External-feedback laser cooling of molecular gases

We analyze the laser cooling of a gas of polarizable particles by continuous dispersive position detection and active feedback. Microkelvin temperatures can be attained inside an optical resonator, while in free space cooling requires wavelength-size beams. The maximum cooling flux is set by the thermal Doppler width, with typical values between ${10}^5$ and ${10}^9\text{molecules}\text{per}\text{second}$.

Figure 1

(Color online) Schematic of continuous-feedback cooling. The moving molecule periodically detunes the standing-wave resonator by ${\delta }_m\left(t\right)$. The observed fractional change in cavity transmission $\epsilon \left(t\right)$ acts as an error signal that is applied to change the input power $P_i$ by a fraction $-{\epsilon }_{\mathrm{fb}}\left(t\right)$. The feedback open-loop gain $H\left(s\right)$ is set by an external electronic circuit.

Figure 2

(Color online) Cooling force versus molecular velocity $v$ for conventional Doppler cooling (a), and for feedback cooling with loop gain $H\left(s\right)=s$ (differentiator, b), $H\left(s\right)=s(1+s∕10)∕(1+8s∕10)$ (c), and $H\left(s\right)=s(1+s∕10)(1+s∕100)∕\left[(1+8s∕10)(1+s∕20)\right]$ (d). Here $s=iv∕u$, where $u$ sets the unity gain frequency for the differentiator loop.

Figure 3

(Color online) Differentiator cooling flux (cooling power divided by $k_{B}T$) versus temperature $T$ for various free-space scattering rates ${\Gamma }_{\mathrm{sc}}$ and sample sizes $N$, calculated for CaH with $\lambda =760\mathrm{nm}$ inside a bad cavity $(\eta =0.05)$. The parameters are (a) $N={10}^6$, ${\Gamma }_{\mathrm{sc}}={10}^7{\mathrm{s}}^{-1}$ (thin solid line), (b) $N={10}^5$, ${\Gamma }_{\mathrm{sc}}={10}^7{\mathrm{s}}^{-1}$ (dashed), (c) $N={10}^8$, ${\Gamma }_{\mathrm{sc}}={10}^2{\mathrm{s}}^{-1}$ (dotted), and (d) $N={10}^8$, ${\Gamma }_{\mathrm{sc}}=10{\mathrm{s}}^{-1}$ (dash-dotted). The thick solid line is the optimum rate with ${\Gamma }_{\mathrm{sc}}$ adjusted to maintain ${\overline{\Gamma }}_{\mathrm{cav}}=1$.