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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 $\varphi$ 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 giveHere $I_b$ 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 $\hslash /(2e)$ puts the undriven equation into the mechanical formThe effective inertia is a coefficient for a dimensionless quantum phase, not a literal material mass moving through the insulating layer. The capacitor's energy, $C{V}^2/2=M_{\varphi }{\dot{\varphi }}^2/2$, 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.^[ref]

The current-phase relation of a superconducting weak link gives its pair electric current $I_s$ as a function of the Gauge-invariant Josephson phase $\varphi$, at specified temperature and other conditions. For the ideal junction used below, $I_s=I_c\sin \varphi$ with $I_c>0$. More general periodic relations can contain higher Fourier harmonics; their maximum magnitude defines the corresponding superconducting critical current.

In the ideal sinusoidal Josephson junction, the pair electric current and quantum phase obeyHere $e>0$ is the magnitude of the Electron electric charge, $\hslash =h/(2\pi )$ uses the Planck constant, and $I_c$ is the junction's superconducting critical current under the specified conditions. Define the signs as follows: positive conventional electric current flows from terminal 2 to terminal 1, and $V$ is the voltage of terminal 2 relative to terminal 1. The Gauge-invariant Josephson phase below fixes the matching orientation of $\varphi$. The DC Josephson effect and AC Josephson effect follow from these relations. ^[ref]

A Josephson junction couples two regions with superconductivity through a weak link. A tunnel barrier is one implementation.

The remarkable possibility is a steady electric current across an insulating barrier at zero electric potential difference. This is the DC Josephson effect, carried by coherent transfer of Cooper pairs. It differs from single-electron superconducting tunneling, which probes the quasiparticle excitation spectrum and produces a dissipative current-voltage characteristic. Both contributions can occur in the same superconducting tunnel device. Brian D. Josephson predicted the pair contribution in 1962; his paper treats transfer that leaves the quasiparticle distribution unchanged. ^[ref]

The Josephson effect turns a difference of quantum phases into a measurable electric current, and an applied electric potential difference into a rate of phase advance. The Giaever gap experiment instead turns an electric potential difference into an excitation energy scale. These complementary measurements expose two aspects of superconductivity: its coherent order parameter and its gapped quasiparticle spectrum. The microscopic pairing calculation belongs to BCS theory and the BCS gap equation. A tunnel diode uses a different device, a semiconductor p-n junction.

Adding capacitance and an external bias makes the Gauge-invariant Josephson phase dynamical. Current-biased Josephson phase dynamics explains how its trapped energy levels and escape can be measured electrically.

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 intoThis is the effective potential energy for an unwrapped Gauge-invariant Josephson phase; it includes the bias source and decreases by $2\pi s{E}_J$ on advancing one period. A local minimum and the adjacent downhill barrier areTo verify the barrier, solve $U^{\prime }=E_J(\sin \varphi -s)=0$, select $U^{\prime \prime }({\varphi }_m)>0$ and $U^{\prime \prime }({\varphi }_b)<0$, and subtract their potential energies. The small-oscillation angular frequency is fixed by the curvature:Thus the barrier and oscillation frequency both vanish as $s\to 1^{-}$, but with different powers. For $ϵ=1-s\ll 1$, expanding $\arccos (1-ϵ)$ and $\sqrt{2ϵ-{ϵ}^2}$ givesThe first approximation is for a nearly critical bias. At $s=0$ the exact downhill barrier is $2E_J$; at $s\ge 1$ there is no confining local minimum in this model.

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 $\Delta U/(\hslash {\omega }_p)=10.2784$. 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.