High resource of azimuthal entanglement in terms of Cartesian variables of noncollinear biphotons

Single-particle and coincidence distributions of photons are analyzed for the noncollinear frequency-degenerate type-I regime of spontaneous parametric down-conversion. Noncollinearity itself is shown to provide a new mechanism of strong broadening of the single-particle distributions in Cartesian components of the photon's transverse wave vectors. Related to this, the degree of entanglement appears to be very high in agreement with the earlier performed analysis in the formalism of spherical angles characterizing photon's wave vectors [Phys. Rev. A 93, 033830 (2016)]. In Cartesian variables, very broad curves of single-particle distributions are found to have a rather unusual and peculiar shape. In theory, the key reason for these effects is the reduction of the total wave function of two photons over one of two orthogonal degrees of freedom. In the suggested and discussed experimental schemes this means that all photons of the emission cone have to be taken into account rather than only photons propagating in one given plane which is a common practice in many experiments.

Figure 1

The difference of the refractive indices $\mathrm{\Delta }n\left({\phi }_0\right)$ for a BBO crystal and the pump wavelength ${\lambda }_p=0.4047\mu \mathrm{m}$.

Figure 2

The cone opening angle ${\theta }_0$ as a function of the angle ${\phi }_0$ between the optical axis of a crystal and the $z$ axis.

Figure 3

The function $F\left(k_{-x}\right)$ (15) in dependence on the dimensionless variable ${\lambda }_p{k}_{-x}/\pi$ at ${\theta }_0=0.28\mathrm{rad}$ and $L=0.5\mathrm{cm}$.

Figure 4

Exact [solid line, Eq. (15)] and approximate [dashed line, Eq. (16)] functions $F\left(k_{-x}\right)$ in a small vicinity of the point $2{\theta }_0$.

Figure 5

Coincidence and single-particle distributions in $k_{1x}$(in units of $\pi /{\lambda }_p$) for the pump and crystal parameters $w=L={10}^{-1}\mathrm{cm},{\phi }_0=0.5275\mathrm{rad}$, and ${\theta }_0=0.1\mathrm{rad}$.

Figure 6

Single-particle distribution in $k_{1x}$ in the cases ${\theta }_0=0.04\left(a\right);{\theta }_0=0.02\left(b\right)$; and ${\theta }_0=0\left(c\right)$.

Figure 7

Section of the emission cone by a plane $\left(xy\right)\perp 0z;r_0$ and $\delta r$ are the radius and thickness of the ring, and the dashed line is its diameter; $D_1^{(a,b)}$ and $D_2$ are detectors; vertical lines indicate directions of scanning the detectors $D_1^{(a,b)}$; position of vertical lines is shown to be parametrized by varying values of $k_{1x};l^{(a,b)}$ are length ways at which photons are registered and summed during scanning.

Figure 8

Expected single-particle distribution in the case of registration only photons propagating in the plane $(x,z)$, with all other photons ignored.