Non-monotonicity in the quantum-classical transition for the harmonic oscillator

The negative volume of a Wigner function can be considered a measure the degree of quantumness of a system. In the absence of decoherence, this measure of quantumness of an harmonic oscillator increases with the energy eigenstates $n$. This seeming disagreement with the correspondence principle is of course resolved when considering decoherence. Interestingly, when any decoherence is introduced, an eigenstate $n_{peak}$ with `maximal quantumness' results so that the quantum-classical transition is non-monotonic. The value of $n_{peak}$ decreases with time and strength of environmental coupling.