Quantum system dynamics with a weakly nonlinear Josephson junction bath

We investigate the influence of a weakly nonlinear Josephson bath consisting chain junctions on dynamics small quantum system (LC oscillator). Focusing regime where charging energy is largest scale, we perturbatively calculate correlation function to leading order in divided by while keeping cosine potential exactly. When variation along ensures fast decay function, LC oscillator that and capacitively coupled can be solved through Markovian master equation. establish duality relation for between regimes large energies, respectively. The results applied cases either nonuniformly engineered or disordered chain. Furthermore, find may become non-Markovian when temperature increased beyond zero-temperature limit gets shifted constant does not with time.