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Journal of Physics B Atomic Molecular and Optical Physics

Fokker-Planck analysis of atomic beam cooling by frequency chirp methods

Hartmut Wallis, Wolfgang A. Ertmer

Published Sep 14, 1988• 8 citations
Publisher

Abstract

Atomic beam deceleration and cooling by frequency sweep techniques is analysed by means of a Fokker-Planck equation (FPE) with time dependent drift and diffusion coefficients. The effect of compression of the velocity distribution is discussed with respect to theoretical and experimental limitations of atomic beam cooling imposed by the photon recoil. The cooling limit derived from the linear approximation of the FPE fails, if the broadening of the cooled velocity distribution peak by diffusion causes the loss of atoms from the resonant velocity range. This phenomenon is studied by numerical solutions of the FPE. The occurrence of higher final temperatures in previous Monte-Carlo calculations and in experiments is discussed. The anomalous diffusion coefficient caused by the antibunching of the scattered photons is taken into account. The possibility of transverse compression of the atomic beam is investigated by means of a reduced FPE for the transverse degrees of freedom.

DOI
10.1088/0953-4075/21/17/013

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Atomic beam deceleration and cooling by frequency sweep techniques is analysed by means of a Fokker-Planck equation (FPE) with time dependent drift and diffusion coefficients. The effect of compression of the velocity distribution is discussed with respect to theoretical and experimental limitations of atomic beam cooling imposed by the photon recoil. The cooling limit derived from the linear approximation of the FPE fails, if the broadening of the cooled velocity distribution peak by diffusion causes the loss of atoms from the resonant velocity range. This phenomenon is studied by numerical solutions of the FPE. The occurrence of higher final temperatures in previous Monte-Carlo calculations and in experiments is discussed. The anomalous diffusion coefficient caused by the antibunching of the scattered photons is taken into account. The possibility of transverse compression of the atomic beam is investigated by means of a reduced FPE for the transverse degrees of freedom.
The theory of the resonance radiation force is studied as it applies to the slowing of a two-level atom using a swept frequency laser. The theory is developed in detail for this case. A single partial differential equation called the Fokker-Planck equation is found to describe the atomic motion for many cases. This equation is solved numerically for a one dimensional geometry. Results from this numerical solution are found to be within 10% of experimental results. Analysis of this problem in one dimension leads to the prediction that for fixed amounts of frequency sweeping, a faster scan rate will yield smaller full width at half maximum (FWHM) spread in the velocity distribution for the atomic beam. This program also predicts that the atoms can be slowed arbitrarily close to 0 m/sec using this technique. The program used for this analysis is included. Keywords: Resonance radiation force; Slowing an atomic beam; Laser cooling of atoms; Fokker Planck equation; Thesis.
We develop an analytical approach to Doppler cooling of atoms by one- or two-photon transitions when the natural width of the excited level is so small that the process leads to a Doppler temperature comparable with the recoil temperature. In such a limit the usual Fokker–Planck equation for Doppler cooling does not correctly describe the cooling process. We propose a generalization of the Fokker–Planck approach and derive a new model that is able to reproduce the numerical results correctly, even very close to the recoil limit. Two cases of practical interest, one-photon Doppler cooling of strontium and two-photon Doppler cooling of hydrogen, are considered. We also treat the possibility of adding a controlled quenching of the sharp line to optimize the cooling process.
Based on the momentum method the Fokker-Planck equation is derived for a velocity distribution perturbed by monomode laser excitation (hole burning). If the used collision kernel applies to marked forward scattering and if the collision partner is assumed to be infinitely heavy, the FPE degenerates into the Legendre equation. While a proportional term accounts for the decay of atoms, an inhomogeneity accounts for their creation. The considered species is metastable argon ions. These ions are probably created by electronic ionization, while both electronic deexcitation and metastability exchange interfere in their destruction. The laser keeps the solution of the FPE at zero level for any velocity whose component along the laser beam upsilon z,0 follows from the laser frequency by the Doppler condition. We treat this constraint as a boundary condition for the solution. The latter is put up as a superposition of the Legendre function which solves the homogeneous part, with a special solution of the inhomogeneous equation. Free parameters are adjusted to the boundary condition. The thus obtained solution is finally integrated over all the velocity groups of atoms which under our assumptions are not coupled collisionally, to yield the expected shape of the signal. The velocity groups were assumed to be distributed according to Maxwell. We devised two types of experiments to measure the shape of the hole. Both are suited to measure holes on the wings as well as on the centre of the line. Remarkable similarity of the measured and the calculated signals is found.
The semiclassical dynamics of atoms, when the atoms are confined inside a standing-wave high-finesse resonator, is theoretically studied. The atoms are cooled by scattering processes in which the photons of a transverse laser are coherently scattered into the cavity mode. We derive a Fokker-Planck equation for the atomic center-of-mass variables which allows us to determine the equations of motion in the semiclassical limit for any value of the intensity of the laser field. We extract its prediction for the dynamics when the resonator is essentially in the vacuum state and the atoms are cooled by scattering photons into the cavity mode, which then decays. Its predictions for the stationary atomic distribution are compared with the ones of the Fokker-Planck equation by Domokos et al. [J. Phys. B 34, 187 (2001)], which has been derived under different assumptions. We find full agreement in the considered parameter regime.
We derive a Fokker-Planck equation (FPE) for the center-of-mass Wigner function of an atom with an arbitrary level structure, moving in a given three-dimensional light field. This FPE is obtained by eliminating adiabatically the internal degrees of freedom from the expansion of the equation of motion of the atomic density operator into a power series of photon momentum. The force and the diffusion tensor in the FPE may be computed by solving a set of differential equations closely related to the time-evolution equations of the density matrix for the internal state of the atom, as encountered in the conventional recoilless spectroscopy. By studying spontaneous, induced, and free-evolution terms in the equations of motion of the density operator for various level configurations, we devise algorithms that enable a computer to build all the required equations automatically for an arbitrary atomic level scheme.