This page is among our 'okay articles'. It should be further expanded but we hope that it will still be useful in its present stage.Relation between ac Josephson effect and double-well Bose-Einstein-condensate oscillations. L. Radzihovsky and V. Gurarie in Phys. Rev. A 81:063609 (2010).  What the paper says!?

This studies the relationship between the original and genuine Josephson effect and a condensate trapped in two wells. The comparison is based on the frequencies of oscillatins of the ac effect. A key insight is that the superconducting Josephson effect has a frequency that depends on the voltage $V$ only through fundamental constants (Josephson constant): $$\omega_J={2e\over\hbar}V$$ Besides, the phase difference grows exactly lineary with time.

In contrast, the Bosonic Josephson effect, which typically takes place in finite systems, has frequency that depends on the coupling strength: $$\omega_R={2\sqrt{J^2+\delta^2}\over\hbar}$$ R is for Rabi.

To make the connection, one must go to the thermodynamic limit $L\to\infty$ ($L$ the width of the wells) with as well the number of particles $N\to\infty$.

The fundamental questions are then paused in these terms:

The Authors explain that the tunnelling $J$ scales like $1/L$ and thus a formal analogy is recovered in that case.

The exact meaning and consequences are not too clear to me, in particular if deviations for small system sizes are a shortcoming or richer physics.

Note the ugly typo on Gross-Petaevskii.