Calculation of threshold curves for Josephson quantum interference devices

In many applications of Josephson quantum interference devices, or interferometers, it is important to know the threshold curve which relates the maximum zero-voltage current of the device to the externally applied magnetic field. Generally there is no analytic expression for this curve available and numerical calculation is required. Several such techniques are discussed emphasizing one found to be most rapid in execution. This latter technique treats the problem as that of maximizing the zero-voltage current subject to the constraints that the sum of the phase differences of the superconducting order parameter across junctions and inductances in each interferometer loop be an integral multiple of 2π. Lagrange multipliers are used to formulate the constraints and to obtain simultaneous nonlinear equations whose solution yields the threshold curve. These equations are solved using the multi-dimensional Newton-Raphson iteration procedure. An implementation of this technique has been programmed in APL and has calculated 50 point threshold curves in times as small as 2 minutes. Aspects of this implementation are discussed and examples of threshold curves are given.