  Industrial, metrological, and medical applications provide a strong technological pull for advanced nanoscale sensors exploiting the unique sensitivity of quantum coherent systems to their environments. Essential to the functionality of these devices is the availability of control protocols which shape the sensor's response to the environment in frequency space. However, a key challenge in these applications is that common control routines result in out-of-band spectral leakage which complicates interpretation of the sensor's signal. In this work we demonstrate provably optimal narrowband control protocols ideally suited to quantum sensing. Our results, based on experiments with trapped ions using modulation in the form of discrete prolate spheroidal sequences (aka Slepian functions), demonstrate reduction of spectral leakage by orders of magnitude over conventional controls. We tune the narrowband sensitivity using concepts from RF engineering and experimentally reconstruct complex noise spectra using engineered noise for quantitative performance evaluation. We then deploy these techniques to identify previously immeasurable frequency-resolved amplitude noise in our qubit synthesis chain with calibrated sensitivity better than 0.001 dB. 
Application of optimal band-limited control protocols to quantum noise   sensing