In this paper, we discuss the macroscopic quantum behavior of simple superconducting circuits. Starting from a Lagrangian for electromagnetic field with broken gauge symmetry, we construct a quantum circuit model for a superconducting weak link (SQUID) ring, together with the appropriate canonical commutation relations. We demonstrate that this model can be used to describe macroscopic excitations of the superconducting condensate and the localized charge states found in some ultrasmall-capacitance weak-link devices.

The statistical properties of a single quantum object and an ensemble of independent such objects are considered in detail for two-level systems. Computer simulations of dynamic zero-point quantum fluctuations for a single quantum object are reported and compared with analytic solutions for the ensemble case.