  Nuclear magnetic resonance gyroscopes that detect rotation as a shift in the
precession frequency of nuclear spins, have attracted a lot of attentions.
Under a feedback-generated drive, the precession frequency is supposed to be
dependent only on the angular momentum and an applied magnetic field. However,
nuclei with spins larger than 1/2, experience electric quadrupole interaction
with electric field gradients at the cell walls. This quadrupole interaction
shifts the precession frequencies of the nuclear spins, which brings inaccuracy
to the rotation measurement as the quadrupole interaction constant $C_q$ is
difficult to precisely measure. In this work, the effects of quadrupole
interaction on nuclear magnetic resonance gyroscopes is theoretically studied.
We find that, when the constant $C_q$ is small compared to the characteristic
decay rate of the system, as the strength of the feedback-driving field
increases, the quadrupole shift monotonically decreases, regardless of the sign
of $C_q$. In large $C_q$ regime, more than one precession frequency exists, and
the nuclear spins may precess with a single frequency or multi-frequencies
depending on initial conditions. In this regime, with large driving amplitudes,
the nuclear spin can restore the single-frequency-precession. These results are
obtained by solving an effective master equation of the nuclear spins in a
rotating frame, from which both the steady-state solutions and dynamics of the
system are shown.
