= Solution
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> is
$$
\delta=\phi_2-\phi_1-\frac q\hbar\int_1^2\mathbf A\cdot d\mathbf l.
$$
With a consistent directed-current convention, the weak-link relation is $I_s=I_c\sin\delta$. A voltage $V=U_2-U_1$ gives $\dot\delta=2eV/\hbar$, 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, $\hbar\dot\delta=\mu_1-\mu_2$ 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 gives
$$
\boxed{I_{c,\rm loop}(\Phi)=2I_0\left|\cos\frac{\pi\Phi}{\Phi_0}\right|.}
$$
For unequal junctions the corresponding result is $\sqrt{I_1^2+I_2^2+2I_1I_2\cos(2\pi\Phi/\Phi_0)}$. 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.<Quantum Interference Effects in Josephson Tunneling by R. C. Jaklevic and others>[1964 primary report]{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.<Direct Observation of Tunneling and Nonlinear Self-Trapping in a Single Bosonic Josephson Junction by Michael Albiez and others>[2005 abstract: double-well implementation, density measurement, phase from fringes and nonlinear self-trapping]{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.
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