The interaction between ultrashort light pulses and non-absorbing materials is dominated by impulsive stimulated Raman scattering (ISRS). The description of ISRS in the context of pump&probe experiments is based on effective classical models describing the interaction between the phonon and pulsed electromagnetic fields. Here we report a theoretical description of ISRS where we do not make any semi-classical approximation and we treat both photonic and phononic degrees of freedom at the quantum level. The results of the quantum model are compared with semiclassical results and validated by means of spectrally resolved pump&probe measurements on α-quartz.