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DISCUSSIONS ON THE RATE EQUATION OF LEVEL POPULATION DENSITY

Zheng Shaotang

Published Jan 1, 1984• 0 citations
OpenAlex

Abstract

In this paper we discuss the approximations introduced in the average-ion model usually employed for level population calculations. An improved system of equations of higher-order approximation is given. It is proved that in electric dipole approximation the level population probability Pnlj is independent of the quantum number j. leading to a great reduction of the number of equations to be solved. Finally, a method feasible for calculating these rate equations is suggested.

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In this paper we discuss the approximations introduced in the average-ion model usually employed for level population calculations. An improved system of equations of higher-order approximation is given. It is proved that in electric dipole approximation the level population probability Pnlj is independent of the quantum number j. leading to a great reduction of the number of equations to be solved. Finally, a method feasible for calculating these rate equations is suggested.
The rate-equation approximation is one of the most fundamental and universally employed simplifications in laser analyses. The accuracy and regions of applicability of this approximation are explored in comparisons with more rigorous semiclassical models. Higher-order rate-equation approximations are also developed, and these improved models can yield much better accuracy than conventional rate equations with little added complexity. The modified adiabatic elimination methods reported here would also be useful in reducing the mathematical models governing other physical systems.
The shifted frequency approximation is used to solve the equations of motion for the density matrix describing a quantum mechanical two-level system, with decay, in an applied monochromatic electric field. Expressions are obtained for the transient and the long-term linear electric dipole moment and occupation probability, and the long-term third-order dipole moment. The results are valid across the frequency spectrum, including resonance, provided the phenomenological decay times for the system are not too large or too close in value.
Flip-flop processes due to magnetic dipole-dipole interaction between neighboring ions in crystals doped by rare-earth ions are one of the mechanisms of relaxation between hyperfine levels. Modeling of this mechanism has so far been macroscopic, characterized by an average rate describing the relaxation of all ions. Here, however, we present a microscopic model of flip-flop interactions between individual nuclear spins of dopant ions. Every ion is situated in a unique local environment in the crystal, where each ion has different distances and a unique orientation relative to its nearest neighbors, as determined by the lattice structure. Thus each ion has a unique flip-flop rate and the collective relaxation dynamics of all ions in a bulk crystal is a sum of many exponential decays, giving rise to a distribution of rates rather than a single average decay rate. We employ this model to calculate flip-flop rates in Pr3+{\mathrm{Pr}}^{3+}Pr3+:Y2SiO5{\mathrm{Y}}_{2}{\mathrm{SiO}}_{5}Y2​SiO5​ and show experimental measurements of population decay of the ground state hyperfine levels at $\ensuremath{\sim}2$ K. We also present a method to measure rates of individual transitions from hole burning spectra that requires significantly fewer fitting parameters in theoretical rate equations compared to earlier work. Furthermore, we measure the effect of external magnetic field on the flip-flop rates and observe that the rates slow down by two orders of magnitude in a field of 5--10 mT.