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...

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.

The power law 1/f(alpha) in the power spectrum characterizes the fluctuating observables of many complex natural systems. Considering the energy levels of a quantum system as a discrete time series where the energy plays the role of time, the level fluctuations can be characterized by the power spectrum. Using a family of quantum billiards, we analyze the order-to-chaos transition in terms of this power spectrum. A power law 1/f(alpha) is found at all the transition stages, and it is shown that the exponent alpha is related to the chaotic component of the classical phase space of the quantum system.

Sequences of energy levels in nuclei are often plagued with missing levels whose number and position are unknown. It is also quite usual that all the quantum numbers of certain levels cannot be experimentally determined, and thus levels of different symmetries are mixed in the same sequence. The analysis of these imperfect spectra (from the point of view of spectral statistics) is unavoidable if one wants to extract some statistical information. The power spectrum of the δ q statistic has emerged in recent years as an important tool for the study of quantum chaos and spectral statistics. We derive analytical expressions for the observed power spectrum in terms of the fraction of observed levels and the number of mixed sequences. These expressions are tested with large shell model spectra simulating realistic experimental situations. A good estimation of the number of mixed symmetries and the fraction of missing levels is obtained by means of a least-squares fit in a wide set of different situations.

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