Revealing inadvertent periodic modulation of qubit frequency

The paper describes the means to reveal and characterize slow periodic modulation of qubit frequency. Such modulation can come from different sources and can impact qubit stability. We show that the modulation leads to very sharp peaks in the power spectrum of outcomes of periodically repeated Ramsey measurements. The positions and shapes of the peaks allow finding both the frequency and the amplitude of the modulation. We also explore how additional slow fluctuations of the qubit frequency and fluctuations of the modulation frequency affect the spectrum. The analytical results are in excellent agreement with extensive simulations.