Zitterbewegung

The Zitterbewegung (ZB) was first regarded as a relativistic effect rooted in the Dirac equation and related to a ‘trembling’ or oscillatory motion of the center of a free wave packet. The ZB is caused by the interference between the positive and negative energy states in the wave packet; the characteristic frequency of this motion is determined by the gap between the two states. It was believed that the experimental observation of the effect is impossible since one would confine the electron to a scale of the Compton wavelength ℏ(c m₀)^-1, where m₀ is the bare mass of the electron.

However, the ZB is not a strictly relativistic effect: it can appear even for a nonrelativistic particle moving in a crystal or for quasiparticles governed by the Bogoliubov–de Gennes equations in superconductors. Most recently, Schliemann et al. predicted the ZB in spintronic systems where the experimental observation of the effect is more realistic due to the much smaller frequency of the oscillatory motion. In these semiconductor nanostructures spin-orbit coupling generates an oscillatory motion of the wave packet.

In the first paper we present a unified description of the ZB in the systems mentioned above. Our approach makes it possible to calculate with simple algebra (without using operator algebra) the time dependence of the position operator of the particle for a wide class of systems. We also easily verify the results first obtained by Schliemann et al. for spintronic systems. Our result directly shows that the ZB is not necessarily a relativistic effect but it is related to the coupling between the components of the eigenstates of the system. This phenomenon in graphene is thus the direct consequence of the pseudo-spin degree of freedom.

In the second paper we presented a general theory for Zitterbewegung and derived a general and simple expression for the position operator x(t) in Heisenberg picture and in momentum representation, and for a given system it can easily be calculated. In contrast to systems studied in the literature the Zitterbewegung is a universal phenomenon and it always appears in the quantum dynamics of a system of quasi-particle with more than one degree of freedom. Our main result shows that the Zitterbewegung, in general, is a multifrequency beating effect in Heisenberg picture and has a close relation to the Berry connection. We believe that our work presented here provides a better understanding and experimental guide for the Zitterbewegung studied intensively in the literature.

In order to understand the basic concept of Zitterbewegung, one has to look at the problem upside-down: it is no question, why an extra oscillation is superposed on the uniform linear motion of a free particle, but the real question sounds like this: under what circumstances do the all time present oscillations of the quantum theory result in a uniform linear motion?

Publications on this topic

[1] J CSERTI, Gy DÁVID: "Unified Description of the Zitterbewegung for Spintronic, Graphene, and Superconducting Systems" In Physical Review B, 74, 172305, (2006) DOI.

[2] Gy DÁVID, J CSERTI: "General Theory of the Zitterbewegung" In Physical Review B, 81, 121417(R) (2010) DOI.