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We propose a quantum mechanical method of constraining non-Newtonian gravityat the nanometer range. In this method, a hybrid electro-optomechanical systemis employed. Applying a strong driving field, we can obtain normal modesplitting of the electromechanical subsystem which is related to the resonancefrequency of the mechanical oscillator. Moreover, we investigate therelationship between the variance of normal mode splitting and the resonancefrequency shift induced by the gradient of exotic forces provided that oursystem is operated at critical points. Furthermore, via suppressing the Casimirbackground, we set a constraint on the non-Newtonian gravity which improves theprevious bounds by about a factor of 7 at 1 nanometer range. Our resultsindicate that our method could be put into consideration in relevantexperimental searches.

Proposal for constraining non-Newtonian gravity at nm range via criticality enhanced measurement of resonance frequency shift