Abstract

A high-speed four-state continuous-variable quantum key distribution (CV-QKD) system, enabled by wavelength-division multiplexing, polarization multiplexing, and orbital angular momentum (OAM) multiplexing, is studied in the presence of atmospheric turbulence. The atmospheric turbulence channel is emulated by two spatial light modulators (SLMs) on which two randomly generated azimuthal phase patterns yielding Andrews’ spectrum are recorded. The phase noise is mitigated by the phase noise cancellation (PNC) stage, and channel transmittance can be monitored directly by the D.C. level in our PNC stage. After the system calibration, a total SKR of >1.68 Gbit/s can be reached in the ideal system, featured with lossless channel and free of excess noise. In our experiment, based on commercial photodetectors, the minimum transmittances of 0.21 and 0.29 are required for OAM states of 2 (or −2) and 6 (or −6), respectively, to guarantee the secure transmission, while a total SKR of 120 Mbit/s can be obtained in case of mean transmittances.

© 2017 Optical Society of America

1. Introduction

Quantum key distribution (QKD) is one of the most mature applications of quantum cryptography, which enables two legitimated parties, called “Alice” and “Bob”, to establish shared cryptographic keys [1,2]. QKD has been recognized as the only approach so far to realize secure communications, guaranteed by the laws of quantum mechanics. QKD implementations can be built via optical fibers [3,4] or free-space optical (FSO) links [5–7]. The latter one has attracted increasing attentions because of its great flexibility for infrastructure establishment. However, the unknown transmittance fluctuation caused by atmospheric turbulence may bring into the picture the excessive post-processing noise [8].

Numerous QKD protocols have been proposed since the invention of BB84 protocol, most of which are involved with two ineluctable topics: proof of security and improvement of secret key rate (SKR). It is well-known that both discrete-variable protocols and continuous-variable QKD (CV-QKD) protocols based on Gaussian distribution can be established with unconditional security [9,10]. CV-QKD protocols are compatible with fully developed optical telecommunication technologies, which have been extensively studied recently because they potentially enable a high-speed QKD systems.

In order to further increase the SKR, noise control and reconciliation efficiency improvement in principle are the most straightforward methods. Self-homodyne detection has been commonly implemented in CV-QKD systems to control phase noise, where the brighter local oscillator (LO) light is emitted at Alice’s side, multiplexed and co-propagated along with the quantum keys through the quantum channels, and then mixed in the coherent detector at Bob’s side [11]. This kind of schemes offer the high SKR brought by phase noise mitigation, but potentially open a security loophole for Eve, as discussed in [12]. In addition, cooled avalanched photodiodes (APDs) are also usually applied in current CV-QKD systems [13]. A high SKR is enabled by the electrical noise suppression in these systems, but with the price of limited bandwidth and high cost. Given that most of the state-of-art error correction codes are developed for discrete modulation, it is necessary to bridge the gap between CV-QKD and discrete modulation to facilitate the reconciliation efficiency [14–16]. CV-QKD protocols based on discrete modulation, e.g., the four-state protocol, have been proved to be secure against collective attacks [17]. Strictly speaking, some controversies concerning the limits and assumptions in real implementations have existed since the birth of these protocols.

In this work, we propose a high-speed RF-assisted four-state CV-QKD system, which is enabled by three-dimensional (3D) multiplexing based on polarization, wavelength, and orbital angular momentum (OAM) degrees of freedom; and evaluate it in the presence of atmospheric turbulence. The atmospheric turbulence channel is emulated by two spatial light modulators (SLMs) on which two randomly generated azimuthal phase patterns yielding Andrews’ spectrum [18] are recorded. The validity of our polarization-insensitive atmospheric turbulence emulator is verified in terms of on-axis intensity probability density function (PDF) and irradiance correlation function (ICF). We demonstrate CV-QKD FSO system enabled by 24 multiplexed/demultiplexed QPSK channels, based on two polarization states, six wavelengths, and four OAM modes. Wavelength interleaving is introduced to mitigate the crosstalk arising from the adjacent wavelength channels. Each channel is encoded by the quantum keys with four-state discrete modulation, i.e., quaternary phase shift keying (QPSK) modulation, which are prepared and sent from Alice to Bob; the phase noise cancellation (PNC) stage is installed at Bob’s side after classical coherent detection to control the excess noise [4]. In addition, the D.C. level in this stage is used to monitor the channel transmittance. As a result, the post-processing noise can be effectively avoided. After the system calibration, transmittance fluctuation monitoring, and residual excess noise measurement, a total maximum SKR of >1.68 Gbit/s can be reached in the ideal system, which is featured by lossless channel and completely eliminated excess noise. In our experiment, the minimum channel transmittances of 0.21 and 0.29 are required for OAM states of 2 (or −2) and 6 (or −6), respectively, to guarantee the secure transmission, while a total SKR of 120 Mbit/s can be obtained, with commercial photodetectors, in case of mean transmittances.

The remainder of this paper is organized as follows. In Section 2, the proposed system CV-QKD system is described along with the SKR derivation. The verification of the polarization- insensitive atmospheric turbulence emulator and the monitored transmittance fluctuation are presented in Section 3. In Section 4, the excess noise levels are measured, followed by the obtained SKR results analysis. Finally, the conclusions are summarized in Section 5.

2. Proposed CV-QKD system description

The schematic of the proposed 3D multiplexed four-state CV-QKD system is depicted in Fig. 1. At Alice’s side, the continuous wave (CW) light beams on a 50-GHz grid (193.55-193.30THz) are generated with the help of six lasers with a linewidth less than 10 kHz each. The six wavelength channels are multiplexed together and sent to an I/Q modulator. A sequence of QPSK symbols is generated with a key rate of 2.5 Gbit/s, and up-converted to the RF domain centered at 5 GHz, which is implemented with the help of an arbitrary waveform generator. The generated RF signals are then used as inputs to the I/Q modulator, which is operated at the quadrature point in both in-phase and quadrature branches. In order to reduce the excess noise caused by the bias fluctuation, a modulator bias controller (MBC) is used to track and lock to the optimum bias. Notice that small signal modulation is required during the electro-optical conversion. A typical optical spectrum after I/Q modulation is shown as an inset of Fig. 1(a). A 50/100-GHz optical interleaver (IL) is employed to separate the odd and even sub-channels, which are then de-correlated by several hundreds of symbols via additional optical fiber. The modulation variance VA of the signals, expressed in shot-noise units, is adjusted by tuning variable optical attenuators (VOAs). Each beam is collimated and converted to OAM beams by a reflective phase-only spatial light modulator (SLM), which is featured with 1920 × 1080 pixels’ resolution, 15.36 × 8.64 mm active area, and 60 Hz image frame rate. The left and right halves of the SLM1 screen are loaded with different phase patterns, which then convert the incoming Gaussian beams from the odd and even wavelength sub-channels to the target OAM beams. Specifically, the Gaussian beam multiplexed by three odd wavelength sub-channels is converted to the superstition of OAM modes with states of 2 and −6, while the other Gaussian beam multiplexed by three even wavelength sub-channels is converted to the superstition of OAM modes with states of −2 and 6. A blazed “fork” phase pattern is used to separate the desirable OAM beams from the Gaussian beam. The wavelength and OAM multiplexed beams are combined by a beam splitter (BS), which is followed by a half-wave plate (HWP) to adjust the polarization. Then these beams are split into two branches, one of which is delayed before combined with the other one via a polarization beam splitter (PBS). The concept of the 3D multiplexing, i.e., wavelength division multiplexing (WDM), OAM multiplexing, and polarization division multiplexing (PDM) is illustrated as the inset in Fig. 1(b).

 figure: Fig. 1

Fig. 1 The schematic of the proposed 3D multiplexed four-state CV-QKD system. Inset (a) the optical spectrum after I/Q modulation; Inset (b) the concept of 3D multiplexing. PBS: polarization beam splitter, BS: beam splitter, MR: mirror, HWP: half-wave plate, IL: optical interleaver, OTF: optical tunable filter, LO: local oscillator laser, PNC: phase noise cancellation stage.

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The beams sent from Alice’s side then pass through the polarization-insensitive turbulent emulator, which is configured by a BS, a PBS, a HWP, two mirrors, and two polarization-sensitive phase-only SLMs (SLM2 and SLM3). For more detailed descriptions, please refer to [19]. Both SLM screens are loaded with randomly generated azimuthal phase patterns following Andrews’ spectrum. The time-varying atmospheric turbulence is emulated by constantly upgrading both patterns at 50 Hz rate. Our turbulent model is built based on the Rytov variance of σR2=0.2, and both polarizations are affected by the same turbulence effects. [The Rytov variance, defined as σR2=1.23Cn2(2π/λ)7/6L11/6, is a relevant parameter to characterize the turbulence strength because it takes the propagation distance L, the operating wavelength λ, and the refractive structure parameter Cn2 into account.]

At Bob’s side, the distorted beams are collected by a compressing telescope, followed by a HWP to select one polarization for OAM de-multiplexing. An inverse spiral-phase pattern is loaded onto SLM4 to convert the target OAM beam back to a Gaussian-like beam, which is then coupled to a 2-meter AR-coated single-mode fiber (SMF). This patch of SMF plays as a mode sorter, where high-order OAM beams fade off soon except for zero-order Gaussian beam. One wavelength channel is selected by the optical tunable filter (OTF) and then mixes with the LO light-beam in the coherent detector. This coherent detector includes an optical 90° hybrid, two 9 GHz PIN photodiodes, and a real-time oscilloscope with 100 GSa/s and 33 GHz analog bandwidth. Notice that any polarization, OAM mode, and wavelength channel can be effectively separated and de-multiplexed for coherent detection by switching the HWP, spiral-phase pattern, and the OTF.

The captured in-phase and quadrature photo-currents after the coherent detection are then passed through the PNC stage that incorporates two square operators, one addition operator, one digital D.C. cancellation block, and one digital down-converter. For more information about the working principle of such PNC stage, please refer to [4]. In order to further specify the new-found relationship between the D.C. level in the PNC stage and transmittance fluctuation over the turbulent channel, we present the equation after the addition operator as:

iC(t)(1+A2)T22AI(t)cos(w1tπ4)+2AQ(t)cos(w1tπ4)+nC
where A is the modulation index applied to the I/Q modulator at Alice’s side, I(t) and Q(t) are the base-band QPSK symbols generated at Alice’s side, and w1 is the RF angular frequency used for up-conversion at Alice’s side, T denotes the channel transmittance, and nC represents the resulting noise. Notice that high-frequency cross-terms are not shown in Eq. (1), since they will be filtered out easily in the following digital down-converter. As we can see from Eq. (1), the D.C. level, i.e., (1+A2)T2, can be used to accurately measure the channel transmittance. In principle, “D.C. level” is not an accurate term considering the transmittance fluctuation; while the transmittance fluctuation rate is normally less than the order of kHz (a maximum of 100Hz can be realized in our experiment), as a result, more than Megabits can be transmitted during the channel coherence time. Therefore, we denote the first term in Eq. (1) as the “D.C. level”.

The final SKR definition based on reverse reconciliation is given as [4],

ΔI=βIABχBE
where β is the reconciliation efficiency, IAB is the Shannon mutual information between Alice and Bob, χBE is the Holevo bound, and they can be identified as

IAB=log2(V+χtot1+χtot).
χBE=G(λ112)+G(λ212)G(λ312)G(λ412)
withV=VA+1,χtot=χline+χhet/T,χline=1/T+ϵ1,χhet=(2+2Velη)/η,G(x)=(x+1)log2(x+1)xlog2(x),λ1,2=12(A±A24B),λ3,4=12(C±C24D),A=V2+T2(V+χline)22TZ42,B=(TV2+TVχlineTZ42)2,C=Aχhet2+B+1+2χhet[VB+T(V+χline)]+2TZ42[T(V+χtot)]2,D=(V+χhetB)2[T(V+χtot)]2ξ0,2=1/2eVA/2[cosh(VA/2±cos(VA/2))],ξ1,3=1/2eVA/2[sinh(VA/2±sin(VA/2))].

In the above expressions, VA is Alice’s modulation variance, ϵ denotes the excess noise, Vel represent the electrical noise at Bob’s setup, and η is the global detection efficiency of Bob’s apparatus, which mainly contributed by the conversion efficiency from OAM to Gaussian mode, the coupling efficiency from free-space to SMF, and the detection efficiency of the coherent detector. Notice that VA, ϵ, and Vel are expressed in shot-noise units. After the system calibration [20], Bob’s apparatus yields an electrical noise of Vel=0.13, a global detection efficiency of η=0.2 for the detection of OAM states of ± 2, and a global detection efficiency of η=0.11 for the detection of OAM states of ± 6. Notice that the global detection efficiency is measured when only one OAM mode is transmitted from Alice and de-multiplexed at Bob’ side in the absence of atmospheric turbulence.

3. Polarization-insensitive atmospheric turbulence emulator

Our FSO channel in the presence of atmospheric turbulence is emulated indoor, so it is necessary to validate the atmospheric turbulence emulator first. Two statistical figures of merit are selected for analyzing the accuracy of our model, i.e., on-axis received intensity PDF and ICF. In our experiment, a collimated Gaussian beam with a diameter of 2.4 mm is transmitted through our turbulent emulator and captured by a power sensor with an aperture size of <50 μm, which is exactly centered with the beam axis. The recorded on-axis intensity PDF in principle should yield Gamma-Gamma distribution. Figure 2(a) shows the experimentally obtained on-axis intensity PDF and the corresponding analytical Gamma-Gamma distribution. An excellent agreement is found between the analytical Gamma-Gamma distribution and the experimental data. The ICF is related to the irradiance distribution captured after the turbulent model [21], which can be expressed as Γ (r1,r2, L)=E{[I1E(I1)][I2E(I2)]}, where I1 and I2 are the captured irradiances of positions r and r2 at distance L, respectively; the operator E() denotes an ensemble average of the quantity inside the parenthesis. The target continuous FSO channel is numerically configured by 100 equidistant slabs, which is featured by the path length of L = 2.6 km and turbulence refractive structure parameter of Cn2=1.76×1015m2/3. The ICF based on two discrete phase patterns is recorded and compared with the corresponding ICF based on the continuous path. Figure 2(b) shows the ICFs obtained from our experiment and the target continuous turbulence model. We can see that our experimental results agree well with the numerical curve. Therefore, the accuracy of our turbulent model has been validated in terms of on-axis intensity PDF and ICF.

 figure: Fig. 2

Fig. 2 (a) PDF of the obtained on-axis intensity fluctuation and the analytical Gamma-Gamma distribution. (b) ICFs based on the target continuous path and our turbulent model.

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After the verification of our turbulent model, the turbulence introduced transmittance fluctuation has to be monitored accurately to avoid post-processing noise. Figure 3 shows the monitored statistical distribution of the channel transmittance for OAM states of ± 2 and ± 6. The probability distributions in Figs. 3(a) and 3(b) are quite similar, and the mean transmittance is measured to be 0.57. Similarly, the histograms between Figs. 3(c) and 3(d) are comparable, and the mean transmittance is obtained as 0.43. Notice that the intrinsic transmittance loss in our turbulent model, e.g., the reflectivity of the SLM screens, and 3-dB loss of the BS, is not considered here. This is because such intrinsic loss will not occur in real FSO channels. In addition, we need to point out that the lower mean transmittance and wider distribution can be found in cases of OAM states of ± 6, which is not only because the OAM modes with larger state indices are more sensitive to turbulent effects, but also due to the limited size of our SLM screens, which introduce the boundary effects.

 figure: Fig. 3

Fig. 3 (a) Probability distribution of the fluctuating channel transmittances for (a) OAM state of −2; (b) OAM state of 2;(c) OAM state of −6; (d) OAM state of 6.

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4. Experimental results and analysis

The excess noise is essential to be measured accurately before the SKR calculations. When the modulation variance VA is set to 0.3, the figure below shows the excess noise of the proposed system as a function of time in cases of de-multiplexing OAM states of 2 and 6. Notice that there is no difference between de-multiplexing OAM states of 2 (or 6) and −2 (or −6) in term of excess noise. The mean excess noises are measured to be 0.017 and 0.02 in cases of OAM states of 2 and 6, respectively. Each point is measured with a block size of 106 points. In principle, the phase noise introduced excess noise can be totally eliminated by the proposed PNC stage. The excess noise caused by the channel crosstalk can also be removed via wavelength interleaving. We believe the excess noises are mainly arising from the imperfect phase patterns loaded onto the SLM screen at Alice’s side, as a result, the generated OAM modes are not perfectly pure. In addition, the excess noise fluctuation is caused by the bias fluctuation applied to the I/Q modulator, even though the MBC is used.

After all noise sources have been measured, we calculate the SKRs with a reconciliation efficiency of β=0.9 in both ideal case and in the experimental setup. In the ideal situation, the excess noise will be completely compensated for and the maximum achievable channel transmittance will be T = 1. Figure 4 shows calculated SKRs for OAM states of 2 and 6 under ideal situation, where the OAM modes are located at horizontal polarization and wavelength channel is centered at 193.40 THz. These results can be considered as the upper bound of SKRs for the proposed CV-QKD system. The system performance will not vary with the polarizations, wavelength channels or the conjugated OAM modes (OAM states of −2 and −6). We can find from Figs. 4(a) and 4(b) that the SKRs of >50 Mbit/s and 90 Mbit/s are achievable in cases of OAM states of 6 and 2, respectively. When we take into account both polarizations, six wavelength channels, and all four OAM modes, a maximum SKR of >1.68 Gbit/s (140 × 2 × 2 × 3 Mbit/s) can be reached in a lossless channel. Given that only one coherent receiver is available in our lab, we detect the target wavelength, polarization, and OAM mode at a given time. Considering the 3D multiplexed channels can be treated as independent channels, the overall SKR can be obtained by adding the SKRs of individual channels.

 figure: Fig. 4

Fig. 4 Measured excess noise in cases of (a) OAM state of 2; (b) OAM state of 6.

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In our experimental setup, lossless channel is impossible, and the residual excess noise can’t be neglected. The average SKRs in our experiment are summarized below as a function of channel transmittance and modulation variance in cases of OAM states of 2 and 6. Compared to the SKRs shown in Fig. 5, the SKR performances achieved in Figs. 6 (a) and (b) have been degraded due to the presence of excess noise. To guarantee a secure CV-QKD transmission, the minimum channel transmittances of 0.21 and 0.29 are required for OAM states of 2 and 6, respectively. A maximum SKR of > 45 Mbit/s can be obtain in case of OAM state of 2, when the channel transmittance T is 0.9, and the modulation variance VA is 0.4; similarly, more than 16 Mbit/s SKR can be reached for OAM state of 6 in the scenario of T = 0.8, and VA = 0.38. Since the mean channel transmittances have been monitored to be 0.57 and 0.43 for OAM states of 2 and 6, respectively, it is meaningful to point out the corresponding maximum SKRs, which are about 6 Mbit/s and 4 Mbit/s. By summing the SKRs obtained from six wavelength channels, two polarization states, and four OAM, the total SKR of 120 Mbit/s can be achieved at the mean transmittances.

 figure: Fig. 5

Fig. 5 Calculated SKRs in the ideal situation as a function of the modulation variance and channel transmittance in cases of (a) OAM state of 6; (b) OAM state of 2.

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 figure: Fig. 6

Fig. 6 Experimental SKRs as a function of the modulation variance and channel transmittance in cases of (a) OAM state of 2; (b) OAM state of 6.

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5. Concluding remarks

We have studied a 3D multiplexed four-state CV-QKD system in the presence of atmospheric turbulence. The 3D multiplexing scheme, i.e., WDM, PDM, and OAM, enables a high-speed CV-QKD system. The phase noise introduced excess noise has been effectively mitigated by the proposed PNC stage. The turbulence channel has been emulated by two SLMs on which two randomly generated azimuthal phase patterns yielding Andrews’ spectrum have been recorded. The validity of such model has been verified by reproducing the on-axis intensity PDF and ICF. The resulting transmittance fluctuation can be monitored directly by the D.C. level in our PNC stage, which means that our CV-QKD system is free of the post-processing noise. After the system calibration, transmittance monitoring and the residual excess noise measurement, a total maximum SKR of >1.68 Gbit/s can be reached in the ideal system, characterized by the lossless channel and free of excess noise, which represents the upper bound of SKR in proposed CV-QKD system. In our experiment, the minimum channel transmittances of 0.21 and 0.29 are required for OAM states of 2 (or −2) and 6 (or −6), respectively, to guarantee the secure transmission, while a total SKR of 120 Mbit/s can be obtained in case of mean transmittances, when commercial photodetectors are used.

It is worthwhile to notice that any fiber for spatial division multiplexing (SDM), such as few-mode, few-core, or few-mode-few-core fiber, can be employed in the 3D multiplexed CV-QKD system. However, the signal propagated over the fiber for SDM will be affected by chromatic dispersion, mode coupling, and the mode dispersion effects, which will limit the transmission distance. Moreover, the polarization mode dispersion must be compensated in optical domain so that polarization multiplexing can be used as a degree of freedom. In principle, various mode coupling and dispersion effects have to be compensated in optical domain to guarantee a reliable CV-QKD system.

Funding

Office of Naval Research (ONR) MURI program (N00014-13-1-0627).

References and links

1. C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing (IEEE, 1984), pp. 175–179.

2. C. H. Bennett, “Quantum cryptography using any two nonorthogonal states,” Phys. Rev. Lett. 68(21), 3121–3124 (1992). [CrossRef]   [PubMed]  

3. Q. Dinh Xuan, Z. Zhang, and P. L. Voss, “A 24 km fiber-based discretely signaled continuous variable quantum key distribution system,” Opt. Express 17(26), 24244–24249 (2009). [CrossRef]   [PubMed]  

4. Z. Qu, I. B. Djordjevic, and M. A. Neifeld, “RF-subcarrier-assisted Four-state continuous-variable QKD Based on Coherent Detection,” Opt. Lett. 41(23), 5507–5510 (2016). [CrossRef]   [PubMed]  

5. B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16(11), 113018 (2014). [CrossRef]  

6. X. Sun, I. B. Djordjevic, and M. A. Neifeld, “Multiple spatial modes based QKD over marine free-space optical channels in the presence of atmospheric turbulence,” Opt. Express 24(24), 27663–27673 (2016). [CrossRef]   [PubMed]  

7. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005). [CrossRef]   [PubMed]  

8. A. A. Semenov, F. Toppel, D. Yu. Vasylyev, H. V. Gomonay, and W. Vogel, “Homodyne detection for atmosphere channels,” Phys. Rev. A 85(1), 013826 (2012). [CrossRef]  

9. A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282(5389), 706–709 (1998). [CrossRef]   [PubMed]  

10. F. Grosshans, “Collective attacks and unconditional security in continuous variable quantum key distribution,” Phys. Rev. Lett. 94(2), 020504 (2005). [CrossRef]   [PubMed]  

11. T. C .Ralph, “Continuous variable quantum cryptography,” Phys. Rev. A 61, 010303(R) (1999). [CrossRef]  

12. P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys. Rev. A 87(6), 062313 (2013). [CrossRef]  

13. D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6(1), 19201 (2016). [CrossRef]   [PubMed]  

14. Y. Zhao, M. Heid, J. Rigas, and N. Lutkenhaus, “Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks,” Phys. Rev. A 79(1), 012307 (2009). [CrossRef]  

15. H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A 86(2), 022338 (2012). [CrossRef]  

16. Z. Zhang and P. L. Voss, “Security of a discretely signaled continuous variable quantum key distribution protocol for high rate systems,” Opt. Express 17(14), 12090–12108 (2009). [CrossRef]   [PubMed]  

17. A. Leverrier and P. Grangier, “Unconditional security proof of long-distance continuous-variable quantum key distribution with discrete modulation,” Phys. Rev. Lett. 102(18), 180504 (2009). [CrossRef]   [PubMed]  

18. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).

19. J. Liu and J. Wang, “Polarization-insensitive PAM-4-carrying free-space orbital angular momentum (OAM) communications,” Opt. Express 24(4), 4258–4269 (2016). [CrossRef]   [PubMed]  

20. S. Fossier, E. Diamanti, T. Debuisschert, A. Villing, R. Tualle-Brouri, and P. Grangier, “Field test of a continuous-variable quantum key distribution prototype,” New J. Phys. 11(4), 045023 (2009). [CrossRef]  

21. Z. Qu and I. B. Djordjevic, “500 Gb/s free-space optical transmission over strong atmospheric turbulence channels,” Opt. Lett. 41(14), 3285–3288 (2016). [CrossRef]   [PubMed]  

References

  • View by:

  1. C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing (IEEE, 1984), pp. 175–179.
  2. C. H. Bennett, “Quantum cryptography using any two nonorthogonal states,” Phys. Rev. Lett. 68(21), 3121–3124 (1992).
    [Crossref] [PubMed]
  3. Q. Dinh Xuan, Z. Zhang, and P. L. Voss, “A 24 km fiber-based discretely signaled continuous variable quantum key distribution system,” Opt. Express 17(26), 24244–24249 (2009).
    [Crossref] [PubMed]
  4. Z. Qu, I. B. Djordjevic, and M. A. Neifeld, “RF-subcarrier-assisted Four-state continuous-variable QKD Based on Coherent Detection,” Opt. Lett. 41(23), 5507–5510 (2016).
    [Crossref] [PubMed]
  5. B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16(11), 113018 (2014).
    [Crossref]
  6. X. Sun, I. B. Djordjevic, and M. A. Neifeld, “Multiple spatial modes based QKD over marine free-space optical channels in the presence of atmospheric turbulence,” Opt. Express 24(24), 27663–27673 (2016).
    [Crossref] [PubMed]
  7. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
    [Crossref] [PubMed]
  8. A. A. Semenov, F. Toppel, D. Yu. Vasylyev, H. V. Gomonay, and W. Vogel, “Homodyne detection for atmosphere channels,” Phys. Rev. A 85(1), 013826 (2012).
    [Crossref]
  9. A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282(5389), 706–709 (1998).
    [Crossref] [PubMed]
  10. F. Grosshans, “Collective attacks and unconditional security in continuous variable quantum key distribution,” Phys. Rev. Lett. 94(2), 020504 (2005).
    [Crossref] [PubMed]
  11. T. C .Ralph, “Continuous variable quantum cryptography,” Phys. Rev. A 61, 010303(R) (1999).
    [Crossref]
  12. P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys. Rev. A 87(6), 062313 (2013).
    [Crossref]
  13. D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6(1), 19201 (2016).
    [Crossref] [PubMed]
  14. Y. Zhao, M. Heid, J. Rigas, and N. Lutkenhaus, “Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks,” Phys. Rev. A 79(1), 012307 (2009).
    [Crossref]
  15. H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A 86(2), 022338 (2012).
    [Crossref]
  16. Z. Zhang and P. L. Voss, “Security of a discretely signaled continuous variable quantum key distribution protocol for high rate systems,” Opt. Express 17(14), 12090–12108 (2009).
    [Crossref] [PubMed]
  17. A. Leverrier and P. Grangier, “Unconditional security proof of long-distance continuous-variable quantum key distribution with discrete modulation,” Phys. Rev. Lett. 102(18), 180504 (2009).
    [Crossref] [PubMed]
  18. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
  19. J. Liu and J. Wang, “Polarization-insensitive PAM-4-carrying free-space orbital angular momentum (OAM) communications,” Opt. Express 24(4), 4258–4269 (2016).
    [Crossref] [PubMed]
  20. S. Fossier, E. Diamanti, T. Debuisschert, A. Villing, R. Tualle-Brouri, and P. Grangier, “Field test of a continuous-variable quantum key distribution prototype,” New J. Phys. 11(4), 045023 (2009).
    [Crossref]
  21. Z. Qu and I. B. Djordjevic, “500 Gb/s free-space optical transmission over strong atmospheric turbulence channels,” Opt. Lett. 41(14), 3285–3288 (2016).
    [Crossref] [PubMed]

2016 (5)

2014 (1)

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16(11), 113018 (2014).
[Crossref]

2013 (1)

P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys. Rev. A 87(6), 062313 (2013).
[Crossref]

2012 (2)

H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A 86(2), 022338 (2012).
[Crossref]

A. A. Semenov, F. Toppel, D. Yu. Vasylyev, H. V. Gomonay, and W. Vogel, “Homodyne detection for atmosphere channels,” Phys. Rev. A 85(1), 013826 (2012).
[Crossref]

2009 (5)

Q. Dinh Xuan, Z. Zhang, and P. L. Voss, “A 24 km fiber-based discretely signaled continuous variable quantum key distribution system,” Opt. Express 17(26), 24244–24249 (2009).
[Crossref] [PubMed]

Z. Zhang and P. L. Voss, “Security of a discretely signaled continuous variable quantum key distribution protocol for high rate systems,” Opt. Express 17(14), 12090–12108 (2009).
[Crossref] [PubMed]

A. Leverrier and P. Grangier, “Unconditional security proof of long-distance continuous-variable quantum key distribution with discrete modulation,” Phys. Rev. Lett. 102(18), 180504 (2009).
[Crossref] [PubMed]

S. Fossier, E. Diamanti, T. Debuisschert, A. Villing, R. Tualle-Brouri, and P. Grangier, “Field test of a continuous-variable quantum key distribution prototype,” New J. Phys. 11(4), 045023 (2009).
[Crossref]

Y. Zhao, M. Heid, J. Rigas, and N. Lutkenhaus, “Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks,” Phys. Rev. A 79(1), 012307 (2009).
[Crossref]

2005 (2)

F. Grosshans, “Collective attacks and unconditional security in continuous variable quantum key distribution,” Phys. Rev. Lett. 94(2), 020504 (2005).
[Crossref] [PubMed]

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

1998 (1)

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282(5389), 706–709 (1998).
[Crossref] [PubMed]

1992 (1)

C. H. Bennett, “Quantum cryptography using any two nonorthogonal states,” Phys. Rev. Lett. 68(21), 3121–3124 (1992).
[Crossref] [PubMed]

Bennett, C. H.

C. H. Bennett, “Quantum cryptography using any two nonorthogonal states,” Phys. Rev. Lett. 68(21), 3121–3124 (1992).
[Crossref] [PubMed]

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing (IEEE, 1984), pp. 175–179.

Brassard, G.

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing (IEEE, 1984), pp. 175–179.

Braunstein, S. L.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282(5389), 706–709 (1998).
[Crossref] [PubMed]

Debuisschert, T.

S. Fossier, E. Diamanti, T. Debuisschert, A. Villing, R. Tualle-Brouri, and P. Grangier, “Field test of a continuous-variable quantum key distribution prototype,” New J. Phys. 11(4), 045023 (2009).
[Crossref]

Diamanti, E.

P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys. Rev. A 87(6), 062313 (2013).
[Crossref]

S. Fossier, E. Diamanti, T. Debuisschert, A. Villing, R. Tualle-Brouri, and P. Grangier, “Field test of a continuous-variable quantum key distribution prototype,” New J. Phys. 11(4), 045023 (2009).
[Crossref]

Dinh Xuan, Q.

Djordjevic, I. B.

Fang, J.

H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A 86(2), 022338 (2012).
[Crossref]

Fossier, S.

S. Fossier, E. Diamanti, T. Debuisschert, A. Villing, R. Tualle-Brouri, and P. Grangier, “Field test of a continuous-variable quantum key distribution prototype,” New J. Phys. 11(4), 045023 (2009).
[Crossref]

Fuchs, C. A.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282(5389), 706–709 (1998).
[Crossref] [PubMed]

Furusawa, A.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282(5389), 706–709 (1998).
[Crossref] [PubMed]

Gomonay, H. V.

A. A. Semenov, F. Toppel, D. Yu. Vasylyev, H. V. Gomonay, and W. Vogel, “Homodyne detection for atmosphere channels,” Phys. Rev. A 85(1), 013826 (2012).
[Crossref]

Grangier, P.

A. Leverrier and P. Grangier, “Unconditional security proof of long-distance continuous-variable quantum key distribution with discrete modulation,” Phys. Rev. Lett. 102(18), 180504 (2009).
[Crossref] [PubMed]

S. Fossier, E. Diamanti, T. Debuisschert, A. Villing, R. Tualle-Brouri, and P. Grangier, “Field test of a continuous-variable quantum key distribution prototype,” New J. Phys. 11(4), 045023 (2009).
[Crossref]

Grosshans, F.

F. Grosshans, “Collective attacks and unconditional security in continuous variable quantum key distribution,” Phys. Rev. Lett. 94(2), 020504 (2005).
[Crossref] [PubMed]

He, G.

H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A 86(2), 022338 (2012).
[Crossref]

Heid, M.

Y. Zhao, M. Heid, J. Rigas, and N. Lutkenhaus, “Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks,” Phys. Rev. A 79(1), 012307 (2009).
[Crossref]

Heim, B.

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16(11), 113018 (2014).
[Crossref]

Huang, D.

D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6(1), 19201 (2016).
[Crossref] [PubMed]

Huang, P.

D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6(1), 19201 (2016).
[Crossref] [PubMed]

Jouguet, P.

P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys. Rev. A 87(6), 062313 (2013).
[Crossref]

Khan, I.

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16(11), 113018 (2014).
[Crossref]

Killoran, N.

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16(11), 113018 (2014).
[Crossref]

Kimble, H. J.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282(5389), 706–709 (1998).
[Crossref] [PubMed]

Kunz-Jacques, S.

P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys. Rev. A 87(6), 062313 (2013).
[Crossref]

Leuchs, G.

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16(11), 113018 (2014).
[Crossref]

Leverrier, A.

A. Leverrier and P. Grangier, “Unconditional security proof of long-distance continuous-variable quantum key distribution with discrete modulation,” Phys. Rev. Lett. 102(18), 180504 (2009).
[Crossref] [PubMed]

Lin, D.

D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6(1), 19201 (2016).
[Crossref] [PubMed]

Liu, J.

Lutkenhaus, N.

Y. Zhao, M. Heid, J. Rigas, and N. Lutkenhaus, “Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks,” Phys. Rev. A 79(1), 012307 (2009).
[Crossref]

Marquardt, Ch.

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16(11), 113018 (2014).
[Crossref]

Neifeld, M. A.

Paterson, C.

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

Peuntinger, C.

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16(11), 113018 (2014).
[Crossref]

Polzik, E. S.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282(5389), 706–709 (1998).
[Crossref] [PubMed]

Qu, Z.

Rigas, J.

Y. Zhao, M. Heid, J. Rigas, and N. Lutkenhaus, “Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks,” Phys. Rev. A 79(1), 012307 (2009).
[Crossref]

Semenov, A. A.

A. A. Semenov, F. Toppel, D. Yu. Vasylyev, H. V. Gomonay, and W. Vogel, “Homodyne detection for atmosphere channels,” Phys. Rev. A 85(1), 013826 (2012).
[Crossref]

Sørensen, J. L.

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282(5389), 706–709 (1998).
[Crossref] [PubMed]

Sun, X.

Toppel, F.

A. A. Semenov, F. Toppel, D. Yu. Vasylyev, H. V. Gomonay, and W. Vogel, “Homodyne detection for atmosphere channels,” Phys. Rev. A 85(1), 013826 (2012).
[Crossref]

Tualle-Brouri, R.

S. Fossier, E. Diamanti, T. Debuisschert, A. Villing, R. Tualle-Brouri, and P. Grangier, “Field test of a continuous-variable quantum key distribution prototype,” New J. Phys. 11(4), 045023 (2009).
[Crossref]

Vasylyev, D. Yu.

A. A. Semenov, F. Toppel, D. Yu. Vasylyev, H. V. Gomonay, and W. Vogel, “Homodyne detection for atmosphere channels,” Phys. Rev. A 85(1), 013826 (2012).
[Crossref]

Villing, A.

S. Fossier, E. Diamanti, T. Debuisschert, A. Villing, R. Tualle-Brouri, and P. Grangier, “Field test of a continuous-variable quantum key distribution prototype,” New J. Phys. 11(4), 045023 (2009).
[Crossref]

Vogel, W.

A. A. Semenov, F. Toppel, D. Yu. Vasylyev, H. V. Gomonay, and W. Vogel, “Homodyne detection for atmosphere channels,” Phys. Rev. A 85(1), 013826 (2012).
[Crossref]

Voss, P. L.

Wang, J.

Wittmann, C.

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16(11), 113018 (2014).
[Crossref]

Zeng, G.

D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6(1), 19201 (2016).
[Crossref] [PubMed]

Zhang, H.

H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A 86(2), 022338 (2012).
[Crossref]

Zhang, Z.

Zhao, Y.

Y. Zhao, M. Heid, J. Rigas, and N. Lutkenhaus, “Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks,” Phys. Rev. A 79(1), 012307 (2009).
[Crossref]

New J. Phys. (2)

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16(11), 113018 (2014).
[Crossref]

S. Fossier, E. Diamanti, T. Debuisschert, A. Villing, R. Tualle-Brouri, and P. Grangier, “Field test of a continuous-variable quantum key distribution prototype,” New J. Phys. 11(4), 045023 (2009).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. A (4)

A. A. Semenov, F. Toppel, D. Yu. Vasylyev, H. V. Gomonay, and W. Vogel, “Homodyne detection for atmosphere channels,” Phys. Rev. A 85(1), 013826 (2012).
[Crossref]

Y. Zhao, M. Heid, J. Rigas, and N. Lutkenhaus, “Asymptotic security of binary modulated continuous-variable quantum key distribution under collective attacks,” Phys. Rev. A 79(1), 012307 (2009).
[Crossref]

H. Zhang, J. Fang, and G. He, “Improving the performance of the four-state continuous-variable quantum key distribution by using optical amplifiers,” Phys. Rev. A 86(2), 022338 (2012).
[Crossref]

P. Jouguet, S. Kunz-Jacques, and E. Diamanti, “Preventing calibration attacks on the local oscillator in continuous-variable quantum key distribution,” Phys. Rev. A 87(6), 062313 (2013).
[Crossref]

Phys. Rev. Lett. (4)

F. Grosshans, “Collective attacks and unconditional security in continuous variable quantum key distribution,” Phys. Rev. Lett. 94(2), 020504 (2005).
[Crossref] [PubMed]

A. Leverrier and P. Grangier, “Unconditional security proof of long-distance continuous-variable quantum key distribution with discrete modulation,” Phys. Rev. Lett. 102(18), 180504 (2009).
[Crossref] [PubMed]

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

C. H. Bennett, “Quantum cryptography using any two nonorthogonal states,” Phys. Rev. Lett. 68(21), 3121–3124 (1992).
[Crossref] [PubMed]

Sci. Rep. (1)

D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6(1), 19201 (2016).
[Crossref] [PubMed]

Science (1)

A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282(5389), 706–709 (1998).
[Crossref] [PubMed]

Other (3)

C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing (IEEE, 1984), pp. 175–179.

T. C .Ralph, “Continuous variable quantum cryptography,” Phys. Rev. A 61, 010303(R) (1999).
[Crossref]

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).

Cited By

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Figures (6)

Fig. 1
Fig. 1 The schematic of the proposed 3D multiplexed four-state CV-QKD system. Inset (a) the optical spectrum after I/Q modulation; Inset (b) the concept of 3D multiplexing. PBS: polarization beam splitter, BS: beam splitter, MR: mirror, HWP: half-wave plate, IL: optical interleaver, OTF: optical tunable filter, LO: local oscillator laser, PNC: phase noise cancellation stage.
Fig. 2
Fig. 2 (a) PDF of the obtained on-axis intensity fluctuation and the analytical Gamma-Gamma distribution. (b) ICFs based on the target continuous path and our turbulent model.
Fig. 3
Fig. 3 (a) Probability distribution of the fluctuating channel transmittances for (a) OAM state of −2; (b) OAM state of 2;(c) OAM state of −6; (d) OAM state of 6.
Fig. 4
Fig. 4 Measured excess noise in cases of (a) OAM state of 2; (b) OAM state of 6.
Fig. 5
Fig. 5 Calculated SKRs in the ideal situation as a function of the modulation variance and channel transmittance in cases of (a) OAM state of 6; (b) OAM state of 2.
Fig. 6
Fig. 6 Experimental SKRs as a function of the modulation variance and channel transmittance in cases of (a) OAM state of 2; (b) OAM state of 6.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

i C ( t )( 1+ A 2 ) T 2 2 AI(t)cos( w 1 t π 4 )+ 2 AQ(t)cos( w 1 t π 4 )+ n C
ΔI=β I AB χ BE
I AB = log 2 ( V+ χ tot 1+ χ tot ).
χ BE =G( λ 1 1 2 )+G( λ 2 1 2 )G( λ 3 1 2 )G( λ 4 1 2 )
with V= V A +1, χ tot = χ line + χ het /T, χ line =1/T+ϵ1, χ het =(2+2 V el η)/η, G( x )=( x+1 ) log 2 ( x+1 )x log 2 (x), λ 1,2 = 1 2 (A± A 2 4B ) , λ 3,4 = 1 2 (C± C 2 4D ), A= V 2 + T 2 (V+ χ line ) 2 2T Z 4 2 , B= (T V 2 +TV χ line T Z 4 2 ) 2 , C= A χ het 2 +B+1+2 χ het [ V B +T( V+ χ line ) ]+2T Z 4 2 [T( V+ χ tot )] 2 , D= (V+ χ het B ) 2 [T( V+ χ tot )] 2 ξ 0,2 =1/2 e V A /2 [cosh( V A /2±cos( V A /2))], ξ 1,3 =1/2 e V A /2 [sinh( V A /2±sin( V A /2))].

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