Simulation of Zitterbewegung by modelling the Dirac equation in metamaterials

Abstract

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...

Analytic Solutions to Coherent Control of the Dirac Equation

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