Digital noise spectroscopy with a quantum sensor

Characterizing the environmental noise in quantum systems is an essential task in both fundamental physics and quantum applications. Typical protocols for quantum noise spectroscopy rely on periodic dynamical decoupling sequences that yield narrow frequency filters, and sample the noise spectrum in frequency space using the $\delta$-filter approximation. Here we propose a digital noise spectroscopy protocol based on Walsh-function dynamical decoupling. By measuring the decoherence of a qubit under a set of Walsh modulation sequences, the auto-correlation of a stationary Gaussian noise is directly reconstructed, while its discrete Fourier transform gives the corresponding noise spectrum. In comparison to previous noise spectroscopy methods, the accuracy of our method is only limited by the sampling in time-space. We then perform a proof-of-principle demonstration by using a single nitrogen-vacancy center in diamond to characterize its environmental noise dominated by the surrounding $^{13}$C nuclear spins, and discuss practical limitations of the reconstruction accuracy and avenues for its improvement.