We introduce the instantaneous eigenstate method to study the evolution of quantum states in non-dispersive, homogeneous media with arbitrary time-varying permittivity and permeability. This method leverages the Heisenberg equation to bypass the Schrödinger equation, which leads to a complicated infinite set of coupled differential equations. Instead, the method allows the computation of the state evolution by solving only two coupled differential equations. Using this approach, we draw general conclusions about photon statistics in time-varying media. Our findings reveal that the maximum probability of generating a single photon pair from vacuum in such media is $25\%$, while Bell states can be created with a maximum probability of $84\%$. Additionally, we demonstrate that the spectral response of emitted photons depends on the temporal profiles of permittivity and permeability and provide a method to determine the required profiles for specified spectral profiles. These results provide deep insights into photon state manipulation in time-varying media. Furthermore, the instantaneous eigenstate method opens new opportunities to study state evolution in other systems where the Heisenberg equation offers a more tractable solution than the Schrödinger equation.