Supercooling of Atoms in an Optical Resonator

We investigate laser cooling of an ensemble of atoms in an optical cavity. We demonstrate that when atomic dipoles are synchronized in the regime of steady-state superradiance, the motion of the atoms may be subject to a giant frictional force leading to potentially very low temperatures. The ultimate temperature limits are determined by a modified atomic linewidth, which can be orders of magnitude smaller than the cavity linewidth. The cooling rate is enhanced by the superradiant emission into the cavity mode allowing reasonable cooling rates even for dipolar transitions with ultranarrow linewidth.

Figure 1

Atoms with ultranarrow transition $|g⟩\leftrightarrow |e⟩$ are confined to the axis of a standing-wave mode of an optical cavity. Different implementations of pumping may be considered [25, 27]. In the simplest scenario shown, a transition is driven from the ground state $|g⟩$ to an auxiliary state $|a⟩$ that rapidly decays to the excited state $|e⟩$. In this way $|a⟩$ can be adiabatically eliminated and a two-state pseudospin description in the $\{|g⟩,|e⟩\}$ subspace used, with repumping corresponding to an effective rate $w$ from $|g⟩$ to $|e⟩$. If the repumping laser is directed normal to the cavity axis, the absorption does not modify the momentum. Momentum recoil is induced by the on-axis component of the wave vector $\overrightarrow{k^{\prime }}$ of the dipole radiation pattern for the $|a⟩\leftrightarrow |e⟩$ transition.

Figure 2

Time evolution of the average momentum square (red dots) evaluated from 4000 trajectories simulated by integrating Eqs. (6) and (8) for 1 (a), 20 (b), and 60 atoms (c). The blue solid line is a fit to an exponential decay. The parameters are $\mathrm{\Delta }=\kappa /2=100$, ${\mathrm{\Gamma }}_C=0.1$, and ${\omega }_{\mathrm{r}}=0.25$. The repumping rates are chosen such that the average atomic population inversion in all cases is the same [$w=0.15$ (a), 0.28 (b), 1.3 (c)]. Insets show the momentum statistics. The blue solid line is a fit to a Gaussian distribution.

Figure 3

(a) Cooling rate (in units of the single atom cooling rate $R_S$) as a function of atom number. (b) Final momentum width ($\mathrm{\Delta }p=\sqrt{⟨p^2⟩}$, blue squares) and spin-spin correlation (red dots) as a function of atom number. The parameters are the same as those in Fig. 2.

Figure 4

(a) Final momentum width as a function of ${\mathrm{\Gamma }}_C$ for 40 atoms. The parameters are $\mathrm{\Delta }=\kappa /2=200$, $w=N{\mathrm{\Gamma }}_C/4$, and ${\omega }_{\mathrm{r}}=0.25$. (b) Final momentum width as a function of repumping strength for 40 atoms without ($k^{\prime }=0$, blue squares) and with recoil associated with repumping ($k^{\prime }=k$, red dots). The parameters are $\mathrm{\Delta }=\kappa /2=200$, ${\mathrm{\Gamma }}_C=0.5$, and ${\omega }_r=0.25$.