Long-lived Microwave Electromechanical Systems Enabled by Cubic Silicon-Carbide Membrane Crystals

Cubic silicon-carbide crystals, known for their high thermal conductivity and in-plane stress, hold significant promise for the development of high-quality ($Q$) mechanical oscillators. Enabling coherent electrical manipulation of long-lived mechanical resonators would be instrumental in advancing the development of phononic memories, repeaters, and transducers for microwave quantum states. In this study, we demonstrate the compatibility of high-stress and crystalline (3C-phase) silicon-carbide membranes with superconducting microwave circuits. We establish a coherent electromechanical interface for long-lived phonons, allowing precise control over the electromechanical cooperativity. This interface enables tunable slow-light time with group delays extending up to an impressive duration of \emph{an hour}. We then investigate a phononic memory based on the high-$Q$ ($10^{8}$) silicon-carbide membrane, capable of storing and retrieving microwave coherent states \emph{on-demand}. The thermal and coherent components can be distinguished through state tomography in quadrature phase space, which shows an exponential increase and decay trend respectively as the storage time increases. The electromechanical interface and phononic memory made from crystalline silicon-carbide membrane possess enticing attributes, including low microwave-induced mechanical heating, phase coherence, an energy decay time of $T_{1}=19.9$~s, and it acquires less than one quantum noise within $\tau_{\textrm{coh}}=41.3$~ms storage period. These findings underscore the unique opportunities provided by cubic silicon-carbide membrane crystals for the storage and transfer of quantum information across distinct components of hybrid quantum systems.