We theoretically investigate transitions in a two-level system, which are induced by a sweep through an avoided crossing in the presence of coupling to a single, excited bosonic mode. This allows us to propose the initial bosonic excitation into a number state as a new possible control parameter for the survival probability at long times. The expansion of number states in terms of coherent states centered around points on a circle in phase space makes a multi-Davydov-Ansatz the method of choice to perform the required numerical calculations. It is revealed that the starting time of the transition greatly affects the final transition probabilities. In addition, we found that the mixing angle, which is tuning between the diagonal and off-diagonal coupling, is decisive for the ability to control the transition via number state excitation. For a mixing angle of π/4, we found the maximal effect of number state excitation on the transition probability.Published under license by AIP Publishing. https://doi. ARTICLEscitation.org/journal/jcp FIG. 7. Transition probabilities as a function of n: θ = 0 (blue), θ = π/4 (red), and θ = π/2 (green). APPENDIX: PROPERTIES OF THE DISPLACEMENT OPERATOR AND THE LAGUERRE POLYNOMIALSIn the following, we collect properties of the displacement operator and its action on number states that are frequently needed in

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Due to the exponential increase of the numerical effort with the number of degrees of freedom, moving basis functions have a long history in quantum dynamics. In addition, spawning of new basis functions is routinely applied. Here we advocate the opposite process: the programmed removal of motional freedom of selected basis functions. This is a necessity for converged numerical results wrt the size of a non-orthogonal basis, because generically two or more states approach each other too closely early on, rendering unstable the matrix inversion, required to make the equations of motion explicit. An application to the sub-Ohmic spin-boson model demonstrates the power of the proposed methodology.