  This work introduces the so-called synchrosqueezed wavelet transform, to shed
light on the dipolar fluctuations of a thermal ensemble of nuclear spins in a
diamond crystal structure, hyperfine-coupled to a central spin. The raw time
series of the nuclear spin bath coherent dynamics is acquired through the
two-point correlation function computed using the cluster correlation expansion
method. The dynamics can be conveniently analyzed according to zero-, single-,
and double-quantum transitions derived from the dipolar pairwise spin flips. We
show that in the early-time behavior when the coherence is preserved in the
spin ensemble, the Overhauser field fluctuations are modulated by
dipole-dipole-induced small inhomogeneous detunings of nearly resonant
transitions within the bath. The resulting beating extending over relatively
longer time intervals is featured on the scalograms where both temporal and
spectral behaviors of nuclear spin noise are unveiled simultaneously. Moreover,
a second kind of beating that affects faster dynamics is readily discernible,
originating from the inhomogeneous spread of the hyperfine coupling of each
nucleus with the central spin. Additionally, any quadrupolar nuclei within the
bath imprint as beating residing in the zero-quantum channel. The nuclear spin
environment can be directionally probed by orienting the hyperfine axis.
Thereby, crucial spatial information about the closely separated spin clusters
surrounding the central spin are accessible. Thus, a wavelet-based post
processing can facilitate the identification of proximal nuclear spins as
revealed by their unique beating patterns on the scalograms. Finally, when
these features are overwhelmed by either weakly or strongly coupled classical
noise sources, we demonstrate the efficacy of thresholding techniques in the
wavelet domain in denoising contaminated scalograms.
