We have developed a mean-field model to describe the dynamics of a non-equilibrium BoseEinstein condensate of exciton-polaritons in a semiconductor microcavity. The spectrum of elementary excitations around the stationary state is analytically studied in different geometries. A diffusive behaviour of the Goldstone mode is found in the spatially homogeneous case and new features are predicted for the Josephson effect in a two-well geometry. [4,5]. The system under investigation consists of a semiconductor microcavity containing a few quantum wells with an excitonic transition strongly coupled to the cavity photon mode. In this strong coupling regime, the basic excitations of the system are excitonpolaritons, i.e. linear superpositions of a quantum well exciton and a cavity photon. As compared to other examples of BEC, namely in liquid 4 He and ultracold atomic gases, the main novelty of the present polariton system is its intrinsic non-equilibrium nature due to the finite lifetime of polaritons. The condensate has in fact to be continuously replenished from the relaxation of optically injected high energy excitations (e.g. free carriers or hot polaritons), and its steady state results from a dynamical equilibrium between pumping and losses. This makes the present system a unique candidate for the study of the BEC phase transition in a non-equilibrium context. Recent theoretical work [6] has suggested that the non-equilibrium condition is responsible for dramatic changes in the dispersion of low-lying excitations of incoherently pumped polariton condensates: the sound mode of equilibrium condensates is replaced by a diffusive mode with flat dispersion, as it typically happens in coherently driven pattern forming systems, such as Benard cells in heat convection [7] or optical parametric oscillators [8].The present Letter is devoted to the development of a simple and generic model of a non-equilibrium condensate which does not involve the microscopic physics of the polariton, and can be used to describe the dynamics independently of the details of the specific pumping scheme. Our model is inspired by classical treatments of laser operation [9], and closely resembles the generic model of atom lasers developed in [10]. In this way, we are able not only to confirm the conclusions of Ref.[6] but also to analytically relate the elementary excitation spectrum to experimentally accessible quantities. The same model is then applied to the Josephson effect [11,12,13] in a system of two weakly coupled polaritonic condensates: predictions are given for the frequency and the intrinsic damping rate of Josephson oscillations, and overdamped behavior is anticipated in the case of strong damping.The experimental scheme used to create the polariton condensate is sketched in Fig.1a: under a continuouswave high energy illumination, hot free carriers are generated in the semiconductor material forming the microcavity. Their cooling down by phonon emission leads the formation of a incoherent gas of bound excitons in the quantum wells, ...

One of the most striking quantum effects in an interacting Bose gas at low temperature is superfluidity. First observed in liquid 4 He, this phenomenon has been intensively studied in a variety of systems for its remarkable features such as the persistence of superflows and the proliferation of quantized vortices 1 . The achievement of Bose-Einstein condensation in dilute atomic gases 2 provided the opportunity to observe and study superfluidity in an extremely clean and well-controlled environment. In the solid state, Bose-Einstein condensation of exciton polaritons has been reported recently 3-6 . Polaritons are strongly interacting light-matter quasiparticles that occur naturally in semiconductor microcavities in the strong-coupling regime and constitute an interesting example of composite bosons. Here, we report the observation of spontaneous formation of pinned quantized vortices in the Bose-condensed phase of a polariton fluid. Theoretical insight into the possible origin of such vortices is presented in terms of a generalized Gross-Pitaevskii equation. Whereas the observation of quantized vortices is, in itself, not sufficient for establishing the superfluid nature of the non-equilibrium polariton condensate, it suggests parallels between our system and conventional superfluids.Vortices in superfluids carry quantized phase winding and circulation of the superfluid particles around their core. By definition, vortices are characterized by (1) a rotation of the phase around the vortex by an integer multiple of 2π, commonly known as the topological charge of the vortex and (2) the vanishing of the superfluid population at their core. Owing to their major importance for the understanding of superfluidity, they have been intensively studied theoretically 7 and experimentally 8-10 in disorder-free, stirred three-dimensional Bose-Einstein condensates (BECs) of dilute atomic gases and in quasi-two-dimensional BECs where they spontaneously emerge from thermal fluctuations 11,12 and are strictly related to the Berezinskii-Kosterlitz-Thouless phase transition [13][14][15] . Here, we observed the spontaneous appearance of pinned singly quantized vortices as an intrinsic feature of non-equilibrium polariton BECs in the presence of disorder. The same planar CdTe microcavity sample was used as in our previous studies 3, 16,17 . The polariton condensate was created by means of non-resonant continuous-wave optical excitation, the intensity of which is used to drive the polaritons throughout the phase transition, as demonstrated by the condensate emission energy being located close to the bottom of the polariton dispersion. The condensate steady state is determined by a dynamical balance between the incoming and the outgoing flow of polaritons: in contrast to atomic BECs, the polariton condensate is in an intrinsically non-equilibrium condition. From this point of view, it is therefore closer to a laser, but fundamental differences are still to be noted with respect to a standard photon laser: the bosonic particles under in...

We develop a mean-field theory of the spatial profile and the spectral properties of polariton condensates in nonresonantly pumped semiconductor microcavities in the strong coupling regime. Predictions are obtained for both the continuous-wave and the pulsed excitation regimes and the specific signatures of the non-equilibrium character of the condensation process are pointed out. A striking sensitivity of the condensate shape on the optical pump spot size is demonstrated by analytical and numerical calculations, in good quantitative agreement with recent experimental observations. PACS numbers: 03.75. Kk, 71.36.+c, 42.65.Sf, 05.70.Ln First evidences of Bose-Einstein condensation (BEC) in a solid-state system have been recently reported in a gas of exciton-polaritons in a semiconductor microcavity in the strong coupling regime [1,2,3,4,5,6]. In addition of being a remarkable example of an exciton condensate, this system opens interesting perspectives towards the study of the BEC phenomenon in completely new regimes. Polariton condensates differ in several fundamental aspects from the ideal case generally considered in textbooks: polariton-polariton interactions are significant and the system is far from thermodynamical equilibrium. While the separate effect on condensation of the interactions and of the non-equilibrium condition is already well understood from either ultracold atom [7] or laser [8] theory, not much is yet known about the interplay of the two effects when simultaneously present. In this case, the Bose gas is in fact a quantum degenerate, interacting many-body system whose stationary state does not correspond to a thermal equilibrium state, but rather originates from a dynamical balance of pumping and losses [9,10,11] Some striking consequences of the non-equilibrium condition on the elementary excitations have been recently predicted, the propagating sound mode of equilibrium condensate being e.g. replaced by a diffusive mode [10,11]. Even more remarkably, an unexpectedly rich behavior has been observed in recent experiments in the spatial and spectral shapes of the condensate depending on the size of the pump spot [1,2]. While at equilibrium, BEC generally occurs at zero momentum and the trapping potential is only responsible for the k-space broadening due to finite size [7], the first experimental studies of polariton condensates performed with a relatively small pump laser spot showed BoseEinstein condensation into a ring of momentum states with a non-zero wave vector [1]; mutual coherence of different k-states was however interferometrically demonstrated, which proved that a true condensate was created, and not a fragmented one [12]. Standard condensation around k = 0 was then recovered in later experiments using a much wider excitation spot [2,3]. So far, most of these experimental observations have challenged theoretical understanding [13]: the purpose of the present Letter is to propose a complete and unified theoretical model able to explain them in a simple and physically transparent ...

Coherent manipulation of spin ensembles is a key issue in the development of spintronics. In particular, multivalued spin switching may lead to new schemes of logic gating and memories. This phenomenon has been studied with atom vapours 30 years ago, but is still awaited in the solid state. Here, we demonstrate spin multistability with microcavity polaritons in a trap. Owing to the spinor nature of these light-matter quasiparticles and to the anisotropy of their interactions, we can optically control the spin state of a single confined level by tuning the excitation power, frequency and polarization. First, we realize high-efficiency power-dependent polarization switching. Then, at constant excitation power, we evidence polarization hysteresis and determine the conditions for realizing multivalued spin switching. Finally, we demonstrate an unexpected regime, where our system behaves as a high-contrast spin trigger. These results open new pathways to the development of advanced spintronics devices and to the realization of multivalued logic circuits. Spin manipulation is the object of an intense research activity in a great variety of solid-state systems [1][2][3] . Owing to significant advances in tunability and miniaturization, semiconductor nanostructures have turned into ideal laboratories to address spintronics challenges 4 . In this respect, microcavity polaritons hold great potential 5,6 . Arising from the normal-mode coupling between cavity photons and quantum-well excitons, polaritons behave as bosons and possess unique coherence properties that have led to the demonstration of Bose-Einstein condensation and superfluidity [7][8][9] . A great advantage of polaritons is the one-to-one correspondence between the polariton spin and the polarization of the emitted light. This allowed the observations of the optical spin Hall effect 10 , or of half-quantum vortices 11 , which have shown that polaritons exhibit remarkable spin carrier properties. Finally, recent realizations of optical bistability 12,13 and electrical injection in polariton diodes 14 allow the implementation of low-power polaritronic devices working at room temperature 15,16 . Spin multistability refers to the possibility for a system to present three or more stable spin states for a given excitation condition. It requires precise control of coherence and interactions and is therefore difficult to realize. The only successful studies of multistability with a spinor system were carried out with atomic vapours 30 years ago 17,18 . Its demonstration in the solid state would clearly lead to new schemes of spin-based logic devices 19,20 . Microcavity polaritons were recently predicted to be promising candidates to explore spin multistability 21 . This phenomenon rapidly emerged as an innovative solution for the design of spin memory elements 22 , and for the realization of logic gates based on the selective transport of spin-polarized polaritons 23,24 . Such developments first require an experimental demonstration of spin multistability in a pattern...