  We discuss a parameter estimation problem using quantum decoherece in the
double-slit interferometer. We consider a particle coupled to a massive scalar
field after the particle passing through the double slit and solve the dynamics
non-perturbatively for the coupling by the WKB approximation. This allows us to
analyze the estimation problem which cannot be treated by master equation used
in the research of quantum probe. In this model, the scalar field reduces the
interference fringes of the particle and the fringe pattern depends on the
field mass and coupling. To evaluate the contrast and the estimation precision
obtained from the pattern, we introduce the interferometric visibility and the
Fisher information matrix of the field mass and coupling. For the fringe
pattern observed on the distant screen, we derive a simple relation between the
visibility and the Fisher matrix. Also, focusing on the estimation precision of
the mass, we find that the Fisher information characterizes the wave-particle
duality in the double-slit interferometer.
