Konstantin N. Pichugin

Conductance of Open Quantum Billiards and Classical Trajectories

Abstract We analyse the transport phenomena of 2D quantum billiards with convexboundary of different shape. The quantum mechanical analysis is performedby means of the poles of the S matrix while the classical analysis is based onthe motion of a free particle inside the cavity along trajectories with a differentnumber of bounces at the boundary. The value of the conductance depends onthe manner the leads are attached to the cavity. The Fourier transform of thetransmission amplitudes is compared with the length of the classical paths.There is good agreement between classical and quantum mechanical resultswhen the conductance is achieved mainly by special short-lived states such aswhispering gallery modes (WGM) and bouncing ball modes (BBM). In thesecases, also the localization of the wave functions agrees with the picture ofthe classical paths. The S matrix is calculated classically and compared withthe transmission coefficients of the quantum mechanical calculations for fivemodes in each lead. The number of modes coupled to the special states iseffectively reduced.I. INTRODUCTION

Whispering gallery modes in open quantum billiards

The poles of the S matrix and the wave functions of open two-dimensional quantum billiards with convex boundary of different shape are calculated by using the method of complex scaling. Two leads are attached to the cavities. The conductance of the cavities is calculated at energies with one, two, and three open channels in each lead. Bands of overlapping resonance states appear that are localized along the convex boundary of the cavities and contribute coherently to the conductance. These bands correspond to the whispering gallery modes known from classical calculations.

Bound states in the continuum in open Aharonov-Bohm rings

Using the formalism of the effective Hamiltonian, we consider bound states in a continuum (BIC). They are nonhermitian effective Hamiltonian eigenstates that have real eigenvalues. 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 a transport solution with an arbitrary coefficient and exist in a 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 the energy and magnetic flux; however, it’s realization strongly depends on the way to the BIC point.

INTERFERING RESONANCES IN A QUANTUM BILLIARD

We present a method for numerically obtaining the positions, widths and wavefunctions of resonance states in a two dimensional billiard connected to a waveguide. For a rectangular billiard, we study the dynamics of three resonance poles lying separated from the other ones. As a function of increasing coupling strength between the waveguide and the billiard two of the states become trapped while the width of the third one continues to increase for all coupling strengths. This behavior of the resonance poles is reflected in the time delay function which can be studied experimentally.

Multiple bound states in scissor-shaped waveguides

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

Spectroscopic studies in open quantum systems

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Light induced Josephson like current between two coupled nonlinear cavities coupled with a symmetrically positioned photonic crystal waveguide.

Such a four-terminal junction with a tunable

Signatures of quantum chaos in the nodal points and streamlines in electron transport through billiards

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Frequency comb generation by symmetry-protected bound state in the continuum

can realized experimentally if a right-angle structure is filled by a ferrite. It is known that for

Channel dropping via bound states in the continuum in a system of two nonlinear cavities between two linear waveguides

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