  We present a method for the direct measurement of the Wigner characteristic
function of a damped 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 non-unitary time evolution of the target harmonic oscillator,
describing its damping by 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.
