Modeling sympathetic cooling of molecules by ultracold atoms

We model sympathetic cooling of ground-state CaF molecules by ultracold Li and Rb atoms. The molecules are moving in a microwave trap, while the atoms are trapped magnetically. We calculate the differential elastic cross sections for CaF-Li and CaF-Rb collisions, using model Lennard-Jones potentials adjusted to give typical values for the $s$-wave scattering length. Together with trajectory calculations, these differential cross sections are used to simulate the cooling of the molecules, the heating of the atoms, and the loss of atoms from the trap. We show that a hard-sphere collision model based on an energy-dependent momentum transport cross section accurately predicts the molecule cooling rate but underestimates the rates of atom heating and loss. Our simulations suggest that Rb is a more effective coolant than Li for ground-state molecules, and that the cooling dynamics is less sensitive to the exact value of the $s$-wave scattering length when Rb is used. Using realistic experimental parameters, we find that molecules can be sympathetically cooled to $100\mu \mathrm{K}$ in about 10 s. By applying evaporative cooling to the atoms, the cooling rate can be increased and the final temperature of the molecules can be reduced to $1 \mu \mathrm{K}$ within 30 s.

Figure 1

Total elastic cross section ${\sigma }_{\text{el}}$ (solid lines) and transport cross section ${\sigma }_{\eta }^{\left(1\right)}$ (dashed lines) for positive (black) and negative (red/gray) signs of the scattering length. (a) CaF+${}^{87}\mathrm{Rb},\left|a\right|=1.5\overline{a}$; (b) CaF+${}^{87}\mathrm{Rb},\left|a\right|=0.5\overline{a}$; (c) CaF+${}^7\mathrm{Li},\left|a\right|=1.5\overline{a}$; (d) CaF+${}^7\mathrm{Li},\left|a\right|=0.5\overline{a}$.

Figure 2

Results of various collision models: (i) hard-sphere model with energy-independent cross section $4\pi {\overline{a}}^2$; (ii) full energy-dependent differential cross-section model; (iii) hard-sphere model with ${\sigma }_{\text{el}}\left(E^{\text{cm}}\right)$; (iv) hard-sphere model with ${\sigma }_{\eta }^{\left(1\right)}\left(E^{\text{cm}}\right)$; (v) hard-sphere model with classical approximation to ${\sigma }_{\eta }^{\left(1\right)}\left(E^{\text{cm}}\right)$. The graphs show (a) cross section versus collision energy; (b) fraction of molecules with kinetic energy below 10 mK versus time; (c) mean kinetic energy of that fraction versus time. The coolant is Rb and $a=+1.5\overline{a}$.

Figure 3

Cooling rate of molecules as a function of their kinetic energy, estimated from Eq. (9), when the coolant is (a) Rb and (b) Li, and for various values of the $s$-wave scattering length: $a=+1.5\overline{a}$ (red solid line), $a=+0.5\overline{a}$ (blue dashed-dotted line), $a=-0.5\overline{a}$ (green dotted line), and $a=-1.5\overline{a}$ (black dashed line).

Figure 4

Time evolution of the phase-space distribution of molecules in the $x$ direction. The cooling times are (a) 0, (b) 2, (c) 10, and (d) 20 s. The coolant is Rb and $a=+1.5\overline{a}$.

Figure 5

Kinetic energy distributions after 2, 10, and 20 s. The coolant is Rb. Left panels have $a=+1.5\overline{a}$ while right panels have $a=-1.5\overline{a}$.

Figure 6

Kinetic energy distributions after 2, 10, 20, and 40 s. The coolant is Li. Left panels have $a=+1.5\overline{a}$ while right panels have $a=-1.5\overline{a}$.

Figure 7

Fraction of molecules with kinetic energy below 10 mK as a function of time for (a) Rb and (b) Li, for four different values of the scattering lengths: $a=+1.5\overline{a}$ (red), $a=+0.5\overline{a}$ (blue), $a=-0.5\overline{a}$ (green), and $a=-1.5\overline{a}$ (black).

Figure 8

Mean kinetic energy of the cold fraction as a function of time when the coolant is (a) Rb and (b) Li, and for various values of the $s$-wave scattering length: $a=+1.5\overline{a}$ (red), $a=+0.5\overline{a}$ (blue), $a=-0.5\overline{a}$ (green), and $a=-1.5\overline{a}$ (black).

Figure 9

Fraction of cold molecules as a function of time for the initial temperatures of $T_{\text{i}}=20\mathrm{mK}$ (red solid line) and $T_{\text{i}}=70\mathrm{mK}$ (black dashed line). The coolant is Rb and $a=+1.5\overline{a}$.

Figure 10

(a) Atom heating rate per molecule and (b) atom loss rate per molecule for the EDT-HS model (dashed line) and the full DCS model (solid line). The coolant is Rb, $a=+1.5\overline{a}$, and the molecules have an initial temperature of 70 mK. There are ${10}^5$ molecules and ${10}^9$ atoms.

Figure 11

Differential cross sections and their contributions to heating and loss. The solid black line shows the full quantum mechanical $d\sigma /d\omega$, while the solid red line shows $(1-cos\mathrm{\Theta })d\sigma /d\omega$; the dashed lines show the corresponding quantities for the EDT-HS model. The vertical line shows the value of ${\mathrm{\Theta }}_{\text{crit}}$. (a) $E_{\text{CaF}}^{\text{lab}}=2\mathrm{mK}$. (b) $E_{\text{CaF}}^{\text{lab}}=20\mathrm{mK}$. The coolant is Rb and $a=+1.5\overline{a}$.

Figure 12

Loss (red) and heating (blue) cross sections as a function of CaF laboratory energy for the EDT-HS model (dashed lines) and the full DCS model (solid lines). ${\sigma }_{\text{el}}$ (solid black line) and ${\sigma }_{\eta }^{\left(1\right)}$ (dashed black line) are shown for comparison. The coolant is Rb and $a=+1.5\overline{a}$.

Figure 13

Kinetic energy distributions at four different times (2, 10, 20, and 50 s) and for two values of the evaporative cooling parameter: (a) $\eta =5.52$, (b) $\eta =8.14$. For comparison, a Maxwell-Boltzmann distribution at $1 \mu \mathrm{K}$ is shown by a red line. The coolant is Rb and $a=+1.5\overline{a}$.

Figure 14

Sympathetic cooling of molecules with evaporative cooling applied to the atoms. Graphs show the time evolution of (a) the fraction of molecules with kinetic energy below 1 mK, when $a=+1.5\overline{a}$; (b) the mean kinetic energy of the ultracold fraction when $a=+1.5\overline{a}$; (c) the mean kinetic energy of the ultracold fraction when $a=-0.5\overline{a}$. (i, black) $\eta =5.52$, (ii, red) $\eta =6.67$, (iii, blue) $\eta =8.14$. In (b) and (c), the dashed lines show how the atomic temperature evolves. The long-dashed green line shows the atom temperature without evaporative cooling.