  Transient absorption (TA) spectroscopy has long been an invaluable tool for
determining the energetics and dynamics of excited states in atomic, molecular,
and solid-state systems. When pump pulse intensities are sufficiently high, the
resulting TA spectra include both the generally desired third-order response of
the studied material as well as responses that are higher order in the electric
field amplitudes of the pulses. It has recently been shown that
pump-intensity-dependent TA measurements allow separating the various orders of
response of the TA signal, but the information content available in those
higher orders has not been described. We give a general framework, intuition,
and nomenclature for understanding the information contained in high-order TA
spectra. Standard TA spectra are generally interpreted in terms of three
fundamental processes: ground-state bleach (GSB), stimulated emission (SE), and
excited state absorption (ESA), and we extend those concepts to higher order.
Each order introduces two new processes: SE and ESA from highly excited states
that were not accessible in lower orders. In addition, each order contain
negations of lower-order processes, just as GSB is a negation of the linear
absorption. We show the new spectral and dynamical information that is
introduced at each order and show how the relative signs of the signals in
different orders can be used to identify which processes are dominant.
