  Interaction among harmonic oscillators described by a trilinear Hamiltonian
$\hbar \xi (a^{\dagger} b c + a b^{\dagger} c^{\dagger}$) is one of the most
fundamental models in quantum optics. By employing the anharmonicity of the
Coublomb potential in a linear trapped three-ion crystal, we experimentally
implement it among three normal modes of motion in the strong-coupling regime,
where the coupling strength is much larger than the decoherence rate of the
ions motion. We use it to simulate the interaction of atom and light as
described by the Tavis-Cummings model and the process of nondegenerate
parametric down conversion in the regime of depleted pump.
