Prospects for the cavity-assisted laser cooling of molecules

Cooling of molecules via free-space dissipative scattering of photons is thought not to be practicable due to the inherently large number of Raman loss channels available to molecules and the prohibitive expense of building multiple-repumping laser systems. The use of an optical cavity to enhance coherent Rayleigh scattering into a decaying cavity mode has been suggested as a potential method to mitigate Raman loss, thereby enabling the laser cooling of molecules to ultracold temperatures. We discuss the possibility of cavity-assisted laser cooling of particles without closed transitions, identify conditions necessary to achieve efficient cooling, and suggest solutions given experimental constraints. Specifically, it is shown that cooperativities much greater than unity are required for cooling without loss, and that this could be achieved via the superradiant scattering associated with intracavity self-localization of the molecules. Particular emphasis is given to the polar hydroxyl radical (OH), cold samples of which are readily obtained from Stark deceleration.

Figure 1

(Color online) (a) Fabry-Pérot cavity with resonance frequency ${\omega }_c$ driven by a laser of frequency ${\omega }_d$ and drive strength ${\Omega }_d$, defined in Eq. (A3). In steady state, there are $n_c=⟨a^{†}a⟩$ intracavity photons which escape the cavity via the mirrors at the rate $2\kappa$. (b) Three-level representation of the $N$ intracavity molecules. A classical laser field of Rabi frequency ${\Omega }_p$ and frequency ${\omega }_p$ couples the ground state $|a⟩$ to the electronically excited state $|e⟩$, whose frequency difference is ${\omega }_a$. Excitations decay at the total rate $\gamma$. Assuming $|e⟩$ is unsaturated, population is elastically (Rayleigh) scattered back to $|a⟩$ at the rate ${\gamma }_{\text{Ry}}$. Population is lost to the myriad molecular states, represented collectively by $|b⟩$, at the Raman scattering rate ${\gamma }_{\text{Rn}}$. The difference between ${\omega }_a$ and the transition frequency ${\omega }_b$ between $|b⟩$ and $e$ is much larger than ${\Delta }_{pa}$. (c) The molecule-cavity system may be pumped by the cavity drive laser, a transverse pump beam (shown), or both. The transverse pump beam of frequency ${\omega }_p$ and Rabi frequency ${\Omega }_p$ is typically red detuned from both the cavity and the molecular resonance $\left({\omega }_p<{\omega }_c⪡{\omega }_a\right)$. The $N$ molecules may be trapped or transiently passing through the cavity mode.

Figure 2

(Color online) Ratio of the cooling rate to the spontaneous emission rate, $\beta /\left(\gamma ⟨{\widehat{\sigma }}_{+}{\widehat{\sigma }}_{-}⟩\right)$, in which a negative-valued ratio indicates cooling. Parameters chosen for an OH molecule electronically excited in the cavity described near the end of Sec. 3 which has a finesse of 5000 and length of 2 cm. The detuning ${\Delta }_{pc}$ is set such that the pump field is $-\kappa$ detuned from the lower dressed state.

Figure 3

(Color online) (a) Cartoon of multiple intracavity particle scattering and self-localization. Green dots represent the scatterers and the blue sinusoidal line is the intracavity field $E$. There are two antinodes per wavelength, distinguished by opposite even or odd phases. The even are represented as the upper antinodes while the odd are the lower. (b) Below a threshold transverse pump Rabi frequency ${\Omega }_p<{\Omega }_{\text{th}}$, the molecules at the even and odd sites experience an identical dipole trapping force, proportional to ${\left|E\right|}^2$. The random nature of the molecule position implies that at any given time there could be $\sqrt{N}$ more molecules in the odd sites than the even. (c) This leads to an increase in scattered light from the unpaired odd molecules, and the interference with the pump beam creates an ever deeper optical potential for molecules in the odd sites. (d) The even sites are depopulated over the odd, leading to a phase transition from particles located at every antinode to a $\lambda$ spacing. Concomitantly, superradiance ensues, which increases the per molecule cooperativity by $N$. The choice of odd over even is arbitrary and the symmetry is spontaneously broken in favor of one versus the other.

Figure 4

(Color online) Approximate experimental OH Stark-decelerator efficiency curve—density versus final velocity—for the JILA apparatus [14, 51, 52]. Inset shows the approximate volume of the OH packet in zones II and III. The packet is larger in the pre-Stark-decelerator zone I, and expands to fill the magnetic or electric trap volume in zone IV. Expected improvements in magnetic traps for ground-state molecules should provide samples of density greater than ${10}^5{\text{cm}}^{-3}$, as shown by the dotted line in zone IV.

Figure 5

(Color online) Dressed-state spectroscopy for $N$ three-level atoms (molecules) coupled to a cavity mode in the presence of a transverse pump field ${\Omega }_p$. The detunings are set to $\left[{\Delta }_{pa},{\Delta }_{pc}\right]<0$ for cavity cooling, and the pump field connects the absolute ground state of the joint system $|g⟩$ with $|e,0⟩$. For efficient cavity cooling, the magnitude of ${\Omega }_p$ is set so that the population of $|e,0⟩$ is low, suppressing both incoherent scattering to $|g⟩$ and Raman scattering to the secondary ground states, collectively represented by $|b,0⟩$. With these detunings, the molecule(s)-cavity coupling $g$ mixes the excited states to produce the dressed states $|\pm ⟩$. The lower dressed state $|-⟩$ may be viewed as a polariton excitation, as it is mostly comprised of the cavity excitation.

Figure 6

OH level diagram depicting the $P_1\left(1\right)$, $Q_1\left(1\right)$, and $Q_{21}\left(1\right)$ electronic transitions. The first excited level $A{{}^2\Sigma }^{+}$ and the two $X{}^2\Pi$ ground states are shown. For clarity, only the $P_1\left(1\right)$ decay channels are shown: $P_{12}\left(1\right)$, $O_{12}\left(2\right)$, and the ${\nu }^{\prime }=0\to {\nu }^{″}\ne 0$ transitions.