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# Resonance statistics in a microwave cavity with a thin antenna

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*in*Physics Letters A 228(3) · February 1997

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Abstract

We propose a model for scattering in a flat resonator with a thin antenna. The results are applied to rectangular microwave cavities. We compute the resonance spacing distribution and show that it agrees well with experimental data provided the antenna radius is much smaller than wavelengths of the resonance wavefunctions.

- ... We now discuss the general idea of the approach used in Ref. [11] which allows us to treat the presence of scatterers and antennas in a simple way assuming that the wavelength of the field is much larger than the characteristic size of the area of the attachment. Figure 1 illustrates the principal setup to measure spectra of a cavity. ...... We substantially follow paper [11]. Basically the idea of the approach is to construct a self-adjoint extension of the Hamiltonian in the presence of point-like perturbations. ...... We call this area Ω ǫ . Following the general procedure [17] [9] [11] we consider two quadratic forms ...Article
- Mar 2008

The theoretical interpretation of measurements of "wavefunctions" and spectra in electromagnetic cavities excited by antennas is considered. Assuming that the characteristic wavelength of the field inside the cavity is much larger than the radius of the antenna, we describe antennas as "point-like perturbations". This approach strongly simplifies the problem reducing the whole information on the antenna to four effective constants. In the framework of this approach we overcame the divergency of series of the phenomenological scattering theory and justify assumptions lying at the heart of "wavefunction measurements". This selfconsistent approach allowed us to go beyond the one-pole approximation, in particular, to treat the experiments with degenerated states. The central idea of the approach is to introduce ``renormalized'' Green function, which contains the information on boundary reflections and has no singularity inside the cavity. Comment: 23 pages, 6 figures - ... The choice of the coupling depends on particular properties of the junction it models, of course, but one would like to select a subclass representing a " natural " coupling. One way to achieve this goal was suggested in [23]: comparing the scattering matrix of the junction given by (4.1) with the low–energy behavior of scattering in the system of a plane to which a cylindrical " tube " is attached, and taking into account the condition (4.2), we arrive at the identification ...... where ρ is the contact radius. Physical relevance of these conditions was illustrated in [23] by explaining the experimentally observed distribution of resonances in a microwave resonator with a thin antenna. Motivated by this, we will use in the following (4.1) and (4.3) to describe the coupling between the leads and the scatterers. ...... The constant depends only on the manifold G we will neglect it in the following, because its nonzero value means just a coupling constant renormalization: D j has to be changed to D j + 2πc(G) . For a flat rectangular G we found in [23] an agreement with the experiment using c(G) = 0 . ...Article
- Sep 2001
- J MATH PHYS

We study transport in quantum systems consisting of a finite array of N identical single-channel scatterers. A general expression of the S matrix in terms of the individual-element data obtained recently for potential scattering is rederived in this wider context. It shows in particular how the band spectrum of the infinite periodic system arises in the limit N→∞. We illustrate the result on two kinds of examples. The first are serial graphs obtained by chaining loops or T-junctions. Another example concerns geometric scatterers where the individual element consists of a surface with a pair of leads; we show that apart from the resonances coming from the decoupled-surface eigenvalues, such scatterers exhibit the high-energy behavior typical for the δ′ interaction for the physically interesting couplings. © 2001 American Institute of Physics. - ... We note that the proper definition of the scattering length was missing in Refs. [1, 12]. This has led to the deficiencies discussed above. ...Article
- Oct 2009

In the paper we analyze the "singular statistics" of pseudointegrable Seba billiards and show that taking into account growing number of resonances one observes the transition from "semi-Poissonian"-like statistics to Poissonian. This observation is in agreement with an argument that a classical particle does not feel a point perturbation. However, our findings contradict results reported earlier (P. Seba, Phys. Rev. Lett. 64, 1855 (1990)). Comment: 22 pages, 11 figures; submitted to New Journal of Physics - ... In this paper we consider the simplest situation when we have a single connected manifold to which a finite number of semiinfinite leads are attached — one is especially interested in transport in such a system. Particular models of this type have been studied, e.g., in [7, 8, 18, 20, 25]. Again for the sake of simplicity we limit ourselves mostly to the situation when there are no external fields; the Hamiltonian will act as the negative second derivative on the halflines representing the leads and as Laplace-Beltrami operator on the manifold. ...ArticleFull-text available
- Feb 2013

We discuss resonances for a nonrelativistic and spinless quantum particle confined to a two- or three-dimensional Riemannian manifold to which a finite number of semiinfinite leads is attached. Resolvent and scattering resonances are shown to coincide in this situation. Next we consider the resonances together with embedded eigenvalues and ask about the high-energy asymptotics of such a family. For the case when all the halflines are attached at a single point we prove that all resonances are in the momentum plane confined to a strip parallel to the real axis, in contrast to the analogous asymptotics in some metric quantum graphs; we illustrate it on several simple examples. On the other hand, the resonance behaviour can be influenced by a magnetic field. We provide an example of such a `hedgehog' manifold at which a suitable Aharonov-Bohm flux leads to absence of any true resonance, i.e. that corresponding to a pole outside the real axis. - ... The starting point is the general expression for the scattering matrix (S-matrix) for such systems, that we derived in the previous paper [10], called Part I. It considered the main properties of cavities with point-like perturbations, following the ideas of [20]. In the derived expression the whole information on the antenna construction was reduced to four effective constants. ...Article
- Nov 2010

[For Part I see the authors and R. Höhmann, ibid. 41, No. 27, Article ID 275101, 22 p. (2008; Zbl 1149.81023).] We consider an application of a general theory for cavities with point-like perturbations for a rectangular shape. Hereby, we concentrate on experimental wave patterns obtained for nearly degenerate states. The nodal lines in these patterns may be broken, which is an effect coming only from the experimental determination of the patterns. We show that a wavefunction measurement based on a movable point-like perturbation has an intrinsic limit of resolution. When shifts of resonances become comparable with level spacings, the most pronounced effect of the unavoidable mixing of eigenfunctions is a rearrangement of the experimentally obtained nodal line structure. These findings are explained within a framework of the developed theory. - ... However, if the system is integrable, then the distribution is of Poisson type. There is a similar conjecture for resonance distribution [13]. Obviously, an SG of any finite order is integrable; however, it is still unknown whether or not any SG of infinite b) Dependence of the distance between opposite peaks on the value of perturbation. ...A continuous model of the Sierpinski gasket (SG) is suggested. The Laplace operator on the SG is defined. An effective computational algorithm for solving the scattering problem is suggested. The self-similarity of the graph of transmission coefficient via the wave number k is observed. A violation of symmetry of the SG is considered. The results are compared with the discrete SG model.
- ... Linear point interactions arise as a particular, but relevant, application of the more general theory of self-adjoint extension of symmetric operators; a theory that has gained new popularity in recent years also for the application to the study of evolution equations in non-standard domains, such as quantum graphs (see e.g., [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]) and quantum hybrids (see e.g., [27][28][29][30][31]). ...ArticleFull-text available
- Apr 2018

We present an introduction to the nonlinear Schr\"odinger equation (NLSE) with concentrated nonlinearities in $\mathbb{R}^2$. Precisely, taking a cue from the linear problem, we sketch the main challenges and the typical difficulties that arise in the two-dimensional case, and mention some recent results obtained by the authors on local and global well-posedness. - ArticleFull-text available
- Apr 2018

- Article
- Aug 1997
- PHYS LETT A

We examine the quantum motion of two particles interacting through a contact force which are confined in a rectangular domain in two and three dimensions. When there is a difference in the mass scale of two particles, adiabatic separation of the fast and slow variables can be performed. Appearance of the Berry phase and magnetic flux is pointed out. The system is reduced to a one-particle Aharonov-Bohm billiard in two-dimensional case. In three dimension, the problem effectively becomes the motion of a particle in the presence of closed flux string in a box billiard. - ArticleFull-text available
- Mar 2008

The purpose of this text is to set up a few basic notions concerning quantum graphs, to indicate some areas addressed in the quantum graph research, and to provide some pointers to the literature. The pointers in many cases are secondary, i.e. they refer to other surveys. - We investigate a charged two-dimensional particle in a homogeneous magnetic field interacting with a periodic array of point obstacles. We show that while Landau levels remain to be infinitely degenerate eigenvalues, between them the system has bands of absolutely continuous spectrum and exhibits thus a transport along the array. We also compute the band functions and the corresponding probability current. The fact that the presence of boundaries can induce a transport in a system with a homogeneous magnetic field is known for long [1, 2] and has numerous consequences in solid state physics. The mentioned pioneering papers were followed by tremendous number of studies in which the magnetic transport was analyzed numerically in various models as well as experimentally. The obstacles at which the particle “bounces ” need not be hard walls but also objects with openings such as various antidot lattices; for a sample of literature
- Article
- May 2002
- J MATH PHYS

The quantum-mechanical scattering on a compact Riemannian manifold with semi-axes attached to it (hedgehog-shaped manifold) is considered. The complete description of the spectral structure of Schroedinger operators on such a manifold is done, the proof of existence and uniqueness of scattering states is presented, an explicit form for the scattering matrix is obtained and unitary nature of this matrix is proven. It is shown that the positive part of the spectrum of the Schroedinger operator on the initial compact manifold as well as the spectrum of a point perturbation of such an operator may be recovered from the scattering amplitude for one attached half-line. Moreover, the positive part of the spectrum of the initial Schroedinger operator is fully determined by the conductance properties of an "electronic device" consisting of the initial manifold and two "wires" attached to it. - Article
- Jan 2003
- J Phys Math Gen

We study a free quantum motion on periodically structured manifolds composed of elementary two-dimensional "cells" connected either by linear segments or through points where the two cells touch. The general theory is illustrated with numerous examples in which the elementary components are spherical surfaces arranged into chains in a straight or zigzag way, or two-dimensional square-lattice "carpets". We show that the spectra of such systems have an infinite number of gaps and that the latter dominate the spectrum at high energies. 1 - ArticleFull-text available
- Sep 2010

We discuss spectral and resonance properties of a Hamiltonian describing motion of an electron moving on a "hybrid surface" consisting on a halfline attached by its endpoints to a plane under influence of a constant magnetic field which interacts with its spin through a Rashba-type term. Comment: 16 pages, 2 pdf figures - Article
- Jun 2012
- J MATH PHYS

We study the transmission of a quantum particle along a straight input--output line to which a graph $\Gamma$ is attached at a point. In the point of contact we impose a singularity represented by a certain properly chosen F\"ul\"op--Tsutsui coupling with a coupling parameter $\alpha$. We show that the probability of transmission along the line as a function of the particle energy tends to the indicator function of the energy spectrum of $\Gamma$ as $\alpha\to\infty$. This effect can be used for a spectral analysis of the given graph $\Gamma$. Its applications include a control of a transmission along the line and spectral filtering. The result is illustrated with an example where $\Gamma$ is a loop exposed to a magnetic field. Two more quantum devices are designed using other special F\"ul\"op--Tsutsui vertex couplings. They can serve as a band-stop filter and as a spectral separator, respectively. - Article
- Sep 2005
- ACTA PHYS POL A

We study approximations of billiard systems by lattice graphs. It is demonstrated that under natural assumptions the graph wave functions approximate solutions of the Schrödinger equation with energy resealed by the billiard dimension. As an example, we analyze a Sinai billiard and a scattering system obtained by attaching a pair of external leads to it. The results illustrate emergence of global structures in large quantum graphs. - Article
- May 2012

Resonance and decay phenomena are ubiquitous in the quantum world. To understand them in their complexity it is useful to study solvable models in a wide sense, that is, systems which can be treated by analytical means. The present review offers a survey of such models starting the classical Friedrichs result and carrying further to recent developments in the theory of quantum graphs. Our attention concentrates on dynamical mechanism underlying resonance effects and at time evolution of the related unstable systems. - Article
- Feb 2005
- REP MATH PHYS

We show how the von Neumann theory of self-adjoint extensions can be used to investigatequantum systems the configuration space of which can be decomposed into parts of different dimensionalities. The method can be applied in many situations; we illustrate it on examples including point contact spectroscopy, nanotube systems, microwave resonators, or spin conductance oscillations. - Article
- Dec 2010
- NEW J PHYS

In this paper, we analyze the 'singular statistics' of pseudointegrable Šeba billiards, i.e. billiards perturbed by zero-range perturbations. We have shown that the computation of a spectrum is reduced to the calculation of the uniquely defined renormalized Green's function. We relate a spectrum of the billiard to the scattering length, which is the only parameter describing the perturbation. We show that taking into account the growing number of resonances, one observes a transition from 'semi-Poissonian'-like statistics to Poissonian. This observation is in agreement with the argument that a classical particle does not feel a point perturbation. - Article
- Apr 2011
- REP MATH PHYS

We discuss spectral and resonance properties of a Hamiltonian describing motion of an electron moving on a "hybrid surface" consisting on a halfline attached by its endpoints to a plane under influence of a constant magnetic field which interacts with its spin through a Rashba-type term. - Article
- Apr 2011
- REP MATH PHYS

We discuss spectral and resonance properties of a Hamiltonian describing motion of an electron moving on a "hybrid surface" consisting on a halfline attached by its endpoint to a plane under influence of a constant magnetic field which interacts with its spin through a Rashba-type term. - ArticleFull-text available
- May 2014

We investigate non-equilibrium particle transport in the system consisting of a geometric scatterer and two leads coupled to heat baths with different chemical potentials. We derive expression for the corresponding current the carriers of which are fermions and analyze numerically its dependence of the model parameters in examples, where the scatterer has a rectangular or triangular shape. - Book
- Jan 2013

This volume presents self-contained survey articles on modern research areas written by experts in their fields. The topics are located at the interface of spectral theory, theory of partial differential operators, stochastic analysis, and mathematical physics. The articles are accessible to graduate students and researches from other fields of mathematics or physics while also being of value to experts, as they report on the state of the art in the respective fields. - Article
- Dec 2016
- PHYS LETT A

We study the spectrum, resonances and scattering matrix of a quantum hamiltonian on a "hybrid surface" consisting of a half-line attached by its endpoint to the vertex of a concave planar wedge. At the boundary of the wedge, outside the vertex, Dirichlet boundary are imposed. The system is tunable by varying the measure of the angle at the vertex. - In this paper we attempt to reconstruct one of the last projects of Volodya Geyler which remained unfinished. We study motion of a quantum particle in the plane to which a halfline lead is attached assuming that the particle has spin $\frac12$ and the plane component of the Hamiltonian contains a spin-orbit interaction of either Rashba or Dresselhaus type. We construct the class of admissible Hamiltonians and derive an explicit expression for the Green function applying it to the scattering in such a system. Comment: LaTeX, 9 pages; in memoriam Vladimir A. Geyler (1943-2007)

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The signatures of classical chaos and the role of periodic orbits in the wave-mechanical eigenvalue spectra of two-dimensional billiards are studied experimentally in microwave cavities. The survival probability for all the chaotic cavity data shows a ``correlation hole,'' in agreement with theory, that is absent for the integrable cavity. The spectral rigidity Delta3(L), which is a measure of long-range correlation, is shown to be particularly sensitive to the presence of marginally stable periodic orbits. Agreement with random-matrix theory is achieved only after excluding such orbits, which we do by constructing a special geometry, the Sinai stadium. Pseudointegrable geometries are also studied, and are found to display intermediate behavior. - Article
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We develop a statistical theory of the amplitude of Coulomb blockade oscillations in semiconductor quantum dots based on the hypothesis that chaotic dynamics in the dot potential leads to behavior described by random-matrix theory. Breaking time-reversal symmetry is predicted to cause an experimentally observable change in the distribution of amplitudes. The theory is tested numerically and good agreement is found. - The complete sequence of 1060 eigenmodes with frequencies between 0.75 and 17.5 GHz of a quasi-two-dimensional superconducting microwave resonator shaped like a quarter of a stadium billiard with a Q value of Q~=105-107 was measured for the first time. The semiclassical analysis is in good agreement with the experimental data, and provides a new scheme for the statistical analysis and comparison with predictions based on the Gaussian orthogonal ensemble.
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We present a complete analytical theory of conductance fluctuations in quantum dots in the regime of chaotic dynamics. Using the supersymmetry method we calculate for the first time exactly the conductance distribution function for a system of noninteracting electrons. Knowledge of this quantity enables us to obtain information about statistical properties of wave functions of a closed dot. The far tail of the distribution function describes fluctuations of resonance conductance in the Coulomb blockade regime. - The complete set of resonance parameters for 950 resonances of a superconducting microwave cavity connected to three antennas has been measured. This cavity simulates the quantum mechanics of a particle in a Bunimovich stadium. The partial widths are found to follow a Porter-Thomas distribution. The Fourier transforms of the $S$-matrix autocorrelation functions decay algebraically (nonexponentially) in time. These results agree perfectly with the predictions of random-matrix theory. They constitute one of the most stringent tests ever of this expected connection between chaotic dynamics and randommatrix theory.
- We report measurements of mesoscopic fluctuations of Coulomb blockade peaks in a shape- deformable GaAs quantum dot. Distributions of peak heights agree with predicted universal functions for both zero and nonzero magnetic fields. Parametric fluctuations of peak height and position, measured using a two-dimensional sweep over gate voltage and magnetic field, yield autocorrelations of height fluctuations consistent with a predicted Lorentzian-squared form for the unitary ensemble. We discuss the dependence of the correlation field on temperature and coupling to the leads as the dot is opened.
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We consider "geometric" scattering for a Laplace-Beltrami operator on a compact Riemannian manifold inserted between "wires," that is, two half-lines. We discuss applicability and correctness of this model. With an example, we show that such a scattering problem may exhibit unusual properties: the transition coefficient has a sequence of sharp peaks which become more and more distant at high energy and otherwise turns to zero. Introduction In this paper we consider certain boundary value problems, namely, the one-dimensional scattering on two- or three-dimensional compact Riemannian manifolds. The motivation for studying this problems is twofold. The first comes from the paper of J. Avron, P. Exner and Y. Last [2], who dicussed the problem of approximating the scattering properties of ffi 0 - interaction by certain graphs. By ffi 0 - interaction we mean the boundary condition of the type u 0 (+a) = u 0 (Gammaa); u(+a) Gamma u(Gammaa) = ffu 0 (a) imposed on the funct...