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

We present a detailed scheme for the analog quantum simulation of $\mathbb{Z}_2$ gauge theories in crystals of trapped ions, which exploits a more efficient hybrid encoding of the gauge and matter fields using the native internal and motional degrees of freedom. We introduce a versatile toolbox based on parametric excitations corresponding to different spin-motion-coupling schemes that induce a tunneling of the ions vibrational excitations conditioned to their internal qubit state. This building block, when implemented with a single trapped ion, corresponds to a minimal $\mathbb{Z}_2$ gauge theory, where the qubit plays the role of the gauge field on a synthetic link, and the vibrational excitations along different trap axes mimic the dynamical matter fields two synthetic sites, each carrying a $\mathbb{Z}_2$ charge. To evaluate their feasibility, we perform numerical simulations of the state-dependent tunneling using realistic parameters, and identify the leading sources of error in future experiments. We discuss how to generalise this minimal case to more complex settings by increasing the number of ions, moving from a single link to a $\mathbb{Z}_2$ plaquette, and to an entire $\mathbb{Z}_2$ chain. We present analytical expressions for the gauge-invariant dynamics and the corresponding confinement, which are benchmarked using matrix product state simulations.