= Tilted Josephson washboard potential
{title2=$U(\phi)=-E_J(\cos\phi+s\phi)$}
At fixed positive bias ratio $s=I_b/I_c$ with $0<s<1$, the <Josephson coupling energy> and work of the ideal bias source combine into
$$
U(\phi)=-E_J(\cos\phi+s\phi),\qquad E_J=\frac{\hbar I_c}{2e}.
$$
This is the effective <potential energy> for an unwrapped <Gauge-invariant Josephson phase>; it includes the bias source and decreases by $2\pi sE_J$ on advancing one period. A local minimum and the adjacent downhill barrier are
$$
\phi_m=\arcsin s,\qquad \phi_b=\pi-\arcsin s,\qquad
\Delta U=U(\phi_b)-U(\phi_m)=2E_J\left[\sqrt{1-s^2}-s\arccos s\right].
$$
To verify the barrier, solve $U'=E_J(\sin\phi-s)=0$, select $U''(\phi_m)>0$ and $U''(\phi_b)<0$, and subtract their <potential energies>. The small-oscillation angular <frequency> is fixed by the curvature:
$$
U''(\phi_m)=E_J\sqrt{1-s^2},\qquad
\omega_p^2=\frac{U''(\phi_m)}{M_\phi},\qquad
\omega_p=\sqrt{\frac{2eI_c}{\hbar C}}(1-s^2)^{1/4}.
$$
Thus the barrier and oscillation <frequency> both vanish as $s\to1^-$, but with different powers. For $\epsilon=1-s\ll1$, expanding $\arccos(1-\epsilon)$ and $\sqrt{2\epsilon-\epsilon^2}$ gives
$$
\Delta U\simeq\frac{4\sqrt2}{3}E_J\epsilon^{3/2},\qquad
\omega_p\simeq\sqrt{\frac{2eI_c}{\hbar C}}(2\epsilon)^{1/4}.
$$
The first approximation is for a nearly critical bias. At $s=0$ the exact downhill barrier is $2E_J$; at $s\geq1$ there is no confining local minimum in this model.
\Image[/josephson-washboard-phase-levels.png]
{title=A tilted phase potential confines low energy levels and permits escape}
{description=Original chosen model with $I_c=1\,\mathrm{\mu A}$, $C=4\,\mathrm{pF}$ and $I_b=0.90I_c$. Three horizontal lines are <quantum harmonic oscillator> approximations $E_n-U(\phi_m)=(n+1/2)\hbar\omega_p$, for $n=0,1,2$; they are not exact <energy levels> or measured data. The horizontal axis is a <quantum phase>, not distance through an insulator. Arrows distinguish over-barrier <thermal activation> and below-barrier <quantum tunnelling>.}
For those stipulated parameters, direct evaluation gives $E_J/h=496.684\,\mathrm{GHz}$, $\Delta U/h=29.7670\,\mathrm{GHz}$, and $\omega_p/(2\pi)=2.89608\,\mathrm{GHz}$. Hence $\Delta U/(\hbar\omega_p)=10.2784$. The nearly critical barrier approximation gives $29.6165\,\mathrm{GHz}$, about $0.51\%$ below the exact result. These are original circuit calculations, not parameters assigned to the historical samples.
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