  Josephson junctions constructed from superconductor-quantum
dot-superconductor (S-QD-S) heterostructures have been used to realize a
variety of voltage-tunable superconducting quantum devices, including qubits
and parametric amplifiers. In such devices, the interplay between the charge
degree of freedom associated with the quantum dot and its environment must be
considered for faithful modeling of circuit dynamics. Here we describe the
self-consistent quantization of a capacitively-shunted S-QD-S junction via
path-integral formulation. In the effective Hamiltonian, the Josephson
potential for the Andreev bound states reproduces earlier results for static
phase bias, whereas the charging energy term has new features: (i) the system's
capacitance is renormalized by the junction gate voltage, an effect which
depends on the strength of the tunneling rates between the dot and its
superconducting leads as well, and (ii) an additional charge offset appears for
asymmetric junctions. These results are important to understand future
experiments and quantum devices incorporating S-QD-S junctions in arbitrary
impedance environments.
