<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Publishing DTD v1.3 20210610//EN" "JATS-journalpublishing1-3.dtd">
<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">10.17586/2220-8054-2025-16-3-333-342</article-id><article-id custom-type="elpub" pub-id-type="custom">najo-322</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>NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="ru"><subject>НАНОСИСТЕМЫ: ФИЗИКА, ХИМИЯ, МАТЕМАТИКА</subject></subj-group></article-categories><title-group><article-title>Sub-Poissonian light in fluctuating thermal-loss bosonic channels</article-title><trans-title-group xml:lang="ru"><trans-title>Субпуассоновский свет в бозонных каналах с флуктуирующими тепловыми потерями</trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Степанов</surname><given-names>И.</given-names></name><name name-style="western" xml:lang="en"><surname>Stepanov</surname><given-names>I.</given-names></name></name-alternatives><bio xml:lang="en"><p>Ilia Stepanov</p><p>Kadetskaya Line, 3, St. Petersburg, 199034</p></bio><email xlink:type="simple">i.stepanov@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/0000-0002-9081-8900</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Гончаров</surname><given-names>Р.</given-names></name><name name-style="western" xml:lang="en"><surname>Goncharov</surname><given-names>R.</given-names></name></name-alternatives><bio xml:lang="en"><p>Roman Goncharov</p><p>Kadetskaya Line, 3, St. Petersburg, 199034</p><p>Birzhevaya Line, 16, Saint Petersburg, 199034</p><p>6th Vasilyevskogo Ostrova Line, 59, Saint Petersburg, 199178</p></bio><email xlink:type="simple">rkgoncharov@itmo.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/0000-0002-1023-3284</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Киселев</surname><given-names>А. Д.</given-names></name><name name-style="western" xml:lang="en"><surname>Kiselev</surname><given-names>A. D.</given-names></name></name-alternatives><bio xml:lang="en"><p>Alexei D. Kiselev</p><p>Birzhevaya Line, 16, Saint Petersburg, 199034</p><p>Kadetskaya Line 3b, Saint Petersburg 199034</p></bio><email xlink:type="simple">adkiselev@itmo.ru</email><xref ref-type="aff" rid="aff-3"/></contrib></contrib-group><aff xml:lang="en" id="aff-1"><institution>Quantum Information Laboratory, ITMO University</institution><country>Russian Federation</country></aff><aff xml:lang="en" id="aff-2"><institution>Quantum Information Laboratory, ITMO University; Laboratory for Quantum Communications, ITMO University; SMARTS-Quanttelecom LLC</institution><country>Russian Federation</country></aff><aff xml:lang="en" id="aff-3"><institution>Laboratory for Quantum Communications, ITMO University; Laboratory of Quantum Processes and Measurements, ITMO University</institution><country>Russian Federation</country></aff><pub-date pub-type="collection"><year>2025</year></pub-date><pub-date pub-type="epub"><day>29</day><month>06</month><year>2025</year></pub-date><volume>16</volume><issue>3</issue><fpage>333</fpage><lpage>342</lpage><permissions><copyright-statement>Copyright &amp;#x00A9; Stepanov I., Goncharov R., Kiselev A.D., 2025</copyright-statement><copyright-year>2025</copyright-year><copyright-holder xml:lang="ru">Степанов И., Гончаров Р., Киселев А.Д.</copyright-holder><copyright-holder xml:lang="en">Stepanov I., Goncharov R., Kiselev A.D.</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/322">https://nanojournal.ifmo.ru/jour/article/view/322</self-uri><abstract><p>We study the photon statistics of a single-mode sub-Poissonian light propagating in the lossy thermal bosonic channel with fluctuating transmittance which can be regarded as a temperature-dependent model of the turbulent atmosphere. By assuming that the variance of the transmittance can be expressed in terms of the fluctuation strength parameter we show that the photon statistics of the light remains sub-Poissonian provided the averaged transmittance exceeds its critical value. The critical transmittance is analytically computed as a function of the input states’ parameters, temperature, and the fluctuation strength. The results are applied to study special cases of the one-mode squeezed states and the odd optical Shr¨odinger cats.</p></abstract><trans-abstract xml:lang="ru"><p>Мы изучаем статистику фотонов одномодового субпуассоновского света, распространяющегося в тепловом бозонном канале с потерями и флуктуирующим пропусканием, который можно рассматривать как зависящую от температуры модель турбулентной атмосферы. Предполагая, что дисперсия пропускания может быть выражена через параметр силы флуктуации, мы показываем, что статистика фотонов света остается субпуассоновской при условии, что усредненный коэффициент пропускания превышает свое критическое значение. Критический коэффициент пропускания аналитически вычисляется как функция параметров входных состояний, температуры и силы флуктуации. Результаты применяются для изучения особых случаев одномодовых сжатых состояний и нечетных оптических состояний кота Шредингера.</p></trans-abstract><kwd-group xml:lang="ru"><kwd>субпуассоновский свет</kwd><kwd>флуктуации</kwd><kwd>бозонный канал</kwd></kwd-group><kwd-group xml:lang="en"><kwd>Sub-Poissonian light</kwd><kwd>fluctuations</kwd><kwd>bosonic channel</kwd></kwd-group><funding-group><funding-statement xml:lang="en">This work was supported by the Russian Science Foundation (project No. 24–29–00786).</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">Kenfack A. and Zyczkowski K. Negativity of the Wigner function as an indicator of non-classicality. Journal of Optics B: Quantum and Semiclassical Optics, 2004, 6(10), P. 396.</mixed-citation><mixed-citation xml:lang="en">Kenfack A. and Zyczkowski K. Negativity of the Wigner function as an indicator of non-classicality. Journal of Optics B: Quantum and Semiclassical Optics, 2004, 6(10), P. 396.</mixed-citation></citation-alternatives></ref><ref id="cit2"><label>2</label><citation-alternatives><mixed-citation xml:lang="ru">Teich M.C. and Saleh B.E.A. Squeezed state of light. Quantum Optics: Journal of the European Optical Society Part B, 1989, 1(2), P. 153–191.</mixed-citation><mixed-citation xml:lang="en">Teich M.C. and Saleh B.E.A. Squeezed state of light. Quantum Optics: Journal of the European Optical Society Part B, 1989, 1(2), P. 153–191.</mixed-citation></citation-alternatives></ref><ref id="cit3"><label>3</label><citation-alternatives><mixed-citation xml:lang="ru">Lvovsky A.I. Squeezed Light, ch. 5, P. 121–163. John Wiley &amp; Sons, Ltd, 2015.</mixed-citation><mixed-citation xml:lang="en">Lvovsky A.I. Squeezed Light, ch. 5, P. 121–163. John Wiley &amp; Sons, Ltd, 2015.</mixed-citation></citation-alternatives></ref><ref id="cit4"><label>4</label><citation-alternatives><mixed-citation xml:lang="ru">Davidovich L. Sub-Poissonian processes in quantum optics. Rev. Mod. Phys., 1996, 68, P. 127–173.</mixed-citation><mixed-citation xml:lang="en">Davidovich L. Sub-Poissonian processes in quantum optics. Rev. Mod. Phys., 1996, 68, P. 127–173.</mixed-citation></citation-alternatives></ref><ref id="cit5"><label>5</label><citation-alternatives><mixed-citation xml:lang="ru">Goldberg A.Z., Klimov A.B., Grassl M., Leuchs G., and S´anchez-Soto L.L. Extremal quantum states. AVS Quantum Science, 2020, 2(4), P. 044701.</mixed-citation><mixed-citation xml:lang="en">Goldberg A.Z., Klimov A.B., Grassl M., Leuchs G., and S´anchez-Soto L.L. Extremal quantum states. AVS Quantum Science, 2020, 2(4), P. 044701.</mixed-citation></citation-alternatives></ref><ref id="cit6"><label>6</label><citation-alternatives><mixed-citation xml:lang="ru">Tan K.C. and Jeong H. Nonclassical light and metrological power: An introductory review. AVS Quantum Science, 2019, 1(1) P. 014701.</mixed-citation><mixed-citation xml:lang="en">Tan K.C. and Jeong H. Nonclassical light and metrological power: An introductory review. AVS Quantum Science, 2019, 1(1) P. 014701.</mixed-citation></citation-alternatives></ref><ref id="cit7"><label>7</label><citation-alternatives><mixed-citation xml:lang="ru">Li R.-D., Choi S.-K., Kim C., and Kumar P. Generation of sub-Poissonian pulses of light. Phys. Rev. A, 1995, 51, P. R3429–R3432.</mixed-citation><mixed-citation xml:lang="en">Li R.-D., Choi S.-K., Kim C., and Kumar P. Generation of sub-Poissonian pulses of light. Phys. Rev. A, 1995, 51, P. R3429–R3432.</mixed-citation></citation-alternatives></ref><ref id="cit8"><label>8</label><citation-alternatives><mixed-citation xml:lang="ru">Peˇrina J.,Mich´alek V., Machulka R., and Haderka O. Two-beam light with simultaneous anticorrelations in photon-number fluctuations and sub-Poissonian statistics. Phys. Rev. A, 2021, 104, P. 013712.</mixed-citation><mixed-citation xml:lang="en">Peˇrina J.,Mich´alek V., Machulka R., and Haderka O. Two-beam light with simultaneous anticorrelations in photon-number fluctuations and sub-Poissonian statistics. Phys. Rev. A, 2021, 104, P. 013712.</mixed-citation></citation-alternatives></ref><ref id="cit9"><label>9</label><citation-alternatives><mixed-citation xml:lang="ru">Iskhakov T.S., Usenko V.C., Andersen U.L., Filip R., Chekhova M.V., and Leuchs G. Heralded source of bright multi-mode mesoscopic sub- Poissonian light. Opt. Lett., 2016, 41, P. 2149–2152.</mixed-citation><mixed-citation xml:lang="en">Iskhakov T.S., Usenko V.C., Andersen U.L., Filip R., Chekhova M.V., and Leuchs G. Heralded source of bright multi-mode mesoscopic sub- Poissonian light. Opt. Lett., 2016, 41, P. 2149–2152.</mixed-citation></citation-alternatives></ref><ref id="cit10"><label>10</label><citation-alternatives><mixed-citation xml:lang="ru">Lal N., Shajilal B., Anwar A., Perumangatt C., and Singh R.P., Observing sub-Poissonian statistics of twisted single photons using oscilloscope. Review of Scientific Instruments, 2019, 90(11), P. 113104.</mixed-citation><mixed-citation xml:lang="en">Lal N., Shajilal B., Anwar A., Perumangatt C., and Singh R.P., Observing sub-Poissonian statistics of twisted single photons using oscilloscope. Review of Scientific Instruments, 2019, 90(11), P. 113104.</mixed-citation></citation-alternatives></ref><ref id="cit11"><label>11</label><citation-alternatives><mixed-citation xml:lang="ru">Samedov V.V. Discovery of sub-Poissonian statistics of light photons in scintillators that did not take place. Physics of Atomic Nuclei, 2019, 84(11), P. 2048–2054.</mixed-citation><mixed-citation xml:lang="en">Samedov V.V. Discovery of sub-Poissonian statistics of light photons in scintillators that did not take place. Physics of Atomic Nuclei, 2019, 84(11), P. 2048–2054.</mixed-citation></citation-alternatives></ref><ref id="cit12"><label>12</label><citation-alternatives><mixed-citation xml:lang="ru">Berchera I.R. and Degiovanni I.P. Quantum imaging with sub-Poissonian light: challenges and perspectives in optical metrology. Metrologia, 2019, 56, P. 024001.</mixed-citation><mixed-citation xml:lang="en">Berchera I.R. and Degiovanni I.P. Quantum imaging with sub-Poissonian light: challenges and perspectives in optical metrology. Metrologia, 2019, 56, P. 024001.</mixed-citation></citation-alternatives></ref><ref id="cit13"><label>13</label><citation-alternatives><mixed-citation xml:lang="ru">Erenso D., Vyas R., and Singh S. Higher-order sub-Poissonian photon statistics in terms of factorial moments. J. Opt. Soc. Am. B, 2002, 19, P. 1471–1475.</mixed-citation><mixed-citation xml:lang="en">Erenso D., Vyas R., and Singh S. Higher-order sub-Poissonian photon statistics in terms of factorial moments. J. Opt. Soc. Am. B, 2002, 19, P. 1471–1475.</mixed-citation></citation-alternatives></ref><ref id="cit14"><label>14</label><citation-alternatives><mixed-citation xml:lang="ru">Peˇrina J., Mich´alek V., and Haderka O. Higher-order sub-Poissonian-like nonclassical fields: Theoretical and experimental comparison. Phys. Rev. A, 2017, 96, P. 033852.</mixed-citation><mixed-citation xml:lang="en">Peˇrina J., Mich´alek V., and Haderka O. Higher-order sub-Poissonian-like nonclassical fields: Theoretical and experimental comparison. Phys. Rev. A, 2017, 96, P. 033852.</mixed-citation></citation-alternatives></ref><ref id="cit15"><label>15</label><citation-alternatives><mixed-citation xml:lang="ru">Waks E., Santori C., and Yamamoto Y. Security aspects of quantum key distribution with sub-Poisson light. Phys. Rev. A, 2002, 66, P. 042315.</mixed-citation><mixed-citation xml:lang="en">Waks E., Santori C., and Yamamoto Y. Security aspects of quantum key distribution with sub-Poisson light. Phys. Rev. A, 2002, 66, P. 042315.</mixed-citation></citation-alternatives></ref><ref id="cit16"><label>16</label><citation-alternatives><mixed-citation xml:lang="ru">Kupko T., M. von Helversen, Rickert L., Schulze J.-H., Strittmatter A., Gschrey M., Rodt S., Reitzenstein S., and Heindel T. Tools for the performance optimization of single-photon quantum key distribution. npj Quantum Information, 2020, 6(1), P. 29.</mixed-citation><mixed-citation xml:lang="en">Kupko T., M. von Helversen, Rickert L., Schulze J.-H., Strittmatter A., Gschrey M., Rodt S., Reitzenstein S., and Heindel T. Tools for the performance optimization of single-photon quantum key distribution. npj Quantum Information, 2020, 6(1), P. 29.</mixed-citation></citation-alternatives></ref><ref id="cit17"><label>17</label><citation-alternatives><mixed-citation xml:lang="ru">Pirandola S., Andersen U.L., Banchi L., Berta M., Bunandar D., Colbeck R., Englund D., Gehring T., Lupo C., Ottaviani C., Pereira J.L., Razavi M., Shaari J.S., Tomamichel M., Usenko V.C., Vallone G., Villoresi P., and Wallden P., Advances in quantum cryptography. Adv. Opt. Photon., 2020, 12, P. 1012–1236.</mixed-citation><mixed-citation xml:lang="en">Pirandola S., Andersen U.L., Banchi L., Berta M., Bunandar D., Colbeck R., Englund D., Gehring T., Lupo C., Ottaviani C., Pereira J.L., Razavi M., Shaari J.S., Tomamichel M., Usenko V.C., Vallone G., Villoresi P., and Wallden P., Advances in quantum cryptography. Adv. Opt. Photon., 2020, 12, P. 1012–1236.</mixed-citation></citation-alternatives></ref><ref id="cit18"><label>18</label><citation-alternatives><mixed-citation xml:lang="ru">Goncharov R.K., Kiselev A.D., Samsonov E.O., and Egorov V.I. Continuous-variable quantum key distribution: security analysis with trusted hard-ware noise against general attacks. Nanosystems: Physics, Chemistry, Mathematics, 2022, 13, P. 372–391.</mixed-citation><mixed-citation xml:lang="en">Goncharov R.K., Kiselev A.D., Samsonov E.O., and Egorov V.I. Continuous-variable quantum key distribution: security analysis with trusted hard-ware noise against general attacks. Nanosystems: Physics, Chemistry, Mathematics, 2022, 13, P. 372–391.</mixed-citation></citation-alternatives></ref><ref id="cit19"><label>19</label><citation-alternatives><mixed-citation xml:lang="ru">Peuntinger C., Heim B., M¨uller C.R., Gabriel C., Marquardt C., and Leuchs G. Distribution of squeezed states through an atmospheric channel. Phys. Rev. Lett., 2014, 113, P. 060502.</mixed-citation><mixed-citation xml:lang="en">Peuntinger C., Heim B., M¨uller C.R., Gabriel C., Marquardt C., and Leuchs G. Distribution of squeezed states through an atmospheric channel. Phys. Rev. Lett., 2014, 113, P. 060502.</mixed-citation></citation-alternatives></ref><ref id="cit20"><label>20</label><citation-alternatives><mixed-citation xml:lang="ru">Heim B., Peuntinger C., Killoran N., Khan I., Wittmann C., Marquardt C., and Leuchs G. Atmospheric continuous-variable quantum communication. New Journal of Physics, 2014, 16, P. 113018.</mixed-citation><mixed-citation xml:lang="en">Heim B., Peuntinger C., Killoran N., Khan I., Wittmann C., Marquardt C., and Leuchs G. Atmospheric continuous-variable quantum communication. New Journal of Physics, 2014, 16, P. 113018.</mixed-citation></citation-alternatives></ref><ref id="cit21"><label>21</label><citation-alternatives><mixed-citation xml:lang="ru">Derkach I., Usenko V.C., and Filip R. Squeezing-enhanced quantum key distribution over atmospheric channels. New Journal of Physics, 2020, 22, P. 053006.</mixed-citation><mixed-citation xml:lang="en">Derkach I., Usenko V.C., and Filip R. Squeezing-enhanced quantum key distribution over atmospheric channels. New Journal of Physics, 2020, 22, P. 053006.</mixed-citation></citation-alternatives></ref><ref id="cit22"><label>22</label><citation-alternatives><mixed-citation xml:lang="ru">Pirandola S. Limits and security of free-space quantum communications. Phys. Rev. Res., 2021, 3, P. 013279.</mixed-citation><mixed-citation xml:lang="en">Pirandola S. Limits and security of free-space quantum communications. Phys. Rev. Res., 2021, 3, P. 013279.</mixed-citation></citation-alternatives></ref><ref id="cit23"><label>23</label><citation-alternatives><mixed-citation xml:lang="ru">Kravtsov Y.A. Propagation of electromagnetic waves through a turbulent atmosphere. Reports on Progress in Physics, 1992, 55, P. 39–112.</mixed-citation><mixed-citation xml:lang="en">Kravtsov Y.A. Propagation of electromagnetic waves through a turbulent atmosphere. Reports on Progress in Physics, 1992, 55, P. 39–112.</mixed-citation></citation-alternatives></ref><ref id="cit24"><label>24</label><citation-alternatives><mixed-citation xml:lang="ru">Milonni P.W., Carter J.H., Peterson C.G., and Hughes R.J. Effects of propagation through atmospheric turbulence on photon statistics. Journal of Optics B: Quantum and Semiclassical Optics, 2004, 6, P. S742.</mixed-citation><mixed-citation xml:lang="en">Milonni P.W., Carter J.H., Peterson C.G., and Hughes R.J. Effects of propagation through atmospheric turbulence on photon statistics. Journal of Optics B: Quantum and Semiclassical Optics, 2004, 6, P. S742.</mixed-citation></citation-alternatives></ref><ref id="cit25"><label>25</label><citation-alternatives><mixed-citation xml:lang="ru">Kiesel T., Vogel W., Parigi V., Zavatta A., and Bellini M. Experimental determination of a nonclassical Glauber-Sudarshan P function. Phys. Rev. A, 2008, 78, P. 021804.</mixed-citation><mixed-citation xml:lang="en">Kiesel T., Vogel W., Parigi V., Zavatta A., and Bellini M. Experimental determination of a nonclassical Glauber-Sudarshan P function. Phys. Rev. A, 2008, 78, P. 021804.</mixed-citation></citation-alternatives></ref><ref id="cit26"><label>26</label><citation-alternatives><mixed-citation xml:lang="ru">Semenov A.A. and Vogel W. Quantum light in the turbulent atmosphere. Phys. Rev. A, 2009, 80, P. 021802.</mixed-citation><mixed-citation xml:lang="en">Semenov A.A. and Vogel W. Quantum light in the turbulent atmosphere. Phys. Rev. A, 2009, 80, P. 021802.</mixed-citation></citation-alternatives></ref><ref id="cit27"><label>27</label><citation-alternatives><mixed-citation xml:lang="ru">Vasylyev D.Y., Semenov A.A., and VogelW. Toward global quantum communication: Beam wandering preserves nonclassicality. Phys. Rev. Lett., 2012, 108, P. 220501.</mixed-citation><mixed-citation xml:lang="en">Vasylyev D.Y., Semenov A.A., and VogelW. Toward global quantum communication: Beam wandering preserves nonclassicality. Phys. Rev. Lett., 2012, 108, P. 220501.</mixed-citation></citation-alternatives></ref><ref id="cit28"><label>28</label><citation-alternatives><mixed-citation xml:lang="ru">Gumberidze M.O., Semenov A.A., Vasylyev D., and Vogel W., Bell nonlocality in the turbulent atmosphere. Phys. Rev. A, 2016, 94, P. 053801.</mixed-citation><mixed-citation xml:lang="en">Gumberidze M.O., Semenov A.A., Vasylyev D., and Vogel W., Bell nonlocality in the turbulent atmosphere. Phys. Rev. A, 2016, 94, P. 053801.</mixed-citation></citation-alternatives></ref><ref id="cit29"><label>29</label><citation-alternatives><mixed-citation xml:lang="ru">Vasylyev D., Semenov A.A., and VogelW. Atmospheric quantum channels with weak and strong turbulence. Phys. Rev. Lett., 2016, 117, P. 090501.</mixed-citation><mixed-citation xml:lang="en">Vasylyev D., Semenov A.A., and VogelW. Atmospheric quantum channels with weak and strong turbulence. Phys. Rev. Lett., 2016, 117, P. 090501.</mixed-citation></citation-alternatives></ref><ref id="cit30"><label>30</label><citation-alternatives><mixed-citation xml:lang="ru">Bohmann M., Semenov A.A., Sperling J., and Vogel W. Gaussian entanglement in the turbulent atmosphere. Phys. Rev. A, 2016, 94, P. 010302.</mixed-citation><mixed-citation xml:lang="en">Bohmann M., Semenov A.A., Sperling J., and Vogel W. Gaussian entanglement in the turbulent atmosphere. Phys. Rev. A, 2016, 94, P. 010302.</mixed-citation></citation-alternatives></ref><ref id="cit31"><label>31</label><citation-alternatives><mixed-citation xml:lang="ru">Avetisyan H. and Monken C.H. Higher order correlation beams in atmosphere under strong turbulence conditions. Opt. Express, 2016, 24, P. 2318–2335.</mixed-citation><mixed-citation xml:lang="en">Avetisyan H. and Monken C.H. Higher order correlation beams in atmosphere under strong turbulence conditions. Opt. Express, 2016, 24, P. 2318–2335.</mixed-citation></citation-alternatives></ref><ref id="cit32"><label>32</label><citation-alternatives><mixed-citation xml:lang="ru">Bohmann M., Sperling J., Semenov A.A., and Vogel W. Higher-order nonclassical effects in fluctuating-loss channels. Phys. Rev. A, 2017, 95, P. 012324.</mixed-citation><mixed-citation xml:lang="en">Bohmann M., Sperling J., Semenov A.A., and Vogel W. Higher-order nonclassical effects in fluctuating-loss channels. Phys. Rev. A, 2017, 95, P. 012324.</mixed-citation></citation-alternatives></ref><ref id="cit33"><label>33</label><citation-alternatives><mixed-citation xml:lang="ru">Vasylyev D., Vogel W., and Semenov A.A. Theory of atmospheric quantum channels based on the law of total probability. Phys. Rev. A, 2018, 97, P. 063852.</mixed-citation><mixed-citation xml:lang="en">Vasylyev D., Vogel W., and Semenov A.A. Theory of atmospheric quantum channels based on the law of total probability. Phys. Rev. A, 2018, 97, P. 063852.</mixed-citation></citation-alternatives></ref><ref id="cit34"><label>34</label><citation-alternatives><mixed-citation xml:lang="ru">Vasylyev D., Vogel W., and Moll F. Satellite-mediated quantum atmospheric links. Phys. Rev. A, 2019, 99, P. 053830.</mixed-citation><mixed-citation xml:lang="en">Vasylyev D., Vogel W., and Moll F. Satellite-mediated quantum atmospheric links. Phys. Rev. A, 2019, 99, P. 053830.</mixed-citation></citation-alternatives></ref><ref id="cit35"><label>35</label><citation-alternatives><mixed-citation xml:lang="ru">Klen M. and Semenov A.A. Numerical simulations of atmospheric quantum channels. Phys. Rev. A, 2023, 108, P. 033718.</mixed-citation><mixed-citation xml:lang="en">Klen M. and Semenov A.A. Numerical simulations of atmospheric quantum channels. Phys. Rev. A, 2023, 108, P. 033718.</mixed-citation></citation-alternatives></ref><ref id="cit36"><label>36</label><citation-alternatives><mixed-citation xml:lang="ru">Kiselev A.D., Ali R., and Rybin A.V. Lindblad dynamics and disentanglement in multi-mode bosonic systems. Entropy, 2021, 23(11), P. 1409.</mixed-citation><mixed-citation xml:lang="en">Kiselev A.D., Ali R., and Rybin A.V. Lindblad dynamics and disentanglement in multi-mode bosonic systems. Entropy, 2021, 23(11), P. 1409.</mixed-citation></citation-alternatives></ref><ref id="cit37"><label>37</label><citation-alternatives><mixed-citation xml:lang="ru">Mandel L. and Wolf E. Optical Coherence and Quantum Optics. Cambridge, Cambridge University Press, 1995.</mixed-citation><mixed-citation xml:lang="en">Mandel L. and Wolf E. Optical Coherence and Quantum Optics. Cambridge, Cambridge University Press, 1995.</mixed-citation></citation-alternatives></ref><ref id="cit38"><label>38</label><citation-alternatives><mixed-citation xml:lang="ru">Mandel L. Sub-Poissonian photon statistics in resonance fluorescence. Opt. Lett., 1979, 4, P. 205–207.</mixed-citation><mixed-citation xml:lang="en">Mandel L. Sub-Poissonian photon statistics in resonance fluorescence. Opt. Lett., 1979, 4, P. 205–207.</mixed-citation></citation-alternatives></ref><ref id="cit39"><label>39</label><citation-alternatives><mixed-citation xml:lang="ru">Short R. and Mandel L. Observation of sub-Poissonian photon statistics. Phys. Rev. Lett., 1983, 51, P. 384–387.</mixed-citation><mixed-citation xml:lang="en">Short R. and Mandel L. Observation of sub-Poissonian photon statistics. Phys. Rev. Lett., 1983, 51, P. 384–387.</mixed-citation></citation-alternatives></ref><ref id="cit40"><label>40</label><citation-alternatives><mixed-citation xml:lang="ru">Banakh V.A. and Mironov V.L. Phase approximation of the Huygens–Kirchhoff method in problems of laser-beam propagation in the turbulent atmosphere. Opt. Lett., 1977, 1(5), P. 172–174.</mixed-citation><mixed-citation xml:lang="en">Banakh V.A. and Mironov V.L. Phase approximation of the Huygens–Kirchhoff method in problems of laser-beam propagation in the turbulent atmosphere. Opt. Lett., 1977, 1(5), P. 172–174.</mixed-citation></citation-alternatives></ref><ref id="cit41"><label>41</label><citation-alternatives><mixed-citation xml:lang="ru">Banakh V.A. and Mironov V.L. Phase approximation of the Huygens–Kirchhoff method in problems of space-limited optical-beam propagation in turbulent atmosphere. Opt. Lett., 1979, 4(8), P. 259–261.</mixed-citation><mixed-citation xml:lang="en">Banakh V.A. and Mironov V.L. Phase approximation of the Huygens–Kirchhoff method in problems of space-limited optical-beam propagation in turbulent atmosphere. Opt. Lett., 1979, 4(8), P. 259–261.</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>
