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Physical Review A

Three-dimensional analysis of the magneto-optical trap for (1+3)-level atoms

Sang-Kyung Choi, Sang Eon Park, Jingbiao Chen, Vladimir G. Minogin

Published Jan 30, 2008• 18 citations
Publisher

Abstract

We describe a three-dimensional model of the magneto-optical trap (MOT) for the simplest case of (1+3)-level atoms. The atomic density matrix equations are solved up to eighth order in the Rabi frequency and compared with experimental observations. We propose that coherences between the excited state sublevels can largely explain the dependence of the atomic temperature on detuning and light intensity.

DOI
10.1103/physreva.77.015405

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We describe a three-dimensional model of the magneto-optical trap (MOT) for the simplest case of (1+3)-level atoms. The atomic density matrix equations are solved up to eighth order in the Rabi frequency and compared with experimental observations. We propose that coherences between the excited state sublevels can largely explain the dependence of the atomic temperature on detuning and light intensity.
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.
The kinetics of atoms in a three-dimensional magneto-optical trap (MOT) is considered. A three-dimensional MOT model has been constructed for an atom with the optical transition J g = 0 → J e = 1 (J g, e is the total angular momentum in the ground and excited states) in the semiclassical approximation by taking into account the influence of the relative phases of light fields on the kinetics of atoms. We show that the influence of the relative phases can be neglected only in the limit of low light field intensities. Generally, the choice of relative phases can have a strong influence on the kinetics of atoms in a MOT.
Since the first proposal and the first experiment on magneto-optical trapping of atoms, magneto-optical traps (MOT) became the most popular traps for cold atoms. Despite the numerous successful applications of MOT it is still unclear what ultimate parameters could be expected for atoms trapped in MOT. We develop the theory of MOT for cold atoms under the multilevel dipole interaction schemes. Theory is based on consideration of one-dimensional motion of atoms which have three ground-state sublevels (F/sub g/=1) and five upper-state sublevels (F/sub e/=2) and are in inhomogeneous magnetic field and in the field of two counter-propagating circular polarized laser waves. It is shown that in multilevel interaction schemes two-photon processes are responsible for formation of a narrow velocity and spatial structure in the radiation force. It is found that two-photon coherent process enhances considerably the friction force and the oscillation frequency at negative detuning. Single-photon and two-photon processes are shown to produce the two-component structure of the MOT potential and accordingly the two-component coordinate distribution.
We present a theoretical study of magneto-optical trapping (MOT) force exerted on magnesium monofluoride (MgF) with three-dimensional rate equations, in which we have considered the complex vibrational and rotational levels and the effects of small internal splittings and degeneracies, including fine and hyperfine structures and the magnetic quantum numbers. We investigate the feasibility of MOT for MgF with a very small excited-state ggg factor $(\phantom{\rule{4pt}{0ex}}{g}_{e}=\ensuremath{-}0.0002)$ and a large radiative decay rate $(\mathrm{\ensuremath{\Gamma}}=2\ensuremath{\pi}\ifmmode\times\else\texttimes\fi{}22\phantom{\rule{0.16em}{0ex}}\mathrm{MHz})$ for the electronic transition of $X{\phantom{\rule{0.16em}{0ex}}}^{2}{\mathrm{\ensuremath{\Sigma}}}^{+}$ to $A{\phantom{\rule{0.16em}{0ex}}}^{2}{\mathrm{\ensuremath{\Pi}}}_{1/2}$ states. We also optimize the MOT with reference to the three-, four-, and more-frequency component models with various polarization configurations and detunings. By applying the dual-frequency arrangement to more than one hyperfine level, we suggest a configuration of the (3+1)(3+1)(3+1)-frequency components for achieving the MOT of MgF.
We have derived ab initioanalytical results for the shape of the double-structure potential well for a (3+5)-level atoms in the magneto-optical trap (MOT). We show that for a case of a (3+5)-level atom and a one-dimensional ${\ensuremath{\sigma}}^{+}\ensuremath{-}{\ensuremath{\sigma}}^{\ensuremath{-}}$ laser field, the sharp bottom part of the MOT potential well results from the two-photon processes between the atomic ground-state magnetic sublevels. These nonlinear processes are, in particular, responsible for atomic temperatures below the Doppler limit which is due to the one-photon processes in the MOT. We also find that in more complicated atomic schemes, even-order multiphoton processes contribute to the sharp and narrow MOT potential well.
A magneto-optical trap (MOT) of atoms or molecules is studied when two lasers of different detunings and polarization are used. Especially for $J\ensuremath{\rightarrow}J,J\ensuremath{-}1$ transitions, a scheme using more than one frequency per transition and different polarization is required to create a significant force. Calculations have been performed with the simplest forms of the $J\ensuremath{\rightarrow}J\ensuremath{-}1$ case (i.e., ${J}^{\ensuremath{'}\ensuremath{'}}=1\ensuremath{\rightarrow}{J}^{\ensuremath{'}}=0$) and $J\ensuremath{\rightarrow}J$ case (i.e., ${J}^{\ensuremath{'}\ensuremath{'}}=1/2\ensuremath{\rightarrow}{J}^{\ensuremath{'}}=1/2$). A one-dimensional (1D) model is presented and a complete 3D simulation using rate equations confirms the results. Even in the absence of Zeeman effect in the excited state, where no force is expected in the single laser field configuration, we show that efficient cooling and trapping forces are restored in our configuration. We study this mechanism for the ${\mathrm{C}}_{2}^{\ensuremath{-}}$ molecular anion as a typical example of the interplay between the two simple transitions $J\ensuremath{\rightarrow}J,J\ensuremath{-}1$.