Nonequilibrium Model of Photon Condensation

We develop a nonequilibrium model of condensation and lasing of photons in a dye filled microcavity. We examine in detail the nature of the thermalization process induced by absorption and emission of photons by the dye molecules, and investigate when the photons are able to reach a thermal equilibrium Bose-Einstein distribution. At low temperatures, or large cavity losses, the absorption and emission rates are too small to allow the photons to reach thermal equilibrium and the behavior becomes more like that of a conventional laser.

Figure 1

Cartoon of the system showing the decay processes included in Eq. (2). The zoomed in view shows the energy level structure of the dye molecules. The graph shows the characteristic behavior of the emission rate, $\Gamma (-{\delta }_m)$ (dashed line) and the absorption rate, $\Gamma ({\delta }_m)$ (solid line) described in the main text. The vertical (red) lines show the typical spacing of the photon modes confined in the cavity.

Figure 2

Mode populations $g_m{n}_m$ vs detuning ${\delta }_m$ for various pump strengths. Crosses are results of the nonequilibrium model, and lines show Bose-Einstein distributions fitted to the tail of the numerical results. Insets show the pump powers, ${\Gamma }_{\uparrow }/{\Gamma }_{\downarrow }={\mathrm{e}}^{\beta {\mu }_{\mathrm{eff}}}$, (dashed lines) compared to the ratio $\Gamma (-{\delta }_m)/\Gamma ({\delta }_m)\simeq {\mathrm{e}}^{\beta {\delta }_m}$ (solid, black line) which gives a good approximation to the threshold. Panel (a) corresponds to experimental losses, $\kappa =10\mathrm{MHz}$. Panel (b) shows $\kappa =5\mathrm{GHz}$ where losses prevent thermalization. Other parameters are: $\gamma =100\mathrm{THz}$, ${\Gamma }_{\downarrow }=1\mathrm{GHz}$, $S=0.5$, $\Omega =1\mathrm{THz}$, $N={10}^{11}$, $g=0.1\mathrm{GHz}$, $T=300\mathrm{K}$, ${\delta }_0=-200\mathrm{THz}$, and the mode spacing $ϵ=10\mathrm{THz}$.

Figure 3

(a) Total number of photons in the cavity $N_{\mathrm{tot}}$ vs pump power. The dashed vertical lines show the threshold [defined by $\max (n_m)=T/ϵ$]. (b) Threshold pump power vs temperature, for various cavity loss rates as indicated. The dashed (black) lines show the same threshold calculated for the equilibrium theory. (c) Population $N_{\mathrm{tot}}$ at threshold vs temperature for the same loss rates shown in (b). The dashed (black) line is the equilibrium prediction of critical density. (d) Index of the mode which gains a macroscopic occupation for the same loss rates as in (b). All other parameters are the same as in Fig. 2.