Path Integrals: From Quantum Mechanics to Photonics

The path integral formulation of quantum mechanics, i.e., the idea that the evolution of a quantum system is determined as a sum over all the possible trajectories that would take the system from the initial to its final state of its dynamical evolution, is perhaps the most elegant and universal framework developed in theoretical physics, second only to the Standard Model of particle physics. In this tutorial, we retrace the steps that led to the creation of such a remarkable framework, discuss its foundations, and present some of the classical examples of problems that can be solved using the path integral formalism, as a way to introduce the readers to the topic, and help them get familiar with the formalism. Then, we focus our attention on the use of path integrals in optics and photonics, and discuss in detail how they have been used in the past to approach several problems, ranging from the propagation of light in inhomogeneous media, to parametric amplification, and quantum nonlinear optics in arbitrary media. To complement this, we also briefly present the Path Integral Monte Carlo (PIMC) method, as a valuable computational resource for condensed matter physics, and discuss its potential applications and advantages if used in photonics.