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<article article-type="research-article" dtd-version="1.3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xml:lang="en"><front><journal-meta><journal-id journal-id-type="publisher-id">najo</journal-id><journal-title-group><journal-title xml:lang="en">Nanosystems: Physics, Chemistry, Mathematics</journal-title><trans-title-group xml:lang="ru"><trans-title>Наносистемы: физика, химия, математика</trans-title></trans-title-group></journal-title-group><issn pub-type="ppub">2220-8054</issn><issn pub-type="epub">2305-7971</issn><publisher><publisher-name>Университет ИТМО</publisher-name></publisher></journal-meta><article-meta><article-id custom-type="elpub" pub-id-type="custom">najo-1490</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="en"><subject>PHYSICS</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="ru"><subject>ФИЗИКА</subject></subj-group></article-categories><title-group><article-title>Heterodyne Detection Method Of Multimode States for Subcarrier Wave Continuous Variable Quantum Key Distribution</article-title><trans-title-group xml:lang="ru"><trans-title></trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0003-4564-8284</contrib-id><name-alternatives><name name-style="western" xml:lang="en"><surname>Filipov</surname><given-names>Ilya Maksimovich</given-names></name></name-alternatives><email xlink:type="simple">imfilipov@itmo.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-3919-4151</contrib-id><name-alternatives><name name-style="western" xml:lang="en"><surname>Dashkov</surname><given-names>Michael Victorovich</given-names></name></name-alternatives><email xlink:type="simple">m.dashkov@psuti.ru</email><xref ref-type="aff" rid="aff-2"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="western" xml:lang="en"><surname>Bogdanova</surname><given-names>Ekaterina Igorevna</given-names></name></name-alternatives><email xlink:type="simple">ei.bogdanova@psuti.ru</email><xref ref-type="aff" rid="aff-2"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0003-0789-998X</contrib-id><name-alternatives><name name-style="western" xml:lang="en"><surname>Zinovev</surname><given-names>Alexandr Vyacheslavovich</given-names></name></name-alternatives><email xlink:type="simple">avzinovev@itmo.ru</email><xref ref-type="aff" rid="aff-3"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-2414-3490</contrib-id><name-alternatives><name name-style="western" xml:lang="en"><surname>Chistiakov</surname><given-names>Vladimir Viktorovich</given-names></name></name-alternatives><email xlink:type="simple">v_chistyakov@itmo.ru</email><xref ref-type="aff" rid="aff-3"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-3894-511X</contrib-id><name-alternatives><name name-style="western" xml:lang="en"><surname>Kiselev</surname><given-names>Fedor Dmitrievich</given-names></name></name-alternatives><email xlink:type="simple">fdkiselev@itmo.ru</email><xref ref-type="aff" rid="aff-3"/></contrib></contrib-group><aff-alternatives id="aff-1"><aff xml:lang="ru">ITMO University<country>Россия</country></aff><aff xml:lang="en">ITMO University, 199034, Saint Petersburg, Kadetskaya Line 3k2, Russia; SMARTS-Quanttelecom LLC, 199178, Saint Petersburg, Vasilievsky island 6 Line 59, Russia<country>Russian Federation</country></aff></aff-alternatives><aff-alternatives id="aff-2"><aff xml:lang="en">Povolzhskiy State University of Telecommunications and Informatics (PSUTI), 443010, Samara, Moskovskoe Shosse St., 77, Russia<country>Russian Federation</country></aff></aff-alternatives><aff-alternatives id="aff-3"><aff xml:lang="en">ITMO University, 199034, Saint Petersburg, Kadetskaya Line 3k2, Russia</aff></aff-alternatives><pub-date pub-type="collection"><year>2025</year></pub-date><pub-date pub-type="epub"><day>05</day><month>02</month><year>2026</year></pub-date><volume>16</volume><issue>6</issue><elocation-id>1490</elocation-id><permissions><copyright-statement>Copyright &amp;#x00A9; Filipov I.M., Dashkov M.V., Bogdanova E.I., Zinovev A.V., Chistiakov V.V., Kiselev F.D., 2026</copyright-statement><copyright-year>2026</copyright-year><copyright-holder xml:lang="ru">Filipov I.M., Dashkov M.V., Bogdanova E.I., Zinovev A.V., Chistiakov V.V., Kiselev F.D.</copyright-holder><copyright-holder xml:lang="en">Filipov I.M., Dashkov M.V., Bogdanova E.I., Zinovev A.V., Chistiakov V.V., Kiselev F.D.</copyright-holder><license license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>This work is licensed under a Creative Commons Attribution 4.0 License.</license-p></license></permissions><self-uri xlink:href="https://nanojournal.ifmo.ru/jour/article/view/1490">https://nanojournal.ifmo.ru/jour/article/view/1490</self-uri><abstract><p>A novel coherent detection method for subcarrier wave (SCW) quantum states applied to continuous-variable quantum key distribution (CV-QKD) is presented. The proposed approach relies on repeated phase modulation at the receiver and spatial separation of the carrier and subcarrier frequency components. The resulting output is an intermediate frequency determined by the difference between the sender’s and receiver’s modulation frequencies. An analytical model of the detection output is developed, and a comparative analysis with alternative heterodyne detection methods is provided. Experimental validation confirms the linear dependence of the output signal on the receiver’s modulation frequency and the sender’s modulation index in the small-modulation regime. Furthermore, the feasibility of the proposed method is demonstrated through the detection of discretely modulated signals using quadrature phase-shift keying (QPSK).</p></abstract><kwd-group xml:lang="en"><kwd>coherent detection</kwd><kwd>subcarrier wave</kwd><kwd>continuous variable</kwd><kwd>quantum key distribution</kwd></kwd-group><funding-group xml:lang="en"><funding-statement>Russian Science Foundation</funding-statement></funding-group></article-meta></front><back><ref-list><title>References</title><ref id="cit1"><label>1</label><citation-alternatives><mixed-citation xml:lang="ru">S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, et al., “Advances in</mixed-citation><mixed-citation xml:lang="en">S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, et al., “Advances in</mixed-citation></citation-alternatives></ref><ref id="cit2"><label>2</label><citation-alternatives><mixed-citation xml:lang="ru">quantum cryptography,” Advances in optics and photonics, vol. 12, no. 4, pp. 1012–1236, 2020.</mixed-citation><mixed-citation xml:lang="en">quantum cryptography,” Advances in optics and photonics, vol. 12, no. 4, pp. 1012–1236, 2020.</mixed-citation></citation-alternatives></ref><ref id="cit3"><label>3</label><citation-alternatives><mixed-citation xml:lang="ru">G. Zhang, J. Y. Haw, H. Cai, F. Xu, S. Assad, J. F. Fitzsimons, X. Zhou, Y. Zhang, S. Yu, J. Wu, et al., “An integrated silicon photonic chip platform</mixed-citation><mixed-citation xml:lang="en">G. Zhang, J. Y. Haw, H. Cai, F. Xu, S. Assad, J. F. Fitzsimons, X. Zhou, Y. Zhang, S. Yu, J. Wu, et al., “An integrated silicon photonic chip platform</mixed-citation></citation-alternatives></ref><ref id="cit4"><label>4</label><citation-alternatives><mixed-citation xml:lang="ru">for continuous-variable quantum key distribution,” Nature Photonics, vol. 13, no. 12, pp. 839–842, 2019.</mixed-citation><mixed-citation xml:lang="en">for continuous-variable quantum key distribution,” Nature Photonics, vol. 13, no. 12, pp. 839–842, 2019.</mixed-citation></citation-alternatives></ref><ref id="cit5"><label>5</label><citation-alternatives><mixed-citation xml:lang="ru">A. A. Hajomer, C. Bruynsteen, I. Derkach, N. Jain, A. Bomhals, S. Bastiaens, U. L. Andersen, X. Yin, and T. Gehring, “Continuous-variable</mixed-citation><mixed-citation xml:lang="en">A. A. Hajomer, C. Bruynsteen, I. Derkach, N. Jain, A. Bomhals, S. Bastiaens, U. L. Andersen, X. Yin, and T. Gehring, “Continuous-variable</mixed-citation></citation-alternatives></ref><ref id="cit6"><label>6</label><citation-alternatives><mixed-citation xml:lang="ru">quantum key distribution at 10 gbaud using an integrated photonic-electronic receiver,” Optica, vol. 11, no. 9, pp. 1197–1204, 2024.</mixed-citation><mixed-citation xml:lang="en">quantum key distribution at 10 gbaud using an integrated photonic-electronic receiver,” Optica, vol. 11, no. 9, pp. 1197–1204, 2024.</mixed-citation></citation-alternatives></ref><ref id="cit7"><label>7</label><citation-alternatives><mixed-citation xml:lang="ru">Y. Pi´etri, L. Trigo Vidarte, M. Schiavon, L. Vivien, P. Grangier, A. Rhouni, and E. Diamanti, “Experimental demonstration of continuous-variable</mixed-citation><mixed-citation xml:lang="en">Y. Pi´etri, L. Trigo Vidarte, M. Schiavon, L. Vivien, P. Grangier, A. Rhouni, and E. Diamanti, “Experimental demonstration of continuous-variable</mixed-citation></citation-alternatives></ref><ref id="cit8"><label>8</label><citation-alternatives><mixed-citation xml:lang="ru">quantum key distribution with a silicon photonics integrated receiver,” Optica Quantum, vol. 2, no. 6, pp. 428–437, 2024.</mixed-citation><mixed-citation xml:lang="en">quantum key distribution with a silicon photonics integrated receiver,” Optica Quantum, vol. 2, no. 6, pp. 428–437, 2024.</mixed-citation></citation-alternatives></ref><ref id="cit9"><label>9</label><citation-alternatives><mixed-citation xml:lang="ru">A. A. Hajomer, I. Derkach, V. C. Usenko, U. L. Andersen, and T. Gehring, “Coexistence of continuous-variable quantum key distribution and</mixed-citation><mixed-citation xml:lang="en">A. A. Hajomer, I. Derkach, V. C. Usenko, U. L. Andersen, and T. Gehring, “Coexistence of continuous-variable quantum key distribution and</mixed-citation></citation-alternatives></ref><ref id="cit10"><label>10</label><citation-alternatives><mixed-citation xml:lang="ru">classical data over 120-km fiber,” arXiv preprint arXiv:2502.17388, 2025.</mixed-citation><mixed-citation xml:lang="en">classical data over 120-km fiber,” arXiv preprint arXiv:2502.17388, 2025.</mixed-citation></citation-alternatives></ref><ref id="cit11"><label>11</label><citation-alternatives><mixed-citation xml:lang="ru">D. Milovanˇcev, N. Voki´c, F. Laudenbach, C. Pacher, H. H¨ubel, and B. Schrenk, “High rate cv-qkd secured mobile wdm fronthaul for dense 5g</mixed-citation><mixed-citation xml:lang="en">D. Milovanˇcev, N. Voki´c, F. Laudenbach, C. Pacher, H. H¨ubel, and B. Schrenk, “High rate cv-qkd secured mobile wdm fronthaul for dense 5g</mixed-citation></citation-alternatives></ref><ref id="cit12"><label>12</label><citation-alternatives><mixed-citation xml:lang="ru">radio networks,” Journal of Lightwave Technology, vol. 39, no. 11, pp. 3445–3457, 2021.</mixed-citation><mixed-citation xml:lang="en">radio networks,” Journal of Lightwave Technology, vol. 39, no. 11, pp. 3445–3457, 2021.</mixed-citation></citation-alternatives></ref><ref id="cit13"><label>13</label><citation-alternatives><mixed-citation xml:lang="ru">Y. Zhang, Y. Bian, Z. Li, S. Yu, and H. Guo, “Continuous-variable quantum key distribution system: Past, present, and future,” Applied Physics</mixed-citation><mixed-citation xml:lang="en">Y. Zhang, Y. Bian, Z. Li, S. Yu, and H. Guo, “Continuous-variable quantum key distribution system: Past, present, and future,” Applied Physics</mixed-citation></citation-alternatives></ref><ref id="cit14"><label>14</label><citation-alternatives><mixed-citation xml:lang="ru">Reviews, vol. 11, no. 1, 2024.</mixed-citation><mixed-citation xml:lang="en">Reviews, vol. 11, no. 1, 2024.</mixed-citation></citation-alternatives></ref><ref id="cit15"><label>15</label><citation-alternatives><mixed-citation xml:lang="ru">E. Samsonov, R. Goncharov, A. Gaidash, A. Kozubov, V. Egorov, and A. Gleim, “Subcarrier wave continuous variable quantum key distribution</mixed-citation><mixed-citation xml:lang="en">E. Samsonov, R. Goncharov, A. Gaidash, A. Kozubov, V. Egorov, and A. Gleim, “Subcarrier wave continuous variable quantum key distribution</mixed-citation></citation-alternatives></ref><ref id="cit16"><label>16</label><citation-alternatives><mixed-citation xml:lang="ru">with discrete modulation: mathematical model and finite-key analysis,” Scientific Reports, vol. 10, no. 1, p. 10034, 2020.</mixed-citation><mixed-citation xml:lang="en">with discrete modulation: mathematical model and finite-key analysis,” Scientific Reports, vol. 10, no. 1, p. 10034, 2020.</mixed-citation></citation-alternatives></ref><ref id="cit17"><label>17</label><citation-alternatives><mixed-citation xml:lang="ru">Z. Su, jintao Wang, D. Cai, X. Guo, D. Wang, and Z. Li, “Experimental demonstration of phase sensitive multimode continuous variable quantum</mixed-citation><mixed-citation xml:lang="en">Z. Su, jintao Wang, D. Cai, X. Guo, D. Wang, and Z. Li, “Experimental demonstration of phase sensitive multimode continuous variable quantum</mixed-citation></citation-alternatives></ref><ref id="cit18"><label>18</label><citation-alternatives><mixed-citation xml:lang="ru">key distribution with improved secure key rate,” Photonics Research, vol. 11, no. 11, 2023.</mixed-citation><mixed-citation xml:lang="en">key distribution with improved secure key rate,” Photonics Research, vol. 11, no. 11, 2023.</mixed-citation></citation-alternatives></ref><ref id="cit19"><label>19</label><citation-alternatives><mixed-citation xml:lang="ru">E. Samsonov, R. Goncharov, M. Fadeev, A. Zinoviev, D. Kirichenko, B. Nasedkin, A. Kiselev, and V. Egorov, “Coherent detection schemes for</mixed-citation><mixed-citation xml:lang="en">E. Samsonov, R. Goncharov, M. Fadeev, A. Zinoviev, D. Kirichenko, B. Nasedkin, A. Kiselev, and V. Egorov, “Coherent detection schemes for</mixed-citation></citation-alternatives></ref><ref id="cit20"><label>20</label><citation-alternatives><mixed-citation xml:lang="ru">subcarrier wave continuous variable quantum key distribution,” Journal of the Optical Society of America B: Optical Physics, vol. 38, no. 7,</mixed-citation><mixed-citation xml:lang="en">subcarrier wave continuous variable quantum key distribution,” Journal of the Optical Society of America B: Optical Physics, vol. 38, no. 7,</mixed-citation></citation-alternatives></ref><ref id="cit21"><label>21</label><citation-alternatives><mixed-citation xml:lang="ru">pp. 2215–2222, 2021.</mixed-citation><mixed-citation xml:lang="en">pp. 2215–2222, 2021.</mixed-citation></citation-alternatives></ref><ref id="cit22"><label>22</label><citation-alternatives><mixed-citation xml:lang="ru">R. Goncharov, E. Samsonov, and A. D. Kiselev, “Subcarrier wave quantum key distribution system with gaussian modulation,” Journal of Physics: Conference Series, vol. 2103, no. 1, 2021.</mixed-citation><mixed-citation xml:lang="en">R. Goncharov, E. Samsonov, and A. D. Kiselev, “Subcarrier wave quantum key distribution system with gaussian modulation,” Journal of Physics: Conference Series, vol. 2103, no. 1, 2021.</mixed-citation></citation-alternatives></ref><ref id="cit23"><label>23</label><citation-alternatives><mixed-citation xml:lang="ru">D. Pereira, M. Almeida, M. Fac˜ao, A. N. Pinto, and N. A. Silva, “Probabilistic shaped 128-apsk cv-qkd transmission system over optical fibres,” Optics Letters, vol. 47, no. 15, pp. 3948–3951, 2022.</mixed-citation><mixed-citation xml:lang="en">D. Pereira, M. Almeida, M. Fac˜ao, A. N. Pinto, and N. A. Silva, “Probabilistic shaped 128-apsk cv-qkd transmission system over optical fibres,” Optics Letters, vol. 47, no. 15, pp. 3948–3951, 2022.</mixed-citation></citation-alternatives></ref><ref id="cit24"><label>24</label><citation-alternatives><mixed-citation xml:lang="ru">F. Roumestan, A. Ghazisaeidi, J. Renaudier, L. T. Vidarte, A. Leverrier, E. Diamanti, and P. Grangier, “Shaped constellation continuous variable</mixed-citation><mixed-citation xml:lang="en">F. Roumestan, A. Ghazisaeidi, J. Renaudier, L. T. Vidarte, A. Leverrier, E. Diamanti, and P. Grangier, “Shaped constellation continuous variable</mixed-citation></citation-alternatives></ref><ref id="cit25"><label>25</label><citation-alternatives><mixed-citation xml:lang="ru">quantum key distribution: Concepts, methods and experimental validation,” Journal of Lightwave Technology, 2024.</mixed-citation><mixed-citation xml:lang="en">quantum key distribution: Concepts, methods and experimental validation,” Journal of Lightwave Technology, 2024.</mixed-citation></citation-alternatives></ref><ref id="cit26"><label>26</label><citation-alternatives><mixed-citation xml:lang="ru">V. L. da Silva, M. A. Dias, N. A. F. Neto, and A. B. Tacla, “From coherent communications to quantum security: Modern techniques in cv-qkd,”</mixed-citation><mixed-citation xml:lang="en">V. L. da Silva, M. A. Dias, N. A. F. Neto, and A. B. Tacla, “From coherent communications to quantum security: Modern techniques in cv-qkd,”</mixed-citation></citation-alternatives></ref><ref id="cit27"><label>27</label><citation-alternatives><mixed-citation xml:lang="ru">in 2024 SBFoton International Optics and Photonics Conference (SBFoton IOPC), pp. 1–5, IEEE, 2024.</mixed-citation><mixed-citation xml:lang="en">in 2024 SBFoton International Optics and Photonics Conference (SBFoton IOPC), pp. 1–5, IEEE, 2024.</mixed-citation></citation-alternatives></ref><ref id="cit28"><label>28</label><citation-alternatives><mixed-citation xml:lang="ru">B. Qi, L.-L. Huang, L. Qian, and H.-K. Lo, “Experimental study on the gaussian-modulated coherent-state quantum key distribution over standard telecommunication fibers,” Physical Review A—Atomic, Molecular, and Optical Physics, vol. 76, no. 5, p. 052323, 2007.</mixed-citation><mixed-citation xml:lang="en">B. Qi, L.-L. Huang, L. Qian, and H.-K. Lo, “Experimental study on the gaussian-modulated coherent-state quantum key distribution over standard telecommunication fibers,” Physical Review A—Atomic, Molecular, and Optical Physics, vol. 76, no. 5, p. 052323, 2007.</mixed-citation></citation-alternatives></ref><ref id="cit29"><label>29</label><citation-alternatives><mixed-citation xml:lang="ru">B. Nasedkin, R. Goncharov, P. Morozova, I. Filipov, V. Chistiakov, E. Samsonov, and V. Egorov, “Quantum hacking on the technical implementa-</mixed-citation><mixed-citation xml:lang="en">B. Nasedkin, R. Goncharov, P. Morozova, I. Filipov, V. Chistiakov, E. Samsonov, and V. Egorov, “Quantum hacking on the technical implementa-</mixed-citation></citation-alternatives></ref><ref id="cit30"><label>30</label><citation-alternatives><mixed-citation xml:lang="ru">tion of continuous-variable quantum key distribution systems,” Radiophysics and Quantum Electronics, pp. 1–15. , 2025</mixed-citation><mixed-citation xml:lang="en">tion of continuous-variable quantum key distribution systems,” Radiophysics and Quantum Electronics, pp. 1–15. , 2025</mixed-citation></citation-alternatives></ref></ref-list><fn-group><fn fn-type="conflict"><p>The authors declare that there are no conflicts of interest present.</p></fn></fn-group></back></article>
