  Coupling with an external environment inevitably affects the dynamics of a
quantum system. Here, we consider how charging performances of a quantum
battery, modelled as a two level system, are influenced by the presence of an
Ohmic thermal reservoir. The latter is coupled to both longitudinal and
transverse spin components of the quantum battery including decoherence and
pure dephasing mechanisms. Charging and discharging dynamics of the quantum
battery, subjected to a static driving, are obtained exploiting a proper
mapping into the so-called spin-boson model. Analytic expressions for the time
evolution of the energy stored in the weak coupling regime are presented
relying on a systematic weak damping expansion. Here, decoherence and pure
dephasing dissipative coupling are discussed in details. We argue that the
former results in better charging performances, showing also interesting
features reminiscent of the Lamb shift level splitting renormalization induced
by the presence of the reservoir. Charging stability is also addressed, by
monitoring the energy behaviour after the charging protocol has been switched
off. This study presents a general framework to investigate relaxation effects,
able to include also non Markovian effects, and it reveals the importance of
controlling and, possibly, engineering system-bath coupling in the realization
of quantum batteries.
