Exciton-polariton topological insulator.

Topological insulators-materials that are insulating in the bulk but allow electrons to flow on their surface-are striking examples of materials in which topological invariants are manifested in robustness against perturbations such as defects and disorder1. Their most prominent feature is the emergence of edge states at the boundary between areas with different topological properties. The observable physical effect is unidirectional robust transport of these edge states. Topological insulators were originally observed in the integer quantum Hall effect2 (in which conductance is quantized in a strong magnetic field) and subsequently suggested3-5 and observed6 to exist without a magnetic field, by virtue of other effects such as strong spin-orbit interaction. These were systems of correlated electrons. During the past decade, the concepts of topological physics have been introduced into other fields, including microwaves7,8, photonic systems9,10, cold atoms11,12, acoustics13,14 and even mechanics15. Recently, topological insulators were suggested to be possible in exciton-polariton systems16-18 organized as honeycomb (graphene-like) lattices, under the influence of a magnetic field. Exciton-polaritons are part-light, part-matter quasiparticles that emerge from strong coupling of quantum-well excitons and cavity photons19. Accordingly, the predicted topological effects differ from all those demonstrated thus far. Here we demonstrate experimentally an exciton-polariton topological insulator. Our lattice of coupled semiconductor microcavities is excited non-resonantly by a laser, and an applied magnetic field leads to the unidirectional flow of a polariton wavepacket around the edge of the array. This chiral edge mode is populated by a polariton condensation mechanism. We use scanning imaging techniques in real space and Fourier space to measure photoluminescence and thus visualize the mode as it propagates. We demonstrate that the topological edge mode goes around defects, and that its propagation direction can be reversed by inverting the applied magnetic field. Our exciton-polariton topological insulator paves the way for topological phenomena that involve light-matter interaction, amplification and the interaction of exciton-polaritons as a nonlinear many-body system.

Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard.

Exciton-polaritons are hybrid light-matter quasiparticles formed by strongly interacting photons and excitons (electron-hole pairs) in semiconductor microcavities. They have emerged as a robust solid-state platform for next-generation optoelectronic applications as well as for fundamental studies of quantum many-body physics. Importantly, exciton-polaritons are a profoundly open (that is, non-Hermitian) quantum system, which requires constant pumping of energy and continuously decays, releasing coherent radiation. Thus, the exciton-polaritons always exist in a balanced potential landscape of gain and loss. However, the inherent non-Hermitian nature of this potential has so far been largely ignored in exciton-polariton physics. Here we demonstrate that non-Hermiticity dramatically modifies the structure of modes and spectral degeneracies in exciton-polariton systems, and, therefore, will affect their quantum transport, localization and dynamical properties. Using a spatially structured optical pump, we create a chaotic exciton-polariton billiard--a two-dimensional area enclosed by a curved potential barrier. Eigenmodes of this billiard exhibit multiple non-Hermitian spectral degeneracies, known as exceptional points. Such points can cause remarkable wave phenomena, such as unidirectional transport, anomalous lasing/absorption and chiral modes. By varying parameters of the billiard, we observe crossing and anti-crossing of energy levels and reveal the non-trivial topological modal structure exclusive to non-Hermitian systems. We also observe mode switching and a topological Berry phase for a parameter loop encircling the exceptional point. Our findings pave the way to studies of non-Hermitian quantum dynamics of exciton-polaritons, which may uncover novel operating principles for polariton-based devices.

Nonreciprocal Transport of Exciton Polaritons in a Non-Hermitian Chain.

We consider exciton polaritons in a zigzag chain of coupled elliptical micropillars subjected to incoherent excitation. The driven-dissipative nature of the system along with the naturally present polarization splitting inside the pillars gives rise to nonreciprocal dynamics, which eventually leads to the non-Hermitian skin effect, where all the modes of the system collapse to one edge. As a result, the polaritons propagate only in one direction along the chain, independent of the excitation position, and the propagation in the opposite direction is suppressed. The system shows robustness against disorder and, using the bistable nature of polaritons to encode information, we show one-way information transfer. This paves the way for compact and robust feedback-free one-dimensional polariton transmission channels without the need for external magnetic field, which are compatible with proposals for polaritonic circuits.

Coupling between Exciton-Polariton Corner Modes through Edge States.

Recently realized higher order topological insulators have taken a surge of interest among the theoretical and experimental condensed matter community. The two-dimensional second order topological insulators give rise to zero-dimensional localized corner modes that reside within the band gap of the system along with edge modes that inhabit a band edge next to bulk modes. Thanks to the topological nature, information can be trapped at the corners of these systems, which will be unhampered even in the presence of disorder. Being localized at the corners, the exchange of information among the corner states is an issue. Here we show that the nonlinearity in an exciton polariton system can allow the coupling between the different corners through the edge states based on optical parametric scattering, realizing a system of multiple connectible topological modes.

Optically induced transparency in bosonic cascade lasers.

Bosonic cascade lasers are terahertz (THz) lasers based on stimulated radiative transitions between bosonic condensates of excitons or exciton-polaritons confined in a trap. We study the interaction of an incoming THz pulse resonant in frequency with the transitions between neighboring energy levels of the cascade. We show that at certain optical pump conditions the cascade becomes transparent to the incident pulse: it neither absorbs nor amplifies it in the mean-field approximation. The populations of intermediate levels of the bosonic cascade change as the THz pulse passes, nevertheless. In comparison, a fermionic cascade laser does not reveal any of these properties.

Design for a Nanoscale Single-Photon Spin Splitter for Modes with Orbital Angular Momentum.

We propose using the effective spin-orbit coupling of light in Bragg-modulated cylindrical waveguides for the efficient separation of spin-up and spin-down photons emitted by a single photon emitter. Because of the spin and directional dependence of photonic stop bands in the waveguides, spin-up (-down) photon propagation in the negative (positive) direction along the waveguide axis is blocked while the same photon freely propagates in the opposite direction. Frequency shifts of photonic band structures induced by the spin-orbit coupling are verified by finite-difference time-domain numerical simulations.

Chiral Modes at Exceptional Points in Exciton-Polariton Quantum Fluids.

We demonstrate the generation of chiral modes-vortex flows with fixed handedness in exciton-polariton quantum fluids. The chiral modes arise in the vicinity of exceptional points (non-Hermitian spectral degeneracies) in an optically induced resonator for exciton polaritons. In particular, a vortex is generated by driving two dipole modes of the non-Hermitian ring resonator into degeneracy. Transition through the exceptional point in the space of the system's parameters is enabled by precise manipulation of real and imaginary parts of the closed-wall potential forming the resonator. As the system is driven to the vicinity of the exceptional point, we observe the formation of a vortex state with a fixed orbital angular momentum (topological charge). This method can be extended to generate higher-order orbital angular momentum states through coalescence of multiple non-Hermitian spectral degeneracies. Our Letter demonstrates the possibility of exploiting nontrivial and counterintuitive properties of waves near exceptional points in macroscopic quantum systems.

Spontaneous Polariton Currents in Periodic Lateral Chains.

We predict spontaneous generation of superfluid polariton currents in planar microcavities with lateral periodic modulation of both the potential and decay rate. A spontaneous breaking of spatial inversion symmetry of a polariton condensate emerges at a critical pumping, and the current direction is stochastically chosen. We analyze the stability of the current with respect to the fluctuations of the condensate. A peculiar spatial current domain structure emerges, where the current direction is switched at the domain walls, and the characteristic domain size and lifetime scale with the pumping power.

Spin Order and Phase Transitions in Chains of Polariton Condensates.

We demonstrate that multiply coupled spinor polariton condensates can be optically tuned through a sequence of spin-ordered phases by changing the coupling strength between nearest neighbors. For closed four-condensate chains these phases span from ferromagnetic (FM) to antiferromagnetic (AFM), separated by an unexpected crossover phase. This crossover phase is composed of alternating FM-AFM bonds. For larger eight-condensate chains, we show the critical role of spatial inhomogeneities and demonstrate a scheme to overcome them and prepare any desired spin state. Our observations thus demonstrate a fully controllable nonequilibrium spin lattice.

Cellular automata in photonic cavity arrays.

We propose theoretically a photonic Turing machine based on cellular automata in arrays of nonlinear cavities coupled with artificial gauge fields. The state of the system is recorded making use of the bistability of driven cavities, in which losses are fully compensated by an external continuous drive. The sequential update of the automaton layers is achieved automatically, by the local switching of bistable states, without requiring any additional synchronization or temporal control.

Perceptrons with Hebbian learning based on wave ensembles in spatially patterned potentials.

A general scheme to realize a perceptron for hardware neural networks is presented, where multiple interconnections are achieved by a superposition of Schrödinger waves. Spatially patterned potentials process information by coupling different points of reciprocal space. The necessary potential shape is obtained from the Hebbian learning rule, either through exact calculation or construction from a superposition of known optical inputs. This allows implementation in a wide range of compact optical systems, including (1) any nonlinear optical system, (2) optical systems patterned by optical lithography, and (3) exciton-polariton systems with phonon or nuclear spin interactions.

Optomechanics with cavity polaritons: dissipative coupling and unconventional bistability.

We study a hybrid system formed from an optomechanical resonator and a cavity mode strongly coupled to an excitonic transition inside a quantum well. We show that due to the mixing of cavity photon and exciton states, the emergent quasiparticles-polaritons-possess coupling to the mechanical mode of both a dispersive and dissipative nature. We calculate the occupancies of polariton modes and reveal bistable behavior, which deviates from conventional Kerr nonlinearity or dispersive coupling cases due to the dissipative coupling. The described system serves as a good candidate for future polaritonic devices.

Nonlinear effects in multi-photon polaritonics.

We consider theoretically nonlinear effects in a semiconductor quantum well embedded inside a photonic microcavity. Two-photon absorption by a 2p exciton state is considered and investigated; the matrix element of two-photon absorption is calculated. We compute the emission spectrum of the sample and demonstrate that under coherent pumping the nonlinearity of the two photon absorption process gives rise to bistability.

Stochastic Gross-Pitaevskii equation for the dynamical thermalization of Bose-Einstein condensates.

We present a theory for the description of energy relaxation in a nonequilibrium condensate of bosonic particles. The approach is based on coupling to a thermal bath of other particles (e.g., phonons in a crystal, or noncondensed atoms in a cold atom system), which are treated with a Monte Carlo type approach. Together with a full account of particle-particle interactions, dynamic driving, and particle loss, this offers a complete description of recent experiments in which Bose-Einstein condensates are seen to relax their energy as they propagate in real space and time. As an example, we apply the theory to the solid-state system of microcavity exciton polaritons, in which nonequilibrium effects are particularly prominent.

Proposal for a bosonic cascade laser.

We propose a concept of a quantum cascade laser based on transitions of bosonic quasiparticles (excitons) in a parabolic potential trap in a semiconductor microcavity. This laser would emit terahertz radiation due to bosonic stimulation of excitonic transitions. The dynamics of a bosonic cascade is strongly different from the dynamics of a conventional fermionic cascade laser. We show that populations of excitonic ladders are parity dependent and quantized if the laser operates without an external terahertz cavity.

Spin currents in a coherent exciton gas.

We report the observation of spin currents in a coherent gas of indirect excitons. The realized long-range spin currents originate from the formation of a coherent gas of bosonic pairs--a new mechanism to suppress the spin relaxation. The spin currents result in the appearance of a variety of polarization patterns, including helical patterns, four-leaf patterns, spiral patterns, bell patterns, and periodic patterns. We demonstrate control of the spin currents by a magnetic field. We also present a theory of coherent exciton spin transport that describes the observed exciton polarization patterns and indicates the trajectories of the spin currents.

Nonlinear optical spin Hall effect and long-range spin transport in polariton lasers.

We report on the experimental observation of the nonlinear analogue of the optical spin Hall effect under highly nonresonant circularly polarized excitation of an exciton-polariton condensate in a GaAs/AlGaAs microcavity. The circularly polarized polariton condensates propagate over macroscopic distances, while the collective condensate spins coherently precess around an effective magnetic field in the sample plane performing up to four complete revolutions.

Spontaneous symmetry breaking in a polariton and photon laser.

We report on the simultaneous observation of spontaneous symmetry breaking and long-range spatial coherence both in the strong- and the weak-coupling regime in a semiconductor microcavity. Under pulsed excitation, the formation of a stochastic order parameter is observed in polariton and photon lasing regimes. Single-shot measurements of the Stokes vector of the emission exhibit the buildup of stochastic polarization. Below threshold, the polarization noise does not exceed 10%, while above threshold we observe a total polarization of up to 50% after each excitation pulse, while the polarization averaged over the ensemble of pulses remains nearly zero. In both polariton and photon lasing regimes, the stochastic polarization buildup is accompanied by the buildup of spatial coherence. We find that the Landau criterion of spontaneous symmetry breaking and Penrose-Onsager criterion of long-range order for Bose-Einstein condensation are met in both polariton and photon lasing regimes.

Propagation and amplification dynamics of 1D polariton condensates.

The dynamics of propagating polariton condensates in one-dimensional microcavities is investigated through time resolved experiments. We find a strong increase in the condensate intensity when it travels through the nonresonantly excited area. This amplification is shown to come from bosonic stimulated relaxation of reservoir excitons into the polariton condensate, allowing for the repopulation of the condensate through nonresonant pumping. Thus, we experimentally demonstrate a polariton amplifier with a large band width, opening the way towards the transport of polaritons with high densities over macroscopic distances.