Pairwise Decoherence in Coupled Spin Qubit Networks

Experiments involving phase coherent dynamics of networks of spins, such as echo experiments, will only work if decoherence can be suppressed. We show here, by analyzing the particular example of a crystalline network of ${\mathrm{Fe}}_8$ molecules, that most decoherence typically comes from pairwise interactions (particularly dipolar interactions) between the spins, which cause “correlated errors.” However, at very low $T$ these are strongly suppressed. These results have important implications for the design of quantum information processing systems using electronic spins.

Figure 1

In a strong transverse field ${\mathbf{H}}_{\perp }$, the two potential wells of an easy axis spin system approach each other on the Bloch sphere. (a) The spin anisotropy energy for a ${\mathrm{Fe}}_8$ molecular spin with ${\mathbf{H}}_{\perp }$ along $\widehat{y}$, easy axis $\widehat{z}$ and hard axis $\widehat{x}$, when ${\mu }_0{H}_y=2.5\mathrm{T}$. The low-lying states $|{\mathcal{Z}}_{\pm }⟩$ are approximately localized in the two potential wells. The quantum-mechanical eigenstates are symmetric and antisymmetric superpositions of $|{\mathcal{Z}}_{\pm }⟩$ (see text), separated by the tunneling gap $2{\Delta }_0$. (b) The resonance experimental setup and the spin states on the Bloch sphere. (c) At $T\ll {\Delta }_0/k_B$ only the lowest-energy eigenstate is populated, $|\mathcal{S}⟩=2^{-1/2}(|{\mathcal{Z}}_{+}⟩+|{\mathcal{Z}}_{-}⟩)$. A short $\mu$ wave pulse prepares the system in the $|{\mathcal{Z}}_{+}⟩$ state ($\pi /2$ rotation, corresponding to $1/4$ of a Rabi oscillation). The spin then tunnels coherently between $|{\mathcal{Z}}_{+}⟩$ and $|{\mathcal{Z}}_{-}⟩$ at a frequency $2{\Delta }_0/\hslash$. The effect of static inhomogeneities in ${\Delta }_0$ can be compensated by a $\pi$ pulse in a spin-echo sequence (not shown).

Figure 2

${\mathrm{Fe}}_8$ in a transverse field $H_y$: (a) tunneling splitting $2{\Delta }_0$ of the lowest doublet, as a function of $H_y$. (b) Effective $g$ factors for ${\mathrm{Fe}}_8$ as a function of the ratio ${\Delta }_0(H_y)/{\Omega }_0$. In both figures, the values at ${\mu }_0{H}_y=2.5\mathrm{T}$ are indicated by a black arrow. Our decoherence rate calculations are valid when ${\Delta }_0\gg U_d$, and do not apply in the gray areas shown.

Figure 3

Feynman self-energy graphs for the $\mathbf{q}=0$ uniform precession mode, interacting with magnon excitations. We depict (a) a 3-magnon process, which is ruled out by energy conservation, (b) a 4-magnon process, and (c) a 6-magnon process.

Figure 4

Dimensionless decoherence rates ${\gamma }_{\varphi }=\hslash /T_2{\Delta }_0$ as a function of tunneling gap $2{\Delta }_0$ in ${\mathrm{Fe}}_8$, at the indicated $T$. Thin lines: ${\gamma }_{\varphi }^{\mathrm{vV}}$ arising from pair-flip processes, Eq. (4a). We omit ${\gamma }_{\varphi }^{\mathrm{vV}}$ at $T=0.05\mathrm{K}\ll U_d/k_B$. Thick lines: ${\gamma }_{\varphi }^m$ from magnon scattering, Eq. (5). The gap between the ${\gamma }_{\varphi }^{\mathrm{vV}}$ and ${\gamma }_{\varphi }^m$ lines is the crossover region between the validity of the two methods. The dashed and dotted lines show, respectively, the phonon [${\gamma }_{\varphi }^{\mathrm{ph}}$, Eq. (7)] and nuclear (${\gamma }_{\varphi }^{\mathrm{NS}}$) decoherence rates at $T=0.05\mathrm{K}$. The arrow indicates the optimal operation point of the ${\mathrm{Fe}}_8$ spin qubit at $T=0.05\mathrm{K}$. Inset: ${\gamma }_{\varphi }^{\mathrm{vV}}$ and ${\gamma }_{\varphi }^m$ for an isotropic spin-$1/2$ on the same ${\mathrm{Fe}}_8$ lattice, as a function of the Zeeman gap $g_e{\mu }_B{\mu }_0{H}_y$.