{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T01:23:51Z","timestamp":1779153831447,"version":"3.51.4"},"reference-count":58,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,8,30]],"date-time":"2024-08-30T00:00:00Z","timestamp":1724976000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"ART020-Quantum Technologies Project"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We study the chaotic motion of a semi-classical optomechanical system coupled to a non-Markovian environment with a finite correlation time. By studying the emergence of chaos using the Lyapunov exponent with the changing non-Markovian parameter, we show that the non-Markovian environment can significantly enhance chaos. It is observed that a non-Markovian environment characterized by the Ornstein\u2013Uhlenbeck type noise can modify the generation of chaos with different environmental memory times. As a comparison, the crossover properties from Markov to non-Markovian regimes are also discussed. Our findings indicate that the quantum memory effects on the onset of chaos may become a useful property to be investigated in quantum manipulations and control.<\/jats:p>","DOI":"10.3390\/e26090742","type":"journal-article","created":{"date-parts":[[2024,8,30]],"date-time":"2024-08-30T09:39:32Z","timestamp":1725010772000},"page":"742","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Chaos in Optomechanical Systems Coupled to a Non-Markovian Environment"],"prefix":"10.3390","volume":"26","author":[{"given":"Pengju","family":"Chen","sequence":"first","affiliation":[{"name":"Center for Quantum Science and Engineering, Department of Physics, Stevens Institute of Technology, Hoboken, NJ 07030, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nan","family":"Yang","sequence":"additional","affiliation":[{"name":"Center for Quantum Science and Engineering, Department of Physics, Stevens Institute of Technology, Hoboken, NJ 07030, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Austen","family":"Couvertier","sequence":"additional","affiliation":[{"name":"Center for Quantum Science and Engineering, Department of Physics, Stevens Institute of Technology, Hoboken, NJ 07030, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Quanzhen","family":"Ding","sequence":"additional","affiliation":[{"name":"Center for Quantum Science and Engineering, Department of Physics, Stevens Institute of Technology, Hoboken, NJ 07030, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rupak","family":"Chatterjee","sequence":"additional","affiliation":[{"name":"Deptartment of Applied Physics, New York University Tandon School of Engineering, Brooklyn, NY 11201, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ting","family":"Yu","sequence":"additional","affiliation":[{"name":"Center for Quantum Science and Engineering, Department of Physics, Stevens Institute of Technology, Hoboken, NJ 07030, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1391","DOI":"10.1103\/RevModPhys.86.1391","article-title":"Cavity optomechanics","volume":"86","author":"Aspelmeyer","year":"2014","journal-title":"Rev. Mod. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"167203","DOI":"10.1103\/PhysRevLett.98.167203","article-title":"Chaotic Quivering of Micron-Scaled On-Chip Resonators Excited by Centrifugal Optical Pressure","volume":"98","author":"Carmon","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"033812","DOI":"10.1103\/PhysRevA.92.033812","article-title":"Noise suppression of on-chip mechanical resonators by chaotic coherent feedback","volume":"92","author":"Yang","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1126\/science.aao0763","article-title":"Chaos-assisted broadband momentum transformation in optical microresonators","volume":"358","author":"Jiang","year":"2017","journal-title":"Science"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1038\/nphoton.2016.73","article-title":"Optomechanically induced stochastic resonance and chaos transfer between optical fields","volume":"10","author":"Monifi","year":"2016","journal-title":"Nat. Photon."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"021804(R)","DOI":"10.1103\/PhysRevA.84.021804","article-title":"Photonic Josephson effect, phase transitions, and chaos in optomechanical systems","volume":"84","author":"Larson","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"110406","DOI":"10.1103\/PhysRevLett.112.110406","article-title":"Nonlinear Dynamics and Quantum Entanglement in Optomechanical Systems","volume":"112","author":"Wang","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"015205","DOI":"10.1088\/1612-202X\/aca97a","article-title":"Macroscopic entanglement generation in optomechanical system embedded in non-Markovian environment","volume":"20","author":"Yang","year":"2022","journal-title":"Laser Phys. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"113029","DOI":"10.1088\/1367-2630\/18\/11\/113029","article-title":"Classical dynamical gauge fields in optomechanics","volume":"18","author":"Walter","year":"2016","journal-title":"New J. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"253601","DOI":"10.1103\/PhysRevLett.114.253601","article-title":"PT-Symmetry-Breaking Chaos in Optomechanics","volume":"114","author":"Jing","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"366","DOI":"10.1038\/nphoton.2016.67","article-title":"Vibrations copying optical chaos","volume":"10","author":"Sciamanna","year":"2016","journal-title":"Nat. Photon."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"14965","DOI":"10.1038\/ncomms14965","article-title":"Nonlinear dynamics and chaos in an optomechanical beam","volume":"8","author":"Capuj","year":"2017","journal-title":"Nat. Commun."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"134101","DOI":"10.1103\/PhysRevLett.103.134101","article-title":"Observation of an Exceptional Point in a Chaotic Optical Microcavity","volume":"103","author":"Lee","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3543","DOI":"10.1364\/OL.39.003543","article-title":"Chaotic oscillations of coupled nanobeam cavities with tailored optomechanical potentials","volume":"39","author":"Sun","year":"2014","journal-title":"Opt. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"163601","DOI":"10.1103\/PhysRevLett.115.163601","article-title":"Self-Ordered Limit Cycles, Chaos, and Phase Slippage with a Superfluid inside an Optical Resonator","volume":"115","author":"Piazza","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"053851","DOI":"10.1103\/PhysRevA.101.053851","article-title":"Intermittent chaos in cavity optomechanics","volume":"101","author":"Zhang","year":"2020","journal-title":"Phys. Rev. A"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Wang, G.L., Lai, Y.C., and Grebogi, C. (2016). Transient chaos\u2014A resolution of breakdown of quantum-classical correspondence in optomechanics. Sci. Rep., 6.","DOI":"10.1038\/srep35381"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"013601","DOI":"10.1103\/PhysRevLett.114.013601","article-title":"Route to chaos in optomechanics","volume":"114","author":"Bakemeier","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","unstructured":"Nakamura, K. (1993). Quantum Chaos: A New Paradigm of Nonlinear Dynamics, Cambridge University Press."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1515","DOI":"10.1103\/PhysRevLett.53.1515","article-title":"Bound-state eigenfunctions of classically chaotic Hamiltonian systems: Scars of periodic orbits","volume":"53","author":"Heller","year":"1984","journal-title":"Phys. Rev. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Heller, E.J. (2018). The Semiclassical Way to Dynamics and Spectroscopy, Princeton University Press.","DOI":"10.23943\/9781400890293"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Gutzwiller, M.C. (1990). Chaos in Classical and Quantum Mechanics, Springer.","DOI":"10.1007\/978-1-4612-0983-6"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"026001","DOI":"10.1088\/0034-4885\/71\/2\/026001","article-title":"Many-body physics and quantum chaos","volume":"71","author":"Ullmo","year":"2008","journal-title":"Rep. Prog. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Wright, M., and Weaver, R. (2010). New Directions in Linear Acoustics and Vibration: Quantum Chaos, Random Matrix Theory and Complexity, Cambridge University Press.","DOI":"10.1017\/CBO9780511781520"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Garc\u00eda-Mata, I., Jalabert, R.A., and Wisniacki, D.A. (2022). Out-of-time-order correlators and quantum chaos. arXiv.","DOI":"10.4249\/scholarpedia.55237"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"054220","DOI":"10.1103\/PhysRevE.107.054220","article-title":"Relative asymptotic oscillations of the out-of-time-ordered correlator as a quantum chaos indicator","volume":"107","year":"2023","journal-title":"Phys. Rev. E"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"013049","DOI":"10.1088\/1367-2630\/ab6522","article-title":"Nonlinear dynamics of weakly dissipative optomechanical systems","volume":"22","author":"Roque","year":"2020","journal-title":"New J. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Breuer, H., and Petruccione, F. (2002). The Theory of Open Quantum Systems, Oxford University Press.","DOI":"10.1007\/3-540-44874-8_4"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Weiss, U. (2012). Quantum Dissipative Systems, World Scientific.","DOI":"10.1142\/9789814374927"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"015001","DOI":"10.1103\/RevModPhys.89.015001","article-title":"Dynamics of non-Markovian open quantum systems","volume":"89","author":"Alonso","year":"2017","journal-title":"Rev. Mod. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"032410","DOI":"10.1103\/PhysRevA.107.032410","article-title":"Generating quantum entanglement between macroscopic objects with continuous measurement and feedback control","volume":"107","author":"Miki","year":"2023","journal-title":"Phys. Rev. A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"230312","DOI":"10.1007\/s11433-022-2043-3","article-title":"Phase-controlled asymmetric optomechanical entanglement against optical backscattering","volume":"66","author":"Liu","year":"2023","journal-title":"Sci. China Phys. Mech. Astron."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1801","DOI":"10.1103\/PhysRevLett.82.1801","article-title":"Open system dynamics with non-Markovian quantum trajectories","volume":"82","author":"Strunz","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1103\/PhysRevA.60.91","article-title":"Non-Markovian quantum-state diffusion: Perturbation approach","volume":"60","author":"Yu","year":"1999","journal-title":"Phys. Rev. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1699","DOI":"10.1103\/PhysRevA.58.1699","article-title":"Non-Markovian quantum state diffusion","volume":"58","author":"Gisin","year":"1998","journal-title":"Phys. Rev. A"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"052115","DOI":"10.1103\/PhysRevA.69.052115","article-title":"Convolutionless non-Markovian master equations and quantum trajectories: Brownian motion","volume":"69","author":"Strunz","year":"2004","journal-title":"Phys. Rev. A"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"062107","DOI":"10.1103\/PhysRevA.69.062107","article-title":"Non-Markovian quantum trajectories versus master equations: Finite-temperature heat bath","volume":"69","author":"Yu","year":"2004","journal-title":"Phys. Rev. A"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"240403","DOI":"10.1103\/PhysRevLett.105.240403","article-title":"Non-Markovian relaxation of a three-level system: Quantum trajectory approach","volume":"105","author":"Jing","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"040101","DOI":"10.1103\/PhysRevA.85.040101","article-title":"Nonadiabatic elimination of auxiliary modes in continuous quantum measurements","volume":"85","author":"Yang","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"052104","DOI":"10.1103\/PhysRevA.90.052104","article-title":"Exact non-Markovian master equations for multiple qubit systems: Quantum-trajectory approach","volume":"90","author":"Chen","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"435301","DOI":"10.1088\/1751-8113\/47\/43\/435301","article-title":"Perturbation methods for the non-Markovian quantum state diffusion equation","volume":"47","author":"Xu","year":"2014","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"580","DOI":"10.1103\/PhysRevLett.68.580","article-title":"Wave-function approach to dissipative processes in quantum optics","volume":"68","author":"Dalibard","year":"1992","journal-title":"Phys. Rev. Lett."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"5677","DOI":"10.1088\/0305-4470\/25\/21\/023","article-title":"The quantum-state diffusion model applied to open systems","volume":"25","author":"Gisin","year":"1992","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Carmichael, H. (1993). An Open Systems Approach to Quantum Optics Springer-Verlag, Springer.","DOI":"10.1007\/978-3-540-47620-7"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1652","DOI":"10.1103\/PhysRevA.47.1652","article-title":"Interpretation of quantum jump and diffusion processes illustrated on the Bloch sphere","volume":"47","author":"Wiseman","year":"1993","journal-title":"Phys. Rev. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1103\/RevModPhys.70.101","article-title":"The quantum-jump approach to dissipative dynamics in quantum optics","volume":"70","author":"Plenio","year":"1998","journal-title":"Rev. Mod. Phys."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"931","DOI":"10.1038\/nphys2085","article-title":"Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems","volume":"7","author":"Liu","year":"2011","journal-title":"Nat. Phys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"103901","DOI":"10.1103\/PhysRevLett.96.103901","article-title":"Dynamical Multistability Induced by Radiation Pressure in High-Finesse Micromechanical Optical Cavities","volume":"96","author":"Marquardt","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"095013","DOI":"10.1088\/1367-2630\/10\/9\/095013","article-title":"The optomechanical instability in the quantum regime","volume":"10","author":"Ludwig","year":"2008","journal-title":"New J. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/0167-2789(85)90011-9","article-title":"Determining Lyapunov exponents from a time series","volume":"16","author":"Wolf","year":"1985","journal-title":"Phys. D"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"012334","DOI":"10.1103\/PhysRevA.94.012334","article-title":"Memory-effect-induced macroscopic-microscopic entanglement","volume":"94","author":"Mu","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"031803","DOI":"10.1103\/PhysRevA.83.031803","article-title":"Fast nondeterministic random-bit generation using on-chip chaos lasers","volume":"83","author":"Harayama","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"4590","DOI":"10.1038\/s41467-023-40152-w","article-title":"Harnessing microcomb-based parallel chaos for random number generation and optical decision making","volume":"14","author":"Shen","year":"2023","journal-title":"Nat. Commun."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1900270","DOI":"10.1002\/andp.201900270","article-title":"Out-of-Time-Order Correlators and Quantum Phase Transitions in the Rabi and Dicke Models","volume":"532","author":"Sun","year":"2020","journal-title":"Ann. Phys."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"024101","DOI":"10.1103\/PhysRevLett.122.024101","article-title":"Quantum and classical Lyapunov exponents in atom-field interaction systems","volume":"122","author":"Santos","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"080601","DOI":"10.1103\/PhysRevLett.118.080601","article-title":"Nonergodicity in the anisotropic dicke model","volume":"118","author":"Buijsman","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"013722","DOI":"10.1103\/PhysRevA.104.013722","article-title":"Beyond the Rabi model: Light interactions with polar atomic systems in a cavity","volume":"104","author":"Scala","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Burgarth, D., Facchi, P., Hillier, R., and Ligab\u00f2, M. (2023). Taming the Rotating Wave Approximation. arXiv.","DOI":"10.22331\/q-2024-02-21-1262"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/9\/742\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:45:50Z","timestamp":1760111150000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/9\/742"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,30]]},"references-count":58,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["e26090742"],"URL":"https:\/\/doi.org\/10.3390\/e26090742","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,30]]}}}