Concatenated continuous driving of silicon qubit by amplitude and phase modulation

The rate of coherence loss is lower for a qubit under the Rabi drive than a freely evolving qubit $T_{2}^{\rm{Rabi}}>T_{2}^*$. Building on this principle, concatenated continuous driving (CCD) keeps the qubit under continuous drive to suppress noise and manipulate dressed states by either phase or amplitude modulation. In this work, we propose a variant of CCD which simultaneously modulates both the amplitude and phase of the driving field to generate a circularly polarized field in the rotating frame of the carrier frequency. This circular-modulated CCD(CMCCD) cancels the counter-rotating term in the second rotating frame, eliminating a systematic pulse-area error that arises from an imperfect rotating wave approximation for fast gates. Numerical simulations demonstrate that the proposed CMCCD achieves higher gate fidelity than conventional CCD schemes. We further implement and compare different CCD protocols using an electron spin-qubit in an isotopically purified $^{28}$Si-MOS quantum dot and evaluate its robustness by applying static detuning and Rabi frequency errors. The robustness is significantly improved compared with the standard Rabi drive, showing the effectiveness of this scheme for qubit arrays with variation in qubit frequency, coupling to the Rabi drive, and low-frequency noise. The proposed scheme can be applied to various physical systems, including trapped atoms, cold atoms, superconducting qubits, and NV centers.