Tuning the Josephson current in carbon nanotubes with the Kondo effect

We investigate the Josephson current in a single wall carbon nanotube connected to superconducting electrodes. We focus on the parameter regime in which transport is dominated by Kondo physics. A sizeable supercurrent is observed for odd number of electrons on the nanotube when the Kondo temperature T-K is sufficiently large compared to the superconducting gap. On the other hand when, in the center of the Kondo ridge, T-K is slightly smaller than the superconducting gap, the supercurrent is found to be extremely sensitive to the gate voltage V-BG. Whereas it is largely suppressed at the center of the ridge, it shows a sharp increase at a finite value of V-BG. This increase can be attributed to a doublet-singlet transition of the spin state of the nanotube island leading to a pi shift in the current phase relation. This transition is very sensitive to the asymmetry of the contacts and is in good agreement with theoretical predictions.