1. Introduction
Recently, there has been increased attention on high-dimensional orbital angular momentum (OAM) entanglement due to its numerous advantages, including higher information capacity and enhanced security in quantum cryptography [1–4]. Quantum entanglement is a powerful and enigmatic resource in the field of quantum information processing. However, the OAM entangled state is very fragile when propagating through free space [5–7]. The decoherence of OAM entanglement caused by atmospheric turbulence is a significant issue that limits its application in quantum communication systems. Therefore, it is of great significance to study the decoherence of OAM entanglement in atmospheric turbulence.
So far, numerous theoretical and experimental studies have extensively investigated the effects of turbulence on OAM entanglement [4, 6, 8–17]. The study of the evolution of turbulence-induced OAM entanglement of a two-qubit state reveals that OAM entanglement with higher OAM values is more robust under weak scintillation conditions [8–10]. All four symmetric Bell states demonstrate a consistent trend of entanglement deterioration when passing through turbulence [14]. However, recent study has demonstrated no discernible trend in the evolution of high-dimensional OAM entanglement in atmospheric turbulence based on the azimuthal mode [4]. Note that the evolution of OAM entanglement typically relies on the original state [3]. The different OAM modes do not couple equally, and the coupling between nearest neighbors is stronger than between next-nearest neighbors. Therefore, to better understand the evolution of high-dimensional OAM entanglement in atmospheric turbulence, it is necessary to investigate the decay of entanglement in other high-dimensional OAM states. Furthermore, past research has shown that optical turbulence in the free atmosphere, specifically in the stable layered stratosphere, is primarily characterized by unevenness and directional variation [18–21]. However, little attention has been paid to the impact of anisotropic turbulence on the evolution of high-dimensional OAM entanglement.
In this study, the evolution of high-dimensional OAM entanglement in an anisotropic non-Kolmogorov turbulent atmosphere is numerically investigated. We analyze the decay of high-dimensional OAM entanglement in turbulence using a multiple-phase screen approximation. Moreover, the entanglement negativity and fidelity are employed to quantify entanglement in a high-dimensional OAM state [8, 22]. The paper is structured as follows: section 2 introduces the numerical procedure that was used, while section 3 presents the numerical results and analysis. Finally, section 4 presents the conclusions.
2. Numerical procedure
Assuming a maximally entangled qutrit state as the initial quantum state for numerical simulations, without loss of generality. This state is encoded by Laguerre–Gaussian (LG) modes in a three-dimensional ($3\times 3$ ) space, where the coupling between one OAM mode and another is stronger because the two OAM values are closer to each other.
$\begin{eqnarray}\begin{array}{rcl}{\left|\psi \right\rangle }_{{\rm{in}}} & = & \left({\left|{\ell },{\ell }-1\right\rangle }_{{\rm{AB}}}+{\left|{\ell }-1,{\ell }\right\rangle }_{{\rm{AB}}}\right.\\ & & \left.+\,{\left|{\ell }+1,{\ell }+1\right\rangle }_{{\rm{AB}}}\right)/\sqrt{3},\end{array}\end{eqnarray}$
where ${\ell }$ is the azimuthal mode. The footnotes A and B label the respective subsystems of two photons. The LG mode can be expressed as follows, considering normalized coordinates [5] $\begin{eqnarray}r\left|{\ell }\right\rangle =\displaystyle \frac{1}{\sqrt{2\pi }}{R}_{{\ell },p}\left(r,z\right)\exp \left({\rm{i}}{\ell }\theta \right),\end{eqnarray}$
where $\theta $ is the azimuthal angle, ${r}^{2}={x}^{2}+{y}^{2},$ and ${R}_{{\ell },p}\left(r,z\right)$ represents the LG function given by [23, 24] $\begin{eqnarray}\begin{array}{rcl}{R}_{{\ell },p}\left(r,\,z\right) & = & {\left(\displaystyle \frac{p!{2}^{\left|{\ell }\right|}}{\pi \left(p+\left|{\ell }\right|\right)!}\right)}^{1/2}\displaystyle \frac{{\left(\sqrt{2}r\right)}^{\left|{\ell }\right|}}{{w}^{\left|{\ell }\right|+1}\left(z\right)}{L}_{p}^{\left|{\ell }\right|}\left(\displaystyle \frac{2{r}^{2}}{{w}^{2}\left(z\right)}\right)\\ & & \times \,\exp \left(-\displaystyle \frac{{r}^{2}}{{w}^{2}\left(z\right)}\right)\exp \left({\rm{i}}\left({\ell }+1\right){\tan }^{-1}\left({z/z}_{R}\right)\right.\\ & & \left.-\,\displaystyle \frac{{\rm{i}}k{r}^{2}}{2z\left[1+{\left({z}_{R}/z\right)}^{2}\right]}\right),\end{array}\end{eqnarray}$
where the parameter $w\left(z\right)={w}_{0}\sqrt{1+{\left(\tfrac{z}{{z}_{R}}\right)}^{2}}$ is associated with the propagation distance $z,$ initial beam radius ${w}_{0},$ and the Rayleigh range ${z}_{R}=\tfrac{k{w}_{0}^{2}}{2}.$ The wave number $k=\tfrac{2\pi }{\lambda }$ is correlated with the wavelength $\lambda ,$ and ${L}_{p}^{\left|{\ell }\right|}$ represents the associated Laguerre polynomials. In the present study, the cases ${\ell }=1,3,5$ and $p=0$ are considered.When OAM entangled photon pairs propagate through a turbulent atmosphere modeled by a series of random phase screen, the OAM beam will undergo substantial wavefront distortion as a result of the stochastic phase fluctuations induced by the turbulent atmospheric conditions [3, 8–17]. The relationship between random phase fluctuation $\phi \left({\boldsymbol{x}}\right)$ and random refractive index fluctuation $\delta n\left({\boldsymbol{x}}\right)$ is given by the following formula
$\begin{eqnarray}\phi \left({\boldsymbol{x}}\right)={k}_{0}{\int }_{0}^{{\rm{\Delta }}z}\delta n\left({\boldsymbol{x}}\right){\rm{d}}z,\end{eqnarray}$
where $\delta n\left({\boldsymbol{x}}\right)=n-1,$ $n$ is the refractive index of a turbulent medium. ${k}_{0}$ is the wave number in a vacuum. ${\rm{\Delta }}z$ is a distance between consecutive phase screen. The phase distortion of the OAM beam directly causes increased crosstalk among adjacent OAM modes, consequently leading to the degradation of OAM entanglement.The properties of random fluctuations in the refractive index are determined by a turbulent medium. Here, the generalized anisotropic von Karman spectrum, which considers the asymmetric properties of turbulence eddies in the orthogonal xy-plane along the path, is used to obtain the random phase modulations [25]5 ) reduces to the conventional isotropic non-Kolmogorov spectrum model.
$\begin{eqnarray}\displaystyle \begin{array}{l}{{\rm{\Phi }}}_{n}\left({\boldsymbol{\kappa }}\right)=A\left(\alpha \right){\tilde{C}}_{n}^{2}{\xi }_{x}{\xi }_{y}\\ \,\times \,\frac{\exp \left[-\left({\xi }_{x}^{2}{\kappa }_{x}^{2}+{\xi }_{y}^{2}{\kappa }_{y}^{2}+{\kappa }_{z}^{2}\right)/{\kappa }_{l}^{2}\right]}{{\left({\xi }_{x}^{2}{\kappa }_{x}^{2}+{\xi }_{y}^{2}{\kappa }_{y}^{2}+{\kappa }_{z}^{2}+{\kappa }_{0}^{2}\right)}^{\alpha /2}},\,\left(3\lt \alpha \lt 4\right),\end{array}\end{eqnarray}$
${\boldsymbol{\kappa }}=\left({\kappa }_{x},{\kappa }_{y},{\kappa }_{z}\right)$ representing a vector spatial frequency, $\kappa =\sqrt{{\kappa }_{x}^{2}+{\kappa }_{y}^{2}+{\kappa }_{z}^{2}},$ ${\xi }_{x}$ and ${\xi }_{y}$ being two anisotropy parameters representing scale-dependent stretching along the $x$ and $y$ directions, respectively. ${\tilde{C}}_{n}^{2}$ (in units of ${m}^{3-\alpha }$ ) representing a generalized refractive-index structure parameter. α being the generalized exponent. ${\kappa }_{l}=c\left(\alpha \right)/{l}_{0},$ ${\kappa }_{0}=2\pi /{L}_{0},$ ${l}_{0}$ and ${L}_{0}$ being inner scale and outer scale of turbulence, respectively. The parameters $A\left(\alpha \right)$ and $c\left(\alpha \right)$ are given by $\begin{eqnarray}A\left(\alpha \right)={\rm{\Gamma }}\left(\alpha -1\right)\cos \left(\pi \alpha /2\right)/\left(4{\pi }^{2}\right),\end{eqnarray}$
$\begin{eqnarray}c\left(\alpha \right)={\left\{A\left(\alpha \right){\rm{\Gamma }}\left[\left(5-\alpha \right)/2\right]2\pi /3\right\}}^{1/\left(\alpha -5\right)},\end{eqnarray}$
where symbol ${\rm{\Gamma }}$ is the gamma function. If ${\xi }_{x}={\xi }_{y}=1$ is satisfied, equation (The turbulent phase screens can be generated using the Fast Fourier Transform (FFT) in conjunction with the power density spectrum [26]. The final desired turbulent phase screen is the sum of the FFT-based screen and the low-spatial screen. The FFT-based turbulent phase screen can be expressed as
$\begin{eqnarray}\displaystyle \begin{array}{l}{\phi }_{\mathrm{FFT}}\left(m,n\right)=\displaystyle \sum _{m\text{'}=-N/2}^{N/2-1}\displaystyle \sum _{n\text{'}=-N/2}^{N/2-1}h\left(m\text{'},n\text{'}\right)\\ \,\times \,\sqrt{2\pi {k}^{2}{\Phi }_{n}\left({\kappa }_{x},{\kappa }_{y},0\right){\rm{\Delta }}z}{{\rm{e}}}^{{\rm{i}}\left(mm^{\prime} /N+nn^{\prime} /N\right)}{\rm{\Delta }}\kappa ,\end{array}\end{eqnarray}$
where $m,n,m^{\prime} ,n^{\prime} =-N/2,\mathrm{..}.,N/2-1$ are the integer indices, $h\left(m^{\prime} ,n^{\prime} \right)$ is a zero-mean normally distributed random complex valued function, ${\rm{\Delta }}\kappa $ is the sampling intervals in spacing frequency domain. The low-spatial screen can be expressed as [27] $\begin{eqnarray}\begin{array}{l}{\phi }_{\mathrm{Low}}\left(m,n\right)=\displaystyle \sum _{m\text{'}=-{N}_{x}/2}^{{N}_{x}/2-1}\displaystyle \sum _{n\text{'}=-{N}_{y}/2}^{{N}_{y}/2-1}h\left(m\text{'},n\text{'}\right)\\ \,\times \,\sqrt{2\pi {k}^{2}{\Phi }_{n}\left({\kappa }_{x},{\kappa }_{y},0\right){\rm{\Delta }}z}{{\rm{e}}}^{{\rm{i}}\left(mm^{\prime} /{N}_{x}N+nn^{\prime} /{N}_{y}N\right)}{\rm{\Delta }}\kappa .\end{array}\end{eqnarray}$
The refractive index fluctuations in a turbulent atmosphere lead to crosstalk of OAM modes among adjacent photon modes, causing the photon’s state to become a superposition of multiple states. In the following discussion, we extract only the information contained in the modes where ${\ell }\pm 1$ and ${\ell }.$ After the entangled photon propagating over a distance of ${\rm{\Delta }}z$ through turbulence atmosphere, each photon state will be transformed into
$\begin{eqnarray}\begin{array}{rcl}{\left|{\ell }+1\right\rangle }_{{\rm{A}}} & \to & {a}_{1}{\left|{\ell }+1\right\rangle }_{{\rm{A}}}+{a}_{0}{\left|{\ell }\right\rangle }_{{\rm{A}}}+{a}_{-1}{\left|{\ell }-1\right\rangle }_{{\rm{A}}},\\ {\left|{\ell }\right\rangle }_{A} & \to & {b}_{1}{\left|{\ell }+1\right\rangle }_{A}+{b}_{0}{\left|{\ell }\right\rangle }_{A}+{b}_{-1}{\left|{\ell }-1\right\rangle }_{A},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\left|{\ell }-1\right\rangle }_{{\rm{A}}} & \to & {c}_{1}{\left|{\ell }+1\right\rangle }_{{\rm{A}}}+{c}_{0}{\left|{\ell }\right\rangle }_{{\rm{A}}}+{c}_{-1}{\left|{\ell }-1\right\rangle }_{{\rm{A}}},\\ {\left|{\ell }+1\right\rangle }_{B} & \to & {d}_{1}{\left|{\ell }+1\right\rangle }_{B}+{d}_{0}{\left|{\ell }\right\rangle }_{B}+{d}_{-1}{\left|{\ell }-1\right\rangle }_{B},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\left|{\ell }\right\rangle }_{{\rm{B}}} & \to & {e}_{1}{\left|{\ell }+1\right\rangle }_{{\rm{B}}}+{e}_{0}{\left|{\ell }\right\rangle }_{{\rm{B}}}+{e}_{-1}{\left|{\ell }-1\right\rangle }_{{\rm{B}}},\\ {\left|{\ell }-1\right\rangle }_{B} & \to & {f}_{1}{\left|{\ell }+1\right\rangle }_{B}+{f}_{0}{\left|{\ell }\right\rangle }_{B}+{f}_{-1}{\left|{\ell }-1\right\rangle }_{B},\end{array}\end{eqnarray}$
where ${a}_{1}={}_{{\rm{A}}}\left\langle {\ell }+1\right|{U}_{{\rm{\Delta }}z}{\left|{\ell }+1\right\rangle }_{{\rm{A}}},$ ${a}_{0}={\,}_{{\rm{A}}}\left\langle {\ell }\right|{U}_{{\rm{\Delta }}z}{\left|{\ell }+1\right\rangle }_{{\rm{A}}},$ ${a}_{-1}={\,}_{{\rm{A}}}\left\langle {\ell }-1\right|{U}_{{\rm{\Delta }}z}{\left|{\ell }+1\right\rangle }_{{\rm{A}}},$ etc are the complex coefficients in the expansion of the distorted state in terms of the OAM basis. ${U}_{{\rm{\Delta }}z}$ represents the unitary operator that describes propagation through turbulence over a distance of ${\rm{\Delta }}z$ [10] .After the initial maximally entangled qutrit state propagates through atmospheric turbulence, it will be transformed into
$\begin{eqnarray}\begin{array}{rcl}{\left|\psi \right\rangle }_{{\rm{in}}}\to {\left|\psi \right\rangle }_{{\rm{out}}} & = & {C}_{1}{\left|{\ell }+1,{\ell }+1\right\rangle }_{{\rm{AB}}}+{C}_{2}{\left|{\ell }+1,{\ell }\right\rangle }_{{\rm{AB}}}\\ & & +\,{C}_{3}{\left|{\ell }+1,{\ell }-1\right\rangle }_{{\rm{AB}}}+{C}_{4}{\left|{\ell },{\ell }+1\right\rangle }_{{\rm{AB}}}\\ & & +\,{C}_{5}{\left|{\ell },{\ell }\right\rangle }_{{\rm{AB}}}+{C}_{6}{\left|{\ell },{\ell }-1\right\rangle }_{{\rm{AB}}}+{C}_{7}{\left|{\ell }-1,{\ell }+1\right\rangle }_{{\rm{AB}}}\\ & & +\,{C}_{8}{\left|{\ell }-1,{\ell }\right\rangle }_{{\rm{AB}}}+{C}_{9}{\left|{\ell }-1,{\ell }-1\right\rangle }_{{\rm{AB}}},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{c}{C}_{1}=\displaystyle \frac{1}{\sqrt{3}}\left({b}_{1}{f}_{1}+{c}_{1}{e}_{1}+{a}_{1}{d}_{1}\right),\\ {C}_{2}=\displaystyle \frac{1}{\sqrt{3}}\left({b}_{1}{f}_{0}+{c}_{1}{e}_{0}+{a}_{1}{d}_{0}\right),\\ {C}_{3}=\displaystyle \frac{1}{\sqrt{3}}\left({b}_{1}{f}_{-1}+{c}_{1}{e}_{-1}+{a}_{1}{d}_{-1}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{c}{C}_{4}=\displaystyle \frac{1}{\sqrt{3}}\left({b}_{0}{f}_{1}+{c}_{0}{e}_{1}+{a}_{0}{d}_{1}\right),\\ {C}_{5}=\displaystyle \frac{1}{\sqrt{3}}\left({b}_{0}{f}_{0}+{c}_{0}{e}_{0}+{a}_{0}{d}_{0}\right),\\ {C}_{6}=\displaystyle \frac{1}{\sqrt{3}}\left({b}_{0}{f}_{-1}+{c}_{0}{e}_{-1}+{a}_{0}{d}_{-1}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{c}{C}_{7}=\displaystyle \frac{1}{\sqrt{3}}\left({b}_{-1}{f}_{1}+{c}_{-1}{e}_{1}+{a}_{-1}{d}_{1}\right),\\ {C}_{8}=\displaystyle \frac{1}{\sqrt{3}}\left({b}_{-1}{f}_{0}+{c}_{-1}{e}_{0}+{a}_{-1}{d}_{0}\right),\\ {C}_{9}=\displaystyle \frac{1}{\sqrt{3}}\left({b}_{-1}{f}_{-1}+{c}_{-1}{e}_{-1}+{a}_{-1}{d}_{-1}\right).\end{array}\end{eqnarray}$
In order to acquire a precise depiction of the evolution of the input OAM entangled state, it is necessary to calculate the average of the density matrices across all potential scenarios within the turbulence medium. The output density matrix is determined as follows3 ). A total of $N=500$ such runs are performed. The simulated total transmission distance is dependent on the parameters of turbulence and OAM beam. Since the overall propagation distances in the simulations can vary significantly based on the different parameters, different values of ${\rm{\Delta }}z$ are used in different simulations.
$\begin{eqnarray}{\rho }_{{\rm{out}}}=\displaystyle \frac{\displaystyle {\sum }_{n}^{N}\left|{\psi }_{n}\right\rangle \left\langle {\psi }_{n}\right|}{{\rm{Tr}}\left\{\displaystyle {\sum }_{n}^{N}\left|{\psi }_{n}\right\rangle \left\langle {\psi }_{n}\right|\right\}},\end{eqnarray}$
where $\left|{\psi }_{n}\right\rangle $ represents the state of the qubit after the photons propagate through the nth turbulence phase screen. In the study, it is necessary to utilize six optical fields in order to simulate each iteration of the input state. Each optical field is represented by a $256\times 256$ array of samples of the complex-valued function for the mode specified in equation (The entanglement negativity and fidelity are employed to quantify entanglement in a high-dimensional OAM state. The entanglement negativity is defined as [22]
$\begin{eqnarray}\varepsilon =\displaystyle \frac{1}{2}\displaystyle \displaystyle \sum _{n}\left(\left|{\lambda }_{n}\right|-{\lambda }_{n}\right),\end{eqnarray}$
where ${\lambda }_{n}$ represents the eigenvalues of the partial transpose of the density matrix ${\rho }_{{\rm{AB}}}^{{\rm{T}}}.$ ${\rho }_{{\rm{AB}}}^{{\rm{T}}}$ denotes the partial transpose of ${\rho }_{{\rm{out}}}$ with respect to A photon’s subsystem, and the entanglement fidelity is given by [8] $\begin{eqnarray}F\left({\rho }_{{\rm{in}}},{\rho }_{{\rm{out}}}\right)=\left\langle {\psi }_{{\rm{in}}}\right|{\rho }_{{\rm{out}}}\left|{\psi }_{{\rm{in}}}\right\rangle .\end{eqnarray}$
In these studies, the entanglement negativity and fidelity are plotted as function of a single dimensionless parameter, ${w}_{0}/{\rho }_{0}$ and ${\rho }_{0}$ is the coherence parameter, which can be expressed by [28]
$\begin{eqnarray}{\rho }_{0}={\left(A\left(\alpha \right)B\left(\alpha \right){\tilde{C}}_{n}^{2}{k}^{2}z\right)}^{-\displaystyle \frac{1}{\alpha -2}},\end{eqnarray}$
where $B\left(\alpha \right)=\tfrac{-{(2)}^{4-\alpha }{\pi }^{2}{\rm{\Gamma }}\left(\left(2-\alpha \right)/2\right)}{\left(2{\left(8/\left(\alpha -2\right){\rm{\Gamma }}\left(2/\left(\alpha -2\right)\right)\right)}^{\displaystyle \tfrac{\alpha -2}{2}}{\rm{\Gamma }}\left(\alpha /2\right)\right)}.$ Nearly all of the dimension parameters of the system (the initial beam radius ${w}_{0},$ the wavelength $\lambda ,$ the propagation distance $z,$ the refractive index structure constant ${\tilde{C}}_{n}^{2}$ and the generalized exponent $\alpha $ ) are combined into ${w}_{0}/{\rho }_{0}.$ 3. Numerical results and analysis
Here, the impact of anisotropic non-Kolmogorov turbulence on a high-dimensional OAM entangled state is investigated through numerical simulation. In the following numerical examples, anisotropy coefficient ${\xi }_{x}=1,$ ${\xi }_{y}=2,$ generalized exponent α = 3.8, a generalized refractive-index structure parameter ${\tilde{C}}_{n}^{2}=1\times {10}^{-15}{{\rm{m}}}^{3-\alpha }.$ The outer scale and inner scale of turbulence are set to be ${L}_{0}=10\,{\rm{m}},$ ${l}_{0}=1\,{\rm{mm}},$ respectively. The beam parameters are set to be ${\ell }=1,$ ${w}_{0}=0.3\,{\rm{m}},$ $\lambda =600\,{\rm{nm}},$ unless otherwise specified. The results are shown in figures 1–3.
Figure 1. Negativity as a function of ratio ${w}_{0}/{\rho }_{0}$ for the different values of the azimuthal modes. |
Figure 2. Negativity as a function of ratio ${w}_{0}/{\rho }_{0}$ for different turbulence parameters. For different anisotropy coefficient (b) of generalized exponent, (b) of turbulence inner scale, and (d) of turbulence outer scale. |
Figure 3. Fidelity as a function of ratio ${w}_{0}/{\rho }_{0}$ for the different values of the azimuthal modes. |
The error bars represent the dispersion of each run from the mean.
Figure 1 illustrates the entanglement negativity plotted against the ratio ${w}_{0}/{\rho }_{0}$ for different values of the azimuthal modes. As shown in figure 1, the decay of negativity is slower for high-dimensional OAM entanglement with a lower azimuthal mode. The observation deviates from the original trend that OAM entanglement with higher OAM modes exhibits stronger robustness in weak scintillation [8–13]. This can be understood as follows. For different initial higher-dimensional OAM entangled states, the neighboring modes are all separated by $\left|{\rm{\Delta }}{\ell }\right|=1$ and $\left|{\rm{\Delta }}{\ell }\right|=2.$ That is to say, the coupling strength between two neighboring OAM modes in different initial OAM entangled states is the same. Additionally, a beam with a higher-order OAM mode experiences significant crosstalk than one with a smaller OAM mode. Hence, the negativity entanglement persists longer for lower values of $\left|{\ell }\right|.$
It should be noted, however, that previous studies have indicated that there is no universal decay trend of entanglement in higher-dimensional OAM state in turbulence, based on the azimuthal mode [4]. Additionally, the negativity curves intersect with each other. The study of our results indicates that adjusting the input state enhances the robustness and prolongs the decay of negativity entanglement for high-dimensional OAM entanglement with a lower azimuthal mode under weak scintillation conditions. As a result, it is crucial to propose a robust input state in higher dimensions that can effectively mitigate the impact of turbulence.
Figure 2 presents that the entanglement negativity is plotted against the ratio ${w}_{0}/{\rho }_{0}$ for different turbulence parameters. It can be seen in figure 2(a) that as ${\xi }_{y}$ increases, the negative entanglement persists longer. This can be understood as follows. An alternative explanation for this phenomenon is the change in the curvature of the anisotropic turbulence eddies [29, 30]. With the increase of ${\xi }_{y},$ anisotropic turbulent eddies can be treated as lenses with longer radii of curvature. The OAM beam passing through an anisotropic turbulent eddy will experience less deviation from the propagation direction. This results in an OAM beam with a smaller radius and improved performance. Therefore, as ${\xi }_{y}$ increases, the influence of anisotropic turbulence eddies on high-dimensional OAM entanglement decreases. It can be observed from figure 2(b) that the entanglement negativity gradually decreases in anisotropic turbulence. Nevertheless, the entanglement of high-dimensional OAM lasts longer in turbulent conditions with a larger generalized exponent. This means that the impact of anisotropic turbulence on the evolution of entanglement in the high-dimensional OAM state becomes smaller when the refractive index generalized exponent increases. It is reasonable, since the normalized intensity profiles of OAM beams spread more rapidly when the generalized exponent of the refractive index is smaller [29]. Figure 2(c) shows that the curves illustrating the evolution of entanglement negativity are identical for all values of the turbulence inner scale. This suggests that the impact of the inner scale of anisotropic turbulence on high-dimensional OAM entanglement is minimal and can be ignored. This can be understood as follows: the main effect of the inner scale on the OAM beam is to cause the beam intensity fluctuations (scintillation) in the beam. The intensity fluctuations of OAM beam are weak in the weak scintillation regime. Thus, the influence of inner scale on the evolution of high-dimensional OAM entanglement is negligible. Figure 2(d) illustrates that high-dimensional OAM entanglement is less affected by anisotropic turbulence with a smaller outer scale. This implies that turbulence with a smaller outer scale has less influence on the evolution of high-dimensional OAM entanglement. This can be understood as follows: The outer scale forms the upper limit of the inertial range. A decrease in the outer scale is equivalent to an increase in the strength of turbulence. In this case, OAM-entangled photons will encounter more turbulent eddies along its propagation paths, resulting in a more significant impact on high-dimensional OAM entanglement.
Figure 3 illustrates the fidelity is plotted against the ratio ${w}_{0}/{\rho }_{0}$ for different azimuthal modes. It can be observed from figure 3 that the fidelity gradually decreases in a turbulent atmosphere. However, the fidelity of a high-dimensional OAM entangled state with a lower azimuthal mode and a smaller beam radius decays slowly. This indicates that a high-dimensional OAM entangled state with a smaller beam radius and a smaller azimuthal mode will be more robust to turbulence.
4. Conclusions
In summary, the transmission of pairs of photons carrying high-dimensional OAM entanglement across an anisotropic non-Kolmogorov turbulent atmosphere is investigated using the multiple phase-screen simulation method. Here, the initial high-dimensional OAM state is assumed to be a maximally entangled qutrit state, which has a stronger coupling between adjacent OAM modes. Two photons are considered to have both passed through the anisotropic turbulent atmosphere. The effects of OAM beam parameters and turbulence parameters on the evolution of high-dimensional OAM entanglement in weak scintillation regime are evaluated. The numerical findings suggest that in the anisotropic non-Kolmogorov turbulent atmosphere, the OAM entanglement exhibits a consistent decrease. However, high-dimensional OAM entangled states with lower azimuthal mode values and a smaller beam radius are more robust in a turbulent atmosphere. This suggests that utilizing a high-dimensional OAM entangled state with a smaller azimuthal mode value and a smaller beam radius can improve the performance of the quantum communication system. Furthermore, high-dimensional OAM entanglement decreases monotonically as the ratio of the anisotropy parameter in the $x$ direction to that in the $y$ direction, i.e. ${\xi }_{x}/{\xi }_{y},$ decreases, which mean anisotropy in turbulence generally plays a positive role in OAM entanglement-based communication. Anisotropic non-Kolmogorov turbulence with a larger generalized exponent and a small turbulence outer scale has a reduced impact on the evolution of high-dimensional OAM entanglement. In addition, the influence of the inner scale of anisotropic turbulence on the evolution of high-dimensional OAM entanglement is minimal and can be ignored. Our findings will provide valuable insights into the behavior of high-dimensional OAM entanglement in turbulent atmosphere and have significant implications for secure and efficient quantum communication in practical applications.
Acknowledgments
This work was supported by the Project of the Hubei Provincial Department of Science and Technology (Grant Nos. 2022CFB957, 2022CFB475) and the National Natural Science Foundation of China (Grant No. 11847118).