Simultaneous Spectral Estimation of Dephasing and Amplitude Noise on a Qubit Sensor via Optimally Band-Limited Control

The fragility of quantum systems makes them ideally suited for sensing applications at the nanoscale. However, interpreting the output signal of a qubit-based sensor is generally complicated by background clutter due to out-of-band spectral leakage, as well as ambiguity in signal origin when the sensor is operated with noisy hardware. We present a sensing protocol based on optimally band-limited ``Slepian functions'' that can overcome these challenges, by providing narrowband sensing of ambient dephasing noise, coupling additively to the sensor along the $z$ axis, while permitting isolation of the target noise spectrum from other contributions coupling along a different axis. This is achieved by introducing a finite-difference control modulation, which linearizes the sensor's response and affords tunable band-limited ``windowing'' in frequency. Building on these techniques, we experimentally demonstrate two spectral estimation capabilities using a trapped-ion qubit sensor. We first perform efficient experimental reconstruction of a ``mixed'' dephasing spectrum, composed of a broadband $1/f$-type spectrum with discrete spurs. We then demonstrate the simultaneous reconstruction of overlapping dephasing and control noise spectra from a single set of measurements, in a setting where the two noise sources contribute equally to the sensor's response. Our approach provides a direct means to augment quantum-sensor performance in the presence of both complex broadband noise environments and imperfect control signals, by optimally complying with realistic time-bandwidth constraints.