Optimal conditions for the Bell test using spontaneous parametric down-conversion sources

We theoretically and experimentally investigate the optimal conditions for the Bell experiment using spontaneous parametric down-conversion (SPDC) sources. In theory, we show that a relatively large average photon number (typically $\sim 0.5$) is desirable to observe the maximum violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. Moreover, we show that the violation converges to 0.27 in the limit of large average photon number. In experiment, we perform the Bell experiment without postselection using polarization-entangled photon pairs at 1550-nm telecommunication wavelength generated from SPDC sources. While the violation of the CHSH inequality is not directly observed due to the overall detection efficiencies of our system, the experimental values agree well with those obtained by the theory with experimental imperfections. Furthermore, in the range of small average photon numbers ($\le 0.1$), we propose and demonstrate a method to estimate the ideal CHSH value intrinsically contained in the tested state from the lossy experimental data without assuming the input quantum state.

Figure 1

(a) The schematic diagram for the Bell experiment. Alice and Bob share a pair of particles and choose the measurement settings $X_i\in \{X_1,X_2\}$ and $Y_j\in \{Y_1,Y_2\}$, respectively. The measurement outcomes are binary, i.e., $a_i,b_j\in \{-1,+1\}$. (b) The realistic model for the Bell experiment. An entangled photon pair is generated by means of a pair of two-mode squeezed vacua (TMSV) over polarization modes. The polarization measurement is realized by the polarization mixing followed by the on-off type, single-photon detectors with dark counts.

Figure 2

(a) Log-linear plot of the average photon number $\lambda$ vs $S$. For each point we fix $\lambda$ and optimize the other parameters. (b) The overall detection efficiency $\eta$ vs $S$ for the three different ranges of $\lambda$. For each $\eta$ we fix the ranges of $\lambda$ and perform optimizations.

Figure 3

(a) The overall detection efficiency $\eta$ vs the optimal average photon numbers (${\lambda }_1$ and ${\lambda }_2$). (b) $\eta$ vs ${\lambda }_1/{\lambda }_2$. (c) $\eta$ vs the optimal angles of the polarizers. In the simulations, we assume that $\eta :={\eta }_1={\eta }_2={\eta }_3={\eta }_4$ and $\nu =0$.

Figure 4

The setup for the Bell experiment. To generate entangled photon pairs by SPDC, we used counterpropagating pump pulses to excite the PPKTP crystal in the Sagnac loop interferometer. Alice and Bob choose the measurement angles $\{{\theta }_{A1},{\theta }_{A2}\}$ and $\{{\theta }_{B1},{\theta }_{B2}\}$, respectively, and assign +1 or −1 for each detection event to calculate the $S$ value. DFB: distributed feedback laser, EDFA: erbium-doped fiber amplifier, PPLN/W: periodically poled lithium niobate waveguide, PPKTP: periodically poled potassium titanyl phosphate, QWP: quarter-waveplate, HWP: half-waveplate, DM: dichroic mirror, PBS: polarization beamsplitter, FPBS: fiber-based PBS.

Figure 5

(a) The $S$ values obtained by the theory with unity detection efficiencies (red square), the theory with experimental parameters (yellow circle), and the experimental results (blue triangle) for the various values of $\lambda$. The black diamonds represent the $S$ values obtained by compensating the losses of the system in the range of $\lambda \le 0.1$. (b) The enlarged figure of the enclosed part.