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arXiv:1204.3049 · Physical Review A

Response of a particle in a one-dimensional lattice to an applied force: Dynamics of the effective mass

Federico Duque-Gomez, John E. Sipe

Published May 11, 2012• 9 citations
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Abstract

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.

DOI
10.1103/physreva.85.053412

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cond-mat.mes-hallCross-listedarXiv:2602.23965•Mar 2, 2026
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 response of a particle in a periodic potential to an applied force is commonly described by an effective mass, which accounts for the detailed interaction between the particle and the surrounding potential. Using a Bose-Einstein condensate of (87)Rb atoms initially in the ground band of an optical lattice, we experimentally show that the initial response of a particle to an applied force is in fact characterized by the bare mass. Subsequently, the particle response undergoes rapid oscillations and only over time scales that are long compared to those of the interband dynamics is the effective mass observed to be an appropriate description. Our results elucidate the role of the effective mass on short time scales, which is relevant for example in the interaction of few-cycle laser pulses with dielectric and semiconductor materials.
We study the dynamics of interacting superfluid bosons in a one-dimensional vertical optical lattice after a sudden increase of the lattice potential depth. We show that this system can be exploited to investigate the effects of strong interactions on Bloch oscillations. We perform theoretical modeling of this system, identify experimental challenges, and explore a regime of Bloch oscillations characterized by interaction-induced matter-wave collapse and revivals which modify the Bloch oscillations dynamics. In addition, we study three dephasing mechanisms: finite value of tunneling, effective three-body interactions, and a background harmonic potential. We also find that the center-of-mass motion in the presence of finite tunneling goes through collapse and revivals, giving an example of quantum transport where interaction-induced revivals are important. We quantify the effects of residual harmonic trapping on the momentum distribution dynamics and show the occurrence of an interaction-modified temporal Talbot effect. Finally, we analyze the prospects and challenges of exploiting Bloch oscillations of cold atoms in the strongly interacting regime for precision measurement of the gravitational acceleration ggg.
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
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 report experimental results on the properties of Bose-Einstein condensates in one-dimensional optical lattices. By accelerating the lattice, we observed Bloch oscillations of the condensate in the lowest band, as well as Landau-Zener (LZ) tunneling into higher bands when the lattice depth was reduced and/or the acceleration of the lattice was increased. The dependence of the LZ tunneling rate on the condensate density was then related to mean-field effects modifying the effective potential acting on the condensate, yielding good agreement with recent theoretical work. We also present several methods for measuring the lattice depth and discuss the effects of the micromotion in the time-orbiting-potential trap on our experimental results.
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.