= Quantum escape of a current-biased Josephson junction
{title2=$\Gamma$}
<Quantum escape of a current-biased Josephson junction> is decay of a trapped collective <quantum phase> state through a barrier in the <tilted Josephson washboard potential>. This barrier lies in the <potential energy> of an <electrical circuit> coordinate. Transfer of <Cooper pairs> across the physical insulating layer already supplies the <Josephson coupling energy>; it does not by itself demonstrate this further collective escape. <Single-electron superconducting tunneling> is a third, distinct process.
At higher <temperature> $T$, <thermal activation> instead lifts the coordinate over the barrier. In the rare-escape regime its characteristic form is the <Kramers escape rate>
$$
\Gamma_{\rm th}\simeq a_{\rm th}\frac{\omega_p}{2\pi}\exp\left(-\frac{\Delta U}{k_BT}\right),
$$
where $k_B$ is the <Boltzmann constant> and the prefactor depends on <damping> and thermal equilibration. Cooling suppresses this route. For a quasistationary <energy level> at <energy> $E$, the leading conservative <WKB approximation> instead has
$$
\Gamma_E\ \propto\ \exp\left[-\frac{2}{\hbar}\int_{\phi_1}^{\phi_2}\sqrt{2M_\phi\left(U(\phi)-E\right)}\,d\phi\right],
$$
with the two turning points bracketing the forbidden barrier. This is an exponential estimate requiring a large action compared with $\hbar$, slow escape compared with intrawell motion, and weak environmental corrections. It is not a complete rate prediction near the barrier top or with substantial <damping>.
The familiar cubic-barrier exponent can be checked independently. Choose $V(x)=M_\phi\omega_p^2x^2(1-x/a)/2$, with $a>0$. Its top is at $2a/3$ and $\Delta U=2M_\phi\omega_p^2a^2/27$. At the well-bottom reference <energy> $E=0$,
$$
B=\frac{2M_\phi\omega_p}{\hbar}\int_0^a x\sqrt{1-x/a}\,dx
=\frac{8M_\phi\omega_pa^2}{15\hbar}
=\frac{36\Delta U}{5\hbar\omega_p}.
$$
Substitution $u=x/a$ reduces the dimensionless <integral> to $\int_0^1u\sqrt{1-u}\,du=4/15$. This leading semiclassical action uses a cubic potential and a well-bottom reference; actual ground-state escape also needs the zero-point correction, prefactor and environment. In the chosen washboard example it gives $B\simeq74.0$ after matching the exact barrier and curvature. This matching is an illustrative approximation, not an exact action integral for the cosine potential.
Escape need not yield a sustained voltage: the coordinate can become trapped again. The running-state readout assumes sufficient <underdamping> and bias to avoid such retrapping. The parallel model has <quality factor> $Q=\omega_pRC$; large $Q$ makes energy loss per oscillation small. Low-temperature rate saturation alone cannot distinguish <quantum tunnelling> from unwanted heating or external noise.
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