DOI: 10.5281/zenodo.15156783
title: Data and code associated with the paper 'How to use the dispersion in the χ(3) tensor for broadband generation of polarization-entangled photons'

description: Data and code associated with the following paper V. Vento, et al., Phys. Rev. Res. (2025)

A thorough explanation of the experiment performed is available there.

The name of each sub-folder and file in S-AS_chi3_code.zip indicates the corresponding figure number ("FIG #") and the type of content ("Data", "Analysis" or "Model").

FIG 1 (a) Sketched amplitude of the two relevant \chi^{(3)} tensor elements in diamond under vertically polarized pump at \omega_p; \Omega_\nu is the Raman-active optical phonon frequency. (c) Stokes spectrum measured by single-photon fiber spectroscopy.

FIG 2 (a) Theoretical amplitude and phase of \chi^{(3)}_{VVVV} (solid blue) and \chi^{(3)}_{HVVH} (solid red). The Raman and the electronic contributions to the latter are shown as dashed and dotted gray lines, respectively. At the frequencies where \chi^{(3)}_{VVVV} and \chi^{(3)}_{HVVH} have the same amplitude (black circles), the maximally entangled states \ket{\Psi_\pm} are generated. (b) Frequency-resolved anti-Stokes--Stokes coincidences measured after selecting V (black squares) or H (black circles) polarization in the two detection paths. The data are acquired in 1 hour. The solid blue curve is a polynomial fit that reflects the setup polarization response function since no resonances are expected in this configuration. The solid red curve represents the fit function A\log(I_H(\omega)\ast G(\omega))+b multiplied by the setup response, where the free parameters \Delta = 130.59 cm^{-1}, \sigma_G=24.80 cm^{-1}, A=0.82, b=1.51 are extracted by fitting within the gray region. The dashed red line shows the fit function without convolution, corresponding to the red line in panel (a).

FIG 3 (a-b) Symbols represent the measured CHSH parameters S^{\Psi+}(\omega) (a) and S^{\Psi-}(\omega) (b) extracted from a set of 16 coincidence measurements of 1 hour each. The error bars are calculated by propagating the standard deviation of the coincidence counts in the region [630, 890] cm^{-1}. The shaded domain is delimited by the upper classical bound of 2 and the upper quantum bound of 2\sqrt{2}. The dotted black vertical line indicates the Raman resonance, solid lines are theoretical curves; dashed blue lines are computed without the Raman contribution. In (b) the dashed black curve is computed without accounting for the setup response.

FIG 4 Bell test for different pump polarizations. Shaded gray circles represent the measured CHSH parameter S^{\Psi+}(\omega) for linear pump polarization at \phi = \frac{\pi}{4}. The gray line is the corresponding theoretical curve. The black line represents the theoretical S^{\Psi+}(\omega) for linear pump polarization at \phi=0 (vertical) as in Fig.3(b). The red line represents the theoretical CHSH parameter S^{\Phi-}(\omega) for circular pump polarization, i.e. \phi=\frac{\pi}{4} and \xi=\frac{\pi}{2}. The dotted black vertical line indicates the Raman resonance. The blue region spans between the classical upper bound of 2 and the quantum upper bound of 2\sqrt{2}. 

FIG S1 Time-shift calibration. The red and blue lines indicate the arrival times of the Stokes photons T_S and of the anti-Stokes photons T_A, respectively, as a function of the absolute Raman shift. Black crosses represent the calculated coincidence times T_C; the black line is the corresponding fit. The gray area denotes the spectral region in which correlated Stokes and anti-Stokes photons can be detected, and it's limited by the fibers' bandwidths and the spectral filters.

FIG S2 Dependence of \Delta and \Delta' on the excitation wavelength. The red and blue circles represent, respectively, the measurements of \Delta and \Delta' reported in [M. D. Levenson and N. Bloembergen, Phys. Rev. B 10, 4447, 1974]. The black lines are the corresponding linear regressions. The red cross indicates the value \Delta=130.59 cm^{-1} that we measure at 781 nm. The blue cross indicates the value \Delta'=47.49\text{~cm}^{-1}, calculated by inserting the measured \Delta and the parameters F=2 and F'=1.25 in Eq.2, showing that the same linear trend vs. wavelength as in [Phys. Rev. B 10, 4447, 1974] is satisfied.

FIG S3 Bell test measurements and two-photon interference. (a) Measured values of E_{\theta_1,\theta_2}(\omega) for a vertically polarized pump are shown as dark cyan circles for the angles (\theta_1,\theta_2) = (0,\frac{\pi}{8}), light cyan for (0,-\frac{\pi}{8}), yellow for (\frac{\pi}{4},\frac{\pi}{8}) and orange for (\frac{\pi}{4},-\frac{\pi}{8}). The dotted black vertical line indicates the Raman resonance. The error bars are calculated by propagating the standard deviation of the coincidence counts in the region [630, 890] cm^{-1}. (b) Measured E_{\theta_1,\theta_2} at 910 cm^{-1}$ (dotted red line in (a)) for \theta_1=0 (circles) and \theta_1=\frac{\pi}{4} (triangles) and different values of \theta_2. Colored points for the same angles as (a). In all panels, the solid lines are theoretical curves. 

FIG S4 Measured CAR under vertically polarized pump after selecting horizontally polarized photons in the two detection paths. The dashed line represents the classical limit of 2 set by the Cauchy-Schwarz inequality.