Mitigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution using real-time adaptive optics with phase unwrapping
Zhiwei Tao, Yichong Ren, Azezigul Abdukirim, Shiwei Liu, Ruizhong Rao
MMitigating the effect of atmospheric turbulence onorbital angular momentum-based quantum keydistribution by real-time adaptive optics
Zhiwei Tao , Yichong Ren † Azezigul Abdukirim ShiweiLiu , Ruizhong Rao School of Environmental Science and Optoelectronic Technology, University ofScience and Technology of China, Hefei 230022, China Key Laboratory of Atmospheric Optics, Anhui Institute of Optics and FineMechanics, Chinese Academy of Sciences, Hefei 230031, ChinaE-mail: [email protected]
February 2021
Abstract.
Quantum key distribution (QKD) employed orbital angular momentum(OAM) for encoding enhances the system security and information capacity betweentwo communication parties. However, such advantages significantly degrade becauseof the fragility of OAM states in atmospheric turbulence. Unlike previous researches,we first illustrate the performance degradation of OAM-based QKD by infinite longphase screen (ILPS), which offers a feasible way for dynamically correcting realisticturbulence through adaptive optics (AO). Secondly, considering the failure of AO whileencountering the branch cuts, we evaluate the quality enhancement of OAM-basedQKD under moderate turbulent conditions by AO after implementing the wrappedcuts elimination. Finally, we demonstrate that in a realistic environment, real-timeAO can still mitigate the impact of atmospheric turbulence on OAM-based QKD evenin the high wind velocity regime.
1. Introduction
Quantum key distribution (QKD) [1–6] provides unconditionally secure random numbers[7–12] between two authenticated distant parties (Alice and Bob) based on theunclonable principle of quantum physics [13,14]. In conventional QKD schemes, photonscarrying spin angular momentum are usually used to encode information [15–20].However, the information capacity of this binary system is limited to 1 bit per photon,making it more vulnerable to eavesdroppers’ attacks. Unlike the encodings of qubits,the freedom of orbital angular momentum (OAM) [21], associated with each photonin a beam has azimuthal dependence phase e ilφ , where l represents the topologicalcharge or helicity, offer an alternative way to encode information in an infinitely largeHilbert space [22], moreover, the system with higher-order possess higher security a r X i v : . [ qu a n t - ph ] F e b itigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution by real-time adaptive optics m [38] . Unfortunately, the presence ofatmospheric turbulence remains the greatest chanllege to the effective implementationof OAM-based QKD in free-space communication. Random air refractive indexfluctuations caused by atmospheric turbulence disrupt the phase alignment of theoriginal transmitted optical fields, split the optical vortex into several individual vorticesand create the photonic OAM pairs during the propagation [43–45], all combinationsof which results in mode scrambling at receiver [46–52], leading the high dimensionalencoding lose its unique advantages.To mitigate to the adverse effects of atmospheric turbulence, the most commoncompensation strategy is to use adaptive optics (AO) [38, 53–57]. Recently, manynumerical experiments have been performed using AO correction in quantum levels[58–60]. However, all these scenarios are simulated by random phase screens, whichcan only be applied if the turbulence satisfies Taylor’s hypothesis [61]. (i.e., when thelaser pulse width is significantly narrower than the time scale of random refractive indexfluctuations, atmospheric turbulence is assumed to be stationary during this tiny timefraction.) To overcome this barrier, we adopt the infinitely long phase screen (ILPS) [62]to dynamically simulate the evolution of the average quantum bit error rate (QBER) ofOAM-based QKD.On the other hand, the occurrence of branch points [63] caused by intensitymodulation is accompanied by the implementation of atmospheric propagation,broadening the OAM spectrum distribution [60, 64] as well as leading to a reducedperformance of OAM-based QKD. These phase discontinuities complicate the phasecorrections onto a deformable mirror in a realistic AO system. Considering theinability of traditional AO while encountering the branch cuts [65–67], we evaluate theperformance enhancement of OAM-based QKD by AO after implementing the wrappedcuts elimination. We also demonstrate that such local area manipulation is physicallypossible and brings a significant improvement previously poorly corrected.However, it should be noted that this approach brings the drawback that it reducesthe response rate of the AO system [53,68] and tends to affect the correction performanceof the AO system while the turbulence is changing rapidly [69]. Concretely, the slope ofthe beacon light originally measured by the Hartmann wavefront sensor is not directlyconverted into the voltage signal of a deformable mirror, but the reconstruction of beaconlaser phase. Secondly, the reconstructed phase is repaired by phase unwrapping andthen converted into the voltage signal of a deformable mirror. To examine our schemefeasibility, we re-evaluate the capability of AO and conclude that the phase-unwrappedAO is still able to mitigate the impact of atmospheric turbulence on OAM-based QKD,even in the high wind velocity regime.This paper is structured as follows. Sec. 2 first briefly introduces the two itigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution by real-time adaptive optics
2. Physical model
A photon’s degrees of freedom, such as time-energy, OAM, and position momentum, aretypically physical sources employed to implement high-dimensional encoding. For OAM-based QKD protocol, we study the two mutually unbiased bases (MUBs) that consistof a group of successive OAM photon states and its Fourier conjugate angular basis | j (cid:105) = 1 / √ d (cid:80) Ll = − L | l (cid:105) exp ( − i πjl/ (2 L + 1)), where | l (cid:105) represents a single-photon stateof Laguerre-Gaussian (LG) modes [70] LG ,l ( r, φ, d = 2 L + 1 is the dimension of theencoding space. L is the maximum azimuthal indices used in this protocol. Without lossof generality, the LG modes at output plane z can be described in normalized cylindricalcoordinates by LG p,l ( r, φ, z ) = Aw z (cid:32) √ rw z (cid:33) | l | L | l | p (cid:18) r w z (cid:19) × exp (cid:20) − r w z + i (cid:18) kr R z + lφ − (2 p + | l | + 1) ϕ g (cid:19)(cid:21) (1)with A = (cid:112) p ! /π ( p + | l | )! representing the normalization constant and radial quantumnumber p , where L | l | p ( · ) is the generalized Laguerre polynomial, w z = w (cid:113) z/z R ) is beam waist with w being the beam waist at input plane. z R = πw /λ and k = 2 π/λ respectively denotes the Rayleigh range and the wave number, λ is the wavelength, R z = z (cid:2) z R /z ) (cid:3) is the radius of curvature and ϕ g stand for the Gouy phaseassociated with propagation phase in this protocol. A photonic OAM state | l (cid:105) is distorted while propagating through the atmosphericturbulence can be considered as a pure phase perturbation [46, 47] at weak scintillationscale. To investigate the influence of turbulence on OAM-based QKD under arbitraryscintillation conditions, the received field of propagated OAM states simulated by the itigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution by real-time adaptive optics Figure 1.
The source generates two mutually unbiased bases and sends throughatmospheric turbulence (modeled by a series of trapezoidal phase screens) toward adetector. split-step method [76] is substituted to the pure phase perturbation approximation. Asshown in Fig. 1, multiple turbulent cells of ILPS are arranged consecutively to forma phase-screen array for simulating two MUB propagation across the turbulence [77].For the convenience of presentation, the procedure representing the i -th realization ofphotonic state traveling through the turbulent air is represented as unitary operator (cid:98) U ( i ) turb instead of the extended Huygens-Fresnel integral [79]. Since the LG modes withthe same w form an orthonormal basis, after undergoing the unitary transformation,the received state becomes a superposition of several LG modes, described using ket-branotation as follows (cid:12)(cid:12)(cid:12) ψ ( i ) l (cid:69) = (cid:98) U ( i ) turb | l (cid:105) = ∞ (cid:88) p =0 ∞ (cid:88) l = −∞ c ( i ) pl | pl (cid:105) (2)with | pl (cid:105) corresponding to a single photon state of LG modes LG p,l ( r, φ, z ), where c ( i ) l,p = (cid:104) pl | (cid:98) U ( i ) turb | l (cid:105) . Notably, to attribute the distribution of intermodal crosstalk to theimpact of turbulence, the waist of LG basis is modified to w z [80] during the spectraldecomposition. The probability of finding one photon in the sent OAM state with index l at the receiving plane is written as [81] p ( i ) l = ∞ (cid:88) p =0 (cid:12)(cid:12)(cid:12) (cid:104) pl | (cid:98) U ( i ) turb | l (cid:105) (cid:12)(cid:12)(cid:12) (3)In our numerical experiments, since Alice randomly chooses her photonic OAMstates from two MUBs, we only concern about the spectral broadening of the initiallytransmitted modes within the subspace of the primary encoding basis. The modescrambling outside the subspace is regarded as atmospheric losses during the simulation.Besides, to get the projection measured matrix of OAM-based QKD, we normalize thecrosstalk probability of each incident OAM state. Under the coordinate representation,the projection measurement is expressed by [71] p ( i ) l k → l m = (cid:80) ∞ p =0 (cid:82) R (cid:82) π (cid:12)(cid:12)(cid:12) ψ ( i ) l k ( r, φ, z ) LG ∗ p,l m ( r, φ, z ) (cid:12)(cid:12)(cid:12) rdrdφ (cid:80) Ll m = − L (cid:80) ∞ p =0 (cid:82) R (cid:82) π (cid:12)(cid:12)(cid:12) ψ ( i ) l k ( r, φ, z ) LG ∗ p,l m ( r, φ, z ) (cid:12)(cid:12)(cid:12) rdrdφ (4) itigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution by real-time adaptive optics p ( i ) l k → l m represents the probability of measurement outcomes with OAM component l m when the incident OAM value equals to l k , and R is the radius of receiving aperture.In order to explore the impact of atmospheric turbulence on the secure key rateof OAM-based QKD, this simulation ignores the noise arising from the detector andexperimental instruments, such as dark counts [29], afterpulsing effect [82] et al. Hence,the bit error arises from the turbulent channel can be quantified by the average quantumbits error rate (QBER). For the given two MUBs, the average QBER in OAM basis [83] Q ( i ) OAM = 1 d L (cid:88) l k ,l m = − L,k (cid:54) = m ∞ (cid:88) p =0 (cid:12)(cid:12)(cid:12) (cid:104) pl m | (cid:98) U ( i ) turb | l k (cid:105) (cid:12)(cid:12)(cid:12) (5)Likewise, the average QBER in ANG basis Q ( i ) ANG can be also acquired from the aboveprocedure. By averaging the QBER over two MUBs, the total turbulent errors causedby atmospheric turbulence are described as Q ( i ) = 12 (cid:16) Q ( i ) OAM + Q ( i ) ANG (cid:17) (6)Finally, we evaluate how the information that is securely exchanged between bothparties before the postselection for OAM-based QKD by the minimum secret key rate,which can be calculated as [25, 83] r ( i )min = log d + 2 (cid:20) Q ( i ) log Q ( i ) d − (cid:0) − Q ( i ) (cid:1) log (cid:0) − Q ( i ) (cid:1)(cid:21) (7)
3. Numerical methods
The simplest method to simulate atmospheric turbulence is to use the random phasescreen [84] combined with subharmonic compensation [85]. However, the random phasescreens are based on the fact that the laser pulse width is significantly narrower thanthe time scale of random refractive index fluctuations [61]. Therefore, the dynamicalchallenges of AO correction for incident light traveling through turbulent air cannot beinvestigated within this method, such as simulating the execution rate of AO system isless than the variety time of turbulence. Moreover, the static assumption will lose itsusefulness when studying the influence of wind velocity on OAM-based QKD capability.The ILPS method gives a feasible solution to simulate the dynamic process of AOcorrection and the OAM photonic states propagating through the turbulent channel.Based on the initially generated random phase screen, the principle of the extensionalgorithm is summarized as two steps [62]: firstly, the new row or column X added inthe next iteration is generated from the part of data Z in the original phase screen. Theformula is as follows X = AZ + Bβ (8)where Z represents the last N col rows/columns of the phase screen, β is a Gaussianrandom vector with zero mean and its covariance equal to unity. The matrix of A and itigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution by real-time adaptive optics Figure 2.
Two independent infinite long phase screens, with (a) θ = π/ v = 1 m/s (b) θ = 3 π/ v = 1 m/s . B stand for the combination of covariance of the X and Z vectors. More details for thematrix A and B and their dimensional information can be seen in [62].Secondly, the covariance of X and Z vectors including the matrices (cid:10) ZZ T (cid:11) , (cid:10) XZ T (cid:11) , (cid:10) ZX T (cid:11) and (cid:10) XX T (cid:11) can be achieved from constructing the distance matrix L i,j ≡ R ( X i , Z j ), where X i and Z j is the i -th element in X and the j -th element in Z , and acting them on the phase covariance function, following the von-Karman rulespresented as below C ϕ ( r ) = (cid:18) πrL (cid:19) (cid:18) L r (cid:19) Γ (11 / / π / (cid:20)
245 Γ (cid:18) (cid:19)(cid:21) K / (cid:18) πrL (cid:19) (9)with the outer scale L and the Fried parameter r , where K / ( · ) is the McDonaldfunction and Γ ( · ) is the gamma function.It is worth noting that the conventional ILPS can only simulate the effects ofwind velocity changes in both horizontal and vertical directions, which significantlyincreases the limitation of utilizing the algorithm. For this reason, we modified theILPS algorithm proposed in [62], which can be divided into two parts: on the onehand, the wind velocity in arbitrary direction is decomposed into the superposition ofhorizontal and vertical directions in each iteration. For ensuring the simultaneity of themovement, the algorithm first detects whether the need demand for a new column inthe horizontal direction, followed by the vertical direction. It subtracts one pixel fromthe total displaced pixel number after each movement. On the other hand, after eachiteration of horizontal and vertical movement, the remaining movement margin of lessthan one-pixel number in both directions is automatically counted as a part of the totalmoved pixel number in the next iteration [86] (see examples presented in Fig. 2). itigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution by real-time adaptive optics A well-developed AO system comprises three primaryassignments: wavefront measurement, reconstruction, and beam stabilization. TheHartmann wavefront sensor measures the slope information of the beacon laser.Subsequently, it inverts the voltage signal through the reconstructor, which is employedto generate the phase profile of a deformable mirror. For the closed-loop system,the corrected phase is also employed to compensate the slope information [87], bothcombinations that continuously improve the phase reconstruction accuracy.Since the corrected phase generated by AO system is only an approximation of thebeacon laser phase after propagation, the correction capability of an entire system cancommonly be determined by the coefficient number of the beacon laser phase after theZernike polynomial decomposition [56, 57]. (e.g., for an ideal AO system, the order ofreconstruction can reach infinity, which usually considers as an upper boundary for theperformance enhancement of AO). For a realistic AO system with N -th order correctioncapability, the phase estimated from the deformable mirror can be expressed as (cid:101) ϕ ( i ) ( r, φ, z ) = N (cid:88) n =1 a ( i ) n ( z ) Z n (cid:16) rR , φ (cid:17) (10)where (cid:101) ϕ ( i ) ( r, φ, z ) is the i -th reconstruction phase of the beacon laser, Z n ( · ) is theZernike polynomial with n -th order and the coefficients a ( i ) n ( z ) are given by the overlapintegral a ( i ) n ( z ) = (cid:90) R (cid:90) π ϕ ( i ) ( r, φ, z ) Z n (cid:16) rR , φ (cid:17) rdrdφ (11)with ϕ ( i ) ( r, φ, z ) as the phase of the propagated beacon laser. After the reconstruction,the i -th realization of the corrected phase will imprint on the single OAM photon state.we express the action of AO by unitary operator (cid:98) U ( i ) AO ≡ exp (cid:8) − i (cid:101) ϕ ( i ) ( r, φ, z ) (cid:9) withket-bra notation (cid:12)(cid:12)(cid:12) (cid:101) ψ ( i ) l (cid:69) = (cid:98) U ( i ) AO (cid:98) U ( i ) turb | l (cid:105) (12)Finally, we emphasize that in our AO-aided OAM-based QKD scheme, the signalpart of the incident modes consists of photonic OAM states and their conjugate basisANG states. Besides, we employ the platform beam as a probe state to detectthe turbulent conditions [88]. Our simulations ensure that two beams are emittedsimultaneously and propagated coaxially to provide a better correction for the signalmodes. A realistic AO system has proven to be ineffectiveat compensating for the branch points that occur at the places of zero amplitude inoptical field [89]. These unavoidable phase cuts [63] degrade the performance of OAM-based QKD due to the inability of a continuous surface deformable mirror to correctthe discontinuous phase [65–67]. Besides, some previously correlated researches have itigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution by real-time adaptive optics . ϕ ( i ) ( r, φ, z )can be accomplished by adding an integer multiple of 2 π at each pixel of wrapped phase,which is mathematically expressed as [94]Φ ( i ) ( r, φ, z ) = (cid:40) ϕ ( i ) ( r, φ, z ) + 2 πk ( r, φ, z ) ( r, φ ) ∈ Gϕ ( i ) ( r, φ, z ) ( r, φ ) / ∈ G (13)where Φ ( i ) ( r, φ, z ) is the unwrapped perturbed phase, and k ( r, φ, z ) is the integer thatneeds to be solved, regarded as a multi-class classification problem, G stands for the setof minimum neighborhood containing the wrapped cuts. Considering the perturbedphase varies with a period of 2 π , we indicate that the above operation is entirelyphysically feasible. After implementing phase unwrapping, the decomposition coefficientis modified and can be re-evaluated by A ( i ) n ( z ) = (cid:90) R (cid:90) π Φ ( i ) ( r, φ, z ) Z n (cid:16) rR , φ (cid:17) rdrdφ (14)which leads to a significant performance enhancement compared with the previouspoorly correction.
4. Results
To investigate the undesirable impact of atmospheric turbulence on OAM-based QKDand its partially corrected by AO, the evolution of a single OAM state perturbed whiletraveling across the turbulent channel, including the specific order correction by AO,are illustrated. In Fig. 3, we plot the phase profiles of a state l = 3 for a singlepropagated realization with no correction (Fig. 3(a) and 3(d)), realistic correction [95](Fig. 3(b) and 3(e)) and ideal correction (Fig. 3(c) and 3(f)), where the upper andlower three diagrams respectively represent the results realized under weak and strongscintillation [96].Fig. 3(a) shows that the phase profiles of the quantum state are slightly distortedafter propagation. This circumstance becomes more severe under strong scintillation(Fig. 3(d)), which leads the ordered structure of phase distribution completelydisrupted. As shown in Fig. 3(a), we observe that the initial vortex splits into threeindividual vortices accompanied by the vortex-antivortex pairs regeneration near the itigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution by real-time adaptive optics Figure 3.
Phase distribitions for a single OAM state with azimuthal index l = 3, fora single atmospheric propagation with no correction ((a) and (d)), realistic correction((b) and (e)) and ideal correction ((c) and (f)), the upper and lower three plotsrespectively represent the realization under weak and strong scintillation correspondingto σ R = 2 .
235 and 0 . original vortex. Besides, the turbulence also causes a longer discontinuous cut in thephase distribution (For a detailed explanation, we refer the reader to Refs. [63, 64]).Fig. 3(b) and (e) show the phase profile repaired by realistic AO realization ((b) 30-order realization (e) 50-order realization). It is noteworthy that, in the weak scintillationregime, the longest phase cuts that occured in the perturbed phase is eliminated, leadingthe unwrapped phase is more approach to the unperturbed one. In the contrary, in thestrong scintillation regime, the corrected phase remains almost unchanged to the phaseprofile in the presence of turbulence. Fortunately, such circumstance is significantlyalleviated in the case of the ideal correction.The poorly correction is likely because the length of branch cuts becomes a crucialfactor affecting the AO correction effectiveness. In other words, in the weak scintillationregime, only a few branch cuts are generated such that these can be ignored when weimplement AO correction (see more discussions in Refs. [45]). However, although weeliminate all wrapped cuts in the strong scintillation regime, the accumulation of branchcuts eventually destroys the phase profile of the beacon laser.To quantitatively evaluate the capability of AO, the crosstalk probabilitydistribution (including the radial number scrambling) averaged over 500 realizationsof turbulence are presented in Fig. 4. We observe that in the weak scintillation regime, itigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution by real-time adaptive optics Figure 4.
OAM spectra, including radial number scrambling, averaged over 500realizations of turbulence with azimuthal index l = 3, for different circumstances ofcorrection (a), (d) no AO; (b), (e) realistic AO; (c), (f) ideal AO, the upper and lowerthree plots respectively represent the realization under weak and strong scintillation.All parameters are same as in Fig. 3. our enhanced AO scheme effectively restores the power into the initial OAM state evenin the presence of branch points. Conversely, in the strong scintillation regime, wenote that our AO loses its advantage to mitigate turbulence-induced crosstalk betweendifferent OAM modes with the accumulation of branch points. Further, we also findthe well-known result [60, 97, 98] that the spectral broadening peaks around the initiallytransmitted state. In the strong scintillation regime, the crosstalk distribution has twopeaks around l = 3 and l = −
3, the combination of which results in the average OAMof received states is small than the initial one [43].
From the above analysis of the propagation characteristics of a single OAM state, weemploy the AO correction to improve the quality of OAM-based QKD (without (a)and with (b) increasing the mode spacing in encoding subspace). The average QBERwithout, with realistic and with ideal AO corrected variation curve of the QKD systemusing 5-dimensional OAM state encoding recorded over half an hour are illustrated inFig. 5. we set the realistic correction capability of AO to 30-order and turbulent strengthto 2 . × − m − / during this simulation [99].As illustrated in Fig. 5(a), we find that without increasing mode spacing (i.e., using |− (cid:105) , |− (cid:105) , | (cid:105) , | (cid:105) and | (cid:105) for encoding), the average QBER can be mitigated from41 .
9% with a standard deviation of 6 .
14% to 11 .
1% with a standard deviation of 2 . . |− (cid:105) , |− (cid:105) , | (cid:105) , | (cid:105) and itigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution by real-time adaptive optics Figure 5.
Average QBERs of the QKD system without, with realistic and withideal AO corrected realization using 5-dimensional OAM state encoding over half anhour. Panel (a) and (b) are realized without and with increasing mode spacing inencoding subspace respectively. The inset is a QBER frequency histogram achievedfrom the variation curves of the average QBER. The dashed, dash-dotted and dottedlines in (a), (b) denote the mean value of the average QBER without ((a) Q = 41 . Q = 33 . Q = 11 .
1% (b) Q = 6 . Q = 2 .
3% (b) Q = 0 . C n = 2 . × − m − / . | (cid:105) for encoding, shown in Fig. 5(b)), we can alleviate the average QBER from 41 . .
1% with no correction, 11 .
1% to 6 .
9% with realistic correction and 2 .
3% to 0 . itigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution by real-time adaptive optics Figure 6.
Measured crosstalk matrix of the OAM basis (a), (d) without, (b), (e)with realistic, and (c), (f) with ideal correction respectively. The upper and lowerthree plots represent the realization without and with increasing mode spacing. Allparameters are same as in Fig. 5.
We describe how the average QBER and the secure key rate of OAM-based QKD changeswith respect to r under different dimensions of encoding in Fig. 7, where each columnfrom left to right represents the results without (Fig. 7(a) and (d)), with realistic (Fig.7(b) and (e)) and with ideal (Fig. 7(c) and (f)) AO correction respectively. The dashedvertical lines located at r ≈ . m in all six plots correspond to the onset of weakscintillation.Based on the results presented in Fig. 7(a), we observe that the average QBERgradually decreases as the turbulence becomes weaker. For OAM-based QKD withhigher dimension, the average QBER is overall superior to the lower dimensionalencoding in the strong scintillation regime, which is likely explained by the OAMeigenstates with higher azimuthal indices have larger beam size so that they are moresusceptible to atmospheric turbulence. Further, in the weak scintillation regime, allthese differences between different curves become smaller with the decreasing turbulentstrength. Besides, as illustrated in Fig. 7(d), we see that no positive key rate canbe achieved by OAM-based QKD of arbitrary dimensions under moderate to strongturbulence (i.e., r ranges from 0 . m to 0 . m ), which leads that both parties cannotbuild a secure communication link under this circumstance. Generally, we expect toimprove the security and communication rate of OAM-based QKD when implementingthe high dimensional encoding. However, we find in Fig. 7(d) that a lower secret keyrate is achieved in spite of a higher encoding subspace is employed when r ≈ . m itigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution by real-time adaptive optics Figure 7.
Average QBER (a)–(c) and secret key rate (d)–(f) of OAM-based QKDas a function of r with different dimensions of encoding. Different circumstances ofcorrection are considered: (a), (d) no AO; (b), (e) realistic AO; (c), (f) ideal AO,averaged over 300 realizations of turbulence. The dashed vertical lines at r ≈ . m in all six plots correspond to σ R = 1. The error bars represent the standrad error. Thecorrection capability of AO and mode spacing is set to 30-order and 1 in both cases. (i.e., a more secure communication link guaranteed by the OAM-based QKD of highdimensional encoding is destroyed by the atmospheric turbulence). Fortunately, highdimensional OAM-based QKD will reveal its unique advantages as the turbulent strengthgradually decreases. Therefore, it should be noted that for a given turbulent strength,selecting an appropriate encoded dimension is especially critical to acquire a betterperformance of OAM-based QKD (e.g., we choose d = 3 for r ≈ . m and d = 7 for r ≈ . m ).The central and last columns illustrated in Fig. 7 compares the QKD performanceachieved with realistic and ideal AO correction. Based on the comparison between Fig.7(a) and (b), we see that in the weak scintillation regime, the average QBER significantlydecreases with the weakening of turbulent strength, which implied that our enhancedAO is effective for improving the quality of OAM-based QKD. Specifically, we noticethat realistic AO can achieve the best performance under moderate turbulent condition(e.g., when r = 0 . m and d = 5, the average QBER decreases from 44% to 14%and the secret key improves from no positive keys to 0 .
605 bits/photon). In Fig. 7(e),we re-evaluate how the secure key rate changes with respect to r under the realisticAO correction. Compared with Fig. 7(d), we observe that the realistic AO correctioncan lead to the positive key rate in moderate turbulence and transform the evolutionof secret key rate becomes the horizontally panning version of the curves without AOcorrection. Further, with the help of ideal correction, we also achieve in Fig. 7(c) and (f)an upper boundary for the performance of OAM-based QKD, which leads the positive itigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution by real-time adaptive optics Figure 8.
Average QBER (a)–(c) and secret key rate (d)–(f) of OAM-based QKDas a function of r with different dimensions of encoding. Different circumstances ofcorrection are considered: (a), (d) no AO; (b), (e) realistic AO; (c), (f) ideal AO,averaged over 300 realizations of turbulence. The dashed vertical lines at r ≈ . m in all six plots correspond to σ R = 1. The error bars represent the standrad error. Thecorrection capability of AO and mode spacing is set to 30-order and 2 in both cases. key rate in all considered turbulent strengths (e.g., when r = 0 . m and d = 5, thesecret key rate improves from no positive keys to 1 .
04 bits/photon).We also enhance the performance of OAM-based QKD through increasing modespacing in encoding subspace. Comparing the results achieved between the twostrategies (Fig. 7 and 8), we note that in the strong scintillation regime, when themode spacing becomes 2, the average QBER of OAM-based QKD counterintuitivelyincreases compared with successive encoding (see the comparison between in Fig. 7(a)and 8(a), e.g., when r = 0 . m , the average QBER without AO correction improvesfrom 75 .
32% to 80 .
32% for d = 7, 68 .
98% to 73 .
95% for d = 5 and 56 .
49% to 60 . d = 3, more explanations are discussed in Refs. [38]). However, such performancedegradation does not appear when employing the ideal correction (see the comparisonbetween Fig. 7(c) and 8(c)), which is a compromise result between the degradation andcorrection. Besides, in the weak scintillation regime, we conclude that increasing themode spacing can partly improve the quality of OAM-based QKD, especially with thehelp of AO (e.g., when r = 0 . m and d = 5, the secret key rate improves from 1 . . .
04 bits/photonto 2 .
27 bits/photon with ideal AO correction). itigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution by real-time adaptive optics Figure 9.
Average QBER (a), (b) and secret key rate (c), (d) of OAM-based QKDas a function of correction order with different dimensions of encoding. Panel (a), (c)and panel (b), (d) represent the results realized without and with increasing modespacing respectively. All calculations in four plots are 300 realizations of turbulence.The error bars represent the standrad error. The turbulence level in four plots is C n = 1 . × − m − / . We evaluate how the performance of AO-aided OAM-based QKD changes with respectto the correction order under different dimensions of encoding in Fig. 9. We set C n = 1 . × − m − / during the simulation. It can be seen from the diagram thatthe quality of OAM-based QKD improves significantly as the correction order of AOincreases. However, such significant performance enhancement is only in the lowerorder correction range. The primary reason due to the turbulence-induced aberrationsis almost entirely concentrated in the low-frequency part. Since we set the turbulentstrength in the medium to strong turbulence range, we notice that when the correctionorder is larger than 100-order, the quality of OAM-based QKD still improves somewhatwith increasing order. We anticipate that the quality of OAM-based QKD might not besignificantly enhanced with increasing the correction order of AO in the weak turbulenceregime. In Fig. 9(b), we illustrate the results of increasing mode spacing in encodingsubspace. We observe that only a minor performance enhancement appears comparedto the successive encoding.Another feature can be seen in Fig. 9(c) and (d) is that when we increase thecorrection order of AO and mode spacing in encoding subspace, we can recover theadvantage of high-dimensional coding to some extent, but choosing 3-dimensional OAMstates for encoding is still the best choice under such turbulent condition. itigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution by real-time adaptive optics Figure 10.
Average QBER (a)–(c) and secret key rate (d)–(f) of 5-dimensional OAM-based QKD as a function of r with different sizes of receiving aperture. Differentcircumstances of correction are considered: (a), (d) no AO; (b), (e) realistic AO; (c),(f) ideal AO, averaged over 300 realizations of turbulence. The dashed vertical linesat r ≈ . m in all six plots correspond to σ R = 1. The error bars represent thestandrad error. All other parameters are same as in Fig. 7. The state-dependent loss [71, 103] caused by the diffraction effectssignificantly degrades the performance of OAM-based QKD in long-distancepropagation. In the two MUBs, an OAM eigenstate with a larger azimuthal numbersuffers from more significant loss and acquires more propagation phase [78] due to thelimited size of receiving aperture, leading to a reduced quality of QKD in the absence ofturbulence. In order to evaluate the impact of state-dependent loss on the performance ofOAM-based QKD, we re-examine how the average QBER and the secure key rate changewith respect to r under different sizes of receiving aperture. Here, we only consider theperformance degradation in five-dimensional OAM-based QKD. the adoption of otherparameters employed during the simulation is the same as that in subsection 3.As illustrated in Fig. 10(a), we observe that in the absence of AO correction and thechannel is under strong turbulent strength condition, reducing the receiving aperturecan partly mitigate the impact of atmospheric turbulence (e.g., when r = 0 . m , theaverage QBER decreases from 53 .
4% to 44 . itigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution by real-time adaptive optics Figure 11.
Average QBER of 5-dimensional OAM-based QKD as a function of r with different wind velocities, averaged over 300 realizations of turbulence. The dottedhorizontal line represents the ten times time servo lag of AO if assuming the bandwidthof AO is f AO = 200 Hz . The dash-dotted lines denotes the Greenwood time accordingto Eq. 15, Refs. [38,57] the crossing point of this dash-dotted lines and the dotted linerepresents the bandwidth of AO is large enough to provide a comprehensive correction.The correction capability of AO is same as in Fig. 7. The error bars represent thestandrad error. QBER (Fig. 10(b) and (c), e.g., when r = 0 . m , the average QBER increases from23 .
77% to 35 .
93% with ideal AO correction) and decrease to secret key rate (Fig. 10(e)and (f), e.g., when r = 0 . m , the secret key rate decreases from 2 .
89 bits/photonto 1 .
32 bits/photon with ideal AO correction). Such an observation likely due to afterimplementing AO correction, the aperture loss becomes the primary source of bit errorfor OAM-based QKD.
The mean-square residual phase error [53, 68] due to servo time lagbetween the wavefront measurement and its correction can also significantly affect thequality enhancement of OAM-based QKD achieved by AO. In the above simulation,we investigate the quality enhancement based on the assumption that AO correctionis implemented in real-time. However, such implementation can not be realized exceptthe Greenwood frequency is small enough for the servo bandwidth of AO system [107].In order to re-evaluate the impact of performance degradation caused by the time delayeffect on OAM-based QKD, we set the bandwidth of AO as one-fifth of the samplingfrequency of Hartmann camera [108] (i.e., if we assume the camera sample frequencyequals to 1
KHz , then f AO = 200 Hz ), and change the Greenwood frequency f G byadjusting the wind velocity through the following relationship [53] f G = 0 . vr (15)In Fig. 11, we consider how the average QBER changes with respect to r under different wind velocities. We observe that when the wind velocity is small(i.e., v = 1 m/s ), the evolution of average QBER almost coincides with the resultsachieved without time delay. On the contrary, with the increase of wind velocity, the itigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution by real-time adaptive optics r = 0 . m , the averageQBER increases from 28 .
34% to 40 .
54% for v = 5 m/s , from 28 .
34% to 49 .
78% for v = 10 m/s ). Besides, we also notice that when the turbulent strength becomes weaker,the deviation integrally inclines to be zeros. To explain the above phenomena, Fig.11 is horizontally divided into two part [109], we observe that when v = 5 m/s and theGreenwood time ( τ G = 1 /f G ) passes this threshold, the bandwidth of AO is large enoughto provide a comprehensive correction (see more discussions in Refs. [38]). However,such conclusions are not established for v = 10 m/s , which implies that in the highwind velocity regime, the bandwidth of AO needs to be enhanced to satisfy the fasttime-varying wavefront sampling and compensation. Finally, by comparing with theresults obtained without AO correction, we see that AO remains able to mitigate theperformance degradation even though it works in the high wind velocity environment.
5. Discussion and conclusion
In this paper, we employ the modified ILPS method to simulate the performancedegradation of OAM-based QKD in arbitrary turbulent strengths. The main ideabehind ILPS is to use the partial data from the previous phase screen to generatethe next iteration results, which implies the originally established phase screen with nocorrelation between two iterations is connected. An advantage of this design is thatthe wind velocity of turbulence can be simulated by adjusting the amount of motionin each iteration, which provides a feasible solution for simulating the employmentof AO to correct the turbulence-induced aberrations in real-time. However, we haveto highlight the shortcoming of our modified ILPS, which uses the spatial correlationof the phase screen between two iterations instead of the temporal correlation of theturbulence variations (e.g., Wind velocity variation increases the Greenwood frequencyof the turbulence [53, 69], however, ILPS does not perform well in the face of turbulencevariation in the windless environment), a topic needs to be addressed in the future.Considering the inability of traditional AO while encountering the phase cuts[65–67], we demonstrate the feasibility of AO that includes the wrapped cuts eliminationto alleviate the impact of atmospheric turbulence on OAM-based QKD significantly.We show that in the weak scintillation regime, realistic AO adoption for correctingturbulence aberration can lead to approximately 30% performance enhancement.However, in the strong scintillation regime, we observe a rapid reduction of the capabilityof AO and suggest that if we wish to obtain a high-quality performance enhancement,an advanced AO system is necessary for the realistic experiments [38].Since the additional noise contributions (such as state-dependent loss, time delayeffect of AO, etc.) degrade the high-quality enhancement achieved by AO, we re-evaluate the capability of AO and conclude that AO correction can rebuild the securecommunication channel and recover the high information capacity, even in the increasedloss and wind velocity environment. However, for the real-time corrected consideration, itigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution by real-time adaptive optics
Acknowledgments
Zhiwei Tao would like to thank Dr. Yichong Ren and Dr. Ruizhong Rao for their carefulreading and insightful suggestments of our manuscript. The work was financially sup-ported by the Anhui Provincial Natural Science Foundation (Grant No. 1908085QA37),National Natural Science Foundation of China (Grant No. 11904369), and State KeyLaboratory of Pulsed Power Laser Technology Supported by Open Research Fund ofState Key Laboratory of Pulsed Power Laser Technology (Grant No. 2019ZR07).
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