Conditional Control of Donor Nuclear Spins in Silicon Using Stark Shifts

Electric fields can be used to tune donor spins in silicon using the Stark shift, whereby the donor electron wave function is displaced by an electric field, modifying the hyperfine coupling between the electron spin and the donor nuclear spin. We present a technique based on dynamic decoupling of the electron spin to accurately determine the Stark shift, and illustrate this using antimony donors in isotopically purified silicon-28. We then demonstrate two different methods to use a dc electric field combined with an applied resonant radio-frequency (rf) field to conditionally control donor nuclear spins. The first method combines an electric-field induced conditional phase gate with standard rf pulses, and the second one simply detunes the spins off resonance. Finally, we consider different strategies to reduce the effect of electric field inhomogeneities and obtain above 90% process fidelities.

Figure 1

Measurement of the Stark shift in ${}^{28}\mathrm{Si}:{}^{121}\mathrm{Sb}$ using dynamical decoupling. (a) Cartoon of an antimony donor under the effect of an electric field. In light blue, the electron wave function is shifted from the donor nucleus. (b) Uhrig dynamical decoupling (UDD) sequence with four refocusing pulses. As each $\pi$ pulse reverses the phase acquisition (see signs in sequence), the dc electric field is applied in between alternating pairs of $\pi$ pulses. Using bipolar (positive and negative) voltage pulses, the linear Stark shift contribution is eliminated and only the quadratic part remains. The last two ESR pulses were phase cycled to remove any stimulated echoes. (c) Electron spin phase evolution measurement using the $m_I=-5/2$ ESR transition for different electric fields. (d) Fourier transform showing the frequency shift distribution in the sample with a Lorentzian fit.

Figure 2

Electrically controlled nuclear phase gate. (a) Measurement pulse sequence where a Hahn echo is used to refocus the inhomogeneous broadening happening during the application of the voltage pulse. (b) Bloch sphere representation of the nuclear spin rotations: in red, $X$ rotation from the rf pulse. In green, $Z$ rotation from the voltage pulse. (c) The duration of the voltage and rf pulses are independently swept in a 2D map, resulting in spin rotations about the $X$ and $Z$ axes. The measured electron spin echo intensity is $-1$ for an unperturbed nuclear spin, and 0 when the nuclear spin population is fully inverted, as is typical for Davies ENDOR measurements. (d) Projection of the 2D map for ${\tau }_V=0$ (“off”) and 0.2 ms (“on”), showing how the nuclear spin can be made effectively insensitive to the applied rf field for an appropriate duration of the (150 V) voltage pulse.

Figure 3

Electrically tuning the nuclear spin transition frequency. (a) Davies ENDOR sequence with square wave voltage pulses. (b) Bloch sphere representation of the nuclear spin evolution in the rotating frame at the rf frequency which is resonant under a pure quadratic Stark shift. The linear Stark shift component adds additional off-resonant evolution, which can be compensated by applying a square-wave voltage pulse. (c) ENDOR spectra measured with and without 150 V pulses, where ${\tau }_{\text{rf}}=1\mathrm{ms}$ corresponding to a $\pi$ pulse. For a single unipolar voltage pulse, the shifted line is broadened by the linear Stark shift. The shifted ENDOR line can be narrowed by increasing the frequency of a square wave bipolar voltage pulse, up to a point, when it becomes limited by electric field inhomogeneity. (d)–(f) Nuclear spin Rabi oscillations, where: (d) the voltage is constantly applied; (e) the voltage is applied only within the light blue window which starts when the nuclear spin is in an eigenstate; (f) the voltage is applied within the light blue window which starts when the nuclear spin is in a coherent superposition state.

Figure 4

(a) Sequence taken from Ref. [29] used for transferring an electron state to a nuclear state. This is used to produce any input state ($\pm X$, $\pm Y$, $\pm Z$ and identity) due to limited rf phase control in our experiment. The $\pi$ rf gate in the middle is a large bandwidth ($\approx 30\mathrm{kHz}$) and refocuses any dephasing when the voltage is on. (b) Rabi oscillations with and without voltages for an input coherent superposition. The oscillations decay quite fast for negative amplitudes as they correspond to even number of $\pi$ pulses which prevent refocusing of the nuclear coherence. (c) Quantum process tomography ${\chi }_{\exp }$ matrices with voltage on (fidelity compared to a $Y$ rotation) and voltage off (fidelity compared to the identity matrix). Only the real part is shown as the imaginary part is negligible.