We apply the fermion commutation technique for composite bosons to polariton-polariton scattering in semiconductor planar microcavities. Derivations are presented in a simple and physically transparent fashion. A procedure of orthogonolization of the initial and final two-exciton state wavefunctions is used to calculate the effective scattering matrix elements and the scattering rates. We show how the bosonic stimulation of the scattering appears in this full fermionic approach whose equivalence to the bosonization method is thus demonstrated in the regime of low exciton density. We find an additional contribution to polariton-polariton scattering due to the exciton oscillator strength saturation, which we analyze as well. We present a theory of the polariton-polariton scattering with opposite spin orientations and show that this scattering process takes place mainly via dark excitonic states. Analytical estimations of the effective scattering amplitudes are given.

We present a theory of topological edge states in one-dimensional resonant photonic crystals with a compound unit cell. Contrary to the traditional electronic topological states the states under consideration are radiative, i.e., they decay in time due to the light escape through the structure boundaries. We demonstrate that the states survive despite their radiative decay and can be detected both in time-and frequency-dependent light reflection.PACS numbers: 78.67. Pt,78.47.jg Introduction. Topological insulator is an electronic material that has a band gap in its interior like an ordinary insulator but possesses conducting states on its edge or surface. The surface states of topological insulators have been extensively studied both in two-and threedimensional materials [1]. Recently an untrivial link has been revealed between such seemingly distinct systems as topological insulators, one-dimensional (1D) quasicrystals, and periodic 1D crystals with compound unit cell [2][3][4]. Particularly, it has been demonstrated that the 1D Aubry-André-Harper (AAH) model, or a "bichromatic" system (both incommensurate and commensurate), exhibits topological properties similar to those attributed to systems of a higher dimension [2][3][4]. This model allows states at sharp boundaries between two distinct topological systems. The system is described by a 1D tightbinding Hamiltonian with nearest-neighbor hopping and an on-site potential [5]. In the generalized AAH model both the hopping terms and the on-site potential are cosine modulated. It is the modulation phase that adds the second degree of freedom and permits one to relate the descendent 1D model with a 2D "ancestor" system which has a 2D band structure and quantized Chern numbers. In this Letter, instead of quasiparticles which tunnel from one site to another, we consider a 1D sequence of sites with resonant excitations long-range coupled through an electromagnetic field [6]. Such system is open, its eigenfrequencies are complex and its eigenstates are quasistationary due to the radiative decay. Hence, the resonant optical lattice stands out of the standard classification of topological insulators, developed for conservative and Hermitian electronic problems [7]. Nevertheless, we show here that this 1D bichromatic resonant photonic crystal demonstrates the topological properties in spite of being open and formulate general condition for the edge state existence. We also demonstrate how the radiative character of the problem opens new pathways to optical detection of the edge states. This provides an important insight into the rapidly expanding field of the electromagnetic topological states in photonic crystals [8,9], coupled cavities [10], waveguide arrays [11][12][13], and metamaterials [14].

Topological concepts open many new horizons for photonic devices, from integrated optics to lasers [1][2][3]. The complexity of large scale topological devices asks for an effective solution of the inverse problem: how best to engineer the topology for a specific application? We introduce a novel machine learning approach to the topological inverse problem [4][5][6]. We train a neural network system with the band structure of the Aubry-Andre-Harper model and then adopt the network for solving the inverse problem. Our application is able to identify the parameters of a complex topological insulator in order to obtain protected edge states at target frequencies. One challenging aspect is handling the multivalued branches of the direct problem and discarding unphysical solutions. We overcome this problem by adopting a self-consistent method to only select physically relevant solutions. We demonstrate our technique in a realistic topological laser design and by resorting to the widely available open-source TensorFlow library [7]. Our results are general and scalable to thousands of topological components. This new inverse design technique based on machine learning potentially extends the applications of topological photonics, for example, to frequency combs, quantum sources, neuromorphic computing and metrology.

We exploit topological edge states in resonant photonic crystals to attain strongly localized resonances and demonstrate lasing in these modes upon optical excitation. The use of virtually lossless topologically isolated edge states may lead to a class of thresholdless lasers operating without inversion. One needs, however, to understand whether topological states may be coupled to external radiation and act as active cavities. We study a two-level topological insulator and show that self-induced transparency pulses can directly excite edge states. We simulate laser emission by a suitably designed topological cavity and show that it can emit tunable radiation. For a configuration of sites following the off-diagonal Aubry-Andre-Harper model, we solve the Maxwell-Bloch equations in the time domain and provide a first-principles confirmation of topological lasers. Our results open the road to a class of light emitters with topological protection for applications ranging from low-cost energetically effective integrated laser sources, also including silicon photonics, to strong-coupling devices for studying ultrafast quantum processes with engineered vacuum