Testing Hardy’s nonlocality proof with genuine energy-time entanglement

We show two experimental realizations of Hardy’s ladder test of quantum nonlocality using energy-time correlated photons, following the scheme proposed by Cabello et al.[Phys. Rev. Lett. 102, 040401 (2009)]. Unlike previous energy-time Bell experiments, these tests require precisely tailored nonmaximally entangled states. One of them is equivalent to the two-setting and two-outcome Bell test requiring a minimum detection efficiency. The reported experiments are still affected by the locality and detection loopholes, but are free of the post-selection loophole of previous energy-time and time-bin Bell tests.

Figure 1

Ladder proof schemes for $K=1$ and $K=2$.

Figure 2

Scheme needed to implement a Hardy’s test using energy-time entanglement. VBS: variable beamsplitter.

Figure 3

Experimental setup. A step-by-step translation stage allows us to create the indistinguishability condition ($L_A-S_A=L_B-S_B$). Coincidence windows are $3\mathit{ns}$ and, due to continuous wave pumping, the probability of two photon-pair events is negligible. Interference filters select a bandwidth of $3.5$ nm. The radiation is coupled into single-mode optical fibers and sent to pigtailed avalanche photo-counting modules connected to a circuit used to record the single and the coincidence counts. The phase of the interferometer can be controlled using a piezoelectric stage on which PBS2${}_A$ is assembled.

Figure 4

(a) Generation of nonmaximally entangled state and (b) projection on the measurement basis. This setup is repeated on both modes $A$ and $B$, and is used to generate a nonmaximally entangled state and project on the desired base, as described in the text.

Figure 5

Experimental results for $K=1$. These two graphs show the result obtained for the one-step ladder proof. The top graph shows the agreement between the data and the theoretical prediction (continuous line) for the probability $P(a_1,b_1)$. The bottom graph shows the obtained values of ${\mathcal{S}}_1$ compared to the theory (continuous line). The dotted line represents the classical bound: when ${\mathcal{S}}_1$ is greater than 0, the inequality (1) is violated.

Figure 6

Experimental results for $K=2$. The results obtained for a two-step ladder proof are shown. As in the previous figure, the top graph shows the probability $P(a_2,b_2)$ along with the theoretical function, while the bottom graph shows the inequality violation. When ${\mathcal{S}}_2$ is greater than 0, the inequality (5) is violated.