  Coupling a quantum many-body system to a cavity can create bifurcation points
in its phase diagram, where the ground state makes sudden switchings between
different phases. Here we study the dynamical quantum phase transition of a
transverse field Ising model coupled to a cavity. We show that an infinitesimal
quench of the cavity driving at the bifurcation points induces gradual
evolution of the Ising model to pass across the quantum critical point and
excites quasiparticles. Meanwhile, when the driving is slowly ramped through
the bifurcation points, the adiabaticity of the evolution and the number of
quasiparticle excitations are strongly affected by cavity-induced nonlinearity.
Introducing and manipulating cavity-induced nonlinearity hence provide an
effective approach to control the dynamics and the adiabaticity of adiabatic
quantum processes. Our model can be implemented with superconducting quantum
circuits.
