  The co-localization of density modulations and particle polarization is a
characteristic emergent feature of motile active matter in activity gradients.
It can therefore play the role of a smoking gun for the mesoscale detection of
intrinsic microscopic activity.
  We employ the active-Brownian-particle (ABP) model to derive precise
analytical expressions for the density and polarization profiles of a single
Janus-type swimmer in the vicinity of an abrupt activity step, including
situations with an orientation-dependent propulsion speed. Our results agree
well with measurement data for a hot microswimmer presented in the companion
paper [1] and can serve as a template for more complex applications, e.g., to
motility-induced phase separation or studies of physical boundaries.
  The essential physics behind our formal results is robustly captured and
elucidated by a schematic two-species ``run-and-tumble'' model.

Soft Condensed Matter;Statistical Mechanics