Quantum Complementarity for the Superconducting Condensate and the Resulting Electrodynamic Duality

Experiments with small capacitance Josephson junction arrays are described that demonstrate the quantum behavior of the phase of the superconducting condensate. This quantum behavior is based on the complementarity of phase and number for a condensate state of many bosons. A key factor to observation of this quantum behavior is the coupling of the superconducting condensate state to dissipation in the electrodynamic environment of the Josephson junction. It is shown how one-dimensional Josephson junction arrays can be used to design an environment with very weak coupling to dissipation, allowing for measurement of a condensate with well defined number. The well defined number is manifest in the Coulomb blockade of Cooper pair tunneling. PACS numbers: 85.25.C, 85.35, 03.67