Fluctuation dynamics of an open photon Bose-Einstein condensate

Bosonic gases coupled to a particle reservoir have proven to support a regime of operation where Bose-Einstein condensation coexists with unusually large particle-number fluctuations. Experimentally, this situation has been realized with two-dimensional photon gases in a dye-filled optical microcavity. Here we investigate theoretically and experimentally the open-system dynamics of a grand canonical Bose-Einstein condensate of photons. We identify a regime with temporal oscillations of the second-order coherence function $g^{\left(2\right)}\left(\tau \right)$, even though the energy spectrum closely matches the predictions for an equilibrium Bose-Einstein distribution and the system is operated deeply in the regime of weak light-matter coupling. The observed temporal oscillations are attributed to the nonlinear, weakly driven dissipative nature of the system, which leads to time-reversal symmetry breaking.

Figure 1

Photon flow through the condensed photon gas in the dye-filled microcavity. The dye molecules, which are excited with a pump rate ${\mathrm{\Gamma }}_{\uparrow }$, constitute a heat and particle reservoir for the condensed photons. The photon dispersion in the cavity in terms of the transverse wave vector is $h\nu =\sqrt{{\left(h{\nu }_c\right)}^2+{\left(\hslash c{k}_r\right)}^2}$, with ${\nu }_c={\omega }_c/2\pi$ the cavity cutoff frequency and $c$ the light velocity in the dye medium. The photon loss rate of the condensate through the cavity mirrors is represented by $\kappa$.

Figure 2

Optical spectra (arb. units) as a function of wavelength $\lambda$, taken from cavity emission through the mirrors at the cutoff wavelength of ${\lambda }_c=571.3$ nm, where the experiment was conducted. The solid lines are fits of the Bose-Einstein distribution at $300\mathrm{K}$ (broadened by the experimental resolution) to the experimental data, with the steady-state average condensate photon number ${\langle n\rangle }_{\infty }$ as the only adjustable parameter. The fitted values of ${\langle n\rangle }_{\infty }$ are shown in the legend. The spectra at different ${\langle n\rangle }_{\infty }$ are vertically shifted for clarity.

Figure 3

Typical results for the time dependence of the second-order correlation function for two different photon numbers in the condensate, as indicated. The solid lines are fits to the model function (19). The envelope of the observed correlation signal $g^{\left(2\right)}\left(\tau \right)$ temporally decays with a typical relaxation time of about 4 ns.

Figure 4

Oscillation frequency ${\omega }^{\left(2\right)}$ of the second-order correlation function $g^{\left(2\right)}\left(\tau \right)$ as a function of the average photon number ${\langle n\rangle }_{\infty }$ in the condensate, as measured in the experiment (circles) and predicted by a nonlinear rate equation model (solid line). See the text for details. The parameter values are $M=5.17\times {10}^9, \kappa =2.33\mathrm{GHz}, B_{\text{em}}=2.50\times {10}^{-5}\mathrm{GHz}, B_{\text{em}}/B_{\text{abs}}\approx 57$ (Kennard-Stepanov relation corresponding to the cutoff wavelength of ${\lambda }_c=571.3$ nm), and ${\mathrm{\Gamma }}_{\downarrow }=0$.