SNS junctions along the BCS-BEC crossover

We present a theory of SNS junctions, where a normal metal is in contact with two superconductors, along the crossover from the BCS to the BEC regime. Such systems can be realized experimentally by gate-controlled doping of thin film superconductors. We show that Andreev levels exist even in the BEC regime, as long as there is sufficient density of states in the normal metal. We derive the Josephson current through one-dimensional Sharvin contacts. We find that the Josephson current at zero temperature has no peak along the BCS-BEC crossover at $T=0\phantom{\rule{4.pt}{0ex}}\text{K}$ unless the order paramter $\mathrm{\ensuremath{\Delta}}$ is peaked, as it is the case for superconductors with one-dimensional density of states. At finite temperature, however, we find that the Josephson current is maximal at the BCS-BEC crossover for any dimension. We identify the mechanism for this phenomenon to be a decrease in the number of Andreev levels, accompanied by an increase in excitation energy to the unoccupied levels, along the BCS-BEC crossover. This makes it less likely that these states are thermally occupied. As a consequence, the Josephson current is more strongly reduced on the BCS side of the crossover, resulting in a maximal Josephson current at the BCS-BEC crossover. Furthermore, we find that the current phase relation changes along the BCE-BEC crossover with a slope which has a pronounced maximum on the BEC-side of the BCS-BEC crossover. This maximum becomes larger and sharper as the temperature is lowered.