Observation of chiral state transfer without encircling an exceptional point.

The adiabatic theorem, a corollary of the Schrödinger equation, manifests itself in a profoundly different way in non-Hermitian arrangements, resulting in counterintuitive state transfer schemes that have no counterpart in closed quantum systems. In particular, the dynamical encirclement of exceptional points (EPs) in parameter space has been shown to lead to a chiral phase accumulation, non-adiabatic jumps and topological mode conversion1-8. Recent theoretical studies, however, have shown that contrary to previously established demonstrations, this behaviour is not strictly a result of winding around a non-Hermitian degeneracy9. Instead, it seems to be mostly attributed to the non-trivial landscape of the Riemann surfaces, sometimes because of the presence of an EP in the vicinity9-11. Here, in an effort to bring this counterintuitive aspect of non-Hermitian systems to light and confirm this hypothesis, we provide a set of experiments to directly observe the field evolution and chiral state conversion in an EP-excluding cycle in a slowly varying non-Hermitian system. To do so, a versatile yet unique fibre-based photonic emulator is realized that utilizes the polarization degrees of freedom in a quasi-common-path single-ring arrangement. Our observations may open up new avenues for light manipulation and state conversion, as well as providing a foundation for understanding the intricacies of the adiabatic theorem in non-Hermitian systems.

Non-Hermitian ring laser gyroscopes with enhanced Sagnac sensitivity.

Gyroscopes are essential to many diverse applications associated with navigation, positioning and inertial sensing1. In general, most optical gyroscopes rely on the Sagnac effect-a relativistically induced phase shift that scales linearly with the rotational velocity2,3. In ring laser gyroscopes (RLGs), this shift manifests as a resonance splitting in the emission spectrum, which can be detected as a beat frequency4. The need for ever more precise RLGs has fuelled research activities aimed at boosting the sensitivity of RLGs beyond the limits dictated by geometrical constraints, including attempts to use either dispersive or nonlinear effects5-8. Here we establish and experimentally demonstrate a method using non-Hermitian singularities, or exceptional points, to enhance the Sagnac scale factor9-13. By exploiting the increased rotational sensitivity of RLGs in the vicinity of an exceptional point, we enhance the resonance splitting by up to a factor of 20. Our results pave the way towards the next generation of ultrasensitive and compact RLGs and provide a practical approach for the development of other classes of integrated sensor.

Nonlinear Optics: feature issue introduction.

This joint issue of Optics Express and Optical Materials Express showcases 29 articles that report the latest advancements in nonlinear optics. These articles include contributions from authors who participated in the Optica Nonlinear Optics Topical Meeting, which took place in Honolulu, Hawaii, from July 10th to July 14th, 2023. The conference was organized by Optica (formerly known as OSA). As an introduction, the editors provide a summary of these articles, which cover a broad range of topics in nonlinear optics, spanning from fundamental nonlinear optical concepts to novel nonlinear effects, and from innovative nonlinear materials to topics such as ultrafast optics, machine learning empowered nonlinear optics, and unconventional applications. This diverse array of contributions reflects the dynamic and interdisciplinary nature of contemporary research in the field of nonlinear optics while showcasing some of the most recent developments.

Dalton's law of partial optical thermodynamic pressures in highly multimoded nonlinear photonic systems.

We show that in highly multimoded nonlinear photonic systems, the optical thermodynamic pressures emerging from different species of the optical field obey Dalton's law of partial pressures. In multimode settings, the optical thermodynamic pressure is defined as the conjugate to the extensive variable associated with the system's total number of modes and is directly related to the actual electrodynamic radiation forces exerted at the physical boundaries of the system. Here, we extend this notion to photonic configuration supporting different species of the optical field. Under thermal equilibrium conditions, we formally derive an equation that establishes a direct link between the partial thermodynamic pressures and the electrodynamic radiation pressures exerted by each polarization species. Our theoretical framework provides a straightforward approach for quantifying the total radiation pressures through the system's thermodynamic variables without invoking the Maxwell stress tensor formalism. In essence, we show that the total electrodynamic pressure in such arrangements can be obtained in an effortless manner from initial excitation conditions, thus avoiding time-consuming simulations of the utterly complex multimode dynamics. To illustrate the validity of our results, we carry out numerical simulations in multimoded nonlinear optical structures supporting two polarization species and demonstrate excellent agreement with the Maxwell stress tensor method.

Bimorphic Floquet topological insulators.

Topological theories have established a unique set of rules that govern the transport properties in a wide variety of wave-mechanical settings. In a marked departure from the established approaches that induce Floquet topological phases by specifically tailored discrete coupling protocols or helical lattice motions, we introduce a class of bimorphic Floquet topological insulators that leverage connective chains with periodically modulated on-site potentials to reveal rich topological features in the system. In exploring a 'chain-driven' generalization of the archetypical Floquet honeycomb lattice, we identify a rich phase structure that can host multiple non-trivial topological phases associated simultaneously with both Chern-type and anomalous chiral states. Experiments carried out in photonic waveguide lattices reveal a strongly confined helical edge state that, owing to its origin in bulk flat bands, can be set into motion in a topologically protected fashion, or halted at will, without compromising its adherence to individual lattice sites.

Coherence properties of light in highly multimoded nonlinear parabolic fibers under optical equilibrium conditions.

We study the coherence characteristics of light propagating in nonlinear graded-index (GRIN) multimode fibers after attaining optical thermal equilibrium conditions. The role of optical temperature on the spatial mutual coherence function and the associated correlation area is systematically investigated. In this respect, we show that the coherence properties of the field at the output of a multimode nonlinear fiber can be controlled through its optical thermodynamic properties.

Thermalization of the Ablowitz-Ladik lattice in the presence of non-integrable perturbations.

We investigate the statistical mechanics of the photonic Ablowitz-Ladik lattice, the integrable version of the discrete nonlinear Schrödinger equation. In this regard, we demonstrate that in the presence of perturbations, the complex response of this system can be accurately captured within the framework of optical thermodynamics. Along these lines, we shed light on the true relevance of chaos in the thermalization of the Ablowitz-Ladik system. Our results indicate that when linear and nonlinear perturbations are incorporated, this weakly nonlinear lattice will thermalize into a proper Rayleigh-Jeans distribution with a well-defined temperature and chemical potential, in spite of the fact that the underlying nonlinearity is non-local and hence does not have a multi-wave mixing representation. This result illustrates that in the supermode basis, a non-local and non-Hermitian nonlinearity can in fact properly thermalize this periodic array in the presence of two quasi-conserved quantities.

Nature of Optical Thermodynamic Pressure Exerted in Highly Multimoded Nonlinear Systems.

The theory of optical thermodynamics provides a comprehensive framework that enables a self-consistent description of the intricate dynamics of nonlinear multimoded photonic systems. This theory, among others, predicts a pressurelike intensive quantity (p[over ^]) that is conjugate to the system's total number of modes (M)-its corresponding extensive variable. Yet at this point, the nature of this intensive quantity is still nebulous. In this Letter, we elucidate the physical origin of the optical thermodynamic pressure and demonstrate its dual essence. In this context, we rigorously derive an expression that splits p[over ^] into two distinct components, a term that is explicitly tied to the electrodynamic radiation pressure and a second entropic part that is responsible for the entropy change. We utilize this result to establish a formalism that simplifies the quantification of radiation pressure under nonlinear equilibrium conditions, thus eliminating the need for a tedious evaluation of the Maxwell stress tensor. Our theoretical analysis is corroborated by numerical simulations carried out in highly multimoded nonlinear optical structures. These results may provide a novel way in predicting and controlling radiation pressure processes in a variety of nonlinear electromagnetic settings.

Enhanced sensitivity at higher-order exceptional points.

Non-Hermitian degeneracies, also known as exceptional points, have recently emerged as a new way to engineer the response of open physical systems, that is, those that interact with the environment. They correspond to points in parameter space at which the eigenvalues of the underlying system and the corresponding eigenvectors simultaneously coalesce. In optics, the abrupt nature of the phase transitions that are encountered around exceptional points has been shown to lead to many intriguing phenomena, such as loss-induced transparency, unidirectional invisibility, band merging, topological chirality and laser mode selectivity. Recently, it has been shown that the bifurcation properties of second-order non-Hermitian degeneracies can provide a means of enhancing the sensitivity (frequency shifts) of resonant optical structures to external perturbations. Of particular interest is the use of even higher-order exceptional points (greater than second order), which in principle could further amplify the effect of perturbations, leading to even greater sensitivity. Although a growing number of theoretical studies have been devoted to such higher-order degeneracies, their experimental demonstration in the optical domain has so far remained elusive. Here we report the observation of higher-order exceptional points in a coupled cavity arrangement-specifically, a ternary, parity-time-symmetric photonic laser molecule-with a carefully tailored gain-loss distribution. We study the system in the spectral domain and find that the frequency response associated with this system follows a cube-root dependence on induced perturbations in the refractive index. Our work paves the way for utilizing non-Hermitian degeneracies in fields including photonics, optomechanics, microwaves and atomic physics.

Erratum: Enhanced sensitivity at higher-order exceptional points.

This corrects the article DOI: 10.1038/nature23280.

Direct modulation of electrically pumped coupled microring lasers.

We demonstrate how the presence of gain-loss contrast between two coupled identical resonators can be used as a new degree of freedom to enhance the modulation frequency response of laser diodes. An electrically pumped microring laser system with a bending radius of 50 µm is fabricated on an InAlGaAs/InP MQW p-i-n structure. The room temperature continuous wave (CW) laser threshold current of the device is 27 mA. By adjusting the ratio between the injection current levels in the two coupled microrings, our experimental results clearly show a bandwidth improvement by up to 1.63 times the fundamental resonant frequency of the individual device. This matches well with our rate equation simulation model.

Extraction of silver losses at cryogenic temperatures through the optical characterization of silver-coated plasmonic nanolasers.

We report on the extraction of silver losses in the range 10 K-180 K by performing temperature-dependent micro-photoluminescence measurements in conjunction with numerical simulations on silver-coated nanolasers around near-infrared telecommunication wavelengths. By mapping changes in the quality factor of nanolasers into silver-loss variations, the imaginary part of silver permittivity is extracted at cryogenic temperatures. The latter is estimated to reach values an order of magnitude lower than room-temperature values. Temperature-dependent values for the thermo-optic coefficient of III-V semiconductors occupying the cavity are estimated as well. This data is missing from the literature and is crucial for precise device modeling. Our results can be useful for device designing, the theoretical validation of experimental observations as well as the evaluation of thermal effects in silver-coated nanophotonic structures.

Network analysis of Weyl semimetal photogalvanic systems.

We develop a general methodology capable of analyzing the response of Weyl semimetal (WSM) photogalvanic networks. Both single-port and multiport configurations are investigated via extended versions of Norton's theorem. An equivalent circuit model is provided where the photogalvanic currents induced in these gapless topological materials can be treated as polarization-dependent sources. To illustrate our approach, we carry out transport simulations in arbitrarily shaped configurations involving pertinent WSMs. Our analysis indicates that the photogalvanic currents collected in a multi-electrode system directly depend on the geometry of the structure as well as on the excitation and polarization pattern of the incident light. Our results could be helpful in designing novel optoelectronic systems that make use of the intriguing features associated with WSMs.

Thermalization Dynamics of Nonlinear Non-Hermitian Optical Lattices.

We develop a rigorous theoretical framework based on principles from statistical mechanics that allows one to predict the equilibrium response of classical non-Hermitian arrangements in the weakly nonlinear regime. In this respect, we demonstrate that a pseudo-Hermitian configuration can always be driven into thermal equilibrium when a proper nonlinear operator is paired with the linear Hamiltonian of the system. We show that, in this case, the system will thermodynamically settle into an irregular pattern that does not resemble any known statistical distribution. Interestingly, this stable equilibrium response is associated with a Rayleigh-Jeans law when viewed within an appropriately transformed space that displays unitary dynamics. By considering a non-Hermitian Su-Schrieffer-Heeger chain, our results indicate that the thermodynamic equilibrium will always favor the edge modes instead of the ground state, in stark contrast to conventional nonlinear Hermitian configurations. Moreover, non-Hermitian lattices are shown to exhibit unusually high heat capacities, potentially acting as optical heat reservoirs to other Hermitian systems, by employing only a small number of sites and low power levels.

Realizing spin Hamiltonians in nanoscale active photonic lattices.

Spin models arise in the microscopic description of magnetic materials and have been recently used to map certain classes of optimization problems involving large degrees of freedom. In this regard, various optical implementations of such Hamiltonians have been demonstrated to quickly converge to the global minimum in the energy landscape. Yet, so far, an integrated nanophotonic platform capable of emulating complex magnetic materials is still missing. Here, we show that the cooperative interplay among vectorial electromagnetic modes in coupled metallic nanolasers can be utilized to implement certain types of spin Hamiltonians. Depending on the topology/geometry of the arrays, these structures can be governed by a classical XY Hamiltonian that exhibits ferromagnetic and antiferromagnetic couplings, as well as geometrical frustration. Our results pave the way towards a scalable nanophotonic platform to study spin exchange interactions and could address a variety of optimization problems.

Enhanced modulation characteristics in broken symmetric coupled microring lasers.

The dynamical behavior of broken symmetric coupled cavity lasers is theoretically investigated. The frequency response of this class of lasers is obtained using small signal analysis under direct modulation. Our model predicts a modulation bandwidth enhancement as a broken symmetric laser, operating in the parity-time (PT) symmetry and non-PT symmetry domains. This theoretical prediction is numerically examined in a laser system based on an InGaAs quantum dot platform. Our results clearly show that in these structures, in addition to the injection current, the gain-loss contrast can be used as a new degree of freedom in order to control the characteristic poles of the frequency response function.

Measuring the frequency response of optically pumped metal-clad nanolasers.

We report on our initial attempt to characterize the intrinsic frequency response of metal-clad nanolasers. The probed nanolaser is optically biased and modulated, allowing the emitted signal to be detected using a high-speed photodiode at each modulation frequency. Based on this technique, the prospect of high-speed operation of nanolasers is evaluated by measuring the D-factor, which is the ratio of the resonance frequency to the square root of its output power(fR/Pout1/2). Our measurements show that for nanolasers, this factor is an order of magnitude greater than that of other state-of-the-art directly modulated semiconductor lasers. The theoretical analysis, based on the rate equation model and finite element method simulations of the cavity is in full agreement with the measurement results.

Unidirectional light emission in PT-symmetric microring lasers.

The synergetic use of gain and loss in parity-time symmetric coupled resonators has been shown to lead to single-mode lasing operation. However, at the corresponding resonance frequency, an ideal ring resonator tends to support two degenerate eigenmodes, traveling along the cavity in opposite directions. Here, we show a unidirectional single-moded parity-time symmetric laser by incorporating active S-bend structures with opposite chirality in the respective ring resonators. Such chiral elements break the rotation symmetry of the ring cavities by providing an asymmetric coupling between the clockwise (CW) and the counterclockwise (CCW) traveling modes, hence creating a new type of exceptional point. This property, consequently, leads to the suppression of one of the counter-propagating modes. In this paper, we first measure the extinction ratio between the CW and CCW modes in a single ring resonator in the presence of an S-bend waveguide. We then experimentally investigate the unidirectional emission in PT-symmetric systems below and above the exceptional point. Finally, unidirectional emission will be shown in systems of two S-bend ring resonators coupled through a link structure.

Edge-Mode Lasing in 1D Topological Active Arrays.

We report the first observation of lasing topological edge states in a 1D Su-Schrieffer-Heeger active array of microring resonators. We show that the judicious use of non-Hermiticity can promote single edge-mode lasing in such arrays. Our experimental and theoretical results demonstrate that, in the presence of chiral-time symmetry, this non-Hermitian topological structure can experience phase transitions that are dictated by a complex geometric phase. Our work may pave the way towards understanding the fundamental aspects associated with the interplay among non-Hermiticity, nonlinearity, and topology in active systems.

Dynamically Encircling Exceptional Points: Exact Evolution and Polarization State Conversion.

We show that a two-level non-Hermitian Hamiltonian with constant off-diagonal exchange elements can be analyzed exactly when the underlying exceptional point is perfectly encircled in the complex plane. The state evolution of this system is explicitly obtained in terms of an ensuing transfer matrix, even for large encirclements, regardless of adiabatic conditions. Our results clearly explain the direction-dependent nature of this process and why in the adiabatic limit its outcome is dominated by a specific eigenstate-irrespective of initial conditions. Moreover, numerical simulations suggest that this mechanism can still persist in the presence of nonlinear effects. We further show that this robust process can be harnessed to realize an optical omnipolarizer: a configuration that generates a desired polarization output regardless of the input polarization state, while from the opposite direction it always produces the counterpart eigenstate.