  Vibrational degrees of freedom in trapped-ion systems have recently been
gaining attention as a quantum resource, beyond the role as a mediator for
entangling quantum operations on internal degrees of freedom, because of the
large available Hilbert space. The vibrational modes can be represented as
quantum harmonic oscillators and thus offer a Hilbert space with infinite
dimension. Here we review recent theoretical and experimental progress in the
coherent manipulation of the vibrational modes, including bosonic encoding
schemes in quantum information, reliable and efficient measurement techniques,
and quantum operations that allow various quantum simulations and quantum
computation algorithms. We describe experiments using the vibrational modes,
including the preparation of non-classical states, molecular vibronic sampling,
and applications in quantum thermodynamics. We finally discuss the potential
prospects and challenges of trapped-ion vibrational-mode quantum information
processing.
