{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,22]],"date-time":"2025-11-22T17:05:02Z","timestamp":1763831102430,"version":"build-2065373602"},"reference-count":87,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,3,26]],"date-time":"2020-03-26T00:00:00Z","timestamp":1585180800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We examine the spectral structure and many-body dynamics of two and three repulsively interacting bosons trapped in a one-dimensional double-well, for variable barrier height, inter-particle interaction strength, and initial conditions. By exact diagonalization of the many-particle Hamiltonian, we specifically explore the dynamical behavior of the particles launched either at the single-particle ground state or saddle-point energy, in a time-independent potential. We complement these results by a characterization of the cross-over from diabatic to quasi-adiabatic evolution under finite-time switching of the potential barrier, via the associated time evolution of a single particle\u2019s von Neumann entropy. This is achieved with the help of the multiconfigurational time-dependent Hartree method for indistinguishable particles (MCTDH-X)\u2014which also allows us to extrapolate our results for increasing particle numbers.<\/jats:p>","DOI":"10.3390\/e22040382","type":"journal-article","created":{"date-parts":[[2020,3,27]],"date-time":"2020-03-27T09:04:38Z","timestamp":1585299878000},"page":"382","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double-Well"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2684-4984","authenticated-orcid":false,"given":"Frank","family":"Sch\u00e4fer","sequence":"first","affiliation":[{"name":"Physikalisches Institut, Albert-Ludwigs-Universit\u00e4t Freiburg, Hermann-Herder-Stra\u00dfe 3, D-79104 Freiburg, Germany"},{"name":"Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1552-4101","authenticated-orcid":false,"given":"Miguel","family":"Bastarrachea-Magnani","sequence":"additional","affiliation":[{"name":"Physikalisches Institut, Albert-Ludwigs-Universit\u00e4t Freiburg, Hermann-Herder-Stra\u00dfe 3, D-79104 Freiburg, Germany"},{"name":"Department of Physics and Astronomy, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8859-1655","authenticated-orcid":false,"given":"Axel","family":"Lode","sequence":"additional","affiliation":[{"name":"Physikalisches Institut, Albert-Ludwigs-Universit\u00e4t Freiburg, Hermann-Herder-Stra\u00dfe 3, D-79104 Freiburg, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Laurent","family":"de Parny","sequence":"additional","affiliation":[{"name":"Physikalisches Institut, Albert-Ludwigs-Universit\u00e4t Freiburg, Hermann-Herder-Stra\u00dfe 3, D-79104 Freiburg, Germany"},{"name":"ACRI-ST, 260 route du Pin Montard, 06904 Sophia Antipolis CEDEX, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andreas","family":"Buchleitner","sequence":"additional","affiliation":[{"name":"Physikalisches Institut, Albert-Ludwigs-Universit\u00e4t Freiburg, Hermann-Herder-Stra\u00dfe 3, D-79104 Freiburg, Germany"},{"name":"Freiburg Institute for Advanced Studies (FRIAS), Albert-Ludwigs-Universit\u00e4t Freiburg, Albertstr. 19, D-79104 Freiburg, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1126\/science.1201351","article-title":"Deterministic preparation of a tunable few-fermion system","volume":"332","author":"Serwane","year":"2011","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"080402","DOI":"10.1103\/PhysRevLett.114.080402","article-title":"Two Fermions in a Double Well: Exploring a Fundamental Building Block of the Hubbard Model","volume":"114","author":"Murmann","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1103\/RevModPhys.78.179","article-title":"Dynamics of Bose-Einstein condensates in optical lattices","volume":"78","author":"Morsch","year":"2006","journal-title":"Rev. Mod. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"010402","DOI":"10.1103\/PhysRevLett.95.010402","article-title":"Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction","volume":"95","author":"Albiez","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1038\/nature07244","article-title":"A Mott insulator of fermionic atoms in an optical lattice","volume":"455","author":"Strohmaier","year":"2008","journal-title":"Nature"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1038\/415039a","article-title":"Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms","volume":"415","author":"Greiner","year":"2002","journal-title":"Nature"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S1049-250X(05)52001-9","article-title":"Exploring quantum matter with ultracold atoms in optical lattices","volume":"52","author":"Bloch","year":"2005","journal-title":"Adv. At. Mol. Opt. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"885","DOI":"10.1103\/RevModPhys.80.885","article-title":"Many-body Physics with ultracold gases","volume":"80","author":"Bloch","year":"2008","journal-title":"Rev. Mod. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1016","DOI":"10.1038\/nature07126","article-title":"Quantum coherence and entanglement with ultracold atoms in optical lattices","volume":"453","author":"Bloch","year":"2008","journal-title":"Nature"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"995","DOI":"10.1126\/science.aal3837","article-title":"Quantum simulations with ultracold atoms in optical lattices","volume":"357","author":"Gross","year":"2017","journal-title":"Science"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1103\/RevModPhys.77.259","article-title":"The density-matrix renormalization group","volume":"77","year":"2005","journal-title":"Rev. Mod. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.aop.2010.09.012","article-title":"The density-matrix renormalization group in the age of matrix product states","volume":"326","year":"2011","journal-title":"Ann. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"125015","DOI":"10.1088\/1367-2630\/14\/12\/125015","article-title":"Out-of-equilibrium dynamics with matrix product states","volume":"14","author":"Wall","year":"2012","journal-title":"New J. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"033613","DOI":"10.1103\/PhysRevA.77.033613","article-title":"Multiconfigurational time-dependent Hartree method for bosons: Many-body dynamics of bosonic systems","volume":"77","author":"Alon","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"011001","DOI":"10.1103\/RevModPhys.92.011001","article-title":"Colloquium: Multiconfigurational time-dependent Hartree approaches for indistinguishable particles","volume":"92","author":"Lode","year":"2020","journal-title":"Rev. Mod. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"L393","DOI":"10.1088\/0953-4075\/36\/21\/L04","article-title":"Time delay between singly and doubly ionizing wavepackets in laser-driven helium","volume":"36","author":"Parker","year":"2003","journal-title":"J. Phys. B At. Mol. Opt. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"253002","DOI":"10.1103\/PhysRevLett.91.253002","article-title":"Interaction-induced decoherence of atomic Bloch oscillations","volume":"91","author":"Buchleitner","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"170403","DOI":"10.1103\/PhysRevLett.118.170403","article-title":"Anderson localization of ultracold atoms: Where is the mobility edge?","volume":"118","author":"Pasek","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","unstructured":"Davies, E.B. (1976). Quantum Theory of Open Systems, Academic Press."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Alicki, R., and Lendi, K. (2007). Quantum Dynamical Semigroups and Applications, Springer.","DOI":"10.1016\/B0-12-512666-2\/00384-9"},{"key":"ref_21","unstructured":"Gardiner, C., and Zoller, P. (2004). Quantum Noise: A Handbook of Markovian and non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, Springer. [3rd ed.]."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Breuer, H.P., and Petruccione, F. (2002). The Theory of Open Quantum Systems, Oxford University Press.","DOI":"10.1007\/3-540-44874-8_4"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"215302","DOI":"10.1103\/PhysRevLett.123.215302","article-title":"Enhancement of Many-Body Quantum Interference in Chaotic Bosonic Systems: The Role of Symmetry and Dynamics","volume":"123","author":"Schlagheck","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/S0370-1573(97)00088-4","article-title":"Random-matrix theories in quantum physics: Common concepts","volume":"299","author":"Guhr","year":"1998","journal-title":"Phys. Rep."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"020104","DOI":"10.1103\/PhysRevA.94.020104","article-title":"From many-particle interference to correlation spectroscopy","volume":"94","author":"Walschaers","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"106603","DOI":"10.1103\/PhysRevLett.122.106603","article-title":"Many-Body Multifractality throughout Bosonic Superfluid and Mott Insulator Phases","volume":"122","author":"Lindinger","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"063606","DOI":"10.1103\/PhysRevA.86.063606","article-title":"Numerically exact quantum dynamics of bosons with time-dependent interactions of harmonic type","volume":"86","author":"Lode","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"033635","DOI":"10.1103\/PhysRevA.93.033635","article-title":"Multiconfigurational time-dependent Hartree method for fermions: Implementation, exactness, and few-fermion tunneling to open space","volume":"93","author":"Fasshauer","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1146\/annurev-conmatphys-031115-011327","article-title":"Quantum transport on disordered and noisy networks: An interplay of structural complexity and uncertainty","volume":"7","author":"Walschaers","year":"2016","journal-title":"Annu. Rev. Condens. Matter Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"081201","DOI":"10.1103\/PhysRevB.99.081201","article-title":"Resolution of the exponent puzzle for the Anderson transition in doped semiconductors","volume":"99","author":"Carnio","year":"2019","journal-title":"Phys. Rev. B"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1029","DOI":"10.1038\/nature06112","article-title":"Direct observation of second-order atom tunnelling","volume":"448","author":"Trotzky","year":"2007","journal-title":"Nature"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"4318","DOI":"10.1103\/PhysRevA.55.4318","article-title":"Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential","volume":"55","author":"Milburn","year":"1997","journal-title":"Phys. Rev. A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"4950","DOI":"10.1103\/PhysRevLett.79.4950","article-title":"Quantum coherent atomic tunneling between two trapped Bose-Einstein condensates","volume":"79","author":"Smerzi","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"023601","DOI":"10.1103\/PhysRevA.63.023601","article-title":"Dynamic splitting of a Bose-Einstein condensate","volume":"63","author":"Menotti","year":"2001","journal-title":"Phys. Rev. A"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"023615","DOI":"10.1103\/PhysRevA.71.023615","article-title":"Quantum phase-space picture of Bose-Einstein condensates in a double well","volume":"71","author":"Mahmud","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1140\/epjd\/e2007-00224-4","article-title":"Quantum dynamics of bosons in a double-well potential: Josephson oscillations, self-trapping and ultralong tunneling times","volume":"44","author":"Salgueiro","year":"2007","journal-title":"Eur. Phys. J. D"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"053616","DOI":"10.1103\/PhysRevA.76.053616","article-title":"Boson pairs in a one-dimensional split trap","volume":"76","author":"Murphy","year":"2007","journal-title":"Phys. Rev. A"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"063413","DOI":"10.1103\/PhysRevA.77.063413","article-title":"Low-energy excitations of a boson pair in a double-well trap","volume":"77","author":"Murphy","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"030402","DOI":"10.1103\/PhysRevLett.99.030402","article-title":"Role of excited states in the splitting of a trapped interacting Bose-Einstein condensate by a time-dependent barrier","volume":"99","author":"Streltsov","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"220601","DOI":"10.1103\/PhysRevLett.103.220601","article-title":"Exact Quantum Dynamics of a Bosonic Josephson Junction","volume":"103","author":"Sakmann","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"053612","DOI":"10.1103\/PhysRevA.74.053612","article-title":"Ultracold few-boson systems in a double-well trap","volume":"74","author":"Meyer","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"043608","DOI":"10.1103\/PhysRevA.75.043608","article-title":"Excitations of few-boson systems in one-dimensional harmonic and double wells","volume":"75","author":"Meyer","year":"2007","journal-title":"Phys. Rev. A"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"013621","DOI":"10.1103\/PhysRevA.78.013621","article-title":"Tunneling dynamics of a few bosons in a double well","volume":"78","author":"Meyer","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"040401","DOI":"10.1103\/PhysRevLett.100.040401","article-title":"Few-Boson dynamics in double wells: From single-atom to correlated pair tunneling","volume":"100","author":"Meyer","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"053622","DOI":"10.1103\/PhysRevA.94.053622","article-title":"Many-body excitations and deexcitations in trapped ultracold bosonic clouds","volume":"94","author":"Theisen","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1140\/epjd\/e2016-70016-x","article-title":"Exact dynamics of two ultra-cold bosons confined in a one-dimensional double-well potential","volume":"70","author":"Dobrzyniecki","year":"2016","journal-title":"Eur. Phys. J. D"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"230403","DOI":"10.1103\/PhysRevLett.118.230403","article-title":"Crossing over from attractive to repulsive interactions in a tunneling bosonic josephson junction","volume":"118","author":"Spagnolli","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"032119","DOI":"10.1103\/PhysRevA.88.032119","article-title":"Two-band Bose-Hubbard model for many-body resonant tunneling in the Wannier-Stark system","volume":"88","author":"Madronero","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"056213","DOI":"10.1103\/PhysRevE.68.056213","article-title":"Floquet-Bloch operator for the Bose-Hubbard model with static field","volume":"68","author":"Kolovsky","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"632","DOI":"10.1209\/epl\/i2004-10265-7","article-title":"Quantum chaos in the Bose-Hubbard model","volume":"68","author":"Kolovsky","year":"2004","journal-title":"Europhys. Lett."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"063601","DOI":"10.1103\/PhysRevA.93.063601","article-title":"Multiconfigurational time-dependent Hartree method for bosons with internal degrees of freedom: Theory and composite fragmentation of multicomponent Bose-Einstein condensates","volume":"93","author":"Lode","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_52","unstructured":"Lode, A.U.J., Tsatsos, M.C., Fasshauer, E., Lin, R., Papariello, L., Molignini, P., L\u00e9v\u00eaque, C., and Weiner, S.E. (2020, March 24). MCTDH-X: The Time-Dependent Multiconfigurational Hartree for Indistinguishable Particles Software. Available online: http:\/\/ultracold.org."},{"key":"ref_53","first-page":"011052","article-title":"Parametric excitation of a Bose-Einstein condensate: From Faraday waves to granulation","volume":"9","author":"Nguyen","year":"2019","journal-title":"Phys. Rev. X"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"938","DOI":"10.1103\/PhysRevLett.81.938","article-title":"Atomic scattering in the presence of an external confinement and a gas of impenetrable bosons","volume":"81","author":"Olshanii","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"043626","DOI":"10.1103\/PhysRevA.87.043626","article-title":"Tunneling decay of two interacting bosons in an asymmetric double-well potential: A spectral approach","volume":"87","author":"Hunn","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_56","unstructured":"Sch\u00e4fer, F. (March 2018). Dynamics and Spectral Structure of Strongly Interacting Bosons in a Bouble Well. [Master\u2019s Thesis, Albert-Ludwigs-Universit\u00e4t Freiburg]."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"042113","DOI":"10.1103\/PhysRevA.66.042113","article-title":"Dynamics of entanglement between two trapped atoms","volume":"66","author":"Mack","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"012336","DOI":"10.1103\/PhysRevA.73.012336","article-title":"Entanglement between two interacting atoms in a one-dimensional harmonic trap","volume":"73","author":"Sun","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"053631","DOI":"10.1103\/PhysRevA.82.053631","article-title":"Dynamics and decoherence of two cold bosons in a one-dimensional harmonic trap","volume":"82","author":"Brewczyk","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/prop.200310051","article-title":"Entanglement and properties","volume":"51","author":"Ghirardi","year":"2003","journal-title":"Fortschritte der Physik"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"012109","DOI":"10.1103\/PhysRevA.70.012109","article-title":"General criterion for the entanglement of two indistinguishable particles","volume":"70","author":"Ghirardi","year":"2004","journal-title":"Phys. Rev. A"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"1045","DOI":"10.1002\/prop.200410173","article-title":"Criteria for the entanglement of composite systems with identical particles","volume":"52","author":"Ghirardi","year":"2004","journal-title":"Fortschritte der Physik"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"091001","DOI":"10.1088\/0953-4075\/44\/9\/091001","article-title":"Entanglement and squeezing with identical particles: Ultracold atom quantum metrology","volume":"44","author":"Benatti","year":"2011","journal-title":"J. Phys. B"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1002\/prop.201200079","article-title":"Entanglement of identical particles and the detection process","volume":"61","author":"Tichy","year":"2013","journal-title":"Fortschritte der Physik"},{"key":"ref_65","unstructured":"Hunn, S. (September 2013). Microscopic Theory of Decaying Many-Particle Systems. [Ph.D. Thesis, Albert-Ludwigs-Universit\u00e4t Freiburg]."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1023\/A:1018705520999","article-title":"Two cold atoms in a harmonic trap","volume":"28","author":"Busch","year":"1998","journal-title":"Found. Phys."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"023615","DOI":"10.1103\/PhysRevA.78.023615","article-title":"Reduced density matrices and coherence of trapped interacting bosons","volume":"78","author":"Sakmann","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"023602","DOI":"10.1103\/PhysRevA.89.023602","article-title":"Universality of fragmentation in the Schr\u00f6dinger dynamics of bosonic Josephson junctions","volume":"89","author":"Sakmann","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1016\/j.physleta.2017.12.027","article-title":"Effective two-mode description of a few ultra-cold bosons in a double-well potential","volume":"382","author":"Dobrzyniecki","year":"2018","journal-title":"Phys. Lett. A"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"013609","DOI":"10.1103\/PhysRevA.97.013609","article-title":"Effective three-body interactions for bosons in a double-well confinement","volume":"97","author":"Dobrzyniecki","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"050405","DOI":"10.1103\/PhysRevLett.92.050405","article-title":"Atom interferometry with Bose-Einstein condensates in a double-well potential","volume":"92","author":"Shin","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"2386","DOI":"10.1126\/science.1058149","article-title":"Squeezed states in a Bose-Einstein condensate","volume":"291","author":"Orzel","year":"2001","journal-title":"Science"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1002\/andp.201500365","article-title":"Two cold atoms in a time-dependent harmonic trap in one dimension","volume":"528","author":"Ebert","year":"2016","journal-title":"Annalen der Physik"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"043604","DOI":"10.1103\/PhysRevA.83.043604","article-title":"Accurate multi-boson long-time dynamics in triple-well periodic traps","volume":"83","author":"Streltsov","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_75","doi-asserted-by":"crossref","unstructured":"Alon, O.E. (2019). Analysis of a Trapped Bose\u2013Einstein Condensate in Terms of Position, Momentum, and Angular-Momentum Variance. Symmetry, 11.","DOI":"10.3390\/sym11111344"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"3571","DOI":"10.1063\/1.456888","article-title":"The Fourier grid Hamiltonian method for bound state eigenvalues and eigenfunctions","volume":"91","author":"Marston","year":"1989","journal-title":"J. Chem. Phys."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/0010-4655(91)90023-E","article-title":"Two computer programs for solving the Schr\u00f6dinger equation for bound-state eigenvalues and eigenfunctions using the Fourier grid Hamiltonian method","volume":"67","author":"Ward","year":"1991","journal-title":"Comput. Phys. Commun."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1145\/363707.363723","article-title":"Pracniques: Further remarks on reducing truncation errors","volume":"8","author":"Kahan","year":"1965","journal-title":"Commun. ACM"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"013602","DOI":"10.1103\/PhysRevA.82.013602","article-title":"Discretized versus continuous models of p-wave interacting fermions in one dimension","volume":"82","author":"Muth","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_80","unstructured":"Polizzi, E., and Kestyn, J. (2012). FEAST Eigenvalue Solver v3. 0 User Guide. arXiv."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"115112","DOI":"10.1103\/PhysRevB.79.115112","article-title":"Density-matrix-based algorithm for solving eigenvalue problems","volume":"79","author":"Polizzi","year":"2009","journal-title":"Phys. Rev. B"},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"013616","DOI":"10.1103\/PhysRevA.94.013616","article-title":"Dynamics of Hubbard Hamiltonians with the multiconfigurational time-dependent Hartree method for indistinguishable particles","volume":"94","author":"Lode","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/s10909-015-1285-y","article-title":"Vortex reconnections in anisotropic trapped three-dimensional Bose\u2013Einstein condensates","volume":"180","author":"Wells","year":"2015","journal-title":"J. Low Temp. Phys."},{"key":"ref_84","doi-asserted-by":"crossref","unstructured":"Kramer, P., and Saraceno, M. (1981). Geometry of the Time-Dependent Variational Principle in Quantum Mechanics, Springer.","DOI":"10.1007\/3-540-10579-4"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1051\/m2an\/2010018","article-title":"An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics","volume":"44","author":"Conte","year":"2010","journal-title":"ESAIM Math. Model. Numer. Anal."},{"key":"ref_86","doi-asserted-by":"crossref","unstructured":"Lin, R., Molignini, P., Papariello, L., Tsatsos, M.C., Leveque, C., Weiner, S.E., Fasshauer, E., Chitra, R., and Lode, A.U.J. (2020). MCTDH-X: The multiconfigurational time-dependent Hartree method for indistinguishable particles software. Quantum Sci. Technol.","DOI":"10.1088\/2058-9565\/ab788b"},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0370-1573(99)00047-2","article-title":"The multiconfiguration time-dependent Hartree (MCTDH) method: A highly efficient algorithm for propagating wavepackets","volume":"324","author":"Beck","year":"2000","journal-title":"Phys. Rep."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/4\/382\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:12:01Z","timestamp":1760173921000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/4\/382"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,26]]},"references-count":87,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["e22040382"],"URL":"https:\/\/doi.org\/10.3390\/e22040382","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2020,3,26]]}}}