{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:07:00Z","timestamp":1753888020949,"version":"3.41.2"},"reference-count":51,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2021,12,17]],"date-time":"2021-12-17T00:00:00Z","timestamp":1639699200000},"content-version":"vor","delay-in-days":350,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003621","name":"Ministry of Science, ICT and Future Planning","doi-asserted-by":"publisher","award":["NRF-2021R1F1A1062849"],"award-info":[{"award-number":["NRF-2021R1F1A1062849"]}],"id":[{"id":"10.13039\/501100003621","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>We investigate quantum features of three coupled dissipative nano\u2010optomechanical oscillators. The Hamiltonian of the system is somewhat complicated due not only to the coupling of the optomechanical oscillators but to the dissipation in the system as well. In order to simplify the problem, a spatial unitary transformation approach and a matrix\u2010diagonalization method are used. From such procedures, the Hamiltonian is eventually diagonalized. In other words, the complicated original Hamiltonian is transformed to a simple one which is associated to three independent simple harmonic oscillators. By utilizing such a simplification of the Hamiltonian, complete solutions (wave functions) of the Schr\u00f6dinger equation for the optomechanical system are obtained. We confirm that the probability density converges to the origin of the coordinate in a symmetric manner as the optomechanical energy dissipates. The wave functions that we have derived can be used as a basic tool for evaluating diverse quantum consequences of the system, such as quadrature fluctuations, entanglement entropy, energy evolution, transition probability, and the Wigner function.<\/jats:p>","DOI":"10.1155\/2021\/1103419","type":"journal-article","created":{"date-parts":[[2021,12,18]],"date-time":"2021-12-18T03:05:05Z","timestamp":1639796705000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Quantum Approach to Damped Three Coupled Nano\u2010Optomechanical Oscillators"],"prefix":"10.1155","volume":"2021","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4416-4408","authenticated-orcid":false,"given":"Jeong Ryeol","family":"Choi","sequence":"first","affiliation":[]},{"given":"Salah","family":"Menouar","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2021,12,17]]},"reference":[{"key":"e_1_2_9_1_2","doi-asserted-by":"publisher","DOI":"10.1103\/physreva.91.063836"},{"key":"e_1_2_9_2_2","doi-asserted-by":"publisher","DOI":"10.3390\/nano9010020"},{"key":"e_1_2_9_3_2","doi-asserted-by":"publisher","DOI":"10.1038\/nature08171"},{"key":"e_1_2_9_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10946-012-9288-5"},{"key":"e_1_2_9_5_2","doi-asserted-by":"publisher","DOI":"10.1006\/aphy.1996.0137"},{"key":"e_1_2_9_6_2","doi-asserted-by":"publisher","DOI":"10.1364\/josab.23.001146"},{"key":"e_1_2_9_7_2","doi-asserted-by":"publisher","DOI":"10.1126\/science.1137307"},{"key":"e_1_2_9_8_2","doi-asserted-by":"publisher","DOI":"10.1038\/nature09721"},{"key":"e_1_2_9_9_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.473189"},{"key":"e_1_2_9_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.chemphys.2006.05.007"},{"key":"e_1_2_9_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/s0167-2789(00)00071-3"},{"key":"e_1_2_9_12_2","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/29\/9\/015"},{"key":"e_1_2_9_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/s0370-1573(01)00077-1"},{"key":"e_1_2_9_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2015.12.007"},{"key":"e_1_2_9_15_2","doi-asserted-by":"publisher","DOI":"10.1103\/physreva.99.022130"},{"key":"e_1_2_9_16_2","doi-asserted-by":"publisher","DOI":"10.1088\/0253-6102\/45\/1\/027"},{"key":"e_1_2_9_17_2","doi-asserted-by":"publisher","DOI":"10.1023\/a:1025931630841"},{"key":"e_1_2_9_18_2","doi-asserted-by":"publisher","DOI":"10.1038\/nature05027"},{"key":"e_1_2_9_19_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.4916728"},{"key":"e_1_2_9_20_2","doi-asserted-by":"publisher","DOI":"10.1364\/ol.32.001611"},{"key":"e_1_2_9_21_2","doi-asserted-by":"publisher","DOI":"10.1103\/physreva.81.041803"},{"key":"e_1_2_9_22_2","doi-asserted-by":"publisher","DOI":"10.1103\/physreva.92.033841"},{"key":"e_1_2_9_23_2","doi-asserted-by":"publisher","DOI":"10.1038\/nphoton.2010.5"},{"key":"e_1_2_9_24_2","doi-asserted-by":"publisher","DOI":"10.1038\/srep33404"},{"key":"e_1_2_9_25_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.98.030405"},{"key":"e_1_2_9_26_2","doi-asserted-by":"publisher","DOI":"10.1080\/00018730902831009"},{"key":"e_1_2_9_27_2","doi-asserted-by":"publisher","DOI":"10.1103\/physrevlett.109.190502"},{"key":"e_1_2_9_28_2","doi-asserted-by":"publisher","DOI":"10.1103\/physrevlett.95.260502"},{"key":"e_1_2_9_29_2","doi-asserted-by":"publisher","DOI":"10.3390\/e23070837"},{"key":"e_1_2_9_30_2","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/34\/37\/321"},{"key":"e_1_2_9_31_2","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/30\/8\/011"},{"key":"e_1_2_9_32_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.rinp.2013.06.003"},{"key":"e_1_2_9_33_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02581073"},{"key":"e_1_2_9_34_2","first-page":"594","article-title":"Quantum theory of the harmonic oscillator in nonconservative systems","volume":"41","author":"Um C.-I.","year":"2002","journal-title":"Journal of the Korean Physical Society"},{"key":"e_1_2_9_35_2","doi-asserted-by":"publisher","DOI":"10.1103\/physreva.87.032114"},{"key":"e_1_2_9_36_2","doi-asserted-by":"publisher","DOI":"10.1016\/s0375-9601(01)00600-4"},{"key":"e_1_2_9_37_2","doi-asserted-by":"publisher","DOI":"10.1016\/s0375-9601(02)00062-2"},{"key":"e_1_2_9_38_2","doi-asserted-by":"publisher","DOI":"10.1103\/physreva.49.2301"},{"key":"e_1_2_9_39_2","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8113\/42\/3\/035307"},{"key":"e_1_2_9_40_2","doi-asserted-by":"publisher","DOI":"10.1103\/physrevlett.66.527"},{"key":"e_1_2_9_41_2","doi-asserted-by":"publisher","DOI":"10.1103\/physreva.62.014103"},{"key":"e_1_2_9_42_2","doi-asserted-by":"publisher","DOI":"10.1103\/physreva.68.062308"},{"key":"e_1_2_9_43_2","unstructured":"KronenburgM. 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