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Most theoretical treatments of inhomogeneous superconductivity have so farbeen based on the Bogoliubov-deGennes equations, which implement the standardBCS mean-field decoupling in the presence of spatial inhomogeneities. Thisapproach is reliable even at finite temperature for weak inter-particleattraction, when the Cooper pair size is much larger than the averageinter-particle distance, but looses accuracy for increasing attraction when theCooper pair size becomes comparable or even smaller than the averageinter-particle distance, in particular when finite-temperature effects areconsidered. In these cases, inclusion of pairing fluctuations beyond mean fieldis unavoidably required, a task which, however, turns out to be especiallydifficult in the presence of inhomogeneities. Here, we implement the inclusionof pairing fluctuations directly on a coarse-graining version of theBogoliubov-deGennes equations, which makes it simpler and faster to obtain asolution over the whole sector of the temperature-coupling phase diagramcorresponding to the BCS-BEC crossover in the broken-symmetry phase. Wespecifically apply this method in the presence of a super-current flow, suchthat problems related to the Josephson effect throughout the BCS-BEC crossovercan be addressed under a variety of circumstances. This is relevant especiallyin the view of recent experimental data with ultra-cold Fermi atoms, to whichthe results of the present approach are shown to favorably compare in thecompanion Letter [1]. [1] V. Piselli, L. Pisani, and G. Calvanese Strinati,Josephson current flowing through a nontrivial geometry: The role of pairingfluctuations across the BCS-BEC crossover, companion Letter.

Inclusion of pairing fluctuations in the differential equation for the  gap parameter for superfluid fermions in the presence of nontrivial spatial  constraints