Anomalous weak values and the violation of a multiple-measurement Leggett-Garg inequality

Quantum mechanics presents peculiar properties that, on the one hand, have been the subject of several theoretical and experimental studies about its very foundations and, on the other hand, provide tools for developing new technologies, the so-called quantum technologies. The nonclassicality pointed out by Leggett-Garg inequalities has represented, with Bell inequalities, one of the most investigated subjects. In this article we study the connection of Leggett-Garg inequalities with a new emerging field of quantum measurement, the weak values in the case of a series of sequential measurements on a single object. In detail, we perform an experimental study of the four-time-correlator Leggett-Garg test, by exploiting single and sequential weak measurements performed on heralded single photons.

Figure 1

Experimental setup. Heralded single photons are produced by downconversion in a 5-mm-long ${\mathrm{LiIO}}_3$ nonlinear crystal (NLC) cut for type-I phase matching; the pump beam, obtained by second harmonic generation (SHG) of a mode-locked laser (rep. rate, 76 MHz), produces idler (${\lambda }_i=920$ nm) and signal (${\lambda }_s=702$ nm) photons, which are then coupled in a single-mode fiber (SMF). The idler photons are detected by means of a single-photon avalanche diode (SPAD), which imparts a trigger to the signal detection system. Signal photons are prepared in the polarization state $|{\psi }_A\rangle$ by means of a polarizing beam splitter (PBS) and a half-wave plate (HWP) ($I_A=1$); then they pass through a birefringent system $B{C}_y$ that shifts them in the transverse $y$ direction, depending on their polarization, thus measuring $I_C$ weakly. $B{C}_y$ consists of two birefringent crystals: the first one realizes the weak interaction, while the second one compensates temporal walk-off and decoherence effects. A similar system, $B{C}_x$, performs in cascade a weak measurement of $I_B$ by shifting the photons in the $x$ direction; this is placed after a HWP allowing one to measure along an arbitrary linear polarization axis. A further HWP is used to counter the basis change and decide the observable $I_D$ which determines the postselection. The photons are finally detected by means of a spatial-resolving $32\times 32$ SPAD array.

Figure 2

Behavior of the quantity ${\mathcal{B}}_4$ in Eq. (7) vs the parameters $\alpha$ (related to the state $|{\psi }_A\rangle$) and $\delta$ (determining the states $|{\psi }_D\rangle$ and $|{\psi }_D^{\perp }\rangle$), both in $\pi$ units, for four different values of the parameter $\gamma$ defining the polarization operator $I_B$: $\gamma =0.1\pi$ for plot (a), $\gamma =0.4\pi$ for plot (b), $\gamma =0.5\pi$ for plot (c), and $\gamma =0.95\pi$ for plot (d). In each of these plots, the yellow part of the surface indicates the nonviolation area ($-2\le {\mathcal{B}}_4\le 2$), while in orange and magenta are highlighted respectively the positive (${\mathcal{B}}_4>2$) and negative (${\mathcal{B}}_4<-2$) violation areas. In each plot, the blue arrow indicates the point for which the violation was experimentally checked.