## Abstract

In this paper, we describe the study of the faithful propagation of entangled orbital angular momentum states of light under atmospheric turbulence. The spatial mode is encoded in the Ince-Gauss modes that constitute a complete family of exact and orthogonal solutions of the paraxial wave equation in an elliptic coordinate system. Adaptive optics is employed to protect the entanglement from degradation, in which the threshold of turbulence strength could be enhanced for a reliable entanglement distribution. We find that the evolution of entanglements relies on ellipticity and shows the opposite trend when adopting adaptive optics. The turbulence strengths, at which the concurrences of various entangled states become zero, are different without adaptive optics but almost the same with adaptive optics. The trace of the density matrix is independent of the different ellipticity with or without adaptive optics. We believe that this investigation is useful for long-distance quantum communications and quantum networks using orbital angular momentum as information carriers.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Spatial modes of optical field are currently the focus of intense research within the quantum information community [1–3]. Not only are they of fundamental interest, but they are also practically useful owing to their intrinsic orthogonality and discrete infinite Hilbert spaces [4,5]. Over the past decades, spatial modes has been extensively investigated in free-space optical communication in both classical and quantum cases. For example, classical communication using spatial modes has been demonstrated successfully over a distance of 143 km [6] and a transmission rate of 100 Tbits [7]. In the quantum region, the quantum key distribution [8] over a distance of $d=210 m$ and entanglement distribution over an intra-city link of 3 km have also been reported [9].

In free-space quantum communications optical links, one of the main challenges for transmitting spatially-structured photons faithfully is atmospheric turbulence, which distorts the wavefront and causes fluctuations in the intensity of the light beam. Theoretically, the dimensions of the Hilbert space of spatial modes are infinite, and turbulence usually maps photons prepared in a certain spatial mode onto a linear combination of multiple modes on an orthogonal basis, which leads to a loss of information [10]. Compared with a system using polarization encoding, this is a drawback for any system encoded with the spatial degree of freedom, as usually the entire Hilbert space is used in polarization encoding. To address the challenge of turbulence elimination of spatial modes, several theoretical and experimental approaches have been proposed in recent years [11–13].

In order to achieve a longer propagation distance of spatial photons, a compensation scheme, such as adaptive optics (AO), can be used to mitigate errors caused by turbulence [14]. Based on a wavefront sensor, deformable, and tip-tilt mirrors controlled by a real-time feedback loop, AO has developed into an indispensable part in large astronomical telescopes [15]. In recent years, AO has also been frequently applied in the study of quantum communication. Specifically, recent work also demonstrates the potential applications of AO in increasing the fidelity and reducing the entanglement losses of spatial states [10,16].

Commonly used spaces for spatial modes include the Laguerre-Gaussian (LG) basis, which denotes a state in a circular cylindrical coordinate system, and the Hermite-Gaussian (HG) basis, which denotes a state in the Cartesian coordinate system. LG modes are denoted as $LG_{n,l}$ where $l$ is the orbital angular momentum (OAM) index and $p$ is the radial index and HG modes $HG_{n_x,n_y}$ are described by the indices $n_x$ and $n_y$. Here, we study entangled states based on a class of continuous spaces of the so-called Ince-Gauss (IG) mode [17,18], which denotes spatial modes in the elliptic coordinate system and has both, LG and HG modes as special cases. It is found that the effect of turbulence on entangled states is ellipticity dependent. Entangled IG states with larger ellipticity are less likely to be affected by turbulence compared with those with smaller ellipticity. In addition, we applied AO during optical transmission, which greatly reduces entanglement attenuation. By employing AO, the opposite tendency is presented, that is, entangled IG states with smaller ellipticity turn out to be more robust. However, turbulence strength for states totally disentangled with adaptive optics is independent of ellipticity. In both cases, traces for different states degrade almost consistently. These results are counter intuitive because IG modes with different ellipticity span the same space. These phenomena have never been reported, and we believe that the findings of this study could lead to some new insights and novel techniques.

## 2. Theoretical background

#### 2.1 Ince-Gauss mode

LG and HG modes are well known as the exact solutions of the free-space paraxial wave equation (PWE)($\nabla ^2_t+2ik\partial /{\partial z})\Psi (r)=0$ in circular cylindrical and Cartesian coordinate systems, respectively. In addition to these two modes, Bandres introduced another, complete basis of the transverse eigenmodes IG modes [17], which are exact solutions to the paraxial wave equation in an elliptic coordinate system. Further, even IG modes are expressed as

In the extreme case of $\epsilon \rightarrow 0$, the IG modes will be transformed into LG modes, and the transition from IG modes to HG modes occurs when $\epsilon \rightarrow \infty$ [17,20]. These features make IG modes a promising candidate for enabling an understanding of continuous basis and basis-dependent effects.

Figure 1 shows the intensity distributions of several IG modes, including the special cases of IG modes tending to LG and HG modes, respectively.

#### 2.2 Model of turbulence

The refractive index of the atmosphere changes randomly with space and time, which causes random deformations of light beams during propagation. Statistical analysis must be employed as it is almost impossible to describe the refractive index exactly. Many current models for free-space channels can be traced back to the work of Kolmogorov [21,22], David L. Fried [23], and further extensions [24,25]. In 1941, Kolmogorov developed a statistical theory that illustrated that the statistics of the refractive index could be controlled by power spectral density [21]. This establishes a relation between the phase fluctuation and the power spectral density $\Phi (\kappa )$. In our simulations, we choose the modified von Kármán spectrum, which takes both the influence of an inner and outer scale into consideration [25,26]. The modified von Kármán spectrum could be described as

By applying the multi-phase screen method, the turbulence is expressed by equally spaced phase screens separated by a distance of $\Delta Z$ [10,28]. Each of them is represented by a matrix generated by the sub-harmonic method [27] and introduces phase fluctuation to the incident light beam. Following the experimentally demonstrated model [29], the propagation path is broken up into 10 $m$ cells. In each phase screen, the Fried parameter is given as $r_{0_{plane}}\approx (\frac {\Delta Z}{Z})^{-3/5 }r_0$, so that accumulated over all phase screens, it yields a channel that is approximately equivalent to $r_0$. After passing an individual phase screen, the light beam experiences a vacuum propagation performed with an angular spectrum propagator until the next phase screen [30].

#### 2.3 Adaptive optics

The concept of AO was originally developed for astronomy, but it has been applied to OAM-based classical communication [31,32]. Typically, a beacon Gaussian beam and signal beam are prepared in orthogonal polarization states. The Gaussian beam is then coupled with the signal light and sent along the same propagation path as the signal light. On the receiver side, the Gaussian beam is separated from the signal by a polarization beam splitter and then sent to the wavefront sensor. The wavefront sensor extracts the phase correction $\phi _c(r,\theta )$ from the Gaussian beam. Therefore, the corrected state is given as $\tilde {\psi }(r,\theta )=exp(-i\phi _c(r,\theta ))\psi (r,\theta )$, where $\psi (r,\theta )$ is the received signal beam [16].

The scheme described above is known as post-compensation, which corrects the wavefront at the receiver side. Furthermore, AO correction has also been applied at the sender side (pre-compensation) [33] and on both sides [32]. It has been claimed that pre-compensation offers some advantages over post-compensation [29]. As the light beam diverges over the propagation, this increase in beam diameter limits the effectiveness of post-compensation. However, the technique of pre-compensation has not yet been fully developed, as a rapid control method and precise alignment are required. Here, we focus on the most common post-compensation scheme.

## 3. Result and discussion

Without loss of generality, we set the initial state to be one of the Bell states

We assume a scenario in which both photons are transmitted through the turbulence. Then, the evolution of the entangled state can be described as $|\Psi '\rangle =U_{turb}|\Psi _0\rangle$ where $U_{turb}$ denotes the effects of the turbulence on an input spatial mode, while the corrected state can be described as $|\tilde {\Psi '}\rangle =exp(-i\phi _c(r,\theta ))U_{turb}|\Psi _0\rangle$ in the presence of AO.

As we do not have detailed knowledge of the turbulence, it is necessary to compute the ensemble average of the density matrix. Upon computing the ensemble average, the propagated state turns into a mixed state, as described by the density matrix

where $\mathcal {T}=Tr[\sum _{i=1}^{N}|\Psi _{i}\rangle \langle \Psi _{i}|]$ renormalizes the state, and subscript $i$ denotes the state for each run.Given the density matrix, entanglement can be quantified by concurrence [34], which is given by

Each run gives a pure state representing the evolution of Eq. (4) after propagation. A total of 400 runs were performed for each turbulence strength. The ensemble average is computed for these 400 states. All simulations are carried out on a $0.25m\times 0.25m$ square grid with $1024\times 1024$ points for a wavelength of 710 nm and an initial beam waist of 0.025 m. As shown in Fig. 2(a), concurrence is plotted against the dimensionless scintillation strength $w_0/r_0$ where $w_0$ is the beam waist and the Fried parameter $r_0=(0.423C^2_nk^2Z)^{-3/5}$ [22]. Here, we choose to fix the distance to be 1 km and the beam waist while changing the structure parameter $C^2_n$ to achieve different turbulence strengths.

Furthermore, as the intensity distribution of the Gaussian beam is mainly localized in a fixed area while IG modes have a broader intensity profile with higher modes, a beacon laser would no longer cover the entire area of the higher modes. Due to the fact that an IG mode $IG_{p,m,\epsilon }$ disperses with $\sqrt {p+1}$, we choose the beam waist of the Gaussian beam in AO to be 2.45 times ($p=5$) that of the waist of the IG beams to ensure that the beacon can overlap the area of the required modes.As metioned above, one of the main features of the IG modes $IG_{p,m,\epsilon }$ is that the basis with various $\epsilon$ is mathematically different, but physically equivalent. However, the role of ellipticity in turbulent transmission for an IG state is not completely clear. In [36], it was demonstrated that the influence of turbulence is basis dependent, whereby the fidelity difference of $IG_{4,0,0}$ and $IG_{4,0,4}$ can be up to $7\%$. This indicates that single HG modes, corresponding to the case when $\epsilon \rightarrow \infty$, are more robust against turbulence than LG modes. This effect stems from the symmetry of the phase profile of the HG modes [37,38]. The phase profile of HG modes is an eigenfunction of the displacement operator $\hat {T}(\Delta \vec r)$ which is given by $\hat {T}(\Delta \vec r)=exp(i\Delta \vec r\cdot \hat {p}/\hbar )$. $\Delta \vec r=\Delta x\hat {x}+\Delta y\hat {y}$, $\hat {p}=-i\hbar ((\partial /\partial x)\hat {x}+(\partial /\partial y)\hat {y})$ is the momentum operator, and $\hat {x}$ and $\hat {y}$ are unit vectors in $x$ and $y$ directions, respectively. Thus, the phase profile does not change when subjected to lateral displacement along the axis of symmetry caused by turbulence. With an increasing ellipticity, an IG mode becomes more symmetric with lateral displacement which allows the beam to experience less turbulence-induced loss than the mode with lower ellipticity. Here, we find that this is also the case for entangled IG states with different $\epsilon$. Figure 2(a) shows the evolution of entangled IG states with different ellipticity and establishes the result that states with higher $\epsilon$ are less likely to be affected by turbulence.

However, taking AO into consideration proved to be interesting. For all entangled states, a correction with AO significantly improves concurrence. The reason that the AO cannot fully recover concurrence can be partly attributed to intensity fluctuations or scintillations [39]. AO can compensate for phase distortions but not for scintillations in the strong turbulence regime. Actually, scintillation imposes a limit on the efficiency of AO.

Contrary to the case of entangled states with higher ellipticity being more robust in stronger turbulence, AO inverts the trend and the concurrence of states with small ellipticity is higher than those with large ellipticity. In [10], it was shown that AO is less effective for higher order LG modes owing to different beam geometries, as the beacon Gaussian beam has its maximum intensity at the center, while vortex beams have central singularities. It can be easily determined from Fig. 1 that states with large $\epsilon$ have more patterns with vanishing intensities. Thus, $IG_{p,m,0.0001}$ bears more similarities in intensity distribution with a Gaussian beam than $IG_{p,m,4}$ and $IG_{p,m,1000}$. This difference in geometry mitigates the wave-front correction from a Gaussian beam.

Another interesting result is that we denote the Fried parameter when the concurrence becomes zero as $r_{0end}$; those $r_{0end}$s are very close to each other in the presence of AO. This means that the entangled states become totally separable under the same strength of turbulence. However, those $r_{0end}$ without AO, shown in Fig. 2(a), increase with ellipticity. As AO could only compensate for the phase distortion but not the intensity fluctuation, the end of concurrence is mainly determined by the strength of the scintillation. Therefore, an intuitive conjecture is that the effects of scintillation are independent of ellipticity.

Photon loss is another key effect in quantum communication. This is related to the trace evolution of the density matrix in an open quantum system. The decay of the trace represents a turbulence-induced mode cross-talk outside the encoding subspace of $IG_{5,\pm 1}$ in our simulation. As shown in Fig. 2(b), the curves of the trace are very close to each other, and they look similar to a single curve. AO also significantly enhances the trace in all cases, and thus could improve the signal-to-noise ratio of the communication process. In both cases, all curves of the trace evolution coincide well. All IG modes are normalized and they can be mapped to other modes by a unitary transformation. Such a unitary transformation would not affect the trace of the density matrix which leads to the consistency of trace evolution. To note that the photon loss is determined by the strength of scintillation. The trace evolution confirms the conjecture that the effect of scintillation are independent of ellipticity.

The effects of turbulence cannot be fully compensated by AO. Thus, we present the phase and intensity fluctuations resulting in mode crosstalk, as shown in Fig. 3. This explains why AO enhances the concurrence and traces with different efficiencies. The crosstalk between $IG_{5,1,\epsilon }$ and $IG_{5,-1,\epsilon }$ is lower than that outside the encoding subspace, leading to a weak reduction in concurrence. On the contrary, the relatively high crosstalk outside the encoding space signifies a strong trace degradation.Finally, we provide an example of applying AO in quantum key distribution based on IG modes. We denote $IG^+_{5,1,\epsilon }$ and $IG^-_{5,1,\epsilon }$ as $|0\rangle$ and $|1\rangle$, respectively. The key rate of the BB84 protocol is given as $K=Q*(1-H_2(\delta _z)-H_2(\delta _x))$ [40], where Q is the photon loss, and $H_2(*)$ denotes the binary entropy. $\delta _z$ and $\delta _x$ are the error rates in the corresponding measurement basis. Figures 4(a) and 4(b) show the dependence of the quantum bit error rate (QBER) and the key rate on the turbulence strength, respectively. IG modes with larger ellipticity could provide a higher key rate and lower QBER which are natural results of stronger robustness against the turbulence. A dramatic improvement in key rate could be obtained in the presence of AO, which allows the quantum key distribution protocol to be implemented under stronger turbulence and over a further distance. The curves of the QBER without AO are not monotonic because the strong turbulence would increase the crosstalk outside the encoding subspace and cause a relatively small population of IG modes in the encoded modes. It should be noted that the low key rate in the absence of AO is mainly caused by photon loss but not QBER. The key rate in Fig. 4(a) is computed based on the qubit case, whereby only two spatial modes are encoded. Higher key rates can be obtained by encoding photons as qudits.

## 4. Summary

In summary, we use an empirical multi-phase screening method to investigate the propagation of a continuous class of spatial modes in free space. It turns out that the effects of turbulence on entangled spatial modes depend on ellipticity. Entangled states with higher ellipticity are more robust than those with lower ellipticity. Another quantity related to entanglement is the trace that marks the photon loss in the encoding space. The evolution of the trace which is determined by the scintillation, unlike concurrence, is independent of ellipticity.

Moreover, we also test the effectiveness of AO with regard to ellipticity. Although the efficiency of adaptive optics on the entangled spatial states has been revealed [16,31], we uncover some unforeseen features in this study. Concurrence in different bases shows an opposite trend compared with the case of without AO. That is, AO performs better with lower ellipticity. Another remarkable feature is that in the presence of AO, entangled states in different bases become separable under the same turbulence strength. The trace of the density matrix remains independent of ellipticity.

The phenomena described above are beyond our expectations. This could be a topic for future research to deepen our understanding of turbulence, as well as the spatial modes of light.

## Appendix

Here we present the effects of grid numbers which denotes sampling accuracy. The simulation is performed for single $IG_{5,1,\epsilon }$ photons propagation and a Fried parameter $r_0=1.98$. The results are presented in Fig. 5 where each data point is obtained by averaging 500 iterations and the fidelity shows convergence for both modes within the statistic uncertainty when $N\geq 550$ . Therefore the grid number in the simulation is large enough to represent the faithful propagation.

## Acknowledgement

We would like to thank Xuemei Gu and Anton Zeilinger for illuminating discussions and acknowledge the support of the National Key Research and Development Program of China under Grant No. 2017YFA0303700. The Key Research and Development Program of Guangdong province under Grant No. 2018B030325002, the National Natural Science Foundation of China under Grant No. 11974205, and the Beijing Advanced Innovation Center for Future Chip (ICFC).

## Disclosures

The authors declare no conflicts of interest.

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