Synthetic $${{\mathbb{Z}}}_{2}$$ gauge theories based on parametric excitations of trapped ions

Resource efficient schemes for the quantum simulation of lattice gauge theories can benefit from hybrid encodings of gauge and matter fields that use the native degrees of freedom, such as internal qubits and motional phonons in trapped-ion devices. We propose to use a parametric scheme to induce a tunneling of the phonons conditioned to the internal qubit state which, when implemented with a single trapped ion, corresponds to a minimal $${{\mathbb{Z}}}_{2}$$ gauge theory. To evaluate the feasibility of this scheme, 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 generalize 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. An outstanding question for gauge-theory quantum simulators is to find viable schemes that allow one to move from the initial prototypes towards the large-scale regime. In this work, the authors present a detailed toolbox for the quantum simulator of Z2 lattice gauge theories coupled to dynamical matter using trapped-ion systems that can overcome these limitations.