  We study quantum effects of strong driving field applied to dissipative
hybrid qubit-cavity system which are relevant for a realization of quantum
gates in superconducting quantum metamaterials. We demonstrate that effects of
strong and non-stationary drivings have significantly quantum nature and can
not be treated by means of mean-field approximation. This is shown from a
comparison of steady state solution of the standard Maxwell-Bloch equations and
numerical solution of Lindblad equation on a density matrix. We show that
mean-field approach provides very good agreement with the density matrix
solution at not very strong drivings $f<f^*$ but at $f>f^*$ a growing value of
quantum correlations between fluctuations in qubit and photon sectors changes a
behavior of the system. We show that in regime of non-adiabatic switching on of
the driving such a quantum correlations influence a dynamics of qubit and
photons even at weak $f$.
