Oscillatory multiband dynamics of free particles: The ubiquity of zitterbewegung effects

TL;DR

This paper explores the widespread occurrence of zitterbewegung-like oscillations across various Hamiltonians in relativistic and condensed matter systems, revealing their fundamental connection to quantum uncertainties and magnetic interactions.

Contribution

It demonstrates that zitterbewegung phenomena are common in diverse Hamiltonians, including gapped, spin-split, and Landau systems, and links oscillation amplitudes to measurement uncertainties.

Findings

Oscillation amplitude equals the uncertainty in velocity components.

Zitterbewegung features are shared by Landau and Rashba Hamiltonians.

Oscillatory dynamics arise from interplay of spin splitting and magnetic fields.

Abstract

In the Dirac theory for the motion of free relativistic electrons, highly oscillatory components appear in the time evolution of physical observables such as position, velocity, and spin angular momentum. This effect is known as zitterbewegung. We present a theoretical analysis of rather different Hamiltonians with gapped and/or spin-split energy spectrum (including the Rashba, Luttinger, and Kane Hamiltonians) that exhibit analogs of zitterbewegung as a common feature. We find that the amplitude of oscillations of the Heisenberg velocity operator v(t) generally equals the uncertainty for a simultaneous measurement of two linearly independent components of v. It is also shown that many features of zitterbewegung are shared by the simple and well-known Landau Hamiltonian describing the dynamics of two-dimensional (2D) electron systems in the presence of a magnetic field perpendicular to the plane. Finally, we also discuss the oscillatory dynamics of 2D electrons arising from the interplay of Rashba spin splitting and a perpendicular magnetic field.