Laser cooling and magneto-optical trapping of molecules analyzed using optical Bloch equations and the Fokker-Planck-Kramers equation

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Abstract
We study theoretically the behavior of laser-cooled calcium monofluoride (CaF) molecules in an optical molasses and magneto-optical trap (MOT), and compare our results to recent experiments. We use multi-level optical Bloch equations to estimate the force and the diffusion constant, followed by a Fokker-Planck-Kramers equation to calculate the time-evolution of the velocity distribution. The calculations are done in three-dimensions, and we include all the relevant energy levels of the molecule and all the relevant frequency components of the light. Similar to simpler model systems, the velocity-dependent force curve exhibits Doppler and polarization-gradient forces of opposite signs. We show that the temperature of the MOT is governed mainly by the balance of these two forces. Our calculated MOT temperatures and photon scattering rates are in broad agreement with those measured experimentally over a wide range of parameters. In a blue-detuned molasses, the temperature is determined by the balance of polarization gradient cooling, and heating due to momentum diffusion, with no significant contribution from Doppler heating. In the molasses, we calculate a damping rate similar to the measured one, and steady-state temperatures that have the same dependence on laser intensity and applied magnetic field as measured experimentally, but are consistently a few times smaller than measured. We attribute the higher temperatures in the experiments to fluctuations of the dipole force which are not captured by our model. We show that the photon scattering rate is strongly influenced by the presence of dark states in the system, but that the scattering rate does not go to zero even for stationary molecules because of the transient nature of the dark states.

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