Molecular cooling via Sisyphus processes

We present a study of Sisyphus cooling of molecules: The scattering of a single photon removes a substantial amount of the molecular kinetic energy and an optical pumping step allows one to repeat the process. A review of the produced cold molecules so far indicates that the method can be implemented for most of them, making it a promising method able to produce a large sample of molecules at sub-mK temperatures. Considerations of the required experimental parameters, for instance the laser power and linewidth or the trap anisotropy and dimensionality, are given. Rate equations, as well as scattering and dipolar forces, are solved using kinetic Monte Carlo methods for several lasers and several levels. For NH molecules, such detailed simulation predicts a 1000-fold temperature reduction and an increase of the phase-space density by a factor of 10${}^7$. Even in the case of molecules with both low Franck-Condon coefficients and a nonclosed pumping scheme, 60% of trapped molecules can be cooled from 100 mK to sub-mK temperatures in a few seconds. Additionally, these methods can be applied to continuously decelerate and cool a molecular beam.

Figure 1

Principle of Sisyphus cooling of molecules: (1) removal of kinetic energy through motion in an external potential; (2) “Sisyphus transfer step”: a dissipative process avoids the reverse motion; (3) this “one-way” (or “single-photon”) process can be repeated by bringing the molecule back in position using the trapping potential; and (4) in its original internal state using light absorption in a “repumping step” (symbolized by the shaped spectrum for amplitude selection). $r$ designs the radial coordinate. In 2D or 3D, due to the angular momentum the particles can miss the center, therefore $r_{\mathrm{repump}}$ could be nonzero.

Figure 2

Illustration of possible strategies to remove the kinetic energy from the sample. The upper part illustrates the process in a 1D picture, whereas the lower part indicates it in a 2D picture. The transfer position is indicated by a hatched area. Two kinds of trajectories, but with the same kinetic energy, are shown: the dashed red line corresponds to an isotropic trap case ${\omega }_x={\omega }_y$ and the solid line with ${\omega }_x=0.8{\omega }_y$. Left panel: The selection is done spatially (in the hatched region shown by the line or box) to catch the molecule where it has almost zero kinetic energy. Center panel: The selection is done spectrally depending on the potential energy of the particle when it has almost zero kinetic energy. Right panel: The selection is done depending on the velocity through the lifetime of the (absorption or spontaneous emission) transition.

Figure 3

Principle of NH Sisyphus cooling of molecules. The widths of the arrows are proportional to the branching ratios.

Figure 4

Results of NH Sisyphus cooling of molecules in a linear (quadrupole) trap with energy degeneracy at center. Molecules number $N$ (left axis) in the $M^{\prime \prime }=1$ (square) and $M^{\prime \prime }=0$ (circle) states as well as the temperature $T$ (triangle, right axis) in a function of time.

Figure 5

Results of NH Sisyphus cooling of molecules in a quadratic (Ioffe-Pritchard) anisotropic trap. Molecules number $N$ in the $M^{\prime \prime }=1$ and $M^{\prime \prime }=0$ states and temperature $T$ are displayed on the left axes. The relative phase-space density $D$ is displayed on the right axes in a function of time.

Figure 6

Branching ratios, proportional to the broadening of the blue lines, between vibrational (left) and rotational (right) levels after transfer by the narrow-band Sisyphus laser (dashed black) and the spectrally shaped amplitude selection broadband optical pumping (green hatched). $J^{\prime \prime }=2$ is not pumped for $v^{\prime \prime }=0$.

Figure 7

Time evolution of the (stacked) number $N$ of particles in the $|v^{\prime \prime }{J}^{\prime \prime }\rangle$ state and the (three axis) averaged temperature $T$ of the particles in ${|022\rangle }^{\prime \prime }$. The sequence is repeated every 1 s.

Figure 8

Illustration of optical deceleration of a molecular beam using Sisyphus cooling when molecules travel through magnetic coils. Left panel: illustration of the principle. Right panel: Simulation using pairs of 3-mm-radius Helmholtz coils separated by 1.5 cm and creating a 1 T axial field. The circularly polarized lasers are 5 W focused with 1.5 mm waist, 1 GHz laser linewidth. The monokinetic molecular beams cover 1 mm radially.

Figure 9

Effect of the trap anisotropy, on the minimum kinetic $E_{\mathrm{kin}}$ (lower panel for a quadratic trap) and potential energy $E_{\mathrm{pot}}$ (higher panel for a linear trap) reached after 5 ms (black square) or 20 ms (red circle) evolution time of $N=1000$ molecules initially at 100 mK. The worst energy case among the $N=1000$ molecules is plotted, in temperature units, in a function of $\alpha$. $1-\alpha$ can be seen as the aspect ratio of the trap (see text for details).

Figure 10

Typical evolution of number of molecules $N$, temperature $T$ (or energy $E$), phase-space density $D$, and time required $t$, for the sample under different $d$-dimensional cooling strategies (“Pos.” means a selection in position as in Fig. 2, case 1, “Ener.” a selection in energy as in Fig. 2, case 2, and “Speed” a selection with rate proportional to velocity as in Fig. 2, case 3). The values are given in the harmonic trap cases with angular frequencies initial ${\omega }_{\mathrm{in}}$ and final ${\omega }_{\mathrm{fin}}$. The partial one-way ones take only part of the population (with a box of given energy or spatial range). The full one-way strategies move the previous box to catch all particles. The first Sisyphus step (step 3 of Fig. 1) put the population back in the original potential. The Sisyphus is thus simply the repetition of this step. The parameters, except the photon numbers, are given in the power of $N_0=k_{\mathrm{B}}T/\Delta E$, which is a parameter indicating the number of slices of the sample over which we catch particles by spatial (or energy) steps. The time evolution is given, in the “$t$” line, in the function of the typical oscillation time $T_1$ along the fastest axis motion: $d-1$ indicates, for instance, that the time evolves like $t\propto T_1{N}_0^{d-1}$.

Figure 11

Schematics of levels $i$ and $j$ interacting with a broadband laser and with other levels.