Experimental violation of a Bell inequality with two different degrees of freedom of entangled particle pairs

We demonstrate hybrid entanglement of photon pairs via the experimental violation of a Bell inequality with two different degrees of freedom (DOF), namely, the path (linear momentum) of one photon and the polarization of the other photon. Hybrid entangled photon pairs are created by spontaneous parametric down conversion and coherent polarization to path conversion for one photon. For that photon, path superposition is analyzed, and polarization superposition for its twin photon. The correlations between these two measurements give an $S$ parameter of $S=2.653\pm 0.027$ in a Clauser-Horne-Shimony-Holt inequality and thus violate local realism for two different DOF by more than 24 standard deviations. This experimentally supports the idea that entanglement is a fundamental concept which is indifferent to the specific physical realization of Hilbert space.

Figure 1

(Color online) Experimental Setup. Polarization-entangled photon pairs are generated in an EPR-Bell state via spontaneous parametric down conversion (SPDC). A picosecond-pulsed Nd:vanadate laser emitting light at the wavelength of 355 nm after frequency tripling (repetition rate of 76 MHz and average power of 200 mW) pumps a $\beta$-barium borate $\left(\beta \text{-BBO}\right)$ crystal in a cross rings type-II scheme of SPDC [24]. Good spectrum and spatial mode overlapping is achieved by using interference filters with 1 nm bandwidth centered around 710 nm and by collecting the entangled photon, pairs into single-mode fibers [25]. In order to create the hybrid-entangled state (1), the source also consists of a polarizing beam splitter, two in-line polarization controllers (in-PCs) and an additional linear polarizer (Pol1) [see main text for details]. The photon in spatial mode A is directed toward the interferometric path measurement setup. We combine both paths on a single-mode fiber beam splitter and the length of the whole interferometer is about 2 m. The phase scanner is realized via position change $\left(x\right)$ of the PBS. The photon in spatial mode B is directed toward the polarization measurement setup. It consists of a quarter-wave plate (QWP2) and a linear polarizer (Pol2) with the transmission axis oriented along angle $\varphi$, which together allow to project photon B into the desired polarization states. Both photon A and photon B are detected by multimode fiber coupled silicon avalanche photodiodes (Det 1, 2, 3). Photon A is analyzed in the superposition of the two path states along ${\alpha }_1$, ${\alpha }_2$, and their orthogonal directions on its Bloch sphere shown in the inset (a). Photon B is analyzed in the superposition of the polarization states along directions ${\beta }_1$, ${\beta }_2$, and their orthogonal directions on its Bloch sphere shown in the inset (b).

Figure 2

(Color online) Experimental results. (a) The single counts of Det1 (red square dots) and Det2 (black circular dots) are fitted with sinusoidal curves (red dash and black solid lines for Det1 and Det2, respectively) at the beginning and the end in order to calibrate the local phase of the Mach-Zehnder interferometer. (b) The coincidence counts between Det1 and Det3 (green square dots) and Det2 with Det3 (blue circular dots) and the corresponding sinusoidal fits (green dash and blue solid lines, respectively). They are used to construct the correlation coefficients in order to violate the Bell inequality. Alternating color (gray) shadings are designating the different settings of Pol2. The actions of removing and reinserting Pol1 are identified with arrows. The error bars are the square roots of the corresponding counts.

Figure 3

Four correlation functions of the CHSH inequality for four different settings. Operationally, the setting on photon A side is given by the position of the phase scanner $x$ and on photon B side it is the orientation of the polarizer $\varphi$. This manifests the hybrid nature of the entangled photon pair. The error bars represent its statistical errors.