Trapped-atom cooling beyond the Lamb-Dicke limit using electromagnetically induced transparency

We investigate the cooling of trapped atoms by electromagnetically induced transparency under conditions of weak confinement and beyond the Lamb-Dicke limit, i.e., the spontaneous decay width is large compared to the trap oscillation frequency and the recoil energy is a substantial fraction of the vibrational energy spacing of the trap. Numerical solutions of the Liouville equation for a density matrix describing states of vibrational and electronic degrees of freedom show that vibrational cooling is feasible at even substantial values of the Lamb-Dicke parameter and under conditions of weak confinement, a situation where sideband pumping is inefficient. Our approach permits us to predict cooling efficiency and cooling rates under realistic experimental conditions for neutral atoms in optical dipole traps.

Figure 1

Electronic configuration of the three-level atom. The thin lines signify trap vibrational levels. For the definition of the sign of the detunings $\Delta$ and $\delta$ see Eq. (7).

Figure 2

Absolute values of the Franck-Condon factors $\left|F\left(n,m,\eta \right)\right|$ from Eq. (12) are shown in the upper row for two values of $\eta$. The relative strength of sideband transitions originating from $n=20$ is given in the lower row.

Figure 3

The imbalance between sideband cooling $\left({\delta }_{\text{eff}}/\omega >0\right)$ and sideband heating $\left({\delta }_{\text{eff}}/\omega <0\right)$ is shown for two values of the detuning $\Delta$. The full line gives the three-level EIT line shape. The bars indicate the strength of sideband transitions (summed over the lowest 100 vibrational levels of the HO) which are active when the two-photon detuning is set to $\delta =0$, the EIT minimum. ($\eta =0.25$, $\Gamma =2\pi \times 6\text{MHz}$, $g_1=2\pi \times 1.4\text{MHz}$, and $g_2=2\pi \times 0.1\text{MHz}$).

Figure 4

Predictions for stationary vibrational population in EIT cooling limit at various values of $\eta$ as a function of detuning $\Delta /\Gamma$. Parameters are $\omega =\Gamma /10$, $g_1=\Gamma$, $g_2=\Gamma /10$.

Figure 5

Predictions for stationary vibrational population in EIT cooling limit at various values of $\eta$ as a function of the Rabi frequency of the coupling laser $g_1/\Gamma$. Parameters are $\omega =\Gamma /10$, $\Delta =2.5\Gamma$, $g_2=\Gamma /10$.

Figure 6

Cooling limit as a function trap frequency, in comparison with the prediction in the Lamb-Dicke limit. Parameters are $\Delta =15\text{MHz}$, $\Gamma =6\text{MHz}$, $g_1=1.4\text{MHz}$, $g_2=100\text{kHz}$.

Figure 7

Temporal development of cooling for $\eta =0.25$ (left-hand side) and $\eta =0.55$ (right-hand side). The full lines give the population in $n=0$, the dashed lines in $n=1$, while the dotted lines give the sum of populations in vibrational levels $n\ge 2$. Parameters are $\Delta =20\text{MHz}$, $\Gamma =6\text{MHz}$, $g_2=100\text{kHz}$, and (left-hand side) $\omega =58\text{kHz}$, $g_1=3.2\text{MHz}$, $\delta =58\text{kHz}$ and (right-hand side) $\omega =12\text{kHz}$, $g_1=1.5\text{MHz}$, $\delta =6\text{kHz}$.

Figure 8

The frequency sensitivity of the cooling window is apparent from this plot which gives the stationary population in the vibrational levels in the electronic state $|2⟩$ as a function of the two-photon detuning $\delta$. The plot on the left-hand side is for $\eta =0.25$, the plot on the right-hand side is for $\eta =0.55$. The full lines give the population in $n=0$, the dashed lines in $n=1$, while the dotted lines give the sum of populations in vibrational levels $n\ge 2$. Parameters are the same as those in Fig. 7.