Zitterbewegung of nearly-free and tightly bound electrons in solids

We show theoretically that nonrelativistic nearly-free electrons in solids should experience a trembling motion (Zitterbewegung, ZB) in absence of external fields, similarly to relativistic electrons in vacuum. The Zitterbewegung is directly related to the influence of periodic potential on the free electron motion. The frequency of ZB is $\omega\approx E_g/\hbar$, where $E_g$ is the energy gap. The amplitude of ZB is determined by the strength of periodic potential and the lattice period and it can be of the order of nanometers. We show that the amplitude of ZB does not depend much on the width of the wave packet representing an electron in real space. An analogue of the Foldy-Wouthuysen transformation, known from relativistic quantum mechanics, is introduced in order to decouple electron states in various bands. We demonstrate that, after the bands are decoupled, electrons should be treated as particles of a finite size. In contrast to nearly-free electrons we consider a two-band model of tightly bound electrons. We show that also in this case the electrons should experience the trembling motion. It is concluded that the phenomenon of Zitterbewegung of electrons in crystalline solids is a rule rather than an exception.Comment: 22 pages, 6 figures Published version, minor changes mad

Zitterbewegung of relativistic electrons in a magnetic field and its simulation by trapped ions

One-electron 3+1 and 2+1 Dirac equations are used to calculate the motion of a relativistic electron in a vacuum in the presence of an external magnetic field. First, calculations are carried on an operator level and exact analytical results are obtained for the electron trajectories which contain both intraband frequency components, identified as the cyclotron motion, as well as interband frequency components, identified as the trembling motion (Zitterbewegung, ZB). Next, time-dependent Heisenberg operators are used for the same problem to compute average values of electron position and velocity employing Gaussian wave packets. It is shown that the presence of a magnetic field and the resulting quantization of the energy spectrum has pronounced effects on the electron Zitterbewegung: it introduces intraband frequency components into the motion, influences all the frequencies and makes the motion stationary (not decaying in time) in case of the 2+1 Dirac equation. Finally, simulations of the 2+1 Dirac equation and the resulting electron ZB in the presence of a magnetic field are proposed and described employing trapped ions and laser excitations. Using simulation parameters achieved in recent experiments of Gerritsma and coworkers we show that the effects of the simulated magnetic field on ZB are considerable and can certainly be observed.Comment: 19 pages, 9 figures, published versio

Temperature dependence of the electron spin g factor in GaAs

The temperature dependence of the electron spin $g$ factor in GaAs is investigated experimentally and theoretically. Experimentally, the $g$ factor was measured using time-resolved Faraday rotation due to Larmor precession of electron spins in the temperature range between 4.5 K and 190 K. The experiment shows an almost linear increase of the $g$ value with the temperature. This result is in good agreement with other measurements based on photoluminescence quantum beats and time-resolved Kerr rotation up to room temperature. The experimental data are described theoretically taking into account a diminishing fundamental energy gap in GaAs due to lattice thermal dilatation and nonparabolicity of the conduction band calculated using a five-level kp model. At higher temperatures electrons populate higher Landau levels and the average $g$ factor is obtained from a summation over many levels. A very good description of the experimental data is obtained indicating that the observed increase of the spin $g$ factor with the temperature is predominantly due to band's nonparabolicity.Comment: 6 pages 4 figure

Zitterbewegung (trembling motion) of electrons in narrow gap semiconductors

Theory of trembling motion [Zitterbewegung (ZB)] of charge carriers in various narrow-gap materials is reviewed. Nearly free electrons in a periodic potential, InSb-type semiconductors, bilayer graphene, monolayer graphene and carbon nanotubes are considered. General features of ZB are emphasized. It is shown that, when the charge carriers are prepared in the form of Gaussian wave packets, the ZB has a transient character with the decay time of femtoseconds in graphene and picoseconds in nanotubes. Zitterbewegung of electrons in graphene in the presence of an external magnetic field is mentioned. A similarity of ZB in semiconductors to that of relativistic electrons in a vacuum is stressed. Possible ways of observing the trembling motion in solids are mentioned.Comment: 8 pages, 5 figure

The Quantum Hall Effect and Inter-edge State Tunneling Within a Barrier

We have introduced a controllable nano-scale incursion into a potential barrier imposed across a two-dimensional electron gas, and report on the phenomena that we observe as the incursion develops. In the quantum Hall regime, the conductance of this system displays quantized plateaus, broad minima and oscillations. We explain these features and their evolution with electrostatic potential geometry and magnetic field as a progression of current patterns formed by tunneling between edge and localized states within the barrier.Comment: RevTeX + 4 postscript figures. Self-unpacking uuencoded files. Unpacking instructions are at the beginning of the files. To appear in Physical Review

A laser based accelerator for ultracold atoms

We present first results on our implementation of a laser based accelerator for ultracold atoms. Atoms cooled to a temperature of 420 nK are confined and accelerated by means of laser tweezer beams and the atomic scattering is directly observed in laser absorption imaging. The optical collider has been characterized using Rb87 atoms in the |F=2,mF=2> state, but the scheme is not restricted to atoms in any particular magnetic substates and can readily be extended to other atomic species as well.Comment: (c) 2012 The Optical Society, 3 pages, 4 figures, 1 movie lin