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Two weakly linked Bose-Einstein condensates form a bosonic Josephson junction. Their relative phase controls a particle current through the weak link, analogous to the pair current of a superconducting Josephson junction. For neutral reservoirs the phase evolution follows a chemical-potential difference rather than an electric voltage. Interactions can make the tunneling dynamics nonlinear, including self-trapping at sufficiently large imbalance.

An experimental condensed matter physicist who coauthored a phase-sensitive test of d-wave superconductivity using a tricrystal ring. A ring containing three grain-boundary Josephson junctions exhibited spontaneous half-quantum magnetic flux, while two-junction control rings did not. The geometry converts a sign change of the pairing order parameter into a measurable magnetic signal.^[ref]

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]

A SQUID with two Josephson junctions in a superconducting loop. For equal junctions, negligible self-inductance and no intrinsic phase offset, adding their sinusoidal currents gives the displayed flux-dependent critical current. An rf SQUID instead has one junction and uses loop inductance.

A spatially uniform Josephson junction in a uniform transverse in-plane magnetic field has a linear gauge-invariant phase profile across its width. Integrating its sinusoidal local current-phase relation of a superconducting weak link gives a signed sinc coefficient multiplying the sine of the central phase. Maximizing over that phase gives the displayed nonnegative critical-current envelope. Zeros occur at nonzero integer multiples of the superconducting flux quantum. The relevant magnetic thickness includes electrode penetration when appreciable, and self-fields or nonuniform local critical current density alter the simple pattern.

John Clarke coauthored experiments testing quantum mechanics through a macroscopic superconducting electrical circuit. In 1985, a current-biased Josephson junction exhibited quantized levels of its phase difference, with positions predicted from independently measured classical circuit parameters. The experiment connected precision electrical measurement with the discrete spectrum of a collective circuit coordinate.^[ref]

A physicist researching superconducting quantum interference devices and quantum-limited measurements.^[ref]

John M. Martinis coauthored a 1985 observation of quantized energy levels in a current-biased Josephson junction. The relevant coordinate was the phase difference across an electrical circuit element, rather than the position of one isolated particle. The measured levels agreed with a quantum mechanics calculation using junction parameters measured in the classical regime. This showed that a fabricated superconducting circuit could possess a discrete excitation spectrum suitable for controlled quantum experiments.^[ref]

A physicist researching superconducting devices and quantum computers based on Josephson junctions.^[ref]

The ideal Josephson junction has a phase-dependent energyDifferentiating and using the voltage–phase relation yieldsThus the ideal pair element stores and returns energy. At $V=0$ it carries a static electric current without electrical power loss. Under a nonzero periodic drive its instantaneous power need not vanish, but its energy returns after a full phase cycle. Real junctions also have a capacitance, quasiparticle transport and an external circuit; their total losses cannot be inferred from this conservative pair term alone.

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.

The 1985 spectroscopy experiment used niobium-based tunnel Josephson junctions on silicon, thermally anchored with final bias low-pass filters to a dilution refrigerator. Calibrated germanium and cobalt thermometers measured temperature. A filtered coaxial cable injected microwaves; repeated switches to nonzero electric potential difference measured escape with irradiation on and off. Bias scans at fixed microwave frequency resolved separate enhancements matching calculated unequal energy level spacings. Raising temperature merged the structure toward classical response. Junction parameters were calibrated in the classical regime; reducing the superconducting critical current with a magnetic field checked thermal readout. Parameter uncertainties limited absolute resonance positions.^[ref]

The companion escape experiment mounted a tunnel Josephson junction on an attenuating coaxial cable, cold-anchored its bias filters, ramped the electric current, and counted switches to nonzero electric potential difference. Rapid current shutoff limited heating. Classical escape and microwave resonance calibrated the superconducting critical current, capacitance and environmental loading. Low-temperature escape agreed with the quantum prediction using these calibrated parameters. A magnetic field reduced the superconducting critical current to keep the same apparatus thermally activated at lower temperature, testing excess noise. Frequency-dependent loading contributed uncertainty; the predicted damping correction was smaller than combined errors, so the experiment did not resolve that correction.^[ref]

Together these tests connect spectral structure and escape of the same kind of collective coordinate. They do not constitute a demonstration of coherent gate operations on a qubit, or prove that every macroscopic coordinate retains quantum behavior under arbitrary environmental coupling. The attraction is more concrete: a fabricated electrical circuit admits both an independently measurable classical model and quantitative quantum tests of its phase dynamics.

Michel H. Devoret coauthored a 1985 measurement of macroscopic quantum tunnelling in a current-biased Josephson junction. At higher temperatures, escape from the zero-voltage state characterized the junction in a thermal regime. At low temperatures, the escape rate stopped depending on temperature and agreed with the predicted quantum escape rate using those independently determined parameters. The comparison supplied quantitative evidence for tunnelling of the collective circuit coordinate, rather than merely observing an unexplained low-temperature switching rate.^[ref]

Identity source: commons.wikimedia.org/wiki/File:Michel_H._Devoret,_French_American_physicist.jpg

quantum tunnelling is transmission through a finite region in which the classical kinetic energy $E-V$ is negative. The wavefunction need not vanish there: the Schrödinger equation permits evanescent solutions. The transmitted particle retains its conserved energy; it does not borrow energy to surmount the barrier.

For a rectangular barrier of height $V_b$ on $0<x<d$, with $0<E<V_b$ and zero exterior potential energy, define $k=\sqrt{2mE}/\hslash$ and $\kappa =\sqrt{2m(V_b-E)}/\hslash$. A left-incident scattering solution has the three piecesContinuity of $\psi ,{\psi }^{\prime }$ at both boundaries imposesEliminating $b,c,r$ givesEqual incident and exit velocities justify $T=|t{|}^2$; different exterior potential energies require the velocity ratio. Probability-current conservation gives $R+T=1$. Although either real exponential alone has zero current, their complex-coefficient combination can carry the current crossing the barrier.

For $\kappa d\gg 1$, $T\simeq 16E(V_b-E)V_b^{-2}{e}^{-2\kappa d}$. A slowly varying barrier generalizes this to the WKB approximation $T\sim \exp [-2{\int }_{x_1}^{x_2}\sqrt{2m(V-E)}dx/\hslash ]$, with matching prefactors and turning-point limitations. This exponential sensitivity explains why very small geometric or energy changes can produce large changes in a tunnelling rate.

In alpha decay, a quasibound alpha particle penetrates the nuclear Coulomb barrier; penetration probability, attempt frequency and nuclear preformation determine the rate. In a scanning tunnelling microscope, a bias between a conducting tip and sample gives current approximately proportional to $e^{-2\kappa d}$. Piezoelectric positioning and constant-current feedback probe surfaces, while vibration isolation and electronic-state dependence limit a purely geometric interpretation. Atomic-scale steps and surface reconstruction were resolved in the original surface-imaging study.^[ref] In a Josephson junction, coherent tunnelling of Cooper pairs permits $I=I_c\sin \phi$, a phase-dependent supercurrent through a weak link. Thus the same barrier physics explains natural decay and underpins sensitive surface and superconducting devices.

In a classically forbidden region, the Schrödinger equation has evanescent solutions rather than forcing the wavefunction to vanish. Matching at both sides of a finite barrier permits nonzero transmission. For a broad barrier the WKB approximation giveswhere $C$ is a matching prefactor. Transmission is exponentially sensitive to barrier width and height. A single decaying exponential function carries no probability current; the matched scattering solution includes the combination needed for transmitted current. energy is conserved throughout.

In alpha decay, a quasibound alpha particle penetrates the nuclear Coulomb barrier. Multiplying the penetration probability by an assault frequency estimates the decay rate; the strong dependence on alpha energy explains the wide range of lifetimes. In a scanning tunnelling microscope, a voltage drives electron quantum tunnelling between a conducting tip and sample. Approximately $I\propto V{e}^{-2\kappa d}$, so constant-current feedback resolves changes in gap width, with electronic density of states also affecting the image. Piezoelectric motion, vibration isolation and stable voltage are necessary; a bright feature is not automatically a taller atom. Atomic-scale imaging was demonstrated with a vacuum-tunnelling microscope in the original surface study.^[ref] In a Josephson junction, Cooper pair quantum tunnelling through a weak link gives $I=I_c\sin \phi$, allowing supercurrent without an applied voltage and phase-sensitive interference. These are distinct applications of barrier penetration and coupling with quantum coherence, not particles borrowing energy to climb over a barrier.

Quantum tunnelling is transmission through a finite classically forbidden region, where $E<V(x)$. The stationary Schrödinger equation permits an evanescent wavefunction inside such a barrier rather than forcing it to vanish at its entrance. Matching to propagating states beyond a barrier of finite width gives a nonzero transmitted probability current.

For a rectangular barrier of height $V_0$ and width $a$, with equal exterior potentials and $0<E<V_0$, write $\kappa =\sqrt{2m(V_0-E)}/\hslash$. The exact rectangular-barrier tunnelling probability isFor an opaque barrier its dominant dependence is $e^{-2\kappa a}$. For a slowly varying barrier, the WKB approximation generalizes this to $T\sim \exp [-(2/\hslash ){\int }_{x_1}^{x_2}\sqrt{2m(V(x)-E)}dx]$, apart from matching prefactors. Barrier width, particle mass and energy therefore have a strong influence. Energy is conserved in a static barrier: tunnelling does not require the particle to borrow energy or acquire a classical path with negative kinetic energy.

In alpha decay, a nuclear alpha-particle configuration is confined by the short-range nuclear interaction but separated from the exterior by a Coulomb barrier. Its energy can be positive relative to infinity while still lying below the barrier maximum. Escape by tunnelling produces alpha particles at that conserved energy, with an exponentially small decay probability per nuclear encounter. The pronounced dependence of half-life on alpha energy is a characteristic consequence of the barrier exponent, unlike a simple classical escape threshold.

In scanning tunneling microscopy, a conducting tip is brought within a nanometer-scale vacuum gap of a conducting surface and biased weakly. Electron tunnelling gives a current that depends approximately exponentially on separation, $I\propto V{e}^{-2\kappa s}$ in a simple low-bias model. Feedback adjusts the tip height to keep current fixed while scanning, producing a surface map sensitive to both geometry and electronic states. Finite current through the non-contact gap, and its steep dependence on separation, are the direct tunnelling signatures; current is not a direct measurement of topographic height alone.

A Josephson junction provides a collective superconducting example. Two superconductors separated by a thin barrier can carry a coherent pair current $I=I_c\sin \phi$ even at zero voltage. At constant voltage the AC Josephson effect gives $d\phi /dt=2eV/\hslash$ and oscillation frequency $2eV/h$. The coupled condensate quantum phases and pair transfer distinguish this from independent normal-electron transmission. These examples involve different microscopic barriers, but all require quantum coupling across a region that suppresses classical passage. Comparing weak-temperature tunnelling behavior with thermally activated escape helps distinguish the mechanisms experimentally.

A Josephson junction weakly couples two superconducting condensates. Its coupling energy is $-E_J\cos \phi$, with a gauge-invariant phase difference across the junction. Choose current and voltage orientations consistently so thatThe DC Josephson effect permits a stationary zero-voltage supercurrent up to $I_c$. A constant nonzero voltage produces the AC Josephson effect, with frequency $f=2e|V|/h$. Resistive and capacitive currents are additional circuit contributions, not part of the ideal sinusoidal supercurrent alone.

In a symmetric DC SQUID with negligible loop self-inductance, the two junction phases differ by $2\pi \Phi /{\Phi }_0$. Adding their currents and maximizing over the common phase givesFlux changes therefore modulate the critical current or a suitably biased output voltage. This converts phase interference into sensitive flux measurement. Finite loop inductance, unequal junctions, noise and readout conditions alter the simple modulation; the loop flux then includes screening currents. A SQUID is useful in magnetometry, but achieving sensitivity requires both a suitable circuit and controlled environmental noise.

A SQUID converts magnetic flux into a periodic interference response of superconducting weak links. A DC SQUID has two Josephson junctions in a loop. For equal junction critical currents $I_0$ and negligible loop self-inductance, their Gauge-invariant Josephson phases differ by $2\pi \Phi /{\Phi }_0$, with ${\Phi }_0=h/(2e)$. Adding $I_0\sin {\delta }_1+I_0\sin {\delta }_2$ and maximizing over the common phase givesThus the critical current repeats for every added flux quantum. A suitable current bias and dissipative junction dynamics convert the changing critical current into an observable voltage modulation. Junction asymmetry prevents exact zero minima, and appreciable self-inductance makes the flux include the circulating current's contribution; these require a modified response model. The Jaklevic Josephson interference experiment supplies an early experimental realization of the interference principle.

An RF SQUID instead uses one junction in a loop inductively coupled to a radio-frequency resonator. In either design, a pickup coil and flux transformer can couple a small external field into the sensor, and flux feedback can keep it near a sensitive operating point over a larger measurement range. Applications include weak-field magnetometry, measurements of magnetic susceptibility and persistent currents, geophysical surveys and precision low-current detection. Gradiometers suppress common external fields, while thermal, junction and environmental noise set practical sensitivity. The instrument measures a flux-dependent electrical response; sensitivity is not established solely by the small size of ${\Phi }_0$.

A Josephson junction converts a relative condensate phase into a measurable current. For superconductors the relevant coherent object is a Cooper pair, with electric charge $q=-2e$. Its Gauge-invariant Josephson phase isWith a consistent directed-current convention, the weak-link relation is $I_s=I_c\sin \delta$. A voltage $V=U_2-U_1$ gives $\dot{\delta }=2eV/\hslash$, so a constant voltage produces the AC Josephson effect at frequency $2e|V|/h$. For neutral atom reservoirs the phase evolution instead follows their chemical-potential difference, $\hslash \dot{\delta }={\mu }_1-{\mu }_2$ with the displayed phase labeling. The current transfers atoms rather than electric charge, but still probes coherent matter-wave amplitudes.

A DC SQUID provides two coherent paths around a superconducting loop. Neglect loop self-inductance, take equal junction critical currents $I_0$, and let $\Phi$ be the enclosed flux. The relative junction phase differs by $2\pi \Phi /{\Phi }_0$, with superconducting flux quantum ${\Phi }_0=h/(2e)$. Adding the two sinusoidal currents and maximizing over their common phase givesFor unequal junctions the corresponding result is $\sqrt{I_1^2+I_2^2+2I_1{I}_2\cos (2\pi \Phi /{\Phi }_0)}$. Finite inductance and junction asymmetry alter the ideal modulation, so a measured minimum need not be zero. Flux changes a phase difference between the paths; the detector reads the resulting critical-current or voltage modulation. The Jaklevic Josephson interference experiment established superconducting interference experimentally.^[ref]

A bosonic Josephson junction uses two weakly linked Bose-Einstein condensates. In the Albiez double-well Josephson experiment, the population distribution was measured in situ and the relative phase inferred from interference fringes. The apparatus therefore tests both number transfer and phase coherence, rather than interpreting a density oscillation alone as a phase measurement. Interactions make the dynamics nonlinear: large imbalance can produce self-trapping instead of simple tunneling oscillations.^[ref]

The usefulness is interferometric sensitivity: a small phase perturbation can change a measurable current. It requires a coherent weak link, controlled phase noise and the appropriate particle's phase evolution; a generic insulating barrier between incoherent reservoirs is not already a matter-wave interferometer.

The superconducting gap function is a momentum-dependent pairing amplitude $\Delta (\mathbf{k})$, including a phase as well as a magnitude. A fully gapped isotropic state and a state with symmetry-required nodes can have different low-temperature responses. Heat capacity, thermal transport, London penetration depth and tunnelling spectra probe the low-energy quasiparticle spectrum. Activated behavior suggests a nonzero minimum gap; power laws can indicate nodes. For example, clean line nodes give a penetration-depth correction linear in temperature, whereas disorder can change the apparent power.

Such magnitude-sensitive probes require care: impurities, several bands, anisotropic full gaps and surface states can imitate or modify simple expectations. A node does not by itself establish a particular sign pattern, and the absence of a coherence peak is not a complete symmetry determination. The strongest identification combines momentum-sensitive spectra with phase-sensitive superconducting gap experiments.

For an illustrative $d_{x^2-y^2}$ state, $\Delta (\phi )\propto \cos 2\phi$ changes sign under a $90^{\circ }$ rotation. A Josephson junction to a conventional s-wave reference can turn that sign into an intrinsic phase shift. In a corner DC SQUID with junctions on orthogonal crystal faces, one branch can acquire an extra $\pi$ relative to the other. For equal junctions and negligible loop inductance,Changing ${\delta }_{\mathrm{int}}$ from zero to $\pi$ shifts the interference pattern by half a flux period. The 1993 YBCO–Pb experiment compared phase-coherent junction geometries and found evidence for that sign reversal, while explicitly discussing SQUID asymmetry and trapped vortices. Those controls matter: an arbitrary shifted trace is not by itself a proof of pairing symmetry.^[ref]

A tricrystal half-flux experiment arranges grain-boundary junctions so that a sign-changing order parameter produces an odd total $\pi$ shift around a ring. Fluxoid consistency then favors a half-integer flux state, with the detailed spontaneous flux depending on loop inductance and junction strengths. Choosing comparison geometries without that frustration and controlling trapped external flux separates the pairing-phase effect from ordinary flux trapping.^[ref]

The conclusion is strongest when the same candidate symmetry explains nodes, crystal-axis dependence and these phase-sensitive interference results. The phase-sensitive observation distinguishes a sign-changing gap from a merely anisotropic positive gap; no single temperature power law supplies that distinction.

A Josephson junction weakly couples two superconducting condensates. In a consistent orientation convention, its Gauge-invariant Josephson phase $\delta$ obeysThe DC Josephson effect permits a current at zero voltage; the AC Josephson effect gives oscillations at frequency $2eV/h$. The junction's energy is $-E_J\cos \delta$. Adding a capacitance gives a charging scale $E_C=e^2/(2C)$ and, in the simplest isolated-island model, Hamiltonian $4E_C(n-n_g{)}^2-E_J\cos \delta$. Competition between phase locking and charge fluctuations permits quantized circuit dynamics, rather than an ordinary linear resistor.

Two junctions in a loop form a DC SQUID. When junctions are equal and loop self-inductance is negligible, their phase difference is fixed by external flux and their currents interfere:This supplies sensitive magnetometry and flux-tunable circuit elements. Unequal junctions leave nonzero minima; finite inductance and screening change the simple pattern. The phase-sensitive gap test in the preceding alternative adds an internal pairing-phase offset to this same interference principle.

Matter-wave interference also compares coherent complex amplitudes. For overlapping condensates with local densities $n_1,n_2$,An atomic interferometer can split a trapped condensate, evolve its two components, release them and image the overlapping density by absorption of a probe beam. For two initially separated sources a distance $d$ apart, free expansion for time $t$ gives an approximate far-field fringe spacing $ht/(md)$. The measured fringe phase encodes relative phase and accumulated dynamical phase; population imbalance lowers visibility. Interference between expanding condensates was an important demonstration of atomic coherence.^[ref]

Josephson oscillation requires an ongoing weak link that exchanges particles; interference after release need not have such a link. Independent condensates can show fringes with a different relative phase in each realization, so a single image is not evidence of a reproducible phase reference over many runs. In both settings the useful object is relative phase, but superconducting circuits also couple electrically to magnetic flux and charge, while neutral-atom devices accumulate matter-wave phase from motion and external potentials.