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We study the effect of external mechanical pumps on the second-order sideband generation (SSG) efficiency in a double-cavity optomechanical system. The SSG efficiency can be tuned by changing the amplitude and phase of the mechanical pumps. When external mechanical pumps are applied to both mechanical resonators, the influence on the SSG efficiency is a superposition of the two mechanical pumps driven separately. The SSG efficiency can be increased up to three times with the application of mechanical pumps compared to the case without external mechanical pumps. The present investigation proposes a method that can be applied to more cavity cascades to achieve more desirable SSG efficiency and optimize nonlinear optical properties.

Modulation of the second-order sideband efficiency of a double-cavity optomechanical system by mechanical pumps