Quantum Fisher information and photon statistics from shift current shot noise.

Quantum Fisher information (QFI) sets the ultimate precision of optical phase measurements and can reveal multiphoton entanglement. Yet the photon-number fluctuations that encode this sensitivity are difficult to convert into an electrical signal. We theoretically predict that a photodetector utilizing the shot noise of the quantum-geometric shift current of exciton polaritons can measure photon-number statistics and, for pure states, infer the QFI. By solving the Lindblad equation, we obtain the time-dependent nonlinear photocurrent for an arbitrary initial photon state. It turns out that, regardless of the quantum state of the incident light, the integrated current depends only on the mean photon number. In contrast, the shot noise retains information about photon-number fluctuations: Its Fano factor is proportional to the photon-number variance. Numerical calculations confirm these relations for optical Schrödinger cat and squeezed-vacuum states. A photodetector based on shift current noise can therefore provide a solid-state platform for high-precision measurements of quantum-optical statistics without relying on mobile photocarriers.