Milena Davidović

Understanding interference experiments with polarized light through photon trajectories

Abstract Bohmian mechanics allows to visualize and understand the quantum-mechanical behavior of massive particles in terms of trajectories. As shown by Bialynicki-Birula, Electromagnetism also admits a hydrodynamical formulation when the existence of a wave function for photons (properly defined) is assumed. This formulation thus provides an alternative interpretation of optical phenomena in terms of photon trajectories, whose flow yields a pictorial view of the evolution of the electromagnetic energy density in configuration space. This trajectory-based theoretical framework is considered here to study and analyze the outcome from Young-type diffraction experiments within the context of the Arago–Fresnel laws. More specifically, photon trajectories in the region behind the two slits are obtained in the case where the slits are illuminated by a polarized monochromatic plane wave. Expressions to determine electromagnetic energy flow lines and photon trajectories within this scenario are provided, as well as a procedure to compute them in the particular case of gratings totally transparent inside the slits and completely absorbing outside them. As is shown, the electromagnetic energy flow lines obtained allow to monitor at each point of space the behavior of the electromagnetic energy flow and, therefore, to evaluate the effects caused on it by the presence (right behind each slit) of polarizers with the same or different polarization axes. This leads to a trajectory-based picture of the Arago–Fresnel laws for the interference of polarized light.

Electromagnetic energy flow lines as possible paths of photons

Motivated by recent experiments where interference patterns behind a grating are obtained by accumulating single photon events, we provide here an electromagnetic energy flow-line description to explain the emergence of such patterns. We find and discuss an analogy between the equation describing these energy flow lines and the equation of Bohmian trajectories used to describe the motion of massive particles.

Should particle trajectories comply with the transverse momentum distribution

Abstract.The momentum distributions associated with both the wave function of a particle behind a grating and the corresponding Bohmian trajectories are investigated and compared. Near the grating, it is observed that the former does not depend on the distance from the grating, while the latter changes with this distance. However, as one moves further apart from the grating, in the far field, both distributions become identical.

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

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.

Trajectory-based interpretation of Young's experiment, the Arago–Fresnel laws and the Poisson–Arago spot for photons and massive particles

We present a trajectory-based interpretation of Youngs experiment, the Arago–Fresnel laws and the Poisson–Arago spot. This approach is based on the equation of the trajectory associated with the quantum probability current density in the case of massive particles, and the Poynting vector for the electromagnetic field in the case of photons. Both the form and properties of the evaluated photon trajectories are in good agreement with the averaged trajectories of single photons observed recently in Youngs experiment by Steinbergs group at the University of Toronto. In the case of the Arago–Fresnel laws for polarized light, the trajectory interpretation presented here differs from interpretations based on the concept of ‘which-way’ (or ‘which-slit’) information and quantum erasure. More specifically, the observers information about the slit that the photons went through is not relevant to the existence of interference; what is relevant is the form of the electromagnetic energy density and its evolution, which will model consequently the distribution of trajectories and their topology. Finally, we also show that the distributions of end points of a large number of evaluated photon trajectories are in agreement with the distributions measured at the screen behind a circular disc, clearly giving rise to the Poisson–Arago spot.

Coherence loss and revivals in atomic interferometry: a quantum-recoil analysis

The coherence effects induced by external photons coupled to matter waves inside a Mach–Zehnder three-grating interferometer are analyzed. Alternatively to atom–photon entanglement scenarios, the model considered here only relies on the atomic wavefunction and the momentum shift induced in it by the photon scattering events. A functional dependence is thus found between the observables, namely the fringe visibility and the phase shift, and the transversal momentum transfer distribution. Good quantitative agreement is found when comparing the results obtained from our model with the experimental data.

Description of Classical and Quantum Interference in View of the Concept of Flow Line

Bohmian mechanics, a hydrodynamic formulation of quantum mechanics, relies on the concept of trajectory, which evolves in time in compliance with dynamical information conveyed by the wave function. Here, this appealing idea is considered to analyze both classical and quantum interference, thus providing an alternative and more intuitive framework to understand the time evolution of waves either in terms of the flow of energy (for instance, for mechanical waves, sound waves, and electromagnetic waves) or analogously the flow of probability (quantum waves). Furthermore, this procedure also supplies a more robust explanation of interference phenomena, which currently is only based on the superposition principle. That is, while this principle only describes how different waves combine and what effects these combinations may lead to, flow lines provide a more precise explanation on how the energy or probability propagates in space before, during, and after the combination of such waves, without dealing with them separately (i.e., the combination or superposition is taken as a whole). In this sense, concepts such as constructive and destructive interference, typically associated with the superposition principle, physically correspond to more or less dense swarms of (energy or probability) flow lines. A direct consequence of this description is that, when considering the distribution of electromagnetic energy flow lines behind two slits, each one covered by a differently oriented polarizer, it is naturally found that external observers’ information on the slit crossed by single photons (understood as energy parcels) is totally irrelevant for the existence of interference fringes, in striking contrast to what is commonly stated and taught.

Image positions of a vertical rod in a liquid-filled cylindrical container

We describe a rather simple optical experiment, which many students can easily carry out, but the theoretical explanation of which requires far-from-simple mathematical analysis and application of numerical methods. A thin vertical rod, partially immersed in a liquid-filled transparent cylindrical container, is moved from the centre toward the wall and observed laterally. As it moves, the part seen through the air and the part seen through the water start to separate, so that at a certain distance from the centre the lower part of the rod becomes invisible. We show that this happens at the distance for which the refracted ray that reaches the observers eye is along the tangent to the surface of a cylinder. We derive the expression for this distance as a function of the index of refraction and ratio d/r, where d is the distance of the observer from the surface and r is the radius of the container. The locus of image positions is determined by evaluating the intersections of pairs of close rays from the rod which reach the observers eye after refraction at the surface of the container.

Evolution of the wave function of an atom hit by a photon in a three-grating interferometer

In 1995, Chapman et al. (1995 Phys. Rev. Lett. 75 2783) showed experimentally that the interference contrast in a three-grating atom interferometer does not vanish under the presence of scattering events with photons, as required by the complementarity principle. In this work we provide an analytical study of this experiment, determining the evolution of the atom wave function along the three-grating Mach-Zehnder interferometer under the assumption that the atom is hit by a photon after passing through the first grating. The consideration of a transverse wave function in momentum representation is essential in this study. As is shown, the number of atoms transmitted through the third grating is given by a simple periodic function of the lateral shift along this grating, both in the absence and in the presence of photon scattering. Moreover, the relative contrast (laser on/laser off) is shown to be a simple analytical function of the ratio d_p/\lambda_i, where d_p is the distance between atomic paths at the scattering locus and \lambda_i the scattered photon wavelength. We argue that this dependence, being in agreement with experimental results, can be regarded to show compatibility of the wave and corpuscle properties of atoms.

Derivation of the Husimi symbols without antinormal ordering, scale transformation and uncertainty relations

We propose a new method for the derivation of Husimi symbols, for operators that are given in the form of products of an arbitrary number of coordinates, and momentum operators, in an arbitrary order. For such an operator, in the standard approach, one expresses coordinate and momentum operators as a linear combination of the creation and annihilation operators, and then uses the antinormal ordering to obtain the final form of the symbol. In our method, one obtains the Husimi symbol in a much more straightforward fashion, departing directly from operator explicit form without transforming it through creation and annihilation operators. With this method the mean values of some operators are found. It is shown how the Heisenberg and the Schrodinger–Robertson uncertainty relations, for position and momentum, are transformed under scale transformation . The physical sense of some states which can be constructed with this transformation is also discussed.