  We study the statistics of the work done in a zero temperature quench of the
coupling constant in the Dicke model describing the interaction between a gas
of two level atoms and a single electromagnetic cavity mode. When either the
final or the initial coupling constants approach the critical coupling
$\lambda_c$ that separates the normal and superradiant phases of the system,
the probability distribution of the work done displays singular behavior. The
average work tends to diverge as the initial coupling parameter is brought
closer to the critical value $\lambda_c$. In contrast, for quenches ending
close to criticality, the distribution of work has finite moments but displays
a sequence of edge singularities. This contrasting behavior is related to the
difference between the processes of compression and expansion of a particle
subject to a sudden change of its confining potential. We confirm this by
studying in detail the time dependent statistics of other observables, such as
the quadratures of the photons and the total occupation of the bosonic modes.
