Full-field mapping of spatially varying polarization entanglement generated from spontaneous parametric down-conversion

Abstract

Two-photon states generated from spontaneous parametric down-conversion (SPDC) can display entanglement in all degrees of freedom (DoFs) of light, including spatial, temporal, and polarization. The coupling between different DoFs of a two-photon state has been shown to display rich structures that enable novel and robust information processing schemes. While existing literature has studied these couplings by post-selecting the SPDC field, a comprehensive understanding of the inherent spatial-polarization coupling produced in the SPDC process is still lacking. This work produces a full spatial map of the polarization entanglement generated across the entire SPDC field. We observe an entire class of near-maximally polarization-entangled states with an average concurrence of $0.8303\pm0.0004$, which, together with a certified spatial dimensionality of 148, could potentially offer access to a 251-dimensional hyperentanglement. The spatial-polarization coupling manifests as radially or linearly varying polarization-entangled states, whose wavefunctions are dependent on the transverse momenta of the down-converted photons and the pump beam, respectively. Our study lays important groundwork for further exploiting the coupling between entanglement in different DoFs for future quantum technologies.

Figures (6)

• Figure 1: (a) Schematic of the experimental setup. The $\beta$-barium borate (BBO) double-crystal produces a two-photon state, which exhibits spatially varying polarization entanglement. The Tpx3Cam captures the two-photon field in the momentum or position basis through different lens configurations and in different polarization bases through combinations of the quarter-wave plate (QWP), half-wave plate (HWP), and polarizing beam splitter (PBS). TC: temporal compensator, SPF: short-pass filter with cutoff wavelength at 500 nm, DM: dichroic mirror, BS: beam splitter, BPF: band-pass filter centered at 800 nm with a bandwidth of 40 nm. $f_{1}$ = 50 mm, $f_{2}$ = 100 mm. $f_{3}$ = 150 mm for far-field measurements or 75 mm for near-field measurements. Dashed lines represent the intermediate planes imaging the near-field of the crystal using $f_{1}$ and $f_{2}$. (b-c) Time-stamp histogram of photons detected in the far-field and near-field of the crystal, representing projection onto the momentum (p) and position (q) bases of the two-photon state.
• Figure 2: (a) Concurrence $C$ and (b) two-photon polarization phase $\phi$ of the two-photon states measured between momentum-correlated superpixels. The pixel coordinates correspond to the central position of the superpixels for signal photons. In the colormap, the saturation depicts the normalized pair generation rate, $I$, and the hue depicts $C$ and $\phi$ in (a) and (b), respectively. (c) Real and imaginary parts of the density matrix reconstructed at the center of the SPDC field, the corresponding signal pixel is marked with a black square in (a) and (b).
• Figure 3: Schematic depiction of (a) the discrete momentum basis in the far-field and (c) the discrete position basis in the near-field. The corresponding superpixels are depicted in a brightened color scale. Cross-correlations between the spatial modes of signal and idler photons in the (b) momentum basis and (d) position basis.
• Figure 4: (a) Schematic of the modified experimental setup. PM: prism mirror. Focal lengths of the lenses are $f_{0}$ = 100 mm, $f_{1}$ = 50 mm, $f_{2}$ = 100 mm $f_{3}$ = 150 mm and $f_{4}$ = 75 mm. The angular width of the pump beam is changed by inserting $f_{0}$ into the beam path. A CCD camera images the pump beam in the Fourier plane to analyze its angular width. (b) Far-field image of the SPDC field taken by the Tpx3Cam. (c) Concurrence and (d) relative phase of the two-photon states measured between momentum-correlated superpixels. Pixel coordinates correspond to the central position of the superpixels for signal photons.
• Figure 5: Pump beam profile (greyscale image) and two-photon momentum correlation profile (colored image) for (a) collimated and (b) weakly focused pump beam. The inset captions state the beam width measured at the camera sensor plane. (c-d) Concurrence and (e-f) the two-photon polarization phase measured between a fixed signal superpixel and all momentum-correlated idler superpixels, where (c) and (e) correspond to collimated pumped beam while (d) and (f) correspond to weakly focused pump beam. Pixel coordinates indicate the transverse momenta of the pump beam $\textbf{p}_\text{p} = \textbf{p}_s + \textbf{p}_i$, in which the geometrical centers of two-photon momentum correlation profile are denoted $\textbf{p}_\text{px} = \textbf{p}_\text{py} = 0$