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In recent years, notions drawn from non-Hermitian physics and parity–time (PT) symmetry have attracted considerable attention. In particular, the realization that the interplay between gain and loss can lead to entirely new and unexpected features has initiated an intense research effort to explore non-Hermitian systems both theoretically and experimentally. Here we review recent progress in this emerging field, and provide an outlook to future directions and developments. This Review Article outlines the exploration of the interplay between parity–time symmetry and non-Hermitian physics in optics, plasmonics and optomechanics.

Non-Hermitian physics and PT symmetry

The development of new artificial structures and materials is today one of the major research challenges in optics. In most studies so far, the design of such structures has been based on the judicious manipulation of their refractive index properties. Recently, the prospect of simultaneously using gain and loss was suggested as a new way of achieving optical behaviour that is at present unattainable with standard arrangements. What facilitated these quests is the recently developed notion of 'parity-time symmetry' in optical systems, which allows a controlled interplay between gain and loss. Here we report the experimental observation of light transport in large-scale temporal lattices that are parity-time symmetric. In addition, we demonstrate that periodic structures respecting this symmetry can act as unidirectional invisible media when operated near their exceptional points. Our experimental results represent a step in the application of concepts from parity-time symmetry to a new generation of multifunctional optical devices and networks.

Parity–time synthetic photonic lattices

Journal of Statistical Mechanics: theory and experiment

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.

Exceptional points in optics and photonics.

When quantum heat engines or refrigerators operate in the over-thermalization regime, quantum effects are not expected as the thermal baths eliminates all coherences. In this work we show that even in this seemingly non-quantum regime, it is possible to harvest coherence before it is erased, and obtain non classical effects. While the engine is not affected by the coherence harvesting, the extracted coherence can be used to cool external particles and reduce their entropy (the energy of the auxiliary particles is not modified). On top of cooling the extracted coherence can activate passive states. Coherence extraction also has an impact on the basic theory of heat machines. With coherence extraction, the standard expression for the total entropy production -Qc/Tc-Qh/Th is replaced by a simple relative entropy expression, that is orders of magnitude smaller in the regime of interest. Consequently, coherence extraction enables reversibility that is otherwise impossible. In addition, we present a collective engine operation scheme where coherence is recycled and shared between engines. This leads to an interesting performance boost where the total power scales quadratically with the number of engines, rather than linearly. Finally, a scheme for resource accounting is presented. It is demonstrated that coherence extraction and coherence injection extend the scope of known quantum thermal machines and add new modes of operation.

Coherence recycling, collective operation, and coherence induced reversibility in quantum heat engines

Non-unitary operations generated by an effective non-Hermitian Hamiltonian can be used to create quantum state manipulations which are impossible in Hermitian quantum mechanics. These operations include state preparation (or cooling) and non-orthogonal state discrimination. In this work we put a lower bound on the resources needed for the construction of some given non-unitary evolution. Passive systems are studied in detail and a general feature of such a system is derived. After interpreting our results using the singular value decomposition, several examples are studied analytically. In particular, we put a lower bound on the resources needed for non-Hermitian state preparation and non-orthogonal state discrimination.

Resources needed for non-unitary quantum operations

The Floquet resonance states of a periodically driven system are obtained using a special weighted power method which forces the dynamical symmetry into regular wave propagation methods. We demonstrate the utility of this method for the calculation of high harmonic generation spectra in rare gases. By using this method the resonance Floquet state which dominates the spectrum is obtained directly at a low computational cost. This may allow the calculation of resonance Floquet states in parameter regimes not accessible before. Implementation involves only slight adjustments to the standard wave propagation codes. The method's unique advantage is explained and demonstrated.

Calculation of high-order harmonic generation spectrum and Floquet resonance states using the dynamical symmetry

Unambiguous-nonorthogonal-state discrimination has a fundamental importance in quantum information. Moreover, it can be used for entanglement distillation and secure communication. The discrimination is carried out by a positive operator-valued measure (POVM) generalized measurement, which is typically implemented by coupling the system to an ancilla. We find a trade-off between the needed energy resources and the evolution time needed to implement the POVM and express it in terms of an actionlike cost inequality. We find the realization that minimizes this actionlike cost and show that, in this case, the cost is determined by the maximal population transfer from the system to the ancilla. We demonstrate our findings in an example of a three-level system coupled to a laser.

Time-energy trade-off in unambiguous-state-discrimination positive operator-valued measures

Quantum error mitigation (QEM) comprises methods for suppressing noise in quantum computers without involving the presently impractical hardware overhead associated with quantum error correction codes. Unfortunately, current QEM techniques are limited to weak noise or lack scalability. We develop a QEM method called KIK that adapts to the noise strength of the target device and therefore can handle moderate-to-strong noise. The implementation of the method is experimentally simple, and the required number of quantum circuits is independent of the size of the system. Furthermore, we show that it can be integrated with randomized compiling for handling both incoherent and coherent noise. We demonstrate our findings in the IBM quantum computers and through numerical simulations.

Raam Uzdin

Papers

Coherence recycling, collective operation, and coherence induced reversibility in quantum heat engines

Resources needed for non-unitary quantum operations

Calculation of high-order harmonic generation spectrum and Floquet resonance states using the dynamical symmetry

Time-energy trade-off in unambiguous-state-discrimination positive operator-valued measures

Adaptive quantum error mitigation using pulse-based inverse evolutions