Entanglement is a fundamental principle in quantum mechanics, and is known for its crucial part in quantum information science and quantum communication, among others. In attosecond science, the possible effects of entanglement in light-matter interaction are receiving more and more attention. In this research field, typically radiation in the extreme ultraviolet (XUV) or x-ray regime is used to study time-dependent dynamics in time-resolved spectroscopy experiments. The high photon energy of this radiation leads to ionization of any sample placed in the way, thus creating multicomponent quantum systems with possible entangled subsystems, e.g. ion and photoelectron. Entanglement between the subsystems can have measurable effects on the outcome of an experiment, and can even prevent the observation of time-dependent observables. This notion is most prominent if the experiment includes only measurements in one of the subsystems, which means for example in the case of molecular photoionization, that only the ion or photoelectron is measured. This setting leads to interesting questions about the role of entanglement in attosecond photoionization processes. In particular, what is the role of entanglement between the ion and departing photoelectron regarding the observability of coherence-based dynamics in the ion? Furthermore, is it possible to control the degree of coherence in the ion and the degree of ion+photoelectron entanglement, respectively, by alternating the pulse properties of the ionizing pulses? To answer these questions, an experimental protocol is described in this thesis, which utilizes two phase-locked XUV pulses together with a near-infrared pulse (NIR) to dissociatively ionize hydrogen molecules. The main experiments presented here investigate the impact of entanglement between the cation and photo- electron, first on vibrational coherence, and subsequently on electronic coherence created in H2+. In the first experiment, a pair of phase-locked XUV pulses creates a vibrational wave packet in the 1sσg state of H2+ , which is subsequently probed by the NIR pulse dissociating the ion. The spectral properties of the ionizing pulses are tailored by alternating the delay between the two XUV pulses, thus controlling the degree of vibrational coherence and the degree of ion+photoelectron entanglement, respectively. In the second experiment, the two XUV pulses create an entangled ion+photoelectron system during the dissociative ionization of H2. The NIR probe pulse can subsequently project the initially entangled system into a coherent superposition of the first gerade (1sσg) and ungerade (2pσu) electronic states of the H2+ cation. The ability of the NIR pulse to convert the initially entangled system into a coherent superposition of electronic states is controlled by changing the time delay between the ionizing XUV pulses. These studies show the crucial role of ion+photoelectron entanglement in attosecond science, especially in attosecond pump-probe experiments, and they present a first attempt to link ultrafast science with quantum information theory.
