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The response of superfluids to the external rotation, evidenced by emergence of quantised vortices, distinguishes them from conventional fluids. In this work, we demonstrate that the number of vortices in a stirred polariton condensate depends on the characteristic size of the employed rotating potential induced by the nonresonant laser excitation. For smaller sizes, a single vortex with a topological charge of +-1 corresponding to the stirring direction is formed. However, for larger optical traps, clusters of two or three co-rotating vortices appear in the narrow range of GHz stirring speed.

Topological bound states in the continuum are confined wave-mechanical objects that offer advantageous ways to enhance light-matter interactions in compact photonic devices. In particular, their large quality factor in the strong-coupling regime has recently enabled the demonstration of Bose-Einstein condensation of bound-state-in-the-continuum polaritons. Here, we show that condensation into a negative-mass bound state in the continuum exhibits interaction-induced state confinement, opening opportunities for optically reprogrammable molecular arrays of quantum fluids of light. We exploit this optical trapping mechanism to demonstrate that such molecular complexes show hybridization with macroscopic modes with unusual topological charge multiplicity. Additionally, we demonstrate the scalability of our technique by constructing extended mono- and diatomic chains of bound-state-in-the-continuum polariton fluids that display non-Hermitian band formation and the opening of a minigap. Our findings offer insights into large-scale, reprogrammable, driven, dissipative many-body systems in the strong-coupling regime.

We report on experimental observation of next-nearest-neighbour coupling between ballistically expanding spinor exciton-polariton condensates in a planar semiconductor microcavity. All-optical control over the coupling strength between neighbouring condensates is demonstrated through distance-periodic pseudospin screening of their ballistic particle outflow due to the inherent splitting of the planar cavity transverse-electric (TE) and transverse-magnetic (TM) modes. By screening the nearest-neighbour coupling we overcome the conventional spatial coupling hierarchy between condensates. This offers a promising route towards creating unconventional non-planar many-body Hamiltonians using networks of ballistically expanding spinor exciton-polariton condensates.

The appearance of quantised vortices in the classical ``rotating bucket'' experiments of liquid helium and ultracold dilute gases provides the means for fundamental and comparative studies of different superfluids. Here, we realize the ``rotating bucket'' experiment for optically trapped quantum fluid of light based on exciton-polariton Bose-Einstein condensate in semiconductor microcavity. We utilise the beating note of two frequency-stabilized single-mode lasers to generate an asymmetric time-periodic rotating, non-resonant excitation profile that both injects and stirs the condensate through its interaction with a background exciton reservoir. The pump-induced external rotation of the condensate results in the appearance of a co-rotating quantised vortex. We investigate the rotation-frequency dependence and reveal the range of stirring frequencies (from 1 to 4 GHz) which favors quantised vortex formation. We describe the phenomenology using the generalised Gross-Pitaevskii equation. Our results enable the study of polariton superfluidity on a par with other superfluids, as well as deterministic, all-optical control over structured nonlinear light.

We propose a method to enhance the spatial coupling between ballistic exciton-polariton condensates in a semiconductor microcavity based on available spatial light modulator technologies. Our method, verified by numerically solving a generalized Gross-Pitaevskii model, exploits the strong nonequilibrium nature of exciton-polariton condensation driven by localized nonresonant optical excitation. Tailoring the excitation beam profile from a Gaussian into a polygonal shape results in refracted and focused radial streams of outflowing polaritons from the excited condensate which can be directed towards nearest neighbors. Our method can be used to lower the threshold power needed to achieve polariton condensation and increase spatial coherence in extended systems, paving the way towards creating extremely large-scale quantum fluids of light.

We study the modification of the spatial coupling parameter between interacting ballistic exciton-polariton condensates in the presence of photonic spin orbit coupling appearing from TE-TM splitting in planar semiconductor microcavities. We propose a strategy to make the coupling strength between next-nearest-neighbours stronger than between nearest-neighbour, which inverts the conventional idea of the spatial coupling hierarchy between sites. Our strategy relies on the dominantly populated high-momentum components in the ballistic condensates which, in the presence of TE-TM splitting, lead to rapid radial precession of the polariton pseudospin. As a consequence, condensate pairs experience distance-periodic screening of their interaction strength, severely modifying their synchronization and condensation threshold solutions.

We demonstrate spontaneous formation of a nonlinear vortex cluster state in a microcavity exciton-polariton condensate with time-periodic sign flipping of its topological charges at the GHz scale. When optically pumped with a ring-shaped nonresonant laser, the trapped condensate experiences intricate high-order mode competition and fractures into two distinct trap levels. The resulting mode interference leads to robust condensate density beatings with periodic appearance of orderly arranged phase singularities. Our work opens new perspectives on creating structured free-evolving light, and singular optics in the strong light-matter coupling regime.

We investigate an all-optical microscale planar lensing technique based on coherent fluids of semiconductor cavity exciton-polariton condensates. Our theoretical analysis underpins the potential in using state-of-the-art spatial light modulation of nonresonant excitation beams to guide and focus polariton condensates away from their pumping region. The nonresonant excitation profile generates an excitonic reservoir that blueshifts the polariton mode and provides gain, which can be spatially tailored into lens shapes at the microscale to refract condensate waves. We propose several different avenues in controlling the condensate fluid, and demonstrate formation of highly enhanced and localised condensates away from the pumped reservoirs. This opens new perspectives in guiding quantum fluids of light and generating polariton condensates that are shielded from detrimental reservoir dephasing effects.

Polaritonic lattices offer a unique testbed for studying nonlinear driven-dissipative physics. They show qualitative changes of a steady state as a function of system parameters, which resemble non-equilibrium phase transitions. Unlike their equilibrium counterparts, these transitions cannot be characterised by conventional statistical physics methods. Here, we study a lattice of square-arranged polariton condensates with nearest-neighbour coupling, and simulate the polarisation (pseudo-spin) dynamics of the polariton lattice, observing regions with distinct steady-state polarisation patterns. We classify these patterns using machine learning methods and determine the boundaries separating different regions. First, we use unsupervised data mining techniques to sketch the boundaries of phase transitions. We then apply learning by confusion, a neural network-based method for learning labels in the dataset, and extract the polaritonic phase diagram. Our work takes a step towards AI-enabled studies of polaritonic systems.

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 implement full polarization tomography on the photon correlations in a spinor exciton-polariton condensate. Our measurements reveal condensate pseudospin mean-field dynamics spanning from stochastic switching between linear polarization components, limit cycles, and stable fixed points, and their intrinsic relation to the condensate photon statistics. We optically harness the cavity birefringence, polariton interactions, and the optical orientation of the photoexcited exciton background to engineer photon statistics with precise control. Our results demonstrate a smooth transition from a highly coherent to a super-thermal state of the condensate polarization components.

We demonstrate spin polarized jets in extended systems of ballistic exciton-polariton condensates in semiconductor microcavities using optical non-resonant excitation geometries. The structure of the spin jets is determined by the digitally reprogrammable, spatially non-uniform, degree of circular polarization of the excitation laser. The presence of the laser excitation, strong particle interactions, and spin-relaxation leads to a tunable spin-dependent potential landscape for polaritons, with the appearance of intricate polarization patterns due to coherent matter-wave interference. Our work realizes polarization-structured coherent light sources in the absence of gauge fields.

A condensed matter platform for analogue simulation of complex two-dimensional molecular bonding configurations, based on optically trapped exciton-polariton condensates is proposed. The stable occupation of polariton condensates in the excited states of their optically configurable potential traps permits emulation of excited atomic orbitals. A classical mean field model describing the dissipative coupling mechanism between p-orbital condensates is derived, identifying lowest threshold condensation solutions as a function of trap parameters corresponding to bound and antibound $\pi$ and $\sigma$ bonding configurations, similar to those in quantum chemistry.

We investigate a photonic device consisting of two coupled optical cavities possessing Rashba-Dresselhaus spin-orbit coupling, TE-TM splitting, and linear polarisation splitting that opens a tuneable energy gap at the diabolic points of the photon dispersion; giving rise to an actively addressable local Berry curvature. The proposed architecture stems from recent advancements in the design of artificial photonic gauge fields in liquid crystal cavities [K. Rechcińska et al., Science 366, 727 (2019)]. Our study opens new perspectives for topological photonics, room-temperature spinoptronics, and studies on the quantum geometrical structure of photonic bands in extreme settings.

We demonstrate a regime in which matter-wave condensates of exciton-polaritons trapped in an elliptically shaped two-dimensional potential appear as a coherent mixture of ground and first-excited state of the quantum harmonic oscillator. This system resembles an optically controllable two-level system and produces near terahertz harmonic oscillations of the condensate's center of mass along the major axis of the elliptical trapping potential. The population ratio between the two trap levels is tunable through the excitation laser power and is shown to follow Lotka-Volterra dynamics. We demonstrate coherence formation between two spatially displaced trapped condensate oscillators - the polaritonic analogue of Huygen's clock synchronization for coupled condensate oscillators.

Artificial lattices of coherently coupled macroscopic states are at the heart of applications ranging from solving hard combinatorial optimisation problems to simulating complex many-body physical systems. The size and complexity of the problems scales with the extent of coherence across the lattice. Although the fundamental limit of spatial coherence depends on the nature of the couplings and lattice parameters, it is usually engineering constrains that define the size of the system. Here, we engineer polariton condensate lattices with active control on the spatial arrangement and condensate density that result in near-diffraction limited emission, and spatial coherence that exceeds by nearly two orders of magnitude the size of each individual condensate. We utilise these advancements to unravel the dependence of spatial correlations between polariton condensates on the lattice geometry.

One of the recently established paradigms in condensed matter physics is examining a system's behaviour in artificially constructed potentials, giving insight into physical phenomena of quantum fluids in hard-to-reach settings. A prominent example is the matter-wave scatterer lattice, also known as the barrier lattice or repulsive Dirac comb. There, high energy matter waves undergo transmission and reflection through narrow width barriers leading to stringent phase matching conditions with subsequent lattice band formation. It is one of the most well taught system in quantum mechanics but its realisation for macroscopic matter-wave fluids has remained elusive, in contrast to evanescently coupled lattice sites or waveguides. Here, we implement and study a system of exciton-polariton condensates in a non-Hermitian Lieb lattice of scatterer potentials by optically injecting incoherent exciton clouds which both emit, and interact with traveling polariton waves. By fine tuning the lattice parameters, we reveal a nonequilibrium phase transition between two distinct regimes of polariton condensation: a scatterer lattice of gain guided polaritons condensing on the lattice potential maxima, and trapped polaritons condensing in the lattice potential minima. The transition is characterised by multimodal condensation due to gain competition between the two regimes. Energy tomography on the polariton emission enables us to measure the intricate band structure of the optically induced lattices. Our results pave the way towards unexplored physics of non-Hermitian fluids in non-stationary mixtures of confined and freely expanding waves.

First order coherence measurements of a polariton condensate, reveal a regime where the condensate pseudo-spin precesses persistently within the driving optical pulse. Within a single 20 $\mu$s optical pulse the condensate pseudo-spin performs over $10^5$ precessions with striking frequency stability. The condensate maintains its phase coherence even after a complete precession of the spin vector, making the observed state by a definition a spin coherent state. The emergence of the precession is traced to the polariton interactions that give rise to a self-induced out-of-plane magnetic field that in turn drives the spin dynamics. We find that the Larmor oscillation frequency scales with the condensate density, enabling external tuning of this effect by optical means. The stability of the system allows for the realization of integrated optical magnetometry devices with the use of materials with enhanced exciton $g$-factor and can facilitate spin squeezing effects and active coherent control on the Bloch sphere in polariton condensates.

We investigate the optical orientation, polarization pinning, and depolarization of optically confined semiconductor exciton-polariton condensates. We perform a complete mapping of the condensate polarization as a function of incident nonresonant excitation polarization and power. We utilize a ring-shaped excitation pattern to generate an exciton-induced potential that spatially confines polariton condensates into a single mode. We observe that formation of circular polarization in the condensate persists even for a weakly cocircularly polarized pump. By varying the excitation ring diameter we realize a transition from the condensate polarization being pinned along the coordinate-dependent cavity-strain axes, to a regime of zero degree of condensate polarization. Analysis through the driven-dissipative stochastic Gross-Pitaevskii equation reveals that this depolarization stems from a competition between sample induced in-plane polarization splitting and the condensate-reservoir overlap. An increase in the role of the latter results in weakening of the condensate fixed-point phase space attractors, and enhanced random phase space walk and appearance of limit cycle trajectories, reducing the degree of time-integrated polarization.

Synthetic crystal lattices provide ideal environments for simulating and exploring the band structure of solid-state materials in clean and controlled experimental settings. Physical realisations have, so far, dominantly focused on implementing irreversible patterning of the system, or interference techniques such as optical lattices of cold atoms. Here, we realise reprogrammable synthetic band-structure engineering in an all optical exciton-polariton lattice. We demonstrate polariton condensation into excited states of linear one-dimensional lattices, periodic rings, dimerised non-trivial topological phases, and defect modes utilising malleable optically imprinted non-Hermitian potential landscapes. The stable excited nature of the condensate lattice with strong interactions between sites results in an actively tuneable non-Hermitian analogue of the Su-Schrieffer-Heeger system.

We demonstrate tunable dissipative interactions between optically trapped exciton-polariton condensates. We apply annular shaped nonresonant optical beams to both generate and confine each condensate to their respective traps, pinning their natural frequencies. Coupling between condensates is realized through the finite escape rate of coherent polaritons from the traps leading to robust phase locking with neighboring condensates. The coupling is controlled by adjusting the polariton propagation distance between neighbors. This permits us to map out regimes of both strong and weak dissipative coupling, with the former characterized by clear in-phase and anti-phase synchronization of the condensates. With robust single-energy occupation governed by dissipative coupling of optically-trapped polariton condensates, we present a system which offers a potential optical platform for the optimization of randomly connected $XY$ Hamiltonians.

We demonstrate deterministic control of the nearest and next-nearest neighbor coupling in the unit cell of a square lattice of microcavity exciton-polariton condensates. We tune the coupling in a continuous and reversible manner by optically imprinting potential barriers of variable height, in the form of spatially localized incoherent exciton reservoirs that modify the particle flow between condensates. By controlling the couplings in a $2\times2$ polariton cluster, we realize ferromagnetic, anti-ferromagnetic and paired ferromagnetic phases. Our approach paves the way towards simulating complex condensed matter phases through precise control of the individual couplings in networks of optical nonlinear oscillators.

We theoretically explore nonresonantly pumped polaritonic graphene, a system consisting of a honeycomb lattice of micropillars in the regime of strong light-matter coupling. We demonstrate that, depending on the parameters of the structure, such as intensity of the pump and coupling strength between the pillars, the system shows rich variety of macroscopic ordering, including analogs of ferromagnetic, antiferromagnetic, and resonant valence bond phases. Transitions between these phases are associated with dramatic reshaping of the spectrum of the system connected with spontaneous appearance of topological order.

We report stable orthogonally polarised domains in high-density polariton solitons propagating in a semiconductor microcavity wire. This effect arises from spin dependent polariton-polariton interactions and pump-induced imbalance of polariton spin populations. The interactions result in an effective magnetic field acting on polariton spin across the soliton profile, leading to the formation of polarisation domains. Our experimental findings are in excellent agreement with theoretical modelling taking into account these effects.

We analyse nonequilibrium phase transitions in microcavity polariton condensates trapped in optically induced annular potentials. We develop an analytic model for annular optical traps, which gives an intuitive interpretation for recent experimental observations on the polariton spatial mode switching with variation of the trap size. In the vicinity of polariton lasing threshold we then develop a nonlinear mean-field model accounting for interactions and gain saturation, and identify several bifurcation scenarios leading to formation of high angular momentum quantum vortices. For experimentally relevant parameters we predict the emergence of spatially and temporally ordered polariton condensates (time crystals), which can be witnessed by frequency combs in the polariton lasing spectrum or by direct time-resolved optical emission measurements. In contrast to previous realizations, our polaritonic time crystal is spontaneously formed from an incoherent excitonic bath and does not inherit its frequency from any periodic driving field.

A lattice of locally bistable driven-dissipative cavity polaritons is found theoretically to effectively simulate the Ising model, also enabling an effective transverse field. We benchmark the system performance for spin glass problems, and study the scaling of the ground state energy deviation and success probability as a function of system size. As particular examples we consider NP-hard problems embedded in the Ising model, namely graph partitioning and the knapsack problem. We find that locally bistable polariton networks act as classical simulators for solving optimization problems, which can potentially present an improvement within the exponential complexity class.

We analyze theoretically a network of all-to-all coupled polariton modes, realized by a trapped polariton condensate excited by a comb of different frequencies. In the low-density regime the system dynamically finds a state with maximal gain defined by the average intensities (weights) of the excitation beams, analogous to active mode locking in lasers, and thus solves a maximum eigenvalue problem set by the matrix of weights. The method opens the possibility to tailor a superposition of populated bosonic modes in the trapped condensate by appropriate choice of drive.

We explore nonlinear transitions of polariton wavepackets, first, to a soliton and then to a standing wave polariton condensate in a multi-mode microwire system. At low polariton density we observe ballistic propagation of the multi-mode polariton wavepackets arising from the interference between different transverse modes. With increasing polariton density, the wavepackets transform into single mode bright solitons due to effects of both inter-modal and intra-modal polariton-polariton scattering. Further increase of the excitation density increases thermalisation speed leading to relaxation of the polariton density distribution in momentum space with the resultant formation of a non-equilibrium condensate manifested by a standing wave pattern across the whole sample.

We present a theoretical scheme for multistability in planar microcavity exciton-polariton condensates under nonresonant driving. Using an excitation profile resulting in a spatially patterned condensate, we observe organized phase locking which can abruptly reorganize as a result of pump induced instability made possible by nonlinear interactions. For $\pi/2$ symmetric systems this reorganization can be regarded as a parity transition and is found to be a fingerprint of multistable regimes existing over a finite range of excitation strengths. The natural degeneracy of the planar equations of motion gives rise to parity bifurcation points where the condensate, as a function of excitation intensity, bifurcates into one of two anisotropic degenerate solutions. Deterministic transitions between multistable states are made possible using controlled nonresonant pulses, perturbing the solution from one attractor to another.

Vorticity is a key ingredient to a broad variety of fluid phenomena, and its quantised version is considered to be the hallmark of superfluidity. Circulating flows that correspond to vortices of a large topological charge, termed giant vortices, are notoriously difficult to realise and even when externally imprinted, they are unstable, breaking into many vortices of a single charge. In spite of many theoretical proposals on the formation and stabilisation of giant vortices in ultra-cold atomic Bose-Einstein condensates and other superfluid systems, their experimental realisation remains elusive. Polariton condensates stand out from other superfluid systems due to their particularly strong interparticle interactions combined with their non-equilibrium nature, and as such provide an alternative testbed for the study of vortices. Here, we non-resonantly excite an odd number of polariton condensates at the vertices of a regular polygon and we observe the formation of a stable discrete vortex state with a large topological charge as a consequence of antibonding frustration between nearest neighbouring 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 4-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 8 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 non-equilibrium spin lattice.

We present a scheme of interaction-induced topological bandstructures based on the spin anisotropy of exciton-polaritons in semiconductor microcavities. We predict theoretically that this scheme allows the engineering of topological gaps, without requiring a magnetic field or strong spin-orbit interaction (transverse electric-transverse magnetic splitting). Under non-resonant pumping, we find that an initially topologically trivial system undergoes a topological transition upon the spontaneous breaking of phase symmetry associated with polariton condensation. Under resonant coherent pumping, we find that it is also possible to engineer a topological dispersion that is linear in wavevector -- a property associated with polariton superfluidity.

We study analytically and numerically the condensation of a driven-dissipative exciton-polariton system using symmetric nonresonant pumping geometries. We show that the lowest condensation threshold solution carries a definite parity as a consequence of the symmetric excitation profile. At higher pump intensities competition between the two parities can result in critical quenching of one and saturation of the other. Using long pump channels, we show that the competition of the condensate parities gives rise to a different type of topologically stable defect propagating indefinitely along the condensate. The defects display repulsive interactions and are characterized by a sustained wavepacket carrying a pair of opposite parity domain walls in the condensate channel.

An infinite chain of driven-dissipative condensate spins with uniform nearest-neighbor coherent coupling is solved analytically and investigated numerically. Above a critical occupation threshold the condensates undergo spontaneous spin bifurcation (becoming magnetized) forming a binary chain of spin-up or spin-down states. Minimization of the bifurcation threshold determines the magnetic order as a function of the coupling strength. This allows control of multiple magnetic orders via adiabatic (slow ramping of) pumping. In addition to ferromagnetic and anti-ferromagnetic ordered states we show the formation of a paired-spin ordered state $\left|\dots \uparrow \uparrow \downarrow \downarrow \dots \right. \rangle$ as a consequence of the phase degree of freedom between condensates.

We report on the observation of spin whirls in a radially expanding polariton condensate formed under non-resonant optical excitation. Real space imaging of polarization- and time-resolved photoluminescence reveal a spiralling polarization pattern in the plane of the microcavity. Simulations of the spatiotemporal dynamics of a spinor condensate reveal the crucial role of polariton interactions with a spinor exciton reservoir. Harnessing spin dependent interactions between the exciton reservoir and polariton condensates allows for the manipulation of spin currents and the realization of dynamic collective spin effects in solid state systems.

We study the polarization dynamics of a spatially expanding polariton condensate under nonresonant linearly polarized optical excitation. The spatially and temporally resolved polariton emission reveals the formation of non-trivial spin textures in the form of a quadruplet polarization pattern both in the linear and circular Stokes parameters, and an octuplet in the diagonal Stokes parameter. The continuous rotation of the polariton pseudospin vector through the condensate due to TE-TM splitting exhibits an ordered pattern of half-skyrmions associated with a half-integer topological number. A theoretical model based on a driven-dissipative Gross-Pitaevskii equation coupled with an exciton reservoir describes the dynamics of the nontrivial spin textures through the optical spin-Hall effect.

We show theoretically that an open-dissipative polariton condensate confined within a trapping potential and driven by an incoherent pumping scheme gives rise to bistability between odd and even modes of the potential. Switching from one state to the other can be controlled via incoherent pulsing which becomes an important step towards construction of low-powered opto-electronic devices. The origin of the effect comes from modulational instability between odd and even states of the trapping potential governed by the nonlinear polariton-polariton interactions.

We show theoretically that the strong coupling of circularly polarized photons to an exciton in ring-like semiconductor nanostructures results in physical nonequivalence of clockwise and counterclockwise exciton rotations in the ring. As a consequence, the stationary energy splitting of exciton states corresponding to these mutually opposite rotations appears. This excitonic Aharonov-Bohm effect depends on the intensity and frequency of the circularly polarized field and can be detected in state-of-the-art optical experiments.

We show theoretically that strong electron coupling to circularly polarized photons in non-singly-connected nanostructures results in the appearance of an artificial gauge field that changes the electron phase. The effect arises from the breaking of time-reversal symmetry and is analogous to the well-known Aharonov-Bohm phase effect. It can manifest itself in the oscillations of conductance as a function of the intensity and frequency of the illumination. The theory of the effect is elaborated for mesoscopic rings in both ballistic and diffusive regimes.

We show that in a non-equilibrium system of an exciton-polariton condensate, where polaritons are generated from incoherent pumping, a ring-shaped pump allows for stationary vortex memory elements of topological charge $m = 1$ or $m = -1$. Using simple potential guides we can choose whether to copy the same charge or invert it onto another spatially separate ring pump. Such manipulation of binary information opens the possibility of a new type processing using vortices as topologically protected memory components.

We study theoretically the ground states of topological defects in a spinor four-component condensate of cold indirect excitons. We analyze possible ground state solutions for different configurations of vortices and half-vortices. We show that if only Rashba or Dreselhaus spin-orbit interaction (SOI) for electrons is present the stable states of topological defects can represent a cylindrically symmetric half-vortex or half vortex-antivortex pairs, or a non-trivial pattern with warped vortices. In the presence of both of Rashba and Dresselhaus SOI the ground state of a condensate represents a stripe phase and vortex type solutions become unstable.