Gal Harari

Topological insulator laser: Experiments

Topological protection for lasers Ideas based on topology, initially developed in mathematics to describe the properties of geometric space under deformations, are now finding application in materials, electronics, and optics. The main driver is topological protection, a property that provides stability to a system even in the presence of defects. Harari et al. outline a theoretical proposal that carries such ideas over to geometrically designed laser cavities. The lasing mode is confined to the topological edge state of the cavity structure. Bandres et al. implemented those ideas to fabricate a topological insulator laser with an array of ring resonators. The results demonstrate a powerful platform for developing new laser systems. Science, this issue p. eaar4003, p. eaar4005 Lasing is observed in an edge mode of a designed optical topological insulator. INTRODUCTION Physical systems that exhibit topological invariants are naturally endowed with robustness against perturbations, as was recently demonstrated in many settings in condensed matter, photonics, cold atoms, acoustics, and more. The most prominent manifestations of topological systems are topological insulators, which exhibit scatter-free edge-state transport, immune to perturbations and disorder. Recent years have witnessed intense efforts toward exploiting these physical phenomena in the optical domain, with new ideas ranging from topology-driven unidirectional devices to topological protection of path entanglement. But perhaps more technologically relevant than all topological photonic settings studied thus far is, as proposed by the accompanying theoretical paper by Harari et al., an all-dielectric magnet-free topological insulator laser, with desirable properties stemming from the topological transport of light in the laser cavity. RATIONALE We demonstrate nonmagnetic topological insulator lasers. The topological properties of the laser system give rise to single-mode lasing, robustness against fabrication defects, and notably higher slope efficiencies compared to those of the topologically trivial counterparts. We further exploit the properties of the active topological platform by assembling topological insulator lasers from S-chiral microresonators that enforce predetermined unidirectional lasing even in the absence of magnetic fields. RESULTS Our topological insulator laser system is an aperiodic array of 10 unit cell–by–10 unit cell coupled ring resonators on an InGaAsP quantum wells platform. The active lattice uses the topological architecture suggested in the accompanying theoretical paper. This two-dimensional setting is composed of a square lattice of ring resonators coupled to each other by means of link rings. The intermediary links are judiciously spatially shifted to introduce a set of hopping phases, establishing a synthetic magnetic field and two topological band gaps. The gain in this laser system is provided by optical pumping. To promote lasing of the topologically protected edge modes, we pump the outer perimeter of the array while leaving the interior lossy. We find that this topological insulator laser operates in single mode even considerably above threshold, whereas the corresponding topologically trivial realizations lase in multiple modes. Moreover, the topological laser displays a slope efficiency that is considerably higher than that in the corresponding trivial realizations. We further demonstrate the topological features of this laser by observing that in the topological array, all sites emit coherently at the same wavelength, whereas in the trivial array, lasing occurs in localized regions, each at a different frequency. Also, by pumping only part of the topological array, we demonstrate that the topological edge mode always travels along the perimeter and emits light through the output coupler. By contrast, when we pump the trivial array far from the output coupler, no light is extracted from the coupler because the lasing occurs at stationary modes. We also observe that, even in the presence of defects, the topological protection always leads to more efficient lasing compared to that of the trivial counterpart. Finally, to show the potential of this active system, we assemble a topological system based on S-chiral resonators, which can provide new avenues to control the topological features. CONCLUSION We have experimentally demonstrated an all-dielectric topological insulator laser and found that the topological features enhance the lasing performance of a two-dimensional array of microresonators, making them lase in unison in an extended topologically protected scatter-free edge mode. The observed single longitudinal-mode operation leads to a considerably higher slope efficiency as compared to that of a corresponding topologically trivial system. Our results pave the way toward a new class of active topological photonic devices, such as laser arrays, that can operate in a coherent fashion with high efficiencies. Topological insulator laser. (A) Top-view image of the lasing pattern (topological edge mode) in a 10 unit cell–by–10 unit cell array of topologically coupled resonators and the output ports. (B) Output intensity versus pump intensity for a topological insulator laser and its trivial counterpart. The enhancement of the slope efficiency is about threefold. Comparing the power emitted in the single mode of the topological array to that of the highest power mode in the trivial array, the topological outperforms the trivial by more than a factor of 10

Topological Optical Waveguiding in Silicon and the Transition between Topological and Trivial Defect States

One-dimensional models with topological band structures represent a simple and versatile platform to demonstrate novel topological concepts. Here we experimentally study topologically protected states in silicon at the interface between two dimer chains with different Zak phases. Furthermore, we propose and demonstrate that, in a system where topological and trivial defect modes coexist, we can probe them independently. Tuning the configuration of the interface, we observe the transition between a single topological defect and a compound trivial defect state. These results provide a new paradigm for topologically protected waveguiding in a complementary metal-oxide-semiconductor compatible platform and highlight the novel concept of isolating topological and trivial defect modes in the same system that can have important implications in topological physics.

Topological insulator laser: Theory

Topological protection for lasers Ideas based on topology, initially developed in mathematics to describe the properties of geometric space under deformations, are now finding application in materials, electronics, and optics. The main driver is topological protection, a property that provides stability to a system even in the presence of defects. Harari et al. outline a theoretical proposal that carries such ideas over to geometrically designed laser cavities. The lasing mode is confined to the topological edge state of the cavity structure. Bandres et al. implemented those ideas to fabricate a topological insulator laser with an array of ring resonators. The results demonstrate a powerful platform for developing new laser systems. Science, this issue p. eaar4003, p. eaar4005 Lasing is observed in an edge mode of a designed optical topological insulator. INTRODUCTION Topological insulators emerged in condensed matter physics and constitute a new phase of matter, with insulating bulk and robust edge conductance that is immune to imperfections and disorder. To date, topological protection is known to be a ubiquitous phenomenon, occurring in many physical settings, ranging from photonics and cold atoms to acoustic, mechanical, and elastic systems. So far, however, most of these studies were carried out in entirely passive, linear, and conservative settings. RATIONALE We propose topological insulator lasers: lasers whose lasing mode exhibits topologically protected transport without magnetic fields. Extending topological physics to lasers is far from natural. In fact, lasers are built on foundations that are seemingly inconsistent with the essence of topological insulators: They require gain (and thus are non-Hermitian), they are nonlinear entities because the gain must be saturable, and they are open systems because they emit light. These properties, common to all lasers, cast major doubts on the possibility of harnessing topological features to make a topological insulator laser. Despite this common mindset, we show that the use of topological properties leads to highly efficient lasers, robust to defects and disorder, with single-mode lasing even at conditions high above the laser threshold. RESULTS We demonstrate that topological insulator lasers are theoretically possible and experimentally feasible. We consider two configurations involving planar arrays of coupled active resonators. The first is based on the Haldane model, archetypical for topological systems. The second model, geared toward experiment, constitutes an aperiodic array architecture creating an artificial magnetic field. We show that by introducing saturable gain and loss, it is possible to make these systems lase in a topological edge state. In this way, the lasing mode exhibits topologically protected transport; the light propagates unidirectionally along the edges of the cavity, immune to scattering and disorder, unaffected by the shape of the edges. Moreover, we show that the underlying topological properties not only make the system robust to fabrication and operational disorder and defects, they also lead to a highly efficient single-mode lasing that remains single-mode even at gain values high above the laser threshold. The figure describes the geometry and features of a topological insulator laser based on the Haldane model while adding saturable gain, loss, and an output port. The cavity is a planar honeycomb lattice of coupled microring resonators, pumped at the perimeter with a lossy interior. We show that under these conditions, lasing occurs at the topological edge mode, which has unidirectional flux and is extended around the perimeter with almost-uniform intensity. The topological cavities exhibit higher efficiency than the trivial cavity, even under strong disorder. For the topological laser with a small gap, the topological protection holds as long as the disorder level is smaller than the gap size. DISCUSSION The concept of the topological insulator laser alters current understanding of the interplay between disorder and lasing, and opens exciting possibilities at the interface of topological physics and laser science, such as topologically protected transport in systems with gain. We show here that the laser system based on the archetypal Haldane model exhibits topologically protected transport, with features similar to those of its passive counterpart. This behavior means that this system is likely to have topological invariants, despite the nonhermiticity. Technologically, the topological insulator laser offers an avenue to make many semiconductor lasers operate as one single-mode high-power laser. The topological insulator laser constructed from an aperiodic array of resonators was realized experimentally in an all-dielectric platform, as described in the accompanying experimental paper by Bandres et al. Topological insulator laser based on the Haldane model and its efficiency. (A) Planar honeycomb lattice of coupled microring resonators pumped at the perimeter. The topological lasing mode has unidirectional flux with almost-uniform intensity, which builds up as the mode circulates and drops when passing the output coupler. (B) Slope efficiency (in arbitrary units) versus disorder strength for three cases differing only in the Haldane phase (of the next-to-nearest neighbor coupling): a topological laser with the maximum gap (blue; Haldane phase of π/2), one with a small topological gap (red; Haldane phase of π/8), and a topologically trivial laser with no gap (black; Haldane phase of 0). Topological insulators are phases of matter characterized by topological edge states that propagate in a unidirectional manner that is robust to imperfections and disorder. These attributes make topological insulator systems ideal candidates for enabling applications in quantum computation and spintronics. We propose a concept that exploits topological effects in a unique way: the topological insulator laser. These are lasers whose lasing mode exhibits topologically protected transport without magnetic fields. The underlying topological properties lead to a highly efficient laser, robust to defects and disorder, with single-mode lasing even at very high gain values. The topological insulator laser alters current understanding of the interplay between disorder and lasing, and at the same time opens exciting possibilities in topological physics, such as topologically protected transport in systems with gain. On the technological side, the topological insulator laser provides a route to arrays of semiconductor lasers that operate as one single-mode high-power laser coupled efficiently into an output port.

Quantum Čerenkov Radiation: Spectral Cutoffs and the Role of Spin and Orbital Angular Momentum

We show that the well-known Cerenkov effect contains new phenomena arising from the quantum nature of charged particles. The Cerenkov transition amplitudes allow coupling between the charged particle and the emitted photon through their orbital angular momentum and spin, by scattering into preferred angles and polarizations. Importantly, the spectral response reveals a discontinuity immediately below a frequency cutoff that can occur in the optical region. Near this cutoff, the intensity of the conventional Cerenkov radiation (CR) is very small but still finite, while our quantum calculation predicts exactly zero intensity above the cutoff. Below that cutoff, with proper shaping of electron beams (ebeams), we predict that the traditional CR angle splits into two distinctive cones of photonic shockwaves. One of the shockwaves can move along a backward cone, otherwise considered impossible for conventional CR in ordinary matter. Our findings are observable for ebeams with realistic parameters, offering new applications including novel quantum optics sources, and opening a new realm for Cerenkov detectors involving the spin and orbital angular momentum of charged particles.

Chiral state conversion without encircling an exceptional point

Dynamically varying system parameters along a path enclosing an exceptional point is known to lead to chiral mode conversion. But is it necessary to include this non-Hermitian degeneracy inside the contour for this process to take place? We show that a sufficiently slow variation of parameters, even away from the systems exceptional point, can also lead to a robust asymmetric state exchange. To study this process, we consider a prototypical two-level non-Hermitian Hamiltonian with a constant coupling between elements. Closed-form solutions are obtained when the amplification or attenuation coefficients in this arrangement are varied in conjunction with the resonance detuning along a circular contour. Using asymptotic expansions, this input-independent mode conversion is theoretically proven to take place irrespective of whether the exceptional point is enclosed upon encirclement. Our results significantly broaden the range of parameter space required for the experimental realization of such chiral mode conversion processes.

Ultrasensitive micro-scale parity-time-symmetric ring laser gyroscope.

We propose a new scheme for ultrasensitive laser gyroscopes that utilizes the physics of exceptional points. By exploiting the properties of such non-Hermitian degeneracies, we show that the rotation-induced frequency splitting becomes proportional to the square root of the gyration speed (Ω), thus enhancing the sensitivity to low angular rotations by orders of magnitudes. In addition, at its maximum sensitivity limit, the measurable spectral splitting is independent of the radius of the rings involved. This Letter paves the way toward a new class of ultrasensitive miniature ring laser gyroscopes on chip.

Propagator for the general time-dependent harmonic oscillator with application to an ion trap

We present the simplest possible formula for the propagator of the general time-dependent quadratic Hamiltonian, including linear terms. The method is based on the use of a linear time-dependent invariant and requires only the solution of a linear homogeneous second-order ordinary differential equation corresponding to the classical quadratic Hamiltonian. We give an example for the case of the Paul trap.

Non-diffracting multi-electron vortex beams balancing their electron–electron interactions

The wave-like nature of electrons has been known for almost a century, but only in recent years has the ability to shape the wavefunction of EBeams (Electron-Beams) become experimentally accessible. Various EBeam wavefunctions have been demonstrated, such as vortex, self-accelerating, Bessel EBeams etc. However, none has attempted to manipulate multi-electron beams, because the repulsion between electrons rapidly alters the beam shape. Here, we show how interference effects of the quantum wavefunction describing multiple electrons can be used to exactly balance both the repulsion and diffraction-broadening. We propose non-diffracting wavepackets of multiple electrons, which can also carry orbital angular momentum. Such wavefunction shaping facilitates the use of multi-electron beams in electron microscopy with higher current without compromising on spatial resolution. Simulating the quantum evolution in three-dimensions and time, we show that imprinting such wavefunctions on electron pulses leads to shape-preserving multi-electrons ultrashort pulses. Our scheme applies to any beams of charged particles, such as protons and ion beams.Vortex electron beams are generated using single electrons but their low beam-density is a limitation in electron microscopy. Here the authors propose a scheme for the realization of non-diffracting electron beams by shaping wavepackets of multiple electrons and including electron–electron interactions.

Non-linear shape preserving electron-beams

We show that shaping the initial wavefunction of a multi-electron system can lead to electron beams displaying shape-preserving propagation in spite of the inherent repulsion among electrons. This idea suggests applications in microscopy and lithography.

Ultrasensitive micro-scale parity-time-symmetric ring laser gyroscope (Conference Presentation)

A new scheme for ultrasensitive laser gyroscopes that utilizes the physics of exceptional points will be presented. By exploiting the properties of such non-Hermitian degeneracies, we show that the rotation-induced frequency splitting becomes proportional to the square root of the gyration speed (√𝛀)- thus enhancing the sensitivity to low angular rotations by orders of magnitudes. In addition, at its maximum sensitivity limit, the measurable spectral splitting is independent of the radius of the rings involved. Our work paves the way towards a new class of ultrasensitive miniature ring laser gyroscopes on chip.