Large Velocity Capture Range and Low Temperatures with Cavities

There are interesting modifications to the Doppler force when atoms strongly couple to an optical cavity. In particular, there is the possibility to increase the velocity capture range while maintaining a final temperature close to the Doppler limit. The mechanism is based on the multiple absorption emissions of each cavity photon. A previously reported counterintuitive Doppler effect is clarified.

Figure 1

An atom experiences a Doppler force which is modified due to the presence of cavity mirrors: (a) ring cavity; (b) linear cavity. The amount of light collected at the output of a mirror highly depends on the velocity of the particle.

Figure 2

(a) Example of the force ${\mathit{F}}_{k_L}$ as a function of velocity, for ($\kappa =0.1\gamma$, $g=4\gamma$, ${\Delta }_c={\Delta }_a=2\gamma$). The scaling is arbitrary. The Doppler shift tunes the photons to be resonant with the eigenstates $|+⟩$ and $|-⟩$ of the atom-cavity system when ${\mathit{k}}_L·{\mathit{v}}_{\pm }=-{\Delta }_{\pm }\approx -{\Delta }_a\mp g$, (${\Delta }_{\pm }={\omega }_{\pm }-{\omega }_L$). (b) Resonance frequencies ${\omega }_{\pm }$ of the coupled atom-cavity system as a function of the atom’s position orthogonal to the cavity.

Figure 3

Spatially averaged force ${\overline{\mathit{F}}}_{k_L}$ for a laser standing wave as a function of velocity, for $\kappa =8\gamma$, ${\Delta }_c=1.8\kappa$): (a) ($g=1.3\kappa$, ${\Delta }_a=10\gamma$); (b) ($g=\kappa$, ${\Delta }_a=6.3\gamma$). For decreasing $g$ or increasing ${\Delta }_c$, the lowest peaks (state $|+⟩$) disappear and one recovers the narrow Doppler peaks (dashed line). In both figures, the force is effective over a large range of velocities $\Delta v_c\approx 16\gamma /k_L$, but the temperature is close to the Doppler limit: (a) $k_B{T}_{k_L}\approx 2.1\hslash \gamma$; (b) $k_B{T}_{k_L}\approx 1.7\hslash \gamma$.

Figure 4

Spatially averaged temperature as a function of the cavity decay rate $\kappa$, for ($g=\kappa$, ${\Delta }_c=1.8\kappa$) and ${\Delta }_a$ such that ${\Delta }_{\mathrm{eff}}(0)={\gamma }_{\mathrm{eff}}(0)$. The temperature $k_B{T}_{k_L}$ (solid line) [and $k_B{T}_{k_L}^{\mathrm{app}}$ (dashed line)] is close to the Doppler limit $\hslash \gamma$ (horizontal line). For $\kappa =16\gamma \gg \gamma$ it is only $k_B{T}_{k_L}\approx 2.5\hslash \gamma$. (b) Sub-Doppler cooling: Force (solid line) and the Doppler force (dashed line), ($\kappa =0.2\gamma$, $g=2\gamma$, ${\Delta }_c=0.9\gamma$, ${\Delta }_a=7\gamma$). The narrow peak around $v=0$ is responsible for sub-Doppler cooling $\kappa \ll \gamma$.