The exponential growth of the out-of-time-ordered correlator (OTOC) has been proposed as a quantum signature of classical chaos. The growth rate is expected to coincide with the classical Lyapunov exponent. This quantum-classical correspondence has been corroborated for the kicked rotor and the stadium billiard, which are one-body chaotic systems. The conjecture has not yet been validated for realistic systems with interactions. We make progress in this direction by studying the OTOC in the Dicke model, where two-level atoms cooperatively interact with a quantized radiation field. For parameters where the model is chaotic in the classical limit, the OTOC increases exponentially in time with a rate that closely follows the classical Lyapunov exponent.Quantum chaos tries to bridge quantum and classical mechanics. The search for quantum signatures of classical chaos has ranged from level statistics [1,2] and the structure of the eigenstates [3,4] to the exponential increase of complexity [5,6] and the exponential decay of the overlap of two wave packets [7][8][9][10]. Recently, the pursuit of exponential instabilities in the quantum domain has been revived by the conjecture of a bound on the rate growth of the out-of-time-ordered correlator (OTOC) [11,12]. First introduced in the context of superconductivity [13], the OTOC is now presented as a measure of quantum chaos, with its growth rate being associated with the classical Lyapunov exponent. The OTOC is not only a theoretical quantity, but has also been measured experimentally via nuclear magnetic resonance techniques [14][15][16][17].The correspondence between the OTOC growth rate and the classical Lyapunov exponent has been explicitly shown in two cases of one-body chaotic systems, the kicked-rotor [18] and, after a first unsuccessful attempt [19], the stadium billiard [20]. It was also achieved for chaotic maps [21]. For interacting many-body systems, while exponential behaviors for the OTOC have been found for the Sachdev-Ye-Kitaev model [11,22] and for the Bose-Hubbard model [23,24], a direct demonstration of the quantum-classical correspondence has not yet been made. Studies in this direction include [6,[25][26][27][28][29] and [30].Here, we investigate the OTOC for the Dicke model [31,32]. Comparing with one-body systems, the model is a step up toward an explicit quantum-classical correspondence for interacting many-body systems, since it contains N atoms interacting with a quantized field.The Dicke model was originally proposed to explain the collective phenomenon of superradiance: the field mediates interatomic interactions, which causes the atoms to act collectively [31,33]. Superradiance has been experimentally studied with ultracold atoms in optical cavities [34][35][36][37][38][39]. The model has also found applications beyond superradiance in various different fields. It has been employed, for instance, in studies of ground-state and excited-state quantum phase tran-sitions [33,[40][41][42][43][44], entanglement creation [45], nonequilibrium dynamic...

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The relation between the onset of chaos and critical phenomena, like quantum phase transitions (QPTs) and excited-state quantum phase transitions (ESQPTs), is analyzed for atom-field systems. While it has been speculated that the onset of hard chaos is associated with ESQPTs based in the resonant case, the off-resonant cases, and a close look at the vicinity of the QPTs in resonance, show clearly that both phenomena, ESQPTs and chaos, respond to different mechanisms. The results are supported in a detailed numerical study of the dynamics of the semiclassical Hamiltonian of the Dicke model. The appearance of chaos is quantified calculating the largest Lyapunov exponent for a wide sample of initial conditions in the whole available phase space for a given energy. The percentage of the available phase space with chaotic trajectories is evaluated as a function of energy and coupling between the qubit and bosonic part, allowing us to obtain maps in the space of coupling and energy, where ergodic properties are observed in the model. Different sets of Hamiltonian parameters are considered, including resonant and off-resonant cases.

Employing efficient diagonalization techniques, we perform a detailed quantitative study of the regular and chaotic regions in phase space in the simplest non-integrable atom-field system, the Dicke model. A close correlation between the classical Lyapunov exponents and the quantum Participation Ratio of coherent states on the eigenenergy basis is exhibited for different points in the phase space. It is also shown that the Participation Ratio scales linearly with the number of atoms in chaotic regions, and with its square root in the regular ones.

Using coherent states as initial states, we investigate the quantum dynamics of the Lipkin-Meshkov-Glick (LMG) and Dicke models in the semi-classical limit. They are representative models of bounded systems with one-and two-degrees of freedom, respectively. The first model is integrable, while the second one has both regular and chaotic regimes. Our analysis is based on the survival probability. Within the regular regime, the energy distribution of the initial coherent states consists of quasi-harmonic sub-sequences of energies with Gaussian weights. This allows for the derivation of analytical expressions that accurately describe the entire evolution of the survival probability, from t = 0 to the saturation of the dynamics. The evolution shows decaying oscillations with a rate that depends on the anharmonicity of the spectrum and, in the case of the Dicke model, on interference terms coming from the simultaneous excitation of its two-degrees of freedom. As we move away from the regular regime, the complexity of the survival probability is shown to be closely connected with the properties of the corresponding classical phase space. Our approach has broad applicability, since its central assumptions are not particular of the studied models. + |z|2 J e zĴ+ |J, −J , and |α = e −|α| 2 /2 e αâ † |0 , are used as initial states for the LMG, |Ψ(0) = |z 0 , and the Dicke, |Ψ(0) = |z 0 ⊗ |α 0 , models. Likewise they are used to define classical corresponding Hamiltonians [38,50]: h LM G = z|H LM G |z /J and h D = z| ⊗ α|H D |α ⊗ |z /J.