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 Ωrf and 0.5\% inhomogeneity in the the light shift we can prepare an arbitrary Haar-random state in a time T=4.5π/Ωrf with average fidelity ⟨Fψ⟩=0.9992, and an arbitrary Haar-random SU(10) map in a time T=24π/Ωrf with average fidelity ⟨FU⟩=0.9923.
@article{arxiv.2106.13705,
title = {Quantum Optimal Control of Nuclear Spin Qudecimals in $^{87}\text{Sr}$},
author = {Sivaprasad Omanakuttan and Anupam Mitra and Michael J. Martin and Ivan H Deutsch},
journal= {arXiv preprint arXiv:2106.13705},
year = {2022}
}