  Multiphoton absorption is of vital importance in many spectroscopic,
microscopic or lithographic applications. However, given that it is an
inherently weak process, the detection of multiphoton absorption signals
typically requires large field intensities, hindering its applicability in many
practical situations. In this work, we show that placing a multiphoton
absorbent inside an imbalanced nonlinear interferometer can enhance the
precision of multiphoton cross-section estimation with respect to strategies
based on direct transmission measurements by coherent or even squeezed light.
In particular, the power scaling of the sensitivity with photon flux can be
increased by an order of magnitude compared to transmission measurements of the
sample with coherent light, meaning that a signal could be observed at
substantially reduced excitation intensities. Furthermore, we show that this
enhanced measurement precision is robust against experimental imperfections
leading to photon losses, which usually tend to degrade the detection
sensitivity. We trace the origin of this enhancement to an optimal degree of
squeezing which has to be generated in a nonlinear SU(1,1)-interferometer.
