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