Spontaneous parametric down-conversion: Revisiting the parameters of transverse entanglement outside the near zone

The reduced density matrix ${\rho }_r(x,x^{\prime })$ of the transverse biphoton state beyond the near zone is found to be real on and only on diagonals in the plane of the photon's transverse coordinates $(x,x^{\prime })$. This remarkable feature is found to occur at any distances of photon propagation with diffraction spreading completely taken into account. The functions of the reduced density matrix at two orthogonal diagonals, ${\rho }_r(x,x)$ and ${\rho }_r(x,-x)$, are considered as the only two complementary single-particle distributions generated by the reduced density matrix. The ratio of widths of these two distributions is interpreted as the new entanglement parameter $R_{\mathrm{diag}}$ which is found to be very close to and completely compatible with the Schmidt entanglement parameter $K$.

Figure 1

The main ($\mathrm{m}.\mathrm{d}.$) and side ($\mathrm{s}.\mathrm{d}.$) diagonals in the momentum $(k_{1x},k_{1x}^{\prime })$ and coordinate $(x_1,x_1^{\prime })$ planes.

Figure 2

The functions $exp(-u^2)$ and ${\mathrm{sinc}}^2\left(2u^2\right)$ and coincidence of their widths at half maxima.

Figure 3

Found with the use of the wave function ${\tilde{\mathrm{\Psi }}}_{2\mathrm{G}}(x_1,x_2;\zeta )$ (45) widths of the unconditional and conditional single-particle distributions and their ratio parameter ${\tilde{R}}_{2\mathrm{G}}\left(\zeta \right)$ (50) vs $\zeta /ab$; $\mathrm{\Delta }{x}_1^{(\mathrm{u}.\mathrm{c}.)}$ and $\mathrm{\Delta }{x}_1^{\left(c\right)}$ are in units of $\sqrt{a^2+b^2}$ and $a/b=3$; the dashed line is the Schmidt entanglement parameter $K_{2\mathrm{G}}$ (33).

Figure 4

The widths $\mathrm{\Delta }{x}_1^{(\mathrm{s}.\mathrm{d}.)}$ and $\mathrm{\Delta }{x}_1^{(\mathrm{m}.\mathrm{d}.)}$ vs $\zeta$ and their ratio $R_{\text{diag}2\mathrm{G}}=\mathrm{\Delta }{x}_1^{(\mathrm{s}.\mathrm{d}.)}/\mathrm{\Delta }{x}_1^{(\mathrm{m}.\mathrm{d}.)}$; for comparison the dashed line shows the usual conditional distribution width $\mathrm{\Delta }{x}_1^{(\mathrm{c}.)}\left(\zeta \right)$ (49) which differs significantly from $\mathrm{\Delta }{x}_1^{(\mathrm{m}.\mathrm{d}.)}\left(\zeta \right)$; in Fig. 2, the ratio $a/b$ is taken equal to 3, and all widths are in units of $\sqrt{a^2+b^2}$.

Figure 5

Generated by the Gauss-sinc wave function (57) with $s=0.85$, numerically calculated dependencies on $\eta$ (solid line) and on $1/\eta$ (dashed line) of the Schmidt entanglement parameter $K_{G-\mathrm{sinc}}$, the momentum representation.

Figure 6

Transverse-entanglement parameters in the momentum representation vs the control parameter $\eta$ (31): The Schmidt parameter $K$, the diagonal and the width-ratio parameters $R_{\mathrm{diag}}$ and $R$, and, for comparison, the $2\mathrm{G}$ Schmidt parameter $K_{2\mathrm{G}}$ (33).

Figure 7

The widths of the conditional and unconditional distributions in the coordinate representation vs the dimensionless propagation distance, and the parameter $\tilde{R}(\zeta /ab)$.

Figure 8

Widths of the conditional and unconditional distributions in the coordinate representation vs the propagation distance, and the parameter ${\tilde{R}}_{\mathrm{diag}}(\zeta /ab)$.

Figure 9

A scheme for measuring the phase $\phi (x_1,x_2;\zeta )$ of the wave function $\tilde{\mathrm{\Psi }}(x_1,x_2;\zeta )$ (part 1); KEP is the knife-edge prism splitting the pump ($p$) for two equal counterpropagating parts; Cr and DM denote two identical nonlinear crystals and dichroic mirrors; in principle, the KEP can be replaced by a beamsplitter and a couple of additional mirrors.

Figure 10

A scheme for extracting photons and measuring the phase $\phi (x_1,x_2;\zeta )\equiv \phi$ (part 2); C.S.1, C.S.2, and C.S.3 are coincidence schemes; $D_1$–$D_4$ are detectors; PBS is the polarization beamsplitter turned ${45}^{\circ }$ around the $x$ axis; HWP and QWP are the half-wave and quarter-wave phase plates, with the latter to be installed into the lower channel for the second series of measurements; $s.t.l.$ are the signal transmission lines from C.S.1 to SC.S.2 and C.S.3.

Figure 11

Indicated by dashed lines with arrows, solutions of Eqs. (65) and (70) in the cases $\delta >0,\tilde{\delta }>0$, and $\delta <0,\tilde{\delta }<0$, where $\delta =w_{{45}^{\circ }}-w_{{135}^{\circ }}$ and $\tilde{\delta }={\tilde{w}}_{{45}^{\circ }}-{\tilde{w}}_{{135}^{\circ }}$.