We show that the complex $\cal PT$-symmetric periodic potential $V(x) = - ({\rm i} ξ\sin 2x + N)^2$, where $ξ$ is real and $N$ is a positive integer, is quasi-exactly solvable. For odd values of $N \ge 3$, it may lead to exceptional points depending upon the strength of the coupling parameter $ξ$. The corresponding Schrödinger equation is also shown to go over to the Mathieu equation asymptotically. The limiting value of the exceptional points derived in our scheme is consistent with known branch-point singularities of the Mathieu equation.
Starting with the Gaudin-like Bethe ansatz equations associated with the quasi-exactly solved (QES) exceptional points of the asymmetric quantum Rabi model (AQRM) a spectral equivalence is established with QES hyperbolic Schrödinger potentials on the line. This leads to particular QES Pöschl-Teller potentials. The complete spectral equivalence is then established between the AQRM and generalised Pöschl-Teller potentials. This result extends a previous mapping between the symmetric quantum Rabi model and a QES Pöschl-Teller potential. The complete spectral equivalence between the two systems su…
The dynamics of an isolated quantum system is coherent and unitary. Weak coupling to the environment leads to decoherence, which is traditionally modeled with a Lindblad equation for the system's density matrix. Starting from a pure state, such a system approaches a steady state (mixed or otherwise) in an underdamped or overdamped manner. This transition occurs at an eigenvalue degeneracy of a Lindblad superoperator, called an exceptional point (EP), where corresponding eigenvectors coalesce. Recent years have seen an explosion of interest in creating exceptional points in a truly quantum doma…
Recently, there has been a lot of activity in the research field of topological non-Hermitian physics, partly driven by fundamental interests and partly driven by applications in photonics. However, despite these activities, a general classification and characterization of non-Hermitian Dirac models that describe the experimental systems is missing. Here, we present a systematic investigation of massive Dirac models on periodic lattices, perturbed by general non-Hermitian terms. We find that there are three different types of non-Hermitian terms. For each case we determine the bulk exceptional…
Anomalous transport in quasiperiodic lattices: emergent exceptional points at band edges and log-periodic oscillations
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所凝聚态物理国家实验室）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）
Tuning topological phases and exceptional points in a non-Hermitian Rice Mele model beyond nearest neighbors
Donostia International Physics Center (DIPC)（圣塞巴斯蒂安国际物理中心）；Chemistry Faculty (UPV/EHU)（巴斯克地区理工大学化学学院）；IKERBASQUE, Basque Foundation for Science（伊克拉巴斯基奎巴斯克科学基金会）；Wrocław University of Science and Technology（弗罗茨瓦夫科技大学）
We study optomechanically induced transparency in a microresonator coupled with nanoparticles. By tuning the relative angle of the nanoparticles, exceptional points (EPs) emerge periodically in this system and thus strongly modify both the transmission rate and the group delay of the signal. As a result, controllable slow-to-fast light switching can be achieved by manipulating external nanoparticles. This provides a new way to engineer EP-assisted optomechanical devices for applications in optical communications and signal processing.
We calculate the exceptional points of the eigenvalues of several parameter-dependent Hamiltonian operators of mathematical and physical interest. We show that the calculation is greatly facilitated by the application of the discriminant to the secular determinant. In this way the problem reduces to finding the roots of a polynomial function of just one variable, the parameter in the Hamiltonian operator. As illustrative examples we consider a particle in a one-dimensional box with a polynomial potential, the periodic Mathieu equation, the Stark effect in a polar rigid rotor and in a polar sym…
Parity-time (PT)-symmetric Hamiltonians have widespread significance in non-Hermitian physics. A PT-symmetric Hamiltonian can exhibit distinct phases with either real or complex eigenspectrum, while the transition points in between, the so-called exceptional points, give rise to a host of critical behaviors that holds great promise for applications. For spatially periodic non-Hermitian systems, PT symmetries are commonly characterized and observed in line with the Bloch band theory, with exceptional points dwelling in the Brillouin zone. Here, in nonunitary quantum walks of single photons, we …
Periodically driven non-Hermitian systems can exhibit rich topological band structure and non-Hermitian skin effect, without analogs in their static or Hermitian counterparts. In this work we investigate the exceptional band-touching points in the Floquet quasi-energy bands, the topological characterization of such exceptions points and the Floquet non-Hermitian skin effect (FNHSE). Specifically, we exploit the simplicity of periodically quenched two-band systems in one dimension or two dimensions to analytically obtain the Floquet effective Hamiltonian as well as locations of the many excepti…
Exceptional points (EPs), at which more than one eigenvalue and eigenvector coalesce, are unique spectral features of Non-Hermiticity (NH) systems. They exist widely in open systems with complex energy spectra. We experimentally demonstrate the appearance of paired EPs in a periodical driven degenerate optical cavity along the synthetic orbital angular momentum (OAM) dimension with a tunable parameter. The complex-energy band structures and the key features of EPs, i.e. their Fermi arcs, parity-time symmetry breaking transition, energy swapping, and half-integer band windings are directly obse…
Minimal, open quantum systems that are governed by non-Hermitian Hamiltonians have been realized across multiple platforms in the past two years. Here we investigate the dynamics of open systems with Hermitian or anti-Hermitian Hamiltonians, both of which can be implemented in such platforms. For a single system subject to unitary and thermal dynamics in a periodic manner, we show that the corresponding Floquet Hamiltonian has a rich phase diagram with numerous exceptional-point (EP) degeneracy contours. This protocol can be used to realize a quantum Hatano-Nelson model that is characterized b…
We report the first experimental observation of multiple transitions showing the emergence and disappearance of slowly decaying eigenmodes in a dissipative, Floquet electronic system with synthetic components. Conventional wisdom has it that such transitions occur at exceptional points, and avoided-level-crossing driven phenomena in purely dissipative systems are formerly unexplored. Remarkably, in our system, the slowly decaying eigenmodes emerge without exceptional points. Our experimental setup makes use of an LC oscillator inductively coupled to an RLC oscillator, where the time-periodic (…
We study the behavior of the out-of-time-ordered correlator (OTOC) in a non-Hermitian quantum Ising system. We show that the OTOC can diagnose not only the ground state exceptional point, which hosts the Yang-Lee edge singularity, but also the \textit{dynamical} exceptional point at the excited state. We find that the evolution of the OTOC in the parity-time symmetric phase can be divided into two stages: in the short-time stage, the OTOC oscillates periodically, and when the parameter is near the ground state exceptional point, this oscillation behavior can be described by both the scaling th…
Emergence of chaos with exceptional points in reset-driven Floquet dynamics
Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California（南加州大学明何电机工程学院）；Department of Physics and Astronomy, University of Southern California（南加州大学物理与天文学系）
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 -symmetry non-Hermitian SSH model with open boundary condition based on exact approximate solution. We show that the long-wave length standing-wave modes can be achieved within the linear dispersion region when the system is tuned at the exceptional point (EP). The whole Hilbert space can be decomposed into two quasi-Hermitian subspaces, which are consisted of positive and negative energy le…
Certain real parameters of a Hamiltonian, when continued to complex values, can give rise to singular points called exceptional points ($EP$'s), where two or more eigenvalues coincide and the complexified Hamiltonian becomes non-diagonalizable. We show that for a generic $d$-dimensional topological superconductor / superfluid with a chiral symmetry, one can find $EP$'s associated with the chiral zero energy Majorana fermions bound to a topological defect / edge. Exploiting the chiral symmetry, we propose a formula for counting the number ($n$) of such chiral zero modes. We also establish the c…
Parity-time ($\mathcal{PT}$) symmetry plays an important role both in non-Hermitian and topological systems. In non-Hermitian systems $\mathcal{PT}$ symmetry can lead to an entirely real energy spectrum, while in topological systems $\mathcal{PT}$ symmetry gives rise to stable and protected Dirac points. Here, we study a $\mathcal{PT}$-symmetric system which is both non-Hermitian and topological, namely a $\mathcal{PT}$-symmetric Dirac semimetal with non-Hermitian perturbations in three dimensions. We find that, in general, there are only two types of symmetry allowed non-Hermitian perturbatio…
Exceptional-point-constrained locking of boundary-sensitive topological transitions in chiral non-Hermitian SSH-type lattices
Institute of Theoretical Physics and State Key Laboratory of Quantum Optics Technologies and Devices, Shanxi University（山西大学理论物理研究所与量子光学技术与器件国家重点实验室）；Department of Physics, Fudan University（复旦大学物理系）
Non-Hermitian, tight-binding $\mathcal{PT}$-symmetric models are extensively studied in the literature. Here, we investigate two forms of non-Hermitian Hamiltonians to study the $\mathcal{PT}$-symmetry breaking thresholds and features of corresponding surfaces of exceptional points (EPs). They include one-dimensional chains with uniform or 2-periodic tunnelling amplitudes, one pair of balanced gain and loss potentials $Δ\pmıγ$ at parity-symmetric sites, and periodic or open boundary conditions. By introducing a Hermitian detuning potential, we obtain the dependence of the $\mathcal{PT}$-thresh…
The analytically tractable zoo of similarity-induced exceptional structures
Max Planck Institute for the Science of Light（马克斯·普朗克光科学研究所）；Friedrich-Alexander Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡弗里德里希·亚历山大大学）；University of Ottawa（渥太华大学）；Utrecht University（乌得勒支大学）；Uppsala University（乌普萨拉大学）
