arXiv subjects
Publications and source records attributed to Almas F. Sadreev.
Similar to the Berry conjecture of quantum chaos we consider elastic analogue which incorporates longitudinal and transverse elastic displacements with corresponding wave vectors. Based on that we derive the correlation functions for amplitudes and intensities of elastic displacements. Comparison to numerics in a quarter Bunimovich stadium demonstrates excellent agreement.
Using formalism of effective Hamiltonian we consider bound states in continuum (BIC). They are those eigen states of non-hermitian effective Hamiltonian which have real eigen values. It is shown that BICs are orthogonal to open channels of the leads, i.e. disconnected from the continuum. As a result BICs can be superposed to transport solution with arbitrary coefficient and exist in propagation band. The one-dimensional Aharonov-Bohm rings that are opened by attaching single-channel leads to them allow exact consideration of BICs. BICs occur at discrete values of energy and magnetic flux however it's realization strongly depend on a way to the BIC's point.
We formulate two types of electric RLC resonance network equivalent to quantum billiards. In the network of inductors grounded by capacitors squared resonant frequencies are eigenvalues of the quantum billiard. In the network of capacitors grounded by inductors squared resonant frequencies are given by inverse eigen values of the billiard. In both cases local voltages play role of the wave function of the quantum billiard. However as different from quantum billiards there is a heat power because of resistance of the inductors. In the equivalent chaotic billiards we derive the distribution of the heat power which well describes numerical statistics.
We consider infinitely long electromagnetic wave guide filled with a ferrite. The wave guide has arbitrary but constant cross-section $. We show that Maxwell equations are equivalent to the Schrödinger equation for single electron in the two-dimensional quantum dot of the form D with account of the Rashba spin-orbit interaction. The spin-orbit constant is determining by components of magnetic permeability of the ferrite. The upper component of electron spinor function corresponds to the z-th component electric field, while the down component $χ$ related to the z-th component of magnetic field by relation (30).
In the tight-binding approximation we consider multi-channel transmission through a billiard coupled to leads. Following Dittes we derive the coupling matrix, the scattering matrix and the effective Hamiltonian, but take into account the energy restriction of the conductance band. The complex eigenvalues of the effective Hamiltonian define the poles of the scattering matrix. For some simple cases, we present exact values for the poles. We derive also the condition for the appearance of double poles.
We study bound states of the two-dimensional Helmholtz equations with Dirichlet boundary conditions in an open geometry given by two straight leads of the same width which cross at an angle $θ$. Such a four-terminal junction with a tunable $θ$ can realized experimentally if a right-angle structure is filled by a ferrite. It is known that for $θ=90^o$ there is one proper bound state and one eigenvalue embedded in the continuum. We show that the number of eigenvalues becomes larger with increasing asymmetry and the bound-state energies are increasing as functions of $θ$ in the interval $(0,90^o)$. Moreover, states which are sufficiently strongly bent exist in pairs with a small energy difference and opposite parities. Finally, we discuss how with increasing $θ$ the bound states transform into the quasi-bound states with a complex wave vector.
The probability current statistics of two-dimensional open chaotic ballistic billiards is studied both analytically and numerically. Assuming that the real and imaginary parts of the scattering wave function are both random Gaussian fields, we find a universal distribution function for the probability current. In by-passing we recover previous analytic forms for wave function statistics. The expressions bridge the entire region from GOE to GUE type statistics. Our analytic expressions are verified numerically by explicit quantum-mechanical calculations of transport through a Bunimovich billiard.
The rectangular microwave resonator filed by a ferrite with uniaxial magnetic anisotropy is considered. It is shown that this task can be reduced to an empty rhombus resonator with vertex angle defined by external magnetic field provided the magnetic anisotropy of the ferrite is strong. Therefore statistics of eigen frequencies for TM modes is described by the Brody or semi-Poisson distribution with some exceptional cases.
According to Berry a wave-chaotic state may be viewed as a superposition of monochromatic plane waves with random phases and amplitudes. Here we consider the distribution of nodal points associated with this state. Using the property that both the real and imaginary parts of the wave function are random Gaussian fields we analyze the correlation function and densities of the nodal points. Using two approaches (the Poisson and Bernoulli) we derive the distribution of nearest neighbor separations. Furthermore the distribution functions for nodal points with specific chirality are found. Comparison is made with results from from numerical calculations for the Berry wave function.
Streamlines and distributions of nodal points are used as signatures of chaos in coherent electron transport through three types of billiards, Sinai, Bunimovich and rectangular. Numerical averaged distribution functions of nearest distances between nodal points are presented. We find the same form for the Sinai and Bunimovich billiards and suggest that there is a universal form that can be used as a signature of quantum chaos for electron transport in open billiards. The universal distribution function is found to be insensitive to the way avaraging is performed (over positions of leads, over an energy interval with a few conductance fluctuations, or both). The integrable rectangular billiard, on the other hand, displays nonuniversal distribution with a central peak related to partial order of nodal points for the case of symmetric attachment of leads. However cases with nonsymmetric leads tend to the universal form. Also it is shown how nodal points in rectangular billiard can lead to "channeling of quantum flows" while disorder in nodal points in the Sinai billiard gives rise to unstable irregular behavior of the flow.