Conditional Control of Donor Nuclear Spins in Silicon Using Stark Shifts

Abstract

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.The manipulation of donor spins in silicon is mainly realised by the application of resonant AC magnetic fields. These can be used to globally control a large ensemble of spins with high fidelity [1], and also to control a single spin [2]. However, applying local AC magnetic fields to donors in an interacting array is practically challenging, especially considering the number of high frequency, high power microwave lines in a mature device. For such reasons, the majority of scalable quantum computing architectures based on donors in silicon [3][4][5], combine globally applied AC magnetic fields with the ability to electrically tune the donor spin, via the Stark effect which primarily modulates the hyperfine interaction between the electron and nuclear spin.The Stark effect arises from a perturbation of the electron wave function as it is pulled away from the nucleus, mixing its ground state energy level with the higher orbital excited states [6]. Following the original proposal by Kane [3], several theoretical studies have examined Stark-tuning of both donor spins [6][7][8][9], while experiments have focused on measuring the Stark shift parameters using electron spin resonance (ESR) of the donor [10][11][12]. These measurements showed a typical Stark-shift induced change in the hyperfine coupling on the order of kHz (for electric fields around 0.1 V/µm). This falls well within the ESR linewidth (12 MHz in natural silicon, or typically 100 kHz in silicon-28 limited by the magnetic field homogeneity), making it impossible to Stark-shift the electron spin by more than a linewidth.The nuclear spin, on the other hand, typically has much smaller spin resonance linewidth compared to the electron spin (γ n /γ e =∼ 4 × 10 −4 , where γ n/e is the gyromagnetic ratio of the nuclear/electron spin) and yet the effect of the Stark-tuned hyperfine shift on the nuclear spin resonance frequency is nearly the same as for the electron spin. As a consequence, using the Stark effect to tune spins in and out of resonance with globally applied fields becomes more readily achievable. Furthermore, the coherence times of the nuclear spin can reach...

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