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Two weakly coupled condensates in a double-well potential realize one bosonic Josephson junction. The 2005 experiment measured the density distribution in situ and inferred the evolving relative phase from interference fringes. It distinguished tunneling oscillations from nonlinear suppression of large-amplitude transfer, called macroscopic quantum self-trapping.^[ref]

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.

Macroscopic quantum self-trapping is a nonlinear regime of a bosonic Josephson junction in which an initial population imbalance does not undergo the full sign-changing tunneling oscillations expected from a simple linear two-state model. Interactions alter the relation between relative phase and population flow.

An experiment with two weakly linked condensates in a double-well potential measured both the density distribution and the relative phase inferred from interference fringes. Their combination distinguished nonlinear self-trapping from merely observing unequal populations at one instant.^[ref]

An experimental atomic physicist who coauthored the realization of a single bosonic Josephson junction from two weakly coupled Bose-Einstein condensates in a double-well potential. The experiment measured the density in each well and inferred the relative phase from interference fringes. It distinguished oscillatory transfer from interaction-induced macroscopic quantum self-trapping, where an initial population imbalance persists despite the coupling.^[ref]

The printed identity has the wrong sign for the specified relative phase. Differentiating the energy found above and evaluating the operator commutator giveThus the phase derivative of a two-mode condensate energy isA concrete counterexample to the printed sign is $N=100$, $\theta =\pi /4$, $\chi =\pi /2$ and $t>0$: its left side is $100t$, whereas its printed right side is $-100t$, independently of $w$ and $\mu$.

There is also a direct derivation without inserting the Hamiltonian operator. Since $|\theta ,\chi ;N⟩=e^{i\chi {\hat{n}}_R}|\theta ,0;N⟩$, differentiating this unitary operator gives ${\partial }_{\chi }E=i⟨[\hat{H},{\hat{n}}_R]⟩$. Conservation of the total particle number, $[\hat{H},{\hat{n}}_L+{\hat{n}}_R]=0$, then supplies the minus sign for the left well.

By the Heisenberg equation of motion, $\hslash d⟨{\hat{n}}_L⟩/dt=i⟨[\hat{H},{\hat{n}}_L]⟩$. Define the bosonic Josephson junction particle current from left to right as $J_{L\to R}=-d⟨{\hat{n}}_L⟩/dt$. The corrected identity meansThe energy slope with respect to the relative phase drives particle transfer; an extremum with respect to that phase has zero instantaneous particle current. The printed sign would work with $e^{-i\chi }$ in the right-well orbital, or with $[{\hat{n}}_L,\hat{H}]$ in place of $[\hat{H},{\hat{n}}_L]$.

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.