Fig. 2: Design of experiments.
From: Experimental exchange of grins between quantum Cheshire cats
a Concept of weak value extraction. Top: rigorous weak measurement. The system and pointer are entangled by coupling evolution \({\mathcal{U}}=\exp (-i{\mathcal{H}}t)\) with the interaction Hamiltonian being \({\mathcal{H}}={O}_{{\rm{s}}}\otimes P\). The demon cast judicious choices of post-selection to obtain the weak value \({\left\langle {O}_{{\rm{s}}}\right\rangle }_{{w}}\). Mid: interpretation of weak value without pointer. The system's evolution \({{\mathcal{U}}}_{s}\) due to a perturbation is characterised by weak value in the weak interaction regime23. Bottom: weak value extraction based on perturbation. A linear relationship between the post-selection probability and the interaction time of ITE can be established, whose incline gives the real part of the weak value \({\left\langle O\right\rangle }_{{w}}\). b Photon source. Biphoton (central wavelength 813.4 nm), four-qubit hyperentangled state is generated via type-I spontaneous parametric down-conversion (SPDC) process by pumping a β-barium borate (BBO) crystal twice in a confocal structure24,25. CM concave mirror, QWP quarter-wave plate. c The main setup to exchange grins. Neutral density filter (ND) and polarisation-sensitive density filter (PD) implement the perturbation on the path observable Πand the conditional spin observable σz ⊗ Π, respectively. A polarisation independent BS, followed by four polarisation analysers, each composed of a set of quarter- or half-wave plates (QWP, HWP) and a PBS, implements post-selection. Four 0.2-mm-thick glass plates (GPs) are inserted in the four arms with one of them rotatable to compensate the biphoton phase. The photons are filtered by 3-nm-bandwidth interference filters (IF) with a central wavelength of 813.4 nm before collected by four single-mode fibres and guided to single-photon detectors. d A PD is constructed from two polarisation beam displacers (BDs) and an ND filter inserted in one arm.
