Response of a classical mesoscopic oscillator to a two-level quantum system

We investigate the dynamics of a classical mechanical oscillator coupled to the simplest quantum system, a single qubit. Using the Feynman-Vernon influence functional formalism, we show that the qubit's influence manifests as both deterministic and stochastic forces on the oscillator. These forces are highly dependent on the qubit's initial quantum state, imprinting unique measurable signatures onto the oscillator's response. The present results provide a direct pathway to quantum state reconstruction through classical noise spectroscopy. By employing the Fisher Information Matrix, we quantify the efficacy of estimating the initial qubit state from the continuous classical record, revealing a fundamental temporal asymmetry between population and phase estimation. This framework has potential applications to mesoscopic optomechanical experiments, quantum metrology, and tabletop tests of the quantum nature of gravity.