Exceptional points of arbitrary high orders induced by non-Markovian dynamics

TL;DR

This work shows that non-Markovian dynamics in a finite-size reservoir enable effective exceptional points of arbitrarily high order in a single harmonic oscillator system, beyond the system's nominal degrees of freedom. By modeling the oscillator coupled to a finite equidistant reservoir and deriving the non-Hermitian evolution, the authors reveal revival dynamics described by $ a_n(t) = e^{-\,\gamma t} L_n^{(-1)}(2\gamma t) $, with revivals occurring at $ T_R = \frac{2\pi}{\delta\omega} $ and each revival corresponding to an increasingly higher-order EP. The analysis yields a transcendental eigenfrequency condition and Lorentzian-distributed eigenstate weights, linking non-exponential decay to spectral features that include additional peaks whose count scales with the EP order. The findings suggest that by selecting observation times and measuring revival amplitudes, one can observe EPs of arbitrary order in platforms such as qubits coupled to waveguides or single-mode cavities, highlighting a practical route to high-sensitivity non-Hermitian systems induced by memory effects.

Abstract

Exceptional points are singularities in the spectrum of non-Hermitian systems in which several eigenvectors are linearly dependent and their eigenvalues are equal to each other. Usually it is assumed that the order of the exceptional point is limited by the number of degrees of freedom of a non-Hermitian system. In this letter, we refute this common opinion and show that non-Markovian effects can lead to dynamics characteristic of systems with exceptional points of higher orders than the number of degrees of freedom in the system. This takes place when the energy returns from reservoir to the system such that the dynamics of the system are divided into intervals in which it describes by the product of the exponential and a polynomial function of ever-increasing order. We demonstrate that by choosing the observation time, it is possible to observe exceptional points of arbitrary high orders.

Figures (2)

• Figure 1: Dynamics of $|a(t)|^2$ (the dashed black line) and the absolute values of the revival amplitudes, $a_n(t)$, determined by the Eq. (\ref{['eq:9']}). The blue line shows $a_0 (t) \theta (t)$; the red line shows $a_1 (t-\frac{2\pi}{\delta\omega}) \theta (t-\frac{2\pi}{\delta\omega})$; and the green line shows $a_2 (t-\frac{4\pi}{\delta\omega}) \theta (t-\frac{4\pi}{\delta\omega})$. Here $T_R=\frac{2\pi}{\delta\omega}$ indicates the time of the first revival, $\delta\omega=0.002 \omega_0$, $\gamma \approx 0.0035 \omega_0$ and $\omega_0=1$.
• Figure 2: Dynamics of the third $|a_3 (t)|^2$ (a) and fifth $|a_5 (t)|^2$ (c) revivals and their frequency spectra $|S_3(\omega)|^2$ (b) and $|S_5 (\omega)|^2$ (d). The black dashed line corresponds to spectrum of the initial evolution $a_0(t)$, namely, the Lorentz distribution $S_0 (\omega)=\frac{\gamma}{\pi}\frac{1}{\omega^2 + \gamma^2}$ . Here $\delta\omega=0.002 \omega_0$, $\gamma \approx 0.0071 \omega_0$, and $\omega_0=1$.