Mode-mixing quantum gates and entanglement without particle creation in periodically accelerated cavities

We show that mode-mixing quantum gates can be produced by non-uniform relativistic acceleration. Periodic motion in cavities exhibits a series of resonant conditions producing entangling quantum gates between different frequency modes. The resonant condition associated with particle creation is the main feature of the dynamical Casimir effect which has been recently demonstrated in superconducting circuits. We show that a second resonance, which has attracted less attention since it implies negligible particle production, produces a beam splitting quantum gate leading to a resonant enhancement of entanglement which can be used as the first evidence of acceleration effects in mechanical oscillators. We propose a desktop experiment where the frequencies associated with this second resonance can be produced mechanically.