{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:29:53Z","timestamp":1760146193600,"version":"build-2065373602"},"reference-count":96,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,3]],"date-time":"2024-10-03T00:00:00Z","timestamp":1727913600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work, we study the time-dependent behavior of quantum correlations of a system of an inverted oscillator governed by out-of-equilibrium dynamics using the well-known Schwinger\u2013Keldysh formalism in the presence of quantum mechanical quench. Considering a generalized structure of a time-dependent Hamiltonian for an inverted oscillator system, we use the invariant operator method to obtain its eigenstate and continuous energy eigenvalues. Using the expression for the eigenstate, we further derive the most general expression for the generating function as well as the out-of-time-ordered correlators (OTOCs) for the given system using this formalism. Further, considering the time-dependent coupling and frequency of the quantum inverted oscillator characterized by quench parameters, we comment on the dynamical behavior, specifically the early, intermediate and late time-dependent features of the OTOC for the quenched quantum inverted oscillator. Next, we study a specific case, where the system of an inverted oscillator exhibits chaotic behavior by computing the quantum Lyapunov exponent from the time-dependent behavior of OTOCs in the presence of the given quench profile.<\/jats:p>","DOI":"10.3390\/sym16101308","type":"journal-article","created":{"date-parts":[[2024,10,3]],"date-time":"2024-10-03T10:31:13Z","timestamp":1727951473000},"page":"1308","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Schwinger\u2013Keldysh Path Integral Formalism for a Quenched Quantum Inverted Oscillator"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0459-3873","authenticated-orcid":false,"given":"Sayantan","family":"Choudhury","sequence":"first","affiliation":[{"name":"Centre For Cosmology and Science Popularization (CCSP), SGT University, Gurugram 122505, Haryana, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1029-3807","authenticated-orcid":false,"given":"Suman","family":"Dey","sequence":"additional","affiliation":[{"name":"Department of Physics, Visva-Bharati University, Santiniketan 731235, West Bengal, India"}]},{"given":"Rakshit Mandish","family":"Gharat","sequence":"additional","affiliation":[{"name":"Department of Physics, National Institute of Technology Karnataka, Surathkal 575025, Karnataka, India"}]},{"given":"Saptarshi","family":"Mandal","sequence":"additional","affiliation":[{"name":"Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur 721302, West Bengal, India"}]},{"given":"Nilesh","family":"Pandey","sequence":"additional","affiliation":[{"name":"Centre For Cosmology and Science Popularization (CCSP), SGT University, Gurugram 122505, Haryana, India"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1063\/1.1703727","article-title":"Brownian Motion of a Quantum Oscillator","volume":"2","author":"Schwinger","year":"1961","journal-title":"J. Math. Phys."},{"key":"ref_2","first-page":"1018","article-title":"Diagram technique for nonequilibrium processes","volume":"20","author":"Keldysh","year":"1965","journal-title":"Sov. Phys. JETP"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"069","DOI":"10.1007\/JHEP06(2017)069","article-title":"Schwinger-Keldysh formalism. Part I: BRST symmetries and superspace","volume":"6","author":"Haehl","year":"2017","journal-title":"JHEP"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"070","DOI":"10.1007\/JHEP06(2017)070","article-title":"Schwinger-Keldysh formalism. Part II: Thermal equivariant cohomology","volume":"6","author":"Haehl","year":"2017","journal-title":"JHEP"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"105023","DOI":"10.1103\/PhysRevD.97.105023","article-title":"Schwinger-Keldysh superspace in quantum mechanics","volume":"97","author":"Geracie","year":"2018","journal-title":"Phys. Rev. D"},{"key":"ref_6","unstructured":"Bentov, Y. (2021). Schwinger-Keldysh path integral for the quantum harmonic oscillator. arXiv."},{"key":"ref_7","unstructured":"Bohra, H., Choudhury, S., Chauhan, P., Narayan, P., Panda, S., and Swain, A. (2019). Relating the curvature of De Sitter Universe to Open Quantum Lamb Shift Spectroscopy. arXiv."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"134307","DOI":"10.1103\/PhysRevB.92.134307","article-title":"Thermodynamic Equilibrium as a Symmetry of the Schwinger-Keldysh Action","volume":"92","author":"Sieberer","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Georgii, H.O. (2011). Gibbs Measures and Phase Transitions, De Gruyter.","DOI":"10.1515\/9783110250329"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1007\/BF02832514","article-title":"On the generalized ward identity","volume":"6","author":"Takahashi","year":"1957","journal-title":"Nuovo C."},{"key":"ref_11","unstructured":"Peskin, M., and Schroeder, D. (1995). An Introduction to Quantum Field Theory, Westview Press."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1140\/epjc\/s10052-019-6751-2","article-title":"Quantum Out-of-Equilibrium Cosmology","volume":"79","author":"Choudhury","year":"2019","journal-title":"Eur. Phys. J. C"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"748","DOI":"10.1140\/epjc\/s10052-020-8302-2","article-title":"Open Quantum Entanglement: A study of two atomic system in static patch of de Sitter space","volume":"80","author":"Akhtar","year":"2020","journal-title":"Eur. Phys. J. C"},{"key":"ref_14","first-page":"e006","article-title":"Schwinger-Keldysh Diagrammatics for Primordial Perturbations","volume":"12","author":"Chen","year":"2017","journal-title":"JCAP"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2769","DOI":"10.1103\/PhysRevD.49.2769","article-title":"Nonequilibrium evolution of scalar fields in FRW cosmologies I","volume":"49","author":"Boyanovsky","year":"1994","journal-title":"Phys. Rev. D"},{"key":"ref_16","unstructured":"Glorioso, P., Crossley, M., and Liu, H. (2018). A prescription for holographic Schwinger-Keldysh contour in non-equilibrium systems. arXiv."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1007\/JHEP01(2016)184","article-title":"The Fluid Manifesto: Emergent symmetries, hydrodynamics, and black holes","volume":"1","author":"Haehl","year":"2016","journal-title":"JHEP"},{"key":"ref_18","first-page":"e046","article-title":"Schwinger-Keldysh propagators from AdS\/CFT correspondence","volume":"3","author":"Herzog","year":"2003","journal-title":"JHEP"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"057","DOI":"10.1088\/1126-6708\/2009\/10\/057","article-title":"Fermionic Schwinger-Keldysh Propagators from AdS\/CFT","volume":"10","author":"Giecold","year":"2009","journal-title":"JHEP"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1140\/epjc\/s10052-017-5503-4","article-title":"Entangled de Sitter from stringy axionic Bell pair I: An analysis using Bunch\u2013Davies vacuum","volume":"78","author":"Choudhury","year":"2018","journal-title":"Eur. Phys. J. C"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"114606","DOI":"10.1016\/j.nuclphysb.2019.03.018","article-title":"Quantum entanglement in de Sitter space from stringy axion: An analysis using \u03b1 vacua","volume":"943","author":"Choudhury","year":"2019","journal-title":"Nucl. Phys. B"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"025002","DOI":"10.1103\/PhysRevD.106.025002","article-title":"Entanglement in interacting quenched two-body coupled oscillator system","volume":"106","author":"Choudhury","year":"2022","journal-title":"Phys. Rev. D"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1458","DOI":"10.1063\/1.1664991","article-title":"An Exact Quantum Theory of the Time-Dependent Harmonic Oscillator and of a Charged Particle in a Time-Dependent Electromagnetic Field","volume":"10","author":"Lewis","year":"1969","journal-title":"J. Math. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/s11128-022-03440-w","article-title":"Dynamics of entropy and information of time-dependent quantum systems: Exact results","volume":"21","author":"Andrzejewski","year":"2022","journal-title":"Quant. Inf. Proc."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"258","DOI":"10.1140\/epjp\/i2017-11524-7","article-title":"On the invariant method for the time-dependent non-Hermitian Hamiltonians","volume":"132","author":"Khantoul","year":"2017","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_26","unstructured":"Choudhury, S. (2021). Cosmological Geometric Phase From Pure Quantum States: A study without\/with having Bell\u2019s inequality violation. arXiv."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2113","DOI":"10.1023\/B:IJTP.0000049014.09671.e2","article-title":"Coherent and squeezed states for light in homogeneous conducting linear media by an invariant operator method","volume":"43","author":"Choi","year":"2004","journal-title":"Int. J. Theor. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"949","DOI":"10.1143\/ptp\/93.5.949","article-title":"Systematic Treatment of General Time-Dependent Harmonic Oscillator in Classical and Quantum Mechanics","volume":"93","author":"Kanasugi","year":"1995","journal-title":"Prog. Theor. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"018","DOI":"10.1088\/1475-7516\/2012\/04\/018","article-title":"Fourth level MSSM inflation from new flat directions","volume":"4","author":"Choudhury","year":"2012","journal-title":"JCAP"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.nuclphysb.2013.05.010","article-title":"DBI Galileon inflation in background SUGRA","volume":"874","author":"Choudhury","year":"2013","journal-title":"Nucl. Phys. B"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"043529","DOI":"10.1103\/PhysRevD.85.043529","article-title":"Brane inflation in background supergravity","volume":"85","author":"Choudhury","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/j.nuclphysb.2014.01.002","article-title":"Higgs inflation from new K\u00e4hler potential","volume":"880","author":"Choudhury","year":"2014","journal-title":"Nucl. Phys. B"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1140\/epjc\/s10052-016-4072-2","article-title":"COSMOS-e\u2019-GTachyon from string theory","volume":"76","author":"Choudhury","year":"2016","journal-title":"Eur. Phys. J. C"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"033273","DOI":"10.1103\/PhysRevResearch.2.033273","article-title":"Rise of cosmological complexity: Saturation of growth and chaos","volume":"2","author":"Bhattacharyya","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1140\/epjc\/s10052-017-5001-8","article-title":"COSMOS-e\u2032-Soft Higgsotic attractors","volume":"77","author":"Choudhury","year":"2017","journal-title":"Eur. Phys. J. C"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Choudhury, S., Panda, S., and Singh, R. (2017). Bell violation in primordial cosmology. Universe, 3.","DOI":"10.3390\/universe3010013"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1140\/epjc\/s10052-016-4553-3","article-title":"Bell violation in the Sky","volume":"77","author":"Choudhury","year":"2017","journal-title":"Eur. Phys. J. C"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"026021","DOI":"10.1103\/PhysRevD.101.026021","article-title":"Chaos and Complexity in Quantum Mechanics","volume":"101","author":"Ali","year":"2020","journal-title":"Phys. Rev. D"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"106020","DOI":"10.1103\/PhysRevD.101.106020","article-title":"Cosmological Complexity","volume":"101","author":"Bhattacharyya","year":"2020","journal-title":"Phys. Rev. D"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"064002","DOI":"10.1103\/PhysRevD.68.064002","article-title":"Squeezed states in the de Sitter vacuum","volume":"68","author":"Einhorn","year":"2003","journal-title":"Phys. Rev. D"},{"key":"ref_41","first-page":"e022","article-title":"Canonical transformations and squeezing formalism in cosmology","volume":"2","author":"Grain","year":"2020","journal-title":"JCAP"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1440","DOI":"10.1103\/PhysRevD.46.1440","article-title":"Generation of squeezed radiation from vacuum in the cosmos and the laboratory","volume":"46","author":"Grishchuk","year":"1992","journal-title":"Phys. Rev. D"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"e026","DOI":"10.21468\/SciPostPhysCore.4.4.026","article-title":"Quantum aspects of chaos and complexity from bouncing cosmology: A study with two-mode single field squeezed state formalism","volume":"4","author":"Bhargava","year":"2021","journal-title":"SciPost Phys. Core"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Choudhury, S., Chowdhury, S., Gupta, N., Mishara, A., Selvam, S.P., Panda, S., Pasquino, G.D., Singha, C., and Swain, A. (2021). Circuit Complexity from Cosmological Islands. Symmetry, 13.","DOI":"10.20944\/preprints202105.0626.v1"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"065002","DOI":"10.1103\/PhysRevD.104.065002","article-title":"Circuit complexity as a novel probe of quantum entanglement: A study with black hole gas in arbitrary dimensions","volume":"104","author":"Adhikari","year":"2021","journal-title":"Phys. Rev. D"},{"key":"ref_46","unstructured":"Choudhury, S., Mukherjee, A., Pandey, N., and Roy, A. (2021). Causality Constraint on Circuit Complexity from COSMOEFT. arXiv."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"e036","DOI":"10.1088\/1475-7516\/2021\/10\/036","article-title":"Real-space entanglement in the Cosmic Microwave Background","volume":"2021","author":"Martin","year":"2021","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_48","first-page":"e036","article-title":"Four-mode squeezed states in de Sitter space: A study with two field interacting quantum system","volume":"70","author":"Choudhury","year":"2022","journal-title":"Prog. Phys."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"016801","DOI":"10.1103\/PhysRevLett.121.016801","article-title":"Detecting Equilibrium and Dynamical Quantum Phase Transitions in Ising Chains via Out-of-Time-Ordered Correlators","volume":"121","author":"Heyl","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1007\/JHEP07(2019)006","article-title":"Probing out-of-time-order correlators","volume":"2019","author":"Chaudhuri","year":"2019","journal-title":"J. High Energy Phys."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1007\/JHEP12(2017)154","article-title":"Thermal out-of-time-order correlators, KMS relations, and spectral functions","volume":"2017","author":"Haehl","year":"2017","journal-title":"J. High Energy Phys."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1007\/JHEP07(2019)102","article-title":"Out of time ordered quantum dissipation","volume":"2019","author":"Chakrabarty","year":"2019","journal-title":"J. High Energy Phys."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1007\/JHEP02(2019)018","article-title":"Spectral representation of thermal OTO correlators","volume":"2019","author":"Chaudhuri","year":"2019","journal-title":"J. High Energy Phys."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Choudhury, S. (2020). The Cosmological OTOC: Formulating new cosmological micro-canonical correlation functions for random chaotic fluctuations in Out-of-Equilibrium Quantum Statistical Field Theory. Symmetry, 12.","DOI":"10.20944\/preprints202007.0038.v1"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1007\/JHEP10(2017)138","article-title":"Out-of-time-order correlators in quantum mechanics","volume":"2017","author":"Hashimoto","year":"2017","journal-title":"J. High Energy Phys."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Haehl, F.M., Loganayagam, R., Narayan, P., and Rangamani, M. (2017). Classification of out-of-time-order correlators. arXiv.","DOI":"10.1007\/JHEP12(2017)154"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"013","DOI":"10.21468\/SciPostPhys.7.1.013","article-title":"Out of time ordered effective dynamics of a quartic oscillator","volume":"7","author":"Chakrabarty","year":"2019","journal-title":"SciPost Phys."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"067","DOI":"10.1007\/JHEP03(2014)067","article-title":"Black holes and the butterfly effect","volume":"3","author":"Shenker","year":"2014","journal-title":"JHEP"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/JHEP05(2018)183","article-title":"The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual","volume":"5","author":"Kitaev","year":"2018","journal-title":"JHEP"},{"key":"ref_60","first-page":"1200","article-title":"Quasiclassical Method in the Theory of Superconductivity","volume":"28","author":"Larkin","year":"1969","journal-title":"Sov. J. Exp. Theor. Phys."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/0550-3213(85)90525-5","article-title":"The gravitational shock wave of a massless particle","volume":"253","author":"Dray","year":"1985","journal-title":"Nucl. Phys. B"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1016\/0550-3213(90)90174-C","article-title":"The black hole interpretation of string theory","volume":"335","year":"1990","journal-title":"Nucl. Phys. B"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1103\/RevModPhys.83.863","article-title":"Colloquium: Nonequilibrium dynamics of closed interacting quantum systems","volume":"83","author":"Polkovnikov","year":"2011","journal-title":"Rev. Mod. Phys."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"056001","DOI":"10.1088\/0034-4885\/79\/5\/056001","article-title":"Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems","volume":"79","author":"Gogolin","year":"2016","journal-title":"Rep. Prog. Phys."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"064001","DOI":"10.1088\/1742-5468\/2016\/06\/064001","article-title":"Introduction to \u2018Quantum Integrability in Out of Equilibrium Systems\u2019","volume":"2016","author":"Calabrese","year":"2016","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"064009","DOI":"10.1088\/1742-5468\/2016\/06\/064009","article-title":"Prethermalization and universal dynamics in near-integrable quantum systems","volume":"2016","author":"Langen","year":"2016","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"900","DOI":"10.1038\/nature04693","article-title":"A quantum Newton\u2019s cradle","volume":"440","author":"Kinoshita","year":"2006","journal-title":"Nature"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1038\/nature06149","article-title":"Non-equilibrium coherence dynamics in one-dimensional Bose gases","volume":"449","author":"Hofferberth","year":"2007","journal-title":"Nature"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1038\/nphys2232","article-title":"Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas","volume":"8","author":"Trotzky","year":"2012","journal-title":"Nat. Phys."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"1318","DOI":"10.1126\/science.1224953","article-title":"Relaxation and Prethermalization in an Isolated Quantum System","volume":"337","author":"Gring","year":"2012","journal-title":"Science"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1038\/nature10748","article-title":"Light-cone-like spreading of correlations in a quantum many-body system","volume":"481","author":"Cheneau","year":"2012","journal-title":"Nature"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"053003","DOI":"10.1103\/PhysRevLett.111.053003","article-title":"Quantum Quench in an Atomic One-Dimensional Ising Chain","volume":"111","author":"Meinert","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"640","DOI":"10.1038\/nphys2739","article-title":"Local emergence of thermal correlations in an isolated quantum many-body system","volume":"9","author":"Langen","year":"2013","journal-title":"Nat. Phys."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1038\/nature12541","article-title":"Microscopic observation of magnon bound states and their dynamics","volume":"502","author":"Fukuhara","year":"2013","journal-title":"Nature"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1038\/nphys2561","article-title":"Quantum dynamics of a mobile spin impurity","volume":"9","author":"Fukuhara","year":"2013","journal-title":"Nat. Phys."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"052143","DOI":"10.1103\/PhysRevE.99.052143","article-title":"Timescales in the quench dynamics of many-body quantum systems: Participation ratio versus out-of-time ordered correlator","volume":"99","author":"Borgonovi","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"060303","DOI":"10.1103\/PhysRevB.97.060303","article-title":"Generic dynamical features of quenched interacting quantum systems: Survival probability, density imbalance, and out-of-time-ordered correlator","volume":"97","author":"Santos","year":"2018","journal-title":"Phys. Rev. B"},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1016\/j.scib.2017.04.011","article-title":"Out-of-time-order correlation for many-body localization","volume":"62","author":"Fan","year":"2017","journal-title":"Sci. Bull."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"012215","DOI":"10.1103\/PhysRevE.100.012215","article-title":"Exact relaxation dynamics and quantum information scrambling in multiply quenched harmonic chains","volume":"100","author":"Ghosh","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"214402","DOI":"10.1103\/PhysRevB.103.214402","article-title":"Dynamical crossover in the transient quench dynamics of short-range transverse-field Ising models","volume":"103","author":"Sun","year":"2021","journal-title":"Phys. Rev. B"},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"841","DOI":"10.1139\/cjp-2014-0553","article-title":"Gaussian wave packet states of a generalized inverted harmonic oscillator with time-dependent mass and frequency","volume":"93","author":"Pedrosa","year":"2015","journal-title":"Can. J. Phys."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1007\/s10714-018-2438-5","article-title":"Inverting a normal harmonic oscillator: Physical interpretation and applications","volume":"50","author":"Rajeev","year":"2018","journal-title":"Gen. Rel. Grav."},{"key":"ref_83","doi-asserted-by":"crossref","unstructured":"Bhagat, K.Y., Bose, B., Choudhury, S., Chowdhury, S., Das, R.N., Dastider, S.G., Gupta, N., Maji, A., Pasquino, G.D., and Paul, S. (2020). The Generalized OTOC from Supersymmetric Quantum Mechanics\u2014Study of Random Fluctuations from Eigenstate Representation of Correlation Functions. Symmetry, 13.","DOI":"10.3390\/sym13010044"},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1007\/JHEP08(2016)106","article-title":"A bound on chaos","volume":"8","author":"Maldacena","year":"2016","journal-title":"JHEP"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"058","DOI":"10.1007\/JHEP02(2019)058","article-title":"How to Build the Thermofield Double State","volume":"2","author":"Cottrell","year":"2019","journal-title":"JHEP"},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1007\/JHEP05(2015)132","article-title":"Stringy effects in scrambling","volume":"5","author":"Shenker","year":"2015","journal-title":"JHEP"},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"051","DOI":"10.1007\/JHEP03(2015)051","article-title":"Localized shocks","volume":"3","author":"Roberts","year":"2015","journal-title":"JHEP"},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"131603","DOI":"10.1103\/PhysRevLett.115.131603","article-title":"Two-dimensional conformal field theory and the butterfly effect","volume":"115","author":"Roberts","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"126007","DOI":"10.1103\/PhysRevD.90.126007","article-title":"Complexity and Shock Wave Geometries","volume":"90","author":"Stanford","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_90","doi-asserted-by":"crossref","unstructured":"Stanford, D. (2014). Black Holes and the Butterfly Effect. [Ph.D. Thesis, Stanford University].","DOI":"10.1007\/JHEP03(2014)067"},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"046","DOI":"10.1007\/JHEP12(2014)046","article-title":"Multiple Shocks","volume":"12","author":"Shenker","year":"2014","journal-title":"JHEP"},{"key":"ref_92","doi-asserted-by":"crossref","unstructured":"Choudhury, S. (2021). The Cosmological OTOC: A New Proposal for Quantifying Auto-correlated Random Non-chaotic Primordial Fluctuations. Symmetry, 13.","DOI":"10.20944\/preprints202102.0616.v1"},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"027","DOI":"10.1007\/JHEP09(2020)027","article-title":"Quantum quench and thermalization to GGE in arbitrary dimensions and the odd-even effect","volume":"9","author":"Banerjee","year":"2020","journal-title":"JHEP"},{"key":"ref_94","first-page":"e027","article-title":"Thermalization in 2D critical quench and UV\/IR mixing","volume":"1","author":"Mandal","year":"2018","journal-title":"JHEP"},{"key":"ref_95","unstructured":"Paranjape, S., and Sorokhaibam, N. (2016). Exact Growth of Entanglement and Dynamical Phase Transition in Global Fermionic Quench. arXiv."},{"key":"ref_96","unstructured":"Banerjee, S., Choudhury, S., Chowdhury, S., Knaute, J., Panda, S., and Shirish, K. (2021). Thermalization in Quenched De Sitter Space. arXiv."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/10\/1308\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:10:12Z","timestamp":1760112612000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/10\/1308"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,3]]},"references-count":96,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["sym16101308"],"URL":"https:\/\/doi.org\/10.3390\/sym16101308","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,10,3]]}}}