1. INTRODUCTION

The orbital angular momentum (OAM) photon has attracted much attention, because in principle it has an infinite number of eigenstates, which not only can store and process quantum states in a high-dimensional Hilbert space, but also hold promise for many potential applications in quantum information processing [1–5]. However, when entangled OAM photons propagate across turbulence, the beam wandering, wavefront distortions, and scintillation destroy the coherence of OAM entanglement [6–10], which is the biggest challenge for realizing OAM-based quantum communication. Therefore, it is more imperative to describe the OAM entanglement evolution in turbulence so the OAM modes can be successfully applied to the free-space quantum communication.

Recently, the effect of atmospheric turbulence on entangled OAM photons has been addressed in many theoretical and experimental studies [7,11–20]. Most of them are based on the Kolmogorov power spectrum, which is widely accepted and applied to researching optical wave propagation in the atmosphere. Until now, it has been experimentally shown that, generally, atmospheric turbulence might possess a different structure from the conventional Kolmogorov one [21]. According to several observations, there have been significant deviations from the Kolmogorov turbulence model in the upper troposphere and stratosphere, as well as along the non-homogeneous path [22–24]. Besides, to our knowledge, in the Kolmogorov power spectrum, the influence of the inner and outer scales, which may be very important to entanglement evolution, was neglected in most previous studies.

In this paper, we have numerically studied the propagation of two OAM entangled photons through non-Kolmogorov turbulence. Through the use of the multiphase screen model, the influences of the azimuthal mode, the generalized exponent, the turbulent outer scale, and the turbulent inner scale on entanglement evolution in the weak scintillation regime are investigated in detail. Meanwhile, the results drawn are compared with the Smith and Raymer theories (S&R) [16] and the result in Ref. [11]. The paper is organized as follows: the numerical procedure is introduced in Section 2. In Section 3, the numerical results are presented. The conclusion is presented in Section 4.

2. NUMERICAL PROCEDURE

The numerical setup is shown in Fig. 1. Without loss of generality, it is assumed that a source field produces a pair of photons, whose input state is a Bell state encoded by LG modes with the opposite azimuthal quantum number

We first consider the random refractive index fluctuation $\delta n(\mathbf{x})$, which in an inhomogeneous turbulent medium is presented as

 

Fig. 1. Source produces a pairs of OAM-entangled qubits, whose wave fronts get deteriorated as they propagate through a series of phase screens along the (horizontal) $z$ axis toward a detector.

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In the following discussions, the paper employs the non-Kolmogorov power spectral for the refractive index fluctuations, given by [22,23]

Generally, the turbulent phase screens can be realized by the spectral method [27,28], in which the phase screens are randomly generated by the fast Fourier transform (FFT) associated with the power density spectrum. It is given by

Because the turbulence phase screen is the sum of Eqs. (9) and (10), when $N_x$ and $N_y$ are large, the straightforward evaluation of Eq. (10) is time consuming in the case of no FFT. If $N_x=N_y=N$ is satisfied, then Eq. (10) becomes a fractional Fourier transform, which can be evaluated using the fast algorithm that has a complexity proportional to that of the FFT algorithm [30].

For the Kolmogorov turbulence statistics with a finite outer scale, the phase structure function can be written as [31]

 

Fig. 2. Comparisons between Eq. (11) and statistical mean of the structure function obtained from samples of 1000 phase screens generated with the spectral method. The black color is the results of Eq. (11), while the red represents the sample average results. The value of the Fried parameter is $r_0=3.18\mathrm{cm}$.

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Now let us consider the matrix elements of the output OAM state. Due to the random refractive index fluctuations in a turbulent atmosphere, the matrix elements of the output state are scattered into infinite-dimensional OAM space. However, during the measurement process, only the information that is contained in a finite OAM subspace can be extracted. As a result, the paper has post-selected the transmitted state within the desired OAM subspace (i.e., ${|\ell ⟩}_A{|\ell ⟩}_B$, ${|\ell ⟩}_A{|-\ell ⟩}_B$, ${|-\ell ⟩}_A{|\ell ⟩}_B{|-\ell ⟩}_A{|-\ell ⟩}_B$), and calculated the post-selected density matrices. In addition, to describe the evolution of bi-photon OAM states accurately, the ensemble averages of the post-selected density matrices corresponding to $M$ different instances of the turbulent medium are computed [12,13].

Finally, we normalize the post-selected density matrix and examine the decay of entanglement by calculating the concurrence $C(\rho )$ [32] from the normalized density matrix. The concurrence is plotted as a function, i.e., ${\omega }_0/{\rho }_0$, and ${\rho }_0$ is the coherence parameter, which can be expressed by [23,33]

It should be mentioned that the four optical fields are required to simulate each run 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 given in Eq. (2). Furthermore, in the paper, a number of $M=500$ such runs have been performed to obtain the post-selected density matrices.

3. NUMERICAL RESULTS

First, the numerical result is compared with the S&R theory [16] and the result in Ref. [11]. Here, the paper only considers the case in which both photons pass through atmospheric turbulence with wavelength $\lambda =750\mathrm{nm}$, radial index $p=0$, waist width $\omega =0.1\mathrm{m}$, generalized exponent $\alpha =11/3$, inner scale $l_0=5\mathrm{mm}$, outer scale $L_0=500\mathrm{m}$, and refraction index structure constant $C_n^2=2\times {10}^{-15}{\mathrm{m}}^{-2/3}$. This, of course, means that the evolution of the OAM entanglement is in the weak scintillation regime, where one dimensionless parameter (${\omega }_0/{\rho }_0$) is required to describe the evolution of the concurrence [13].

As shown in Fig. 3, the numerical result agrees well with the result in Ref. [11] but disagree with the S&R theory curve. This difference comes from the phase structure function employing a quadratic approximation in the S&R theory.

 

Fig. 3. Concurrence as a function of ratio ${\omega }_0/{\rho }_0$ when both the photons propagate through turbulence. In (a) $l=1$, in (b) $l=3$, in (c) $l=5$, and in (d) $l=7$.

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Now the entanglement evolution of OAM photons propagating in non-Kolmogorov turbulence is investigated by the numerical simulations presented in the previous section. Additionally, the case where both photons pass through atmospheric turbulence is also considered. The results are shown in Figs. 4–7.

 

Fig. 4. Concurrence as a function of ratio ${\omega }_0/{\rho }_0$ for different values of the azimuthal modes.

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Fig. 5. Concurrence as a function of ratio ${\omega }_0/{\rho }_0$ for different values of the generalized exponent.

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Fig. 6. Concurrence as a function of ratio ${\omega }_0/{\rho }_0$ for different values of the turbulence outer scale with $\alpha =3.6$.

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Fig. 7. Concurrence as a function of ratio ${\omega }_0/{\rho }_0$ for different values of the turbulence inner scale with $\alpha =3.6$.

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Figure 4 shows that the concurrence is plotted against the ratio ${\omega }_0/{\rho }_0$ for different values of the azimuthal mode $\ell$ with waist width $\omega =0.1\mathrm{m}$, radial index $p=0$, and wavelength $\lambda =750\mathrm{nm}$. The non-Kolmogorov spectrum parameters are set as $\alpha =3.6$, $l_0=5\mathrm{mm}$, $L_0=10\mathrm{m}$, and $C_n^2=2\times {10}^{-15}{\mathrm{m}}^{-3/5}$. As indicated by Fig. 4, the concurrence decays through the non-Kolmogorov turbulence, but the decay of concurrence becomes slower with the increase of the azimuthal mode, which means the entangled OAM photons with the larger azimuthal mode are less affected by turbulence. This can be understood as follows. With the azimuthal mode increasing, the OAM beam widens, and its phase front oscillates more rapidly, which implies that its spatial phase structure gets finer [11].

Figure 5 shows that the concurrence is plotted against the ratio ${\omega }_0/{\rho }_0$ for different values of the generalized exponent $\alpha$ with $\omega =0.1\mathrm{m}$, $\lambda =750\mathrm{nm}$, $p=0$, $\ell =1$, $l_0=5\mathrm{mm}$, $L_0=10\mathrm{m}$, and $C_n^2=2\times {10}^{-15}{\mathrm{m}}^{3-\alpha }$. As seen from Fig. 5, the concurrence decays slower in turbulence with the increase of the generalized exponent $\alpha$, thus indicating less influence on the OAM entanglement by the turbulence with the higher generalized exponent $\alpha$.

In Fig. 6, the concurrence is plotted against the ratio ${\omega }_0/{\rho }_0$ for different values of the turbulence outer scale with $\omega =0.1\mathrm{m}$, $\lambda =750\mathrm{nm}$, $p=0$, $\ell =1$, $l_0=5\mathrm{mm}$, and $C_n^2=2\times {10}^{-15}{\mathrm{m}}^{3-\alpha }$. It is deduced from Fig. 6 that the concurrence can survive longer as the turbulence outer scale decreases. This reveals that the OAM entanglement will be less affected by turbulence with a smaller outer scale. For $L_0=1\mathrm{m}$, shown in Fig. 6, the curve for the concurrence decays as a function of the ratio ${\omega }_0/{\rho }_0$ increases, but at the tail there is a fluctuation behavior, which might be due to fewer statistical samples.

Figure 7 shows the concurrence plotted against the ratio ${\omega }_0/{\rho }_0$ for different values of the turbulence inner scale with $\omega =0.1\mathrm{m}$, $\lambda =750\mathrm{nm}$, $p=0$, $\ell =1$, $L_0=10\mathrm{m}$, and $C_n^2=2\times {10}^{-15}{\mathrm{m}}^{3-\alpha }$. One finds from Fig. 7 that all the decay curves are nearly identical for different values of the turbulence inner scale, which means the influence of the turbulence inner scale on OAM entanglement can be ignored.

4. CONCLUSION

In conclusion, using the multiphase screen model, the paper numerically investigated the entanglement evolution of OAM photons propagating in non-Kolmogorov turbulence, especially focusing on the influences of the azimuthal mode and turbulence parameters on entanglement evolution in the weak scintillation regime. The case in which both photons passed through turbulence was considered. Our numerical results show that the entanglement decays through non-Kolmogorov turbulence, but for larger azimuthal modes, it decays more slowly under the same turbulence condition. Meanwhile, when the other parameters are fixed, the OAM entanglement is less affected by turbulence with a higher generalized exponent or a smaller turbulence outer scale. Moreover, the influence of the turbulence inner scale on OAM entanglement can be ignored. It is believed that these findings may be useful in applications in free-space optical communications.