Single Qudit Control in $^{87}$Sr via Optical Nuclear Electric Resonance

TL;DR

This work shows that optical nuclear electric resonance (ONER) can be extended from qubit to qudit control in $^{87}$Sr, exploiting the $I=9/2$ nuclear-spin manifold ($d=10$) to perform fast, high-fidelity single-qudit rotations. Using an amplitude-modulated laser on the $^1S_0 ightarrow ^3P_1$ transition and careful laser tuning between excited-state transitions, the authors demonstrate, via Lindblad-master-equation simulations, that π-gate fidelities exceeding $99.9\%$ are achievable for multiple one-level transitions, with nuclear Rabi frequencies in the tens of kHz and gate times in the microsecond regime. The protocol remains robust against realistic fluctuations in laser amplitude, detuning, polarization, modulation period, and magnetic field, and spontaneous scattering is kept negligible under the chosen parameters. The results establish ONER as a practical, scalable route for high-dimensional quantum information processing with neutral atoms, including the potential integration with two-qudit gates and non-destructive readout, and motivate future exploration of high-field Rydberg interactions for entangling operations.

Abstract

Optical nuclear electric resonance (ONER) was recently proposed as a fast and robust single-qubit gate mechanism in $^{87}$Sr. Here, we demonstrate through numerical simulations that ONER can be extended to single-qudit control, addressing multiple one-level hyperfine transitions within the ten-dimensional nuclear-spin manifold. We identify suitable operating regimes and show that ONER enables high-fidelity spin manipulations, with simulated $π$-gate fidelities exceeding 99.9\%, while maintaining coherence under realistic parameter fluctuations. These results establish a proof-of-principle for optical qudit control in $^{87}$Sr and delineate practical parameter ranges for future experiments, highlighting ONER as a promising pathway toward high-dimensional quantum information processing.

Figures (5)

• Figure 1: Schematic illustration of the optical nuclear electric resonance (ONER) protocol applied to the level structure of the (5s$^2$) $^1S_0 \rightarrow$ (5s5p) $^3P_1$ optical transition in $^{87}$Sr for qudit control. Amplitude-modulated laser fields with periods $T_{m_I\leftrightarrow m_I+1}$, frequencies $\omega_{m_I\leftrightarrow m_I+1}$ and electronic Rabi frequency $\Omega_\mathrm{E}/2\pi$ drive the system, resulting in an adiabatically modulated occupation of the electronically excited state. In the $^1S_0$ ground state, the magnetic nuclear spin quantum number $m_I$ remains well-defined. In the $^3P_1$ excited states, the non-zero hyperfine interaction ($Q,A \neq 0$) mixes nuclear spin states. Which states are mixed can be selected via the laser frequency $\omega_{m_I\leftrightarrow m_I+1}$, which is chosen to be in between the respective spin levels in the $m_J = -1$ manifold. The amplitude modulation is adjusted to the specific transition, enabling several hyperfine nuclear spin transitions in the $^1S_0$ ground state.
• Figure 2: (a) Amplitude modulation period scan and corresponding nuclear Rabi frequencies of $m_I \leftrightarrow m_I+1$ transitions. We show the $\pi$-pulse fidelity at the first flip $P_{m_I\leftrightarrow m_I+1}$ for all $\Delta m_I = \pm 1$ transitions at $B_0 = 3000\;\mathrm{G}$, $\Omega_{E,0} / 2\pi = 30\;\mathrm{MHz}$, $\theta_0 = 60\;\mathrm{deg}$ and $\Delta_{m_I \leftrightarrow m_I+1}$ as specified in Equation \ref{['eq: set laser frequency']}. Full transitions occur at different amplitude modulation periods $T_{m_I \leftrightarrow m_I+1}$ for different levels $m_I$. (b) Rabi oscillations of all $\Delta m_I = \pm 1$ at the optimal amplitude modulation period $T_{m_I \leftrightarrow m_I+1}$.
• Figure 3: Amplitude modulation period scan and corresponding nuclear Rabi frequencies. We show the $\pi$-pulse fidelity at the first flip $P_{m_I\leftrightarrow m_I+1}$ for all $\Delta m_I = \pm 1$ transitions at parameters $B_0 = 3000\;\mathrm{G}$, $\Omega_{E,0} / 2\pi = 30\;\mathrm{MHz}$, $\theta_0 = 60\;\mathrm{deg}$ and $\omega_{m_I \leftrightarrow m_I+1}$ as specified in the main text. Full transitions occur at different amplitude modulation periods $T_{m_I \leftrightarrow m_I+1}$ for different levels $m_I$. Moreover, multiple equidistantly spaced transitions occur for each level.
• Figure 4: (a) Amplitude modulation period scan and corresponding nuclear Rabi frequencies. We show the $\pi$-pulse fidelity at the first flip $P_{m_I\leftrightarrow m_I+1}$ for all $\Delta m_I = \pm 1$ transitions at parameters $B_0 = 1000\;\mathrm{G}$, $\Omega_{E,0} / 2\pi = 15\;\mathrm{MHz}$, $\theta_0 = 75\;\mathrm{deg}$ and $\omega_{m_I \leftrightarrow m_I+1}$ as specified in the main text. Full transitions occur at different amplitude modulation periods $T_{m_I \leftrightarrow m_I+1}$ for different levels $m_I$. (b) Rabi oscillations of all $\Delta m_I = \pm 1$ at the optimal amplitude modulation period $T_{m_I \leftrightarrow m_I+1}$.
• Figure 5: (a) Amplitude modulation period scan and corresponding nuclear Rabi frequencies. We show the $\pi$-pulse fidelity at the first flip $P_{m_I\leftrightarrow m_I+1}$ for all $\Delta m_I = \pm 1$ transitions at parameters $B_0 = 1500\;\mathrm{G}$, $\Omega_{E,0} / 2\pi = 20\;\mathrm{MHz}$, $\theta_0 = 75\;\mathrm{deg}$ and $\omega_{m_I \leftrightarrow m_I+1}$ as specified in the main text. Full transitions occur at different amplitude modulation periods $T_{m_I \leftrightarrow m_I+1}$ for different levels $m_I$. (b) Rabi oscillations of all $\Delta m_I = \pm 1$ at the optimal amplitude modulation period $T_{m_I \leftrightarrow m_I+1}$.