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

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 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.