Solution of the Bogoliubov-deGennes Equations at Zero Temperature Throughout the BCS-BEC Crossover: Josephson and Related Effects

The BCS-BEC crossover has received much attention lately, owing especially to its experimental realization with trapped ultracold Fermi atoms. Theoretically, the two limiting situations, of paired fermions described by BCS theory in weak coupling and of composite bosons undergoing BoseEinstein condensation (BEC) in strong coupling, can be connected with continuity throughout the crossover. This evolution encompasses the unitary limit at intermediate values of the coupling, where the scattering length for two-fermion scattering diverges. Several quantities have been measured experimentally and calculated theoretically in this context over the last several years, with the notable exception of the Josephson and related effects. This is in spite of the fact that the Josephson effect is intimately associated with the spontaneous breaking of the phase of the complex order parameter which unifies superconductivity and superfluidity. In the present paper, we aim at filling (at least partially) this gap and investigate the evolution of the Josephson and related effects throughout the BCSBEC crossover, by performing a systematic numerical solution of the (time-independent) Bogoliubovde Gennes (BdG) equations at zero temperature in a fully self-consistent fashion. We consider a stationary and uniform current flowing in the presence of a three-dimensional barrier with a slab geometry. This extended geometry is specifically required to reach the BEC limit of the crossover, where the formation of composite bosons in terms of their fermionic constituents requires consideration of wave vectors with components along all three dimensions. In addition, we regard the fermionic attraction to extend unmodified over the barrier region, a situation that typically applies to ultracold Fermi atoms. The fully selfconsistent solution of the BdG equations in such an extended geometry and coupling range represents a non-trivial numerical calculation. The numerical strategies and algorithms we have adopted will therefore be described in detail, with the aim of easing further independent studies. Several results are obtained by the present calculation. The profiles of the magnitude and phase of the gap parameter across the barrier are determined under a variety of conditions. We find that the Josephson current is considerably enhanced at about unitarity for all barriers we have considered. A related enhancement is also found in the contribution to the total current from the Andreev bound states, which stem from the depression of the gap profile about the barrier. The Josephson currentphase characteristics (relating the total current J to the phase difference across the barrier) turn out to evolve from the standard J / sin relation to J / cos , when the height of the barrier is decreased at fixed coupling or the coupling is decreased for a given barrier. For vanishing barrier height, we find that the critical Josephson current approaches the limiting value predicted by the Landau criterion, which is determined by either pair-breaking or soundmode excitations depending on the coupling value. In the BCS limit, we reveal the presence of Friedel oscillations in the oscillatory modulations of the gap and density profiles. In this limit, we also emphasize the special role played by the Andreev bound state in determining the critical Josephson current in the presence of a barrier. Finally, the stability of the two branches, out of which the Josephson characteristics are composed, is analyzed by calculating the energy required to produce a given spatial profile of the gap parameter.