The effect of quantum statistics in quantum gases and liquids results in observable collective properties among many-particle systems. One prime example is Bose-Einstein condensation, whose onset in a quantum liquid leads to phenomena such as superfluidity and superconductivity. A Bose-Einstein condensate is generally defined as a macroscopic occupation of a single-particle quantum state, a phenomenon technically referred to as off-diagonal long-range order due to non-vanishing off-diagonal components of the single-particle density matrix. The wavefunction of the condensate is an order parameter whose phase is essential in characterizing the coherence and superfluid phenomena. The long-range spatial coherence leads to the existence of phase-locked multiple condensates in an array of superfluid helium, superconducting Josephson junctions or atomic Bose-Einstein condensates. Under certain circumstances, a quantum phase difference of pi is predicted to develop among weakly coupled Josephson junctions. Such a meta-stable pi-state was discovered in a weak link of superfluid 3He, which is characterized by a 'p-wave' order parameter. The possible existence of such a pi-state in weakly coupled atomic Bose-Einstein condensates has also been proposed, but remains undiscovered. Here we report the observation of spontaneous build-up of in-phase ('zero-state') and antiphase ('pi-state') 'superfluid' states in a solid-state system; an array of exciton-polariton condensates connected by weak periodic potential barriers within a semiconductor microcavity. These in-phase and antiphase states reflect the band structure of the one-dimensional polariton array and the dynamic characteristics of metastable exciton-polariton condensates.

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In a homogeneous two-dimensional system at non-zero temperature there can be no ordering of infinite range 1,2 . However, for a Bose liquid under such conditions, a superfluid phase is predicted 3-5 . Bound vortex-antivortex pairs dominate the thermodynamics and phase coherence properties in this superfluid regime. It is believed that several systems share this behaviour when the parameter describing their ordered state has two degrees of freedom 6 . This theory has been tested for some of them 7-12 , but there has been no direct experimental observation of a quasi-condensate that includes a bound vortex-antivortex pair. Here we present an experimental technique that can identify a single vortex-antivortex pair in a two-dimensional exciton-polariton condensate. The pair is generated through the inhomogeneous spot profile of the pumping laser, and is revealed in the time-integrated phase maps acquired using Michelson interferometry. Numerical modelling based on the open-dissipative Gross-Pitaevskii equation suggests that the pair evolution is distinctly different in this non-equilibrium system compared with atomic condensates 13 .Microcavity exciton polaritons 14 behave as a system of strictly two-dimensional bosons when their density is below the exciton saturation density. As a result of their half-light half-matter nature, their effective mass is extremely small, so that quantum many-body effects are important at relatively high temperatures, even up to room temperature 15 . In particular, dynamic polariton condensation is observed [16][17][18] , and its signatures are similar to Bose-Einstein condensation, namely massive occupation of the ground state and phase coherence up to large distances. However, the short lifetime allows only the formation of a quasi-equilibrium steady state, in which polaritons escaping from the cavity are continuously replenished by the external pump through an intermediate reservoir state.Here, we show that single vortex-antivortex pairs can be observed in an exciton-polariton condensate. We have created a pumping spot that generates a minimum of the condensate density at the centre. A zero in density can be thought of as a superposition of a vortex and antivortex that can be separated by an external perturbation 19 . Thus, the centre of the condensate acts as a source of vortex-antivortex pairs. We have found that for a particular condensate size, there is on average one pair at any time. A Michelson interferometer is used to reconstruct the time-integrated phase map of the system. When the sample disorder potential is stronger than the blueshift induced by polaritonpolariton interactions, pinned pairs appear at certain locations. When the sample disorder potential is weak, on the other hand, they are mobile. They appear along a fixed axis, because of a small asymmetry in the pumping spot, and are created with a random polarization, namely the vortex can appear on the right side of the spot and the antivortex on the left, or vice versa. In the time-integrated measurement, two disti...

We create a large exciton-polariton condensate and employ a Michelson interferometer setup to characterize the short-and longdistance behavior of the first order spatial correlation function. Our experimental results show distinct features of both the twodimensional and nonequilibrium characters of the condensate. We find that the gaussian short-distance decay is followed by a powerlaw decay at longer distances, as expected for a two-dimensional condensate. The exponent of the power law is measured in the range 0.9-1.2, larger than is possible in equilibrium. We compare the experimental results to a theoretical model to understand the features required to observe a power law and to clarify the influence of external noise on spatial coherence in nonequilibrium phase transitions. Our results indicate that Berezinskii-KosterlitzThouless-like phase order survives in open-dissipative systems.quantum well excitons | semiconductor microcavities T he spatial correlation function quantifies the coherence properties of a system (1). In a 3D Bose-condensed gas, long range order is observed, and the correlation function decays toward a plateau at large distances (2, 3). In the homogeneous 2D Bose gas (4), however, no long range order can be established (5). Instead, Berezinskii-Kosterlitz-Thouless (BKT) theory of the equilibrium interacting gas predicts a transition to a low-temperature superfluid phase, which shows a power-law decay of the correlation function (6, 7). Unfortunately, it is frequently hard to directly measure this, and only very recently (8) was indication of the power-law decay of coherence seen in a 2D atomic gas. It has been theoretically predicted (9, 10) that power-law decay of coherence survives in the nonequilibrium problem, and it is this prediction that the current experiment sets out to test.Exciton polaritons are short-lived quasiparticles formed in a semiconductor quantum well strongly coupled to a planar microcavity (11). Each one is a superposition of a quantum well exciton and a microcavity photon, and they behave as 2D bosons below the Mott density. Above a threshold particle density, condensation is observed (12). Due to the nonequilibrium nature of polariton condensation, understanding its coherence properties is quite revealing regarding the different roles of fluctuations in the equilibrium and nonequilibrium problems.Previous measurements on polariton condensates have demonstrated coherence at large distances but were limited by large experimental uncertainties (13) or highly disordered samples (14,15), and the long-distance behavior could not be fully extracted. Recently, the correlation function at large distances was studied in 1D condensates confined in a quantum wire (16) and in a valley of the disorder potential (17). In ref. 16, the data was energy-resolved so that excited states were filtered out, while in ref. 17 a rare area on the sample was chosen in which a single mode condensate is seen. The purpose of both those experiments was to investigate how long the coherence length o...

In this letter, we present a study of the condensation of exciton-polaritons in large etched pillar structures that exhibit shallow edge trapping. The ≈ 100 µm ×100 µm pillars were fabricated using photolithography and a BCl 3 /Cl 2 reactive ion etch. A low energy region emerged along the etched edge, with the minima ≈ 7 µm from the outer edge. The depth of the trap was 0.5 − 1.5 meV relative to the level central region, with the deepest trapping at the corners. We were able to produce a Bose-Einstein condensate in the trap near the edges and corners by pumping non-resonantly in the middle of the pillar. This condensate began as a set of disconnected condensates at various points along the edges, but then became a single mono-energetic condensate as the polariton density was increased. Similar edge traps could be used to produce shallow 1D traps along edges or other more complex traps using various etch geometries and scales. In the past two decades, many experiments have used polaritons resulting from strong coupling between trapped microcavity photons and quantum well (QW) excitons. These bosonic particles have a very light mass (∼ 10 −4 m e) due to being partially photonic, but also strong particle-particle interactions from being partially excitonic. 1 This combination of a light mass and strong interactions leads to the formation of Bose-Einstein con-densates (BECs) at relatively high temperatures (∼ 10 K). 2-4 Polaritons provide a promising system for studying bosonic particles at even higher temperatures, and polariton lasing has been observed at room temperature in both GaN 5 and organic 6 systems. Many methods of confinement have been used to study polariton dynamics in a variety of geometries. Applying stress to a thin (≈ 100 µm) GaAs sample can be used to shift the exciton energy, resulting in a harmonic trap. 4,7 Pumping such a stress trap non-resonantly in the center forms a repulsive barrier and can be used to form a ring geometry. 8 Complex pumping geometries can also be used to confine polaritons, including the use of two or more pump spots in various arrangements or using a ring-shaped pump spot. 9-13 More permanent methods of confinement include producing a spacer in certain regions of the cavity during the growth process, 14-16 using sub-wavelength gratings as the top mirror, 17,18 depositing metal strips onto the top mirror, 19 and etching the sample after growth to form 1D wires, 2D pillars, and 2D arrays of coupled pillars. 20-25 While optically induced trapping potentials have the advantage of being easily reconfigured, etched trapping allows the confinement to be somewhat independent of the pump laser. Post-growth etching also produces much higher potential barriers at the etched edges than the a) Electronic mail: dmm154@pitt.edu b) J. K. Wuenschell is now at Physical Sciences Laboratory, The Aerospace Corporation, El Segundo, CA 90245, USA deposition of metal strips, and it is compatible with our existing sample materials and growth methods, unlike sub-wavelength gratings or modulating t...