 Solution

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.