  We study the time dynamics of random density matrices generated by evolving
the same pure state using a Gaussian orthogonal ensemble (GOE) of Hamiltonians.
We show that the spectral statistics of the resulting mixed state is well
described by random matrix theory (RMT) and undergoes a crossover from the GOE
to the Gaussian unitary ensemble (GUE) for short and large times respectively.
Using a semi-analytical treatment relying on a power series of the density
matrix as a function of time, we find that the crossover occurs in a
characteristic time that scales as the inverse of the Hilbert space dimension.
The RMT results are contrasted with a paradigmatic model of many-body
localization in the chaotic regime, where the GUE statistics is reached at
large times, while for short times the statistics strongly depends on the
peculiarity of the considered subspace.
