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arXiv:1108.1883 · Physical Review B

Generalized acceleration theorem for spatiotemporal Bloch waves

Stephan Arlinghaus, Martin Holthaus

Published Aug 8, 2011• 33 citations
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Abstract

A representation is put forward for wave functions of quantum particles in periodic lattice potentials subjected to homogeneous time-periodic forcing, based on an expansion with respect to Bloch-like states which embody both the spatial and the temporal periodicity. It is shown that there exists a generalization of Bloch's famous acceleration theorem which grows out of this representation and captures the effect of a weak probe force applied in addition to a strong dressing force. Taken together, these elements point at a ``dressing and probing'' strategy for coherent wave-packet manipulation, which could be implemented in present experiments with optical lattices.

DOI
10.1103/physrevb.84.054301

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cond-mat.mes-hallCross-listedarXiv:2602.23965•Mar 2, 2026
A representation is put forward for wave functions of quantum particles in periodic lattice potentials subjected to homogeneous time-periodic forcing, based on an expansion with respect to Bloch-like states which embody both the spatial and the temporal periodicity. It is shown that there exists a generalization of Bloch's famous acceleration theorem which grows out of this representation and captures the effect of a weak probe force applied in addition to a strong dressing force. Taken together, these elements point at a ``dressing and probing'' strategy for coherent wave-packet manipulation, which could be implemented in present experiments with optical lattices.
We study the behavior of the expectation value of the acceleration of a particle in a one-dimensional periodic potential when an external homogeneous force is suddenly applied. The theory is formulated in terms of modified Bloch states that include the interband mixing induced by the force. This approach allows us to understand the behavior of the wave packet, which responds with a mass that is initially the bare mass, and subsequently oscillates around the value predicted by the effective mass. If Zener tunneling can be neglected, the expression obtained for the acceleration of the particle is valid over time scales of the order of a Bloch oscillation, which are of interest for experiments with cold atoms in optical lattices. We discuss how these oscillations can be tuned in an optical lattice for experimental detection.
The scope of the present publication-based thesis on periodically forced optical lattices is twofold: In the first part, the reader is introduced to the main concepts required to formulate the central results of this thesis, and a review is given of landmark papers on coherently controlling ultracold atoms by subjecting them to periodic forces. The second part provides the author's own results of original research on the subject and consists of four publications and a preprint. It is shown how ultracold atoms in optical lattices can be used as strong-field simulators, dynamic localization in optical lattices are described in detail, and multiphoton-like processes are studied. A generalized acceleration theorem for spatiotemporal Bloch waves is developed and used to enable a controlled wave-packet manipulation in optical lattices.
The acceleration theorem for wave packet propagation in periodic potentials disentangles the kkk-space dynamics and real space dynamics. This is well known and understood for Bloch oscillations and super Bloch oscillations in the presence of position-independent forces. Here, we analyze the dynamics of a model system in which the kkk-space dynamics and the real space dynamics are inextricably intertwined due to a position-dependent force provided by a parabolic trap. We demonstrate that this coupling gives rise to significantly modified and rich dynamics when the lattice is shaken by a modulated parabolic potential. The dynamics range from chirped Bloch-harmonic oscillations to asymmetric spreading oscillations. We analyze these findings by tracing the spatiotemporal dynamics in real space and by visualizing the relative phase in the kkk-space dynamics which leads to an accurate explanation of the obtained phenomena. We also compare our numerical results to a local acceleration model and obtain very good agreement for the case of coherent oscillations, however, deviations for oscillations with spreading dynamics which altogether supports the interpretations of our findings.
Motivated by recent experimental progress achieved with ultracold atoms in kilohertz-driven optical lattices, we provide a theoretical discussion of mechanisms governing the response of a particle in a cosine lattice potential to strong forcing pulses with smooth envelope. Such pulses effectuate adiabatic motion of a wave packet's momentum distribution on quasienergy surfaces created by spatiotemporal Bloch waves. Deviations from adiabaticity can then be deliberately exploited for exerting coherent control and for reaching target states which may not be accessible by other means. As one particular example, we consider an analog of the $\ensuremath{\pi}$ pulses known from optical resonance. We also suggest adapting further techniques previously developed for controlling atomic and molecular dynamics by laser pulses to the coherent control of matter waves in shaken optical lattices.
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.
The transient picture for a Bloch electron accelerating in an arbitrarily time-dependent homogeneous electric field is developed. The temporal sequence for the analysis includes the instant after electron injection, followed by the time required for a small change in electron wavenumber away from initial injection, leading to the final time evolution over many Bloch periods. The time-dependent behavior is studied using the properties of the Schrödinger equation. The electric field is described through the vector potential gauge, and the instantaneous eigenstates of the Bloch, electric-field-dependent Hamiltonian are used as basis states in describing the Bloch dynamics in the electric field. For each temporal sequence considered, the solution to the Schrödinger equation is established and comparatively discussed. The expectation value of the momentum is obtained for the special case of first order in a constant electric field; the resulting velocity derived is a field-dependent generalization of the natural Zitterbewegung-like behavior discussed in the recent literature. The early-time and long-time limits of the momentum expectation value and its time derivative demonstrate that the resistance to Bloch acceleration after initial band injection varies from real mass to effective mass dynamics as the electron accelerates through the band under the influence of electric field. This changing inertia from early injection of a free-mass electron is the result of the {\it real mass} electron {\it dressing-up} into the states of the crystal to become {\it an effective mass} electron. The ramifications of this temporal {\it dressing} behavior are discussed in considering the general dynamics of Bloch electrons subject to ultrastrong electric fields.