  In-situ Raman experiments together with transport measurements have been
carried out on carbon nanotubes as a function of gate voltage. In metallic
tubes, a large increase in the Raman frequency of the $G^-$ band, accompanied
by a substantial decrease of its line-width, is observed with electron or hole
doping. In addition, we see an increase in Raman frequency of the $G^+$ band in
semiconducting tubes. These results are quantitatively explained using
ab-initio calculations that take into account effects beyond the adiabatic
approximation. Our results imply that Raman spectroscopy can be used as an
accurate measure of the doping of both metallic and semiconducting nanotubes.
