Elsevier

Annals of Physics

Volume 353, February 2015, Pages 205-221
Annals of Physics

Full quantum mechanical analysis of atomic three-grating Mach–Zehnder interferometry

https://doi.org/10.1016/j.aop.2014.11.012Get rights and content

Highlights

  • A simple model is proposed to analyze experiments based on atomic Mach–Zehnder interferometry.

  • The model can be easily handled both analytically and computationally.

  • A theoretical analysis based on the combination of the position and momentum representations is considered.

  • Wave and particle aspects are shown to coexist within the same experiment, thus removing the old wave-corpuscle dichotomy.

  • A good agreement between numerical simulations and experimental data is found without appealing to best-fit procedures.

Abstract

Atomic three-grating Mach–Zehnder interferometry constitutes an important tool to probe fundamental aspects of the quantum theory. There is, however, a remarkable gap in the literature between the oversimplified models and robust numerical simulations considered to describe the corresponding experiments. Consequently, the former usually lead to paradoxical scenarios, such as the wave–particle dual behavior of atoms, while the latter make difficult the data analysis in simple terms. Here these issues are tackled by means of a simple grating working model consisting of evenly-spaced Gaussian slits. As is shown, this model suffices to explore and explain such experiments both analytically and numerically, giving a good account of the full atomic journey inside the interferometer, and hence contributing to make less mystic the physics involved. More specifically, it provides a clear and unambiguous picture of the wavefront splitting that takes place inside the interferometer, illustrating how the momentum along each emerging diffraction order is well defined even though the wave function itself still displays a rather complex shape. To this end, the local transverse momentum is also introduced in this context as a reliable analytical tool. The splitting, apart from being a key issue to understand atomic Mach–Zehnder interferometry, also demonstrates at a fundamental level how wave and particle aspects are always present in the experiment, without incurring in any contradiction or interpretive paradox. On the other hand, at a practical level, the generality and versatility of the model and methodology presented, makes them suitable to attack analogous problems in a simple manner after a convenient tuning.

Introduction

Matter-wave interferometry constitutes an important application of quantum interference with both fundamental and practical interests [1], [2], [3], [4], [5], [6]. In analogy to optics, this sensitive technique allows us to determine properties of the diffracted particles as well as of any other element acting on them during their transit through the interferometer. It was at the beginning of the 1990s when Kasevich and Chu [7] showed that matter-wave Mach–Zehnder interferometry can be achieved by using the same basic ideas of its optical analog: if the atomic wave function can be coherently split up, and later on each diffracted wave is conveniently redirected in order to eventually achieve their recombination on some space region, then an interference pattern will arise on that spot. For neutral atoms this can be done by means of periodic gratings, which play the role of optical beam splitters. This property, exploited in different diffraction experiments with fundamental purposes  [8], [9], [10], [11], [12], gave rise to the former experimental implementations of atomic Mach–Zehnder interferometers in the early 1990s, first with transmission gratings [13] and then with light standing waves  [14]. These interferometers are based on a very efficient production of spatially separated coherent waves  [15]. Similar interferometers are also used for large molecules [16], [17], [18], although due to their relatively smaller thermal wavelength, they work within the near field or Fresnel regime, benefiting from the grating self-imaging produced by the so-called Talbot–Lau effect  [19], [20], a combination of the Talbot  [21], [22], [23], [24] and Lau  [25], [26], [27] effects.

To describe and analyze this kind of experiments different analytical and numerical treatments have been proposed in the literature  [2], [3], [4], [28]. Nonetheless, there is a substantial gap between simple models and exact numerical simulations, which makes difficult getting a unified view of the physics involved in these experiments. Consequently, many times our understanding of them is oversimplified, which leads us to emphasize “paradoxical” aspects of quantum mechanics. One of these aspects is, for example, the commonly assumed wave–particle dual nature of quantum systems. Depending on how the experiment is performed, one “decides” between one or the other, which manifests as one or another type of outcome. Now, leaving aside ontological issues, a pragmatic, accurate description of the experiment requires the use of a wave function. The evolution of this wave function in the course of the experiment is strongly influenced by the boundary conditions associated with such an experiment, as well as by any other physical effect that might take place (e.g., presence of photon scattering events  [4]), which will unavoidably lead to different outcomes. That is, any trace of paradox disappears if we just focus on the wave function and all the factors that affect it during the performance of the experiment—we obtain what should be obtained, leaving not much room for speculating about dual behaviors. This is precisely the scenario posed by atomic Mach–Zehnder interferometers, where it is common to describe the evolution between consecutive gratings in terms of classical-like paths (see Fig. 1), although their recombination (and, actually, also their emergence) has to do with a pure wave-like behavior.

Moved by these facts and their important implications, here we revisit the problem with a working model consisting of a set of three gratings with Gaussian slits (slits characterized by a Gaussian transmission function), while its analysis is conducted by means of a combination of the position and momentum representations of the (atomic) wave function. As is shown, the synergy between analytical results and numerical simulations obtained in this way helps to describe and understand the functioning of these interferometers in a relatively simple manner. Specifically, we provide a clear picture of the wavefront splitting process that takes place at the gratings, showing how the typical path-like picture of the interferometer  [29] (see Fig. 1) coexists with the complex interference patterns exhibited by the wave function between consecutive gratings. This is a key point to understand the simplified models commonly used in the literature to explain this type of interferometers, where the particular shape of the wave function is neglected and only the momentum carried along the paths associated with each involved diffraction order is considered. In this regard, we have introduced the concept of local transverse momentum, borrowed from the Bohmian formulation of quantum mechanics [30], as analytical tool. By means of this quantity it is possible to properly quantify the local value of the momentum (not to be confused with the usual momentum expectation value) at each point of the transverse coordinate, which is related to the quantum flux  [31], [32] evaluated on that point. Furthermore, we would also like to stress the practical side of this model as an efficient tool to attack analogous problems with presence of incoherence sources and/or decoherent events in a simple manner.

This work has been organized as follows. The general theoretical elements involved in our analysis of the atomic three-grating Mach–Zehnder interferometer are described in Section  2, including the Gaussian grating model considered. Analytical results obtained from this model in the far field are presented and discussed in Section  3. The outcomes from the numerical simulations illustrating different aspects of the wave-function full evolution between consecutive gratings are analyzed and discussed in Section  4. The conclusions from this work are summarized in Section  5.

Section snippets

General aspects of grating diffraction

Atomic three-grating Mach–Zehnder interferometers consist of three evenly-spaced and parallel periodic gratings, as it is illustrated in Fig. 1. In this sketch the slits are parallel to the y axis and span a relatively long distance (larger than the cross-section of the incident beam). This implies translational invariance along the y axis, which makes diffraction to be independent of this coordinate and allows a reduction of the problem dimensionality to the transverse (x) and longitudinal (z)

Beam splitting and subsequent recombination

The paths depicted in Fig. 1 behind G1 or G2 constitute a convenient and simplified representation of the tracks followed by the different diffraction orders that develop from the transmitted atomic wave function. To understand the origin of these paths we need to focus on how this wave function evolves in the far field, where its shape only depends on the aspect ratio x/z, as it is indicated by Eq. (16). With this in mind, let us first consider the transit between G1 and G2. Provided the

Methodology

How close are the previous analytical results to the actual evolution of the atomic wave function inside the interferometer? In order to investigate this question, we decided to perform a series of numerical simulations that illustrate both the full evolution of the wave function between gratings. We would like to stress that the model that we are using is fully analytical (the integration in time of each diffracted Gaussian wave packet ψj is fully analytical  [49]), and therefore the numerical

Concluding remarks

We have shown that by using the relationship between the configuration and momentum representations of a wave function it is possible a simple analysis of atomic three-grating Mach–Zehnder interferometers. This analysis shows how three gratings behave in the same way as the set of two beam-splitters and two mirrors does in conventional optical Mach–Zehnder interferometers. As a convenient working model we have considered a Gaussian grating given its analytical and computational advantages. On

Acknowledgments

Support from the Ministerio de Economía y Competitividad (Spain) under Project No. FIS2011-29596-C02-01 (AS) as well as a “Ramón y Cajal” Research Fellowship with Ref. RYC-2010-05768 (AS), and the Ministry of Education, Science and Technological Development of Serbia under Projects Nos. OI171005 (MB), OI171028 (MD), and III45016 (MB, MD) is acknowledged.

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