Quantum Optimal Control of Nuclear Spin Qudecimals in $^{87}\text{Sr}$

TL;DR

This paper demonstrates the feasibility of controlling a 10-dimensional nuclear spin system in extsuperscript{87}Sr using quantum optimal control, achieving high-fidelity state preparation and unitary maps despite decoherence and inhomogeneities.

Contribution

It introduces a method for implementing arbitrary quantum control on nuclear spin qudecimals in extsuperscript{87}Sr with high fidelity, including robustness against dephasing.

Findings

High-fidelity arbitrary state preparation (0.9992) within 4.5 Rabi periods.

Implementation of arbitrary SU(10) maps with fidelity 0.9923 in 24 Rabi periods.

Robust control techniques mitigate dephasing from light-shift inhomogeneities.

Abstract

We study the ability to implement unitary maps on states of the $I=9/2$ nuclear spin in \textsuperscript{87}Sr, a $d=10$ dimensional (qudecimal) Hilbert space, using quantum optimal control. Through a combination of nuclear spin-resonance and a tensor AC-Stark shift, by solely modulating the phase of a radio-frequency magnetic field, the system is quantum controllable. Alkaline earth atoms, such as \textsuperscript{87}Sr, have a very favorable figure-of-merit for such control due to narrow intercombination lines and the large hyperfine splitting in the excited states. We numerically study the quantum speed-limit, optimal parameters, and the fidelity of arbitrary state preparation and full SU(10) maps, including the presence of decoherence due to optical pumping induced by the light-shifting laser. We also study the use of robust control to mitigate some dephasing due to inhomogeneities in the light shift. We find that with an rf-Rabi frequency of $\Omega_\text{rf}$ and 0.5\% inhomogeneity in the the light shift we can prepare an arbitrary Haar-random state in a time $T={4.5}\pi/\Omega_\text{rf}$ with average fidelity $\langle \mathcal{F}_\psi \rangle =0.9992$, and an arbitrary Haar-random SU(10) map in a time $T=24\pi/\Omega_\text{rf}$ with average fidelity $\langle \mathcal{F}_U \rangle = 0.9923$.