Thermalization and breakdown of thermalization in photon condensates

We examine in detail the mechanisms behind thermalization and Bose-Einstein condensation (BEC) of a gas of photons in a dye-filled microcavity. We derive a microscopic quantum model, based on that of a standard laser, and show how this model can reproduce the behavior of recent experiments. Using the rate-equation approximation of this model, we show how a thermal distribution of photons arises. We go on to describe how the nonequilibrium effects in our model can cause thermalization to break down as one moves away from the experimental parameter values. In particular, we examine the effects of changing cavity length, and of altering the vibrational spectrum of the dye molecules. We are able to identify two measures which quantify whether the system is in thermal equilibrium. Using these, we plot “phase diagrams” distinguishing BEC and standard lasing regimes. Going beyond the rate-equation approximation, our quantum model allows us to investigate both the second-order coherence $g^{\left(2\right)}$ and the linewidth of the emission from the cavity. We show how the linewidth collapses as the system transitions to a Bose condensed state, and compare the results to the Schawlow-Townes linewidth.

Figure 1

Cartoon of the system showing the decay processes included in Eq. (6). The zoomed-in view shows the energy-level structure of the dye molecules.

Figure 2

The effective absorption (solid) and emission (dashed) rates $\Gamma (\pm \delta )$ for different vibrational mode structures. In (a) we show the case of one mode with parameters $S=0.5$, $\Omega =5\mathrm{THz}$, $\gamma =50\mathrm{THz}$. In (b) we also include a second mode with $S=0.5$, $\Omega =60\mathrm{THz}$, $\gamma =30\mathrm{THz}.$

Figure 3

Steady-state behavior of our model as it is pumped through the BEC transition. (a) shows the total number of photons in the cavity. (b) gives the fraction of molecules in the excited state and (c) shows the effective chemical potential. In (c) we also plot (as the red, dashed curve) the simple below threshold expression ${\mu }_0=k_{B}Tln{\Gamma }_{\uparrow }^{\mathrm{tot}}\left(0\right)/{\Gamma }_{\downarrow }^{\mathrm{tot}}\left(0\right)$. The parameters used are $S=0.5$, $\gamma =50\mathrm{THz}$, $\Omega =5\mathrm{THz}$, $\kappa =100\mathrm{MHz}$, $T=300\mathrm{K}$, $N={10}^9$, $g=1\mathrm{GHz}$, ${\delta }_0=-200\mathrm{THz}$, ${\Gamma }_{\downarrow }=1\mathrm{GHz}$.

Figure 4

(a) Photon distribution just above threshold pump power, illustrating the effect of decreasing the energy of the lowest cavity mode ${\delta }_0$. From left to right, the green curve is ${\delta }_0=-300\mathrm{THz}$, blue is ${\delta }_0=-200\mathrm{THz}$, and red is ${\delta }_0=-100\mathrm{THz}$. The dashed lines show Bose-Einstein fits to the tails of the data. (b) The absorption (solid) and emission (dashed) rates for the same parameters. The shaded regions show which modes are included for the same colored curves as in (a). All parameters except ${\delta }_0$ are the same as in Fig. 3.

Figure 5

Comparison between fitted chemical potential, energy of maximally populated mode, and cavity cutoff, as the cutoff is varied. Above threshold in equilibrium, the maximally occupied mode and effective chemical potential both lock to the cavity cutoff (dashed line). The solid (red) and dotted-dashed (blue) lines correspond, respectively, to the fitted chemical potential, and energy of the maximally populated mode as extracted from the steady-state distribution for the same parameters as given in Fig. 3.

Figure 6

The effect of a peaked absorption and emission spectrum. In (a) we show the form of $\Gamma (\pm {\delta }_m)$ when we included a vibrational mode which is not overdamped (solid line shows absorption, dashed line emission). In (b) and (c) we show the effect of increasing the cavity losses for this spectrum below (${\Gamma }_{\uparrow }=0.5{\Gamma }_{\mathrm{thresh}}$) and above (${\Gamma }_{\uparrow }=2.5{\Gamma }_{\mathrm{thresh}}$) threshold, respectively. In each case we plot the behavior for two cavity loss rates: $\kappa =1\mathrm{THz}$ (red) and $\kappa =1\mathrm{GHz}$ (blue). The dashed lines are the fit to a Bose-Einstein distribution. Other parameter values are the same as Fig. 3 except as follows: The two vibrational modes are characterized by $S_1=0.1$, ${\Omega }_1=5\mathrm{THz}$, ${\gamma }_1=50\mathrm{THz}$, $S_2=0.5$, ${\Omega }_2=30\mathrm{THz}$, ${\gamma }_2=5\mathrm{THz}$.

Figure 7

Criteria for thermalization vs $S$ and $T$. In (a) the color shows the index of the mode which gains a macroscopic occupation, the solid line separates the region where the ground mode condenses (hashed) from the region where excited cavity modes are macroscopically occupied. In (b) we show a similar diagram but with the “order parameter”'$\beta (\mu -{\delta }_0)$.

Figure 8

Criteria for thermalization vs $\gamma$ and $T$. Panel (a) shows the index of the mode which is macroscopically occupied above threshold. Panel (b) shows the chemical potential of the Bose-Einstein fit to the data $\beta (\mu -{\delta }_0)$.

Figure 9

Phase diagram for condensation and lasing, showing the required pump power for a mode to go above threshold (blue) compared to the equilibrium prediction (red).

Figure 10

Time evolution towards thermal equilibrium both below (${\Gamma }_{\uparrow }=0.2{\Gamma }_{\mathrm{thresh}}$) (a), (b) and above (${\Gamma }_{\uparrow }=10{\Gamma }_{\mathrm{thresh}}$) (c), (d) threshold. In (a) and (c) we show the photon distribution at different times, the dashed red curves show the steady-state solution. In (b) and (d) we show the mean frequency of the photon intensity $\langle {\delta }_m\rangle$ as a function of time. The squares show the locations at which the traces in (a) and (c) are taken. Parameters are the same as in Fig. 3.

Figure 11

Second-order coherence function $g^{\left(2\right)}$ as we sweep through threshold. The solid red curve shows the results of a full numerical calculation of the probability distribution for $N=100$ while the red dashed curve shows the result of the calculation based on second-order cumulants described in the text. The blue curves show similar results for $N=500$. The other parameters used are the same as those in Fig. 3 except we directly specify $\Gamma (-\delta )=1\mathrm{GHz}$, $\Gamma \left(\delta \right)=50\mathrm{MHz}$.

Figure 12

The emission spectra below (a) and above (b) the transition point, calculated for $N=100$. The exact solution is shown as the solid (black) lines while the Lorentzian fits are shown as dashed (red) curves. Panel (c) shows the linewidth of the emission spectrum. In the small system size limit ($N=100$), we show the full numerics as the solid black line while the mean field approximation is given by the red dashed line. In the thermodynamic limit ($N={10}^9$) we show the full mean field calculation as the blue solid line, while the Schawlow-Townes result is the green dashed curve. The points marked a and b indicate the locations at which the spectra in the other panels were calculated. The other parameters used are the same as those in Fig. 3 except we directly specify $\Gamma (-\delta )=1\mathrm{GHz}$, $\Gamma \left(\delta \right)=50\mathrm{MHz}$.