Ultimate sensitivity of multiparameter estimation in quantum sensing with undetected photons

Quantum sensing with undetected photons is a technique where photons of one wavelength probe a sample, but information is extracted by measuring photons of another wavelength that never interacts with the sample. This has seen significant experimental advances in applications such as spectroscopy, microscopy, and bio-sensing. However, a detailed theoretical analysis using the tools of quantum metrology is currently lacking. Thus it is unclear how far away current schemes are from fundamental limits, and what the optimal measurement strategies are. We apply a multiparameter quantum estimation framework to quantify the error when estimating the unknown transmission and phase shift of a sample. The optimal measurement scheme is shown to require only a single controllable phase shift, easily implementable in existing setups. We also study how to use multipass interactions to maximise information gain. In general the optimum number of passes scales inversely with the log of the transmission of the sample. This work clarifies the metrological power of quantum sensing with undetected photons, and provides guidance for the design of experiments requiring high sensitivity.