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Physical Review E

Emulation of quantum mechanical billiards by electrical resonance circuits

Olof Bengtsson, Johan E Larsson, Karl‐Fredrik Berggren

Published May 13, 2005• 12 citations
Publisher

Abstract

We propose that a two-dimensional electric network may be used for fundamental studies of wave function properties, transport, and related statistics. Using Kirchhoff's current law and the j omega method we find that the network is analogous to a discretized Schrödinger equation for quantum billiards and dots. Thus complex electric potentials play the role of quantum mechanical wave functions. Ways of realizing the electric network are discussed briefly. The role of symmetries is outlined, and a direct way of selecting states with a given symmetry is presented.

DOI
10.1103/physreve.71.056206

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We propose that a two-dimensional electric network may be used for fundamental studies of wave function properties, transport, and related statistics. Using Kirchhoff's current law and the j omega method we find that the network is analogous to a discretized Schrödinger equation for quantum billiards and dots. Thus complex electric potentials play the role of quantum mechanical wave functions. Ways of realizing the electric network are discussed briefly. The role of symmetries is outlined, and a direct way of selecting states with a given symmetry is presented.
Several key advances in the understanding of quantum chaos have resulted from the study of billiard systems, in which a particle is confined in a two‐dimensional potential well. There has been many numerical studies of wave function characteristics, nodal line and nodal point features and current distributions and correlations. One finds that chaotic states may often be represented by Gaussian random waves which consist of a superposition of monochromatic plane waves traveling in different directions.Direct studies of quantum billiards like 2D mesoscopic ballistic semiconductor quantum dots are, however, unfeasible because of their smallness and because they are normally imbedded deeply in a semiconductor structure. The 2D quantum billiard may, however, be mapped onto a flat microwave cavity, which has allowed for direct measurements of wave functions, current distributions etc. Here we discuss an alternative unconventional realization of a quantum billiard. Thus we outline how electrical networks may be used for fundamental studies of wave function properties and transport in general and, more specifically, their mapping onto open quantum dots. In addition to the scientific case the network model has obvious pedagogical merits.
We consider two electric RLC resonance networks that are equivalent to quantum billiards. In a network of inductors grounded by capacitors, the eigenvalues of the quantum billiard correspond to the squared resonant frequencies. In a network of capacitors grounded by inductors, the eigenvalues of the billiard are given by the inverse of the squared resonant frequencies. In both cases, the local voltages play the role of the wave function of the quantum billiard. However, unlike for quantum billiards, there is a heat power because of the resistance of the inductors. In the equivalent chaotic billiards, we derive a distribution of the heat power which describes well the numerical statistics.
A heuristic model for particle states and current flow in open ballistic two-dimensional (2D) quantum dots/wave billiards is proposed. The model makes use of complex potentials first introduced in phenomenological nuclear inelastic scattering theory (the optical model). Here we assume that external input and output leads connecting the system to the source and the drain regions may be represented by complex potentials. In this way, a current may be set up between the two 'pseudo-leads'. Probability densities and current flows for an open quantum dot are analyzed here numerically and the results are compared with the microwave measurements used to emulate the system. The model is of conceptual as well as practical interest. In addition to quantum billiards, it may be used as a tool per se to analyze transport in classical wave analogues, such as microwave resonators, acoustic resonators, effects of leakage on such systems, etc.
Equivalent circuits are developed to represent the Schr\"odinger amplitude equation for one, two, and three independent variables in orthogonal curvilinear coordinate systems. The networks allow the assumption of any arbitrary potential energy and may be solved, within any desired degree of accuracy, either by an a.c. network analyzer, or by numerical and analytical circuit methods. It is shown that by varying the impressed frequency on a network of inductors and capacitors (or by keeping the frequency constant and varying the capacitors), it is possible to find by measurements the eigenvalues, eigenfunctions, and the statistical mean of various operators belonging to the system represented. The electrical model may, of course, be replaced by an analogous mechanical model containing moving masses and springs. At first the network for the one-dimensional wave equation for a single particle in Cartesian coordinates is developed in detail, then the general case. A companion paper contains results of a study made on an a.c. network analyzer of one-dimensional problems: a potential well, a double barrier, the harmonic oscillator, and the rigid rotator. The curves show good agreement, within the accuracy of the instruments, with the known eigenvalues, eigenfunctions, and "tunnel" effects.
Two-dimensional quantum billiards are one of the most important paradigms for exploring the connection between quantum and classical worlds. Researchers are mainly focused on nonintegrable and irregular shapes to understand the quantum characteristics of chaotic billiards. The emergence of the scarred modes relevant to unstable periodic orbits (POs) is one intriguing finding in nonintegrable quantum billiards. On the other hand, stable POs are abundant in integrable billiards. The quantum wavefunctions associated with stable POs have been shown to play a key role in ballistic transport. A variety of physical systems, such as microwave cavities, optical fibers, optical resonators, vibrating plates, acoustic waves, and liquid surface waves, are used to analogously simulate the wave properties of quantum billiards. This article gives a comprehensive review for the subtle connection between the quantum level clustering and the classical POs for three integrable billiards including square, equilateral triangle, and circular billiards.