Synthetic Non-Hermitian Photonic Structures: Recent Results and Future Challenges

Optical sources in the long wavelength regime is scarce. Quantum cascade lasers have become the only tunable alternative. Achieving high power and single mode sources in the long wavelength regime remains a challenge. The inducing of higher order modes significantly reduces the laser beam quality. Various methods are designed to filter these modes. Implementing the PT symmetry approach is one of the effective methods for filtering these modes. By proper choice of loss and gain, it is possible to achieve desired optical properties. One of the important feature of the PT symmetric optical systems is that it provides an method for both transverse and longitudinal mode control. In this work, we design an symmetric directional coupler to achieve PT symmetric single transverse mode long wavelength QCL lasers using an active QCL waveguide and a waveguide which has equal amount of loss in the gain of active waveguide. Numerical simulation results and implementation of the design will be discussed.

The existence of non-Hermitian defect states in optical systems is known for coupled resonators with asymmetric backscattering. Here, we demonstrate that defect states in open optical systems can exist due to lifetime differences of counterpropagating modes without the need for asymmetric backscattering within the single resonator. We apply our findings to a finite system of coupled resonators perturbed by nanoparticles, in which we create an interface by inverting the orientation of the resonators in half of the chain. We compare a tight-binding approximation to a full-wave numerical simulation, showing that a system with spectrally isolated defect states can be implemented in a photonic device.

We unveil the geometrical meaning of winding number and utilize it to characterize the topological phases in one-dimensional chiral non-Hermitian systems. While chiral symmetry ensures the winding number of Hermitian systems being integers, it can take half integers for non-Hermitian systems. We give a geometrical interpretation of the half integers by demonstrating that the winding number $\nu$ of a non-Hermitian system is equal to half of the summation of two winding numbers $\nu_1$ and $\nu_2$ associated with two exceptional points respectively. The winding numbers $\nu_1$ and $\nu_2$ represent the times \of real part of the Hamiltonian in momentum space encircling the exceptional points and can only take integers. We further find that the difference of $\nu_1$ and $\nu_2$ is related to the second winding number or energy vorticity. By applying our scheme to a non-Hermitian Su-Schrieffer-Heeger model and an extended version of it, we show that the topologically different phases can be well characterized by winding numbers. Furthermore, we demonstrate that the existence of left and right zero-mode edge states is closely related to the winding number $\nu_1$ and $\nu_2$.

In previous works [1, 2], the 2-dimensional charge transport with parallel (in plane) magnetic field was considered from the theoretical point of view showing explicitly that the specific form of the emergent equation enforces the respective field solution to fulfil the Majorana condition. In this talk we review, explain and analyze these important results in the context of the generated physical effects, namely, the quantum ring as spin filter, the quantum Hall effect and a new one of pure topological origin (as the described by the Aharonov–Casher theorems). The link with supersymmetrical models as with the chiral effects in Weyl semimetals are briefly discussed. [1]D. J. Cirilo-Lombardo, Phys. Part. Nucl. Lett. 13, 26–31 (2016). [2]D. J. Cirilo-Lombardo, Int. J. Geom. Methods Mod. Phys. 12, 1550088 (2015).

The bulk-boundary correspondence in non-Hermitian systems is a matter of current intense and controversial debate. Here we study higher order topological states in non-Hermitian systems in light of the recently discussed biorthogonal bulk-boundary correspondence.

We will extend the notion of coherent perfect absorbers(CPA) in the case of periodically modulated media. We will show that this extra(time modulation) degree of freedom allows for on-the-fly reconfigurability while at the same time while at the same time reduces dramatically the amount of losses that are needed in order to observe CPA. Finally we show that specific driving schemes can lead to the realization of effective mirrors which, in turn, results in the realization of unidirectional CPAs.

We theoretically investigate how $2+1$ dimensional symmetry-protected Floquet topological insulators can be realized in photonic lattices of evanescently coupled waveguides. On a periodically driven array of waveguides arranged in a quadratic lattice, eight distinct symmetry operations can be implemented. A small subset of these eight symmetry operators allows for symmetry-protected topological boundary states that propagate in opposite directions. Remarkbly, different symmetry operations do not necessitate different waveguide structures. In this way, we implement a driving protocol on the quadratic lattice that can switch between time-reversal, chiral and particle-hole symmetry, and can be directly realized experimentally. For all three symmetries, the occurrence of counterpropagating edge transport is demonstrated. Especially for time-reversal symmetry, scatter-free edge transport is observed in agreement with Kramers degeneracy.

Photons are ideal carriers of quantum information. Generating pure single photons is therefore important for practical implementations of quantum information tasks. A standard way of generating single photons begins with spontaneous generation of a pair of photons in a pumped nonlinear medium; detection of one of the photons in the pair announces the presence of the other photon. As they are being generated, these photons have correlated spectral properties determined by the shape of the pumped light and by the momentum conservation of the light in the medium. Algorithms were developed in the last few years to control those correlation properties by shaping the poling structure of crystals. Customized poling structures can help create a specific photon distribution output by tracking the generated field amplitude as it progresses through the structure. However, these poling techniques can't directly be used to solve problems involving realistic structures with changing dispersion relations. We solved this problem by rethinking an algorithm which keeps track of the dispersion relations as they change in the crystals. This allowed us to find an optimal poling structure for an experimentally designed waveguide with a systematic variation in its dispersion relations for different poling orientations, and to find solutions to a few theoretical cases of pathological crystals. While we applied our algorithm to non-classical optics, the results are applicable as well to classical non-linear optics scenarios.

The Möbius strip, a long sheet of paper whose ends are glued together after a $180^{\circ}$ twist, has remarkable geometric and topological properties. Here, we consider dielectric Möbius strips of finite width and investigate the interplay between geometric properties and resonant light propagation. We show how the polarization dynamics of the electromagnetic wave depends on the topological properties, and demonstrate how the geometric phase can be manipulated between $0$ and $\pi$ through the system geometry. The loss of the Möbius character in thick cavities and for small twist segment lengths allows one to manipulate the polarization dynamics and the far-field emission, and opens the venue for applications.

A striking signature of the non-Hermitian physics in open systems are exceptional points in parameter space where not only the eigenvalues (frequencies) but simultaneously also the corresponding eigenstates (modes) coalesce. We report on exceptional points in whispering-gallery cavities where three modes coalesce. These exceptional points of third order are formed without external perturbations such as gain and loss or external scatterers. Rather an adjustment of the cavity's refractive index profile and/or the boundary shape is used to generate the exceptional points of third order.

One of the main features of topological phases of matter is bulk-boundary correspondence, which links the presence of topological boundary states to a non-trivial topological bulk invariant. Recently, it was realized that this famed bulk-boundary correspondence generally breaks down in topological systems that are described by non-Hermitian Hamiltonians. While the Bloch bands of the periodic Hamiltonian thus fail to provide useful information for the open system, we show that such models can be accurately quantified by making use of so-called biorthogonal quantum mechanics leading to the concept of biorthogonal bulk-boundary correspondence, which is formulated directly in the open system. On my poster, I present our findings by showing explicit examples of the non-Hermitian SSH chain and a non-Hermitian Chern insulator by making use of a generic method with which we can find exact solutions for the right and left eigenvectors of the boundary modes. This poster is based on Phys. Rev. Lett. 121, 026808 (2018).

Chains of chiral-coupled microresonators with two internal, asymmetrically coupled modes, have been shown previously [1] to display topological defect states which have no Hermitian counterpart. In this work, I will shown how two dimensional arrays of these resonators form complex bulk Fermi-arcs in the dispersion, which can be further tuned to display purely real or imaginary branches. Furthermore I will also show that at the interface between resonators where the asymmetry has been reversed, one dimensional edge-states which spectrally connected real and imaginary bulk bands form. These complex arcs between real and imaginary bulk bands can then be made free standing by controlling the anisotropy (real space distances between resonators in the x-direction and y-directions respectively) of the array. [1] - S. Malzard, C. Poli, and H. Schomerus, Phys. Rev. Lett. $\textbf{115}$, 200402 (2015)

We experimentally demonstrate that simultaneous subtraction of photons from two-mode squeezed vacuum states leads to the generation of entangled states with a larger average photon number. This technique allows for the engineering of a novel family of entangled multiphoton states with tunable mean photon numbers and degrees of entanglement.

A non-Hermitian degeneracy or exceptional point (EP) is a singular point in parameter space of a non-Hermitian operator where the eigenvalues and eigenvectors coalesce. By varying system parameters of the underlying Hamiltonian along a closed contour in the vicinity of an EP (it does not necessarily have to be enclosed) the final state depends asymmetrically on the encircling direction (chiral mode conversion). In a non-Hermitian PT-symmetric scattering system the EP marks the symmetry-breaking transition of the scattering matrix. We show that the EP of the scattering matrix is strongly related to the point where the chiral mode conversion breaks down and that the chiral mode conversion corresponds to an asymmetric behavior of the reflectivities. In this context we connect the scattering properties of $\mathcal{P}\mathcal{T}$-symmetric potentials with the chiral mode switch in the vicinity of an exceptional point of the Hamiltonian.

We numerically analyzed the supratransmission phenomenon in the discrete nonlinear Schrödinger equation with the cubic-quintic nonlinearity. We perform in the case of competing self-focusing cubic and defocusing quintic nonlinearities the novel scenarios of supratranmission such as, the generation of the train of dark soliton carried by a travelling kink for certain driving amplitude and generation of travelling kink.

Topological edge states appear at the interface of two topologically distinct Hermitian insulators. We study the extension of this idea to non-Hermitian systems. We consider PT-symmetric and topologically distinct non-Hermitian insulators with real spectra and study topological edge states at the interface of them. We show that PT symmetry is spontaneously broken at the interface during the topological phase transition. Therefore, topological edge states with complex energy eigenvalues appear at the interface. We apply our idea to a complex extension of the Su-Schrieffer-Heeger model.