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arXiv:1201.1389 · Physical Review A

Huygens-Fresnel-Kirchhoff construction for quantum propagators with application to diffraction in space and time

Arseni Goussev

Published Jan 18, 2012• 20 citations
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Abstract

We address the phenomenon of diffraction of nonrelativistic matter waves on openings in absorbing screens. To this end, we expand the full quantum propagator, connecting two points on the opposite sides of the screen, in terms of the free particle propagator and spatiotemporal properties of the opening. Our construction, based on the Huygens-Fresnel principle, describes the quantum phenomena of diffraction in space and diffraction in time, as well as the interplay between the two. We illustrate the method by calculating diffraction patterns for localized wave packets passing through various time-dependent openings in one and two spatial dimensions.

DOI
10.1103/physreva.85.013626

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We address the phenomenon of diffraction of nonrelativistic matter waves on openings in absorbing screens. To this end, we expand the full quantum propagator, connecting two points on the opposite sides of the screen, in terms of the free particle propagator and spatiotemporal properties of the opening. Our construction, based on the Huygens-Fresnel principle, describes the quantum phenomena of diffraction in space and diffraction in time, as well as the interplay between the two. We illustrate the method by calculating diffraction patterns for localized wave packets passing through various time-dependent openings in one and two spatial dimensions.
A theoretical model of diffraction based on the concept of quantum measurement is presented. It provides a general expression of the state vector of a particle after its passage through an aperture of any shape in a plane screen (diaphragm). This model meets the double requirement of compatibility with the Huygens-Fresnel principle and with the kinematics of the particle-diaphragm interaction. This led to assume that the diaphragm is a device for measuring the three spatial coordinates of the particle and that the transition from the initial state (incident wave) to the final state (diffracted wave) consists of two specific projections which occur successively in a sequence involving a transitional state. In the case of the diffraction at infinity (Fraunhofer diffraction), the model predicts the intensity of the diffracted wave over the whole angular range of diffraction (0 ∘ - 90 ∘ ). The predictions of the quantum model and of the theories of Fresnel-Kirchhoff (FK) and Rayleigh-Sommerfeld (RS1 and RS2) are close at small diffraction angles but significantly different at large angles, a region for which specific experimental studies are lacking. A measurement of the particle flow intensity in this region should make it possible to test the FK and RS1-2 theories and the suggested quantum model.
In this study, the scattering of matter waves that accompany to the quantum particles by an absorbing half-plane is examined. A quantum shutter, which is opened at an initial time, is proposed to close the semi-infinite complementary aperture of the half-screen. Hence the matter waves, confined in a determined region of space, transmits to the second region by scattering from the semi-infinite aperture. The mathematical expressions of the scattered fields are plotted numerically and interpreted according to the diffraction theory of Young.
Using the Fresnel–Kirchhoff integral, we express the state of the light that diffracts from an aperture to a point, by superposition of two or more waves. The amplitudes of the waves are determined by the distances of the point from the ray optics borders of the emerging beams. The interference approach to the Fresnel diffraction leads to a universal formulation of the normalized intensity distribution on the diffraction pattern that determines the state of the incident light, the parameters of the aperture, and its distance from the observation plane, uniquely. In the photon approach, according to the uncertainty principle, confining the position of a photon to an aperture changes the probable propagation direction of the photon in an interval that leads to the diffraction of the photon or light. Thus, the normalized intensity distribution on the diffraction pattern is the probability distribution of a photon. To this probability distribution, a probability amplitude distribution is associated that diffracts similar to a wave in Fresnel diffraction. Applying the introduced approach to the study of the diffraction, from different simple apertures and phase steps, we deduce general behaviors of the Fresnel diffraction and show it is involved in all aspects of light phenomena.
A theoretical model of diffraction based on quantum mechanics is presented. It provides a general expression of the quantum state of a particle after its passage through an aperture in an opaque plane (diaphragm). In this model, the diaphragm is considered as a measurement device not only of the transverse coordinates but also of the longitudinal coordinate of the particle. Moreover, the change of state from the initial state (incident wave) to the final state (diffracted wave) results from two successive projections involving a transitional state which corresponds to the state of the particle when it is localized inside the aperture. These features of the model results from the need to ensure compatibility with the Huygens-Fresnel principle and with the kinematics constraints of the particle-diaphragm interaction. In the case of the diffraction at infinity (Fraunhofer diffraction), the predictions of the quantum model and of the classical theories based on the integrals of Fresnel-Kirchhoff (FK) and Rayleigh-Sommerfeld (RS1 and RS2) are different. They are close at small diffraction angles but contradictory at large angles. In this latter region, only the quantum and RS1 models predict the probable decrease in the intensity of the diffracted wave to zero when the diffraction angle reaches 90{\deg}. However, this decrease is different according to the model and it turns out that the discrepancies become significant beyond 60{\deg}. A measurement of the intensity of the particle flow in this region should allow to test the two theories.
In recent years, matter-wave interferometry has attracted growing attention due to its unique suitability for high-precision measurements and the 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 of diffraction in time. It describes time evolution of an arbitrary initial quantum state in the presence of a time-dependent absorbing barrier, governed by an arbitrary aperture function. Our results enable a quantitative description of diffraction and interference patterns in a large variety of setups, and may be used to devise new diffraction and interference experiments with atoms and molecules.