Relativistic Rabi Dynamics and Scalings of Highly charged Hydrogen-like Atoms

Abstract

Interactions between light and matter is one way to study both particles whose small spatial sizes render them irresolvable to our eyes and events where the characteristic time scale is too short for conventional recording and measurement techniques. The advent of free electron lasers allow laser pulses of high intensity and small temporal duration such that ultra-fast dynamics in matter systems and resulting non-linear effects can be resolved experimentally. This thesis discusses some theories to mathematically model such light-matter interactions and presents results based both on analytical expressions—where such expressions are available— from the Schrödinger and the Dirac equations as well as numerical calculations. Some models presented are a semi-classical model where light is treated classically and the atomic system quantum-mechanically and a statistical model which describes the light as quantized to feature effects like spontaneous emission and thermal excitation. The main theme of the thesis is relativistic effects and scalings of interaction effects with nuclear charge Z.

The results discussed in the work shows that the equivalent characteristics of the dynamics of highly ionized hydrogen-like atoms predicted by the Schrödinger equation fail when relativistic effects are taken into account. This is shown to be a result of the quantities such as energy and mean radial distance from the nucleus not being exactly proportional to powers of Z in Dirac’s description of quantum mechanics. Further, deviations occur due to the necessary detuning of the driving laser field when spin-orbit interactions (a relativistic effect) split states of non-zero orbital momentum. Also, it is shown that as the characteristic time scales of interactions and inherent effects such as spontaneous emission scale differently with Z, the characteristic dynamics are not invariant to atomic number and it is not possible to rescale quantities to make them so.

This work provides a guide for future Free-electron laser experiments on highly charged ions at X-ray wave-
lengths.

Popular Abstract

What implications do Albert Einstein’s theory of relativity have in atomic physics and how does observable quantities change in experiments as we move down the periodic table to larger atoms? These questions are some of the major themes in the thesis.

One of the few ways we can measure and understand atoms is through their interaction with light. The light that an electron in an atom can absorb or emit has an energy that matches the difference between the “states” the electron occupies before and after the interaction. It is therefore of great interest to understand these states, as well as the likelihood of transitions between them.

By placing an atom in a laser field with an energy corresponding to the energy difference between two states, one can drive the electron population from one state to the other and back again, producing an oscillatory behavior much like a pendulum swinging from one side to the other. These oscillations are called Rabi oscillations, and they are the reason the name Rabi appears in the title of the thesis. Despite this simple description, additional features appear when several states lie close in energy, allowing the laser to drive multiple transitions. To explain why this happens, and why it is of interest, some additional background is useful.

Hydrogen consists of one positively charged proton and one negatively charged electron, whereas ionized helium has two positively charged protons and one electron. A common misconception—one even suggested by the famous Schrödinger equation—is that ionized helium, as an atomic system, should display the same characteristics as hydrogen, merely scaled to account for the increased nuclear charge.
However, when relativistic effects are taken into account, this turns out not to be the case. For a proper theoretical description of the behavior of highly charged ions in experiments, such effects must therefore be included. In the thesis, these relativistic dynamics and scalings are discussed. Interestingly, the previously mentioned states split into two as a consequence of intrinsic properties of the electron. As a result, the laser becomes detuned from the exact transition energies of the states, which affects the oscillation dynamics.
A helpful analogy to understand this—while not to be taken too literally, but still intuitively useful—is to imagine the electron as the Earth and the atomic nucleus as the Sun.
The Earth exhibits two kinds of motion: it rotates about its own axis and revolves around the Sun in its orbit. Whether the Earth rotates in the same direction as it revolves, or in the opposite direction, has a counterpart in atomic physics. Similarly, an electron has slightly different energies depending on whether its intrinsic and orbital angular momenta are aligned or opposed. This is a relativistic effect that is overlooked in the Schrödinger equation.

It is discussed in the thesis that the essential implication of this is that, when a laser drives the population from an initial state, there are two very closely spaced final states available. The oscillating system—recalling the pendulum analogy—thus becomes a so-called three-level system. From classical (that is, non-relativistic) quantum mechanics, one would expect the behavior of such a three-level system to be equivalent for all ionized atoms with a single electron, provided that quantities such as the time scale and laser energy are appropriately scaled. A key result of relativity, however, is that these scalings are not linear.
Furthermore, the energy difference between aligned and anti-aligned configurations increases drastically for heavier elements. One of the main results discussed in the thesis is that this three-level system exhibits nonregular oscillations, arising primarily from laser detuning. This behavior is both theoretically derived and numerically simulated

The subjects and results of this thesis, as well as of related work, are important for interpreting experiments on highly charged ions. This is a highly active and contemporary area of atomic physics, where the advent and continued development of so-called free-electron lasers have dramatically extended the limits of how short laser pulses can be and how high their intensities can reach. Such experiments are crucial for probing the fundamental properties of atoms, which ultimately make up everything we observe.

• author: Salbrink, Filip
• supervisor: Marcus Dahlström
• organization: Mathematical Physics; Department of Physics
• alternative title: Relativistisk Rabidynamik och Skalningar i Starkt Laddade Vätelika Atomer
• course: PHYM01 20252
• year: 2026
• type: H2 - Master's Degree (Two Years)
• subject: Physics and Astronomy
• keywords: Light-matter Interactions, Hydrogen-like Atoms, Dirac-equation, Highly Charged Ions, Relativistic Scalings, Free-electron Laser, X-ray Spectroscopy, Non-linear Dynamics, Quantum Optics
• language: English
• id: 9220182
• date added to LUP: 2026-01-22 11:16:15
• date last changed: 2026-01-22 11:16:15