Decoherence of an exchange qubit by hyperfine interaction

We study three-electron-spin decoherence in a semiconductor triple quantum dot with a linear geometry. The three electron spins are coupled by exchange interactions $J_{12}$ and $J_{23}$, and we clarify inhomogeneous and homogeneous dephasing dynamics for a logical qubit encoded in the $\left(S=1/2,S^z=1/2\right)$ subspace. We first justify that qubit leakage via the fluctuating Overhauser field can be effectively suppressed by sufficiently large Zeeman and exchange splittings. In both $J_{12}=J_{23}$ and $J_{12}\ne J_{23}$ regimes, we construct an effective pure dephasing Hamiltonian with the Zeeman splitting $E_Z\gg J_{12},J_{23}$. Both effective Hamiltonians have the same order of magnitude as that for a single-spin qubit, and the relevant dephasing time scales ($T_2^{*},T^{\mathrm{FID}}$, etc.) are of the same order as those for a single spin. We provide estimates of the dynamics of three-spin free induction decay, the decay of a Hahn spin echo, and the decay of echoes from a CPMG pulse sequence for GaAs quantum dots.

Figure 1

Schematics of a linear triple dot, with one electron per dot. The interdot spin couplings are of the Heisenberg exchange type, labeled by $J_{12},J_{23}$, and $J_{13}$. In several recent experimental implementations [22, 25, 26], $J_{13}\simeq 0$.

Figure 2

Energy spectrum of three linearly coupled electron spins in a uniform magnetic field, following the labeling scheme indicated in Table 1, for two limiting cases: (a) $J\gg E_Z$, and (b) $E_Z\gg J$. For GaAs quantum dots, with a negative $g$ factor $\left(g=-0.44\right)$, $|Q_{3/2}\rangle$ is the ground state.

Figure 3

Possible qubit leakage dynamics $Q_{D^{\prime }}\left(t\right)$ in GaAs (red solid line). Both curves assume the same field gradients. (See the main text for the definitions of $\mu ,{\theta }_{12}$, and ${\theta }_{32}$.) The blue dashed curve corresponds to the case of a mean gradient ${\overline{B}}_{2\overline{1}2}^z\ne 0$ [40], while the red curve corresponds to ${\overline{B}}_{2\overline{1}2}^z=0$.

Figure 4

The coherence loss $\{1-\text{Re}\left[W^{\text{nFID}}\left(t\right)\right]\}$ (solid lines) and $\{1-\text{Re}\left[W^{\text{s,nFID}}\left(t\right)\right]\}$ (dotted lines) as a function of time, in log-log scale, based on Eqs. (21) and (22) for GaAs at (a) $B=0.1$ T and (b) $B=0.5$ T.

Figure 5

The nFID coherence function as a function of time, based on Eqs. (21) and (22) for GaAs. (a) $|W^{\text{nFID}}\left(t\right)|$ (black solid line), $\text{Re}\left[W^{\text{nFID}}\left(t\right)\right]$ (red dashed line), and $\text{Im}\left[W^{\text{nFID}}\left(t\right)\right]$ (green dotted line) at $B=0.5$ T with $N={10}^6$. (b) The comparison of $\text{Re}\left[W\right(t\left)\right]$ between the encoding and single-spin qubit made at $B=0.1$ T (blue solid line for the nFID and blue dotted line for the single-spin nFID) and at $B=0.5$ T (red dashed line for the nFID and red dash-dotted line for the single-spin nFID), with $N={10}^6$.

Figure 6

The coherence functions with the application of (a) Hahn echo and (b) 2-pulse CPMG sequence, for GaAs with $N={10}^6$ at $B=0.1$ T (blue solid line), $B=0.5$ T (red dashed line), and $B=1$ T (green dotted line). In both panels, $N={10}^6$.

Figure 7

The coherence loss $\{1-\text{Re}[W\left(t\right)\left]\right\}$ for a GaAs triple dot at $B=0.1$ T in the case of nFID (black solid line, with the smallest slope) and with the application of a Hahn echo pulse (red solid line) and 2-pulse CPMG sequence (green solid line, with the largest slope). The dashed lines correspond to the relevant coherence loss for a single-spin qubit. We set $N={10}^6$ for all data.

Figure 8

$\text{Re}\left[W^{\text{nFID}}\left(t\right)\right]$ with $\Delta =0$ (black solid line), $\Delta =-J/2$ (red dashed line), and $\Delta =-0.73J$ (blue dotted line) at $B=0.5$ T for GaAs with $N={10}^6$. These $\Delta$ values correspond to the cases of $J_{23}=J_{12},J_{23}=J_{12}/2$, and $J_{23}=0.27J_{12}$. The inset shows $|W^{\text{nFID}}\left(t\right)|$ at short times for these three $\Delta$ values, which are very similar.