  Three-wave mixing in second-order nonlinear optical processes cannot occur in
atomic systems due to the electric-dipole selection rules. In contrast, we
demonstrate that second-order nonlinear processes can occur in a
superconducting quantum circuit (i.e., a superconducting artificial atom) when
the inversion symmetry of the potential energy is broken by simply changing the
applied magnetic flux. In particular, we show that difference- and
sum-frequencies (and second harmonics) can be generated in the microwave regime
in a controllable manner by using a single three-level superconducting flux
quantum circuit (SFQC). For our proposed parameters, the frequency tunability
of this circuit can be achieved in the range of about 17 GHz for the
sum-frequency generation, and around 42 GHz (or 26 GHz) for the
difference-frequency generation. Our proposal provides a simple method to
generate second-order nonlinear processes within current experimental
parameters of SFQCs.
