Quantum multifractality as a probe of phase space in the Dicke model.

We study the multifractal behavior of coherent states projected in the energy eigenbasis of the spin-boson Dicke Hamiltonian, a paradigmatic model describing the collective interaction between a single bosonic mode and a set of two-level systems. By examining the linear approximation and parabolic correction to the mass exponents, we find ergodic and multifractal coherent states and show that they reflect details of the structure of the classical phase space, including chaos, regularity, and features of localization. The analysis of multifractality stands as a sensitive tool to detect changes and structures in phase space, complementary to classical tools to investigate it. We also address the difficulties involved in the multifractal analyses of systems with unbounded Hilbert spaces.

Avoided crossings and dynamical tunneling close to excited-state quantum phase transitions.

Using the Wehrl entropy, we study the delocalization in phase space of energy eigenstates in the vicinity of avoided crossings in the Lipkin-Meshkov-Glick model. These avoided crossings, appearing at intermediate energies in a certain parameter region of the model, originate classically from pairs of trajectories lying in different phase-space regions which, contrary to the low-energy regime, are not connected by the discrete parity symmetry of the model. As coupling parameters are varied, a sudden increase of the Wehrl entropy is observed for eigenstates participating in avoided crossings that are close to the critical energy of the excited-state quantum phase transition. This allows us to detect when an avoided crossing is accompanied by a superposition of the pair of classical trajectories in the Husimi function of eigenstates. This superposition yields an enhancement of dynamical tunneling, which is observed by considering initial Bloch states that evolve partially into the partner region of the paired classical trajectories, thus breaking the quantum-classical correspondence in the evolution of observables.

Quantum localization measures in phase space.

Measuring the degree of localization of quantum states in phase space is essential for the description of the dynamics and equilibration of quantum systems, but this topic is far from being understood. There is no unique way to measure localization, and individual measures can reflect different aspects of the same quantum state. Here we present a general scheme to define localization in measure spaces, which is based on what we call Rényi occupations, from which any measure of localization can be derived. We apply this scheme to the four-dimensional unbounded phase space of the interacting spin-boson Dicke model. In particular, we make a detailed comparison of two localization measures based on the Husimi function in the regime where the model is chaotic, namely, one that projects the Husimi function over the finite phase space of the spin and another that uses the Husimi function defined over classical energy shells. We elucidate the origin of their differences, showing that in unbounded spaces the definition of maximal delocalization requires a bounded reference subspace, with different selections leading to contextual answers.

Dynamical signatures of quantum chaos and relaxation time scales in a spin-boson system.

Quantum systems whose classical counterparts are chaotic typically have highly correlated eigenvalues and level statistics that coincide with those from ensembles of full random matrices. A dynamical manifestation of these correlations comes in the form of the so-called correlation hole, which is a dip below the saturation point of the survival probability's time evolution. In this work, we study the correlation hole in the spin-boson (Dicke) model, which presents a chaotic regime and can be realized in experiments with ultracold atoms and ion traps. We derive an analytical expression that describes the entire evolution of the survival probability and allows us to determine the time scales of its relaxation to equilibrium. This expression shows remarkable agreement with our numerical results. While the initial decay and the time to reach the minimum of the correlation hole depend on the initial state, the dynamics beyond the hole up to equilibration is universal. We find that the relaxation time of the survival probability for the Dicke model increases linearly with system size.

Classical chaos in atom-field systems.

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.

Delocalization and quantum chaos in atom-field systems.

Employing efficient diagonalization techniques, we perform a detailed quantitative study of the regular and chaotic regions in phase space in the simplest nonintegrable 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.