1. Introduction

Quantum communications, as a new and special type of optical communications, can provide extremely high security because of the fundamental nature of quantum mechanics [1,2], and has been a topic of intense interest for over three decades. Compared with fiber-based quantum communications, free-space quantum communications could provide a more appealing solution for secure communications over much larger distances [3,4], and can be applied to a global quantum communications network. Moreover, satellite-to-ground quantum communications are expected to be the best choice for building a global secure communications network [5,6].

Typically relying on polarization-coding protocols, such as BB84 [7] (see Sec. 3.1), free-space quantum communications use photon polarization to carry information, in which the sender (Alice) and receiver (Bob) must share common reference frames with respect to the polarization. Unfortunately, in free-space quantum communications, there may be moving stations, such as satellites. During communication, the telescopes of Alice and Bob will move with the motion of the satellite in order to maintain the link. Consequently, this will produce deflection and pitching, and will change the reference frame of the responding terminal [8]. Once the reference frame of one end of the link has changed, there will be reference-frame deviation between the two ends of the link; we call this basis deviation. In the case that Alice sends signals to Bob according to her reference frame, and Bob detects the signal with respect to his own reference frame without compensating for the basis deviation, the quantum-bit error ratio (QBER) may become intolerably high [9]. Compensating for the basis deviation is clearly a key issue that must be settled in satellite-to-ground quantum communications.

Although transmission of quantum bits (qubit) without alignment between sender and receiver being required has been realized by using larger alphabets [10], such approaches require the sender and receiver to maintain constant locations. They may therefore be more suitable for ground-to-ground quantum communications but not for satellite-to-ground quantum communications. Another approach to realize quantum communication without a shared reference frame is using entangled states [11]; however, entangled states are more difficult to generate and measure than single-photon states. Recently, a new implementation of the BB84 protocol that employs a d-dimensional Hilbert space was presented to communicate qubits without reference frame alignment [12]. Employing a d-dimensional Hilbert space means increasing the bits per photon that can be sent; using this protocol may increase the key generation rate, but doubling or tripling the key generation rate may not be so simple using current technology.

One approach to rectify this problem is the polarization-basis tracking scheme recently proposed by Toyoshima [13]. It adjusts the two basis vectors (the horizontal axes of the reference frames) of the two platforms (send and receive) to a common axis related to the reference-laser polarization at the receiver side. This means the system needs to add a reference laser and a polarization modulator on the transmitter. Such an approach will consume excessive resources in terms of weight, space, and energy. In practice, in free-space quantum communication, especially in satellite-to-ground quantum communication, resources are too restricted in a satellite to implement the polarization-basis tracking scheme.

In this paper, we propose a basis-deviation compensation scheme that can overcome the inconvenient and burdensome restrictions discussed above, and can be better suited for ground-to-satellite quantum communications. This scheme is similar to the time-multiplexing compensation technique [14]; however, the probe beam used here is one of the sender’s quantum signals, instead of an independent probe pulse. Therefore, we can use a BB84 decoding module to detect the change of polarization of the probe beam, i.e., the basis deviation. We also use a half-wave plate to provide basis compensation. The compensation accuracy obtained is ~2°. Furthermore, the BB84 decoding module can also be used to monitor the accuracy of the compensation scheme.

This paper is organized as follows. In Sec. 2, we introduce the basis deviation produced by a moving satellite and analyze the QBER caused by basis deviation. In Sec. 3, we develop an equation to calculate the basis deviation with the BB84 decoding module and implement a laboratory experiment to verify the detection accuracy of the BB84 decoding module. Subsequently, we develop a basis-deviation compensation system in Sec. 4, and test it on a voyage experiment. We draw conclusions in Sec. 5.

2. Analysis of basis deviation

2.1 Basis deviation resulting from satellite motion

A working ground-to-satellite quantum communications system involving a low-Earth orbiting satellite (LEOS) and an optical ground station (OGS) necessitates an acquisition, tracking, and pointing (ATP) system [15] in order to create an optical communications link. In the ATP procedure, shown in Fig. 1(a), the satellite first adjusts its telescope to point towards the OGS and emits a beacon light. Then, the OGS rotates its telescope to catch the beacon light and replies with a response signal. When the satellite acquires the response signal, the optical communications link is created. Afterwards, both of their telescopes will remain pointing towards each other while the satellite moves. Through the stable optical communications link, the satellite sender (Alice) transmits polarized qubits according to her reference frame to the OGS receiver (Bob). Bob deciphers the qubits in his own reference frame. In the case of a moving satellite, a problem arises in that the basis vectors of Alice and Bob will change as a result of the movement of the rotating telescopes (e.g., deflecting and pitching), as shown in Fig. 1(b). This implies a basis deviation occurring between their reference frames.

 

Fig. 1 Influence of satellite motion on basis vectors in a satellite-to-ground quantum communications link. (a) The ATP process for a satellite communicating with the OGS while the satellite executes its orbit. Points A and B are two specific points on the orbit. (b) The basis vectors of the satellite and the OGS change with the rotation of the telescope. H1A and H2A represent the basis vectors of the satellite and OGS, respectively, at Point A, and H1B and H2B represent the deflected basis vectors of the satellite and OGS, respectively, at Point B.

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To describe the basis deviation clearly, we consider two points (Points A and B) on the satellite orbit, as shown in Fig. 1. We assume that at Point A the satellite and the OGS share the same reference frame, which means the basis vectors of the satellite and the OGS (H1A and H2A, respectively, in Fig. 1(b)) are in the same direction and no basis deviation occurs between them. As the satellite travels to Point B, subject to the control of the ATP system, the telescopes of the satellite and the OGS will rotate to face each other so that the communication link will be maintained. Then the OGS basis vector will be deflected from H1A to H1B with deflection angle ${\theta }_1$; correspondingly, the satellite basis vector will be deflected from H2A to H2B with deflection angle ${\theta }_2$ (${\theta }_1$ and ${\theta }_2$ are vector angles with the positive direction anticlockwise along the optical axis). In that case, at Point B Alice and Bob no longer share the same reference frame, but have a basis deviation given by:

In practice, a basis deviation of $\theta$ is not so easy to obtain as suggested by Eq. (1); ${\theta }_1$ and ${\theta }_2$ are both determined by the rotation angle (both horizontal deflection and vertical pitching) of the OGS and the satellite telescopes. Calculating these quantities is not trivial and depends on the parameters of specific telescopes and optical systems; every mirror and beam must be taken into consideration. Furthermore, we must find the position of the satellite to calculate ${\theta }_2$. Auxiliary equipment (e.g., GPS, inclinometer, and radio) must be fixed on the satellite to acquire the posture parameters and transfer them to the OGS. In addition to the increased burden that this equipment places on the satellite, the transmission will take a certain time before arriving at the OGS. When the OGS obtains the value of ${\theta }_2$, the satellite has already moved into another position with a different ${\theta }_2$.

Fortunately, it is unnecessary to be concerned with all the problems above. We can directly detect the value of the basis deviation with the BB84 decoding module; the details of this method will be discussed in Sec. 2.3. First, we must determine the fraction of QBER due to basis deviation.

2.2 QBER caused by basis deviation

Owing to the intrinsic non-classical features of quantum mechanics [16], it is not possible to perform a measurement on a quantum system without modifying it; therefore, an eavesdropper will unavoidably introduce errors in the transmission of data when they intrude. According to the BB84 protocol, the QBER is the criteria for judging the presence of an eavesdropper. Alice and Bob will use a suitable subset of data to estimate the QBER after the transmission, and proceed with using the remaining data to establish a secret key only if the QBER is below a certain threshold. Otherwise, they will discard the transmission data and begin a new data transmission while concluding that an eavesdropper is monitoring the communications.

In actual implementation, however, inevitable systematic errors, such as channel loss, polarization error, and background photon noise, will increase the QBER and increase the difficulty in observing an eavesdropper. Thus, the threshold of QBER for observing an eavesdropper must be restricted to within a reasonable range, else the security of the quantum communication will be threatened.

There are two main factors in the systematic polarization error: phase delay and basis deviation. Phase delay is caused by problems in the optical elements (i.e., faulty film coatings and unsuitable materials will cause serious phase delay) and in the free-space channel (i.e., atmospheric conditions; however, this effect is secondary with respect to the contribution from the optical elements) [17]. Phase delay will lead to elliptic polarization [18] and decrease the signal’s extinction ratio (ER), which is the ratio of the greatest energy versus the smallest energy when a light penetrates a polarizer.

In elliptic-polarized light (see Fig. 2(a)), the phase delay can be recorded as elliptic angle $\alpha$, and the ER can be described as the quadratic power ratio of major axis $a$ versus minor axis $b$:

 

Fig. 2 The ER of elliptic-polarized light: (a) the profile of elliptic-polarized light detected by the H/V base and (b) the profile of elliptic-polarized light with a basis deviation of $\theta$ detected by the H/V base.

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Additionally, the basis deviation will cause the polarized signal to diverge from its original orientation and will further increase the QBER. We assume an H-polarized signal has a basis deviation of $\theta$ related to Bob’s frame (refer to Fig. 2(b)). If Bob uses his H/V base to detect the signal, then the ratio of the energies detected by the H basis versus the V basis would be:

From Eq. (5), we can easily draw the conclusion that the fraction of QBER caused by polarization error is the sum of the QBERs produced by the phase delay and by the basis deviation. Thus, the QBER produced by the basis deviation is:

 

Fig. 3 The results for (a) the fraction of QBER caused by polarization error $N\text{'}$ and (b) the fraction caused by a basis deviation of $\Delta N$ as functions of basis deviation and ER.

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In Fig. 3(a), we can see that the smaller the ER is, the larger that the QBER grows for the same basis deviation, and that the growths of the QBER with basis deviation for different ERs are similar, which we can also see clearly in Fig. 3(b). A quantum communication system maintains a relatively fixed ER, while the basis deviation continues to change with the rotation of the satellite. Therefore, we must maintain the ER as high as possible and reduce the basis deviation via compensation to a low level to ensure a low QBER.

In our experiment, the sending and receiving terminals can ensure an ER of the transmitted signal greater than 200, in which the QBER produced by phase delay is below 0.5%. Furthermore, we want to compensate for the basis deviation so that $\Delta N$ is below 0.5% as well. According to Eq. (6) and Fig. 3(b), in the case that $ER=200$ and $\Delta N<0.5\%$, basis deviation should be reduced via compensation to less than 4°. In the following section, we provide a basis-deviation detection scheme and a compensation approach, and test their precisions in a tabletop experiment and a voyage experiment, respectively.

3. Basis-deviation detection scheme and experimental verification

3.1 Detection scheme for basis deviation

The BB84 decoding module is used in most polarization-encoded quantum communication systems, and is used as a polarization decoder in the receiver. It can divide an incident signal into quadriform signals with specific polarizations (that is, 0° (H), 90° (V), + 45° ( + ) and −45° (−) polarizations) according to the receiver’s reference frame, and it can also obtain their photon-counts [19]. Through the Stokes characterization of the quadriform signals, we can deduce the polarization of the incident signal. If we transmit a probe beam with known polarization according to the sender’s base to the receiver, the polarization can be calculated according to the receiver’s base. By comparing the calculated polarization with the original polarization, we can obtain the basis deviation between the sender and receiver.

A standard BB84 decoding module is shown in Fig. 4. A beam splitter (BS) splits the incident signal (IS) into reflection and transmission paths. In the BS’s reflected path, the polarizing beam splitter (PBS) PBS1 separates the signal into transmission and reflection paths according to the H and V polarizations, respectively. In the BS’s transmission path, the submodule consisting of PBS2 and a half-wave plate (HWP), the fast axis of which is at 22.5°, rotates the signal from a +/− ( + 45° and −45° polarizations) base into a H/V base so that the PBS2 can separate the + and − polarizations into the transmission and reflection paths, respectively. The quadriform signals are focused by coupling lenses into optical fibers and transmitted to four avalanche-diode photo-detectors (APD).

 

Fig. 4 The standard BB84 decoding module. IS: incident signal; BS: beam splitter; HWP: half-wave plate; PBS: polarization beam splitter; P: polarizer; FL: focusing lens; APD: avalanche-diode photo-detector.

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In reality, the optical components are not ideal, especially the optical coatings. For example, the coating on the BS will cause a phase delay and efficiency differences between the H and V components in both the reflected and transmitted paths. We can test the BS to determine those efficiency differences and phase delays. We denote the efficiencies by $R_H$, $R_V$, $T_H$, and $T_V$, and the phase delays by ${\delta }_1$ (reflected phase delay) and ${\delta }_2$ (transmitted phase delay). The ERs of the reflected and transmitted components of both PBSs are always greater than 1000; thus, we can take the phase delays of the two PBSs to be 0. However, the energy efficiencies of the transmission and reflection paths of the PBSs are different. By testing the energy efficiencies of the PBSs together with the focusing lens and APDs, we obtain the energy efficiencies of the four paths, denoted as ${\tau }_1$, ${\tau }_2$, ${\tau }_3$, and ${\tau }_4$.

The four APDs can acquire the photon-counts of the quadriform signals, enabling us to obtain $E_H$, $E_V$, $E_{\text{+}}$, and $E_{\text{-}}$. Through the Stokes vector of BB84 decoding module, we can obtain the polarization angle $\theta$ of the incident signal using the photon-counts of the quadriform signals [20]:

Having obtained this basic result, we now apply it to a typical scenario. We assume that Alice (located on the satellite) sends a probe beam (an H-polarized signal, for example) based on her H/V basis to Bob (located on the ground). Owing to the satellite motion, the signal does not have a polarization of H with respect to Bob’s H/V basis, but instead has a deviation of $\theta$. Nonetheless, Eq. (7) enables Bob to compute the deviated polarization of the signal according to his H/V basis. Using the calculated polarization, subtracting the original polarization gives the basis deviation between Bob and Alice; this is how the BB84 decoding module can detect basis deviations.

3.2 Laboratory experiment

In order to demonstrate the feasibility of this approach, we built a tabletop BB84 decoding module with discrete optical elements (Fig. 5). We attenuated a light source (LS) into a quantum signal. We used a polarizer, P0, to polarize the signal with a polarization angle of ${\theta }_r$ in order to simulate different basis-deviation angles. The quarter-wave plate HP0 rotated with P0 to produce a phase delay in order to simulate the ER of an actual signal. The ER of the signal was maintained in the range between 800 and 180. Four APDs acquired the photon-counts of the signal and transmitted them to a computer in order to calculate the polarization ${\theta }_c$ of the signal.

 

Fig. 5 The tabletop detection system with the BB84 decoding module.

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Comparing ${\theta }_c$ with ${\theta }_r$ gives the precision of the BB84 decoding module in detecting polarization:

The calculation error ${\eta }_c^{}$ is caused by the variations in the APDs, including variations in dark counts, $C_d$, and in the variation of photon detection efficiency (PDE), $P_v$. In the case that the photon count detected by an APD is C, we find that the total variation of the APD is:

We assume that the actual PDEs of the four APDs are $P_H$, $P_V$, $P_{+}$, and $P_{-}$, so the energies deduced from the counts detected by the four APDs in Eq. (8) are actually $P_H{E}_H$, $P_V{E}_V$, $P_{\text{+}}{E}_{\text{+}}$, and $P_{-}{E}_{\text{-}}$. In order to simplify Eq. (7), we assume all the other conditions are ideal, with exactly $R_H=R_V=T_H=T_V=0.5$, ${\delta }_2=0$, and ${\tau }_1={\tau }_2={\tau }_3={\tau }_4=0.5$. We also note that $k=P_H/P_V$, $x\text{=}{E}_H/E_V$, $q=P_{\text{+}}/P_{\text{-}}$, and $y\text{=}{E}_{\text{+}}/E_{\text{-}}$. Then, the calculated polarization angle is:

 

Fig. 6 (a) The relationship between calculation error and signal polarization angle. (b) Calculation angle as a function of q and k when the signal polarization angle is 45°.

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Figure 6(a) shows that the calculation error ${\eta }_c^{}$ reaches its greatest value when $\theta =45°$, and has a repeating period of 45°. We performed many simulations to verify this result, including changing the value of the ER and$\Delta P$. Figure 6 is plotted with $ER=200$ and $\Delta P=10\%$. With this result, we can analyze the calculation angle $\theta \text{'}$ as a function of q and k when the signal polarization angle is 45°, and find that $\theta \text{'}$ reaches its maximum value when k is smallest and q is greatest, and that $\theta \text{'}$ reaches its minimum value when k is greatest and q is smallest. Therefore, ${\eta }_c=\theta {\text{'}}_{max}-\theta {\text{'}}_{min}$ is exactly:

According to the scale precision of P0 and the installation precision of the HWP and PBS, we get ${\eta }_r=0.2$, ${\eta }_h^{}=0.4$, and ${\eta }_p^{}=0.2$. Because the BS cannot affect the polarization of a signal, we ignore its error. By bringing these parameters into Eq. (10), we find $\eta =0.78°$. Therefore, the detection precision of the BB84 decoding module, $\Delta \theta$, should be closed to 0.78°.

In order to test $\Delta \theta$, we use P0 to polarize the incident signal over a 90° range in 5° increments. In addition, the polarization angle ${\theta }_r$ can be directly transmitted to the computer through the motor-stage mounting of P0. Then the APDs of the BB84 decoding module acquire the photon-counts and transmit them to the computer in order to calculate the polarization angle ${\theta }_c$. Employing Eq. (9), we arrive at the detection precision of the BB84 decoding module. Figure 7 gives the results for (a) the calculated polarization and (b) the detection accuracy as functions of basis deviation.

 

Fig. 7 The results for (a) the calculated polarization and (b) the detection accuracy as functions of basis deviation.

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Based on Fig. (6), we conclude that the BB84 decoding module can detect basis deviation effectively. The mean value of $\Delta \theta$ is ~0.09° and its root-mean-square (RMS) value is 0.22. The RMS of $\Delta \theta$ is somewhat large, indicating that the detection precision of the BB84 decoding module is not quite stable. As we can see in Fig. 6, $\Delta \theta$ is quite scattered in the range of $-0.3~0.43°$, and its variation is ~0.73°. This is very close to the system error of the module, $\eta =0.78°$. Therefore, we can conclude that the detection precision of the BB84 decoding module is ~0.73°.

Owing to the intrinsic characteristics of the BB84 decoding module, particularly the variation of the APD, the accuracy of the module can barely be improved. However, according to Eq. (6), with an accuracy of 0.73°, the fraction of QBER caused by basis deviation after compensation is just 0.016%. Moreover, in actual applications, there is no P0 in the detection system; thus, the accuracy may in fact be higher. Therefore, we can conclude that this detection scheme is suitable for adaptation in a basis-deviation compensation approach.

4. Basis-deviation compensation system and the Zhoushan voyage experiment

4.1 Compensation system for basis deviation

As we can reliably measure the basis deviation, compensating for it will be straightforward. We use the HWP to rotate the polarization of the incident light in order to provide the compensation. When the fast-axis orientation of the HWP is at angle $\beta$, the Jones matrix of the HWP can be written as [18]:

 

Fig. 8 The light-rotation characteristics of the HWP, showing (a) the incident light and (b) the output light.

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To make the system lighter and more convenient, we developed a novel basis-deviation compensation scheme to guarantee that the receiver obtains the correct polarization of the signal. As we discussed in Sec. 3.1, the BB84 decoding module can detect the basis deviation via a probe beam; in that case, we can use a HWP mounted on a motor stage to compensate for the basis deviation, as shown in Fig. 9.The probe beam can be any of the four quantum photons (H, V, + , −) sent by Alice, and shares the same communication channel of the quantum photons. This will tend to reduce the weight and simplify the sender and receiver of a satellite-to-ground quantum communication system and will satisfactorily compensate for the basis deviation.

 

Fig. 9 The basis-deviation compensation system in Bob’s terminal.

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The system operates as follows. Alice and Bob first locate each other’s positions through a classical communication system, and create the quantum communication channel using the ATP system. Alice then sends a probe beam to Bob; we use H-polarized photons by way of example. The signal passes through the HWP and enters the BB84 decoding module. Bob acquires the counts from the four APDs and calculates the polarization of the signal. If the calculated polarization is H, then the HWP maintains its position; otherwise, Bob will control the HWP, rotating it to the appropriate angle to compensate for the deviation of the probe beam until the test result is H.

During this time, the BB84 decoding module is continuously detecting the probe beam to see if its polarization direction is parallel to H. Thus, the BB84 decoding module has an additional capability other than basis-deviation detection—that is, measuring the precision of the basis-deviation compensation system.

We applied the basis-deviation compensation system to the Zhoushan voyage experiment and the results detected by the BB84 decoding module demonstrate its practical utility for satellite-to-ground quantum communications.

4.2 Zhoushan voyage experiment

Typical problems associated with satellites, such as satellites’ motion, gesture swing, and atmospheric turbulence, must be dealt with before we can put the satellite-to-ground quantum communications system into practice. However, to comprehensively overcome such problems requires the use of a real satellite. Therefore, we adopted a hot-air balloon to simulate a satellite and successfully characterized the problems of satellite gesture swing and atmospheric turbulence [3]; however, a hot-air balloon does not move in a fashion that adequately simulates the motion of a satellite. Therefore, we carried out the Zhoushan voyage experiment to simulate a satellite more comprehensively.

The Zhoushan voyage experiment is the improved version of the hot-air balloon experiment. We located the receiving terminal on the top of a building in an airport, and fixed the transmitting terminal on a helicopter, as shown in Fig. 10. The helicopter flies along two routes at a speed of 100 km/h. One route has a length of 18 km with a nearest distance to Bob of 7.5 km, and the other route has a length of 10 km with a nearest distance to Bob of 2.5 km. In the first route, Alice moves such that her rotation is over a range of 1.75 rad with a maximum rotational speed of 3.7 mrad/s, while in the second route Alice moves over a range of 2.2 rad with a maximum rotational speed of 11 mrad/s, which is the maximum rotational speed that most LEOSs can achieve. Moreover, the helicopter can turn on its side to simulate the gesture swing of the satellite.

 

Fig. 10 The Zhoushan voyage experiment. The sender (Alice) is mounted on a helicopter while the receiver (Bob) is mounted on the top floor of a building in an airport. Here, H, V, + , and −: pulse lasers with polarizations of H, V, + , and −, respectively; ATP: the acquisition, tracking, and pointing system; CCU: the central control unit; CCS: the classic communication system; HWP: the half-wave plate; and MS: motor stage.

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The Zhoushan voyage experiment was intend to test the ATP mechanisms along with quantum transmission on a satellite-like platform. The basis-deviation compensation system could be easily implemented in this airborne experiment, so we tested its performance too. We installed a motor-stage driving the HWP before the BB84 decoding module and appropriately modified the code in the central control unit (CCU) software, transforming the receiver terminal into a basis-deviation compensation system, as shown in Fig. 10.

During the first several minutes after the helicopter takes off, Alice and Bob create the quantum communication channel through the ATP system following the procedure elaborated in Ref [3]. Subsequently, quantum communication and basis-deviation compensation can be performed, but they cannot work simultaneously, as the quantum signal and probe beam share the same channel. Time-multiplexing techniques [14] provide a good choice for their transmission. Our aim here, however, is to test the compensation scheme, so Alice only sends a probe beam here.

According to Sec. 3.2, if we want to decrease the error in basis deviation, we must increase the average counts of the APD. For example, when the detection error of the BB84 decoding module is 1°, from Eqs. (10) and (12), we obtain $k\text{=}1.03$. Therefore, the total variation of the APD is 3%, which indicates the dark count variation of the APD is ~1%, giving the average counts of the APD as 2000 c/s. Therefore, the photon counts must be greater than 2000 c/s to guarantee the 1° detection accuracy of the system. This make the total photon count for the four APDs greater than 8000 c/s. Note that because of atmospheric attenuation, the quantum signal will be left with only a few thousand counts per second when arriving at Bob. We therefore decreased the attenuation of Alice to guarantee enough power in the probe beam.

We used a H-polarized pulsed laser at Alice to generate the standard light at a repetition rate of 1 MHz. After attenuation, the average photon number per pulse was $0.1<\mu <0.2$ when arriving at Bob. At 850 nm, the PDE of the APD is ~50%; this gives a total photon count of ~10⁵ c/s detected by the BB84 decoding module. As we reported in Sec. 4.1, we use the BB84 decoding module to detect the basis deviation as well as to record the results of compensation-accuracy, and the HWP will automatically compensate for the basis deviation according to the detected basis deviation.

The results are shown in Fig. 11, with two sets of values taken from the 10-km route presented. Figures 11(a) and 11(c) indicate the total counts detected by the four APDs, while Figs. 11(b) and 11(d) indicate the corresponding basis deviation (or compensated polarization) detected by the BB84 decoding module. The time between obtaining the two sets of values is about 5 min, which means the helicopter has navigated over a range of ~2 rad, i.e., ~118°.

 

Fig. 11 The total counts of the four APDs ((a) and (c)) and the polarization detected by the receiver module ((b) and (d)) after basis-deviation compensation. The two sets of data ((a)-(b) and (c)-(d)) were acquired in the 10-km route.

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Note that the basis deviations in Figs. 11(b) and 11(d) are mostly between −1° and 1°, despite several points at −5°. At these points, the corresponding count is ~0. This is because the helicopter produced a very strong airflow, which perturbed the helicopter terminal so violently that the ATP system did not work well at such times. The signal was interrupted once control of the ATP system was lost. At such times, the BB84 decoding module could not obtain a signal; the total counts were 0 and the calculated basis deviation went out of range. Since the motor stage of the HWP (see Fig. (9)) may become stuck if the basis-deviation angle becomes too large, we defined that a basis deviation outside the range ± 5° corresponds to a loss of system control—this is the origin of the −5° accuracy when the ATP failed to create a quantum channel owing to the airflow.

Fortunately, there is no strong airflow in space where the satellite operates. Despite the disturbance of airflow in our terrestrial demonstration, the basis-deviation compensation system works well with a system precision of ~2° and an RMS of ~0.3°. The RMS is somewhat high as a result of the variation of the APD, which indicates that the basis-deviation compensation system is not quite as stable as desired. Nevertheless, According to Eq. (6), the fraction of QBER caused by basis deviation is ~0.12%. This compensation system can be well adopted in satellite-to-ground quantum communications in order to reduce the basis deviation arising from the satellite motion.

5. Conclusions

It is necessary that in polarization-encoded quantum communication all the polarization states must be transmitted and received correctly. However, unlike fixed-station quantum communications, the moving station in satellite-to-ground quantum communications will cause basis deviation between the sender and the receiver, and will subsequently increase the quantum-bit error ratio (QBER), so that compensating for basis deviation is a crucial issue.

This paper reports a simple and convenient solution for detecting and compensating for the basis deviation, which can be easily adopted in satellite-to-ground quantum communication without placing any additional burden on the satellite. We analyzed in detail the effects of phase delay and basis deviation on QBER and deduced an equation for calculating QBER. The basis-deviation detection scheme was tested experimentally, demonstrating an accuracy of 1°. The compensation scheme was also tested in the Zhoushan voyage experiment; it was shown to have a 2° accuracy and to reduce the fraction of QBER caused by basis deviation to 0.12%.

Although using the BB84 decoding module to detect basis deviation is convenient, it occupies the quantum communication source, which means that we cannot transmit qubits during basis-deviation compensation. Time-multiplexing techniques could provide a potential solution to this problem. Furthermore, the accuracy and stability of this compensation scheme cannot be greatly improved owing to the restricted characteristics of avalanche-diode photo-detectors (APD). The basis-deviation compensation scheme might be more accurate if we upgrade the photon detectors, such as by replacing the APD with a superconducting nanowire single-photon detector (SNSPD) [22].

Otherwise, we are currently working on using the beacon laser of the acquisition, tracking, and pointing system as a probe beam. Using this scheme, we can potentially detect basis deviation by using classical light. However, this scheme is not easily realized and remains to be implemented.