  In this work, we propose a method to investigate controllable qubit-resonator
interactions in a Dicke model with driven biased term. The nonlinearity of
spectrum, which can be induced by qubit-resonator interactions, plays an
important role in such controllable interactions. To gain insight into
mechanism of the nonlinearity, we perform a unitary transformation to the
Hamiltonian. The results show that the nonlinearity of the transformed
Hamiltonian depends on the qubit-resonator coupling strength. The general forms
of the effective Hamiltonians are discussed in detail based on the frequency
modulation approach. The dynamical evolution can be switched on and off by
adjusting the modulation parameters. By utilizing such controllable
interactions, we discuss the creation of Dicke states and arbitrary
superposition of Dicke states. We also consider the nonlinearity of energy
level for the limit of large qubit numbers. In the thermodynamics limit, the
kerr type nonlinearity is induced from "magnon"-resonator coupling, and the
selective preparation of "magnon" Fock states can be studied under "magnon"
scenario.
