Probing the edge between integrability and quantum chaos in interacting few-atom systems

Thomás Fogarty1, Miguel Ángel García-March2, Lea F. Santos3, and Nathan L. Harshman4

1Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan
2Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, E-46022 València, Spain
3Department of Physics, Yeshiva University, New York, New York 10016, USA
4Department of Physics, American University, 4400 Massachusetts Ave. NW, Washington, DC 20016, USA

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


Interacting quantum systems in the chaotic domain are at the core of various ongoing studies of many-body physics, ranging from the scrambling of quantum information to the onset of thermalization. We propose a minimum model for chaos that can be experimentally realized with cold atoms trapped in one-dimensional multi-well potentials. We explore the emergence of chaos as the number of particles is increased, starting with as few as two, and as the number of wells is increased, ranging from a double well to a multi-well Kronig-Penney-like system. In this way, we illuminate the narrow boundary between integrability and chaos in a highly tunable few-body system. We show that the competition between the particle interactions and the periodic structure of the confining potential reveals subtle indications of quantum chaos for 3 particles, while for 4 particles stronger signatures are seen. The analysis is performed for bosonic particles and could also be extended to distinguishable fermions.

Ultracold atoms trapped in light allow for exquisite experimental control and may form the working units of future quantum information processing devices. However, we show that even systems with a few interacting atoms in a simple trap could manifest signatures of chaotic behavior. Our research on this simple model probes the boundary where the constraints of symmetry become too weak, and the system descends into chaos.

The model we study confines the atoms into a one-dimensional infinite square well broken into sub-wells by thin barriers. This model is solvable in the extreme limiting cases of no interactions and barriers, and infinite interaction and barriers. In the intermediate parameter range of finite barriers and interactions we use numerical methods to find the energy spectrum and to calculate some well-known indicators of quantum chaos. What we find is that for two particles, the system shows no signs of chaos and for four particles the system is chaotic for a wide range of parameters. The intermediate case of three particles can exhibit a wide range of behaviors depending on the number of subwells and the strengths of the interactions and barriers.

Our results indicate that is experimentally feasible to explore the transition to quantum chaos in these small, simple systems. The implications of quantum chaos for control have relevance for applications to quantum information processing. Knowing where this boundary lies means that one could avoid chaotic regimes, or alternatively exploit the randomness of chaos for novel protocols.

► BibTeX data

► References

[1] Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn. Colloquium: Many-body localization, thermalization, and entanglement. Rev. Mod. Phys., 91: 021001, May 2019. 10.1103/​RevModPhys.91.021001. URL https:/​/​​doi/​10.1103/​RevModPhys.91.021001.

[2] Y. Alhassid and R. D. Levine. Spectral autocorrelation function in the statistical theory of energy levels. Phys. Rev. A, 46: 4650–4653, 1992. 10.1103/​PhysRevA.46.4650. URL https:/​/​​doi/​10.1103/​PhysRevA.46.4650.

[3] G. E. Astrakharchik, J. Boronat, J. Casulleras, and S. Giorgini. Beyond the Tonks-Girardeau gas: Strongly correlated regime in quasi-one-dimensional bose gases. Phys. Rev. Lett., 95: 190407, Nov 2005. 10.1103/​PhysRevLett.95.190407. URL https:/​/​​doi/​10.1103/​PhysRevLett.95.190407.

[4] Daniel Barredo, Sylvain de Léséleuc, Vincent Lienhard, Thierry Lahaye, and Antoine Browaeys. An atom-by-atom assembler of defect-free arbitrary two-dimensional atomic arrays. Science, 354 (6315): 1021–1023, 2016. 10.1126/​science.aah3778.

[5] M T Batchelor, M Bortz, X W Guan, and N Oelkers. Evidence for the super Tonks-Girardeau gas. J. Stat. Mech., 2005 (10): L10001–L10001, oct 2005. 10.1088/​1742-5468/​2005/​10/​l10001. URL https:/​/​​10.1088/​1742-5468/​2005/​10/​l10001.

[6] J. H. Becher, E. Sindici, R. Klemt, S. Jochim, A. J. Daley, and P. M. Preiss. Measurement of identical particle entanglement and the influence of antisymmetrization. Phys. Rev. Lett., 125: 180402, Oct 2020. 10.1103/​PhysRevLett.125.180402. URL https:/​/​​doi/​10.1103/​PhysRevLett.125.180402.

[7] Andrea Bergschneider, Vincent M Klinkhamer, Jan Hendrik Becher, Ralf Klemt, Lukas Palm, Gerhard Zürn, Selim Jochim, and Philipp M Preiss. Experimental characterization of two-particle entanglement through position and momentum correlations. Nat. Phys., 15 (7): 640–644, 2019. 10.1038/​s41567-019-0508-6.

[8] Hannes Bernien, Sylvain Schwartz, Alexander Keesling, Harry Levine, Ahmed Omran, Hannes Pichler, Soonwon Choi, Alexander S. Zibrov, Manuel Endres, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin. Probing many-body dynamics on a 51-atom quantum simulator. Nature, 551 (7682): 579–584, 2017. ISSN 1476-4687. 10.1038/​nature24622. URL https:/​/​​10.1038/​nature24622.

[9] M. V. Berry. Quantizing a classically ergodic system: Sinai's billiard and the KKR method. Ann. Phys., 131 (1): 163–216, January 1981. ISSN 0003-4916. 10.1016/​0003-4916(81)90189-5. URL https:/​/​​science/​article/​pii/​0003491681901895.

[10] M. V. Berry and M. Tabor. Level clustering in the regular spectrum. Proc. R. Soc. Lond. A, 356: 375 – 394, 1977. 10.1098/​rspa.1977.0140.

[11] M. V. Berry and M. Wilkinson. Diabolical Points in the Spectra of Triangles. Proc. R. Soc. Lond. A, 392 (1802): 15–43, 1984. ISSN 0080-4630. 10.1098/​rspa.1984.0022. URL https:/​/​​stable/​2397740.

[12] B. Bertini, F. Heidrich-Meisner, C. Karrasch, T. Prosen, R. Steinigeweg, and M. Žnidarič. Finite-temperature transport in one-dimensional quantum lattice models. Rev. Mod. Phys., 93: 025003, May 2021. 10.1103/​RevModPhys.93.025003. URL https:/​/​​doi/​10.1103/​RevModPhys.93.025003.

[13] Wouter Beugeling, Roderich Moessner, and Masudul Haque. Off-diagonal matrix elements of local operators in many-body quantum systems. Phys. Rev. E, 91: 012144, Jan 2015. 10.1103/​PhysRevE.91.012144. URL https:/​/​​doi/​10.1103/​PhysRevE.91.012144.

[14] Immanuel Bloch, Jean Dalibard, and Wilhelm Zwerger. Many-body physics with ultracold gases. Rev. Mod. Phys., 80: 885–964, Jul 2008. 10.1103/​RevModPhys.80.885. URL https:/​/​​doi/​10.1103/​RevModPhys.80.885.

[15] D Blume. Few-body physics with ultracold atomic and molecular systems in traps. Rep. Prog. Phys., 75 (4): 046401, mar 2012. 10.1088/​0034-4885/​75/​4/​046401. URL https:/​/​​10.1088/​0034-4885/​75/​4/​046401.

[16] Doerte Blume. Jumping from two and three particles to infinitely many. Physics, 3: 74, 2010. 10.1103/​Physics.3.74.

[17] F. Borgonovi, F. M. Izrailev, L. F. Santos, and V. G. Zelevinsky. Quantum chaos and thermalization in isolated systems of interacting particles. Phys. Rep., 626: 1, 2016. 10.1016/​j.physrep.2016.02.005.

[18] R. Bourgain, J. Pellegrino, A. Fuhrmanek, Y. R. P. Sortais, and A. Browaeys. Evaporative cooling of a small number of atoms in a single-beam microscopic dipole trap. Phys. Rev. A, 88: 023428, Aug 2013. 10.1103/​PhysRevA.88.023428. URL https:/​/​​doi/​10.1103/​PhysRevA.88.023428.

[19] Marlon Brenes, John Goold, and Marcos Rigol. Low-frequency behavior of off-diagonal matrix elements in the integrable XXZ chain and in a locally perturbed quantum-chaotic XXZ chain. Phys. Rev. B, 102: 075127, Aug 2020a. 10.1103/​PhysRevB.102.075127. URL https:/​/​​doi/​10.1103/​PhysRevB.102.075127.

[20] Marlon Brenes, Tyler LeBlond, John Goold, and Marcos Rigol. Eigenstate thermalization in a locally perturbed integrable system. Phys. Rev. Lett., 125: 070605, Aug 2020b. https:/​/​​10.1103/​PhysRevLett.125.070605. URL https:/​/​​doi/​10.1103/​PhysRevLett.125.070605.

[21] Marlon Brenes, Silvia Pappalardi, Mark T. Mitchison, John Goold, and Alessandro Silva. Out-of-time-order correlations and the fine structure of eigenstate thermalisation. arXiv:2103.01161, 2021. URL https:/​/​​abs/​2103.01161.

[22] T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, and S. S. M. Wong. Random-matrix physics: spectrum and strength fluctuations. Rev. Mod. Phys., 53: 385, 1981. 10.1103/​RevModPhys.53.385. URL https:/​/​​doi/​10.1103/​RevModPhys.53.385.

[23] Steve Campbell, Miguel Ángel García-March, Thomás Fogarty, and Thomas Busch. Quenching small quantum gases: Genesis of the orthogonality catastrophe. Phys. Rev. A, 90: 013617, Jul 2014. 10.1103/​PhysRevA.90.013617. URL https:/​/​​doi/​10.1103/​PhysRevA.90.013617.

[24] Marko Cetina, Michael Jag, Rianne S. Lous, Isabella Fritsche, Jook T. M. Walraven, Rudolf Grimm, Jesper Levinsen, Meera M. Parish, Richard Schmidt, Michael Knap, and Eugene Demler. Ultrafast many-body interferometry of impurities coupled to a Fermi sea. Science, 354 (6308): 96–99, 2016. ISSN 0036-8075. 10.1126/​science.aaf5134.

[25] P. Cheinet, S. Trotzky, M. Feld, U. Schnorrberger, M. Moreno-Cardoner, S. Fölling, and I. Bloch. Counting atoms using interaction blockade in an optical superlattice. Phys. Rev. Lett., 101: 090404, Aug 2008. 10.1103/​PhysRevLett.101.090404. URL https:/​/​​doi/​10.1103/​PhysRevLett.101.090404.

[26] Aurélia Chenu, Javier Molina-Vilaplana, and Adolfo del Campo. Work statistics, Loschmidt echo and information scrambling in chaotic quantum systems. Quantum, 3: 127, March 2019. ISSN 2521-327X. 10.22331/​q-2019-03-04-127. URL https:/​/​​10.22331/​q-2019-03-04-127.

[27] J. Cotler, N. Hunter-Jones, J. Liu, and B. Yoshida. Chaos, complexity, and random matrices. J. High Energy Phys., 2017 (11): 48, Nov 2017. 10.1007/​JHEP11(2017)048. URL https:/​/​​10.1007/​JHEP11(2017)048.

[28] L. D'Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol. From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics. Adv. Phys., 65 (3): 239–362, 2016. 10.1080/​00018732.2016.1198134. URL https:/​/​​10.1080/​00018732.2016.1198134.

[29] Javier de la Cruz, Sergio Lerma-Hernández, and Jorge G. Hirsch. Quantum chaos in a system with high degree of symmetries. Phys. Rev. E, 102: 032208, Sep 2020. 10.1103/​PhysRevE.102.032208. URL https:/​/​​doi/​10.1103/​PhysRevE.102.032208.

[30] Manuel Endres, Hannes Bernien, Alexander Keesling, Harry Levine, Eric R. Anschuetz, Alexandre Krajenbrink, Crystal Senko, Vladan Vuletic, Markus Greiner, and Mikhail D. Lukin. Atom-by-atom assembly of defect-free one-dimensional cold atom arrays. Science, 354 (6315): 1024–1027, 2016. 10.1126/​science.aah3752.

[31] Hans Feldmeier and Jürgen Schnack. Molecular dynamics for fermions. Rev. Mod. Phys., 72: 655–688, Jul 2000. 10.1103/​RevModPhys.72.655. URL https:/​/​​doi/​10.1103/​RevModPhys.72.655.

[32] V. V. Flambaum and F. M. Izrailev. Statistical theory of finite Fermi systems based on the structure of chaotic eigenstates. Phys. Rev. E, 56: 5144, 1997. 10.1103/​PhysRevE.56.5144.

[33] V. V. Flambaum, A. A. Gribakina, G. F. Gribakin, and M. G. Kozlov. Structure of compound states in the chaotic spectrum of the Ce atom: Localization properties, matrix elements, and enhancement of weak perturbations. Phys. Rev. A, 50: 267–296, 1994. 10.1103/​PhysRevA.50.267.

[34] V. V. Flambaum, F. M. Izrailev, and G. Casati. Towards a statistical theory of finite Fermi systems and compound states: Random two-body interaction approach. Phys. Rev. E, 54: 2136–2139, Aug 1996. 10.1103/​PhysRevE.54.2136. URL https:/​/​​doi/​10.1103/​PhysRevE.54.2136.

[35] Thomás Fogarty, Sebastian Deffner, Thomas Busch, and Steve Campbell. Orthogonality catastrophe as a consequence of the quantum speed limit. Phys. Rev. Lett., 124: 110601, Mar 2020. 10.1103/​PhysRevLett.124.110601. URL https:/​/​​doi/​10.1103/​PhysRevLett.124.110601.

[36] M. A. García-March, B. Juliá-Díaz, G. E. Astrakharchik, J. Boronat, and A. Polls. Distinguishability, degeneracy, and correlations in three harmonically trapped bosons in one dimension. Phys. Rev. A, 90: 063605, Dec 2014. 10.1103/​PhysRevA.90.063605. URL https:/​/​​doi/​10.1103/​PhysRevA.90.063605.

[37] M. A. García-March, A. Yuste, B. Juliá-Díaz, and A. Polls. Mesoscopic superpositions of Tonks-Girardeau states and the Bose-Fermi mapping. Phys. Rev. A, 92: 033621, Sep 2015. 10.1103/​PhysRevA.92.033621. URL https:/​/​​doi/​10.1103/​PhysRevA.92.033621.

[38] M A Garcia-March, S van Frank, M Bonneau, J Schmiedmayer, M Lewenstein, and Lea F Santos. Relaxation, chaos, and thermalization in a three-mode model of a Bose–Einstein condensate. New J. Phys., 20 (11): 113039, nov 2018. 10.1088/​1367-2630/​aaed68. URL https:/​/​​10.1088/​1367-2630/​aaed68.

[39] M. Gaudin. Boundary energy of a Bose gas in one dimension. Physical Review A, 4 (1): 386–394, July 1971. 10.1103/​PhysRevA.4.386. URL http:/​/​​doi/​10.1103/​PhysRevA.4.386.

[40] Michel Gaudin. The Bethe Wavefunction. Cambridge University Press, Trans. Caux, Jean-Sébastien edition, April 2014. ISBN 978-1-107-04585-9. 10.1017/​CBO9781107053885.

[41] Marvin Girardeau. Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension. Journal of Mathematical Physics, 1 (6): 516–523, November 1960. ISSN 00222488. 10.1063/​1.1703687. URL http:/​/​​resource/​1/​jmapaq/​v1/​i6/​p516_s1?isAuthorized=no.

[42] J. Goold, T. Fogarty, N. Lo Gullo, M. Paternostro, and Th. Busch. Orthogonality catastrophe as a consequence of qubit embedding in an ultracold Fermi gas. Phys. Rev. A, 84: 063632, Dec 2011. 10.1103/​PhysRevA.84.063632. URL https:/​/​​doi/​10.1103/​PhysRevA.84.063632.

[43] T. Guhr and H.A. Weidenmüller. Correlations in anticrossing spectra and scattering theory. Analytical aspects. Chem. Phys., 146: 21 – 38, 1990. 10.1016/​0301-0104(90)90003-R. URL http:/​/​​science/​article/​pii/​030101049090003R.

[44] T. Guhr, A. Mueller-Gröeling, and H. A. Weidenmüller. Random matrix theories in quantum physics: Common concepts. Phys. Rep., 299: 189, 1998. 10.1016/​S0370-1573(97)00088-4.

[45] Elmar Haller, Mattias Gustavsson, Manfred J. Mark, Johann G. Danzl, Russell Hart, Guido Pupillo, and Hanns-Christoph Nägerl. Realization of an excited, strongly correlated quantum gas phase. Science, 325 (5945): 1224–1227, 2009. 10.1126/​science.1175850.

[46] N. L. Harshman. Symmetries of three harmonically trapped particles in one dimension. Phys. Rev. A, 86: 052122, Nov 2012. 10.1103/​PhysRevA.86.052122. URL https:/​/​​doi/​10.1103/​PhysRevA.86.052122.

[47] N. L. Harshman. One-dimensional traps, two-body interactions, few-body symmetries: II. $N$ particles. Few-Body Systems, 57 (1): 45–69, 2016a. ISSN 0177-7963, 1432-5411. 10.1007/​s00601-015-1025-5. URL http:/​/​​article/​10.1007/​s00601-015-1025-5.

[48] N. L. Harshman. Infinite barriers and symmetries for a few trapped particles in one dimension. Phys. Rev. A, 95 (5): 053616, May 2017. 10.1103/​PhysRevA.95.053616. URL https:/​/​​doi/​10.1103/​PhysRevA.95.053616.

[49] N. L. Harshman, Maxim Olshanii, A. S. Dehkharghani, A. G. Volosniev, Steven Glenn Jackson, and N. T. Zinner. Integrable families of hard-core particles with unequal masses in a one-dimensional harmonic trap. Phys. Rev. X, 7: 041001, Oct 2017. 10.1103/​PhysRevX.7.041001. URL https:/​/​​doi/​10.1103/​PhysRevX.7.041001.

[50] N.L. Harshman. One-dimensional traps, two-body interactions, few-body symmetries: I. One, two, and three particles. Few-Body Systems, 57 (1): 11–43, 2016b. 10.1007/​s00601-015-1024-6.

[51] U. Hartmann, H.A. Weidenmüller, and T. Guhr. Correlations in anticrossing spectra and scattering theory: Numerical simulations. Chem. Phys., 150 (3): 311 – 320, 1991. ISSN 0301-0104. http:/​/​​10.1016/​0301-0104(91)87105-5. URL http:/​/​​science/​article/​pii/​0301010491871055.

[52] Xiaodong He, Peng Xu, Jin Wang, and Mingsheng Zhan. High efficient loading of two atoms into a microscopic optical trap by dynamically reshaping the trap with a spatial light modulator. Opt. Express, 18 (13): 13586–13592, 2010. 10.1364/​OE.18.013586.

[53] Ole J. Heilmann and Elliott H. Lieb. Violation of the noncrossing rule: The Hubbard Hamiltonian for benzene. Ann. N.Y. Acad. Sci., 172 (15): 584–617, 1971. ISSN 1749-6632. 10.1111/​j.1749-6632.1971.tb34956.x. URL https:/​/​​doi/​abs/​10.1111/​j.1749-6632.1971.tb34956.x.

[54] David Huber, Oleksandr V. Marchukov, Hans-Werner Hammer, and Artem G. Volosniev. Morphology of three-body quantum states from machine learning. New Journal of Physics, 2021. 10.1088/​1367-2630/​ac0576. URL http:/​/​​article/​10.1088/​1367-2630/​ac0576.

[55] Marko Žnidarič. Weak integrability breaking: Chaos with integrability signature in coherent diffusion. Phys. Rev. Lett., 125: 180605, Oct 2020. 10.1103/​PhysRevLett.125.180605. URL https:/​/​​doi/​10.1103/​PhysRevLett.125.180605.

[56] Victor Ivrii. 100 years of Weyl's law. Bull. Math. Sci., 6 (3): 379–452, October 2016. ISSN 1664-3607, 1664-3615. 10.1007/​s13373-016-0089-y. URL https:/​/​​article/​10.1007/​s13373-016-0089-y.

[57] F. M. Izrailev. Quantum-classical correspondence for isolated systems of interacting particles: Localization and ergodicity in energy space. Phys. Scr., T90: 95–104, 2001. 10.1238/​Physica.Topical.090a00095.

[58] Felix M. Izrailev. Simple models of quantum chaos: Spectrum and eigenfunctions. Phys. Rep., 196 (5): 299 – 392, 1990. ISSN 0370-1573. https:/​/​​10.1016/​0370-1573(90)90067-C. URL http:/​/​​science/​article/​pii/​037015739090067C.

[59] Sudhir R. Jain. Exact solution of the Schrödinger equation for a particle in a regular N-simplex. Phys. Lett. A, 372 (12): 1978–1981, March 2008. ISSN 0375-9601. 10.1016/​j.physleta.2007.11.016. URL https:/​/​​science/​article/​pii/​S0375960107016027.

[60] A. M. Kaufman, B. J. Lester, and C. A. Regal. Cooling a single atom in an optical tweezer to its quantum ground state. Phys. Rev. X, 2: 041014, Nov 2012. 10.1103/​PhysRevX.2.041014.

[61] A. M. Kaufman, B. J. Lester, C. M. Reynolds, M. L. Wall, M. Foss-Feig, K. R. A. Hazzard, A. M. Rey, and C. A. Regal. Two-particle quantum interference in tunnel-coupled optical tweezers. Science, 345 (6194): 306–309, 2014. 10.1126/​science.1250057.

[62] Tim Keller and Thomás Fogarty. Probing the out-of-equilibrium dynamics of two interacting atoms. Phys. Rev. A, 94: 063620, Dec 2016. 10.1103/​PhysRevA.94.063620. URL https:/​/​​doi/​10.1103/​PhysRevA.94.063620.

[63] Michael Knap, Aditya Shashi, Yusuke Nishida, Adilet Imambekov, Dmitry A. Abanin, and Eugene Demler. Time-dependent impurity in ultracold fermions: Orthogonality catastrophe and beyond. Phys. Rev. X, 2: 041020, Dec 2012. 10.1103/​PhysRevX.2.041020. URL https:/​/​​doi/​10.1103/​PhysRevX.2.041020.

[64] H. R. Krishnamurthy, H. S. Mani, and H. C. Verma. Exact solution of the schrodinger equation for a particle in a tetrahedral box. J. Phys. A, 15 (7): 2131–2137, July 1982. ISSN 0305-4470. 10.1088/​0305-4470/​15/​7/​024. URL https:/​/​​10.1088/​0305-4470/​15/​7/​024.

[65] M. Łacki, M. A. Baranov, H. Pichler, and P. Zoller. Nanoscale ``dark state'' optical potentials for cold atoms. Phys. Rev. Lett., 117: 233001, Nov 2016. 10.1103/​PhysRevLett.117.233001. URL https:/​/​​doi/​10.1103/​PhysRevLett.117.233001.

[66] Tyler LeBlond, Krishnanand Mallayya, Lev Vidmar, and Marcos Rigol. Entanglement and matrix elements of observables in interacting integrable systems. Phys. Rev. E, 100: 062134, Dec 2019. 10.1103/​PhysRevE.100.062134. URL https:/​/​​doi/​10.1103/​PhysRevE.100.062134.

[67] S. Lerma-Hernández, D. Villaseñor, M. A. Bastarrachea-Magnani, E. J. Torres-Herrera, L. F. Santos, and J. G. Hirsch. Dynamical signatures of quantum chaos and relaxation time scales in a spin-boson system. Phys. Rev. E, 100: 012218, Jul 2019. 10.1103/​PhysRevE.100.012218. URL https:/​/​​doi/​10.1103/​PhysRevE.100.012218.

[68] Brian J. Lester, Niclas Luick, Adam M. Kaufman, Collin M. Reynolds, and Cindy A. Regal. Rapid production of uniformly filled arrays of neutral atoms. Phys. Rev. Lett., 115: 073003, Aug 2015. 10.1103/​PhysRevLett.115.073003. URL https:/​/​​doi/​10.1103/​PhysRevLett.115.073003.

[69] Luc Leviandier, Maurice Lombardi, Rémi Jost, and Jean Paul Pique. Fourier transform: A tool to measure statistical level properties in very complex spectra. Phys. Rev. Lett., 56: 2449–2452, Jun 1986. 10.1103/​PhysRevLett.56.2449.

[70] Maciej Lewenstein, Anna Sanpera, and Veronica Ahufinger. Ultracold Atoms in Optical Lattices: Simulating quantum many-body systems. Oxford University Press, 2012. 10.1093/​acprof:oso/​9780199573127.001.0001.

[71] F Leyvraz, A García, H Kohler, and T H Seligman. Fidelity under isospectral perturbations: a random matrix study. J. Phys. A, 46 (27): 275303, 2013. 10.1088/​1751-8113/​46/​27/​275303. URL http:/​/​​1751-8121/​46/​i=27/​a=275303.

[72] Elliott H. Lieb and Werner Liniger. Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Phys. Rev., 130: 1605–1616, May 1963. 10.1103/​PhysRev.130.1605. URL https:/​/​​doi/​10.1103/​PhysRev.130.1605.

[73] M. Lombardi and T. H. Seligman. Universal and nonuniversal statistical properties of levels and intensities for chaotic Rydberg molecules. Phys. Rev. A, 47: 3571–3586, May 1993. 10.1103/​PhysRevA.47.3571. URL http:/​/​​doi/​10.1103/​PhysRevA.47.3571.

[74] D. Luitz and Y. Bar Lev. The ergodic side of the many-body localization transition. Ann. Phys.(Berlin), 529 (7): 1600350, 2017. 10.1002/​andp.201600350.

[75] Juan Maldacena, Stephen H. Shenker, and Douglas Stanford. A bound on chaos. J. High Energy Phys., 2016 (8): 106, 2016. 10.1007/​JHEP08(2016)106. URL http:/​/​​10.1007/​JHEP08(2016)106.

[76] M. L. Mehta. Random Matrices. Academic Press, Boston, USA, 1991.

[77] Nicolás Mirkin and Diego Wisniacki. Quantum chaos, equilibration, and control in extremely short spin chains. Phys. Rev. E, 103: L020201, Feb 2021. 10.1103/​PhysRevE.103.L020201. URL https:/​/​​doi/​10.1103/​PhysRevE.103.L020201.

[78] Simon Murmann, Andrea Bergschneider, Vincent M Klinkhamer, Gerhard Zürn, Thomas Lompe, and Selim Jochim. Two fermions in a double well: Exploring a fundamental building block of the Hubbard model. Phys. Rev. Lett., 114 (8): 080402, 2015. 10.1103/​PhysRevLett.114.080402.

[79] R. Nandkishore and D.A. Huse. Many-body localization and thermalization in quantum statistical mechanics. Annu. Rev. Condens. Matter Phys., 6: 15, 2015. 10.1146/​annurev-conmatphys-031214-014726.

[80] N. Oelkers, M. T. Batchelor, M. Bortz, and X.-W. Guan. Bethe ansatz study of one-dimensional Bose and Fermi gases with periodic and hard wall boundary conditions. J. Phys. A, 39 (5): 1073, 2006. ISSN 0305-4470. 10.1088/​0305-4470/​39/​5/​005. URL http:/​/​​0305-4470/​39/​i=5/​a=005.

[81] M. Olshanii. Atomic scattering in the presence of an external confinement and a gas of impenetrable bosons. Phys. Rev. Lett., 81: 938–941, Aug 1998. 10.1103/​PhysRevLett.81.938. URL https:/​/​​doi/​10.1103/​PhysRevLett.81.938.

[82] Maxim Olshanii and Steven G. Jackson. An exactly solvable quantum four-body problem associated with the symmetries of an octacube. New J. Phys., 17 (10): 105005, 2015. ISSN 1367-2630. 10.1088/​1367-2630/​17/​10/​105005. URL http:/​/​​1367-2630/​17/​i=10/​a=105005.

[83] Maxim Olshanii, Thibault Scoquart, Dmitry Yampolsky, Vanja Dunjko, and Steven Glenn Jackson. Creating entanglement using integrals of motion. Phys. Rev. A, 97 (1): 013630, January 2018. 10.1103/​PhysRevA.97.013630. URL https:/​/​​doi/​10.1103/​PhysRevA.97.013630.

[84] Guido Pagano, Marco Mancini, Giacomo Cappellini, Pietro Lombardi, Florian Schäfer, Hui Hu, Xia-Ji Liu, Jacopo Catani, Carlo Sias, Massimo Inguscio, and Leonardo Fallani. A one-dimensional liquid of fermions with tunable spin. Nat. Phys., 10 (3): 198–201, 2014. 10.1038/​nphys2878. URL https:/​/​​10.1038/​nphys2878.

[85] Akhilesh Pandey and Ramakrishna Ramaswamy. Level spacings for harmonic-oscillator systems. Phys. Rev. A, 43: 4237–4243, Apr 1991. 10.1103/​PhysRevA.43.4237. URL http:/​/​​doi/​10.1103/​PhysRevA.43.4237.

[86] J. P. Pique, Y. Chen, R. W. Field, and J. L. Kinsey. Chaos and dynamics on 0.5–300 ps time scales in vibrationally excited acetylene: Fourier transform of stimulated-emission pumping spectrum. Phys. Rev. Lett., 58: 475–478, Feb 1987. 10.1103/​PhysRevLett.58.475. URL http:/​/​​doi/​10.1103/​PhysRevLett.58.475.

[87] R. E. Prange. The spectral form factor is not self-averaging. Phys. Rev. Lett., 78: 2280, 1997. 10.1103/​PhysRevLett.78.2280. URL https:/​/​​doi/​10.1103/​PhysRevLett.78.2280.

[88] Cindy Regal. Bringing order to neutral atom arrays. Science, 354 (6315): 972–973, 2016. 10.1126/​science.aaj2145.

[89] Irina Reshodko, Albert Benseny, Judit Romhányi, and Thomas Busch. Topological states in the Kronig–Penney model with arbitrary scattering potentials. New J. Phys., 21 (1): 013010, jan 2019. 10.1088/​1367-2630/​aaf9bf. URL https:/​/​​10.1088/​1367-2630/​aaf9bf.

[90] Marcos Rigol and Alejandro Muramatsu. Ground-state properties of hard-core bosons confined on one-dimensional optical lattices. Phys. Rev. A, 72: 013604, Jul 2005. 10.1103/​PhysRevA.72.013604. URL https:/​/​​doi/​10.1103/​PhysRevA.72.013604.

[91] Daniel A. Roberts, Douglas Stanford, and Leonard Susskind. Localized shocks. JHEP, 2015 (3): 51, 2015. ISSN 1029-8479. 10.1007/​JHEP03(2015)051. URL http:/​/​​10.1007/​JHEP03(2015)051.

[92] Lea F. Santos, Francisco Pérez-Bernal, and E. Jonathan Torres-Herrera. Speck of chaos. Phys. Rev. Research, 2: 043034, Oct 2020. 10.1103/​PhysRevResearch.2.043034. URL https:/​/​​doi/​10.1103/​PhysRevResearch.2.043034.

[93] S. Saskin, J. T. Wilson, B. Grinkemeyer, and J. D. Thompson. Narrow-line cooling and imaging of ytterbium atoms in an optical tweezer array. Phys. Rev. Lett., 122: 143002, Apr 2019. 10.1103/​PhysRevLett.122.143002. URL https:/​/​​doi/​10.1103/​PhysRevLett.122.143002.

[94] M. Schiulaz, M. Távora, and L. F. Santos. From few- to many-body quantum systems. Quantum Sci. Technol., 3: 044006, 2018. 10.1088/​2058-9565/​aad913.

[95] Mauro Schiulaz, E. Jonathan Torres-Herrera, and Lea F. Santos. Thouless and relaxation time scales in many-body quantum systems. Phys. Rev. B, 99: 174313, May 2019. 10.1103/​PhysRevB.99.174313. URL https:/​/​​doi/​10.1103/​PhysRevB.99.174313.

[96] Mauro Schiulaz, E. Jonathan Torres-Herrera, Francisco Pérez-Bernal, and Lea F. Santos. Self-averaging in many-body quantum systems out of equilibrium: Chaotic systems. Phys. Rev. B, 101: 174312, May 2020. 10.1103/​PhysRevB.101.174312. URL https:/​/​​doi/​10.1103/​PhysRevB.101.174312.

[97] J. Schnack and H. Feldmeier. Statistical properties of fermionic molecular dynamics. Nuc. Phys. A, 601 (2): 181 – 194, 1996. ISSN 0375-9474. 10.1016/​0375-9474(95)00505-6. URL http:/​/​​science/​article/​pii/​0375947495005056.

[98] M. Schreiber, S. S. Hodgman, Pr. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch. Observation of many-body localization of interacting fermions in a quasirandom optical lattice. Science, 349 (6250): 842–845, 2015. ISSN 0036-8075. 10.1126/​science.aaa7432. URL http:/​/​​content/​349/​6250/​842.

[99] Kai-Niklas Schymik, Vincent Lienhard, Daniel Barredo, Pascal Scholl, Hannah Williams, Antoine Browaeys, and Thierry Lahaye. Enhanced atom-by-atom assembly of arbitrary tweezer arrays. Phys. Rev. A, 102: 063107, Dec 2020. 10.1103/​PhysRevA.102.063107.

[100] Friedhelm Serwane, Gerhard Zürn, Thomas Lompe, TB Ottenstein, AN Wenz, and S Jochim. Deterministic preparation of a tunable few-fermion system. Science, 332 (6027): 336–338, 2011. 10.1126/​science.1201351.

[101] G. B. Shaw. Degeneracy in the particle-in-a-box problem. J. Phys. A, 7 (13): 1537–1546, September 1974. ISSN 0301-0015. 10.1088/​0305-4470/​7/​13/​008. URL https:/​/​​10.1088/​0305-4470/​7/​13/​008.

[102] Tomasz Sowiński and Miguel Ángel García-March. One-dimensional mixtures of several ultracold atoms: a review. Rep. Progr. Phys., 82 (10): 104401, 2019. 10.1088/​1361-6633/​ab3a80.

[103] H-J. Stöckmann. Quantum Chaos: an introduction. Cambridge University Press, Cambridge, UK, 2006. 10.1017/​CBO9780511524622.

[104] E. J. Torres-Herrera and Lea F. Santos. Extended nonergodic states in disordered many-body quantum systems. Ann. Phys. (Berlin), 529: 1600284, 2017a. ISSN 1521-3889. 10.1002/​andp.201600284. URL http:/​/​​10.1002/​andp.201600284.

[105] E. J. Torres-Herrera and Lea F. Santos. Dynamical manifestations of quantum chaos: correlation hole and bulge. Philos. Trans. Royal Soc. A, 375 (2108): 20160434, 2017b. 10.1098/​rsta.2016.0434.

[106] E. J. Torres-Herrera, Antonio M. García-García, and Lea F. Santos. Generic dynamical features of quenched interacting quantum systems: Survival probability, density imbalance, and out-of-time-ordered correlator. Phys. Rev. B, 97: 060303, Feb 2018. 10.1103/​PhysRevB.97.060303. URL https:/​/​​doi/​10.1103/​PhysRevB.97.060303.

[107] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić. Weak ergodicity breaking from quantum many-body scars. Nat. Phys., 14 (7): 745–749, 2018. ISSN 1745-2481. 10.1038/​s41567-018-0137-5. URL https:/​/​​10.1038/​s41567-018-0137-5.

[108] J. W. Turner. On the quantum particle in a polyhedral box. J. Phys. A, 17 (14): 2791–2797, October 1984. ISSN 0305-4470. 10.1088/​0305-4470/​17/​14/​022. URL https:/​/​​10.1088/​0305-4470/​17/​14/​022.

[109] Y. Wang, S. Subhankar, P. Bienias, M. Łacki, T-C. Tsui, M. A. Baranov, A. V. Gorshkov, P. Zoller, J. V. Porto, and S. L. Rolston. Dark state optical lattice with a subwavelength spatial structure. Phys. Rev. Lett., 120: 083601, Feb 2018. 10.1103/​PhysRevLett.120.083601. URL https:/​/​​doi/​10.1103/​PhysRevLett.120.083601.

[110] AN Wenz, G Zürn, Simon Murmann, I Brouzos, T Lompe, and S Jochim. From few to many: Observing the formation of a Fermi sea one atom at a time. Science, 342 (6157): 457–460, 2013. 10.1126/​science.1240516.

[111] Joshua Wilkie and Paul Brumer. Time-dependent manifestations of quantum chaos. Phys. Rev. Lett., 67: 1185–1188, Sep 1991. 10.1103/​PhysRevLett.67.1185. URL https:/​/​​doi/​10.1103/​PhysRevLett.67.1185.

[112] Sebastian Will, Thorsten Best, Ulrich Schneider, Lucia Hackermüller, Dirk-Sören Lühmann, and Immanuel Bloch. Time-resolved observation of coherent multi-body interactions in quantum phase revivals. Nature, 465 (7295): 197–201, May 2010. ISSN 1476-4687. 10.1038/​nature09036. URL https:/​/​​10.1038/​nature09036.

[113] Pablo R. Zangara, Axel D. Dente, E. J. Torres-Herrera, Horacio M. Pastawski, A. Iucci, and Lea F. Santos. Time fluctuations in isolated quantum systems of interacting particles. Phys. Rev. E, 88: 032913, 2013. 10.1103/​PhysRevE.88.032913.

[114] V. Zelevinsky, B. A. Brown, N. Frazier, and M. Horoi. The nuclear shell model as a testing ground for many-body quantum chaos. Phys. Rep., 276: 85–176, 1996. 10.1016/​S0370-1573(96)00007-5.

[115] Guy Zisling, Lea F. Santos, and Yevgeny Bar Lev. How many particles make up a chaotic many-body quantum system? SciPost Phys., 10: 88, 2021. 10.21468/​SciPostPhys.10.4.088. URL https:/​/​​10.21468/​SciPostPhys.10.4.088.

Cited by

[1] E. R. Castro, Jorge Chávez-Carlos, I. Roditi, Lea F. Santos, and Jorge G. Hirsch, "Quantum-classical correspondence of a system of interacting bosons in a triple-well potential", Quantum 5, 563 (2021).

[2] Karin Wittmann W., E. R. Castro, Angela Foerster, and Lea F. Santos, "Interacting bosons in a triple well: Preface of many-body quantum chaos", Physical Review E 105 3, 034204 (2022).

[3] S.I. Mistakidis, A.G. Volosniev, R.E. Barfknecht, T. Fogarty, Th. Busch, A. Foerster, P. Schmelcher, and N.T. Zinner, "Few-body Bose gases in low dimensions—A laboratory for quantum dynamics", Physics Reports 1042, 1 (2023).

[4] Liang Xiang, Jiachen Chen, Zitian Zhu, Zixuan Song, Zehang Bao, Xuhao Zhu, Feitong Jin, Ke Wang, Shibo Xu, Yiren Zou, Hekang Li, Zhen Wang, Chao Song, Alexander Yue, Justine Partridge, Qiujiang Guo, Rubem Mondaini, H. Wang, and Richard T. Scalettar, "Enhanced quantum state transfer by circumventing quantum chaotic behavior", Nature Communications 15 1, 4918 (2024).

[5] Carlos F. Destefani and Xavier Oriols, "Kinetic energy equipartition: A tool to characterize quantum thermalization", Physical Review Research 5 3, 033168 (2023).

[6] Vir B. Bulchandani, David A. Huse, and Sarang Gopalakrishnan, "Onset of many-body quantum chaos due to breaking integrability", Physical Review B 105 21, 214308 (2022).

[7] Qian Wang and Marko Robnik, "Statistics of phase space localization measures and quantum chaos in the kicked top model", Physical Review E 107 5, 054213 (2023).

[8] Qian Wang, "Quantum Chaos in the Extended Dicke Model", Entropy 24 10, 1415 (2022).

[9] Patrycja Łydżba and Tomasz Sowiński, "Signatures of quantum chaos in low-energy mixtures of few fermions", Physical Review A 106 1, 013301 (2022).

[10] Tran Duong Anh-Tai, Mathias Mikkelsen, Thomas Busch, and Thomás Fogarty, "Quantum chaos in interacting Bose-Bose mixtures", SciPost Physics 15 2, 048 (2023).

[11] Carlos F. Destefani and Xavier Oriols, "Assessing quantum thermalization in physical and configuration spaces via many-body weak values", Physical Review A 107 1, 012213 (2023).

[12] J. Pawłowski, M. Panfil, J. Herbrych, and M. Mierzejewski, "Long-living prethermalization in nearly integrable spin ladders", Physical Review B 109 16, L161109 (2024).

[13] Christoph Schönle, David Jansen, Fabian Heidrich-Meisner, and Lev Vidmar, "Eigenstate thermalization hypothesis through the lens of autocorrelation functions", Physical Review B 103 23, 235137 (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-22 04:03:59) and SAO/NASA ADS (last updated successfully 2024-06-22 04:03:59). The list may be incomplete as not all publishers provide suitable and complete citation data.