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abstract: Exceptional points in optics Many complex systems operate with loss. Mathematically, these systems can be described as non Hermitian. A property of such a system is that there can exist certain conditions exceptional points where gain and loss can be perfectly balanced and exotic behavior is predicted to occur. Optical systems generally possess gain and loss and so are ideal systems for exploring exceptional point physics. Miri and Alu review the topic of exceptional points in photonics and explore some of the possible exotic behavior that might be expected from engineering such systems. Science, this issue p. eaar7709 BACKGROUND Singularities are critical points for which the behavior of a mathematical model governing a physical system is of a fundamentally different nature compared to the neighboring points. Exceptional points are spectral singularities in the parameter space of a system in which two or more eigenvalues, and their corresponding eigenvectors, simultaneously coalesce. Such degeneracies are peculiar features of nonconservative systems that exchange energy with their surrounding environment. In the past two decades, there has been a growing interest in investigating such nonconservative systems, particularly in connection with the quantum mechanics notions of parity time symmetry, after the realization that some non Hermitian Hamiltonians exhibit entirely real spectra. Lately, non Hermitian systems have raised considerable attention in photonics, given that optical gain and loss can be integrated as nonconservative ingredients to create artificial materials and structures with altogether new optical properties. ADVANCES As we introduce gain and loss in a nanophotonic system, the emergence of exceptional point singularities dramatically alters the overall response, leading to a range of exotic functionalities associated with abrupt phase transitions in the eigenvalue spectrum. Even though such a peculiar effect has been known theoretically for several years, its controllable realization has not been made possible until recently and with advances in exploiting gain and loss in guided wave photonic systems. As shown in a range of recent theoretical and experimental works, this property creates opportunities for ultrasensitive measurements and for manipulating the modal content of multimode lasers. In addition, adiabatic parametric evolution around exceptional points provides interesting schemes for topological energy transfer and designing mode and polarization converters in photonics. Lately, non Hermitian degeneracies have also been exploited for the design of laser systems, new nonlinear optics phenomena, and exotic scattering features in open systems. OUTLOOK Thus far, non Hermitian systems have been largely disregarded owing to the dominance of the Hermitian theories in most areas of physics. Recent advances in the theory of non Hermitian systems in connection with exceptional point singularities has revolutionized our understanding of such complex systems. In the context of optics and photonics, in particular, this topic is highly important because of the ubiquity of nonconservative elements of gain and loss. In this regard, the theoretical developments in the field of non Hermitian physics have allowed us to revisit some of the well established platforms with a new angle of utilizing gain and loss as new degrees of freedom, in stark contrast with the traditional approach of avoiding these elements. On the experimental front, progress in fabrication technologies has allowed for harnessing gain and loss in chip scale photonic systems. These theoretical and experimental developments have put forward new schemes for controlling the functionality of micro and nanophotonic devices. This is mainly based on the anomalous parameter dependence in the response of non Hermitian systems when operating around exceptional point singularities. Such effects can have important ramifications in controlling light in new nanophotonic device designs, which are fundamentally based on engineering the interplay of coupling and dissipation and amplification mechanisms in multimode systems. Potential applications of such designs reside in coupled cavity laser sources with better coherence properties, coupled nonlinear resonators with engineered dispersion, compact polarization and spatial mode converters, and highly efficient reconfigurable diffraction surfaces. In addition, the notion of the exceptional point provides opportunities to take advantage of the inevitable dissipation in environments such as plasmonic and semiconductor materials, which play a key role in optoelectronics. Finally, emerging platforms such as optomechanical cavities provide opportunities to investigate exceptional points and their associated phenomena in multiphysics systems. Ubiquity of non Hermitian systems, supporting exceptional points, in photonics. (A) A generic non Hermitian optical system involving two coupled modes with different detuning, o1,2, and gain loss values, g1,2, coupled at rate of m. The real part of the associated eigenvalues in a two dimensional parameter space of the system, revealing the emergence of an exceptional point (EP) singularity. a1 and a2 are the modal amplitudes. (B to E) A range of different photonic systems, which are all governed by the coupled mode equations. (B) Two coupled lasers pumped at different rates. (C) Dynamical interaction between optical and mechanical degrees of freedom in an optomechanical cavity. (D) A resonator with counter rotating whispering gallery modes. CW, clockwise; CCW, counterclockwise. (E) A thin metasurface composed of coupled nanoantennas as building blocks. CREDITS: IMAGE IN (A) BASED ON A CONCEPT FROM H. HODAEI ET AL. SCIENCE 346, 975 (2014) IMAGE IN (D) BASED ON CONCEPTS FROM W. CHEN ET AL. NATURE 548, 192 (2017) Exceptional points are branch point singularities in the parameter space of a system at which two or more eigenvalues, and their corresponding eigenvectors, coalesce and become degenerate. Such peculiar degeneracies are distinct features of non Hermitian systems, which do not obey conservation laws because they exchange energy with the surrounding environment. Non Hermiticity has been of great interest in recent years, particularly in connection with the quantum mechanical notion of parity time symmetry, after the realization that Hamiltonians satisfying this special symmetry can exhibit entirely real spectra. These concepts have become of particular interest in photonics because optical gain and loss can be integrated and controlled with high resolution in nanoscale structures, realizing an ideal playground for non Hermitian physics, parity time symmetry, and exceptional points. As we control dissipation and amplification in a nanophotonic system, the emergence of exceptional point singularities dramatically alters their overall response, leading to a range of exotic optical functionalities associated with abrupt phase transitions in the eigenvalue spectrum. These concepts enable ultrasensitive measurements, superior manipulation of the modal content of multimode lasers, and adiabatic control of topological energy transfer for mode and polarization conversion. Non Hermitian degeneracies have also been exploited in exotic laser systems, new nonlinear optics schemes, and exotic scattering features in open systems. Here we review the opportunities offered by exceptional point physics in photonics, discuss recent developments in theoretical and experimental research based on photonic exceptional points, and examine future opportunities in this area from basic science to applied technology.
author_names: Mohammad-Ali Miri

Andrea Alu
title: Exceptional points in optics and photonics
venue: Science
year: 2019

abstract: Optics Many complex systems operate with loss. Mathematically, these systems can be described as non Hermitian. A property of such a system is that there can exist certain conditions exceptional points where gain and loss can be perfectly balanced and exotic behavior is predicted to occur. Optical systems generally possess gain and loss and so are ideal systems for exploring exceptional point physics. Miri and Alu review the topic of exceptional points in photonics and explore some of the possible exotic behavior that might be expected from engineering such systems. Science this issue p. [eaar7709][1] [1] /lookup/doi/10.1126/science.aar7709
author_names: Ian S Osborne
title: Exceptional points in optics
year: 2019

abstract: The introduction of parity time (PT) symmetry in optics and photonics has initiated intense activities exploring the exotic properties of these structures, eventually leading to the more general notion of non Hermitian photonics. Efforts to understand the behavior of these systems have revealed a host of distinct features originating from the unusual character of their eigenspectra and eigenstates. These include for example, spontaneous symmetry breaking, bandgap merging, laser self termination, unidirectional invisibility, and ultra sensitivity to external perturbations. A central notion pertinent to all these effects is the concept of exceptional points (EPs) Also known as branch points, EPs are non Hermitian spectral singularities that arise when two (or more) eigenvalues and their corresponding eigenstates become identical. While exceptional points of order two have been studied thoroughly at both the theoretical and experimental level, higher order EPs are attracting attention only recently. Here we discuss a systematic approach based on a recursive bosonic quantization scheme for generating discrete photonic networks that exhibit exceptional points of any arbitrary order. We also discuss the spectral properties and the extreme dynamics near these singularities as well as their physical implementation in various photonic platforms.
author_names: Mohammad H Teimourpour

Q Zhong

Mercedeh Khajavikhan

Ramy El-Ganainy
title: Higher Order Exceptional Points in Discrete Photonics Platforms
year: 2018

abstract: Non conservative physical systems admit a special kind of spectral degeneracy, known as exceptional point (EP) at which eigenvalues and eigenvectors of the corresponding non Hermitian Hamiltonian coalesce. Dynamical parametric encircling of the EP can lead to non adiabatic evolution associated with a state flip, a sharp transition between the resonant modes. Physical consequences of the dynamical encircling of EPs in open dissipative systems have been explored in optics and photonics. Building on the recent progress in understanding the parity time \mathscr{P}\mathscr{T} PT symmetric dynamics in spin systems, we use topological properties of EPs to implement chiral non reciprocal transmission of a spin through the material with non uniform magnetization, like helical magnet. We consider an exemplary system, spin torque driven single spin described by the time dependent non Hermitian Hamiltonian. We show that encircling individual EPs in a parameter space results in non reciprocal spin dynamics and find the range of optimal protocol parameters for high efficiency asymmetric spin filter based on this effect. Our findings offer a platform for non reciprocal spin devices for spintronics and magnonics.
author_names: Alexey Galda

Valerii M Vinokur
title: Exceptional points in classical spin dynamics
venue: Scientific Reports
year: 2019

abstract: In the past decade, the concept of parity time \mathcal{PT} symmetry, originally introduced in non Hermitian extensions of quantum mechanical theories, has come into thinking of photonics, providing a fertile ground for studying, observing, and utilizing some of the peculiar aspects of \mathcal{PT} symmetry in optics. Together with related concepts of non Hermitian physics of open quantum systems, such as non Hermitian degeneracies (exceptional points) and spectral singularities, \mathcal{PT} symmetry represents one among the most fruitful ideas introduced in optics in the past few years. Judicious tailoring of optical gain and loss in integrated photonic structures has emerged as a new paradigm in shaping the flow of light in unprecedented ways, with major applications encompassing laser science and technology, optical sensing, and optical material engineering. In this perspective, I review some of the main achievements and emerging areas of \mathcal{PT} symmetric and non Hermtian photonics, and provide an outline of challenges and directions for future research in one of the fastest growing research area of photonics.
author_names: Stefano Longhi
title: Parity Time Symmetry meets Photonics: A New Twist in non Hermitian Optics
year: 2017

abstract: The prospect of judiciously utilizing both optical gain and loss has been recently suggested as a means to control the flow of light. This proposition makes use of some newly developed concepts based on nonHermiticity and parity time (PT) symmetry ideas first conceived within quantum field theories. By harnessing such notions, recent works indicate that novel synthetic structures and devices with counterintuitive properties can be realized, potentially enabling new possibilities in the field of optics and integrated photonics. Non Hermitian degeneracies, also known as exceptional points (EPs) have also emerged as a new paradigm for engineering the response of optical systems. Such non Hermitian degeneracies are by themselves interesting entities. As opposed to standard degeneracies, at an EP, not only do the eigenvalues coalesce but so do the corresponding eigenstates. In fact, at such bifurcations, the relevant eigenvectors collapse on each other and as a result, the dimensionality of the system is abruptly reduced. This in turn has a profound effect on how the system responds to a perturbation. In this case one can show that when a perturbation of strength e acts on an Nth order EP (when N eigenvalues and eigenvectors merge) the resulting eigenvalue (frequency) splitting is now proportional to e. This indicates that the sensitivity of a set up can be enhanced by several orders of magnitude by exploiting the physics of EPs. Among many different non conservative photonic configurations, parity time (PT) symmetric arrangements are of particular interest since they provide an excellent platform to explore the physics of EPs for enhanced sensing applications. The enhanced response experimentally observed in a PTternary sensing system is shown in Fig. 1. Another intriguing property of EPs is the way they respond if the system parameters vary in such a way so as to either enclose or not enclose such points. If the contour does encircle an EP in a quasi static fashion, one can then show that the eigenvalues swap with each other and as a result, the eigenvectors follow suit. What makes this possible is the structure of the Riemann sheets in the complex domain upon which the eigenvalues move (Fig. 2) The situation is entirely different under dynamic conditions. If the parameters (coupling, detuning, birefringence etc. of a two level nonHermitian system are dynamically varied in a judicious manner, in such a way that the contour enclosing the exceptional point follows a clockwise path, then only one of the local eigenvectors will appear at the output while the other eigenstate will be drastically suppressed. This adiabatic arrangement can faithfully convert any arbitrary mixture of states into one of the local states with a nearly unity transmission coefficient. In this talk, we provide an overview of recent developments in this field. The use of other type symmetries in photonics will be also discussed. Fig. 1. (a) A ternary PT symmetric laser sensor. (b) its corresponding enhanced response. Images from Hodaei et al. Nature 548, 187 191 (2017)
author_names: Demetrios N Christodoulides
title: Parity Time and other Symmetries in Optics
year: 2018

abstract: Advances in topological photonics and non Hermitian optics have drastically changed our perception on how interdisciplinary concepts may empower unprecedented applications. Bridging the two areas could uncover the reciprocity between topology and non Hermiticity in complex systems. So far, such endeavors have focused mainly on linear optics regime. Here, we establish a nonlinear non Hermitian topological platform for control of parity time (PT) symmetry and topological edge states. Experimentally, we demonstrate that optical nonlinearity effectively modulates the gain and loss of a topological interface waveguide in a non Hermitian Su Schrieffer Heeger lattice, leading to switching between PT and non PT symmetric regimes accompanied by destruction and restoration of topological zero modes. Theoretically, we examine the fundamental issue of the interplay between two antagonistic effects: the sensitivity close to exceptional points and the robustness of non Hermitian topological modes. Realizing single channel control of global PT symmetry via local nonlinearity may herald new possibilities for light manipulation and unconventional device applications.
author_names: Shiqi Xia

Dimitrios Kaltsas

Daohong Song

Ioannis Komis

Jingjun Xu

Alexander Szameit

Hrvoje Buljan

Konstantinos G Makris

Zhigang Chen
title: Nonlinear control of PT symmetry and non Hermitian topological states
year: 2020

abstract: The ability to generate new counter intuitive optical phenomenon through the control of gain, loss and modal coupling in non Hermitian photonic systems has launched a new paradigm for optics and photonics research [1] Particularly interesting is the role of so called 'exceptional points' of the nonHermitian Hamiltonian, a non Hermitian analogue of a quantum degeneracy. To date, most of the achievements in this area have relied on precise, rigidly controlled system design. In order, to take full advantage of the possibilities presented by this new field, it would be beneficial to incorporate materials with more variable control and flexibility. Colloidal Quantum Dots (CQD) are such a material class: their properties, such as gain and loss or spectral gain position and bandwidth, may be controlled by chemical design or by manipulation of the solid state packing arrangement. Meanwhile the inherent ease in processing and scalability of these materials allows for multiple photonic elements to be fabricated. In this talk, I will present investigations on circular microdisk lasers based on core/alloyedshell CdSe/Cd1 xZnxSe1 ySy CQDs [2] Please attend the lecture to learn more.
author_names: Evan Lafalce
title: UNIVERSITY OF UTAH Non Hermitian Laser Mode Dynamics in Coupled Quantum Dot Microdisks
year: 2018

abstract: Non Hermitian quantum mechanics with parity time (PT) symmetry is presently gaining great interest, especially within the fields of photonics and optics. Here, we give a brief overview of low dimensional semiconductor nanodevices using the example of a quantum dot with input and output leads, which are mimicked by imaginary potentials for gain and loss, and how wave functions, particle flow, coalescence of levels and associated breaking of PT symmetry may be analysed within such a framework. Special attention is given to the presence of exceptional points and symmetry breaking. Related features for musical string instruments and 'wolf notes' are outlined briefly with suggestions for further experiments.
author_names: K -F Berggren

Felix Tellander

Irina I Yakimenko
title: Basic modelling of transport in 2D wave mechanical nanodots and billiards with balanced gain and loss mediated by complex potentials.
venue: Journal of physics. Condensed matter an Institute of Physics journal
year: 2018

abstract: In recent years, the concept of parity time (PT) symmetry has received considerable attention in the field of optics and photonics. In PT symmetric arrangements, the interaction between gain/loss contrast and coupling leads to the formation of exceptional points in parameter space. At these junctures, not only the eigenvalues but also the eigenvectors tend to merge, resulting in a sudden reduction of the dimensionality of the eigen space. Consequently, in the vicinity of such points, the eigenfrequencies are strongly affected by external perturbationsas the system regains its original dimensionality. This unique behavior can be utilized to fundamentally enhance the sensitivity of micro resonators. Here, we experimentally investigate this effect in integrated semiconductor PT symmetric microring lasers that are biased at exceptional points. Using this arrangement, we demonstrate >10 fold enhancement in sensitivity. Our results also show that unlike standard microcavities, the parity time symmetric system responds to the square root of the perturbation. Our work provides a new avenue for enhancing the sensitivity of optical integrated sensors.
author_names: H Hodaei

A U Hassan

Hipolito Garcia-Gracia

William E Hayenga

Demetrios N Christodoulides

Mercedeh Khajavikhan
title: Enhanced sensitivity in PT symmetric coupled resonators
venue: LASE
year: 2017