Quantum formalism of Fraunhofer diffraction is obtained. The state of the diffraction optical field is connected with the state of the incident optical field by a diffraction factor. Based on this formalism, correlations of the diffraction modes are calculated with different kinds of incident optical fields. Influence of correlations of the incident modes on the diffraction pattern is analyzed and an explanation of the ''ghost'' diffraction is proposed.
We present one- and two-photon diffraction and interference experiments involving parametric down-converted photon pairs. By controlling the divergence of the pump beam in parametric down-conversion, the diffraction-interference pattern produced by an object changes from a quantum (perfectly correlated) case to a classical (uncorrelated) one. The observed diffraction and interference patterns are accurately reproduced by Fourier-optical analysis taking into account the quantum spatial correlation. We show that the relation between the spatial correlation and the object size plays a crucial rol…
We demonstrate one- and two-photon diffraction and interference experiments utilizing parametric down-converted photon pairs (biphotons) and a transmission grating. With two-photon detection, the biphoton exhibits a diffraction-interference pattern equivalent to that of an effective single particle that is associated with half the wavelength of the constituent photons. With one-photon detection, however no diffraction-interference pattern is observed. We show that these phenomena originate from the spatial quantum correlation between the down-converted photons.
A simple scheme is proposed for observing the ghost interference and diffraction. The signal and the idler beams are produced by a beam splitter with the incident light being in a thermal state. A slit is inserted into the signal beam. We derive rigorously that interference-diffraction patterns can be observed in the first-order correlation by scanning the probe in the idler beam.
In this paper we will be concerned with the explanation of the interference and diffraction patterns observed as an outcome of the Young double slit experiment. We will show that such explanation may be given {\it only} in terms of a corpuscular theory, which has been our approach since the first paper of this series. This explanation will be accomplished with an extension we make here of the domain of applicability of the Born-Sommerfeld rules that we derived in paper XI of this series.
Interference and diffraction of two-identical-particles are considered in the context of open quantum systems. This theoretical study is carried out within two approaches, the effective time-dependent Hamiltonian due to Caldirola-Kanai (CK) and the Caldeira-Leggett (CL) one where a master equation for the reduced density matrix is used under the presence of dissipation and temperature of the environment. Two simple but very illustrative examples are considered, diffraction by a single and two Gaussian slits by analyzing the mean square separation between particles, single-particle probability …
Quantum history states were recently formulated by extending the consistent histories approach of Griffiths to the entangled superposition of evolution paths and were then experimented with Greenberger--Horne--Zeilinger states. Tensor product structure of history-dependent correlations was also recently exploited as a quantum computing resource in simple linear optical setups performing multiplane diffraction (MPD) of fermionic and bosonic particles with remarkable promises. This significantly motivates the definition of quantum histories of MPD as entanglement resources with the inherent capa…
A double cavity with a quantum mechanical and a classical field is located immediately behind of a double-slit in order to analyse the wave-particle duality. Both fields have common nodes and antinodes through which a three-level atom passes after crossing the double-slit. The atom-field interaction is maximum when the atom crosses a common antinode and path-information can be recorded on the phase of the quantum field. On other hand, if the atom crosses a common node, the interaction is null and no path-information is stored. A quadrature measurement on the quantum field can reveal the path f…
We show that single and multislit experiments involving matter waves may be constructed to assess correlations between the position and momentum of a single free particle. These correlations give rise to position dependent phases which develop dynamically and may play an important role in the interference patterns. For large enough transverse coherence lenght such interference patterns are noticeably different from those of a classical dispersion free wave.
We show that periodically doped, flat surfaces can act as reflective diffraction gratings for atomic and molecular matter waves. The diffraction element is realized by exploiting that charged dopants locally suppress quantum reflection from the Casimir-Polder potential. We present a general quantum scattering theory for reflection off periodically charged surfaces and discuss the requirements for the observation of multiple diffraction peaks.
Run-and-tumble particles, frequently considered today for modeling bacterial locomotion, naturally appear outside a biological context as well, e.g. for producing waves in the telegraph process. Here, we use a wave function to drive their propulsion and tumbling. Such quantum-active motion realizes a jittery motion of Dirac electrons (as in the famous Zitterbewegung): the Dirac electron is a run-and-tumble particle, where the tumbling is between chiralities. We visualize the trajectories in diffraction and double slit experiments for electrons. In particular, that yields the time-of-arrival st…
We employ the broken-link model to create a barrier with slits in a two-dimensional lattice. The diffraction and interference patterns of the probability distribution of quantum walkers passing through the slits are analyzed. Simulations were performed using the main types of coins, and display diffraction and interference patterns that depend on the choice of coins.
The loss of contrast in double-slit electron-diffraction due to dephasing and decoherence processes is studied. It is shown that the spatial correlation function of diffraction patterns can be used to distinguish between dephasing and decoherence. This establishes a measure of time-reversibility that does not require the determination of coherence terms of the density matrix, while von Neumann entropy, another measure of time-reversibility, does require coherence terms. This technique is exciting in view of the need to understand and control the detrimental experimental effects of contrast los…
Matter-wave interferometry has been largely studied in the last few years. Usually, the main problem in the analysis of the diffraction experiments is to establish the causes for the loss of coherence observed in the interference pattern. In this work, we use different type of environmental couplings to model a two slit diffraction experiment with massive particles. For each model, we study the effects of decoherence on the interference pattern and define a visibility function that measures the loss of contrast of the interference fringes on a distant screen. Finally, we apply our results to t…
Scattering cold particles on an $N$-slit grating is shown to reproduce an interference pattern, that manifests itself in the near-field region as the fractal Talbot carpet. In the far-field region the pattern is transformed to an ordinary diffraction, where principal beams are partitioned from each other by ($N-2$) weak ones. A probability density plot of the wave function, to be represented by a gaussian wavepacket, is calculated both in the near-field region and in the far-field one. Bohmian (geodesic) trajectories, to be calculated by a guidance equation, are superimposed on the probability…
Quantum backflow is an interference effect in which a matter-wave packet comprised of only plane waves with non-negative momenta exhibits negative probability flux. Here we show that this effect is mathematically equivalent to the appearance of classically-forbidden probability flux when a matter-wave packet, initially confined to a semi-infinite line, expands in free space.
Quantum Origin of Diffraction from Bright and Dark States
School of Science, Xi’an University of Posts and Telecommunications（西安邮电大学理学院）；School of Physics and Information Technology, Shaanxi Normal University（陕西师范大学物理学与信息技术学院）
Double-slit diffraction is a corner stone of quantum mechanics. It illustrates key features of quantum mechanics: interference and the particle-wave duality of matter. In 1965, Richard Feynman presented a thought experiment to show these features. Here we demonstrate the full realization of his famous thought experiment. By placing a movable mask in front of a double-slit to control the transmission through the individual slits, probability distributions for single- and double-slit arrangements were observed. Also, by recording single electron detection events diffracting through a double-slit…
The concept of quantum superposition is reconsidered and discussed from the viewpoint of Bohmian mechanics, the hydrodynamic formulation of quantum mechanics, in order to elucidate some physical consequences that go beyond the simple mathematical idea of linearly combining vectors in a Hilbert space. Specifically, the discussion turns around the connection between symmetries characterizing the wave function and the behavior in time displayed by the quantum flux when the latter is analyzed in terms of streamlines (Bohmian trajectories). This is illustrated with a series of analytical results an…
In the recent years, mater-wave interferometry has attracted growing attention due to its unique suitability for high-precision measurements and study of fundamental aspects of quantum theory. Diffraction and interference of matter waves can be observed not only at a spatial aperture (such as a screen edge, slit, or grating), but also at a time-domain aperture (such as an absorbing barrier, or "shutter", that is being periodically switched on and off). The wave phenomenon of the latter type is commonly referred to as "diffraction in time". Here, we introduce a versatile, exactly solvable model…
We analyze the diffraction regime of the Kapitza-Dirac effect for particles entangled in momentum. The detection patterns show two-particle interferences. In the single-mode case we identify a discontinuity in the set of joint detection probabilities, associated with the disconnected character of the space of non-separable states. For Gaussian multi-mode states we derive the diffraction patterns, providing an example of the dependence of the light-matter interaction on entanglement. When the particles are identical, we can explore the relation between exchange and entanglement effects. We find…
In 1924 Louis de Broglie introduced the concept of wave-particle duality: the Planck constant $h$ relates the momentum $p$ of a massive particle to its de Broglie wavelength $λ=h/p$. The superposition principle is one of the main postulates of quantum mechanics; diffraction and interference phenomena are therefore predicted and have been observed on objects of increasing complexity, from electrons to neutrons and molecules. Beyond the early electron diffraction experiments, the demonstration of single-electron double-slit-like interference was a highly sought-after result. Initially proposed b…
Diffraction in time (DIT) is a fundamental phenomenon in quantum dynamics due to time-dependent obstacles and slits. It is formally analogous to diffraction of light, and is expected to play an increasing role to design coherent matter wave sources, as in the atom laser, to analyze time-of-flight information and emission from ultrafast pulsed excitations, and in applications of coherent matter waves in integrated atom-optical circuits. We demonstrate that DIT emerges robustly in quantum waves emitted by an exponentially decaying source and provide a simple explanation of the phenomenon, as an …
We study the possibility of creating spatial patterns having subwavelength size by using the so-called dark states formed by the interaction between atoms and optical fields. These optical fields have a specified spatial distribution. Our experiments in Rb vapor display spatial patterns that are smaller than the length determined by the diffraction limit of the optical system used in the experiment. This approach may have applications to interference lithography and might be used in coherent Raman spectroscopy to create patterns with subwavelength spatial resolution.
The wave-particle duality of massive objects is a cornerstone of quantum physics and a key property of many modern tools such as electron microscopy, neutron diffraction or atom interferometry. Here we report on the first experimental demonstration of quantum interference lithography with complex molecules. Molecular matter-wave interference patterns are deposited onto a reconstructed Si(111) 7x7 surface and imaged using scanning tunneling microscopy. Thereby both the particle and the quantum wave character of the molecules can be visualized in one and the same image. This new approach to nano…
In this study, we solve analytically the Schrodinger equation for a macroscopic quantum oscillator as a central system coupled to two environmental micro-oscillating particles. Then, the double-slit interference patterns are investigated in two limiting cases, considering the limits of uncertainty in the position probability distribution. Moreover, we analyze the interference patterns based on a recent proposal called stochastic electrodynamics with spin. Our results show that when the quantum character of the macro-system is decreased, the diffraction pattern becomes more similar to a classic…
In the Kapitza-Dirac effect, atoms, molecules, or swift electrons are diffracted off a standing wave grating of the light intensity created by two counter-propagating laser fields. In ultrafast electron optics, such a coherent beam splitter offers interesting perspectives for ultrafast beam shaping. Here, we study, both analytically and numerically, the effect of the inclination angle between two laser fields on the diffraction of pulsed, low-energy electron beams. For sufficiently high light intensities, we observe a rich variety of complex diffraction patterns. These do not only reflect inte…
We analyze Niels Bohr's proposed two-slit interference experiment with highly charged particles that argues that the consistency of elementary quantum mechanics requires that the electromagnetic field must be quantized. In the experiment a particle's path through the slits is determined by measuring the Coulomb field that it produces at large distances; under these conditions the interference pattern must be suppressed. The key is that as the particle's trajectory is bent in diffraction by the slits it must radiate and the radiation must carry away phase information. Thus the radiation field m…
Time-resolved femtosecond x-ray diffraction patterns from laser-excited molecular iodine are used to create a movie of intramolecular motion with time and space resolution of $30~$fs and $0.3$ Å. The high spatial fidelity is due to interference between the moving excitation and the static initial charge distribution. This x-ray interference has not been employed to image internal motion in molecules before. The initial state is used as the local oscillator for heterodyne amplification of the excited charge distribution to retrieve real-space movies of atomic motion on Ångstrom and femtosecond …
We report on the theoretical investigation of Wigner distribution function (WDF) reconstruction of the motional quantum state of large molecules in de Broglie interference. De Broglie interference of fullerenes and as the like already proves the wavelike behaviour of these heavy particles, while we aim to extract more quantitative information about the superposition quantum state in motion. We simulate the reconstruction of the WDF numerically based on an analytic probability distribution and investigate its properties by variation of parameters, which are relevant for the experiment. Even tho…
The smallest spot in optical lithography and microscopy is generally limited by diffraction. Quantum lithography, which utilizes interference between groups of N entangled photons, was recently proposed to beat the diffraction limit by a factor N. Here we propose a simple method to obtain N photons interference with classical pulses that excite a narrow multiphoton transition, thus shifting the "quantum weight" from the electromagnetic field to the lithographic material. We show how a practical complete lithographic scheme can be developed and demonstrate the underlying principles experimental…
Sources of broadband quantum correlated photons present a valuable resource for quantum metrology, sensing, and communication. Here, we report the generation of spectrally broadband correlated photons from frequency nondegenerate spontaneous parametric down-conversion in a custom-designed lithium niobate superlattice. The superlattice induces a nonlinear interference, resulting in an experimentally observed comb-like emission spanning 0.060 um and 1.4 um of spectral bandwidth at 0.647 um and 3.0 um wavelengths, respectively. In addition to useful technological applications, our concept offers …
We explore the hydrodynamic analogues of quantum wave-particle duality in the context of a bouncing droplet system which we model in such a way as to promote comparisons to the de Broglie-Bohm interpretation of quantum mechanics. Through numerical means we obtain single-slit diffraction and double-slit interference patterns that strongly resemble those reported in experiment and that reflect a striking resemblance to quantum diffraction and interference on a phenomenological level. We, however, identify evident differences from quantum mechanics which arise from the governing equations at the …
In this article we present an analytic solution of the famous problem of diffraction and interference of electrons through one and two slits (for simplicity, only the one-dimensional case is considered). In addition to exact formulas, we exhibit various approximations of the electron distribution which facilitate the interpretation of the results. Our derivation is based on the Feynman path integral formula and this work could therefore also serve as an interesting pedagogical introduction to Feynman's formulation of quantum mechanics for university students dealing with the foundations of qua…
We present a simplified model for dynamical diffraction of particles through a periodic thick perfect crystal based on repeated application of a coherent beam splitting unitary at coarse-grained lattice sites. By demanding translational invariance and a computationally tractable number of sites in the coarse-graining we show how this approach reproduces many results typical of dynamical diffraction theory and experiments. This approach has the benefit of being applicable in the thick, thin, and intermediate crystal regimes. The method is applied to a three-blade neutron interferometer to predi…
Fourth-order interference is an information processing primitive for photonic quantum technologies. When used in conjunction with post-selection, it forms the basis of photonic controlled logic gates, entangling measurements, and can be used to produce quantum correlations. Here, using classical weak coherent states as inputs, we study fourth-order interference in novel $4 \times 4$ multi-port beam splitters built within multi-core optical fibers. Using two mutually incoherent weak laser pulses as inputs, we observe high-quality fourth order interference between photons from different cores, a…
Mass-angle correlation of fission fragments has been understood as manifestation of quasifission. We show that this is not so: the effect can originate from correlation between fusion-fission amplitudes with different total spins signifying matter-wave interference in compound nucleus processes. This resolves the well-known puzzle with the mass-angle correlation in the complete fusion sub-barrier reaction $^{16}$O+$^{238}$U. Our finding is important for more reliable predictions of production cross sections for superheavy elements. Matter-wave interference also produces quantum-classical trans…
We consider various effects that are encountered in matter wave interference experiments with massive nanoparticles. The text-book example of far-field interference at a grating is compared with diffraction into the dark field behind an opaque aperture, commonly designated as Poisson's spot or the spot of Arago. Our estimates indicate that both phenomena may still be observed in a mass range exceeding present-day experiments by at least two orders of magnitude. They both require, however, the development of sufficiently cold, intense and coherent cluster beams. While the observation of Poisson…
Interference is one of the most fundamental features which characterizes quantum systems. Here we provide an exhaustive analysis of the interfere dynamics associated with wave-packet superpositions from both the standard quantum-mechanical perspective and the Bohmian one. From this analysis, clear and insightful pictures of the physics involved in this kind of processes are obtained, which are of general validity (i.e., regardless of the type of wave packets considered) in the understanding of more complex cases where interference is crucial (e.g., scattering problems, slit diffraction, quantu…
Spatial and momentum correlations are important in the analysis of the quantum states and different phases of trapped ultracold atom systems as a function of the strength of interatomic interactions. Identification and understanding of spin-resolved patterns exhibited in two-point correlations, accessible directly by experiments, are key for uncovering the symmetry and structure of the many-body wave functions of the trapped system. Using the configuration interaction method for exact diagonalization of the many-body Hamiltonian of $N=2-4$ fermionic atoms trapped in single, double, triple, and…
We show how robust statistical features of a many-particle quantum state's two-point correlations after transmission through a multi-mode random scatterer can be used as a sensitive probe of the injected particles' mutual indistinguishability. This generalises Hong-Ou-Mandel interference as a diagnostic tool for many-particle transmission signals across multi-mode random scatterers. Furthermore, we show how, from such statistical features of the many-particle interference pattern, information can be deduced on the temporal structure of the many-particle input state, by inspection of the many-p…
