  We study the equilibrium dynamics of the relative phase in a superconducting
Josephson link taking into account the quantum fluctuations of the
electromagnetic vacuum. The photons act as a superohmic heat bath on the
relative Cooper pair number and thus, indirectly, on the macroscopic phase
difference \phi. This leads to an enhancement of the mean square <\phi^2> that
adds to the spread due to the Coulomb interaction carried by the longitudinal
electromagnetic field. We also include the interaction with the electronic
degrees of freedom due to quasiparticle tunneling, which couple to the phase
and only indirectly to the particle number. The simultaneous inclusion of both
the radiation field fluctuations and quasiparticle tunneling leads to a novel
type of particle--bath Hamiltonian in which the quantum particle couples
through its position and momentum to two independent bosonic heat baths. We
study the interplay between the two mechanisms in the present context and find
interference contributions to the quantum fluctuations of the phase. We explore
the observability of the QED effects discussed here.
