Engineering walk-off-induced orbital angular momentum spectrum in spontaneous parametric downconversion

Abstract

Spontaneous parametric downconversion (SPDC) has been considered as a reliable source of high- dimensional entangled states in orbital angular momentum (OAM) basis. In real-world experiments, the spatial walk-off of the pump often degrades the fidelity of the generated quantum state. Since the walk-off effect breaks the rotational symmetry of the system, the conservation of total OAM is violated. Although the compensation of walk-off effects has become a well-established experimental technique, a systematic modal analysis of the spatial walk-off effect is still incomplete for SPDC. Here, we quantitatively analyze the violation of OAM conservation due to the pump walk-off effect in SPDC processes. We have derived a scaling law of the total OAM distribution with respect to the pump walk-off angle. We have also explored the feasibility of using the spatial walk-off as a mechanism to engineer the generated quantum state. Our study has provided guidelines for the generation of OAM-entangled state under realistic experimental conditions.

Figures (4)

• Figure 1: Far-field intensity profiles of generated signal field at 710 nm. The type-I SPDC is driven by a Gaussian pump field with a waist of 0.2 mm at 355 nm. The direction of the walk-off is assumed to be along the $x$-axis. a. Intensity profile of downconverted signal at collinear phase matching with no pump walk-off ($\rho = 0$). b. Intensity profile of downconverted signal at collinear phase matching with a pump spatial walk-off of $\rho = 3^{\circ}$. c. Intensity profile of downconverted signal at non-collinear phase matching with no pump spatial walk-off ($\rho = 0$). d. Intensity profile of downconverted signal at non-collinear phase matching with a spatial walk-off of $\rho = 3^{\circ}$.
• Figure 2: Numerical dependence of $f_{leak}$ on the focusing parameter of the pump beam $\sqrt{L/z_R}$ for multiple pump walk-off angles under a. collinear phase matching condition and b. non-collinear phase matching condition. The probability of measuring a change in total OAM (i.e., violation of OAM conservation) becomes more pronounced for long crystals and a focused pump. Furthermore, we observe that the non-collinear geometry exhibits greater resilience to the walk-off effect.
• Figure 3: Total OAM distribution and its scaling with the pump spatial walk-off. The type-I SPDC is pumped by a Gaussian beam with $l_{tot} = l_p = 0$. a-c. Probability distribution of the total OAM for a walk-off angle of a. 1$^\circ$, b. 3$^\circ$ and c. 5$^\circ$ under the collinear phase matching condition. Note that the coupling to non-zero OAM modes becomes significant at larger walk-off angles, suggesting a stronger effect of broken rotational symmetry. d. Scaling of the first three total OAM sidebands ($\left|l_{tot}\right| = 1, 2,3$) under the small walk-off approximation. The probability of having a change of $m\hbar$ in the total OAM scales with $m$-th order azimuthal harmonic in the Jacobi-Anger expansion: $S_{l_s,\text{ } m-l_s} \propto (\text{tan} \rho)^{2\left|m\right|} \approx \rho^{2\left|m\right|}$. The purple, maroon and the orange lines show the polynomial dependence of $(\text{tan} \rho)^2$, $(\text{tan} \rho)^4$ and $(\text{tan} \rho)^6$ respectively.
• Figure 4: Total OAM distribution of the downconverted signal-idler pair corrected by pump astigmatism. The downconverted photon pair is generated in a 3-mm-long type-I BBO with an estimated pump walk-off of $\rho \approx 4.6^\circ$ determined from $n_e \text{tan} \rho = -\partial n_e/\partial \theta$eimerl1987optical. The applied astigmatism suppresses the coupling of total OAM to odd-order azimuthal harmonics and enhances even-order changes due to its $\text{cos}(2\phi)$ dependence.