Wannier Stark Research Articles

Orthonormal and periodic levels for quantum cascade laser simulation

A Python package to evaluate Wannier, Wannier–Stark, and EZ (both energy and location Z resolved) levels for quantum cascade lasers is presented. We provide the underlying theory in detail with a focus on the orthonormality and periodicity of the generated states.

Floquet analysis perspective of driven light–matter interaction models

In this paper, we analyze the harmonically driven Jaynes–Cummings and Lipkin–Meshkov–Glick models using both numerical integration of time-dependent Hamiltonians and Floquet theory. For a separation of time scales between the drive and intrinsic Rabi oscillations in the former model, the driving results in an effective periodic reversal of time. The corresponding Floquet Hamiltonian is a Wannier–Stark model, which can be analytically solved. Despite the chaotic nature of the driven Lipkin–Meshkov–Glick model, moderate system sizes can display qualitatively different behaviors under varying system parameters. Ergodicity arises in systems that are neither adiabatic nor diabatic, owing to repeated multi-level Landau–Zener transitions. Chaotic behavior, observed in slow driving, manifests as random jumps in the magnetization, suggesting potential utility as a random number generator. Furthermore, we discuss both models in terms of a Floquet Fock state lattice.

Wannier–Stark localization of electronic states in 4H-SiC MOS inversion layer

The electronic states in 4H-SiC MOS inversion layers are theoretically analyzed using the empirical pseudopotential method (EPM). The analysis shows that the Wannier–Stark localization occurs, which is absent in an effective mass approximation (EMA). The Wannier–Stark localization modifies the electronic states in the MOS inversion layers. A model is proposed to describe the in-plane dispersion of subbands affected by the Wannier–Stark localization. The differences between the EPM and EMA results for the subband energy levels and the in-plane effective masses are discussed.

Generalized framework for the step response of semiconducting materials to optically switched electrical bias input in terahertz emission spectroscopy

In this paper, a generalized framework for the step response of semiconducting materials to optically switched electrical bias input in terahertz emission spectroscopy was developed, introducing a complex response function into the frequency domain. A comprehensive formula was obtained for calculating the transient current from the poles of the response function under bias field. This formula was found to give transient currents of an exponentially saturated, overshooting, or oscillating nature for three different regimes of charge transport in an isolated energy band. Furthermore, it was clarified how the transient terahertz emission is linked to the response function and its retarded contribution. When either true current or polarization current is linear with respect to bias field and dominates the transient current, the retarded contribution is identified as the small-signal ac complex conductivity. Two different spectral examples of the retarded contribution were given for actual terahertz Bloch oscillations in biased semiconductor superlattices, describing the small-signal response of polarization current under the Wannier–Stark localization and the large-signal response of true current under interminiband mixing.

Non-Floquet engineering in periodically driven dissipative open quantum systems

Floquet engineering plays a key role in realizing novel dynamical topological states. The conventional Floquet engineering, however, only applies to time-periodic non-dissipative Hermitian systems, and for the open quantum systems, non-Hermitian processes usually occur. So far, it remains unclear how to characterize the topological phases of time-periodic open quantum systems via the frequency space Floquet Hamiltonian. Here, we propose the non-Floquet theory to solve the problem and illustrate it by a continuously time-periodic non-Hermitian bipartite chain. In non-Floquet theory, a temporal non-unitary transformation is exercised on the Floquet states, and the transformed Floquet spectrum restores the form of the Wannier–Stark ladder. Besides, we also show that different choices of the starting points of the driving period can result in different localization behavior, effects of which can reversely be utilized to design quantum detectors of phases in dissipative oscillating fields. Our methods are capable of describing topological features in dynamical open quantum systems with various driving types and can find its applications to construct new types of dynamical topological materials.

Electronic states underlying peculiar Bloch oscillations in a biased semiconductor superlattice with interminiband mixing

We simulate the energy eigenstates of electrons and holes, which underlie terahertz Bloch oscillations observed recently with a peculiar phase shift, in a biased semiconductor superlattice with interminiband mixing. Electron wavefunctions are extended downstream, while some of them have a moderately first-miniband-like nature that partly resembles Wannier–Stark localization upstream. When a typical heavy-hole wavefunction is chosen, a few first-miniband-like electron wavefunctions prove to have large overlap integrals with it under relatively low bias fields. These electron wavefunctions are thus essential for the optical excitation of the Bloch oscillations, supported experimentally by the peak patterns of dc photocurrent spectra.

Observation of the Wannier-Stark ladder in plasmonic waveguide arrays.

Evanescently coupled waveguides are a powerful platform to study and visualize the wave dynamics in tight-binding systems. Here, we investigate the propagation of surface plasmon polaritons in arrays of dielectric loaded surface plasmon polariton waveguides with a propagation constant gradient acting as an effective external potential. Using leakage radiation microscopy, we observe in real-space for single site excitation a periodic breathing of the wave packet and an oscillatory motion in the case of Gaussian excitation of multiple waveguides. The corresponding momentum-resolved spectra are composed of sets of equally spaced modes. We interpret these observations as the plasmonic analogs of Bloch oscillations and the Wannier-Stark ladder, respectively.

Dynamics of Negativity of a Wannier–Stark Many‐Body Localized System Coupled to a Bath

An interacting system subjected to a strong linear potential can host a many‐body localized (MBL) phase when being slightly perturbed. This so‐called Wannier–Stark or “tilted‐field” MBL phase inherits many properties from the well‐investigated disordered MBL phase, and provides an alternative route to experimentally engineer interacting localized systems without quenched disorder. Herein, the dynamics of entanglement in a Wannier–Stark MBL system coupled to a dephasing environment is investigated. As an accessible entanglement proxy, the third Rényi negativity is used, which reduces to the third Rényi entropy in case the system is isolated from the environment. This measure captures the characteristic logarithmic growth of interacting localized phases in the intermediate‐time regime, where the effects of the coupling to the environment are not yet dominating the dynamics. Thus, it forms a tool to distinguish Wannier–Stark MBL from noninteracting Wannier–Stark localization up to intermediate time‐scales, and to quantify quantum correlations in mixed‐state dynamics.

Low-field onset of Wannier-Stark localization in a polycrystalline hybrid organic inorganic perovskite

Methylammonium lead iodide perovskite (MAPbI3) is renowned for an impressive power conversion efficiency rise and cost-effective fabrication for photovoltaics. In this work, we demonstrate that polycrystalline MAPbI3s undergo drastic changes in optical properties at moderate field strengths with an ultrafast response time, via transient Wannier Stark localization. The distinct band structure of this material - the large lattice periodicity, the narrow electronic energy bandwidths, and the coincidence of these two along the same high-symmetry direction – enables relatively weak fields to bring this material into the Wannier Stark regime. Its polycrystalline nature is not detrimental to the optical switching performance of the material, since the least dispersive direction of the band structure dominates the contribution to the optical response, which favors low-cost fabrication. Together with the outstanding photophysical properties of MAPbI3, this finding highlights the great potential of this material in ultrafast light modulation and novel photonic applications.

Dephasing of terahertz Bloch oscillations in a GaAs-based narrow-minigap superlattice excited by tunable pump photon energy

We investigate peculiar Bloch oscillations under interminiband mixing in a GaAs-based narrow-minigap superlattice by terahertz emission spectroscopy. The dephasing time decreases with increasing pump photon energy and drops substantially at a certain energy, in contrast to that reported for Wannier–Stark ladder states in a nearly isolated miniband. This behavior can be ascribed to the LO phonon emission of electrons that are excited with in-plane excess energies and accelerated under dc bias electric field in the conduction first miniband. Our findings support the validity of the miniband transport model employed recently to account for an accompanying π/2 shift in the oscillation phase.

Wannier-Stark flatbands in Bravais lattices

We systematically construct flatbands (FB) for tight-binding models on simple Bravais lattices in space dimension $d \geq 2$ in the presence of a static uniform DC field. Commensurate DC field directions yield irreducible Wannier-Stark (WS) bands in perpendicular dimension $d - 1$ with $d$-dimensional eigenfunctions. The irreducible bands turn into dispersionless flatbands in the absence of nearest neighbor hoppings between lattice sites in any direction perpendicular to the DC field. The number of commensurate directions which yield flatbands is of measure one. We arrive at a complete halt of transport, with the DC field prohibiting transport along the field direction, and the flatbands prohibiting transport in all perpendicular directions as well. The anisotropic flatband eigenstates are localizing at least factorially (faster than exponential).

Chiral Bloch–Zener oscillations of spin–orbit coupled cold atoms in an optical superlattice

We study the Bloch oscillation (BO) dynamics of spin–orbit coupled cold atoms in a one-dimensional Su–Schrieffer–Heeger optical lattice. The Hamiltonian of our system can be mapped into a tilted SSH ladder pierced by an effective magnetic flux. Based on the calculation of Wannier–Stark states, we investigate dynamical properties of BOs with the chiral character of spin-momentum locking. Importantly, the chiral Bloch–Zener oscillations can be induced by adjusting system parameters. The corresponding Landau–Zener (LZ) tunneling probability is analytically derived by the LZ model and we found a good agreement with the numerical simulation.

Geometric phase of Wannier–Stark ladders in alkaline-earth(-like) atoms

We discuss the geometric phase of Wanner-Stark ladders generated by periodically driven clock states in alkaline-earth(-like) atoms. Using $^{171}$Yb atoms as a concrete example, we show that clock states driven by two detuned clock lasers can be mapped to two-band Wannier-Stark ladders, where dynamics of the system along the ladder is mapped to Bloch oscillations in a one-dimensional topological lattice. When the adiabatic condition is satisfied, the geometric phase accumulated in one period of the oscillation is quantized, and reveals the change of band topology as the laser parameters are tuned. We show how the geometric phase can be experimentally detected through interference between different nuclear spin states. Our study sheds light on the engineering of exotic band structures in Floquet dynamics.

Wannier-Stark ladder spectrum of Bloch oscillations of magneto-dipole spin waves in graded 1D magnonic crystals

We have used the method of Wannier functions to calculate the frequencies and profiles of spin waves localized in one-dimensional magnonic crystals due to a gradient in the bias magnetic field. This localization of spin waves is analogous to the phenomenon of Bloch oscillations of quantum-mechanical electrons in crystals in a uniform electric field. As a convenient yet realistic model, we consider backward volume magnetostatic spin waves in a film of yttrium-iron garnet in a bias magnetic field comprising spatially uniform, cosine and gradient contributions. The spin-wave spectrum is shown to have the characteristic form of a Wannier–Stark ladder. The analytical results are verified using those obtained using numerical micromagnetic simulations. The physics of spin-wave Bloch oscillations combines the topics of magnonic crystals and graded magnonic index — the two cornerstones of modern magnonics.

Anomalous transport in a topological Wannier-Stark ladder

A dc (e.g. electric) field with commensurate lattice direction turns a single particle band structure in $d=3$ dimensions into an infinite set of equally spaced irreducible $(d-1)=2$-dimensional Wannier-Stark (WS) band structures that are spatially localized along the field direction. Particle transport is expected to be suppressed once the WS bands are gapped in energy. The topological character of the irreducible band structure leads to one-dimensional sets of boundary states which fill the energy gaps. As a result, eigenmodes are smoothly connected in energy and space and yield anomalous particle transport throughout the ladder. The number of chiral boundary modes can be tuned by the dc field strength and manifests through the distribution of dissipated energy and spatial motion, and the temperature dependence of angular momentum carried by particles.

In English

One of the promising materials for the development of 2D photonic structures is macroporous silicon obtained using photoanodic etching. Presence of periodically located cylindrical pores divided by silicon columns provides large effective surface of samples and enhanced optical and photo-physical characteristics of macroporous silicon structures. In this paper, the near-IR light absorption oscillations of 2D macroporous silicon structures with microporous silicon layers, SiO 2 nanocoatings and CdTe, ZnO surface nanocrystals are studded taking into account the Wannier–Stark electro-optical effect. An analysis of the experimental absorption spectra was carried out within the model of the resonant electron scattering with infinite amplitude on impurity states in strong electric field, with difference between two resonant energies equaled to the step of Wannier–Stark ladder. The constant oscillation period specifies the realization of Wannier–Stark effect on the randomly distributed surface bonds on silicon-nanocoating interface. We compared the IR absorption oscillations in 2D macroporous silicon with surface nanocoatings, analyzed shifts and deviations of oscillation peaks. The coherence of the IR spectra oscillations increases with the decrease of surface state concentration and with the optimal area of contact of nanocrystals to the macropore surface. Thus, the shift of oscillations for ZnO nanoparticles with optimal size of nanocrystals (3.7–4.4 nm) leads to deviations of the oscillation peaks within 0.26–0.42 meV, the oscillation coherence reaches 0.25–0.4 %. The small broadening parameter of the Wannier–Stark ladder levels G = 0.3¸0.8 cm –1 equals to that for surface phonon polaritons measured in thin films of II-VI semiconductors. Controllability and coherence of the investigated system are determined by forming of coherent Wannier levels in a narrow triangular potential well formed by an electric field at the silicon-nanocoating interface. In addition, we have proposed a high coherent optical quantum computer based on the quantum Wannier-Stark electro-optical effect on a silicon matrix with macropores and a layer of nanocrystals on the macropore surface.

The Bloch Electron Response to Electric Fields: Application to Graphene

The theory of Bloch electron dynamics for carriers in homogeneous electric fields of arbitrary time dependence is developed in consideration of the electronic transport properties in graphene. The general approach is to use the accelerated Bloch state representation (ABR) as a basis so that the dependence upon the electric field, including Zener tunneling, is treated exactly; also, the electric field is described in the vector potential gauge. Within the ABR, the instantaneous eigenstates for the central Hamiltonian are described and utilized to develop both the time‐dependent wave functions and the single‐particle density matrix pictures with explicit application to intrinsic graphene. The Bloch electron analysis for graphene in a constant electric field reveals the explicit manifestation of electron–hole pair creation, Bloch oscillations, and Wannier–Stark localization. The average velocity and acceleration are established as an explicit function of the electric field, and their behavior is characterized on short and long time scales. Further, it is shown that the average acceleration in a nonvanishing electric field gives rise to an electric field induced dynamical effective mass tensor in the direction of the field.

Topological order versus many-body localization in periodically modulated spin chains

In this paper, we study periodically modulated $s=1/2$ spin chain in a linear gradient potential (LP) that is generated by an external magnetic field. In the absence of the LP, the system has topological states that exhibit a magnetization plateau for a uniform external magnetic field. These topological states have a finite integer Chern number and their stability is clarified by an equivalent spinless fermion system derived by a Jordan-Wigner transformation. We show that the LP, which is nothing but a constant electric field in the spinless fermion system, destabilizes the topological states, because it induces localization called Wannier-Stark (WS) localization. We clarify the phase diagram in the presence of the LP and on-site diagonal disorder. To this end, we carefully study edge excitations under the open boundary condition, which are a hallmark of the topological order. We find a very interesting phenomenon indicating existence of a quasi-edge modes that take the place of the genuine edge modes in certain parameter regions. This is a precursor of the WS localization realized in topological states. Finally, we investigate many-body localization induced by a sufficiently strong LP or disorder. To this end, we study the energy-level statistics for whole energy levels, and find unexpected extended-state regimes located in intermediate potential-gradient and weak on-site disorder regimes. We verify this phenomenon by calculating variance of the entanglement entropy. The present system is closely related to quantum Hall state in two dimensions, and therefore our findings can be observed not only in experiments on ultra-cold atomic gases but also quantum Hall physics.

Optical control of the layer degree of freedom through Wannier–Stark states in polar 3R MoS2

Electrons in two-dimensional layered crystals gain a discrete positional degree of freedom over layers. We propose the two-dimensional transition metal dichalcogenide homostructure with polar symmetry as a prototypical platform where the degrees of freedom for the layers and valleys can be independently controlled through an optical method. In 3R MoS2, a model system, the presence of the spontaneous polarization and built-in electric field along the stacking axis is theoretically proven by the density functional theory. The K valley states under the electric field exhibit Wannier–Stark type localization with atomic-scale confinement driven by double group symmetry. The simple interlayer-dynamics-selection rule of the valley carriers in 3R homostructure enables a binary operation, upward or downward motion, using visible and infrared light sources. Together with the valley-index, a 2 2 states/cell device using a dual-frequency polarized light source is suggested.

From Bloch oscillations to many-body localization in clean interacting systems

In this work we demonstrate that nonrandom mechanisms that lead to single-particle localization may also lead to many-body localization, even in the absence of disorder. In particular, we consider interacting spins and fermions in the presence of a linear potential. In the noninteracting limit, these models show the well-known Wannier-Stark localization. We analyze the fate of this localization in the presence of interactions. Remarkably, we find that beyond a critical value of the potential gradient these models exhibit nonergodic behavior as indicated by their spectral and dynamical properties. These models, therefore, constitute a class of generic nonrandom models that fail to thermalize. As such, they suggest new directions for experimentally exploring and understanding the phenomena of many-body localization. We supplement our work by showing that by using machine-learning techniques the level statistics of a system may be calculated without generating and diagonalizing the Hamiltonian, which allows a generation of large statistics.