The standard theory of dynamical Coulomb blockade [$P(E)$ theory] in ultra-small tunnel junctions has been formulated on the basis of phase-phase correlations by several authors. It was recently extended by several experimental and theoretical works to account for novel features such as electromagnetic environment-based renormalization effects. Despite this progress, aspects of the theory remain elusive especially in the case of linear arrays. Here, we apply path integral formalism to re-derive the Cooper-pair current and the BCS quasi-particle current in single small Josephson junctions and extend it to include long Josephson junction arrays as effective single junctions. We consider renormalization effects of applied oscillating voltages due to the impedance environment of a single junction as well as its implication to the array. As is the case in the single junction, we find that the amplitude of applied oscillating electromagnetic fields is renormalized by the same complex-valued weight $Xi(omega) = |Xi(omega)|exp ieta(omega)$ that rescales the environmental impedance in the $P(E)$ function. This weight acts as a linear response function for applied oscillating electromagnetic fields driving the quantum circuit, leading to a mass gap in the thermal spectrum of the electromagnetic field. The mass gap can be modeled as a pair of exotic `particle excitation with quantum statistics determined by the argument $eta(omega)$. In the case of the array, this pair corresponds to a bosonic charge soliton/anti-soliton pair injected into the array by the electromagnetic field. Possible application of these results is in dynamical Coulomb blockade experiments where long arrays are used as electromagnetic power detectors.
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