Stark ladder resonances for small electric fields

We prove the existence of resonances in the semi-classical regime of smallh for Stark ladder Hamiltonians in one-dimension. The potentialv is a real periodic function with period which is the restriction to of a function analytic in a strip about . The electric field strengthF satisfies the bounds |v|>F>0. In general, the imaginary part of the resonances are bounded above by, for some 0<1, where _T h ^-1 is the single barrier tunneling distance in the Agmon metric forv+Fx. In the regime where the distance between resonant wells is , we prove that there is at least one resonance whose width is bounded above byce ^–/F , for some ,c>0 independent ofh andF forh sufficiently small. This is an extension of the Oppenheimer formula for the Stark effect to the case of periodic potentials.Partially supported by NSF Grant DMS-8911242     

Nonlinear delocalization on disordered Stark ladder

We study effects of weak nonlinearity on localization of waves in disordered Stark ladder corresponding to propagation in presence of disorder and a static field. Our numerical results show that nonlinearity leads to delocalization with subdiffusive spreading along the ladder. The exponent of spreading remains close to its value in absence of the static field. The delocalization implies the existence of statistical entanglement between far away parts of the spreading wave packet indicating importance of long-range effects.     

Stark resonances in disordered systems

We study further the metastable behavior of Metropolis dynamics for the two-dimensional nearest neighbor ferromagnetic Ising model, with positive and small external field, in the limit as the temperature vanishes (see [NS]). We focus on the typical features of the escape (nucleation) from the (metastable) configuration with all spins –1, to the (stable) configuration with all spins +1. Using the reversibility of the process as the main tool, we prove (for the discrete time version of the model) that the first step of a typical escaping path is the time reverse of a typical time evolution of a shrinking subcritical rectangular droplet, which is one slice smaller than a critical droplet. This subcritical droplet then evolves in a time of order 1 to a critical droplet, which finally grows with features described in [NS].Work partially supported by the Brazilian CNPq and by the American NSF, under grant DMS91-00725     

Stark ladder laser with a coherent electron subsystem

A theory of generation in a two-subband “Stark ladder” with a coherent electron subsystem is developed. In the proposed model, electrons reach the upper level of a quantum well due to resonant tunneling and pass to the lower level of the well (vertical transitions), emitting a photon ?ω, then tunnel resonantly to the upper level of a neighboring well, performing a radiative transition, and so on until electrons leave the lower level of the last well. A static electric field applied to the superlattice shifts the levels so that the lower level of the nth well coincides with the upper level of the (n+1)th well. Analytic expressions are derived for the wave functions and polarization currents of an N-well structure. The possibility of bulk oscillation of the N-well structure in the optimal mode with an efficiency close to unity, weak reflection, and a linear dependence of the power on the pumping current is demonstrated. The total generation power is proportional to the number of wells. For structures with an even number of wells, the energy of electrons from the emitter must simply coincide with the resonance energy for any laser fields; i.e., the energy tuning which is necessary in a single-well structure is not required. Universal relations are derived for parameters of the N-well structure, which ensure the simultaneous fulfillment of resonance conditions in all the wells. The possibility of coherent lasing in a one-subband Stark ladder with a lower gain is also indicated.     

Geometric theory of Stark resonances in multielectron systems

In this paper we consider a class of many-body systems in a weak homogeneous electric field. This class includes atoms and molecules with infinitely heavy nuclei. It follows from one of the results of this paper and a result of [S 3] that the bound states of such systems in the absence of electric field turn into resonances (which we call the Stark resonances) as soon as the electric field is switched on. (The stability part of this result was earlier proven in [HeSi] (see also [Hu 2]) under an assumption of dilation analyticity.) The main result of this paper is exponential bounds on the width (and therefore the lower exponential bounds on the life-time) of the Stark resonances. These bounds are given in terms of the Stark instanton action. In contrast to the usual (one body) action the latter is not entirely classical but incorporates certain quantum data (like ionization energies). The bounds give a partial generalization to the many electron case of the well-known Oppenheimer formula for the hydrogen.Research support by NSERC under grant NA 7901     

Asymptotics of resonances for 1D Stark operators

We consider the Stark operator perturbed by a compactly supported potential on the real line. We determine the forbidden domain for resonances, asymptotics of resonances at high energy and asymptotics of the resonance counting function for large radius.     

Stark resonances: Asymptotics and distributional Borel Sum

We prove that the Stark effect perturbation theory of a class of bound states uniquely determines the position and the width of the resonances by Distributional Borel Sum. In particular the small field asymptotics of the width is uniquely related to the large order asymptotics of the perturbation coefficients. Similar results apply to all the resonances of the anharmonic and double well oscillators.Partially supported by Ministero della Università e della Ricerca Scientifica     

Existence of resonances in three dimensions

IfP is an elliptic self-adjoint perturbation of the Laplacian on ³, and the coefficients ofP– decay super-exponentially, then we show thatP has infinitely many resonances. The resonances are defined here as the poles of the meromorphic continuation of (P–²)^–1.     

Stark ladders of resonances: Wannier ladders and perturbation theory

LetH _B be any fixed one-dimensional Bloch Hamiltonian with only the firstm gaps open andH _F=H_B+F_x be the corresponding Stark Hamiltonian. For any positiveF small enoughH _F has onlym ladders of sharp resonances given by the analytic translation method, the decoupled band approximation and the regular perturbation theroy. This way, the Wannier conjecture becomes a definite regular perturbation theory for the Stark ladders as eigenvalues of the translated Hamiltonian.     

A note on the modulation of Stark field induced resonances

Clusters of resonances have been observed to modulate in a regular manner in the Stark spectra of the potassium atom. This phenomenon is due to the manifold structure of the atom in an electric field. The energy spacings of the regular features are related to the modulation caused by the shortest precessing trajectories but the correspondence is not exactly one-to-one.     

Lower bounds on width of Stark resonances in one dimension

We prove a lower bound for the width of Stark resonances in one dimension.     

Loss and gain in Bloch oscillating super-superlattices: THz Stark ladder spectroscopy

Bloch oscillation in electrically biased semiconductor superlattices offer broadband terahertz gain from DC up to the Bloch frequency or Stark splitting. Useful gain up to 2–3 THz can provide a basis for solid-state electronic oscillators operating at 10 times the frequency of existing devices.A major stumbling block is the inherent instability of the electrically biased doped superlattices to the formation of static or dynamic electric field domains. To circumvent this, we have fabricated super-superlattices in which a large superlattice is punctuated with heavily doped regions. The short superlattice sections have subcritical “nL” products.Room temperature, terahertz photon-assisted transport in short InGaAs/InAlAs superlattice cells allows us to determine the Stark ladder splitting as the superlattice is electrically biased and confirms the absence of electric field domains in short structures.Absorption of radiation from 1.5 to 2.5 THz by electrically biased InAs/AlSb super-superlattices exhibit a crossover from loss to gain as the Stark ladder is opened. Measurements are carried out at room temperature in a novel planar terahertz waveguide defined by photonic band gap sidewalls and loaded with an array of electrically biased super-superlattices. The frequency-dependent crossover voltage indicates 80% participation of the super-superlattice.     

Wannier–Stark resonances in Zener tunneling diodes with GaAs/AlAs superlattices

We present DC transport measurements of the valence to conduction band (Zener) tunneling current in ap–i–ndiode with an ultrathin intrinsic layer containing a (GaAs)₅/(AlAs)₂multi-quantum well structure. According to recent theoretical predictions, the DC current should show maxima as a function of the reverse bias voltage that reflect the formation of Wannier–Stark resonances. So far, Wannier–Stark resonances have only been observed optically and never in a regime of strong Zener tunneling. Experimentally, we find the second derivative of the current-voltage characteristics to show a weak oscillatory structure indeed, indicating the existence of Wannier–Stark resonances in Zener tunneling.     

Existence and Borel summability of resonances in hydrogen stark effect

Existence of resonances in hydrogen Stark effect is proved. It is also proved that the divergent time-independent perturbation expansions are Borel summable to the resonances, and a simple application of the Borel-Padé method for computing their position and width is indicated.Partially supported by G.N.F.M., C.N.R.Partially supported by I.N.F.N., Sezione di Bologna.     

Effect of high magnetic fields on the conductivity of a quantum cylinder under Stark ladder conditions

The conductivity of a quantum cylinder with a parabolic lateral confinement potential and a superstructure is studied under conditions where uniform static quantizing electric and magnetic fields are applied along the cylinder axis. The charge carriers are assumed to be scattered by optical phonons. The dependence of the current density along the superlattice axis on the dc magnetic field is obtained. It is shown that, under certain conditions, the so-called Stark-hybrid-phonon resonance appears due to the hybridization of the electronic energy spectrum. In turn, this gives rise to a sharply nonmonotonic magnetic-field dependence of the current density.     

The derivation of explicit solutions to Wannier–Stark resonances for electrons on coupled chains

Electrons on infinite coupled chains with nearest neighbour couplings under uniform electric and magnetic fields can be expressed as conditionally solvable systems, which concerns both discrete coordinate and wavenumber representations. We then have to account for multiparameter extra-conditions relying on the single chain phase of the system, which amounts to perform a suitable selection of parameters. The implicit plots provided by such conditions exhibit both regular and irregular patterns. This results in the onset of a finite number of Wannier–Stark resonances, now by performing rescalings needed. However, this time the resonance width is sensitive to the quantum number characterizing the Stark-ladder. Bound-state limits, rescalings and approximations proceeding irrespective of the wavenumber have also been presented.