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arXiv:1005.0890

Matter waves in the Talbot-Lau interferometry

Valeriy I. Sbitnev

Published May 6, 2010• 7 citations
arXivHTMLPDF

Abstract

Paths of particles, emitted from distributed sources and passing out through slits of two gratings, $G_{0}$ and $G_{1}$, up to detectors, have been computed in details by the path integral method. The slits are represented by Gaussian functions that simulate fuzzy edges of the slits. Waves of matter be computed by this method show perfect interference patterns both between the gratings and behind the second grating. Coherent and noncoherent the distributed particle sources reproducing the interference patterns are discussed in details. Paraxial approximation stems from the wave function when removing the distributed sources onto infinity. The more hard-edged slits of the grating $G_{1}$ are examined by simulating those slits by a superposition of more hard the Gaussian functions. As for the particles here we consider fullerene molecules. De Broglie wavelength of the molecules is adopted equal to 5 pm.

DOI
10.48550/arxiv.1005.0890

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Paths of particles, emitted from distributed sources and passing out through slits of two gratings, G0G_{0}G0​ and G1G_{1}G1​, up to detectors, have been computed in details by the path integral method. The slits are represented by Gaussian functions that simulate fuzzy edges of the slits. Waves of matter be computed by this method show perfect interference patterns both between the gratings and behind the second grating. Coherent and noncoherent the distributed particle sources reproducing the interference patterns are discussed in details. Paraxial approximation stems from the wave function when removing the distributed sources onto infinity. The more hard-edged slits of the grating G1G_{1}G1​ are examined by simulating those slits by a superposition of more hard the Gaussian functions. As for the particles here we consider fullerene molecules. De Broglie wavelength of the molecules is adopted equal to 5 pm.
Particle paths, emitted from distributed sources and passing out through slits of two gratings, G0 and G1, up to detectors, have been computed in detail by the path integral method. The particles under consideration are fullerene molecules with a De Broglie wavelength equal to 5 pm. The slits are Gaussian functions that simulate fuzzy edges of the slits. Waves of the matter computed by this method show perfect interference patterns both between the gratings and behind the second grating. Coherent and non-coherent distributed particle sources reproducing the interference patterns are discussed in detail. Paraxial approximation results from removing the distributed sources onto innity. This approximation gives a wave function reproducing an exact copy of the Talbot carpet. PACS numbers: 03.75.-b, 03.75.Dg, 42.25.Hz
Path integrals give a possibility to compute in details routes of particles from particle sources through slit gratings and further to detectors. The path integral for a particle passing through the Gaussian slit results in the Gaussian wavepacket. The wavepackets prepared on N slits and superposed together give rise to interference pattern in the near-field zone. It transforms to diffraction in the far-field zone represented by divergent principal rays, at that all rays are partitioned from each other by (N-2) subsidiary rays. The Bohmian trajectories in the near-field zone of N-slit gratings show wavy behavior. And they become straight in the far-field zone. The trajectories show zigzag behavior on the interference Talbot carpet (ratio of particle wavelength to a distance between slits are much smaller than 1 and N>>1). Interference from the the N-slit gratings is simulated by scattering monochromatic neutrons (wavelength=0.5 nm). Also we have considered simulation of interference fringes arising at scattering on an N-slit grating of fullerene molecules (according to the real experiment stated in e-print 1001.0468).
For various experiments to investigate quantum effects, matter-wave diffraction is an apparent example. Particle diffractions have been measured even at the large molecular level, such as the C 60 molecules. In order to describe the matter-wave diffraction of C 60 beam, we assume the initial wave functions behind a grating in form of the Fourier series with a Gaussian distribution function. By applying the Feynman path integral, the exact wave functions of the molecule diffraction in the near-field regime can be derived analytically. The obtained probability density distributions are corresponding to interference fringes as found in the Fresnel diffraction. The probability of finding C 60 molecules, behind the grating at the Talbot length, consistency with the mentioned experimental data of C 60 diffractions.
We show our progress in setting up a three grating near-field Talbot-Lau interferometer (TLI) for large molecules such as C60. The TLI is the preferred tool for showing the wave nature of large particles, as it accepts beams of poor spatial coherence and because of its scaling properties with increasing mass of the particles. The spatial coherence is prepared by the first grating, it can be seen as an array of many collimation slits. Our setup is highly flexible in terms of the possibility of changing most of the parameters (such as distances between the gratings and grating angles) from the outside of the UHV-chamber. As the interferometer and the molecular beam source are place in separate vacuum chambers, refilling the oven and adjusting the setup parameters to different molecules is possible without opening the interferometer chamber. This is necessary, as the pressure in the interferometer chamber has to be lower than 5.10-7 mbar in order to observe high interference contrast. The TLI is highly sensitive to vibrations, especially in the range of 1 kHz (the inverse time-of-flight of the molecules through the interferometer), which makes vibration damping crucial for the success of observing interference. We observed vibration amplitudes of up to 75 nm at low frequencies (158 Hz). Compared to the inverse time-of-flight of the molecules, these vibrations are not 'seen' by the molecules and are non-critical. We observed interference of C60 with a contrast of 27 %, by scanning the self-image of the diffraction grating. In the present setup, an interference contrast of 60 % should be observed with perfect alignment of the gratings. Time-of-flight measurements showed small modulations of the velocity distribution at for interference expected distances of 2.5 ms. We also show the possibility of reconstructing the Wigner distribution function out of measurements for the case of Talbot interference under experimental conditions.
We present a theoretical framework to describe the effects of decoherence on matter waves in Talbot-Lau interferometry. Using a Wigner description of the stationary beam the loss of interference contrast can be calculated in closed form. The formulation includes both the decohering coupling to the environment and the coherent interaction with the grating walls. It facilitates the quantitative distinction of genuine quantum interference from the expectations of classical mechanics. We provide realistic microscopic descriptions of the experimentally relevant interactions in terms of the bulk properties of the particles and show that the treatment is equivalent to solving the corresponding master equation in paraxial approximation.
The de Broglie wavelength λ= h/mv of simple particles (photon, electron, neutron) atoms (helium, neon) and molecules (fullerene, fluorous porphyrin) deduced from the de Broglie (dB) relation and those measured by double slits interferometry are compared.The de Broglie relation is verified in very large domains of velocity v and mass m of the particles, respectively 8 and 13 orders of magnitude.In this paper one analyses the interference spectra of photons and electrons and give the Intensity Relations IR(II-DI) between the interference intensity (II) by two slits and the diffracted intensity (DI) by one slit.The double slits interference in the de Broglie-Bohm interpretation is an interference of particle and pilot waves, internal and external waves each passing by a different slit.The velocities of these waves are discussed.From the universal de Broglie relation and the relation vp = vw between the particle and pilot waves one shows that only in the domain of the maximum of the light spectrum (λm~ 550 µm) the dB relation leads to the Plank-Einstein (PE) relation hν=mc 2 .A measure of the mass of the photon m* = 4 10 -36 Kg is deduced.The de Broglie relation for different wave lengths shows that the PE relation must be modified, the velocity of light is not constant as shown by various experiments: Global Positioning Systems GPS, red shift in astronomy etc.This has been discussed by Marmet in years 2000.The same conclusion was given by Sato in 2010 for electromagnetic waves.The trajectories of electrons observed by Jönsson by double slits interferometry and then calculated by Gondran according to the de Broglie-Bohm interpretation are then analyzed in detail.The correlations observed between the properties of these trajectories at short and long distances from the slits permit to give the general equation of the trajectories.The origin of these properties is discussed in the framework model of De Broglie and Bohm, assuming that at any point the intensity of the two waves (particle and pilot wave) passing by the 2 slits are deceasing functions of the distance to the slits.The changes of direction of the trajectories are explained.