Quantum limit for the measurement of a classical force coupled to a noisy quantum-mechanical oscillator

Precise measurements of tiny forces and displacements play an important role in science and technology. The precision of recent experiments, while beginning to reach the limits imposed by quantum mechanics, is necessarily spoiled by the unavoidable influence of noise. Here we obtain a quantum limit for the uncertainty in the estimation of the amplitude of a classical force with arbitrary time dependence, acting on a noisy quantum-mechanical oscillator. We determine the best initial state of the oscillator and the best measurement procedure, thus getting a rigorous and useful benchmark for experiments aiming to detect extremely small forces.