The implementation of weak-value amplification requires the pre-and post-selection of states of a quantum system, followed by the observation of the response of the meter, which interacts weakly with the system. Data acquisition from the meter is conditioned to successful post-selection events. Here we derive an optimal post-selection procedure for estimating the coupling constant between system and meter, and show that it leads both to weak-value amplification and to the saturation of the quantum Fisher information, under conditions fulfilled by all previously reported experiments on the amplification of weak signals. For most of the pre-selected states, full information on the coupling constant can be extracted from the meter data set alone, while for a small fraction of the space of pre-selected states, it must be obtained from the post-selection statistics.

We experimentally investigate the relative advantages of implementing weak-value-based metrology versus standard methods. While the techniques outlined herein apply more generally, we measure small optical beam deflections both using a Sagnac interferometer with a monitored dark port (the weak-value-based technique), and by focusing the entire beam to a split detector (the standard technique). By introducing controlled external transverse detector modulations and transverse beam deflection momentum modulations, we quantify the mitigation of these sources in the weak-value-based experiment versus the standard focusing experiment. The experiments are compared using a combination of deterministic and stochastic methods. In all cases, the weak-values technique performs the same or better than the standard technique by up to two orders of magnitude in precision for our parameters. We further measure the statistical efficiency of the weak-values-based technique. By postselecting on $1\%$ of the photons, we obtain $99\%$ of the available Fisher information of the beam deflection parameter

Post-selection strategies have been proposed with the aim of amplifying weak signals, which may help to overcome detection thresholds associated with technical noise in high-precision measurements. Here we use an optical setup to experimentally explore two different post-selection protocols for the estimation of a small parameter: a weak-value amplification procedure and an alternative method, that does not provide amplification, but nonetheless is shown to be more robust for the sake of parameter estimation. Each technique leads approximately to the saturation of quantum limits for the estimation precision, expressed by the Cramér-Rao bound. For both situations, we show that information on the parameter is obtained jointly from the measuring device and the post-selection statistics.

We investigate the transverse mode structure of the down-converted beams generated by a type-II optical parametric oscillator (OPO) driven by a structured pump. Our analysis focus on the selection rules imposed by the spatial overlap between the transverse modes of the three fields involved in the non-linear interaction. These rules imply a hierarchy of oscillation thresholds that determine the possible transverse modes generated by the OPO, as remarkably confirmed with experimental results.

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