  Driven quantum systems coupled to an environment typically exhibit
effectively thermal behavior with relaxational dynamics near criticality.
However, a different qualitative behavior might be expected in the weakly
dissipative limit due to the competition between coherent dynamics and weak
dissipation. In this work, we investigate a driven-dissipative infinite-range
Ising model in the presence of individual atomic dissipation, a model that
emerges from the paradigmatic open Dicke model in the large-detuning limit. We
show that the system undergoes a dynamical crossover from relaxational
dynamics, with a characteristic dynamical exponent $\zeta=1/2$, to underdamped
critical dynamics governed by the exponent $\zeta=1/4$ in the weakly
dissipative regime; a behavior that is markedly distinct from that of
equilibrium. Finally, utilizing an exact diagrammatic representation, we
demonstrate that the dynamical crossover to underdamped criticality is not an
artifact of the mean-field nature of the model and persists even in the
presence of short-range perturbations.
