High Purity OAM Entangled Photons from SPDC with Reduced Spatial Spectral Correlations

Abstract

Entanglement generated by Spontaneous Parametric Down Conversion (SPDC) involves multiple, often mutually correlated degrees of freedom. These degrees of freedom are often treated independently, overlooking the intrinsic correlation between them. We focus on the spatial spectral correlations that, if left uncontrolled, introduce distinguishability and reduce coherence, undermining applications such as high-dimensional OAM encoding. We analyze the spatio spectral structure of the biphoton and identify source configurations enabling a strong reduction of such correlations. We then quantify how spatial spectral coupling degrades OAM spatial purity, mapping high-purity regions as functions of OAM order, crystal length, and pump/collection waists. The resulting design parameters enable engineering bright, high purity OAM entangled sources, reducing the need for loss-introducing filtering and therefore supporting scalable high-dimensional photonic quantum technologies.

Figures (6)

• Figure 1: Spatial-spectral structure of the two-photon state for type-I SPDC. A pump photon with wavelength $\lambda_p = 400\,\mathrm{nm}$ and beam width $w_p = 28\,\mathrm{\mu m}$ passes through a LiIO$_3$ crystal of length $L = 5\,\mathrm{mm}$mikhailova2008valencia2007. Top row: \ref{['fig:subfig3']}, Gaussian spatial-spectral pump function, \ref{['fig:subfig4']} general sinc phase-matching function (PMF) computed from Eq. (\ref{['sincsum']}), and \ref{['fig:subfig5']} resulting intensity distribution. Bottom row: \ref{['fig:subfig6']} pump function, \ref{['fig:subfig7']} double-sinc approximation of the phase-matching function from Eq. (\ref{['sincmult2']}), and \ref{['fig:subfig8']} corresponding intensity distribution. All plots are shown as a function of the signal wavelength $\lambda_s$ and transverse momentum $q_s$ (restricted to the $x$-direction), with fixed idler wavelength $\lambda_i = 800~\mathrm{nm}$ and transverse momentum $q_i = 0.01~\mu\mathrm{m}^{-1}$. The pump function defines a narrow horizontal band centered at $\lambda_s \approx 800~\mathrm{nm}$ and extended along the spatial axis. The phase-matching function \ref{['fig:subfig4']} exhibits a V-shape (see Appendix \ref{['app:c']}), while its approximation \ref{['fig:subfig7']} reduces to a rectangular-like modulation. The resulting joint intensity distributions [\ref{['fig:subfig5']} and \ref{['fig:subfig8']}] show that the approximation reproduces the localization around $\lambda_s \approx 800~\mathrm{nm}$ within a $\sim 1~\mathrm{nm}$ spectral window, but does not enforce strong selection in the transverse momentum.
• Figure 2: Caption: The normalized spatial purity $\mathrm{Tr}(\hat{\rho}^2_{s})$ for the SPDC general model Eq. (\ref{['eq:phi-sinc']}) (curves with symbols) and for the four-Gaussian model Eq. (\ref{['4g']}) (solid curve) are plotted as a function of the ratio between the as spatial collection mode waist $w_{s}$ ($w_{s}=w_{i}$) and the pump beam waist $w_p$. The purity provided by the spatial-spectral coupled wavefunction is compared with the one of the separable Gaussian model varying \ref{['fig:subfigw']} the OAM number $\ell_s=-\ell_i=\ell$, \ref{['fig:subfigl']} crystal length, \ref{['fig:subfigll']} pump beam waist and pump pulse duration \ref{['fig:subfigt']}.
• Figure 3: Phase matching function for semi-general model represented by Eq. (\ref{['sincsum']}) (solid blue line) and its double-sinc approximation represented by Eq. (\ref{['sincmult2']})-(\ref{['sincmult']}) (dashed orange line). The double-sinc function is a good approximation of the phase matching function for a broad range of wavelengths \ref{['fig:subfigaa']}, if the relative transverse momentum between the photons is small $|q_s-q_i| \approx 0.02 \,\mathrm{\mu m^{-1}}$. The double-sinc approximation also fits well the behavior of the general PMF in the spatial domain; for wavelength degenerated photons \ref{['fig:subfigdd']}, the approximation covers the same wide range of transverse momenta allowed by the general phase-matching conditions.
• Figure 4: The squared modulus of the SPDC type-I general wavefunction Eq. (\ref{['eq:phi-sinc']})-(\ref{['sincsum']}) (solid blue curve) and of its four-Gaussian approximation Eq. (\ref{['4g']}) (dashed orange curve). \ref{['fig:subfigag']} When plotted as a function of the signal position, the four-Gaussian gives a reasonable estimate of the general wavefunction for photons with the same wavelength without discriminating a specific photon's position. The general wavefunction requires wavelength degenerated photons when the pump pulse duration is long \ref{['fig:subfigbg']}, and allows a certain bandwidth for shorter pump pulse duration \ref{['fig:subfigcg']}. Plots obtained for a SPDC type-I photons exiting a $L=0.5\,\mathrm{mm}$ LiIO3 crystal, pump wavelength $\lambda_p=400\,\mathrm{nm}$ and beam waist $w_p=28\,\mathrm{\mu m}$, .
• Figure 5: Squared modulus of the general type-II wavefunction Eq. (\ref{['generalII']})-(\ref{['sincsII']}) (solid blue curve) with its four-Gaussian approximation Eq. (\ref{['4f2']}) (dashed orange curve) as a function of the signal wavelength. Different pump pulse regimes are analyzed, long ($\tau = 50\,\mathrm{ps}$) \ref{['fig:subfigbII']} and short pulse duration ($\tau = 50\,\mathrm{fs}$) \ref{['fig:subfigcII']}. As in the type-I case, a longer pulse constrains the spectral bandwidth of the twin-photons, while a shorter pulse broadens it. The type-II spectrum is slightly wider than type-I, a feature well captured by the four-Gaussian approximation with the proper $\beta$ adjustment. For photons generated from a $L=0.5\,\mathrm{mm}$ BBO crystal with pump wavelength $\lambda_p = 400\,\mathrm{nm}$, beam waist $w_p = 28\,\mathrm{\mu m}$, group velocities $v_{g, p}=c/1.708$, $v_{g, s}=c/1.626$ and $v_{g, i}=c/1.684$, group velocity dispersion $\text{GVD}_p=180\,\mathrm{fs}^2/\,\mathrm{mm}$, $\text{GVD}_s=61.7\,\mathrm{fs}^2/\,\mathrm{mm}$ and $\text{GVD}_p=75.1\,\mathrm{fs}^2/\,\mathrm{mm}$.