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Josephson equations

nosplit "Josephson effect" words: 195
Two equations derived from first principles by Brian Josephson that characterize the device, somewhat like an I-V curve: $$I(t)=Ic​sin(φ(t))dtdφ(t)​=ℏ2eV(t)​ (2)$$ where:
• : Josephson current
• : the Josephson phase, a function defined by the second equation plus initial conditions
• : input voltage of the system
• : current across the junction, determined by the input voltage
Note how these equations are not a typical I-V curve, as they are not an instantaneous dependency between voltage and current: the history of the voltage matters! Or in other words, the system has an internal state, represented by the Josephson phase at a given point in time.
They give the I-V curve of the device, i.e. what current you get when you apply a given voltage across the device.
To understand them better, it is important to look at some important cases separately: