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...

We analyze excited-state quantum phase transitions (ESQPTs) in three schematic (integrable and nonintegrable) models describing a single-mode bosonic field coupled to a collection of atoms. It is shown that the presence of the ESQPT in these models affects the quantum relaxation processes following an abrupt quench in the control parameter. Clear-cut evidence of the ESQPT effects is presented in integrable models, while in a nonintegrable model the evidence is blurred due to chaotic behavior of the system in the region around the critical energy.

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We study the critical behavior of excited states and its relation to order and chaos in the Jaynes-Cummings and Dicke models of quantum optics. We show that both models exhibit a chain of excited-state quantum phase transitions demarcating the upper edge of the superradiant phase. For the Dicke model, the signatures of criticality in excited states are blurred by the onset of quantum chaos. We show that the emergence of quantum chaos is caused by the precursors of the excited-state quantum phase transition.