One of the fundamental axioms of quantum mechanics is associated with the Hermiticity of physical observables 1 . In the case of the Hamiltonian operator, this requirement not only implies real eigenenergies but also guarantees probability conservation. Interestingly, a wide class of non-Hermitian Hamiltonians can still show entirely real spectra. Among these are Hamiltonians respecting parity-time (PT) symmetry 2-7 . Even though the Hermiticity of quantum observables was never in doubt, such concepts have motivated discussions on several fronts in physics, including quantum field theories 8 , nonHermitian Anderson models 9 and open quantum systems 10,11 , to mention a few. Although the impact of PT symmetry in these fields is still debated, it has been recently realized that optics can provide a fertile ground where PT-related notions can be implemented and experimentally investigated [12][13][14][15] . In this letter we report the first observation of the behaviour of a PT optical coupled system that judiciously involves a complex index potential. We observe both spontaneous PT symmetry breaking and power oscillations violating left-right symmetry. Our results may pave the way towards a new class of PT-synthetic materials with intriguing and unexpected properties that rely on non-reciprocal light propagation and tailored transverse energy flow.Before we introduce the concept of spacetime reflection in optics, we first briefly outline some of the basic aspects of this symmetry within the context of quantum mechanics. In general, a HamiltonianĤ =p 2 /2m + V (x) (wherex andp are position and momentum operators respectively, m is mass and V is the potential) is considered to be PT symmetric, PTĤ =Ĥ PT , provided that it shares common eigenfunctions with the PT operator 1,[16][17][18][19][20][21] . This condition corresponds to an exact or unbroken PT symmetry, as opposed to that of broken PT symmetry, where, even though PTĤ =Ĥ PT is still valid,Ĥ and PT (or any other antilinear operator) possess different eigenvectors 22 . For the case considered here, given that the action of the parity P and time T operators is defined asp → −p,x → −x andp → −p,x →x, i → −i, respectively, it then follows that a necessary (but not sufficient) condition for a Hamiltonian to be PT symmetric is V (x) = V * (−x). In other words, PT symmetry requires that the real part of the potential V is an even function of position x, whereas the imaginary part is odd; that is, the Hamiltonian must have the formĤ =p 2 /2m + V R (x) + iεV I (x), where V R,I are the symmetric and antisymmetric components of V , respectively 12-14 . Clearly, if ε = 0, this Hamiltonian is Hermitian. It turns out that, even if the antisymmetric imaginary component is finite, this class of potentials can still allow for both bound and radiation states, all with entirely real spectra. This is possible as long as ε is below some threshold, ε < ε th . If, on the other hand, this limit is crossed (ε > ε th ), the spectrum ceases to be real and starts to involve imaginary ei...

The topological insulator is a fundamentally new phase of matter, with the striking property that the conduction of electrons occurs only on its surface, not within the bulk, and that conduction is topologically protected. Topological protection, the total lack of scattering of electron waves by disorder, is perhaps the most fascinating

One of the most interesting phenomena in solid-state physics is Anderson localization, which predicts that an electron may become immobile when placed in a disordered lattice. The origin of localization is interference between multiple scatterings of the electron by random defects in the potential, altering the eigenmodes from being extended (Bloch waves) to exponentially localized. As a result, the material is transformed from a conductor to an insulator. Anderson's work dates back to 1958, yet strong localization has never been observed in atomic crystals, because localization occurs only if the potential (the periodic lattice and the fluctuations superimposed on it) is time-independent. However, in atomic crystals important deviations from the Anderson model always occur, because of thermally excited phonons and electron-electron interactions. Realizing that Anderson localization is a wave phenomenon relying on interference, these concepts were extended to optics. Indeed, both weak and strong localization effects were experimentally demonstrated, traditionally by studying the transmission properties of randomly distributed optical scatterers (typically suspensions or powders of dielectric materials). However, in these studies the potential was fully random, rather than being 'frozen' fluctuations on a periodic potential, as the Anderson model assumes. Here we report the experimental observation of Anderson localization in a perturbed periodic potential: the transverse localization of light caused by random fluctuations on a two-dimensional photonic lattice. We demonstrate how ballistic transport becomes diffusive in the presence of disorder, and that crossover to Anderson localization occurs at a higher level of disorder. Finally, we study how nonlinearities affect Anderson localization. As Anderson localization is a universal phenomenon, the ideas presented here could also be implemented in other systems (for example, matter waves), thereby making it feasible to explore experimentally long-sought fundamental concepts, and bringing up a variety of intriguing questions related to the interplay between disorder and nonlinearity.

Physical systems exhibiting topological invariants are naturally endowed with robustness against perturbations, as manifested in topological insulators-materials exhibiting robust electron transport, immune from scattering by defects and disorder. Recent years have witnessed intense efforts toward exploiting these phenomena in photonics. Here we demonstrate a nonmagnetic topological insulator laser system exhibiting topologically protected transport in the cavity. Its topological properties give rise to single-mode lasing, robustness against defects, and considerably higher slope efficiencies compared to the topologically trivial counterparts. We further exploit the properties of active topological platforms by assembling the system from -chiral microresonators, enforcing predetermined unidirectional lasing without magnetic fields. This work paves the way toward active topological devices with exciting properties and functionalities.

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