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 TK is sufficiently large compared to the superconducting gap. On the other hand when, in the center of the Kondo ridge, TK is slightly smaller than the superconducting gap, the supercurrent is found to be extremely sensitive to the gate voltage VBG. Whereas it is largely suppressed at the center of the ridge, it shows a sharp increase at a finite value of VBG. This increase can be attributed to a doublet-singlet transition of the spin state of the nanotube island leading to a phase 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.