Quantum materials exhibit exotic properties and intriguing collective phenomena arising from quantum degrees of freedom and strong many-body correlations, offering significant potential for novel technological applications. Photon-based spectroscopies, including angle-resolved photoemission spectroscopy (ARPES) and resonant inelastic X-ray spectroscopy (RIXS), have become essential tools for investigating these properties both in equilibrium and out-of-equilibrium.

This thesis develops advanced theoretical methods for photon-based spectroscopies. Chapter 2 introduces the Wannier-ARPES formalism, in order to calculate photoemission matrix elements from Wannier-function based slab tight-binding models. A key contribution is a microscopic theory for circular dichroism ARPES (CD-ARPES), enabling the mapping of wavefunction-related properties like orbital textures and band topology. Using Wannier-ARPES formalism, experimental CD-ARPES results can be accurately simulated, clearly distinguishing intra-atomic terms --which reflect local orbital angular momentum (OAM) -- and inter-atomic interference terms, which introduce universal photon-energy dependencies.

Chapter 3 explores dynamics observed through time-resolved ARPES (tr-ARPES), with particular focus on Floquet engineering. The chapter highlights the relation between Floquet states and multiphoton photoemission (mPP) with varying field strengths. In the intermediate field strength, photoemission probes Floquet quasienergy splitting. In a stronger field, complex non-adiabatic dynamics among Floquet states emerge, directly influencing observed mPP features. These insights are critical for realizing Floquet engineering and provide valuable views into nonlinear driven dynamics.

Chapter 4 focuses on resonant inelastic X-ray spectroscopy (RIXS) for strongly correlated systems, where various elementary excitations can be probed. A cluster model approach, combining exact diagonalization and an effective Anderson impurity model derived from \textit{ab-initio} parameters, is used to describe many-body excitations and the RIXS spectrum. The method successfully captures intensity variations of d-d excitations across magnetic phase transitions in YBaCuFeO5.

Overall, this thesis bridges theoretical advancements with photon-based spectroscopy experiments, providing tools not only to interpret complex spectroscopic data but also to guide future experimental studies for deeper insights into quantum materials.