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Physica Scripta

Analysis of electron wave functions that exhibit oscillatory behavior at the Bloch frequency

John N. Churchill, F.E. Holmstrom

Published Sep 1, 1994• 2 citations
Publisher

Abstract

Energy eigenstates are superimposed, using the Stark ladder energy spectrum, to construct exact, closed-form wave packets for an electron in a uniform dc electric field and a virtual lattice. We find that (1) perfectly periodic temporal variation at the Bloch frequency does not necessarily imply that there is a spatial oscillation and (2) perfectly periodic variation in both spatial and temporal behavior at the Bloch frequency does not imply that the motion is related in any way to the equation, d k /d t = − eE/h .

DOI
10.1088/0031-8949/50/3/016

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Energy eigenstates are superimposed, using the Stark ladder energy spectrum, to construct exact, closed-form wave packets for an electron in a uniform dc electric field and a virtual lattice. We find that (1) perfectly periodic temporal variation at the Bloch frequency does not necessarily imply that there is a spatial oscillation and (2) perfectly periodic variation in both spatial and temporal behavior at the Bloch frequency does not imply that the motion is related in any way to the equation, d k /d t = − eE/h .
The time-evolution of a wave packet in a perfectly periodic lattice potential is considered for an electron that is in a zero-field Bloch energy eigenstate prior to the adiabatic application of a uniform d.c. electric field. Power series expansions in time are constructed for both the exact wave function and the Houston wave function. A comparison of the coefficients in these two expansions shows that the Houston function diverges instantly and drastically from the exact wave function. A special-case example is presented for which the exact wave function is pure-real for all values of time whereas the corresponding Houston function is not.
The early quantum theory of electrical conductivity in crystal lattices by Bloch and Zener [1,2] led to the striking prediction that a homogeneous static electric field induces an oscillatory rather than uniform motion of the electrons. These so-called Bloch oscillations have never been observed in natural crystals because the scattering time of the electrons by the lattice defects is much shorter than the Bloch period. In semiconductor superlattices the larger spatial period leads to a much shorter Bloch period (,600 fs) and Bloch oscillations have recently been observed through the emission of THz radiation by the electrons [3]. Here we present Bloch oscillations of atoms in the fundamental energy band of a periodic optical potential. We directly measure the atomic momentum distribution evolving in time under the influence of a constant inertial force for various potential depths. We experimentally observe oscillation periods in the millisecond range as well as positive and negative effective masses. Bloch oscillations are a pure quantum effect which can be explained in a simple one-dimensional model. The periodicity of the lattice (period d) leads to a band structure (Fig. 1) of the energy spectrum of the particle and the corresponding eigenenergies Ensqd and eigenstates jn, ql (Bloch states) are labeled by the discrete band index n and the continuous quasimomentum q; Ensqd and jn, ql are periodic functions of q with period 2pyd and q is conventionally restricted to the first Brillouin zone g2pyd, pydg [4]. Under the influence of a constant external force F, weak enough not to induce interband transitions, a given Bloch state jn, qs0dl evolves (up to a phase factor) into the state jn, qstdl according to qstd ­ qs0d 1 Ftyh . (1) This evolution is periodic with a period tB ­ hyjFjd corresponding to the time required for the quasimomentum to scan a full Brillouin zone. The mean velocity in jn, qstdl
Energy eigenstates are superimposed in order to form the wave packet for an electron propagating in one dimension under the influence of a uniform, time-independent electric field. Flux velocity, phase velocity, and envelope velocity are considered. Asymptotic expressions are derived which describe the motion when the electric field is small and/or the wave packet is far away from the classical turning point. A graphical method is presented by which one can obtain both the position and shape of the envelope at a time t if the position and shape are known at some initial time.
A quantum particle subjected to a constant force undergoes an accelerated motion following a parabolic path, which differs from the classical motion just because of wave packet spreading (quantum diffusion). However, when a periodic potential is added (such as in a crystal) the particle undergoes Bragg scattering and an oscillatory (rather than accelerated) motion is found, corresponding to the famous Bloch oscillations (BOs). Here, we introduce an exactly-solvable quantum Hamiltonian model, corresponding to a generalized Wannier–Stark Hamiltonian [Formula: see text], in which a quantum particle shows an intermediate dynamical behavior, namely an oscillatory motion superimposed to an accelerated one. Such a novel dynamical behavior is referred to as accelerated BOs. Analytical expressions of the spectrum, improper eigenfunctions and propagator of the generalized Wannier–Stark Hamiltonian [Formula: see text] are derived. Finally, it is shown that acceleration and quantum diffusion in the generalized Wannier–Stark Hamiltonian are prevented for Airy wave packets, which undergo a periodic breathing dynamics that can be referred to as Airy–Bloch oscillations.
We study the dynamics of a quantum walker simultaneously subjected to time-independent and -dependent phases. Such dynamics emulates a charged quantum particle in a lattice subjected to a superposition of static and harmonic electric fields. With proper settings, we investigate the possibility to induce Bloch-like superoscillations, resulting from a close tuning of the frequency of the harmonic phase $\ensuremath{\omega}$ and that associated with the regular Bloch-like oscillations ${\ensuremath{\omega}}_{B}$. By exploring the frequency spectra of the wave-packet centroid, we are able to distinguish the regimes on which regular and super-Bloch oscillations are predominant. Furthermore, we show that under exact resonant conditions $\ensuremath{\omega}={\ensuremath{\omega}}_{B}$ unidirectional motion is established with the wave-packet average velocity being a function of the quantum coin operator parameter, the relative strengths of the static and harmonic terms, as well as the own phase of the harmonic field. We show that the average drift velocity can be well described within a continuous-time analogous model.
Bloch oscillation (BO), i.e., the oscillatory motion of a quantum particle in a periodic potential, is one of the most striking effects of coherent quantum transport in matter. In the semiclassical picture, it is well known that BOs can be explained owing to the periodic band structure of the crystal and the so-called acceleration theorem: since in the momentum space the particle wave packet drifts with a constant speed without being distorted, in real space the probability distribution of the particle undergoes a periodic motion following a trajectory which exactly reproduces the shape of the lattice band. In non-Hermitian lattices with a complex (i.e., not real) energy band, extension of the semiclassical model is not intuitive. Here we show that the acceleration theorem holds for non-Hermitian lattices with a complex energy band only on average, and that the periodic wave-packet motion of the particle in real space is described by a trajectory in the complex plane, i.e., it generally corresponds to reshaping and breathing of the wave packet in addition to a transverse oscillatory motion. The concept of BOs involving complex trajectories is exemplified by considering two examples of non-Hermitian lattices with a complex band dispersion relation, including the Hatano-Nelson tight-binding Hamiltonian describing the hopping motion of a quantum particle on a linear lattice with an imaginary vector potential and a tight-binding lattice with imaginary hopping rates.