Phonon-to-spin mapping in a system of a trapped ion via optimal control

We propose a protocol for measurement of the phonon number distribution of a harmonic oscillator based on selective mapping to a discrete spin-1/2 degree of freedom. We consider a system of a harmonically trapped ion, where a transition between two long-lived states can be driven with resolved motional sidebands. The required unitary transforms are generated by amplitude-modulated polychromatic radiation fields, where the time-domain ramps are obtained from numerical optimization by application of the chopped random basis algorithm (CRAB). We provide a detailed analysis of the scaling behavior of the attainable fidelities and required times for the mapping transform with respect to the size of the Hilbert space. As one application we show how the mapping can be employed as a building block for experiments which require measurement of the work distribution of a quantum process.