Trilinear hamiltonian with trapped ions and its applications

The model of three harmonic oscillators coupled by the trilinear Hamiltonian of the form $a^{\dagger} b c + a b^{\dagger} c^{\dagger}$ can describe wide range of physical processes. We experimentally realize such interaction between three modes of motion in the system of 3 trapped Yb ions. We discuss several application of this coupling, including implementation of the quantum absorption refrigerator, simulation of the interaction between light and atoms described by a Tavis-Cummings model, simulation of the non-degenerate parametric down conversion process in the fully quantum regime and studies of a simple model of Hawking radiation.