In this work, we study quantum chaos by focusing on the evolution of initially close states in the dynamics of the Quantum Kicked Rotor (QKR). We propose a novel measure, the Quantum Lyapunov Exponent (QLE), to quantify the degree of chaos in this quantum system, analogous to its classical counterpart. We begin by modeling the momentum space and then the QLE is computed through analyzing the fidelity between evolving states, offering insights into the quantum chaotic behavior. Furthermore, we extend our investigations to various initial states: localized, uniform, spreading, contracting and oscillating in momentum space. Our results unveil a diverse range of dynamical behaviors, highlighting the complex nature of quantum chaos. Finally, we propose an innovative optimization framework to represent a complex state as a superposition of the aforementioned states, which has potential implications for visualizing and understanding the dynamics of multifaceted quantum systems.
In this work, we study quantum chaos by focusing on the evolution of initially close states in the dynamics of the Quantum Kicked Rotor (QKR). We propose a novel measure, the Quantum Lyapunov Exponent (QLE), to quantify the degree of chaos in this quantum system, analogous to its classical counterpart. We begin by modeling the momentum space and then the QLE is computed through analyzing the fidelity between evolving states, offering insights into the quantum chaotic behavior. Furthermore, we extend our investigations to various initial states: localized, uniform, spreading, contracting and oscillating in momentum space. Our results unveil a diverse range of dynamical behaviors, highlighting the complex nature of quantum chaos. Finally, we propose an innovative optimization framework to represent a complex state as a superposition of the aforementioned states, which has potential implications for visualizing and understanding the dynamics of multifaceted quantum systems.
We study two classes of quantum phenomena associated with classical chaos in a variety of quantum models: (i) dynamical localization and its extension and generalization to interacting few- and many-body systems and (ii) quantum exponential divergences in high-order correlators and other diagnostics of quantum chaos. Dynamical localization (DL) is a subtle phenomenon related to Anderson localization. It hinges on quantum interference and is typically destroyed in presence of interactions. DL often manifests as a failure of a driven system to heat up, violating the foundations of statistical physics. Kicked rotor (KR) is a prototypical chaotic classical model that exhibits linear energy growth with time. The quantum kicked rotor (QKR) features DL instead: its energy saturates. Multiple attempts of many-body generalizations faced difficulties in preserving DL. Recently, DL was shown in a special integrable many-body model. We study non-integrable models of few- and many-body QKR-like systems and provide direct evidence that DL can persist there. In addition, we show how a novel related concept of localization landscape can be applied to study transport in rippled channels. Out-of-time-ordered correlator (OTOC) was proposed as an indicator of quantum chaos, since in the semiclassical limit, this correlator's possible exponential growth rate (CGR) resembles the classical Lyapunov exponent (LE). We show that the CGR in QKR is related, but distinct from the LE in KR. We also show a singularity in the OTOC at the Ehrenfest time tᴱ due to a delay in the onset of quantum interference. Next, we study scaling of OTOC beyond tᴱ. We then explore how the OTOC-based approach to quantum chaos relates to the random-matrix-theoretical description by introducing an operator we dub the Lyapunovian. Its level statistics is calculated for quantum stadium billiard, a seminal model of quantum chaos, and aligns perfectly with the Wigner-Dyson surmise. In the semiclassical limit, the Lyapunovian reduces to the matrix of uncorrelated finite-time Lyapunov exponents, connecting the CGR at early times, when the quantum effects are weak, to universal level repulsion that hinges on strong quantum interference. Finally, we consider quantum polygonal billiards: their classical counterparts are non-chaotic. We show exponential growth of the OTOCs in these systems, sharply contrasted with the classical behavior even before quantum interference develops.
Chaos in classical systems can be characterized by Lyapunov exponents that measure the exponential divergence of nearby trajectories, but directly extending this framework to quantum mechanics has been a persistent challenge. The wavelike nature of quantum states and the non-commutative geometry of quantum phase space obstruct a straightforward generalization of classical chaos theory. Here we develop a rigorous approach to quantum chaos by leveraging quantum optimal transport theory, which provides the missing geometric foundation for measuring distances between extended quantum distributions. We define quantum Lyapunov exponents that naturally avoid divergences encountered when naïvely generalizing classical exponents, and show that in the semiclassical limit they recover the classical global expansion rate, and hence the usual maximal Lyapunov exponent when they coincide. Our framework provides a tight connection between the divergence of classical trajectories and semiclassical phase space evolution, and additionally clarifies the role of out-of-time-order correlators as diagnostics of quantum chaos. These results establish quantum optimal transport as a unifying mathematical foundation for quantum chaos theory, providing new tools to characterize dynamical behavior across the full range of quantum dynamics from simple few-body models to complex many-body systems.
Despite the periodic kicks, a linear kicked rotor (LKR) is an integrable and exactly solvable model in which the kinetic energy term is linear in momentum. It was recently shown that spatially interacting LKRs are also integrable, and results in dynamical localization in the corresponding quantum regime. Similar localized phases exist in other nonintegrable models such as the coupled relativistic kicked rotors. This work, using a two-body LKR, demonstrates two main results; first, it is shown that chaos can be induced in the integrable linear kicked rotor through interactions between the momenta of rotors. An analytical estimate of its Lyapunov exponent is obtained. Second, the quantum dynamics of this chaotic model, upon variation of kicking and interaction strengths, is shown to exhibit a variety of phases: classically induced localization, dynamical localization, subdiffusive, and diffusive phases. We point out the signatures of these phases from the perspective of entanglement production in this system. By defining an effective Hilbert space dimension, the entanglement growth rate can be understood using appropriate random matrix averages.
We show that there is genuine chaos in quantum dynamics by introducing a physical distance between two quantum states. Qualitatively different from existing distances for quantum states, for example, the Fubini-Study distance, the physical distance between two mutually orthogonal quantum states, can be very small. As a result, two quantum states, which are initially very close by physical distance, can diverge from each other during the ensuing quantum dynamical evolution. We are able to use physical distance to define the quantum Lyapunov exponent and the quantum chaos measure. The latter leads to a quantum analog of the classical Poincaré section, which maps out the regions where quantum dynamics is regular and the regions where it is chaotic. Three different systems-a kicked rotor, the three-site Bose-Hubbard model, and the spin-1/2 XXZ model-are used to illustrate our results.
The kicked rotor provides a simple yet powerful model for introducing many of the central concepts of classical and quantum chaos. Despite its apparent simplicity, it exhibits rich dynamical behavior and has found applications across a wide range of fields, including atomic and optical physics, condensed matter physics, and emerging quantum technologies. This chapter begins by exploring foundational ideas using the kicked rotor as a unifying framework. We first discuss the transition from regular to chaotic motion in the classical system, and then introduce key quantum phenomena such as dynamical localization and quantum resonances. Special attention is devoted to the emergence of characteristic time scales and their role in the quantum-classical correspondence. To make these ideas more concrete, we also provide a brief overview of experimental realizations of the kicked rotor and its variants, illustrating how theoretical concepts are implemented in practice. In the second part of the chapter, we guide the reader toward more recent and advanced developments. Topics include near-resonant dynamics, topological features of kicked systems, the emergence of quantum dynamical phases inferred from classical transport properties, and extensions to non-Hermitian physics. We conclude with a discussion of open problems and future perspectives, outlining directions in which the kicked rotor continues to offer valuable insights.
Generating high-quality random numbers is essential for cryptography, statistical simulations, and cybersecurity. While pseudo-random number generators (PRNGs) are computationally efficient, their deterministic nature introduces potential security vulnerabilities. Recent research has explored chaotic systems as entropy sources due to their sensitivity to initial conditions-a key property for randomness generation. An emerging direction involves leveraging chaos in quantum systems. This paper proposes a quantum-inspired PRNG based on the Quantum Kicked Rotor (QKR), a well-known model in quantum chaos exhibiting complex dynamics such as dynamical localization. The method simulates the QKR and extracts pseudo-random sequences through momentum basis measurements. The generated sequences are evaluated for statistical quality, including the NIST 800-22 standard, which passed all 15 test. The sequences exhibit strong non-periodicity, a critical property for secure cryptographic applications. The proposed approach combines the unpredictability of quantum chaos with the scalability of classical simulation, offering a practical and efficient alternative for secure random number generation.