Probability Amplitude Fluctuations in Experimental Wave Chaotic Eigenmodes with and Without Time-Reversal Symmetry

Abstract

We have measured the probability density jc͑r͒j 2 in the semiclassical limit of a classically chaotic square well potential with and without time reversal symmetry, and compared our findings with theoretical predictions. We find that wave functions with time-reversal symmetry have larger fluctuations than those without time-reversal symmetry. To quantify the degree of these fluctuations, eigenmodes both with and without time-reversal symmetry are statistically analyzed and the two-point spatial correlation function and the probability density distribution function of the eigenmodes are found to agree with theoretical predictions. [S0031-9007(98)07259-7] PACS numbers: 05.45. + b, 03.65.Ge, 85.70.Ge The quantum mechanical behavior of nonintegrable systems has been a very intriguing subject for many decades and is still one of active research [1]. This is not only because of the interesting fundamental physics of "quantum chaos" but also because of the strong analogy between the quantum mechanical behavior of mesoscopic systems and the wave chaotic behavior of nonintegrable systems [2]. Hence the subject can pave the way to understanding the statistical properties of electronic eigenstates and eigenfunctions of mesoscopic systems, such as quantum dots and quantum wires, which will be increasingly important for future technological applications.Theories and numerical simulations suggest that for integrable (nonchaotic) systems a large degree of degeneracy is allowed in the eigenvalues of the system; therefore the spacing between neighboring eigenvalues obeys Poisson statistics. For nonintegrable systems the existence of classical chaos breaks the degeneracies, and therefore the eigenvalue spacing statistics are clearly no longer Poisson. Theories propose that the statistics of the eigenvalue spacing are governed not only by the integrability but also by the time-reversal symmetry of the system. One expects the spectral properties of a chaotic system with time-reversal symmetry to follow the statistics of a Gaussian orthogonal ensemble (GOE) of random matrices, while a chaotic system with broken time-reversal symmetry is expected to follow the statistics of a Gaussian unitary ensemble (GUE) of random matrices. In recent years several experiments similar to those described here have demonstrated that these theoretical predictions are indeed correct [3][4][5].An even more intriguing subject is the investigation of the eigenfunctions of wave chaotic systems. Although some of the interesting behavior of eigenfunctions in systems with GOE symmetry has been explored by several groups [6][7][8], there has been no experimental examination of the time-reversal symmetry dependence of the eigenfunction behavior [9]. Investigating the detailed behavior of both time-reversal symmetric (TRS) and time-reversal symmetry broken (TRSB) wave functions is also imperative as it will give insights into the behavior of electronic wave functions of nanoscale structures that have these symmetries.In this paper we report our experimental re...

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