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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 pub-id-type="doi">10.17586/2220-8054-2022-13-4-392-403</article-id><article-id custom-type="elpub" pub-id-type="custom">najo-250</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="ru"><subject>Статьи</subject></subj-group></article-categories><title-group><article-title>Comparison of Gaussian and vortex beams in free-space QKD with phase encoding in turbulent atmosphere</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"><name-alternatives><name name-style="western" xml:lang="en"><surname>Adam</surname><given-names>I. A.</given-names></name></name-alternatives><email xlink:type="simple">adam_yura@mail.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="western" xml:lang="en"><surname>Yashin</surname><given-names>D. A.</given-names></name></name-alternatives><email xlink:type="simple">dayashin@itmo.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="western" xml:lang="en"><surname>Kargina</surname><given-names>D. A.</given-names></name></name-alternatives><email xlink:type="simple">dakargina17@gmail.com</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="western" xml:lang="en"><surname>Nasedkin</surname><given-names>B. A.</given-names></name></name-alternatives><email xlink:type="simple">b.nasedkin@bk.ru</email><xref ref-type="aff" rid="aff-1"/></contrib></contrib-group><aff xml:lang="en" id="aff-1"><institution>ITMO University</institution><country>Russian Federation</country></aff><pub-date pub-type="collection"><year>2022</year></pub-date><pub-date pub-type="epub"><day>06</day><month>06</month><year>2025</year></pub-date><volume>13</volume><issue>4</issue><fpage>392</fpage><lpage>403</lpage><permissions><copyright-statement>Copyright &amp;#x00A9; Adam I.A., Yashin D.A., Kargina D.A., Nasedkin B.A., 2025</copyright-statement><copyright-year>2025</copyright-year><copyright-holder xml:lang="ru">Adam I.A., Yashin D.A., Kargina D.A., Nasedkin B.A.</copyright-holder><copyright-holder xml:lang="en">Adam I.A., Yashin D.A., Kargina D.A., Nasedkin B.A.</copyright-holder><license xml:lang="ru" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>Данная работа распространяется под лицензией Creative Commons Attribution 4.0.</license-p></license><license xml:lang="en" 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/250">https://nanojournal.ifmo.ru/jour/article/view/250</self-uri><abstract><p>At present, free-space QKD systems are being actively researched and developed. The main limitation of these systems remains the strong influence of atmospheric turbulence and weather conditions on the propagating Gaussian beam. In turn, a number of works have shown that vortex beams are more stable in a turbulent atmosphere. Thus, in this work, the use of vortex beams in the free-space QKD system with phase encoding under the condition of a turbulent atmosphere and their comparison with Gaussian beams are studied. The possibility of phase modulation preservation with additional modulation and demodulation of the vortex beam is also investigated.</p></abstract><kwd-group xml:lang="en"><kwd>spectral gap</kwd><kwd>quantum graph</kwd><kwd>Schrodinger operator</kwd><kwd>discrete spectrum</kwd></kwd-group></article-meta></front><back><ref-list><title>References</title><ref id="cit1"><label>1</label><citation-alternatives><mixed-citation xml:lang="ru">Wootters W.K., Zurek W.H., The no-cloning theorem, Physics Today, 2009, 62(2), P. 76-77.</mixed-citation><mixed-citation xml:lang="en">Wootters W.K., Zurek W.H., The no-cloning theorem, Physics Today, 2009, 62(2), P. 76-77.</mixed-citation></citation-alternatives></ref><ref id="cit2"><label>2</label><citation-alternatives><mixed-citation xml:lang="ru">Bennett C.H., Brassard G., Quantum cryptography: public key distribution and coin tossing. 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