Synthetic $\mathbb{Z}_2$ gauge theories based on parametric excitations of trapped ions

O. Băzăvan, S. Saner, E. Tirrito, G. Araneda, R. Srinivas, A. Bermudez

Published 2023-05-15Version 1

We present a detailed scheme for the implementation of $\mathbb{Z}_2$ gauge theories with dynamical bosonic matter using analog quantum simulators based on crystals of trapped ions. We introduce a versatile toolbox based on a state-dependent parametric excitation, which can be implemented using different interactions that couple the ions' internal qubit states to their motion, and induces a tunneling of the vibrational excitations of the crystal mediated by the trapped-ion qubits. To evaluate the feasibility of this toolbox, we perform numerical simulations of the considered schemes using realistic experimental parameters. This building block, when implemented with a single trapped ion, corresponds to a minimal $\mathbb{Z}_2$ gauge theory on a synthetic link where the qubit resides, playing the role of the gauge field. The vibrational excitations of the ion along different trap axes mimic the dynamical matter fields carrying a $\mathbb{Z}_2$ charge. We discuss how to generalise this minimal case to more complex settings by increasing the number of ions. We describe various possibilities which allow us to move from a single $\mathbb{Z}_2$ plaquette to full $\mathbb{Z}_2$ gauge chains. We present analytical expressions for the gauge-invariant dynamics and confinement, which are benchmarked using matrix product state simulations.