  We demonstrate that electron-phonon interaction in quantum dots embedded in
one-dimensional systems leads to pronounced, non-Markovian decoherence of
optical transitions. The experiments we present focus on the lineshape of
photoluminescence from low-temperature axially localized carbon nanotube
excitons. The independent boson model that we use to model the phonon
interactions reproduces with very high accuracy the broad and asymmetric
emission lines and the weak red-detuned radial breathing mode replicas observed
in the experiments. The intrinsic phonon-induced pure-dephasing of the
zero-phonon line is two orders of magnitude larger than the lifetime broadening
and is a hallmark of the reduced dimensionality of the phonon bath. The
non-Markovian nature of this decoherence mechanism may have adverse
consequences for applications of one-dimensional systems in quantum information
processing.
