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Abstract
A quantum bit error rate (QBER) based algorithm for polarization random drift compensation is proposed. For a transmission window of 8 ms, for instance in aerial fiber installations, the algorithm overhead is below 1%. In an extreme turbulent situation, where the transmission window is as shorter as 0.8 ms, the overhead is still below 10%. Besides being able to operate smoothly, even in a very extreme situation, the algorithm overhead is also insensitive to the length of the communication system. It is upper layer protocol agnostic, and it is based on the mapping of the QBER on the Poincaré sphere. The algorithm finds the polarization reversal operator, which results in much lower overhead contrary to the blind methods currently used. The algorithm reverts the polarization random drift performing two QBER estimations and applying three rotations, at most. The uncertainty on the two QBER estimations defines an area over the sphere surface that is related with upper-layer protocols QBER threshold.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quantum communications (QC) have been implemented using different degrees-of-freedom of single-photons, such as the polarization or phase. The use of polarization to encode and decode quantum information appears natural for the exchange of qubits over optical links [1,2]. Nevertheless, standard single-mode optical fibers do not preserve the state of polarization (SOP), and therefore an active polarization basis alignment (PBA) method is required, to preserve the quantum information [3]. In order to allow the large deployment of polarization encoding QC systems, the PBA scheme must be efficient, simple, upper- layer protocol agnostic, and able to operate in a large variety of environment conditions.
We can consider two generic approaches for PBA, interrupting and real-timing methods. In the interrupting methods basis alignment stops the transmission of quantum information. In real-timing methods auxiliary channels tend to be needed. In [3], the authors quantitatively analyze both methods, considering the polarization drift-time and the tracking speed. They concluded that the interrupting methods should be fast enough to revert the polarization in a time interval much shorter than the drift time. In [4], it was reported that the random polarization drift can induce a QBER exceeding $2.5\%$ in less than 7 ms for aerial-fibers. In [5], also for aerial-fibers, transmission windows as short as 1 ms have been reported in polarization encoding quantum communication systems due to random polarization drifts. The real-timing methods tend to be less critical in terms of time polarization linewidth [6], but generally require extra hardware to support the exchange of out-of-band control information [3]. A method that avoids the exchange of out-of-band information, but even still fast enough to operate under heavy external conditions it is clearly desirable. [3]. In real-time scenarios, two different approaches have been presented: wavelength-division multiplexing polarization basis alignment (WDM-PBA) [4,7,8], and time-division multiplexing polarization basis alignment (TDM-PBA) [9–12]. In [4], SOP tracking is performed using a hill-climbing algorithm in conjugation with a WDM polarization tracking scheme. In [8], a protocol-agnostic scheme is proposed using WDM-PBA in aerial fibers. In [12], it is shown that the achievable reach can be increased by using TDM-PBA based schemes. TDM-PBA may be implemented using classical [9,10] or quantum reference signals [11]. In classical based TDM-PBA the co-propagation can produce a strong degradation in the weak quantum signals [9]. In [11], a TDM quantum reference signal is transmitted along with the quantum data signal, also avoiding the need of using both classical and quantum receivers. In [13], a protocol-dependent real-time scheme, free of reference signals is presented, where QKD unveiled bits are used to feed the algorithm to compensate random polarization drifts. That method has the advantage of not add additional overhead, but it is not protocol-agnostic, which can limit its practical implementation. In [14], an accurate QBER estimation method is proposed, and a QBER based PBA method is presented. That method is simple, upper-layer protocol agnostic and able to operate under different external conditions [14]. However, it uses a blind algorithm to align the polarization basis, which makes it quite inefficient, namely under large external condition perturbations [4,8]. This method uses $12.5\%$ of overhead on average in a laboratory environment, where the polarization remains stable for much longer than in aerial optical fiber installations [14]. In [6], a theoretical polarization drift model, which is able to describe random polarization rotations for installed fibers under different external conditions based on a single parameter is presented. This parameter, named polarization linewidth, quantifies how fast the SOP changes with time [6]. This parameter takes into account the installation of the optical fiber and the external perturbations, and it allows to model the polarization random drift speed.
In this paper, we develop a method to compensate the polarization random drift in optical fibers by mapping the estimated QBER on the Poincaré sphere. This method solves the problem of finding the appropriate polarization reversal operator. We show that polarization random drift can be reversed by applying appropriate polarization rotations on the Poincaré sphere, in three iterations at most. This method is able to operate under different external perturbations and it is upper-layer protocol agnostic, it does not need auxiliary classical signals, extra spectral bands, nor additional hardware, and provides polarization basis alignment in less than tens of microseconds, with very low overhead.
This paper contains five sections. In Section 2, the proposed algorithm is detailed. In Section 3, the QBER estimation impact on polarization compensation algorithm efficiency is discussed. In Section 4, the algorithm behaviour is assessed considering a realistic situation. Finally, in Section 5, the main conclusions of this work are summarized.
2. Algorithm description
The 3D-Stokes space is a mathematically convenient alternative to represent and easily visualize the SOP of an optical field [6]. We start by showing that it is also possible to map the QBER on the Poincaré sphere, and then find the appropriate polarization reversal operator. We assume the polarization-based quantum communication system is composed by a transmitter, which emits weak laser pulses (approximated single-photon source) from a highly attenuated laser with a well defined SOP. After fiber propagation, SOP of the single photons are measured at the detection stage, comprised by an electronic polarization controller (EPC) followed by a polarization beam splitter (PBS), and a receiver with two single-photon detectors, see Fig. 1. Without any loss of generality, we are going to assume that an error at the receiver occurs when for instance a horizontally polarized photon at the transmitter output follows the vertical path of the PBS at the receiver, inducing a click on the detector V, see Fig. 1. Note that any initial polarization state can be reduced to the previous case by a solid rotation of the Poincaré sphere, in the same way that any SOP can be used as a reference for the null QBER.
Fig. 1. Horizontal SOP evolution throughout a quantum channel (optical fiber) which induces random polarization rotations, and detection probabilities at receiver ($P_V$ and $P_H$). EPC: Electronic Polarization Controller. PBS: Polarization Beam Splitter. V: Single-photon detector in the PBS vertical port. H: Single-photon detector in the PBS horizontal port. $V_1$, $V_2$ and $V_3$: Voltages applied on the EPC to induce a certain rotation.
In detail, when a photon reaches the PBS it has the probability $P_H$ to follow the horizontal path, and the probability $P_V$ of follow the vertical path [12],
where angles $\theta /2$ and $\varphi /2$ correspond to the orientation and ellipticity angles of the arriving photon SOP on Poincaré sphere representation, respectively [15]. Considering a horizontal state at the fiber input, we can write Therefore, a QBER specifies a set of possible orientation and ellipticity angles. This set of values define a circle of a sphere on the Poincaré sphere, which corresponds to a QBER with reference to a given initial SOP. In the present case, apart from an horizontal polarized photon at the input of the quantum communication channel, we are also assuming fully polarized light. Therefore, the normalized Stokes parameter $s_1$ can be written as where $\theta \in [0, 2\pi ]$, $\varphi \in [-\pi /2, \pi /2]$ [16]. The QBER can also be written in terms of $s_1$ as Thus, the circle of a sphere resulting from a QBER value defines a set of possible SOP locations, which are at the same distance from the reference point, As we can see in Fig. 2(a), a single value of QBER has more than one possible polarization reversal operator associated with it, even though a single received SOP leads to a single QBER value. Let us assume that a particular polarization rotation leads to a QBER of $10\%$, see Fig. 2(a). Looking into Fig. 2(a), we can see that the polarization reversal operator still remains unknown, although it is restricted to rotations that lead the SOP from $(s_1,s_2,s_3)^T=(1,0,0)^T$, i.e. a horizontal initial SOP, to a point on the circle of the sphere that represents the $10\%$ QBER. A subsequent deterministic rotation in conjunction with a new QBER calculation allows to reduce the number of possible polarization reversal operators to only two possibilities, see Fig. 2(b). Note that a rotation can be characterized by the two angles, $\theta$ and $\varphi$,Fig. 2. (a) Circle of a sphere with all possible states on Poincaré sphere that correspond to the $\textrm {QBER}=10\%$. (b) Circle of a sphere that corresponds to the $\textrm {QBER}=10\%$ rotated considering $\theta _{\textrm {max}}$ and $\varphi _{\textrm {max}}$, and circle of a sphere with all possible states on Poincaré sphere that corresponds to the $\textrm {QBER}$ after the previous rotation. The two symbols $\bullet$ represent the intersection points that correspond to the two possible SOP locations.
Once a rotation has been performed, as shown in Fig. 2(b), another calculation of QBER is done. Let us assume the performed rotation was done using the orientation angle $\theta _{\textrm {max}}/4$ and ellipticity angle $\varphi _{\textrm {max}}/4$, where $\theta _{\textrm {max}}$ and $\varphi _{\textrm {max}}$ are the maximum angles defined by the circle of the sphere corresponding to the first QBER value calculated, see Fig. 2(a). A new value for QBER allows to draw another circle of a sphere on Poincaré sphere, which intercepts the previous rotated circle in two points, which are shown in Fig. 2(b) with circle marks. Therefore, the initial infinite number of possible polarization reversal operators is reduced to only two possibilities, which correspond to the reversal operator of the two intersection points. In order to obtain an analytical expression for the two intersection points, we can consider the parametric equations of a 3D circle, see Eq. (8). Note that $m$ takes the value $1$ for the initial QBER rotated circle, and $2$ for the circle of a sphere after the QBER re-calculation,
In Eq. (8), $( x_c^{(m)}, y_c^{(m)}, z_c^{(m)} )$ are the center coordinates, and $r_m$ is the radius of each circle $m$. Note that after measuring the QBER, a circle of a sphere is defined. From Eq. (2), $( x_c^{(m)}, y_c^{(m)}, z_c^{(m)} )$, $r_m$, and the plane containing the circle defined by the orthogonal vector $\vec {n} = \vec {v}_{m1} \times \vec {v}_{m2}$, where $\vec {v}_{m1}=(x_{m1},y_{m1},z_{m1})$ and $\vec {v}_{m2}=(x_{m2},y_{m2},z_{m2})$, can be readily obtained.
The two measured QBER values define two circles of a sphere that intersect in two points, which can be obtained from
Fig. 3. Description of the algorithm to find the reversal operator and compensate the polarization random drifts.
Note that the voltages applied on the EPC, $V_1$, $V_2$, $V_3$, can be written in terms of angles $\theta$ and $\varphi$. These voltages induce a certain phase shift on the wave-plates by changing its orientation, which implies a rotation of the SOP based on a set of rotation angles, $\chi _1$, $\chi _2$, $\chi _3$. These angles can be written in terms of the orientation and ellipticity angles, $\theta$ and $\varphi$. Looking into Fig. 1, a random SOP $\hat {s}_i$ inputs the EPC facing the first quarter-wave-plate(QWP) that outputs, in turn, the SOP $\hat {s}_j$,
where $\textrm {M}_{\lambda /4}$ is the QWP matrix [16] and $\textrm {R} = {\textrm {R}_3}(2\chi _1)$ is the rotation matrix of the wave-plate. The angle $\chi _1$ is given by [16], The second wave-plate is a half-wave-plate (HWP), and transforms the linear SOP $\hat {s}_j$ into another linear SOP [16], which in practice means a rotation by $\theta$ around $S_3$ when $\varphi =0$, where, $\textrm {M}_{\lambda /2}$ is the HWP matrix, and $\hat {s}_j=(s_{1j} , s_{2j}, 0)^T$ defined by Eq. (10) in [16]. In this way, where, $s_{1j}$, $s_{2j}$ are defined by Eq. (16) and Eq. (17), respectively. In addition, $s_{1i}=\sin {\theta }\cos {\phi }$ and $s_{2i} = \cos {\theta }\sin {\varphi }$, see Eq. (10). Finally, the EPC output SOP is defined as, where, From a practical implementation point of view, we can not use all the qubits to estimate the QBER. In a practical scenario, one expects to use as few as possible number of qubits to compensate the random polarization rotations inside the optical fiber, leaving the most number of qubits for quantum communication purpose. In this scenario, the QBER is estimated taken into account a certain number of received qubits, $N_r$ using where $e_r$ is the number of errors in $N_r$ qubits [14].This estimation is performed with a certain confidence interval which depends on the number of qubits that we use to perform it. In this section, we are assuming that $N_r$ is large enough to provide an accurate estimation of the QBER. In the next section, we are going to consider and assess the impact of $N_r$ during the algorithm’s running. We also assume that our quantum communication system can operate with a QBER threshold of $\textrm {QBER}_{\textrm {th}}~=~ 5\%$. Above this threshold value, the quantum communication system cannot operate. The goal of the presented algorithm is to force the QBER to be below the threshold. Figure 4 shows eight different cases corresponding to different initial conditions, where it is shown that regardless the respective initial QBER, the final QBER is always below the threshold. The inset in Fig. 4 shows the location of the different SOPs on the Poincaré sphere, where each one is on the circle of a sphere corresponding to the QBER measured. In Fig. 4 every case starts from an initial QBER estimation, i.e the first marker. The algorithm starts from this initial QBER estimation and performs the first rotation. The second marker is the QBER estimation after the first rotation. Here, the algorithm chooses one of the two intersection points, see Eq. (9). After that, we wait for a 5 errors event, or for 100 qubits received to estimate the new QBER. Note that more than 5 errors implies an estimated QBER larger than the threshold, in this case where a $5\%$ threshold was assumed. When it wrongly chooses the intersection point, the next marker is a QBER above the $\textrm {QBER}_{\textrm {th}}$. On the other hand, when it rightly chooses the intersection point, the next marker is a QBER below threshold, and the following. A final marker with high accuracy on QBER estimation is also included in Fig. 4, to show the proper operation of the algorithm. In Fig. 4, we have shown that the algorithm finds the appropriate polarization reversal operator and compensates any polarization random drift leading the initial QBER to a value below the threshold after two or three rotations, at maximum.
Fig. 4. QBER evolution during the random polarization random drift compensation algorithm running. Markers represent QBER measurements for different initial QBER values. The initial SOP are represented on the Poincaré sphere shown in the inset, where the reference SOP is represented as a blue dot.
A complete and general polarization compensation algorithm should be able to find and reverse any SOP drift. We have described a method to compensate a particular SOP, nonetheless, our SOP drift compensation method can be generalized to any polarization drift suffered by an arbitrary SOP. In order to generalize the operation of the algorithm for an arbitrary channel input SOP, at least two canonically conjugated non-orthogonal states should be compensated [7,9]. In this way, the algorithm continuously looks into two values of QBER for two non-orthogonal states of polarization, for instance the linear states $|H \rangle$ and $ |45^{\circ }\rangle $. Based on those two values of QBER, i.e. $\textrm {QBER}_{|H \rangle}$ and $\textrm {QBER}_{|45^{\circ }\rangle}$, the algorithm starts the polarization compensation procedure as soon as one of them rises above the defined threshold. Subsequently, the algorithm actuates in the other non-orthogonal state. Let us assume a generic EPC with four wave-plates. Those wave-plates are linear retarders with two different orientations for the fast axis. The matrix for this generic EPC can be written as
3. Impact of QBER estimation accuracy
In order to assess the impact of the QBER estimation accuracy in the algorithm performance, we are going to use a new coordinate $\gamma$, such that
where $\gamma$ is the angle between the axis $S_1$, and the Stokes vector of the SOP. In this way, the QBER in Eq. (2) can be written in terms of $\gamma$ as The number of qubits required for each algorithm iteration is the number of the qubits used for each QBER estimation, occurring in stages (i), (iii) and (vi), see Fig. 3. The last QBER estimation, at stage (vi), does not lead to any rotation, and therefore does not require high accuracy. In this way, we can assume that where $n_1$, $n_2$ and $n_3$ are the number of qubits used in QBER estimations at (i), (iii), and (vi) defined in Fig. 3, respectively. Therefore, the total number of qubits required to compensate the polarization random drift can be written as, see Eq. (15) from [14],Note that the QBER estimation uncertainty, at stages (i) and (iii), can be written as
Fig. 5. Representation of the area defined by the uncertainties of the first and second QBER estimations on the Poincaré sphere surface, $A_i$. This area is preserved after the final rotation, i.e. $A_i=A_f$. Inset shows a zoom in of the area resulted from the uncertainties of the two QBER estimations.
From Eqs. (23) and (26), the uncertainty $\Delta \textrm {QBER}_i$ at stages (i) and (iii) can be related to the corresponding $\gamma _i$, as well as to $\delta \gamma _i$, using
The induced rotation into the SOP associated with the QBER estimations, at stage (v), is going to place the uncertainty area, $A_i$, around the target SOP, preserving its shape, $A_f$, as shown in Fig. 5. Note that the final QBER, $\textrm {QBER}_f$, is null at $\gamma _f = 0$, therefore from Eq. (23) we obtain The final QBER estimation uncertainty depends on the first and second QBER estimations, and as a consequence, the $\Delta \textrm {QBER}_f$, see Eq. (28), depends on the $\delta \gamma _1$ and $\delta \gamma _2$. Note that the uncertainties $\Delta \textrm {QBER}_1$ and $\Delta \textrm {QBER}_2$ define an area that remains constant after the final rotation, see Fig. 5. In the worst case scenario, $\delta \gamma _f$ will be the sum of both uncertainties $\delta \gamma _1$ and $\delta \gamma _2$, For a given confidence interval ($1-\alpha$), the algorithm satisfies Following this discussion, we can calculate the number of qubits required for QBER estimation at stages (i) and (iii) of the algorithm, so that the polarization control random drift algorithm assures a $\textrm {QBER}_f$ below a certain QBER threshold, $\textrm {QBER}_{\textrm {th}}$, with a certain probability. Note that in a small rotation regime in stage (ii), we can also assume that $\delta \gamma _1\approx \delta \gamma _2\approx \delta \gamma$, and $\gamma _1 \approx \gamma _2 \approx \gamma$, which implies $\Delta \textrm {QBER}_1 \approx \Delta \textrm {QBER}_2$. Therefore $n_1 \approx n_2 \approx n$, and so that the total number of required qubits will be $n_b = 2n$.Table 1 shows the number of qubits used to perform each QBER estimation, $n$, calculated given a certain $\textrm {QBER}_{\textrm {th}}$, using Eqs. (28) and (29) to calculate $\delta \gamma$, and using Eq. (23) to obtain $\gamma$ in order to obtain the initial QBER uncertainty. Using the initial QBER uncertainty, the initial QBER, and for a given confidence level using Eq. (25), we can obtain the total number of required qubits, $n_b$, and subsequently $n$, the number of qubits to estimate QBER in stage (i) and (iii). The final QBER threshold has a high impact on the number of qubits required to estimate QBER during protocol execution, since the number of qubits is inversely proportional to the QBER threshold, as one can see in Table 1, i.e for smaller $\textrm {QBER}_{\textrm {th}}$ a larger number of qubits is required [14]. This indicates that, in order to achieve a lowest QBER the performed rotation must be precise, and so that the estimated QBER should be as accurate as possible, which is obtained using a large number of qubits for the estimate. In order to assess the algorithm performance, we perform a simulation for a QBER threshold of $3\%$, considering two initial values for QBER, $10\%$ and $40\%$. The SOP at the receiver input is randomly chosen between all possible SOP on the circle of a sphere corresponding with the desirable initial QBER. Following Table 1, and considering the $3\%$ threshold, we use 243 qubits to estimate each QBER, at stages (i) and (iii) of the algorithm. We run 1000 simulations for each initial QBER. Moreover, the reached QBER estimation was performed with a high accuracy, 3500 qubits were used. Note that this estimation is not part of the algorithm, and it was only performed here to assess the algorithm performance. The obtained results show that for an initial QBER of $10\%$ and $40\%$, the reached QBER is above the $\textrm {QBER}_{\textrm {th}}$ only in $1.3\%$ and $1.8\%$ of the cases, respectively, which is in good agreement with Eq. (30), considering a confidence level of $99\%$, i.e. $2\alpha =2\%$.
Table 1. Number of qubits required for estimation of QBER in terms of $\textrm {QBER}_{\textrm {th}}$ for a confidence level of $99\%$.
4. Algorithm overhead
We performed numerical simulations, where the proposed algorithm was applied to a system developed to model a quantum communication system using polarization encoding. We assumed that the system operates at $100 \textrm { MQubit/s}$, considering current avalanche photo-diodes based on single-photon detectors technology [17].
4.1 With perfect devices
In a first instance, we only assess the algorithm actuation impact on a quantum communication system considering perfect devices, i.e. an EPC that actuates instantaneously, perfect single-photon sources, and single-photon detectors with unitary efficiency. The random polarization rotations were induced numerically simulating a polarization scrambler, based on [6]. We assume a QBER threshold imposed by the upper-layer protocols equal to $3\%$. This value should allow current quantum communication protocols to operate smoothly [18,19].
We consider two scenarios for the impact of polarization drift. A first case, with an average transmission window of $0.8$ ms, and another with a $8$ ms average transmission window. We should note that transmission windows of $1$ ms have been reported for very turbulent aerial fibers [5]. Buried fibers typically present transmission windows in the order of at least tens of seconds [3], and in the laboratory results have been reported with transmission windows of several minutes [14]. Therefore, both considered scenarios, $0.8$ and $8$ ms, can be seen as "worst case" scenarios. To model the polarization drift, we follow [6]. To obtain the desired transmission windows we use a polarization linewidth, $\Delta _p$ [6], of $20 \textrm { }\mu$Hz and $0.2\textrm { }\mu$Hz, respectively. Note that we refer to polarization linewidth as the parameter used to measure the speed of the drift suffered by the SOP, and it has units of $s^{-1}$.
The polarization control system comprises two operation modes: a monitoring mode and an actuation mode. In the monitoring mode the QBER is estimated every 100 control received qubits and with a maximum sliding window of one thousand qubits. For polarization linewidth of $20\textrm { }\mu$Hz and $0.2\textrm { }\mu$Hz, we assume 1 control qubit per 100, and per 500 transmitted qubits, respectively. From the defined QBER threshold and using Table 1, an actuation QBER can be obtained such that the upper-boundary QBER estimation does not exceed the user defined threshold. This QBER actuation value leads to the commutation between the monitoring and the actuation mode. We use a $2\%$ value for the actuation QBER. When the algorithm enters in the actuation mode, it follows the steps presented in Section III. In this mode, all transmitted qubits are used for polarization control. After the algorithm actuation, the QBER estimation window is reset.
In order to assess the algorithm’s performance, we measured the algorithm’s overhead, the actuation time, actuation frequency, average QBER, and maximum QBER for both situations, corresponding to the polarization linewidth of $20$ and $0.2 \textrm { }\mu$Hz. The algorithm’s overhead is defined as the ratio between the number of qubits used for polarization monitoring and control, and all transmitted qubits. The algorithm’s actuation time is the average time that the algorithm takes to compensate the polarization drift, and leads the QBER to a value below the actuation QBER. The algorithm’s actuation frequency is defined as the number of times that the algorithm actuates per unit of time. The average and maximum QBER are calculated considering the data qubits. To assess the algorithm performance, we run simulations during $20 \textrm { ms}$ time windows on the specified scenarios.
Figures 6(a) and 6(b) show the evolution of QBER for the two considered scenarios using the proposed algorithm to find the polarization reversal operator to compensate the drift. As it is shown, whenever the QBER rises above the actuation QBER, $2.0\%$, the algorithm actuates being able to reestablish the qubits data transmission in $12\textrm { }\mu$s on average for polarization linewidth of $20 \textrm { }\mu$Hz, see Fig. 6(a), and in $7.39\textrm { }\mu$s on average for $0.2 \textrm { }\mu$Hz, see Fig. 6(b). The actuation times are represented by vertical lines in the plots, where the width of the lines correspond to the algorithm actuation time. On average, the algorithm actuates 1.15 times per millisecond, imposing a transmission window of $0.8$ ms on average with an overhead of $2.54 \%$ in the case represented in Fig. 6(a). For the case presented in Fig. 6(b) the algorithm actuates 0.15 times per millisecond, imposing a transmission window of $8$ ms on average with an overhead of $0.31\%$. Note that during monitoring mode, the QBER estimation demands $1\%$ and $0.2\%$ overhead in $20\textrm { }\mu$Hz and $0.2 \textrm { }\mu$Hz polarization linewidth scenarios, respectively. The remaining overhead is used by actuation mode. The average QBER during data qubits transmission remained below the $3\%$ threshold, and a maximum QBER of $2.1\%$ was obtained in both scenarios.
Fig. 6. (a) - QBER monitoring with actuation of the proposed algorithm for polarization random drift compensation in an extreme scenario, where polarization linewidth is $20\textrm { }\mu$Hz. (b) - QBER monitoring with actuation of the proposed algorithm for polarization random drift compensation in scenario considering a polarization linewidth of $0.2 \textrm { }\mu$Hz. Vertical black lines represent the actuation time that the algorithm takes to find the polarization reversal operator and reverse the polarization drift.
4.2 With imperfect devices
Now, we will assess the algorithm’s performance considering off the shelf imperfect devices and its technical limitations. We consider a highly attenuate laser, i.e. a source with a Poisson statistics, a $0.2$ dB/km fiber channel attenuation [20], and $25 \%$ single-photon detectors efficiency [21]. In order to overcome the issues related with no-click events, and reduce its impact on algorithm’s performance, the number of photons in control qubits is optimized. By increasing the number of photons in each control qubit, we also increase the double-click events when the photons polarization is not perfectly aligned due to polarization random drift. Double-clicks can also be caused by the detectors dark-counts, which we assumed a value of $5 \times 10^{-4}$ for each detector [21]. Both, no-click and double-click events will impact the overhead used by the algorithm, since the qubits measured in that situation are discarded, and not taken into account for QBER estimation. An average number of photons per pulse of $0.1$ at the transmitter output was assumed [22]. Two optical channel lengths were defined to perform this analysis, $40$ km and $80$ km. Figure 7 shows the overhead according with the average number of photons per control qubit at the transmitter output. We change the number of photons in the control qubits in order to find the optimum number photons at the receiver input, which we found to be around 5. Note that, for $40$ km optical channel length, the number of photons needed at transmitter output is much lower than for $80$ km optical channel length. The four curves placed above $4 \%$ overhead correspond to a $20 \textrm { }\mu$Hz polarization linewidth, where the transmission window is on average $0.8$ ms, and it is shown that the overhead does not depend significantly on the optical channel length, as well as the optimum number of photons per control qubits.
Fig. 7. Overhead for different average number of photons per control pulse, considering two different optical channel lengths (40 and 80 km). For each fiber length, the overhead was measured considering both a perfect and an imperfect EPC. Two values for polarization linewidth were considered, $0.2 \textrm { } \mu$Hz and $20\textrm { }\mu$Hz. Dashed and dashed-dot lines represent the intrinsic overhead values considering perfect devices for $20$ and $0.2 \textrm { }\mu$Hz values of polarization linewidth, respectively.
Due to technological limitations, the EPC does not induce an instantaneous rotation, and it demands a certain time interval to stabilize the output SOP. In this way, the EPC actuation time should be considered in the system assessment. An EPC actuation time of $20$ $\mu \textrm {s}$ was assumed [8], which corresponds to increase the overhead by $2000$ qubits in each performed rotation for a transmission rate of $100$ MQubits/s. Figure 7 also shows the overhead resulted from adding the EPC actuation time for both optical channel lengths. Even though the overhead increases due this technological limitation, it remains below $9\%$ even for the $0.8$ transmission window, which nevertheless is lower than the value presented in [14] for a larger transmission window.
The overhead obtained for both ideal scenarios previously considered is also shown in Fig. 7 corresponding to $2.54 \%$ and $0.31 \%$ for a polarization linewidth of $20\textrm { }\mu$Hz and $0.2\textrm { }\mu$Hz, respectively. Moreover, according with the number of photons per control qubit the overhead was also calculated for both optical channel lengths considering a $0.2\textrm { }\mu$Hz polarization linewidth, that imposes a transmission window of $8$ ms on average. As shown in Fig. 7, the overhead remains below $1.5 \%$ for every number of photons considered, even taken into account imperfect devices.
Figures 8(a) and 8(b) show the QBER monitoring for a polarization linewidth of $20\textrm { }\mu$Hz considering both lengths for the communication channel, 40 km and 80 km, respectively. In this case, the number of photons per control pulse was adjusted aiming to achieve the minimum overhead. In this way, according with Fig. 7, we chose 32 photons for the 40 km situation and 200 photons for the 80 km situation. We consider that the EPC takes $10 \mu s$ to apply a single rotation. Comparing Fig. 8(a) with Fig. 6(a), we can see that now the vertical black lines tend to be wider, due to EPC actuation time. Even so, it is able to reverse the drift in $20 \mu s$, in average, and with an overhead lower than $3 \%$ in both cases.
Fig. 8. QBER monitoring taking into account the actuation time of the EPC besides the detector imperfections. Data obtained for a $20\textrm { }\mu$Hz polarization linewidth. Vertical black lines represent the actuation time that the algorithm takes to find the polarization reversal operator and reverse the polarization drift. (a) 40 km optical fiber length. (b) 80 km optical fiber length.
5. Conclusion
We presented an algorithm to automatically compensate the polarization random drift in polarization-encoding based quantum transmission systems. This algorithm is based on QBER estimation and on its representation on the Poincaré sphere, allowing to find the appropriate polarization reversal operator in order to minimize the algorithm overhead. From the estimated QBER, a circle on the Poincaré sphere is defined leading to a set of possible polarization states. By performing a deterministic rotation, the algorithm reduces this set of polarization states to only two possible SOP. From this two possible states of polarization the algorithm is able to compensate the polarization random drift in a very short time.
It was shown that the proposed algorithm is always able to force the QBER to a value below a user-defined threshold in three iterations, at most. In addition, the uncertainty in the final QBER was related with an area calculated in the Poincaré sphere surface based on two QBER estimation uncertainties. From this area, we obtain the number of qubits required in the QBER estimations to guarantee a final QBER below the threshold.
Moreover, the proposed algorithm was assessed considering two different scenarios. It was assumed two values for polarization linewidth, $20\textrm { } \mu$Hz, which imposes a transmission window of around $0.8$ ms, and $0.2 \textrm { }\mu$Hz, which imposes a transmission window of around $8$ ms. In both situations, the algorithm was capable of maintain the QBER below the $3\%$ threshold using only $2.54\%$ of overhead, and $0.31 \%$ of overhead, respectively. Furthermore, when imperfect devices limitations are taken into account, namely the EPC actuation time, a highly attenuated laser source, optical channel attenuation, and single-photon detectors efficiency, the overhead slightly increases but remains well below the values reported by blind algorithms. For a transmission window of $8$ ms, the overhead is still below $1\%$, even considering the impact due to device imperfections.
The proposed method actively aligns the polarization basis with very low overhead, and without using out-of-band signals. Due to its low overhead, it can be used even in scenarios where the fiber is subjected to heavy external perturbations, such as buried fibers in highway, railways, or even aerial fibers, where transmission windows are very short due to the random polarization drift. The presented results show the possibility to revert the polarization drifts in tens of microseconds.
Funding
Fundação para a Ciência e a Tecnologia (SFRH/BD/145670/2019, UID/EEA/50008/2019); Fundo Regional para a Ciência e Tecnologia (POCI-01-0145- FEDER-029405, POCI-01-0145-FEDER-031826, UIDB/50008/2020).
Disclosures
The authors declare no conflicts of interest.
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