Collapse of resonance in quasi-one-dimensional quantum channels

We study the resonance structure of the conductance (transmissivity) of a quasi-one-dimensional channel that contains an attractive impurity of finite dimensions and derive an exact expression for the scattering matrix. We show that an impurity of finite dimensions may cause a set of Fano resonances to appear in the transmissivity. We also find that due to the coherent interaction the Fano resonances can collapse and discrete levels may appear in the continuum. Finally, we establish the wave function of the discrete levels and study the channel transmissivity in the critical regime. Zh. éksp. Teor. Fiz. 116, 263–275 (July 1999)     

Crossings and Anticrossings of Unbound States

We investigate the characteristic crossings and anticrossings of energies and widths of a doublet of resonances, observed in the vicinity of, and at a degeneracy of unbound states, when the control parameters of the system are varied. This characteristic behavior is explained in terms of the local, topological structure of the surfaces that represent the complex energy eigenvalues in parameter space in the vicinity of a degeneracy point. In the simple but illustrative case of the scattering of a beam of particles by a double barrier potential well with two regions of trapping, we solved numerically the implicit, transcendental equation that defines the eigenwave numbers of a degenerate isolated doublet of resonances as functions of the real, control parameters of the system. We found that, at a degeneracy of unbound states, the surface representing the resonance eigenwave numbers as functions of the control parameters has an algebraic branch point of rank one. Unfolding the degeneracy point, crossings and anticrossings of energies and widths are obtained as projections of sections of the eigenwave number surfaces.     

Electronic resonances in a quantum well having a periodically uneven boundary

The spectrum of the electronic states in an infinitely deep two-dimensional potential well, where one wall is periodically uneven, is investigated theoretically. It is shown that in non-Bragg type resonances — standing electron wave resonances, which are modes of different spatial harmonics of the electron wave function — arise in such a well. The resonances occur in a wide range of energies, starting at values close to the bottom in each 2D subband. The resonance interaction splits the energy spectrum and results in the appearance of gaps, giving the electron spectrum a miniband character. The properties of the electron gas vary substantially in accordance with the new characteristics of the spectrum. Fiz. Tverd. Tela (St. Petersburg) 41, 1867–1870 (October 1999)     

Degeneracy of Resonances: Branch Point and Branch Cuts in Parameter Space

The rich phenomenology of crossings and anticrossings of energies and widths, observed in an isolated doublet of resonances when one control parameter is varied, is fully explained in terms of the topological properties of the energy hypersurfaces close to the degeneracy point. The hypersurface representing the complex resonance eigenvalues, as functions of the control parameters, has an algebraic branch point of rank one, and branch cuts in its real and imaginary parts, in parameter space. Associated with this singularity in parameter space, the scattering matrix, S _ℓ (E), and the Green’s function, G _ℓ ⁽⁺⁾(k; r,r'), have one double pole in the unphysical sheet of the complex energy plane. We characterize the universal unfolding or deformation of any degeneracy point of two unbound states in parameter space by means of a universal 2-parameter family of functions which is contact equivalent to the pole position function of the isolated doublet of resonances at the exceptional point and includes all small perturbations of the degeneracy condition up to contact equivalence.     

On Algebraic Integrability of the Deformed Elliptic Calogero-Moser Problem

Abstract

We discuss stationary solutions of the discrete nonlinear Schrödinger equation (DNSE) with a potential of the ? ⁴ type which is generically applicable to several quantum spin, electron and classical lattice systems. We show that there may arise chaotic spatial structures in the form of incommensurate or irregular quantum states. As a first (typical) example we consider a single electron which is strongly coupled with phonons on a 1D chain of atoms — the (Rashba)–Holstein polaron model. In the adiabatic approximation this system is conventionally described by the DNSE. Another relevant example is that of superconducting states in layered superconductors described by the same DNSE. Amongst many other applications the typical example for a classical lattice is a system of coupled nonlinear oscillators. We present the exact energy spectrum of this model in the strong coupling limit and the corresponding wave function. Using this as a starting point we go on to calculate the wave function for moderate coupling and find that the energy eigenvalue of these structures of the wave function is in exquisite agreement with the exact strong coupling result. This procedure allows us to obtain (numerically) exact solutions of the DNSE directly. When applied to our typical example we find that the wave function of an electron on a deformable lattice (and other quantum or classical discrete systems) may exhibit incommensurate and irregular structures. These states are analogous to the periodic, quasiperiodic and chaotic structures found in classical chaotic dynamics.     

Semiclassical energy levels of electrons in metals with band degeneracy lines

We show that in calculating the semiclassical energy levels of electrons in metals located in a magnetic field, one must determine whether or not the corresponding electron paths in the space of wave vectors k are attached to a band degeneracy line. Calculations in the two possible cases, i.e., with and without such attachment, differ by |e|ℏ/2m*c, where e is the electron charge and m* is the cyclotron mass of the electron. This shift in the energy levels is of a topological nature, and its existence depends neither on the specific form of the electron dispersion relation ε(k) near the electron path nor on the shape or size of this path. The reason for this shift lies in the fact that the electron orbit is attached to the band degeneracy line, which is the line of singular points of the Bloch wave functions. In many respects this effect is similar to the Aharonov-Bohm effect if the band degeneracy line is considered an infinitely thin “solenoid.” This shift in energy levels should become apparent in studies of oscillation phenomena in metals. We give examples of metals in which the conditions for observing the shift is probably the most favorable. Zh. éksp. Teor. Fiz. 114, 1375–1392 (October 1998)     

A Model of Wavefunction Collapse in Discrete Space-Time

We give a new argument supporting a gravitational role in quantum collapse. It is demonstrated that the discreteness of space-time, which results from the proper combination of quantum theory and general relativity, may inevitably result in the dynamical collapse of the wave function. Moreover, the minimum size of discrete space-time yields a plausible collapse criterion consistent with experiments. By assuming that the source to collapse the wave function is the inherent random motion of particles described by the wave function, we further propose a concrete model of wavefunction collapse in the discrete space-time. It is shown that the model is consistent with the existing experiments and macroscopic experiences. PACS numbers: 0365B 0460     

Interference of quantum states in electronic waveguides with impurities

Effects of interference between propagating and localized states in quasi-one-dimensional electronic waveguides containing finite-size attracting impurities (quantum dots) are investigated. The electron scattering matrix is calculated in the framework of the Feshbach theory [H. Feshbach, Ann. Phys. 5, 357 (1958); Ann. Phys. 19, 287 (1962)], when resonant states in closed channels are taken into account exactly, while non-resonant states are taken into account in perturbation theory. It is shown that finite-size attracting impurities may generate a series of asymmetric Fano resonances in the waveguide transmission. As a result of interference of electron states, the characteristics of resonances may oscillate upon a change in the impurity parameters. The conditions are determined under which the interference of an electron wave leads to a “collapse” and “swing” of Fano resonances.     

Effect of phasons and magnetic fields on the electronic spectrum of a three-dimensional quasicrystal

The effect of phasons and magnetic fields on the electronic spectrum of an icosahedral quasicrystal is investigated in the tight-binding approximation. Phasons smooth the singular spectrum and produce a greater delocalization of the critical wave functions. A magnetic field shifts the limits of the spectrum, smooths the spectrum, lifts the degeneracy, and also delocalizes the wave functions. Pis’ma Zh. éksp. Teor. Fiz. 69, No. 9, 659–661 (10 May 1999)     

Interaction effects on persistent current of ballistic cylindrical nanostructures

We investigate clean cylindrical nanostructures with an applied longitudinal static magnetic field. The ground state of these systems becomes degenerate for particular values of the field due to Aharonov-Bohm effect. The Coulomb interaction introduces couplings between the electronic configurations. Consequently, depending on particular selection rules, the ground state may become, in the interacting case, a many body state at the degeneracy points: a gap is then opened. To study this problem, we propose a variational multireference wave function which goes beyond the Hartree-Fock approximation. Using this ansatz, in addition to the replacements of some crossings by avoided crossings, two other important effects of the electron-electron interaction are pointed out: (i) the long-range part of the Coulomb potential tends to shift the position in magnetic field of the crossing or avoided crossing points and, (ii) at the points of degeneracy or near degeneracy, the interaction can drive the system from a singlet to a triplet state inducing new real crossing points in the ground state energy curve as function of the field. In any case, the crossing points that are due to either orbital or spin effects, should manifest themselves in various experiments as sudden changes in the response of the system (magnetoconductance, magnetopolarisability, ...) when the magnetic field is tuned.     

Electromagnetically induced transparency in degenerate two-level systems

The evolution of an electromagnetic wave with slowly varying polarization, which interacts resonantly with the medium formed by degenerate two-level atoms, is studied using the wave function approach under the conditions of electromagnetically induced transparency. It is shown that the amplitude of the wave field propagates at the velocity of light in such a medium. The equation obtained for the polarization parameter has a solution in the form of a simple wave. The breaking length is determined. It is shown that the velocity of propagation of polarization waves may be much smaller than the velocity of light. The proposed approach is common for two-level systems with an arbitrary degeneracy. The case of a system with Zeeman degeneracy is analyzed in detail. The dependence of the velocity of propagation of the polarization structure on the amplitude and polarization is determined for an arbitrary level degeneracy. The evolution of the polarization structure in such a medium is discussed.     

Excitation of 8Be monopole resonances under αα scattering

A new approach to the realization of the hyperspherical function method is developed. An expansion of the eight-nucleon wave function in the hyperspherical basis is applied to the investigation of the continuous spectrum of two-α-particle states. Narrow monopole resonances of ⁸Be at energies from 30 to 60MeV are discovered. The problem is treated on the one-open-channel approximation. The influence of the channels, which are not taken into account in this approximation, on the formation of these resonances is discussed.     

Effect of a weak magnetic field on resonant features of conductance in an open circular billiard with Dresselhaus spin-orbit interaction

The effect of a weak magnetic field on the transport properties of an open circular billiard with attached channels in the presence of Dresselhaus spin-orbit interaction are considered. It is shown that the application of a magnetic field leads to splitting of the Fano resonances, which were found earlier on the energy dependence of conductance, into pairs of resonances with half lower amplitudes. The relationship between the energy values to which these resonances collapse when the spin-orbit interaction is absent, and the levels of the energy spectrum in the corresponding closed billiard has been established. It is shown also that the applied magnetic field induces a qualitative change in the spin polarization of a wave in the output channel.     

Parametric instability and Hamiltonian chaos in cavity semiclassical electrodynamics

We study the nonlinear dynamics of the interaction of two-level atoms and a selected mode of a high-Q cavity with frequency modulation analytically and numerically. In the absence of modulation, the corresponding semiclassical Heisenberg equations for the expectation values of the collective atomic observables and the field-mode amplitudes allow, in the rotating wave approximation and in the strong-coupling limit, an exact solution with arbitrary detuning. Using this solution, we detect the coherent effect of trapping of the population of atomic levels and of trapping of the number of photons in the cavity. The explanation for this effect lies in the destructive interference of the atomic dipoles and the field mode. The integrable version of the system of equations exhibits a separatrix near which a stochastic layer is formed when modulation is introduced. The width of the layer is found to gradually increase with degree of modulation, and finally it fills the entire energy-permissible volume of the phase space. We show that the rotating wave approximation does not hinder the formation of Hamiltonian chaos in cavity semiclassical electrodynamics. The calculation of the maximum Lyapunov indices of nonlinear (in this approximation) equations of motion as functions of the modulation frequency δ and the frequency of natural Rabi oscillations of the atom-field system, Ω, suggests that Hamiltonian chaos appears first in the area of the fundamental parametric resonance, δ/2Ω≃1. Parametric instability increases with increasing modulation and decreasing detuning from the atom-field resonance, generating at exact resonance new areas of chaos corresponding to multiple parametric resonances. The results of numerical experiments and estimates of the characteristic parameters show that Rydberg atoms placed in a high-Q microwave cavity are possible objects for observing parametric instability and dynamical chaos. Zh. éksp. Teor. Fiz. 115, 740–753 (February 1999)     

Third-order diamagnetic susceptibilities of hydrogenlike atoms

We develop a method for calculating diamagnetic susceptibilities based on higher-order perturbation theory for the wave function and energy of the excited states of the hydrogen atom with degeneracy of arbitrary multiplicity. We derive analytical expressions for third-order matrix elements in the spherical states |nlm〉 with fixed principal quantum number n and magnetic quantum number m. The formulas for the susceptibilities of doubly degenerate levels are represented in the form of radical-fractional relationships containing polynomials in the principal quantum number. We establish the existence of a monotonic interdependence between the absolute values of susceptibilities of the first three orders. We also present the results of numerical calculations for the states with n⩽6 and m⩽3 mixed by the field. Finally, for Rydberg states with large n and small m we detect the existence of a discontinuity in the interdependence of the susceptibilities at the boundary between the doublet and equidistant parts of the spectrum of diamagnetic sublevels with opposite parities. Zh. éksp. Teor. Fiz. 116, 838–857 (September 1999)     

Parametrically excited helicopter ground resonance dynamics with high blade asymmetries

The present work is aimed at verifying the influence of high asymmetries in the variation of in-plane lead-lag stiffness of one blade on the ground resonance phenomenon in helicopters. The periodical equations of motions are analyzed by using Floquet's Theory (FM) and the boundaries of instabilities predicted. The stability chart obtained as a function of asymmetry parameters and rotor speed reveals a complex evolution of critical zones and the existence of bifurcation points at low rotor speed values. Additionally, it is known that when treated as parametric excitations; periodic terms may cause parametric resonances in dynamic systems, some of which can become unstable. Therefore, the helicopter is later considered as a parametrically excited system and the equations are treated analytically by applying the Method of Multiple Scales (MMS). A stability analysis is used to verify the existence of unstable parametric resonances with first and second-order sets of equations. The results are compared and validated with those obtained by Floquet's Theory. Moreover, an explanation is given for the presence of unstable motion at low rotor speeds due to parametric instabilities of the second order.     

Transverse instability threshold in counterpropagating light beams for a nonlinear medium with local photorefractive response

We derive the threshold conditions for the instability of counterpropagating waves in a nonlinear medium with local photorefractive response against the excitation of transverse small-angle structures. These conditions allow for all the important types of diffraction from refractive-index reflection gratings and are not limited to the case of strict frequency degeneracy of the waves. We study the dependence of the crystal-thickness threshold and the secondary wave emission angle on the crystal parameters and the pump conditions. We show that when the pump wave intensities differ considerably, excitation of standing light structures is replaced by excitation of traveling structures. Finally, we discuss the applications of the theory to experiments with the photorefractive crystals LiNbO₃ and LiTaO₃. Zh. éksp. Teor. Fiz. 111, 1611–1623 (May 1977)     

Accurate transport cross sections for the Lennard-Jones potential

It is well-known that in open quantum systems resonances can coalesce at an exceptional point, where both the energies and the wave functions coincide. In contrast to the usual behaviour of the scattering amplitude at one resonance, the coalescence of two resonances invokes a pole of second order in the Green’s function, in addition to the usual first order pole. We show that the interference due to the two pole terms of different order gives rise to patterns in the scattering cross section which closely resemble Fano-Feshbach resonances. We demonstrate this by extending previous work on the analogy of Fano-Feshbach resonances to classical resonances in a system of two driven coupled damped harmonic oscillators.