Author Correction: Ultrafast hole spin qubit with gate-tunable spin–orbit switch functionality

Correction to: Nature Nanotechnology https://doi.org/10.1038/s41565-020-00828-6, published online 11 January 2021.

In the version of this Article originally published, the dependence of the Rabi frequency of the hole spin qubit on the g-factor was stated incorrectly. We wrote that the Rabi frequency is proportional to the g-factor perpendicular to the external magnetic field (g_⊥), but, in fact, it is the g-factor parallel to the external magnetic field (g_||) as shown in ref. 25 (Golovach, V. N, Borhani, M. & Loss, D.). Thus, the following sentence has been amended from “This effective field B_eff drives the Rabi oscillations, with Rabi frequency f_Rabi = g_⊥μ_BB_eff(t)/2h, with g_⊥ the g-factor along the direction of B_eff and thus perpendicular to B_ext, and h Planck’s constant.” to “This effective field B_eff drives the Rabi oscillations, with Rabi frequency f_Rabi = g_||μ_BB_eff(t)/2h, with g_|| the g-factor along the direction of B_ext, and h Planck’s constant.” Accordingly, the phrase “the g-factor along B_ext,” has been deleted from the sentence starting “Here, f_MW and P_MW are kept fixed...”.

When measuring at a constant value of the external magnetic field B_ext, f_Rabi and l_so still depend on the g-factor (as in the original version). However, when measuring at a fixed resonant microwave frequency f_MW = g_||µ_BB_ext/h, any change of the g-factor is compensated for by an inversely proportional change of B_ext, and the apparent g-factor dependence vanishes from the equation of the Rabi frequency, thus making the observation of a gate-voltage-dependent spin–orbit coupling strength more robust. This is now reflected in the corrected Article, as described below.

The following changes have been made: (1) in the sentence beginning “We extracted a spin–orbit length...”, ‘3 nm’ has been changed to ‘4 nm’; (2) in the sentence beginning “For SOI-mediated spin rotations”, ‘g_⊥’ has been changed to ‘g_||’; (3) in the sentence beginning “We found remarkably short...”, ‘28 nm’ and ‘3 nm’ have been changed to ‘23 nm’ and ‘4 nm’, respectively. Also, the following paragraph has been amended from

“Therefore, the Rabi frequency depends on l_so, the a.c. electric field amplitude E_MW(t) created through the periodic gate voltage modulation, B_ext, the quantum dot confinement Δ_orb and the g-factor. Despite an observed change of g_∥ with V_M (refs. ^18,43) by a factor of 1.5 (Fig. 3c), the effect on the Rabi frequency was small: at a constant Zeeman energy, we can write f_Rabi ∝ f_MW g_⊥/g_∥. Hence, if the g-factor anisotropy g_⊥/g_∥ is only weakly affected by gate voltages, as observed here (Supplementary Fig. 3d,e), then the Rabi frequency change is correspondingly small.”

to

“For measurements at a fixed microwave frequency f_MW, as done here, the apparent g-factor dependence vanishes and the Rabi frequency depends only on the spin–orbit length l_so, the quantum dot orbital level splitting Δ_orb and the a.c. electric field E_MW(t) created through the periodic gate voltage modulation (see Supplementary equation (1).”

Subsequent to the above change, reference 43 now becomes reference 45, and has now been cited as follows: “Moreover, the variation of g_∥ (ref. 45) in Fig. 3c...” References 44 and 45 have been renumbered to 43 and 44, respectively.

Also, in the Supplementary Information, the correction to the g-factor results in its elimination from Supplementary equation (1), making the Rabi frequency independent of the g-factor for our measurements where the microwave frequency was held fixed, as previously discussed; the corrected equation is:

$$\it f_{{\mathrm{Rabi}}} = f_{{\mathrm{MW}}} \cdot \frac{{l_{{\mathrm{dot}}}}}{{l_{{\mathrm{so}}}}} \cdot \frac{{e\left| {{\mathbf{E}}_{{\mathrm{MW}}}\left( t \right)} \right|l_{{\mathrm{dot}}}}}{{{{\Delta }}_{{\mathrm{orb}}}}}.$$

The paragraph following Supplementary equation (1) has been updated to read “Here, we used f_Rabi = g_||μ_B|B_eff(t)|/(2h) with |B_eff(t)| as defined in Eq. 1 of the main text, and |B_ext(t)| is converted to f_MW via the resonance condition g_||μ_B|B_ext(t)| = hf_MW, since f_MW is held fixed. In order to stay on resonance, we compensate the variation of g_|| (shown in Figure 3 c of the main text) with a proportional change of |B_ext(t)| in the opposite direction, effectively making f_Rabi independent of g_||. We note that $\it l_{dot} = \hbar {\mathrm{/}}\sqrt {{{\Delta }}_{{\mathrm{orb}}}m_{{\mathrm{eff}}}}$ with the effective hole mass m_eff.”

The related discussion in the Supplementary Information has also been updated as follows: (1) the paragraph beginning “Lastly, the Rabi frequency...” has been changed to “In conclusion, we estimate the contributions from changes in electric feld amplitude (no change) and orbital level splitting to account for a total of not more than a 1.5-fold change of the Rabi frequency. The very large change of the Rabi frequency by a factor of 7 can therefore be attributed mainly to an electrically tunable spin-orbit interaction, as expected from a direct Rashba type of spin-orbit interaction”; (2) the sentence beginning “In order to see the effect of...” has been changed to “We also study the evolution of the g-factor in other directions as a function of V_M.”; (3) the sentence beginning “Figure S3 c shows these...” has been changed to “Figure S3 c shows g-factors in two directions that are perpendicular to the orientation of B_ext as function of V_M, extracted from measurements of the Larmor frequency.”; (4) the sentences beginning “These two directions...” and “While these changes in anisotropy...” have been deleted; (5) in the sentence beginning “The corresponding values...”, “|E(t)|, g, Δ_orb” has been changed to “|E(t)|, Δ_orb”; (6) the sentence beginning “For the g-factor...” has been changed to “For Δ_orb we take the values shown in Figure S3 b, which were obtained from measurements of bias triangles”; (6) in the sentence beginning “For the measurement with V_M...” ‘3 nm (20 nm)’ has been changed to ‘4 nm (26 nm); (7) in the Figure S4 caption, “measured values of g, Δ_orb” has been changed to “measured values of Δ_orb”.

The conclusions of the Article are unaffected.

The online versions of the Article have been amended and the Supplementary Information file replaced.