  We present a method for the direct measurement of the Wigner characteristic
function of a thermalizing harmonic oscillator that is completely inaccessible
for control or measurement. The strategy employs a recently proposed
probe-measurement-based scheme [Phys. Rev. Lett. 122, 110406 (2019)] which
relies on the pulsed control of a two-level probe. We generalize this scheme to
the case of a nonunitary time evolution of the target harmonic oscillator,
describing its thermalization through contact to a finite-temperature
environment, given in the form of a Lindblad master equation. This
generalization is achieved using a superoperator formalism and yields
analytical expressions for the direct measurement of the characteristic
function, accounting for the decoherence during the measurement process.
