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Exploring the non-Hermitian properties of semiconductor materials for optical applications is at the forefront of photonic research. However, the selection of appropriate systems to implement such photonic devices remains a topic of debate. In this work, we demonstrate that a perovskite crystal, characterized by its easy and low-cost manufacturing, when placed between two distributed Bragg reflectors with an air gap, can form a natural double microcavity. This construction shows promising properties for the realisation of novel, tunable non-Hermitian photonic devices through strong light-matter coupling. We reveal that such a system exhibits double-coupled polariton modes with dispersion including multiple inflection points. Owing to its non-Hermiticity, our system exhibits nonreciprocal properties and allows for the observation of exceptional points. Our experimental studies are in agreement with the theoretical analysis based on coupled mode theory and calculations based on transfer matrix method.

The emergence of spatial coherence in a confined two-dimensional Bose gas of exciton-polaritons with tuneable interactions offers a unique opportunity to explore the role of interactions in a phase transition in a driven-dissipative quantum system, where both the phase transition and thermalisation are mediated by interactions. We investigate, experimentally and numerically, the phase correlations and steady-state properties of the gas over a wide range of interaction strengths by varying the photonic/excitonic fraction of the polaritons and their density. We find that the first order spatial coherence function exhibits algebraic decay consistent with the Berezinskii-Kosterlitz-Thouless (BKT) phase transition. Surprisingly, the exponent of the algebraic decay is inversely proportional to the coherent density of polaritons, in analogy to equilibrium superfluids above the BKT transition, but with a different proportionality constant. Our work paves the way for future investigations of the phenomenon of phase transitions and superfluidity in a driven-dissipative setting

We investigate the turbulent properties of a confined driven-dissipative polariton quantum fluid after a pulsed excitation. Using numerical simulations, we provide insight into the vortex clustering processes that emerge during the relaxation dynamics of the initially injected vortex cloud. A confrontation between conservative and non-conservative dynamics reveals that the onset of clusterization strongly depends on the interplay between the different characteristic system lengths and time scales at stake, with an additional time scale due to dissipation in the non-conservative case. Quantification of the clustering observables allows us to numerically characterize the optimal conditions for observing Onsager condensation in decaying polariton systems, demonstrating its experimental reachability under pulse excitation. These findings hold significance for exploring the onset of turbulent dynamics in open systems, spanning both classical and quantum domains.

Recent study demonstrated that steady states of a polariton system may show a first-order dissipative phase transition with an exceptional point that appears as an endpoint of the phase boundary [R. Hanai et al., Phys. Rev. Lett. 122, 185301 (2019)]. Here, we show that this phase transition is strictly related to the stability of solutions. In general, the exceptional point does not correspond to the endpoint of a phase transition, but rather it is the point where stable and unstable solutions coalesce. Moreover, we show that the transition may occur also in the weak coupling regime, which was excluded previously. In a certain range of parameters, we demonstrate permanent Rabi-like oscillations between light and matter fields. Our results contribute to the understanding of nonequilibrium light-matter systems, but can be generalized to any two-component oscillatory systems with gain and loss.

Polaritons are quasiparticles resulting from strong quantum coupling of light and matter. Peculiar properties of polaritons are a mixture of physics usually restricted to one of these realms, making them interesting for study not only from the fundamental point of view but also for applications. In recent years, many studies have been devoted to the potential use of exciton-polaritons for computing. Very recently, it has been shown experimentally that they can be harnessed not only for digital computing, but also for optical neural networks and for optimization related to hard computational problems. Here, we provide a brief review of recent studies and most important results in this area. We focus our attention in particular on the emerging concepts of non-von-Neumann computing schemes and their realizations in exciton-polariton systems.

A setup that simulates ground states of spin graphs would allow one to solve computationally hard optimisation problems efficiently. Current optical setups to this goal have difficulties decoupling the amplitude and phase degrees of freedom of each effective spin; risking to yield the mapping invalid, a problem known as amplitude heterogeneity. Here, we propose a setup with coupled active optical cavity modes, where this problem is eliminated through their particular geometric arrangement. Acting as an effective Monte Carlo solver, the ground state can be found exactly. By tuning a parameter, the setup solves XY or Ising problems.

We observe natural exceptional points in the excitation spectrum of an exciton-polariton system by optically tuning the light-matter interactions. The observed exceptional points do not require any spatial or polarization degrees of freedom and result solely from the transition from weak to strong light-matter coupling. We demonstrate that they do not coincide with the threshold for photon lasing, confirming previous theoretical predictions [Phys. Rev. Lett. 122, 185301 (2019), Optica 7, 1015 (2020) ]. Using a technique where a strong coherent laser pump induces up-converted excitations, we encircle the exceptional point in the parameter space of coupling strength and particle momentum. Our method of local optical control of light-matter coupling paves the way to investigation of fundamental phenomena including dissipative phase transitions and non-Hermitian topological states.

We propose that light-matter coupling can be used to realize synthetic lattices. In particular, we consider a one-dimensional chain of exciton-photon sites to create a comb lattice that exhibits a transition from a flat band to a finite mass dispersion by tuning site-dependent light-matter coupling. Moreover, in a non-Hermitian system with gain and loss, the flat band phase is much more robust and the transition is accompanied by the appearance of exceptional points in the complex energy spectrum. We demonstrate that by engineering the light-matter coupling in the synthetic lattice, one can explore various phases in the lasing regime. Our proposal paves the way for studying non-Hermitian systems in higher dimensions.

We investigate the spin polarization of localized exciton-polariton condensates. We demonstrate the presence of an effective magnetic field leading to the formation of elliptically polarized condensates. We show that this synthetic field has an entirely photonic origin, which we believe is unique for the CdTe-based microcavities. Moreover, the degree of spin polarization of localized polariton condensates in samples with magnetic ions depends on the excitation power or polarization of the non-resonant excitation laser. In an external magnetic field, the semimagnetic condensate spontaneously builds up strong spin polarization. Based on the magnetic field behavior of the condensate in the presence of magnetic ions, we apply a model that allows us to estimate the polariton-polariton interaction strength in a CdTe-system to approx. 0.8 $\mu \text{eV}\!\cdot\!\mu \text{m}^2$.

Turbulent phenomena are among the most striking effects that both classical and quantum fluids can exhibit. While classical turbulence is ubiquitous in nature, the observation of quantum turbulence requires the precise manipulation of quantum fluids such as superfluid helium or atomic Bose-Einstein condensates. In this work we demonstrate the turbulent dynamics of a 2D quantum fluid of exciton-polaritons, hybrid light-matter quasiparticles, both by measuring the kinetic energy spectrum and showing the onset of vortex clustering. We demonstrate that the formation of clusters of quantum vortices is triggered by the increase of the incompressible kinetic energy per vortex, showing the tendency of the vortex-gas towards highly excited configurations despite the dissipative nature of our system. These results lay the basis for the investigations of quantum turbulence in two-dimensional fluids of light.

We demonstrate that time-delayed nonlinear effects in exciton-polaritons can be used to construct neural networks where information is coded in optical pulses arriving consecutively on the sample. The highly nonlinear effects are induced by time-dependent interactions with the excitonic reservoir. These nonlinearities allow to create a nonlinear XOR logic gate that can perform operations on the picosecond timescale. An optoelectronic neural network based on the constructed logic gate performs classification of spoken digits with a high accuracy rate.

In contrast to software simulations of neural networks, hardware implementations have often limited or no tunability. While such networks promise great improvements in terms of speed and energy efficiency, their performance is limited by the difficulty to apply efficient training. We propose and realize experimentally an optical system where highly efficient backpropagation training can be applied through an array of highly nonlinear, non-tunable nodes. The system includes exciton-polariton nodes realizing nonlinear activation functions. We demonstrate a high classification accuracy in the MNIST handwritten digit benchmark in a single hidden layer system.

Spin-orbit interactions which couple spin of a particle with its momentum degrees of freedom lie at the center of spintronic applications. Of special interest in semiconductor physics are Rashba and Dresselhaus spin-orbit coupling (SOC). When equal in strength, the Rashba and Dresselhaus fields result in SU(2) spin rotation symmetry and emergence of the persistent spin helix (PSH) only investigated for charge carriers in semiconductor quantum wells. Recently, a synthetic Rashba-Dresselhaus Hamiltonian was shown to describe cavity photons confined in a microcavity filled with optically anisotropic liquid crystal. In this work, we present a purely optical realisation of two types of spin patterns corresponding to PSH and the Stern-Gerlach experiment in such a cavity. We show how the symmetry of the Hamiltonian results in spatial oscillations of the spin orientation of photons travelling in the plane of the cavity.

We investigate the phase diagram of a two-dimensional driven-dissipative system of polaritons coupled to the excitonic reservoir. We find that two critical points exists. The first corresponds to the quasi-condensation and the second to a first-order phase transition from the non-uniform state with spatially modulated density to a uniform state. The latter is related to the modulational instability of a homogeneous state due to the repulsive interactions with the non-condensed reservoir. The first-order character of the transition is evidenced by a discontinuity in the density and the correlation length as well as the phase coexistence and metastability. Moreover, we show that a signature of a Berezinskii-Kosterlitz-Thouless-like transition can be observed in the non-uniform phase.

We demonstrate the controlled coherent transfer of topological domain walls in a one-dimensional non-Hermitian chain of interacting Bose-Einstein condensates. The topological protection stems from a spatially patterned pump in an open-dissipative system. As a testbed setup of the proposed phenomenon, we consider a chain of coupled micropillars with embedded quantum wells, possessing exciton-polariton resonances. The transfer of a domain wall is driven by spatially localised, adiabatic pump modulation in the vicinity of the domain wall. The stochastic calculations prove the coherent nature of the domain wall transfer. For appropriate system parameters the coherence degree is preserved after multiple transitions, paving the way towards long-range transfer of a coherent quantum state.

We predict the existence of non-Hermitian topologically protected end states in a one-dimensional exciton-polariton condensate lattice, where topological transitions are driven by the laser pump pattern. We show that the number of end states can be described by a Chern number and a topological invariant based on the Wilson loop. We find that such transitions arise due to \it enforced exceptional points which can be predicted directly from the bulk Bloch wave functions. This allows us to establish a new type of bulk-boundary correspondence for non-Hermitian systems and to compute the phase diagram of an open chain analytically. Finally, we demonstrate topological lasing of a single end-mode in a realistic model of a microcavity lattice.

Machine learning software applications are nowadays ubiquitous in many fields of science and society for their outstanding capability of solving computationally vast problems like the recognition of patterns and regularities in big datasets. One of the main goals of research is the realization of a physical neural network able to perform data processing in a much faster and energy-efficient way than the state-of-the-art technology. Here we show that lattices of exciton-polariton condensates accomplish neuromorphic computing using fast optical nonlinearities and with lower error rate than any previous hardware implementation. We demonstrate that our neural network significantly increases the recognition efficiency compared to the linear classification algorithms on one of the most widely used benchmarks, the MNIST problem, showing a concrete advantage from the integration of optical systems in reservoir computing architectures.

We study nonresonantly pumped exciton-polariton system in the vicinity of the dynamical instability threshold. We find that the system exhibits unique and rich dynamics, which leads to spatiotemporal pattern formation. The patterns have a tree-like structure and are reminiscent of structures that appear in a variety of soft matter systems. Within the approximation of slow and fast time scales, we show that the polariton model exhibits self-replication point in analogy to reaction-diffusion systems.

Quantum fluids of light are an emerging platform for energy efficient signal processing, ultra-sensitive interferometry and quantum simulators at elevated temperatures. Here we demonstrate the optical control of the topological excitations induced in a large polariton condensate, realising the bosonic analog of a long Josephson junction and reporting the first observation of bosonic Josephson vortices. When a phase difference is imposed at the boundaries of the condensate, two extended regions become separated by a sharp $\pi$-slippage of the phase and a solitonic depletion of the density, forming an insulating barrier with a suppressed order parameter. The superfluid behavior, that is a smooth phase gradient across the system instead of the sharp phase jump, is recovered at higher polariton densities and it is mediated by the nucleation of Josephson vortices within the barrier. Our results contribute to the understanding of dissipation and stability of elementary excitations in macroscopic quantum systems.

Bosonic condensates of exciton polaritons (light-matter quasiparticles in a semiconductor) provide a solid-state platform for studies of non-equilibrium quantum systems with a spontaneous macroscopic coherence. These driven, dissipative condensates typically coexist and interact with an incoherent reservoir, which undermines measurements of key parameters of the condensate. Here, we overcome this limitation by creating a high-density exciton-polariton condensate in an optically-induced "box" trap. In this so-called Thomas-Fermi regime, the condensate is fully separated from the reservoir and its behaviour is dominated by interparticle interactions. We use this regime to directly measure the polariton-polariton interaction strength, and reduce the existing uncertainty in its value from four orders of magnitude to within three times the theoretical prediction. The Thomas-Fermi regime has previously been demonstrated only in ultracold atomic gases in thermal equilibrium. In a non-equilibrium exciton-polariton system, this regime offers a novel opportunity to study interaction-driven effects unmasked by an incoherent reservoir.

The availability of large amounts of data and the necessity to process it efficiently have led to rapid development of machine learning techniques. To name a few examples, artificial neural network architectures are commonly used for financial forecasting, speech and image recognition, robotics, medicine, and even research. Direct hardware for neural networks is highly sought for overcoming the von Neumann bottleneck of software implementations. Reservoir computing (RC) is a recent and increasingly popular bio-inspired computing scheme which holds promise for an efficient temporal information processing. We demonstrate the applicability and performance of reservoir computing in a general complex Ginzburg-Landau lattice model, which adequately describes dynamics of a wide class of systems, including coherent photonic devices. In particular, we propose that the concept can be readily applied in exciton-polariton lattices, which are characterized by unprecedented photonic nonlinearity, opening the way to signal processing at rates of the order of 1 Tbit (1/s).

The suppression of Zeeman energy splitting due to spin-dependent interactions within a Bose-Einstein condensate (the spin Meissner effect) was predicted to occur up to a certain value of magnetic field strength. We report a clear observation of this effect in semimagnetic microcavities which exhibit the giant Zeeman energy splitting between two spin-polarised polariton states as high as 2 meV, and demonstrate that partial suppression of energy difference occurs already in the uncondensed phase in a striking similarity to the up-critical superconductors in the fluctuation dominated regime. These observations are explained quantitatively by a kinetic model accounting for both the condensed and uncondensed polaritons and taking into account the non-equilibrium character of the system.

An exciton-polariton microcavity that incorporates magnetic ions can exhibit a spontaneous self-trapping phenomenon which is an analog of the classical polaron effect. We investigate in detail the full model of a polariton condensate that includes pumping and losses, the spin degree of freedom, external magnetic field and energy relaxation. In the quasi-one-dimensional case, we show that the polaron effect can give rise to a spontaneous lattice of perfectly arranged polarization domains in an antiferromagnetic configuration. We find that partial polarization of the condensate at moderate magnetic field strengths facilitates the formation of such "polaron lattices", which are qualitatively different from self-trapped polarons that appear in a fully polarized condensate. Within the Bogoliubov-de Gennes approximation, we calculate the instability condition which marks the appearance of the patterns. Surprisingly, we find that the stability condition displays a discontinuity at the point of partial-full polarization threshold.

We provide an analytical and numerical description of relaxation oscillations in the nonresonantly pumped polariton condensate. The presented considerations are based on the open dissipative Gross-Pitaevskii equation coupled to a pair of rate equations. The evolution of the condensate density can be explained qualitatively by studying the topology of the trajectory in phase space. We use a fixed points analysis for the classification of the different regimes of condensate dynamics, including fast stabilization, slow oscillations and ultrashort pulse emission. We obtain an analytical condition for the occurrence of relaxation oscillations. Continuous and pulsed condensate excitation considered and we demonstrate that in the latter case the existence of the second reservoir is necessary for the emergence of oscillations. We show that relaxation oscillations should be expected to occur in systems with relatively short polariton lifetime.

We study the emergence of hysteresis during the process of quantum phase transition from an antiferromagnetic to a phase-separated state in a spin-1 Bose Einstein condensate of ultracold atoms. We explicitly demonstrate the appearance of a hysteresis loop with various quench times showing that it is rate-independent for large magnetizations only. In other cases scaling of the hysteresis loop area is observed, which we explain by using the Kibble-Zurek theory in the limit of small magnetization. The effect of an external harmonic trapping potential is also discussed.

We study theoretically the emission of dark solitons induced by a moving defect in a nonresonantly pumped exciton-polariton condensate. The number of created dark solitons per unit of time is found to be strongly dependent on the pump power. We relate the observed dynamics of this process to the oscillations of the drag force experienced by the condensate. We investigate the stability of the polariton quantum fluid and present various types of dynamics depending on the condensate and moving obstacle parameters. Furthermore, we provide analytical expressions for dark soliton dynamics using the variational method adapted to the non-equilibrium polariton system. The determined dynamical equations are found to be in excellent agreement with the results of numerical simulations.

Owing to their integer spin, exciton-polaritons in microcavities can be used for observation of non-equilibrium Bose-Einstein condensation in solid state. However, spin-related phenomena of such condensates are difficult to explore due to the relatively small Zeeman effect of standard semiconductor microcavity systems and the strong tendency to sustain an equal population of two spin components, which precludes the observation of condensates with a well defined spin projection along the axis of the system. The enhancement of the Zeeman splitting can be achieved by introducing magnetic ions to the quantum wells, and consequently forming semimagnetic polaritons. In this system, increasing magnetic field can induce polariton condensation at constant excitation power. Here we evidence the spin polarization of a semimagnetic polaritons condensate exhibiting a circularly polarized emission over 95% even in a moderate magnetic field of about 3 T. Furthermore, we show that unlike nonmagnetic polaritons, an increase on excitation power results in an increase of the semimagnetic polaritons condensate spin polarization. These properties open new possibilities for testing theoretically predicted phenomena of spin polarized condensate.

We consider a condensate of exciton-polaritons in a diluted magnetic semiconductor microcavity. Such system may exhibit magnetic self-trapping in the case of sufficiently strong coupling between polaritons and magnetic ions embedded in the semiconductor. We investigate the effect of the nonequilibrium nature of exciton-polaritons on the physics of the resulting self-trapped magnetic polarons. We find that multiple polarons can exist at the same time, and derive a critical condition for self-trapping which is different to the one predicted previously in the equilibrium case. Using the Bogoliubov-de Gennes approximation, we calculate the excitation spectrum and provide a physical explanation in terms of the effective magnetic attraction between polaritons, mediated by the ion subsystem.

Bose-Einstein condensate of exciton polaritons in a semiconductor microcavity is a macroscopically populated coherent quantum state subject to concurrent pumping and decay. Debates about the fundamental nature of the condensed phase in this open quantum system still persist. Here, we gain a new insight into the spontaneous condensation process by imaging long-lifetime exciton polaritons in a high-quality inorganic microcavity in the single-shot optical excitation regime, without averaging over multiple condensate realisations. In this highly non-stationary regime, a condensate is strongly influenced by the `hot' incoherent reservoir, and reservoir depletion is critical for the transition to the ground energy and momentum state. Condensates formed by more photonic exciton polaritons exhibit dramatic reservoir-induced density filamentation and shot-to-shot fluctuations. In contrast, condensates of more excitonic quasiparticles display smooth density and are second-order coherent. Our observations show that the single-shot measurements offer a unique opportunity to study formation of macroscopic phase coherence during a quantum phase transition in a solid state system.

We demonstrate the existence of the excited state of an exciton-polariton in a semiconductor microcavity. The strong coupling of the quantum well heavy-hole exciton in an excited 2s state to the cavity photon is observed in non-zero magnetic field due to surprisingly fast increase of Rabi energy of the 2s exciton-polariton in magnetic field. This effect is explained by a strong modification of the wave-function of the relative electron-hole motion for the 2s exciton state.

We demonstrate the existence of a novel quasiparticle: an exciton in a semiconductor doubly dressed with two photons of different wavelengths: near infrared cavity photon and terahertz (THz) photon, with the THz coupling strength approaching the ultra-strong coupling regime. This quasiparticle is composed of three different bosons, being a mixture of a matter-light quasiparticle. Our observations are confirmed by a detailed theoretical analysis, treating quantum mechanically all three bosonic fields. The doubly dressed quasiparticles retain the bosonic nature of their constituents, but their internal quantum structure strongly depends on the intensity of the applied terahertz field.

In this work we experimentally demonstrate for the first time spontaneous generation of two-dimensional exciton-polariton X-waves. X-waves belong to the family of localized packets, which are capable of sustaining their shape with no spreading even in the linear regime. This allows to keep the packet shape and size for very low densities and very long times compared, for instance, to soliton waves, which always necessitate a nonlinearity to compensate the diffusion. Here we exploit the polariton nonlinearity and unique structured dispersion, comprising both positive- and negative-mass curvatures, to trigger an asymmetric four wave mixing in the momentum space. This ultimately enables self-formation of a spatial X-wave front. By means of ultrafast imaging experiments we observe the early reshaping of the initial Gaussian packet into the X-pulse and its propagation even for vanishing small densities. This allows us to outline the crucial effects and parameters driving the phenomena and to tune the degree of peak superluminal propagation, which we found to be in a good agreement with numerical simulations.

We investigate the process of coarsening via annihilation of vortex-antivortex pairs, following the quench to the condensate phase in a nonresonantly pumped polariton system. We find that the late-time dynamics is an example of universal phase ordering kinetics, characterized by scaling of correlation functions in time. Depending on the parameters of the system, the evolution of the characteristic length scale L(t) can be the same as for the two-dimensional XY model, described by a power law with the dynamical exponent z \aprox 2 and a logarithmic correction, or z \aprox 1 which agrees with previous studies of conservative superfluids.

We study a system of interacting matter quasiparticles strongly coupled to photons inside an optomechanical cavity. The resulting normal modes of the system are represented by hybrid polaritonic quasiparticles, which acquire effective nonlinearity. Its strength is influenced by the presence of the mechanical mode and depends on the resonance frequency of the cavity. This leads to an interactive type of optomechanical coupling, being distinct from the previously studied dispersive and dissipative couplings in optomechanical systems. The emergent interactive coupling is shown to generate effective optical nonlinearity terms of high order, being quartic in the polariton number. We consider particular systems of exciton-polaritons and dipolaritons, and show that the induced effective optical nonlinearity due to the interactive coupling can exceed in magnitude the strength of Kerr nonlinear terms, such as those arising from polariton-polariton interactions. As applications, we show that the higher order terms give rise to localized bright flat top solitons, which may form spontaneously in polariton condensates.

We study the effect of an external harmonic trapping potential on an outcome of the non-adiabatic quantum phase transition from an antiferromagnetic to a phase-separated state in a spin-1 atomic condensate. Previously, we demonstrated that the dynamics of an untrapped system exhibits double universality with two different scaling laws appearing due to conservation of magnetization. We show that in the presence of a trap double universality persists. However, the corresponding scaling exponents are strongly modified by transfer of local magnetization across the system. The values of these exponents cannot be explained by the effect of causality alone, as in the spinless case. We derive the appropriate scaling laws based on a slow diffusive-drift relaxation process in the local density approximation.

The observation of spin-related phenomena of microcavity polaritons has been limited due to weak Zeeman effect of non-magnetic semiconductors. We demonstrate that the incorporation of magnetic ions into quantum wells placed in a non-magnetic microcavity results in enhanced effects of magnetic field on exciton-polaritons. We show that in such a structure the Zeeman splitting of exciton-polaritons strongly depends on the photon - exciton detuning and polariton wavevector. Our experimental data are explained by a model where the impact of magnetic field on the lower polariton state is directly inherited from the excitonic component, and the coupling strength to cavity photon is modified by external magnetic field.

By imaging single-shot realizations of an organic polariton quantum fluid, we observe the long-sought dynamical instability of non-equilibrium condensates. Without any free parameters, we find an excellent agreement between the experimental data and a numerical simulation of the open-dissipative Gross-Pitaevskii equation, which allows us to draw several important conclusions about the physics of the system. We find that the reservoir dynamics are in the strongly nonadiabatic regime, which renders the complex Ginzburg-Landau description invalid. The observed transition from stable to unstable fluid can only be explained by taking into account the specific form of reservoir-mediated instability as well as particle currents induced by the finite extent of the pump spot.

A novel, optimized numerical method of modeling of an exciton-polariton superfluid in a semiconductor microcavity was proposed. Exciton-polaritons are spin-carrying quasiparticles formed from photons strongly coupled to excitons. They possess unique properties, interesting from the point of view of fundamental research as well as numerous potential applications. However, their numerical modeling is challenging due to the structure of nonlinear differential equations describing their evolution. In this paper, we propose to solve the equations with a modified Runge--Kutta method of 4th order, further optimized for efficient computations. The algorithms were implemented in form of C++ programs fitted for parallel environments and utilizing vector instructions. The programs form the EPCGP suite which have been used for theoretical investigation of exciton-polaritons.

We investigate the possibility of creating X-waves, or localized wave packets, in resonantly excited exciton-polariton superfluids. We demonstrate the existence of X-wave traveling solutions in the coupled exciton-photon system past the inflection point, where the effective mass of lower polaritons is negative in the direction perpendicular to the wavevector of the pumping beam. Contrary to the case of bright solitons, X-waves do not require nonlinearity for sustaining their shape. Nevertheless, we show that nonlinearity is important for their dynamics, as it allows for their spontaneous formation from an initial Gaussian wave packet. Unique properties of exciton-polaritons may lead to applications of their X-waves in long-distance signal propagation inside novel integrated optoelectronic circuits based on excitons.

We study the relation between the models commonly used to describe the dynamics of nonresonantly pumped exciton-polariton condensates, namely the ones described by the complex Ginzburg-Landau equation, and by the open-dissipative Gross-Pitaevskii equation including a separate equation for the reservoir density. In particular, we focus on the validity of the adiabatic approximation that allows to reduce the coupled condensate-reservoir dynamics to a single partial differential equation. We find that the adiabatic approximation consists of three independent analytical conditions that have to be fulfilled simultaneously. By investigating stochastic versions of the two corresponding models, we verify that the breakdown of the adiabatic approximation can lead to discrepancies in correlation lengths and distributions of fluctuations. Additionally, we consider the phase diffusion and number fluctuations of a condensate in a box, and show that self-consistent description requires treatment beyond the typical Bogoliubov approximation.

We develop a method for investigating nonequilibrium dynamics of an ultracold system that is initially at thermal equilibrium. Our procedure is based on the classical fields approximation with appropriately prepared initial state. As an application of the method, we investigate the influence of thermal fluctuations on the quantum phase transition from an antiferromagnetic to phase separated ground state in a spin-1 Bose-Einstein condensate of ultracold atoms. We find that at temperatures significantly lower than the critical condensation temperature $T_c$ the scaling law for the number of created spin defects remains intact.

We study, theoretically and numerically, the onset and development of modulational instability in an incoherently pumped spatially homogeneous polariton condensate. Within the framework of mean-field theory, we identify regimes of modulational instability in two cases: 1) Strong feedback between the condensate and reservoir, which may occur in scalar condensates, and 2) Parametric scattering in the presence of polarization splitting in spinor condensates. In both cases we investigate the instability induced textures in space and time including non-equilibrium dynamics of phase dislocations and vortices. In particular we discuss the mechanism of vortex destabilization and formation of spiraling waves. We also identify the presence of topological defects, which take the form of half-vortex pairs in the spinor case, giving an "eyelet" structure in intensity and dipole type structure in the spin polarization. In the modulationally stable parameter domains, we observe formation of the phase defects in the process of condensate formation from an initially spatially incoherent low-density state. In analogy to the Kibble-Zurek type scaling for nonequilibrium phase transitions, we find that the defect density scales with the pumping rate.

The family of one-dimensional localized solutions to dissipative nonlinear equations includes a variety of objects such as sources, sinks, shocks (kinks), and pulses. These states are in general accompanied by nontrivial density currents, which are not necessarily related to the movement of the object itself. We investigate the existence and physical properties of sink-type solutions in nonresonantly pumped exciton-polariton condensates modeled by an open-dissipative Gross-Pitaevskii equation. While sinks possess density profiles similar to bright solitons, they are qualitatively different objects as they exist in the case of repulsive interactions and represent a heteroclinic solution. We show that sinks can be created in realistic systems with appropriately designed pumping profiles. We also conclude that in two-dimensional configurations, due to the proliferation of vortices, sinks do not appear.

We detail the influence of a magnetic field on exciton-polaritons inside a semiconductor microcavity. Magnetic field can be used as a tuning parameter for exciton and photon resonances. We discuss the change of the exciton energy, the oscillator strength and redistribution of the polariton density along the dispersion curves due to the magnetically-induced detuning. We have observed that field-induced shrinkage of the exciton wave function has a direct influence not only on the exciton oscillator strength, which is observed to increase with the magnetic field, but also on the polariton linewidth. We discuss the effect of the Zeeman splitting on polaritons which magnitude changes with the exciton Hopfield coefficient and can be modelled by independent coupling of the two spin components of excitons with cavity photons.

We investigate the stability and coherence properties of one-dimensional exciton-polariton condensates under nonresonant pumping. We model the condensate dynamics using the open-dissipative Gross-Pitaevskii equation. In the case of spatially homogeneous pumping, we find that the instability of the steady state leads to significant eduction of the coherence length. We consider two effects that can lead to the stabilization of the steady state, i.e. the polariton energy relaxation and the influence of an inhomogeneous pumping profile. We find that, while the former has little effect on the stability, the latter is very effective in stabilizing the condensate which results in a large coherence length.

We predict the existence of a self-localized solution in a nonresonantly pumped exciton-polariton condensate. The solution has a shape resembling the well-known hyperbolic tangent profile of the dark soliton, but exhibits several distinct features. We find that it performs small oscillations, which are transformed into 'soliton explosions' at lower pumping intensities. Moreover, after hundreds or thousands of picoseconds of apparently stable evolution the soliton decays abruptly, which is explained by the acceleration instability found previously in the Bekki-Nozaki hole solutions of the complex Ginzburg-Landau equation. We show that the soliton can be formed spontaneously from a small seed in the polariton field or by using spatial modulation of the pumping profile.

We investigate the dynamics and outcome of a quantum phase transition from an antiferromagnetic to phase separated ground state in a spin-1 Bose-Einstein condensate of ultracold atoms. We explicitly demonstrate double universality in dynamics within experiments with various quench time. Furthermore, we show that spin domains created in the nonequilibrium transition constitute a set of mutually incoherent quasicondensates. The quasicondensates appear to be positioned in a semi-regular fashion, which is a result of the conservation of local magnetization during the post-selection dynamics.

Polaritons in microcavities are versatile quasi-2D bosonic particles with a high degree of coherence and strong nonlinearities, thanks to their hybrid light-matter character. In their condensed form, they display striking quantum hydrodynamic features analogous to atomic Bose-Einstein condensates, such as long-range order coherence, superfluidity and quantized vorticity. Their variegated dispersive and dissipative properties, however, set significant differences from their atomic counterpart. In this work, we report the unique phenomenology that is observed when a pulse of light impacts the polariton vacuum: the condensate that is instantaneously formed does not splash in real space but instead coheres into an enigmatic structure, featuring concentric rings and, most notably, a sharp and bright peak at the center. Using a state-of-the-art ultrafast imaging with 50 fs time steps, we are able to track the dynamics of the polariton mean-field wavefunction in both real and reciprocal space. The observation of the real-space collapse of the condensate into an extremely localized---resolution limited---peak is at odd with the repulsive interactions of polaritons and their positive effective mass. An unconventional mechanism is therefore at play to account for our observations. Our modeling suggests that self-trapping due to a local heating of the crystal lattice---that can be described as a collective polaron formed by a polariton condensate---could be involved. These observations hint at the fascinating fluid dynamics of polaritons in conditions of extreme intensities and ultrafast times.

We investigate the process of condensation of exciton-polaritons in a one-dimensional nanowire, predicting spontaneous formation of domains of uncondensed excitons and condensed polaritons. We demonstrate a universal scaling law for the density of domains, which results from the competition between characteristic timescales present in the system. However, we find that the system does not follow the standard Kibble-Żurek scenario of a nonequilibrium phase transition.

We consider a phase transition from antiferromagnetic to phase separated ground state in a spin-1 Bose-Einstein condensate of ultracold atoms. We demonstrate the occurrence of two scaling laws, for the number of spin fluctuations just after the phase transition, and for the number of spin domains in the final, stable configuration. Only the first scaling can be explained by the standard Kibble-Żurek mechanism. We explain the occurrence of two scaling laws by a model including post-selection of spin domains due to the conservation of condensate magnetization.

We demonstrate a method of controlled creation of spin domains in spin-1 antiferromagnetic Bose-Einstein condensates. The method exploits the phenomenon of phase separation of spin components in an external potential. By using an appropriate time dependent potential, a composition of spin domains can be created, as we demonstrate in the particular cases of a double well and a periodic potential. In contrast to other methods, which rely on spatially inhomogeneous magnetic fields, here the domain structure is completely determined by the optical fields, which makes the method versatile and reconfigurable. It allows for creation of domains of various sizes, with the spatial resolution limited by the spin healing length only.

We investigate the ground states of a Bose-Einstein condensate of indirect excitons coupled to an electron gas. We show that in a properly designed system, the crossing of a roton minimum into the negative energy domain can result in the appearance of the supersolid phase, characterized by periodicity in both real and reciprocal space. Accounting for the spin-dependent exchange interaction of excitons we obtain ferromagnetic supersolid domains. The Fourier spectra of excitations of weakly perturbed supersolids show pronounced diffraction maxima which may be detected experimentally.

We consider a spin-1 Bose-Einstein condensate trapped in a harmonic potential under the influence of a homogeneous magnetic field. We investigate spatial and spin structure of the mean-field ground states under constraints on the number of atoms and the total magnetization. We show that the trapping potential can make the antiferromagnetic condensate separate into three, and ferromagnetic condensate into two distinct phases. In the ferromagnetic case, the magnetization is located in the center of the harmonic trap, while in the antiferromagnetic case magnetized phases appear in the outer regions. We describe how the transition from the Thomas-Fermi regime to the single-mode approximation regime with decreasing number of atoms results in the disappearance of the domains. We suggest that the ground states can be created in experiment by adiabatically changing the magnetic field strength.

We show that metastable phases of an antiferromagnetic spin-1 condensate in a simple model with pure contact interactions can exhibit a rotonlike minimum in the excitation spectrum. The introduction of magnetic field gives rise to the instability of roton modes, which can lead to spontaneous emergence of regular periodic, polygonal, polyhedral or crystalline patterns, as shown in numerical simulations within the truncated Wigner approximation. An explanation of the occurrence of rotonlike instability is given based on the energy and spin conservation laws.

We analyze the structure of spin-1 Bose-Einstein condensates in the presence of a homogenous magnetic field. We classify the homogenous stationary states and study their existence, bifurcations, and energy spectra. We reveal that the phase separation can occur in the ground state of polar condensates, while the spin components of the ferromagnetic condensates are always miscible and no phase separation occurs. Our theoretical model, confirmed by numerical simulations, explains that this phenomenon takes place when the energy of the lowest homogenous state is a concave function of the magnetization. In particular, we predict that phase separation can be observed in a $^{23}$Na condensate confined in a highly elongated harmonic trap. Finally, we discuss the phenomena of dynamical instability and spin domain formation.

Antiferromagnetic condensates are generally believed not to display modulational instability and subsequent spin-domain formation. Here we demonstrate that in he presence of a homogeneous magnetic field antiferromagnetic spin-1 Bose-Einstein condensates can undergo spatial modulational instability followed by the subsequent generation of spin domains. Employing numerical simulations for realistic conditions, we show how this novel effect can be observed in sodium condensates confined in an optical trap. Finally, we link this instability and spin-domain formation with stationary modes of the condensate.

We introduce a two dimensional model for the Bose-Einstein condensate with both attractive and repulsive nonlinearities. We assume a combination of a double well potential in one direction, and an optical lattice along the perpendicular coordinate. We look for dual core solitons in this model, focusing on their symmetry-breaking bifurcations. The analysis employs a variational approximation, which is verified by numerical results. The bifurcation which transforms antisymmetric gap solitons into asymmetric ones is of supercritical type in the case of repulsion; in the attraction model, increase of the optical latttice strength leads to a gradual transition from subcritical bifurcation (for symmetric solitons) to a supercritical one.

We consider effects of inter-species attraction on two-component gap solitons (GSs) in the binary BEC with intra-species repulsion, trapped in the one-dimensional optical lattice (OL). Systematic simulations of the coupled Gross-Pitaevskii equations (GPEs) corroborate an assumption that, because the effective mass of GSs is negative, the inter-species attraction may \emphsplit the two-component soliton. Two critical values, $\kappa_{1} $ and $\kappa_{2}$, of the OL strength ($\kappa $) are identified. Two-species GSs with fully overlapping wave functions are stable in strong lattices ($\kappa >\kappa_{1}$). In an intermediate region, $\kappa_{1}>\kappa >\kappa_{2}$, the soliton splits into a double-humped state with separated components. Finally, in weak lattices ($\kappa <\kappa_{2}$%), the splitting generates a pair of freely moving single-species GSs. We present and explain the dependence of $\kappa_{1}$ and $\kappa_{2}$ on thenumber of atoms (total norm), and on the relative strength of the competing inter-species attraction and intra-species repulsion. The splitting of asymmetric solitons, with unequal norms of the two species, is briefly considered too. It is found and explained that the splitting threshold grows with the increase of the asymmetry.

We suggest an efficient method for generating matter-wave gap solitons in a repulsive Bose-Einstein condensate, when the gap soliton is formed from a condensate cloud in a harmonic trap after turning on a one-dimensional optical lattice. We demonstrate numerically that this approach does not require preparing the initial atomic wave packet in a specific state corresponding to the edge of the Brillouin zone of the spectrum, and losses that occur during the soliton generation process can be suppressed by an appropriate adiabatic switching of the optical lattice.

We investigate the stability properties of breather soliton trains in a three-dimensional Bose-Einstein Condensate with Feshbach Resonance Management of the scattering length. This is done so as to generate both attractive and repulsive interaction. The condensate is con ned only by a one dimensional optical lattice and we consider both strong, moderate, and weak con nement. By strong con nement we mean a situation in which a quasi two dimensional soliton is created. Moderate con nement admits a fully three dimensional soliton. Weak con nement allows individual solitons to interact. Stability properties are investigated by several theoretical methods such as a variational analysis, treatment of motion in e ective potential wells, and collapse dynamics. Armed with all the information forthcoming from these methods, we then undertake a numerical calculation. Our theoretical predictions are fully con rmed, perhaps to a higher degree than expected. We compare regions of stability in parameter space obtained from a fully 3D analysis with those from a quasi two-dimensional treatment, when the dynamics in one direction are frozen. We nd that in the 3D case the stability region splits into two parts. However, as we tighten the con nement, one of the islands of stability moves toward higher frequencies and the lower frequency region becomes more and more like that for quasi 2D. We demonstrate these solutions in direct numerical simulations and, importantly, suggest a way of creating robust 3D solitons in experiments in a Bose Einstein Condensate in a one-dimensional lattice.

We investigate the stability properties of breather solitons in a three-dimensional Bose-Einstein Condensate with Feshbach Resonance Management of the scattering length and con ned only by a one dimensional optical lattice. We compare regions of stability in parameter space obtained from a fully 3D analysis with those from a quasi two-dimensional treatment. For moderate con nement we discover a new island of stability in the 3D case, not present in the quasi 2D treatment. Stable solutions from this region have nontrivial dynamics in the lattice direction, hence they describe fully 3D breather solitons. We demonstrate these solutions in direct numerical simulations and outline a possible way of creating robust 3D solitons in experiments in a Bose Einstein Condensate in a one-dimensional lattice. We point other possible applications.