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Polarization compensation method based on the wave plate group in phase mismatch for freespace quantum key distribution
EPJ Quantum Technology volume 10, Article number: 6 (2023)
Abstract
Maintaining the polarization state in communication terminals is vital for polarizationencoding freespace quantum key distribution (QKD). Wave plate group phase mismatch caused by manufacturing errors, complex environmental effects, and the working wavelength deviation can reduce the polarization compensation effect. We found in theoretical analysis, that increasing phase mismatch of wave plates leads to the compensation method failure and reduces robustness. We propose a complementary polarization compensation method, which can effectively improve the robustness. Experimental results show that this method can improve the compensation effect by 50% at a slight phase mismatch, and realize a polarization extinction ratio exceeding 250:1 at the ergodic area even if the phase deviates to 0.27π. This method is beneficial to the highstability design of freespace QKD systems and has the potential to be applied to QKD systems operating at multiple wavelengths.
1 Introduction
Quantum communication provides a new communication method that is only intrinsically unconditional secure in theory, and quantum key distribution (QKD) is the most practical in the area of quantum communication [1]. Longdistance QKD has become the focus of fierce international competition in recent years [2–5]. The current distance of QKD in optical fibers can be up to 830 km [6], while further breakthroughs in this limitation of distance and realizing QKD in a wide area has reached a bottleneck. Compared to optical fiber links, freespace QKD can ignore the limitations of terrains. Particularly, satellitebased freespace QKD can work in an extremely long distance owing to the ultralow transmission loss in vacuum. Based on this fact, freespace QKD has become a key part of the establishment of the global quantum communication network [7, 8].
In freespace QKD, the most accepted encoding scheme is to use the polarization state of photons, such as the BB84 protocol [9] and entanglementbased protocol [10]. The influence of the optical system on the polarization state increases the bit error rate because the carrier of quantum information is the polarization state of a single photon. The transmission through optical devices (such as optical fibers and mirrors) leads to the degradation of the polarization state [11, 12]. Therefore, polarization maintenance in the optical system becomes a key factor for the successful realization of freespace QKD.
In relative researches on polarization maintenance, polarization control at optical terminals is usually realized through polarizationmaintaining coating, phase compensation, and wave plate group compensation [13–18]. Wave plate group compensation is the most widely used method. In Ref. [14], a method of polarization compensation using a combination of three wave plates was proposed for the first time, and the polarization compensation for a fiber with polarization degradation was realized. After compensation, the polarization extinction ratio (PER) reached 200:1. In another Ref. [15], researchers calculated the rotation angles of wave plates with the stochastic parallel gradient descent algorithm and compensated the transformation matrix of an optical fiber; the PER successfully reached 500:1. The studies mentioned above focus only on the compensation for polarization degradation devices, whereas in the study by Micius [16], a systemwide polarizationmaintaining method for related terminals was introduced in detail. Polarizationmaintaining coating, phase compensation, wave plate group compensation, and other methods were comprehensively adopted, and the PER of the entire quantum communication terminal could exceed 500:1. Ref. [17] introduced a method of polarization maintenance for a ground transmitting antenna of the groundtosatellite entanglement distribution. The researchers successfully maintained the polarization state of the entire optical system using the methods of coating and wave plate group and compensated for the basis vector deviation caused by the satellite motion; the PER of the system reached 400:1.
The polarizationmaintaining method used in Ref. [16] and Ref. [17] has relatively high requirements for the polarizationmaintaining coating of optical components. In addition, once these optical components are degraded by the complex environment, this will directly affect the system performance. In order to solve the aforementioned problems, a realtime polarization compensation method was proposed in Ref. [18]. After obtaining the parameters of a polarization degradation optical element, the polarization compensation for the system can be directly performed by changing the angles of wave plates. Such a realtime polarization compensation method can greatly reduce the requirement of polarizationmaintaining coating for a single optical element.
The methods of polarization compensation mentioned above assume that the phase of wave plates matches the theoretical value. However, in reality, because of various factors such as manufacturing errors, complex environmental effects [19], and the working wavelength deviation, the used wave plates deviate from the ideal phase, resulting in unsatisfying results or failure in the polarization compensation. In addition, based on literature, there is no relative research on the polarization compensation under wave plate phase mismatch.
In this study, we extend the compensation method of the idealphase wave plate group in Ref. [14] to the case of the phase mismatch of wave plates, thereby improving the robustness and compensation results. In Sect. 2, we introduce the polarization compensation principle under the phase mismatch of wave plates. Furthermore, the influence of the phase mismatch on the robustness of the traditional compensation method and the robustness improvement method are studied. In Sect. 3, the performed experiments, obtained results, and relative discussions are presented. The experimental results show that the PER of the method has been improved to a certain extent compared with the traditional method. Almost all the ergodic points are compensated with a PER of more than 250:1 when the phase is near the mismatch boundary, which verifies the feasibility of the proposed method. In Sect. 4, conclusions are drawn. The successful implementation of the method will contribute to the highstability design of QKD systems. This method can also be applied in the polarizationmaintaining design of freespace QKD systems operating at multiple wavelengths, helping to reduce the cost and volume and providing a reference for the design of polarizationsensitive coherent optical communication terminals.
2 Method and numerical analysis
2.1 Method of polarization compensation
The polarization state of any optical system with unitary transformation can be compensated and restored by the wave plate group (1/4 wave plate, 1/4 wave plate, and 1/2 wave plate). As described in Ref. [14], the degenerated righthanded circularly polarized light can be restored by rotating the first two 1/4 wave plates. The righthanded light is always orthogonal to the linearly polarized light in Poincaré sphere because the optical system is usually a unitary transformation system. This is because the transmission of an optical fiber [14], rotation transformation, and phase delay of a mirror [18] can be considered as unitary transformations. The linearly polarized light is restored to the equatorial plane when the righthanded light is restored. Finally, the linearly polarized light is restored by rotating the 1/2 wave plate.
As shown in Fig. 1, if the 1/4 wave plates are replaced with wave plates with any phase retardance, the phase retardants of wave plate R1 and wave plate R2 are assumed to be \(\delta _{1}\) and \(\delta _{2}\) respectively, and the compensation angles are assumed to be \(\theta _{1}\) and \(\theta _{2}\) respectively. The following equation should be satisfied when the righthanded circularly polarized light is restored:
where \(R_{d} = [1,SR_{1},SR_{2},SR_{3}]^{T}\) is the degenerated righthanded circularly polarized light, \(R = [1,0,0,1]^{T}\) is the compensated ideal righthanded circularly polarized light, and \(M_{\delta _{i},\theta _{i}}\) can be expressed as the Mueller matrix [20]:
When the condition of Eq. (1) is satisfied, righthanded circularly polarized light R is restored with
When the righthanded polarized light is restored, there is still the basis vector deviation between the linearly polarized light in the Poincaré sphere equatorial plane and the anticipated value. In freespace QKD, a basis vector deviation between the transmitter and the receiver is inevitable [11, 21, 22]. Generally, the receiver is equipped with a nearly perfect 1/2 wave plate for basis vector correction [22, 23], which can help to achieve highfidelity transmission of quantum keys from the transmitter to the receiver.
Substituting expressions of \(R_{d}\) and R and Eq. (2) into Eq. (3), the following expression is obtained:
From Eq. (4), we obtain
where the following condition needs to be satisfied:
Equation (6) can also be expressed as
When \(\delta _{1} = \pm \delta _{2} = \frac{\pi}{2} + k\pi \), \(k = \cdots,  2,  1,0,1,2,\ldots \) , \( 1 \le SR_{3} \le 1\). In other words, when the standard 1/4 wave plates are used, \(R_{d}\) can be compensated at any position of the Poincare sphere. When the phases are mismatched, \(R_{d}\) can only be compensated at the limited position of the Poincaré sphere, as shown in Fig. 2(a), thus reducing the method robustness.
In order to improve the reduced robustness caused by the phase mismatch, when the righthanded circularly polarized light cannot be restored to the original position through the wave plate, \(R_{d}\) can be rotated to the lefthanded circularly polarized light position, as shown in Fig. 1. At this time, the linearly polarized light can be restored to the equatorial plane of the Poincaré sphere, whereas a phase delay of π is present between ±45° linearly polarized light causing state interchanging. In this case, QKD can still be performed by changing the channel. In the process of correcting the basis vector deviation [22], interchanged state can be easily observed. The code correction can be realized by publishing this information through the public channel.
When \(R_{d}\) is rotated to the lefthanded circularly polarized position \(L = [1,0,0,1]^{T}\), Eq. (5) becomes
where the following condition needs to be satisfied:
Equation (9) can also be expressed as
It is worth noting that, in this complementary method, the equations can be solved when either Eq. (7) or Eq. (10) is satisfied. They can be integrated as
Because \( 1 \le SR_{3} \le 1\), the system will have complete solutions only when \(\delta _{2} = \delta _{1}\) or \(\delta _{2} =  \delta _{1}\). At this time, it has
In the complementary case, both equation groups have solutions when the following conditions are satisfied:
If the equation group is 100% solvable, the following conditions need to be satisfied:
In this situation, \( 1 \le SR_{3} \le 1\). All the points on the Poincaré sphere can be satisfied, and the robustness is the best.
2.2 Numerical simulation analysis
The solution existence regions (blue and black) at certain phases in the noncomplementary and complementary methods according to Eq. (8) and Eq. (11) are shown in Fig. 2(a) and (b), respectively. Before the complementary method, owing to the phase mismatch, the robustness is obviously reduced. After the complementary method, the solution existence area increases, and the two phases are equal or have opposite signs to achieve the best robustness.
Figures 2(c) and (d) show the ratio of the solution existence area over the total area of the Poincaré sphere under the certain phases calculated by traversing each point (Mongo Carlo) according to the Eq. (7) and Eq. (10), respectively. Obviously, the complementary method can greatly improve the robustness, and the range with the best robustness is consistent with the range described by Eq. (12), which further verifies the effectiveness of our method. Therefore, as long as the phase retardants of the wave plates are guaranteed to be equal and within this range, the \(R_{d}\) traversing the entire Poincaré sphere can be compensated.
It is worth noting that, in practical engineering applications, the 1/4 wave plates used in the same system are usually of the same batch, and the phases of the two wave plates are basically the same. Furthermore, the installation positions of the two wave plates and the environment are almost the same; thus, the environmental effects are relatively close. Therefore, in practical engineering applications, generation of two wave plates with the same phase retardance is relatively easy. In this case, we can easily reach high robustness when using the polarization compensation method in this study.
3 Experiments and results
3.1 Robustness verification experiment
To verify the robustness of our method, we designed as experimental setup shown in Fig. 3(a). It includes a \(R_{d}\) generation system (\(R_{d}\) generator), a polarization compensation system (compensator), and a polarization analysis system (polarization analyzer). By changing the angles of P and QWP, we can obtain the \(R_{d}\) traversing the Poincaré sphere. After degradation polarization light \(R_{d}\) is compensated by two phasemismatched 1/4 wave plates R1 and R2, the polarization analysis system composed of a polarizer and power detector detects the effect of polarization compensation.
In this experiment, we used two 850 nm zeroorder quartz 1/4 wave plates. The phase retardances of the two wave plates at different wavelengths were also different. At 850 nm, the phases were 0.496π and 0.493π, respectively, at 1064 nm – were 0.400π and 0.394π, and at 1550 nm – were 0.269π and 0.269π. The phases of the two wave plates within the 850–1550 nm band were both within the range of 0.25π–0.75π. According to the numerical analysis, the corresponding Poincaré sphere solution existence regions at 850, 1064, and 1550 nm are shown in Fig. 4.
We used two 850 nm wave plates to perform compensation on arbitrary righthanded circular degradation polarization light at 1064 nm. The \(R_{d}\) generator was composed of a polarizer and nearideal 1064 nm QWP. A total of 20 × 15 ergodic points on the Poincaré sphere were generated by rotating the polarizer and the QWP. The corresponding relationship between the sequence of each sampling point and the parameters of Stokes vector is shown in Fig. 5(a).
After theoretical calculation and experimental compensation, the compensated PER of the degradation polarization light at each point on the Poincaré sphere is shown in Fig. 6(a). The absolute value of the fourth term of the Stokes parameters after compensation is shown in Fig. 6(b). The PER after compensation at each point is less than 1.3, and the corresponding fourth term of the Stoke parameters is close to 1, being close to the ideal circularly polarized state. The results near the poles of the Poincaré sphere are consistent with the results at other points. Thus, this method can help to improve the robustness of polarization compensation.
3.2 Demonstration of the phase mismatch compensation system in a QKD experiment
The results of the abovementioned robustness experiments show that, using our method, the nonideal wave plate has the same polarization compensation ability as the ideal wave plate. This polarization compensation method can be applied to quantum communication. The wavelength of QKD is extended to 1550 nm based on the requirements of daytime quantum communication [24, 25]. In order to balance the needs of the high bit rate of quantum communication at night and the needs of allday time, nextgeneration quantum communication terminals will be equipped with both 850 nm and 1550 nm quantum QKD.
To this end, we designed an experiment with an optical system capable of simultaneously compensating polarization degradation at 850 nm and 1550 nm to verify the feasibility of the method for practical system compensation. As shown in Fig. 3(b), the experimental setup consists of a BB84 module (different BB84 modules are used for different wavelengths), polarization degradation simulator, polarization compensation system, and polarization analysis system. The fourchannel laser of the BB84 module generates H, V, +, − linearly polarized light through a polarizer, PBS, and BS. The optical polarization degradation simulator is composed of two nearperfect 1/4 wave plates, which can simulate the unitary matrix of the optical system with arbitrary polarization degradation. The compensation wave plate adopts the same 850 nm zeroorder quartz 1/4 wave as in the robust experiment.
According to the \(R_{d}\) traversing the Poincaré sphere, we can reverse the corresponding angles of the two QWPs of the degradation simulator at each point. We sampled 16 × 10 points for each wavelength. The corresponding relationship between the sequence of sampling points and the Stokes parameters is similar to Fig. 5(a). The \(R_{d}\) on the Poincaré sphere corresponding to the sampling point is shown in Fig. 5(b), and the corresponding H, V, +, and − degradation polarization states caused by the degradation simulator are shown in Fig. 5(c). The experimental results are shown in Fig. 7.
Compared with the traditional method (850 mt), our method (850 mis) has a certain increase in PER from an average of 2089:1 to 3068:1. The traditional method does not consider the phase error, which leads to a decrease in accuracy, whereas our method solves this problem.
The 850 nm PER of our method is higher than 250:1 at every point, and the PER at 1550 nm is lower than 250:1 at only two points (but also nearby), which is comparable with the result reported in Ref. [14]. The polarization angle is consistent with the theoretically calculated result. After the basis vector correction, compared with Fig. 5(c), the disorderly distributed polarized light is compensated to the expected position. In the region where the solution does not exist in the traditional method, there are no obvious differences in PER compared to the other regions, and the polarization angle is also consistent with the theoretical value. The results above reveal that the proposed method has good robustness and compensation effect.
Even if the phase reaches up to 0.269π, our method still has a good compensation effect. It can be inferred that, within the phase range of 0.25π–0.75π, we can still obtain a satisfying compensation effect, while the phase of the wave plates with a phase error is sufficient to be included in this range.
4 Conclusions
In this study, we extend the compensation method of the idealphase wave plate group proposed in Ref. [15] to the case of phase mismatch of wave plates. We explore the phase mismatch influence on the robustness of the proposed and traditional compensation methods. The experimental results show that the proposed method can improve the polarization extinction ratio (PER) by 50%. Almost all ergodic points are compensated with a PER of more than 250:1 when the phase is near the mismatch boundary, which verifies the method feasibility. The successful implementation of this method will contribute to the highstability design of freespace QKD systems.
The method can be applied in the polarizationmaintaining design of freespace QKD systems operating at multiple wavelengths, reducing the cost and volume and providing a reference for the design of polarizationsensitive coherent optical communication terminals.
However, challenges remain in the application of this method to practical systems, for example, insitu highprecision measurement on the wave plate parameters and unitary matrix of an optical system and compensation of complex dynamic systems, which will inevitably affect the compensation effect of the method. These challenges will be the focus of future research.
Availability of data and materials
Raw experimental data and calculations can be obtained from the corresponding author upon a reasonable request.
Abbreviations
 QKD:

quantum key distribution
 PER:

polarization extinction ratio
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Funding
National Science Fund for Distinguished Young Scholars (62125505), National Key R&D Program of China (2020YFB2205900),the Shanghai Science and Technology Major Project (No. 2019SHZDZX01), and the Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDB35000000).
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Yongjian Tan: Conceptualization, Methodology, Investigation, Writing Original Draft, and Experiments. Jincai Wu: Conceptualization, Formal analysis, Writing Original Draft, Supervision, and Methodology. Zhiping He: Supervision, Writing  Review & Editing, and Funding acquisition. Liang Zhang: Project administration, Funding acquisition. Tianxing Sun: Writing  Original Draft. Zhihua Song: Validation.
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Tan, Y., Zhang, L., Sun, T. et al. Polarization compensation method based on the wave plate group in phase mismatch for freespace quantum key distribution. EPJ Quantum Technol. 10, 6 (2023). https://doi.org/10.1140/epjqt/s40507023001634
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DOI: https://doi.org/10.1140/epjqt/s40507023001634