Non-linear trapped-ion spin-motion coupling with orthogonal geometric phase gates

Non-linear boson interactions are essential for quantum metrology and continuous variable quantum computation (CVQC). In trapped ions, subsets of such interactions can be generated for phonons through higher-order spatial derivatives of applied fields. Unfortunately, no scheme exists that enables a broad set of non-linear interactions in an "all-in-one" experimental setup. Here, we propose a new method of generating such interactions that uses two (linear) non-commuting geometric phase gates. We show how this can be used to produce various types non-linear interactions including, among others, one and two mode squeezing, beam splitting, and trisqueezing operations.