  Condensed-matter emitters offer enriched cavity quantum electrodynamical
effects due to the coupling to external degrees of freedom. In the case of
carbon nanotubes a very peculiar coupling between localized excitons and the
one-dimensional acoustic phonon modes can be achieved, which gives rise to
pronounced phonon wings in the luminescence spectrum. By coupling an individual
nanotube to a tunable optical micro-cavity, we show that this peculiar
exciton-phonon coupling is a valuable resource to enlarge the tuning range of
the single-photon source while keeping an excellent exciton-photon coupling
efficiency and spectral purity. Using the unique flexibility of our scanning
fiber cavity, we are able to measure the efficiency spectrum of the very same
nanotube in the Purcell regime for several mode volumes. Whereas this
efficiency spectrum looks very much like the free-space luminescence spectrum
when the Purcell factor is small (large mode volume), we show that the
deformation of this spectrum at lower mode volumes can be traced back to the
strength of the exciton-photon coupling. It shows an enhanced efficiency on the
red wing that arises from the asymmetry of the incoherent energy exchange
processes between the exciton and the cavity. This allows us to obtain a tuning
range up to several hundred times the spectral width of the source.
