  The magnetic state of a quantum dot attached to superconducting leads is
experimentally shown to be controlled by the superconducting phase difference
across the dot. This is done by probing the relation between the Josephson
current and the superconducting phase difference of a carbon nanotube junction
whose Kondo energy and superconducting gap are of comparable size. It exhibits
distinctively anharmonic behavior, revealing a phase mediated singlet to
doublet transition. We obtain an excellent quantitative agreement with
numerically exact quantum Monte Carlo calculations. This provides strong
support that we indeed observed the finite temperature signatures of the phase
controlled zero temperature level-crossing transition originating from strong
local electronic correlations.
