Democritus theorised atoms as the infinitely small & indivisible parts of matter; mathematically we tend to think them of points; physically not so; Aristotle agreed with this - he was against the possibility of the infinitely small being points; since points need to cohere somehow they must have extension; in mathematics where we do have points this cohesion is provided by topology.
Democritus claimed everything was made of atoms; but his theory of the perception of light strangely went against this dictum; he theorised infinitely thin films - eidolon - carry the image being seen and enter the eye; this sounds bizarre but isn't in fact far from wavefronts entering an eye; Newton when he read of Democritus's theory in Lucretious's De Rerum Natura corrected this discrepency by inventing the notion of light corpuscules.
In general, qualitative terms the atomic theory saturates the modern outlook in physics, mathematics, economics and the Anglo-American tradition of philosophy; where they go under the name of atomic theory of matter (classical) & the quanta of energy (Planck), the analytic theory of sets (Georg Cantor), the rational self-interested actor of classical economics (Adam Smith) and logical atomism (Wittgenstein); this side of modern thinking is also called reductonism.
Does anyone apply it to qualitative experience, and ask if that is divided up into instants?
All of the above isn't really applicable to qualitative experience; as experience requires the self; Kant, however, does have a theory how experience is synthesised from intuitions (qualia); though this isn't theorised as infinitely small - it is qualitively so - along the lines that de Craema pointed out.
It seems to me that the infinitely small could not be like something, but I can think of no conclusive reason to suppose that it cannot be like something.
If the infinitely small was also infinitesimal there are arguments first marshalled by Aristotle that this cannot be possible; cohesion is required, thus for a time like thing we need duration, and for a spacelike thing we need extension; though it is perhaps a little strange it is possible to have an infinitely small, but not infinitesimal 'object'; and this can be demonstrated quite rigourously by whats called synthetic geometry.