[Paper Review] Bogolyubov's averaging theorem applied to the Kramers-Henneberger Hamiltonian

Elena Floriani, Jonathan Dubois+1 more|arXiv (Cornell University)|Jul 5, 2021

Laser-Matter Interactions and Applications25 references4 citations

TL;DR

This paper applies a modified version of Bogolyubov's averaging theorem to the Kramers-Henneberger (KH) Hamiltonian for an electron in a laser field, showing that the amended theorem provides significantly more accurate estimates of trajectory differences than the original formulation. The key contribution is that the improved estimates confirm the validity of the KH approximation away from the ionic core, but fail near the potential minimum—where the KH atom is defined—limiting the theoretical justification for stable laser-dressed states.

ABSTRACT

We apply Bogolyubov's averaging theorem to the motion of an electron of an atom driven by a linearly polarized laser field in the Kramers-Henneberger frame. We provide estimates of the differences between the original trajectories and the trajectories associated with the averaged system as a function of the parameters of the laser field and the region of phase space. We formulate a modified Bogolyubov averaging theorem based on the Hamiltonian properties of the system, and show that this version is better suited for these systems. From these estimates, we discuss the validity of the Kramers-Henneberger approximation.

Motivation & Objective

• To rigorously assess the validity of the Kramers-Henneberger (KH) approximation in strong-field atomic physics using averaging theory.
• To identify the limitations of the original Bogolyubov averaging theorem when applied to time-dependent Hamiltonian systems with long-range Coulomb potentials.
• To develop and apply a modified averaging theorem tailored to Hamiltonian dynamics that yields tighter estimates of trajectory divergence between the original and averaged systems.
• To determine under which laser parameters and phase space regions the KH approximation remains valid, especially near the ionic core.
• To clarify why the KH approximation fails to justify the existence of stable laser-dressed states at the potential minimum despite its widespread use.

Proposed method

• Apply a canonical transformation to shift into the Kramers-Henneberger frame, transforming the time-dependent Hamiltonian into a form amenable to averaging.
• Use a soft-Coulomb potential to model the electron-ion interaction, preserving long-range effects critical near the core.
• Formulate a modified version of Bogolyubov’s averaging theorem that accounts for Hamiltonian structure, improving error estimates in phase space.
• Derive explicit bounds on the difference between trajectories of the original and averaged systems using the modified theorem.
• Compare the new estimates with those from the original Bogolyubov theorem and Proposition 1, analyzing their dependence on laser intensity, frequency, and distance from the core.
• Use numerical simulations (via a MATLAB script) to validate theoretical bounds and visualize trajectory divergence in phase space.

Experimental results

Research questions

• RQ1 Under what conditions is the Kramers-Henneberger approximation valid for electron dynamics in a strong laser field?
• RQ2 How do the error estimates from Bogolyubov’s averaging theorem compare with those from the modified Hamiltonian-adapted version?
• RQ3 Why does the original averaging theorem fail to provide reliable estimates near the ionic core, and can a revised version overcome this?
• RQ4 To what extent can the KH approximation justify the existence of stable, laser-dressed bound states near the potential minimum?
• RQ5 How does the trajectory divergence depend on laser parameters (intensity, frequency) and the region of phase space visited by the electron?

Key findings

• The modified Bogolyubov theorem provides significantly tighter bounds on trajectory divergence than the original formulation, especially in the intermediate and small-distance regimes.
• Estimates from the modified theorem decay faster with distance from the core, indicating better validity of the KH approximation in regions farther from the ionic core.
• The exponential growth in trajectory error is reduced from μ⁻³ to μ⁻³/² dependence in the exponent, slowing divergence as the electron approaches the core.
• The KH approximation remains valid for trajectories far from the core (at least one quiver radius away), but fails to provide reliable estimates near the potential minimum where the KH atom is defined.
• Numerical simulations confirm that the theoretical bounds are consistent with observed trajectory behavior, though the dynamics remains more complex than predicted by simple averaging.
• The original Bogolyubov theorem and its proposition-based variant yield overly pessimistic estimates near the core, while the modified theorem shows improved accuracy but still cannot validate the approximation at the minimum of the KH potential.