Theory of Bose condensation of light via laser cooling of atoms

A Bose-Einstein condensate (BEC) is a quantum phase of matter achieved at low temperatures. Photons, one of the most prominent species of bosons, do not typically condense due to the lack of a particle number conservation. We recently described a photon thermalization mechanism which gives rise to a grand canonical ensemble of light with effective photon number conservation between a subsystem and a particle reservoir. This mechanism occurs during Doppler laser cooling of atoms where the atoms serve as a temperature reservoir while the cooling laser photons serve as a particle reservoir. In contrast to typical discussions of BEC, our system is better treated with a controlled chemical potential rather than a controlled particle number, and is subject to energy-dependent loss. Here, we address the question of the possibility of a BEC of photons in this laser cooling photon thermalization scenario and theoretically demonstrate that a Bose condensation of photons can be realized by cooling an ensemble of two-level atoms (realizable with alkaline-earth atoms) inside a Fabry-Pérot cavity.

Figure 1

(a) Schematic of an ensemble of two-level atoms which are Doppler cooled by laser fields (green arrows) and free-space photon modes (red arrows), while also interacting with cavity-photon modes (blue arrows within the light-blue region). (b) The state degeneracy $g\left(E\right)$ (equivalent to the density of states) of the transverse cavity modes is equivalent to that of a 2D massive particle in a harmonic trapping potential with a lower cutoff energy $\hslash {\omega }_c=\hslash {\omega }_{q_{\parallel }00}$ including polarization. The green line shows the energy of a single laser photon $\hslash {\omega }_L$. (c) The scattering process in which an atom is excited by the laser field and then emits a cavity photon. (d) The scattering process in which an atom absorbs a cavity photon and then scatters back into the laser field.

Figure 2

(a), (b) The condensate fraction $n_0/n_{\mathrm{tot}}$ [shown in (a)] and the total average number of photons $n_{\mathrm{tot}}$ [shown in (b)] at a fixed temperature as a function of the loss-shifted chemical potential $\tilde{\mu }$ plotted on a logarithmic scale. The ideal grand canonical ensemble result is shown in black lines, and the colored lines represent the modified results with two different Rabi frequencies. The critical points representing the onset of the BEC phase are identified by dots. We take parameters for the ${}^{1}S_0\text{−}{}^{3}P_1$ Yb intercombination transition [57], ${\omega }_L/2\pi \approx {\omega }_A/2\pi =539$ THz, $\mathrm{\Gamma }/2\pi =180$ kHz, $E_{r,{\mathit{k}}_L}/h=3.74$ kHz, and assume $|{\mathit{k}}_L-\mathit{q}|\approx \sqrt{2}\left|{\mathit{k}}_L\right|$, ${\overline{\mathrm{\Delta }}}_L\approx -157\mathrm{\Gamma }$, and a cavity trapping frequency ${\omega }_T/2\pi =231$ kHz. (c) The condensate fraction $n_0/n_{\mathrm{tot}}$ as a function of temperature while keeping the total number of photons fixed. The ideal grand canonical distribution is shown in black lines while the colored lines indicate loss-modified results with two different Rabi frequencies. We take parameters for the Yb intercombination transition with ${\omega }_T/2\pi =231$ kHz for $n_{\mathrm{tot}}={10}^3$ and ${\omega }_T/2\pi =73$ kHz for $n_{\mathrm{tot}}={10}^4$ such that $n_{\mathrm{tot}}{({\omega }_T/2\pi )}^2=5.33\times {10}^{13}{\text{s}}^{-2}$, proportional to the 2D number density, is fixed, which leaves the critical temperature essentially unchanged.

Figure 3

Calculated phase diagram of cavity photons as a function of $\mathrm{\Omega }$ and $\mu$. At high power, photon generation can exceed loss leading to gain (green region) and possibly lasing; cavity photons can be described as a grand canonical ensemble (yellow) at equilibrium, and we find the formation of a photon BEC (orange) within the GCE area and near the gain boundary. For low power, photon loss prevents equilibration of photons to a single temperature corresponding to that of the atomic motion, and only quasithermal light (blue) is expected. In this diagram we use the same parameters as in Fig. 2.

Figure 4

Simulations of the far-field photonic “time-of-flight” images and corresponding cross sections through the center, where $r_{\perp }$ is the far-field transverse position. The parameters are the same as in Fig. 2 with $\mathrm{\Omega }=0.3|{\overline{\mathrm{\Delta }}}_L|$.