  Birth of objective properties from subjective quantum world has been one of
the key questions in the quantum-to-classical transition. Basing on recent
results in the field, we study it in a quantum mechanical model of a
boson-boson interaction-quantum Brownian motion. Using various simplifications
we prove a formation for thermal environments of, so called, spectrum broadcast
structures, responsible for perceived objectivity. In the quantum measurement
limit we prove that this structure is always formed, providing the
characteristic timescales. Including self-Hamiltonians of the environment, we
show the exponential scaling of the effect with the size of the environment.
Finally, in the full model we numerically study the influence of squeezing in
the initial state of the environment, showing broader regions of formation than
for non-squeezed thermal states.
