1. E. L. Ginzton Laboratory, Stanford University, Stanford, California 94305, USA
2. Institute of Industrial Science, University of Tokyo, Meguro-ku, Tokyo 153-8505, Japan
3. National Institute of Informatics, Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
4. NTT basic research laboratories, NTT Corporation, Atugi, Kanagawa 243-0198, Japan

Correspondence to: C. W. Lai^1,2,3Y. Yamamoto^1,3 Correspondence and requests for materials should be addressed to C.W.L. (Email: cwlai@stanford.edu) or Y.Y. (Email: yyamamoto@stanford.edu).

Abstract

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^{1, 2, 3}. The wavefunction of the condensate is an order parameter whose phase is essential in characterizing the coherence and superfluid phenomena^{4, 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 junctions^{13, 14, 15} or atomic Bose–Einstein condensates^{15, 16, 17, 18}. Under certain circumstances, a quantum phase difference of is predicted to develop among weakly coupled Josephson junctions¹⁹. Such a meta-stable -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 -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 ('-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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