Photon thermalization via laser cooling of atoms

Laser cooling of atomic motion enables a wide variety of technological and scientific explorations using cold atoms. Here we focus on the effect of laser cooling on the photons instead of on the atoms. Specifically, we show that noninteracting photons can thermalize with the atoms to a grand canonical ensemble with a nonzero chemical potential. This thermalization is accomplished via scattering of light between different optical modes, mediated by the laser-cooling process. While optically thin modes lead to traditional laser cooling of the atoms, the dynamics of multiple scattering in optically thick modes has been more challenging to describe. We find that in an appropriate set of limits, multiple scattering leads to thermalization of the light with the atomic motion in a manner that approximately conserves total photon number between the laser beams and optically thick modes. In this regime, the subsystem corresponding to the thermalized modes is describable by a grand canonical ensemble with a chemical potential nearly equal to the energy of a single laser photon. We consider realization of this regime using two-level atoms in Doppler cooling, and find physically realistic conditions for rare-earth atoms. With the addition of photon-photon interactions, this system could provide a platform for exploring many-body physics.

Figure 1

(a) Schematic of an ensemble of Doppler-cooled two-level atoms interacting with long-lived cavity (system) photon modes $a_{\mathit{q}}$ [blue (dark-gray) wavy arrows within the light-blue (light-gray) region] and lossy (bath) photon modes $b_{\mathit{k}}$ [red (gray) wavy arrows]. Traditional laser cooling arises via loss into the bath modes, while scattering into and out of the blue modes leads to our projected regime of photon thermalization. (b), (c) The dominant atom-photon scattering processes that lead to a grand canonical ensemble of system photons. (b) The atom is excited by the laser field then emits a system photon. (c) The atom absorbs a system photon then scatters back into the laser field. Effective system plus pump photon number conservation applies due to adiabatic elimination of the atomic excited state and the rotating wave approximation. (d) Characterization of system photon regimes for a single mode cavity with laser Rabi frequency ($\mathrm{\Omega }$) and the laser detuning from the system photon energy ($\hslash ({\omega }_L-{\omega }_{\mathit{q}})$) as parameters. At higher powers, photon generation can exceed loss as per Eq. (1), leading to either gain [green (gray)] and possibly lasing, or the formation of a grand canonical ensemble for light [yellow (light gray)]. For low powers or large laser detunings from the system photon, photon loss prevents detailed balance with the atomic motion and only quasithermal light is expected [blue (dark gray)]. In this diagram we use the physical parameters for the Yb ${}^1{S}_0{-}^3{P}_1$ narrow cooling transition [39] with ${\omega }_A/2\pi =539$ THz, $\mathrm{\Gamma }/2\pi =180$ kHz, ${\overline{\mathrm{\Delta }}}_L\approx -157\mathrm{\Gamma }$, and assume $|{\mathit{k}}_L-\mathit{q}|=\sqrt{2}{k}_L$.

Figure 2

Diagrams for laser-induced scattering between atomic ground states with different momenta to lowest order in $\mathrm{\Omega }$. (a) The dressed atomic excited-state propagator (double dashed line) is defined by including nonperturbative effects due to the coupling to the bath photon modes $b_{\mathit{k}}$ [red (gray) wavy arrow] and neglecting the effect of the system photons assuming ${\mathrm{\Gamma }}_b\approx \mathrm{\Gamma }\gg {\mathrm{\Gamma }}_a$. (b) The diagrammatic representation of the scattering amplitude of a ground-state atom (solid black line) from an initial momentum state $\mathit{p}$ to a final state $\mathit{p}+\hslash {\mathit{k}}_L-\hslash \mathit{k}$ by absorbing a pump photon [green (gray) straight arrow] and emitting a bath photon. This process is associated with an effective coupling $R_{\mathit{k}}\left(\mathit{p}\right)$ between momentum states $|g,\mathit{p}\rangle$ and $|g,\mathit{p}+\hslash {\mathit{k}}_L-\hslash \mathit{k}\rangle$. (c) The scattering amplitude from $|g,\mathit{p}\rangle$ to $|g,\mathit{p}+\hslash {\mathit{k}}_L-\hslash \mathit{q}\rangle$ by absorbing a pump photon and emitting a system photon $a_{\mathit{q}}$ [blue (dark-gray) wavy arrow], associated with a coupling $R_{\mathit{q}}^{+}\left(\mathit{p}\right)$. (d) The scattering amplitude from $|g,\mathit{p}\rangle$ to $|g,\mathit{p}+\hslash \mathit{q}-\hslash {\mathit{k}}_L\rangle$ by absorbing a system photon $a_{\mathit{q}}$ and emitting a pump photon, associated with a coupling $R_{\mathit{q}}^{-}\left(\mathit{p}\right)$. Not shown is the process in which a system photon is rescattered to a bath photon, which is treated in Fig. 4.

Figure 3

Diagrammatic comparison between Fermi's golden rule and self-consistent Fermi's golden rule. (a) The dressed state picture of the atomic-ground-state propagator (double black line) is defined by including nonperturbative effects due to the ground-state scattering induced by laser. The double dashed line is the atomic excited-state propagator dressed by the bath modes $b_{\mathit{k}}$ as shown in Fig. 2. (b) For reference, we give the diagrammatic representation of the (regular) Fermi's golden rule scattering amplitude that involves a system photon $a_{\mathit{q}}$ emission, using the dressed excited-state propagators which leads to a standard FGR result. (c) The diagrammatic representation of the self-consistent Fermi's golden rule scattering amplitude, in which we also replace the atomic-ground-state propagators with dressed ones to account for the finite lifetime of the original and final atomic motional states set by rapid emission processes into bath modes.

Figure 4

The diagrammatic representation of the scattering amplitude for four possible processes associated with transitions out of the initial state $|g,\mathit{p}\rangle$ into final atomic states with a change in the ground-state momentum. (a) Scattering process that absorbs a laser photon and spontaneously decays into the bath modes. (b) Scattering process that absorbs a laser photon and emits a system photon. (c) Scattering process that absorbs a system photon and scatters back into the laser mode. (d) Scattering process that absorbs a system photon and scatters into the bath modes.

Figure 5

Grand canonical ensemble realization for a tunable-frequency single-mode cavity with a Yb gas. (a) Equilibrium system photon occupation number (${\overline{n}}_{\mathit{q}}$) as a function of the laser detuning from the system photon frequency (${\omega }_L-{\omega }_{\mathit{q}}$). The ideal grand canonical ensemble result is plotted as the dotted black line, the standard detuning with ${\overline{\mathrm{\Delta }}}_L\approx -\frac{\mathrm{\Gamma }}{2}$ and $\mathrm{\Omega }=0.15\mathrm{\Gamma }$ is shown as the purple dashed line, and for large detuning with ${\overline{\mathrm{\Delta }}}_L\approx -157\mathrm{\Gamma }$ and varying $\mathrm{\Omega }$ are shown with colored solid lines. (b) The observed shift in the equilibrium-to-gain chemical potential ($\delta \mu$) as a function of the laser intensity ($\mathrm{\Omega }$) with ${\overline{\mathrm{\Delta }}}_L\approx -157\mathrm{\Gamma }$. (c) The effective temperature at the equilibrium-to-gain transition ($T_o$) as a function of the laser intensity ($\mathrm{\Omega }$) with ${\overline{\mathrm{\Delta }}}_L\approx -157\mathrm{\Gamma }$. (d) The ratio between the effective temperature $T_{\mathrm{eff}}$ and $T_o$ as a function of the shifted laser detuning from the system photon frequency (${\omega }_L-{\omega }_{\mathit{q}}-\delta \mu /\hslash$) and the laser intensity ($\mathrm{\Omega }$) with ${\overline{\mathrm{\Delta }}}_L\approx -157\mathrm{\Gamma }$. The $y$ axis has a lower cutoff ${\mathrm{\Omega }}_c/\left|{\overline{\mathrm{\Delta }}}_L\right|=0.04576$, which also corresponds to the end point (dot) in Figs. 5–5. In these plots we assume $|{\mathit{k}}_L-\mathit{q}|\approx \sqrt{2}\left|{\mathit{k}}_L\right|$ and take the parameters for the Yb intercombination transition to ${}^{3}P_1$, ${\omega }_A/2\pi =539$ THz, $\mathrm{\Gamma }/2\pi =180$ kHz, $E_{r,{\mathit{k}}_L}/h=3.74$ kHz.