  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.
