10 - Berezinskii-Kosterlitz-Thouless Phase of a Driven-Dissipative Condensate

Summary

Microcavity exciton-polaritons are interacting Bose particles which are confined in a two-dimensional (2D) system suitable for studying coherence properties in an inherently nonequilibrium condition. A primary question of interest here is whether a true long-range order exists among the 2D exciton-polaritons in a driven open system. We give an overview of theoretical and experimental works concerning this question, and we summarize the current understanding of coherence properties in the context of Berezinskii-Kosterlitz-Thouless transition.

Introduction

Strange but striking phenomena, which are accessed by advanced experimental techniques, become a fuel to stimulate both experimental and theoretical research. Experimentalists concoct new tools for sophisticated measurements, and theorists establish models in order to explain the surprising observation, ultimately expanding our knowledge boundary. A classic example of the seed to the knowledge expansion is the feature of abnormally high heat conductivity in liquid helium reported by Kapitza and Allen's group, who used cryogenic liquefaction techniques in 1938 [1, 2]. It is a precursor to a “resistance-less flow” a new phase of matter, coined as superfluidity in the He-II phase. Immediately after this discovery, London conceived a brilliant insight between superfluidity and Bose-Einstein condensation (BEC) of noninteracting ideal Bose gases [3], which has led to establish the concept of coherence as off-diagonal long-range order emerging in the exotic states of matter. Since then, it is one of the core themes in equilibrium Bose systems to elucidate the intimate link of superfluidity and BEC in natural and artificial materials, where dimensionality and interaction play a crucial role in determining the system phase.

Let us consider the noninteracting ideal Bose gases whose particle number N is fixed in a three-dimensional box with a volume V. According to the Bose-Einstein statistics, the average occupation number N_i in the state i with energy is given by with the chemical potential and a temperature parameter (Boltzmann constant k_B and temperature T). For the positive real number of is restricted to be smaller than, and the ground-state particle number N₀ diverges as approaches the lowest energy. Its thermodynamic phase transition refers to BEC, in which the macroscopic occupation in the ground state is represented by the classical field operator, where is the particle density and is the phase.