Lifshitz-like Metastability and Optimal Dephasing in Dissipative Bosonic Lattices

In dissipative bosonic systems, dephasing is typically expected to accelerate relaxation and suppress coherent dynamics. However, we show that in networks of coherently coupled bosonic modes with non-uniform local dissipation, the presence of quasi-dark states leads to a nontrivial response to dephasing: while weak dephasing facilitates equilibration, moderate to strong dephasing induces a pronounced slowdown of relaxation, revealing the existence of an optimal dephasing rate that enhances equilibration. Using exact dynamical equations for second-order moments, we demonstrate that dephasing suppresses coherent transport and gives rise to long-lived collective modes that dominate the system's late-time behavior. This phenomenon bears striking similarities to Lifshitz-tail states, which are known in disordered systems to cause anomalously slow relaxation. Our results uncover a counterintuitive mechanism by which dephasing, rather than promoting equilibration, can dynamically decouple specific modes from dissipation, thereby protecting excitations. These findings highlight how non-Hermitian physics in open bosonic systems can give rise to unexpected dynamical regimes, paving the way for new strategies to control relaxation and decoherence in bosonic quantum systems, with broad implications for both experimental and theoretical quantum science.