Reconstructing weak values without weak measurements

Maurice A. de Gosson

Yang Gao, Wenqiang Luo, Qiang Yang +3 more

The wave function is important for understanding and predicting the behavior of photons. Recent studies have shown that wave function can be reconstructed through weak measurement. But currently, the relationships of wave function, weak value, and probe are single and not complete. Here, we propose a wave function reconstruction method based on complex weak values and complex shift probes. Four relationships of them are built and the wave function can be reconstructed by position and momentum shift probes. The complex weak values can be obtained by adjusting the pre- and post-selection in weak measurement, while the complex shift probes originate from the complex nature of the Fresnel coefficient in total internal reflection. To demonstrate the feasibility of the proposed scheme, an experiment of phase distribution reconstruction is achieved by a real weak value and a momentum shift probe; the experimental results agree well with the theory and show a high contrast. Therefore, the wave function can be conveniently reconstructed by our scheme and the relationships between the wave function, weak values, and probes become more complete. This work provides a valuable reference for the application of weak measurement in image processing, quantitative phase imaging, and label-free biological imaging.

Erkka Haapasalo, PEKKA J. LAHTI, Jussi Schultz

We generalize the concept of the weak value of a quantum observable to cover arbitrary real positive operator measures. We show that the definition is operationally meaningful in the sense that it can be understood within the quantum theory of sequential measurements. In particular, we show that the weak value can be obtained from a single measurement scheme. We then present a detailed analysis of the recent experiment [J. S. Lundeen et al., Nature (London) 474, 188 (2011)] concerning the reconstruction of the state of a photon using weak measurements. We compare their method with the reconstruction method through informationally complete phase space measurements. In particular, we show that unlike with phase space measurements, the reconstruction of a completely unknown state is not always possible using the method of weak measurements.

Ruth E. Kastner

Wenxian Luo, Qiang Yang, Yan Wang +2 more

Phase distribution is crucial for obtaining the whole information of phase object in imaging. However, due to the transparency of phase objects, the phase distribution cannot be obtained in traditional bright-field imaging. Here, we propose a phase distribution reconstruction based on the imaginary value of weak measurement, which is achieved by a Pancharatnam–Berry phase metasurface. Moreover, by adjusting the weak value to a real weak value, amplitude reconstruction can also be achieved. Different parts of weak values can be obtained by controlling the preselection and postselection states. The weak interaction is achieved by the metasurface, which is convenient to control the weak coupling and reduce the complexity of the experimental setup at the same time. Furthermore, the edge contour of objects can also be acquired by the weak measurement, and the edge sharpness is related to types of the weak values, which plays an important role in image resolution. Our method paves a way for quantitative phase imaging and unlabeled biological imaging.

Jeff Tollaksen, Yakir Aharonov