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Journal of Physics B Atomic Molecular and Optical Physics

Atomic momentum diffusion in a σ+-σ-laser configuration: influence of an internal sublevel structure

Yvan Castin, Klaus Mølmer

Published Nov 28, 1990• 26 citations
Publisher

Abstract

In relation to the newly discovered laser cooling mechanisms, the authors have studied the momentum diffusion of an atom with a j (ground)-j+1 (excited) state transition (j not=0), interacting with two counterpropagating sigma +- sigma - plane waves. A correlation between successive photon absorptions via the multi-sublevel internal atomic structure leads to a large increase of the momentum diffusion coefficient. In a simple probabilistic picture this enhanced diffusion is interpreted as a random walk with effective steps much larger than the one-photon recoil momentum k. For j=1 the authors report the results of a complete calculation (i.e. with a quantum description of the internal atomic state) showing a marked dependence upon the velocity of the atom as well as on the parameters of the laser field (detuning and power).

DOI
10.1088/0953-4075/23/22/012

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Journal of Physics B Atomic Molecular and Optical Physics•Oct 28, 1994•5 citations
In relation to the newly discovered laser cooling mechanisms, the authors have studied the momentum diffusion of an atom with a j (ground)-j+1 (excited) state transition (j not=0), interacting with two counterpropagating sigma +- sigma - plane waves. A correlation between successive photon absorptions via the multi-sublevel internal atomic structure leads to a large increase of the momentum diffusion coefficient. In a simple probabilistic picture this enhanced diffusion is interpreted as a random walk with effective steps much larger than the one-photon recoil momentum k. For j=1 the authors report the results of a complete calculation (i.e. with a quantum description of the internal atomic state) showing a marked dependence upon the velocity of the atom as well as on the parameters of the laser field (detuning and power).
In the process of cooling and trapping neutral atoms, the authors investigate a new laser configuration, which consists of two counterpropagating laser beams with orthogonal sigma + and sigma - polarisations. They show that such a configuration looks more promising than an ordinary standing wave (where the two counterpropagating waves have the same polarisation), and they explain this result as being due to angular momentum conservation which prevents any coherent redistribution of photons between the two waves. The authors' conclusions are based on a quantitative calculation of the various parameters (potential depth, friction coefficient, diffusion coefficient) describing the mean value and the fluctuations of the radiative forces experienced, in such a laser configuration, by an atom with a J=0 ground state and a J=1 excited state.
We extend the kinetic theory of deep laser cooling of multilevel atoms by counterpropagating laser waves to a case of linearly polarized waves and compare the results with those for circularly polarized waves. Considering the excitation of a $(3+5)\ensuremath{-}\mathrm{level}$ atom by two counterpropagating waves with orthogonal linear polarizations $(\mathrm{lin}\ensuremath{\perp}\mathrm{lin}$ configuration), we show that even-order stepwise processes composed of the two-photon optical processes are responsible for narrow structures near zero velocity in the ground-state atomic coherence, atomic populations, and optical coherences. The stepwise processes composed of the two-photon structures are shown to produce velocity structures much narrower than the two-photon structures produced by a single ${\ensuremath{\sigma}}^{+}\ensuremath{-}{\ensuremath{\sigma}}^{\ensuremath{-}}$ configuration [Phys. Rev. A 60, 3148 (1999)]. A Fokker-Planck type kinetic equation is derived for a model of a $(3+5)\ensuremath{-}\mathrm{level}$ atom interacting with two linearly polarized waves in a $\mathrm{lin}\ensuremath{\perp}\mathrm{lin}$ configuration. It is found that stepwise multiphoton coherent processes enhance considerably the radiation force at low velocities and the friction produced by the force at a negative detuning. Contrary to a case of a ${\ensuremath{\sigma}}^{+}\ensuremath{-}{\ensuremath{\sigma}}^{\ensuremath{-}}$ configuration the stepwise processes are found to increase the diffusion coefficient. The stepwise even-order processes caused by two linearly polarized waves are identified as the basic physical mechanism responsible for deep laser cooling of atoms in a $\mathrm{lin}\ensuremath{\perp}\mathrm{lin}$ configuration. The temperature of laser-cooled atoms is derived for a model of a $(3+5)\ensuremath{-}\mathrm{level}$ atom excited by a $\mathrm{lin}\ensuremath{\perp}\mathrm{lin}$ configuration and found to be asymptotically close to that for a case of ${\ensuremath{\sigma}}^{+}\ensuremath{-}{\ensuremath{\sigma}}^{\ensuremath{-}}$ configuration at large detuning.
The momentum diffusion coefficient, as a function of position and velocity, is discussed for two-state atoms travelling through a standing-wave laser field. Equations leading to the force and diffusion coefficient have been obtained by Minogin. The authors present an alternative derivation, based on moments of the Wigner function and give numerical solutions in terms of matrix continued fractions. In strong fields, the discussion is supplemented by a derivation of the diffusion coefficient within the dressed-atom basis, and a picture of the diffusion mechanism emerges which is quite different from the one at low intensities. Calculated forces and diffusion coefficients are used in a Fokker-Planck equation to obtain stationary atomic velocity distributions.
We extend the kinetic theory of the laser cooling of atoms to four multilevel atomic schemes: (3+5)-, (5+7)-, (7+9)-, and (9+11)-level atom. In all four atomic schemes the atoms are considered to be excited by two counterpropagating circularly polarized laser waves. A comparison of four multilevel atomic schemes shows that the even-order multiphoton processes are responsible for the narrow structures near zero velocity in the atomic coherences, atomic populations, dipole radiation forces, and the diffusion coefficients. The even-order multiphoton processes in a multilevel atom with a big number of magnetic sublevels are shown to produce the velocity structures, which are much narrower than that produced by the two-photon processes in a (3+5)-level atom. It is found that the multiphoton processes enhance the radiation force at low velocities and the friction produced by the force at a negative detuning. The even-order multiphoton processes caused by the laser waves are identified as the basic physical mechanism responsible for the sub-Doppler laser cooling of atoms. The temperature of laser-cooled atoms is derived for all four models and found to be lower for atomic schemes with a larger number of magnetic sublevels.
A theory of deep laser cooling of atoms by counterpropagating circular polarized laser waves in multilevel interaction schemes is developed. Using a (3+5)-level atom as an example, we show that in multilevel interaction schemes two-photon processes are responsible for creating a narrow structure near zero velocity in the ground-state atomic coherence. This structure is in turn responsible for narrow velocity structures in atomic populations and optical coherences. A Fokker-Planck-type kinetic equation is derived for a model of a (3+5)-level interaction scheme. It is found that two-photon coherent processes considerably enhance the radiation force at low velocities and the friction produced by the force at a negative detuning, and considerably decrease the diffusion coefficient. Two-photon processes in multilevel interaction schemes are identified as the basic physical mechanism responsible for deep laser cooling of atoms. The temperature of laser-cooled atoms is derived for a model (3+5)-level atom, and found to be in good agreement with the experimentally observed temperature for laser-cooled cesium atoms.