From Lasers to Photon Bose--Einstein Condensates: A Unified Description via an Open-Dissipative Bose--Einstein Distribution

TL;DR

The paper addresses how the open, driven-dissipative nature of photon Bose–Einstein condensation modifies condensation in a dye-filled microcavity. It develops a mean-field rate-equation framework derived from a Lindblad master equation and derives a steady-state open-dissipative Bose–Einstein distribution with a self-consistent chemical potential $μ$. The results show that the condensation threshold $N_c$ is significantly altered by pumping, cavity losses, and non-radiative decay compared to a closed system, while the condensed-phase chemical potential $μ_c$ remains only mildly sensitive to open-system parameters; in the single-mode limit a laser-like regime is recovered only when $B_{12}=B_{21}$ and $ω=ω_ ext{ZPL}$. The study demonstrates convergence to the thermodynamic limit with many photon modes and reveals a strong linear dependence of $N_c$ on cavity losses, highlighting the necessity of open-dissipative statistics for accurate descriptions and offering experimentally testable predictions (e.g., ≈10% differences in $N_c$) to probe non-equilibrium thermodynamics in photonic systems.

Abstract

Photon condensation was first experimentally realized in 2010 within a dye-filled microcavity at room temperature. Since then, interest in the field has increased significantly, as a photon Bose-Einstein condensate (BEC) represents a prototypical driven-dissipative system. Here, we investigate how its inherent open nature influences the condensation process both quantitatively and qualitatively. To this end, we consider a mean-field model, which can be derived microscopically from a Lindblad master equation. The underlying rate equations depend on various external parameters such as emission and absorption rates of the dye molecules as well as the cavity photon loss rate. In steady state, we obtain an open-dissipative Bose-Einstein distribution for the mode occupations. The chemical potential of this distribution depends on the occupations of the dye molecules in both their ground and excited state and must therefore be determined self-consistently. We find that the resulting photon distribution is strongly influenced by the driven-dissipative parameters. Based on this result, we identify the main differences between a photonic BEC, an atomic BEC, and a laser.

Figures (6)

• Figure 1: Panel (a) shows for the single-mode model the photon number $N$ as a function of the pumping strength $p$. Dotted and dashed lines represent small and large pumping limits, respectively. Blue curves correspond to the photon BEC case of unequal absorption and emission rates $B_{12} \neq B_{21}$, whereas red lines represent the laser case $B_{12}=B_{21}$. Panel (b) depicts the relative population inversion from Eq. \ref{['eq:population_inversion']}, for the same two cases as in panel (a), with the dashed line highlighting equal population. In both panels the following parameter values were used for the purpose of illustration: $M=10^5$, $\Gamma_{\rm c} = 10^4\,\text{Hz}$, $\Gamma_{\rm nr} = 5\,\text{Hz}$ as well as $B_{12} = 0.5\,\text{Hz}$, $B_{21} = 1\,\text{Hz}$ for the photon BEC case and $B_{12} = 1\,\text{Hz} = B_{21}$ in the laser limit, respectively.
• Figure 2: Experimental data reported in Ref. Schmitt_Data2024. Panel (a) displays the measured emission (red) and absorption (blue) rates $B_{21}(\omega)$ and $B_{12}(\omega)$ together with fits obtained from \ref{['eq:spectra_fit_funca']} and \ref{['eq:spectra_fit_funcb']}. The vertical dotted line indicates the cavity cut-off frequency $\omega_\text{cut}=2\pi\cdot 500\,\text{THz}$, while the dashed line marks the zero-phonon line $\omega_\text{ZPL}=2\pi\cdot 550\,\text{THz}$. Panel (b) shows the cavity loss rates. The inset magnifies the frequency region, which is most relevant for subsequent numerical simulations and includes the fit from Eq. \ref{['eq:loss_fit']}. The horizontal dashed line denotes the approximate value derived from Eq. \ref{['eq:loss_approx']}.
• Figure 3: Panel (a) shows the total photon number as a function of the pumping strength. The inset displays the occupation for selected modes. Panel (b) represents the occupations of the ground state and of the thermal states. The inset shows the critical particle number obtained independently from ground-state (red) and thermal-state (blue) occupations. The dashed horizontal black line indicates the numerically calculated thermodynamic-limit value, whereas the dotted horizontal black line depicts the thermodynamic-limit value of the closed system calculated from Eq. \ref{['eq:Nc_closed']}. The color coding is identical in both panels.
• Figure 4: Panel (a) and (b) show the occupation of the ground state and excited state molecules, whereas panel (c) depicts the chemical potential, calculated via Eq. \ref{['eq:chemical_pot']}.
• Figure 5: Panel (a) depicts the photon occupation of ground state and thermal states for varying cavity losses $\Gamma_{\rm c}$. The inset illustrates the corresponding dependence of the critical particle number. Panel (b) presents the chemical potential. Its inset highlights the value in the condensed phase, where dots indicate numerical results, the red curve depicts the exact expression of Eq. \ref{['eq:chem_pot_cond']}, and the dotted black line shows the expansion of Eq. \ref{['eq:chem_pot_cond']} valid for small cavity losses according to Eq. \ref{['eq:mu_cond_expansion']}.