  The preparation of mechanical quadrature-squeezed states holds significant
importance in cavity optomechanics because the squeezed states have extensive
applications in understanding fundamental quantum mechanics and exploiting
modern quantum technonogy. Here, we propose a reliable scheme for generating
mechanical quadrature squeezing in a typical cavity optomechanical system via
seeking for optimal cavity-field driving pulses using the gradient-descent
algorithm. We realize strong quadrature squeezing in a mechanical resonator
that exceeds the 3 dB steady-state limit, even with a thermal phonon occupancy
of one hundred. Furthermore, the mechanical squeezing can be ultrafastly
created within one mechanical oscillation period. We also obtain the optimal
pulsed drivings associated with the created mechanical squeezings and analyze
the mechanism for mechanical squeezing generation. This work will promote the
application of optimal quantum control in quantum optics and quantum
information science.
