Quantum phenomena explored with neutrons

Neutrons are suitable tools for quantum experiments since they are massive, experience nuclear, electromagnetic and gravitational interaction and are easy to manipulate and to detect. Perfect crystal interferometry opened new possibilities to explore quantum phenomena on a new ground. The 4π-symmetry of spinor wave functions, the spin-superposition law, topological quantum phases and various gravitational effects have been examined using this method. Experiments exploiting contextuality and Kochen–Specker phenomena exhibit intrinsic entanglement in single particle systems. This may have consequences for a deeper understanding of quantum physics and for applications in future quantum communication systems.


Introduction
Duality in quantum mechanics attaches matter and wave properties to any physical system. Neutrons, as uncharged elementary particles, are proper tools for the investigation of both fundamental properties since they carry well-known particle and wave properties as shown in table 1. Free neutrons can be produced by various nuclear reactions such as fission, fusion or spallation and slowed down in a moderator material. They can then be detected with high efficiency by various detectors based also on nuclear reaction, e.g. BF 3 or He-3 detectors. Wave and particle features are connected by the de Broglie relation and the Schrödinger equation where p represents the canonical momentum and |ψ( r , t)| 2 the probability of finding the particle within a certain region of space and time interval near r and t, respectively. In the stationary situation, the total energy is conserved and therefore the time-independent form can be used. ∇ 2 ψ(r ) +V (r ) ψ(r ) = E ψ, (1.3) V denotes the space averaged interaction potential, which can be caused by a nuclear, electromagnetic or gravitational interaction. The related potentials are (e.g. Sears 1989)

4)
V m = −µ σ · B(r ), (1.5) where b c and N denote the coherent scattering length and the particle number, respectively; B is the magnetic field strength, g the gravitational acceleration, g the angular velocity of the Earth (|¯ g | = 0.727 × 10 −4 rad s −1 ) and L the angular momentum of the neutrons relative to the center of the Earth. Since it makes a difference how a quantum system experiences an excursion in parameter space, a topological effect can be expected as well (Berry 1984, Shapere andWilczek 1989). In the interaction-free region the solution of the Schrödinger equation is given by a coherent superposition of plane waves resulting in a wave packet ψ( r , t) = (2π ) −3/2 a( k, ω) e i( k r −ωt) d 3 k. (1.7) The density of states g( k, ω) = |a( k, ω)| 2 with ω = ω k =hk 2 /2m and the coherence function, with a characteristic width c (coherence length), can be defined in analogy to standard optical concepts (Glauber 1963, Mandel andWolf 1995).
The basic particle properties of the neutron have been exploited in the past as well. Properties of the nucleon-nucleon force can be investigated by few-body experiments, the weak interaction by measuring the β-decay products of neutrons and the search for an electric dipole moment is intended to discover new physics beyond the standard model of particle physics (Abele 2008). In this paper, we will focus on recent neutron interferometry experiments but also comment on related ultra-cold neutron and gravity quantization experiments (Nesvizhevsky et al 2002, Jenke et al 2011.

Coherence experiments
Photon and particle beams can be split coherently in ordinary or momentum space, which provides the basis for interferometry experiments where the relative phase becomes a measurable quantity. A wide, spatially coherent separation of neutron beams is feasible in perfect crystal interferometers (Rauch et al 1974). In this case nuclear, magnetic and gravitational phase shifts can be created and measured precisely (Rauch and Werner 2000). Coherent beam separation in momentum space is used in spin-echo systems (Mezei 1972(Mezei , 1980 and has recently been used for interferometric measurements as well (Bouwman et al 2008).
Perfect crystal interferometers are based on perfect atomic arrangement in silicon crystals. A monolithic design and a stable environment provide the parallelity of the lattice planes throughout the interferometer. Figure 1 shows different interferometers that have been created and used in the course of our experiments. The wave function behind the interferometer in the forward (O) direction is composed of wave functions arising from beam paths I and II where beam path I is transmitted-reflected-reflected and beam path II is reflected-reflected-transmitted. In the case of zero absorption both wave functions have to be equal due to symmetry principles (ψ I = ψ II ) and when a phase shift (χ = k ds = (1 − n)k D) is applied we have ψ II = e iχ ψ I and an intensity (D denotes the optical path length) I ∝ |ψ I + ψ II | 2 ∝ |ψ I | 2 (1 + cos χ). (2.1) When the wave packet form of the incident beam (equation (1.7)) and unavoidable small disturbances in the crystal and its geometry are taken into account, one has where A, B and are characteristic parameters, which depend on all imperfections of a specific setup. High-contrast and high-order interference pattern can be observed as shown in figure 2. The periodicity of the interference pattern gives the phase shift χ and the envelope gives the coherence function ( ). Equation (2.2) indicates that each interference fringe is slightly different from all the others indicating individuality even in the case of a quasi-periodic phenomenon.
The different interactions mentioned in section 1 cause different phase shifts 3) where D denotes the distance in which the interaction acts on the system, A 0 the area encircled by the coherent beams, α the angle of rotation of the interferometer around the incident beam axis, L the colatitude angle where the experiment is carried out and t the solid angle of the excursion seen from the center of the Bloch sphere. The Sagnac term in equation (2.5) holds in this form when the incident beam lies in the North-South direction. The topological phase χ t depends on the solid angle C of the excursion cycle seen from the center of the Bloch sphere. Measurements of χ n give accurate values for neutron scattering lengths of various elements and isotopes (e.g. Kaiser et al 1979), χ m measurements demonstrated for the first time the 4π -symmetry of spinor wave functions (Rauch et al 1975 and the spin superposition law (Summhammer et al 1983) and χ g measurements demonstrated the gravitational law  and the Sagnac effect for elementary particles (Werner et al 1979). The most direct interferometric verification of a topological quantum phase was obtained by rotation of a spin flipper within an interferometer (Allman et al 1997). More details can be found in the book by Rauch and Werner (2000).

Post-selection and dephasing experiments
Such measurements show that much more information can be extracted even behind the interferometer when advanced post-selection procedures are applied (figure 3). Here, we deal with post-selection in momentum space but other parameter spaces can be used as well. At high interference order ( c ) the interference pattern disappears due to the finite width of the coherence function (equation (2.2)), but at the same time a modulation of the momentum distribution appears (figure 3, Rauch 1993, Jacobson et al 1994. This modulation can be measured with an additional analyzer crystal behind the interferometer. In this case, Schrödinger cat-like non-classical quantum states are produced, which are rather fragile against any fluctuation and dissipation process, and related experiments may help us to understand the transition from a quantum to a classical state (Giulini et al 1996, Haroche andRaimond 2006).
Magnetic noise fields can simulate related decoherence effects even when they are not inherently irreversible and have to be taken rather as dephasing components although the time dependence of the noise field causes multi-photon exchange processes between neutron and field (Summhammer et al 1995). The dephasing can be observed at low order c by the reduction of the interference contrast and at high order ( c ) from the smearing of the momentum distribution.
The experimental setup is shown in figure 4 and typical results measured at large phase shifts are shown in figure 5. There is an ongoing discussion whether such noise-induced decoherence phenomena can simulate quantum decoherence processes where an entanglement to the environment is required. Although the noise field causes multi-photon exchange all processes can be retrieved by opposite noise fields or the same field in the other beam. It is still an open question how an inherently irreversible decoherence process can be realized.
It should be mentioned that the topological (geometric) phases, as discussed in the next section, behave completely differently with respect to noisy fields.

Topological phases
Geometric or topological phases are of interest since they are caused by forceless interactions when the quantum system experiences various excursions in phase space. The field has been  opened by Pancharatnam (1956) and applied to adiabatic, non-adiabatic, cyclic and noncyclic excursions in phase space (Berry 1984, Aharonov and Anandan 1987, Shapere and Wilczek 1989. The phase shift in these cases is given by a dynamical (α) and a geometric term (φ g ) Figure 6. Double-loop interferometer for the measurement of a non-adiabatic and non-cyclic quantum phase (Filipp et al 2005(Filipp et al , 2009a. The closure line C 2 -C 1 follows the geodesic between these points. Measured and calculated values are shown (lower right panel).
with |φ(t) = e iφ |ψ(t) . In the case of an adiabatic excursion the geometric phase becomes half of the solid angle ( ) of the excursion seen by the center of the Bloch sphere. This has been verified with neutrons with a rather high accuracy (φ g = −0.51(1) ) (Allman et al 1997, Filipp et al 2009a. Non-cyclic and non-adiabatic phase excursions have attracted attention recently. In this case the end-point of the excursion has to be connected with the start point by a geodesic line (Samuel and Bhandari 1988). Related experiments have been done by means of a double-loop interferometer, a phase shifter with different thicknesses (SP2) and an absorber sheet (A and transmission T ) where excursions like those shown in figure 6 can be realized (Filipp et al 2005). The dynamical phase can be canceled when the condition χ 1 + χ 2 T = const is fulfilled.
The topological phases may become important for advanced quantum communication, last but not least because they are rather robust against any disturbances as predicted by DeChiara and Palma (2003) and tested experimentally with ultra-cold neutrons by Filipp et al (2009a). This has been done with ultra-cold polarized neutrons trapped within a storage bottle surrounded by Helmholtz coils, which permit a controlled adiabatic rotation of the neutron spin and the application of a noisy field (Filipp et al 2009b). Rotating the spin in the upper and lower hemispheres in opposite directions balances the dynamical phase (spin-echo method) and doubles the dynamical one. First an accurate measurement of the geometric phase has been made without a noisy field (χ t = −0.51(1) t , see equation (2.6)) and then the topological phase has been measured within noisy fields (figure 7). A comparison of the dephasing effect of the dynamical and geometric phases shows completely different behavior (figures 5 and 7). Whereas Figure 7. Measurement of the geometric phase with ultra-cold neutrons using a spin-echo method (above) and the dephasing of this phase in noisy magnetic fields (below) (Filipp et al 2009a).
the dephasing of the dynamical phase increases with the strength and duration of the noisy field, the geometric phase improves with it.

Confinement-induced phase
When a quantum system becomes confined within a potential caused by some wall material, the transverse momentum becomes quantized, k n⊥ = nπ/a, and this causes a change of the longitudinal momentum as well, k ,n = 2m(E in − E n⊥ )/h 2 , and concomitant phase shifts, χ n,conf = (k ,n − k)D ∼ = −k 2 ,n D/2k = −π n 2 λD/4a 2 . Related experiments have been proposed (Levy-Leblond 1987, Greenberger 1988) and realized with neutron interferometry (Rauch et al 2002). These experiments have been performed with a stack of silicon slits with a slit width of a = 22.1 µm, a length of D = 20 mm and a neutron wavelength λ = 0.189(3) nm. The energy levels excited within these slits have energies E 1, = 0.4172 peV, E 2, = 1.669 peV, etc. There are about 360 levels within the potential and their excitation depends on the angle of the incident beam component relative to the surface of the walls. The main contribution to a measurable phase shift comes from the low-lying levels. The situation and typical results are shown in figure 8. As a final result, a phase shift of χ conf = 2.8(4) • has been reported, whereas the theoretical value is 2.5 • . The low contrast of the interference pattern with the slits inserted can be explained by the beam attenuation, the variation of the slit width and the mixing of different quantum states. More recent experiments show even larger phase shifts and a stronger discrepancy with calculated values, which will be the subject of further investigations. This is a worthwhile endeavor because the effect is a purely quantum phenomenon and free from van der Waal interactions and Casimir forces (Casimir and Polder 1948).

Neutron interferometry: violation of a Bell-like inequality
An EPR-Bell argument (Einstein, Podolsky andRosen 1935, Bell 1964) is ideal to focus on the conflict between local realistic theories and quantum mechanics. While a number of experimental tests of the violation of Bell's inequalities (Einstein et al 1935, Bell 1964) have been performed with correlated photon pairs (Tittel et al 1998, Bertlmann andZeilinger 2002 and references therein), a single-neutron system provides a more interesting subject for such tests (Hasegawa et al 2003, Sponar et al 2010a. A class of hidden-variable theories, larger than the local one, is known as the non-contextual hidden-variable theories (NCHVT), where the measured value v[A] of an observable A is assumed predetermined and not affected by a joint (or simultaneous) measurement of an observable B compatible with A (Mermin 1993). Kochen and Specker started studies of non-contextual theories to demonstrate the conflict between these theories and quantum mechanics (Kochen and Specker 1967). An experimentally feasible test consists of joint measurements of commuting observables of single neutrons in an appropriately generated non-factorizable state. The current experiment is an improved version of our previous experiment (Hasegawa et al 2003): a newly developed spin flipper leads to a lower decoherence rate and, consequently, to higher contrast of the interference fringes. As a result, a larger violation than in the previous experiment is observed.
3.1.1. Theory. In the interferometer experiment with polarized neutrons the total wave function represents the entanglement between the spatial part and the spinor part. The normalized total wave function | Bell can be written as a Bell-like state where |↑ and |↓ denote the up-and down-spin states and |I and |II denote the two beam paths in the interferometer. A Bell-like inequality for a single-neutron experiment is given in terms of expectation values E(α; gχ ) for the joint measurement of the spin state and the path by (Basu et al 2001) −2 S 2, where S := E(α 1 , χ 1 ) + E(α 1 , χ 2 ) − E(α 2 , χ 1 ) + E(α 2 , χ 2 ). (3.1) Here, α and χ are associated with the projectors to the spin states, 1 √ 2 (|↑ ± e iα |↓ ), and the path states, 1 √ 2 (|I ± e iχ |II ).
Quantum theory predicts sinusoidal behavior for the count rate such as N (α, χ ) = 1 2 [1 + cos(α + χ)], and similar behavior is also expected for the resulting expectation value: E(α; χ) = cos(α + χ ). According to this dependence of E on α and χ, the Bell-like inequality is not obeyed for various sets of α and χ values. In particular, the theoretical maximum violation, S = 2 √ 2 ≈ 2.82 > 2, is expected for the set α 1 = 0, α 2 = π/2, χ 1 = −π/4 and χ 2 = π/4. In a real experiment, one always encounters unavoidable environmental disturbances, imperfect polarization, limited alignments of optical components and so on. All these factors reduce the visibility of the sinusoidal behavior of N (αsχ), which is characterized by the contrast of the sinusoidal oscillation. The obtained value of S decreases in proportion to these contrasts. That is, the mean contrast should be larger than √ 2/2 (≈70.7%) in order to allow a demonstration of the violation of the Bell-like inequality.

Experiment.
Our experimental demonstration of the violation of a Bell-like inequality by means of interferometry with polarized neutrons consists of three steps: (i) generation of a Bell-like neutron state given by | Bell = 1 √ 2 (|← ⊗ |I + |→ ⊗ |II ), (ii) manipulation of this state, where the parameters α and χ are adjusted to select neutrons with certain properties for detection, and (iii) detection of neutrons realized after a polarization analysis in the O-beam to obtain correlation coefficients. In the present experiment a new method is applied to generate the Bell-like state. A schematic view of the experimental setup is shown in figure 9.
In previous experiments by Hasegawa et al (2003Hasegawa et al ( , 2006Hasegawa et al ( , 2007 a spin-up polarized neutron beam | enters the interferometer and, after splitting into two beam paths at the first plate of the interferometer, its spin was rotated by π/2 in one path and by −π/2 in the other. As a result, the spinor in path I changes to |← , while that in path II changes to |→ , thus yielding a Bell-like state. The new method to generate the Bell-like state consists of two steps: (i) the spin is rotated by π/2 before entering the interferometer (the spinor |↑ is rotated to |→ ) and (ii) the azimuthal angle of the spin in path I is turned by π relative to that in path II. To achieve this, a new spin turner in the interferometer has been developed. In particular, a spin turner for step (ii) is realized by reducing locally the strength of the guide field in beam path I by using appropriate magnetic shielding. A cylindrical tube of Mu metal is used with both ends open, where the neutron beam passes in the axial direction t, without touching any material (see figure 9). This soft-magnetic tube weakens the guide field inside and thus reduces the Larmor precession in this region. In our previous experiments, the neutrons in paths I and II passed through a Mu-metal sheet (magnetized perpendicularly to the guide field), which considerably reduced contrast of the interference fringes due to a dephasing effect due to the passage through this sheet. In contrast, no material is placed in the beam with the new method: the new Mu-metal tube indeed caused much less loss of interference contrast.
The experiment was carried out at the high flux reactor of the Institute Laue-Langevin (ILL) (Kroupa et al 2000). The wavelength of the incident beam was tuned to λ 0 = 1.92 Å. The beam was polarized vertically by magnetic-prism refractions. A parallel-sided Al plate was used as a phase shifter, to vary χ. A pair of water-cooled Helmholtz coils produced a fairly uniform magnetic guide field B 0ẑ around the interferometer. A super-mirror together with a dc spin rotator, to adjust α, enabled us to select neutrons in certain spin directions for detection. The Mu-metal tube was placed in path I of the interferometer. The dimensions of the tube were: length = 13.0 mm, mean diameter = 15.5 mm and wall thickness = 0.10 mm. The measured difference of the azimuthal angle between the spins in paths I and II after the tube was 0.98 ± 0.05π, with a guide field B 0 = 2.19 mT. The contrasts of the interferometer itself (in empty scans) were nearly 90%.
Since a maximum violation of the Bell-like inequality is expected at the values α = 0, π/2, the spin-analysis parameter was tuned to α = 0, π/2, π and −π/2. The contrasts in the χ scans for these α values were C = 0.813(5), 0.717(5), 0.859(5) and 0.726(5), respectively. A typical set of intensity oscillations is shown in figure 10. An overall mean contrast of C = 0.778(3) was achieved, which significantly exceeds the value in the previous experiment (Hasegawa et al 2003). After fitting the measured count rates to a sinusoidal dependence by the least-squares method, the expectation values were determined. The same measurements were repeated seven Figure 10. Typical set of sinusoidal oscillations, obtained by scanning path phase χ at spinor rotation angles α = 0, π/2, π and −π/2. times to reduce statistical errors. We obtained for S in the Bell-like inequality equation (3.1) a value of The error includes statistical and systematic errors, where the main reason for systematic errors was phase instability of the interferometer and inaccuracies in the adjustment of optical elements. This violates the Bell-like inequality by ∼29 standard deviations and so clearly confirms quantum contextuality, while rejecting NCHVT at the same time.

Neutron polarimetry: falsification of a contextual modelà la Leggett
In 2003, Leggett proposed a class of nonlocal (crypto-non-local) hidden-variable theories and proved that his model is incompatible with quantum predictions (Leggett 2003). Experimental tests confirm the incompatibility of such theories (Gröblacher et al 2007a, b, Paterek et al 2007, Branciard et al 2007, 2008. Until now, however, crypto-nonlocal modelsà la Leggett have been tested only with photon systems. Here we report experiments for a crypto-contextual model with matter-waves of neutrons (Hasegawa et al 2011). We exploit entanglement between degrees of freedom, i.e. energy-spin, of neutrons and use a neutron polarimeter (Sponar et al 2010a(Sponar et al , 2010b). This apparatus enables neutron optical experiments with high-efficiency manipulations and insensitivity to ambient disturbances.

Theory.
For our polarimetric test, we follow the criteria used in the first experimental investigation by Gröblacher et al (2007aGröblacher et al ( , 2007b. We define the crypto-contextual theory to be tested here as based on the following assumptions: Whereas assumptions (i) and (ii) are common to ordinary non-contextuality assumptions, assumption (iii), namely Malus' law, demands correlations even between measurements of compatible observables, which is a particular point of crypto-contextual modelsà la Leggett. Following the path used in previous works (Gröblacher et al 2007a, 2007b, Branciard et al 2007 and assuming full rotational symmetry, an inequality can be used to test cryptocontextual hidden variable (CCHV) models in our experiments. Denoting the measurement settings of observables A and B by a 1 , a 2 and b 1 , b 2 on the Bloch sphere (see figure 11), respectively, the crypto-contextual Leggett-like inequality is expressed by where E j ( a j ; φ) represent the expectation value of joint (correlation) measurements at the settings a j and b j with the relative angle φ. For a singlet state, quantum mechanics predicts for the expectation values E j ( a j ; φ) = − a j · b j = − cos φ which gives for the S-function S QM (φ)= 2|1 + cos φ|. The maximum violation is expected at φ max ∼ 0.1π: a bound of the Leggettlike inequality is given by S CCHV = 3.797, whereas quantum mechanics predicts a value of S QM = 3.899. It is worth noting here that the difference between predictions by the model and quantum mechanics is very small, S = 0.102: this calls for extremely high contrast, or rather high correlation, values, i.e. higher than 97.4%, in the measurements and thus makes the experiments non-trivial and challenging.

Experiment.
The experimental setup, together with a Bloch sphere description of the spin and the energy degree of freedom, is depicted in figure 11. The measurement is based on joint measurements of two commuting observables, A spin for the spin and B energy for the total energy degree of freedom of neutrons. In the experiment, a maximally entangled Bell-like state with the spin basis, |↑ and |↓ , as well as the energy basis, |E 0 and |E 0 −hω , is generated, and followed by successive energy and spin measurements. The experiment was carried out at the research reactor facility TRIGA Mark II of the Vienna University of Technology. The incident beam from the reactor is monochromatized to λ = 1.96 Å by a highly oriented pyrolytic graphite (PG) monochromator and propagates in the +y-direction. Reflected from a bent Co-Ti super-mirror array, the beam is highly polarized (more than 99%). The same technique is employed for analyzing the polarization downstream. A high-efficiency (>99%) BF 3 detector is used for measuring the expectation values appearing in equation (3.3) Two identical radiofrequency (RF) spin rotators are employed, each producing a sinusoidally oscillating magnetic field (∼1 G for a π/2 rotation) with ω = 40 kHz. They are about 20 cm long and made of enameled copper wire wound on PVD pipes (diameter ∼4 cm). Both put in a homogeneous and static magnetic guide field (∼13 G) supplied by two rectangular coils in Helmholtz geometry. After tuning the strength of the guide field, scans of the magnetic field amplitude of the RF spin rotators exhibit sinusoidal intensity modulations with more than 99% contrast: highefficiency manipulation is ensured. The second RF spin rotator (RF2) downstream is mounted on a translation table, which enables precise adjustment of the neutron's flight-time between the two spin rotators. The maximally entangled Bell-like state N Bell is generated by tuning the rotation angle of RF1 to π/2. The amplitude and the phase of the oscillating magnetic field tuned by RF2 are directly associated with the parameters of the measurement. Additionally, the adjustment of the position of RF2 enables precise tuning of the relative phase between the two energy eigenstates. The spin analyzer filters out the down-component of the spin. As a result, the needed measurement settings on the equator of the Bloch sphere, for the energy, and arbitrary directions for the spin are realized by RF2 together with the analyzer.
For the measurement of the Leggett-like inequality (equation (3.3)) correlation measurements between settings outside the single plane are needed. The parameter φ is varied, which represents the latitudinal deviation from corresponding points on the equator by varying the phase of RF2 (see figure 11). The polar angle setting is π/2 for the measurement directions in the (equatorial) plane and π/2 − φ for the direction b 2 . We obtained final sinusoidal oscillations with mean contrasts of 98.5%, which is the highest correlation ever obtained between commuting observables of massive particles.
The maximum discrepancy between the crypto-contextual Leggett-model and quantum mechanics is expected at the angle φ max ∼ 0.1π. We have carried out the measurement by setting φ = 0.104π. The S CCHV -value of the crypto-contextual Leggett-like inequality is determined as S CCHV = 3.8387(61), which is clearly larger than the boundary 3.7921. The violation is more than 7.6 standard deviations. In order to see the tendency of the violations, the parameter φ is varied around the point of maximum violation: eight different φ-values are chosen between 0 and 0.226π. Figure 12 is the plot of the experimentally determined S CCHV together with the bound of the contextual model and the quantum mechanical prediction, where a 99% contrast of S-values as a function of the derivation angle φ to test alternative quantum theoryà la Leggett. The red curve is the prediction of quantum mechanics and the broken line gives the boundary provided by the cryptocontextual Leggett model. The S CCHV -value is determined as 3.8387(61) at φ = 0.104π, which is clearly larger than the boundary 3.7921 (Hasegawa et al 2011).
the correlation is taken into account. The obtained values fully follow the quantum mechanical predictions. All errors include statistical and systematic errors: the systematic errors of the present experiments are found to be much smaller than those for the interferometric experiments (Hasegawa et al 2011, 2003, Bartosik et al 2009 where such errors mainly result from the phase instability of the interferograms. The present experiment clearly confirms the violation of the crypto-contextual model.

Concluding remarks
The wave-particle duality is manifest in neutron interference experiments and many textbook experiments of quantum mechanics have been realized with this technique. To a large extent neutron quantum optics paved the way for optics with even heavier particles. The results are well described by standard quantum mechanics although their epistemological understanding is still under discussion. Expectation values are measured but individual events remain random and the physical phenomenon how the quantum world converts into a classical one remains an open question. Here various dephasing experiments with noisy magnetic fields are described which mimic real decoherence effects, which are seen as the cause for this transition. Employing single-particle interferences, intrinsic entanglements between different degrees of freedom have been observed and various Bell-like inequalities have been tested. These results demonstrate quantum contextuality as an important feature of quantum theory, and the strength of entanglement phenomena in physics. From these results various hidden variable theories can be rejected. Multi-entanglement experiments are related to the Kochen-Specker theorem, which may help to further broaden our understanding of quantum phenomena. It may even become relevant for modern quantum communication systems, mainly because in the entanglement feature additional information can be stored.