Impact of valley phase and splitting on readout of silicon spin qubits

We investigate the effect of the valley degree of freedom on Pauli-spin blockade readout of spin qubits in silicon. The valley splitting energy sets the singlet-triplet splitting and thereby constrains the detuning range. The valley phase difference controls the relative strength of the intra- and intervalley tunnel couplings, which, in the proposed Pauli-spin blockade readout scheme, couple singlets and polarized triplets, respectively. We find that high conversion fidelity is possible for a wide range of phase differences, while taking into account experimentally observed valley splittings and tunnel couplings. We also show that the control of the valley splitting together with the optimization of the readout detuning can compensate the effect of the valley phase difference. To increase the measurement fidelity and extend the relaxation time we propose a latching protocol that requires a triple quantum dot and exploits weak long-range tunnel coupling. These opportunities are promising for scaling spin qubit systems and improving qubit readout fidelity.

Figure 1

(a) Spin-valley single-particle energy levels of a silicon double quantum dot. The valley ground states are shown in red, while the excited valley states, separated by the dot-dependent valley splitting $E_v^{L\left(R\right)}$, are in blue. A dot- and valley-dependent Zeeman energy (e.g., $E_Z^{L,-}$) splits the spin states. The constant-color arrows represent the intravalley tunnel coupling $t_{\pm \pm }$, while intervalley coupling $t_{\pm \mp }$ arrows have a color gradient. (b) Top: Cross section of a schematic device. The confinement gate (C) defines the dots, plunger gates (G) accumulate the electrons and control the out-of-plane electric field, while the barrier gates (B) tune the tunnel coupling $t$. Bottom: Schematic stability diagram of a double quantum dot. Green and orange lines mark the dot-lead transitions. Interdot intervalley tunneling occurs along the dashed line, separated from the ground state line by the right dot valley splitting $E_v^R$.

Figure 2

(a) Energy levels of a multivalley double quantum dot with $E_v>E_Z$. We distinguish three separate branches of energy states: the valley ground states (red), the valley excited states (blue), and the valley mixed states (light blue and red). The difference in valley splitting causes the splitting between the $(+,-)$ and $(-,+)$ subbranches. The simulation parameters were $U_o^R=30\mathrm{meV}, t=1.5$ GHz, $\mathrm{\Delta }\varphi =\pi /2, E_v^{R\left(L\right)}=105$ (100) $\mu \mathrm{eV}, E_o^R=10 \mathrm{meV}$, and $E_Z^{R\left(L\right),-}=28.65\left(28.5\right)\mu \mathrm{eV}$. The two-spin states have a similar order for the three branches and are shown for the lowest branch only. (b) Zoom-in at the intravalley anticrossing. Increasing the phase difference from 0 to $\pi$ changes the tunneling from pure intravalley to pure intervalley. The light brown rectangle highlights the high-fidelity detuning range. (c) Schematic of Pauli-spin blockade readout sequence. The states used in the readout protocol are shown with the same colors as in (d). The double quantum dot is initialized either in $|\uparrow ,\downarrow \rangle$ (in blue) or $|\downarrow ,\downarrow \rangle$ (in red). The detuning is consequently changed linearly with an adiabatic pulse. Here $E_v^{R\left(L\right)}=300\left(305\right)\mu \mathrm{eV}$ and $\delta E_Z^{-}=20.7$ neV. (d) Results of time evolution simulations, using the same parameters as in (c). Inside the high-fidelity region $F$ is higher than 99.9%. The oscillations at negative detuning in the top panel are the fingerprint of a singlet (1,1). The pulse starts at the symmetry point and the pulse duration is set to achieve a high degree of adiabaticity (see Appendix pp2 for more details).

Figure 3

Fidelity obtained by pulsing from $\epsilon =U^R-1$ meV to $U^R+E_v^R$ in $1\mu \mathrm{s}$ with $\mathrm{\Delta }t=1$ ps, $t=1.5$ GHz, and $\delta E_Z^{-}=5$ MHz, for a range of experimentally achieved valley splitting energies (here $E_v^R=E_v^L$). Contour lines are shown in white. The decrease in $F$ for $\mathrm{\Delta }\varphi$ approaching $\pi$ (i.e., $t_{\pm \pm }\to 0$) is caused by an increase in diabaticity, due to the constant pulse duration, absent in adiabatic evolution (red). For each point of the map, we have plotted the maximum achievable fidelity by taking the optimal detuning.

Figure 4

Fidelity obtained by time evolution simulations (color map with white contour lines) and perfectly adiabatic pulses (red). The difference between the detuning position of maximum fidelity obtained from time evolution simulations (dots) and adiabatic pulses (dashed line) is due to the finite speed of the pulse. As a consequence of the chosen parameters (e.g., $\delta E_Z^{-}=5$ MHz, $E_v^R=300\mu \mathrm{eV}, t=1.5$ GHz, and pulsing from $U^R-2E_v^R$ to $U^R+2E_v^R$) a gap where $F<99.9\%$ opens for intermediate $\mathrm{\Delta }\varphi$. The maximum fidelity has a nonmonotonic dependence on $\mathrm{\Delta }\varphi$, visualized here by the color map of the dots; a minimum is observed at $\pi /2$ (light green), a local maximum at $\pi$ (blue), and the maximum at 0 (dark blue).

Figure 5

Fidelity obtained by time evolution simulations (color map with white contour lines) and adiabatic pulses (red). A realistic value of 0.3 meV enables reaching $F>99\%$ for $t$ up to 4.5 GHz in a wide range of phase differences (here up to $\mathrm{\Delta }\varphi =0.8\pi$). The reduction in fidelity at low $t$ is caused by higher diabaticity of the pulse. In the top right panel it prevents reaching $F>99.9\%$. As in Fig. 4 $\delta E_Z^{-}=5$ MHz and the pulse extremes are $U^R\pm 2E_v^R$.

Figure 6

(a) Schematic of a triple-dot device with negligibly weak long-range coupling. (b) Left panel: Hysteretic stability diagram of a tripled quantum dot with the readout pulse scheme. The first step [from (I) to (II)] is a standard double-quantum-dot Pauli-spin blockade readout pulse. In the second step [from (II) to (III)], charge moves from the target to the left ancilla qubit only if the target qubit is initially spin blocked (${\mathrm{III}}^{\prime }$). Transitions between nearest-neighboring dots are denoted by two-color interdot transition lines. The edges of the hysteresis regions are marked by three-color lines. The line color reflects the involved dots. Right panels: Energy level diagrams showing the triple-dot occupation at (I), (II), (III), and (${\mathrm{III}}^{\prime }$) positions depending on the initial spin state. (c) Left panel: The readout is performed by oscillating the middle and left dot energy levels forcing the system to oscillates between the (1,0,1) and (0,1,1) charge states. Right panel: The charge state mixing leads to a capacitive term measured via rf gate-based dispersive readout. For simplicity, the charge occupation of the right dot is dropped since it remains constant.

Figure 7

(a) The speed-normalized “angular velocity” ${\tilde{\alpha }}_1$ of the ground state as a function of the detuning. At negative detuning it reduces to ${\tilde{\alpha }}_{12}$, while at positive to ${\tilde{\alpha }}_{13}$. Here $t=1.5$ GHz, $\delta E_Z=10$ MHz, and $\mathrm{\Delta }\varphi =0$. The peak at the anticrossing corresponds to the contribution of a pure two-level system, while the broad peak is due to the fact that $|\uparrow ,\downarrow \rangle$ and $|\downarrow ,\uparrow \rangle$ are coupled to each other only via the singlet (0,2). (b) The “99.9% adiabatic probability” local speed as a function of detuning. The lower and upper horizontal lines are the global speed obtained using Eq. (B2) and the Landau-Zener formula, respectively. The middle one stems from the global 99.8% probability speed.