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Coherent zero-state and π-state in an exciton–polariton condensate array


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 matrix1,2,3. The wavefunction of the condensate is an order parameter whose phase is essential in characterizing the coherence and superfluid phenomena4,5,6,7,8,9,10,11. The long-range spatial coherence leads to the existence of phase-locked multiple condensates in an array of superfluid helium12, superconducting Josephson junctions13,14,15 or atomic Bose–Einstein condensates15,16,17,18. Under certain circumstances, a quantum phase difference of π is predicted to develop among weakly coupled Josephson junctions19. Such a meta-stable π-state was discovered in a weak link of superfluid 3He, which is characterized by a ‘p-wave’ order parameter20. The possible existence of such a π-state in weakly coupled atomic Bose–Einstein condensates has also been proposed21, but remains undiscovered. Here we report the observation of spontaneous build-up of in-phase (‘zero-state’) and antiphase (‘π-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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Figure 1: Formation of an exciton–polariton array.
Figure 2: Imaging and spectroscopy of exciton–polariton distribution in coordinate and momentum space.
Figure 3: Band structure and ‘superfluid’ states in an exciton–polariton condensate array.
Figure 4: Evolution of the ‘zero-state’ and ‘π-state’.


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This work was supported by the JST/SORST programme and by Special Coordination Funds for Promoting Science and Technology in Japan. The high-quality microcavity cavity sample is courtesy of G. S. Solomon, R. Hey, K. Ploog and A. Forchel. We thank T. Maruyama for support and S. Sasaki for the device fabrication. C.W.L. thanks Y. R. Shen for comments and discussions.

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Correspondence to C. W. Lai or Y. Yamamoto.

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The file contains Supplementary Notes with Supplementary Figures S1-S6 and additional references. The document describes 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. (PDF 3416 kb)

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Lai, C., Kim, N., Utsunomiya, S. et al. Coherent zero-state and π-state in an exciton–polariton condensate array. Nature 450, 529–532 (2007).

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