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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. 

There is no unique way to quantify the degree of delocalization of quantum states in unbounded continuous spaces. In this work, we explore a recently introduced localization measure that quantifies the portion of the classical phase space occupied by a quantum state. The measure is based on the α -moments of the Husimi function and is known as the Rényi occupation of order α . With this quantity and random pure states, we find a general expression to identify states that are maximally delocalized in phase space. Using this expression and the Dicke model, which is an interacting spin-boson model with an unbounded four-dimensional phase space, we show that the Rényi occupations with α > 1 are highly effective at revealing quantum scars. Furthermore, by analyzing the high moments ( α > 1 ) of the Husimi function, we are able to identify qualitatively and quantitatively the unstable periodic orbits that scar some of the eigenstates of the model. 

Abstract In a classically chaotic system that is ergodic, any trajectory will be arbitrarily close to any point of the available phase space after a long time, filling it uniformly. Using Born’s rules to connect quantum states with probabilities, one might then expect that all quantum states in the chaotic regime should be uniformly distributed in phase space. This simplified picture was shaken by the discovery of quantum scarring, where some eigenstates are concentrated along unstable periodic orbits. Despite that, it is widely accepted that most eigenstates of chaotic models are indeed ergodic. Our results show instead that all eigenstates of the chaotic Dicke model are actually scarred. They also show that even the most random states of this interacting atom-photon system never occupy more than half of the available phase space. Quantum ergodicity is achievable only as an ensemble property, after temporal averages are performed. 

Abstract As the name indicates, a periodic orbit is a solution for a dynamical system that repeats itself in time. In the regular regime, periodic orbits are stable, while in the chaotic regime, they become unstable. The presence of unstable periodic orbits is directly associated with the phenomenon of quantum scarring, which restricts the degree of delocalization of the eigenstates and leads to revivals in the dynamics. Here, we study the Dicke model in the superradiant phase and identify two sets of fundamental periodic orbits. This experimentally realizable atom–photon model is regular at low energies and chaotic at high energies. We study the effects of the periodic orbits in the structure of the eigenstates in both regular and chaotic regimes and obtain their quantized energies. We also introduce a measure to quantify how much scarred an eigenstate gets by each family of periodic orbits and compare the dynamics of initial coherent states close and away from those orbits.