Path integrals: From quantum mechanics to photonics

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 method, as a valuable computational resource for condensed matter physics, and discuss its potential applications and advantages if used in photonics.

Perturbation approach in Heisenberg equations for lasers

“…So each set of random frequencies corresponds to particular realization of the random process. Such a realization may be an analog of the path integral [31,32]. Mean Appendix B: Spectrum of the operator product It is sufficient to know power spectra in order to describe the system in the stationary case.…”

“…It lets to find mean values of high-order correlations of products of operators, but it does not calculate spectra of optical fields. Path integral formalism can be used in some problems of nonlinear and quantum optics [31,32]. However, it is applied mostly to systems with quadratic Hamiltonians, i.e.…”

Diamagnetic susceptibility from a nonadiabatic path-integral simulation of few-electron systems

“…The PIMC approach enables statistically exact simulation of thermal density matrices [15] but also various response properties [16][17][18] of real few-body Coulomb systems, as long as the particles are distinguishable. Much larger systems [21] with indistinguishable fermions can also be simulated exactly, but only with exponentially decreasing numerical efficiency.…”