Exact State-Independent Recurrences Found in Finite-Dimensional Floquet Systems

The question of how systems return to their starting conditions lies at the heart of physics, and recent work by Amit Anand, Dinesh Valluri, and Jack Davis, alongside Shohini Ghose and colleagues at the Institute for Quantum Computing and DIENS, École Normale Supérieure, offers a new perspective on this phenomenon in periodically driven quantum systems. Building on the established Poincaré recurrence theorem, the team investigates the conditions under which exact, state-independent recurrences occur, moving beyond approximate measures of return. They demonstrate a powerful connection between the underlying mathematical structure of these systems and the possibility of recurrence, constructing an arithmetic framework that predicts all potential recurrence times by analysing the spectrum of the system’s evolution. This approach not only identifies scenarios where exact recurrences are possible, but also rigorously proves their absence for specific system parameters, revealing that rational parameters do not guarantee recurrence and deepening our understanding of the relationship between recurrence and chaotic behaviour.

Quantum Recurrence in Driven Chaotic Systems

This research investigates quantum recurrence, a phenomenon where a quantum system returns to its initial state after a certain time, within systems subjected to periodic forces. The work explores how this recurrence relates to concepts of chaos and order, particularly in systems like the quantum kicked top, a model rotator experiencing repeated impulses. Researchers examine the conditions necessary for recurrence, the timescales involved, and its connection to other quantum properties like entanglement and the tendency towards thermal equilibrium. The study extends to many-body systems, exploring how interactions between multiple quantum particles influence recurrence and the emergence of “quantum scars”, special states that resist the usual tendency to reach thermal equilibrium. The findings have potential applications in areas like quantum metrology, enhancing the precision of measurements, and quantum information processing, and draws connections to other fields, including arithmetical chaos and the intriguing concept of time crystals.

Floquet Unitary Splitting Field Analysis Confirms Recurrence

Researchers have developed a new method to definitively identify or rule out exact recurrences in periodically driven quantum systems, moving beyond previous approaches that relied on approximate estimations. Recognizing the limitations of earlier techniques, the team applied tools from algebraic field theory, focusing on the cyclotomic structure of the Floquet unitary, the mathematical operator describing the system’s evolution. This allows them to determine whether a system will precisely return to its initial state after a specific duration. The method analyzes the “splitting field” of the Floquet unitary’s characteristic polynomial, revealing the underlying algebraic relationships governing the system’s behavior.

By examining the degree of “cyclotomic extensions” within this field, researchers can connect the system’s potential for temporal periodicity to specific mathematical properties. This generates a finite, checkable set of candidate recurrence times, and crucially, if none satisfy the recurrence condition, the method definitively proves that no exact recurrence exists, offering a level of certainty rarely achieved in previous studies. Researchers demonstrated the effectiveness of this approach by applying it to the quantum kicked top model, leveraging its conserved angular momentum to refine the process.

Rational Parameters and Exact Quantum Recurrence

This research presents a novel method for identifying exact, state-independent recurrences in a broad class of finite-dimensional Floquet systems, which are periodically driven quantum systems. By leveraging techniques from algebraic field theory, the authors developed an arithmetic framework that determines all possible recurrence times by analyzing the spectral properties of the Floquet unitary operator. This approach provides both positive identification of potential recurrence times and definitive proof of their absence for given system parameters, offering a rigorous way to investigate long-time dynamics. The findings demonstrate that rational Hamiltonian parameters do not automatically guarantee exact recurrence, revealing a nuanced relationship between system parameters and the potential for periodic behavior. Applying this method to the quantum kicked top model, a well-studied angular momentum system, the researchers investigated recurrence for specific parameter values and system sizes, identifying both known and previously undetected periodicities. Importantly, the presence of exact recurrences serves as a clear indicator against chaotic behavior for those specific parameters, and the method is particularly efficient for analyzing Floquet systems with a single tunable parameter.