High-Fidelity Raman Spin-Dependent Kicks in the Presence of Micromotion

TL;DR

The paper tackles fast, high-fidelity two-qubit gates in trapped-ion processors by implementing spin-dependent kicks (SDKs) with a continuous-wave (CW) Raman scheme driven by nanosecond pulses, designed to be robust to intrinsic micromotion. It introduces a general amplitude-modulated CW approach with a tunable beat frequency, and develops a three-stage analytical and numerical framework to isolate micromotion, suppress backward kicks, and identify optimal RF phase and frequency. A key result is the micromotion phase-matching condition $\omega_\mathrm{R} (t_0 + \frac{\tau}{2}) + \phi_\mathrm{R} = (2n+1)\frac{\pi}{2}$ under the fast-SDK limit, which minimizes micromotion-induced errors, and simulations show infidelities as low as $10^{-9}$ in the absence of micromotion and below $5\times 10^{-5}$ with micromotion. The work also compares CW and pulsed schemes, finds comparable high-fidelity performance with substantially lower peak powers for CW, and establishes a practical path toward sub-trap-period, high-fidelity SDK-based entangling gates and scalable trapped-ion quantum processors.

Abstract

We propose high-fidelity single-qubit spin-dependent kicks (SDKs) for trapped ions using nanosecond Raman pulses via amplitude modulation of a continuous-wave laser with a tunable beat frequency. We develop a general method for maintaining SDK performance in the presence of micromotion by identifying optimal choices of the RF phase and frequency that suppress unwanted backward kicks. The proposed scheme enables SDK infidelities as low as $10^{-9}$ in the absence of micromotion, and below $10^{-5}$ with micromotion. This study lays the foundation for the realization of sub-trap-period and high-fidelity two-qubit gates based on SDKs.

Figures (4)

• Figure 1: (a) Schematic of the Raman spin-dependent kick (SDK). Two counterpropagating laser beams ($E_1$ and $E_2$) with lin $\perp$ lin polarization interact with a single ion in a linear ion chain aligned along the $x$ axis (dashed line). The ions are confined in a linear Paul trap with RF (shaded) and DC (not shown) electrodes. The quantization axis is set by a static magnetic field $B$ along the $z$ axis. (b) A generic level diagram of the Raman transition. An ion initially in the spin ground state $\ket{0}$ experiences a forward kick when it absorbs a photon from $E_1$ and emits into $E_2$, gaining momentum in the $+z$ direction. Here $\Delta$ denotes the detuning of the Raman beams from the excited states, $\omega_\mathrm{a}$ is the qubit frequency splitting, and $\Delta\omega$ is the frequency difference between the two Raman beams.
• Figure 2: Population dynamics and pulse envelopes of continuous-wave (CW) and pulsed SDKs as a function of time without including micromotion. (a) For a $5~\mathrm{ns}$ constant-amplitude pulse (bottom), both analytical and simulation calculations (top) yield an infidelity of $1 - \mathcal{F} = 1.9 \times 10^{-5}$ at the exact resonant condition $\Delta\omega = \omega_\mathrm{a}$. (b) For a $5~\mathrm{ns}$ sine-shaped pulse (bottom), the infidelity is reduced to $1.4 \times 10^{-9}$ when the Raman beat frequency is optimized to $\Delta\omega = \qty(1 + 5 \times 10^{-5}) \omega_\mathrm{a}$. Both CW SDKs in (a) and (b) employ a total pulse area of $\theta = \pi$ and a peak Rabi frequency $\Omega_\mathrm{max} \gtrsim 2\pi \times 100~\mathrm{MHz}$. (c) Pulsed SDKs can achieve comparable $10^{-9}$ infidelity by approximately sampling the sine envelope and optimizing the repetition rate $\omega_\mathrm{rep}$ and Raman beat $\Delta\omega$. For a sequence of ten $10~\mathrm{ps}$ pulses of optimized amplitudes within $5.6~\mathrm{ns}$ (bottom), the infidelity of a single pulsed SDK is $3.0 \times 10^{-9}$, with $\Delta\omega = 0.027 \omega_\mathrm{a}$ and $\omega_\mathrm{rep} = 2\pi \times 1.9~\mathrm{GHz}$. The corresponding peak Rabi frequency exceeds $2\pi \times 8~\mathrm{GHz}$.
• Figure 3: Infidelity landscape of sine-shaped CW SDKs as a function of the RF frequency $\omega_\mathrm{R}$ and RF phase $\phi_\mathrm{R}$ using a full quantum simulation that incorporates both secular motion and micromotion effects. The regions where micromotion effects are suppressed agree with the analytical prediction under the fast SDK approximation (black dotted line). White boxed regions indicate parameter regimes yielding SDK infidelity below $5 \times 10^{-5}$ even in the presence of micromotion. The parameters used are identical to those in Fig. \ref{['fig: 2']}(b), with additional realistic Mathieu parameters in the $z$ direction, $a_z = 0$, $q_z = 0.15$, and the Lamb-Dicke parameter $\eta = 0.1$.
• Figure 4: Infidelity of a sine-shaped CW SDK as a function of the errors in pulse area $\delta\theta$ and Raman beat frequency $\delta\qty[\Delta\omega]$, respectively. We consider up to $1\%$ error in pulse area from $\theta = \pi$ and up to $0.1\%$ error in Raman beat from $\Delta\omega = \qty(1 + 5 \times 10^{-5}) \omega_\mathrm{a}$. The infidelity remains below $10^{-3}$ across the pulse area sweep and below $10^{-2}$ across the Raman beat sweep. These calculations assume one set of optimal RF parameters identified in Fig. \ref{['fig: 3']} with $\omega_\mathrm{R} = 2\pi \times 33.64~\mathrm{MHz}$ and $\phi_\mathrm{R} = 2\pi \times 0.17$.