  Pure dephasing of an exciton in a small quantum dot by optical and acoustic
phonons is calculated using the ``independent boson model''. Considering the
case of zero temperature the dephasing is shown to be only partial which
manifests itself in the polarization decaying to a finite value. Typical
dephasing times can be assigned even though the spectra exhibits strongly
non-Lorentzian line shapes. We show that the dephasing from LO phonon
scattering, occurs on a much larger time scale than that of dephasing due to
acoustic phonons which for low temperatures are also a more efficient dephasing
mechanism. The typical dephasing time is shown to strongly depend on the
quantum dot size whereas the electron phonon ``coupling strength'' and external
electric fields tend mostly to effect the residual coherence. The relevance of
the dephasing times for current quantum information processing implementation
schemes in quantum dots is discussed.
