Optimal protocol to increase the lifetime of a harmonic oscillator via a squeezing reservoir

Reservoir engineering makes it possible to adjust the dissipation of quantum systems by driving the reservoir. This inspires investigations on how to slow a system's damping via squeezing reservoirs. In this paper we characterize damping by the quantum reliability of quantum states. We analyze the reliability loss of the harmonic oscillators in a squeezed thermal reservoir, where the quantum states under consideration have garnered significant interest in recent experiments, such as the coherent state, the squeezed coherent state, and the cat state. We explore the optimal protocol to extend the lifetime of these quantum states by adjusting the squeezed parameters of the squeezed thermal reservoir in both the high-damping and low-damping regimes. In the low-damping regime, our findings suggest that the unique optimal protocol follows the phase-matching condition and is insensitive to the reservoir's temperature. We also evaluate the optimal condition for the large damping regime. Furthermore, we show that when the state parameters satisfy a specific condition, there exists an interval for the damping rate, within which a bifurcation of the optimal protocol occurs.