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arXiv:2307.05675 · Physical review. E

Analysis of chaos and regularity in the open Dicke model

David Villaseñor, Pablo Barberis-Blostein

Published Jan 5, 2024• 5 citations
arXivHTMLPDF

Abstract

We present an analysis of chaos and regularity in the open Dicke model, when dissipation is due to cavity losses. Due to the infinite Liouville space of this model, we also introduce a criterion to numerically find a complex spectrum which approximately represents the system spectrum. The isolated Dicke model has a well-defined classical limit with two degrees of freedom. We select two case studies where the classical isolated system shows regularity and where chaos appears. To characterize the open system as regular or chaotic, we study regions of the complex spectrum taking windows over the absolute value of its eigenvalues. Our results for this infinite-dimensional system agree with the Grobe-Haake-Sommers (GHS) conjecture for Markovian dissipative open quantum systems, finding the expected 2D Poisson distribution for regular regimes, and the distribution of the Ginibre unitary ensemble (GinUE) for the chaotic ones, respectively.

DOI
10.1103/physreve.109.014206

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We present an analysis of chaos and regularity in the open Dicke model, when dissipation is due to cavity losses. Due to the infinite Liouville space of this model, we also introduce a criterion to numerically find a complex spectrum which approximately represents the system spectrum. The isolated Dicke model has a well-defined classical limit with two degrees of freedom. We select two case studies where the classical isolated system shows regularity and where chaos appears. To characterize the open system as regular or chaotic, we study regions of the complex spectrum taking windows over the absolute value of its eigenvalues. Our results for this infinite-dimensional system agree with the Grobe-Haake-Sommers (GHS) conjecture for Markovian dissipative open quantum systems, finding the expected 2D Poisson distribution for regular regimes, and the distribution of the Ginibre unitary ensemble (GinUE) for the chaotic ones, respectively.
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.
Abstract The interaction of a quantized electromagnetic field in a cavity with a set of two-level atoms inside it can be described with algebraic Hamiltonians of increasing complexity, from the Rabi to the Dicke models. Their algebraic character allows, through the use of coherent states, a semiclassical description in phase space, where the non-integrable Dicke model has regions associated with regular and chaotic motion. The appearance of classical chaos can be quantified calculating the largest Lyapunov exponent over the whole available phase space for a given energy. In the quantum regime, employing efficient diagonalization techniques, we are able to perform a detailed quantitative study of the regular and chaotic regions, where the quantum participation ratio ( P R ) of coherent states on the eigenenergy basis plays a role equivalent to the Lyapunov exponent. It is noted that, in the thermodynamic limit, dividing the participation ratio by the number of atoms leads to a positive value in chaotic regions, while it tends to zero in the regular ones.
The emergence of chaos in an atom-field system is studied employing both semiclassical and numerical quantum techniques, taking advantage of the algebraic character of the Hamiltonian. A semiclassical Hamiltonian is obtained by considering the expectation value of the quantum Hamiltonian in Glauber (for the field) and Bloch (for the atoms) coherent states. Regular and chaotic regions are identified by looking at the Poincaré sections for different energies and parameter values. An analytical expression for the semiclassical energy density of states is obtained by integrating the available phase space, which provides an exact unfolding to extract the fluctuations in the level statistics. Quantum chaos is recognized in these fluctuations, as a function of the coupling strength, for different regions in the energy spectrum, evaluating the Anderson–Darling (A–D) parameter, which distinguishes the Wigner- or Poisson-like distributions. Peres lattices play a role similar to the Poincaré section for quantum states. They are calculated employing efficient numerical solutions and are a powerful visual tool to identify individual states belonging to a regular or chaotic region, classified by utilizing the Poincaré sections and the A–D parameter. Finally, the quantum Husimi function for selected excited states is shown to have a noticeable similitude with the Poincaré sections at the same energy.
We explore the connections between dissipative quantum phase transitions and non-Hermitian random matrix theory. For this, we work in the framework of the dissipative Dicke model which is archetypal of symmetry-breaking phase transitions in open quantum systems. We establish that the Liouvillian describing the quantum dynamics exhibits distinct spectral features of integrable and chaotic character on the two sides of the critical point. We follow the distribution of the spacings of the complex Liouvillian eigenvalues across the critical point. In the normal and superradiant phases, the distributions are two-dimensional Poisson and that of the Ginibre unitary random matrix ensemble, respectively. Our results are corroborated by computing a recently introduced complex-plane generalization of the consecutive level-spacing ratio distribution. Our approach can be readily adapted for classifying the nature of quantum dynamics across dissipative critical points in other 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.
We present a detailed analysis of the connection between chaos and the onset of thermalization in the spin-boson Dicke model. This system has a well-defined classical limit with two degrees of freedom, and it presents both regular and chaotic regions. Our studies of the eigenstate expectation values and the distributions of the off-diagonal elements of the number of photons and the number of excited atoms validate the diagonal and off-diagonal eigenstate thermalization hypothesis (ETH) in the chaotic region, thus ensuring thermalization. The validity of the ETH reflects the chaotic structure of the eigenstates, which we corroborate using the von Neumann entanglement entropy and the Shannon entropy. Our results for the Shannon entropy also make evident the advantages of the so-called "efficient basis" over the widespread employed Fock basis when investigating the unbounded spectrum of the Dicke model. The efficient basis gives us access to a larger number of converged states than what can be reached with the Fock basis.