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

TL;DR

This paper derives analytical expressions for the survival probability of coherent states in regular regimes of the LMG and Dicke models, linking quantum dynamics with classical phase space properties.

Contribution

It provides a general analytical framework for understanding survival probability evolution in regular regimes of bounded quantum systems.

Findings

Analytical formulas accurately describe survival probability evolution.

Decay oscillations depend on spectral anharmonicity and model-specific interference.

Complexity increases as systems move away from 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.