Interference effects in isolated Josephson junction arrays with geometric symmetries

Reducing the size of the junctions in a Josephson junction array takes the system from the classical to the quantum regime and turns the condensate phases on the islands into periodic quantum variables. Under suitable conditions such quantum arrays exhibit Aharonov-Bohm-Casher-type interference effects which change the spectrum of this macroscopic object by the addition of one Cooper pair to the array. We discuss the case of an isolated symmetric loop of N identical junctions encircling one half superconducting quantum of magnetic flux. We argue that the ground state is nondegenerate if the total number of Cooper pairs on the array is divisible by N, and doubly degenerate otherwise (after the stray charges are compensated by the gate voltages). This result is first established in the two opposite limits of dominating Josephson and charging energies, and is further explained using symmetry arguments. As long as the ground-state symmetry is preserved, its degeneracy remains unchanged in the entire range of charging-to-Josephson energy ratio. Such interference effects may be relevant in applications of superconducting junction arrays for quantum computing.