Thermalization kinetics of light: From laser dynamics to equilibrium condensation of photons

We report a time-resolved study of the thermalization dynamics and the lasing to photon Bose-Einstein condensation crossover by in situ monitoring the photon kinetics in a dye microcavity. When the equilibration of the light to the dye temperature by absorption and reemission is faster than photon loss in the cavity, the optical spectrum becomes Bose-Einstein distributed and photons accumulate at low-energy states, forming a Bose-Einstein condensate. The thermalization of the photon gas and its evolution from nonequilibrium initial distributions to condensation is monitored in real time. In contrast, if photons leave the cavity before they thermalize, the system operates as a laser.

Figure 1

(a) Optical microresonator filled with dye solution. While trapped in the resonator, photons thermalize by dye absorption reemission cycles. Light leaking from the cavity is monitored spatially or spectrally using a streak camera. (b) Measured, normalized spectral profiles of dye absorption $\alpha \left(\lambda \right)$, fluorescence $f\left(\lambda \right)$ (top panel), and cavity loss rate $\mathrm{\Gamma }\left(\lambda \right)$ (middle panel). The bottom panel schematically shows the cavity transverse mode structure with the cutoff wavelength ${\lambda }_{\mathrm{c}}$, indicating the linearly increasing degeneracy of the excited transverse modes at lower wavelengths. (c) Quadratic photon dispersion in the cavity along with a Bose-Einstein condensed (filled) and a nonequilibrium spectral photon distribution (hatched).

Figure 2

(a) Spectral distribution for different times after the pump pulse (dots, vertically shifted) for different cutoff wavelengths ${\lambda }_{\mathrm{c}}$, along with 300 K Bose-Einstein distributions (solid lines). The low-wavelength part of the experimental spectra (shaded) is fitted with a linear function, and in (b) (top panel) the evolution of the slope of the thermal wings (circles) is compared to that of a fully thermalized spectrum with corresponding chemical potential (solid line). For $601\mathrm{nm}$ (left) no thermalization is observed, while for a cutoff closer to the dye absorption maximum photons thermalize within the cavity lifetime (middle and right). The total cavity emission (bottom panel) yields the time scales for chemical equilibration between photons and dye excitations ${\tau }_{\mathrm{ch}}=\left\{450\mathrm{ps};560\mathrm{ps};>350\mathrm{ps}\right\}$ (note the different scales on the time axis). (c) Observed photon gas thermalization time ${\tau }_{\mathrm{th}}$ (data points) versus dye reabsorption time compared to theory (solid line), showing the transition from nonequilibrium conditions to thermalization for larger dye absorption and longer cavity lifetimes [27]. Experimental parameters: ${\lambda }_{\mathrm{c}}=\left\{601\mathrm{nm};585\mathrm{nm};577\mathrm{nm};571\mathrm{nm}\right\}$ (corresponding cavity photon lifetimes $\left\{26\mathrm{ps};162\mathrm{ps};331\mathrm{ps};540\mathrm{ps}\right\}$) at $\rho ={10}^{-4}\mathrm{mol}/\mathrm{l}$, and ${\lambda }_{\mathrm{c}}=585\mathrm{nm}$ at $\rho ={10}^{-3}\mathrm{mol}/\mathrm{l}$.

Figure 3

(a) Spectral evolution of the photon gas dynamics (line-normalized) for off-center pumping with a $80\text{−}\mu \mathrm{m}$-diameter pump spot $160\mu \mathrm{m}$ offset from the cavity axis for different cutoff settings. The off-center pumping locally inverts the dye molecules, which decay by stimulated emission of photons into highly excited transverse modes overlapping with the pump region. The blue line on top indicates the harmonic trap potential. For a cutoff wavelength far red from the dye absorption peak no thermalization is observed (top, left). Photons partially thermalize to lower energies for larger dye absorption (top, right) and finally collapse into a bimodal distribution with a Bose-Einstein condensate peak (bottom graphs, with ${\lambda }_{\mathrm{c}}=574$ and $567\mathrm{nm}$, corresponding to ${\left[\rho \sigma \left({\lambda }_{\mathrm{c}}\right)c\right]}^{-1}=96$ and $33\mathrm{ps}$, respectively). As an indication for the redistribution towards longer wavelength, the dashed white line illustrates the evolution of the maximum of the emission not in the ground mode. (b) Spectra measured shortly after photon injection (green squares) and near the end of the streak camera time trace (red circles), averaged over the shaded areas in (a). $\rho =2.5\times {10}^{-4}\mathrm{mol}/\mathrm{l}$.

Figure 4

(a) Mode-locked laser operation and photon condensate. Evolution of the spatial profile (line-normalized) of the radiation transmitted through one cavity mirror for three cutoff wavelengths. A pump beam with $27\mu \mathrm{m}$ diameter spatially displaced by $50\mu \mathrm{m}$ from the trap center excites an optical wave packet oscillating in the harmonic trapping potential (indicated on top). For the data recorded with largest cutoff wavelength (left) and correspondingly low dye absorption the oscillating mode-locked laser operation persists (classical oscillation indicated by dashed line), with the majority of the emission leaking out of the resonator at the turning points, while the rest is near the instrumental detection limit. In contrast, for larger reabsorption photons thermalize and a condensate in the trap center builds up (right). The middle graph depicts results for the crossover region between lasing and condensation. (b) Left panel: Variation of the observed intensity in (a) (right), close to the oscillator reversal points near $x=\pm 50\mu \mathrm{m}$ (purple and green, respectively), their mean intensity (red), and (right panel) relative intensity in the condensate mode. $\rho ={10}^{-4}\mathrm{mol}/\mathrm{l}$.