Cavity quantum electrodynamics in finite-bandwidth squeezed reservoir

Light-matter interaction with squeezed vacuum has received much interest for the ability to enhance the native interaction strength between an atom and a photon with a reservoir assumed to have an infinite bandwidth. Here, we study a model of parametrically driven cavity quantum electrodynamics (cavity QED) for enhancing light-matter interaction while subjected to a finite-bandwidth squeezed vacuum drive. Our method is capable of unveiling the effect of relative bandwidth as well as squeezing required to observe the anticipated anti-crossing spectrum and enhanced cooperativity without the ideal squeezed bath assumption. Furthermore, we analyze the practicality of said models when including intrinsic photon loss due to resonators imperfection. With these results, we outline the requirements for experimentally implementing an effectively squeezed bath in solid-state platforms such as InAs quantum dot cavity QED such that \textit{in situ} control and enhancement of light-matter interaction could be realized.