We study theoretically the emph{return probability experiment}, used to measure the dephasing time $T_2^*$, in a double quantum dot (DQD) in semiconducting carbon nanotubes (CNTs) with spin-orbit coupling and disorder induced valley mixing. Dephasing
is due to hyperfine interaction with the spins of the ${}^{13}$C nuclei. Due to the valley and spin degrees of freedom four bounded states exist for any given longitudinal mode in the quantum dot. At zero magnetic field the spin-orbit coupling and the valley mixing split those four states into two Kramers doublets. The valley mixing term for a given dot is determined by the intra-dot disorder and therefore the states in the Kramers doublets belonging to different dots are different. We show how nonzero single-particle interdot tunneling amplitudes between states belonging to different doublets give rise to new avoided crossings, as a function of detuning, in the relevant two particle spectrum, crossing over from the two electrons in one dot states configuration, $(0,2)$, to the one electron in each dot configuration, $(1,1)$. In contrast to the clean system, multiple Landau-Zener processes affect the separation and the joining stages of each single-shot measurement and they affect the outcome of the measurement in a way that strongly depends on the initial state. We find that a well-defined return probability experiment is realized when, at each single-shot cycle, the (0,2) ground state is prepared. In this case, valley mixing increases the saturation value of the measured return probability, whereas the probability to return to the (0,2) ground state remains unchanged. Finally, we study the effect of the valley mixing in the high magnetic field limit; for a parallel magnetic field the predictions coincide with a clean nanotube, while the disorder effect is always relevant with a magnetic field perpendicular to the nanotube axis.
We consider theoretically ${}^{13}$C-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 proba
bility after separation into the two dots, which have random nuclear-spin configurations. We focus on the long time saturation value of the return probability, $P_infty$. Because of the valley degree of freedom, the analysis is more complex than in, for example, GaAs quantum dots, which have two distinct $P_infty$ values depending on the magnetic field. Here the prepared state and the measured state is non-unique 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, on the external magnetic field---both Zeeman and orbital effects - and on the spin-orbit splitting. The lowest saturation value, being $P_infty$=1/3, occurs at zero magnetic field for nanotubes with spin-orbit coupling and the initial state being the groundstate, this situation is equivalent to double dots without the valley degree of freedom. In total, we report nine dynamically different situations that give $P_infty$=1/3, 3/8, 2/5, 1/2 and for valley anti-symmetric prepared states in an axial magnetic field, $P_infty$=1. When the groundstate 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 $P_infty$=1/3 to 3/8.
