Fröhlich Condensation of Bosons: Graph texture of curl flux network for nonequilibrium properties

Abstract

Nonequilibrium condensates of bosons subject to energy pump and dissipation are investigated, manifesting the Fröhlich 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 Fröhlich 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.

Figures (3)

• Figure 1: (a) Schematic illustration of phonon dispersion and relaxation towards the lowest-energy mode. (b) Level structure for bosons subject to external energy pump and dissipation. (c) 2D hexagonal-grid graph for nonequilibrium bosons mapped from Eq.(\ref{['EOMP']}), with nonvanishing net currents. (d) 1D tree graph for the BEC phase where the net currents vanish. (e) Steady-state number distribution $P_{n,N}$ when pump is above the threshold; (f) $P_{n,N}$ when pump is below the threshold. (g,up) Condensation fraction $f$ against pump rate $[R = 3S]$; (g,down) Pearson correlation between $n$ and $N$ against pump rate $[R = 3S]$.
• Figure 2: Curl flux (white arrows) ${\rm J}_{\rm c}$ for (a) FC phase in the above-threshold regime, (b) thermal phase in the below-threshold regime, where the flux is calculated from Eq.(\ref{['Jc']}). Obviously, no loops are presented in thermal phase. (c) Current network (white arrows) obtained from Eq.(\ref{['Jnet']}), showing the curl nature consistent with (a). (d) Modulus $|{\rm J}_{\rm c}|$ revealing a summit-crater landscape with a ring ridge; green loop on the ridge locates the maximum curl flux. Black square spot denotes the singularity $z_{\rm m}$ of ${\rm J}_{\rm c}$.
• Figure 3: Illustration of the loop affinity $\Phi$. (a) The affinities for $\triangledown,\vartriangle$ such that $\Phi_{\vartriangle} = -\Phi_{\triangledown}$ indicate opposite cycles (purple arrows). An edge flow is presented thereby (big arrows). Small panel: loop affinity vs. pump rate. (b) Loop affinity in Eq.(\ref{['Ast']}) is regardless of deformations of the graph.