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

TL;DR

This work addresses the challenge of quantifying error bounds for effective Hamiltonians in periodically driven quantum systems, where the Floquet–Magnus expansion may fail to converge. It introduces a nonperturbative, integration-by-parts framework that reproduces the Floquet–Magnus results while providing explicit, order-by-order error bounds that hold irrespective of convergence, with Ω = 2π/T governing the high-frequency scaling. Applied to the semiclassical Rabi model, the method yields the RWA and Bloch–Siegert Hamiltonians at second order and a third-order effective Hamiltonian that matches the Floquet–Magnus expansion up to 1/ω^{2}, accompanied by rigorous bounds on the distance between exact and approximate evolutions. Numerical comparisons reveal that higher-order descriptions can outperform lower-order ones over long times, and the third-order model offers the best long-time accuracy within the studied regime. The framework thus furnishes a robust, general tool for principled, quantitatively controlled approximations in driven quantum dynamics, with potential extensions to unbounded Hamiltonians and broader quantum-technological applications.

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.

Figures (5)

• Figure 1: Numerical distance (in the operator norm) between the exact evolution generated by the semiclassical Rabi Hamiltonian, and the evolutions of the RWA (orange) and Bloch--Siegert (blue) Hamiltonians, for large times. We set $g=1$ and $\omega = 5$ for the numerics.
• Figure 2: Numerical distance (in the operator norm) between the exact evolution generated by the semiclassical Rabi Hamiltonian, and the evolutions of the RWA (orange) and Bloch--Siegert (blue) Hamiltonians, for short times. We set $g=1$ and $\omega = 5$ for the numerics.
• Figure 3: Numerical distance (in the operator norm) between the exact evolution generated by the semiclassical Rabi Hamiltonian, and the evolutions of the RWA (green), Bloch--Siegert (blue), and third-order (red) effective Hamiltonians, for large times. We set $g=1$ and $\omega = 5$ for the numerics. Additionally, the bounds for each of these effective Hamiltonians are shown as dashed lines.
• Figure 4: Log-log plot of the distance (in the operator norm) between the exact evolution generated by the semiclassical Rabi Hamiltonian, and the evolutions generated by the RWA (green), Bloch--Siegert (blue), and third-order (red) effective Hamiltonians, for large times. We set $g=1$ and $t = 100$ for the numerics. Additionally, the bounds for each of these effective Hamiltonians are shown as dashed lines.
• Figure 5: Numerical distance (in the operator norm) between the exact evolution generated by the semiclassical Rabi Hamiltonian, and the evolutions of the RWA (green), Bloch--Siegert (blue), and third-order (red) effective Hamiltonians, for times multiple of the period. We set $g=1$ and $\omega = 5$ for the numerics.