Time-Dependent Logarithmic Perturbation Theory for Quantum Dynamics: Formulation and Applications

TL;DR

This paper develops a time-dependent logarithmic perturbation theory for quantum systems, providing a new analytical framework for calculating energy shifts and observables under time-dependent perturbations.

Contribution

It introduces a novel extension of logarithmic perturbation theory to time-dependent quantum dynamics, preserving the integral structure and enabling accurate computation of physical quantities.

Findings

Exact solutions for harmonic oscillator under laser driving

Application to hydrogen atom with high accuracy results

Framework relates energy shifts to AC Stark effects and polarizabilities

Abstract

We present a time-dependent extension of logarithmic perturbation theory for nonrelativistic quantum dynamics governed by the Schr\"odinger equation, in which the logarithm of the wave function is expanded in powers of a coupling constant. The resulting hierarchy of equations defining the perturbative corrections is governed by a gauge-rotated Hamiltonian of the unperturbed system and leads to closed-integral expressions for the time-dependent corrections based on Duhamel's formula. This closed-integral structure of corrections is a hallmark of time-independent logarithmic perturbation theory and is preserved in the present extension. This structure, in particular, provides a computable expression for the instantaneous energy shift. Furthermore, dynamic energy shifts arise naturally within this framework in the form of time-averaged expectation values of pseudopotentials and can be related, for example, to AC Stark shifts and electric polarizabilities. As an illustration, we apply the method to the harmonic oscillator and the hydrogen atom, both driven by a time-dependent laser field. The harmonic oscillator provides a proof of principle for which the exact solution is recovered, while the hydrogen atom illustrates the method applied to atomic systems. Supported by numerical simulations, we demonstrate the applicability to obtain relevant physical observables with high accuracy. The present approach offers a promising alternative for analytical studies of time-dependent multi-photon processes in the perturbative regime.