Relative phase and Josephson dynamics between weakly coupled Richardson models

We consider two weakly coupled Richardson models to study the formation of a relative phase and the Josephson dynamics between two mesoscopic attractively interacting fermionic systems. Our results apply to superconducting properties of coupled ultrasmall metallic grains and to Cooper-pairing superfluidity in neutral systems with a finite number of fermions. We discuss how a definite relative phase between the two systems emerges and how it can be conveniently extracted from the many-body wave function, finding that a definite relative phase difference emerges even for very small numbers of pairs (similar to 10). The Josephson dynamics and the current-phase characteristics are then investigated, showing that the critical current has a maximum at the BCS-BEC crossover. For the considered initial conditions a two-state model gives a good description of the dynamics and of the current-phase characteristics.