Mathematically Modeling the Josephson Junction
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Josephson junctions are widely used in applications where a very precise voltage source is required, one example being metrology, where they are used to help define the values of several fundamental constants of nature. Modeling the Josephson junction furthers our understanding of the behavior of the junction which has had profound impacts in the metrological world. This junction and its voltage can be described by a nonlinear differential equation similar to that of the simple pendulum. Differential equations of this type are difficult to understand and very difficult if not impossible to solve explicitly. After deriving this nonlinear differential equation using quantum mechanical arguments, the solutions are found independently using both analytical and numerical approaches. The qualitative behavior of solutions change at a critical current, resulting in either no voltage across the junction or voltage asymptotically approaching the classical Ohm's law. Further modeling of this system could include modeling specific behaviors including the AC and DC Josephson effects.