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

TL;DR

This paper proposes a method for simulating $ ext{Z}_2$ gauge theories using trapped ions, leveraging their internal and motional states, and demonstrates its feasibility through numerical simulations and error analysis.

Contribution

It introduces a hybrid encoding scheme for $ ext{Z}_2$ gauge theories in trapped ions and provides a detailed protocol for their analog quantum simulation.

Findings

Numerical simulations show feasible state-dependent tunneling with realistic parameters.

Analytical expressions for gauge-invariant dynamics and confinement are derived and validated.

The scheme can be scaled from a single link to complex $ ext{Z}_2$ lattice structures.

Abstract

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.