Projected dipole moments of individual two-level defects extracted using circuit quantum electrodynamics

Material-based two-level systems (TLSs), appearing as defects in low-temperature devices including superconducting qubits and photon detectors, are difficult to characterize. In this study we apply a uniform dc-electric field across a film to tune the energies of TLSs within. The film is embedded in a superconducting resonator such that it forms a circuit quantum electrodynamical (cQED) system. The energy of individual TLSs is observed as a function of the known tuning field. By studying TLSs for which we can determine the tunneling energy, the actual $p_z$, dipole moments projected along the uniform field direction, are individually obtained. A distribution is created with 60 $p_z$. We describe the distribution using a model with two dipole moment magnitudes, and a fit yields the corresponding values $p=p_1= 2.8\pm 0.2$ Debye and $p=p_2=8.3\pm0.4$ Debye. For a strong-coupled TLS the vacuum-Rabi splitting can be obtained with $p_z$ and tunneling energy. This allows a measurement of the circuit's zero-point electric field fluctuations, in a method that does not need the electric-field volume.