Optics and interferometry with matter waves is the art of coherently manipulating the translational motion of particles like neutrons, atoms and molecules. Coherent atom optics is an extension of techniques that were developed for manipulating internal quantum states. Applying these ideas to translational motion required the development of techniques to localize atoms and transfer population coherently between distant localities. In this view position and momentum are (continuous) quantum mechanical degrees of freedom analogous to discrete internal quantum states. In our contribution we start with an introduction into matter wave optics in sect. 1, discuss coherent atom optics and atom interferometry techniques for molecular beams in sect. 2 and for trapped atoms in sect. 3. In sect. 4 we then describe tools and experiments that allow us to probe the evolution of quantum states of many-body systems by atom interference.
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