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Linear entropy growth (left panel) and Loschmidt echo decay (right) for various values of ǫ, ranging from 0.001 to 0.1 and for the PAC with N = 450, k = 0.005. Both Lyapunov and Ruelle regimes can be seen when the rates saturate at an ǫ-independent value.
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The relationship between the spectral properties of diffusive open quantum maps and the classical spectrum of Ruelle-Pollicott resonances was studied. These resonances determine the asymptotic time regime for several quantities of interest, the linear entropy, the Loschmidt echo and the correlations of the initial state. The development of a numeri...
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Citations
... For some kinds of strongly classically chaotic systems, a correspondence between the spectrum of the (Gaussian coarse-grained) propagator of the density matrix and the RPRs can be established, upon appropriately taking the semiclassical and zero coarse-graining limits [85]. Then using an iterative method [86,87], the largest RPRs can be efficiently obtained from the quantum propagator (e. g. the ones computed for Figs. 3 and 5). The influence of RPRs in chaotic systems can also be found in other related quantities like the purity (when considering bipartite systems, or open systems) and the Loschmidt echo. ...
... Ideed, after the Ehrenfest time, these quantities saturate to a constant value, typically dependent on the Hilbert space dimension. The way they approach to equilibrium is exponential and depends on the RPRs [87][88][89]. ...
Quantum Chaos has originally emerged as the field which studies how the properties of classical chaotic systems arise in their quantum counterparts. The growing interest in quantum many-body systems, with no obvious classical meaning has led to consider time-dependent quantities that can help to characterize and redefine Quantum Chaos. This article reviews the prominent role that the out of time ordered correlator (OTOC) plays to achieve such goal.
... They exhibit ergodicity, mixing, exponential sensitivity to variation of the initial conditions (the positivity of the Lyapunov exponent), and the positivity of the Kolmogorov-Sinai entropy [48]. Detailed understanding of dynamics of cat maps is important also for the much richer world of nonlinear hyperbolic toral automorphisms, see [16,24,47] for examples. ...
... x 1 = g 11 m 1 + g 11 x 0 + g 11 x 2 , with = 1, g 11 = 1/s verifying the general Green's function formula (24). For single symbol m 1 ≡ m the set of inequalities (28) thus reduces to ...
The dynamics of an extended, spatiotemporally chaotic system might appear extremely complex. Nevertheless, the local dynamics, observed through a finite spatiotemporal window, can often be thought of as a visitation sequence of a finite repertoire of finite patterns. To make statistical predictions about the system, one needs to know how often a given pattern occurs. Here we address this fundamental question within a spatiotemporal cat, a one-dimensional spatial lattice of coupled cat maps evolving in time. In spatiotemporal cat, any spatiotemporal state is labeled by a unique two-dimensional lattice of symbols from a finite alphabet, with the lattice states and their symbolic representation related linearly (hence ‘linear encoding’). We show that the state of the system over a finite spatiotemporal domain can be described with exponentially increasing precision by a finite pattern of symbols, and we provide a systematic, lattice Green’s function methodology to calculate the frequency (i.e., the measure) of such states.
... This led to the discovery of Ruelle-Pollicott resonances, where mixing and the decay of correlations in chaotic systems were related to the point spectrum of the Ruelle-Perron-Frobenius operator [12][13][14]. Indeed, the power spectra of chaotic systems are still actively used to analyze the behavior of everything from open quantum systems [15,16] to climate models [17]. ...
... Accordingly, our findings bear on modern applications of Ruelle-Pollicott resonance theory. These applications are leading, for example, to better understanding of sensitivities in climate models [17] and the dynamics of open quantum systems via their correspondence to classical chaotic dynamical systems [15,16]. Our results provide full analytical correspondence between observed correlation and the spectral properties of nonunitary models. ...
Power spectral densities are a common, convenient, and powerful way to analyze signals, so much so that they are now broadly deployed across the sciences and engineering—from quantum physics to cosmology and from crystallography to neuroscience to speech recognition. The features they reveal not only identify prominent signal frequencies but also hint at mechanisms that generate correlation and lead to resonance. Despite their near-centuries-long run of successes in signal analysis, here we show that flat power spectra can be generated by highly complex processes, effectively hiding all inherent structure in complex signals. Historically, this circumstance has been widely misinterpreted, being taken as the renowned signature of “structureless” white noise—the benchmark of randomness. We argue, in contrast, to the extent that most real-world complex systems exhibit correlations beyond pairwise statistics their structures evade power spectra and other pairwise statistical measures. As concrete physical examples, we demonstrate that fraudulent white noise hides the predictable structure of both entangled quantum systems and chaotic crystals. To make these words of warning operational, we present constructive results that explore how this situation comes about and the high toll it takes in understanding complex mechanisms. First, we give the closed-form solution for the power spectrum of a very broad class of structurally complex signal generators. Second, we demonstrate the close relationship between eigenspectra of evolution operators and power spectra. Third, we characterize the minimal generative structure implied by any power spectrum. Fourth, we show how to construct arbitrarily complex processes with flat power spectra. Finally, leveraging this diagnosis of the problem, we point the way to developing more incisive tools for discovering structure in complex signals.
... They exhibit ergodicity, mixing, exponential sensitivity to variation of the initial conditions (the positivity of the Lyapunov exponent), and the positivity of the Kolmogorov-Sinai entropy [46]. Detailed understanding of dynamics of cat maps is important also for the much richer world of nonlinear hyperbolic toral automorphisms, see [15,23,45] for examples. ...
The dynamics of an extended, spatiotemporally chaotic system might appear extremely complex. Nevertheless, the local dynamics, observed through a finite spatiotemporal window, can often be thought of as a visitation sequence of a finite repertoire of finite patterns. To make statistical predictions about the system, one needs to know how often a given pattern occurs. Here we address this fundamental question within a spatiotemporal cat, a 1-dimensional spatial lattice of coupled cat maps evolving in time. In spatiotemporal cat, any spatiotemporal state is labeled by a unique 2-dimensional lattice of symbols from a finite alphabet, with the lattice states and their symbolic representation related linearly (hence "linear encoding"). We show that the state of the system over a finite spatiotemporal domain can be described with exponentially increasing precision by a finite pattern of symbols, and we provide a systematic, lattice Green's function methodology to calculate the frequency (i.e., the measure) of such states.
... In the classical setting, the dynamical map acting on the probability density function is the Frobenius-Perron (FP) operator, and the eigenvalues are known as the Pollicott-Ruelle (PR) resonances. Nonnenmacher [82] and García-Mata et al [83] consider quantum maps with well-defined classical limits, and found that the PR resonances governed the asymptotic decay rates of the quantum system. In [82], it was argued that a spectral gap in the quantum map is symptomatic of ergodicity and mixing of its classical limit, whereas a closing spectral gap appeared to coincide with integrability. ...
We study the mixing behavior of random Lindblad generators with no symmetries, using the dynamical map or propagator of the dissipative evolution. In particular, we determine the long-time behavior of a dissipative form factor, which is the trace of the propagator, and use this as a diagnostic for the existence or absence of a spectral gap in the distribution of eigenvalues of the Lindblad generator. We find that simple generators with a single jump operator are slowly mixing, and relax algebraically in time, due to the closing of the spectral gap in the thermodynamic limit. Introducing additional jump operators or a Hamiltonian opens up a spectral gap which remains finite in the thermodynamic limit, leading to exponential relaxation and thus rapid mixing. We use the method of moments and introduce a novel diagrammatic expansion to determine exactly the form factor to leading order in Hilbert space dimension N. We also present numerical support for our main results.
... This led to the discovery of Ruelle-Pollicott resonances, where mixing and the decay of correlations in chaotic systems were related to the point spectrum of the Ruelle-Perron-Frobenius operator [11][12][13]. Indeed, the power spectra of chaotic systems are still actively used to analyze the behavior of everything from open quantum systems [14,15] to climate models [16]. ...
... Accordingly, our findings are relevant to modern applications of Ruelle-Pollicott resonance theory. These applications are leading, for example, to better understanding of sensitivities in climate models [16] and the dynamics of open quantum systems via their correspondence to classical chaotic dynamical systems [14,15]. Our results provide full analytical correspondence between observed correlation and the spectral properties of nonunitary models. ...
Power spectral densities are a common, convenient, and powerful way to analyze signals. So much so that they are now broadly deployed across the sciences and engineering---from quantum physics to cosmology, and from crystallography to neuroscience to speech recognition. The features they reveal not only identify prominent signal-frequencies but also hint at mechanisms that generate correlation and lead to resonance. Despite their near-centuries-long run of successes in signal analysis, here we show that flat power spectra can be generated by highly complex processes, effectively hiding all inherent structure in complex signals. Historically, this circumstance has been widely misinterpreted, being taken as the renowned signature of "structureless" white noise---the benchmark of randomness. We argue, in contrast, to the extent that most real-world complex systems exhibit correlations beyond pairwise statistics their structures evade power spectra and other pairwise statistical measures. To make these words of warning operational, we present constructive results that explore how this situation comes about and the high toll it takes in understanding complex mechanisms. First, we give the closed-form solution for the power spectrum of a very broad class of structurally-complex signal generators. Second, we demonstrate the close relationship between eigen-spectra of evolution operators and power spectra. Third, we characterize the minimal generative structure implied by any power spectrum. Fourth, we show how to construct arbitrarily complex processes with flat power spectra. Finally, leveraging this diagnosis of the problem, we point the way to developing more incisive tools for discovering structure in complex signals.
... Along these lines, Refs. [86] and [87] consider quantum maps with well-defined classical limits, and found that the PR resonances governed the asymptotic decay rates of the quantum system. In [86], it was argued that a spectral gap in the quantum map is symptomatic of ergodicity and mixing of its classical limit, whereas a closing spectral gap appeared to coincide with integrability. ...
We study the mixing behavior of random Lindblad generators with no symmetries, using the dynamical map or propagator of the dissipative evolution. In particular, we determine the long-time behavior of a dissipative form factor, which is the trace of the propagator, and use this as a diagnostic for the existence or absence of a spectral gap in the distribution of eigenvalues of the Lindblad generator. We find that simple generators with a single jump operator are slowly mixing, and relax algebraically in time, due to the closing of the spectral gap in the thermodynamic limit. Introducing additional jump operators or a Hamiltonian opens up a spectral gap which remains finite in the thermodynamic limit, leading to exponential relaxation and thus rapid mixing. We use the method of moments and introduce a novel diagrammatic expansion to determine exactly the form factor to leading order in Hilbert space dimension $N$. We also present numerical support for our main results.
... The deep connection between the quantum propagator and the RPRs was established as a type of spectral quantum-classical correspondence, by introducing a coarse-grained propagator [36,[43][44][45]. Such a propagator can be defined as a two-step superoperator: ...
... At each (discrete) time t the unitary mapÛ is followed by an incoherent sum of all the possible translations in phase space with a (quasi-)Gaussian weight c ϵ ðξÞ centered at ξ ¼ 0. For convenience (see Ref. [44]), c ϵ ðξÞ is defined as the two-dimensional discrete Fourier transform of cðμ; νÞ ¼ e −ðϵN=πÞ½sin 2 ðπμ=NÞþsin 2 ðπν=NÞ=2 . It is approximately Gaussian with width proportional to 1=ϵ, and ϵ therefore characterizing the size of the coarse graining. ...
... Classical Ulam partitioning is not any easier. Rather, we profit from the fact that only the largest (in modulus) eigenvalue is needed and an iterative method based on Lanczos power iteration can be used [44,46,47]. It essentially consists of iterating an initial arbitrary state forward and back, building a matrix, and solving a generalized eigenvalue problem. ...
Two properties are needed for a classical system to be chaotic: exponential stretching and mixing. Recently, out-of-time order correlators were proposed as a measure of chaos in a wide range of physical systems. While most of the attention has previously been devoted to the short time stretching aspect of chaos, characterized by the Lyapunov exponent, we show for quantum maps that the out-of-time correlator approaches its stationary value exponentially with a rate determined by the Ruelle-Pollicot resonances. This property constitutes clear evidence of the dual role of the underlying classical chaos dictating the behavior of the correlator at different timescales.
... This is an approximation that works exactly in some cases, e.g. a billiard that has elastic collisions on the walls and diffusion in the free evolution between collisions. This kind of two-step model has been used to study quantum to classical correspondence and the emergence of classical properties from the quantum dynamics (Nonnenmacher 2003;García-Mata & Saraceno 2004) The effect of decoherence can be characterized by using the purity ...
The Boltzmann echo (BE) is a measure of irreversibility and sensitivity to perturbations for non-isolated systems. Recently, different regimes of this quantity were described for chaotic systems. There is a perturbative regime where the BE decays with a rate given by the sum of a term depending on the accuracy with which the system is time reversed and a term depending on the coupling between the system and the environment. In addition, a parameter-independent regime, characterized by the classical Lyapunov exponent, is expected. In this paper, we study the behaviour of the BE in hyperbolic maps that are in contact with different environments. We analyse the emergence of the different regimes and show that the behaviour of the decay rate of the BE is strongly dependent on the type of environment.
... a few examples of their significance, they play a role in transport properties (e.g. the drift of particles in self-similar billiards [6] and the diffusion coefficient [23,25,48]), the escape rate of open systems [24,26], and the asymptotic behavior of entropy [20,22,24] and the Loschmidt echo 1 [8,17,20,22]. They are also known to describe deviations of the statistical distribution of quantum eigenvalues from the distribution predicted by Random Matrix Theory (in other words, they describe nonuniversal behavior of quantum systems) [1]. ...
... a few examples of their significance, they play a role in transport properties (e.g. the drift of particles in self-similar billiards [6] and the diffusion coefficient [23,25,48]), the escape rate of open systems [24,26], and the asymptotic behavior of entropy [20,22,24] and the Loschmidt echo 1 [8,17,20,22]. They are also known to describe deviations of the statistical distribution of quantum eigenvalues from the distribution predicted by Random Matrix Theory (in other words, they describe nonuniversal behavior of quantum systems) [1]. ...
... A recent paper [37] provides rigorous results demonstrating that the spectrum of the quantum coarsegrained propagator converges to that of the classical operator in the classical limit (N → ∞). Numerical studies have also supported the conclusion that quantum resonances match closely with classical values [20,21,22,36]. The RP resonances thus provide a firm basis for quantum-classical correspondence and may be found from either classical systems or their quantum counterparts. ...
In this dissertation I discuss methods of calculating the Ruelle-Pollicott resonances of the perturbed cat map. The Ruelle-Pollicott (RP) resonances are the stable eigenvalues of the coarse-grained Frobenius-Perron (FP) operator (which governs the classical dynamics of chaotic systems) in the semiclassical limit. The RP resonances describe the decay of correlation functions and are important in the link between classical and quantum chaotic systems. They are in general very hard to calculate, however, and for most systems there are no exact analytical expressions. The need for numerical approximations motivates the present work. I describe two different methods of calculating the classical RP resonances using the perturbed cat map with a linear perturbation as a toy model to test them. The first method makes use of a trace formula to connect the spectral determinant of the FP operator to the periodic orbits of the map. The second method involves truncating the infinite-dimensional FP operator to a finite dimension. The matrix elements of this truncated FP operator are constructed using trigonometric basis states, and the eigenvalues of the reduced matrix are calculated. The eigenvalues that remain "frozen" in place in the limit N --> infinity are identified as the RP resonances. We had the most success with the second method. In addition, we show by analyzing the fractal dimension of the map that there is a lower bound to the resonance spectrum. In conclusion I discuss the quantization of the perturbed cat map and the quantum RP resonances. Several methods for calculating the quantum resonances were attempted, none of which yielded accurate results. This is an area open to future work.
