Nonrelativistic zitterbewegung in two-band systems

Ferrari, L.; Russo, Giovanna

doi:10.1103/physrevb.42.7454

Cited by 52 publications

(45 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The factor k D is a measure for the summation in momentum space and is considered to be analogous to the measure dk of an integral. Since momentum space is spaced equidistantly in our approach, we write down a sum in (14) with measure Δ k and focus on the plane wave solution in this more fundamental theory part. The spacing of Δ k and its relation to position space is discussed in the section on numerical implementation 4.…”

Section: Time Evolution Of Maxwell Equationsmentioning

confidence: 99%

“…Nevertheless, the Zitterbewegung can be modelled in various other effective systems and its appearance has been subject of diverse investigations. It has been discussed for superconductors [12] in 1970, for onedimensional chains [13] in 1990 and many considerations set in on semiconductors [14][15][16][17][18][19][20]. In 2005 renewed discussions on Zitterbewegung in Spintronics [21,22], Graphene [23,24] and carbon nanotubes [25,26] were raised, (see also [27]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Simulation of Zitterbewegung by modelling the Dirac equation in metamaterials

et al. 2015

View full text Add to dashboard Cite

Strong field processes which occur in intense laser fields are a test area for relativistic quantum field theory, but difficult to study experimentally and theoretically. Thus, modelling relativistic quantum dynamics and therewith the Dirac equation can help to understand quantum field theory. We develop a dynamic description of an effective Dirac theory in metamaterials in which the wave function is modelled by the corresponding electric and magnetic field in the metamaterial. This electromagnetic field can be probed in the experimental setup, which means that the wave function of the effective theory is directly accessible by measurement. Our model is based on a plane wave expansion which establishes the identification of Dirac spinors with single-frequency excitations of the electromagnetic field in the metamaterial. We check the validity of our relativistic quantum dynamics simulation by demonstrating the emergence of Zitterbewegung and verifying it with an analytic solution.OPEN ACCESS RECEIVED waveguide operates in the microwave regime, the simulated wave function can be directly probed in the experiment. But only quasi-stationary field configurations have been investigated and the question arises, whether the time-independent description in terms of frequency-eigensolutions is capable of forming dynamics which correspond to the Dirac equation.Here, we extend the quasi-static theory to a dynamic description of an effective Dirac equation (section 2) and use it for the demonstration of the Zitterbewegung in theory. In the sections 2.1 and 2.2 we repeat the already established description [38] of the Maxwell equations in metamaterials and show how solutions of the Dirac equation can be deduced. Next, we formulate a formal solution of the Maxwell equations by an expansion in plane-wave solutions in section 2.3. This solution can be identified with the unitary time-evolution of the Dirac equation, which is discussed in section 2.4, in which we also give an explicit mapping between the electromagnetic field and the Dirac wave-function in frequency-and momentum space. The effective parameters for the mass and the speed of light of the emulated Dirac equation depend on the metamaterial and are derived in section 2.5. In section 3 we refer back to the Maxwell equations for deriving boundary conditions, with which electromagnetic input pulses can be injected at the metamaterial interfaces in our simulation.After the introduction of the theory foundations, we describe the numerical implementation in section 4, in which we also consider the properties of periodic boundary conditions which are implied by our simulation method.In the results section 5, we present the metamaterial simulations in three kinds of dynamical scenarios, in which we first consider the easiest possible setup for a Zitterbewegung with a Gaussian wave packet excitation 5.1. This setup is modified into a counterpropagating excitation, such that an experimental realization might be more feasible 5.2 and finally we also account for the influence of t...

show abstract

Section: Time Evolution Of Maxwell Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Simulation of Zitterbewegung by modelling the Dirac equation in metamaterials

et al. 2015

View full text Add to dashboard Cite

show abstract

“…Dirac cones and Dirac points owe their names to the formal analogy existing between the two-dimensional Helmholtz equation describing the electromagnetic behavior of photons near these points and the Dirac equation for the motion of relativistic, free electrons in the absence of external fields. The subject is well-rooted in solid state physics, in which Dirac cones appear in a great variety of scenarios such as non-relativistic motion of particles in a crystal 8 and semiconductor nanostructures for spintronic applications. 9 Particularly important in this framework is the recent discovery of graphene, 10 a purely two-dimensional electronic system in which the conduction band and the valence band touch each other at the Dirac point, leading to remarkable electronic transport properties.…”

Section: Introductionmentioning

confidence: 99%

Field localization and enhancement near the Dirac point of a finite defectless photonic crystal

et al. 2013

View full text Add to dashboard Cite

We use a rigorous electromagnetic approach to show the existence of strongly localized modes in the stop band of a linear, two-dimensional, finite photonic crystal near its Dirac point. At normal incidence, the crystal exhibits a Dirac point with 100% transmission. At angles slightly off the normal, where the crystal is 100% reflective, instead of exponentially decaying fields as in a photonic stop band, the field becomes strongly localized and enhanced inside the crystal. We explain that this anomalous localization is due to guided mode resonances that are the foundation of the Dirac point itself and also shape its adjacent band gap. Besides shedding new light on the physical origin of Dirac points in finite photonic crystals, our results could have applications in many nonlinear light-matter interaction phenomena in which it is crucial to achieve a high degree of light localization. DOI: 10.1103/PhysRevB.87.08513

show abstract

“…The theoretical existence of the quivering motion has been evidenced by numerical simulations of the Dirac equation and of quantum field theory. While Zitterbewegung oscillations cannot be directly observed by current experimental techniques for a Dirac electron since the amplitude should by very small (equal to the Compton wavelengthh/mc with m the rest mass of the relativistic particle, namely ≈ 10 −12 m for an electron), solid state and atomic physics provide physical hardware to simulate the phenomenon [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52], and they have recently boosted a renewed interest in the Dirac equation features.…”

Section: Introductionmentioning

confidence: 99%

Discrete Time Dirac Quantum Walk in 3+1 Dimensions

D’Ariano

Mosco

Perinotti

et al. 2016

Entropy

View full text Add to dashboard Cite

Abstract:In this paper we consider quantum walks whose evolution converges to the Dirac equation in the limit of small wave-vectors. We show exact Fast Fourier implementation of the Dirac quantum walks in one, two, and three space dimensions. The behaviour of particle states-defined as states smoothly peaked in some wave-vector eigenstate of the walk-is described by an approximated dispersive differential equation that for small wave-vectors gives the usual Dirac particle and antiparticle kinematics. The accuracy of the approximation is provided in terms of a lower bound on the fidelity between the exactly evolved state and the approximated one. The jittering of the position operator expectation value for states having both a particle and an antiparticle component is analytically derived and observed in the numerical implementations.

show abstract

Nonrelativistic zitterbewegung in two-band systems

Cited by 52 publications

References 14 publications

Simulation of Zitterbewegung by modelling the Dirac equation in metamaterials

Simulation of Zitterbewegung by modelling the Dirac equation in metamaterials

Field localization and enhancement near the Dirac point of a finite defectless photonic crystal

Discrete Time Dirac Quantum Walk in 3+1 Dimensions

Contact Info

Product

Resources

About