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Formal equivalence between Maxwell equations and the de Broglie-Bohm theory for two-dimensional optical microcavities

Aurélien Drezet, Bernard Michael Nabet

TL;DR

This work establishes a formal link between Maxwellian energy transport and Bohmian hydrodynamics for photons in a 2D planar microcavity. By deriving a paraxial, effective Schrödinger equation from Maxwell's equations and showing that the in-plane Bohmian velocity satisfies $\mathbf{v}_{dBB,||}=\frac{1}{\varepsilon m}\mathrm{Im}[\frac{\boldsymbol{\nabla}_{||}\Psi}{\Psi}]$, the authors connect energy flow $\mathbf{S}$ and energy density $u$ via $\mathbf{v}=\mathbf{S}/u$, validating the Bohmian interpretation in 2D optics. They extend the framework to lossy cavities with a non-Hermitian Hamiltonian, introduce spin-1/2 Pauli spinors with orbital and spin contributions to the current, and discuss stochastic Bohmian mechanics for absorption, showing that leakage enables a nonzero velocity in evanescent fields. The analysis tackles Klaers2025, arguing that radiative leakage and finite-source effects reconcile Bohmian trajectories with experimental observations, and highlights broader implications for microphotonics and polarization-enabled photonic devices.

Abstract

We analyze the formal equivalence between the electromagnetic energy conservation law derived from Maxwell's equations in an optical microcavity and the conservation of a probability fluid associated with the de Broglie-Bohm theory for an effective massive particle describing a photon in this cavity. This work is part of a critical analysis of recent experiments [Nature \textbf{643}, 67-72 (2025)] carried out with a view to refuting the de Broglie-Bohm theory. Furthermore, the consequences of our analysis for microphotonics go far beyond these experiments. In particular, extensions that take into account photon spin and stochastic aspects associated with radiative or absorption losses are considered.

Formal equivalence between Maxwell equations and the de Broglie-Bohm theory for two-dimensional optical microcavities

TL;DR

This work establishes a formal link between Maxwellian energy transport and Bohmian hydrodynamics for photons in a 2D planar microcavity. By deriving a paraxial, effective Schrödinger equation from Maxwell's equations and showing that the in-plane Bohmian velocity satisfies , the authors connect energy flow and energy density via , validating the Bohmian interpretation in 2D optics. They extend the framework to lossy cavities with a non-Hermitian Hamiltonian, introduce spin-1/2 Pauli spinors with orbital and spin contributions to the current, and discuss stochastic Bohmian mechanics for absorption, showing that leakage enables a nonzero velocity in evanescent fields. The analysis tackles Klaers2025, arguing that radiative leakage and finite-source effects reconcile Bohmian trajectories with experimental observations, and highlights broader implications for microphotonics and polarization-enabled photonic devices.

Abstract

We analyze the formal equivalence between the electromagnetic energy conservation law derived from Maxwell's equations in an optical microcavity and the conservation of a probability fluid associated with the de Broglie-Bohm theory for an effective massive particle describing a photon in this cavity. This work is part of a critical analysis of recent experiments [Nature \textbf{643}, 67-72 (2025)] carried out with a view to refuting the de Broglie-Bohm theory. Furthermore, the consequences of our analysis for microphotonics go far beyond these experiments. In particular, extensions that take into account photon spin and stochastic aspects associated with radiative or absorption losses are considered.

Paper Structure

This paper contains 8 sections, 74 equations, 1 figure.

Table of Contents

  1. The Poynting theorem for complex effective electromagnetic fields
  2. Bohmian formalism for an ideal two-dimensional optical microcavity
  3. Bohmian formalism for a lossy two-dimensional optical microcavity
  4. Derivation from Maxwell's equations
  5. Generalization for particles with spin-$1/2$
  6. Conservation of probability and stochastic Bohmian mechanics
  7. Final remarks and conclusion
  8. Leakage radiation and finite source

Figures (1)

  • Figure 1: Typical Bohmian trajectories for light inside a lossy optical microcavity. The intensity map $|\Psi(x,z)|^2$ as a function of $x$ and $z$ shows the decaying of the leaky wave along the $+x$ axis of the planar wave guide and the (very weak) growing of the wave along the $+z$ axis normal to the cavity. White lines are averaged Bohmian paths and red curves corresponds to actual energy flow curves associated with the local Poynting vector.