Detecting quantum noise of a solid-state spin ensemble with dispersive measurement

We theoretically explore protocols for measuring the spin polarization of an ensemble of solid-state spins, with precision at or below the standard quantum limit. Such measurements in the solid state are challenging, as standard approaches based on optical fluorescence are often limited by poor readout fidelity. Indirect microwave-resonator-mediated measurements provide an attractive alternative, though a full analysis of relevant sources of measurement noise is lacking. In this work, we study dispersive readout of an inhomogeneously broadened spin ensemble via coupling to a driven resonator measured via homodyne detection. We derive generic analytic conditions for when the homodyne measurement can be limited by the fundamental spin-projection noise, as opposed to microwave-drive shot noise or resonator phase noise. By studying fluctuations of the measurement record in detail, we also propose an experimental protocol for directly detecting spin squeezing, i.e., a reduction of the spin ensemble’s intrinsic projection noise from entanglement. Our protocol provides a method for benchmarking entangled states for quantum-enhanced metrology.