The coupling between microwave fields and atoms (or atom-like systems) is inherently weaker than for optical fields, making microwave signal manipulation for applications like quantum information processing technically challenging. In order to better understand this coupling and to develop tools for measuring it, we explore the microwave coupling to atoms using the atomic candle technique, and push it beyond the bounds of the small-signal regime in order to deliver a larger signal. In familiar two-level systems, responses beyond the usual Rabi oscillations can arise when a single-tone drive is phase-modulated, causing the steady-state populations to oscillate at integer harmonics of the modulation frequency. Resonant behavior of the first two harmonics for frequencies near the Rabi frequency, known as $alpha$ and $beta$ Rabi resonances, is widely used for microwave-field magnetometry and as a power standard known as the atomic candle. Here, we explore Rabi resonances beyond the small-signal approximation and report upon experimental observations of higher-harmonic population response for microwave hyperfine transitions in cold $^{87}rm Rb$ atoms, which we compare to numerical simulations.
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