  We report on a time-domain study of parametric synchronization in a magnetic
tunnel junction based spin torque nano-oscillator (STNO). Time-domain
measurements of the instantaneous frequency ($f_{i}$) of a parametrically
synchronized STNO show random short-term unlocking of the STNO signal for low
injected radio-frequency (RF) power, which cannot be revealed in time-averaged
frequency domain measurements. Macrospin simulations reproduce the experimental
results and reveal that the random unlocking during synchronization is driven
by thermal fluctuations. We show that by using a high injected RF power, random
unlocking of the STNO can be avoided. However, a perfect synchronization
characterized by complete suppression of phase noise, so-called phase noise
squeezing, can be obtained only at a significantly higher RF power. Our
macrospin simulations suggest that a lower temperature and a higher positive
ratio of the field-like torque to the spin transfer torque reduce the threshold
RF power required for phase noise squeezing under parametric synchronization.
