We investigate the connection between classical and quantum chaos in a dissipative system of two coupled collective spins. Although the classical dynamics possesses a stable fixed-point attractor, we find pronounced signatures of chaos in the transient dynamics preceding relaxation. In the quantum model, such transient chaos is accompanied by a transition to random-matrix statistics in the relevant part of the Liouvillian spectrum and by a qualitative increase in the number of Liouvillian modes contributing to expectation values of physical observables. The transition occurs before the bifurcation at which the stable fixed point disappears and a chaotic attractor emerges. Our results show that Liouvillian quantum chaos can be associated with transient classical chaos, rather than requiring a chaotic asymptotic attractor, and provide a refinement of the Grobe-Haake-Sommers conjecture.
We investigate the connection between classical and quantum chaos in a dissipative system of two coupled collective spins. Although the classical dynamics possesses a stable fixed-point attractor, we find pronounced signatures of chaos in the transient dynamics preceding relaxation. In the quantum model, such transient chaos is accompanied by a transition to random-matrix statistics in the relevant part of the Liouvillian spectrum and by a qualitative increase in the number of Liouvillian modes contributing to expectation values of physical observables. The transition occurs before the bifurcation at which the stable fixed point disappears and a chaotic attractor emerges. Our results show that Liouvillian quantum chaos can be associated with transient classical chaos, rather than requiring a chaotic asymptotic attractor, and provide a refinement of the Grobe-Haake-Sommers conjecture.
Quantum chaos has recently received increasing attention due to its relationship with experimental and theoretical studies of nonequilibrium quantum dynamics, thermalization, and the scrambling of quantum information. In an isolated system, quantum chaos refers to properties of the spectrum that emerge when the classical counterpart of the system is chaotic. However, despite experimental progress leading to longer coherence times, interactions with an environment can never be neglected, which calls for a definition of quantum chaos in dissipative systems. Advances in this direction were brought by the Grobe-Haake-Sommers (GHS) conjecture, which connects chaos in a dissipative classical system with cubic repulsion of the eigenvalues of the quantum counterpart and regularity with linear level repulsion. Here, we show that the GHS conjecture does not hold for the open Dicke model, which is a spin-boson model of experimental interest. We show that the onset of cubic level repulsion in the open quantum model is not always related with chaotic structures in the classical limit. This result challenges the universality of the GHS conjecture and raises the question of what is the source of spectral correlations in open quantum systems.
Dissipative quantum chaos plays a central role in the characterization and control of information scrambling, nonunitary evolution, and thermalization, but it still lacks a precise definition. The Grobe-Haake-Sommers conjecture, which links Ginibre level repulsion to classical chaotic dynamics, was recently shown to fail [Phys. Rev. Lett. 133, 240404 (2024)PRLTAO0031-900710.1103/PhysRevLett.133.240404]. We properly restore the quantum-classical correspondence through a dynamical approach based on the von Neumann entropy (VNE) and out-of-time-order correlators (OTOCs), which reveal signatures of chaos beyond spectral statistics. Focusing on the open anisotropic Dicke model, we identify two distinct regimes: transient chaos, marked by rapid early-time growth of VNE and OTOCs followed by low saturation values, and steady-state chaos, characterized by high long-time values. We introduce a random matrix toy model and show that Ginibre spectral statistics signals short-time chaos rather than steady-state chaos. Our results establish VNE dynamics and OTOCs as reliable diagnostics of dissipative quantum chaos across different timescales.
The effect of a large environment on a finite-size quantum mechanical system is two-fold: It brings dissipation, but also fluctuations of thermal and quantum origin. While dissipation tends to stabilize the dynamics, we question if and how environmental quantum fluctuations can generate chaos in an otherwise classically non-chaotic system. We work out a paradigmatic model of quantum optics: the dissipative Dicke model, where a large spin interacts with a dissipative harmonic mode. We dial in the classical/quantum correspondence by working in the semiclassical regime at large but finite spin. We demonstrate that, starting from a classically regular phase space in the superradiant regime, quantum noise can generate a strange attractor with fractal dimension and a positive Lyapunov exponent. We unveil the deep connection with shear-induced chaos that was recently developed in the mathematical community.
We define quantum chaos and integrability in open quantum many-body systems as a dynamical property of single stochastic realizations, referred to as quantum trajectories. This definition relies on the predictions of random matrix theory applied to the subset of the Liouvillian eigenspectrum involved in each quantum trajectory. Our approach, which we name (SSQT), enables a natural distinction between transient and steady-state quantum chaos as general phenomena in open setups. We test the generality and reliability of the SSQT criterion on several dissipative systems, further showing that an open system with a chaotic structure can evolve towards either a chaotic or integrable steady state. We apply our theoretical framework to two driven-dissipative bosonic systems. First, we study the driven-dissipative Bose-Hubbard model, a paradigmatic example of a quantum simulator, clarifying the interplay of integrability, transient, and steady-state chaos across its phase diagram. Our analysis shows the existence of an emergent dissipative quantum chaotic phase, whereas the classical and semiclassical limits display an integrable behavior. In this regime, chaos arises from the quantum and classical fluctuations associated with the dissipation mechanisms. Second, we investigate dissipative quantum chaos in the dispersive readout of a transmon qubit: a measurement technique ubiquitous in superconducting-based quantum hardware. Through the SSQT, we distinguish several regimes where the performance of the measurement instrument can be connected to the integrable or chaotic nature of the underlying driven-dissipative bosonic system. Our work offers a general understanding of the integrable and chaotic dynamics of open quantum systems and paves the way for the investigation of dissipative quantum chaos and its consequences on state-of-the-art noisy intermediate-scale quantum devices.
Understanding the far-from-equilibrium dynamics of dissipative quantum systems, where dissipation and decoherence coexist with unitary dynamics, is an enormous challenge with immense rewards. Often, the only realistic approach is to forgo a detailed microscopic description and search for signatures of universal behavior shared by collections of many distinct, yet sufficiently similar, complex systems. Quantum chaos provides a powerful statistical framework for addressing this question, relying on symmetries to obtain information not accessible otherwise. This thesis examines how to reconcile chaos with dissipation, proceeding along two complementary lines. In Part I, we apply non-Hermitian random matrix theory to open quantum systems with Markovian dissipation and discuss the relaxation timescales and steady states of three representative examples of increasing physical relevance: single-particle Lindbladians and Kraus maps, open free fermions, and dissipative Sachdev-Ye-Kitaev (SYK) models. In Part II, we investigate the symmetries, correlations, and universality of many-body open quantum systems, classifying several models of dissipative quantum matter. From a theoretical viewpoint, this thesis lays out a generic framework for the study of the universal properties of realistic, chaotic, and dissipative quantum systems. From a practical viewpoint, it provides the concrete building blocks of dynamical dissipative evolution constrained by symmetry, with potential technological impact on the fabrication of complex quantum structures. (Full abstract in the thesis.)
In classical mechanics, driven systems with dissipation often exhibit complex, fractal dynamics known as strange attractors. This paper addresses the fundamental question of how such structures manifest in the quantum realm. We investigate the quantum Duffing oscillator, a paradigmatic chaotic system, using the Caldirola-Kanai framework, where dissipation is integrated directly into a time-dependent Hamiltonian. By employing the Husimi distribution to represent the quantum state in phase space, we present the first visualization of a quantum strange attractor within this model. Our simulations demonstrate how an initially simple Gaussian wave packet is stretched, folded, and sculpted by the interplay of chaotic dynamics and energy loss, causing it to localize onto a structure that beautifully mirrors the classical attractor. This quantum "photograph" is inherently smoothed, blurring the infinitely fine fractal details of its classical counterpart as a direct consequence of the uncertainty principle. We supplement this analysis by examining the out-of-time-ordered correlator, which shows that stronger dissipation clarifies the exponential growth associated with the classical Lyapunov exponent, thereby confirming the model's semiclassical behavior. This work offers a compelling geometric perspective on open chaotic quantum systems and sheds new light on the quantum-classical transition.