{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T01:28:14Z","timestamp":1775179694075,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,10,16]],"date-time":"2021-10-16T00:00:00Z","timestamp":1634342400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Active optical media leading to interaction Hamiltonians of the form H=\u03bb\u02dc(a+a\u2020)\u03b6 represent a crucial resource for quantum optical technology. In this paper, we address the characterization of those nonlinear media using quantum probes, as opposed to semiclassical ones. In particular, we investigate how squeezed probes may improve individual and joint estimation of the nonlinear coupling \u03bb\u02dc and of the nonlinearity order \u03b6. Upon using tools from quantum estimation, we show that: (i) the two parameters are compatible, i.e., the may be jointly estimated without additional quantum noise; (ii) the use of squeezed probes improves precision at fixed overall energy of the probe; (iii) for low energy probes, squeezed vacuum represent the most convenient choice, whereas for increasing energy an optimal squeezing fraction may be determined; (iv) using optimized quantum probes, the scaling of the corresponding precision with energy improves, both for individual and joint estimation of the two parameters, compared to semiclassical coherent probes. We conclude that quantum probes represent a resource to enhance precision in the characterization of nonlinear media, and foresee potential applications with current technology.<\/jats:p>","DOI":"10.3390\/e23101353","type":"journal-article","created":{"date-parts":[[2021,10,17]],"date-time":"2021-10-17T23:13:32Z","timestamp":1634512412000},"page":"1353","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":32,"title":["Quantum Probes for the Characterization of Nonlinear Media"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0582-4941","authenticated-orcid":false,"given":"Alessandro","family":"Candeloro","sequence":"first","affiliation":[{"name":"Quantum Technology Lab, Dipartimento di Fisica \u201cAldo Pontremoli\u201d, Universit\u00e0 degli Studi di Milano, I-20133 Milano, Italy"},{"name":"INFN, Sezione di Milano, I-20133 Milano, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7556-9512","authenticated-orcid":false,"given":"Sholeh","family":"Razavian","sequence":"additional","affiliation":[{"name":"Max-Planck-Institut fur Quantenoptik, D-85748 Garching bei Munchen, Germany"},{"name":"Department fur Physik, Ludwig-Maximilians-Universit\u00e4t, D-80799 Munchen, Germany"}]},{"given":"Matteo","family":"Piccolini","sequence":"additional","affiliation":[{"name":"Dipartimento di Ingegneria, Universit\u00e0 di Palermo, I-90128 Palermo, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9280-533X","authenticated-orcid":false,"given":"Berihu","family":"Teklu","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Sciences, Center for Cyber-Physical Systems (C2PS), Khalifa University, Abu Dhabi 127788, United Arab Emirates"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9251-0731","authenticated-orcid":false,"given":"Stefano","family":"Olivares","sequence":"additional","affiliation":[{"name":"Quantum Technology Lab, Dipartimento di Fisica \u201cAldo Pontremoli\u201d, Universit\u00e0 degli Studi di Milano, I-20133 Milano, Italy"},{"name":"INFN, Sezione di Milano, I-20133 Milano, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7523-7289","authenticated-orcid":false,"given":"Matteo G. A.","family":"Paris","sequence":"additional","affiliation":[{"name":"Quantum Technology Lab, Dipartimento di Fisica \u201cAldo Pontremoli\u201d, Universit\u00e0 degli Studi di Milano, I-20133 Milano, Italy"},{"name":"INFN, Sezione di Milano, I-20133 Milano, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1016\/j.pquantelec.2017.06.002","article-title":"From quantum optics to quantum technologies","volume":"54","author":"Browne","year":"2017","journal-title":"Prog. Quantum Electron."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Asselberghs, I., P\u00e9rez-Moreno, J., and Clays, K. (2006). Characterization Techniques of Nonlinear Optical Materials. Non-Linear Optical Properties of Matter: From Molecules to Condensed Phases, Springer.","DOI":"10.1007\/1-4020-4850-5_14"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"10988","DOI":"10.1038\/ncomms10988","article-title":"Nonlinear optomechanical measurement of mechanical motion","volume":"7","author":"Brawley","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"032105","DOI":"10.1103\/PhysRevA.84.032105","article-title":"Qubit thermometry for micromechanical resonators","volume":"84","author":"Brunelli","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Mandel, L., and Wolf, E. (1995). Optical Coherence and Quantum Optics, Cambridge University Press.","DOI":"10.1017\/CBO9781139644105"},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"053001","DOI":"10.1088\/0031-8949\/91\/5\/053001","article-title":"30 years of squeezed light generation","volume":"91","author":"Andersen","year":"2016","journal-title":"Phys. Scr."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"100501","DOI":"10.1103\/PhysRevLett.104.100501","article-title":"Experimental Estimation of Entanglement at the Quantum Limit","volume":"104","author":"Brida","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"032114","DOI":"10.1103\/PhysRevA.89.032114","article-title":"Quantum probes for the spectral properties of a classical environment","volume":"89","author":"Benedetti","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"010302","DOI":"10.1103\/PhysRevA.92.010302","article-title":"Entangled quantum probes for dynamical environmental noise","volume":"92","author":"Rossi","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"012125","DOI":"10.1103\/PhysRevA.97.012125","article-title":"Continuous-variable quantum probes for structured environments","volume":"97","author":"Bina","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"032112","DOI":"10.1103\/PhysRevA.101.032112","article-title":"Two-qubit quantum probes for the temperature of an Ohmic environment","volume":"101","author":"Gebbia","year":"2020","journal-title":"Phys. Rev. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1142\/S0219749909004839","article-title":"Quantum estimation for quantum technology","volume":"07","author":"Paris","year":"2009","journal-title":"Int. J. Quantum Inf."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"126311","DOI":"10.1016\/j.physleta.2020.126311","article-title":"A perspective on multiparameter quantum metrology: From theoretical tools to applications in quantum imaging","volume":"384","author":"Albarelli","year":"2020","journal-title":"Phys. Lett. A"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Nakamura, K. (2019). Quantum Phononics, Springer.","DOI":"10.1007\/978-3-030-11924-9"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"023001","DOI":"10.1088\/1751-8121\/ab5d4d","article-title":"Quantum Fisher information matrix and multiparameter estimation","volume":"53","author":"Liu","year":"2019","journal-title":"J. Phys. Math. Theor."},{"key":"ref_17","unstructured":"Helstrom, C.W. (1976). Quantum Detection and Estimation Theory, Academic Press."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"032104","DOI":"10.1103\/PhysRevA.100.032104","article-title":"Optimal measurements for quantum multiparameter estimation with general states","volume":"100","author":"Yang","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"9852","DOI":"10.1038\/s41598-018-27362-9","article-title":"Uhlmann curvature in dissipative phase transitions","volume":"8","author":"Carollo","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s00220-019-03433-4","article-title":"Attaining the ultimate precision limit in quantum state estimation","volume":"368","author":"Yang","year":"2019","journal-title":"Commun. Math. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1140\/epjst\/e2012-01532-4","article-title":"Quantum optics in the phase space","volume":"203","author":"Olivares","year":"2012","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Serafini, A. (2017). Quantum Continuous Variables: A Primer of Theoretical Methods, CRC Press.","DOI":"10.1201\/9781315118727"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"163601","DOI":"10.1103\/PhysRevLett.124.163601","article-title":"Squeezing Phase Diffusion","volume":"124","author":"Cialdi","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"962","DOI":"10.1063\/1.1705306","article-title":"Exponential Operators and Parameter Differentiation in Quantum Physics","volume":"8","author":"Wilcox","year":"1967","journal-title":"J. Math. Phys."},{"key":"ref_25","unstructured":"Louisell, W.H. (1973). Quantum Statistical Properties of Radiation, Wiley."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/0375-9601(95)00235-U","article-title":"Small amount of squeezing in high-sensitive realistic interferometry","volume":"201i","author":"Paris","year":"1995","journal-title":"Phys. Lett. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"934","DOI":"10.1016\/j.physleta.2009.01.026","article-title":"Squeezed vacuum as a universal quantum probe","volume":"373","author":"Gaiba","year":"2009","journal-title":"Phys. Lett. A"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/10\/1353\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:16:16Z","timestamp":1760166976000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/10\/1353"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,16]]},"references-count":27,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2021,10]]}},"alternative-id":["e23101353"],"URL":"https:\/\/doi.org\/10.3390\/e23101353","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,10,16]]}}}