[Paper Review] Electromagnetic fields in a time-varying medium: Exceptional points and operator symmetries

Theodoros T. Koutserimpas, Romain Fleury|arXiv (Cornell University)|Mar 6, 2020

Quantum Mechanics and Non-Hermitian PhysicsPhysics and Astronomy41 references49 citations

TL;DR

This paper develops a time-transitioning state matrix formalism to analyze electromagnetic wave propagation in time-varying media with permittivity, permeability, and conductivity modulation. By linking energy transitions to exceptional point theory and identifying parity-time symmetric wave states without spatial gain-loss balance, it reveals novel non-Hermitian dynamics in time-modulated systems, offering a new framework for active electromagnetic control and non-reciprocal wave phenomena.

ABSTRACT

In this paper, we study the interactions of electromagnetic waves with a non-dispersive dynamic medium that is temporally dependent. Electromagnetic fields under material time-modulation conserve their momentum but not their energy. We assume a time-variation of the permittivity, permeability and conductivity and derive the appropriate time-domain solutions based on the causality state at a past observation time. We formulate a time-transitioning state matrix and connect the unusual energy transitions of electromagnetic fields in time-varying media with the exceptional point theory. This state-matrix approach allows us to analyze further the electromagnetic waves in terms of parity and time-reversal symmetries and signify parity-time symmetric wave-states without the presence of a spatially symmetric distribution of gain and loss, or any inhomogeneities and material periodicity. This paper provides a useful arsenal to study electromagnetic wave phenomena under time-varying media and points out novel physical insights connecting the resulting energy transitions and electromagnetic modes with exceptional point physics and operator symmetries.

Motivation & Objective

• To develop a time-domain framework for electromagnetic wave solutions in non-dispersive, time-varying media with dynamic permittivity, permeability, and conductivity.
• To connect unusual energy transitions in time-modulated media with exceptional point theory in non-Hermitian systems.
• To identify parity-time (PT) symmetric wave states in time-symmetric media without spatial inhomogeneities or gain-loss engineering.
• To establish the role of causality and past observation time in determining wave stability and energy evolution.
• To provide a unified mathematical formulation using momentum-Fourier transforms and state matrices for time-varying wave systems.

Proposed method

• Derives time-domain solutions using separation of variables and momentum-Fourier integral transforms, assuming causality at a past observation time.
• Constructs a time-transitioning state matrix to model field evolution across temporal discontinuities in material parameters.
• Applies the Liouville-Green approximation to solve second-order differential equations for time-dependent field envelopes.
• Uses the Poynting and momentum conservation theorems to analyze energy and momentum conservation under time modulation.
• Introduces a gauge-fixed potential formulation with a modified Lorentz condition to decouple electric and magnetic field equations.
• Analyzes symmetry properties using parity and time-reversal operators to identify PT-symmetric wave states.

Experimental results

Research questions

• RQ1 How do electromagnetic waves behave in a time-varying medium when energy is not conserved, and what determines their stability?
• RQ2 Can exceptional point physics explain the observed energy transitions in time-modulated electromagnetic systems?
• RQ3 Under what conditions can parity-time symmetric wave states emerge in time-varying media without spatial gain and loss?
• RQ4 How does the choice of past observation time influence the energy and evolution of electromagnetic fields in time-varying media?
• RQ5 What is the role of the time-transitioning state matrix in characterizing wave dynamics across temporal material jumps?

Key findings

• The time-transitioning state matrix successfully models field evolution across temporal discontinuities in permittivity, permeability, and conductivity.
• Energy transitions in time-varying media are linked to exceptional point physics, with stability determined by eigenvalues of the state matrix.
• Parity-time symmetric wave states emerge in time-symmetric media without requiring spatial gain-loss distributions or material periodicity.
• The wavenumber remains constant under time modulation, while frequency shifts due to adiabatic wavelength conversion.
• The system conserves momentum but not energy, with energy transfer quantified by an external power term dependent on time-derivatives of ε and μ.
• Solutions for electric displacement and magnetic induction fields are derived using separation of variables and Liouville-Green approximation, enabling analysis of propagating, amplified, and evanescent modes.