Quantum control of phononic modes via Rabi-sideband resonance

Mechanical resonators usually have many modes with high Q factor and provide a good candidate for multimode quantum memory. However, these modes are usually harmonic; in order to manipulate quantum information, one needs to couple the modes to some nonlinear element such as a qubit. In this work, we study a system that consists of a qubit coupled linearly to a set of phononic modes whose frequencies are comparable to that of the qubit. We consider the coupling strength is smaller than the free spectral range of phonon modes but much stronger than their decay rates which is currently achievable. We show that by Rabi pumping the qubit, one can selectively bring the "dressed" qubit into resonance with one of the phonon modes even though they are off-resonant without pump. The driving-induced coupling allows state swap between the phonon and dressed qubit or simultaneously excites or de-excites them. We also show that by pumping the qubit with two tones whose frequency difference (or sum) equals to that of two phonon modes, one can generate a beamsplitter (or two-mode squeezing) interaction between two phonon modes. The study applies not only to phonon modes but also to multiple microwave cavities or multiple modes in a single cavity coupled to a common superconducting qubit.