Abstract: We discuss number theoretic structure including e.g., modular forms, Eichler forms, elliptic curves, and “Apéry-like numbers” behind certain quantum interactions. These are arising from the special values of the spectral zeta functions for the quantum system. Models where we treat are the (symmetric and asymmetric) quantum Rabi models and non-commutative harmonic oscillators. The former models are the most fundamental models in quantum optics and quantum information theory which describe the light-matter interaction. There is also an interesting connection between former and latter via the confluent process at their respective Heun ODE pictures. If the time allows, we will mention various number theoretic and geometric questions which are open.
