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In today's development of quantum technologies a hybrid integration ofphononic excitations becomes increasingly attractive. As natural quasi-particleexcitations in solid state systems, phonons couple to virtually any otherexcitation and therefore constitute a useful interaction channel betweendifferent building blocks in hybrid quantum systems. This work explores how theefficient light-scattering properties of a single-photon emitter and theappearance of characteristic sidebands in resonance fluorescence spectra, wheninterfaced with an arbitrary phonon quantum state, can be utilized foracousto-optical transduction. Within reasonable approximations, an analyticaldescription for the optical spectra in the low excitation limit is developedwhich can be used to read the number statistics of the initial phonon statefrom a given spectrum. It is shown that the readout is faulty in situationswhere relevant resonant transitions are forbidden due to vanishingFranck-Condon factors, especially when considering spectra with a noisybackground. Two possible solutions to this problem are presented: (A) changingthe detuning of the laser relative to the single-photon emitter which modifiesthe relevant resonant transitions, or (B) increasing dissipation of thesingle-photon emitter to promote off-resonant transitions.

How to read out the phonon number statistics via resonance fluorescence spectroscopy of a single-photon emitter