Hyperfine-induced decoherence in triangular spin-cluster qubits

We theoretically investigate hyperfine-induced decoherence in a triangular spin cluster for different qubit encodings. Electrically controllable eigenstates of spin chirality ($C_z$) show no appreciable decoherence up to ${10}^2\mu$s, while a complete decoherence is estimated for the eigenstates of the total-spin projection ($S_z$) and of the partial spin sum ($S_{12}$) after $10\mu$s. The robustness of chirality is due to its decoupling from both the total- and individual-spin components in the cluster. This results in a suppression of the effective interaction between $C_z$ and the nuclear-spin bath. We finally estimate the reduction of the decoherence time scale for $C_z$, resulting from possible hyperfine contact terms or from the misalignment of the magnetic field.

Figure 1

(a) Schematics of the local spin projections $\langle s_{z,i}\rangle$ in the spin triangle, corresponding to the $|0\rangle$ (red) and $|1\rangle$ (blue) states in the three considered qubit encodings. The logical states of the chirality qubit ($C_z$, left) have identical expectation values $\langle s_{z,i}\rangle$; this is not the case for the other two encodings ($S_{12}$ and $S_z$). (b) Angle ${\alpha }_k$ between the vector $\langle \mathbf{S}\rangle$ and $\widehat{\mathbf{z}}$ for the eigenstates ${|0\rangle }_{C_z}^{\theta }$ (blue) and ${|1\rangle }_{C_z}^{\theta }$ (green), for $\Delta /g{\mu }_B=0.5$ (solid lines) and $2.0$ (dotted). (c) Statistical distribution of the distances $d_{en}$ between the $N_e=3$ electron and the $N_n=200$ nuclear spins with randomly generated positions.

Figure 2

Time dependence of the decoherence factor $r$ for the three qubit encodings: $S_z$ (black), $S_{12}$ (red), and $C_z$ (green for $\theta =0$ and blue for $\theta =\pi /8$). The curves are averaged over $N_{\mathcal{I}}=5\times {10}^4$ randomly generated initial states $|\mathcal{I}\rangle$ of the nuclear bath. Inset: Statistical distribution (squares) of the parameter ${\delta }_{\mathcal{I}}$, and corresponding Gaussian fits (solid lines).

Figure 3

Time evolution of the decoherence factor $r_m$ in the cases of the $S_z$ (black) and $S_{12}$ (red) DOFs. The case of chirality is displayed for small tilting angles $\theta =2\pi k/100$, with $k$ ranging from 0 (upper green curve) to 11 (lower green curves). All curves are averaged over ${10}^2$ randomly generated initial conditions $|\mathcal{I}\rangle$; the spin Hamiltonian parameters are $\Delta =1$K, $B=1$T.

Figure 4

Time dependence of the decoherence factor in the presence of three additional nuclear spins ($N_n=203$) localized at the electron-spin positions ${\mathbf{r}}_i^e$ and coupled to the respective electron spins with a contact coupling (a) $a_p=10$mK or (b) $a_p=1$mK. The solid curves correspond to the cases $S_z$ (black), $S_{12}$ (red), and $C_z$ (figure insets). The dotted lines represent the time dependence of $r_m$ in the absence of the three nuclei with contact couplings.