  Quantum squeezing is an important resource in modern quantum technologies,
such as quantum precision measurement and continuous-variable quantum
information processing. The generation of squeezed states of mechanical modes
is a significant task in cavity optomechanics. Motivated by recent interest in
multimode optomechanics, it becomes an interesting topic to create quadrature
squeezing in multiple mechanical resonators. However, in the
multiple-degenerate-mechanical-mode optomechanical systems, the dark-mode
effect strongly suppresses the quantum effects in mechanical modes. Here we
study the generation of mechanical squeezing in a two-mechanical-mode
optomechanical system by breaking the dark-mode effect with the
synthetic-gauge-field method. We find that when the mechanical modes work at a
finite temperature, the mechanical squeezing is weak or even disappeared due to
the dark-mode effect, while the strong mechanical squeezing can be generated
once the dark-mode effect is broken. In particular, the
thermal-phonon-occupation tolerance of the mechanical squeezing is
approximately three orders of magnitude larger than that without breaking the
dark-mode effect. We also generalize this method to break the dark modes and to
create the mechanical squeezing in a multiple-mechanical-mode optomechanical
system. Our results describe a general physical mechanism and pave the way
towards the generation of noise-resistant quantum resources.
