-\kappa}$ with increasing the time scale $t_s$
of the KZ protocol, where the exponent $\kappa = z/(y_\mu+z)$ depends on the
dynamic exponent $z$ and the renormalization-group dimension $y_\mu$ of the
driving Hamiltonian parameter. Our dynamic scaling arguments are supported by
numerical results for KZ protocols applied to a one-dimensional fermionic wire
undergoing a quantum transition in the same universality class of the quantum
Ising chain, in the presence of dissipative mechanisms which include local
pumping, decay, and dephasing.
