NEUTRONS AS SCHROEDINGER WAVES AND CATS

 

Helmut Rauch

 

Atominstitut der Österreichischen Universitäten

Stadionallee 2, A-1020 Wien, Austria

 

 

Abstract

 

Neutrons are proper tools for testing quantum mechanics because they are massive, they couple to electromagnetic fields due to their magnetic moment and they are subject to all basic interactions and they are sensitive to topological effects, as well. Related experiments will be discussed. Recent neutron interferometry experiments based on postselection methods renewed the discussion about quantum nonlocality and the quantum measuring process. It has been shown that interference phenomena can be revived even when the overall interference pattern has lost its contrast. This indicates a persisting coupling in phase space even in cases of spatially separated Schrödingercat-like situations. These states are extremely fragile and sensitive against any kind of fluctuations and other decoherence processes. More complete quantum experiments also show that a complete retrieval of quantum states behind an interaction volume becomes impossible in principle.

 

 

1    Introduction - basic relations

 

Different kinds of neutron interferometers based on wavefront and amplitude division have been tested in the past [1-4]. The perfect crystal interferometer - first tested in 1974 at our 250 kW TRIGA reactor - provides highest intensity and became the most frequently used neutron interferometer due to its wide beam separation and its universal availability for fundamental-, nuclear- and solid-state physics [5]. It represents a macroscopic quantum device with characteristic dimensions of several centimeters (Fig. 1).

 


 

 

 


Fig. 1.  Sketch of a symmetric perfect crystal neutron interferometer and a typical interference pattern.

 

 

The basis for this kind of neutron interferometry is provided by the undisturbed arrangement of atoms in a monolithic perfect silicon crystal [6,2]. An incident beam is split coherently at the first crystal plate, reflected at the middle plate and coherently superposed at the third plate. From general symmetry considerations follows immediately that the wave functions in both beam paths, which compose the beam in the forward direction behind the interferometer, are equal (), because they are transmitted-reflected-reflected (TRR) and reflected-reflected-transmitted (RRT), respectively. The theoretical treatment of the diffraction process from the perfect crystal is described by the dynamical diffraction theory [7,8]. To preserve the interference properties over the length of the interferometer, the dimensions of the monolithic system have to be accurate on a scale comparable to the so-called Pendellösung length (~50 mm). The whole interferometer crystal has to be placed on a stable goniometer table under conditions avoiding temperature gradients and vibrations. A phase shift between the two coherent beams can be produced by nuclear, magnetic or gravitational interactions. In the first case, the phase shift is most easily calculated using the index of refraction [9,10]:

 

                                               (1)

 

where bc is the coherent scattering length, sr the reaction cross section, and N is the particle density of the phase shifting material. The different k-vector inside the phase shifter causes a spatial shift  of the wave packet which depends on the orientation of the sample surface  and which is related to the scalar phase shift c by

 

                                                                         (2)

 

where c can be written as a path integral of the canonical momentum kc along the beam paths  [11]. Therefore, the intensity behind the interferometer becomes

 

                                                                                   (3)

 

The intensity of the beam in the deviated direction IH follows from particle conservation . Thus, the intensities behind the interferometer vary as a function of the thickness D of the phase shifter, the particle density N or the neutron wavelength .

Neutron optics is a part of quantum optics and many phenomena can be described properly in that terminology where the coherence function plays an important role [12,13]

 

                                                                                                        (4)

 

which is the autocorrelation function of the wave function. Using a wave packet description for the wave functions (amplitudes ) one obtains

 

                (5)

 

where  and  denote the phase shifts at the mean momentum . This gives:

 

 .                                                                                                 (6)

 

Thus the absolute value of the coherence function can be obtained from the fringe visibility  = (IMax - IMin)/(IMax + IMin) or as the Fourier transform of .

The mean square distance related to  defines the coherence length  which is for Gaussian distribution functions directly related to the minimum uncertainty relation . Similar relations can be obtained for time-dependent phenomena where the spectral distribution g(w) and the temporal coherence function come into play.

Any experimental device deviates from the idealized situations: the perfect crystal can have slight deviations from its perfectness, and its dimensions may vary slightly; the phase shifter contributes to such deviations by variations in its thickness and inhomogeneities; and even the neutron beam itself contributes to a deviation from the idealized situation because of its momentum spread k. Therefore, the experimental interference patterns have to be descibed by a generalized relation

 

                                                                                              (7)

 

where A, B and are characteristic parameters of a certain set-up. It should be mentioned, however, that the idealized behaviour described by eq. (3) can nearly be approached by a well balanced set-up (Fig. 1). Phase shifts can be applied in the longitudinal, transverse and vertical directions and the related coherence properties can be measured [14]. In the transverse direction the phase shift becomes wavelength independent (cT = -2dhklNbcD0; dhkl ... reflecting lattice planes), which implies a much larger coherence length in that direction.

All the results of interferometric measurements, obtained up until now can be explained well in terms of the wave picture of quantum mechanics and the complementarity principle of standard quantum mechanics. Nevertheless, one should bear in mind that the neutron also carries well defined particle properties, which have to be transferred through the interferometer. These properties are summarized in Table 1 together with a formulation in the wave picture. Both particle and wave properties are well established and, therefore, neutrons seem to be a proper tool for testing quantum mechanics with massive particles, where the wave-particle dualism becomes very obvious.

All neutron interferometric experiments pertain to the case of self-interference, where during a certain time interval, only one neutron is inside the interferometer, if at all. Usually, at that time the next neutron has not yet been born and is still contained in the uranium nuclei of the reactor fuel. Although there is no interaction between different neutrons, they have a certain common history within predetermined limits which are defined, e.g., by the neutron moderation process, by their movement along the neutron guide tubes, by the monochromator crystal and by the special interferometer set-up. Therefore, any interferometer pattern contains single particle and ensemble properties together.

 

 

2    Classic neutron interference experiments

 

2.1  Gravity experiments

 

The gravitational interaction of neutrons for usual laboratory conditions is of a comparable order of magnitude with the mean nuclear and magnetic interaction, and, therefore, a measurable interference signal is to be expected. The interaction Hamiltonian in this case reads as

 

                                                                                                        (8)

 

where  means the gravitational acceleration directed towards the center of the earth and w the angular rotation frequency of the earth.

The phase shift within the interferometer is calculated by using the path integral with the canonical momentum as mentioned earlier. In this way, after several intermediate steps, one gets the gravitational phase shift

 

                                                                                                       (9)

 

where A is the area enclosed by the coherent beam trajectories in the interferometer, F is the angle at which the interferometer is turned out of the horizontal plane, FL is the latitude of the point at which the experiment takes place, and e is the angle of rotation around the vertical axis. The first expression in the above equation describes the familiar gravitational term and was proven by Colella, Overhauser and Werner [15] by rotating the interferometer around a horizontal axis (COW-experiment). This phase shift can be understood as the difference in gravitational potentials of the two coherent beams, as one travels higher than the other.

 

 

Table 1.  Properties of neutrons

 

PARTICLE PROPERTIES                                                                                  WAVE PROPERTIES

 

m = 1.6748220(25) x 10-27 kg                         CONNECTION                           = 1.319695(20) x 10-15 m

 

s =                                                             de Broglie

= - 9.6491783(18) x 10-27 J/T                                                            for thermal neutrons: = 1.8 A, v = 2200 m/s

= 882.6 (2.7) s                                              Schrödinger                                   = 1.8 x 10-10 m

R = 0.7 fm                                                                            10-8 m

= 12.0 (2.5) x10-4 fm3                                                   &                                                   10-2 m

u - d - d - quark structure                                boundary conditions                     1.942(5) x 106 m

                                                                                                                              0 2 (4)

 

m  mass, s  spin,   magnetic moment,                                                                 c  Compton wavelength, B  de Broglie

  -decay lifetime, R  (magnetic) con-           -mB  __________________                     wavelength, c  coherence length,

finement radius,   electric polarizability;                      ô two level system      p  packet length, k  momentum width,

all other measured quantities like electric          mB  __________________                     t  chopper opening time, v  group velocity,

charge magnetic monopole and magnetic                                                                phase.

dipole moment are compatible with zero.

 

 

The Coriolis or Sagnac term in eq. (9) was observed experimentally for the first time by Werner et al. [16,17], by directing a neutron beam vertically upwards and by turning a perfect crystal interferometer around this vertical axis (see Fig. 2). The result gives an impression of how sensitive the interferometric measuring method actually is. The easiest way to visualize this effect is by imagining the area encompassed by the two coherent beams as a differently oriented flag on the rotating earth. A more detailed discussion can be found by Greenberger [18].

A complimentary investigation to the gravitational measurements was performed by Bonse and Wroblewski [19], who brought the interferometer in a slightly oscillatory motion and, in so doing, also observed a phase shift, this being proportional to the respective acceleration of the interferometer plates. In summary, this proved the validity of the classical transformation laws for non-inertial frames of reference in the quantum limit.

 

 

2.2  Neutron Fizeau effect

 

An additional phase shift arises when there is a relative motion between the beam and the phase shifter. The calculation of this effect can be done on the basis of Galileian transformation because the velocity of the neutron () is much smaller than the velocity of light. Therefore, the momentum of a particle  inside a material which moves with the velocity  is given as

 

 .                                                                                                                (10)

 

The Fizeau phase shift arises from the different phase shifts of a static and moving phase shifter [20].

 

 .                                                 (11)

 


Quantum mechanics predicts that a Fizeau-phase shift occurs only when the boundary is moving relative to the neutron beam which is quite a difference to the optical Fizeau effect which depends on the motional state of the phase shifter material. The first observation of this effect was achieved by Klein et al. [21] with a double slit interferometer and a moving quartz phase shifter.

 


Fig. 2.  Results of the Earth rotational experiment [16,17].

 

2.3  4p-spinor symmetry

 

This is probably one of the most discussed interference experiments. Based on elementary principles of quantum mechanics, the propagation of a wave function can be described by a unitary transformation, given by the relevant Hamiltonian. For magnetic interaction, , the propagation of the two-component spinor wave function, which describes the neutron as a fermion, can be represented as follows:

 

                         (12)

 

where a means the Larmor precession angle

 

 .                                                                                         (13)

 

When inserting the Pauli spin operators, one can easily show that y(a) has a 4p-symmetry, and not the 2p-symmetry which we are used to with respect to expectation values and within the scope of classical physics,

 

 .                                                                                                             (14)

 

These facts, which were not previously regarded as verifiable, can be elucidated very easily with neutron interferometry by observing the intensity modulations, while one of the coherent beams passes through a magnetic field,

 

 .                                                                   (15)

 

 


 

 


Fig. 3.  First verification of the 4p-symmetry of a spinor wavefunction [22].

 

The above relation is valid for polarized as well as for unpolarized neutrons, which points to the inner symmetry properties of fermions. From eqs. (14) and (15) one recognizes that only for a = 4p is the original state reproduced. This was verified, nearly simultaneously, in measurements by Rauch et al. [22] and by Werner et al. [23] (see Fig. 3). Afterwards, this effect was also proven through several other methods and for a series of other fermion systems.

 

 

2.4  Spin-superposition

 

Spin superposition is an often used principle of quantum mechanics. Its curiousity value has been stressed by Wigner [24]. The wave function of both coherent beams is originally polarized in |z>-direction. One beam is then inverted to a polarization in |-z>-direction, whereas the other remains unchanged. Both beams are then superimposed. This spin flip can be produced, for example, by Larmor precession around a magnetic field perpendicular to z-direction. The result for superposition of these two beams, thus prepared, can be obtained by applying the rotation operator (equ. (12)) to the spin-flipped beam for a rotation of 180° in y-direction. If we also allow for a nuclear phase shift, one gets

 

.                              (16)

 

The total wave function  leads to the following polarization of the out-going beam

 

 .                                                                                              (17)

 

Consequently, this polarization lies in the x,y-plane, and is perpendicular to the polarizations of the two superimposed, coherent beams. This implies that a pure quantum state in |z>-direction, e.g. for c = 0, has been transformed into a quantum state in |x>-direction, and, in the sense of self-interference, which definitively applies here, it seems that each neutron has information about the physical situation in both of the widely separated coherent beams. The experiment by Summhammer et al. [25] has fully confirmed this process. Intensity modulations appear only when the polarization analysis is done in the x,y-plane.

This above-mentioned experiment was repeated with a Rabi resonance flipper, which is also routinely used as a spin flipping device in polarized neutron physics. Along with the spin flip, a simultaneous energy exchange is taking place between the resonator system and the neutrons (DE = hwr = 2mB0, where B0 is the strength of the guide field). Here one has to use the time-dependent Schrödinger equation and to take into consideration the exchange of the total energy of the neutron system

 

                                                                                  (18)

 

which leads to the following polarization, when the other unchanged coherent beam |z> is superimposed,

 

 .                                                                                                       (19)

 

The polarization is also in the x,y-plane, however it rotates within this plane synchroneously with the resonance flip-field. It was possible to demonstrate this effect with a stroboscopic measurement, where the polarization in a given direction was measured synchronously with the phase of the flip-field [26].

In connection with these results, the obvious question arises whether the measurement of the energy transfer makes a determination of the beam path possible. One can, however, show that this is impossible, because interference vanishes in the presence of a measurable energy shift (i.e. larger than the energy width of the beam), and because the measurement of the energy change of the flip-field is impossible due to the photon number-phase uncertainty relationship (DfDN > 1).

These energy exchange measurements have been extended by Summhammer et al. [27] to multiphoton exchange experiments. In this case an oscillating magnetic field with a frequency of 7.534 kHz was inserted into one beam and up to five photon emission and absorption processes have been identified from the time resolved interference pattern.

 

 

2.5  Neutron Josephson effect

 

A double coil arrangement can be used for the observation of a new quantum beat effect, which is the magnetic analog to the well-known superconducting electric Josephson effect. If the frequencies of the two coils are chosen to be slightly different, the energy transfer becomes different too . The flipping efficiencies for both coils are always very close to unity (better than 99%). Now, the wave functions change according to

 

                                                                              (20)

 

Therefore, the intensity behind the interferometer exhibit a typical quantum beat effect, given by

 

 .                                                                                          (21)

 

Thus, the intensity behind the interferometer oscillates between the forward and deviated beam without any apparent change inside the interferometer [28]. The time constant of this modulation can reach a macroscopic scale which is correlated to the uncertainty relation . Figure 4 shows the result of an experiment, where the periodicity of the intensity modulation, T = 2/, amounts to T = (47.90 0.15)s caused by a frequency difference of about 0.02 Hz. This corresponds to a mean energy transfer difference E between the two beams, E = 8.610-17 eV, and to an energy sensitivity of 2.710-19 eV, which is by many orders of magnitudes larger than that of other advanced spectroscopic methods. This high resolution is strongly decoupled from the monochromaticity of the neutron beam, which was EB = 5.510-4 eV around a mean energy of the beam EB = 0.023 eV in this case. The quantum beat effect can also be interpreted as the magnetic Josephson effect analog where the phase difference D(t) is driven by the magnetic energy whereas in the well-known Josephson effect in superconducting tunnel junctions [29], the phase of the

Cooper-pairs in both superconductors is driven by the electrical energy.

 

 

 

 


Fig. 4.  Quantum beat effect observed when the frequencies of the two flipper coils differ by about 0.02 Hz around 71.899 79 kHz [28].

 

 

2.6  Stochastic versus deterministic beam path detection

 

A certain beam attenuation can be achieved either by a semi-transparent material or by a proper chopper or slit system. The transmission probalility in the first case is defined by the attenuation cross section a of the material [t = I/I0 = exp(-aN D)]. The change of the wave function is obtained directly from the complex index of refraction (eq.1):

 

                                                               (22)

 

Therefore, the beam modulation behind the interferometer is obtained in the following form

 

                                                                            (23)

 

On the other hand, the transmission probability of a chopper wheel or another shutter system is given by the open to closed ratio, t = topen/(topen + tclosed), and one obtains after straightforward calculations

 

                                                    (24)

 

i.e. the contrast of the interference pattern is proportional to , in the first case, and proportional to t in the second case, although the same number of neutrons are absorbed in both cases. The absorption represents a measuring process in both cases, i.e. a beam path detection, because compound nuclei are produced with an excitation energy of several MeV, which are usually deexcited by capture gamma rays. The measured contrast lies along the lines ”stochastic“ and ”deterministic“ of Fig. 5 [30,31]. The different contrast becomes especially obvious for low transmission probabilities. The discrepancy diverges for t  but it has been shown that in this regime the variations of the transmission due to variations of the thickness or of the density of the absorber plate have to be taken into account which shifts the points below the - (”stochastic“) curve [32].

 


 

 

 


Fig. 5.  Lattice absorber in the interferometer approaching the classical limit when the slits are oriented horizontally and the quantum limit when they are oriented vertically [34].

 

 

The region between the linear and the square root behaviour can be reached by very narrow chopper slits or by narrow transmission lattices, where one starts to loose information of through which individual slit the neutron went. This is exactly the region which shows the transition between a deterministic and a stochastic situation and, therefore, it can be formulated by a Bell-like inequality (, [33]).

The stochastic limit corresponds to the quantum limit when one does not know anymore through which individual slit the neutron went. Which situation exists depends how the slit widths l compares to the coherence lengths in the related direction. In case that the slit widths become comparable to the coherence lengths, the wave function behind the slits show distinct diffraction peaks which correspond to new quantum states (n 0). The creation of the new quantum states means that those labbeled neutrons carry information about the chosen beam path and, therefore, do not contribute to the interference amplitude [34] (Fig. 5). A related experiment has been carried out by rotating an absorption lattice around the beam axis where one changes from  (vertical slits) to  (horizontal slits). Thus, the attenuation factor t has to be generalized including not only nuclear absorption and scattering processes but also lattice diffraction effects if they remove neutrons from the original phase space.

The partial absorption and coherence experiments are closely connected to the quantum duality principle which states that the observation of an interference pattern and the acquisition of which-way informations are mutually exclusive. Various inequalities have been formulated to describe this mutual exclusion principle [35,36]. The most concise formulation reads as

 

                                                                                                                  (25)

 

where V denotes the fringe visibility (equ. (5)) and P is the predictability of the way through the interferometer, which is a quantitative measure of the a priori which-way knowledge.

 

 

 


 


Fig. 6.  Sketch of various postselection methods.

 

 

3    Postselection experiments

 

Various post-selection measurements in neutron interferometry have shown that interference fringes can be restored by proper filtering methods even in cases when the overall beam does not exhibit any interference fringes due to spatial phase shifts larger than the coherence lengths of the interfering beams [37-38]. Postselection procedures can be applied to various parameters of an experiment:

·  spatial postselection

·  momentum postselection

·  counting statistic postselection

·  phase postselection

·  topology postselection

Figure 6 shows some of them schematically. Here we discuss momentum postselection and phase-echo experiments and refer the reader for other methods to the literature [39-41].

 

 

3.1  Postselection of momentum states

 

The experimental arrangement with an indication of the wave packets at different parts of the interference experiment is shown in Figure 6. An additional monochromatization is applied behind the interferometer by means of a single crystals brought into Bragg-position or by time-of-flight systems. Adding the plane wave interference pattern (equ. (3)), the momentum-dependent intensity reads as:

 

 .                                                             (26)

 


The spatial phase shift-dependent intensity is given by eq. (5). The formula show that the overall interference fringes disappear for spatial phase shifts much larger than the coherence lengths . The surprising feature is that I0(k) becomes oscillatory for large phase shifts where the interference fringes described by eq. (5) disappear ([37], see Fig. Fig. 7). This indicates that interference in phase space has to be considered [42].

Fig. 7.  Interference pattern as a function of the relative phase shift (middle) and related wave packets and momentum spectra behind the interferometer for different values of the phase shift [37].

 

 

The amplitude function of the packets arising from beam paths I and II determines the spatial shape of the packets behind the interferometer

 

                                                                                                     (27)

 

which separates for large phase shifts into two peaks (Fig. 7). For an appropriately large displacement (>>c), the related state can be interpreted as a superposition state of two macroscopically distinguishable states, that is a stationary Schrödinger cat-like state [43,44], but here first for massive particles. These states - separated in ordinary space and oscillating in momentum space - seem to be notoriously fragile and sensitive to dephasing effects [44-48].

Measurements of the wavelength spectrum were made with a silicon crystal with a rather narrow mosaic spread which reflects in the parallel position a rather narrow band of neutrons only (k´/k0 0.0003) causing a restored visibility at large phase shifts [38] (Fig. 8). This feature shows that an interference pattern can be revived even behind the interferometer by means of a proper postselection procedure. In this case the overall beam does not show interference fringes anymore and the wave packets originating from the two different beam paths do not overlap. The momentum distribution has been measured by scanning the analyzer crystal through the Bragg-position. These results clearly demonstrate that the predicted spectral modulation (eq. (26)) appears when the interference fringes of the overall beam disappear. The modulation is somehow smeared out due to averaging processes across the beam due to various imperfections, unavoidably existing in any experimental arrangement. The contrast of the empty interferometer was 60%.

 


 

 

 


Fig. 8.  Interference pattern of the unfiltered overall beam (dk/k0 = 0.012, middle) and the filtered beam reflected from a nearly perfect crystal analyzer in the antiparallel position (dk'/k0 = 0.0003, left) and the observed spectral modulation (right) of the outgoing beam for different phase shifter thicknesses [38].

 

 

The new quantum states created behind the interferometer can be analyzed with regard to their uncertainty properties. Analogies between a coherent state behavior and a free but coherently coupled particle motion inside the interferometer have been addressed [31]. In such cases, the dynamical conjugate variables x and p minimize the uncertainty product with identical uncertainties (x)2 = (k)2 = 1/2 (in dimensionless units). Simple calculations show that for (k)2 a value below the coherent state value can be achieved, which in quantum optic terminology means squeezing [49-52]. One emphasizes that a single coherent state does not exhibit squeezing, but a state created by superposition of two coherent states can exhibit a considerable amount of squeezing. Thus highly nonclassical states are made by the power of the quantum mechanical superposition principle.

 

 

3.2  Contrast retrieval by phase echo

 

A large phase shift ( > c) can be applied in one arm of the interferometer, which can be compensated by a negative phase shift acting in the same arm or by the same phase shift applied to the second beam path [53]. Because the phase shift is additative, the coherence function depends on the net phase shift only. Thus, the interference pattern can be restored as it is shown in form of an experimental example in Fig. 9. The phase-echo method can also be applied behind the interferometer loop when multiplate interferometers are used [41]. In this case, the situation becomes even more similar to the situation discussed in the previous section. The experimental results completely confirmed that behaviour. Phase echo is a similar technique to spin echo [3], which is routinely used in neutron spectroscopy and which represents an interference experiment as well.


 

 

 

 


Fig. 9.  Loss of contrast at high interference and its retrieval by an opposite phase shifter inserted into the same beam [53].

 

 

4    Phase space coupling

 

In the previous sections one noticed that coherence phenomena can be exchanged between various parts of the phase space. The appearing modulation of the momentum distribution when the spatial interference pattern disappears may be the most direct evidence of this phenomena (Fig. 8) [38]. In quantum optics, many phenomena are visualized by Wigner quasi-distribution functions, which are defined as [54,13]

 

                                                           (28)

 

 

where in our case

 

                                                                       (29)

 

we arrive at

 

                                 (30)

                       

                       

 

Integration over the momentum variable gives the spatial distribution (eq. (5)) and integration over the spatial variable gives the momentum distribution (eq. (26)). Typical results are shown in Figure 10 [55].

 


 

 

 


Fig. 10.  Wigner functions for various phase shifts without (left) and with (right) fluctuations in the phase shifter [55].

When fluctuations of the phase shifter (dN or dD0) are included, one notices that the wiggle structure between the separated peaks is more sensitive at high interference order than at low order. This causes a decrease of the coherence and a transition from a pure quantum state to a mixture. As a result, upper limits for the separation of massive (Schrödinger-cat) systems due to unavoidable zero-point fluctuations can be derived. It also indicates why the retrieval of a quantum state from one phase space to another one becomes intrinsically more difficult the larger the separations in one phase space happened.

 

 

5    Topological effects

 

Topological and geometrical effects appear in the solution of the Schrödinger equation due to special geometric forms of the interaction [56-58]. Thus they are part of quantum mechanics but they are easily overlooked by a pure intensity experiment. It also shows that a wavefunction often carries more information than those extracted in a standard experiment. A typical example is the spin superposition experiment discussed in Chapt. 2.4 where the exacted result also depends around which axis the spin has been rotated into the opposite direction. Wagh et al. [59] did recently a related experiment and showed clearly the existence of the topological phase. In a similar sense the scalar and the vector Aharonov-Bohm effects of neutrons have been verified by neutron interferometric methods [60,61]. The geometric nature can be seen in Fig. 11 in comparison to the well-known electron Aharonov-Bohm effects.

The concept of topological phases can be extended to the description of absorption phenomena as they are discussed in Chapt. 2.6. Increasing absorption can be attributed to a more particle-like behaviour (see equ. (25)) and the state moves away from the equator to a latitude circle of the Poincaré sphere with a solid angle W related to the geometrical phase fg

 

 .                                                                                              (31)

 


 

 

 

 


Fig. 11.  Comparison of various Aharonov-Bohm phenomena for electrons and neutrons.

This can be detected by a four-plate double loop interferometer as done by Hasegawa et al. [62] (Fig. 12). Good agreement between the theoretical predictions and the experimental results has been achieved.

 

 

6    Discussion

 

It has been shown in the previous sections that more information about a quantum system can be extracted when more experimentally accessible parameters are measured. It becomes obvious that a system remains coupled in phase space even when it becomes separated in any parameter space. Thus, interference properties can be shifted from one parameter space to another one and back again. Related bands of plane wave components which compose the wave packets may be considered as a responsible factor for the understanding of the coupling and non-locality phenomena in quantum mechanics. It looks like that these plane wave components of the wave packets, i.e. narrow bands, interact over much larger distances than the size of the packets. This interaction guides neutrons of certain momentum bands to the 0- or H-beam, respectively. These phenomena throw a new light on the discussion on Schrödinger-cat-like situations in quantum mechanics [63,64]. Spatially separated packets remain entangled in phase space and nonlocality appears as a result of this entanglement. The analogy with optical experiments performed in the time-frequency domain is striking [65].

The summaries drawn for the different experimental situations discussed in this article are followed by a statement that the retrieval of the interference properties by several post-selection procedures become increasingly more difficult the wider the separation of the quantum system happened before. This is also demonstrated by means of the Wigner distributions (Sec. 4, Fig. 10), where it has been shown that the transition from a quantum state to a mixture is always related to some statistical features of the interaction acting on the quantum system. Such fluctuations are, in principle, unavoidable due to residual quantum fluctuations inherent to any physical system.

 


 


 

 


Fig. 12.  Poincaré presentation of the absorption topological phase and the experimental arrangement of the related experiment [62].

Unavoidable fluctuations (even zero-point fluctuations) cause an irreversibility effect which becomes more influential for widely separated Schrödinger-cat like states. All these effects can be described by an increasing entropy inherently associated with any kind of interaction. This also supports the idea that irreversibility is a fundamental property of nature and reversibility, an approximation only, as stated by several authors [66-69,47].

All the results of the neutron interferometric experiments are well described by the formalism of quantum mechanics. According to the complementarity principle of the Copenhagen interpretation, the wave picture has to be used to describe the observed phenomena. The question how the well-defined particle properties of the neutron are transferred through the interferometer, is not a meaningful one within this interpretation, but from the physical point of view it should be an allowed one.

More complete quantum experiments show that a complete retrieval of all wave components behind an interaction the quantum system experienced becomes impossible, in principle. This implies on a high accuracy level a basic non-commutivity of operators A.B  B.A.

 

 

ACKNOWLEDGEMENTS

 

Most of the experimental results discussed in detail have been obtained by our Dortmund-Grenoble-Vienna interferometer group working at the high flux reactor in Grenoble, and some recent ones stem from our cooperation with the Columbia-Missouri group working at the MURR-reactor. The cooperations with these groups and especially the cooperation with colleagues from our Institute, which are cited in the references, are gratefully acknowledged.

 

 

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