Superstatistical Energy Distributions of an Ion in an Ultracold Buffer Gas

An ion in a radio frequency ion trap interacting with a buffer gas of ultracold neutral atoms is a driven dynamical system which has been found to develop a nonthermal energy distribution with a power law tail. The exact analytical form of this distribution is unknown, but has often been represented empirically by q-exponential (Tsallis) functions. Based on the concepts of superstatistics, we introduce a framework for the statistical mechanics of an ion trapped in an rf field subject to collisions with a buffer gas. We derive analytic ion secular energy distributions from first principles both neglecting and including the effects of the thermal energy of the buffer gas. For a buffer gas with a finite temperature, we prove that Tsallis statistics emerges from the combination of a constant heating term and multiplicative energy fluctuations. We show that the resulting distributions essentially depend on experimentally controllable parameters paving the way for an accurate control of the statistical properties of ion-atom hybrid systems.