  Nanoscale Fano resonances, with applications from telecommunications to
ultra-sensitive biosensing, have prompted extensive research. We demonstrate
that a superconducting qubit, jointly coupled to microwave waveguides and an
inter-digital transducer composite device, can exhibit acoustic Fano
resonances. Our analytical framework, leveraging the Taylor series
approximation, elucidates the origins of these quantum acoustic resonances with
periodic Fano-like interference. By analyzing the analytical Fano parameter, we
demonstrate that the Fano resonances and their corresponding Fano widths near
the resonance frequency of a giant atom can be precisely controlled and
manipulated by adjusting the time delay. Moreover, not just the near-resonant
Fano profiles, but the entire periodic Fano resonance features can be precisely
modulated from Lorentz, Fano to quasi-Lorentz shapes by tuning the coupling
strength of the microwave waveguide. Our analytical framework offers insights
into the control and manipulation of periodic Fano resonances in quantum
acoustic waves, thereby presenting significant potential for applications such
as quantum information processing, sensing, and communication.
