Concatenated continuous driving of silicon qubit by amplitude and phase modulation

TL;DR

This work introduces circular-modulated CCD (CMCCD), a dual amplitude-phase modulation scheme that generates a circularly polarized drive in the first rotating frame to cancel the counter-rotating term in the second rotating frame, mitigating systematic pulse-area errors from imperfect RWA. The authors provide a general CMCCD Hamiltonian with modulation parameters and validate the approach experimentally using an isotopically purified $^{28}$Si MOS spin qubit, observing chevron patterns and ladder-like robustness in detuning and Rabi-error tests. Randomized benchmarking indicates enhanced robustness to detuning and, to a lesser extent, to Rabi errors, though base gate fidelity is limited by high-frequency noise; CMCCD nonetheless offers a path to robust, scalable qubit control in arrays with drive and qubit variability. The framework is applicable across multiple qubit platforms, and removing the dependence on the second RWA could enable higher-fidelity gates in systems such as trapped atoms, superconducting qubits, and NV centers. $T_2$-type metrics ($T_{2}^{\rm{Rabi}}$, $T_2^*$) and gate performance are quantified to illustrate the trade-offs between robustness and operation speed under CMCCD.

Abstract

The rate of coherence loss is lower for a qubit under Rabi drive compared to 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 new 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 (CM)-CCD 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 to standard Rabi-drive, showing the effectiveness of this scheme for qubit arrays with variation in qubit frequency, coupling to 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.

Figures (10)

• Figure 1: Polarization of drive field for different CCD schemes. To control the qubit, a drive field resonant with the Larmor frequency $\omega_{\rm{L}}$ is applied. In most cases, the applied drive field is linearly polarized, which is a superposition of co and counter rotating circularly polarized fields. In the first rotating frame, the counter-rotating term results in a second harmonic field that can be neglected using RWA, since $\Omega \ll \omega_L$. For a pure amplitude or phase modulated drive, in the first rotating frame the modulation results in a linearly polarized field along x, y, respectively. In the second rotating frame, the RWA is not so valid as typically $\epsilon_{\rm{m}} < \Omega$, and the counter-rotating term results in a systematic error in the gate. In this work, we propose CMCCD, where the amplitude and phase of the drive field are equally modulated, generating circular polarized field in the first rotating frame. Then in the second-rotating frame, there is no second-harmonic term reducing the gate error.
• Figure 2: Quantum state trajectories of the bare qubit and CCD-protected qubits under (a) detuning error and (b) Rabi error. Modulation parameters are set to $\epsilon_{\rm{m}}=\Omega_0/4$, $\theta_{\rm{m}} = \pi/2$, and $\phi_{\rm{mw}} = 0$. Red lines trace the evolution of the quantum states over a total rotation angle of 20$\pi$, while blue markers indicate the positions corresponding to each $\pi/2$ rotation interval. Note that Bloch spheres for CCD-protected qubits are shown in the second rotating frame, while that for the bare qubit is shown in the first rotating frame. For the case of zero detuning or Rabi error, the AMCCD and PMCCD give a spread in the rotations resulting from integer $\pi/2$-gates, whereas both the bare qubit and CMCCD give perfect integer $\pi/2$-gates. When a static error is introduced, all CCD schemes have a low spread in the state-vector at integer $\pi/2$-gates, whereas the bare qubit suffers a large spread in outcomes.
• Figure 3: Comparison of state infidelity for $Y_\pi$ gate as a function of (a) detuning error and (b) Rabi error. Modulation parameters are set to $\epsilon_{\rm{m}}=\Omega_0/4$, $\theta_{\rm{m}} = \pi/2$, and $\phi_{\rm{mw}} = 0$. The $Y_\pi$ gate performs $Y$ axis rotation from $|0"\rangle$ to $|1"\rangle$ in the CCD schemes with a duration of $\pi/\epsilon_{\rm{m}}$, and from $|0'\rangle$ to $|1'\rangle$ in the bare qubit with a duration of $\pi/\Omega_0$. The state infidelity is defined as $1-|\langle 1"|Y_\pi|0"\rangle|^2$ for CCD schemes and $1-|\langle 1'|Y_\pi|0'\rangle|^2$ for bare qubit. The CMCCD combines the accuracy of a bare Rabi gate at low error, with an improved tolerance to larger errors.
• Figure 4: (a) The top-view scanning electron microscope (SEM) image and (b) its schematic illustration, including a cross-section view along the white dashed line. We operate the electron spin-qubit confined in the left quantum dot (Q1).
• Figure 5: Chevron pattern of different CCD schemes, (a) AMCCD, (b) PMCCD, and (c) CMCCD measured at $\Omega_0 = 2\pi\times$ 3.6 MHz. The left panels show the simulation, and the right panels show the corresponding experimental results. We measure the chevron pattern by varying the $\epsilon_{\rm{m}}$ from 0 to $\Omega_0$. When $\epsilon_{\rm{m}} = 0$, the condition is the same as the bare qubit. As $\epsilon_{\rm{m}}$ increases, the chevron pattern gradually transforms into a ladder-like structure, indicating enhanced robustness against detuning. With further increase toward $\Omega_0$, the ladder-like shape begins to distort, and the robustness correspondingly decreases.