Realization of Hardy’s Thought Experiment with Photons

We present an experimental realization of Hardy’s thought experiment [Phys. Rev. Lett. 68, 2981 (1992)], using photons. The experiment consists of a pair of Mach-Zehnder interferometers that interact through photon bunching at a beam splitter. A striking contradiction is created between the predictions of quantum mechanics and local hidden variables. The contradiction relies on nonmaximally entangled position states of two particles. A Clauser-Horne–type inequality is derived and violated.

Figure 1

Setup for Hardy’s original thought experiment: electrons and positrons enter two interferometers (one on the left, unshaded, and one on the right, shaded) simultaneously. If both particles take arms $w$, they annihilate.

Figure 2

Setup for the implementation of Hardy’s thought experiment using photons. The two interferometers (left: unshaded; right: shaded) share a central beam splitter, where indistinguishable photons taking arms $w^{+}$ and $w^{-}$ bunch. The outermost beam splitters balance the losses from one interferometer to the other. Postselection of the cases in which one photon emerges from each interferometer leads to a state identical to that of Hardy’s thought experiment (Fig. 1).

Figure 3

Top: Interference fringes that characterize the output of each interferometer at the time of the experiment. The probabilities that a photon emerges in the $d$ arm deduced from the fits are given in the plots. Bottom: $N(u^{+}{u}^{-})$ as a function of delay $\Delta$. The probability that the photons are distinguishable $p(\mathrm{disting}.)$ at zero delay is given by $p(\mathrm{disting}.)=N(u^{+}{u}^{-}{)}_{\Delta =0}/N(u^{+}{u}^{-}{)}_{\Delta \gg 0}=8\%$.

Figure 4

The three measurements on the left $N(u^{+}{u}^{-})$, $N(d^{+}{v}^{-})$, and $N(v^{+}{d}^{-})$ lead to an LHV bound on $N(d^{+}{d}^{-})$ [Eq. (5)], shown in the shaded area. The measured $N(d^{+}{d}^{-})$ violates this by 12 standard deviations.