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arXiv:1808.07205 · Physical Review A

Exceptional-point-induced lasing dynamics in a non-Hermitian Su-Schrieffer-Heeger model

K. L. Zhang, P. Wang, Z. Song

Published Apr 10, 2019• 16 citations
arXivHTMLPDF

Abstract

Non-Hermitian systems exhibit many peculiar dynamic behaviors that never showed up in Hermitian systems. The existence of spectral singularity for a non-Hermitian scattering center provides a ``lasing mechanism'' in the context of quantum mechanics. In this paper, we investigate the dynamics of the non-Hermitian Su-Schrieffer-Heeger model around an exceptional point, based on an analysis of parity-time and chiral-time symmetries of the Hamiltonian. We show exactly that a finite system near its exceptional point can support EP-related amplification of cyclic states. It becomes a stationary laser solution with linear-power amplification under certain conditions in an infinite limit. In contrast to the SS lasing mechanism for a scattering system, such an SSH chain acts as an active laser medium at threshold, within which a stationary particle emission can be fired anywhere, rather than a specific location at the non-Hermitian scattering center only. In addition, some relevant peculiar phenomena arising from interference between wave packets are revealed based on the analytical solutions.

DOI
10.1103/physreva.99.042111

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Non-Hermitian systems exhibit many peculiar dynamic behaviors that never showed up in Hermitian systems. The existence of spectral singularity for a non-Hermitian scattering center provides a ``lasing mechanism'' in the context of quantum mechanics. In this paper, we investigate the dynamics of the non-Hermitian Su-Schrieffer-Heeger model around an exceptional point, based on an analysis of parity-time and chiral-time symmetries of the Hamiltonian. We show exactly that a finite system near its exceptional point can support EP-related amplification of cyclic states. It becomes a stationary laser solution with linear-power amplification under certain conditions in an infinite limit. In contrast to the SS lasing mechanism for a scattering system, such an SSH chain acts as an active laser medium at threshold, within which a stationary particle emission can be fired anywhere, rather than a specific location at the non-Hermitian scattering center only. In addition, some relevant peculiar phenomena arising from interference between wave packets are revealed based on the analytical solutions.
The balance of gain and loss in an open system may maintain certain Hermitian dynamical behaviors which can be hardly observed in a popular Hermitian system. In this paper, we systematically study a 1D PT\mathcal{PT}PT-symmetry non-Hermitian Su-Schrieffer-Heeger model with open boundary condition based on an exact approximate solution. We show that the long-wavelength standing-wave modes can be achieved within the linear dispersion region when the system is tuned at the exceptional point. The whole Hilbert space can be decomposed into two quasi-Hermitian subspaces, which are consisted of positive and negative energy levels, respectively. Within each subspace, the system maintains all the features of a Hermitian system. We construct a coherentlike state in a subspace and find that it exhibits perfect simple harmonic motion (SHM). In contrast to a canonical coherent state, the shape of the wave packet deforms periodically rather than entirely by translation, and the amplitude of the SHM is not determined by the initial condition but by the shape of the wave packet. Our result indicates that novel Hermitian dynamics can be realized by a non-Hermitian system.
An exceptional point (EP) is exclusive for non-Hermitian systems and distinct from that at a degeneracy point (DP), supporting intriguing dynamics which can be utilized to probe quantum phase transitions and prepare eigenstates in a Hermitian many-body system. In this paper, we investigate the transition from a DP for a Hermitian system to an EP driven by non-Hermitian terms. We present a theorem on the existence of a transition between a DP and EP for a general system. Specifically, one of twofold degenerate eigenstates of a Hermitian system becomes a coalescing state when a selected non-Hermitian term is added. The obtained EP is robust to the strength of non-Hermitian terms. We illustrate the theorem by an exactly solvable quasi-one-dimensional model, which allows for the existence of a transition between fully degenerate and exceptional spectra driven by non-Hermitian tunnelings in real and kkk spaces, respectively. We also study the EP dynamics for generating coalescing edge modes in Su-Schrieffer-Heeger-like models. This finding reveals the ubiquitous connection between DPs and EPs.
We study a non-Hermitian extension of the Creutz ladder with generic non-reciprocal hopping. By mapping the ladder onto two decoupled non-Hermitian Su--Schrieffer--Heeger (SSH) chains, we uncover a rich structure in parameter space under different boundary conditions. Under periodic boundary conditions, the spectrum admits a fine-tuned line in parameter space with entirely real eigenvalues, while deviations from this line induce a real--complex spectral transition without crossing exceptional points. In contrast, an exact analytical diagonalization under open boundary conditions reveals extended regions in parameter space with purely real or purely imaginary spectra, separated from complex spectral domains by exceptional lines. The intersections of these exceptional lines define triple-junction points where distinct spectral regimes meet, giving rise to a structured phase diagram that is absent under periodic boundary conditions. We further show that flat bands in this system can occur both as Hermitian diabolical points and as non-Hermitian exceptional points, known as exceptional flat bands, where the dynamics is more stringent than in the Hermitian case, leading to distinct spectral and dynamical signatures.
We investigate the interacting non-Hermitian Su-Schrieffer-Heeger (SSH) model, focusing on the interplay between topology and charge ordering. Using a real-space topological marker, charge correlations, and the complex many-body spectrum, we map out the phase diagram under periodic and open boundary conditions. We show that the topological marker remains a robust diagnostic of non-Hermitian topological phases in the presence of interactions and consistently signals their breakdown at the onset of a charge density wave (CDW). We further demonstrate that non-Hermiticity enhances interaction effects: While moderate changes occur under periodic boundary conditions, open boundary conditions lead to a pronounced amplification of staggered charge correlations near exceptional points. This enhancement arises from the accumulation of low-energy states near exceptional points, which promotes electronic instabilities and strengthens CDW tendencies.
Exceptional point (EP) is a spectral singularity in non-Hermitian systems. The passing over the EP leads to a phase transition, which endows the system with unconventional features that find a wide range of applications. However, the need of using the dissipation and amplification limits the possible applications of systems with the EP. In this work, we demonstrate an existence of signature of exceptional point phase transition in Hermitian systems that are free from dissipation and amplification. We consider a composite Hermitian system including both two coupled oscillators and their environment consisting only of several tens of degrees of freedom. We show that the dynamics of such a Hermitian system demonstrate a transition, which occurs at the coupling strength between oscillators corresponding to the EP in the non-Hermitian system. This transition manifests itself even in the non-Markovian regime of the system dynamics in which collapses and revivals of the energy occur. Thus, we demonstrate that the phase transition occurring at the passing over the EP in the non-Hermitian system manifests itself in the Hermitian system at all time. We discuss the experimental scheme to observe the signature of EP phase transition in the non-Markovian regime.
Open photonic systems provide a versatile platform for non-Hermitian physics, enabling control over complex spectra, transport, and light-matter interactions. Exceptional points (EPs), at which eigenvalues and eigenvectors coalesce and the governing operator becomes defective, play a central role because they combine branch-point spectral topology, nonanalytic perturbative response, and controllable eigenstate conversion. This Review provides a unified framework for EP photonics by systematically distinguishing exceptional degeneracies according to the underlying operator, spectral variable, boundary conditions, and experimentally accessible observables. We discuss Hamiltonian EPs, absorbing EPs associated with scattering zeros, real-frequency scattering-matrix and Jones-matrix EPs, Bloch and Floquet EPs, and Liouvillian EPs in open quantum systems. We review their spectral topology, static and dynamical encircling, higher-order exceptional structures, and coexistence with bound states in the continuum, together with applications in sensing, lasing, coherent absorption, directional scattering, polarization and wavefront control, nonlinear optics, optical storage, nonreciprocal photonics, and quantum photonics. We also critically assess the current limitations, practical challenges, and future perspectives of EP-based photonic technologies, with particular attention to robustness, noise, scalability, and experimentally measurable performance.