  Numerical studies are presented to assess error estimates for a separable
(Hartree) approximation for dynamically evolving composite quantum systems
which exhibit distinct scales defined by their mass and frequency ratios. The
relevant error estimates were formally described in our previous work [I.
Burghardt, R. Carles, C. Fermanian Kammerer, B. Lasorne, C. Lasser, J. Phys. A.
54, 414002 (2021)]. Specifically, we consider a representative two-dimensional
tunneling system where a double well and a harmonic coordinate are cubically
coupled. The timedependent Hartree approximation is compared with a fully
correlated solution, for different parameter regimes. The impact of the
coupling and the resulting correlations are quantitatively assessed in terms of
a time-dependent reaction probability along the tunneling coordinate. We show
that the numerical error is correctly predicted on moderate time scales by a
theoretically derived error estimate.
