Quantum Chaos and 1=f Noise A. Relan ˜o, J. M. G. Go ´mez, R. A. Molina, and J. Retamosa Departamento de Fı ´sica Ato ´mica, Molecular y Nuclear, Universidad Complutense de Madrid, E-28040 Madrid, Spain E. Faleiro Departamento de Fı ´sica Aplicada, E.U.I.T. Industrial, Universidad Polite ´cnica de Madrid, E-28012 Madrid, Spain (Received 25 February 2002; published 22 November 2002) 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/PhysRevLett.89.244102 PACS numbers: 05.45.Mt, 05.40.–a, 05.45.Pq, 05.45.Tp The understanding of quantum chaos has greatly ad- vanced 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 pioneer- ing work of Berry and Tabor [1] showed that the spectral fluctuations of a quantum system whose classical ana- logue is fully integrable are well described by Poisson statistics; i.e., the successive energy levels are not corre- lated. In a seminal paper, Bohigas et al. [2] conjectured that the fluctuation properties of generic quantum sys- tems, which in the classical limit are fully chaotic, co- incide 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 develop- ments can be found in [3,4]. We propose in this Letter a different approach to quan- tum 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 se- quence 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 spec- trum. 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 an- other with mean level density equal to one. The actual energy levels E i are mapped into new dimensionless levels i , E i ! i N E i ; i 1; ... ;N; (1) where N is the dimension of the spectrum and N E is given by N E Z E 1 dE 0 E 0 : (2) This function is a smooth approximation to the step function NE that gives the true number of levels up to energy E. The form of the function E can be deter- mined by a best fit of N E to NE. The nearest neighbor spacing sequence is defined by s i i1 i ; i 1; ... ;N 1: (3) For 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 there is no analytical expression for the mean level density [5]. Generally, two suitable statistics are used to study the fluctuation properties of the unfolded spectrum. The nearest neighbor spacing distribution Ps gives informa- tion on the short range correlations among the energy levels. The 3 L statistic makes it possible to study correlations of length L: As we change the L value, we obtain information on the level correlations at all scales. By contrast, in this paper we characterize the spectral fluctuations by the statistic n [6] defined by n X n i1 s i hsi X n i1 w i ; (4) where the index n runs from 1 to N 1. The quantity w i gives the fluctuation of the ith spacing from its mean value hsi 1. The function h 2 n i is closely related to the covariance matrix and thus provides important informa- tion on level correlations. Recently, it has been shown [7] that, under certain assumptions, h 2 n i is a logarithmic function of n for the RMT ensembles. VOLUME 89, NUMBER 24 PHYSICAL REVIEW LETTERS 9DECEMBER 2002 244102-1 0031-9007= 02=89(24)=244102(4)$20.00 2002 The American Physical Society 244102-1