Weak Measurements and Uncertainty Relations in Neutron Optics

Heisenbergs uncertainty principle is, without any doubt, one of the most profound statements in modern quantum physics. It states that pairs of certain properties of a quantum particle cannot be measured with arbitrary accuracy - for instance position and momentum. This is often justified by the notion that every measurement necessarily disturbs the quantum particle, which affects the results of any further measurements. This, however, turns out to be an oversimplification: the uncertainty is partly rooted in the quantum nature of the particle itself. Quantum particles cannot be described as point-like objects with a well- defined velocity. Instead, quantum particles behave like waves, which in general cannot be assigned an exact position. Consequently, a generalized (measurement) uncertainty relation taking both kinds of uncertainty into account - that is from the measurement process and from the quantum behavior - was developed in 2003 and experimentally tested for the first time by us using neutrons. The present research project is aimed to further investigate these types of measurement uncertainty relations. The second topic of the research projects are so called weak values, which are obtained in an experimental procedure usually referred to as weak measurements. The basic idea behind weak measurements is to gain very little information about the observed system, by keeping the disturbance on it, caused by the measurement process, (negligible) small. Weak values have a lot of useful applications. In our case they can be utilized to completely characterize, or more precisely, to reconstruct the state of a quantum system, or express error and disturbance for measurement uncertainty relations. These two phenomena are investigated with neutrons by applying a well-know neutron optical tool, namely the neutron interferometer. Neutron interferometry is based on the wave nature of neutrons. If neutrons are isolated from the atomic nucleus, for example in the fission process of a research reactor, they behave like waves. This is usually refereed the as wave-particle duality. Inside the interferometer a beam of neutrons is split into two sub- beams by reflection from the crystal lattice and recombined coherently afterwards, resulting in interference effects. The resulting high sensitivity makes the neutron interferometer perfectly suited for investigations of uncertainty relations and weak values.

Within the scope of the resrach project fundamental properties of quantum measurements were investigated using neutron optical experiments. Two main topics formed the focus of this research project; on the one hand uncertainty relations (both measurement and preparation uncertainty relations), as well as weak values or weak measurements. The dual nature of neutrons - in some respect a particle, in other respect a wave - rrepresents a suitable starting point for the present research project and thus forms the basis for the investigation of the fundamentals of quantum mechanics with neutrons. Two well-established neutron optical approaches, namely neutron interferometry, where a beam of thermal neutrons is split into two sub-beams and after manipulations on the sub-beams is recombined again, and neutron polarimetry, also referred to as spin interferometry, have been utilized in a series of experiments. In the present research project, experiments to investigate uncertainty relations within the theoretical framework of the operator approach, as well as operational definitions via unsharp measurements in the form of so-called "positive operator value measurements" (POVMs) were successfully carried out. In the present project, we have provided experimental evidence that when performing such general (or unsharp) measurements as in the case of POVMs on a qubit system (here the neutron spin), tighter limits of the uncertainty relation are observed compared to usual projective measurements. Furthermore, we examined various information-theoretic (entropic) approaches to uncertainty relations with regard to both measurement uncertainty relations and preparation uncertainty relations for non-orthogonal observables. Furthermore, the concept of weak values has been used to introduce a new "which-way" observable, which we call "path presence." The measured path presence corresponds to the weak value of the path projector but, in contrast to a normal weak value, is not a statistical average, but applies to each individual neutron. In addition, the commutator relation, which is extremely important for quantum mechanics, was directly checked. This was achived by a weak measurement of the imaginary part of the weak value of a path projector in the neutron interferometer. The project has provided a significant advance in the studies of fundamental questions in quantum physics, since our measurement results are not limited to the neutron's spin and path degree of freedom, but are in fact completely general and can be used for any two qubit (two-level) quantum system.