Analytical description of the survival probability of coherent states in regular regimes

Abstract

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.

Identification of quantum scars via phase-space localization measures

“…The Dicke model [30] was initially introduced to explain the phenomenon of superradiance [36,43,50,80] and has since fostered a wide variety of theoretical studies, including the behavior of out-of-time-ordered correlators [24,59,63], manifestations of quantum scarring [6,29,35,64,65], non-equilibrium dynamics [1,50,51,57,58,78], and measures of quantum localization with respect to phase space [65,79]. The model is also of great interest to experiments with trapped ions [26,71], superconducting circuits [47], and cavity assisted Raman transitions [5,82].…”

Quantum and Classical Lyapunov Exponents in Atom-Field Interaction Systems

“…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 dynamics [46][47][48][49], quantum chaos [50][51][52][53], and monodromy [54,55]. Recently, the model has received revived attention due to new experiments with ion traps [56,57] and the analysis of the OTOC [58,59].In the classical limit, the Dicke model presents regular and chaotic regions depending on the Hamiltonian parameters and excitation energies [53].…”

Eigenstate entanglement in integrable collective spin models

“…We find that the finite size scaling shows that their half-bipartition average EE converges to a universal value, which we calculate analytically for specific parameters in the LMG Hamiltonian, as the average EE over the Dicke basis. An interesting difference with previous results is that, unlike lattice models, the LMG has a well defined classical limit in which it is integrable as per the unequivocal classical criteria [45]. It is also known to be quantum-integrable using Bethe ansatz [46,47] which is one of the main integrability definitions used in quantum mechanics.…”

“…This implies that the eigenvalues of J 2 , j(j + 1) and hence j, are constants of motion. The LMG model has a welldefined classical limit that is integrable, irrespective of the choice of the parameter set (γ x , γ y , h), as per the classical integrability criteria [45].…”