Parameter dependence in the atmospheric decoherence of modally entangled photon pairs

When a pair of photons that are entangled in terms of their transverse modes, such as an orbital angular momentum (OAM) basis, propagates through atmospheric turbulence, the scintillation causes a decay of the entanglement. Here, we use numerical simulations to study how this decoherence process depends on the various dimension parameters of the system. The relevant dimension parameters are the propagation distance, the wavelength, the beam radius, and the refractive index structure constant, indicating the strength of the turbulence. We show that beyond the weak scintillation regime, the entanglement evolution cannot be accurately modeled by a single phase screen that is specified by a single dimensionless parameter. Two dimensionless parameters are necessary to describe the OAM entanglement evolution. Furthermore, it is found that higher OAM modes are not more robust in turbulence beyond the weak scintillation regime.

Figure 1

The source generates two photons that are entangled in OAM. Each photon is then sent through a turbulent atmosphere (modeled by a series of phase screens) toward a detector.

Figure 2

The one-sided channel $(I\otimes \$)$ where only one of the two photons propagates through turbulence.

Figure 3

Comparison of ${\mathcal{C}}_{\mathrm{out}}$ and ${\mathcal{C}}_{\mathrm{ch}}{\mathcal{C}}_{\mathrm{in}}$ as a function of $\mathcal{W}$.

Figure 4

Plot of ${\mathcal{C}}_{\mathrm{out}}$ against ${\mathcal{C}}_{\mathrm{in}}$ for eight different initial states. The error bars represent the dispersion of each run from the mean.

Figure 5

The concurrence against $\mathcal{W}$ for (a) $\ell =1$, (b) $\ell =3$, (c) $\ell =5$, and (d) $\ell =7$. Each graph contains curves for all of the different values of $\mathcal{K}$ given in Table 1.

Figure 6

The difference in the concurrence, $\Delta \mathcal{C}={\mathcal{C}}_{\mathrm{th}}-{\mathcal{C}}_{\mathrm{num}}$, against the Rytov variance ${\sigma }_R^2$ for (a) $\ell =1$, (b) $\ell =3$, (c) $\ell =5$, and (d) $\ell =7$. Each graph contains curves for all of the different values of $\mathcal{K}$ given in Table 1.

Figure 7

The concurrence against $\mathcal{W}$ (left) and against $t$ (right) for (a) $\mathcal{K}=100$, (b) $\mathcal{K}=10$, and (c) $\mathcal{K}=0.06$. Each graph contains curves for $\ell =1,3,5,7$.

Figure 8

Concurrence as a function of $t$ for $\mathcal{K}=0.067$ and $\ell =1$, using the sets of dimension parameters given in Table 2.

Figure 9

The Rytov variance ${\sigma }_R^2$ against the normalized propagation distance $t$. The colored lines show the scintillation strength of optical beams (or photons) as a function of propagation distance under particular conditions, indicated by the value of $\mathcal{K}$. The gray dashed lines are the theoretical bounds (according to the SPS approach) where the concurrence for particular values of $\ell$, as indicated, will go to zero. The red markers represent the points where the concurrence vanishes for the different values of $\ell$ (red square for $\ell =1$, red circle for $\ell =3$, red triangle for $\ell =5$, and red diamond for $\ell =7$). The white markers represent the points where $\Delta \mathcal{C}$ exceeds a value of 0.2 for the different values of $\ell$ (white square for $\ell =1$, white circle for $\ell =3$, white triangle for $\ell =5$, and white diamond for $\ell =7$).