Dynamics of quantum phase transitions in Dicke and Lipkin-Meshkov-Glick models

We consider dynamics of Dicke models, with and without counterrotating terms, under slow variations of parameters which drive the system through a quantum phase transition. The model without counterrotating terms and sweeped detuning is seen in the contexts of a many-body generalization of the Landau-Zener model and the dynamical passage through a second-order quantum phase transition (QPT). Adiabaticity is destroyed when the parameter crosses a critical value. Applying semiclassical analysis based on concepts of classical adiabatic invariants and mapping to the second Painleve equation (PII), we derive a formula which accurately describes particle distributions in the Hilbert space at wide range of parameters and initial conditions of the system. We find striking universal features in the particle distributions which can be probed in an experiment on Feshbach resonance passage or a cavity QED experiment. The dynamics is found to be crucially dependent on the direction of the sweep. The model with counterrotating terms has been realized recently in an experiment with ultracold atomic gases in a cavity. Its semiclassical dynamics is described by a Hamiltonian system with two degrees of freedom. Passage through a QPT corresponds to passage through a bifurcation, and can also be described by PII (after averaging over fast variables), leading to similar universal distributions. Under certain conditions, the Dicke model is reduced to the Lipkin-Meshkov-Glick model.