Coherent quantum engineering of free-space laser cooling

Two distinct lasers are shown to permit controlled cooling of a three-level atomic system to a regime particularly useful for group-II atoms. Alkaline-earth-metal atoms are difficult to laser cool to the micro- or nanokelvin regime, but this technique exhibits encouraging potential to circumvent current roadblocks. Introduction of a sparse-matrix technique permits efficient solution of the master equation for the stationary density matrix, including the quantized atomic momentum. This overcomes long-standing inefficiencies of exact solution methods, and it sidesteps inaccuracies of frequently implemented semiclassical approximations. The realistic theoretical limiting temperatures are optimized over the full parameter space of detunings and intensities. A qualitative interpretation based on the phenomenon of electromagnetically induced transparency reveals dynamical effects due to photon-atom dressing interactions that generate non-Lorentzian line shapes. Through coherent engineering of an asymmetric Fano-type profile, the temperature can be lowered down to the recoil limit range.

Figure 1

Left side: Atomic configuration for the $\Xi$ system in ${}^{24}\mathrm{Mg}$ (see text). Right side: Dressed system, with $s_1\left({\delta }_1\right)=0.001$ and $s_2\left({\delta }_2\right)=1$. Real (top) and imaginary (bottom) parts of the eigenvalues of Eq. (3), with dressed atomic states labeled. The real parts are the energies and the imaginary parts are the effective linewidths of the dressed atomic system. Both are plotted as functions of ${\delta }_2$, with fixed ${\delta }_1=0$.

Figure 2

(Color online) Laser-cooling temperatures for a ${}^{24}\mathrm{Mg}\Xi$ system, as a function of ${\delta }_1$ and ${\delta }_2$, obtained from exact numerical solutions of Eq. (1). The saturation parameter for the lower transition (probe) is $s_1\left({\delta }_1\right)=0.001$ for both plots, and for the upper transition (pump) is $s_2\left({\delta }_2\right)=1$ and 5 in the upper and lower plots, respectively. The temperature is normalized to the 1D Doppler limit for the lower transition $T_D^{\left(1\mathrm{D}\right)}=7\hslash {\Gamma }_1∕40k_B$ [13], which is the optimum temperature expected for cooling with just one laser. The dashed line indicates the location of the bare two-photon resonance.

Figure 3

Upper plot: Absorption spectrum (solid line) as a function of ${\delta }_1$, illustrating the asymmetric line shapes for the dressed system, for fixed ${\delta }_2=-{\Gamma }_1$. The peak of each line shape is located at a dressed eigenenergy, and line shapes at the same energies and widths, but with Lorentzian (symmetric) line shapes are plotted with dotted lines. As an example, optimum-cooling detunings relative to the leftmost resonance are illustrated with a black arrow for the true asymmetric line shape and with a gray arrow for the hypothetical symmetric-line-shape case. The value of the optimum detuning, as well as the slope of the line shape, is seen to be different for these two cases. Lower plot: Solid line, the ratio of the maximum slope of a Lorentzian line shape with width ${\Gamma }_1$ to the slope of the asymmetric line shape, as a function of ${\delta }_1$ with ${\delta }_2=-{\Gamma }_1∕2$. This ratio provides an indication of the expected cooling for the dressed system relative to the Doppler limit for the lower transition. For comparison, fully quantum numerical results are indicated by data points.