Letter

Nature 450, 529-532 (22 November 2007) | doi:10.1038/nature06334; Received 23 May 2007; Accepted 20 September 2007

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

C. W. Lai1,2,3, N. Y. Kim1,2, S. Utsunomiya3,4, G. Roumpos1, H. Deng1, M. D. Fraser1, T. Byrnes2,3, P. Recher1,2, N. Kumada4, T. Fujisawa4 & Y. Yamamoto1,3

  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. Lai1,2,3Y. Yamamoto1,3 Correspondence and requests for materials should be addressed to C.W.L. (Email: cwlai@stanford.edu) or Y.Y. (Email: yyamamoto@stanford.edu).

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 pi is predicted to develop among weakly coupled Josephson junctions19. Such a meta-stable pi-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 pi-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 ('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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