Hyperfine-induced dephasing in three-electron spin qubits

We calculate the pure dephasing time of three-electron exchange-only qubits due to interaction with the nuclear hyperfine field. Within the $S=S_z=1/2$ spin subspace, we derive formulas for the dephasing time as a function of the position within the stability diagram consisting of the $(1,1,1)$ charge region and the neighboring charge sectors coupled by tunneling. The nuclear field and the tunneling are taken into account in a second order approximation. The analytical solutions accurately reproduce the numerical evaluation of the full problem, and in comparison with existing experimental data, we find that the dephasing times are longer but on the same time scale as for single spins.

Figure 1

Charge stability diagram: Lowest-energy charge states as functions of the detunings $\varepsilon$ and ${\varepsilon }_{\mathrm{m}}$ with $U_{\mathrm{c}}=0.2U$. In the absence of tunneling and hyperfine field, states $\left|0\right.〉$, $\left|1\right.〉$, and $\left|2\right.〉$ are degenerate.

Figure 2

Partitioning of the $\varepsilon$–${\varepsilon }_{\mathrm{m}}$ space by the two lowest states with $U_{\mathrm{c}}=0.2U$. Here $\left|0^{\prime }\right.〉$ denotes the lower of the two states we obtain after the hybridization of $\left|0\right.〉$ and $\left|1\right.〉$. The regions of interest of this paper are shaded.

Figure 3

Temporal decay of the coherence $\langle 0\left|\overline{\rho }\left(t\right)\right|1\rangle$. The solid line is obtained by evaluating Eq. (9) with $T_{\phi 01}$ from Eq. (10) and $c=0$, while the dots are calculated using the numerical evaluation of the full problem without approximations by taking the average of density matrices of states evolved by $H$ from the initial state of $(\left|0\right.〉+\left|1\right.〉)/\sqrt{2}$ with 1000 random realizations of the hyperfine field. Here $\varepsilon /U=0.4$, ${\varepsilon }_{\mathrm{m}}/U=0.5$, $U_{\mathrm{c}}/U=0.2$, $t_{\mathrm{l}}/U=0.015$, $t_{\mathrm{r}}/U=0.01$ and $g=2.0$, ${\sigma }_z=15\mu \mathrm{T}$, which corresponds to natural Si [23]. $T_{\phi 01}=656\mathrm{ns}$.

Figure 4

Dephasing time in nanoseconds for realistic input parameters: $U_{\mathrm{c}}/U=0.2$, $t_{\mathrm{l}}/U=0.015$, $t_{\mathrm{r}}/U=0.01$ and $g=2.0$, ${\sigma }_z=15\mu \mathrm{T}$, which corresponds to natural Si [23]. In the upper figure, the analytic formulas for $T_{\phi 0n}$ are evaluated, while in the lower figure, the numerical solution of the full problem is plotted, which is also accurate close to the borders, see the main text.

Figure 5

Dephasing time in nanoseconds for symmetric input parameters: $U_{\mathrm{c}}/U=0.2$, $t_{\mathrm{l}}/t_{\mathrm{r}}=1$ and $g=2.0$, $\sigma =15\mu \mathrm{T}$.