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arXiv:cond-mat/0609005 · Physical Review B

Oscillatory multiband dynamics of free particles: The ubiquity ofzitterbewegungeffects

R. Winkler, Ulrich Zülicke, Jens Bolte

Published May 10, 2007• 70 citations
arXivHTMLPDF

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 $\mathbf{v}(t)$ generally equals the uncertainty for a simultaneous measurement of two linearly independent components of $\mathbf{v}$. 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.

DOI
10.1103/physrevb.75.205314

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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)\mathbf{v}(t)v(t) generally equals the uncertainty for a simultaneous measurement of two linearly independent components of v\mathbf{v}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.
For a free Dirac electron, the Heisenberg equations define an internal dynamical system in the rest frame, isomorphic to a finite three-dimensional oscillator with a compact SO(5) phase space, such that the spin of the electron is the orbital angular momentum of the internal dynamics (Barut and Bracken 1980, 1981a). In the present work, the change in this internal dynamics due to an external magnetic field is studied. In order that the internal motion can be distinguished from the centre of mass motion, the solutions of the corresponding Hamilton and Heisenberg equations for the relativistic classical motion and the relativistic quantum mechanical spinless motion are also presented. The solutions for the electron exhibit the effect of the spin terms both in the internal motion and external motion, and we are able to identify the properties of the Zitterbewegung in the external field.
We explore the dynamics of two-dimensional massless Dirac fermions within a quantum shutter approach, which involves the time evolution of an initial cutoff plane wave. We show that the probability density is governed by an interplay between diffraction in time and Zitterbewegung phenomena typical of relativistic quantum shutter systems with nonzero mass. The time diffraction appears as an oscillatory pattern in the probability density, similar to the effect predicted by Moshinsky in 1952 [Phys. Rev. 88, 625 (1952)] for Schr\"odinger free matter waves. The Zitterbewegung manifests itself as high-frequency oscillations embedded in the time-diffraction profile. We found that these two transient effects are induced by the transverse momentum component of the incident wave ky{k}_{y}ky​ that acts as an effective mass of the system. Furthermore, this effective mass can be manipulated by tuning the incidence angle of the initial quantum state, which allows us to control the frequencies of the transients. In particular, we demonstrate that near a normal incidence condition, the Zitterbewegung appears as a series of quantum beats in the probability density, with a beating frequency 2kyvF2{k}_{y}{v}_{F}2ky​vF​, where vF{v}_{F}vF​ is the Fermi velocity.
We study the relations between massless Dirac fermions in an electromagnetic field and atoms in quantum optics. After getting the solutions for the energy spectrum, we show that it is possible to reproduce the 2D Dirac Hamiltonian, with all its quantum relativistic effects, in a controllable system such as a single trapped ion, through the Jaynes–Cummings and anti-Jaynes–Cummings models. Also we show that under certain conditions, the evolution of the Dirac Hamiltonian provides us with Rashba spin–orbit and linear Dresselhaus couplings. Considering the multimode multiphoton Jaynes–Cummings model interacting with N modes of an electromagnetic field prepared in general pure quantum states, we analyze the Rabi oscillation. Evaluating the time evolution of the Dirac position operator, we determine the Zitterbewegung frequency and the corresponding oscillating term as a function of the electromagnetic field.
Zitterbewegung (ZB, the trembling motion) of free relativistic electrons in a vacuum in the presence of an external magnetic field is calculated. It is shown that the motion of an electron wave packet has intraband frequency components, corresponding to the classical cyclotron motion, and several interband frequency components corresponding to the Zitterbewegung. For a two-dimensional situation, the presence of a magnetic field makes the ZB motion stationary, i.e. not decaying in time. We show how to simulate the ZB in a magnetic field using trapped ions and laser excitations in the spirit of recently observed proof-of-principle ZB simulation by Gerritsma {\it et al.} Nature {\bf 463}, 68 (2010). It is demonstrated that, for the parameters of the Dirac equation simulated by the above experiment, the effect of a magnetic field on the Zitterbewegung is considerable.
The description of the Zitterbewegung at a classical level indicates possible predictions of the electron quantum properties even before quantization: this quivering motion is restricted to a plane, which leads us to only two possible orientations of the corresponding angular momentum, even in the absence of a quantized theory or external fields. Besides, the angular momentum associated with the Zitterbewegung turns out to be proportional to $\ensuremath{\hbar}$. Namely, using a standard constraint analysis we have proven this last result. Thus, assuming that this is an observable phenomenon, we recognize this oscillatory motion as a classical signature of the spin of the electron. We also propose here an interpretation of the Zitterbewegung based on geometrical grounds: it can be seen as the physical degrees of freedom of position variables constrained to a sphere, which enforces the hypothesis of assuming an electron internal structure.
Zitterbewegung of massless particles with an arbitrary spin is analyzed in various representations. Dynamics of the group velocity of a massless particle as a whole and of the corresponding radius vector is determined. This radius vector defines any fixed point of the envelope of the moving wave packet characterizing the particle and its group velocity differs from the group velocities of any points of the wavefront. We consider free massless scalar and Dirac particles, the photon, and massive and massless particles with an arbitrary spin and describe them in different representations. For particles with an arbitrary spin, the generalized Feshbach-Villars representation and the Foldy-Wouthuysen one are used. Zitterbewegung takes place in any representation except for the Foldy-Wouthuysen one. Formulas describing the motion of a ``trembling'' free particle are the same in any representation. In the Foldy-Wouthuysen representation, the operators of the velocity and momentum of a free particle are proportional and Zitterbewegung does not take place. Since the radius vector (position) and velocity operators are the quantum-mechanical counterparts of the classical position and velocity just in the Foldy-Wouthuysen representation, Zitterbewegung is not observable. The same conclusion has been previously made for free massive particles. For relativistic massive particles with spins 0, 1/2, 1 and massless particles with spins 0, 1/2 in arbitrarily strong electromagnetic fields, independent of the external fields Zitterbewegung does not appear in the Foldy-Wouthuysen representation either. This conclusion is made for leading terms in the Hamiltonian proportional to the zero and first powers of the Planck constant and for such terms proportional to ${\ensuremath{\hbar}}^{2}$ which describe contact interactions.