Two low energy photons for one high energy photon in fluorescence possible?

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Offline McKay

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I understand fluorescent materials, in normal lightning conditions, look brighter than other materials because they not only reflect their characteristic color directly, but also convert higher energy photons to its characteristic color, right? And since 1 and a fraction of a photon cant be emitted, then the fraction gets dissipated as heat, right?
But what about cases where the incident radiation is actually twice (or more) as energetic as the materials characteristic  color? Do the materials emit two (or more) photons [plus the fraction thats left as heat] or emit just one photon and rest as heat? Or some materials do one and some the other?