Generation of tunable entanglement and violation of a Bell-like inequality between different degrees of freedom of a single photon

We demonstrate a scheme to generate noncoherent and coherent correlations, i.e., a tunable degree of entanglement, between degrees of freedom of a single photon. Its nature is analogous to the tuning of the purity (first-order coherence) of a single photon forming part of a two-photon state by tailoring the correlations between the paired photons. Therefore, well-known tools such as the Clauser-Horne-Shimony-Holt (CHSH) Bell-like inequality can also be used to characterize entanglement between degrees of freedom. More specifically, CHSH inequality tests are performed, making use of the polarization and the spatial shape of a single photon. The four modes required are two polarization modes and two spatial modes with different orbital angular momentum.

Figure 1

Experimental setup scheme. Laser: Mira 900 (Coherent). Optical system: second harmonic generation (Inspire Blue, Radiantis), spatial filter, linear attenuator, three dichroic mirrors (DM), and short pass filter. ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$: Fourier lenses. BBO: nonlinear crystal. Filtering system: long-pass and band-pass filters. DL: delay line. BS: beam splitter (50:50). ${\mathrm{HWP}}_1$, ${\mathrm{HWP}}_2$, and ${\mathrm{HWP}}_3$: half-wave plates. PBS: polarization beam splitter. ${\mathrm{GT}}_1$ and ${\mathrm{GT}}_2$: Glan-Thompson polarizers. ${\mathrm{QWP}}_1$, ${\mathrm{QWP}}_2$, and ${\mathrm{QWP}}_3$: quarter-wave plates. ${\mathrm{QP}}_1$ and ${\mathrm{QP}}_2$: q plates. ${\mathrm{Det}}_1$ and ${\mathrm{Det}}_2$: single-photon counting modules. C.C.: coincidence-counting electronics.

Figure 2

Coincidence and singles detections as a function of the temporal delay $\tau$ in a Hong-Ou-Mandel (HOM) interferometer. The raw data of the coincidences measured in 10 s are plotted in panel (a), and the singles detected for each detector are shown in panel (b). Closed diamonds (upper curve) correspond to singles detected with detector 1, and closed squares (lower curve) correspond to measurements in detector 2. The compensated and normalized number of coincidences is plotted in panel (c), using the coincidence data of panel (a) and the singles detected with detector 1 shown in panel (b).

Figure 3

Normalized value of the coincidences as a function of the projection angle ${\beta }_2$. Panels (a) and (c): the angle of ${\mathrm{HWP}}_2$ is set to ${\beta }_1=0^{\circ }$; panels (b) and (d): the angle is set to ${\beta }_1={45}^{\circ }$. Curves corresponding to experimental values are shown with error bars. Solid lines are theoretical predictions. Open circles, $|{\Psi }^{-}\rangle$; closed circles, $|{\Psi }^{+}\rangle$; open squares, $|{\Phi }^{-}\rangle$; and closed squares, $|{\Phi }^{+}\rangle$.

Figure 4

Value of the parameter $S$ in a CHSH inequality as a function of the angle $\theta =b_1-a_1$. The colored symbols with error bars represent the experimental data with their standard deviations. The solid colored curves are the theoretical predictions assuming a visibility factor of $V=0.92$. Open circles, $\epsilon =1$; closed squares, $\epsilon =0.8$; closed circles, $\epsilon =0.32$; and open squares, $\epsilon =0.03$. The values of $\epsilon$ correspond to delays of 0 fs ($\epsilon =1$), 200 fs ($\epsilon =0.8$), 400 fs ($\epsilon =0.32$), and 600 fs ($\epsilon =0.03$), as depicted in the HOM dip of Fig. 2. The dashed red line (upper) corresponds to the Tsirelson bound, and the dashed green line (lower) is the CHSH inequality limit.

Figure 5

Value of the CHSH inequality for $\theta =22.5^{\circ }$ as a function of the temporal delay, as depicted in Fig. 2. The solid (blue) curve is the theoretical prediction assuming a visibility factor of $V=0.92$. The dashed red line (upper) corresponds to the Tsirelson bound ($S_{\max }=2\sqrt{2}$), and the dashed green line (lower) is the CHSH inequality limit ($S=2$).