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...
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.
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.
Adiabatic invariants are introduced and shown to provide an approximate second integral of motion for the non-integrable Dicke model, in the energy region where the system exhibits a regular dynamics. This low-energy region is always present and has been described both in a semiclassical and a full quantum analysis. Its Peres lattices exhibit that many observables vary smoothly with energy, along lines which beg for a formal description. It is shown how the adiabatic invariants provide a rationale to their presence in many cases. They are built employing the Born-Oppenheimer approximation, valid when a fast system is coupled to a much slower one. As the Dicke model has a one bosonic and one fermionic degree of freedom, two versions of the approximation are used, depending on which one is the faster. In both cases a noticeably accord with exact numerical results is obtained. The employment of the adiabatic invariants provides a simple and clear theoretical framework to study the physical phenomenology associated to this energy regime, far beyond the energies where the quadratic approximation can be employed. Adiabatic invariants in the Dicke model.For the Dicke model it was first simulated by means of a Bose-Einstein condensate in an optical cavity [21,22]. Also, an open version of the model was realized employing Raman transitions [23], and dynamical non-equilibrium superradiant phase transition has been also observed [24]. Moreover, by means of superconducting QED the Rabi model [25,26] and few-atoms Dicke-like models [27,28] have been explored in this direction.More than ten years ago, C. Emary and T. Brandes built approximate solutions of the Dicke model by means of the Holstein-Primakoff realization of the SU(2) algebra in the thermodynamic limit [29]. This approach describes the behavior of the model for the ground-and low energy-states, exhibiting the superradiant QPT easily and showing the presence of chaos in the spectrum of the Hamiltonian. Besides, the truncated Holstein-Primakoff approach makes it possible to extract the critical exponents for the groundstate energy per particle, the fraction of excited atoms, the number of photons per atom, their fluctuations, and the concurrence [30,31].Enhanced by these experimental and theoretical results, and thanks to its algebraic properties, the Dicke model has become an excellent tool from the theoretical point of view for exploring several features of quantum many-body systems with collective degrees of freedom, e.g., the QPT, the Excited-State Quantum Phase Transitions [32,33,34,35] and their relation with the thermal phase transition [36,37], the onset of quantum chaos and its correspondence to the classical limit [38,39,40,41], quantum quenches [42,43] and the problem of equilibrium and thermalization in isolated manybody quantum systems [44,45], to name a few. This is the motivation for looking for as many analytical or semi-analytical descriptions of the spectra and observables of the model as it is possible. The truncated Holstein-Primakoff approximatio...
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 along 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.
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