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Interference of light and material particles is described with a unified model which does not need to assume the wave-particle duality. A moving particle is associated with a region of spatial correlated points named coherence cone. Its geometry depends on photon or particle momentum and on the parameters of the experimental setup. The final interference pattern is explained as a spatial distribution of particles caused by the coherence cone geometry. In the present context, the wave front superposition principle, wave-particle duality and wave-collapse lose their meaning. Fits of observed single electron and single molecule interference patterns together with the simulation of expected near-field molecule interference (Talbot carpet) demonstrate the model validity.

Interference of light and material particles is usually described with the superposition principle applied to waves. In case of light they are related to the electromagnetic field while, in case of a material particle, to the particle probability density. In spite of the success of this approach, it is a matter of fact that the description of massive objects propagation in terms of waves is a puzzling aspect of quantum mechanics.

Here we describe interference of light and material particles with a unified model which does not need to assume the wave-particle duality of Copenhagen interpretation. The energy propagation of photons or of massive particles is approached by solving the corresponding stationary coupled Helmholtz equations by the Green functions method, which yields two-point correlation functions [

In Section 2 the basic mathematical model is illustrated. Its reliability is demonstrated in Section 3 with, 1) the simulation of the build-up of two-slit single electron interference fringes, 2) the fit of a single electron interference we have observed using a linear transmission grating, 3) the fit of single molecule interference recorded at the Vienna Centre of Quantum Physics and Technology with a sophisticated interferometer, 4) the simulations of near-field, molecule interference figures which are known as Talbot carpets. Finally, a discussion section is reported.

To introduce our approach, let us consider the Schrödinger and Maxwell equations for moving particles and light propagation in field free space, respectively. Taking into account only the stationary solutions

where

Let us consider a space volume delimited by two parallel planes at a distance _{P}) and output plane (O_{P}), _{P} can be expressed as a modal expansion, whose coefficients are given by the two-point correlation at the I_{P}. In turn, the non-paraxial modes, obtained by solving Equation (1) by the Green’s function method [

The reduced coordinates _{P} and _{P}. The region around each position suffixed A includes pairs of points for which the two-point correlation takes on non-null values. It is called the structured support of spatial coherence. Outside the structured support, the two-point correlation nullifies or is negligible. The modal kernel

is a scalar, dimensionless, complex valued and deterministic function, essentially determined by the boundary conditions of the arrangement. _{P}.

By assuming the spatial symmetry of Equation (3) for each point in I_{P}, the modal kernel defines a cone with vertex at the input plane and base at the output plane,

density distribution at I_{P} and O_{P} are the physical observables given by

where

is the contribution of the structured support centred at a given

with

and

given, respectively, by the first and second term of the dimensionless function. From Equation (3) it follows that, in Equation (5a),

with

and

In summary, a model for particle propagation from I_{P} to O_{P} is obtained by considering, 1) two sets of point emitters, with different physical attributes, distributed on I_{P} and, 2) the corresponding cone geometries confined within the volume delimited by I_{P} and O_{P}. The real point emitters represent the photon irradiance or the particle density at a given point

The virtual point emitters are determined by the corresponding two-point correlation over the structured support centred at the emitter position. The correlation cones depend on the shape and size of the experimental setup and the particle momentum according with the modal kernel of Equation (5b). The virtual cones modulate the space between I_{P} and O_{P}, thus contributing to the formation of the final interference pattern. In conclusion, the propagation of particles between two planes is reduced to define two types of cones, i.e. the real and the virtual ones. The mathematical procedure to define the minimum number and distribution of real and virtual point emitters to describe the final interference pattern is analogous to what reported in optics. (see [

Although particle interference has also been described in terms of trajectories (see [

In this section we give a pictorial physical description of the coherence and real particle distribution through the interferometer as derived from the present model. The setup is divided into two stages in which, the output plane of the first stage is the input plane of the second one. With reference to _{P}) and the mask (M_{P}) planes placed at a distance z' to each other. The second stage MD is confined between M_{P} and the detector plane (D_{P}) placed at a distance z. The effective particle source, of size a, is at S_{P}, while a mask with two holes at a distance b, is at M_{P}. A conventional squared modulus detector is at D_{P}.

The behaviour of photons or particles in both stages is described by inserting, in Equations (2)-(6), the corresponding experimental parameters (the integration domains, the distance between planes, etc.). _{P} as in graphs, (a) for high correlation and, (c) for zero correlation. In

Under the high correlation conditions of

In case of zero correlation conditions in the SM-stage, the virtual source is not activated at M_{P} leaving the MD-stage completely uncorrelated,

Let us proof the validity of the model by considering a two-slit or alternatively a double-hole, single electron experiment realized with the Young setup of _{P}. The term “uniform” means that the emission probability is the same for all the source points and any of them emits the same average number of electrons per unit time with the same probability. For a spatial incoherent source _{P} so that, in the SM-stage, Equation (2) becomes

(in optics, the paraxial approach of this expression is given by the Van Cittert-Zernike theorem [_{P}. Accordingly a finite lateral coherence length is defined at a plane, at any distance from the source. This is a substantial different result with respect to the one obtained with a wave approach according to which an infinite transversal coherence is considered. By enlarging the effective source size the number of correlation cones increases. Their superposition in the SM-stage reduces the resulting structured support dimensions (lateral coherence) at M_{P}, as discussed previously. An important consequence is that the effective source size determines the coherence conditions for interference, independently of the electron emission statistics.

The two holes of the Young mask are modelled with mathematical points at

where

with

By taking into account only the fraction S_{0} of electrons which go through the mask, we define P the probability of an electron passing the pinhole at the position

of the complex degree of spatial coherence [_{P} and the condition

Simulations of the build-up of single electron interference patterns with high, partial and very low correlation conditions are shown in ^{−}^{22} kg∙m∙s^{−}^{1}). The two-point correlation at M_{P} is adjusted by changing the distance z' in the SM-stage to obtain three different coherence conditions while, the distance z in the MD-stage is fixed. The resulting detection events of 70,000 electrons are calculated. Detector sensitivity is modeled by adjusting the minimum number of “hits” on a specific pixel, during the integration time, in order to produce the record of electron arrivals. The background and the shot noises of conventional physical detectors are also taken into account. Zero electron loss is assumed so that the emission statistics determines the random electron arrivals to the detector.

Images of _{P} at the pixel indicated at top right. The number of arrivals is reported at top left. The lower frames show the interference figures resulting from the build-up of 70,000 single electrons. The cumulative, single electron detection process is accurately simulated with the present model in accordance with observations [

https://youtu.be/gcKUWLjXvBQ

https://youtu.be/R4zBLL1Wv10

https://youtu.be/wgCb7O9eUqE

The potential of our model is presented to fit the interference pattern obtained with single electrons striking on a line grating (see Appendix 3 for details).

Simulations were carried out by considering a uniform effective source whose size is controlled with the condenser lens system of the microscope [_{P}. Due to the particular construction features of the grating, to obtain a careful fit of the interference peak modulation a Gaussian transmission function was used. Because images were recorded with non-standard electron optical ray path, microscope lens aberrations and non-linear response of the photographic recording were not considered. The excellent agreement with the experimental pattern of

Starting from these promising results, we simulate the single molecule interference patterns of Phthalocyanine (PcH_{2}) and its derivative (F_{24}PcH_{2}) [_{2} and large F_{24}PcH_{2} molecules are, respectively, 1) collimating slit 1 μm wide, collimating slit-grating distance 0.702 m, 2) collimating slit 3 μm wide, collimating slit-grating distance 0.566 m [_{2} and 2.1 pm for F_{24}PcH_{2}, is corrected for vertical dispersion due to gravity (for a more direct comparison of our simulations with those of [

It turns out that the lateral coherence distance at the grating plane is larger for PcH_{2} molecules than for F_{24}PcH_{2} molecules. Because the grating period is the same in both experiments, the number of correlated slits is higher for the small molecules with respect to the bigger ones. It must be pointed out that the numerical fit of the experimental data of

rigorous evaluation of the two-point correlation conditions (Appendix 4), gives directly an excellent fit of the observed results advancing no further hypothesis.

Talbot carpets are near field interference figures which are repeated at specific distances away from the grating plane. The transversal dimension of the maxima and their periods, which depends on the grating distance, can be consistently reduced with respect to the grating slit size. In principle, this peculiarity is worth considering to realize molecular nanostructures on a large scale.

Interference of particles moving in field free space is described with a model that does not associate wave properties to moving corpuscles. Interference fringes result from the combined effect of particle momenta and peculiar configurations of spatial correlated points set out by the two-point correlation function. In particular, we have demonstrated that a careful fit of the experimental data is obtained without assuming that massive molecules propagate as delocalized quantum waves. Therefore, the wave front superposition principle is not needed and, as a consequence, the counter-intuitive features such as wave-particle duality, self-interference and wave- collapse lose their meaning. At present, we are not able to explain the fundamental nature of the spatial correlation but we have reported a physical intuitive and rigorous theory one can calculate with and obtain striking results.

A further remarkable outcome of the model is also obtained. Counter-intuitive infinite spatial coherence (for instance a plane wave) associated with an ideal point source (Dirac delta function) as adopted in Copenhagen interpretation is no longer considered. We have demonstrated that a finite coherence cone is also calculated (in far field conditions) for an ideal point source (see Appendix 5 for details about the definition and size of the structured support).

We hope that these considerations will help to shed new light regarding the role of the space on a basic problem of quantum mechanics.

We wish to thank M. Arndt and his colleagues (Vienna Center of Quantum Science and Technology) for providing the interference curves of their experimental results and F. Navarria, R. Brancaccio and A. Amorosi for

the critical reading of the manuscript. We are indebted to H. Muñoz (Universidad Nacional de Colombia in Medellín) for the realization of the simulations of

RománCastañeda,GiorgioMatteucci,RaffaellaCapelli, (2016) Quantum Interference without Wave-Particle Duality. Journal of Modern Physics,07,375-389. doi: 10.4236/jmp.2016.74038

The conservation of the total wave energy or the total number of particles in the setup is assured by the condition

From Equations (6) it follows

so that

Equations (A1) and (A2) yield

The experimental setup was modelled as a 1D arrangement with a random, uniform emission of single electrons in the SM-stage so that,

Furthermore, _{P}. Consequently, Equations (2) and (4) give

and

respectively, where z' is the axial length of the SM-stage. By assuming that the rms-error between the exact calculation and the paraxial approach for our setup configuration is smaller than 0.5% and by restricting the simulation in the paraxial Fraunhofer domain [

at M_{P}. It means that the Young mask is uniformly illuminated. In addition,

holds in the SM-stage, so that the two-point correlation at M_{P} is given by

_{P}. Fraunhofer paraxial approach was also used in MD-stage. From Equations (7) and (8) it follows

for the physical observable at D_{P}.

We used a Philips EM400 transmission electron microscope (TEM) equipped with a hair-pin filament source operating at 60 keV, giving an electron momentum p = 1.36 × 10^{−}^{22} kg∙m∙s^{−}^{1}. In standard operating conditions, the average distance between consecutive electrons is of the order of 1cm so that, as a reasonable approximation, electrons go through a 30 nm thin film sample one at a time [^{−}^{5} rad, was obtained [

by assuming the grating with

Simulations were carried out under different correlation conditions (

As expected, the profiles show the decrease of the correlation width with the increase of the effective source size. Consequently, the interference fringe contrast is reduced, and a broadening of the maximum angular widths takes place, while the maximum and minimum positions are unaffected. These results show how closely the experimental images can be interpreted by introducing only few approximations for the overall working conditions of the microscope setting. Calculations were implemented by using the conventional Mathematica® platform. They can be also performed in Mathlab®.

The red, solid line curves in _{P} and the grating slits at M_{P}, in the SM-stage. With these entries, the two-point correlation and

the molecule density at M_{P} are calculated in a similar way to the one used in the single electron experiment.

The molecule density at D_{P} for a large number of molecules is given by

as the physical observable molecule density at D_{P} after a great number of single molecule arrivals.

To obtain the redsolid profiles in _{P} and the grating slits at M_{P}, in the SM-stage. Instead of a sinc-function as in Equation (A9), the two-point correlation is now proportional to the Gaussian

experimental results. The physical observable molecule density at D_{P} after a great number of single molecule arrivals is given now by

In order to reproduce the interference profiles of

PcH_{2} | λ (pm) | 4.0 | 4.2 | 4.4 | 4.6 | 4.8 | 5.0 | 5.2 | 5.4 | 5.7 | 6.0 |
---|---|---|---|---|---|---|---|---|---|---|---|

PcH_{2} | WF | 0.06 | 0.12 | 0.24 | 0.35 | 0.47 | 1.0 | 0.71 | 0.53 | 0.29 | 0.18 |

F_{24}PcH_{2} | λ (pm) | 2.0 | 2.1 | 2.2 | 2.3 | 2.4 | 2.5 | 2.6 | 2.7 | 2.8 | 3.0 |

F_{24}PcH_{2} | WF | 0.8 | 1 | 0.8 | 0.7 | 0.6 | 0.5 | 0.4 | 0.3 | 0.2 | 0.1 |

An important question is to define the size of the structured supports of spatial coherence due to different distributions of real point emitters. It is not trivial to establish analytically because of the non-linearity of the expansion kernel of Equation (2), defined in Equation (3). However, the numerical analysis of the behavior of a given correlation cone in the volume delimited by I_{P} and O_{P}, ^{−}^{22} kg∙m∙s^{−}^{1}, centered at

while the correlation cone of the linear array of emitters takes the form

_{P} placed at a distance z = 10^{4} μm from I_{P}, for the single real point emitter and the arrays under the two extreme conditions of spatially coherence and incoherence, respectively. Any partially coherence situation should be contained between these two extremes of spatial coherence.

Although the profiles were calculated for_{A} in such regions. Outside these regions, the geometry of the profiles change but the physical description above remains valid.

In analogy with optics, the cross-section inscribed by the central maximum (i.e. delimited by the first zeroes of _{P}), determines the structured support on O_{P}. However, in case of a source consisting of a single point emitter, the cross-section of the correlation cone is Lorentzian-like at any axial position z, _{P}, the structured support size grows linearly along the z-axis as represented by the dotted lines,

According to

A real point emitter provides a structured support larger than those obtainable with a linear array of point sources. The longer the effective source the smaller the structured support results. As shown in

In accordance to the profiles in