  Understanding how the orbital motion of electrons is coupled to the spin
degree of freedom in nanoscale systems is central for applications in
spin-based electronics and quantum computation. We demonstrate this coupling of
spin and orbit in a carbon nanotube quantum dot in the general multi-electron
regime in presence of finite disorder. Further, we find a strong systematic
dependence of the spin-orbit coupling on the electron occupation of the quantum
dot. This dependence, which even includes a sign change is not demonstrated in
any other system and follows from the curvature-induced spin-orbit split
Dirac-spectrum of the underlying graphene lattice. Our findings unambiguously
show that the spin-orbit coupling is a general property of nanotube quantum
dots which provide a unique platform for the study of spin-orbit effects and
their applications.
