Evolution from BCS superconductivity to Bose condensation: Calculation of the zero-temperature phase coherence length

We consider a fermionic system at zero temperature interacting through an effective nonretarded potential of the type introduced by Nozigrave eres and Schmitt-Rink, and calculate the phase coherence length ξphase (associated with the spatial fluctuations of the superconducting order parameter) by exploiting a functional-integral formulation for the correlation functions and the associated loop expansion. This formulation is especially suited to follow the evolution of the fermionic system from a BCS-type superconductor for weak coupling to a Bose-condensed system for strong coupling, since in the latter limit a direct mapping of the original fermionic system onto an effective system of bosons with a residual boson-boson interaction can be established. Explicit calculations are performed at the one-loop order. The phase coherence length ξphase is compared with the coherence length ξpair for two-electron correlation, which is relevant to distinguish the weak- (kFξpair>1) from the strong- (kFξpair<1) coupling limits (kF being the Fermi wave vector) as well as to follow the crossover in between. It is shown that ξphase coincides with ξpair down to kFξpair≅10, ξpair in turn coinciding with the Pippard coherence length. In the strong-coupling limit we find instead that ξphase>ξpair, with ξpair coinciding with the radius of the bound-electron pair. From the mapping onto an effective system of bosons in the strong-coupling limit we further relate ξpair with the open-quote open-quote range close-quote close-quote of the residual boson-boson interaction, which is physically the only significant length associated with the dynamics of the bosonic system. copyright 1996 The American Physical Society