[Paper Review] Exceptional Points of Degeneracy Induced by Linear Time-Periodic Variation Hamidreza

Kazemi, Hamidreza, Nada, Mohamed Y.+3 more|arXiv (Cornell University)|Apr 3, 2018

Quantum Mechanics and Non-Hermitian Physics60 citations

TL;DR

This paper presents a general theory demonstrating that exceptional points of degeneracy (EPDs) can emerge in linear time-periodic (LTP) systems—including lossless or gainless systems—through time-varying system parameters, not requiring gain or loss. It shows that even a single LC resonator with periodic capacitance variation can host EPDs with purely real resonance frequencies, leading to algebraic energy growth and extreme sensitivity to perturbations, enabling highly sensitive sensing applications.

ABSTRACT

We present a general theory of exceptional points of degeneracy (EPD) in periodically time-variant systems that do not necessarily require the presence of loss or gain, and we show that even a single resonator with a time-periodic component may develop EPDs. An EPD is a special point in a system parameter space at which two or more eigenmodes coalesce in both their eigenvalues and eigenvectors into a single degenerate eigenmode. We demonstrate the conditions for EPDs to exist in time-periodic systems that are either lossless/gainless or with loss and/or gain and we show that a system with zero time-average loss/gain exhibits EPDs with purely real resonance frequencies, yet the resonator energy grows algebraically in time. We show the occurrence of EPDs in a single LC resonator while the introduced concept is general for any time-periodic system. These findings have significant importance in various electromagnetic/photonic systems and pave the way of applications in areas of sensors, amplifiers and modulators. A potential application of this time varying EPD is highlighted as a highly-sensitive sensor.

Motivation & Objective

• To establish a general theoretical framework for exceptional points of degeneracy (EPDs) in linear time-periodic (LTP) systems.
• To demonstrate that EPDs can occur in lossless or gainless LTP systems, challenging the prior assumption that gain/loss is necessary for EPD formation.
• To show that EPDs in LTP systems lead to algebraic energy growth even when resonance frequencies are purely real.
• To explore the extreme sensitivity of EPD systems to small perturbations for sensing applications.
• To validate the theory using a minimal LC resonator model with time-periodic capacitance

Proposed method

• Formulating the time evolution of LTP systems using a first-order differential equation with a time-variant system matrix M(t).
• Using the state transition matrix Φ(t) to describe system evolution over one modulation period Tm, derived from piecewise-constant system matrices.
• Defining the eigenvalue problem via ΦΨ = λΨ, where eigenvalues λ = exp(iωTm) correspond to system eigenfrequencies.
• Identifying EPDs as points where the transition matrix Φ becomes non-diagonalizable and forms a Jordan block, requiring degenerate eigenvalues and coalesced eigenvectors.
• Deriving a sufficient condition for second-order EPDs: tr(Φ)/2 = ±√(det(Φ)), ensuring degenerate eigenvalues and coalescing eigenvectors.
• Applying the theory to a simple LC resonator with time-periodic capacitance, using analytical solutions via Wronskian matrices and state transition matrices

Experimental results

Research questions

• RQ1 Can exceptional points of degeneracy (EPDs) exist in linear time-periodic (LTP) systems without the presence of gain or loss?
• RQ2 What are the sufficient mathematical conditions for EPDs to emerge in LTP systems, particularly in second-order systems?
• RQ3 Can EPDs in LTP systems lead to algebraic energy growth even when resonance frequencies are purely real?
• RQ4 How does the system's response to small perturbations change at an EPD, and can this be exploited for sensing?
• RQ5 Is the phenomenon of EPD formation in LTP systems generalizable beyond specific coupled systems to any time-periodic electromagnetic or RF system?

Key findings

• EPDs can be induced in LTP systems without any gain or loss, including in lossless systems, by periodic variation of system parameters such as capacitance.
• Even in systems with zero time-average loss or gain, EPDs lead to purely real resonance frequencies, yet the resonator energy grows algebraically over time.
• A single LC resonator with time-periodic capacitance can host a second-order EPD, confirmed analytically and numerically.
• The system exhibits extreme sensitivity to perturbations: a 0.1% change in dielectric permittivity (ε = 0.001) causes a 76% change in the quality factor due to a 0.34ω₀ shift in the imaginary part of the resonance frequency.
• The complex resonance frequency shows sharp, asymmetric changes under small perturbations—large shifts in the imaginary part for positive ε, and in the real part for negative ε—highlighting utility in ultra-sensitive sensing.
• The Puiseux series approximation closely matches exact eigenvalue solutions, validating the theoretical model for perturbation analysis