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Numerical studies are presented to assess error estimates for a separable(Hartree) approximation for dynamically evolving composite quantum systemswhich exhibit distinct scales defined by their mass and frequency ratios. Therelevant 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-dimensionaltunneling system where a double well and a harmonic coordinate are cubicallycoupled. The timedependent Hartree approximation is compared with a fullycorrelated solution, for different parameter regimes. The impact of thecoupling and the resulting correlations are quantitatively assessed in terms ofa time-dependent reaction probability along the tunneling coordinate. We showthat the numerical error is correctly predicted on moderate time scales by atheoretically derived error estimate.

Dynamical approximations for composite quantum systems: Assessment of error estimates for a separable ansatz