  Using the exact eigenstates of the inhomogeneous Dicke model obtained by
numerically solving the Bethe equations, we study the decay of bosonic
excitations due to the coupling of the mode to an ensemble of two-level (spin
1/2) systems. We compare the quantum time-evolution of the bosonic mode
population with the mean field description which, for a few bosons agree up to
a relatively long Ehrenfest time. We demonstrate that additional excitations
lead to a dramatic shortening of the period of validity of the mean field
analysis. However, even in the limit where the number of bosons equal the
number of spins, the initial instability remains adequately described by the
mean-field approach leading to a finite, albeit short, Ehrenfest time. Through
finite size analysis, we also present indications that the mean field approach
could still provide an adequate description for thermodynamically large systems
even at long times. However, for mesoscopic systems one cannot expect it to
capture the behavior beyond the initial decay stage in the limit of an
extremely large number of excitations.
