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 $^{13}\mathrm{C}$ nanotube double quantum dot to measure relaxation and dephasing times. The relaxation time ${T}_{1}$ first decreases with a parallel magnetic field and 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 at high field enhances relaxation due to bending-mode phonons. The inhomogeneous dephasing time ${T}_{2}^{*}$ is consistent with previous data on hyperfine coupling strength in $^{13}\mathrm{C}$ nanotubes.