1. Introduction

The well-known application of quantum cryptography is quantum key distribution (QKD) that allows two remote parties, Alice and Bob, to distill a common secret key, despite the presence of a potential eavesdropper, Eve, whose security is based on the fundamental laws of quantum mechanics [1]. Currently, discrete-variable (DV) and continuous-variable (CV) quantum states are both available to perform secure key distribution [2,3]. In CVQKD, the key information is encoded in quadratures of the quantized electromagnetic field, such as coherent states, and hence high secret key rates can be achieved with high-efficiency homodyne (heterodyne) measurements. Since CVQKD can be conveniently integrated into current passive optical network, it has received extensive attentions in theoretical and experimental efforts [4–9]. It has been proved to be secure against general collective attacks, which are optimal in both the asymptotic case [10,11] and the finite-size regime [12–17].

For the last decades, the coexistence of QKD with classical signals has been of great interest for researchers and a series of coexistence schemes have been demonstrated on single-mode fiber (SMF) with wavelength division multiplexing (WDM) [18–23]. It is found that the scattering photons caused by spontaneous Raman scattering (SRS) is the dominant source of noise for QKD in WDM environments [20,22]. Meanwhile, for CVQKD, it can tolerate a much higher level of contamination in WDM environments due to the mode selection of homodyne detector [21]. However, because of the SRS in SMF, the long-distance and high-speed CVQKD systems must be at the expense of the classical launched power [22,23]. There is a trade-off between the performance improvement and the classical launched power, which hinds the practical application of the coexistence scheme for the state-of-the-art technology. Therefore, it is necessary to further explore practical solutions compatible with realistic optical systems.

In the last few years, to avoid capacity crunching in SMF, the space-division multiplexing (SDM) technique has been proposed to increase the data bandwidth. Motivated by the multiplexing in time, wavelength, polarization and phase, another multiplexing dimensionality, spatial dimension, can be elegantly utilized for the data-band widening in few-mode fiber (FMF). The supported spatial modes of the FMF can be exploited as parallel channels for independent signals and hence the data bandwidth can be widened using the SDM technique. It can provide a feasible approach to solve the trade-off mentioned above, as well as the other way of performance improvement of CVQKD. Recently, theoretical and experimental demonstrations have been made for the SDM-based DVQKD on multi-core fiber (MCF) [24–26] and FMF [27]. Subsequently, CVQKD on MCF has also been demonstrated theoretically and experimentally [28,29]. Compared with MCF, FMF has relatively simpler manufacturing process but larger effective core area of mode, which brings about the advantage of a lower SRS noise [27,30]. Unfortunately, there are little results for CVQKD on FMF.

In this paper, using the SDM technique, we demonstrate the feasibility of CVQKD coexisting with classical signals on FMF and make a security analysis of this FMF-based CVQKD system in weak coupling regime. The incremental effects of FMF and its concomitant optical multi/demultiplexing devices are taken into account. These effects include the mode coupling in FMF and in the mode multi/demultiplexer, the differential group delay (DGD) between modes, the Raman crosstalk, the inter-mode four-wave-mixing, and the insert loss of the mode demultiplexer. We find that mode coupling and DGD are the key aspects that should be controlled for the performance improvement of the FMF-based CVQKD system. DGD may become one of the main limits for FMF-based CVQKD towards high-speed system. In addition, to suppress the inter-mode four-wave-mixing, a delicate channel wavelength management is needed to detune the phase match condition. Numerical simulation results show that CVQKD coexisting with classical signals on FMF is feasible, especially for low-DGD system. Moreover, the secret key rate can be increased and the tolerable classical channel power can be improved significantly in metropolitan area compared with that of the single mode case, whereas the maximal secure transmission distance is degraded. Our analysis determines regions of values for important experimental parameters where secret key exchange is possible, which pave a foundation for the experimental FMF-based CVQKD system.

This paper is arranged as follows. In Section 2, we demonstrate the channel characteristic and the excess noise of the FMF-based CVQKD system using the SDM technique. In section 3, we consider the security analysis of the FMF-based CVQKD system and suggest the feasibility of the proposed system with numerical simulation results. Finally, we draw a conclusion in Section 4.

2. FMF-based CVQKD system

Compared with SMF, FMF has an analogous construction but a larger numerical aperture, and thus it can support more modes for communication. In FMF, although modes $EH_{\mu \nu }$ and $HE_{\mu \nu }$ are the true eigenmodes (spatial profiles are invariant to propagation), we usually use the linearly polarized (LP) mode to analyze the supported modes under weakly guiding approximation [31]. The mode $LP_{mn}$ is the linear combination of modes $HE_{m+1,n}$ and $EH_{m-1,n}$ and the mode $LP_{0l}$ is the true $EH_{1l}$ mode (m,n and l are positive integers) [32]. Meanwhile, Modes characterized by almost equal propagation constants are referred as degenerate, implying strong coupling between them. Modes with notably unequal propagation constants are referred as non-degenerate, indicating that the non-degenerate modes are weakly coupled between each other. For a six-mode-based FMF, there are four non-degenerate modes $LP_{01}$, $LP_{11}$, $LP_{21}$ and $LP_{02}$, where the twofold degenerate $LP_{i1}$ mode contains two degenerate modes $LP_{i1a}$ and $LP_{i1b}$ for $i\in \{1,2\}$. The subscripts $a$ and $b$ are to distinguish the two degenerate modes.

As the FMF can simultaneously propagate several modes in one fiber core, it provides a feasible approach to enhance communication capacity. The technology of SDM over FMF (also named mode-division multiplexing) has been employed in several communication scenarios [33,34]. Whether CVQKD can still be compatible with evolving classical communications is worthy of investigation. In this section, we suggest the FMF-based CVQKD system, as depicted in Fig. 1, where the quantum and classical channels are multiplexed by a wavelength division multiplexer followed with a mode division multiplexer. Since all the modes are confined in one fiber core, the interaction between modes becomes stronger, and thus more factors should be considered compared with the single-mode [21,22] and the multi-core case [28,29].

In order to make it convenient for noise analysis, we set the following assumptions. (a) We consider the mode coupling in the weak coupling regime. (b) Without loss of generality, we take a six-mode few-mode fiber as an example, shown in Fig. 1. (c) We consider quantum channels in non-degenerate modes $LP_{01}$ and $LP_{02}$, as the mode coupling will go into strong coupling regime when quantum channels are allocated in degenerate modes $LP_{i1a}$ and $LP_{i1b}$ ($i\in \{1,2\}$). (d) The wavelength of quantum channels are shorter than that of the classical channels in the same mode, and classical channels in the mode having quantum channels are allocated in the optimal wavelength to minimize the SRS noise [22]. (e) The wavelength of quantum channels is disparate in different modes. (f) The wavelength of all classical channels is different from all the quantum channels in different modes so that the crosstalk from strong classical signals can be filtered at the receiver. (g) One of the multiplexed classical channel can be used as clock synchronization signal in the mode in which the quantum signal is launched. (h) The CVQKD system configurations of transmitted LO (TLO) or locally generated LO (LLO) can be referred to Refs. [8,35]. (i) The excess noises analysis of CVQKD coexisting with classical signals on SMF is shown in Appendix A. The security analysis of CVQKD with TLO or LLO can be referred to Refs. [16,36].

 

Fig. 1. The FMF-based CVQKD system involving the SDM. MUX, wavelength division multiplexer; DMUX, wavelength division demultiplexer.

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2.1 Inter-mode crosstalk of quantum channel in the FMF

For the ideal FMF, all modes are orthogonal with each other, and hence there are non crosstalk between modes. Whereas, in practice, it will inevitably generate linear mode coupling between modes caused by random perturbations, which leads to the generation of the inter-mode crosstalk [37]. As the difference of propagation constants between two degenerate modes is nearly equal to 0, the mode coupling between them is very strong, which results the mode coupling falling into the strong coupling regime [37]. Therefore, we only consider the case that quantum channels are allocated in non-degenerate modes. In weak coupling regime, the contaminate contributed from the mode coupling can be seen as a channel-added excess noise. Consequently, the channel-added excess noise generated from the inter-mode crosstalk is now concentrated on two aspects. One is the contaminate from the quantum signal itself, and the other is the crosstalk noise from the phase reference pulse (LO or weak reference pulse). The physical mechanism of both these two noise is that the inter-mode crosstalk in other modes re-coupling back to the original mode. Note, the inter-mode crosstalk that involves different wavelengths can be filtered by the wavelength division demultiplexer and the LO at the receiver.

We firstly analyze the crosstalk noise from the coupling of phase reference signal to quantum signal. This crosstalk noise may be generated due to the crosstalk of the phase reference pulse in other modes re-coupling back to the original mode. Because of the existence of DGD between non-degenerate modes, the crosstalk of the phase reference pulse in other modes will spread temporally and hence may overlap with the quantum signal pulse in time domain, resulting in the time domain overlapping between the quantum signal pulse and the re-coupling inter-mode crosstalk of the phase reference pulse in the quantum channel. As the propagation constant and random polarization evolution between non-degenerate modes are disparate [38], the crosstalk of the phase reference pulse in other modes will undergo different phase and random polarization evolution. While the crosstalk of the phase reference pulse in other modes recouples back to the original mode, the generated re-coupling inter-mode crosstalk is randomized in polarization and phase. Since the mode coupling between modes occurs randomly and constantly along the FMF, this re-coupling inter-mode crosstalk from the phase reference pulse can be treated as the mixture of many sub-pulses that have diverse polarization state and phase. Usually, quantum signal and phase reference signal are polarization multiplexed in the quantum channel. However, the re-coupling inter-mode crosstalk from the phase reference signal is randomized in polarization and is no longer in orthogonal polarization with the quantum signal.

Without loss of generality and to simplify the analysis procedure, we consider the case of the re-coupling inter-mode crosstalk from $LP_{11}$ to $LP_{01}$, as shown in Fig. 2, where the coupling interference of one specific phase reference pulse to the adjacent signal pulse is illustrated. In practice, each phase reference pulse may generate interference to its adjacent quantum signal pulse in the form shown in this figure. Since the transmit speed of signals in the mode $LP_{11}$ is slower than that of signals in the mode $LP_{01}$, the inter-mode crosstalk in the mode $LP_{11}$ will be spread temporally, leading the re-coupling inter-mode crosstalk in the mode $LP_{01}$ spread temporally as well and the temporal width of it is wider than that of the phase reference pulse. The spread time is proportional to the DGD and equal to the time of the group delay between the mode $LP_{01}$ and the mode $LP_{11}$. When the spread time is larger than a certain value, the re-coupling inter-mode crosstalk in $LP_{01}$ will overlap with the quantum signal and contribute an in band excess noise. The excess noise contributed by this procedure is analyzed in Appendix B. It is negligible if the magnitude of excess noise is under $10^{-3}N_0$ ($N_0$ is the shot noise) [22] at the receiver. Here, for a conservative estimation, we ignore the excess noise whose magnitude is lower than $10^{-3}N_0$ referred to the fiber input, and the corresponding $\Delta \tau$ (see Fig. 2) of this excess noise value is the interval threshold $\Delta \tau _{th}$. While $\Delta \tau > \Delta \tau _{th}$, the re-coupling inter-mode crosstalk will be isolated in the time domain and thus can be neglected. Therefore, larger threshold means larger value of $\Delta \tau$, which may prevent systems towards the higher repetition rate.

 

Fig. 2. The mode coupling processes in quantum channel. Process 1: The mode coupling from mode $LP_{01}$ to $LP_{11}$ in the transmission period of the phase reference pulse; Process 2: The mode re-coupling of the inter-mode crosstalk. $\Delta \tau$ is the time interval between the LO pulse and the adjacent quantum signal pulse, which is at the spread side of the inter-mode crosstalk.

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According to Eq. (6), in Fig. 3, we show the results of threshold $\Delta \tau _{th}$ versus the transmission distance $L$ for several different DGD values, where the mode coupling value is $-35$ dB/km, $\Delta l = 0.1$ km and $\langle \hat {N}_{ref}^{out}\rangle = 10^8$ (TLO) or $10^3$ (LLO). Figure 3 demonstrates a low repetition rate system with a $1$ MHz repetition rate and a $50$ns pulse width while Fig. 3(b) a high repetition rate system with a $100$ MHz repetition rate and a $1.2$ns pulse width. We find that the threshold is increased with the DGD and transmission distance. It is stress-free to reach $\Delta \tau > 200$ ns for a $1$ MHz CVQKD system whereas harder to satisfy $\Delta \tau > 5$ ns for a $100$ MHz system. Consequently, it is easy to achieve $\Delta \tau > \Delta \tau _{th}$ for a low repetition rate system, as the magnitude of $\Delta \tau$ is decreased with the increasing of the system’s repetition rate. Since the weak reference pulse is much weaker than the LO pulse, the threshold will be smaller than that of the TLO scheme under the same condition. Therefore, CVQKD system with a LLO scheme is easier to fulfill the condition of $\Delta \tau > \Delta \tau _{th}$ and thus can tolerate a larger mode coupling value and DGD. Although we do not show the impact of the mode coupling value, we can deduce from Eq. (6) that the threshold will also be increased with the mode coupling value. Actually, the re-coupling inter-mode crosstalk will contribute to the in band excess noise is mainly due to the DGD and the mode coupling between modes. Therefore, for practical implementation of the FMF-based CVQKD, these two aspects are two key parameters must be concerned. If the DGD is small enough that the re-coupling inter-mode crosstalk can be isolated temporally, we take this system as low-DGD system. Otherwise, we take the system as high-DGD system.

 

Fig. 3. The negligible threshold $\Delta \tau _{th}$ versus the transmission distance $L$ for different values of the differential group delay. (a) $1$ MHz CVQKD system with 50 $ns$ pulse width. (b) $100$ MHz CVQKD system with $1.2 ns$ pulse width.

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We then focus on the in-band excess noise originated from the mode coupling of the quantum signal itself. The generation process of the re-coupling inter-mode crosstalk of quantum signal itself is similar to the phase reference signal. This re-coupling inter-mode crosstalk of quantum signal is also randomized in polarization and phase for different parts of it, and spread temporally compared with the original quantum signal pulse. Disparate from the re-coupling inter-mode crosstalk of the phase reference signal, the re-coupling inter-mode crosstalk from the quantum signal is overlapped with the quantum signal, and the spread of the re-coupling inter-mode crosstalk in time domain will mitigate this overlap. This is similar to the case that the re-coupling inter-mode crosstalk from the phase reference signal overlaps with the phase reference signal, as shown in Fig. 2. Therefore, in the transmission procedure, quantum signal will coupling from its own mode to other modes and then re-coupling back to the original mode, generating the in-band excess noise generated from the re-coupling inter-mode crosstalk. An insight analysis of this noise in Appendix B indicates that in weak coupling regime, this noise is negligible for a not very large modulation variance.

2.2 Raman crosstalk in the FMF

Non-negligible SRS noise photons will be generated when strong classical signals traveling through the fiber. It has been demonstrated that SRS is the dominant source of noise for CVQKD in a WDM environment over SMF [22]. However, for quantum channels in the FMF, in addition to the SRS noise in its own mode, the SRS photons in other modes may also have noise contributions to quantum channels. In other words, the Raman scatter photons in modes will coupling between each other and may contribute extra excess noise to the quantum channel. The detailed analysis of the Raman crosstalk is shown in Appendix C.

We show the numerical simulation results of the Raman crosstalk for different total classical channel power in modes $LP_{11}$ and $LP_{21}$ in Fig. 4, where the mode coupling value between all modes is $-35$ dB/km and the power of the forward and backward classical channels in $LP_{01}$, $LP_{02}$ are all $0$ dBm. The attenuation coefficients for the modes $LP_{01}$, $LP_{11}$, $LP_{21}$ and $LP_{02}$ are chosen as $-0.218$ dB/km, $-0.215$ dB/km, $-0.21$ dB/km and $-0.21$ dB/km, respectively [39]. In Fig. 4, we show that the Raman crosstalk between modes is non-negligible when the power of classical channels in modes having no quantum channels is much larger than that of the modes having quantum channels. The Raman crosstalk is negligible for identical classical power in all the modes. Besides, as the larger effective area of the mode and the less capture possibility of the SRS photons in the higher-order mode [30], the evaluated value of the above Raman crosstalk is larger than the practical value.

 

Fig. 4. The SRS noise as a function of transmission distance in the mode $LP_{01}$. In the mode $LP_{01}$ and the single mode fiber, the power of forward and backward classical channels are set to be $0$ dBm, respectively. In modes $LP_{11}$ and $LP_{21}$ the total power (the sum of forward and backward channels) are set to be $3$ dBm, $13$ dBm, $18$ dBm and $23$ dBm.

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2.3 Inter-mode crosstalk in the mode multi/demultiplexer

Inter-mode crosstalk may be generated in the mode multiplexer and the demultiplexer because of imperfect modes multiplexing and demultiplexing processes [33]. Namely, signals in one mode will couple into others modes while multiplexing or demultiplexing. This mode coupling can be divided into two processes, i.e. the mode coupling in the mode multiplexer and in the mode demultiplexer. For example, if the inter-mode crosstalk between the mode $LP_{01}$ and the mode $LP_{11}$ is -28 dB [33], we can think of this crosstalk as being -31 dB when multiplexing or demultiplexing. In the mode multiplexer, the mode coupling process involves signals in each mode, whereas in the mode demultiplexer, it involves not only signals but also noise in each mode.

As the wavelength of quantum channels in all non-degenerate modes are different from each other, we can neglect the modal crosstalk from quantum channels and classical channels in other modes. However, due to the coupling in the mode multi/demultiplexer and the transmission process, there may be several kinds of excess noise generated from the quantum channel itself. Without loss of generality, we consider the situation in Fig. 5. In the mode multiplexer, the modal crosstalk with wavelength $\lambda _q$ in the mode $LP_{11}$ will be generated from the phase reference pulse in the mode $LP_{01}$. Through the propagating process, others additional kinds of noise, involving the Raman noise and the inter-mode crosstalk, will be generated in the mode $LP_{11}$ at $\lambda _q$. In the mode demultiplexer, the above-derived additional noise together with the previous modal crosstalk product the modal crosstalk in the mode $LP_{01}$. Similar to the derived result of Eq. (6), the excess noise generated from the processes 4 and 5 in Fig. 5 can be evaluated by

 

Fig. 5. The modal crosstalk in the mode multiplexer and the mode demultiplexer. (a) The modal crosstalk description involving the mode multiplexing and the mode demultiplexing. (b) The excess noise of $\xi _4^{MUX}$ as a function of $\Delta \tau$ and $L$ for $\langle \hat {N}_{ref}^{out}\rangle = 10^8$ and (c) for $\langle \hat {N}_{ref}^{out}\rangle = 10^3$. (d) The excess noise of $\xi _6^{MUX}$ as a function of $\Delta \tau$ for $L$ when $\langle \hat {N}_{ref}^{out}\rangle = 10^3$. Process 3: The mode coupling of the phase reference pulse from mode $LP_{01}$ to $LP_{11}$ in the mode multiplexer; Process 4: The mode crosstalk generated in process 3 re-coupling back to mode $LP_{01}$ in the transmission period; Process 5: The mode crosstalk generated in process 1 re-coupling back to mode $LP_{01}$ in the mode demultiplexer; Process 6: The mode crosstalk generated in process 3 re-coupling back to mode $LP_{01}$ in the mode demultiplexer; Process 7: Raman crosstalk from mode $LP_{11}$ to $LP_{01}$ in the mode demultiplexer; Raman noise in $LP_{01}$ and the back coupling process from mode $LP_{11}$ to $LP_{01}$ are not illustrated in this picture.

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Results of $\xi _4^{MUX}$ and $\xi _6^{MUX}$ are shown in Fig. 5(b)-(d), where we assume that the repetition rate of the CVQKD system is 100 MHz, the differential group delay 0.076 ns/km, the pulse width 1.2 $ns$, $\chi = -35 dB/km$, $\rho = -31 dB$, $\langle \hat {N}_{ref}^{out}\rangle = 10^8$ for (b) and $10^3$ for (c) and (d), $\Delta l = 1 km$. The results of (b) and (c) show that the threshold $\Delta \tau _{th}$ of $\Delta \tau$ will easily beyond 5 $ns$ and the excess noise of $\xi _4^{MUX}$ and $\xi _5^{MUX}$ will increase rapidly when they are larger than $0.001N_0$, especially for the transmitted LO scheme. The excess noise of $\xi _6^{MUX}$ is non-negligible when the modal crosstalk of the mode multiplexer (generated by the process 3) is temporally overlapped with quantum signal in the mode demultiplexer. The excess noise of $\xi ^{MUX}_{SRS}$ can be neglected for $\rho = -31$ dB as the power of $P^{SRS}_{11}$ can not be 30 dB larger than $P^{SRS}_{01}$ in practice. Therefore, for high-DGD system, the crosstalk in the mode multi/demultiplexer will greatly limit the transmission distance of CVQKD, while all these crosstalk in the mode multi/demultiplexer can be neglected for low-DGD system.

2.4 Nonlinear impacts

Inter-mode four-wave-mixing is the main source of nonlinear excess noise. Two pump waves $\omega _i^a$, $\omega _k^c$ and one probe wave $\omega _j^b$ in three modes can create a fourth idler wave $\omega _l^d$, which may just be the quantum channel. They satisfy the general energy conservation relation

We assume that classical channels play the role of pump and probe and take the power of $0$ dBm for each channel. The derived four-wave-mixing power $P_{idler}$ as a function of transmission distance in perfect phase-matching is shown in Fig. 6(a). If we assume the channel bandwidth is 100 GHz and the quantum channel wavelength is 1550 $nm$, the excess noise will become large when fulfilling the phase-matching condition, as shown in Fig. 6(b). For the unperfect phase matching with $\Delta \beta = 10$ $rad$/km, we find that the large phase mismatch can significantly decrease the power of inter-mode four-wave-mixing and thus we can put down this noise to an acceptable degree. Besides, the idler generated by the inter-mode four-wave-mixing between classical channels may justly play the role of the probe so that generate the second idler [40]. If the second idler is just the quantum channel and all of the two four-wave-mixing processes are in perfect phase-matching, this process should also be considered for the FMF-based CVQKD.

 

Fig. 6. (a) The power of the idler as a function of the transmission distance. (b) The resulting excess noise as a function of the transmission distance.

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Therefore, in order to suppress the inter-mode four-wave-mixing effectively, we should take a rational wavelength distribution between quantum and classical channels so that the general energy conservation relation cannot be satisfied or the phase matching factor is large. If doing so, the noise contributed by the inter-mode four-wave-mixing can be neglected and therefore we will not consider this noise for the performance analysis latter.

We note that the above-mentioned results can be similarly extended to the case that quantum channels exist in the mode $LP_{02}$ or simultaneously exist in the mode $LP_{01}$ and $LP_{02}$, which is neglected for simplicity in this section.

3. Performance analysis

In this section, we show the performance of the FMF-based CVQKD in terms of the secret key rate versus transmission distance of GMCS CVQKD protocol, taking into account collective eavesdropping attacks and homodyne detection. Detailed secret key rate calculations can refer to Refs. [16,36]. The global CVQKD parameters that used for numerical simulations are the variance of Alice’s modulation $V_A$, the electronic noise $v_{el}$ and quantum efficiency $\eta _B$ of the homodyne detector, the original system excess noise $\epsilon _0$ and the reconciliation efficiency $\beta$. All the parameters are described as $V_A = 4N_0$, $v_{el} = 0.01N_0$, $\eta _B = 0.6$, $\epsilon _0 = 0.01N_0$, and $\beta = 0.956$ [8]. In addition, the transmittance of the DMUX takes the value of $0.5$ dB [22]. The attenuation coefficient of the FMF’s modes are $0.218$ dB/km, $0.215$ dB/km, $0.21$ dB/km and $0.21$ dB/km for modes $LP_{01}$, $LP_{11}$, $LP_{21}$ and $LP_{02}$ [39]. The mode coupling values of the few-mode fiber and the modal crosstalk of the mode demultiplexer between all the modes are assumed to be equal with each other for simplicity. We don’t consider the noise of inter-mode four-wave-mixing as we assume that it has been suppressed by careful channel wavelength management.

Combining the noise analysis in Section 2, we now show the simulation results of secret key rate and discuss the feasibility of the FMF-based CVQKD system. Multiple parameters are involved to obtain these results, including the mode coupling value, the DGD, the modal crosstalk of the mode demultiplexer and the insert loss of the mode demultiplexer. We assume that only one quantum channel locates in modes $LP_{01}$ and $LP_{02}$ respectively. We first consider the case of low-DGD system, which the DGD is small enough that noises contributed by the re-coupling inter-mode crosstalk and the mode crosstalk of the mode multi/demultiplexer can be temporally isolated and hence can be neglected. Therefore, only the Raman crosstalk and the mode demultiplexer’s insert loss are the main incremental impacts that we should take into account. For example, it is a low-DGD system when we choose the DGD values of Group 1 in Table 1 for a 100 MHz CVQKD system within 100 km transmission distance. We show the results of this case in Fig. 7(a) and 7(b), where the mode coupling value is -35 dB/km and the power of classical channels in $LP_{01}$ and $LP_{02}$ is 0 dBm (both forward and backward channels). The power of classical channels in $LP_{11}$ and $LP_{21}$ is the sum of the forward and backward channels and take2 the values of $3$ $dBm$, $13$ $dBm$, $18$ $dBm$, $23$ $dBm$, respectively. From our results we find that the performance trade-off between CVQKD and classical communication on SMF can be mitigated in short distance for the FMF-based CVQKD system. Larger power of classical channels in modes $LP_{11}$ and $LP_{21}$ generates larger Raman crosstalk noise and thus degrades the performance of CVQKD, while lower insert loss of mode demultiplexer shows better performance. A Large amount and a high power of classical channels in modes $LP_{11}$ and $LP_{21}$ not only contribute larger Raman crosstalk to quantum channels, but also increases the difficulty of the channel wavelength management, which will degrade the effect of the suppressing of the inter-mode four-wave-mixing.

 

Fig. 7. The numerical results of the secret key rate versus the transmission distance for low-DGD and high-DGD system. (a) Low-DGD system with a transmitted LO. (b) Low-DGD system with a locally generated LO. (c) High-DGD system with a transmitted LO. (d) High-DGD system with a locally generated LO. The power in modes $LP_{11}$ and $LP_{21}$ represents the sum of forward and backward classical channel power, both equal to 13 $dBm$ for High-DGD system. Solid lines are the numerical results of the SMF-coexisted CVQKD, which has the same classical channels distribution as the mode $LP_{01}$ but 0.2 dB/km attenuation coefficient. IL: the insert loss of the mode demultiplexer; $\chi$: mode coupling value; $\rho$: mode crosstalk of mode multi/demultiplexer.

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Table 1. DGD vs. $LP_{01}$ (ps/km)

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We then consider the case of the high-DGD CVQKD system. We assume the DGD values of modes are equal as the Group 2 in Table 1. In this case, all the noises analyzed in Section 2 should take into account except the inter-mode four-wave-mixing and the re-coupling inter-mode crosstalk of quantum signal. The results of this case are shown in Fig. 7(c) and 7(d). The power of forward and backward classical channels in modes $LP_{01}$ and $LP_{02}$ is assumed to be 0 dBm, while the power is 10 dBm in modes $LP_{11}$ and $LP_{21}$. It shows that the performance trade-off between CVQKD and classical communication on SMF can also be mitigated in short distance for this case. The impacts result from the mode coupling and the differential group delay can greatly decrease the system’s performance, especially for the TLO scheme. The results have little difference when choose DGD values of Group 2, which is due to the rapid increasing of the noises caused by the re-coupling inter-mode crosstalk and the mode crosstalk of the mode multi/demultiplexer when their values are large enough to impact CVQKD system (see Fig. 5), especially for the TLO scheme. The LLO CVQKD scheme shows potential advantages although its larger phase noise. This is because the strongly transmitted LO may induce much lager excess noise than the weak phase reference pulse through mode coupling. We don’t show the impacts of the mode demultiplexer’s insert loss and the Raman crosstalk here as they have been shown in Fig. 7(a) and 7(b).

Besides, we also study the influence of the signal intensity degradation caused by the mode coupling. Different from SMF, a fraction of the transmitted light in one mode (mode A) will coupling into other modes because of the random modal coupling. Although a fraction of the transmitted light in other modes will couple into the mode A, we should regard these coupled lights as noise in weak coupling regime. Thus, due to the mode coupling, signal transmitting in FMF will suffer an extra loss and hence may decrease the communication quality. To make it comprehensible, we give a example of this phenomenon in Appendix E. However, we find that this extra signal intensity degradation has no influence on the final performance of the FMF-based CVQKD system (thus not shown in Fig. 7). This is mainly due to the significant impact of excess noise so that the influence of the extra intensity degradation is masked.

Our results above show that in weak coupling regime, it is feasible for CVQKD coexisting with high power classical signals on FMF within short distance, when quantum channels are distributed in the non-degenerate modes. The simultaneous transmission of quantum signals in multiple non-degenerate modes makes possible higher secret key of CVQKD, while largely tolerable power of classical channels in degenerate modes guarantees the enough capacity of classical communication. Therefore, compared with SMF-based system, the FMF-based system enables higher performance of CVQKD and much larger power of classical channel in short-distance communication, breaking away the fetter of the performance trade-off between CVQKD and classical communication in SMF. Results show that in weak coupling regime, the mode coupling value and the DGD are the two key parameters to determine the system’s performance and design the FMF-based CVQKD system. A lower mode coupling value and a smaller DGD are preferred for FMF-based system. Moreover, DGD may become one of the main limits for FMF-based CVQKD towards high-speed system, as higher CVQKD system has smaller value of $\Delta \tau$ so that the tolerable DGD value is lower. In addition, the insert loss and the modal crosstalk of the mode multi/demultiplexer as well as the wavelength distribution of classical channels should also be taken into account. A low insert loss and modal crosstalk of mode multi/demultiplexer will bring about better performance of the FMF-based system. To relax the DGD-induced impacts, one possible way is the frequency-multiplexing technique that shifting the frequency of the phase reference pulse so that its crosstalk to quantum signal can be suppressed dramatically (more than 20 dB) [41].

As the strong mode coupling between degenerate modes, quantum signal will strongly coupled with itself in degenerate modes. Therefore, how to recover the quantum signal in degenerate modes still need further research. One possible method is employing the MIMO DSP similar to the classical communication. This approach has not been investigated in CVQKD so far, and its feasibility needs further investigation. It also brings complexity and high cost to the system. Recently, the study of multiple-input multiple-output-free transmission with elliptical core few-mode fiber has been experimentally demonstrated [42]. This elliptical core few-mode fiber can break the degeneracy in space of a mode group and thus make possible the CVQKD in the degenerate modes, such as in modes $LP_{11a}$ and $LP_{11b}$. However, the crosstalk between degenerate modes is still large for CVQKD and the attenuation coefficient, which is up to 0.63 dB/km, is much larger than the standard single-mode fiber. Further research is needed for this elliptical core few-mode fiber. If the attenuation coefficient of it can be as low as 0.21 dB/km and the mode coupling value between all the modes (degenerate and non-degenerate modes) can be controlled within -35 dB/km, the total secret key rate of our FMF-based scheme can be improved significantly within metropolitan area compared with the single-mode case, as shown in Fig. 8.

 

Fig. 8. The numerical results of the secret key rate versus the transmission distance when all the six modes can distribute key. (a) The results of CVQKD with a transmitted LO. (b) The results of CVQKD with a locally generated LO. Here, the mode coupling value $\chi = -35$ dB/km and mode crosstalk of mode demultiplexer $\rho = 31$ dB. The black solid lines represent the results of single-mode case.

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4. Conclusion

We have proposed the FMF-based CVQKD scheme by using SDM and analyzed the noise impact and feasibility of the scheme in weak coupling regime. We demonstrate the effect of the channel characteristic and excess noise on the FMF-based CVQKD with the TLO scheme or LLO scheme. The case that quantum channels in the non-degenerate modes $LP_{01}$ and $LP_{02}$ is investigated as the strong mode coupling between degenerate modes. Analysis results show that the mode coupling of phase reference pulse in the few-mode fiber and in the mode demultiplexer, the Raman crosstalk, the inter-mode four-wave-mixing and the insert loss of the mode demultiplexer have effect on the FMF-based CVQKD system. For the tunable channel parameters, the noise of inter-mode four-wave-mixing can be suppressed effectively. In low-DGD system, the Raman crosstalk and the insert loss of the mode demultiplexer have much effect on the CVQKD system, whereas in high-DGD system, all the factors should be taken into account for the performance analysis. Moreover, DGD may become one of the main limits for FMF-based CVQKD towards high-speed system. Simulation results show that the total secret key rate can be improved within metropolitan area compared with that of the single mode case, whereas the maximal secure transmission distance is degraded. If all the modes can distribute key simultaneously, the total key rate can be increased significantly. These numerical results show that it is feasible for CVQKD coexisting with high power classical signals over FMF. The results may provide theoretical foundation for future CVQKD integrating with classical FMF-based network.

Appendix

A. Excess noise of the SMF-based CVQKD

The excess noises of CVQKD coexisting with classical signals on SMF mainly include the following aspects: (1) the original excess noise of CVQKD system; (2) the SRS noise (see Ref. [22]); (3) phase noise $\varepsilon _{phase}$ [17]. From the experimental results, $\varepsilon _{phase} < 1.2\times 10^{-5}$ can be achieved for CVQKD with the transmitted LO scheme [8]. For the LLO scheme, a phase excess noise of about 0.04 (in SNU), which corresponding to a phase inaccuracy of $\delta \theta \cong 6^{\circ }$, can be achieved in the practical implementation [16]. Therefore, in the main text, we will assume that the phase inaccuracy of the phase compensation $\delta \theta = 6^{\circ }$ and treat this noise as the untrusted excess noise for the LLO scheme.

B. Noise contributions from quantum channel itself

Phase reference pulses The analysis process below is based on the case of Fig. 2. To estimate the excess noise caused by the re-coupling inter-mode crosstalk of phase reference pulse, we firstly subdivide the re-coupling inter-mode crosstalk into a set of Gaussian sub-pulses that each sub-pulse can be seen as the re-coupling inter-mode crosstalk in a short segment $\Delta l$ without spread. The time delay between two adjacent sub-pulses is $\mathrm {DGD}\cdot \Delta l$ and the total crosstalk is the superposition of all the sub-pulses. Then, following the power-coupling model in [37] and the analysis result in [28], the in-band excess noise contributed by the re-coupling inter-mode crosstalk of the phase reference pulse can be evaluated by

(6)$$\begin{aligned} \xi_{XT}^{re,p} &= \frac{1}{2}\cdot2\cdot\frac{TT_MT_W\eta(\frac{1}{2} + \frac{1}{2}e^{{-}2\chi L} - e^{-\chi L})\langle\hat{N}_{ref}^{in}\rangle}{TT_MT_W\eta} \frac{1}{k}\sum_{i = 0}^{k - 1}e^{-\frac{(\mathrm{sign(DGD)}\cdot\Delta\tau - i\cdot\mathrm{DGD}\cdot\Delta l)^2}{2\sigma_t^2}}\\ &= \frac{(\frac{1}{2} + \frac{1}{2}e^{{-}2\chi L} - e^{-\chi L})\langle\hat{N}_{ref}^{out}\rangle}{Tk}\sum_{i = 0}^{k - 1}e^{-\frac{(\mathrm{sign(DGD)}\cdot\Delta\tau - i\cdot \mathrm{DGD}\cdot\Delta l)^2}{2\sigma_t^2}}, \end{aligned}$$

with $T = 10^{-\alpha L/10}$, where $\alpha$ is the quantum channel attenuation coefficient, $T_M$ the transmittance of the mode division demultiplexer, $T_W$ the transmittance of the wavelength division demultiplexer, $\eta$ the quantum efficiency of the coherent detector, $\langle \hat {N}_{ref}^{in}\rangle$ and $\langle \hat {N}_{ref}^{out}\rangle$ are the mean photon number of the phase reference pulse at the fiber input and output respectively, $\sigma _t$ the half-width of the pulse, $k$ the total number of segments, $\mathrm {sign(x)}$ means taking the sign of $x$. According to Eq. (6), we illustrate the estimated excess noise of $\xi _{XT}^{re,p}$ in Fig. 9, where the mode coupling value is $-35$ dB/km, $\Delta l = 0.1$ km, $\langle \hat {N}_{ref}^{out}\rangle = 10^8$ (TLO) or $10^3$ (LLO), the repetition rate of CVQKD system is 100 MHz and the pulse width is 1.2 ns. From the numerical results in Fig. 9, we find that for the TLO scheme, the magnitude of excess noise will increase rapidly when its value starting to higher than 0.001$N_0$. For the LLO scheme, this excess noise is lower than that of the TLO case. For a low speed CVQKD system of 1 MHz repetition rate, a much higher DGD value can be tolerated, and similar diagrams will be obtained.

 

Fig. 9. The re-coupling inter-mode crosstalk noise generated from the phase reference signal as a function of transmission distance. The repetition rate of the CVQKD system is assumed to be 100 MHz. SNU, shot noise units.

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Quantum signal pulse In the transmission procedure, quantum signal will coupling from its own mode to other modes and then re-coupling back to the original mode, generating the in-band excess noise. This excess noise is randomized in phase and polarization and hence can be treat as output from a chaotic source with Bose–Einstein photon statistics [43]. If the average photon number of this noise referred to the input is $\langle N_{XT}^{B,q}\rangle$, then the excess noise contributed by this photons in matched mode referred to the input is given by [21]

(7)$$\xi_{XT}^{re,q} = \frac{1}{2}\cdot 2\cdot \langle N_{XT}^{B,q}\rangle = \langle N_{XT}^{B,q}\rangle,$$

where the factor 1/2 represent the mode selection of the LO. To estimate this noise, it is more convenient to use power coupling model, as it describes the evolution of mode coupling in the way of ensemble average. The noise power at the fiber output can be estimated as $P_{out}^{noise} = P_{out}^{with} - P_{out}^{without}$, where $P_{out}^{with}$ and $P_{out}^{without}$ are the output power of the signal considering or not considering the re-coupling inter-mode crosstalk. $P_{out}^{with}$ can be obtained from the power coupling mode straightforward [37]. $P_{out}^{without}$ can be calculated by modifying the power coupling model as $\frac {dP_i}{dz} = -\alpha _iP_i - \sum _{j\neq i}\chi _{ij}P_i$. Then, $P_{out}^{without}$ is equal to $P_i|_{z = L}$, where $L$ is the transmission distance. Therefore, since signal’s power is proportional to its average number of photons, we can rewrite the above noise power as the excess noise generated from the re-coupling inter-mode crosstalk of the quantum signal itself in the following: $\xi _{XT}^{B,q} = \langle N_{XT}^{B,q}\rangle = (\bar {n}_{out}^{with} - \bar {n}_{out}^{without})/T$ with $T = 10^{-\alpha L/10}$, where $\alpha$ is the fiber attenuation coefficient, $\bar {n}_{out}^{with}$ and $\bar {n}_{out}^{without}$ are the output average photon number of the signal considering or not considering the re-coupling inter-mode crosstalk. To simplify the expression, we now give some assumptions as follows: only one quantum channel in mode $LP_{01}$; the fiber attenuation coefficient for all the four modes is identical; the mode coupling coefficient $\chi$ between modes is identical. Following the above derivation procedure, the excess noise of the inter-mode crosstalk originated from the re-coupling of quantum signal itself can be given by

(8)$$\xi_{XT}^{re,q} = \bar{n}_{in}(\frac{1}{4} + \frac{3}{4}10^{{-}4\chi L/10} - 10^{{-}3\chi L/10}),$$

with $\bar {n}_{in} = (V - 1)/2$, where $V = V_A + 1$ denotes the variance of the quantum signal output Alice’s station, $\chi$ is the mode coupling value, $L$ the transmission distance. $V_A$ is Alice’s modulation variance. We plot the simulation results of $\xi _{XT}^{re,q}$ as a function of $L$ in Fig. 10, where the variance $V$ and mode coupling coefficient $\chi$ take various values. From these results we can deduce that within weak coupling regime, the excess noise from the quantum signal’s re-coupling inter-mode crosstalk can be neglected although the modulation variance is up to 80, as the noise is lower than 0.001$N_0$. This noise should be taken into account in long distance when the mode coupling value is large enough, for instance, -30 $dB/km$. Note, for weak coupling regime, the transmission distance should be lower than 80 $km$ for $\chi = -35$ $dB/km$ (corresponding to a mode coupling strength close to -30 $dB/km$) [44]. In addition, the noise estimation of Eq. (8) should be seen as the upper bound of the re-coupling inter-mode crosstalk noise, as we have neglected the group delay between modes above.

 

Fig. 10. The re-coupling inter-mode crosstalk noise generated from the quantum signal itself as a function of transmission distance. SNU, shot noise units

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C. Evaluation of Raman crosstalk

The photons of Raman scatter in modes will coupling between each other and may contribute extra excess noise to the quantum channel. Assuming that the total input power of forward and backward classical channels in the mode $i$ are $P_{i,f}^{in}$ and $P_{i,b}^{in}$, respectively, where $\quad i \in \{ 1,\ldots ,n\}$ and $n$ is the total mode number. The Raman scatter power $\delta P_{i}^{SRS}(z)$ at wavelength $\lambda$ from a fiber element of length $dz$ at position $z$ is [20]

(9)$$\delta P_{i}^{SRS}(z) = (P_{i,f}^{in}e^{-\alpha_i z} + P_{i,b}^{in}e^{-\alpha_i (L - z)})\kappa(\lambda)\Delta\lambda\cdot dz,$$

where $\kappa (\lambda )$ is the Raman scatter coefficient, $\alpha _i$ is the fiber attenuation coefficient of mode $i$. The Raman scatter power in each mode will couple between each other along the forward transmission from position $z$ to the fiber output. Their coupling process can be modeled by the power coupling model (see [37]), given by

(10)$$\frac{d\left(\delta P_{i}^{SRS}(z')\right)}{dz'} ={-}\alpha_i\delta P_{i}^{SRS}(z')+ \sum_{j\neq i}\nu_{ij} \left( \delta P_{j}^{SRS}(z') - \delta P_{i}^{SRS}(z') \right),$$

where $\nu _{ij}$ is the mode coupling coefficient between mode $i$ and $j$. The initial conditions of Eq. (10) is equal to Eq. (9). Combining all the differential equations above and integrating $z'$ from $z$ to $L$, the Raman scatter power $\delta P_{i}^{SRS}(z)_L$ at the fiber output, which is generated from the fiber element of length $dz$ at position $z$, can be calculated. Then, the Raman scatter power in the mode $i$ at the fiber output can be given by

(11)$$P_{i}^{SRS} = \int_{0}^{L}\delta P_{i}^{SRS}(z)_Ldz.$$

The SRS noise can modeled as output from a chaotic source with Bose-Einstein photon statistics and the in-band noise photon number per spatiotemporal mode referred to the input can be given by [21]

(12)$$\xi_{i}^{SRS} = \frac{1}{2}\frac{2T_MT_W\eta P_{i}^{SRS}}{h\frac{c}{\lambda}N_{mode}T_MT_W\eta T} = \frac{\lambda P_{i}^{SRS}}{hcN_{mode}T}$$

with $N_{mode} = c/\lambda ^2\Delta \lambda$, where $h$ is the Planks constant, $c$ is the speed of light. We assume that the Raman scatter coefficient is $3\times 10^{-9}$ km nm$^{-1}$ in this paper.

D. The evaluation of the inter-mode four-wave-mixing

Without loss of generality and to simplify the analysis procedure, we talk a three-mode FMF as a example. we assume two classical channels in the mode $LP_{11}$ as pumps, one quantum channel and one classical channel in the mode $LP_{01}$ as the idler and the probe. When in the perfect phase-matching condition, the idler power can be obtained from the nonlinear propagation equation in the undepleted pump approximation [40]:

(13)$$\frac{dA_I^{01}(z)}{dz}\cong \gamma(f_{11,01,11,01} + f_{11,11,01,01})A_{p1}^{11}(0)A_{p2}^{11}(0){\bigg(}A_{B}^{01}(0){\bigg)}^*e^{{-}3\alpha z/2} - \frac{\alpha}{2}A_{I}^{01}(z),$$

where $A_I^{01}(z),A_{B}^{01}(z)$ are the field amplitudes for idler and probe respectively, $A_{p1}^{11}(z),A_{p2}^{11}(z)$ are the field amplitudes for first and second pump respectively. $\alpha$ is the loss coefficient of the generated idler, $\gamma$ is the nonlinear coefficient of the $LP_{01}$ mode. $f_{lmnp}$($l,m,n,p$ represent fiber mode) is the nonlinear overlap factors. The power of the idler is $P_{idler} = |A_I^{01}|^2$. For a imperfect phase matching, the power of the idler can be given by $P_{idler} = \eta _{f}|A_I^{01}|^2$, where $\eta _f$ is the four-wave-mixing efficiency [19]. Here, we assume that $\gamma = 1.77$ $\mathrm {W}^{-1}/km$ and $\alpha = 0.2$ dB/km.

E. The extra signal loss caused by the mode coupling

For SMF, there is no extra signal loss contributable to the mode coupling as SMF only support one mode. While signal is transmitted in FMF, a part of its power will couple into other modes, resulting in an extra loss of signal in weak coupling regime. To illustrate this extra signal loss, we assume that only the mode $LP_{01}$ is launched with the signal at the input of a six-mode FMF. Considering power coupling mode [37] and removing the coupling power from other modes to the mode $LP_{01}$ (regarded as noise), the power of the signal in the mode $LP_{01}$ follows the following differential equation:

(14)$$\frac{dP_{01}(z)}{dz} ={-}\alpha'P_{01}(z) - 3\chi P_{01}(z),$$

where $P_{01}(z)$ represents the signal power of the mode $LP_{01}$ at the fiber length of $z$ and $\alpha '$ is the attenuation coefficient of the mode $LP_{01}$. The solution of the above equation can be easily obtained, that is $P_{01}(z) = P_{01}(0)\exp \left [-(\alpha ' + 3\chi )z\right ]$, where $P_{01}(0)$ is the signal power at the fiber input. If the signal is transmitted in a SMF with the same attenuation coefficient $\alpha '$, the signal power at the fiber length of $z$ can be expressed by $P_{SMF}(z) = P_{SMF}(0)\exp \left (-\alpha 'z\right )$, where $P_{SMF}(0)$ is the signal power at the fiber input. Comparing $P_{01}(z)$ with $P_{SMF}(z)$, we find that there is an extra attenuation factor of $\exp \left (-3\chi z\right )$ in $P_{01}(z)$ for $P_{01}(0) = P_{SMF}(0)$. Therefore, compared with SMF, there is a extra signal loss caused by the mode coupling for FMF in weak coupling regime.

Funding

National Natural Science Foundation of China (61801522, 61871407, 61872390); Postgraduate Scientific Research Innovation Project of Hunan Province, China (CX20200209); Postgraduate Independent Exploration and Innovation Project of Central South University, China (2020zzts136).

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61871407, 61872390, 61801522), the Postgraduate Scientific Research Innovation Project of Hunan Province, China (Grant No. CX20200209) and the Postgraduate Independent Exploration and Innovation Project of Central South University, China (Grant No. 2020zzts136).

Disclosures

The authors declare no conflicts of interest.

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