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arXiv:quant-ph/0411100 · Physical Review E

Electric circuit networks equivalent to chaotic quantum billiards

Evgeny N. Bulgakov, Дмитрий Н. Максимов, Almas F. Sadreev

Published Apr 11, 2005• 15 citations
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Abstract

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.

DOI
10.1103/physreve.71.046205

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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.
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.
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.
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This paper reports on a joint theoretical and experimental study of the Pauli quantum-mechanical stress tensor T_{alphabeta}(x,y) for open two-dimensional chaotic billiards. In the case of a finite current flow through the system the interior wave function is expressed as psi=u+iv . With the assumption that u and v are Gaussian random fields we derive analytic expressions for the statistical distributions for the quantum stress tensor components T_{alphabeta} . The Gaussian random field model is tested for a Sinai billiard with two opposite leads by analyzing the scattering wave functions obtained numerically from the corresponding Schrödinger equation. Two-dimensional quantum billiards may be emulated from planar microwave analogs. Hence we report on microwave measurements for an open two-dimensional cavity and how the quantum stress tensor analog is extracted from the recorded electric field. The agreement with the theoretical predictions for the distributions for T_{alphabeta}(x,y) is quite satisfactory for small net currents. However, a distinct difference between experiments and theory is observed at higher net flow, which could be explained using a Gaussian random field, where the net current was taken into account by an additional plane wave with a preferential direction and amplitude.