{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T08:35:45Z","timestamp":1777365345221,"version":"3.51.4"},"reference-count":53,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,11,29]],"date-time":"2022-11-29T00:00:00Z","timestamp":1669680000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["1150403"],"award-info":[{"award-number":["1150403"]}]},{"name":"National Natural Science Foundation of China","award":["61505014"],"award-info":[{"award-number":["61505014"]}]},{"name":"National Natural Science Foundation of China","award":["11704045"],"award-info":[{"award-number":["11704045"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A coherent feedback scheme is used to enhance the degree of squeezing of the output field in a cavity optomechanical system. In the feedback loop, a beam splitter (BS) plays the roles of both a feedback controller and an input\u2013output port. To realize effective enhancement, the output quadrature should take the same form as the input quadrature, and the system should operate at the deamplification situation in the meantime. This can be realized by choosing an appropriate frequency-dependent phase angle for the generalized quadrature. Additionally, both the transmissivity of the BS and the phase factor induced by time delays in the loop affect optical squeezing. For the fixed frequency, the optimal values of transmissivity and phase factor can be used to achieve the enhanced optical squeezing. The effect of optical losses on squeezing is also discussed. Optical squeezing is degraded by the introduced vacuum noise owing to the inefficient transmission in the loop. We show that the enhancement of squeezing is achievable with the parameters of the current experiments.<\/jats:p>","DOI":"10.3390\/e24121741","type":"journal-article","created":{"date-parts":[[2022,11,29]],"date-time":"2022-11-29T02:53:29Z","timestamp":1669690409000},"page":"1741","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Enhancement of Optomechanical Squeezing of Light Using the Optical Coherent Feedback"],"prefix":"10.3390","volume":"24","author":[{"given":"Zhenhua","family":"Wu","sequence":"first","affiliation":[{"name":"School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China"}]},{"given":"Zhen","family":"Yi","sequence":"additional","affiliation":[{"name":"School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6173-6885","authenticated-orcid":false,"given":"Wenju","family":"Gu","sequence":"additional","affiliation":[{"name":"School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0425-7431","authenticated-orcid":false,"given":"Lihui","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5260-0169","authenticated-orcid":false,"given":"Zbigniew","family":"Ficek","sequence":"additional","affiliation":[{"name":"Quantum Optics and Engineering Division, Institute of Physics, University of Zielona G\u00f3ra, Szafrana 4a, 65-516 Zielona G\u00f3ra, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1693","DOI":"10.1103\/PhysRevD.23.1693","article-title":"Quantum-mechanical noise in an interferometer","volume":"23","author":"Caves","year":"1981","journal-title":"Phys. Rev. D"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1209\/0295-5075\/13\/4\/003","article-title":"Quantum limits in interferometric measurements","volume":"13","author":"Jaekel","year":"1990","journal-title":"Europhys. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"022002","DOI":"10.1103\/PhysRevD.65.022002","article-title":"Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and\/or output optics","volume":"65","author":"Kimble","year":"2001","journal-title":"Phys. Rev. D"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"113007","DOI":"10.1088\/1367-2630\/13\/11\/113007","article-title":"Squeezed-state quantum key distribution upon imperfect reconciliation","volume":"13","author":"Usenko","year":"2011","journal-title":"New J. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1103\/RevModPhys.84.621","article-title":"Gaussian quantum information","volume":"84","author":"Weedbrook","year":"2012","journal-title":"Rev. Mod. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"120504","DOI":"10.1103\/PhysRevLett.112.120504","article-title":"Fault-tolerant measurement-based quantum computing with continuous-variable cluster states","volume":"112","author":"Menicucci","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1038\/nphoton.2012.346","article-title":"Biological measurement beyond the quantum limit","volume":"7","author":"Taylor","year":"2013","journal-title":"Nature Photon"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"850","DOI":"10.1364\/OPTICA.5.000850","article-title":"Quantum enhanced optomechanical magnetometry","volume":"5","author":"Li","year":"2018","journal-title":"Optica"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2409","DOI":"10.1103\/PhysRevLett.55.2409","article-title":"Observation of squeezed states generated by four-wave mixing in an optical cavity","volume":"55","author":"Slusher","year":"1985","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1103\/PhysRevLett.57.691","article-title":"Broad-band parametric deamplification of quantum noise in an optical fiber","volume":"57","author":"Shelby","year":"1986","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2520","DOI":"10.1103\/PhysRevLett.57.2520","article-title":"Generation of squeezed states by parametric down conversion","volume":"57","author":"Wu","year":"1986","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"033602","DOI":"10.1103\/PhysRevLett.100.033602","article-title":"Observation of squeezed light with 10-dB quantum noise reduction","volume":"100","author":"Vahlbruch","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"251102","DOI":"10.1103\/PhysRevLett.104.251102","article-title":"Quantum enchancement of the zero-area sagnac interferometer topology for gravitational wave detection","volume":"104","author":"Eberle","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"25763","DOI":"10.1364\/OE.19.025763","article-title":"Squeezed light at 1550 nm with a quantum noise reduction of 12.3 dB","volume":"19","author":"Mehmet","year":"2011","journal-title":"Opt. Express"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"110801","DOI":"10.1103\/PhysRevLett.117.110801","article-title":"Detection of 15 dB squeezed states of light and their application for the absolute calibration of photonelectric quantum efficiency","volume":"117","author":"Vahlbruch","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1126\/science.1231282","article-title":"Observation of radiation pressure shot noise on a macroscopic object","volume":"339","author":"Purdy","year":"2013","journal-title":"Science"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1337","DOI":"10.1103\/PhysRevA.49.1337","article-title":"Quantum-noise reduction using a cavity with a movable mirror","volume":"49","author":"Fabre","year":"1994","journal-title":"Phys. Rev. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4055","DOI":"10.1103\/PhysRevA.49.4055","article-title":"Quantum noise reduction by radiation pressure","volume":"49","author":"Mancini","year":"1994","journal-title":"Phys. Rev. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"103601","DOI":"10.1103\/PhysRevLett.118.103601","article-title":"Noiseless quantum measurement and squeezing of microwave fields ultilizing mechanical vibrations","volume":"118","author":"Pirkkalainen","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"054203","DOI":"10.7498\/aps.69.20191735","article-title":"Research progress in non-classical microwave states preparation based on cavity optomechanical system","volume":"69","author":"Luo","year":"2020","journal-title":"Acta Phys. Sin."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"023801","DOI":"10.1103\/PhysRevA.73.023801","article-title":"Squeezed-state source using radiation-pressure-induced rigidity","volume":"73","author":"Corbitt","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"784","DOI":"10.1038\/s41567-020-0877-x","article-title":"Room-temperature optomechanical suqeezing","volume":"16","author":"Aggarwal","year":"2020","journal-title":"Nat. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"710","DOI":"10.1038\/s41567-019-0770-7","article-title":"Squeezing hots up","volume":"16","author":"Xuereb","year":"2020","journal-title":"Nat. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"013814","DOI":"10.1103\/PhysRevA.81.013814","article-title":"Observation of squeezed states with strong photon-number oscillations","volume":"81","author":"Mehmet","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"233602","DOI":"10.1103\/PhysRevLett.101.233602","article-title":"Phase-sensitive manipulations of a squeezed vacuum field in an optical parametric amplifier inside an optical cavity","volume":"101","author":"Zhang","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"040305(R)","DOI":"10.1103\/PhysRevA.85.040305","article-title":"Cascaded entanglement enhancement","volume":"85","author":"Yan","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"6053","DOI":"10.1364\/OL.43.006053","article-title":"Optomechanical quadrature squeezing in the non-Markovian regime","volume":"43","author":"Xiong","year":"2018","journal-title":"Opt. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"11915","DOI":"10.1364\/OE.26.011915","article-title":"Quantum squeezing in a modulated optomechanical system","volume":"26","author":"Zhang","year":"2018","journal-title":"Opt. Express"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"5545","DOI":"10.1364\/OL.471230","article-title":"Optomechanical squeezing with pulse modulation","volume":"47","author":"Xiong","year":"2022","journal-title":"Opt. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"127824","DOI":"10.1016\/j.physleta.2021.127824","article-title":"Mechanical squeezing induced by Duffing nonlinearity and two driving tones in an optomechanical system","volume":"424","author":"Zhang","year":"2022","journal-title":"Phys. Lett. A"},{"key":"ref_31","unstructured":"Banerjee, P., Kalita, S., and Sarma, A.K. (2022). Mechanical Squeezing in Quadratically-coupled Optomechanical Systems. arXiv."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"476","DOI":"10.1038\/nature11325","article-title":"Non-classical light generated by quantum-noise-driven cavity optomechanics","volume":"488","author":"Brooks","year":"2012","journal-title":"Nature"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1038\/nature12307","article-title":"Squeezed light from a silicon micromechanical resonator","volume":"500","author":"Hill","year":"2013","journal-title":"Nature"},{"key":"ref_34","first-page":"031012","article-title":"Strong optomechanical squeezing of light","volume":"3","author":"Purdy","year":"2013","journal-title":"Phys. Rev. X"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"075403","DOI":"10.1088\/1361-6455\/abecd4","article-title":"Squeezed light generation in cascaded optomechanical systems","volume":"54","author":"Wang","year":"2021","journal-title":"J. Phys. B At. Mol. Opt. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"022108","DOI":"10.1103\/PhysRevA.62.022108","article-title":"Coherent quantum feedback","volume":"62","author":"Lloyd","year":"2000","journal-title":"Phys. Rev. A"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Wiseman, H.M., and Milburn, G.J. (2009). Quantum Measurement and Control, Cambridge University Press.","DOI":"10.1017\/CBO9780511813948"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"042107","DOI":"10.1103\/PhysRevA.80.042107","article-title":"Enhancement of field squeezing using coherent feedback","volume":"80","author":"Gough","year":"2009","journal-title":"Phys. Rev. A"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2530","DOI":"10.1109\/TAC.2009.2031205","article-title":"The series product and its application to quantum feedforward and feedback networks","volume":"54","author":"Gough","year":"2009","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"023804","DOI":"10.1103\/PhysRevA.81.023804","article-title":"Squeezing components in linear quantum feedback networks","volume":"81","author":"Gough","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"4110","DOI":"10.1103\/PhysRevA.49.4110","article-title":"All-optical versus electro-optical quantum-limited feedback","volume":"49","author":"Wiseman","year":"1994","journal-title":"Phys. Rev. A"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"2107","DOI":"10.1109\/TAC.2003.820063","article-title":"Transfer function approach to quantum control\u2014 Part I: Dynamics of quantum feedback systems","volume":"48","author":"Yanagisawa","year":"2003","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2121","DOI":"10.1109\/TAC.2003.820065","article-title":"Transfer function approach toquantum control\u2014Part II: Control concepts and applications","volume":"48","author":"Yanagisawa","year":"2003","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"123601","DOI":"10.1103\/PhysRevLett.105.123601","article-title":"Coherent quantum-noise cancellation for optomechanical sensors","volume":"105","author":"Tsang","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_45","first-page":"041029","article-title":"Coherent versus measurement feedback: Linear systems theory for quantum information","volume":"4","author":"Yamamoto","year":"2014","journal-title":"Phys. Rev. X"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"023509","DOI":"10.1103\/PhysRevA.103.023509","article-title":"Cavity optomechanics assisted by optical coherent feedback","volume":"103","author":"Harwood","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"2317","DOI":"10.1103\/PhysRevD.38.2317","article-title":"Becycling in laser-interferometric gravitational-wave detectors","volume":"38","author":"Meers","year":"1988","journal-title":"Phys. Rev. D"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"211102","DOI":"10.1103\/PhysRevLett.95.211102","article-title":"Demonstration of a squeezed-light-enhanced power- and signal-recycled Michelson interferometer","volume":"95","author":"Vahlbruch","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1038\/s41567-019-0533-5","article-title":"Continuous force and displacement measurement below the standard quantum limit","volume":"15","author":"Mason","year":"2019","journal-title":"Nat. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"776","DOI":"10.1038\/nnano.2017.101","article-title":"Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution","volume":"12","author":"Tsaturyan","year":"2017","journal-title":"Nat. Nanotechnol."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1038\/s41586-018-0643-8","article-title":"Measurement-based quantum control of mechanical motion","volume":"563","author":"Rossi","year":"2018","journal-title":"Nature"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"4321","DOI":"10.1364\/OE.15.004321","article-title":"Observation of \u22129 dB quadrature squeezing with improvement of phase stability in homodyne measurement","volume":"15","author":"Takeno","year":"2007","journal-title":"Opt. Express"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"2045","DOI":"10.1109\/TAC.2012.2195831","article-title":"Experimental demonstration of coherent feedback control on optical field squeezing","volume":"57","author":"Iida","year":"2012","journal-title":"IEEE Trans. Autom. Control"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/12\/1741\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:29:02Z","timestamp":1760146142000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/12\/1741"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,29]]},"references-count":53,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["e24121741"],"URL":"https:\/\/doi.org\/10.3390\/e24121741","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,29]]}}}