Optical resonators store electromagnetic energy. The finite response time of optical resonators provides a feedback mechanism for controlling the dynamics of atomic and mechanical systems and to effectively exchange energy between light and matter. This chapter starts with a derivation of the reflection and transmission coefficients of a confocal optical cavity. The spectrum is characterized by multiple resonances and for most applications a single resonance can be singled out. This leads to the single-mode approximation. We derive the energy stored in the cavity and evaluate the fields of a cavity that is internally excited by a radiating dipole. We calculate the LDOS and derive an expression for cavity-enhanced emission (Purcell effect). We continue with an analysis of microsphere resonators with characteristic whispering-gallery modes and review the effective potential approach, which allows us to cast the problem in form of a Schr\“odinger equation, with parallels to quantum tunneling and radioactive decay. The next section is focused on deriving the cavity perturbation formula, which states that a change in energy is accompanied by a frequency shift. Having established a solid understanding of optical resonators we discuss the interplay of optical and mechanical degrees of freedom within the context of cavity optomechanics. We derive the optomechanical coupling rate and discuss the resolved sideband and the weak-retardation regimes.

The tangent space of a tensor network state is a manifold that parameterizes the variational trajectory of local tensors. It leads to a framework to accommodate the time-dependent variational principle (TDVP), which unifies stationary and time-dependent methods for dealing with tensor networks. It also offers an ideal platform for investigating elementary excitations, including nontrivial topological excitations, in a quantum many-body system under the single-mode approximation. This chapter introduces the tangent-space approaches in the variational determination of MPS and PEPS, starting with a general discussion on the properties of the tangent vectors of uniform MPS. It then exemplifies TDVP by applying it to optimize the ground state MPS. Finally, the methods for calculating the excitation spectra in both one and two dimensions are explored and applied to the antiferromagnetic Heisenberg model on the square lattice.