Phase-Sensitive Heterodyne Detection of MW using EIT Harmonics in Rydberg Atoms

We investigate the generation and characterization of higher-order harmonics in the probe-laser absorption arising from nonlinear interactions in an electromagnetically induced transparency (EIT) ladder system involving Rydberg states and driven by two microwave fields in the heterodyne configuration. We characterize the amplitude and phase of the generated harmonics as a function of the relative frequency and phase of the applied microwave fields. The phase of the $n^{\mathrm{t}\mathrm{h}}$ harmonic follows the relation ${\phi }_n=n\phi$, demonstrating phase multiplication and suggesting that higher-order harmonics may offer an enhanced phase response for phase-sensitive measurements. We further characterize the harmonic amplitudes and bandwidths and find that the measured bandwidths are significantly larger than the intrinsic Rydberg-state linewidth, consistent with power broadening under the experimental conditions. The experimental observations are supported by density-matrix calculations, which reproduce the key features of the measured harmonic response.