non-Hermitian Parity-time Research Articles

Parity-time and anti-parity-time symmetries in heat transfer.

This perspective briefly reviews the recent developments of non-Hermitian parity-time and anti-parity-time physics in dissipative heat transfer systems, highlighting their potentials in novel functional thermal devices in the future.

Observation of energy localization in nonlinear non-Hermitian systems

Energy localization in Hermitian systems has been utilized to generate ultra-sensitivity. Here, we report the interplay between non-Hermitian parity-time (PT) symmetry breaking and the mode localization transition. In our scheme, a PT-symmetric system consists of two coupled LC (inductor–capacitor) resonators: one has a linear loss and the other has a saturated gain described by a nonlinear model. The nonlinear gain is initially set to be slightly higher than the loss, and the system is operated at the exact PT-symmetric phase close to an exceptional point. The capacitance variation applied on the loss resonator, i.e., perturbation, causes PT-symmetry to break, generating complex frequencies. As a result of nonlinear gain, the resonator will grow to reach its steady state and saturate out the gain. This stable oscillation eliminates the complex frequencies, and the mode is ultimately localized at the gain side. We have observed that the voltage amplitude of the gain resonator due to the perturbation has experienced drastic changes. The amplitude ratio before and after the perturbation is sensitive to the perturbation. Our results provide an approach to study perturbation-driven localization phenomena in a PT-symmetric system and pave the way for sensors with ultrahigh sensitivity.

Independent control of circularly polarized light with exceptional topological phase coding metasurfaces

Exceptional points, as degenerate points of non-Hermitian parity-time symmetric systems, have many unique physical properties. Due to its flexible control of electromagnetic waves, a metasurface is frequently used in the field of nanophotonics. In this work, we developed a parity-time symmetric metasurface and implemented the 2π topological phase surrounding an exceptional point. Compared with Pancharatnam-Berry phase, the topological phase around an exceptional point can achieve independent regulation of several circular polarization beams. We combined the Pancharatnam-Berry phase with the exceptional topological phase and proposed a composite coding metasurface to achieve reflection decoupling of different circular polarizations. This work provides a design idea for polarimetric coding metasurfaces in the future.

Wirelessly powered motor operation in dynamic scenarios using non-Hermitian parity-time symmetry

Motors arise as a heart of the mobility society, and wirelessly operated motors may improve our standard of living. Wireless power transfer in the kilohertz and megahertz range has been extensively explored, finding various potential applications in consumer electronics, electric vehicles, and medical implants. However, stable operation of wirelessly powered motors remains challenging due to voltage fluctuations for motors occurring in dynamic scenarios, e.g., the rotating speed of the motors is varied. Here, we theoretically and experimentally demonstrate the operation of a motor, where the power is wirelessly transferred via coils, is robust against the rotating speed by employing the analogy with non-Hermitian parity-time (PT) symmetry. In addition, our system is robust for misalignment of the coils. Our results open up opportunities for the robust operation of motors via wireless power transfer in dynamic scenarios towards autonomous vehicles.

Role of losses and coupling in bright-dark metasurfaces mimicking non-Hermitian parity-time symmetric systems

All real systems are essentially non-conservative in nature, which is non-Hermitian in the fundamental premises of the quantum realm. Such open systems obeying parity-time (PT) symmetry can offer intriguing physics, which has sparked much attention in recent years. The thoughtful interplay of loss and gain in PT-symmetric systems can tune the eigenstates from real to complex space passing through a singular point known as an exceptional point (EP) with concurrently coalescing the discrete eigenstates. In this regard, artificially engineered near-field coupled metasurfaces provide a remarkable podium to introduce as well as control loss and coupling strength simultaneously, manifesting a practical channel to attain EP. Motivated by this notion, we have theoretically explored near-field coupled dark-bright-mode assisted terahertz metasurfaces imitating a typical non-Hermitian PT-symmetric system. Structurally, such metasurfaces are realized by utilizing a pair of orthogonally twisted near-field coupled split-ring resonators (bright-dark-resonators). We have meticulously examined such a system with two distinctive approaches, i.e., the loss parameter and inter-resonator coupling strength. Eventually, our exploration demonstrates that judicious modulation of the loss parameter can invoke an undemanding route to approach the other wisely elusive exceptional point. We believe that this comprehensive study of non-Hermitian quantum systems in a photonic context can lead to the development of accurate and precise ultra-sensitive sensors besides other futuristic photonic devices.

Electromagnetically unclonable functions generated by non-Hermitian absorber-emitter.

Physically unclonable functions (PUFs) are a class of hardware-specific security primitives based on secret keys extracted from integrated circuits, which can protect important information against cyberattacks and reverse engineering. Here, we put forward an emerging type of PUF in the electromagnetic domain by virtue of the self-dual absorber-emitter singularity that uniquely exists in the non-Hermitian parity-time (PT)-symmetric structures. At this self-dual singular point, the reconfigurable emissive and absorptive properties with order-of-magnitude differences in scattered power can respond sensitively to admittance or phase perturbations caused by, for example, manufacturing imperfectness. Consequently, the entropy sourced from inevitable manufacturing variations can be amplified, yielding excellent PUF security metrics in terms of randomness and uniqueness. We show that this electromagnetic PUF can be robust against machine learning-assisted attacks based on the Fourier regression and generative adversarial network. Moreover, the proposed PUF concept is wavelength-scalable in radio frequency, terahertz, infrared, and optical systems, paving a promising avenue toward applications of cryptography and encryption.

Extension of Noether’s theorem in $${\cal P}{\cal T}$$-symmetry systems and its experimental demonstration in an optical setup

Noether's theorem is one of the fundamental laws in physics, relating the symmetry of a physical system to its constant of motion and conservation law. On the other hand, there exist a variety of non-Hermitian parity-time (PT)-symmetric systems, which exhibit novel quantum properties and have attracted increasing interest. In this work, we extend Noether's theorem to a class of significant PT -symmetric systems for which the eigenvalues of the PT-symmetric Hamiltonian H change from purely real numbers to purely imaginary numbers,and introduce a generalized expectation value of an operator based on biorthogonal quantum mechanics. We find that the generalized expectation value of a time-independent operator is a constant of motion when the operator presents a standard symmetry in the PT -symmetry unbroken regime, or a chiral symmetry in the PT-symmetry broken regime. In addition, we experimentally investigate the extended Noether's theorem in PT -symmetric single-qubit and two-qubit systems using an optical setup. Our experiment demonstrates the existence of the constant of motion and reveals how this constant of motion can be used to judge whether the PT -symmetry of a system is broken. Furthermore, a novel phenomenon of masking quantum information is first observed in a PT -symmetric two-qubit system. This study not only contributes to full understanding of the relation between symmetry and conservation law in PT -symmetric physics, but also has potential applications in quantum information theory and quantum communication protocols.

Observation of dark edge states in parity-time-symmetric quantum dynamics.

Topological edge states arise in non-Hermitian parity-time ([Formula: see text])-symmetric systems, and manifest themselves as bright or dark edge states, depending on the imaginary components of their eigenenergies. As the spatial probabilities of dark edge states are suppressed during the non-unitary dynamics, it is a challenge to observe them experimentally. Here we report the experimental detection of dark edge states in photonic quantum walks with spontaneously broken [Formula: see text] symmetry, thus providing a complete description of the topological phenomena therein. We experimentally confirm that the global Berry phase in [Formula: see text]-symmetric quantum-walk dynamics unambiguously defines topological invariants of the system in both the [Formula: see text]-symmetry-unbroken and -broken regimes. Our results establish a unified framework for characterizing topology in [Formula: see text]-symmetric quantum-walk dynamics, and provide a useful method to observe topological phenomena in [Formula: see text]-symmetric non-Hermitian systems in general.

Bifunctional sensing based on an exceptional point with bilayer metasurfaces.

Exceptional points (EPs), the critical phase transition points of non-Hermitian parity-time (PT) systems, exhibit many novel physical properties and associated applications, such as ultra-sensitive detection of perturbations. Here, a bilayer metasurface with two orthogonally oriented split-ring resonators (SRRs) is proposed and a phase transition of the eigenpolarization states is introduced via changing the conductivity of vanadium dioxide (VO2) patch integrated into the gap of one SRR. The metasurface possesses a passive PT symmetry and an EP in polarization space is observed at a certain conductivity of the VO2. Two sensing schemes with the metasurface are proposed to achieve high-sensitivity sensing of temperature and refractive index in the terahertz (THz) range. The metasurface is promising for applications in THz biosensing and polarization manipulation.

Nonreciprocal transmission of electromagnetic waves with nonlinear active plasmonic metasurfaces

Nonreciprocity is important for optical information processing, full-duplex communications, and protection of sensitive laser equipment from unwanted reflections. However, it is very challenging to obtain strong nonreciprocal response in optical frequencies, especially when nanoscale configurations are considered. In this work, we solve this problem by demonstrating a compact bifacial plasmonic metasurface that acts as an ultrathin nonreciprocal transmission filter at near-infrared frequencies. The proposed nanostructure is simple to be practically implemented, since it is composed of two silver nanostripes with different dimensions placed on both sides of an ultrathin active dielectric spacing layer. The introduced gain leads to an exceptional point formation in the linear operation regime, where unidirectional perfect transmission due to loss compensation is achieved from the presented non-Hermitian parity-time symmetric nanoscale system. The demonstrated plasmonic metasurface breaks the Lorentz reciprocity law at the nanoscale when its nonlinear response is considered, mainly due to its spatially asymmetric geometry combined with enhanced nonlinearity. Strong and broadband nonreciprocity with unity transmission contrast from opposite directions is realized under relatively low-input laser intensity values. Furthermore, asymmetric nonlinear third-harmonic generation with a pronounced contrast is also illustrated by the same nonreciprocal metasurface design. The findings of this work can lead to the realization of various self-induced nonreciprocal nanophotonic configurations, such as integrated ultrafast switches, ultrathin protective coatings, and asymmetric directional imaging devices.

Nonlinear all-optical modulator based on non-Hermitian PT symmetry

All-optical modulators with ultrahigh speed are in high demand due to the rapid development of optical interconnection and computation. However, due to weak photon–photon interaction, the advancement of all-optical modulators is consequently hampered by the large footprint and high power consumption. In this work, the enhanced sensitivity around an exceptional point (EP) from parity-time (PT) symmetry theory is initiatively introduced into a nonlinear all-optical modulator design. Further, a non-Hermitian all-optical modulator based on PT symmetry is proposed, which utilizes the large Kerr nonlinearity from indium tin oxide (ITO) in its epsilon-near-zero (ENZ) region. The whole system is expected to operate around EP, giving rise to the advantages of nanoscale integration and large modulation depth. This presented modulator with high efficiency and high-speed all-optical control can be commendably extended to the design methodology of various nanostructures and further prompt the development of all-optical signal processing.

Experimental demonstration of coherence flow in PT- and anti-PT-symmetric systems

Non-Hermitian parity-time ({{{{{{{mathcal{P}}}}}}}}{{{{{{{mathcal{T}}}}}}}}) and anti-parity-time ({{{{{{{mathcal{APT}}}}}}}})-symmetric systems exhibit novel quantum properties and have attracted increasing interest. Although many counterintuitive phenomena in {{{{{{{mathcal{P}}}}}}}}{{{{{{{mathcal{T}}}}}}}}- and {{{{{{{mathcal{APT}}}}}}}}-symmetric systems were previously studied, coherence flow has been rarely investigated. Here, we experimentally demonstrate single-qubit coherence flow in {{{{{{{mathcal{P}}}}}}}}{{{{{{{mathcal{T}}}}}}}}- and {{{{{{{mathcal{APT}}}}}}}}-symmetric systems using an optical setup. In the symmetry unbroken regime, we observe different periodic oscillations of coherence. Particularly, we observe two complete coherence backflows in one period in the {{{{{{{mathcal{P}}}}}}}}{{{{{{{mathcal{T}}}}}}}}-symmetric system, while only one backflow in the {{{{{{{mathcal{APT}}}}}}}}-symmetric system. Moreover, in the symmetry broken regime, we observe the phenomenon of stable value of coherence flow. We derive the analytic proofs of these phenomena and show that most experimental data agree with theoretical results within one standard deviation. This work opens avenues for future study on the dynamics of coherence in {{{{{{{mathcal{P}}}}}}}}{{{{{{{mathcal{T}}}}}}}}- and {{{{{{{mathcal{APT}}}}}}}}-symmetric systems.

Nonorientability-induced PT phase transition in ladder lattices

We study parity-time (PT) phase transitions in the energy spectra of ladder lattices caused by the interplay between non-orientability and non-Hermitian PT symmetry. The energy spectra show level crossings in circular ladder lattices with increasing on-site energy gain-loss because of the orientability of a normal strip. However, the energy levels show PT phase transitions in PT-symmetric Moebius ladder lattices due to the non-orientability of a Moebius strip. In order to understand the level crossings of PT symmetric phases, we generalize the rotational transformation using a complex rotation angle. We also study the modification of resonant tunneling induced by a sharply twisted interface in PT-symmetric ladder lattices. Finally, we find that the perfect transmissions at the zero energy are recovered at the exceptional points of the PT-symmetric system due to the self-orthogonal states.

Ultrarobust Wireless Interrogation for Sensors and Transducers: A Non-Hermitian Telemetry Technique

We herein introduce a non-Hermitian parity-time (PT)-symmetric telemetry platform, capable of wirelessly and robustly acquiring information of passive sensors and chemiresistive or piezoresistive transducers. We theoretically and experimentally demonstrate that the PT-symmetric radio frequency (RF) system, which consists of an active reader and a passive sensor to obey parity (space inversion) and time-reversal symmetries, may provide not only an improved sensitivity but also rapid, accurate, and robust acquisition of sensor data. The unidirectional reflectionless properties arising from the spontaneous PT symmetry-breaking may result in high- $Q$ resonances and thus high resolvability in the measured reflection spectrum. Furthermore, the sensor’s effective resistance (capacitance) can be wirelessly queried by electronically tuning the negative impedance converter (varactors) on the active reader, which allows observation of dual-band reflectionless phenomena. Unlike traditional telemetry systems, the measurement results are robust to the variations in the inductive coupling strength, namely, the distance and alignment between the two coil antennas. The proposed PT-symmetric telemetry paradigm may have an impact on a wide range of wireless sensing and instrumentation applications.

Exceptional point in a metal-graphene hybrid metasurface with tunable asymmetric loss

Observation of exceptional points (EPs) in non-Hermitian parity-time (PT) symmetric systems has led to various nontrivial physics and exotic phenomena. Here, a metal-graphene hybrid non-Hermitian metasurface is proposed in the terahertz regime, whose unit cell is composed of two orthogonally oriented split-ring resonators (SRRs) with identical dimensions but only one SRR containing a graphene patch at the gap. An EP in polarization space is theoretically observed at a certain Fermi level of the graphene patch, where the induced asymmetric loss and the coupling strength between the two SRRs match a certain relation predicted by a coupled mode theory. The numerical fittings using the coupled mode theory agree well with the simulations. Besides, an abrupt phase flip around the EP frequency is observed in the transmission in circular polarization basis, which can be very promising in ultra-sensitive sensing applications.

Exact Floquet solutions in a periodically driven non-Hermitian parity-time symmetric two-level model

We investigate a type of periodically driven non-Hermitian two-level model with parity-time (PT) symmetry. It is shown that in a wide range of the model parameters, this model supports exact Floquet solutions. In particular, under certain parameter conditions, these exact Floquet solutions can characterize the transition from the unbroken PT-symmetric phase to the broken PT-symmetric phase.

Asymmetric scattering of flexural waves in a parity-time symmetric metamaterial beam.

Non-Hermitian parity-time (PT) symmetric systems that possess real eigenvalues have been intensively investigated in quantum mechanics and rapidly extended to optics and acoustics demonstrating a lot of unconventional wave phenomena. Here, a PT symmetric metamaterial beam is designed based on shunted piezoelectric patches and asymmetric wave scattering in the form of flexural waves is demonstrated through analytical and numerical approaches. The gain and loss components in the PT symmetric beam are realized by the introduction of negative and positive resistances into the external shunting circuits, respectively. Effective medium theory and transfer matrix method are employed to determine the effective material parameters and scattering properties of the PT symmetric metamaterial beam. Unidirectional reflectionlessness has been demonstrated analytically and numerically, together with illustrations of the PT phase transition and exceptional points. The tunability of exceptional points is studied by changing the spacing between piezoelectric patches and shunting circuit parameters. The design explores complex material parameters of the beam structure, and could open unique ways to asymmetric wave control, enhanced sensing, amplification, and localization of flexural waves.

Interacting non-Hermitian ultracold atoms in a harmonic trap: Two-body exact solution and a high-order exceptional point

We study interacting ultracold atoms in a three-dimensional (3D) harmonic trap with spin-selective dissipations, which can be effectively described by non-Hermitian parity-time ($\mathcal{PT}$) symmetric Hamiltonians. By exactly solving the non-Hermitian two-body problem of spin-1/2 (spin-1) bosons in a 3D harmonic trap, we find that the system can exhibit third-order (fifth-order) exceptional points (EPs) with ultrasensitive cube-root (fifth-root) spectral response due to interaction anisotropies in spin channels. We also present the general principle for the creation of high-order EPs and their spectral sensitivities with arbitrary particle number $N$ and arbitrary spin $s$. Generally, with spin-independent interactions, the EP order of bosons can be as high as $2Ns+1$, and the spectral response around EP can be as sensitive as $\ensuremath{\sim}{\ensuremath{\epsilon}}^{1/(2ks+1)}$ under a $k$-body interaction anisotropy. Moreover, we propose to detect the ultrasensitive spectral response through the probability dynamics of certain state. These results suggest a convenient route towards more powerful sensor devices in spinor cold atomic systems.

PT phase transitions of edge states at PT symmetric interfaces in non-Hermitian topological insulators

We demonstrate that the non-Hermitian parity-time (PT) symmetric interfaces formed between amplifying and lossy crystals support dissipationless edge states. These PT edge states exhibit gapless spectra in the complex band structure interconnecting complex-valued bulk bands as long as exceptional points (EPs) of edge states exist. As a result, regimes exist where the edge states can spectrally overlap with the bulk continuum without hybridization, and leakage into the bulk states is suppressed due to the PT symmetry. Two exemplary PT symmetric systems, based on valley and quantum hall topological phases, are investigated, and the connection with the corresponding Hermitian systems is established. We find that the edge states smoothly transit to the valley edge states found in Hermitian systems if the magnitude of gain/loss vanishes. The topological nature of the PT edge states can be established within the non-Hermitian Haldane model, where the topological invariance is found to be unaffected by gain or loss. Nonreciprocal PT edge states are discovered at the interfaces between PT-Haldane phases, indicating the interplay between the gain/loss and the magnetic flux. The proposed systems are experimentally feasible to realize in photonics. This has been verified by our rigorous full-wave simulations of edge states in PT-symmetric silicon-based photonic graphene.

Non-Hermitian physics and PT symmetry

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.