  Control over nuclear spin fluctuations is essential for processes that rely
on preserving the quantum state of an embedded system. For this purpose,
squeezing is a viable alternative, so far that has not been properly exploited
for the nuclear spins. Of particular relevance in solids is the electric
quadrupole interaction (QI), which operates on nuclei having spin higher than
1/2. In its general form, QI involves an electric field gradient (EFG)
biaxiality term. Here, we show that as this EFG biaxiality increases, it
enables continuous tuning of single-particle squeezing from the one-axis
twisting to the two-axis countertwisting limits. A detailed analysis of QI
squeezing is provided, exhibiting the intricate consequences of EFG biaxiality.
The initial states over the Bloch sphere are mapped out to identify those
favorable for fast initial squeezing, or for prolonged squeezings. Furthermore,
the evolution of squeezing in the presence of a phase-damping channel and an
external magnetic field are investigated. We observe that dephasing drives
toward an anti-squeezed terminal state, the degree of which increases with the
spin angular momentum. Finally, QI squeezing in the limiting case of a
two-dimensional EFG with a perpendicular magnetic field is discussed, which is
of importance for two-dimensional materials, and the associated beat patterns
in squeezing are revealed.
