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 multilevel 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.
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 multilevel 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.
In red-detuned magneto-optical traps (MOTs) of molecules, sub-Doppler heating competes with Doppler cooling, resulting in high temperature and low density. A solution is offered by the blue-detuned MOT where sub- Doppler cooling dominates and the cloud is compressed. Several blue-detuned molecular MOTs have been implemented. A recent implementation relies on a pair of orthogonally polarized components whose frequency separation is smaller than the transition linewidth. We identify the trapping force in these MOTs. At a certain magnetic field, there is a state that is dark to the laser propagating in one direction, but not to the counterpropagating one. This Zeeman-induced dark state (ZIDS) sets up an imbalance in the photon scattering rate, leading to a restoring force. We also study the role of the moving lattices generated by the closely spaced frequency components of the light. We show that there is a velocity-dependent force that drives the molecules toward the speeds of these moving lattices, and that over a relevant range of magnetic fields this combines with the ZIDS force to transport molecules toward the center of the MOT. Here, gray molasses cooling, assisted by nonadiabatic transitions driven by the time-varying polarization of the light field, cools the molecules toward zero velocity. We study these mechanisms for model systems with simple level structures and then extend them to molecules with ground-state hyperfine structure.
We theoretically investigate the damping and trapping forces in a three-dimensional magneto-optical trap (MOT), by numerically solving the optical Bloch equations.We focus on the case where there are dark states because the atom is driven on a 'type-II' system where the angular momentum of the excited state, F¢, is less than or equal to that of the ground state, F. For these systems we find that the force in a three-dimensional light field has very different behaviour to its one dimensional counterpart.This differs from the more commonly used 'type-I' systems (F F 1 ¢ = + ) where the 1D and 3D behaviours are similar.Unlike type-I systems where, for red-detuned light, both Doppler and sub-Doppler forces damp the atomic motion towards zero velocity, in type-II systems in 3D, the Doppler force and polarization gradient force have opposite signs.As a result, the atom is driven towards a non-zero equilibrium velocity, v 0 , where the two forces cancel.We find that v 0 2 scales linearly with the intensity of the light and is fairly insensitive to the detuning from resonance.We also discover a new magneto-optical force that alters the normal MOT force at low magnetic fields and whose influence is greatest in the type-II systems.We discuss the implications of these findings for the laser cooling and magneto-optical trapping of molecules where type-II transitions are unavoidable in realising closed optical cycling transitions.
We present the properties of a magneto-optical trap (MOT) of CaF molecules. We study the process of loading the MOT from a decelerated buffer-gas-cooled beam, and how best to slow this molecular beam in order to capture the most molecules. We determine how the number of molecules, the photon scattering rate, the oscillation frequency, damping constant, temperature, cloud size and lifetime depend on the key parameters of the MOT, especially the intensity and detuning of the main cooling laser. We compare our results to analytical and numerical models, to the properties of standard atomic MOTs, and to MOTs of SrF molecules. We load up to molecules, and measure a maximum scattering rate of s −1 per molecule, a maximum oscillation frequency of 100 Hz, a maximum damping constant of 500 s −1 , and a minimum MOT rms radius of 1.5 mm. A minimum temperature of 730 μ K is obtained by ramping down the laser intensity to low values. The lifetime, typically about 100 ms, is consistent with a leak out of the cooling cycle with a branching ratio of about . The MOT has a capture velocity of about 11 m s −1 .
Laser cooling and trapping in a magneto-optical trap (MOT) have been essential to the success of cold atom physics in the last decades. Recently, the application of the same techniques to molecules has begun. The complexity of even a simple diatomic molecule makes laser cooling difficult, but promises new applications in many areas of research. In this thesis I describe the development of the first three-dimensional MOT of calcium fluoride (CaF) molecules. First, a cryogenic buffer gas source was set up, producing a pulsed beam of 9.3*10^10 molecules per steradian per pulse with forward velocities around 170 m/s. A similar source for very large molecules was set up during a 5 month internship at the University of Vienna. Next, the molecular pulse was slowed down to the capture velocity of a MOT using chirped laser slowing, resulting in about 7*10^5 CaF molecules passing through the typical MOT volume of 1 cm^3 at velocities of 15+-5 m/s. A new deceleration method, called Zeeman-Sisyphus deceleration, was also investigated. In this method molecules move through a spatially varying magnetic field and are optically pumped between low- and high-field seeking states in such as a way that they are always losing kinetic energy. The method promises to deliver more slow molecules because the molecules are guided transversely as they are decelerated. A small prototype was built and the optical pumping step was tested successfully. Finally, 7.6*10^3 CaF molecules were trapped in a MOT and cooled to a temperature of 8.5 mK. The radial trap frequency is 2 pi*130 Hz and the damping constant is beta=9.5*10^2 s^-1. The lifetime is about 100 ms and depends strongly on the scattering rate. This MOT is an an ideal starting point for a wide range of new experiments with ultracold molecules.
From studies of fundamental physics to quantum technologies the production of ultracold molecules will have a huge impact across a range of applications. For many years laser cooling, which became an invaluable tool in cold atomic physics, was deemed to be too impractical for application to molecules. Nevertheless, laser cooling has now been demonstrated for a few molecular species. Using a frequency-chirped laser slowing technique, the velocity distribution of a pulse of CaF molecules is compressed and slowed from 180 m/s to about 10 m/s. These slow molecules are then captured in a magneto-optical trap. I present measurements that show how the number of molecules, the photon scattering rate, the oscillation frequency, damping constant, temperature, cloud size and lifetime depend on the key parameters of the magneto-optical trap, especially the intensity and detuning of the main cooling laser. The trap contains up to 2*10^4 molecules, the maximum photon scattering rate is 2.5*10^6 s-1 per molecule, the maximum oscillation frequency is 100 Hz, the maximum damping constant is 500 s^-1, and the minimum rms radius of the trapped cloud is 1.5 mm. A minimum temperature of 730 microkelvin is obtained by ramping down the laser intensity to lower values. To reach lower temperatures, the cloud is loaded into a blue-detuned optical molasses, which cools the molecules to 55 microkelvin, well below the Doppler-limiting temperature. I characterise the cooling process and suggest the sub-Doppler mechanisms responsible. These ultracold molecules are the optically pumped into a single quantum state, and coherently transferred between selected hyperfine components of the ground and first-excited rotational states. Finally, the ultracold, state-selected molecules are loaded into a magnetic trap that has a lifetime of about 1 s.
We investigate the effect of a longitudinal magnetic field on the orientational cooling mechanism that operates both in σ+–σ− optical molasses and in a magneto-optical trap. For an atom with different Landé g factors in its ground and excited states, single-photon scattering and stimulated Raman scattering are resonant at different field-dependent velocities. The sharp polarization-gradient feature in the force-versus-velocity curve may therefore be translated to a velocity at which the Doppler force is dominant. In addition to this, we find that the momentum diffusion coefficient can grow as the field is increased. By numerically solving a semiclassical Fokker–Planck equation, we show that these effects lead to a field-induced inhibition of efficient sub-Doppler cooling for atoms with unequal g factors. Our treatment also enables us to discuss other phenomena that are accessible only if both Doppler and polarization-gradient effects are included in the calculation.