= Solution
A <Josephson junction> weakly couples two superconducting condensates. In a consistent orientation convention, its <Gauge-invariant Josephson phase> $\delta$ obeys
$$
I=I_c\sin\delta,\qquad \dot\delta=\frac{2eV}{\hbar},\qquad
E_J=\frac{\hbar I_c}{2e}.
$$
The <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:
$$
\boxed{I_c(\Phi)=2I_0\left|\cos\frac{\pi\Phi}{\Phi_0}\right|.}
$$
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 $n_1,n_2$,
$$
n(\mathbf r)=n_1+n_2+2\sqrt{n_1n_2}\cos\big(\Delta\mathbf k\cdot\mathbf r+\Delta\theta\big).
$$
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.<Nobel lecture: When atoms behave as waves: Bose-Einstein condensation and the atom laser by Wolfgang Ketterle>[Condensate interference and absorption images of overlapping expanding clouds]{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.
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