Numerically optimized amplitude-robust controlled-Z gate for ultracold neutral atoms with individual addressing capability

1. Problem Statement

The paper addresses a critical bottleneck in scaling neutral-atom quantum computers: achieving high-fidelity entangling gates (specifically Controlled-Z or CZ gates) when using individual addressing with tightly focused laser beams.

• Current Limitations: While symmetric gate protocols (where both atoms are illuminated by a global beam) have achieved fidelities >99.7%, universal quantum computing requires individual addressing of atom pairs.
• The Challenge: Individual addressing using tightly focused beams introduces significant Rabi frequency inhomogeneity due to:
  1. Residual thermal motion of trapped atoms (jitter in position).
  2. Beam pointing instability.
  3. Asymmetry: The two atoms in a pair experience different laser intensities ($\Omega_1 \neq \Omega_2$) because they are at different positions within the Gaussian beam profiles.
• Consequence: Standard gate protocols (like the Levine-Pichler or time-optimal gates) are highly sensitive to these variations, leading to a rapid drop in gate fidelity. Existing amplitude-robust proposals offer some improvement but fall short of the robustness required for high-fidelity individual addressing.

2. Methodology

The authors employed a numerical optimization approach to design a new gate protocol that is robust against variations in Rabi frequency.

• Optimization Framework:
  • Utilized a customized version of the RydOpt Python package, leveraging the JAX library and the Adam optimization algorithm.
  • Objective Function: Instead of optimizing for a single Rabi frequency, the algorithm optimized for the average fidelity across a range of Rabi frequencies ($\Omega_0(1 \pm \epsilon)$).
  • Penalty Terms: To ensure robustness, the cost function included penalties for:
    • Fidelity Variation: The variance of fidelity across the Rabi frequency range.
    • Slope: The difference in fidelity between the minimum and maximum Rabi frequencies (ensuring the gate is insensitive to the sign of the variation).
• Phase Profile Parametrization:
  • The laser pulse phase $\xi(t)$ was parametrized using Chopped Random Basis (CRAB) initial profiles augmented with a linear shift.
  • The optimization searched for analytically defined phase profiles that could be applied to different pulse shapes (rectangular, smooth/Bernstein, and smooth-edged).
• Simulation Scenarios:
  • Single-photon excitation: Relevant for Strontium atoms.
  • Two-photon excitation: Relevant for Rubidium and Cesium atoms.
  • Asymmetry Testing: Simulated scenarios where $\Omega_1 \neq \Omega_2$ to mimic individual addressing.
  • Monte-Carlo Simulations: Modeled atomic motion in optical dipole traps at finite temperatures ($1\text{--}10 \mu\text{K}$) to evaluate the impact of thermal jitter on gate fidelity.

3. Key Contributions

1. Novel Gate Protocol: Development of a numerically optimized, amplitude-robust symmetric CZ gate.
2. Order-of-Magnitude Improvement: The new protocol demonstrates robustness to Rabi frequency variations that is nearly an order of magnitude greater than previous proposals (e.g., Ref. [13] and standard time-optimal gates).
3. Individual Addressing Applicability: The protocol is explicitly designed to handle the asymmetry of Rabi frequencies between two individually addressed atoms, a scenario where previous symmetric protocols fail.
4. Excitation Scheme Analysis:
   • Demonstrated superior performance for single-photon excitation (Strontium).
   • Identified that for two-photon excitation (Rubidium), the standard $6P_{3/2}$ intermediate state suffers from mismatched beam waists (due to wavelength differences), but an alternative scheme using the $5P_{3/2}$ state (780 nm + 480 nm) significantly reduces this mismatch and improves fidelity.
5. Analytical Phase Profiles: Provided specific, scalable analytical phase profiles (coefficients listed in Tables I and II) that can be directly implemented in experiments.

4. Key Results

• Robustness to Global Rabi Variation:
  • The optimized gate maintains high fidelity even when the Rabi frequency varies by several percent.
  • In contrast, Levine-Pichler and time-optimal gates show sharp fidelity drops with minor frequency deviations.
• Robustness to Asymmetry ($\Omega_1 \neq \Omega_2$):
  • The gate remains robust when the ratio of Rabi frequencies for the two atoms varies, effectively mitigating errors caused by atomic position jitter within the focused beams.
• Thermal Motion Simulations:
  • Single-Photon (Strontium): The amplitude-robust protocol substantially outperforms the time-optimal scheme at finite temperatures, reducing gate infidelity significantly.
  • Two-Photon (Rubidium):
    • Standard schemes (420 nm + 1013 nm) show marginal improvement over time-optimal gates due to light shifts caused by axial motion and wavelength-dependent Rayleigh lengths.
    • Alternative Scheme (780 nm + 480 nm): Using the $5P_{3/2}$ intermediate state yields substantially better results. The closer wavelengths result in more matched Rayleigh lengths ($4.0 \mu\text{m}$ vs $6.5 \mu\text{m}$), reducing light shifts and Doppler effects.
• Trade-offs: While the $5P_{3/2}$ scheme offers better robustness to beam mismatch, it suffers from a shorter intermediate state lifetime (26 ns vs 118 ns). However, the authors show that high fidelities are still achievable by increasing the detuning and Rabi frequencies, which is feasible with tightly focused beams.

5. Significance

• Enabling Scalable Quantum Computing: This work bridges the gap between high-fidelity global gate experiments and the requirements for universal quantum computing, which necessitates individual qubit control.
• Error Mitigation: By providing a gate protocol that is intrinsically robust to laser intensity fluctuations and atomic motion, the need for extreme cooling or perfect beam stabilization is reduced, making high-fidelity operations more accessible.
• Experimental Guidance: The paper provides concrete experimental parameters (phase profiles, pulse shapes) and suggests specific excitation pathways (e.g., the $5P_{3/2}$ route for Rubidium) that maximize fidelity in individual addressing setups.
• Future Outlook: The authors suggest that this approach, combined with three-photon excitation (which eliminates light shifts entirely), paves the way for high-fidelity entanglement in large-scale neutral atom arrays.

In conclusion, the paper presents a robust, numerically optimized solution to the problem of Rabi frequency inhomogeneity in individual atom addressing, offering a clear path toward high-fidelity quantum gates in scalable neutral-atom quantum processors.