On the derivation of rate equations

R. L. Ingraham

Julie Saikia, Manas Protim Saikia, Rajib Kumar Dubey +3 more

T. Papenbrock, A. N. Salgueiro, Hans A. Weidenmüller

We derive two different sets of rate equations for sympathetic cooling of harmonically trapped bosons or fermions. The rate equations are obtained from a master equation derived earlier by Lewenstein et al. [Phys. Rev. A 51, 4617 (1995)] by means of decoherence and ergodicity arguments. We show analytically that the thermal equilibrium state is a stationary solution of our rate equation. We present analytical results for the rate coefficients which are needed to solve the rate equations, and we give approximate formulas that permit their computation in practice. We solve the two sets of rate equations numerically and compare the results. The cooling times obtained in both approaches agree very well. The equilibration rates show fair agreement.

S. Swain

Starting from the multi-level master equation a non-Markoffian rate equation is derived. Explicit expressions for the transition rates between each pair of atomic levels in this equation are derived at it is shown that these may be interpreted as the sum of one-photon, two-photon,..., etc. transition rates connecting the given pair of levels. The non-Markoffian master equation is exact and so can describe long- or short-time behaviour under monochromatic or broad-band excitation. The exact steady-state solutions are obtained by solving a Markoffian rate equation.

Germán J. de Valcárcel, Eugenio Roldán

Einstein's rate equations are derived from the semiclassical Bloch equations describing the interaction of a classical broadband light field with a two--level system.