  We discuss level splitting and sideband transitions induced by a modulated
coupling between a superconducting quantum circuit and a nanomechanical
resonator. First, we show how to achieve an unconventional time-dependent
longitudinal coupling between a flux (transmon) qubit and the resonator.
Considering a sinusoidal modulation of the coupling strength, we find that a
first-order sideband transition can be split into two. Moreover, under the
driving of a red-detuned field, we discuss the optical response of the qubit
for a resonant probe field. We show that level splitting induced by modulating
this longitudinal coupling can enable two-color electromagnetically induced
transparency (EIT), in addition to single-color EIT. In contrast to standard
predictions of two-color EIT in atomic systems, we apply here only a single
drive (control) field. The monochromatic modulation of the coupling strength is
equivalent to employing two eigenfrequency-tunable mechanical resonators. Both
drive-probe detuning for single-color EIT and the distance between transparent
windows for two-color EIT, can be adjusted by tuning the modulation frequency
of the coupling.
