Our main goal is to prove existence (and uniqueness) of the quantum propagator for time dependent quantum Hamiltonians $\widehat{H}(t)$ when this Hamiltonian is perturbed with a quadratic white noise $\dot{\beta}\widehat{K}$. $\beta$ is a continuous function in time $t$, $\dot{\beta}$ its time derivative and K is a quadratic Hamiltonian. $\widehat{K}$ is the Weyl quantization of $K$. In our approach we use an exact Hermann Kluk formula to deduce a Strichartz estimate for the propagator of $\widehat{H}(t)+\dot{\beta}\widehat{K}$. This is applied to obtain local and global well posedness for solutions for non linear Schrödinger equations with an irregular time dependent linear part.