Quantum and Semi-Classical Signatures of Dissipative Chaos in the Steady State (2506.14961v1)

Abstract: We investigate the quantum-classical correspondence in open quantum many-body systems using the SU(3) Bose-Hubbard trimer as a minimal model. Combining exact diagonalization with semiclassical Langevin dynamics, we establish a direct connection between classical trajectories characterized by fixed-point attractors, limit cycles, or chaos and the spectral and structural properties of the quantum steady state. We show that classical dynamical behavior, as quantified by the sign of the Lyapunov exponent, governs the level statistics of the steady-state density matrix: non-positive exponents associated with regular dynamics yield Poissonian statistics, while positive exponents arising from chaotic dynamics lead to Wigner-Dyson statistics. Strong symmetries constrain the system to lower-dimensional manifolds, suppressing chaos and enforcing localization, while weak symmetries preserve the global structure of the phase space and allow chaotic behavior to persist. To characterize phase-space localization, we introduce the phase-space inverse participation ratio IPR, which defines an effective dimension D of the Husimi distribution's support. We find that the entropy scales as $S \propto \ln N^D$, consistently capturing the classical nature of the underlying dynamics. This semiclassical framework, based on stochastic mixtures of coherent states, successfully reproduces not only observable averages but also finer features such as spectral correlations and localization properties. Our results demonstrate that dissipative quantum chaos is imprinted in the steady-state density matrix, much like in closed systems, and that the interplay between dynamical regimes and symmetry constraints can be systematically probed using spectral and phase-space diagnostics. These tools offer a robust foundation for studying ergodicity, localization, and non-equilibrium phases of open quantum systems.

• The paper establishes a correspondence between classical chaos and quantum level statistics by linking positive Lyapunov exponents with Wigner-Dyson distributions.
• It utilizes exact diagonalization and semiclassical Langevin dynamics to analyze the steady-state behavior of the SU(3) Bose-Hubbard trimer model.
• The study reveals that strong symmetries constrain chaotic behavior while weak symmetries allow quantum ergodicity, providing insights into managing dissipative chaos.

Analyzing Dissipative Chaos in Open Quantum Systems

The paper "Quantum and Semi-Classical Signatures of Dissipative Chaos in the Steady State" undertakes a meticulous examination of the interplay between quantum and classical dynamics in open quantum many-body systems. Focusing on the $SU(3)$ Bose-Hubbard trimer model, this paper sheds light on how classical chaotic behavior translates to quantum properties within open systems, which are influenced by environmental interactions.

Overview and Methodology

In this paper, the authors explore the steady-state properties of open quantum systems using a combined approach of exact diagonalization and semiclassical Langevin dynamics. The model employed, the $SU(3)$ Bose-Hubbard trimer, serves as a critical testbed due to its capacity to exhibit chaos even without external periodic driving. The dynamics are governed by a Lindblad master equation, which facilitates the modeling of dissipations through a set of jump operators.

The paper constructs a link between the classical and quantum realms by establishing a correspondence between Lyapunov exponents, which quantify chaos in classical phase space, and level statistics of the quantum density matrix. The approach categorizes classical trajectories into fixed-point attractors, limit cycles, or chaotic paths, emphasizing the effects of these regimes on quantum observables.

Key Findings

1. Classical and Quantum Correspondence: Classical systems with non-positive Lyapunov exponents (indicative of regular dynamics) show Poissonian level statistics in their quantum steady states. For systems displaying chaotic dynamics with positive Lyapunov exponents, the quantum level statistics align with Wigner-Dyson distributions, characteristic of quantum chaos.
2. Spectral and Structural Properties: The system's quantum steady state is intricately connected to its classical dynamics. In scenarios where classical dynamics are governed by chaos, the quantum density matrices exhibit characteristics of quantum ergodicity, including delocalization in phase space and high von Neumann entropy.
3. Role of Symmetries: The influence of symmetries in quantum dynamics is investigated, revealing that strong symmetries confine systems to lower-dimensional manifolds, mitigating chaotic behavior. This is contrasted with weak symmetries, which allow persistence of chaotic features.
4. Phase-Space Inverse Participation Ratio (IPR$_\phi$): This novel metric quantifies localization in phase space, illustrating the degree of quantum ergodicity. The IPR$_\phi$ effectively distinguishes between localized and delocalized steady states, expanding the toolbox for assessing quantum chaos.

Implications and Future Directions

The paper underscores a significant advancement in understanding the connections between classical chaos and quantum dynamics in open systems. The findings imply that despite environmental dissipation, signatures of classical chaos can leave an imprint on quantum steady states. This opens pathways for further studies into more complex open system models, potentially impacting quantum information processing where environmental interactions are inevitable.

Future research could extend these methods to different quantum models and explore the influence of external driving forces, as well as to investigate how different types of noise influence the chaotic regimes. The semiclassical framework presented could also be exploited to refine quantum control strategies in experimental settings, by leveraging our improved understanding of how chaotic signatures manifest in open quantum systems' steady states.

In summary, this investigation provides an insightful exploration into the nature of dissipative quantum chaos, highlighting a remarkable alignment between classical predictions and quantum reality, thereby enriching our comprehension of open quantum systems' behavior.