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The fragility of quantum systems makes them ideally suited for sensingapplications at the nanoscale. However, interpreting the output signal of aqubit-based sensor is generally complicated by background clutter due toout-of-band spectral leakage, as well as ambiguity in signal origin when thesensor is operated with noisy hardware. We present a sensing protocol based onoptimally band-limited "Slepian functions" that can overcome these challenges,by providing narrowband sensing of ambient dephasing noise, coupling additivelyto the sensor along the ${z}$-axis, while permitting isolation of the targetnoise spectrum from other contributions coupling along a different axis. Thisis achieved by introducing a finite-difference control modulation, whichlinearizes the sensor's response and affords tunable band-limited "windowing"in frequency. Building on these techniques, we experimentally demonstrate twospectral estimation capabilities using a trapped-ion qubit sensor. We firstperform efficient experimental reconstruction of a "mixed" dephasing spectrum,composed of a broadband $1/f$-type spectrum with discrete spurs. We thendemonstrate the simultaneous reconstruction of overlapping dephasing andcontrol noise spectra from a single set of measurements, in a setting where thetwo noise sources contribute equally to the sensor's response. Our approachprovides a direct means to augment quantum-sensor performance in the presenceof both complex broadband noise environments and imperfect control signals, byoptimally complying with realistic time-bandwidth constraints.

Simultaneous spectral estimation of dephasing and amplitude noise on a qubit sensor via optimally band-limited control