  The superfluid to Mott insulator transition and the superradiant transition
are textbook examples for quantum phase transition and coherent quantum optics,
respectively. Recent experiments in ETH and Hamburg succeeded in loading
degenerate bosonic atomic gases in optical lattices inside a cavity, which
enables the first experimental study of the interplay between these two
transitions. In this letter we present the theoretical phase diagram for the
ETH experimental setup, and determine the phase boundaries and the orders of
the phase transitions between the normal superfluid phase, the superfluid with
superradiant light, the normal Mott insulator and the Mott insulator with
superradiant light. We find that in contrast to the second-order superradiant
transition in a weakly interacting Bose condensate, strong correlations in the
superfluid nearby a Mott transition can render the superradiant transition to a
first order one. Our results will stimulate further experimental studies of
interactions between cavity light and strongly interacting quantum matters.
