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We present two approaches capable of describing the dynamics of aninteracting many body system on a lattice coupled globally to a dissipativebosonic mode. Physical realizations are for example ultracold atom gases inoptical lattice coupled to a photonic mode of an optical cavity or electronicgases in solids coupled to THz cavity fields. The first approach, applicablefor large dissipation strengths and any system size, is a variant of themany-body adiabatic elimination method for investigating the long time dynamicsof the system. The second method extends the time-dependent matrix producttechniques to capture the global coupling of the interacting particles to thebosonic mode and its open nature. It gives numerically exact results for smallto intermediate system sizes. As a benchmark for our methods we perform thefull quantum evolution of a Bose-Hubbard chain coupled to a cavity mode. Weshow that important deviations from the mean-field behavior occur whenconsidering the full atoms cavity coupling [1].

Theoretical methods to treat a single dissipative bosonic mode coupled globally to an interacting many body system