  In circuit electromechanics, the coupling strength is usually very small.
Here, replacing the capacitor in circuit electromechanics by a superconducting
flux qubit, we show that the coupling among the qubit and the two resonators
can induce effective electromechanical coupling which can attain the strong
coupling regime at the single photon level with feasible experimental
parameters. We use dispersive couplings among two resonators and the qubit
while the qubit is also driven by an external classical field. These couplings
form a three-wave mixing configuration among the three elements where the qubit
degree of freedom can be adiabatically eliminated, and thus results in the
enhanced coupling between the two resonators. Therefore, our work constitutes
the first step towards studying quantum nonlinear effect in circuit
electromechanics.
