Error bounds for the Floquet-Magnus expansion and their application to the semiclassical quantum Rabi model

• URL: http://arxiv.org/abs/2504.20533v3
• Date: Tue, 04 Nov 2025 08:41:33 GMT
• Title: Error bounds for the Floquet-Magnus expansion and their application to the semiclassical quantum Rabi model
• Abstract summary: We present a nonperturbative method for deriving effective Hamiltonians of arbitrary order for periodically driven systems.<n>We provide explicit error bounds for both the Bloch-Siegert Hamiltonian and its third-order refinement.<n>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.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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