Quantum mechanics is essential for the understanding of many phenomenons in nature – es- pecially in the atomic and subatomic region. But even for macroscopic objects it has to be as- sumed that they are made of atoms, ions and electrons. Only by using this assumption it is possible to describe all objects exactly and completely. Therefore quantum mechanics can be seen as the basis for the description of all phenomenons in nature. During the development of this new theory most physicists had and have problems to accept and understand it within the scope of our classical and nonrelativistic everyday experience. Especially one of the most fun- damental characteristics of quantum mechanics – the entanglement – resulted in lively discussions. It is not possible to describe entangled particles independently and it is certainly not possible to describe an entangled state by using classical physical laws. The peculiarity of such states was described in 1935 by Schrödinger (in the same paper in which he introduced the term en- tanglement) in one sentence: “Best possible knowledge of a whole does not include best pos- sible knowledge of its parts – and that is what keeps coming back to haunt us” . This publication of Schrödinger was triggered by the famous paper of Einstein, Podolsky and Ro- sen . In this paper EPR analyzed the measurement predictions of a two particle system where the particles can not be described independently. They finally argued that quantum mechanics can not be considered complete at least in the view of their requirements for a physical theory, i.e. determinism and locality. The lively discussions about entangled states were purely philosophical up to 1964. In this year Bell presented an experimentally testable inequality that describes bounds on the so-called local hidden variable theory (proposed to make quantum mechanics complete). This bounds are violated by entangled states. To test quantum mechanics many experiments testing Bell’s inequalities have been realized and nearly all experiments have violated the inequali- ties. The proofs of these violations were up to now not possible beyond all points. A combina tion of experimental loopholes is still remaining. Nevertheless it is most probable that the quantum mechanical description (entanglement) is correct. So in turn Bell’s inequalities are a possibility to test experimentally the entanglement of two (or more) particles. Due to fascinating new ideas entanglement gained new interest in the whole physical society. By their independent proposals of the simulation of physics by using quantum mechanical sys- tems Feynman  and Benioff  opened the door in 1982 to the new field of quantum sim- ulators, quantum information processing, quantum computation and quantum cryptography . For all those new ideas entanglement is essential. The goal of our experiment is to show for the first time explicitly the entanglement of two different quantum mechanical systems – of a single photon and a single atom. Therefore it is necessary to think of a way to generate such an entangled state and how to prove the entanglement in experiments. We create entanglement between the spin-state of an atom and the polarization of a photon emitted from the atom. To prove the entanglement we have to test Bell’s inequality for the quantum numbers of the entangled particles. This means we have to carry out polarization measurements of the single photon and state selective measurements (spin measurements) of the single atom. The challenging part of the verification of the entanglement is the state selective measurement. It consists of a state selective transfer to select the measurement basis and a final state detection. The state selective transfer transfers a chosen superposition of the atomic states (the atomic spin of the two states is different, but with degenerated energy) to a second state distinguishable in energy. For the final state measurement a projection measurement on the two levels (hyperfine states) is carried out. In the scope of this diploma thesis laser systems for the state selective transfer are set up. This state selective transfer is realized by an adiabatic population transfer process. To estimate the influence of different experimental parameters numerical calculations based on a three level model are carried out. The parameters of the chosen setup fits to the calculated requirements for an adiabatic transfer. For the hyperfine state measurement of the atom it is necessary to distinguish atoms in two different energetic levels. It is possible to observe fluorescence light from an atom if the wavelength of the light is suitable to an atomic transition. So an idea to detect atoms in a certain energetic level is to observe fluorescence light from atoms in one of the two states. Another possible way is to transfer enough momentum to an atom resonant to an applied light field to kick the atom out of the trap if it is in one particular state and to leave it in the trap, otherwise. By a subsequent detection of the atom it should be possible to distinguish the energetic states. The results from test measurements of those two ideas are presented in this thesis.