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Physical Review A

Three-dimensional theory of the magneto-optical trap: Doppler cooling in the low-intensity limit

Mariusz Gajda, Jan Mostowski

Published Jun 1, 1994• 17 citations
Publisher

Abstract

We derive a Fokker-Planck equation for the center-of-mass Wigner function of an atom moving in a Zeeman neutral-atom trap. The Zeeman structure of atomic sublevels is taken into account. Our analysis is limited to optical transitions between two sets of sublevels corresponding to angular momenta ${\mathit{j}}_{\mathit{b}}$=0 and ${\mathit{j}}_{\mathit{a}}$=1. Expressions for the light pressure, friction, and diffusion forces in the case of low intensity of the laser light are presented. An analytical solution for the stationary center-of-mass Wigner function as well as the size and geometry of the trap are found and discussed.

DOI
10.1103/physreva.49.4864

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We derive a Fokker-Planck equation for the center-of-mass Wigner function of an atom moving in a Zeeman neutral-atom trap. The Zeeman structure of atomic sublevels is taken into account. Our analysis is limited to optical transitions between two sets of sublevels corresponding to angular momenta jb{\mathit{j}}_{\mathit{b}}jb​=0 and ja{\mathit{j}}_{\mathit{a}}ja​=1. Expressions for the light pressure, friction, and diffusion forces in the case of low intensity of the laser light are presented. An analytical solution for the stationary center-of-mass Wigner function as well as the size and geometry of the trap are found and discussed.
In the limit of asymptotically small photon momentum, a Fokker–Planck equation for the center-of-mass Wigner function suffices to describe laser cooling of an atom. We have devised a procedure to compute the light pressure force and the diffusion tensor for an atom with an arbitrary level structure, traveling with an arbitrary velocity in any given three-dimensional light field. The scheme is based on solving a set of differential equations closely related to the density matrix equations of motion encountered in conventional recoilless laser spectroscopy. Computer programs have been written that automatically construct and solve the required equations from broad specifications of the problem, such as energies and angular momenta of the atomic levels. A Langevin equation stochastic simulation has been implemented to solve the Fokker-Planck equation of laser cooling using the computed force and diffusion. Comparisons of optical-molasses experiments and simulations carried out for a model sodium atom with a J = 2→J = 3 transition show qualitative similarities such as sub-Doppler limit temperatures, but the quantitative agreement of theory and experiment is not satisfactory.
We derive an equation of motion for the center-of-mass Wigner function (WF) of a two-level atom interacting with a linearly polarized, monochromatic standing plane wave of light. The derivation is based on a truncation of the spatial Fourier expansion of the WF and is consequently valid only at low intensities of the driving light, but no assumption regarding the size of the recoil velocity need be made. The stationary state of laser cooling is studied analytically and numerically. The asymptotic velocity distribution at large velocities is found to obey a power law, with a power independent of the linewidth of the optical transition.
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.
Derives a Fokker-Planck equation for the Wigner function describing the centre-of-mass motion of a trapped ion in a monochromatic travelling wave of light. The driving term and the coefficients of diffusion are found by solving certain sets of ordinary differential equations. In the low-intensity limit, the author investigates the rate of change of the mechanical energy of the ion and analyse a linearised Fokker-Planck equation valid in the vicinity of the steady state. The quantum mechanical properties of the harmonic oscillator implicit in the equations for the Wigner functions are discussed.
Magnesium atoms are cooled in a magneto-optical trap (MOT) using the 3s(2 1)S(0) --> 3s3p P-1(1) resonance transition. Magnesium is a simple atom which offers a unique possibility for comparison to the Doppler theory of laser cooling. We measure trap parameters as a function of laser-intensity, -detuning and magnetic field gradient. We find the main features well accounted for by Doppler theory, but temperature measurements gave significant higher values than predicted by the Doppler theory. We also observe radiation pressure effects within the MOT that limit the maximum achievable density. Comparing our results to predictions by models in the literature shows a good agreement. Recently we have improved our set-up and now trap about 150 x 10(6) atoms.
Magnesium atoms are cooled in a magneto-optical trap (MOT) using the 3s2 S0 → 3s3p P1 resonance transition. Magnesium is a simple atom which offers a unique possibility for comparison to the Doppler theory of laser cooling. We measure trap parameters as a function of laser-intensity, -detuning and magnetic field gradient. We find the main features well accounted for by Doppler theory, but temperature measurements gave significant higher values than predicted by the Doppler theory. We also observe radiation pressure effects within the MOT that limit the maximum achievable density. Comparing our results to predictions by models in the literature shows a good agreement. Recently we have improved our set-up and now trap about 150 × 106 atoms.