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 $N\to \infty$. 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 $N\approx 1000$. 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 $t\to \infty$ and thermodynamic limit $N\to \infty$ do not commute.

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• Atoms, ions, & molecules in cavities
• Cavity quantum electrodynamics
• Collective effects in atomic physics
• Collective effects in quantum optics
• Entanglement in quantum gases
• Quantum description of light-matter interaction
• Dicke model
• Stochastic differential equations