Fr\"ohlich Condensation of Bosons: Graph texture of curl flux network for nonequilibrium properties

TL;DR

This paper develops a quantum theory to analyze nonequilibrium bosonic condensates, revealing a curl flux network topology and identifying a new order parameter for phase transition to Fr"ohlich condensation.

Contribution

It introduces a graph-based flux network approach and a winding number as a novel order parameter for nonequilibrium Fr"ohlich condensation.

Findings

Identifies a curl flux network topology in nonequilibrium bosonic systems.

Proposes winding number as a new phase transition order parameter.

Connects flux network properties with coherence in cavity polaritons.

Abstract

Nonequilibrium condensates of bosons subject to energy pump and dissipation are investigated, manifesting the Fr\"ohlich coherence proposed in 1968. A quantum theory is developed to capture such a nonequilibrium nature, yielding a certain graphic structure arising from the detailed-balance breaking. The results show a network of probability curl fluxes that reveals a graph topology. The winding number associated with the flux network is thus identified as a new order parameter for the phase transition towards the Fr\"ohlich condensation (FC), not attainable by the symmetry breaking. Our work demonstrates a global property of the FCs, in significant conjunction with the coherence of cavity polaritons that may exhibit robust cooperative phases driven far from equilibrium.