Two-photon Rabi oscillations in hydrogen: A theoretical study of effective Hamiltonian approaches

In this thesis, the effective Hamiltonian formalism is studied and applied to two-photon Rabi oscillations in hydrogen. The process occurs when two photons are absorbed or emitted simultaneously. There exist various approaches to model two-photon transitions. In this thesis, we single out essential quantum states using the projection operator technique and solve for the exact dynamics in this essential subspace of the total Hilbert space. The non-essential states are adiabatically eliminated; their contributions are included perturbatively via the resolvent formalism in form of level-shifts and effective couplings to the essential states. Through this, an effective Hamiltonian in the essential subspace is obtained. In the first part of the thesis, it will be shown how this effective Hamiltonian formalism reveals fascinating ties between the Markov approximation and the pole approximation. Furthermore, higher-order corrections will be discussed. A novel expansion of the resolvent operator is proposed, which in a special case allows for the analytical determination of a second-order effective Hamiltonian. In the second part, two-photon Rabi oscillations in hydrogen are studied using effective Hamiltonians. Rabi oscillations are a coherent process in which a quantum system, driven by monochromatic radiation, periodically oscillates between two states. Thus, complete population transfer between the states is enabled. When driven by two photons, we speak of two-photon Rabi oscillations. While it is known that two-photon Rabi oscillations cannot be driven between the 1s and 2s state due to ionisation, it will be shown that two-photon Rabi oscillations are indeed possible drive between the 1s and 3s/d states. While the 3s state ionises rapidly, the 3d state couples strongly enough to the 1s ground state to facilitate two-photon Rabi oscillations. The mechanism is explained with bright and dark states. With ever stronger free-electron lasers and the near-future prospect of
All matter in our world is made up of atoms. What determines the properties of matter is the behaviour of the electrons in the atoms. This behaviour can be studied and controlled by exposing the electrons to light. An electron bound in an atom can only take on discrete energy values, so-called energy states. By shining light on the electron, we may induce it to jump to a higher energy state; we call this a one-photon transition. How far the electron jumps is determined by the colour of the light. Fascinatingly, if the colour perfectly corresponds to the energy between two energy states, we may observe the electron periodically jumping up and down between two energy states. This is known as Rabi oscillations. What is especially interesting is that the electron goes from being fully in the lower energy state, to being in both states at once, and then to fully inhabiting the upper energy state, before returning down again to repeat the cycle. That the electron at one time is fully in the upper state is not self-evident because the quantum world is based on probabilities. Under special conditions, the electron may not take one single jump to reach an upper state, but may jump twice at once – a so-called two-photon transition. In atoms, energy states that can be reached by one jump cannot be reached by two simultaneous jumps, and vice-versa. This makes two-photon transitions very relevant: We can try to use them to get the electron to energy states which can under normal circumstances not be reached. If the colour and intensity of the light is just right, we may observe two-photon Rabi oscillations of the electron, where the electron at one point fully inhabits the upper energy state. Describing two-photon Rabi oscillations can be challenging since, inbetween the electron’s two jumps, it lands in a third so-called virtual energy state, which we may envision as an invisible trampoline. The probability to find the electron in the virtual state is 0%. As such, we are not in