Exponential-SWAP gate between bosonic motional modes of trapped ions

The exponential-SWAP (eSWAP) gate generates a coherent superposition of identity and SWAP operations, with continuously tunable amplitudes. We experimentally implement this gate between two bosonic motional modes of trapped ions using spin-independent and spin-dependent beam splitter interactions. Measurements of selected slices of the joint characteristic function for distinct input states test the gate and demonstrate continuous control of the identity-SWAP superposition. This implementation provides a tunable interaction for hybrid qubit-bosonic quantum information processing.