  Evolutions of quantum noise, characterized by quadrature squeezing parameter
and Fano factor, and of mixedness, quantified by quantum von Neumann and linear
entropies, of a pumped dissipative non-linear oscillator are studied. The model
can describe a signal mode interacting with a thermal reservoir in a
parametrically pumped cavity with a Kerr non-linearity. It is discussed that
the initial pure states, including coherent states, Fock states, and finite
superpositions of coherent states evolve into the same steady mixed state as
verified by the quantum relative entropy and the Bures metric. It is shown
analytically and verified numerically that the steady state can be well
approximated by a nonclassical Gaussian state exhibiting quadrature squeezing
and sub-Poissonian statistics for the cold thermal reservoir. A rapid increase
is found in the mixedness, especially for the initial Fock states and
superpositions of coherent states, during a very short time interval, and then
for longer evolution times a decrease in the mixedness to the same, for all the
initial states, and relatively low value of the nonclassical Gaussian state.
