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Via the hierarchy of correlations, we study the strongly interactingFermi-Hubbard model in the Mott insulator state and couple it to a Markovianenvironment which constantly monitors the particle numbers $\hatn_\mu^\uparrow$ and $\hat n_\mu^\downarrow$ for each lattice site $\mu$. Asexpected, the environment induces an imaginary part $\gamma$ (i.e., decay rate)of the quasi-particle frequencies$\omega_{\mathbf{k}}\to\omega_{\mathbf{k}}-i\gamma$ and tends to diminish thecorrelations between lattice sites. Surprisingly, the environment does alsosteer the state of the system on intermediate time scales$\mathcal{O}(1/\gamma)$ to a pre-thermalized state very similar to a quantumquench (i.e., suddenly switching on the hopping rate $J$). Full thermalizationoccurs via local on-site heating and takes much longer.

Environment induced pre-thermalization in the Mott-Hubbard model