  Quantum states are important resources and their preparations are essential
prerequisites to all quantum technologies. However, they are extremely fragile
due to the inevitable dissipations. Here, an all-optical generation of a
deterministic squeezed Schr$\ddot{\mathrm{o}}$dinger-cat state based on
dissipation is proposed. Our system is based on the Fredkin-type interaction
between three optical modes, one of which is subject to coherent two-photon
driving and the rest are coherent driving. We show that an effective degenerate
three-wave mixing process can be engineered in our system, which can cause the
simultaneous loss of two photons, resulting in the generation of a
deterministic squeezed Schr$\ddot{\mathrm{o}}$dinger-cat state. More
importantly, by controlling the driving fields in our system, the two-photon
loss can be adjustable, which can accelerate the generation of squeezed
Schr$\ddot{\mathrm{o}}$dinger-cat states. Besides, we exploit the squeezed
Schr$\ddot{\mathrm{o}}$dinger-cat states to estimate the phase in the optical
interferometer, and show that the quantum Fisher information about the phase
can reach the Heisenberg limit in the limit of a large photon number.
Meanwhile, it can have an order of magnitude factor improvement over the
Heisenberg limit in the low-photon-number regime, which is very valuable for
fragile systems that cannot withstand large photon fluxes. This work proposes
an all-optical scheme to deterministically prepare the squeezed
Schr$\ddot{\mathrm{o}}$dinger-cat state with high speed and can also be
generalized to other physical platforms.
