We implemented a joint weak measurement of the trajectories of two photons in a photonic version of Hardy's experiment. The joint weak measurement has been performed via an entangled meter state in polarization degrees of freedom of the two photons. Unlike Hardy's original argument in which the contradiction is inferred by retrodiction, our experiment reveals its paradoxical nature as preposterous values actually read out from the meter. Such a direct observation of a paradox gives us new insights into the spooky action of quantum mechanics.

We consider a case when a weak value is introduced as a physical quantity rather than as an average of weak measurements. The case we treat is a time evolution of a particle by the 1 + 1-dimensional Dirac equation. In particular, in a spontaneous pair production via a supercritical step potential, a quantitative explanation can be given by a weak value for the group velocity of the particle. We also show that the condition for the pair production (supercriticality) corresponds to the condition when the weak value takes a strange value (superluminal velocity). 10

We propose a model for the electric current in graphene in which electric carriers are supplied by virtual particles allowed by the uncertainty relations. The process to make a virtual particle real is described by a weak value of a group velocity: the velocity is requisite for the electric field to give the virtual particle the appropriate changes of both energy and momentum. With the weak value, we approximately estimate the electric current, considering the ballistic transport of the electric carriers. The current shows the quasi-Ohmic with the minimal conductivity of the order of e h / 2 per channel. Crossing a certain ballistic time scale, it is brought to obey the Schwinger mechanism.

When the weak value of a projector is 1, a quantum system behaves as in that eigenstate with probability 1. By definition, however, the weak value may take an anomalous value lying outside the range of probability like −1. From the viewpoint of a physical effect, we show that such a negative weak value of −1 can be regarded as the counterpart of the ordinary value of 1. Using photons, we experimentally verify it as the symmetrical shift in polarization depending on the weak value given by pre-postselection of the path state. Unlike observation of a weak value as an ensemble average via weak measurements, the effect of a weak value is definitely confirmed in Hong-Ou-Mandel effect: the symmetrical shift corresponding to the weak value can be directly observed as the rotation angle of a half wave plate.

The anisotropy of molecular polarizability Da of several nematogens has been determined by depolarized Rayleigh light scattering. The experimentally determined values were found to be noticeably smaller than those obtained by MOPAC calculation. Da values determined from order parameter S and refractive indices show reasonably good agreement with those determined in the present experiment. This result warns us that Da calculated by MOPAC, particularly at a finite wavelength, may be larger than the true value.