Bohmian Chaos and Entanglement in Two-Qubit Systems: Lyapunov Characteristic Number Reveals Chaos Onset

The subtle interplay between order and chaos in quantum systems continues to challenge physicists, and new research from Athanasios C. Tzemos of the Research Center for Astronomy and Applied Mathematics of the Academy of Athens, George Contopoulos also of the Academy of Athens, and Foivos Zanias from the Institute of Physics, University of Amsterdam, sheds light on this fundamental question. The team investigates how chaos emerges within a simple system of two quantum bits, or qubits, by meticulously tracking the behaviour of Bohmian trajectories, which represent the possible paths particles might take. Their work reveals a direct link between the arrangement of critical points in these trajectories and the onset of chaotic behaviour, demonstrating that increasing a specific parameter accelerates the transition to chaos. This detailed understanding of Bohmian dynamics provides valuable insight into the quantum world and offers a new perspective on the emergence of complexity in even the simplest quantum systems.

Unlike standard quantum mechanics, Bohmian mechanics specifies a definite position for particles at all times, raising the question of how this deterministic framework relates to the probabilistic nature of quantum measurements. The study focuses on understanding how the Born rule, which governs the probabilities of measurement outcomes, emerges from the deterministic dynamics of Bohmian mechanics, particularly in entangled systems. Scientists examined two-qubit systems to understand the interplay between determinism and chaos.

The research demonstrates that Bohmian trajectories can indeed be chaotic, even in these relatively simple systems, which is surprising given the underlying Schrödinger equation is typically associated with predictable behaviour. The presence of unstable points, where trajectories are highly sensitive to initial conditions, is identified as a key driver of this chaotic behaviour. Entanglement significantly enhances this chaos, leading to more complex and unpredictable dynamics. The team argues that ergodicity, the tendency of a system to explore all accessible states, plays a crucial role in establishing the Born rule.

If trajectories are ergodic, they will explore all possible states in proportion to the probability density predicted by quantum mechanics, effectively bridging the gap between the deterministic dynamics of Bohmian mechanics and the probabilistic predictions of standard quantum mechanics. Analysis of the quantum potential reveals its role in generating chaotic behaviour. This work is important because it addresses fundamental questions about the interpretation of quantum mechanics, specifically how to reconcile determinism with probability. It provides insights into the emergence of chaos in quantum systems, a relatively unexplored area, and offers a framework for testing the predictions of Bohmian mechanics through numerical simulations. Understanding the dynamics of Bohmian trajectories could also lead to new insights and applications in quantum technologies, contributing to the ongoing debate about the foundations of physics and the nature of reality.

Bohmian Flow, Lyapunov Numbers, and Chaos Analysis

Scientists meticulously investigated the critical points of Bohmian flow in a two-qubit system to understand the onset of chaos. They employed sophisticated numerical methods to track the paths of particles, ensuring high accuracy in their simulations. Researchers calculated a quantity called the stretching number, which measures how quickly nearby trajectories diverge, to analyse the effects of entanglement. They then examined the distribution of the Lyapunov Characteristic Number (LCN), a measure of chaotic behaviour, by calculating it for a large number of trajectories. Analysis revealed that the distribution of LCN values approached a predictable form as the number of trajectories increased, confirming the applicability of a mathematical principle called the Central Limit Theorem to this chaotic system.

The finite-time LCN was computed as an average along each trajectory, effectively sampling many weakly correlated regions of the system’s behaviour. To further quantify how quickly the LCN converged, scientists calculated its mean and standard deviation for varying degrees of entanglement. Results demonstrated that increasing entanglement reduced the time required for chaotic trajectories to converge towards a definitively chaotic state, suggesting a relationship between entanglement and the speed at which chaos emerges.

Bohmian Trajectories and Chaos in Entangled Systems

Scientists have achieved a detailed understanding of Bohmian particle trajectories and the onset of chaos in systems resembling two entangled qubits. The research focuses on critical points within the quantum flow, specifically Y-points and X-points, and how these points influence the paths of particles. Experiments reveal the distances between these critical points and a moving Bohmian particle, with particular attention paid to the times at which chaotic behaviour emerges. The team discovered that ordered and chaotic trajectories can coexist, and identified the mechanism responsible for generating chaos: the interaction of a Bohmian particle with the X-point complex accompanying nodal points.

Results demonstrate that increasing the degree of quantum entanglement reduces the time it takes for chaotic trajectories to converge towards a definitively chaotic state. This means that stronger entanglement accelerates the establishment of chaotic behaviour in the system. Furthermore, the study identified that chaotic trajectories are ergodic, meaning any two initially different chaotic paths eventually exhibit the same distribution of points in space. Scientists quantified this ergodicity using mathematical tools, confirming the uniformity of chaotic trajectories. In contrast, ordered trajectories exhibit a distinct shape, clearly different from the ergodic behaviour of chaotic paths. This work provides a trajectory-based understanding of quantum chaos, offering new insights into the dynamics of complex quantum systems.

Chaos Onset Linked to Nodal Point Dynamics

This research details a comprehensive investigation into the origins of chaos within Bohmian mechanics, specifically examining the critical points of particle trajectories. By meticulously tracking the distances between these critical points and Bohmian particles, scientists have clarified how chaos emerges and why certain trajectories remain ordered. The study reveals that the nature of the nodal point alternates between attracting and repelling, influencing the onset of chaotic behaviour. The team demonstrated that while quantum entanglement does not alter the magnitude of the Lyapunov Characteristic Number, a measure of chaotic sensitivity, it does affect the time it takes for chaotic trajectories to converge towards a definitively chaotic state. Researchers achieved this by numerically modelling the system and analysing the behaviour of trajectories under varying conditions, including cases with different frequencies. This work provides a geometrical understanding of chaos generation through a specific mechanism, offering insights into the dynamics of Bohmian particles.