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Summary We describe the signatures of exciton-polariton condensation without a periodically modulated potential, focusing on the spatial coherence properties and condensation in momentum space. The characteristics of the exciton-polariton condensate form the basis of the study of the exciton-polariton condensate array. We also discuss the diffraction patterns of ‘zero-state’ and ‘π-state’ considering the finite transmittance through the metallic strips and spatial coherence. A rate-equation model is used to describe the dynamics and evolution of the ‘zero-state’ and ‘π-state’ as a function of the pumping rate. Characterization of exciton-polariton ‘condensates’ In two-dimensional (2D) systems with continuous symmetry, the phase fluctuations of the order parameters prevent formation of condensates with off-diagonal long-range order1-3 at finite temperatures. However, a finite-size bosonic 2D fluid can still undergo Bose-Einstein condensation (BEC)30. In addition, quasi-condensates with a topological long-range order31 or long range coherence32 as well as superfluidity enabled by interaction33 (eg. Berezinskii-Kosterlitz-Thouless transition31,34,35) can exist in 2D systems. Above the critical density or below the critical temperature of BEC or BKT transition, the phase can be correlated from region to region as long as the order exists locally. Here we describe experimental observations of the condensation in momentum space and long range spatial coherence of LP condensates in the absence of a periodic potential modulation. Under an elliptical excitation spot with an axial-to-radial aspect ratio of 2:1 (~60 μm × 30 μm), the exciton-polariton momentum distribution is isotropic as shown in Fig. S1. With increasing pumping rates, strong nonlinear LP emission develops from a small area, and the momentum distribution evolves into an anisotropic distribution with a reversed axial-to-radial aspect ratio. Furthermore, the product of the spreads of the LP distribution in momentum and coordinate spaces is only ~2 to 4 times the Heisenberg uncertainty limit36. The anisotropic and narrow momentum distribution above a threshold pumping rate is supportive of a long-range spatial coherence across the condensate. SUPPLEMENTARY INFORMATIONdoi: 10.1038/nature06334www.nature.com/nature 1The effects of condensation in the momentum space are further revealed by the energy versus in-plane momentum LP distribution shown in Fig. S1c. Above the condensation threshold, macro-occupation of a low energy state and sharp reduction of the LP energy linewidth at ||0k = are observed (see also ref. 24,25). The spatial coherence characteristic of the exciton-polariton condensate is quantitatively determined by Young’s double-slit interference experiment36 with double-slits inserted at the conjugate image plane (see the main text, Fig. 2a). The slits are positioned at /2Yd=± with respect to the axis of symmetry (optical axis) and the center of the condensate to ensure equal intensity between two slits, where d is the separation of the two probing slits. The plane of observation is the focal plane of the spectrometer whose entrance slit coincides with the Fourier transform plane (momentum space) of the double slits. Under such an arrangement, the visibility V of the fringes is equal to the degree of coherence37,38 as shown in Fig. S2. The determined spatial coherence is expected to reflect the correlation function, which is proportional to the off-diagonal element of the single-particle density matrix ( ) ( )( )†ˆˆ/2 /2dddρψ ψ=−. Below the pumping threshold the correlation function can be approximated by ( )22exp /Trπ λ−11, where r is the distance between two probed locations and the spatial coherence length is characterized by the thermal de Broglie wavelength */2TBhmkTλπ≡ . At a temperature 10T = K, 2.5Tλ≈ μm for a typical polariton effective *410emm−= . For a homogeneous Bose gas, the correlation function can be expressed as ( ) ( )( )( )2†ˆˆ0exp/Dcrr rηψ ψξ−−+∝−, where D is the dimensionality, η a critical exponent, cξ the correlation length39. The visibility decreases approximately exponentially with increasing slit separations at large separations, where the power low decay is negligible for 2D = . The measured correlation length increase from ~2 μm up to ~18 μm (limited by the pumping spot size) at just above the condensation threshold. At just above the threshold, the visibility decreases from more than 90% to an invariant ~30% until the slits reach the edges of the condensate (Fig. S2). At a high pump rate, the correlation length decreases to ~5 μm, likely due to dephasing of the condensate induced by polariton-polariton interaction. The role of polariton-polariton interaction and types of phase transitions (BEC versus BKT transition) in 2D systems are beyond the scope of this letter. The measured coherence doi: 10.1038/nature06334 SUPPLEMENTARY I N F O R M ATIONwww.nature.com/nature 2length far exceeding the thermal de Broglie wavelength confirms that spatial coherence can develop across the whole condensate. Figure S1| Near-field and far-field imaging and spectroscopy of an exciton-polariton condensate in the absence of metallic strips. The pumping rate is Pth/3, 1.5 Pth, and 6 Pth (Pth ≈ 30mW) from left to right panels. 8T ≈ K and the LP emission wavelength 780λ≈ nm. The condensate is studied at a location with a cavity detuning 6CXEE−≈ meV, near the region with metallic strips where data shown in Fig. 3c and Fig. 4 are measured. a, Pseudo-color (log scale) near-field images of the LP emissions. b, Far-field images showing the LP distribution in momentum space. An anisotropic distribution with a reversed axial-to-radial aspect ratio in coordinate space develops above threshold. c, Energy versus in-plane momentum LP distribution, where a slice of the far-field emission with in-plane momentum within ( )2/sin00.4Xkπλ≈±D is dispersed by the spectrometer and imaged by a CCD. Above the threshold, the LPs exhibit narrow distribution in energy and momentum. doi: 10.1038/nature06334 SUPPLEMENTARY I N F O R M ATIONwww.nature.com/nature 3Figure S2| Spatial coherence measured by Young’s double-slit interference measurement. Young’s interference experiment is conducted in a symmetrical arrangement with slits positioned at an equal distance from the optical axis as shown in Fig. 2a. Two slits with an


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