  Higher-order exceptional points (EPs) in non-Hermitian systems have attracted
great interest due to their advantages in sensitive enhancement and distinct
topological features. However, realization of such EPs is still challenged
because more fine-tuning parameters is generically required in quantum systems,
compared to the second-order EP (EP2). Here, we propose a non-Hermitian
three-mode optomechanical system in the blue-sideband regime for predicting the
third-order EP (EP3). By deriving the pseudo-Hermitian condition for the
proposed system, one cavity with loss and the other one with gain must be
required. Then we show EP3 or EP2 can be observed when the mechanical resonator
(MR) is neutral, loss or gain. For the neutral MR, we find both two degenerate
or two non-degenerate EP3s can be predicted by tuning system parameters in the
parameter space, while four non-degenerate EP2s can be observed when the system
parameters derivate from EP3s, which is distinguished from the previous study
in the red-detuned optomechanical system. For the gain (loss) MR, we find only
two degenerate EP3s or EP2s can be predicted by tuning enhanced coupling
strength. Our proposal provides a potential way to predict higher-order EPs or
multiple EP2s and study multimode quantum squeezing around EPs using the
blue-detuned non-Hermitian optomechanical systems.
