It is shown that the energy spectrum fluctuations of quantum systems can be formally considered as a discrete time series. The power spectrum behavior of such a signal for different systems suggests the following conjecture: The energy spectra of chaotic quantum systems are characterized by 1=f noise. DOI: 10.1103 The understanding of quantum chaos has greatly advanced during the past two decades. It is well known that there is a clear relationship between the energy level fluctuation properties of a quantum system and the large time scale behavior of its classical analogue. The pioneering work of Berry and Tabor [1] showed that the spectral fluctuations of a quantum system whose classical analogue is fully integrable are well described by Poisson statistics; i.e., the successive energy levels are not correlated. In a seminal paper, Bohigas et al. [2] conjectured that the fluctuation properties of generic quantum systems, which in the classical limit are fully chaotic, coincide with those of random matrix theory (RMT). This conjecture is strongly supported by experimental data, many numerical calculations, and analytical work based on semiclassical arguments. A review of later developments can be found in [3,4].We propose in this Letter a different approach to quantum chaos based on traditional methods of time series analysis. The essential feature of chaotic energy spectra in quantum systems is the existence of level repulsion and correlations. To study these correlations, we can consider the energy spectrum as a discrete signal, and the sequence of energy levels as a time series. For example, the sequence of nearest level spacings has similarities with the diffusion process of a particle. But generally we do not need to specify the nature of the analogue time series. As we shall see, examination of the power spectrum of energy level fluctuations reveals very accurate power laws for completely regular or completely chaotic Hamiltonian quantum systems. It turns out that chaotic systems have 1=f noise, in contrast to the Brown noise of regular systems.The first step, previous to any statistical analysis of the spectral fluctuations, is the unfolding of the energy spectrum. Level fluctuation amplitudes are modulated by the value of the mean level density E , and therefore, to compare the fluctuations of different systems, the level density smooth behavior must be removed. The unfolding consists in locally mapping the real spectrum into another with mean level density equal to one. The actual energy levels E i are mapped into new dimensionless levels i ,where N is the dimension of the spectrum and N N E is given byThis function is a smooth approximation to the step function N E that gives the true number of levels up to energy E. The form of the function E can be determined by a best fit of N N E to N E .The nearest neighbor spacing sequence is defined byFor the unfolded levels, the mean level density is equal to 1 and hsi 1. In practical cases, the unfolding procedure can be a difficult task for systems where ther...

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It was recently conjectured that 1/f noise is a fundamental characteristic of spectral fluctuations in chaotic quantum systems. This conjecture is based on the power spectrum behavior of the excitation energy fluctuations, which is different for chaotic and integrable systems. Using random matrix theory, we derive theoretical expressions that explain without free parameters the universal behavior of the excitation energy fluctuations power spectrum. The theory gives excellent agreement with numerical calculations and reproduces to a good approximation the 1/f (1/f(2)) power law characteristic of chaotic (integrable) systems. Moreover, the theoretical results are valid for semiclassical systems as well.

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The main signature of chaos in a quantum system is provided by spectral statistical analysis of the nearestneighbor spacing distribution P(s) and the spectral rigidity given by the ⌬ 3 (L) statistic. It is shown that some standard unfolding procedures, such as local unfolding and Gaussian broadening, lead to a spurious saturation of ⌬ 3 (L) that spoils the relationship of this statistic with the regular or chaotic motion of the system. This effect can also be misinterpreted as Berry's saturation. DOI: 10.1103/PhysRevE.66.036209 PACS number͑s͒: 05.45.Mt, 05.40.Ϫa Quantum chaos has been an active research field since the link between energy level fluctuations and the chaotic or integrable properties of Hamiltonian systems was conjectured ͓1,2͔, providing one of the fundamental signatures of quantum chaos ͓3,4͔ in atoms, molecules, nuclei, quantum dots, etc. The secular or smooth behavior of the level density is a characteristic of each quantum system, while the fluctuations relative to this smooth behavior are related to the regular or chaotic character of the motion in all quantum systems. To achieve the separation of the smooth and fluctuating parts, the energy spectrum is scaled to a sequence with the same local mean spacing along the whole spectrum. This scaling is called unfolding ͓5͔. Although this can be a nontrivial task ͓6͔, the description of the unfolding details of calculations is usually neglected in the literature.In this paper we show that, contrary to common assumptions, the statistics that measure long-range level correlations are strongly dependent on the unfolding procedure utilized, and some standard unfolding methods give very misleading results in regard to the chaoticity of quantum systems. Longrange level correlations are usually measured by means of the Dyson and Mehta ⌬ 3 statistic ͓5͔. On the other hand, we find that short-range correlations, characterized by the nearest-neighbor spacing distribution P(s), are not very sensitive to the unfolding method.Let us consider a rectangular quantum billiard with a size ratio a/bϭ . This is a well known example of a regular system. In general, for regular systems level fluctuations behave like in a sequence of uncorrelated energy levels, and the ⌬ 3 (L) statistic increases linearly with L. However, it was shown by Berry ͓7͔ that the existence of periodic orbits in the phase space of the analogous classical system leads to a saturation of ⌬ 3 (L) for L larger than a certain value L s , related to the period of the shortest periodic orbit. Figure 1 shows the ⌬ 3 behavior for a sequence of 8000 high energy levels of the mentioned quantum billiard, calculated with two different unfolding procedures. The mean level density for this system is given by the Weyl law ͓8͔. Using this density to perform the unfolding, ⌬ 3 follows the straight line of level spacings with Poisson distribution, characteristic of regular systems. In this example, Berry's saturation takes place at L s Ӎ750, that is outside the figure. Let us suppose now that the law giving th...