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Physical Review B

Frequency doubling of Bloch oscillations for interacting electrons in a static electric field

Wandearley S. Dias, Elton Malta Nascimento, M. L. Lyra, F. A. B. F. de Moura

Published Oct 26, 2007• 72 citations
Publisher

Abstract

We study numerically the effect of the on-site Hubbard interaction $U$ on the dynamics of two electrons subjected to an external electric field and restricted to move in a linear chain with open boundaries. We solve the time-dependent Schr\"odinger equation to follow the time evolution of an initially localized two-electron state. For electrons initially far apart, the wave packet develops Bloch oscillations whose characteristic frequency is in agreement with a semiclassical calculation. For initially close electrons in a singlet state, a frequency doubling sets up, which is more pronounced for intermediate couplings. We discuss this effect by revealing the opposite trends the electron-electron coupling produces on the wave-packet components corresponding to bounded and unbounded states.

DOI
10.1103/physrevb.76.155124

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New Journal of Physics•Dec 1, 2021•5 citations
We study numerically the effect of the on-site Hubbard interaction UUU on the dynamics of two electrons subjected to an external electric field and restricted to move in a linear chain with open boundaries. We solve the time-dependent Schr"odinger equation to follow the time evolution of an initially localized two-electron state. For electrons initially far apart, the wave packet develops Bloch oscillations whose characteristic frequency is in agreement with a semiclassical calculation. For initially close electrons in a singlet state, a frequency doubling sets up, which is more pronounced for intermediate couplings. We discuss this effect by revealing the opposite trends the electron-electron coupling produces on the wave-packet components corresponding to bounded and unbounded states.
We investigate the influence of the on-site Hubbard interaction UUU on the eigenstates and dynamics of two electrons restricted to move in a linear chain with long-range correlated disorder. We solve the time-dependent Schr\"odinger equation to follow the time evolution of an initially localized Gaussian two-electron wave packet. In the regime of strongly correlated disorder, for which one-electron extended eigenstates emerge near the band center, the electron-electron coupling promotes the trapping of a finite portion of the wave packet. In the presence of a uniform electric field, the wave packet develops complex Bloch oscillations. The power spectrum of the centroid's velocity trace shows a splitting near the typical semiclassical Bloch frequency, as well as a frequency doubling phenomenon for intermediate couplings which is related to the bounded states components that are present in the wave packet. Finally, we show that localized and extended two-electron eigenstates coexist near the band center with the level spacing distribution showing a universal Poissonian form irrespective to the Hubbard coupling.
The correlated motion of electrons in a one-dimensional system with an externally applied longitudinal electric field is discussed. Within the tight-binding model we show that in addition to the well-known Bloch oscillations, the electron-electron interaction induces time-dependent oscillations of the mobility whose period depends on the strength and range of the coupling only. The oscillations involve transitions between bands of bound and unbound states. The case of two electrons is solved in detail and an extension of the results to more particles is discussed.
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.
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.
We study the dynamics of initially localized and uncorrelated two-particle quantum wave packets evolving in a one-dimensional discrete lattice. In particular, we show that the particles become strongly entangled when directed by a harmonic ac field which is resonant with frequency-doubled Bloch oscillations promoted by a static dc field. Some degree of entanglement is also achieved when the ac field is resonant with the single-particle Bloch oscillations. However, in this case, entanglement is strongly limited by the survival of anticorrelated unbounded states. We further show that the drift velocity of the correlated-particle wave-packet centroid depends on the phase of the ac field. This dependence is similar to the semiclassical prediction for single-particle motion. The drift velocity vanishes in the limit of uncorrelated particles, as well as for Fock-like initial states, which have a null expectation value of the kinetic operator. We reveal that the interparticle interaction influences unbounded- and bounded-state components differently. This leads to a nontrivial nonmonotonic dependence of the drift velocity on the interaction strength.
Abstract Recent developments in qubit engineering make circuit quantum electrodynamics devices promising candidates for the study of Bloch oscillations (BOs) and Landau–Zener (LZ) transitions. In this work, a hybrid circuit chain with alternating site energies under external electric fields is employed to study Bloch–Zener oscillations (BZOs), i.e. coherent superpositions of BOs and LZ transitions. We couple each of the tunable qubits in the chain to dispersionless optical phonons and build an extended Holstein polaron model with the purpose of investigating vibronic effects in the BZOs. We employ an extension of the Davydov ansatz in combination with the Dirac–Frenkel time-dependent variational principle to simulate the dynamics of the qubit chain under the influence of high-frequency quantum harmonic oscillators. Band gaps emerge due to energy differences in site energies at alternating qubit sites, and are shown to play key roles in tuning band structures and time periodic reconstructions of the wave patterns. In the absence of qubit–phonon interactions, the qubits undergo either standard BZOs or breathing modes, depending on whether the initial wave packet is formed by a broad or narrow Gaussian wave packet, respectively. The BZOs can get localized in space if the band gaps are sufficiently large. In the presence of qubit–phonon coupling, the periodic behavior of BZOs can be washed out and undergo dynamic localization. The influence of an ohmic bath on the dynamics of BZOs is investigated by means of a Markovian master equation approach. Finally, we calculate the von Neumann entropy as a measure of the entanglement between qubits and phonons.