= AC Josephson effect
{c}
{title2=$f_J=2e|V|/h$}
For constant nonzero voltage $V$, direct integration gives
$$
\phi(t)=\phi_0+\frac{2eV}{\hbar}t,\qquad
I_s(t)=I_c\sin\!\left(\phi_0+\frac{2eV}{\hbar}t\right),\qquad
f_J=\frac{2e|V|}{h}.
$$
A constant <electric potential difference> therefore produces an oscillating pair <electric current>. The average of this ideal sinusoid over a complete cycle is zero; a measured nonzero steady <electric current> can include <single-electron superconducting tunneling>. Josephson's lecture explicitly separates the ordinary quasiparticle term from the phase-dependent term. <The Discovery of Tunnelling Supercurrents by Brian D. Josephson>[Josephson, Nobel lecture, discussion of I₀(V), the phase-dependent term and Fig. 2]{ref}
For an original numerical example, use the exact SI values $e=1.602176634\times10^{-19}\,\mathrm C$ and $h=6.62607015\times10^{-34}\,\mathrm{J\,s}$. Then
$$
\frac{2e}{h}=483.5978484\ldots\,\mathrm{MHz/\mu V},\qquad
V=10\,\mathrm{\mu V}\ \Longrightarrow\ f_J=4.835978484\ldots\,\mathrm{GHz}.
$$
The corresponding period is about $0.2067834\,\mathrm{ns}$. These values describe an assumed exact applied voltage, not a historical measurement. The factor $2e$ identifies coherent pair transfer; replacing it by $e$ would halve the predicted <frequency>.
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