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arXiv:1904.00439 · Physical review. E

Out-of-time-ordered correlators and quantum walks

Sivaprasad Omanakuttan, Arul Lakshminarayan

Published Jun 24, 2019• 11 citations
arXivHTMLPDF

Abstract

Out-of-time-ordered correlators (OTOCs) have recently attracted significant attention, finding applications in disparate areas, from the physics of many-body systems to quantum black holes, with an exponential growth of the OTOCs indicating quantum chaos. Here we consider OTOCs in the context of coined discrete quantum walks, a well-studied model of quantization of classical random walks with applications to quantum algorithms. Three separate cases of operators, variously localized in the coin and walker spaces, are discussed in this context and it is found that the approximated behavior of the OTOC is well described by simple algebraic functions in all three cases with different timescales of growth. The quadratic increase of OTOCs signals the absence of quantum chaos in these simplest forms of quantum walks.

DOI
10.1103/physreve.99.062128

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Out-of-time-ordered correlators (OTOCs) have recently attracted significant attention, finding applications in disparate areas, from the physics of many-body systems to quantum black holes, with an exponential growth of the OTOCs indicating quantum chaos. Here we consider OTOCs in the context of coined discrete quantum walks, a well-studied model of quantization of classical random walks with applications to quantum algorithms. Three separate cases of operators, variously localized in the coin and walker spaces, are discussed in this context and it is found that the approximated behavior of the OTOC is well described by simple algebraic functions in all three cases with different timescales of growth. The quadratic increase of OTOCs signals the absence of quantum chaos in these simplest forms of quantum walks.
Out-of-time-ordered correlators (OTOCs) have been proposed as a probe of chaos in quantum mechanics, on the basis of their short-time exponential growth found in some particular setups. However, it has been seen that this behavior is not universal. Therefore, we query other quantum chaos manifestations arising from the OTOCs, and we thus study their long-time behavior in systems of completely different nature: quantum maps, which are the simplest chaotic one-body system, and spin chains, which are many-body systems without a classical limit. It is shown that studying the long-time regime of the OTOCs it is possible to detect and gauge the transition between integrability and chaos, and we benchmark the transition with other indicators of quantum chaos based on the spectra and the eigenstates of the systems considered. For systems with a classical analog, we show that the proposed OTOC indicators have a very high accuracy that allow us to detect subtle features along the integrability-to-chaos transition.
Abstract We study numerically and analytically the time dependence and saturation of out-of-time ordered correlators (OTOC) in chaotic few-body quantum-mechanical systems: quantum Henon-Heiles system (weakly chaotic), BMN matrix quantum mechanics (strongly chaotic) and Gaussian random matrix ensembles. The growth pattern of quantum-mechanical OTOC is complex and nonuniversal, with no clear exponential regime at relevant timescales in any of the examples studied (which is not in contradiction to the exponential growth found in the literature for many-body systems, i.e. fields). On the other hand, the plateau (saturated) value of OTOC reached at long times decreases with temperature in a simple and universal way: exp(const./T2) for strong chaos (including random matrices) and exp(const./T) for weak chaos. For small matrices and sufficiently complex operators, there is also another, high-temperature regime where the saturated OTOC grows with temperature. Therefore, the plateau OTOC value is a meaningful indicator of few-body quantum chaos. We also discuss some general consequences of our findings for the AdS/CFT duality.
Recently, the out-of-time-ordered correlator (OTOC) has gained much attention as an indicator of quantum chaos. In the semi-classical limit, its exponential growth rate resembles the classical Lyapunov exponent. The quantum–classical correspondence has been supported for the one-body chaotic systems as well as realistic systems with interactions, as in the Dicke model, a model of multi-two-level atoms and cavity field interactions. To this end, we calculate the OTOC for different variations of the Dicke model in an open quantum system setting. The connection between the superradiant phase transition of the Dicke model and the OTOC is studied. Further, we establish a relation between the OTOC and the second-order coherence function. This becomes important for the experimental studies of the OTOC and quantum chaos in the models of quantum optics.
Out-of-time-ordered correlators (OTOCs) are an effective tool in characterizing black hole chaos, many-body thermalization, and quantum dynamics instability. Previous research findings have shown that the OTOCs' exponential growth (EG) marks the limit for quantum systems. However, we report in this paper a periodically-modulated nonlinear Schr\"odinger system, in which we interestingly find a way of OTOCs' growth: super-EG. We show that the quantum OTOCs, which stems from the quantum chaotic dynamics, will increase in a super-exponential way. We also find that in the classical limit, the hyper-chaos revealed by a linearly-increasing Lyapunov exponent actually triggers the super-EG of classical OTOCs. The results in this paper break the restraints of EG as the limit for quantum systems, which give us insight into the nature of quantum chaos in various fields of physics from black hole to many-body system.
The out-of-time-ordered correlator (OTOC) is a measure of quantum chaos that is being vigorously investigated. Analytically accessible simple models that have long been studied in other contexts could provide insights into such measures. This paper investigates the OTOC in the quantum bakers map which is the quantum version of a simple and exactly solvable model of deterministic chaos that caricatures the action of kneading dough. Exact solutions based on the semiquantum approximation are derived that tracks very well the correlators until the Ehrenfest time. The growth occurs, surprisingly, at the exponential rate of the classical Lyapunov exponent which is half of that expected semiclassically. This exponential growth is modulated by slowly changing coefficients. Beyond this time, saturation occurs at a value close to that of random matrices. Using projectors for observables naturally leads to truncations of the unitary time-t propagator and the growth of their singular values is shown to be intimately related to the growth of the out-of-time-ordered correlators.
In this paper, the out-of-time-order correlators (OTOC) in quantum harmonic oscillators are calculated analytically by second quantization method in perturbative approximation. We consider the coupled harmonic oscillators and anharmonic (quartic) oscillators. The analytic formulas of microcanonical OTOC in the leading order of interaction are obtained. From these results we clearly see that the interactions can enhance the correlation to very large values over time and is proportional to the energy level. The growth of OTOC signatures the quantum chaos therein. The microcanonical OTOC which is an increasing oscillation becomes a simple increasing function in thermal OTOC after sum over all energy level, as different level oscillates with different frequency. The property that, at late time stage OTOC saturates to a constant value, however, does not show in the first-order perturbation studied in this paper.