Wannier-Stark States Research Articles

Kerr nonlinearity effect on the stability of Wannier-Stark states in active optical systems

The paper provides an analytical and numerical investigation of the dynamics of a one-dimensional chain of coupled optical resonators with conservative cubic nonlinearity and the gain saturated by nonlinear losses. The linear dependency of the resonator eigenfrequencies on their indexes makes it possible to use Wannier-Stark states as lasing modes. Numerical simulations have shown that the dependency of the resonant frequencies on the light intensity strongly affects the stability of Wannier-Stark states. To explain the observed destabilization of monochromatic lasing based on Wannier-Stark states a simple perturbation theory has been developed and compared with the data obtained in the numerical simulations.

Hamiltonian engineering of spin-orbit-coupled fermions in a Wannier-Stark optical lattice clock.

Engineering a Hamiltonian system with tunable interactions provides opportunities to optimize performance for quantum sensing and explore emerging phenomena of many-body systems. An optical lattice clock based on partially delocalized Wannier-Stark states in a gravity-tilted shallow lattice supports superior quantum coherence and adjustable interactions via spin-orbit coupling, thus presenting a powerful spin model realization. The relative strength of the on-site and off-site interactions can be tuned to achieve a zero density shift at a “magic” lattice depth. This mechanism, together with a large number of atoms, enables the demonstration of the most stable atomic clock while minimizing a key systematic uncertainty related to atomic density. Interactions can also be maximized by driving off-site Wannier-Stark transitions, realizing a ferromagnetic to paramagnetic dynamical phase transition.

Quantum Enhanced Cavity QED Interferometer with Partially Delocalized Atoms in Lattices.

We propose a quantum enhanced interferometric protocol for gravimetry and force sensing using cold atoms in an optical lattice supported by a standing-wave cavity. By loading the atoms in partially delocalized Wannier-Stark states, it is possible to cancel the undesirable inhomogeneities arising from the mismatch between the lattice and cavity fields and to generate spin squeezed states via a uniform one-axis twisting model. The quantum enhanced sensitivity of the states is combined with the subsequent application of a compound pulse sequence that allows us to separate atoms by several lattice sites. This, together with the capability to load small atomic clouds in the lattice at micrometric distances from a surface, make our setup ideal for sensing short-range forces. We show that for arrays of 10^{4} atoms, our protocol can reduce the required averaging time by a factor of 10 compared to unentangled lattice-based interferometers after accounting for primary sources of decoherence.

Nonequilibrium resonant inelastic x-ray scattering study of an electron-phonon model

We use the nonequilibrium dynamical mean field theory formalism to compute the equilibrium and nonequilibrium resonant inelastic X-ray scattering (RIXS) signal of a strongly interacting fermionic lattice model with a coupling of dispersionless phonons to the total charge on a given site. In the atomic limit, this model produces phonon subbands in the spectral function, but not in the RIXS signal. Electron hopping processes however result in phonon-related modifications of the charge excitation peak. We discuss the equilibrium RIXS spectra and the characteristic features of nonequilibrium states induced by photo-doping and by the application of a static electric field. The latter produces features related to Wannier-Stark states, which are dressed with phonon sidebands. Thanks to the effect of field-induced localization, the phonon features can be clearly resolved even in systems with weak electron-phonon coupling.

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.

Tuning of energy gap and 1D Dirac-like points in artificial graphene and boron nitride monolayer by an external electric field

A simple semianalytical model is proposed to predict the possibilities of effective manipulation of electronic states in a graphene-like and a two dimensional boron nitride-like quantum dot superlattices by means of lateral electric field. Due to the Wannier-Stark localization an one dimensional energy dispersion in the perpendicular to the electric field direction is observed. Interestingly, one dimensional analogs of Dirac points of minibands’ touching are observed in the energy dispersion, for the artificial graphene when electric field is weak enough. However, an energy gap between the minibands is opened when an electric field of a significant strength is applied. On the other hand, it is demonstrated that the appropriate choice of the electric field strength can lead to the recovering of touching points in one dimensional energy dispersion of artificial boron nitride monolayer which are not observable in the absence of electric field. Importantly, the analysis of the electron probability distributions brings out an efficient mechanism of switching between the quantum localized states in different sublattices by means of laterally applied electric field. The obtained results indicate on the possibility of a simple and effective control of the properties of optical and logical devices based on honeycomb semiconductor superlattices . • Wannier-Stark localized states in AG and ABN are considered. • A possibility of control of energy gap and Dirak-like points is predicted. • A switching mechanism between the localized states in different sublattices is proposed.

Theoretical simulation of quantum cascade lasers based on InGaAs/AlInAs and InGaAs/AlAsSb quantum wells

Quantum cascade lasers based on semiconductor InGaAs/AlInAs and InGaAs/AlAsSb quantum wells exhibit electronic intersubband transitions within conduction bands and attract much attention due to their operation at room-temperature short wavelengths. In this work we performed a theoretical simulation of such quantum cascade lasers and studied the influence of their construction and application conditions on optical behaviour. The intersubband absorption processes in these quanta well structures were studied. Electronic properties and the conduction band edge profiles were simulated as well as the probability densities of the Wannier-Stark states were determined. The simulation results showed that with the rise of the temperature the threshold current density increases which also leads to a decrease in the optical gain. An increase of the applied electric field is accompanied by the optical gain rising and decrease of the threshold current density that results in a blue shift of laser frequency.

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.

Nonequilibrium steady states of electric field driven Mott insulators

We present a systematic study of the nonequilibrium steady states (NESS) in Mott insulators driven by DC or AC electric fields, based on the Floquet dynamical mean-field theory. The results are analyzed using a generalized tunneling formula for the current, which is reminiscent of the Meir-Wingreen formula and provides insights into the relevant physical processes. In the DC case, the spectrum of the NESSs exhibits Wannier-Stark (WS) states associated with the lower and upper Hubbard bands. In addition, there emerge WS sidebands from many-body states. Using the tunneling formula, we demonstrate that the tunneling between these WS states leads to peaks or humps in the induced DC current. In the AC case, we cover a wide parameter range of excitation frequencies and field strengths to clarify the crossover from field-induced tunneling behavior in the DC limit to nonequilibrium states dominated by multiphoton absorption in the AC limit. In the crossover regime, the single-particle spectrum is characterized by a coexistence of Floquet sidebands and WS peaks, and the current and double occupation exhibits a nontrivial dependence on the field strength. The tunneling formula works quantitatively well even in the AC case, and we use it to discuss the potential cooperation of tunneling and multi-photon processes in the crossover regime. The tunneling formula and its simplified versions also provide physical insights into the high-harmonic generation in Mott insulators.

Influence of ferroelectric order on the surface electronic structure ofBaTiO3films studied by photoemission spectroscopy

Observation of Bloch oscillations and Wannier-Stark localization of charge carriers is typically impossible in single-crystals, because an electric field higher than the breakdown voltage is required. In BaTiO$_3$ however, high intrinsic electric fields are present due to its ferroelectric properties. With angle-resolved photoemission we directly probe the Wannier-Stark localized surface states of the BaTiO$_3$ film-vacuum interface and show that this effect extends to thin SrTiO$_3$ overlayers. The electrons are found to be localized along the in-plane polarization direction of the BaTiO$_3$ film.

Multi-peak negative differential resistance in silicene superlattice: Toward multi-valued silicene logic devices

Negative differential resistance (NDR) in two dimensional materials has been the subject of strong interest for ultra-low power nanoelectronic applications. Here, we report NDR characteristics of silicene superlattice (SL) at low bias voltages. Transport process manipulated by miniband regime for low bias region and Wannier-Stark (WS) ladders regime with multi-peak NDR for higher bias windows. Local tunneling peaks rise from hybridization of Wannier-Stark rungs in certain bias voltages. The bias position of WS states crossings down shifted with increasing device dimension leading to red shifted NDR window and increases peak to valley ratio (PVR) values. The multiple NDR windows are represented by changing the size and the number of well/barriers in silicene SL. Maximum PVR of 8 obtained for device with low height barriers. Multi-peak NDR with same PVR values in very low bias regime can find key applications in multi-valued memories with low static power dissipation.

Inelastic light scattering from plasmons tunneling between Wannier-Stark states

Using inelastic light scattering, we measure the zone-center electronic excitation modes in a set of multiple quantum wells. The width of the wavefunction barriers was chosen such that it prevents significant coupling of the electron ground states between wells yet is transparent to electron tunneling under an electric field. Under these conditions, we find charge-density-like and spin-density-like plasmons whose energies do not correspond to the excitations calculated for either a single well or a set of Coulomb-coupled wells. The observed energies are proportional to the electric field strength and the lower energy modes agree with predictions for plasmons tunneling between the Wannier-Stark ladder states.Using inelastic light scattering, we measure the zone-center electronic excitation modes in a set of multiple quantum wells. The width of the wavefunction barriers was chosen such that it prevents significant coupling of the electron ground states between wells yet is transparent to electron tunneling under an electric field. Under these conditions, we find charge-density-like and spin-density-like plasmons whose energies do not correspond to the excitations calculated for either a single well or a set of Coulomb-coupled wells. The observed energies are proportional to the electric field strength and the lower energy modes agree with predictions for plasmons tunneling between the Wannier-Stark ladder states.

Identification of a limiting mechanism in GaSb-rich superlattice midwave infrared detector

GaSb-rich superlattice (SL) p-i-n photodiodes grown by molecular beam epitaxy were studied theoretically and experimentally in order to understand the poor dark current characteristics typically obtained. This behavior, independent of the SL-grown material quality, is usually attributed to the presence of defects due to Ga-related bonds, limiting the SL carrier lifetime. By analyzing the photoresponse spectra of reverse-biased photodiodes at 80 K, we have highlighted the presence of an electric field, breaking the minibands into localized Wannier-Stark states. Besides the influence of defects in such GaSb-rich SL structures, this electric field induces a strong tunneling current at low bias which can be the main limiting mechanism explaining the high dark current density of the GaSb-rich SL diode.

Superlattice gain in positive differential conductivity region

We analyze theoretically a superlattice structure proposed by A. Andronov et al. [JETP Lett. 102, 207 (2015)] to give Terahertz gain for an operation point with positive differential conductivity. Here we confirm the existence of gain and show that an optimized structure displays gain above 20 cm−1 at low temperatures, so that lasing may be observable. Comparing a variety of simulations, this gain is found to be strongly affected by elastic scattering. It is shown that the dephasing modifies the nature of the relevant states, so that the common analysis based on Wannier-Stark states is not reliable for a quantitative description of the gain in structures with extremely diagonal transitions.

Semimetallization of dielectrics in strong optical fields

At the heart of ever growing demands for faster signal processing is ultrafast charge transport and control by electromagnetic fields in semiconductors. Intense optical fields have opened fascinating avenues for new phenomena and applications in solids. Because the period of optical fields is on the order of a femtosecond, the current switching and its control by an optical field may pave a way to petahertz optoelectronic devices. Lately, a reversible semimetallization in fused silica on a femtosecond time scale by using a few-cycle strong field (~1 V/Å) is manifested. The strong Wannier-Stark localization and Zener-type tunneling were expected to drive this ultrafast semimetallization. Wider spread of this technology demands better understanding of whether the strong field behavior is universally similar for different dielectrics. Here we employ a carrier-envelope-phase stabilized, few-cycle strong optical field to drive the semimetallization in sapphire, calcium fluoride and quartz and to compare this phenomenon and show its remarkable similarity between them. The similarity in response of these materials, despite the distinguishable differences in their physical properties, suggests the universality of the physical picture explained by the localization of Wannier-Stark states. Our results may blaze a trail to PHz-rate optoelectronics.

Observation of Bloch oscillations in complex PT-symmetric photonic lattices.

Light propagation in periodic environments is often associated with a number of interesting and potentially useful processes. If a crystalline optical potential is also linearly ramped, light can undergo periodic Bloch oscillations, a direct outcome of localized Wannier-Stark states and their equidistant eigenvalue spectrum. Even though these effects have been extensively explored in conservative settings, this is by no means the case in non-Hermitian photonic lattices encompassing both amplification and attenuation. Quite recently, Bloch oscillations have been predicted in parity-time-symmetric structures involving gain and loss in a balanced fashion. While in a complex bulk medium, one intuitively expects that light will typically follow the path of highest amplification, in a periodic system this behavior can be substantially altered by the underlying band structure. Here, we report the first experimental observation of Bloch oscillations in parity-time-symmetric mesh lattices. We show that these revivals exhibit unusual properties like secondary emissions and resonant restoration of PT symmetry. In addition, we present a versatile method for reconstructing the real and imaginary components of the band structure by directly monitoring the light evolution during a cycle of these oscillations.

Wannier-Stark states in double-periodic lattices. II. Two-dimensional lattices

We analyze the Wannier-Stark spectrum of a quantum particle in tilted two-dimensional lattices with the Bloch spectrum consisting of two subbands, which could be either separated by a finite gap or connected by the Dirac points. For rational orientations of the static field given by an arbitrary superposition of the translation vectors, the spectrum is a ladder of energy bands. We obtain asymptotic expressions for the energy bands in the limit of large and weak static fields and study them numerically for intermediate field strength. We show that the structure of energy bands determines the rate of spreading of localized wave packets, which is the quantity measured in laboratory experiments. It is shown that wave-packet dispersion becomes a fractal function of the field orientation in the long-time regime of ballistic spreading.

Wannier-Stark states in double-periodic lattices. I. One-dimensional lattices

We analyze the Wannier-Stark spectrum of a quantum particle in generic one-dimensional double-periodic lattices. In the limit of weak static field the spectrum is shown to be a superposition of two Wannier-Stark ladders originated from two Bloch subbands. As the strength of the field is increased, the spectrum rearranges itself into a single Wannier-Stark ladder. We derive analytical expressions which describe the rearrangement employing the analogy between the Wannier-Stark problem and the driven two-level system in the strong-coupling regime.

Strong-field perspective on high-harmonic radiation from bulk solids.

Mechanisms of high-harmonic generation from crystals are described by treating the electric field of a laser as a quasistatic strong field. Under the quasistatic electric field, electrons in periodic potentials form dressed states, known as Wannier-Stark states. The energy differences between the dressed states determine the frequencies of the radiation. The radiation yield is determined by the magnitudes of the interband and intraband current matrix elements between the dressed states. The generation of attosecond pulses from solids is predicted. Ramifications for strong-field physics are discussed.

Wannier-Stark states of graphene in strong electric field

We find theoretically the energy spectrum of a graphene monolayer in a strong constant electric field using a tight-binding model. Within a single band, we find quantized equidistant energy levels (Wannier-Stark ladder), separated by the Bloch frequency. Singular interband coupling results in mixing of the states of different bands and anticrossing of corresponding levels, which is described analytically near Dirac points and is related to the Pancharatnam-Berry phase. The rate of interband tunneling, which is proportional to the anticrossing gaps in the spectrum, is only inversely proportional to the tunneling distance, in a sharp contrast to conventional solids where this dependence is exponential. This singularity will have major consequences for graphene behavior in strong ultrafast optical fields, in particular, leading to nonadiabaticity of electron excitation dynamics.