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.

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 eigenstateirrespective 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. DOI: 10.1103/PhysRevLett.118.093002 Understanding the dynamics of time-dependent Hamiltonians is key in explaining a wide range of processes in many and diverse physical settings [1]. This ubiquitous class of problems is of significance since it allows one to tailor the evolution of a Hamiltonian towards certain outcomes. If a system is conservative or Hermitian, a cyclic adiabatic change in a multiparameter space can often lead to surprising results such as, for example, the emergence of gauge-invariant geometric phases, as first shown by Berry [2]. Of particular interest is the case where eigenvalue degeneracies are enclosed within the parameter loop. In this latter scenario, the geometric phase is robust against perturbations in the control path since it is related to the flux generated from the degeneracies that act as topological sources.While the Berry phase represents an intuitive and powerful unifying notion, it is by nature based on the adiabatic theorem [3]. Quite recently, a series of studies have critically reexamined these aspects in non-Hermitian environments where it was found that the system behavior can be significantly modified around degeneracies, better known as exceptional points (EPs) [4]. As opposed to conservative systems, in non-Hermitian arrangements both the eigenvalues and the corresponding eigenvectors tend to coalesce at an EP (while unfolding associated vectors of the Jordan form) [5]. In the last few years a number of intriguing possibilities have been realized in structures supporting EPs, including loss-induced transparency [6], single-mode lasing [7], band merging [8], asymmetric diffraction [9], and unidirectional invisibility [10], to mention a few. In other studies, the topological properties associated with the quasistatic encirclement of an EP were also investigated. Under such stationary conditions it was found that the instantaneous eigenstates swap with each other at the end of the parameter cycle with only one acquiring a geometric phase [11,12]. This behavior, attributed to the branch point character of the degeneracy that cause...

Creating optical components that allow light to propagate in only one direction-that is, that allow non-reciprocal propagation or 'isolation' of light-is important for a range of applications. Non-reciprocal propagation of sound can be achieved simply by using mechanical components that spin. Spinning also affects de Broglie waves , so a similar idea could be applied in optics. However, the extreme rotation rates that would be required, owing to light travelling much faster than sound, lead to unwanted wobbling. This wobbling makes it difficult to maintain the separation between the spinning devices and the couplers to within tolerance ranges of several nanometres, which is essential for critical coupling. Consequently, previous applications of optical and optomechanical isolation have used alternative methods. In hard-drive technology, the magnetic read heads of a hard-disk drive fly aerodynamically above the rapidly rotating disk with nanometre precision, separated by a thin film of air with near-zero drag that acts as a lubrication layer . Inspired by this, here we report the fabrication of photonic couplers (tapered fibres that couple light into the resonators) that similarly fly above spherical resonators with a separation of only a few nanometres. The resonators spin fast enough to split their counter-circulating optical modes, making the fibre coupler transparent from one side while simultaneously opaque from the other-that is, generating irreversible transmission. Our setup provides 99.6 per cent isolation of light in standard telecommunication fibres, of the type used for fibre-based quantum interconnects . Unlike flat geometries, such as between a magnetic head and spinning disk, the saddle-like, convex geometry of the fibre and sphere in our setup makes it relatively easy to bring the two closer together, which could enable surface-science studies at nanometre-scale separations.

The quest for ever higher information capacities has brought about a renaissance in multimode optical waveguide systems. This resurgence of interest has recently initiated a flurry of activities in nonlinear multimode fiber optics. The sheer complexity emerging from the presence of a multitude of nonlinearly interacting modes has led not only to new opportunities in observing a host of novel optical effects that are otherwise impossible in single-mode settings, but also to new theoretical challenges in understanding their collective dynamics. In this Article, we present a consistent thermodynamical framework capable of describing in a universal fashion the exceedingly intricate behavior of such nonlinear highly multimoded photonic configurations at thermal equilibrium. By introducing pertinent extensive variables, we derive new equations of state and show that both the "internal energy" and optical power in many-mode arrangements always flow in such a way so as to satisfy the second law of thermodynamics. The laws governing isentropic processes are derived and the prospect for realizing Carnot-like cycles is also presented.In addition to shedding light on fundamental issues, our work may pave the way towards a new generation of high power multimode optical structures and could have ramifications in other manystate nonlinear systems, ranging from Bose-Einstein condensates to optomechanics.

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 (√ )-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. © 2017 Optical Society of America In 1913, Sagnac demonstrated how the rate of rotation associated with an inertial frame of reference can be determined by optical means. In his experiments, the rotation speed was measured through the phase difference between two beams traveling in opposite directions within a loop. Since then, this approach has been successfully used to develop various families of optical rotational sensors [1,2]. A breakthrough in this area came shortly after the discovery of the laser, when Macek and Davis introduced gain in the ring cavity [3]. In this respect, the phase shift between the two counter-propagating beams is effectively converted into a splitting in the resonant frequencies that can in turn be readily measured.In an ideal non-rotating ring laser, the two counterpropagating modes are expected to exhibit the same frequency. On the other hand, if this same system rotates at an angular frequency , the two initially degenerate resonant frequencies split, according to the following expression Δ = 8 Ω.(1) Here and are the enclosed area and the perimeter of the ring, respectively, and is the wavelength within the material associated with this cavity. Ideally, as long as the frequency separation (Δ ) exceeds the quantum limit imposed by the spontaneous emission noise, the rotation speed Ω can be uniquely determined through a heterodyne measurement. For example, for a ring laser with a radius of 10 cm, operating at a wavelength of 1.55 μm, and rotating at a rate of ~10°/hour, one can expect a frequency splitting that is at best on the order of ~ 12 Hz [1]. In many consumer and industrial applications, it is required to detect angular velocities in the range of ~0.1 − 100°/hour -a precision that can be readily attained in state-of-the-art free-space ring laser gyroscopes [4]. Unfortunately however, such sensitivity levels have so far remained practically out-of-reach in fully integrated optical platforms, where the area of the loop is generally smaller, perhaps by several orders of magnitudes. In addition, in on-chip settings, light scattering from the cavity walls can prove detrimental. This is because the ensuing unwanted coupling between the two counter-propagating modes can lead to a lock-in effect, rendering this method ineffective below a certain rotation speed.Currently, several efforts are underway to implement chip-scale laser gyroscopes based on different strategies. One possible ...

A system of two coupled semiconductor-based resonators is studied when lasing around an exceptional point. We show that the presence of nonlinear saturation effects can have important ramifications on the transition behavior of this system. In sharp contrast with linear PT-symmetric configurations, nonlinear processes are capable of reversing the order in which the symmetry breaking occurs. Yet, even in the nonlinear regime, the resulting non-Hermitian states still retain the structural form of the corresponding linear eigenvectors expected above and below the phase transition point. The conclusions of our analysis are in agreement with experimental data.

Single mode lasing is experimentally demonstrated in a transversely multi-moded InP-based semiconductor microring arrangement. In this system, mode discrimination is attained by judiciously utilizing the exceptional points in a parity-time (PT) symmetric double microring configuration. The proposed scheme is versatile, robust to small fabrication errors, and enables the device to operate in a stable manner considerably above threshold while maintaining spatial and spectral purity. The results presented here pave the way towards a new class of chip-scale semiconductor lasers that utilize gain/loss contrast as a primary mechanism for mode selection.Integrated photonic laser systems with larger cross sections are desirable for many applications since they allow for higher energies within the cavities while managing the thermal load and keeping the impact of optical nonlinearities under control. Unfortunately, however, merely enlarging the transverse dimensions of the waveguides inevitably gives rise to competing higher-order spatial modes. This, in turn, compromises the spectral and spatial fidelity of the laser and limits the power allocated within a specific mode [1]. These limitations exist on all scales, and may even be exacerbated in chip-scale semiconductor lasers, where the large gain bandwidths of the active media already pose a challenge in promoting single-mode operation [2].So far, utilizing intra-cavity dispersive elements has been the primary approach for longitudinal mode selection [3], while tapering along the direction of propagation, engineering the refractive index in the cross section, as well as evanescent filtering are some of the extensively explored techniques to enforce single spatial mode operation in such arrangements [4][5][6][7]. Yet, in spite of their success, they are not always compatible with on-chip microcavity structures and in most cases are quite sensitive to small fabrication tolerances. In this respect, it would be desirable to explore alternative avenues to address these issues.Lately, the selective breaking of parity-time (PT) symmetry has been proposed as a viable strategy for obtaining single transverse mode operation [8]. It is suggested that by pairing an active resonator/waveguide with a lossy but otherwise identical partner, it is possible to enforce single-spatial-mode performance even in the presence of strong mode competition in multi-moded laser/amplifier configurations. In general, a structure is considered to be PT-symmetric if it is invariant under the simultaneous action of the (space) and (time) inversion operations [9]. Despite 2 having a non-Hermitian representation, such a system may still support an entirely real spectrum. While originally developed in the context of quantum mechanics [9,10], such notions have recently attracted considerable attention in different areas of optics such as photonic lattices, microresonators, gratings, and lasers [11][12][13][14][15][16][17][18][19][20][21][22]. In optical settings, a structure is PT symmetric if the real part...

The behavior of a parity-time (PT) symmetric coupled microring system is studied when operating in the vicinity of an exceptional point. Using the abrupt phase transition around this point, stable single-mode lasing is demonstrated in spectrally multi-moded micro-ring arrangements.The notion of PT-symmetry was first introduced within the context of mathematical physics [1]. It generally implies that even nonHermitian systems can support entirely real spectra, provided their corresponding Hamiltonians commute with the parity-time (PT) operator. In recent years, several studies have shown that PT symmetric structures can be readily realized in photonic arrangements by exploiting the mathematical isomorphism between the quantum-mechanical Schrödinger equation and the paraxial wave equation of optics [2][3][4][5][6][7]. In this regard, a necessary albeit not sufficient condition for optical PT symmetry to hold is that the refractive index distribution involved should respect the following relationship * (− ) = ( ). In other words, the real part of the complex refractive index profile associated with a photonic PT symmetric structure must be an even function of position, while its imaginary component representing gain or loss must have an odd distribution.By virtue of their non-Hermiticity, PT-symmetric arrangements are capable of supporting exceptional points in their parameter space. Exceptional points, also known as non-Hermitian degeneracies, typically occur in non-conservative physical systems. At these points, the eigenvectors as well as their associated eigenvalues coalesce, resulting in an abrupt phase transition that dramatically changes the behavior of the system [8]. In optical settings, such degeneracies can arise from the interplay of gain and loss in the underlying design. A multitude of recent theoretical and experimental studies has shown different ways of how the presence of exceptional or phase-transition points can be fruitfully utilized to attain new behavior and functionality in non-Hermitian photonic systems, including those respecting PT symmetry [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Thus far, this methodology has been explored in a number of laser studies where a lower threshold and an inverse pump dependence of the output power have been investigated [18][19][20][21][22][23][24]. Lately, the selective breaking of PT symmetry has also been considered for laser mode management applications [23,24]. In particular, in [24], a PT symmetric double ring configuration was proposed as a promising avenue to enforce single longitudinal mode operation in inherently multi-moded semiconductor micro-cavities. This type of system is comprised of two identical ring resonators, where one provides gain while its counterpart is subjected to an appropriate amount of loss.In this work, our goal is to experimentally characterize the behavior of a parity-time symmetric photonic molecule around the exceptional point. We start with developing a mathematical model for a coupled cavity arrangement sh...