We investigate the occurrence of genuine quantum effects and beyond mean-field physics in the balanced and unbalanced open Dicke models with a large yet finite number of atoms 𝑁
. Such driven and dissipative quantum many-body systems have recently been realized in experiments involving ultracold gases inside optical cavities and are known to obey mean-field predictions in the thermodynamic limit 𝑁→∞
. Here we show quantum effects that survive for large but finite 𝑁
, by employing a novel open-system dynamics method that allows us to obtain numerically exact quantum dynamical results for atom numbers up to a mesoscopic 𝑁≈1000
. While we find that beyond-mean-field effects vanish quickly with increasing 𝑁
in the balanced Dicke model, we are able to identify parameter regimes in the unbalanced Dicke model that allow genuine quantum effects to persist even for mesoscopic 𝑁
. They manifest themselves in a strong squeezing of the steady state and a modification of the steady-state phase diagram that cannot be seen in a mean-field description. This is due to the fact that the steady-state limit 𝑡→∞
and thermodynamic limit 𝑁 →∞
do not commute.