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Journal of Physics B Atomic Molecular and Optical Physics

Legendre functions and the theory of hole burning

M. Elbel, H Welp

Published Oct 14, 1993• 2 citations

Abstract

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.

DOI: 10.1088/0953-4075/26/19/015

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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.

Applying the momentum method, the Fokker-Planck equation is derived from the Boltzmann equation for the r -4 potential between atoms and ions. Two earlier attempts are revealed to be limiting cases of our more general equation, namely for very large and very small mass ratio of the collision partners. The Fokker-Planck equation, after being completed with a decay term and a source term, is solved by the Green's function method. The solution turns out to be composed of modes in the velocity space which decay independently towards a stationary spectrum. The amplitudes of the latter depend on the source term which we assume to be shaped like a delta -function if laser pumping is accounted for. Only for very long relaxation times does the lowest mode prevail in the spectrum which then merges into the Maxwell distribution. Several new lines for ion-atom pairs as well as atom-atom pairs have been measured. They are presented along with the fits using the aforementioned formula. As expected, the agreement is satisfactory only in the first case. In the second case the Fokker-Planck approximation no longer applies.

A differential equation for the evolution of Bennet holes in a Maxwell distribution is derived on the basis of the Boltzmann equation. Characteristic forms of these holes are obtained with a sharp peak at the laser frequency and exponential wings attached to it. An experiment to verify the presence of these holes during optical pumping of metastable argon ions is reported here. Electro-optic modulation of the laser light represents a valuable method of overcoming the limit set by hole burning with regard to laser pumping.

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.

Investigations in the field of quantum optics very often requires a molecular simulation of transport phenomena. This paper presents a discrete kinetic model to deal with the quantum optical problem of hole burning. Based on a full discretization of the velocity space, Boltzmann-like equations are established to govern the temporal evolution of heavily disturbed Maxwell distributions of optically excited particles diffusing in a buffer gas. These model equations involve elastic collisions and the interaction of photons with three-level atoms by taking into account the Doppler effect. Important features of the developed model are the realistic simulation of the relaxation process and the existence of a stable and unique equilibrium solution.

Line-shape expressions for two-step resonant excitation in the presence of low-pressure perturbers are derived. These expressions allow for velocity-dependent broadening rates and are well suited for studying the line-shape contributions due to coherent excitation, incoherent excitation (due to dephasing collisions), interference between coherent and incoherent excitation, and velocity-changing collisions. Using the sodium 3S-3P-4D system perturbed by noble-gas atoms as an example and assuming

{C}_{6}

interaction potentials, line shapes at a variety of laser detunings (Doppler selected atomic velocities) are numerically calculated to study variation in the line shape due to competing excitation paths and speed-dependent collision rates. In order to deal with both the line core and the line wing a composite collision kernel consisting of a Lorentzian kernel for small \ensuremath{\Delta}

{v}_{z}

collisions and a Keilson-Storer kernel for large \ensuremath{\Delta}

{v}_{z}

collisions is used. It is also demonstrated how the inelastic (3

{P}_{3/2}

-3

{P}_{1/2}

) velocity-changing collision kernel may be obtained from experimental data at low pressures. Theoretical line shapes and broadening rates are compared with experiment.