Abstract This paper introduces a novel mathematical framework for analyzing bifurcations in quantum–classical hybrid systems, bridging quantum information science with classical nonlinear dynamics. Our approach integrates noncommutative bifurcation theory, quantum center manifold reduction, and measurement-induced transition analysis to provide rigorous mathematical results supported by detailed numerical examples. The proposed model captures bidirectional quantum–classical interactions through coupled differential equations, where quantum measurement backaction and entanglement play fundamental roles in modifying bifurcation behavior. We demonstrate significant advances in understanding quantum–classical interplay, with promising applications in quantum sensing, control, and synchronization. Our method is general and applicable to various nonlinear systems as verified by our numerical examples.
Abstract This paper introduces a novel mathematical framework for analyzing bifurcations in quantum–classical hybrid systems, bridging quantum information science with classical nonlinear dynamics. Our approach integrates noncommutative bifurcation theory, quantum center manifold reduction, and measurement-induced transition analysis to provide rigorous mathematical results supported by detailed numerical examples. The proposed model captures bidirectional quantum–classical interactions through coupled differential equations, where quantum measurement backaction and entanglement play fundamental roles in modifying bifurcation behavior. We demonstrate significant advances in understanding quantum–classical interplay, with promising applications in quantum sensing, control, and synchronization. Our method is general and applicable to various nonlinear systems as verified by our numerical examples.
Abstract Quantum trajectory techniques have been used in the theory of open systems as a starting point for numerical computations and to describe the monitoring of a quantum system in continuous time. We extend this technique to develop a general approach to the dynamics of quantum/classical hybrid systems. By using two coupled stochastic differential equations, we can describe a classical component and a quantum one which have their own intrinsic dynamics and which interact with each other. A mathematically rigorous construction is given, under the restriction of having a Markovian joint dynamics and of involving only bounded operators on the Hilbert space of the quantum component. An important feature is that, if the interaction allows for a flow of information from the quantum component to the classical one, necessarily the dynamics is dissipative. We show also how this theory is connected to a suitable hybrid dynamical semigroup, which reduces to a quantum dynamical semigroup in the purely quantum case and includes Liouville and Kolmogorov–Fokker–Planck equations in the purely classical case. Moreover, this semigroup allows to compare the proposed stochastic dynamics with various other proposals based on hybrid master equations. Some simple examples are constructed in order to show the variety of physical behaviors which can be described; in particular, a model presenting hidden entanglement is introduced.
This paper investigates the development of a dynamic framework for quantum systems, explicitly modeling the interplay between quantum and classical behavior. We propose a 'hybrid' approach, leveraging a 'quantum-classical coherence matrix' to dynamically adjust quantum states in response to classical influences, effectively creating a continuous evolution. The core mechanism centers around maintaining a persistent, evolving state, bridging the gap between the deterministic nature of quantum mechanics and the often-ambiguous behavior of classical systems. The resulting model offers a novel perspective on complex systems, potentially facilitating a deeper understanding of phenomena like phase transitions, emergent behavior, and the limits of classical predictability. This research explores the potential of this framework for applications in quantum computing, materials science, and fundamental physics.
Quantum-induced phase transitions (QIPTs) represent a fascinating and rapidly developing area of research in condensed matter physics and beyond. This paper explores the theoretical framework for understanding and predicting these transitions, leveraging the principles of quantum mechanics to reveal emergent behavior and provide a new perspective on complex system dynamics. We propose a model centered around quantum entanglement and superposition, aiming to bridge the gap between classical and quantum descriptions. The core mechanism involves simulating the evolution of a dynamical system using quantum algorithms, allowing us to identify and characterize the transition points and associated dynamics. The implications of this research extend to diverse fields, including materials science, cosmology, and even fundamental physics. This paper presents a detailed outline of the proposed model, including mathematical formalism and analysis of potential applications. The focus is on establishing a robust theoretical foundation, paving the way for future investigations into the underlying mechanisms driving QIPTs.
Out-of-equilibrium quantum many-body systems stand at the forefront of modern theoretical physics, addressing fundamental questions on thermalization, transport, and universal phenomena. These fields, already well established in condensed matter, statistical physics, quantum optics, and quantum information theory, are progressively gaining even greater relevance with the rapid development of quantum technologies and simulation, which inherently operate in dynamical regimes. In recent years, the traditional paradigm of unitary quantum evolution has been expanded to include measurements, opening new directions in out-of-equilibrium physics. At the core of these advances lie measurement-induced phase transitions (MIPTs), which have emerged as a new class of dynamical critical phenomena characterizing the general behavior of monitored quantum dynamics. When external monitoring intertwines with unitary evolution, many-body quantum correlations change their structure, giving rise to distinct entanglement phases of matter. This discovery has sparked enormous interest in MIPTs, leading to significant advances in open quantum systems, entanglement theory, and more broadly quantum complexity. Despite much progress, a full understanding of monitored many-body dynamics is far from complete, leaving several open questions on the nature of MIPTs, their experimental observability, and the possibilities offered by measurements to enhance control over synthetic quantum matter. These issues persist due to the intrinsic complexity of the problem and the lack of efficient tools to study it, mainly caused by the stochastic character of monitored evolution. This thesis addresses these challenges by expanding the investigation of measurement-induced phenomena in new settings and introducing innovative probes of entanglement and many-body quantum complexity for MIPTs. A core question we investigate is the role of symmetries, non-ergodicity, and especially integrability in measurement-induced criticality, which dramatically affect the non-equilibrium phases. We further explore how these phenomena extend beyond bipartite quantum correlations to multipartite entanglement and quantum non-stabilizerness, highlighting the non-trivial interplay between measurements and complexity notions rooted in quantum information theory. Finally, we focus on the compelling problem of decoherence, modeling how noise spoils entanglement structures. These findings, supported by advanced numerical simulations and theoretical analysis, deepen the current understanding of entanglement, complexity, and integrability in monitored quantum many-body systems, offering new perspectives on their rich behavior. In parallel, we address the experimental problem of dissipation in MIPTs, which is of key relevance for practical implementations. We anticipate the present investigation to foster future research on the nature of monitored dynamical critical phenomena and, more broadly, the applications of measurements in quantum state evolution.
Periodically modulated nonlinear oscillators often display bistability of forced vibrations. This bistability can be used for new types of quantum measurements. They are based on switching between coexisting vibrational states. Since switching is accompanied by a large change of the amplitude and phase of forced vibrations, the measurements are highly sensitive. Quantum and classical noise plays dual role. It imposes a limitation on sensitivity in the familiar regime of a bifurcation amplifier. On the other hand, it makes it possible to use a bistable modulated oscillator in a new regime of a balanced dynamical bridge. We discuss the switching probabilities and show that they display scaling with control parameters. The critical exponents are found for different types of bifurcations and for different types of noise.
The quantum kicked top (QKT) is one of the most widely studied models in quantum chaos, providing a minimal yet powerful framework for exploring the relationship between classical nonlinear dynamics and quantum behavior. Unlike many chaotic systems with infinite-dimensional Hilbert spaces, the QKT possesses a finite-dimensional Hilbert space, making it analytically and numerically controllable while still showing a rich dynamical phenomena. In this chapter, we present a comprehensive introduction to the QKT as a paradigmatic model of quantum chaos. Starting from the classical kicked top, we derive the discrete nonlinear map governing the dynamics on the unit sphere and analyze its phase space structure through fixed points, stability analysis, bifurcations and Lyapunov exponents. We then discuss the role of symmetries, including rotational and time-reversal symmetry, and how their breaking modifies the dynamics. The quantum description is developed using Floquet theory, where the periodically driven spin system is represented by a unitary Floquet operator acting on a (2j+1)-dimensional Hilbert space. Within this framework, signatures of quantum chaos such as spectral statistics, entanglement generation and recurrences are discussed. The model also admits an interpretation as a system of interacting qubits, enabling explicit few-qubit realizations and direct connections with quantum information measures through reduced density matrices and entanglement entropy. By linking classical phase space structures with quantum dynamical indicators, the QKT provides a clear setting to investigate the emergence of chaotic behavior in the semiclassical limit. The chapter, therefore, highlights the quantum kicked top as a bridge between nonlinear classical dynamics, quantum chaos and modern quantum information science.