Exact density oscillations in the Tonks-Girardeau gas and their optical detection

We construct the exact time dependent density profile for a superposition of the ground and singly excited states of a harmonically trapped one dimensional Bose-Einstein condensate in the limit of strongly interacting particles, the Tonks-Girardeau gas. Results of an off resonant light scattering experiment probing the system could allow to determine the number of particles contained in the gas, as well as the coefficients of the superposition.