Exactly solvable model of light-scattering errors in quantum simulations with metastable trapped-ion qubits

We analytically solve a model for light scattering in Ising dynamics of metastable atomic qubits, generalizing the approach of Foss-Feig et al. [Phys. Rev. A 87, 042101 (2013)] to include leakage outside the qubit manifold. We analyze the influence of these fundamental errors in simulations of proposed experiments with metastable levels of ${}^{40}\mathrm{Ca}^{+}$ ions in a Penning trap. We find that “effective magnetic fields” generated by leaked qubits have significant impacts on spin-spin correlation functions for Greenberger-Horne-Zeilinger state preparation or for quantum simulations with strong coupling, while spin squeezing uses a much weaker coupling and is largely insensitive to the simulated leakage errors, even with a few hundred ions. Our theory and results are expected to be useful in modeling a variety of metastable qubit experiments in the future.