= Current-biased Josephson phase dynamics
{title2=$M_\phi=C(\hbar/2e)^2$}
A <Josephson junction> can make a collective <quantum phase> behave as a measurable dynamical coordinate. In <current-biased Josephson phase dynamics>, the <Gauge-invariant Josephson phase> $\phi$ moves in a <tilted Josephson washboard potential>; the junction's <capacitance> supplies its effective inertia. A trapped phase gives zero mean <electric potential difference>, while sustained phase advance produces a nonzero mean <electric potential difference>. <Quantized phase levels in a Josephson junction> and <quantum escape of a current-biased Josephson junction> test <quantum mechanics> for this collective coordinate.
Begin with a stipulated parallel <electrical circuit>: an ideal sinusoidal <Josephson junction>, <capacitance> $C>0$, and effective <electrical resistance> $R>0$. The <Kirchhoff node law> and <Josephson effect> give
$$
I_b+I_\mu(t)+I_N(t)=I_c\sin\phi+\frac{\hbar}{2eR}\dot\phi+\frac{\hbar C}{2e}\ddot\phi.
$$
Here $I_b$ is the constant bias <electric current>, $I_\mu$ is a deliberately applied <microwave> <electric current>, and $I_N$ represents unwanted fluctuating <electric current>. Multiplication by $\hbar/(2e)$ puts the undriven equation into the mechanical form
$$
M_\phi\ddot\phi+\eta\dot\phi+U'(\phi)=0,\qquad
M_\phi=C\left(\frac{\hbar}{2e}\right)^2,\qquad
\eta=\frac{1}{R}\left(\frac{\hbar}{2e}\right)^2.
$$
The effective inertia is a coefficient for a dimensionless <quantum phase>, not a literal material <mass> moving through the insulating layer. The capacitor's <energy>, $CV^2/2=M_\phi\dot\phi^2/2$, provides a direct check of that identification. The effective $R$ includes loading by the external <electrical circuit>; a constant parallel $R$ is an approximation over the relevant <frequency> range.
The useful design freedom is that $I_b$ changes the confinement while $C$ changes its inertia. The same manufactured device can therefore connect a classical oscillation, discrete <energy levels>, and escape to an electrical readout. The <Martinis-Devoret-Clarke circuit experiments> separate these questions experimentally. <John Clarke>, <Michel H. Devoret> and <John M. Martinis> shared the 2025 Nobel Prize in Physics for circuit experiments establishing macroscopic <quantum tunnelling> and discrete <energy levels>.<The Nobel Prize in Physics 2025 — Press release by Royal Swedish Academy of Sciences>[7 October 2025, p.1, award decision]{ref}
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