Emerson Sadurni

First experimental realization of the Dirac oscillator.

We present the first experimental microwave realization of the one-dimensional Dirac oscillator, a paradigm in exactly solvable relativistic systems. The experiment relies on a relation of the Dirac oscillator to a corresponding tight-binding system. This tight-binding system is implemented as a microwave system by a chain of coupled dielectric disks, where the coupling is evanescent and can be adjusted appropriately. The resonances of the finite microwave system yield the spectrum of the one-dimensional Dirac oscillator with and without a mass term. The flexibility of the experimental setup allows the implementation of other one-dimensional Dirac-type equations.

The Dirac‐Moshinsky oscillator: theory and applications

This work summarizes the most important developments in the construction and application of the Dirac‐Moshinsky oscillator (DMO). The literature on the subject is voluminous, mostly because of the avenues that exact solvability opens towards our understanding of relativistic quantum mechanics. Here we make an effort to present the subject in chronological order and also in increasing degree of complexity. We start our discussion with the seminal paper by Moshinsky and Szczepaniak and the immediate implications stemming from it. Then we analyze the extensions of this model to many particles. The one‐particle DMO is revisited in the light of the Jaynes‐Cummings model in quantum optics and exactly solvable extensions are presented. Applications and implementations in hexagonal lattices are given, with a particular emphasis in the emulation of graphene in electromagnetic billiards.

Scattering of a particle with internal structure from a single slit

Classically, rigid objects with elongated shapes can fit through apertures only when properly aligned. Quantum-mechanical particles which have internal structure (e.g. a diatomic molecule) also are affected during attempts to pass through small apertures, but there are interesting differences with classical structured particles. We illustrate here some of these differences for ultra-slow particles. Notably, we predict resonances that correspond to prolonged delays of the rotor within the aperture—a trapping phenomenon not found classically.

Conformal mapping and bound states in bent waveguides

Is it possible to trap a quantum particle in an open geometry? In this work we deal with the boundary value problem of the stationary Schroedinger (or Helmholtz) equation within a waveguide with straight segments and a rectangular bending. The problem can be reduced to a one‐dimensional matrix Schroedinger equation using two descriptions: oblique modes and conformal coordinates. We use a corner‐corrected WKB formalism to find the energies of the one‐dimensional problem. It is shown that the presence of bound states is an effect due to the boundary alone, with no classical counterpart for this geometry. The conformal description proves to be simpler, as the coupling of transversal modes is not essential in this case.

Schematic baryon models, their tight binding description and their microwave realization

A schematic model for baryon excitations is presented in terms of a symmetric Dirac gyroscope, a relativistic model solvable in closed form, that reduces to a rotor in the non-relativistic limit. The model is then mapped on a nearest neighbour tight binding model. In its simplest one-dimensional form this model yields a finite equidistant spectrum. This is experimentally implemented as a chain of dielectric resonators under conditions where their coupling is evanescent and a good agreement with the prediction is achieved.

The Dirac-Moshinsky oscillator coupled to an external field and its connection to quantum optics

The Dirac‐Moshinsky oscillator is an elegant example of an exactly solvable quantum relativistic model that under certain circumstances can be mapped onto the Jaynes‐Cummings model in quantum optics. In this work we show, how to do this in detail. Then we extend it by considering its coupling with an external (isospin) field and find the conditions that maintain solvability. We use this extended system to explore entanglement in relativistic systems and then identify its quantum optical analog: two different atoms interacting with an electromagnetic mode. We show different aspects of entanglement which gain relevance in this last system, which can be used to emulate the former.

Stern-Gerlach splitters for lattice quasispin

We design a Stern-Gerlach apparatus that separates quasispin components on the lattice, without the use of external fields. The effect is engineered using intrinsic parameters, such as hopping amplitudes and on-site potentials. A theoretical description of the apparatus relying on a generalized Foldy-Wouthuysen transformation beyond Dirac points is given. Our results are verified numerically by means of wave-packet evolution, including an analysis of Zitterbewegung on the lattice. The necessary tools for microwave realizations, such as complex hopping amplitudes and chiral effects, are simulated.

Canonical Transformations in Crystals

The space representations of linear canonical transformations were studied by Moshinsky and Quesne in 1972. For a few decades, the bilinear hamiltonian remained as the only exactly solvable representative for such problems. In this work we show that the Mello-Moshinsky equations can be solved exactly for a class of problems with discrete symmetry, leading to exact propagators for Wannier-Stark ladders in one and two dimensional crystals. We give a detailed study for a particle in a triangular lattice under the influence of a time-dependent electric field. A more general set of Mello-Moshinsky equations for arbitrary lattices is presented.

Phase Space Evolution and Discontinuous Schrödinger Waves

The problem of Schrodinger propagation of a discontinuous wavefunction – diffraction in time – is studied under a new light. It is shown that the evolution map in phase space induces a set of affine transformations on discontinuous wavepackets, generating expansions similar to those of wavelet analysis. Such transformations are identified as the cause for the infinitesimal details in diffraction patterns. A simple case of an evolution map, such as SL(2) in a two-dimensional phase space, is shown to produce an infinite set of space-time trajectories of constant probability. The trajectories emerge from a breaking point of the initial wave.

Decoherence at constant excitation

We present a simple exactly solvable extension of the Jaynes–Cummings model by adding dissipation. This is done such that the total number of excitations is conserved. The Liouville operator in the resulting master equation can be reduced to blocks of 4×4 matrices.