Dephasing and hyperfine interaction in carbon nanotube double quantum dots: The clean limit

We consider theoretically 13C-hyperfine interaction-induced dephasing in carbon nanotubes double quantum dots with curvature-induced spin-orbit coupling. For two electrons initially occupying a single dot, we calculate the average return probability after separation into the two dots, which have random nuclear spin configurations. We focus on the long-time saturation value of the return probability, P∞. Because of the valley degree of freedom, the analysis is more complex than in, for example, GaAs quantum dots, which have two distinct P∞ values depending on the magnetic field. Here, the prepared and the measured states are nonunique because two electrons in the same dot are allowed in six different states. Moreover, for one electron in each dot, sixteen states exist and therefore are available for being mixed by the hyperfine field. The return probability experiment is found to be strongly dependent on the prepared state, the external magnetic field—both Zeeman and orbital effects—and the spin-orbit splitting. The lowest saturation value, being P∞ = 1/3, occurs at zero magnetic field for nanotubes with spin-orbit coupling and the initial state being the ground state, this situation is equivalent to double dots without the valley degree of freedom. In total, we report nine dynamically different situations that give P∞ = 1/3, 3/8, 2/5, 1/2, and for valley antisymmetric prepared states in an axial magnetic field, P∞ = 1. When the ground state is prepared, the ratio between the spin-orbit splitting and the Zeeman energy due to a perpendicular magnetic field can tune the effective hyperfine field continuously from being three dimensional to two dimensional giving saturation values from P∞ = 1/3 to 3/8.