Quantum Phase in Nanoscopic Superconductors

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Using the pseudospin representation and the SU(2) phase operators we introduce a complex parameter to characterize both infinite and finite superconducting systems. While in the bulk limit the parameter becomes identical to the conventional order parameter, in the nanoscopic limit its modulus reduces to the number parity effect parameter and its phase takes discrete values. We evaluate the Josephson coupling energy and show that in bulk superconductor it reproduces the conventional expression and in the nanoscopic limit it leads to quantized Josephson effect. Finally, we study the phase flow or dual Josephson effect in a superconductor with fixed number of electrons.