  We analyse particle creation and mode mixing for a quantum field in an
accelerated cavity, assuming small accelerations but allowing arbitrary
velocities, travel times and travel distances, and in particular including the
regime of relativistic velocities. As an application, we identify a desktop
experimental scenario where the mode mixing resonance frequency in linear
sinusoidal motion or in uniform circular motion is significantly below the
particle creation resonance frequencies of the Dynamical Casimir Effect, and
arguably at the threshold of current technology. The mode mixing acts as a
beamsplitter quantum gate, experimentally detectable not only via fluxes or
particle numbers but also via entanglement generation.
