Leakage and dephasing in ${}^{28}\mathrm{Si}$-based exchange-only spin qubits

Exchange-only spin qubits hosted in ${}^{28}\mathrm{Si}$-based triple quantum dots do not suffer from decoherence caused by randomly fluctuating nuclear-spin ensembles and can be relatively robust against electrical noise when operated at a sweet spot. Remaining sources of decoherence are qubit relaxation, leakage out of the qubit subspace, and dephasing due to residual effects of charge noise, the latter two of which are the focus of this work. We investigate spin-orbit-mediated leakage rates to the three-spin ground state accompanied by virtual (i) tunneling, (ii) orbital excitation, and (iii) valley excitation of an electron. We find different power-law dependencies on the applied magnetic field $B$ for the three mechanisms as well as for the two leakage rates, ranging from $\propto B^5$ to $\propto B^{11}$, and identify the sweet spot as a point of minimal leakage. We also revisit the role of electrical noise at the sweet spot and show that it causes a decay of coherent qubit oscillations that follows a power law $\propto 1/t$ (as opposed to the more common exponential decay) and introduces a $\pi /2$ phase shift.

Figure 1

(a) Reference frame of the system. The quantum dots (dashed gray circles) lie in the $xy$ plane. An in-plane magnetic (Zeeman) field $\mathbf{B}$ is applied at an angle $\vartheta$ with the interdot axis, which, in turn, is at an angle $\chi$ with the crystallographic [100] direction. (b) Charge stability diagram, showing the charge ground state of the electrostatic part of the Hamiltonian (1) as a function of $\epsilon$ and $V_m$, using $U_c=0.3U$ and $V_2=2U_c-V_1-V_3$. (c) Qualitative sketch of the lower part of the spectrum of (1) as a function of $\epsilon$ in the regions labeled “RX” in (b), where we assumed $t_{12}=t_{23}$. The gray arrows indicate the leakage processes investigated here. In this plot all variables are in arbitrary units.

Figure 2

Spin-flip-tunneling-assisted leakage rates out of the qubit space across the whole (1,1,1) charge region, from (a) $|1\rangle$ and (b) $|0\rangle$ to $|Q_2\rangle$ in units of ${\gamma }_{\text{ex}}\equiv d^2{\mathrm{\Xi }}_u^2{E}_{\text{Z}}^3/l_{\text{so}}^2{\hslash }^4{v}_t^5{\rho }_0$ (see text for the choice of parameters). (c) ${\mathrm{\Gamma }}_{1,0}$ as a function of $V_m$ for $\epsilon =0$ (circles and crosses), i.e., along the vertical dashed lines in (a) and (b). Solid lines show the analytical results from Eqs. (9) and (10). (d) ${\mathrm{\Gamma }}_{1,0}$ as a function of $\epsilon$ for $V_m=-0.3U$, i.e., along the horizontal dashed lines in (a) and (b).

Figure 3

Numerically calculated time-dependent return probability ${\left|\langle +\left|\psi \right(t)\rangle \right|}^2$ after initializing in $|+\rangle$, averaged over ${10}^5$ different sets of $\delta V_{1,2,3}$ taken from a normal distribution with $\xi =5\mu \mathrm{eV}$ (thin blue line). The thick black line shows the envelope function of the oscillations as predicted by Eq. (26), and the dashed red line shows the best fit obtainable assuming an exponential envelope of the form $\frac{1}{2}+\frac{1}{2}{e}^{-t^2/{\tilde{T}}_{\phi }^2}$.