= Quantized phase levels in a Josephson junction
{title2=$H_\phi=-\hbar^2\partial_\phi^2/(2M_\phi)+U(\phi)$}
For a weakly damped well, quantizing the conservative part of <current-biased Josephson phase dynamics> gives the local <Hamiltonian operator>
$$
H_\phi=-\frac{\hbar^2}{2M_\phi}\frac{d^2}{d\phi^2}+U(\phi).
$$
The tilted <potential energy> is unbounded downhill. Its trapped <energy levels> are therefore long-lived resonances with escape lifetimes, rather than globally bound stationary states. Close to $\phi_m$, set $x=\phi-\phi_m$ and expand
$$
U(\phi_m+x)=U(\phi_m)+\frac12M_\phi\omega_p^2x^2-\frac{E_Js}{6}x^3+O(x^4).
$$
Keeping only the quadratic term gives the <quantum harmonic oscillator> result
$$
E_n-U(\phi_m)\simeq\hbar\omega_p\left(n+\frac12\right),\qquad n=0,1,2,\ldots.
$$
It is useful for states localized near the minimum, with <energy> well below $\Delta U$. It does not count all the resonances accurately near escape. The omitted terms make successive spacings unequal. For the chosen example, the three plotted reference <energies> divided by $h$ are $1.44804$, $4.34412$ and $7.24020\,\mathrm{GHz}$ above the minimum.
A small <microwave> bias contributes $\delta U=-(\hbar/2e)I_\mu(t)\phi$. A one-<photon> transition between <energy levels> is resonant when $hf=E_{n+1}-E_n$. Exciting a higher <energy level> can strongly increase its escape probability. Measuring <microwave> absorption indirectly through escape therefore connects an invisible <quantum phase> state to a visible <electric potential difference>. A classical oscillation can also show resonance, so the mere existence of one peak is insufficient: the bias dependence and resolved unequal spacings are the discriminating predictions.
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