Realization of para-particle oscillators in trapped ions

A para-particle 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. These are captivating not only because of the underlying algebraic structures but also their dynamical properties. Even though these particles are unlikely to be present in nature, a quantum system involving a spin-½ degree of freedom coupled to two bosonic modes yields the Hamiltonian describing para-bosons and para-fermions. In this work, we report the analog quantum simulation of para-particle oscillators with a single trapped ion by tailoring the native couplings to two orthogonal motional modes in the trap. Our system reproduces the well-defined statistics for para-bosons and para-fermions of even order. These results represent the first experimental realization of a quantum simulation of the para-particle dynamics in any physical system and demonstrate the full controllability of para-particle oscillators using a trapped-ion experiment.