Non-linear cooling and control of a mechanical quantum harmonic oscillator

Non-linearities are a key feature allowing non-classical control of quantum harmonic oscillators. However, when non-linearities are strong, designing protocols for control is often difficult, placing a barrier to exploiting these properties fully. Here, using a single trapped-ion oscillator operated in the strongly non-linear regime of the atom-light interaction, we show how to generate localized multi (2, 3, 4, and 5)-component Schrödinger's cat manifolds using a novel form of non-linear reservoir engineering. We then specifically select Hamiltonians which allow us to perform measurements on these state manifolds. To our knowledge, our work is the first experimental use of such high order non-linear processes for control of non-classical states of a quantum harmonic oscillator, opening up a new toolbox which can be applied to bosonic quantum error correction, computation, and sensing.