Full characterization of the quantum spiral bandwidth of entangled biphotons

Spontaneous parametric down-conversion has been shown to be a reliable source of entangled photons. Among the wide range of properties shown to be entangled, it is the orbital angular momentum that is the focus of our study. We investigate, in particular, the bi-photon state generated using a Gaussian pump beam. We derive an expression for the simultaneous correlations in the orbital angular momentum, $\ell$, and radial momentum, $p$, of the down-converted Laguerre-Gaussian beams. Our result allows us, for example, to calculate the spiral bandwidth with no restriction on the geometry of the beams: $\ell$, $p$, and the beam widths are all free parameters. Moreover, we show that, with the usual paraxial and collinear approximations, a fully analytic expression for the correlations can be derived.

Figure 1

Sketch of the SPCD process. A Gaussian pump propagating in the $z$ direction is incident on a short nonlinear crystal. Signal and idler photons are produced at angles ${\vartheta }_s$ and ${\vartheta }_i$ to the pump direction.

Figure 2

Normalized spiral bandwidth for ${\gamma }_s={\gamma }_i=1$ and $p_s=p_i=0$. Results from Eq. (10) in [23] are in red (large dots).

Figure 3

$P_{p_s,p_i}^{0,0}$ for $0<p_i<15$ and $0<p_s<15$, for different pump sizes. The color bar shows the color normalization, which takes place between the minimum (0) and the maximum values in each graph. The same happens in Figs. 4 and 7.

Figure 4

Correlations between $0<p_i<15$ and $0<p_s<15$, for different $\left|\ell \right|$ values and a pump of the same size as signal and idler. Notice how the combination of $p_i$ and $p_s$ with the highest probability of being measured shifts along the diagonal as higher and higher values of $\left|\ell \right|$ are selected. Each graph is normalized to the maximum value, a comparison between the main diagonals for $\left|\ell \right|=4,5,6,7$ is given in Fig. 6.

Figure 5

Three products in the form ${\mathrm{LG}}_0^0{\mathrm{LG}}_p^{\ell }{\mathrm{LG}}_p^{-\ell }$ with $\ell =6$ and $p=0,5,15$ [respectively in blue (right peak), red (middle peak), and brown (left peak)], plotted radially from the origin of the cylindrical coordinate system in $k$ space. The radial coordinate is in units of $1/w_p$.

Figure 6

Overlap between ${\mathrm{LG}}_0^0$ and ${\mathrm{LG}}_p^{\ell }{\mathrm{LG}}_p^{-\ell }$ is best achieved at an optimal value of $p$ (red disks), once $\left|\ell \right|$ is fixed. The red ($\ell =4$) line represents the main diagonal in the plot in Fig. 4c and the blue ($\ell =6$) line represents the main diagonal in the plot in Fig. 4d.

Figure 7

These graphs show that the effect of a width mismatch yields a higher probability of finding a state with different radial indices. If, instead, the value of ${\gamma }_s$ were larger than ${\gamma }_i$ the graphs would be mirrored with respect to the leading diagonal.

Figure 8

These graphs show how the SB varies with the pump width, which increases from top to bottom of the graphs, and its ratio with the signal and idler beams is shown on the left. We can see how the radial numbers influence this dependence by comparing the top with the bottom graphs. The positions where the bandwidths that are shown on the right are taken are marked on the graphs on the left at positions ${\gamma }_s=0.5$ red (medium grey), ${\gamma }_s=2$ green (light grey), and ${\gamma }_s=4$ blue (dark grey). Note that the blue (dark grey) bandwidth on the right is indicated in yellow (${\gamma }_s=4$) on the left.

Figure 9

These graphs show how the difference in the widths of the signal and idler influences the SB. We show the effect for various values of $p_i=p_s$. The positions where the bandwidths that are shown on the right are taken are marked on the graphs on the left at positions ${\gamma }_s=0.5$ red (medium grey), ${\gamma }_s=2$ green (light grey), and ${\gamma }_s=4$ blue (dark grey). Note that the blue (dark grey) bandwidth on the right is indicated in yellow (${\gamma }_s=4$) on the left.

Figure 10

These graphs show how different radial indices in the signal and idler modes (with equal widths) influence the SB. We show the effect for various values of $p_i$, $p_s$. The positions where the bandwidths that are shown on the right are taken are marked on the graphs on the left at positions ${\gamma }_s=0.5$ red (medium grey), ${\gamma }_s=1$ green (light grey), and ${\gamma }_s=2$ blue (dark grey). Note that the blue (dark grey) bandwidth on the right is indicated in yellow (${\gamma }_s=2$) on the left.

Figure 11

Joint effect of having a signal and an idler mode with different radial indices and different widths. The positions where the bandwidths that are shown on the right are taken are marked on the graphs on the left at positions ${\gamma }_s=0.25$ red (medium grey), ${\gamma }_s=2$ green (light grey), and ${\gamma }_s=4$ blue (dark grey). Note that the blue (dark grey) bandwidth on the right is indicated in yellow (${\gamma }_s=4$) on the left.