We consider the problem of two capacitively coupled Josephson junction arrays made of ultrasmall junctions. Each one of the arrays can be in the semiclassical or quantum regimes, depending on their physical parameter values. The former case is dominated by a Cooper-pair superfluid while the quantum one is dominated by dynamic vortices leading to an insulating behavior. We first consider the limit when both arrays are in the semiclassical limit, and next the case when one array is quantum and the other semiclassical. We present WKB and Mean Field theory results for the critical temperature of each array when both are in the semiclassical limit. When one array is in the semiclassical regime and the other one in the quantum fluctuations dominated regimes, we derive a duality transformation between the charged and vortex dominated arrays that involve a gauge vector field, which is proportional to the site coupling capacitance between the arrays. The system considered here has been fabricated and we make some predictions as to possible experimentally measurable quantities that could be compared with theory.
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