Time-Series and Network Analysis in Quantum Dynamics: Comparison with Classical Dynamics
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Time-Series and Network Analysis in Quantum Dynamics: Comparison with Classical Dynamics Pradip Laha1
· S. Lakshmibala1 · V. Balakrishnan1
Received: 3 May 2020 / Accepted: 24 September 2020 / Published online: 14 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Time-series analysis and network analysis are now used extensively in diverse areas of science. In this paper, we apply these techniques to quantum dynamics in an optomechanical system: specifically, the long-time dynamics of the mean photon number in an archetypal tripartite quantum system comprising a single-mode radiation field interacting with a twolevel atom and an oscillating membrane. We also investigate a classical system of interacting Duffing oscillators which effectively mimics several of the features of tripartite quantumoptical systems. In both cases, we examine the manner in which the maximal Lyapunov exponent obtained from a detailed time-series analysis varies with changes in an appropriate tunable parameter of the system. Network analysis is employed in both the quantum and classical models to identify suitable network quantifiers which will reflect these variations with the system parameter. This is a novel approach towards (i) examining how a considerably smaller data set (the network) obtained from a long time series of dynamical variables captures important aspects of the underlying dynamics, and (ii) identifying the differences between classical and quantum dynamics. Keywords Cavity optomechanics · Duffing oscillator · Time-series analysis · Network analysis · Maximal Lyapunov exponent · Recurrence plot
1 Introduction The availability of time-series data in diverse areas such as weather forecasting, climate research and medicine [1–5] has facilitated detailed investigations leading to the extraction of important results on the dynamics of a variety of systems. Several tools have been proposed in time-series analysis to assess the long-time behaviour of complex dynamical systems. The methods used involve the identification and estimation of indicators of the nature of the underlying dynamics such as the maximal Lyapunov exponent (MLE), return maps, return-time distributions, recurrence plots, and so on.
Pradip Laha
[email protected] 1
Department of Physics, IIT Madras, Chennai 600036, India
International Journal of Theoretical Physics (2020) 59:3476–3490
3477
In recent years, the analysis of networks constructed from a long time series has proved to be another important tool that has contributed significantly to the understanding of classical dynamics [6–10]. The problem of handling a large data set is circumvented by reducing it to a considerably smaller optimal set (the network), particularly in the context of machine learning protocols [11, 12]. Different methods have been employed to convert the time series of a classical dynamical variable into an equivalent network, each method capturing specific features of the dynamics encoded in the time series [13–19]. In this paper,
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