  We explore the utility of quantum spin squeezing in quantum magnetometry,
focusing on three-level (qutrit) Nitrogen-Vacancy (NV) centers within diamond,
utilizing a standard Ramsey interferometry pulse protocol. Our investigation
incorporates the effects of dephasing and relaxation on NV centers' dynamics
during Ramsey measurements, modeled via the Lindblad quantum master equation.
We conduct a comparative analysis between the metrological capabilities of a
single NV center and a pair of NV centers, considering Quantum Fisher
Information both with and without spin squeezing. The quantum correlations
between NV centers are assessed through the evaluation of the Kitagawa-Ueda
spin squeezing parameter within a two-level manifold. Additionally, parallel
calculations are conducted using a two-level model (qubit) for NV centers. Our
findings reveal that leveraging qutrits and spin squeezing yields enhanced
magnetometric precision, albeit constrained by dephasing effects. Nevertheless,
even in the absence of dynamical decoupling methods to mitigate environmental
noise, strategic timing of squeezing and free evolution can sustain the
advantages of qutrit-based magnetometry.
