Error bounds for the Floquet-Magnus expansion and their application to the semiclassical quantum Rabi model
arXiv:2504.20533 · doi:10.1103/6bgj-s987
Abstract
We present a general, nonperturbative method for deriving effective Hamiltonians of arbitrary order for periodically driven systems, based on an iterated integration by parts technique. The resulting family of effective Hamiltonians reproduces the well-known Floquet-Magnus expansion, now enhanced with explicit error bounds that quantify the distance between the exact and approximate dynamics at each order, even in cases where the Floquet-Magnus series fails to converge. We apply the method to the semiclassical Rabi model and provide explicit error bounds for both the Bloch-Siegert Hamiltonian and its third-order refinement. Our analysis shows that, while the rotating-wave approximation more accurately captures the true dynamics than the Bloch-Siegert Hamiltonian in most regimes, the third-order approximation ultimately outperforms both.
arXiv comment14 pages, 5 figures. v3: added figure; expanded mathematical explanations; added references
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