Digital and analog simulation of para-particles

A para-oscillator is a parity-deformed harmonic oscillator characterized by an order parameter. This generalizes the standard Fermi-Dirac and Bose-Einstein statistics associated with fermions and bosons to para-particles. We realize a method for simulating and characterizing these alternative particles using a trapped ion experiment. The combination of the Jaynes-Cummings and anti-Jaynes Cummings dynamics present in a trapped ion coupled to multiple modes of motion simultaneously allows us to recover effective Hamiltonians which create a system analogous to para-Fermi or para-Bose oscillators. Trapped ions are a versatile quantum simulator and a main contender for a universal circuit model quantum computer. We use both of these flavors in this project, simulating para-Bosons digitally using Trotterization and para-Fermions directly by tailoring the native ion-mode couplings. We discuss the mapping steps and the latest experimental results.