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We have studied the phase diagram of two capacitively coupled Josephsonjunction arrays with charging energy, $E_c$, and Josephson coupling energy,$E_J$. Our results are obtained using a path integral Quantum Monte Carloalgorithm. The parameter that quantifies the quantum fluctuations in the i-tharray is defined by $\alpha_i\equiv \frac{E_{{c}_i}}{E_{J_i}}$. Depending onthe value of $\alpha_i$, each independent array may be in the semiclassical orin the quantum regime: We find that thermal fluctuations are important when$\alpha \lesssim 1.5 $ and the quantum fluctuations dominate when $2.0 \lesssim\alpha $. We have extensively studied the interplay between vortex and chargedominated individual array phases. The two arrays are coupled via thecapacitance $C_{{\rm inter}}$ at each site of the lattices. We find a {\itreentrant transition} in $\Upsilon(T,\alpha)$, at low temperatures, when one ofthe arrays is in the semiclassical limit (i.e. $\alpha_{1}=0.5 $) and thequantum array has $2.0 \leq\alpha_{2} \leq 2.5$, for the values considered forthe interlayer capacitance. In addition, when $3.0 \leq \alpha_{2} < 4.0$, andfor all the inter-layer couplings considered above, a {\it novel} reentrantphase transition occurs in the charge degrees of freedom, i.e. there is areentrant insulating-conducting transition at low temperatures. We obtain thecorresponding phase diagrams and found some features that resemble those seenin experiments with 2D JJA.Comment: 25 Latex pages including 8 encapsulated poscript figures. Accepted  for publication in Phys. Rev B (Nov. 2004 Issue

Phase and charge re-entrant phase transitions in two capacitively coupled Josephson arrays with ultrasmall junctions