Path Integrals: From Quantum Mechanics to Photonics

TL;DR

This paper reviews the path integral formulation of quantum mechanics, its foundational principles, classical applications, and explores its use in optics and photonics, including computational methods like Path Integral Monte Carlo.

Contribution

It provides a comprehensive tutorial on the path integral formalism and its novel applications in photonics and computational physics.

Findings

Path integrals effectively model light propagation in inhomogeneous media.

Application of path integrals to quantum nonlinear optics.

Potential of Path Integral Monte Carlo in photonics simulations.

Abstract

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.