{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:37:03Z","timestamp":1760233023042,"version":"build-2065373602"},"reference-count":97,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,15]],"date-time":"2022-12-15T00:00:00Z","timestamp":1671062400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NSFC","award":["12205092","12205088","11947069","11905053","22A0507","21B0647","21B0639","20C0495","2022HSKFJJ038"],"award-info":[{"award-number":["12205092","12205088","11947069","11905053","22A0507","21B0647","21B0639","20C0495","2022HSKFJJ038"]}]},{"name":"Scientific Research Fund of Hunan Provincial Education Department","award":["12205092","12205088","11947069","11905053","22A0507","21B0647","21B0639","20C0495","2022HSKFJJ038"],"award-info":[{"award-number":["12205092","12205088","11947069","11905053","22A0507","21B0647","21B0639","20C0495","2022HSKFJJ038"]}]},{"name":"Open fund project of the Key Laboratory of Optoelectronic Control and Detection Technology of University of Hunan Province","award":["12205092","12205088","11947069","11905053","22A0507","21B0647","21B0639","20C0495","2022HSKFJJ038"],"award-info":[{"award-number":["12205092","12205088","11947069","11905053","22A0507","21B0647","21B0639","20C0495","2022HSKFJJ038"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we investigate the effect of the Dicke quantum phase transition on the speed of evolution of the system dynamics. At the phase transition point, the symmetry associated with the system parity operator begins to break down. By comparing the magnitudes of the two types of quantum speed limit times, we find that the quantum speed limit time of the system is described by one of the quantum speed limit times, whether in the normal or superradiant phase. We find that, in the normal phase, the strength of the coupling between the optical field and the atoms has little effect on the dynamical evolution speed of the system. However, in the superradiant phase, a stronger atom\u2013photon coupling strength can accelerate the system dynamics\u2019 evolution. Finally, we investigate the effect of the entanglement of the initial state of the system on the speed of evolution of the system dynamics. We find that in the normal phase, the entanglement of the initial state of the system has almost no effect on the system dynamics\u2019 evolution speed. However, in the superradiant phase, larger entanglement of the system can accelerate the evolution of the system dynamics. Furthermore, we verify the above conclusions by the actual evolution of the system.<\/jats:p>","DOI":"10.3390\/sym14122653","type":"journal-article","created":{"date-parts":[[2022,12,15]],"date-time":"2022-12-15T03:43:49Z","timestamp":1671075829000},"page":"2653","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Quantum Speed-Up Induced by the Quantum Phase Transition in a Nonlinear Dicke Model with Two Impurity Qubits"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3728-4783","authenticated-orcid":false,"given":"Wangjun","family":"Lu","sequence":"first","affiliation":[{"name":"Department of Maths and Physics, Hunan Institute of Engineering, Xiangtan 411104, China"},{"name":"College of Physics and Electronic Engineering, Hengyang Normal University, Hengyang 421002, China"},{"name":"Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou 310027, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cuilu","family":"Zhai","sequence":"additional","affiliation":[{"name":"Department of Maths and Physics, Hunan Institute of Engineering, Xiangtan 411104, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yan","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Physics and Electronic Engineering, Hengyang Normal University, Hengyang 421002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yaju","family":"Song","sequence":"additional","affiliation":[{"name":"College of Physics and Electronic Engineering, Hengyang Normal University, Hengyang 421002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jibing","family":"Yuan","sequence":"additional","affiliation":[{"name":"College of Physics and Electronic Engineering, Hengyang Normal University, Hengyang 421002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Songsong","family":"Li","sequence":"additional","affiliation":[{"name":"College of Physics and Electronic Information, Nanchang Normal University, Nanchang 330032, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shiqing","family":"Tang","sequence":"additional","affiliation":[{"name":"College of Physics and Electronic Engineering, Hengyang Normal University, Hengyang 421002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"453001","DOI":"10.1088\/1751-8121\/aa86c6","article-title":"Quantum speed limits: From Heisenberg\u2019s uncertainty principle to optimal quantum control","volume":"50","author":"Deffner","year":"2017","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"050402","DOI":"10.1103\/PhysRevLett.110.050402","article-title":"Quantum Speed Limit for Physical Processes","volume":"110","author":"Taddei","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"050403","DOI":"10.1103\/PhysRevLett.110.050403","article-title":"Quantum Speed Limits in Open System Dynamics","volume":"110","author":"Egusquiza","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"240501","DOI":"10.1103\/PhysRevLett.103.240501","article-title":"Optimal Control at the Quantum Speed Limit","volume":"103","author":"Caneva","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"070402","DOI":"10.1103\/PhysRevLett.120.070402","article-title":"Quantum Speed Limit is Not Quantum","volume":"120","author":"Okuyama","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"022107","DOI":"10.1103\/PhysRevA.82.022107","article-title":"Geometric derivation of the quantum speed limit","volume":"82","author":"Jones","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"052331","DOI":"10.1103\/PhysRevA.93.052331","article-title":"Quantum speed limits, coherence, and asymmetry","volume":"93","author":"Marvian","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"070401","DOI":"10.1103\/PhysRevLett.120.070401","article-title":"Quantum Speed Limits across the Quantum-to-Classical Transition","volume":"120","author":"Shanahan","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"060409","DOI":"10.1103\/PhysRevLett.120.060409","article-title":"Tightening Quantum Speed Limits for Almost All States","volume":"120","author":"Campaioli","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1103\/PhysRevLett.46.623","article-title":"Energy Cost of Information Transfer","volume":"46","author":"Bekenstein","year":"1981","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1047","DOI":"10.1038\/35023282","article-title":"Ultimate physical limits to computation","volume":"406","author":"Lloyd","year":"2000","journal-title":"Nature"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"030303","DOI":"10.1103\/PhysRevA.74.030303","article-title":"Quantum speed limit for perfect state transfer in one dimension","volume":"74","author":"Yung","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"9798679","DOI":"10.34133\/2022\/9798679","article-title":"Experimental quantum advantage with quantum coupon collector","volume":"2022","author":"Zhou","year":"2022","journal-title":"Research"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"040334","DOI":"10.1103\/PRXQuantum.2.040334","article-title":"Homodyne Detection Quadrature Phase Shift Keying Continuous-Variable Quantum key Distribution with High Excess Noise Tolerance","volume":"2","author":"Liu","year":"2021","journal-title":"PRX Quantum"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"020315","DOI":"10.1103\/PRXQuantum.3.020315","article-title":"Breaking the Rate-Loss Bound of Quantum Key Distribution with Asynchronous Two-Photon Interference","volume":"3","author":"Xie","year":"2022","journal-title":"PRX Quantum"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1007\/BF01857732","article-title":"Physical limitations of rate, depth, and minimum energy in information processing","volume":"21","author":"Levitin","year":"1982","journal-title":"Int. J. Theor. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"237901","DOI":"10.1103\/PhysRevLett.88.237901","article-title":"Computational Capacity of the Universe","volume":"88","author":"Lloyd","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"052109","DOI":"10.1103\/PhysRevA.67.052109","article-title":"Quantum limits to dynamical evolution","volume":"67","author":"Giovannetti","year":"2003","journal-title":"Phys. Rev. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"032305","DOI":"10.1103\/PhysRevA.95.032305","article-title":"Fast quantum computation at arbitrarily low energy","volume":"95","author":"Jordan","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1038\/nphoton.2011.35","article-title":"Advances in quantum metrology","volume":"5","author":"Giovannetti","year":"2011","journal-title":"Nat. Photonics"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"233601","DOI":"10.1103\/PhysRevLett.109.233601","article-title":"Quantum Metrology in Non-Markovian Environments","volume":"109","author":"Chin","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"120405","DOI":"10.1103\/PhysRevLett.112.120405","article-title":"Quantum Metrology in Open Systems: Dissipative Cram\u00e9r-Rao Bound","volume":"112","author":"Alipour","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"073005","DOI":"10.1088\/1367-2630\/15\/7\/073005","article-title":"Quantum metrology with open dynamical systems","volume":"15","author":"Tsang","year":"2013","journal-title":"New J. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"170402","DOI":"10.1103\/PhysRevLett.105.170402","article-title":"Generalized Clausius Inequality for Nonequilibrium Quantum Processes","volume":"105","author":"Deffner","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"022143","DOI":"10.1103\/PhysRevE.87.022143","article-title":"Thermodynamic length for far-from-equilibrium quantum systems","volume":"87","author":"Deffner","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"052104","DOI":"10.1103\/PhysRevA.95.052104","article-title":"Quantum dynamical speedup in a nonequilibrium environment","volume":"95","author":"Cai","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"012312","DOI":"10.1103\/PhysRevA.84.012312","article-title":"Speeding up critical system dynamics through optimized evolution","volume":"84","author":"Caneva","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"260501","DOI":"10.1103\/PhysRevLett.111.260501","article-title":"Driving at the Quantum Speed Limit: Optimal Control of a Two-Level System","volume":"111","author":"Hegerfeldt","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"032110","DOI":"10.1103\/PhysRevA.90.032110","article-title":"High-speed driving of a two-level system","volume":"90","author":"Hegerfeldt","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"010502","DOI":"10.1103\/PhysRevLett.113.010502","article-title":"Information Theoretical Analysis of Quantum Optimal Control","volume":"113","author":"Lloyd","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"062106","DOI":"10.1103\/PhysRevA.92.062106","article-title":"Time-limited optimal dynamics beyond the quantum speed limit","volume":"92","author":"Gajdacz","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"062326","DOI":"10.1103\/PhysRevA.88.062326","article-title":"Speeding up and slowing down the relaxation of a qubit by optimal control","volume":"88","author":"Mukherjee","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"010402","DOI":"10.1103\/PhysRevLett.111.010402","article-title":"Quantum Speed Limit for Non-Markovian Dynamics","volume":"111","author":"Deffner","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"233602","DOI":"10.1103\/PhysRevLett.114.233602","article-title":"Environment-Assisted Speed-up of the Field Evolution in Cavity Quantum Electrodynamics","volume":"114","author":"Cimmarusti","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"012307","DOI":"10.1103\/PhysRevA.89.012307","article-title":"Quantum speedup in a memory environment","volume":"89","author":"Xu","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"4890","DOI":"10.1038\/srep04890","article-title":"Quantum speed limit for arbitrary initial states","volume":"4","author":"Zhang","year":"2014","journal-title":"Sci. Rep."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"032112","DOI":"10.1103\/PhysRevA.91.032112","article-title":"Classical-driving-assisted quantum speed-up","volume":"91","author":"Zhang","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"022102","DOI":"10.1103\/PhysRevA.91.022102","article-title":"Quantum-speed-limit time for multiqubit open systems","volume":"91","author":"Liu","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"8444","DOI":"10.1038\/srep08444","article-title":"Quantum speed limits in open systems: Non-Markovian dynamics without rotating-wave approximation","volume":"5","author":"Sun","year":"2015","journal-title":"Sci. Rep."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2325","DOI":"10.1007\/s11128-016-1291-2","article-title":"Quantum speedup of uncoupled multiqubit open system via dynamical decoupling pulses","volume":"15","author":"Song","year":"2016","journal-title":"Quantum Inf. Process."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"020105","DOI":"10.1103\/PhysRevA.93.020105","article-title":"Mechanism for quantum speedup in open quantum systems","volume":"93","author":"Liu","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"045301","DOI":"10.1088\/1751-8113\/48\/4\/045301","article-title":"The initial state dependence of the quantum speed limit","volume":"48","author":"Wu","year":"2014","journal-title":"J. Phys. A Math. Theor."},{"doi-asserted-by":"crossref","unstructured":"Mandelstam, L., and Tamm, I. (1991). The uncertainty relation between energy and time in non-relativistic quantum mechanics. Selected Papers, Springer.","key":"ref_43","DOI":"10.1007\/978-3-642-74626-0_8"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1016\/S0167-2789(98)00054-2","article-title":"The maximum speed of dynamical evolution","volume":"120","author":"Margolus","year":"1998","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"052125","DOI":"10.1103\/PhysRevA.94.052125","article-title":"Quantum-speed-limit bounds in an open quantum evolution","volume":"94","author":"Mirkin","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"013006","DOI":"10.1088\/1367-2630\/aaf9f5","article-title":"Speed limit for open quantum systems","volume":"21","author":"Funo","year":"2019","journal-title":"New J. Phys."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"022210","DOI":"10.1103\/PhysRevA.103.022210","article-title":"Action quantum speed limits","volume":"103","author":"Guarnieri","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_48","first-page":"021031","article-title":"Generalized Geometric Quantum Speed Limits","volume":"6","author":"Pires","year":"2016","journal-title":"Phys. Rev. X"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"052424","DOI":"10.1103\/PhysRevA.104.052424","article-title":"Control cost and quantum speed limit time in controlled almost-exact state transmission in open systems","volume":"104","author":"Nie","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"042132","DOI":"10.1103\/PhysRevA.98.042132","article-title":"Quantum speed limit for a mixed initial state","volume":"98","author":"Wu","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"210402","DOI":"10.1103\/PhysRevLett.115.210402","article-title":"Quantum Speed Limits for Leakage and Decoherence","volume":"115","author":"Marvian","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"013161","DOI":"10.1103\/PhysRevResearch.2.013161","article-title":"Quantum speed limits and the maximal rate of information production","volume":"2","author":"Deffner","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"168","DOI":"10.22331\/q-2019-08-05-168","article-title":"Tight, robust, and feasible quantum speed limits for open dynamics","volume":"3","author":"Campaioli","year":"2019","journal-title":"Quantum"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"022115","DOI":"10.1103\/PhysRevA.95.022115","article-title":"Improved bound for quantum-speed-limit time in open quantum systems by introducing an alternative fidelity","volume":"95","author":"Ektesabi","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"062204","DOI":"10.1103\/PhysRevA.103.062204","article-title":"Quantum speed limit for thermal states","volume":"103","author":"Lychkovskiy","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"033012","DOI":"10.1088\/1367-2630\/ab099e","article-title":"Quantum speed limits under continuous quantum measurements","volume":"21","year":"2019","journal-title":"New J. Phys."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"100404","DOI":"10.1103\/PhysRevLett.127.100404","article-title":"Quantum Speed Limit Quantified by the Changing Rate of Phase","volume":"127","author":"Sun","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"042606","DOI":"10.1103\/PhysRevA.102.042606","article-title":"Quantum speed limit for robust state characterization and engineering","volume":"102","author":"Kobayashi","year":"2020","journal-title":"Phys. Rev. A"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.physletb.2003.08.076","article-title":"Invariant correlational entropy as a signature of quantum phase transitions in nuclei","volume":"574","author":"Volya","year":"2003","journal-title":"Phys. Lett. B"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"066203","DOI":"10.1103\/PhysRevE.67.066203","article-title":"Chaos and the quantum phase transition in the Dicke model","volume":"67","author":"Emary","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"063039","DOI":"10.1088\/1367-2630\/16\/6\/063039","article-title":"Quantum Fisher information as a signature of the superradiant quantum phase transition","volume":"16","author":"Wang","year":"2014","journal-title":"New J. Phys."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"10482","DOI":"10.1364\/OE.27.010482","article-title":"Quantum critical detector: Amplifying weak signals using discontinuous quantum phase transitions","volume":"27","author":"Yang","year":"2019","journal-title":"Opt. Express"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"052302","DOI":"10.1103\/PhysRevA.77.052302","article-title":"Fidelity between partial states as a signature of quantum phase transitions","volume":"77","author":"Sacramento","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"060102","DOI":"10.1103\/PhysRevA.78.060102","article-title":"Decoherence as a signature of an excited-state quantum phase transition","volume":"78","author":"Arias","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"022112","DOI":"10.1103\/PhysRevA.94.022112","article-title":"Coherence susceptibility as a probe of quantum phase transitions","volume":"94","author":"Chen","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"7404","DOI":"10.1038\/s41598-017-07899-x","article-title":"Single-impurity-induced Dicke quantum phase transition in a cavity-Bose-Einstein condensate","volume":"7","author":"Yuan","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"116401","DOI":"10.1088\/0256-307X\/35\/11\/116401","article-title":"Nonlinear Dicke quantum phase transition and its quantum witness in a cavity-Bose\u2013Einstein-condensate system","volume":"35","author":"Lu","year":"2018","journal-title":"Chin. Phys. Lett."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"032305","DOI":"10.1103\/PhysRevA.101.032305","article-title":"Steered quantum coherence as a signature of quantum phase transitions in spin chains","volume":"101","author":"Hu","year":"2020","journal-title":"Phys. Rev. A"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"184313","DOI":"10.1103\/PhysRevB.100.184313","article-title":"Signature of a nonequilibrium quantum phase transition in the long-time average of the Loschmidt echo","volume":"100","author":"Zhou","year":"2019","journal-title":"Phys. Rev. B"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"034119","DOI":"10.1103\/PhysRevE.104.034119","article-title":"Signatures of excited-state quantum phase transitions in quantum many-body systems: Phase space analysis","volume":"104","author":"Wang","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"140604","DOI":"10.1103\/PhysRevLett.96.140604","article-title":"Decay of Loschmidt Echo Enhanced by Quantum Criticality","volume":"96","author":"Quan","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"3695","DOI":"10.1007\/s11128-015-1186-7","article-title":"Geometric phase, quantum Fisher information, geometric quantum correlation and quantum phase transition in the cavity-Bose\u2013Einstein-condensate system","volume":"15","author":"Wu","year":"2016","journal-title":"Quantum Inf. Process."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"022118","DOI":"10.1103\/PhysRevA.100.022118","article-title":"Excited-state quantum phase transition and the quantum-speed-limit time","volume":"100","author":"Wang","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"060504","DOI":"10.1103\/PhysRevB.95.060504","article-title":"Quenching a quantum critical state by the order parameter: Dynamical quantum phase transitions and quantum speed limits","volume":"95","author":"Heyl","year":"2017","journal-title":"Phys. Rev. B"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"19308","DOI":"10.1038\/srep19308","article-title":"Quantum speed limit and a signal of quantum criticality","volume":"6","author":"Wei","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"043805","DOI":"10.1103\/PhysRevA.98.043805","article-title":"Critical phenomena in an extended Dicke model","volume":"98","author":"Rodriguez","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjd\/e2020-10332-0","article-title":"Quantum phase transition and Berry phase in an extended Dicke model","volume":"74","author":"Guerra","year":"2020","journal-title":"Eur. Phys. J. D"},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"033131","DOI":"10.1103\/PhysRevResearch.2.033131","article-title":"Nonlinear semiclassical dynamics of the unbalanced, open Dicke model","volume":"2","author":"Stitely","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"023299","DOI":"10.1103\/PhysRevResearch.2.023299","article-title":"Operational definition of a quantum speed limit","volume":"2","author":"Shao","year":"2020","journal-title":"Phys. Rev. Res."},{"unstructured":"Shao, L., Zhang, R., Lu, W., Zhang, Z., and Wang, X. (2022). Quantum phase transition in XXZ central spin model. arXiv.","key":"ref_80"},{"doi-asserted-by":"crossref","unstructured":"Wang, Q. (2022). Quantum Chaos in the Extended Dicke Model. Entropy, 24.","key":"ref_81","DOI":"10.3390\/e24101415"},{"doi-asserted-by":"crossref","unstructured":"Lu, W., Zhai, C., Liu, Y., Song, Y., Yuan, J., and Tang, S. (2022). Berry Phase of Two Impurity Qubits as a Signature of Dicke Quantum Phase Transition. Photonics, 9.","key":"ref_82","DOI":"10.3390\/photonics9110844"},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"869","DOI":"10.1103\/PhysRevLett.80.869","article-title":"Teleportation of Continuous Quantum Variables","volume":"80","author":"Braunstein","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"1754","DOI":"10.1103\/PhysRevLett.85.1754","article-title":"Cloning of Continuous Quantum Variables","volume":"85","author":"Cerf","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1126\/science.1163218","article-title":"Cavity Optomechanics with a Bose-Einstein Condensate","volume":"322","author":"Brennecke","year":"2008","journal-title":"Science"},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"052324","DOI":"10.1103\/PhysRevA.83.052324","article-title":"Entanglement detection in hybrid optomechanical systems","volume":"83","author":"Paternostro","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"2608","DOI":"10.1016\/j.physleta.2012.06.037","article-title":"Engineering entanglement mechanically","volume":"376","author":"Asjad","year":"2012","journal-title":"Phys. Lett. A"},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjd\/e2013-40042-5","article-title":"Quantum degenerate Fermi gas entanglement in optomechanics","volume":"67","author":"Asjad","year":"2013","journal-title":"Eur. Phys. J. D"},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"033606","DOI":"10.1103\/PhysRevA.84.033606","article-title":"Steady-state entanglement of a Bose-Einstein condensate and a nanomechanical resonator","volume":"84","author":"Asjad","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"034046","DOI":"10.1103\/PhysRevApplied.16.034046","article-title":"Continuous-Variable Quantum Teleportation Using a Microwave-Enabled Plasmonic Graphene Waveguide","volume":"16","author":"Asjad","year":"2021","journal-title":"Phys. Rev. Appl."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"2930","DOI":"10.1364\/JOSAB.393998","article-title":"Quantum speed limit time, non-Markovianity, and quantum phase transition of the Dicke model","volume":"37","author":"Musadiq","year":"2020","journal-title":"JOSA B"},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1103\/PhysRev.79.845","article-title":"Theory of the Superconducting State. I. The Ground State at the Absolute Zero of Temperature","volume":"79","year":"1950","journal-title":"Phys. Rev."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1080\/00018735500101254","article-title":"Perturbation theory in statistical mechanics","volume":"4","author":"Nakajima","year":"1955","journal-title":"Adv. Phys."},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"1098","DOI":"10.1103\/PhysRev.58.1098","article-title":"Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet","volume":"58","author":"Holstein","year":"1940","journal-title":"Phys. Rev."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"1301","DOI":"10.1038\/nature09009","article-title":"Dicke quantum phase transition with a superfluid gas in an optical cavity","volume":"464","author":"Baumann","year":"2010","journal-title":"Nature"},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"1760","DOI":"10.1016\/j.cpc.2012.02.021","article-title":"QuTiP: An open-source Python framework for the dynamics of open quantum systems","volume":"183","author":"Johansson","year":"2012","journal-title":"Comput. Phys. Commun."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"043606","DOI":"10.1103\/PhysRevA.68.043606","article-title":"Generation of atom\u2013photon entangled states in atomic Bose-Einstein condensate via electromagnetically induced transparency","volume":"68","author":"Kuang","year":"2003","journal-title":"Phys. Rev. A"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/12\/2653\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:41:45Z","timestamp":1760146905000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/12\/2653"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,15]]},"references-count":97,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["sym14122653"],"URL":"https:\/\/doi.org\/10.3390\/sym14122653","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,12,15]]}}}