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Is there a physical interpretation for the characteristic frequency and voltage of a josephson junction?

$$ \omega_c = \frac{2 \pi I_0 R}{\Phi_0} $$

$$ V_c = I_0 R $$

$I_0$ is the critical current of the junction. $R$ is the resistance of non-superconducing electrons going across the junction. $\Phi_0$ is the magnetic flux quantum.

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Since $L_J=\frac{\Phi_0}{2\pi I_0}$ is the inductance of the junction, $1/\omega_c=L_J/R$ is just the $RL$ time constant of the junction. In the resistively-shunted junction model (RSJ), one can interpret $\omega_c$ as the rate at which the junction will switch (undergo a $2\pi$ advance in its phase) when biased just above its critical current.

If $\omega_c=\frac{d\delta}{dt}$ is the rate of change of the phase of the junction ($\delta$), then the second equation is related to the first one via the ac-Josephson equation, $V=\frac{\Phi_0}{2\pi}\frac{d\delta}{dt}$. $V_c$ then is the voltage that will develop across the junction when it switches.

Note that these characteristic scales are mostly useful when the junction is in its "overdamped limit", meaning it is shunted by such a small resistance that the junction capacitance doesn't matter (the junction quality factor $Q<<1$). When this is not the case, a more correct model is the "RCSJ". Then the characteristic timescale for switching is limited by the junction plasma frequency, $\omega_p=\frac{1}{\sqrt{L_JC}}$ where $C$ is the shunt capacitance.

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