  The superconducting coplanar waveguide (SCPW) cavity plays an essential role
in various areas like superconducting qubits, parametric amplifiers, radiation
detectors, and studying magnon-photon and photon-phonon coupling. Despite its
wide-ranging applications, the use of SCPW cavities to study various van der
Waals 2D materials is relatively unexplored. The resonant modes of the SCPW
cavity exquisitely sense the dielectric environment. In this work, we measure
the charge compressibility of bilayer graphene coupled to a half-wavelength
SCPW cavity. Our approach provides a means to detect subtle changes in the
capacitance of the bilayer graphene heterostructure, which depends on the
compressibility of bilayer graphene, manifesting as shifts in the resonant
frequency of the cavity. This method holds promise for exploring a wide class
of van der Waals 2D materials, including transition metal dichalcogenides
(TMDs) and their moir\'e where DC transport measurement is challenging.
