Epidemiologic, Clinical, and Virologic Observations on Dengue in the Kingdom of Tonga

Abstract

We report computational results for the time evolution of the velocity distribution P(V,t) for two-level and multilevel ‘‘Doppler’’ laser cooling. We compare results obtained from the semiclassical (SC) Fokker-Planck equation and from generalized optical Bloch equations applied to density matrices over a basis of products of internal and quantized translational states (QDM). Computer memory requirements are optimized to make large-scale QDM calculations feasible. QDM and SC agree well except for two cases: (a) atoms in the wells of the light-shift potential (with kinetic energy less than the well depth, U0), and (b) atoms with recoil energy ER comparable to or greater than the natural linewidth hG. Transient dips occur in P(V,t) at V=0 in QDM results due to slow cooling of atoms in the light-shift potential wells. Dips in P(V,t) occur at velocity-tuned-resonance (Doppleron) velocities but disappear over long interaction times as atoms accumulate near points where the force is zero. When ER¿hG, sharp peaks occur in P(V,T) at V=±VR from velocity-selective population quasitrapping not previously found in a two-level transition. Sharp features in P(V,t) occur also for J¿J+1 transitions with J>0, small U0/ER, and sufficiently large detuning, from transitions between individual quantum states in the periodic potential.