  We investigate the quantum measurement noise effects on the dynamics of an
atomic Bose lattice gas inside an optical resonator. We describe the dynamics
by means of a hybrid model consisting of a Bose--Hubbard Hamiltonian for the
atoms and a Heisenberg--Langevin equation for the lossy cavity field mode. We
assume that the atoms are prepared initially in the ground state of the lattice
Hamiltonian and then start to interact with the cavity mode. We show that the
cavity field fluctuations originating from the dissipative outcoupling of
photons from the resonator lead to vastly different effects in the different
possible ground state phases, i.e., the superfluid, the supersolid, the Mott-
and the charge-density-wave phases. In the former two phases with the presence
of a superfluid wavefunction, the quantum measurement noise appears as a
driving term leading to excess noise depletion of the ground state. The time
scale for the system to leave the ground scale is determined analytically. For
the latter two incompressible phases, the quantum noise results in the
fluctuation of the chemical potential. We derive an analytical expression for
the corresponding broadening of the quasiparticle resonances.
