Relaxation and dephasing in a two-electron $^{13}$C nanotube double quantum dot

We use charge sensing of Pauli blockade (including spin and isospin) in a two-electron $^{13}$C nanotube double quantum dot to measure relaxation and dephasing times. The relaxation time, $T_1$, first decreases with parallel magnetic field then goes through a minimum in a field of 1.4 T. We attribute both results to the spin-orbit-modified electronic spectrum of carbon nanotubes, which suppresses hyperfine mediated relaxation and enhances relaxation due to soft phonons. The inhomogeneous dephasing time, $T_2^*$, is consistent with previous data on hyperfine coupling strength in $^{13}$C nanotubes. This work was supported by the National Science Foundation under grant no.~NIRT 0210736 and the GRFP, ARO/iARPA, the Department of Defense, and Harvard's Center for Nanoscale Systems.