In a recent Letter, Brunner and Simon proposed an interferometric scheme using imaginary weak values with a frequency-domain analysis to outperform standard interferometry in longitudinal phase shifts [Phys. Rev. Lett105, 010405 (2010)]. Here we demonstrate an interferometric scheme combined with a time-domain analysis to measure longitudinal velocities. The technique employs the near-destructive interference of non-Fourier limited pulses, one Doppler shifted due to a moving mirror in a Michelson interferometer. We achieve a velocity measurement of 400 fm/s and show our estimator to be efficient by reaching its Cramér-Rao bound.

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

We present a parameter estimation technique based on performing joint measurements of a weak interaction away from the weak-value-amplification approximation. Two detectors are used to collect full statistics of the correlations between two weakly entangled degrees of freedom. Without discarding of data, the protocol resembles the anomalous amplification of an imaginary-weak-value-like response. The amplification is induced in the difference signal of both detectors allowing robustness to different sources of technical noise, and offering in addition the advantages of balanced signals for precision metrology. All of the Fisher information about the parameter of interest is collected. A tunable phase controls the strength of the amplification response. We experimentally demonstrate the proposed technique by measuring polarization rotations in a linearly polarized laser pulse. We show that in the presence of technical noise the effective sensitivity and precision of a split detector is increased when compared to a conventional continuous-wave balanced detection technique.Introduction Anomalous amplification [1] has been shown to be advantageous for precision metrology. Such an amplification provides a way to increase a signal while decreasing [2] or retaining the technical-noise floor [3,4]. As a result, the sensitivity and precision of measurements limited by technical noise can be effectively improved, facilitating the saturation of the standard quantum limit. Anomalous amplification was first proposed for metrology with the introduction of the Weak Value (WV) of an observable [1,5], and parameter estimation protocols defined after it are usually known as Weak-ValueAmplification (WVA) techniques. The WV of an observable is obtained by post-selecting the state of a system after a weak interaction with a meter system. In WVA, such measurements in the system induce a discarding of data counts in the measurements of the meter. In addition to the notion that the state of the system is post-selected after the weak interaction, we consider post-selection as the process of selecting and processing desired events, which, for WVA, results in discarding data in the meter. Due to the interference of the pre-and post-selection states of the system the WV can take large complex values outside the eigenvalue spectrum of the observable, which defines the anomalous amplification in WVA. Discussion about the quantum interpretation of such a phenomenon can be found in Refs. [6][7][8][9]. Many recent applications of WVA for metrology have been done in classical optics, where the interference can be understood using standard wave mechanics [10,11].

The technique of almost-balanced weak values amplification (ABWV) was recently proposed [Phys. Rev. Lett. 116: 100803 (2016)]. We demonstrate this technique using a modified Sagnac interferometer, where the counter-propagating beams are spatially separated. The separation between the two beams provides additional amplification, with respect to using colinear beams in a Sagnac interferometer. As a demonstration of the technique, we perform measurements of the angular velocity in one of the mirrors of the interferometer. Within the same setup, the weak-value amplification technique is also performed for comparison. Much higher amplification factors can be obtained using the almost-balanced weak values technique, with the best one achieved in our experiments being as high as 1.2 × 10 7 . In addition, the amplification factor monotonically increases with decreasing post-selection phase for the ABWV case in our experiments, which is not the case for weak-value amplification at small post-selection phases.

We present an interferometric technique for measuring ultrasmall tilts. The information of a tilt in one of the mirrors of a modified Sagnac interferometer is carried by the phase difference between the counter-propagating laser beams. Using a small misalignment of the interferometer, orthogonal to the plane of the tilt, a bimodal (or two-fringe) pattern is induced in the beam's transverse power distribution. By tracking the mean of such a distribution, using a split detector, a sensitive measurement of the phase is performed. With 1.2 mW of continuous-wave laser power, the technique has a shot noise limited sensitivity of 56  frad/Hz and a measured noise floor of 200  frad/Hz for tilt frequencies above 2 Hz. A tilt of 200 frad corresponds to a differential displacement of 4.0 fm in our setup. The novelty of the protocol relies on signal amplification due to the misalignment and on good performance at low frequencies. A noise floor of about 70  prad/Hz is observed between 2 and 100 mHz.