Magneto-optical trapping forces for atoms and molecules with complex level structures

Laser cooling and magneto-optical trapping of molecules typically involves multiple transitions driven by several laser frequencies. We analyze how magneto-optical trapping forces depend on the angular momenta, $F_l$ and $F_u$, and the g-factors, $g_l$ and $g_u$, of the lower and upper states. When $F_l > F_u$ the polarizations must be reversed relative to cases where $F_u \ge F_l$. The correct choice of circular polarization depends on the sign of $g_{u}$ but not on the sign of $g_{l}$. If $g_{u}$ is zero there is no trapping force, and the trapping force is very weak whenever $g_u$ is small compared to $g_l$, which it usually is when the cooling transition is the $^{2}\Sigma$ to $^{2}\Pi_{1/2}$ transition of a molecule. For some molecules, mixing of the excited $^{2}\Pi_{1/2}$ state with a nearby $^{2}\Sigma$ excited state can greatly increase $g_u$, leading to stronger trapping forces. A strong trapping force can also be produced by rapidly and synchronously reversing both the magnetic field and the laser polarizations. We simulate a recent experiment on magneto-optical trapping of SrF molecules, and suggest that an alternative choice of laser beam polarizations will strengthen the trapping force.Comment: 20 pages, 9 figures. Minor changes to text. Part (c) added to figure 1 and first line of table 2 revise

Stark deceleration of lithium hydride molecules

We describe the production of cold, slow-moving LiH molecules. The molecules are produced in the ground state using laser ablation and supersonic expansion, and 68% of the population is transferred to the rotationally excited state using narrowband radiation at the rotational frequency of 444GHz. The molecules are then decelerated from 420m/s to 53m/s using a 100 stage Stark decelerator. We demonstrate and compare two different deceleration modes, one where every stage is used for deceleration, and another where every third stage decelerates and the intervening stages are used to focus the molecules more effectively. We compare our experimental data to the results of simulations and find good agreement. These simulations include the velocity dependence of the detection efficiency and the probability of transitions between the weak-field seeking and strong-field seeking quantum states. Together, the experimental and simulated data provide information about the spatial extent of the source of molecules. We consider the prospects for future trapping and sympathetic cooling experiments.Comment: 14 pages, 6 figures; minor revisions following referee suggestion

Three-dimensional Doppler, polarization-gradient, and magneto-optical forces for atoms and molecules with dark states

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, v₀, where the two forces cancel. We find that v₀² 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

Three-dimensional Doppler, polarization-gradient, and magneto-optical forces for atoms and molecules with dark states

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.Comment: 20 pages, 7 figures. Revised version to correct several small typographical errors and clarify the discussion on page 9. Labeling of figure 1 and colours in figure 5 also changed, and additional information provided for equations 13 and 1

Modeling magneto-optical trapping of CaF molecules

Magneto-optical trapping forces for molecules are far weaker than for alkali atoms because the photon scattering rate is reduced when there are multiple ground states, and because of optical pumping into dark states. The force is further reduced when the upper state has a much smaller Zeeman splitting than the lower state. We use a rate model to estimate the strength of the trapping and damping forces in a magneto-optical trap (MOT) of CaF molecules, using either the A$^{2}\Pi_{1/2}$ - X$^{2}\Sigma^{+}$ transition or the B$^{2}\Sigma^{+}$ - X$^{2}\Sigma^{+}$ transition. We identify a new mechanism of magneto-optical trapping that arises when, in each beam of the MOT, two laser components with opposite polarizations and different detunings address the same transition. This mechanism produces a strong trapping force even when the upper state has little or no Zeeman splitting. It is the main mechanism responsible for the trapping force when the A$^{2}\Pi_{1/2}$ - X$^{2}\Sigma^{+}$ transition is used.Comment: 8 pages, 6 figures. Supplemental Material containing 7 figure

Measurement of the lowest millimetre-wave transition frequency of the CH radical

The CH radical offers a sensitive way to test the hypothesis that fundamental constants measured on earth may differ from those observed in other parts of the universe. The starting point for such a comparison is to have accurate laboratory frequencies. Here we measure the frequency of the lowest millimetre-wave transition of CH, near 535 GHz, with an accuracy of 0.6 kHz. This improves the uncertainty by roughly two orders of magnitude over previous determinations and opens the way for sensitive new tests of varying constants.Comment: 5 pages, 5 figure

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

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

Stark deceleration of CaF molecules in strong- and weak-field seeking states

We report the Stark deceleration of CaF molecules in the strong-field seeking ground state and in a weak-field seeking component of a rotationally-excited state. We use two types of decelerator, a conventional Stark decelerator for the weak-field seekers, and an alternating gradient decelerator for the strong-field seekers, and we compare their relative merits. We also consider the application of laser cooling to increase the phase-space density of decelerated molecules.Comment: 10 pages, 8 figure

Quantum computation in a hybrid array of molecules and Rydberg atoms

We show that an array of polar molecules interacting with Rydberg atoms is a promising hybrid system for scalable quantum computation. Quantum information is stored in long-lived hyperfine or rotational states of molecules which interact indirectly through resonant dipole-dipole interactions with Rydberg atoms. A two-qubit gate based on this interaction has a duration of 1 $\mu$s and an achievable fidelity of 99.9%. The gate is insensitive to the motional states of the particles -- the molecules can be in thermal states, the atoms do not need to be trapped during Rydberg excitation, the gate does not heat the molecules, and heating of the atoms is irrelevant. Within a large, static array, the gate can be applied to arbitrary pairs of molecules separated by tens of micrometres, making the scheme highly scalable. The molecule-atom interaction can also be used for rapid qubit initialization and efficient, non-destructive qubit readout, without driving any molecular transitions. Single qubit gates are driven using microwave pulses alone, exploiting the strong electric dipole transitions between rotational states. Thus, all operations required for large scale quantum computation can be done without moving the molecules or exciting them out of their ground electronic states.Comment: 16 pages, 7 figure