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Our problem on the other hand, is to map classical dynamics to some analogous version in quantum theory.","type":"other"},{"key":"l54l-sff5Cc14R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.shpsa.2022.09.001","type":"journal-article"},{"key":"l54l-sff5Cc15R1","doi-asserted-by":"publisher","DOI":"10.1142\/S0129055X05002376","type":"journal-article"},{"key":"l54l-sff5Cc16R1","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4614-7116-5","type":"book","volume-title":"Quantum Theory for Mathematicians","author":"B.\u00a0C. Hall","year":"2013"},{"key":"l54l-sff5Cc17R1","volume-title":"Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering","author":"S.\u00a0H. Strogatz","year":"2015","type":"book"},{"key":"l54l-sff5Cc18R1","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780199208241.001.0001","type":"book","volume-title":"Nonlinear Ordinary Differential Equations: An introduction for Scientists and Engineers","author":"D.\u00a0W. Jordan","year":"2007"},{"key":"l54l-sff5Cc19R1","unstructured":"The necessity and sufficiency of   (2) can be proved by construction. That is, if we know the system to be Hamiltonian, then we also know how to find H(x,y).","type":"other"},{"key":"l54l-sff5Cc20R1","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1925.0150","type":"journal-article"},{"key":"l54l-sff5Cc21R1","volume-title":"The Principles of Quantum Mechanics","author":"P.\u00a0A.\u00a0M. Dirac","year":"1981","type":"book"},{"key":"l54l-sff5Cc22R1","first-page":"49","type":"journal-article","volume":"63","author":"P.\u00a0A.\u00a0M. Dirac","year":"1963","journal-title":"Proc. R. Ir. Acad. A"},{"key":"l54l-sff5Cc23R1","volume-title":"Quantum Mechanics","author":"J.-L. 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For example, if f(x,y)=xy2, then f\u00af(x\u0302,y\u0302)=(x\u0302y\u03022+y\u03022x\u0302)\/2 and f\u00af(x\u0302,y\u0302)=(x\u0302y\u03022+y\u0302x\u0302y\u0302+y\u03022x\u0302)\/3 are both suitable mappings of f(x,y) (the latter being Weyl ordering   [15, 16]), but not f\u00af(x\u0302,y\u0302)=x\u0302y\u03022.","type":"other"},{"key":"l54l-sff5Cc28R1","unstructured":"Our method works for any sign of \u2202f\/\u2202x+\u2202g\/\u2202y. However, because of the omnipresence of damping forces in practice, the literature has focused mostly on dissipative systems, which are defined by \u2202f\/\u2202x+\u2202g\/\u2202y<0. Aside from designing realistic models, dissipative effects also play an essential role in stabilizing dynamical systems.","type":"other"},{"key":"l54l-sff5Cc29R1","doi-asserted-by":"publisher","DOI":"10.1016\/0370-1573(81)90033-8","type":"journal-article"},{"key":"l54l-sff5Cc30R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.474887","type":"journal-article"},{"key":"l54l-sff5Cc31R1","doi-asserted-by":"publisher","DOI":"10.1016\/S0370-1573(01)00077-1","type":"journal-article"},{"key":"l54l-sff5Cc32R1","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-09649-9","type":"book","volume-title":"Quantum\u2013Classical Correspondence: Dynamical Quantization and the Classical Limit","author":"A.\u00a0O. Bolivar","year":"2004"},{"key":"l54l-sff5Cc33R1","doi-asserted-by":"publisher","DOI":"10.1146\/annurev.physchem.55.091602.094425","type":"journal-article"},{"key":"l54l-sff5Cc34R1","volume-title":"Quantum Mechanics of Non-Hamiltonian and Dissipative Systems","author":"V.\u00a0E. 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Breuer","year":"2002","type":"book"},{"key":"l54l-sff5Cc51R1","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-23354-8","type":"book","volume-title":"Open Quantum Systems, An Introduction","author":"A. 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In this approach, one starts with a Hamiltonian that prescribes how the system interacts with a bath (another system with an infinite number of degrees of freedom, but whose dynamics are not of interest)   [63, 64]. Then, by invoking the rotating-wave and Born\u2013Markov approximations, a physically valid equation\u00a0of motion for the system's state is derived. These steps now constitute the so-called \u201cmicroscopic\u201d derivation of master equations\u00a0in open-systems theory   [50, 51].","type":"other"},{"key":"l54l-sff5Cc63R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.119.670","type":"journal-article"},{"key":"l54l-sff5Cc64R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.124.642","type":"journal-article"},{"key":"l54l-sff5Cc65R1","unstructured":"The word microscopic here has a different meaning to its usage in the previous footnote, where we mentioned the \u201cmicroscopic\u201d derivation of master equations. Here, we are referring to the real inter-particle interactions that occur between the system and its surrounding. In contrast, the term microscopic in the previous footnote merely emphasizes the possibility of deriving a Markovian master equation\u00a0from a system-bath Hamiltonian. Whether the system-bath Hamiltonian corresponds to anything real is generally not of concern. Thus, the \u201cmicroscopic\u201d derivation of master equations\u00a0as we explained in the previous footnote can be understood as a formal procedure   [50, 51], and writing down a system-bath Hamiltonian such that it leads to a desired master equation\u00a0is just as phenomenological as writing down the master equation\u00a0directly.","type":"other"},{"key":"l54l-sff5Cc66R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.53.1890","type":"journal-article"},{"key":"l54l-sff5Cc67R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.aop.2016.02.005","type":"journal-article"},{"key":"l54l-sff5Cc68R1","doi-asserted-by":"crossref","DOI":"10.1201\/b19379","type":"book","volume-title":"Quantum Optomechanics","author":"W.\u00a0P. 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See also Refs.\u00a0  [87, 88, 89] for a historical account.","type":"other"},{"key":"l54l-sff5Cc87R1","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-35.3.367","type":"journal-article"},{"key":"l54l-sff5Cc88R1","volume-title":"Technological Concepts and Mathematical Models in the Evolution of Modern Engineering Systems,","author":"G. Israel","year":"2004","type":"book"},{"key":"l54l-sff5Cc89R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.3670008","type":"journal-article"},{"key":"l54l-sff5Cc90R1","volume-title":"Classical Mechanics","author":"H. Goldstein","year":"2002","type":"book"},{"key":"l54l-sff5Cc91R1","volume-title":"Nonlinear Dynamics and Chaos","author":"J.\u00a0M.\u00a0T. Thompson","year":"2002","type":"book"},{"key":"l54l-sff5Cc92R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.3.013130","type":"journal-article"},{"key":"l54l-sff5Cc93R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevApplied.12.014037","type":"journal-article"},{"key":"l54l-sff5Cc94R1","doi-asserted-by":"publisher","DOI":"10.22331\/q-2020-01-20-224","type":"journal-article"},{"key":"l54l-sff5Cc95R1","unstructured":"Normal ordering is an ambiguous operation unless it is specified explicitly. In particular, it is not a linear operation as can be seen by considering a\u0302a\u0302\u2020. By definition, :a\u0302a\u0302\u2020:=a\u0302\u2020a\u0302. But, since a\u0302a\u0302\u2020=1\u0302+a\u0302\u2020a\u0302, we also have :a\u0302a\u0302\u2020:=:1\u0302+a\u0302\u2020a\u0302:. If normal ordering is linear, and defining :1\u0302:=1\u0302, we then get :a\u0302a\u0302\u2020:=1\u0302+a\u0302\u2020a\u0302\u2260a\u0302\u2020a\u0302. Equation\u00a0  (9) is therefore necessary for   (7) to be well defined. We can also define :h(a\u0302,a\u0302\u2020): by first writing every \u03b1* to the left of \u03b1 in h(\u03b1,\u03b1*), and then letting \u03b1\u27f6a\u0302 and \u03b1*\u27f6a\u0302\u2020.","type":"other"},{"key":"l54l-sff5Cc96R1","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781139035170","type":"book","volume-title":"Quantum Optics","author":"G.\u00a0S. Agarwal","year":"2012"},{"key":"l54l-sff5Cc97R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.156.286","type":"journal-article"},{"key":"l54l-sff5Cc98R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.125.163603","type":"journal-article"},{"key":"l54l-sff5Cc99R1","doi-asserted-by":"publisher","DOI":"10.21468\/SciPostPhys.15.3.121","type":"journal-article"},{"key":"l54l-sff5Cc100R1","doi-asserted-by":"crossref","DOI":"10.1142\/4309","type":"book","volume-title":"The Inverse Variational Problem in Classical Mechanics","author":"J. \u0141opusza\u0144ski","year":"1999"},{"key":"l54l-sff5Cc101R1","doi-asserted-by":"publisher","DOI":"10.1119\/1.11228","type":"journal-article"},{"key":"l54l-sff5Cc102R1","doi-asserted-by":"publisher","DOI":"10.1119\/1.1475331","type":"journal-article"},{"key":"l54l-sff5Cc103R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.2821612","type":"journal-article"},{"key":"l54l-sff5Cc104R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2007.06.076","type":"journal-article"},{"key":"l54l-sff5Cc105R1","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8113\/43\/17\/175205","type":"journal-article"},{"key":"l54l-sff5Cc106R1","doi-asserted-by":"publisher","DOI":"10.3390\/math11122671","type":"journal-article"},{"key":"l54l-sff5Cc107R1","unstructured":"It is well known in classical mechanics that for a single degree of freedom x, the Euler\u2013Lagrange equation\u00a0\u2202L\/\u2202x=(\u2202L\/\u2202x\u2032)\u2032 does not change if L(x,x\u2032,t)\u27f6\u03bbL(x,x\u2032,t) for \u03bb a constant, or, L(x,x\u2032,t)\u27f6L(x,x\u2032,t)+dF\/dt for F an arbitrary function of x and t, but not x\u2032.","type":"other"},{"key":"l54l-sff5Cc108R1","volume-title":"Decoherence and the Quantum-to-Classical Transition","author":"M. Schlosshauer","year":"2007","type":"book"},{"key":"l54l-sff5Cc109R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.physrep.2019.10.001","type":"journal-article"},{"key":"l54l-sff5Cc110R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.103.012118","type":"journal-article"},{"key":"l54l-sff5Cc111R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.111.234101","type":"journal-article"},{"key":"l54l-sff5Cc112R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.112.094102","type":"journal-article"},{"key":"l54l-sff5Cc113R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.90.052904","type":"journal-article"},{"key":"l54l-sff5Cc114R1","doi-asserted-by":"publisher","DOI":"10.1088\/0953-4075\/45\/15\/154001","type":"journal-article"},{"key":"l54l-sff5Cc115R1","volume-title":"Quantum Noise","author":"C.\u00a0W. Gardiner","year":"2004","type":"book"},{"key":"l54l-sff5Cc116R1","doi-asserted-by":"crossref","DOI":"10.1090\/mbk\/082","type":"book","volume-title":"An Introduction to Stochastic Differential Equations","author":"L.\u00a0C. Evans","year":"2013"},{"key":"l54l-sff5Cc117R1","doi-asserted-by":"crossref","DOI":"10.1017\/9781108186735","type":"book","volume-title":"Applied Stochastic Differential Equations","author":"S. S\u00e4rkk\u00e4","year":"2019"},{"key":"l54l-sff5Cc118R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.72.066203","type":"journal-article"},{"key":"l54l-sff5Cc119R1","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8113\/45\/38\/382002","type":"journal-article"},{"key":"l54l-sff5Cc120R1","unstructured":"One possible reason for using a second-order differential equation\u00a0to define a system is Newton's second law. But in nonlinear models, x and y often represent nonmechanical properties and have no direct relation to Newton's second law. For example, nonlinear oscillator models are commonly realised using electrical circuits. In this case x and y obey the familiar circuit laws for voltages and currents. Another example is chemical reactions. In this case, one derives the equations\u00a0of motion for x and y using the principles of chemical kinetics such as the law of mass action. The resulting model then describes how the concentration of reactant molecules evolve. Even more generally, nonlinear models are sometimes introduced phenomenologically without appealing to any physical principles.","type":"other"},{"key":"l54l-sff5Cc121R1","doi-asserted-by":"publisher","DOI":"10.1080\/00107514.2011.642554","type":"journal-article"},{"key":"l54l-sff5Cc122R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.79.3101","type":"journal-article"},{"key":"l54l-sff5Cc122R2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.80.5702","type":"journal-article"},{"key":"l54l-sff5Cc122R3","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.80.5703","type":"journal-article"},{"key":"l54l-sff5Cc123R1","volume-title":"Chemical Oscillations, Waves, and Turbulence","author":"Y. Kuramoto","year":"2003","type":"book"},{"key":"l54l-sff5Cc124R1","unstructured":"One might be tempted to think that a circular limit cycle implies the existence of rotational symmetry in the system, as in the case of the Stuart\u2013Landau oscillator. However, this is not the case. An example of a system with a circular limit cycle that is not rotationally symmetric is x\u2032=y and y\u2032=\u2212x+\u03bby\u2212\u03bb(x2+y2)y for \u03bb a real positive constant   [17]. We will in fact quantize an example of a circular but rotationally asymmetric limit cycle in Sec.\u00a0  4e. We will then see how such quantum limit cycles are represented in quantum phase space by Wigner functions.","type":"other"},{"key":"l54l-sff5Cc125R1","doi-asserted-by":"publisher","DOI":"10.1002\/andp.201400144","type":"journal-article"},{"key":"l54l-sff5Cc126R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.89.022913","type":"journal-article"},{"key":"l54l-sff5Cc127R1","doi-asserted-by":"publisher","DOI":"10.1088\/1742-5468\/2015\/09\/P09009","type":"journal-article"},{"key":"l54l-sff5Cc128R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.117.073601","type":"journal-article"},{"key":"l54l-sff5Cc129R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.96.052210","type":"journal-article"},{"key":"l54l-sff5Cc130R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.120.163601","type":"journal-article"},{"key":"l54l-sff5Cc131R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.97.052203","type":"journal-article"},{"key":"l54l-sff5Cc132R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.102.062205","type":"journal-article"},{"key":"l54l-sff5Cc133R1","doi-asserted-by":"publisher","DOI":"10.1088\/1367-2630\/abf1d7","type":"journal-article"},{"key":"l54l-sff5Cc134R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.104.024214","type":"journal-article"},{"key":"l54l-sff5Cc135R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41598-022-19010-0","type":"journal-article"},{"key":"l54l-sff5Cc136R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.106.024216","type":"journal-article"},{"key":"l54l-sff5Cc137R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.107.024204","type":"journal-article"},{"key":"l54l-sff5Cc138R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.2.013233","type":"journal-article"},{"key":"l54l-sff5Cc139R1","doi-asserted-by":"publisher","DOI":"10.1103\/f97j-474c","type":"journal-article"},{"key":"l54l-sff5Cc141R1","doi-asserted-by":"publisher","DOI":"10.1126\/sciadv.ady5649","type":"journal-article"},{"key":"l54l-sff5Cc142R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.2.033422","type":"journal-article"},{"key":"l54l-sff5Cc143R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.73.023609","type":"journal-article"},{"key":"l54l-sff5Cc144R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.70.022303","type":"journal-article"},{"key":"l54l-sff5Cc145R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.81.043805","type":"journal-article"},{"key":"l54l-sff5Cc146R1","doi-asserted-by":"publisher","DOI":"10.1002\/andp.201600402","type":"journal-article"},{"key":"l54l-sff5Cc147R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41598-019-53526-2","type":"journal-article"},{"key":"l54l-sff5Cc148R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.73.063801","type":"journal-article"},{"key":"l54l-sff5Cc149R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.physleta.2006.01.028","type":"journal-article"},{"key":"l54l-sff5Cc150R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.83.056202","type":"journal-article"},{"key":"l54l-sff5Cc151R1","doi-asserted-by":"publisher","DOI":"10.1140\/epjqt7","type":"journal-article"},{"key":"l54l-sff5Cc152R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41534-017-0019-1","type":"journal-article"},{"key":"l54l-sff5Cc153R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.108.104313","type":"journal-article"},{"key":"l54l-sff5Cc154R1","unstructured":"The steady-state Wigner function being given by   (42) but with \u03c1 replaced by a steady-state density operator \u03c1ss (defined by L\u03c1ss=0). Alternatively, one may also define it by the long-time limit of   (42) if \u03c1 is a time-dependent state.","type":"other"},{"key":"l54l-sff5Cc155R1","doi-asserted-by":"publisher","DOI":"10.1016\/0096-3003(90)90051-4","type":"journal-article"},{"key":"l54l-sff5Cc156R1","volume-title":"Diffusion Processes and Related Problems in Analysis","author":"L. Arnold","year":"1992","type":"book"},{"key":"l54l-sff5Cc157R1","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127496001272","type":"journal-article"},{"key":"l54l-sff5Cc158R1","volume-title":"Stochastic Dynamics","author":"Y. Liang","year":"1999","type":"book"},{"key":"l54l-sff5Cc159R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.81.011106","type":"journal-article"},{"key":"l54l-sff5Cc160R1","volume-title":"Noise-Induced Transitions","author":"W. Horsthemke","year":"2006","type":"book"},{"key":"l54l-sff5Cc161R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cpc.2012.02.021","type":"journal-article"},{"key":"l54l-sff5Cc162R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cpc.2012.11.019","type":"journal-article"},{"key":"l54l-sff5Cc164R1","doi-asserted-by":"publisher","DOI":"10.1103\/PRXQuantum.3.010329","type":"journal-article"},{"key":"l54l-sff5Cc165R1","unstructured":"The direction of phase-space rotation is also different\u2014clockwise in (35), (36), and counterclockwise in (55), (56)\u2014but this is a trivial modification.","type":"other"},{"key":"l54l-sff5Cc166R1","unstructured":"Although the Hamiltonian is unbounded from below, the Lindbladian can be physically realized in a suitable rotating frame.","type":"other"},{"key":"l54l-sff5Cc167R1","unstructured":"Equation\u00a0  (57) describes any limit cycle just after the onset of a supercritical Hopf bifurcation. Since many nonlinear systems other than the van der Pol oscillator have limit cycles created from a Hopf bifurcation, one cannot uniquely associate the van der Pol oscillator with   (57). For this reason, and appropriately so, the nonlinear dynamics literature refers to   (57) as the Stuart\u2013Landau oscillator, which is also the terminology used here. In fact, historically, and as we already hinted in Sec.\u00a0  3b3 under   (37), the van der Pol oscillator is a prototype for relaxation oscillations. These oscillations exist far from the Hopf bifurcation point, and as such, they are noncircular and non-uniformly traversed orbits in phase space. In fact, older publications in physics (as far back as the 1960s to our knowledge) actually considered the difference between the Stuart\u2013Landau and van der Pol oscillators by referring to the former as a rotating-wave van der Pol oscillator   [168, 169, 170]. See Ref.\u00a0  [171] for a pedagogical account of how the rotating-wave approximation is used to reduce the van der Pol oscillator to the Stuart\u2013Landau form. Essentially, the same method goes by the name of Krylov\u2013Bogoliubov time averaging in nonlinear dynamics   [172].","type":"other"},{"key":"l54l-sff5Cc168R1","doi-asserted-by":"publisher","DOI":"10.1109\/JQE.1967.1074446","type":"journal-article"},{"key":"l54l-sff5Cc169R1","volume-title":"Quantum Statistical Properties of Radiation","author":"W.\u00a0H. Louisell","year":"1973","type":"book"},{"key":"l54l-sff5Cc170R1","doi-asserted-by":"crossref","DOI":"10.1093\/acprof:oso\/9780198567769.001.0001","type":"book","volume-title":"Random Processes in Physics and Finance","author":"M. Lax","year":"2006"},{"key":"l54l-sff5Cc171R1","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-61544-3","type":"book","volume-title":"The Fokker\u2013Planck Equation: Methods of Solution and Applications","author":"H. Risken","year":"1996"},{"key":"l54l-sff5Cc172R1","volume-title":"Synchronization: From Simple to Complex","author":"A. Balanov","year":"2009","type":"book"},{"key":"l54l-sff5Cc173R1","volume-title":"Noise in Nonlinear Dynamical Systems","author":"L. Pontryagin","year":"1989","type":"book"},{"key":"l54l-sff5Cc174R1","doi-asserted-by":"publisher","DOI":"10.1016\/0020-7462(96)00030-3","type":"journal-article"},{"key":"l54l-sff5Cc175R1","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6544\/aad208","type":"journal-article"},{"key":"l54l-sff5Cc176R1","doi-asserted-by":"publisher","DOI":"10.1103\/vtt6-qr94","type":"journal-article"},{"key":"l54l-sff5Cc177R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.123.250401","type":"journal-article"},{"key":"l54l-sff5Cc178R1","volume-title":"Noise in Nonlinear Dynamical Systems","year":"1989","type":"book"},{"key":"l54l-sff5Cc179R1","volume-title":"Nonlinear Dynamics of Chaotic and Stochastic Systems","author":"V.\u00a0S. Anishchenko","year":"2007","type":"book"},{"key":"l54l-sff5Cc180R1","volume-title":"An Introduction to Stochastic Dynamics","author":"J. Duan","year":"2015","type":"book"},{"key":"l54l-sff5Cc181R1","doi-asserted-by":"publisher","DOI":"10.1016\/S0006-3495(61)86902-6","type":"journal-article"},{"key":"l54l-sff5Cc182R1","doi-asserted-by":"publisher","DOI":"10.1109\/JRPROC.1962.288235","type":"journal-article"},{"key":"l54l-sff5Cc183R1","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719147","type":"book","volume-title":"Mathematical Models in Biology","author":"L. Edelstein-Keshet","year":"2005"},{"key":"l54l-sff5Cc184R1","volume-title":"Mathematical Models in the Biosciences 1","author":"M. Frame","year":"2021","type":"book"},{"key":"l54l-sff5Cc185R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.79.829","type":"journal-article"},{"key":"l54l-sff5Cc186R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.78.775","type":"journal-article"},{"key":"l54l-sff5Cc187R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.61.6103","type":"journal-article"},{"key":"l54l-sff5Cc188R1","unstructured":"Or in terms of stochastic differential equations, we are assuming \u03a5(t)dt to be of order dt in   (72) and   (75).","type":"other"},{"key":"l54l-sff5Cc189R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01044713","type":"journal-article"},{"key":"l54l-sff5Cc190R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.71.807","type":"journal-article"},{"key":"l54l-sff5Cc191R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.50.3508","type":"journal-article"},{"key":"l54l-sff5Cc192R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.57.3292","type":"journal-article"},{"key":"l54l-sff5Cc193R1","doi-asserted-by":"publisher","DOI":"10.1364\/OE.26.032433","type":"journal-article"},{"key":"l54l-sff5Cc194R1","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511755743","type":"book","volume-title":"Synchronization: A Universal Concept in Nonlinear Sciences","author":"A. Pikovsky","year":"2001"},{"key":"l54l-sff5Cc195R1","unstructured":"The symbol \u2203! (used in mathematics and logic), is the quantifier for unique existence. In words, (iv)\u2013(vi) says V(x) is such that it has exactly one positive root x0; is negative for 0<x<x0; is positive and monotonically increasing for x>x0.","type":"other"},{"key":"l54l-sff5Cc196R1","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-42-00928-1","type":"journal-article"},{"key":"l54l-sff5Cc197R1","unstructured":"We refer the reader to the literature (see e.g., Refs.\u00a0  [196, 198]) for more details as we will not be quantizing such a system here.","type":"other"},{"key":"l54l-sff5Cc198R1","doi-asserted-by":"crossref","DOI":"10.1142\/2778","type":"book","volume-title":"Oscillations in Planar Dynamic Systems","author":"R.\u00a0E. Mickens","year":"1996"},{"key":"l54l-sff5Cc199R1","unstructured":"If we restrict to rotationally symmetric systems then a quantum oscillator with two limit cycles has been studied in Ref.\u00a0  [139].","type":"other"},{"key":"l54l-sff5Cc200R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2020.105234","type":"journal-article"},{"key":"l54l-sff5Cc201R1","unstructured":"The results in Ref.\u00a0  [41] are entirely classical, for which the authors reported the following reasons: (i) Nonlinear and non-Hamiltonian systems are difficult to quantize, and (ii) a working Hamiltonian will most likely be required as a first step towards quantization.","type":"other"},{"key":"l54l-sff5Cc202R1","unstructured":"We note that   (99) and   (100) are in fact special cases of the results in Ref.\u00a0  [41], though this is not important to our critique here.","type":"other"},{"key":"l54l-sff5Cc203R1","unstructured":"See (14) and the commentary under (17) in Ref.\u00a0  [41] for the actual time dependence. We have also simulated   (102)\u2013  (105) directly to check the unboundedness in y and q.","type":"other"},{"key":"l54l-sff5Cc204R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.87.062109","type":"journal-article"},{"key":"l54l-sff5Cc205R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.109.054207","type":"journal-article"},{"key":"l54l-sff5Cc206R1","unstructured":"Nevertheless, there is no contradiction between   (2) and   (110),   (111), because the origin is a center only locally. That is, not all phase-space points follow non-isolated closed paths around the origin. Some in fact tend to infinity. In short, nonconservative systems, as defined by   (2) in Sec.\u00a0  1, need not have attractors.","type":"other"},{"key":"l54l-sff5Cc207R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.27.7547","type":"journal-article"},{"key":"l54l-sff5Cc208R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.31.2294","type":"journal-article"},{"key":"l54l-sff5Cc209R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.109.190403","type":"journal-article"},{"key":"l54l-sff5Cc210R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.120.230404","type":"journal-article"},{"key":"l54l-sff5Cc211R1","unstructured":"A similar technique has been used to quantize a system with a specific y\u0302 dependence in \u2329y\u0302\u232a\u2032   [212].","type":"other"},{"key":"l54l-sff5Cc212R1","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jpclett.6b00498","type":"journal-article"},{"key":"l54l-sff5Cc213R1","doi-asserted-by":"publisher","DOI":"10.1016\/S0375-9601(01)00548-5","type":"journal-article"},{"key":"l54l-sff5Cc214R1","unstructured":"This is the same as (30) of Ref.\u00a0  [213] for n=1, with (q1,p1)=(x,y), \u03b2111=0, and \u03b111=\u03b3. The quantized generator is therefore given by (34) of Ref.\u00a0  [213].","type":"other"},{"key":"l54l-sff5Cc215R1","unstructured":"As with operators, given any superoperator K, we define K\u2020 by \u2329A\u0302,KB\u0302\u232a=\u2329K\u2020A\u0302,B\u0302\u232a for any two operators A\u0302 and B\u0302, and inner product \u2329A\u0302,B\u0302\u232a. To obtain   (128) and   (129), we have used the Hilbert-Schmidt inner product, given by \u2329A\u0302,B\u0302\u232a=Tr[A\u0302\u2020B\u0302].","type":"other"},{"key":"l54l-sff5Cc216R1","unstructured":"Note that cascade quantization operates in phase space, which is always 2n-dimensional.","type":"other"},{"key":"l54l-sff5Cc217R1","doi-asserted-by":"publisher","DOI":"10.1088\/1367-2630\/16\/4\/045014","type":"journal-article"},{"key":"l54l-sff5Cc218R1","doi-asserted-by":"publisher","DOI":"10.1126\/science.aaa2085","type":"journal-article"},{"key":"l54l-sff5Cc219R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-020-0824-x","type":"journal-article"},{"key":"l54l-sff5Cc220R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevX.15.011070","type":"journal-article"},{"key":"l54l-sff5Cc221R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.64.012310","type":"journal-article"},{"key":"l54l-sff5Cc222R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevX.15.011011","type":"journal-article"},{"key":"l54l-sff5Cc224R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-023-38217-x","type":"journal-article"},{"key":"l54l-sff5Cc225R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.98.042118","type":"journal-article"},{"key":"l54l-sff5Cc226R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.58.1098","type":"journal-article"},{"key":"l54l-sff5Cc227R1","doi-asserted-by":"crossref","DOI":"10.1201\/9781003250975","type":"book","volume-title":"Quantum Continuous Variables\u2014A Primer of Theoretical Methods","author":"A. Serafini","year":"2023"},{"key":"l54l-sff5Cc228R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.111.043709","type":"journal-article"},{"key":"l54l-sff5Cc229R1","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-05428-0","type":"book","volume-title":"Quantum Signitures of Chaos","author":"F. Haake","year":"2010"},{"key":"l54l-sff5Cc230R1","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-06343-0","type":"book","volume-title":"Nonlinear Dynamics and Quantum Chaos: An Introduction","author":"S. Wimberger","year":"2014"},{"key":"l54l-sff5Cc231R1","doi-asserted-by":"publisher","DOI":"10.1088\/0034-4885\/67\/1\/R02","type":"journal-article"},{"key":"l54l-sff5Cc232R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.72.1947","type":"journal-article"},{"key":"l54l-sff5Cc233R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.76.1611","type":"journal-article"},{"key":"l54l-sff5Cc234R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.54.1390","type":"journal-article"},{"key":"l54l-sff5Cc235R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.63.022102","type":"journal-article"},{"key":"l54l-sff5Cc236R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-018-0412-5","type":"journal-article"},{"key":"l54l-sff5Cc237R1","doi-asserted-by":"publisher","DOI":"10.1126\/sciadv.abg2616","type":"journal-article"},{"key":"l54l-sff5Cc238R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.6.L042046","type":"journal-article"},{"key":"l54l-sff5Cc239R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.108.032219","type":"journal-article"},{"key":"l54l-sff5Cc240R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.92.062924","type":"journal-article"},{"key":"l54l-sff5Cc241R1","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.17.5.311","type":"journal-article"},{"key":"l54l-sff5Cc242R1","unstructured":"Bateman originally proposed the Lagrangian L=x\u2032q\u2032\u2212\u03b3x\u2032q\u2212\u03c902xq   [37], which omits the xq\u2032 term as in   (A2). Its Euler\u2013Lagrange equation\u00a0can be easily seen to recover   (A1). However, the Lagrangian in the form of   (A2) is commonly found in more modern literature.","type":"other"},{"key":"l54l-sff5Cc243R1","volume-title":"Methods of Theoretical Physics","author":"P.\u00a0M. Morse","year":"1953","type":"book"},{"key":"l54l-sff5Cc244R1","unstructured":"See also Ref.\u00a0  [245], which refers to   (A3) as Feshbach's Hamiltonian.","type":"other"},{"key":"l54l-sff5Cc245R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.523718","type":"journal-article"},{"key":"l54l-sff5Cc246R1","first-page":"67","type":"journal-article","author":"F. Bopp","year":"1973","journal-title":"Bayer. Akad. Wiss. Math.-Natur. Klasse"},{"key":"l54l-sff5Cc247R1","doi-asserted-by":"publisher","DOI":"10.1111\/j.2164-0947.1977.tb02946.x","type":"journal-article"},{"key":"l54l-sff5Cc248R1","unstructured":"It is simple to derive   (A7) by substituting q=xexp(\u03b3t) into Bateman's original version of the dual Lagrangian, L=\u2212x\u2032q\u2032+\u03b3x\u2032q+\u03c902xq, and noting that L and L\/2 have the same Euler\u2013Lagrange equation. The factor of 1\/2 is chosen so that both x2 and y2 are multiplied by 1\/2 in H, as it appears in   (A8).","type":"other"},{"key":"l54l-sff5Cc249R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02960144","type":"journal-article"},{"key":"l54l-sff5Cc250R1","doi-asserted-by":"publisher","DOI":"10.1143\/ptp\/3.4.440","type":"journal-article"},{"key":"l54l-sff5Cc251R1","doi-asserted-by":"publisher","DOI":"10.1088\/0370-1328\/72\/6\/311","type":"journal-article"},{"key":"l54l-sff5Cc252R1","doi-asserted-by":"publisher","DOI":"10.1139\/p58-038","type":"journal-article"},{"key":"l54l-sff5Cc253R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.5045621","type":"journal-article"},{"key":"l54l-sff5Cc254R1","unstructured":"This is a simple consequence of the relationship between the canonical and mechanical momentum already present in the classical theory, given by x\u2032=yexp(\u2212\u03b3t)   [250].","type":"other"},{"key":"l54l-sff5Cc255R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.77.396","type":"journal-article"},{"key":"l54l-sff5Cc256R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.524148","type":"journal-article"},{"key":"l54l-sff5Cc257R1","doi-asserted-by":"publisher","DOI":"10.1119\/1.11767","type":"journal-article"},{"key":"l54l-sff5Cc258R1","unstructured":"A Lagrangian is standard if it has the form of kinetic energy minus the potential energy. Note there is also an ensuing critique of Ray by Kobe and colleagues in Ref.\u00a0  [259].","type":"other"},{"key":"l54l-sff5Cc259R1","doi-asserted-by":"publisher","DOI":"10.1119\/1.14840","type":"journal-article"},{"key":"l54l-sff5Cc260R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.1678812","type":"journal-article"},{"key":"l54l-sff5Cc261R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01010029","type":"journal-article"},{"key":"l54l-sff5Cc262R1","doi-asserted-by":"publisher","DOI":"10.1016\/0370-2693(75)90283-X","type":"journal-article"},{"key":"l54l-sff5Cc263R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.522431","type":"journal-article"},{"key":"l54l-sff5Cc264R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.55.3581","type":"journal-article"},{"key":"l54l-sff5Cc265R1","doi-asserted-by":"publisher","DOI":"10.1088\/0370-1328\/77\/2\/336","type":"journal-article"},{"key":"l54l-sff5Cc266R1","doi-asserted-by":"publisher","DOI":"10.1016\/0375-9601(80)90048-1","type":"journal-article"},{"key":"l54l-sff5Cc267R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01313310","type":"journal-article"},{"key":"l54l-sff5Cc268R1","doi-asserted-by":"publisher","DOI":"10.1016\/0378-4371(79)90057-8","type":"journal-article"},{"key":"l54l-sff5Cc269R1","unstructured":"To preempt our readers who are familiar with quantum-trajectory theory   [70, 270], Dekker's approach does not correspond to the non-Hermitian evolution prescribed by the jump trajectories. The evolution under a non-Hermitian Hamiltonian in quantum trajectories is conditioned on null measurements, and therefore does not have a Lindblad form.","type":"other"},{"key":"l54l-sff5Cc270R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.39.1200","type":"journal-article"},{"key":"l54l-sff5Cc271R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01312934","type":"journal-article"},{"key":"l54l-sff5Cc272R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.16.2126","type":"journal-article"},{"key":"l54l-sff5Cc273R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.5004796","type":"journal-article"},{"key":"l54l-sff5Cc274R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.90.065010","type":"journal-article"},{"key":"l54l-sff5Cc275R1","doi-asserted-by":"publisher","DOI":"10.1063\/5.0039248","type":"journal-article"},{"key":"l54l-sff5Cc276R1","doi-asserted-by":"publisher","DOI":"10.1063\/1.4903991","type":"journal-article"},{"key":"l54l-sff5Cc277R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.aop.2019.02.004","type":"journal-article"},{"key":"l54l-sff5Cc278R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.physleta.2019.06.022","type":"journal-article"},{"key":"l54l-sff5Cc280R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.aop.2020.168091","type":"journal-article"},{"key":"l54l-sff5Cc281R1","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8121\/ab766f","type":"journal-article"},{"key":"l54l-sff5Cc282R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.aop.2005.11.004","type":"journal-article"},{"key":"l54l-sff5Cc283R1","doi-asserted-by":"publisher","DOI":"10.1088\/0034-4885\/63\/12\/201","type":"journal-article"},{"key":"l54l-sff5Cc284R1","unstructured":"An operator A\u0302 is defined by both its domain D(A\u0302), i.e., the set of states A\u0302 is prescribed to act on, in addition to its action on such states. Hence, two operators A\u0302 and B\u0302 are defined to be equivalent, written as A\u0302=B\u0302, if and only if D(A\u0302)=D(B\u0302), and provided they act identically on any |\u03c8\u232a\u2208D(A\u0302). By the same token, A\u0302=A\u0302\u2020, requires A\u0302 and A\u0302\u2020 to have same domain and the same action on their domain. Note that the latter condition defines A\u0302 to be Hermitian, and in physics, it is usually taken to be the same as self-adjointness. But mathematically, self-adjointness is a stronger condition than Hermiticity (since it also requires A\u0302 and A\u0302\u2020 to have the same domain). An example of an operator in quantum mechanics that is Hermitian but not self-adjoint is the momentum operator of a particle confined to the interval [0,1] by an infinite potential function   [283].","type":"other"},{"key":"l54l-sff5Cc285R1","unstructured":"In a closely related paper, Zhu and Klauder investigated the occurrence of non-self-adjoint Hamiltonians in the quantization of nonlinear Hamiltonian systems   [286]. By way of case studies, they examined when a prescribed classical dynamical system will lead to a self-adjoint quantum Hamiltonian, and if not, whether it could be made self-adjoint   [286].","type":"other"},{"key":"l54l-sff5Cc286R1","doi-asserted-by":"publisher","DOI":"10.1119\/1.17221","type":"journal-article"},{"key":"l54l-sff5Cc287R1","doi-asserted-by":"crossref","DOI":"10.1142\/9334","type":"book","volume-title":"Bose, Spin, and Fermi Systems","author":"W.-H. Steeb","year":"2015"},{"key":"l54l-sff5Cc288R1","doi-asserted-by":"publisher","DOI":"10.1016\/S0375-9601(03)00194-4","type":"journal-article"}],"container-title":["Physical Review E"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/accepted\/10.1103\/l54l-sff5","content-type":"application\/pdf","content-version":"am","intended-application":"unspecified"},{"URL":"https:\/\/link.aps.org\/article\/10.1103\/l54l-sff5","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/l54l-sff5\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T20:05:02Z","timestamp":1762459502000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/l54l-sff5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,5]]},"references-count":288,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025,11]]}},"URL":"https:\/\/doi.org\/10.1103\/l54l-sff5","relation":{},"ISSN":["2470-0045","2470-0053"],"issn-type":[{"value":"2470-0045","type":"print"},{"value":"2470-0053","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,5]]},"assertion":[{"value":"2025-04-08","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"\u00a92025 American Physical Society. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright_statement","label":"Copyright statement","group":{"name":"copyright","label":"copyright"}},{"value":"2025","name":"copyright_year","label":"Copyright year","group":{"name":"copyright","label":"copyright"}},{"value":"American Physical Society","name":"copyright_holder","label":"Copyright holder","group":{"name":"copyright","label":"copyright"}}],"article-number":"054206"}}