Abstract

The orbital angular momentum (OAM) of photons is a promising degree of freedom for high-dimensional quantum key distribution (QKD). However, effectively mitigating the adverse effects of atmospheric turbulence is a persistent challenge in OAM QKD systems operating over free-space communication channels. In contrast to previous works focusing on correcting static simulated turbulence, we investigate the performance of OAM QKD in real atmospheric turbulence with real-time adaptive optics (AO) correction. We show that even though our AO system provides a limited correction, it is possible to mitigate the errors induced by weak turbulence and establish a secure channel. The crosstalk induced by turbulence and the performance of AO systems is investigated in two configurations: a lab-scale link with controllable turbulence, and a 340 m long cross-campus link with dynamic atmospheric turbulence. Our experimental results suggest that an advanced AO system with fine beam tracking, reliable beam stabilization, precise wavefront sensing, and accurate wavefront correction is necessary to adequately correct turbulence-induced error. We also propose and demonstrate different solutions to improve the performance of OAM QKD with turbulence, which could enable the possibility of OAM encoding in strong turbulence.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2020 (1)

C. M. Mabena and F. S. Roux, “Quantum channel correction with twisted light using compressive sensing,” Phys. Rev. A 101(1), 013807 (2020).
[Crossref]

2019 (10)

G. Sorelli, N. Leonhard, V. N. Shatokhin, C. Reinlein, and A. Buchleitner, “Entanglement protection of high-dimensional states by adaptive optics,” New J. Phys. 21(2), 023003 (2019).
[Crossref]

C. Liu, K. Pang, Z. Zhao, P. Liao, R. Zhang, H. Song, Y. Cao, J. Du, L. Li, H. Song, Y. Ren, G. Xie, Y. Zhao, J. Zhao, S. M. H. Rafsanjani, A. N. Willner, J. H. Shapiro, R. W. Boyd, M. Tur, and A. E. Willner, “Single-end adaptive optics compensation for emulated turbulence in a bi-directional 10-Mbit/s per channel free-space quantum communication link using orbital-angular-momentum encoding,” Research 2019, 1–10 (2019).
[Crossref]

S. Zhao, W. Li, Y. Shen, Y. Yu, X. Han, H. Zeng, M. Cai, T. Qian, S. Wang, Z. Wang, Y. Xiao, and Y. Gu, “Experimental investigation of quantum key distribution over a water channel,” Appl. Opt. 58(14), 3902–3907 (2019).
[Crossref]

F. Hufnagel, A. Sit, F. Grenapin, F. Bouchard, K. Heshami, D. England, Y. Zhang, B. J. Sussman, R. W. Boyd, G. Leuchs, and E. Karimi, “Characterization of an underwater channel for quantum communications in the Ottawa river,” Opt. Express 27(19), 26346–26354 (2019).
[Crossref]

L. Li, R. Zhang, P. Liao, Y. Cao, H. Song, Y. Zhao, J. Du, Z. Zhao, C. Liu, K. Pang, H. Song, A. Almaiman, D. Starodubov, B. Lynn, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, “Mitigation for turbulence effects in a 40-gbit/s orbital-angular-momentum-multiplexed free-space optical link between a ground station and a retro-reflecting uav using mimo equalization,” Opt. Lett. 44(21), 5181–5184 (2019).
[Crossref]

J. Zhao, M. Mirhosseini, B. Braverman, Y. Zhou, S. M. H. Rafsanjani, Y. Ren, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and R. W. Boyd, “Performance analysis of d-dimensional quantum cryptography under state-dependent diffraction,” Phys. Rev. A 100(3), 032319 (2019).
[Crossref]

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10(1), 1865 (2019).
[Crossref]

Y. Zhou, M. Mirhosseini, S. Oliver, J. Zhao, S. M. H. Rafsanjani, M. P. Lavery, A. E. Willner, and R. W. Boyd, “Using all transverse degrees of freedom in quantum communications based on a generic mode sorter,” Opt. Express 27(7), 10383–10394 (2019).
[Crossref]

J. Jin, J.-P. Bourgoin, R. Tannous, S. Agne, C. J. Pugh, K. B. Kuntz, B. L. Higgins, and T. Jennewein, “Genuine time-bin-encoded quantum key distribution over a turbulent depolarizing free-space channel,” Opt. Express 27(26), 37214–37223 (2019).
[Crossref]

D. Cozzolino, D. Bacco, B. Da Lio, K. Ingerslev, Y. Ding, K. Dalgaard, P. Kristensen, M. Galili, K. Rottwitt, S. Ramachandran, and L. K. Oxenløwe, “Orbital angular momentum states enabling fiber-based high-dimensional quantum communication,” Phys. Rev. Appl. 11(6), 064058 (2019).
[Crossref]

2018 (7)

D. E. B. Flaes, J. Stopka, S. Turtaev, J. F. de Boer, T. Tyc, and T. Čižmár, “Robustness of light-transport processes to bending deformations in graded-index multimode waveguides,” Phys. Rev. Lett. 120(23), 233901 (2018).
[Crossref]

F. Bouchard, K. Heshami, D. England, R. Fickler, R. W. Boyd, B.-G. Englert, L. L. Sánchez-Soto, and E. Karimi, “Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons,” Quantum 2, 111111 (2018).
[Crossref]

F. Bouchard, A. Sit, K. Heshami, R. Fickler, and E. Karimi, “Round-robin differential-phase-shift quantum key distribution with twisted photons,” Phys. Rev. A 98(1), 010301 (2018).
[Crossref]

F. Bouchard, A. Sit, F. Hufnagel, A. Abbas, Y. Zhang, K. Heshami, R. Fickler, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “Quantum cryptography with twisted photons through an outdoor underwater channel,” Opt. Express 26(17), 22563–22573 (2018).
[Crossref]

D. Fu, Y. Zhou, R. Qi, S. Oliver, Y. Wang, S. M. H. Rafsanjani, J. Zhao, M. Mirhosseini, Z. Shi, P. Zhang, and R. W. Boyd, “Realization of a scalable Laguerre–Gaussian mode sorter based on a robust radial mode sorter,” Opt. Express 26(25), 33057–33065 (2018).
[Crossref]

M. P. Lavery, “Vortex instability in turbulent free-space propagation,” New J. Phys. 20(4), 043023 (2018).
[Crossref]

N. Leonhard, G. Sorelli, V. N. Shatokhin, C. Reinlein, and A. Buchleitner, “Protecting the entanglement of twisted photons by adaptive optics,” Phys. Rev. A 97(1), 012321 (2018).
[Crossref]

2017 (5)

Y. Ren, C. Liu, K. Pang, J. Zhao, Y. Cao, G. Xie, L. Li, P. Liao, Z. Zhao, M. Tur, R. W. Boyd, and A. E. Willner, “Spatially multiplexed orbital-angular-momentum-encoded single photon and classical channels in a free-space optical communication link,” Opt. Lett. 42(23), 4881–4884 (2017).
[Crossref]

B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. Lavery, and A. Forbes, “A deterministic detector for vector vortex states,” Sci. Rep. 7(1), 13882–8 (2017).
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L. Ji, J. Gao, A.-L. Yang, Z. Feng, X.-F. Lin, Z.-G. Li, and X.-M. Jin, “Towards quantum communications in free-space seawater,” Opt. Express 25(17), 19795–19806 (2017).
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A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “High-dimensional intracity quantum cryptography with structured photons,” Optica 4(9), 1006–1010 (2017).
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2016 (8)

K. Brádler, M. Mirhosseini, R. Fickler, A. Broadbent, and R. W. Boyd, “Finite-key security analysis for multilevel quantum key distribution,” New J. Phys. 18(7), 073030 (2016).
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Y. Zhang, S. Prabhakar, A. H. Ibrahim, F. S. Roux, A. Forbes, and T. Konrad, “Experimentally observed decay of high-dimensional entanglement through turbulence,” Phys. Rev. A 94(3), 032310 (2016).
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Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
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H.-L. Yin, T.-Y. Chen, Z.-W. Yu, H. Liu, L.-X. You, Y.-H. Zhou, S.-J. Chen, Y. Mao, M.-Q. Huang, W.-J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X.-B. Wang, and J.-W. Pan, “Measurement-device-independent quantum key distribution over a 404 km optical fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
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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).
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R. Ionicioiu, “Sorting quantum systems efficiently,” Sci. Rep. 6(1), 25356 (2016).
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W. Liu, K. Yao, D. Huang, X. Lin, L. Wang, and Y. Lv, “Performance evaluation of coherent free space optical communications with a double-stage fast-steering-mirror adaptive optics system depending on the greenwood frequency,” Opt. Express 24(12), 13288–13302 (2016).
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A. Acín, D. Cavalcanti, E. Passaro, S. Pironio, and P. Skrzypczyk, “Necessary detection efficiencies for secure quantum key distribution and bound randomness,” Phys. Rev. A 93(1), 012319 (2016).
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2015 (2)

B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, “Provably secure and practical quantum key distribution over 307 km of optical fibre,” Nat. Photonics 9(3), 163–168 (2015).
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M. Mirhosseini, O. S. Magana-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (2015).
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2014 (4)

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
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Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, M. P. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre-and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
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B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magana-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the lab with application to orbital angular momentum communication,” New J. Phys. 16(3), 033020 (2014).
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H.-K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2014).
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2013 (3)

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, “Direct and full-scale experimental verifications towards ground–satellite quantum key distribution,” Nat. Photonics 7(5), 387–393 (2013).
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Y. Ren, H. Huang, G. Xie, N. Ahmed, Y. Yan, B. I. Erkmen, N. Chandrasekaran, M. P. Lavery, N. K. Steinhoff, M. Tur, S. Dolinar, M. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Atmospheric turbulence effects on the performance of a free space optical link employing orbital angular momentum multiplexing,” Opt. Lett. 38(20), 4062–4065 (2013).
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M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4(1), 2781 (2013).
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2012 (3)

G. A. Tyler, “Adaptive optics compensation for propagation through deep turbulence: a study of some interesting approaches,” Opt. Eng. 52(2), 021011 (2012).
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F. Steinlechner, P. Trojek, M. Jofre, H. Weier, D. Perez, T. Jennewein, R. Ursin, J. Rarity, M. W. Mitchell, J. P. Torres, H. Weinfurter, and V. Pruneri, “A high-brightness source of polarization-entangled photons optimized for applications in free space,” Opt. Express 20(9), 9640–9649 (2012).
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J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
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2011 (1)

2010 (2)

L. Sheridan and V. Scarani, “Security proof for quantum key distribution using qudit systems,” Phys. Rev. A 82(3), 030301 (2010).
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A. Dousse, J. Suffczyński, A. Beveratos, O. Krebs, A. Lemaître, I. Sagnes, J. Bloch, P. Voisin, and P. Senellart, “Ultrabright source of entangled photon pairs,” Nature 466(7303), 217–220 (2010).
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2009 (2)

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
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G. A. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34(2), 142–144 (2009).
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2007 (1)

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Amer, M. Furst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
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2002 (2)

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88(12), 127902 (2002).
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N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
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1994 (1)

1993 (1)

J. M. Beckers, “Adaptive optics for astronomy: principles, performance, and applications,” Annu. Rev. Astron. Astrophys. 31(1), 13–62 (1993).
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1966 (1)

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R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Amer, M. Furst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3(7), 481–486 (2007).
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C. Liu, K. Pang, Z. Zhao, P. Liao, R. Zhang, H. Song, Y. Cao, J. Du, L. Li, H. Song, Y. Ren, G. Xie, Y. Zhao, J. Zhao, S. M. H. Rafsanjani, A. N. Willner, J. H. Shapiro, R. W. Boyd, M. Tur, and A. E. Willner, “Single-end adaptive optics compensation for emulated turbulence in a bi-directional 10-Mbit/s per channel free-space quantum communication link using orbital-angular-momentum encoding,” Research 2019, 1–10 (2019).
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J. Zhao, M. Mirhosseini, B. Braverman, Y. Zhou, S. M. H. Rafsanjani, Y. Ren, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and R. W. Boyd, “Performance analysis of d-dimensional quantum cryptography under state-dependent diffraction,” Phys. Rev. A 100(3), 032319 (2019).
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Y. Zhou, M. Mirhosseini, S. Oliver, J. Zhao, S. M. H. Rafsanjani, M. P. Lavery, A. E. Willner, and R. W. Boyd, “Using all transverse degrees of freedom in quantum communications based on a generic mode sorter,” Opt. Express 27(7), 10383–10394 (2019).
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D. Fu, Y. Zhou, R. Qi, S. Oliver, Y. Wang, S. M. H. Rafsanjani, J. Zhao, M. Mirhosseini, Z. Shi, P. Zhang, and R. W. Boyd, “Realization of a scalable Laguerre–Gaussian mode sorter based on a robust radial mode sorter,” Opt. Express 26(25), 33057–33065 (2018).
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F. Bouchard, A. Sit, F. Hufnagel, A. Abbas, Y. Zhang, K. Heshami, R. Fickler, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “Quantum cryptography with twisted photons through an outdoor underwater channel,” Opt. Express 26(17), 22563–22573 (2018).
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A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “High-dimensional intracity quantum cryptography with structured photons,” Optica 4(9), 1006–1010 (2017).
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Y. Ren, C. Liu, K. Pang, J. Zhao, Y. Cao, G. Xie, L. Li, P. Liao, Z. Zhao, M. Tur, R. W. Boyd, and A. E. Willner, “Spatially multiplexed orbital-angular-momentum-encoded single photon and classical channels in a free-space optical communication link,” Opt. Lett. 42(23), 4881–4884 (2017).
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K. Brádler, M. Mirhosseini, R. Fickler, A. Broadbent, and R. W. Boyd, “Finite-key security analysis for multilevel quantum key distribution,” New J. Phys. 18(7), 073030 (2016).
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J. Zhao, M. Mirhosseini, B. Braverman, Y. Zhou, S. M. H. Rafsanjani, Y. Ren, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and R. W. Boyd, “Performance analysis of d-dimensional quantum cryptography under state-dependent diffraction,” Phys. Rev. A 100(3), 032319 (2019).
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V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
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Figures (5)
Fig. 1.
Fig. 1. The configuration of the lab-scale link with a controllable turbulence cell. Both signal and beacon beams go through the center of the RH. The size of the signal beam is selected to cover the central part of the DM (3$\times$3 actuators) to avoid the cutoff from the edges. The size of the beacon beam overfills the DM aperture to provide a precise estimation of turbulence across the DM. The polarization of the signal beam is controlled by a polarizer to encode information while the polarization of the beacon beam is fixed in $\vert {H}\rangle$ state.

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Fig. 2.
Fig. 2. Measured fidelity of lab-scale OAM QKD as functions of turbulence strength and time. (a) measured fidelity of the OAM basis. (b) measured fidelity of the ANG basis. (c) measured fidelity of the MUBs which is the average of (a) and (b). The data point with $D/r_0 = 0.01$ corresponds to the no turbulence case. All the yellow and red curves are least-square fitting results of the measured data using the same model with different coefficient $c$. All the error bars are the measured standard deviation of the fidelity distribution over the measurement time. Each error bar is the average standard deviation of all states in the corresponding basis. One example of the fidelity histogram distributions is shown in (d). The standard deviation without AO correction is 11.24$\%$. When AO is turned on, this is reduced to 8.68$\%$. As a comparison, the measurement uncertainty without turbulence is 1.46$\%$, which is mainly induced by the laser power fluctuation. (d) histogram of the fidelity of the OAM state $\ell$ = 1 with and without AO correction. $D/r_0$ is 0.884.

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Fig. 3.
Fig. 3. The configuration of a 340 m long cross-campus link. Both Alice and Bob are on the optical table. Since the turbulence in the cross-campus link is not controllable, the turbulence structure number varies between 5.4$\times 10^{-15}$ $\textrm {m}^{-2/3}$ and 3.2$\times 10^{-14}$ $\textrm {m}^{-2/3}$.

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Fig. 4.
Fig. 4. (a) The prepared and received states in the cross-campus link with different OAM values. All the images acquired under same turbulence but at different timepoints during the night showing the distortions on OAM states. Note that the images of the prepared states have been scaled up by a factor of 3. To clearly show the details of the received states, the images of the received states only show the field in the collection aperture, and the intensities of the received $\vert {\ell = 2, 3}\rangle$ states are enhanced by a factor of 1.5. (b) crosstalk matrix of the OAM basis after AO correction. The average fidelity is found to be 22.57$\%$. (c) The theoretical and measured mode transmission efficiency. The error bars correspond to the standard deviation in transmission efficiencies. The effect of mode-dependent diffraction has been taken into consideration. All the data shown above are measured in turbulence with $D/r_0 = 2.18$.

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Fig. 5.
Fig. 5. (a) and (b) measured crosstalk matrix of the OAM basis in the lab-scale link without and with AO correction respectively. The average fidelity in (a) is 35.53$\%$ and the fidelity in (b) is 43.47$\%$. The mode spacing is 1 in both cases. (c) and (d) measured crosstalk matrix of OAM basis in the lab-scale link without and with AO correction respectively. The average fidelity in (c) is 45.72$\%$ and the fidelity in (d) is 57.54$\%$. The mode spacing is 2 in both cases. The turbulence level in all figures is $D/r_0 = 1.90$.

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Equations (1)

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| j ⟩ = 1 d ∑ ℓ = − L L | ℓ ⟩ exp ( − i 2 π j ℓ / ( 2 L + 1 ) ) ,

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